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--- address: - '$^\ast$Department d’Estructura i Constituents de la MatèriaUniversitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia, Spain' - '$^\dagger$Uji Research CenterYukawa Institute for Theoretical PhysicsKyoto University, Uji 611, Japan' author: - 'Joaquim Gomis[^1][Electronic address: quim@ebubecm1.bitnet]{}and Hiroshi Suzuki[^2][JSPS Junior Scientist Fellow. Also at Department of Physics, Hiroshima University, Higashi-Hiroshima 724, Japan. Electronic address: suzuki@jpnrifp.bitnet]{}' date: = title: 'Covariant Currents in $N=2$ Super-Liouville Theory' --- phyzzx =[92-4UB-ECM-PF 92/4]{} [February 1992]{} Introduction ============ The $N=2$ string or the two-dimensional $N=2$ supergravity introduced by Ademollo [*et al*]{} \[,,,,,,\] has critical dimension $d=2$ and there is no transverse degree of freedom. Very recently, it has been argued that the no-ghost theorem can be established \[\]. The $N=2$ subcritical strings or $N=2$ super-Liouville theory has also been analyzed. Distler, Hlousek, and Kawai \[\] noticed that the local ansatz for the Jacobian, that relates the interacting measure with the free measure, works for any kind of conformal matter or, in terms of strings, for an arbitrary number of space-time dimensions. All those special features of $N=2$ string suggest that this theory is a topological quantum field theory \[,\]. In a previous work \[\], we proved that critical and subcritical $N=2$ strings are topological field theories in the sense that the (super-)coordinate BRST current algebra gives a realization of an $N=2$ superfield extension of the topological conformal algebra \[,\] for arbitrary type of conformal matter. In this paper we want to analyze in detail the appearance of this topological conformal algebra in the case of $N=2$ super-Liouville theory by constructing an effective theory based on anomalous identities associated with the BRST and ghost number symmetries. The finite renormalization of the coupling constant will be interpreted as a one-loop order effect of the BRST invariant measure. The relation with the critical string will be also commented. The organization of the paper is as follows. In Section 2, we compute the BRST and ghost number anomalies in the $N=2$ supergravity in the superconformal gauge, by using the path integral representation \[\]. The $N=2$ super-Liouville theory is constructed in Section 3 with a consideration on the covariant BRST and ghost number supercurrents. In Section 4 we derive the topological conformal algebra. In section 5 there are some conclusions and there is an Appendix about some basic facts of the $N=2$ superfield formalism. BRST and ghost number anomalies in $N=2$ supergravity ===================================================== In this section, we compute anomalies associated with the BRST and the ghost number supercurrents in the two-dimensional $N=2$ supergravity. We follow the path integral prescription for anomaly calculation \[\]. In the case of $N=1$ supergravity, the superfield path integral is known to be an appropriate tool to compute those anomalies \[,\]. For $N=2$ case, however, it is well-known that the action of the matter multiplet (A.4) cannot be written directly in terms of the real scalar superfield (A.8) \[\]. Therefore it is not clear how the superfield path integral is useful in the present context of the anomaly calculation.[^3][For the ghost and anti-ghost multiplet, the action can be written directly by the superfield as in (A.12). Thus the superfield path integral may work.]{} Here we consider the path integral in the component fields. The classical gauge symmetries of the $N=2$ supergravity are the general coordinate, the local Lorentz, the $N=2$ supersymmetry, the Weyl, the super-Weyl, and the chiral transformations \[\]. We assume under our regularization, the Weyl, the super-Weyl and the chiral transformations are anomalous at quantum level. Using the non-anomalous transformations, we can fix the $N=2$ supergravity multiplet in the superconformal gauge as $$\eqalign{ &e_\mu^a=e^{\sigma/2}\delta^a_\mu, \cr &\chi^\pm_\mu=\gamma_\mu\phi^\pm, \cr &A_\mu=\epsilon_{\mu\nu}\partial^\nu\phi, \cr } \ee$$ where $\sigma$ is the Liouville mode and, $\phi^\pm$ and $\phi$ are their $N=2$ superpartners. The BRST supercurrent and ghost number supercurrent anomalies will depend on the Liouville mode $\sigma$ and their $N=2$ superpartners. As we are working in a path integral in the components, we should construct the integration variables depending on $\sigma$, $\phi^\pm$, and $\phi$ such that the integration measure is invariant under the supercoordinate transformations. This program have some difficulties, for $N=1$, see for example \[\]. Here instead we are going to use the following strategy. Let us forget about the anomalous character of the super-Weyl and the chiral transformations. By using these symmetries, we can fix the gauge $$\eqalign{ &e_\mu^a=e^{\sigma/2}\delta^a_\mu, \cr &\chi^\pm_\mu=0, \cr &A_\mu=0. \cr } \ee$$ In this way, the BRST supercurrent and ghost number supercurrent anomalies will depend only on the Liouville mode. When we consider the effect of the super-Weyl and the chiral anomalies, the remaining components of the gravitino and the U(1) gauge field should appear in various anomalies. Here we [*assume*]{} that we are actually using a regularization that is invariant under the super-coordinate and the gauge transformations. Especially, we assume the invariance under the [*global*]{} super transformation: $$\eqalign{ &\delta\sigma=i\left(\alpha^+\phi^--\alpha^-\phi^+\right), \cr &\delta\phi=\alpha^-\phi^++\alpha^+\phi^-, \cr &\delta\phi^\pm =\alpha^\pm\left(\partial\phi\pm i\partial\sigma\right). \cr } \eqn\twotwo$$ To get the dependences on the remaining components of the Liouville superpartners, we will use the invariance (or covariance) under . This strategy is the same as the one for a calculation of the super-Liouville action in \[\]. The partition function of the $N=2$ supergravity in the superconformal gauge is defined by $$\eqalign{ \int\,d\widetilde\mu\,\exp\biggl\{-{1\over2\pi}\int\biggl[\, {1\over2}&\left( -\partial\Xmu\partialbar\Xmu -\partial\Ymu\partialbar\Ymu +\psiplusmu\partialbar\psiminusmu+\psiminusmu\partialbar\psiplusmu \right) \cr &\qquad+b\partialbar c+\betaplus\partialbar\gammaminus +\betaminus\partialbar\gammaplus+\eta\partialbar\xi +({\rm c.\ c.})\,\biggr]\biggr\}, \cr } \eqn\twofour$$ where we defined the integration measure $d\widetilde\mu$ by $$\eqalign{ d\widetilde\mu&= % \Di\bl e^{\sigma/2}\br \Di\bl e^\sigma c\br \Di\bl e^\sigma\overline c\br \Di\bl e^{-\sigma/2}b\br \Di\bl e^{-\sigma/2}\overline b\br \Di\bl e^{3\sigma/4}\gammaplus\br \Di\bl e^{3\sigma/4}\overline\gammaplus\br \cr &\quad\times \Di\bl e^{3\sigma/4}\gammaminus\br \Di\bl e^{3\sigma/4}\overline\gammaminus\br \Di\bl e^{-\sigma/4}\betaplus\br \Di\bl e^{-\sigma/4}\overline\betaplus\br \Di\bl e^{-\sigma/4}\betaminus\br \Di\bl e^{-\sigma/4}\overline\betaminus\br \cr &\quad\times \Di\bl e^{\sigma/2}\xi\br \Di\bl e^{\sigma/2}\overline\xi\br \Di\eta \Di\overline\eta \Di\bl e^{\sigma/4}\psiplusmu\br \Di\bl e^{\sigma/4}\overline\psiplusmu\br \cr &\quad\times \Di\bl e^{\sigma/4}\psiminusmu\br \Di\bl e^{\sigma/4}\overline\psiminusmu\br \Di\bl e^{\sigma/2}\Xmu\br \Di\bl e^{\sigma/2}\Ymu\br \cr &\equiv % \Di\bl e^{\sigma/2}\br \Di\widetilde c\, \Di\widetilde{\overline c}\, \Di\widetilde b\, \Di\widetilde{\overline b}\, \Di\widetilde\gamma^+ \Di\widetilde{\overline\gamma}^+ \Di\widetilde\gamma^- \Di\widetilde{\overline\gamma}^- \Di\widetilde\beta^+ \Di\widetilde{\overline\beta}^+ \Di\widetilde\beta^- \Di\widetilde{\overline\beta}^- \Di\widetilde\xi \Di\widetilde{\overline\xi} \Di\eta \Di\overline\eta \cr &\qquad\times \Di\widetilde\psi^{+\mu} \Di\widetilde{\overline\psi}^{+\mu} \Di\widetilde\psi^{-\mu} \Di\widetilde{\overline\psi}^{-\mu} \Di\widetilde X^\mu \Di\widetilde Y^\mu. \cr } \eqn\twofive$$ The various weight factors ($\exp\sigma$) in the integration measure are determined from the general coordinate (BRST) invariance of the integration measure \[,\]. In the above expression, we did not include the integration of the Liouville (or Weyl) mode $\sigma$. We will turn this point in the next section. Our starting point  and  are the same as the one in \[\]. We also note the above integration measure is invariant under the conformal transformation as noted in \[\]: $$\delta\phi=V\partial\phi+h\left(\partial V\right)\phi,\quad \delta\overline\phi=V\partial\overline\phi, \ee$$ where $h$ is the conformal weight of the generic field $\phi$. In the path integral formulation \[\], the anomaly is generally ascribed to a non-invariance of the integration measure and the Jacobian factor associated with the anomalous transformation gives rise to the anomaly. The Jacobian factor of general conformal fields in a conformally flat background, is analyzed in \[\]. According to \[\], under an infinitesimal change of the integration variable, $\tilde\phi\rightarrow\tilde\phi+\varepsilon(x)\tilde\phi$, a logarithm of the Jacobian factor $J$ is given by $$\ln J=\pm{1\over2\pi}\int d^2x\varepsilon(x) \left[\left(a-b\over3\right)\partial\partialbar\sigma +M^2e^{-(2a+b)\sigma}\right], \eqn\twoeight$$ and, for $\tilde\phi\rightarrow\tilde\phi+\varepsilon(x)\partial\tilde\phi$, $$\eqalign{ &\ln J =\pm{1\over24\pi}\int d^2x\varepsilon(x) \biggl[\left(b^2-4ab\right)\partial\sigma\partialbar\partial\sigma +\left(2a-3b\right)\partialbar\partial^2\sigma \cr &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad -12M^2\left(a+b\right)\partial\sigma e^{-(2a+b)\sigma}\biggr]. \cr } \eqn\twonine$$ In  and , the double sign ($\pm$) corresponds to the statistics of the field $\phi$. The trace operation that is necessary to evaluate the above Jacobian factors is regularized by using an exponential type damping factor $e^{-H/M^2}$ with the regulator $H$, which is defined by $$H\equiv-\Dslash^\dagger\Dslash \equiv-e^{a\sigma}\partial e^{b\sigma}\partialbar e^{a\sigma}. \eqn\twonines$$ where $\Dslash$ is the kinetic operator of the field $\tilde\phi$. By rewriting the action in  in terms of the integration variables in , we can read off the various values of $a$ and $b$ in  for the each fields (see Table 1). To see how our present formulation works, let us first consider the ghost number anomaly \[\]. In $N=2$ case, the ghost number current is known to be anomaly free, due to a cancellation of the background charge \[\]. The ghost number supercurrent \[\] is defined by $$\eqalign{ j_{\rm gh}(Z)&\equiv-BC(Z) \cr &=i\eta c+\thetaminus\left(-\eta\gammaplus+i\betaplus c\right) +\thetaplus\left(\eta\gammaminus+i\betaminus c\right) \cr &\quad+\thetaminus\thetaplus\left(-\eta\xi-\betaminus\gammaplus -\betaplus\gammaminus-bc\right) \cr &\equiv j_{\rm gh}^0(z)+\thetaminus j_{\rm gh}^+(z) +\thetaplus j_{\rm gh}^-(z)+\thetaminus\thetaplus j_{\rm gh}^{+-}(z). \cr } \eqn\twonineprime$$ We can immediately see $$\partialbar\VEV{j_{\rm gh}^0(z)} =\partialbar\VEV{j_{\rm gh}^+(z)} =\partialbar\VEV{j_{\rm gh}^-(z)}=0, \eqn\twooneone$$ i.e., these currents are anomaly free. To show this, let us consider the following infinitesimal change of variables in : $$b\rightarrow b+i\varepsilon(x)\eta,\quad \xi\rightarrow\xi-i\varepsilon(x)c. \eqn\twoeleven$$ Note that the partition function itself does not change under a change of the integration variables. Therefore the variation of the action and the variation of the integration measure should be canceled each other, and we have the following identity: $$-{1\over2\pi}\int d^2x\varepsilon(x) \partialbar\VEV{j_{\rm gh}^0(z)} +\VEV{\ln J}=0, \ee$$ where $J$ is a Jacobian factor associated with the change of variables . However, for , the Jacobian is trivial, i.e., $J=1$ and $\ln J=0$, because the variation of the field is not proportional to the field itself. Therefore $\partialbar\VEV{j_{\rm gh}^0(z)}$ is anomaly free. Similar considerations show other relations in . A potential anomalous term in a vacuum expectation value is thus a product of the equation of motion and the conjugate field, because such a combination is proportional to the Jacobian factor of a change of variable whose variation is proportional to the field itself. In this sense the final combination $\partialbar\VEV{j_{\rm gh}^{+-}(z)}$ is potentially dangerous. Let us consider the following change of variables: $$\eqalign{ &\delta c=\varepsilon(x)c,\quad\delta b=-\varepsilon(x)b, \cr &\delta\gamma^\pm=\varepsilon(x)\gamma^\pm,\quad \delta\beta^\pm=-\varepsilon(x)\beta^\pm, \cr &\delta\xi=\varepsilon(x)\xi,\quad \delta\eta=-\varepsilon(x)\eta, \cr } \eqn\twothirteen$$ (more precisely, we should write down the variation of the tilded integration variables, but for , the variation of the tilded variables is proportional to the one of the untilded variables). Then we have $$-{1\over2\pi}\int d^2x\varepsilon(x) \partialbar\VEV{j_{\rm gh}^{+-}(z)}+\VEV{\ln J}=0, \ee$$ where $J$ is the Jacobian factor associated with the variation . From the master formula , we have $$\ln J=-{1\over4\pi}\int d^2x\varepsilon(x) (-3+2\times2-1)\partialbar\partial\sigma=0, \ee$$ where the contributions from the different sector \[($b$,$c$), ($\beta^\mp$,$\gamma^\pm$), and ($\eta$,$\xi$) respectively\] are separately indicated. We can see that the ghost number anomaly vanishes due to a cancellation of the background charges. Our formulation reproduces the desired answer as is expected. Let us now turn to the anomaly associated with the conservation of the BRST supercurrent. We define the BRST supercurrent as \[\] $$\eqalign{ j_B(Z)&\equiv C\left(T^X+{1\over2}T^{\rm gh}\right) +{1\over4}\Diminus\left[C\left(\Diplus C\right)B\right] +{1\over4}\Diplus\left[C\left(\Diminus C\right)B\right] \cr &\equiv J_B(Z)+\widehat\jmath_B(Z). \cr } \eqn\twosixteen$$ To see the structure of the BRST anomaly, we call the first term in the first line in  as $J_B(Z)$ and the total divergence parts as $\widehat\jmath_B(Z)$. We should note here that we chose the total divergence parts $\widehat\jmath_B(B)$ (which do not affect to the BRST charge $Q_B$) by $$j_B(Z)=-\left\{Q_B,j_{\rm gh}(Z)\right\}, \eqn\twoseventeen$$ to make the BRST current manifestly BRST invariant if $Q_B^2=0$. In this sense the above choice is the most symmetric one and this choice of $\widehat\jmath_B(Z)$ is crucial for our conclusion. We also define the components of the BRST supercurrent as $$\eqalign{ &J_B(Z)\equiv J_B^0(z)+\thetaminus J_B^++\thetaplus J_B^-(z) +\thetaminus\thetaplus J_B^{+-}(z), \cr &\widehat\jmath_B(Z)\equiv\widehat\jmath_B^0(z)+\thetaminus\widehat \jmath_B^+ +\thetaplus\widehat\jmath_B^-(z) +\thetaminus\thetaplus\widehat\jmath_B^{+-}(z). \cr } \ee$$ The explicit form of the component currents of $J_B(Z)$ becomes: $$\eqalign{ &J_B^0(z)={1\over2}c\left[\psiminusmu\psiplusmu -i\partial(c\eta)+{1\over2}\gammaplus\betaminus -{1\over2}\gammaminus\betaplus\right], \cr &J_B^+(z)=-{1\over2}c\biggl[-i\partial\Ymu\psiplusmu +\partial\Xmu\psiplusmu-\partial\left(\gammaplus\eta\right) +i\partial\left(c\betaplus\right)-{1\over2}i\gammaplus b +{1\over2}\gammaplus\partial\eta \cr &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad+{1\over2}\xi\betaplus +{1\over2}i\partial c\betaplus\biggr] \cr &\qquad\qquad-{1\over2}i\gammaplus\left[\psiminusmu\psiplusmu -i\partial(c\eta)+{1\over2}\gammaplus\betaminus -{1\over2}\gammaminus\betaplus\right], \cr &J_B^-(z)=-{1\over2}c\biggl[-i\partial\Ymu\psiminusmu -\partial\Xmu\psiminusmu+\partial\left(\gammaminus\eta\right) +i\partial\left(c\betaminus\right)-{1\over2}i\gammaminus b -{1\over2}\gammaminus\partial\eta \cr &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad-{1\over2}\xi\betaminus +{1\over2}i\partial c\betaminus\biggr] \cr &\qquad\qquad+{1\over2}i\gammaminus\left[\psiminusmu\psiplusmu -i\partial(c\eta)+{1\over2}\gammaplus\betaminus -{1\over2}\gammaminus\betaplus\right], \cr &J_B^{+-}(z)=\widetilde J_B^{+-}(z)+{3\over4} \partial\left(c\gammaplus\betaminus+c\gammaplus\betaminus\right) -{1\over2}\partial(c\xi\eta) \cr &\qquad+{i\over2}\xi\left(\psiminusmu\psiplusmu +\gammaplus\betaminus-\gammaminus\betaplus\right) +{i\over2}\gammaminus\partial\left(\gammaplus\eta\right) -{i\over2}\gammaplus\partial\left(\gammaminus\eta\right) -{1\over2}\gammaminus\gammaplus b. \cr } \ee$$ In the last expression, $\widetilde J_B^{+-}(z)$ is defined by $$\eqalign{ \widetilde J_B^{+-}&\equiv {1\over2}c\left(\partial\Xmu\partial\Xmu+\partial\Ymu\partial\Ymu +\partial\psiminusmu\psiplusmu+\partial\psiplusmu\psiminusmu\right) \cr &\quad+c\left(\partial cb-{1\over2}\gammaminus\partial\betaplus -{3\over2}\partial\gammaminus\betaplus -{1\over2}\gammaplus\partial\betaminus -{3\over2}\partial\gammaplus\betaminus+\partial\xi\eta\right). \cr } \ee$$ Similarly, the components of the hat supercurrent $\widehat\jmath_B(Z)$ are given by $$\eqalign{ &\widehat\jmath_B^0(z)=-{i\over2}\gammaminus\gammaplus\eta +{1\over4}c\left(\gammaplus\betaminus-\gammaminus\betaplus\right), \cr &\widehat\jmath_B^+(z)=-{1\over4}\partial\left(c\gammaplus\eta\right) -{1\over4}c\partial\gammaplus\eta+{i\over2}\gammaplus\xi\eta +{1\over4}\gammaplus\partial c\eta -{i\over4}\gammaplus\gammaplus\betaminus \cr &\qquad\qquad-{1\over4}c\xi\betaplus +{i\over4}c\partial c\betaplus -{i\over4}\gammaplus\gammaminus\betaplus+{i\over4}\gammaplus cb, \cr &\widehat\jmath_B^-(z)={1\over4}\partial\left(c\gammaminus\eta\right) +{1\over4}c\partial\gammaminus\eta+{i\over2}\gammaminus\xi\eta -{1\over4}\gammaminus\partial c\eta -{i\over4}\gammaminus\gammaminus\betaplus \cr &\qquad\qquad+{1\over4}c\xi\betaminus +{i\over4}c\partial c\betaminus -{i\over4}\gammaminus\gammaplus\betaminus+{i\over4}\gammaminus cb, \cr &\widehat\jmath_B^{+-}(z)=-{1\over2}\partial(c\xi\eta) +{1\over4}\partial\left(c\gammaplus\betaminus\right) +{1\over4}\partial\left(c\gammaminus\betaplus\right). \cr } \eqn\twotwentyone$$ Let us start the calculation of the BRST anomaly from the first part, $\partialbar\VEV{J_B^0(z)}$. Noting the dangerous combination, i.e., a product of the equation of motion and the conjugate field, we find $$\partialbar\VEV{J_B^0(z)} =\VEV{{1\over2}c\partialbar\left(\psiminusmu\psiplusmu\right) +{1\over4}c\partialbar\left(\gammaplus\betaminus% -\gammaminus\betaplus\right)}. \ee$$ Therefore we use the following variation of the integration variables $$\delta\psiplusmu=-{1\over2}\varepsilon(x)c\psiplusmu,\quad \delta\psiminusmu={1\over2}\varepsilon(x)c\psiminusmu, \eqn\twotwentythree$$ to get $$-{1\over2\pi}\int d^2x\varepsilon(x) \VEV{{1\over2}c\partialbar\left(\psiminusmu\psiplusmu\right)} +\VEV{\ln J}=0, \eqn\twotwentyfour$$ where $J$ is the Jacobian factor associated with the above transformation . However if we note the fact that, in our present formulation, the Jacobian factor does not depend on the U(1) charge (the superscript $\pm$) but only on the conformal weight, we can see the contributions from $\psiplusmu$ and $\psiminusmu$ cancel each other, i.e., $\ln J=0$ in . Therefore $$\VEV{{1\over2}c\partialbar\left(\psiminusmu\psiplusmu\right)}=0. \eqn\twotwentyfive$$ From the same reason, we have $$\VEV{{1\over4}c\partialbar\left(\gammaplus\betaminus\right)} =\VEV{{1\over4}c\partialbar\left(\gammaminus\betaplus\right)}. \eqn\twotwentysix$$ Combining  and , $$\partialbar\VEV{J_B^0(z)}=0, \eqn\twotwentyseven$$ i.e., $J_B^0(z)$ is anomaly free. For $\partialbar\VEV{J_B^+(z)}$, since $\VEV{-{1\over2}i\gammaplus\partialbar\left(\psiminusmu\psiplusmu \right)}=0$, we can see, $$\partialbar\VEV{J_B^+(z)} =\VEV{{1\over4}i\gammaplus\partialbar(cb) -{1\over4}i\partialbar\left(\gammaplus\gammaplus\betaminus\right) +{1\over4}i\gammaplus\partialbar\left(\gammaminus\betaplus\right)}. \ee$$ We consider the following variations: $$\eqalign{ &\delta c={1\over4}i\varepsilon(x)\gammaplus c,\quad \delta b=-{1\over4}i\varepsilon(x)\gammaplus b, \cr &\delta\gammaplus={1\over4}i\varepsilon(x)\gammaplus\gammaplus,\quad \delta\gammaminus=-{1\over4}i\varepsilon(x)\gammaplus\gammaminus, \cr &\delta\betaplus={1\over4}i\varepsilon(x)\gammaplus\betaplus,\quad \delta\betaminus=-{1\over4}i\varepsilon(x)\gammaplus\betaminus, \cr } \ee$$ and obtain the following identity: $$-{1\over2\pi}\int d^2x\varepsilon(x)\partialbar\VEV{J_B^+(z)} +\VEV{\ln J}=0. \ee$$ From the master formula , we have $$\ln J={1\over2\pi}\int d^2x\varepsilon(x) {i\over8}\gammaplus\partialbar\partial\sigma, \ee$$ and then $$\partialbar\VEV{J_B^+(z)} ={i\over8}\VEV{\gammaplus\partialbar\partial\sigma} ={i\over8}\partialbar\VEV{\gammaplus\partial\sigma}. \eqn\twothirtytwo$$ In deriving the last expression, we used a safe equation of motion, $\VEV{\partialbar\gammaplus}=0$. The equation of motion alone is always safe, i.e., the Schwinger–Dyson equation always is valid. Similarly, for $\partialbar\VEV{J_B^-(z)}$ (by interchanging $+\leftrightarrow-$), we have $$\partialbar\VEV{J_B^-(z)} ={i\over8}\VEV{\gammaminus\partialbar\partial\sigma} ={i\over8}\partialbar\VEV{\gammaminus\partial\sigma}. \eqn\twothirtythree$$ To evaluate $\partialbar\VEV{J_B^{+-}(z)}$, let us first consider $\partialbar\VEV{\widetilde J_B^{+-}(z)}$. We take the following variations: $$\eqalign{ &\delta\sigma=0, \cr &\delta\Xmu=\varepsilon(x)c\partial\Xmu,\quad \delta\Ymu=\varepsilon(x)c\partial\Ymu, \cr &\delta\psi^{\pm\mu} =\varepsilon(x)\left[c\partial\psi^{\pm\mu} +{1\over2}(\partial c)\psi^{\pm\mu}\right], \cr &\delta c=\varepsilon(x)c\partial c, \cr &\delta b=\varepsilon(x)\biggl[ c\partial b+2(\partial c)b \cr &\qquad\qquad +{1\over2}\left(\partial\Xmu\partial\Xmu +\partial\Ymu\partial\Ymu+\partial\psiminusmu\psiplusmu +\partial\psiplusmu\psiminusmu\right) \cr &\qquad\qquad -{1\over2}\gammaminus\partial\betaplus -{3\over2}\partial\gammaminus\betaplus -{1\over2}\gammaplus\partial\betaminus -{3\over2}\partial\gammaplus\betaminus +\partial\xi\eta\biggr], \cr &\delta\gamma^\pm=\varepsilon(x)\left[ c\partial\gamma^\pm-{1\over2}(\partial c)\gamma^\pm\right], \cr &\delta\beta^{\pm}=\varepsilon(x)\left[ c\partial\beta^\pm+{3\over2}(\partial c)\beta^\pm\right], \cr &\delta\xi=\varepsilon(x)c\partial\xi, \cr &\delta\eta=\varepsilon(x)\left[ c\partial\eta+(\partial c)\eta\right]. \cr } \eqn\twothirtyfour$$ Then we have the following identity: $$\eqalign{ &-{1\over2\pi}\int d^2x \biggl\{\varepsilon(x)\partialbar\VEV{\widetilde J_B^{+-}(z)} \cr &\qquad\qquad\quad-\partial\varepsilon(x)c\bigl[ {1\over2}\left(\psiplusmu\partialbar\psiminusmu +\psiminusmu\partialbar\psiplusmu\right) +b\partialbar c+\eta\partialbar\xi \cr &\qquad\qquad\quad% +{3\over2}\betaplus\partialbar\gammaminus +{1\over2}\partialbar\betaplus\gammaminus +{3\over2}\betaminus\partialbar\gammaplus +{1\over2}\partialbar\betaminus\gammaplus\bigr]\biggr\} \cr &\quad+\VEV{\ln J_1}=0, \cr } \eqn\twothirtyfive$$ where $J_1$ is the Jacobian factor associated the variations . The variations  cause the following variations of the integration variables: $$\eqalign{ &\delta\widetilde X^\mu=\varepsilon(x)\left[ c\partial\widetilde X^\mu -{1\over2}c(\partial\sigma)\widetilde X^\mu\right], \cr &\delta\widetilde Y^\mu=\varepsilon(x)\left[ c\partial\widetilde Y^\mu -{1\over2}c(\partial\sigma)\widetilde Y^\mu\right], \cr &\delta\widetilde\psi^{\pm\mu} =\varepsilon(x)\left[c\partial\widetilde\psi^{\pm\mu} -{1\over4}c(\partial\sigma)\widetilde\psi^{\pm\mu} +{1\over2}(\partial c)\widetilde\psi^{\pm\mu}\right], \cr &\delta\widetilde c =\varepsilon(x)e^{-\sigma}\widetilde c\partial\widetilde c, \cr &\delta\widetilde b=\varepsilon(x)\left[ c\partial\widetilde b +{1\over2}c(\partial\sigma)\widetilde b +2(\partial c)\widetilde b\right], \cr &\delta\widetilde\gamma^\pm=\varepsilon(x)\left[ c\partial\widetilde\gamma^\pm -{3\over4}c(\partial\sigma)\widetilde\gamma^\pm -{1\over2}(\partial c)\widetilde\gamma^\pm\right], \cr &\delta\widetilde\beta^{\pm}=\varepsilon(x)\left[ c\partial\widetilde\beta^\pm +{1\over4}c(\partial\sigma)\widetilde\beta^{\pm} +{3\over2}(\partial c)\widetilde\beta^\pm\right], \cr &\delta\widetilde\xi=\varepsilon(x)\left[ c\partial\widetilde\xi -{1\over2}c(\partial\sigma)\widetilde\xi\right], \cr &\delta\eta=\varepsilon(x)\left[ c\partial\eta+(\partial c)\eta\right]. \cr } \ee$$ From the master formula  and  we have $$\eqalign{ &\ln J_1 \cr &={1\over2\pi}\int d^2x\varepsilon(x) \left[-{d\over3}c\partialbar\partial^2\sigma +{d-2\over4}c\partial\sigma\partialbar\partial\sigma -{d+6\over12}\partial c\partialbar\partial\sigma -dM^2\partial\left(ce^\sigma\right)\right]. \cr } \eqn\twothirtyseven$$ For the remaining part in , we can see $$\eqalign{ &-{1\over2\pi}\int d^2x\left(-\partial\varepsilon(x)\right) \biggl\langle c\biggl[ {1\over2}\left(\psiplusmu\partialbar\psiminusmu +\psiminusmu\partialbar\psiplusmu\right) +b\partialbar c+\eta\partialbar\xi \cr &\qquad\qquad\qquad\qquad\qquad +{3\over2}\betaplus\partialbar\gammaminus +{1\over2}\partialbar\betaplus\gammaminus +{3\over2}\betaminus\partialbar\gammaplus +{1\over2}\partialbar\betaminus\gammaplus \biggr]\biggr\rangle \cr &\quad+\VEV{\ln J_2}=0, \cr } \ee$$ where the Jacobian $J_2$ is associated with $$\eqalign{ &\delta\psi^{\pm\mu}=-{1\over2}\partial\varepsilon(x)c\psi^{\pm\mu}, \cr &\delta b=-\partial\varepsilon(x)cb, \cr &\delta\gamma^\pm={1\over2}\partial\varepsilon(x)c\gamma^\pm,\quad \delta\beta^\pm=-{3\over2}\partial\varepsilon(x)c\beta^\pm, \cr &\delta\eta=-\partial\varepsilon(x)c\eta. \cr } \ee$$ By using the master formula, $$\ln J_2={1\over2\pi}\int d^2x\varepsilon(x) \left[-{d-12\over12}\partial\left(c\partialbar\partial\sigma\right) -dM^2\partial\left(ce^\sigma\right)\right]. \eqn\twoforty$$ Combining the above results  and , we have $$\partialbar\VEV{\widetilde J_B^{+-}(z)} ={d-2\over4}\VEV{c\left(\partial\sigma\partialbar\partial\sigma -\partialbar\partial^2\sigma\right)} -{3\over2}\VEV{\partial\left(c\partialbar\partial\sigma\right)}. \ee$$ Finally we have to calculate: $$\eqalign{ &\partialbar\VEV{J_B^{+-}(z)-\widetilde J_B^{+-}(z)} \cr &=\partialbar\VEV{ {3\over4}\partial\left(c\gammaminus\betaplus +c\gammaplus\betaminus\right) -{1\over2}\partial(c\xi\eta)} \cr &=\partial\VEV{{3\over4}c\partialbar\left(\gammaminus\betaplus +\gammaplus\betaminus\right) -{1\over2}c\partialbar(\xi\eta)} \cr &={5\over4}\VEV{\partial\left(c\partialbar\partial\sigma\right)}. \cr } \ee$$ (The calculation is similar to the ghost number anomaly.) Collecting the above considerations, we finally get $$\eqalign{ \partialbar\VEV{J_B^{+-}(z)} &={d-2\over4}\VEV{c\left(\partial\sigma\partialbar\partial\sigma -\partialbar\partial^2\sigma\right)} -{1\over4}\VEV{\partial\left(c\partialbar\partial\sigma\right)} \cr &={d-2\over4}\partialbar\VEV{c \left({1\over2}\partial\sigma\partial\sigma -\partial^2\sigma\right)} -{1\over4}\partialbar\VEV{\partial\left(c\partial\sigma\right)}, \cr } \eqn\twofortythree$$ where, in the final step, we used a safe equation of motion $\VEV{\partialbar c}=0$. For the hat currents in , similar calculations show, $$\eqalign{ &\partialbar\VEV{\widehat\jmath_B^0}=0, \cr &\partialbar\VEV{\widehat\jmath_B^+} =-{i\over8}\VEV{\gammaplus\partialbar\partial\sigma} =-{i\over8}\partialbar\VEV{\gammaplus\partial\sigma}, \cr &\partialbar\VEV{\widehat\jmath_B^-} =-{i\over8}\VEV{\gammaminus\partialbar\partial\sigma} =-{i\over8}\partialbar\VEV{\gammaminus\partial\sigma}, \cr &\partialbar\VEV{\widehat\jmath_B^{+-}} ={1\over4}\VEV{\partial\left(c\partialbar\partial\sigma\right)} ={1\over4}\partialbar\VEV{\partial\left(c\partial\sigma\right)}, \cr } \eqn\twofortyfour$$ where, in the final step, we used safe equations of motion, $\VEV{\partialbar\gamma^\pm}=\VEV{\partialbar c}=0$. One may summarize those identities , , , and , and  in supercurrent forms: $$\eqalign{ &\partialbar\VEV{J_B(Z)} =\partialbar\biggl\langle {i\over8}\thetaminus\gammaplus\partial\sigma +{i\over8}\thetaplus\gammaminus\partial\sigma \cr &\qquad\qquad\qquad\qquad +\thetaminus\thetaplus\left[ {d-2\over4}c\left({1\over2}\partial\sigma\partial\sigma -\partial^2\sigma\right) -{1\over4}\partial\left(c\partial\sigma\right)\right] \biggr\rangle, \cr &\partialbar\VEV{\widehat\jmath_B(Z)} =\partialbar\VEV{ -{i\over8}\thetaminus\gammaplus\partial\sigma -{i\over8}\thetaplus\gammaminus\partial\sigma +{1\over4}\thetaminus\thetaplus \partial\left(c\partial\sigma\right)}. \cr } \eqn\twofortyfive$$ Now, as noted previously, we assumed that our regularization actually preserves the global supersymmetry . In terms of the superfield, this assumption requires the right hand sides of  should behave as covariant supercurrents. Thus here we introduce the super-Liouville field by $$\Phi(Z)=\phi(z)+\theta^-\phi^+(z) +\theta^+\phi^-(z)+i\theta^-\theta^+\partial\sigma(z), \ee$$ where $\phi$ and $\phi^\pm$ are the remaining components of the U(1) gauge field and the gravitino respectively, and $\sigma$ is the Liouville mode. The covariant combinations which reproduce  under a condition $\phi=\phi^\pm=0$ are $$\eqalign{ &\partialbar\VEV{J_B(Z)} ={d-2\over4}\partialbar\VEV{C\left({1\over2}D^-\Phi D^+\Phi +i\partial\Phi\right)} \cr &\qquad\qquad\qquad\qquad-{i\over8}\partialbar \VEV{\Diplus\left(C\Diminus\Phi\right) +\Diminus\left(C\Diplus\Phi\right)}, \cr &\partialbar\VEV{\widehat\jmath_B(Z)} ={i\over8}\partialbar \VEV{\Diplus\left(C\Diminus\Phi\right) +\Diminus\left(C\Diplus\Phi\right)}. \cr } \eqn\twofortyseven$$ The above covariantizations are unique ones. Therefore if we take our definition of the BRST supercurrent , we have $$\partialbar\VEV{j_B(Z)} ={d-2\over4}\partialbar\VEV{C\left({1\over2}D^-\Phi D^+\Phi +i\partial\Phi\right)}. \eqn\twofortyeight$$ Here we should emphasize that the BRST anomaly in  vanishes for $d=2$. In the case of $N=0$ and $N=1$ (super-)gravity, on the other hand, even if one take a BRST invariant BRST current as in , the BRST anomaly contains a additional total divergent piece \[,\], which does not proportional to $d-26$ and $d-10$ respectively. As a consequence, the BRST anomaly in $N=0$ and $N=1$ (super-)gravity does [*not*]{} vanish even in the critical dimension $d=26$ and $d=10$. The origin of the total divergent BRST anomaly is the fact that the BRST invariant path integral measure is invariant under the global BRST transformation, up to a total divergence \[\]. Therefore it generates a total divergent anomaly in general under a [ *localized*]{} BRST transformation, like . In this sense, the absence of a total divergent anomaly in  is an intrinsic feature of the $N=2$ theory and suggests the topological nature of $N=2$ theory as a quantum theory. We can also summarize the ghost number anomaly in terms of the supercurrent: $$\partialbar\VEV{j_{\rm gh}(Z)}=0. \eqn\twofortynine$$ The anomalous identities,  and  will play an important role when we construct the effective covariant supercurrents in $N=2$ super-Liouville theory. $N=2$ super-Liouville theory and the covariant supercurrents ============================================================ In the previous section, we saw that the various anomalies appear through the $\sigma$-dependences in the integration measure . Here we try to construct an effective theory which is supposedly equivalent with the original $N=2$ supergravity , by incorporating the effect of anomalies. Firstly, following a standard procedure to produce the Wess–Zumino term in the string theory \[\], we repeat an infinitesimal transformation of the integration variables. For example, we change $\widetilde X^\mu$ as $$\widetilde X^\mu\longrightarrow \left(1+{\sigma\over2}dt\right)\widetilde X^\mu. \eqn\threeone$$ By repeating this infinitesimal transformation up to a finite $t$, the kinetic operator of $\widetilde X^\mu$ changes to $$e^{-\sigma(1-t)/2}\partial\partialbar e^{-\sigma(1-t)/2}, \ee$$ thus all the $\sigma$ dependences in  and  disappear at $t=1$. On the other hand, from the master formula , the change of variable in  generates the following Jacobian: $$\ln J(t)_{\widetilde X^\mu}={d\over2\pi}dt\int d^2x\sigma\left[ -{1\over12}(1-t)\partialbar\partial\sigma+{1\over2}M^2e^{(1-t)\sigma} \right]. \ee$$ Summing over all the contributions from various fields and by integrating $\ln J(t)$ from $t=0$ to $1$, we have the so-called Liouville action: $$\int_0^1\ln J(t)=-{2-d\over16\pi} \int d^2x\partial\sigma\partialbar\sigma. \eqn\threefour$$ Note that the “Liouville term,” $e^\sigma$, disappears in  because of the supersymmetry of the original model. As noted in the previous section, the Liouville action should invariant under the global super transformation . Therefore, under a supersymmetric regularization, the Liouville action  should have the form \[\], $$S_{\rm Liouville}\equiv-{2-d\over16\pi} \int d^2x\left[\partial\sigma\partialbar\sigma +\partial\phi\partialbar\phi -\phi^+\partialbar\phi^- -\phi^-\partialbar\phi^+ +({\rm c.\ c.})\right]. \eqn\threefive$$ In this stage, since we have extracted the $\sigma$-dependences in the integration measure as the Liouville action , our partition function of the matter and the ghost multiplets in a fixed metric background has the following form: $$\eqalign{ \int\,d\mu\,\exp\biggl\{-{1\over2\pi}\int\biggl[\, {1\over2}&\left( -\partial\Xmu\partialbar\Xmu -\partial\Ymu\partialbar\Ymu +\psiplusmu\partialbar\psiminusmu+\psiminusmu\partialbar\psiplusmu \right) \cr &\qquad+b\partialbar c+\betaplus\partialbar\gammaminus +\betaminus\partialbar\gammaplus+\eta\partialbar\xi +({\rm c.\ c.})\,\biggr]\biggr\}, \cr } \eqn\threesix$$ where $d\mu$ is a “naive” integration measure: $$\eqalign{ d\mu&= \Di c \Di\overline c \Di b \Di\overline b \Di\gammaplus \Di\overline\gammaplus \Di\gammaminus \Di\overline\gammaminus \Di\betaplus \Di\overline\betaplus \Di\betaminus \Di\overline\betaminus \cr &\quad\times \Di\xi \Di\overline\xi \Di\eta \Di\overline\eta \Di\psiplusmu \Di\overline\psiplusmu \Di\psiminusmu \Di\overline\psiminusmu \Di\Xmu \Di\Ymu. \cr } \eqn\threeseven$$ From  and , we have the following correlation functions of $X^\mu(Z)$, $C(Z)$, and $B(Z)$: $$\eqalign{ &\VEV{X^\mu(Z_a)X^\nu(Z_b)}=\eta^{\mu\nu}\ln Z_{ab}, \cr &\VEV{C(Z_a)B(Z_b)}=\VEV{B(Z_a)C(Z_b)}= {{\theta_{ab}^-\theta_{ab}^+}\over{Z_{ab}}}, \cr } \eqn\threeeight$$ where $Z_{ab}$ and $\theta^\pm_{ab}$ are defined by $$\eqalign{ &Z_{ab} =z_a-z_b-\left(\theta_a^+\theta_b^-+\theta_a^-\theta_b^+\right), \cr &\theta^\pm_{ab}=\theta^\pm_a-\theta^\pm_b. \cr } \ee$$ Our next question is the following: What is the correct expression of the ghost number supercurrent and the BRST supercurrent in the partition function ? Note that, in the partition function , we do not have any anomalies and we can always use the naive equations of motion. From the expressions of the BRST anomaly  and the ghost number anomaly , we may take $$\eqalign{ &j_B(Z)\equiv C\left(T^X+{1\over2}T^{\rm gh}\right) +{d-2\over4}C \left({1\over2}\Diminus\Phi\Diplus\Phi+i\partial\Phi\right) \cr &\qquad\qquad +{1\over4}\Diminus\left[C\left(\Diplus C\right)B\right] +{1\over4}\Diplus\left[C\left(\Diminus C\right)B\right], \cr &j_{\rm gh}(Z)\equiv-BC, \cr } \eqn\threeten$$ as the effective covariant supercurrents in the partition function . In , we determined the $\Phi$-dependences as to reproduce  and  under uses of the naive equations of motion of the matter and the ghost fields. This prescription was also applied to $N=0$ and $N=1$ (super-)gravity \[,\]. We emphasize that we obtained the anomalous identities  and  in the BRST invariant path integral framework \[,\], thus those expressions should reflect the (super-)coordinate covariance in the quantum theory. In order to have a complete description of the $N=2$ quantum supergravity, we should quantize the Liouville supermultiplet. We define the partition function of the Liouville supermultiplet as $$\eqalign{ &\int\Di\left(e^{\sigma/2}\right) \Di\left(e^{\sigma/4}\phi^+\right) \Di\left(e^{\sigma/4}\overline\phi^+\right) \Di\left(e^{\sigma/4}\phi^-\right) \Di\left(e^{\sigma/4}\overline\phi^-\right) \Di\left(e^{\sigma/2}\phi\right) \cr &\times\exp\left\{ -{2-d\over16\pi}\int d^2x\left[\partial\sigma\partialbar\sigma +\partial\phi\partialbar\phi -\phi^+\partialbar\phi^- -\phi^-\partialbar\phi^+ +({c.\ c.})\right] \right\}, \cr } \eqn\threeeleven$$ where we have chosen the weight factors ($\exp\sigma$) following the prescription in \[,\]. The full partition function is given by a product of  and . If we apply the same procedure of the derivation of  also to the gravitinos $\phi^\pm$ and the gauge field $\phi$, the partition function  changes to $$\eqalign{ &\int\Di\left(e^{\sigma/2}\right) \Di\phi^+\Di\overline\phi^+\Di\phi^-\Di\overline\phi^-\Di\phi \cr &\times\exp\left[ -\left({2-d\over16\pi}-{1\over24\pi}\right) \int d^2x\partial\sigma\partialbar\sigma -{1\over4\pi}M^2\int d^2xe^\sigma +\cdots \right], \cr } \eqn\threetwelve$$ where we only indicated the $\sigma$-dependence of the action. The integration of the Liouville field in  is, on the other hand, highly non-linear because the integration variable is $e^{\sigma/2}$, not simply $\sigma$. To avoid this problem, here we apply the background field method \[,\] and include the one-loop renormalization effect arising from the non-trivial measure $e^{\sigma/2}$. To do this, we set $e^{\sigma/2}\equiv e^{\sigma_0/2}+\varphi$ and expand the Liouville action in  with respect to $\varphi$ up to the second order. If we assume the $M^2e^\sigma$ term in  is canceled by a suitable counter term, the resulting action for the quantum fluctuation $\varphi$ has the same form of the action of $\widetilde X^\mu(z)$ with replacement $\sigma\rightarrow\sigma_0$. Thus, up to the one-loop order, we have additional contribution from the Liouville part itself, $${1\over16\pi}\int d^2x \partial\sigma_0\partialbar\sigma_0 . \ee$$ We regard this factor as the one-loop finite renormalization effect. Adding this effect to the original contribution from , and after a covariantization, we finally have $$\eqalign{ &\int\Di\sigma \Di\phi^+\Di\overline\phi^+\Di\phi^-\Di\overline\phi^-\Di\phi \cr &\times\exp\left\{ -{1-d\over16\pi}\int d^2x\left[\partial\sigma\partialbar\sigma +\partial\phi\partialbar\phi -\phi^+\partialbar\phi^- -\phi^-\partialbar\phi^+ +({\rm c.\ c.})\right] \right\}, \cr } \eqn\threeforteen$$ where we have rewritten $\sigma_0$ as $\sigma$ and taken a naive integration measure $\sigma$ for the Liouville field, since we already include the (one-loop) quantum effect of $e^{\sigma/2}$. We should note here the coefficient in , $2-d$, changes to $1-d$ in . Since the action in  has the same form as the matter supermultiplet $X^\mu(Z)$, the correlation function of the Liouville superfield $\Phi$ is given by $$\VEV{\Phi\left(Z_a\right)\Phi\left(Z_b\right)} ={4\over d-1}\ln Z_{ab}. \eqn\threefifteen$$ As the effective covariant supercurrents in the partition function , we may take  with a replacement $2-d\rightarrow 1-d$, i.e., $$\eqalign{ &j_B(Z)\equiv C\left(T^X+{1\over2}T^{\rm gh}\right) +{d-1\over4}C \left({1\over2}\Diminus\Phi\Diplus\Phi+i\partial\Phi\right) \cr &\qquad\qquad +{1\over4}\Diminus\left[C\left(\Diplus C\right)B\right] +{1\over4}\Diplus\left[C\left(\Diminus C\right)B\right], \cr &j_{\rm gh}(Z)\equiv-BC. } \eqn\threesixteen$$ We comment on the differences of  from the analogous construction for the $N=0$ and $N=1$ (super-)Liouville cases \[,\]. The differences are i) no appearance of a correction term due to the Liouville field in the ghost number supercurrent $j_{\rm gh}(Z)$ in  and, ii) no appearance of a divergence correction term in the expression of BRST supercurrent $j_B(Z)$ in . The origins of these facts are respectively, i) a vanishing of the ghost number anomaly in $N=2$ theory , ii) no appearance of the BRST anomaly which is not proportional to $d-2$ in . In fact, as is discussed in the following section, these two facts might be related each other. We regard the whole set of the partition function  and , and the effective supercurrents in  as the effective theory of the two-dimensional $N=2$ supergravity, i.e., $N=2$ super-Liouville theory. The advantage of this effective theory is that we can use propagators in a flat space-time, like . Although the replacement $2-d\rightarrow1-d$ in the current operator construction in  is [*ad hoc*]{}, we will check the covariance of those supercurrents by using the operator product expansion (OPE) in the next section. This shift of the parameter, $2-d\rightarrow1-d$ also appeared as the ansatz in \[\]. BRST supercurrent algebra and the topological conformal algebra =============================================================== In this section, we show that our effective supercurrents in  forms a topological conformal algebra \[,\], which appears in two-dimensional topological (conformal) field theories \[,\]. This observation was reported in our previous communication \[\]. Firstly we change the normalization of the Liouville superfield as $$\Phi(Z)\longrightarrow{2\over\sqrt{d-1}}\Phi(Z). \ee$$ Thus the correlation function in  changes to $$\VEV{\Phi\left(Z_a\right)\Phi\left(Z_b\right)} =\ln Z_{ab}, \ee$$ and the effective BRST supercurrent changes to $$\eqalign{ j_B(Z)&=C(Z)\left(T^X+T^{\rm Liouville}+{1\over2}T^{\rm gh}\right) \cr &\quad+{1\over4}D^-\left[C\left(D^+C\right)B\right] +{1\over4}D^+\left[C\left(D^-C\right)B\right]. \cr } \ee$$ In the above expression, we defined the Liouville energy-momentum tensor: $$T^{\rm Liouville}={1\over2}D^-\Phi D^+\Phi+\kappa\partial\Phi, \ee$$ where $\kappa$ satisfies $$\kappa^2={{1-D}\over4}. \ee$$ The BRST charge in the $N=2$ super-Liouville theory thus is given by $$Q_B=\int DZ\,C\left(T^X+T^{\rm Liouville}+{1\over2}T^{\rm gh}\right), \ee$$ and, as we will see, it satisfies $Q_B^2=0$ for any $d$. Furthermore we will also see $$T(Z)=\left\{Q_B,B(Z)\right\}, \eqn\fourseven$$ where the total energy momentum tensor $T$ is defined by $$T=T^X+T^{\rm gh}+T^{\rm Liouville}. \ee$$ At this point we examine the BRST supercurrent algebra in $N=2$ super-Liouville theory. We change the notation as $$\eqalign{ &T(Z)\equiv T(Z), \cr &G(Z)\equiv j_B(Z), \cr &\overline G(Z)\equiv B(Z), \cr &J(Z)\equiv j_{\rm ghost}(Z). \cr } \eqn\fournine$$ For the superconformal properties, the relevant operator product expansion is (for any $d$), $$\eqalign{ T(Z_a)\Psi(Z_b)&\sim h\,{{\theta_{ab}^-\theta_{ab}^+}\over{Z_{ab}^2}}\,\Psi(Z_b) +{1\over{2Z_{ab}}}\left(\theta_{ab}^-D_b^+-\theta_{ab}^+D_b^-\right) \Psi(Z_b) \cr &\quad+{{\theta_{ab}^-\theta_{ab}^+}\over{Z_{ab}}} \,\partial_{z_b}\Psi(Z_b), \cr } \eqn\fourten$$ where $\Psi=T$, $G$, $\overline G$, and $J$ with $h=1$, $0$, $1$, and $0$ respectively. This expression implies those operators are primary fields with the U(1) charge $0$ and the superconformal weight $h$. Especially the case $\Psi=T$ implies a vanishing of the total central charge for any $d$. Moreover the BRST supercurrent $j_B(Z)$ and the ghost number supercurrent $j_{\rm gh}(Z)$ in  are primary fields. In this sense, the supercurrents  in this effective theory are [*covariant*]{} in the quantum level and this desired feature suggests our construction in  is reliable. For another relations between various operators, we have (also for any $d$), $$\eqalign{ &G(Z_a)\overline G(Z_b)\sim {1\over{2Z_{ab}}}\left(\theta_{ab}^-D_b^+-\theta_{ab}^+D_b^-\right) J(Z_b) +{{\theta_{ab}^-\theta_{ab}^+}\over{Z_{ab}}} \,T(Z_b), \cr &G(Z_a)G(Z_b)\sim0, \cr &\overline G(Z_a)\overline G(Z_b)\sim0, \cr &J(Z_a)J(Z_b)\sim0, \cr &J(Z_a)G(Z_b)\sim {{\theta_{ab}^-\theta_{ab}^+}\over{Z_{ab}}}\,G(Z_b), \cr &J(Z_a)\overline G(Z_b)\sim -{{\theta_{ab}^-\theta_{ab}^+}\over{Z_{ab}}}\,\overline G(Z_b). \cr } \eqn\foureleven$$ Surprisingly, in the above operator algebra, no quantum anomalous term appears and it coincides with the classically expected form. In the case of $N=0$ and $N=1$ cases \[,\], on the other hand, the quantum anomalous terms vanish only at $d=-2$ and $d=\pm\infty$ respectively. The algebra in  and  form a kind of the topological conformal algebra or the twisted $N=4$ superconformal algebra \[,\]. Therefore, in the $N=2$ super-Liouville theory, the super-coordinate BRST supercurrent algebra gives a representation of a $N=2$ superfield extension of the topological conformal algebra for any $d$.[^4][By comparing the conformal weight and the statistics of the each component fields, we can see that our algebra in  and  are not the same as the twisted $N=4$ superconformal algebras analyzed by Nojiri \[\].]{} Our observation thus suggests the topological nature of the $N=2$ super-Liouville theory (or the $N=2$ fermionic string theory) as the quantum theory. We note that the first relation in  implies , and the second relation in  implies the BRST invariance of the BRST supercurrent, therefore the BRST charge is nilpotent. The anomaly-free property of the operator algebra in  and  might be understood from the absence of the ghost number anomaly in $N=2$ theory: In our construction, which is analogous to the one in \[,\], the correction of the ghost number current due to the Liouville mode is determined from the ghost number anomaly. In $N=2$ case, since we have no ghost number anomaly in , the form of the effective ghost number supercurrent in  has no correction due to the Liouville mode. From experiences on the $N=0$ and $N=1$ (super-)Liouville theories \[,,\], we know the following relation for the [ *effective*]{} currents: $$j_B=-\left\{Q_B,j_{\rm gh}\right\}, \eqn\fourtwelve$$ and we can see that the $(d-{\rm critical\ dimensions}+1)$ non-proportional correction of the BRST (super-)current in the left hand side is generated from the Liouville correction of the ghost number (super-)current in the right hand side. If we expect the general validity of the relation  in our construction, the BRST supercurrent in the $N=2$ super-Liouville theory will not have $d-1$ non-proportional correction. Actually, we can check  from the explicit form . In the BRST current algebra like  in the $N=0$ and $N=1$ case \[,,\], we can also observe that the anomalous terms in the algebra arise from the above mentioned non-trivial correction of the BRST and ghost number (super-)currents. In the case of $N=2$, therefore, we may expect the anomaly free property of the operator algebra. It is also useful to see how the critical string can be considered as a subcritical string in dimension 1 plus the Liouville superfield, in fact in this situation, $\kappa=0$ and all the operators of the effective theory coincide with the ones of the critical string, since in this case there is no restriction for the possible values of $d$. Conclusion ========== We computed the BRST and the ghost number anomalies in the $N=2$ supergravity in the superconformal gauge, working in a path integral in terms of the component fields. The dependences of the Liouville mode in the anomalies were directly calculated while the dependences of the Liouville superpartners were determined by using the global supersymmetry . The final results were written in terms of superfields. The effective $N=2$ super-Liouville theory was constructed at one loop level and there is a finite renormalization of the coupling constant, i.e., from $2-d$ to $1-d$. The algebra of the operators in  gives rise to an $N=2$ superfield extension of the topological conformal algebra for any value of dimensions $d$ without any anomalous terms. The crucial points for this property are the vanishing of the ghost number anomaly and the definition of the BRST supercurrent. Our observation shows an appearing of a quite simplification in the $N=2$ case and also suggests a topological nature of the $N=2$ super-Liouville theory. The $N=2$ critical string can be considered as an $N=2$ subcritical string in dimension $1$ plus the Liouville superfield. All the above features distinguish $N=2$ string from the $N=0$ and $N=1$ strings. The physical relevance of the topological algebra is under study. Let us recall some basic facts of $N=2$ string in the superfield formalism,[^5][We follow the notation of \[\].]{} The $N=2$ superspace is described in terms of the bosonic $(z,\overline z)$ and the fermionic $(\theta^\pm,\overline\theta^\pm)$ coordinates. We define covariant derivatives by $$D^\pm={\partial\over{\partial\theta^\mp}}+\theta^\pm\partial,\quad \overline D^\pm={\partial\over{\partial\overline\theta^\mp}} +\overline\theta^\pm\overline\partial. \ee$$ The action can be written in terms of two superfields $S^\mu(z,\overline z,\theta^+,\overline\theta^+,\theta^-,\overline\theta^-)$ and $S^{\mu\ast}(z,\overline z,\theta^+,\overline\theta^+,\theta^-,\overline\theta^-)$ satisfying two constraints ($\mu$ runs over $1$ to $d$): $$D^-S^\mu=\overline D^-S^\mu=0, \ee$$ and $$D^+S^{\mu\ast}=\overline D^+S^{\mu\ast}=0. \ee$$ The action is given by \[\] $$A=\int dzd\overline z\int d\theta^+d\overline\theta^+d\theta^-d\overline\theta^- S^{\mu\ast} S^\mu. \ee$$ The solution of the equations of motion $$D^+\overline D^+S^\mu=0,\quad\overline D^-D^-S^{\mu\ast}=0, \ee$$ can be written as $$S^\mu=S_1^\mu+S_2^\mu, \ee$$ where $$\eqalign{ &D^-S_1^\mu=\overline D^-S_1^\mu=\overline D^+S_1^\mu=0, \cr &D^-S_2^\mu=\overline D^-S_2^\mu=D^+S_2^\mu=0. \cr } \ee$$ A real superfield $X^\mu$ is constructed via $$X^\mu\left(z,\theta^+,\theta^-\right) =S_1^\mu\left(z+\theta^-\theta^+,\theta^-\right) +S_1^{\mu\ast}\left(z+\theta^+\theta^-,\theta^+\right). \ee$$ The components of $X^\mu(Z)$ are $$X^\mu(Z)=X^\mu(z)+\theta^-\psi^{+\mu}(z)+\theta^+\psi^{-\mu}(z) +i\theta^-\theta^+\partial Y^\mu(z), \ee$$ where $X^\mu(z)$ and $Y^\mu(z)$ are free bosonic fields and $\psi^{\pm\mu}(z)$ are free fermions. The contribution to the energy momentum tensor from $X^\mu$ is $$T^X(Z)={1\over2}D^-X^\mu D^+X^\mu(Z). \ee$$ The $N=2$ string action is invariant under several local gauge transformations. We are working in the superconformal gauge. The gauge fixing generates a Faddeev–Popov determinant expressible as a superfield action using $N=2$ superfield ghost $C$ and antighost $B$: $$\eqalign{ &C\equiv c+i\theta^+\gamma^--i\theta^-\gamma^ ++i\theta^-\theta^+\xi, \cr &B\equiv-i\eta-i\theta^+\beta^--i\theta^-\beta^+ +\theta^-\theta^+b. \cr } \ee$$ The ghosts $c$ and $b$ are for the $\tau$-$\sigma$ general coordinate invariances, $\gamma^\pm$ and $\beta^\pm$ are the super ghosts for the two local supersymmetry transformations and $\xi$ and $\eta$ are the ghosts associated with the local U(1) symmetry. Their Lagrangians are the first order systems with background charge $Q$ \[\] and statistics $\epsilon$ of $(Q,\epsilon)=(-3,+)$, $(2,-)$ and $(-1,+)$ respectively. Notice that the total background ghost charge vanishes. The ghost action in terms of superfield is given by $$A_{\rm gh}={1\over\pi}\int d^2zd\theta^+d\theta^-B\overline\partial\,C +({\rm c.\ c.}). \ee$$ The ghost energy momentum tensor becomes $$T^{\rm gh}(Z)=\partial(CB)(Z)-{1\over2}D^+CD^-B(Z) -{1\over2}D^-CD^+B(Z). \ee$$ 4em Table 1 [^1]: [^2]: [^3]: [^4]: [^5]:

--- author: - 'Matthieu B[é]{}thermin' - Emanuele Daddi - Georgios Magdis - Claudia Lagos - Mark Sargent - Marcus Albrecht - Hervé Aussel - Frank Bertoldi - Véronique Buat - Maud Galametz - Sébastien Heinis - Olivier Ilbert - Alexander Karim - Anton Koekemoer - Cedric Lacey - 'Emeric Le Floc’h' - Felipe Navarrete - Maurilio Pannella - Corentin Schreiber - Vernesa Smolčić Myrto Symeonidis - Marco Viero bibliography: - 'biblio.bib' date: 'Received 19 September 2014 / Accepted 11 November 2014' title: Evolution of the dust emission of massive galaxies up to z=4 and constraints on their dominant mode of star formation --- Introduction ============ Galaxy properties evolve rapidly across cosmic time. In particular, various studies have shown that the mean star formation rate (SFR) at fixed stellar mass increases by a factor of about 20 between z=0 and z=2 [e.g., @Noeske2007; @Elbaz2007; @Daddi2007; @Pannella2009; @Magdis2010; @Karim2011; @Elbaz2011; @Rodighiero2011; @Whitaker2012; @Heinis2014; @Pannella2014]. This very high SFR can be explained by either larger reservoirs of molecular gas or a higher star formation efficiency (SFE). Large gas reservoirs have been found in massive galaxies at high redshift [e.g., @Daddi2008; @Tacconi2010; @Daddi2010a; @Tacconi2013; @Aravena2013], which could imply high SFRs with SFE similar to that of normal star-forming galaxies in the local Universe. On the other hand, follow-up of bright submillimeter galaxies (SMGs) revealed that their very intense SFR ($\sim$1000M$_\odot$/yr) is also driven by a SFE boosted by a factor of 10 with respect to normal star-forming galaxies in the local Universe [e.g., @Greve2005; @Frayer2008; @Daddi2009a; @Daddi2009b], likely induced by a major merger. This difference can be understood if we consider that galaxies are driven by two types of star formation activity: smooth processes fed by large reservoirs of gas in normal star-forming galaxies and boosted star-formation in gas rich mergers [@Daddi2010b; @Genzel2010].\ Using models based on the existence of this main-sequence of star-forming galaxies, i.e., a tight correlation between SFR and stellar mass, and outliers of this sequence with boosted sSFRs (SFR/M$_\star$) called starbursts hereafter, @Sargent2012 showed that the galaxies with the highest SFR mainly correspond to starbursts, while the bulk of the star formation budget ($\sim$85%) is hosted in normal star-forming galaxies. This approach allows us to better understand the heterogeneous characteristic of distant objects concerning their gas fraction and their SFE [@Sargent2014]. The quick rise of the sSFR would thus be explained by larger gas reservoirs in main-sequence galaxies. However, the most extreme SFRs observed in high-redshift starbursts would be caused by a SFE boosted induced by major mergers.\ At high redshift, the gas mass is difficult to estimate. Two main methods are used. The first is based on the measurement of the intensity of rotational transitions (generally with J$_{\rm upper}<3$) of $^{12}$CO and an assumed CO-to-H$_2$ conversion factor [@Daddi2008; @Tacconi2010; @Saintonge2013; @Tacconi2013]. The main limitation of this method is the uncertainty on this conversion factor, which is expected to be different from the local calibrations in high-redshift galaxies with strongly sub-solar metallicities [@Bothwell2010; @Engel2010; @Genzel2012; @Tan2013; @Genzel2014]. The second method is based on the estimate of the dust mass, which is then converted into gas mass using the locally-calibrated relation between the gas-to-dust ratio and the gas metallicity [e.g., @Munoz-Mateos2009; @Leroy2011; @Remy2014]. The main weakness of this method is the need of an accurate estimate of the gas metallicity and the possible evolution in normalization and scatter of the relation between gas-to-dust ratio and gas metallicity. This method was applied on individual galaxies at high redshift by @Magdis2011 [@Magdis2012b] and @Scoville2014, but also on mean spectral energy distributions (SEDs) measured through a stacking analysis [@Magdis2012b; @Santini2014]. This method has not been applied at redshifts higher than $\sim$2. The aim of this paper is to extend the studies of dust emission and gas fractions derived from dust masses to z$\sim$4 and analyze possible differences in trends as redshift increases.\ In this paper, we combine the information provided by the *Herschel* data and a mass-selected sample of galaxies built from the UltraVISTA data [@Ilbert2013] in COSMOS to study the mean dust emission of galaxies up to z=4 (Sect.\[data\]). We measure the mean SED of galaxies on the main sequence and strong starbursts using a stacking analysis. We then deduce the mean intensity of the radiation field and the mean dust mass in these objects using the @Draine2007 model (Sect.\[stackfit\]). We discuss the observed evolution of these quantities in Sect.\[results\] and the consequences on the nature of star formation processes at high redshift in Sect.\[discussion\]. Throughout this paper, we adopt a $\Lambda$CDM cosmology with $\Omega_m = 0.3$, $\Omega_\Lambda = 0.7$, $H_0 = 70$km/s/Mpc and a @Chabrier2003 initial mass function (IMF).\ Data ==== Stellar mass and photometric redshift catalog using UltraVISTA data {#masscat} ------------------------------------------------------------------- Deep Y, J, H, and K$_{\rm s}$ data (m$_{\rm AB, 5\sigma} \sim$ 25 for the Y band and 24 for the others) were produced by the UltraVISTA survey [@McCracken2012]. The photometric redshift and the stellar mass of the detected galaxies were estimated using Le PHARE [@Arnouts1999; @Ilbert2006] as described in @Ilbert2013. The precision of the photometric redshifts at 1.5$<$z$<$4 is $\sigma_{\rm \Delta z / (1+z)}$ = 0.03. According to @Ilbert2013, this catalog is complete down to $10^{10.26}$M$_\odot$ at z$<$4. X-ray detected active galactic nuclei (AGNs) are also removed from our sample of star-forming galaxies, since the mid-infrared emission of these objects could be strongly affected the AGN. Luminous X-ray obscured AGNs might still be present in the sample. However, their possible presence appear to have limited impact on our work as no mid-infrared excess is observed in the average SEDs measured by stacking (see Fig.\[fig:sedms\] and \[fig:sedsb\] and Sect.\[results\]).\ As this paper studies star-forming galaxies, we focused only on star-forming galaxies selected following the method of @Ilbert2010 based on the rest-frame $\rm NUV-r^{+}$ versus $\rm r^{+}-J$ and similar to the UVJ criterion of [@Williams2009]. The flux densities in each rest-frame band are extrapolated from the closest observer-frame band to minimize potential biases induced by the choice of template library. At z$>$1.5, 40-60% of the objects classified as passive by this color criterion have a sSFR$>10^{-11}$yr$^{-1}$ according to the SED fitting of the optical/near-IR data [@Ilbert2013 their Fig.3]. However, the sSFRs obtained by SED fitting are highly uncertain, because of the degeneracies with the dust attenuation. These peculiar objects are at least 10 times less numerous than the color-selected star-forming sample in all redshift bins. Including them or not in the sample has a negligible impact ($\sim0.25$$\sigma$) on the mean SEDs measured by stacking (see Sect.\[stackfit\]). We thus based our study only on the color-selected population for simplicity.\ *Spitzer*/MIPS data ------------------- The COSMOS field (2 deg$^2$) was observed by *Spitzer* at 24$\mu$m with the multiband imaging photometer (MIPS). A map and a catalog combined with the optical and near-IR data was produced from these observations [@Le_Floch2009]. The 1$\sigma$ point source sensitivity is $\sim$15$\mu$Jy and the full width at half maximum (FWHM) of the point spread function (PSF) is $\sim$6".\ *Herschel*/PACS data -------------------- The PACS (photodetecting array camera and spectrometer, @Poglitsch2010) evolutionary probe survey (PEP, @Lutz2011) mapped the COSMOS field with the *Herschel*[^1] space observatory [@Pilbratt2010] at 100 and 160$\mu$m with a point-source sensitivity of 1.5 mJy and 3.3 mJy and a PSF FWHM of 7.7“ and 12”, respectively. Sources and fluxes of the PEP catalog were extracted using the position of 24$\mu$m sources as a prior. This catalog is used only to select strong starbursts up to z$\sim$3. The 24$\mu$m prior should not induce any incompleteness of the strong-starburst sample, since their minimum expected 24$\mu$m flux is at least 2 times larger than the detection limit at this wavelength [^2].\ *Herschel*/SPIRE data --------------------- We also used *Herschel* data at 250$\mu$m, 350$\mu$m, and 500$\mu$m taken by the spectral and photometric imaging receiver (SPIRE, @Griffin2010) as part of the *Herschel* multitiered extragalactic survey (HerMES, @Oliver2012). The FWHM of the PSF is 18.2“, 24.9”, and 36.3", the 1$\sigma$ instrumental noise is 1.6, 1.3, and 1.9mJy/beam, and the 1$\sigma$ confusion noise is 5.8, 6.3, and 6.8mJy/beam [@Nguyen2010] at 250$\mu$m, 350$\mu$m, and 500$\mu$m, respectively. In this paper, we used the sources catalog extracted using as a prior the positions, the fluxes, the redshifts, and mean colors measured by stacking of 24$\mu$m sources, as described in @Bethermin2012b.\ LABOCA data ----------- The COSMOS field was mapped at 870$\mu$m by the large APEX bolometer Camera (LABOCA) mounted on the Atacama Pathfinder Experiment (APEX) telescope[^3] (PI: Frank Bertoldi, Navarrete et al. in prep.). We retrieved the raw data from the ESO Science Archive facility and reduced them with the publicly available CRUSH (version 2.12–2) pipeline [@Kovacs2006; @Kovacs2008]. We used the algorithm settings optimized for deep field observations[^4]. The output of CRUSH includes an intensity map and a noise map. The mapped area extends over approximately 1.4 square degrees with a non-uniform noise that increases toward the edges of the field. In this work we use the inner $\sim$0.7deg$^2$ of the map where a fairly uniform sensitivity of $\sim$4.3 mJy/beam is reached (Pannella et al. in prep.) with a smoothed beam size of $\sim$27.6". Contrary to SPIRE data, which are confusion limited, LABOCA data are noise limited and the maps are thus beam-smoothed to minimize their RMS. AzTEC data ---------- An area of 0.72deg$^2$ was scanned by the AzTEC bolometer camera mounted on the Atacama submillimeter telescope experiment (ASTE). The sensitivity in the center of the field is 1.23mJy RMS and the PSF FWHM after beam-smoothing is 34" [@Aretxaga2011].\ Methods {#stackfit} ======= Sample selection ---------------- ![\[fig:massdistr\] Stellar mass distribution of our samples of star-forming galaxies in the various redshift bins we used. Only galaxies more massive than our cut of $3\times 10^{10}\,\rm M_\odot$ are represented. The first bin contain fewer objects than the second one because our cut fall at the middle of the first one. The arrows indicate the mean stellar mass in each redshift bin.](Mass_distrib.eps) In this paper, we base our analysis on mass-selected samples of star-forming galaxies (see Sect.\[masscat\]). We chose the same stellar mass cut of $3 \times 10^{10}$M$_\odot$ at all redshifts to be complete up to z$\sim$4. We could have used a lower mass cut at lower redshifts, but we chose this single cut for all redshifts to be able to interpret the observed evolution of the various physical parameters of the galaxies in our sample in an easier way. This cut is slightly higher than the 90% completeness limit at z$\sim$4 cited in @Ilbert2013 [1.8$\times$10$^{10}$M$_\odot$] and implies an high completeness of our sample, which limits potential biases induced by the input catalog on the results of our stacking analysis [e.g., @Heinis2013]. The exact choice of our stellar mass cut has negligible impact on the mean SEDs measured by stacking: we tested a mass cut of $2 \times 10^{10}$M$_\odot$ and $5 \times 10^{10}$M$_\odot$ and found that, after renormalization at the same L$_{\rm IR}$, the SEDs are similar ($\chi_{\rm red}^2$ = 0.57 and 0.79, respectively). These results agree with @Magdis2012b, who did not find any evidence of a dependence of the main-sequence SED on stellar mass at fixed redshift. The mass distribution of star-forming galaxies does not vary significantly with redshift, except in normalization (@Ilbert2013 and Fig.\[fig:massdistr\]). The average stellar mass at all redshifts is between $10^{10.75}$M$_\odot$ and $10^{10.80}$M$_\odot$ (Fig.\[fig:massdistr\] and Table\[tab:physpar\]).\ Star-forming galaxies whose stellar mass is larger than our cut do not correspond to the same populations at z=4 and z=0. The massive objects at z=4 are formed in dense environments, corresponding to the progenitors of today’s clusters and massive groups [e.g., @Conroy2009; @Moster2010; @Behroozi2012a; @Bethermin2013; @Bethermin2014]. Most of these objects are in general quenched between z=4 and z=0 [e.g., @Peng2010]. In contrast, our mass cut at z=0 corresponds to Milky-Way-like galaxies. At all redshift, this cut is just below the mass corresponding to the maximal efficiency of star formation inside halos (defined here as the ratio between stellar mass and halo mass, @Moster2010 [@Behroozi2010; @Bethermin2012b; @Wang2013; @Moster2013]).\ Our stellar mass cut is slightly below the knee of the mass function of star-forming galaxies [@Ilbert2013]. The population we selected thus hosts the majority ($>$50%) of the stellar mass in star-forming galaxies. Since there is a correlation between stellar mass and SFR, we are thus probing the population responsible for a large fraction the star formation (40-65% depending on the redshift according to the @Bethermin2012b model, see also @Karim2011). Our approach is thus different from @Santini2014 who explore in detail how the SEDs evolve at z$<$2.5 in the SFR-M$_\star$ plane using a combination of UV-derived and 24$\mu$m-derived SFRs. We aim to push our analysis to higher redshifts and we thus use this more simple and redshift-invariant selection to allow an easier interpretation and to limit potential selection biases. In addition to this mass selection, we divide our sample by intervals of redshift. The choice of their size is a compromise between large intervals to have a good signal-to-noise ratio at each wavelength and small intervals to limit the broadening of the SEDs because of redshift evolution within the broad redshift bin.\ We also removed strong starbursts from our sample (sSFR$>$10 sSFR$_{\rm MS}$) and studied them separately. These objects are selected using the photometric catalogs described in Sect.\[data\]. For the sources which are detected at 5$\sigma$ at least in two *Herschel* bands, we fitted the SEDs with the template library of @Magdis2012b allowing the mean intensity of the radiation field $\langle U \rangle$ to vary by $\pm$0.6dex (3$\sigma$ of the scatter used in the @Bethermin2012c model). These criteria of two detections at different wavelengths and the high reliability of the detections prevent biasing of the starbursts towards positive fluctuations of the noise in the maps and limit the flux boosting effect. We then estimated the SFR from the infrared luminosity, L$_{\rm IR}$, using the @Kennicutt1998 relation. We performed a first analysis using the same evolution of the main-sequence (sSFR$_{\rm MS}$ versus z) as in @Bethermin2012c to select sSFR$>$10 sSFR$_{\rm MS}$ objects. We then fit the measured evolution of the main-sequence found by a first stacking analysis (see Sect.\[sect:stacking\] and Sect.\[sect:sedfit\]) to prepare the final sample for our analysis. We could have chosen a lower sSFR cut corresponding to 4 times the value at the center of the main-sequence as in @Rodighiero2011, but the sample would be incomplete at z$>$1 because of the flux limit of the infrared catalogs.\ ![\[SBcomp\] The thick red solid line represents the luminosity limit corresponding to a criterion of a 5$\sigma$ detection in at least two *Herschel* bands. The other solid lines are the limits for a detection at only one given wavelength (purple for 100$\mu$m, blue for 160$\mu$m, turquoise for 250$\mu$m, green for 350$\mu$m, orange for 500$\mu$m). The dashed, dot-dash, and three-dot-dash lines indicate the infrared luminosity of a galaxie of $3 \times 10^{10}$M$_\odot$ (our mass cut) at the center of the main sequence, a factor of 4 above it, and a factor of 10 above it, respectively.](LIRlim.eps) Fig.\[SBcomp\] shows the luminosity limit corresponding to a detection at 5$\sigma$ at two wavelengths or more. This was computed using both the starburst and the main-sequence templates of the @Magdis2012b SED library. This library contains different templates for main-sequence and starburst galaxies. The main-sequence template evolves with redshift, but not the starburst one. The lines correspond to the highest luminosity limit found using these two templates for each wavelength, which is the most pessimistic case. We also computed the infrared luminosity associated with a galaxy of $3\times 10^{10}\,\rm M_\odot$, i.e., our mass limit, on the main sequence (dashed line), a factor of 4 above it (dot-dash line), and a factor of 10 above it (three-dot-dash line). All the M$_\star > 3\times 10^{10}\,\rm M_\odot$ strong starbursts (sSFR$>$10 sSFR$_{\rm MS}$) should thus be detected in two or more *Herschel* bands below z=4. There is only one starburst detected in the 3$<$z$<$4 bin. We thus do not analyze this bin, because of its lack of statistical significance. The other bins contain 3, 6, 6, and 8 strong starbursts, respectively, by increasing redshift.\ The sample of main-sequence galaxies is contaminated by the starbursts which have sSFR$<$10 sSFR$_{\rm MS}$ . We expect that this contamination is negligible, since the contribution of all starbursts to the infrared luminosity density is lower than 15% [@Rodighiero2011; @Sargent2012]. To check this hypothesis, we statistically corrected for the contribution of the remaining starbursts with sSFR$<$10 sSFR$_{\rm MS}$ based on the @Bethermin2012b counts model. We assumed both the SED library used for the model and the average SED of strong starbursts found in this study. We found that this statistical subtraction only affected our measurements at most at the 0.2$\sigma$ level. Consequently, we have neglected this contamination in the rest of our study.\ [ccccccccc]{} Redshift & S$_{24}$ & S$_{100}$ & S$_{160}$ & S$_{250}$ & S$_{350}$ & S$_{500}$ & S$_{850}$ & S$_{1100}$\ & $\mu$Jy & mJy& mJy& mJy& mJy& mJy& mJy & mJy\ \ 0.25$<$z$<$0.50 & 410$\pm$23 & 11.87$\pm$0.76 & 23.30$\pm$1.49 & 12.54$\pm$0.97 & 6.43$\pm$0.53 & 2.64$\pm$0.32 & -0.18$\pm$0.23 & 0.21$\pm$0.08\ 0.50$<$z$<$0.75 & 247$\pm$13 & 6.37$\pm$0.43 & 13.82$\pm$0.86 & 9.45$\pm$0.72 & 5.88$\pm$0.46 & 2.57$\pm$0.25 & 0.54$\pm$0.15 & 0.18$\pm$0.06\ 0.75$<$z$<$1.00 & 221$\pm$10 & 4.19$\pm$0.26 & 9.79$\pm$0.60 & 7.75$\pm$0.59 & 5.92$\pm$0.45 & 3.06$\pm$0.25 & 0.53$\pm$0.19 & 0.30$\pm$0.06\ 1.00$<$z$<$1.25 & 144$\pm$7 & 3.31$\pm$0.23 & 8.22$\pm$0.50 & 6.93$\pm$0.53 & 5.78$\pm$0.46 & 3.00$\pm$0.25 & 0.21$\pm$0.15 & 0.30$\pm$0.05\ 1.25$<$z$<$1.50 & 96$\pm$5 & 2.36$\pm$0.14 & 6.70$\pm$0.42 & 5.99$\pm$0.45 & 5.46$\pm$0.41 & 3.17$\pm$0.25 & 0.44$\pm$0.13 & 0.32$\pm$0.04\ 1.50$<$z$<$1.75 & 110$\pm$6 & 1.80$\pm$0.12 & 4.81$\pm$0.33 & 4.79$\pm$0.38 & 4.64$\pm$0.36 & 3.00$\pm$0.25 & 0.54$\pm$0.11 & 0.34$\pm$0.04\ 1.75$<$z$<$2.00 & 113$\pm$5 & 1.31$\pm$0.10 & 3.51$\pm$0.25 & 4.10$\pm$0.32 & 4.11$\pm$0.33 & 2.94$\pm$0.24 & 0.72$\pm$0.12 & 0.32$\pm$0.04\ 2.00$<$z$<$2.50 & 101$\pm$5 & 1.16$\pm$0.08 & 3.28$\pm$0.22 & 4.17$\pm$0.32 & 4.38$\pm$0.34 & 3.25$\pm$0.25 & 0.73$\pm$0.12 & 0.48$\pm$0.04\ 2.50$<$z$<$3.00 & 59$\pm$3 & 0.79$\pm$0.07 & 2.59$\pm$0.22 & 3.41$\pm$0.29 & 3.85$\pm$0.31 & 3.03$\pm$0.26 & 0.87$\pm$0.17 & 0.55$\pm$0.05\ 3.00$<$z$<$3.50 & 47$\pm$5 & 0.61$\pm$0.10 & 2.28$\pm$0.33 & 2.90$\pm$0.30 & 3.65$\pm$0.35 & 2.95$\pm$0.31 & 0.56$\pm$0.18 & 0.44$\pm$0.07\ 3.50$<$z$<$4.00 & 29$\pm$7 & 0.22$\pm$0.20 & 1.68$\pm$0.55 & 2.60$\pm$0.45 & 3.01$\pm$0.51 & 2.52$\pm$0.50 & 0.24$\pm$0.33 & 0.30$\pm$0.14\ \ 0.50$<$z$<$1.00 & 1241$\pm$329 & 57.48$\pm$15.98 & 86.33$\pm$18.31 & 41.57$\pm$7.83 & 16.52$\pm$3.53 & 9.64$\pm$4.73 & 6.91$\pm$5.92 & 2.40$\pm$1.57\ 1.00$<$z$<$1.50 & 264$\pm$77 & 30.59$\pm$3.26 & 64.44$\pm$6.97 & 38.44$\pm$4.92 & 24.79$\pm$3.98 & 13.90$\pm$4.97 & 0.12$\pm$2.62 & 1.36$\pm$0.78\ 1.50$<$z$<$2.00 & 912$\pm$179 & 23.51$\pm$5.04 & 62.46$\pm$13.80 & 42.47$\pm$8.02 & 30.99$\pm$9.27 & 21.46$\pm$7.09 & 2.10$\pm$3.37 & 3.90$\pm$1.16\ 2.00$<$z$<$3.00 & 629$\pm$193 & 13.15$\pm$4.91 & 39.56$\pm$7.77 & 32.25$\pm$4.37 & 35.72$\pm$5.40 & 28.52$\pm$5.20 & 7.98$\pm$2.97 & 5.08$\pm$1.02\ [cccccccc]{} Redshift & log(M$_\star$) & log(L$_{\rm IR}$) & SFR & log(M$_{\rm dust}$) & $\langle U \rangle$ & log(M$_{\rm mol}$) & f$_{\rm mol}$\ & log(M$_\odot$) & log(L$_\odot$) & M$_\odot$/yr & log(M$_\odot$) & & log(M$_\odot$) &\ \ 0.25$<$z$<$0.50 & 10.77 & 10.92$_{-0.04}^{+0.03}$ & 8.3$_{-0.7}^{+0.6}$ & 8.09$_{-0.16}^{+0.12}$ & 5.50$_{-1.50}^{+3.10}$ & 10.04$_{-0.22}^{+0.19}$ & 0.16$_{-0.06}^{+0.07}$\ 0.50$<$z$<$0.75 & 10.76 & 11.19$_{-0.04}^{+0.08}$ & 15.6$_{-1.5}^{+3.3}$ & 8.24$_{-0.15}^{+0.19}$ & 7.23$_{-2.47}^{+3.82}$ & 10.23$_{-0.21}^{+0.24}$ & 0.23$_{-0.07}^{+0.11}$\ 0.75$<$z$<$1.00 & 10.75 & 11.45$_{-0.09}^{+0.07}$ & 27.9$_{-5.4}^{+4.7}$ & 8.44$_{-0.24}^{+0.16}$ & 7.80$_{-2.69}^{+5.44}$ & 10.48$_{-0.28}^{+0.22}$ & 0.35$_{-0.13}^{+0.12}$\ 1.00$<$z$<$1.25 & 10.77 & 11.56$_{-0.04}^{+0.10}$ & 36.4$_{-3.3}^{+9.7}$ & 8.29$_{-0.11}^{+0.28}$ & 15.05$_{-6.68}^{+5.74}$ & 10.34$_{-0.18}^{+0.32}$ & 0.27$_{-0.07}^{+0.16}$\ 1.25$<$z$<$1.50 & 10.76 & 11.69$_{-0.04}^{+0.07}$ & 48.6$_{-4.2}^{+8.9}$ & 8.37$_{-0.10}^{+0.22}$ & 16.52$_{-6.47}^{+5.45}$ & 10.46$_{-0.18}^{+0.26}$ & 0.33$_{-0.08}^{+0.15}$\ 1.50$<$z$<$1.75 & 10.77 & 11.77$_{-0.05}^{+0.05}$ & 58.9$_{-5.9}^{+7.5}$ & 8.45$_{-0.21}^{+0.18}$ & 16.96$_{-6.15}^{+10.90}$ & 10.55$_{-0.26}^{+0.23}$ & 0.37$_{-0.13}^{+0.13}$\ 1.75$<$z$<$2.00 & 10.79 & 11.81$_{-0.03}^{+0.05}$ & 64.4$_{-4.3}^{+8.2}$ & 8.49$_{-0.25}^{+0.18}$ & 16.96$_{-6.15}^{+15.24}$ & 10.63$_{-0.29}^{+0.23}$ & 0.41$_{-0.15}^{+0.13}$\ 2.00$<$z$<$2.50 & 10.79 & 11.99$_{-0.02}^{+0.03}$ & 97.4$_{-5.3}^{+7.7}$ & 8.53$_{-0.19}^{+0.13}$ & 22.58$_{-6.27}^{+14.42}$ & 10.81$_{-0.24}^{+0.20}$ & 0.51$_{-0.13}^{+0.11}$\ 2.50$<$z$<$3.00 & 10.80 & 12.11$_{-0.04}^{+0.03}$ & 130.0$_{-12.6}^{+10.7}$ & 8.48$_{-0.11}^{+0.23}$ & 33.75$_{-14.29}^{+12.85}$ & 10.88$_{-0.18}^{+0.27}$ & 0.55$_{-0.10}^{+0.15}$\ 3.00$<$z$<$3.50 & 10.77 & 12.25$_{-0.05}^{+0.05}$ & 178.5$_{-18.4}^{+22.4}$ & 8.48$_{-0.12}^{+0.10}$ & 48.99$_{-11.32}^{+23.99}$ & 10.99$_{-0.19}^{+0.18}$ & 0.62$_{-0.11}^{+0.09}$\ 3.50$<$z$<$4.00 & 10.80 & 12.34$_{-0.12}^{+0.07}$ & 219.0$_{-54.4}^{+40.2}$ & 8.39$_{-0.50}^{+0.33}$ & 72.98$_{-36.98}^{+167.95}$ & 11.06$_{-0.52}^{+0.36}$ & 0.65$_{-0.29}^{+0.16}$\ \ 0.50$<$z$<$1.00 & 10.57 & 12.25$_{-0.08}^{+0.08}$ & 179.1$_{-150.5}^{+215.0}$ & 8.65$_{-0.04}^{+0.19}$ & 29.80$_{-11.77}^{+9.60}$ & 10.04$_{-0.24}^{+0.30}$ & 0.29$_{-0.10}^{+0.16}$\ 1.00$<$z$<$1.50 & 10.60 & 12.55$_{-0.05}^{+0.03}$ & 350.8$_{-314.5}^{+376.4}$ & 8.99$_{-0.01}^{+0.09}$ & 26.92$_{-6.92}^{+2.88}$ & 10.23$_{-0.23}^{+0.25}$ & 0.45$_{-0.13}^{+0.14}$\ 1.50$<$z$<$2.00 & 10.64 & 12.93$_{-0.18}^{+0.07}$ & 860.1$_{-567.4}^{+1006.8}$ & 9.24$_{-0.09}^{+0.62}$ & 37.68$_{-28.40}^{+11.32}$ & 10.48$_{-0.25}^{+0.66}$ & 0.58$_{-0.14}^{+0.28}$\ 2.00$<$z$<$3.00 & 10.69 & 13.10$_{-0.24}^{+0.07}$ & 1260.0$_{-728.1}^{+1487.1}$ & 9.64$_{-0.47}^{+0.37}$ & 22.22$_{-12.94}^{+50.77}$ & 10.34$_{-0.52}^{+0.44}$ & 0.75$_{-0.28}^{+0.14}$\    Stacking analysis {#sect:stacking} ----------------- We use a similar stacking approach as in @Magdis2012b to measure the mean SEDs of our sub-samples of star-forming galaxies from the mid-infrared to the millimeter domain. Different methods are used at the various wavelength to optimally extract the information depending if the data are confusion or noise limited. At 24$\mu$m, 100$\mu$m, and 160$\mu$m, we produced stacked images using the IAS stacking library [@Bavouzet2008; @Bethermin2010a]. The flux is then measured using aperture photometry with the same parameters and aperture corrections as @Bethermin2010a at 24$\mu$m. At 100$\mu$m and 160$\mu$m, we used a PSF fitting technique. A correction of 10% is applied to take into account the effect of the filtering of the data on the photometric measurements of faint, non-masked sources [@Popesso2012]. At 250$\mu$m, 350$\mu$m, and 500$\mu$m, the photometric uncertainties are not dominated by instrumental noise but by the confusion noise caused by neighboring sources [@Dole2003; @Nguyen2010]. We thus measured the mean flux of the sources computing the mean flux in the pixels centered on a stacked source following @Bethermin2012b. This method minimizes the uncertainties and a potential contamination caused by the clustering of galaxies [@Bethermin2010b]. Finally, we used the same method, but on the beam-convolved map, for LABOCA and AzTEC data as they are noise limited and lower uncertainties are obtained after this beam smoothing. LABOCA and AzTEC maps do not cover the whole area. We thus only stack sources in the covered region to compute the mean flux densities of our various sub-samples. The source selection criteria being exactly the same inside and outside the covered area, this should not introduce any bias.\ These stacking methods can be biased if the stacked sources are strongly clustered or very faint. This bias is caused by the greater probability of finding a source close to another one in the stacked sample compared to a random position. This effect has been discussed in detail by several authors [e.g., @Bavouzet2008; @Bethermin2010b; @Kurczynski2010; @Bethermin2012b; @Bourne2012; @Viero2013b]. In @Magdis2012b, the authors estimated that this bias is lower than the 1$\sigma$ statistical uncertainties and was not corrected. The number of sources to stack in COSMOS compared to the GOODS fields used by @Magdis2012b is much larger and hence the signal-to-noise ratio is much better. The bias caused by clustering is thus non-negligible in COSMOS. Because of the complex edge effects caused by the absence of data around bright stars, the methods using the position of the sources to deblend the contamination caused by the clustering cannot be applied [@Kurczynski2010; @Viero2013b]. Consequently, we developed a method based on realistic simulations of the *Spitzer*, *Herschel*, LABOCA, and AzTEC maps to correct this effect, which induces biases up to 50% at 500$\mu$m around z$\sim$2. The technical details and discussion of these corrections are presented in Appendix\[Annexestacking\].\ The uncertainties on the fluxes are measured using a bootstrap technique [@Jauzac2011]. This method takes into account both the errors coming from the instrumental noise, the confusion, and the sample variance of the galaxy population [@Bethermin2012b]. These uncertainties are combined quadratically with those associated with the calibration and the clustering correction.\ Mean physical properties from SED fitting {#sect:sedfit} ----------------------------------------- We interpreted our measurements of the mean SEDs using the @Draine2007 model as in @Magdis2012b. This model, developed initially to study the interstellar medium in the Milky Way and in nearby galaxies, takes into account the heterogeneity of the intensity of the radiation field. The redshift slices we used have a non-negligible width. To account for this, we convolve the model by the redshift distribution of the galaxies before fitting the data. The majority of the redshifts in our sample are photometric. We thus sum the probability distribution function (PDF) of the redshifts of all the sources in a sub-sample to estimate its intrinsic redshift distribution. The uncertainties on the physical parameters are estimated using the same Monte Carlo method as in @Magdis2012b. The uncertainties on each parameter takes into account the potential degeneracies with the others, i.e., they are the marginalized uncertainties on each individual parameters. Our good sampling of the dust SEDs (8 photometric points between 24$\mu$m and 1.1mm including at least six detections) allows us to break the degeneracy between the dust temperature and the dust mass which is found if only (sub-)mm datapoints are used.\ Instead of using the three parameters describing the distribution of the intensity of the radiation field U of the @Draine2007 model (the minimal radiation field $\rm U_{min}$, the maximal one $\rm U_{max}$, and the slope of the assumed power-law distribution between these two values $\alpha$), we considered only the mean intensity of the radiation field $\langle U \rangle$ for simplicity. The other parameters derived from the fit and used in this paper are the bolometric infrared luminosity integrated between 8 and 1000$\mu$m (L$_{\rm IR}$) and the dust mass (M$_d$). The SFR is derived from L$_{\rm IR}$ using the @Kennicutt1998 conversion factor ($1 \times 10^{-10}$$\rm M_\odot \, yr^{-1} \, L_\odot^{-1}$ after conversion from Salpeter to Chabrier IMF), since the dust-obscured star formation vastly dominates the unobscured component at z$<$4 given the mass-scale considered [@Heinis2013; @Heinis2014; @Pannella2014]. The sSFR is computed using the later SFR and the mean stellar mass extracted from the @Ilbert2013 catalog. The uncertainties on the derived physical parameters presented in the various figures and tables of this paper are the uncertainties on the average values. The dispersion of physical properties inside a population is difficult to measure by stacking and we did not try to compute it in this paper (see Sect.\[discussion\]).\ The residuals of these fits are presented in Appendix\[sect:residuals\]. Tables\[tab:fluxes\] and \[tab:physpar\] summarize the average photometric measurements and the recovered physical parameters, respectively.\ Results ======= Evolution of the mean SED of star-forming galaxies -------------------------------------------------- Figure\[fig:sedobs\] summarizes the results of our stacking analysis. For the main-sequence sample, the flux density varies rapidly with redshift in the PACS 100$\mu$m band, while it is almost constant in the SPIRE 500$\mu$m band. The peak of the flux density distribution in the rest frame moves from $\sim$120$\mu$m to 70$\mu$m between z=0 and z=4. This shift with redshift was already observed at z$\lesssim$2 for mass-selected stacked samples [@Magdis2012b] or a *Herschel*-detected sample [@Lee2013; @Symeonidis2013]. We found no evidence of an evolution of the position of this peak ($\sim$70$\mu$m) for the sample of strong starbursts.\ Figure\[fig:sedms\] and \[fig:sedsb\] show the mean intrinsic luminosity (in $\nu$L$_\nu$ units, the peak of the SEDs is thus shifted toward shorter wavelengths compared with L$_\nu$ units) of our samples of massive star-forming galaxies (since this sample is dominated by main-sequence galaxies, hereafter we call it main-sequence sample) and the fit by the @Draine2007 model. We also observe a strong evolution of the position of the peak of the thermal emission of dust in main-sequence galaxies from $\sim$80$\mu$m at z$\sim$0.4 to $\sim$30$\mu$m at z$\sim$3.75 in $\nu$L$_\nu$ units. The SEDs of strong starbursts have a much more modest evolution (from 50$\mu$m at to 30$\mu$m). The mean luminosity of the galaxies also increases very rapidly with redshift for both main-sequence and strong starburst galaxies.\ At z$>$2, we find that the peak of the dust emission tends to be broader than at lower redshift. The broadening of the mean SEDs induced by the size of the redshift bins has a major impact only on the mid-infrared, where the polycyclic aromatic hydrocarbon (PAH) features are washed out (see black and blue lines in Fig.\[fig:sedms\] and \[fig:sedsb\]), and cannot fully explain why the far-IR peak is broader at higher redshifts. The @Draine2007 model reproduces this broadening by means of a higher $\gamma$ coefficient, i.e., a stronger contribution of regions with a strong heating of the dust. This is consistent with the presence of giant star-forming clumps in high-redshift galaxies [e.g., @Bournaud2007; @Genzel2006]. The best-fit models at high z presents two breaks around 30$\mu$m and 150$\mu$m, which could be artefacts caused by the sharp cuts of the U distribution at its extremal values in the @Draine2007 model.\ Evolution of the specific star formation rate --------------------------------------------- From the fit of the SEDs, we can easily derive the evolution of the mean specific star formation rate of our mass-selected sample with redshift. The results are presented in Fig.\[fig:ssfr\]. The strong starbursts lie about a factor of 10 above the main-sequence, demonstrating that this population is dominated by objects just above our cut of 10 sSFR$_{\rm MS}$. Our results can be fitted by an evolution in redshift as (0.061$\pm$0.006Gyr$^{-1}$)$\times$(1+z)$^{2.82\pm0.12}$ at z$<$2 and as (1+z)$^{2.2\pm0.3}$ at z$>$2. We compared our results with the compilation of measurements of @Sargent2014 at M$_\star = 5 \times 10^{10}$M$_\odot$. At z$<$1.5, our results agree well with the previous measurements. Between z=1.5 and z=3.5, our new measurements follow the lower envelop of the previous measurements. This mild disagreement could have several causes.\ First of all, the clustering effect was not taken into account by the previous analyses based on stacking. This effect is stronger at high redshift, because the bias[^5] of both infrared and mass-selected galaxies increases with redshift [e.g., @Bethermin2013]. In addition, the SEDs peak at a longer wavelength, where the bias is stronger owing to beam size (see Sect.\[sect:simu\]). The tension with the results based on UV-detected galaxies could be explained by a slight incompleteness of the UV-detected samples at low sSFR or a small overestimate of the dust corrections. There could also be effects caused by the different techniques and assumptions used to determine the stellar masses in the various fields (star formation histories, PSF-homogenized photometry or not, presence of nebular emission in the highest redshift bins, template libraries, etc.). Finally, this difference could also be an effect of the variance. These discrepancies on the estimates of sSFRs will be discussed in detail in @Schreiber2014.\ \[fig:sSFR\] ![\[fig:ssfr\] Evolution of the mean sSFR in main-sequence galaxies (blue triangles) and strong starbursts (red squares). The gray diamonds are a compilation of measurements at the same mass performed by @Sargent2014. The blue line is the best fit to our data.](sSFR_z.eps) Evolution of the mean intensity of the radiation field {#sect:U} ------------------------------------------------------ ![\[fig:U\] Evolution of the mean intensity of the radiation field $\langle U \rangle$ in main-sequence galaxies (blue triangles) and strong starbursts (red squares). The black diamonds are the measurements presented in @Magdis2012b based on a similar analysis but in the GOODS fields. The orange asterisk is the mean value found for the local ULIRG sample of @Da_Cunha2008b (see also @Magdis2012b). The black circle is the average value in HRS galaxies [@Ciesla2014]. The solid and dashed lines represent the evolutionary trends expected for a broken and universal FMR, respectively (see Sect.\[sect:U\]). The blue dotted line is the best fit of the evolution of the main-sequence galaxies ($(3.0\pm1.1) \times (1+z)^{1.8 \pm 0.4}$) and the red dotter line the best fit of the strong starburst data by a constant ($31\pm3$).](U_z.eps) The evolution of the mean intensity of the radiation field has different trends in main-sequence galaxies than in strong starbursts (see Fig.\[fig:U\]). This quantity is strongly correlated to the temperature of the dust. We found a rising $\langle U \rangle$ with increasing redshift in main-sequence galaxies up to z=4 ($(3.0\pm1.1) \times (1+z)^{1.8\pm0.4}$), confirming and extending the finding of @Magdis2012b at higher redshift. Other studies [e.g., @Magnelli2013; @Genzel2014] found an increase of the dust temperature with redshift in mass-selected samples.\ The evolution of $\langle U \rangle$ we found can be understood from a few simple assumptions on the evolution of the gas metallicity and the star-formation efficiency (SFE) of galaxies. As shown by @Magdis2012b, $\langle U \rangle$ is proportional to L$_{\rm IR}$/M$_{\rm dust}$. We can also assume that $$L_{\rm IR} \propto \textrm{SFR} \propto M_{\rm mol}^{1/s},$$ where the left-side of the proportionality is the well-established @Kennicutt1998 relation. The right-side of the proportionality is the integrated version of the Schmidt-Kennicutt relation which links the SFR to the mass of molecular gas in a galaxy (M$_{\rm mol}$). @Sargent2014 found a best-fit value for $s$ of 0.83 compiling a large set of public data about low- and high-redshift main-sequence galaxies. The molecular gas mass can also be connected to the gas metallicity Z and the dust mass [e.g., @Leroy2011; @Magdis2012b], $$M_{\rm dust} \propto Z(M_\star, \textrm{SFR}) \times M_{\rm mol},$$ where $Z(M_\star, \textrm{SFR})$ is the gas metallicity which can be connected to M$_\star$ and SFR through the fundamental metallicity relation (FMR, @Mannucci2010). There is recent evidence showing that this relation breaks down at high redshifts. For instance, @Troncoso2014 measured a $\sim$0.5dex lower normalization at z$\sim$3.4 compared to the functional form of the FMR at low redshift. @Amorin2014 found the same offset in a lensed galaxy at z = 3.417. At z$\sim$2.3, @Steidel2014 [see also @Cullen2014] found an offset of 0.34–0.38dex in the mass-metallicity relation and only half of this difference can be explained by the increase of SFR at fixed stellar mass using the FMR. Finally, a break in the metallicity relation is also observed in low mass (log(M$_\star$/M$_\odot$)$\sim$8.5) damped Lyman $\alpha$ absorbers around z=2.6 [@Moller2013]. In our computations, we consider two different relations: a universal FMR where metallicity depends only on M$_\star$ and SFR, and a FMR relation with a correction of $0.30 \times (1.7-z)$dex at z$>$1.7 (hereafter broken FMR), which agrees with the measurements cited previously. Combining these expressions, we can obtain the following evolution: $$\langle U \rangle \propto \frac{L_{\rm IR}}{M_{\rm dust}} \propto \frac{M_{\rm mol}^{\frac{1}{s}-1}}{Z(M_\star, \textrm{SFR})} \propto \frac{\textrm{SFR}^{1-s}}{Z(M_\star, \textrm{SFR})}.\\$$ We computed the expected evolution of $\langle U \rangle$ using the fit to the evolution of sSFR presented in Sect.\[fig:sSFR\] and assuming the mean stellar mass of our sample is $6\times10^{10}$M$_\odot$, the average mass of the main-sequence sample[^6]. We used the value of @Magdis2012b at z=0 to normalize our model. The results are presented in Fig.\[fig:U\] for a universal and a broken FMR. The broken FMR is compatible with all of our data points at 1$\sigma$. The universal FMR implies a significant underestimation of $\langle U \rangle$ at high redshifts (3 and 2$\sigma$ in the two highest redshift bins).\ We checked that the dust heating by the cosmic microwave background (CMB) is not responsible for the quick rise the quick rise in main-sequence galaxies. The CMB temperature at z=4 is 13.5K. The dust temperature that our high-redshift galaxies would have for a virtually z=0 CMB temperature, $T_{\rm dust}^{z=0}$, is estimated following @Da_Cunha2013 $$T_{\rm dust}^{z=0} = \left ( (T_{\rm dust}^{\rm meas})^{4+\beta} - (T_{\rm CMB}^{z=0})^{4+\beta} \left [ (1+z)^{4+\beta} -1 \right ] \right )^{\frac{1}{4+\beta}},$$ where $T_{\rm CMB}^{z=0}$ is the temperature of the CMB at z=0 and $T_{\rm dust}^{\rm meas}$ is the measured dust temperature at high redshift. This temperature is estimated fitting a gray body with an emissivity of $\beta$=1.8 to our photometric measurements at $\lambda_{\rm rest}>$50$\mu$m. The CMB has a relative impact which is lower than 2$\times 10^{-4}$ at all redshifts and thus this effect is negligible. These values are small compared to @Da_Cunha2013, who assumed a dust temperature of 18K. The warmer dust temperatures we measured suggests that the CMB should be less problematic than anticipated.\ Concerning the evolution of $\langle U \rangle$ in strong starbursts, we found no evidence of evolution ($\propto (1+z)^{-0.1\pm1.0}$) and our results can be fitted by a constant $\langle U \rangle$ of 31$\pm$3. Our value of $\langle U \rangle$ at 0.5$<$z$<$3 is similar to the measurements on a sample of local ULIRGs [@Da_Cunha2008b]. This suggests that high-redshift strong starbursts are a more extended version of the nuclei of local ULIRGs, as also suggested by the semi-analytical model of @Lagos2012. At z$\sim$2.5, the main-sequence galaxies and the strong starbursts have similar $\langle U \rangle$ values. However, we do not interpret the origins of these high values of $\langle U \rangle$ in the same way (see Sect.\[sect:mdms\], \[sect:fgas\], and \[discussion\]). At z$>2.5$, we cannot constrain with our analysis if $\langle U \rangle$ in strong starbursts rises as in main-sequence galaxies or stays constant.\ Evolution of the ratio between dust and stellar mass {#sect:mdms} ---------------------------------------------------- We also studied the evolution of the mean ratio between the dust and the stellar mass in the main-sequence galaxies and the strong starbursts. The results are presented Fig.\[fig:MdMs\]. In main-sequence galaxies, this dust-to-stellar-mass ratio rises up to z$\sim$1 and flattens above this redshift. Strong starbursts typically have 5 times higher ratio. Our measurements are compatible within 2$\sigma$ with the slowly rising trend of $(1+z)^{0.05}$ found by @Tan2014 for a compilation of individual starbursts. However, our data favors a steeper slope.\ ![\[fig:MdMs\] Mean ratio between dust and stellar mass as a function of redshift in main-sequence galaxies (blue triangles) and strong starbursts (red squares). The orange asterisk is the mean value found for the local ULIRG sample of @Da_Cunha2008b (see @Magdis2012b). The black circle is the average value in HRS galaxies [@Ciesla2014]. The solid and dashed lines represent the evolutionary trends expected for a broken and universal FMR, respectively (see Sect.\[sect:U\]). The red dot-dashed line is the best-fit of the evolution found for a sample of individually-detected starbursts of @Tan2014. The predictions of the models of @Lagos2012 and @Lacey2014 after applying the same mass cut and sSFR selection are overplotted with a three-dot-dash line and a long-dash line, respectively, with the same color code as the symbols.](Md_Mstar_z.eps) We modeled the evolution of this ratio in main-sequence galaxies using the same simple considerations as in Sect.\[sect:U\]. The evolution of the mean dust-to-stellar-mass ratio can be written as $$\frac{M_{\rm dust}}{M_\star} \propto \frac{Z(M_\star, \textrm{SFR}) \times M_{\rm mol}}{M_\star} \propto \frac{Z(M_\star, \textrm{SFR}) \times \textrm{SFR}^\beta}{M_\star}.$$ One can see that $M_{\rm dust}/M_\star$ is the result of a competition between the rising SFR with increasing redshift and the decreasing gas metallicity. The results are compatible with the broken FMR at 1$\sigma$. The relation obtained with the universal FMR rises too rapidly at high redshift.\ We also compared our results with predictions of two semi-analytical models. The @Lagos2012 and @Lacey2014 models are based on GALFORM. The main difference between these two models is that [@Lagos2012] adopt a universal IMF (a Galactic-like IMF; @Kennicutt1983), while @Lacey2014 adopt a non-universal IMF. In the latter star formation taking place in galaxy disks has a Galactic-like IMF, while starbursts have a more top-heavy IMF. This is done to reproduce the number counts of submillimeter galaxies found by surveys.\ We select galaxies in the models in the same way we do in the observations based on their stellar mass and distance from the main sequence. An important consideration is that to derive stellar masses in the observations we fix the IMF to a Chabrier IMF, which is different to the IMFs adopted in both models. In order to correct for this we multiply stellar masses in the @Lagos2012 model by 1.1 to go from a Kennicutt IMF to a Chabrier IMF. However, this is non-trivial for the @Lacey2014 model, since it adopts two different IMFs. In order to account for this we correct the fraction of the stellar mass that was formed in the disk by the same factor of 1.1, and divide the fraction of stellar mass that was formed during starbursts by 2. The latter factor is taken as an approximation to go from their adopted top-heavy IMF to a Chabrier IMF, but this conversion is not necessarily unique, and it depends on the dust extinction and stellar age (see @Mitchell2013 for details). In this paper we make a unique correction, but warn the reader that a more accurate approach would be to perform SED fitting to the predicted SEDs of galaxies and calculating the stellar mass in the same way we would do for observations.\ Compared to the observations of main-sequence galaxies, the @Lagos2012 model reproduces observations well in the redshift range 1$<$z$<$3, while at $z<1$ and $z>3$ it overpredicts the dust-to-stellar mass ratio. There are different ways to explain the high dust-to-stellar mass ratios: high gas metallicities, high gas masses or stellar masses being too low for the dust masses. In the case of the @Lagos2012 model the high dust-to-stellar mass ratios are most likely coming from massive galaxies being too gas rich since their metallicities are close to solar, which is what we observe in local galaxies of the same stellar mass range. The @Lacey2014 model predicts dust-to-stellar mass ratios that are twice too high compared to the observations in the whole redshift range. In this case this is because the gas metallicities of MS galaxies in the Lacey model are predicted to be supersolar on average (close to twice the solar metallicity, 12+log(O/H)$\sim$9.0), resulting in dust masses that are higher than observed.\ In the case of starbursts, the high values inferred for the dust-to-stellar mass ratio in the observations are difficult to interpret. The @Lagos2012 model underpredicts this quantity by a factor of $\sim$5 and the @Lacey2014 model by a factor of $\sim$2. At first the ratio of 1.5-2% inferred in the observations seems unphysical. However, since the gas fraction (defined here as $\rm M_{\rm mol}/(M_{\rm mol}+M_\star)$) in these high-redshift starbursts is around 50% (see Sect.\[sect:fgas\], but also, e.g., @Riechers2013 and @Fu2013), the high values observed for the dust-to-stellar mass ratio can be reached if the gas-to-dust ratio is 50-67. Values similar to the latter are observed in metal-rich galaxies (12+log(O/H)$\sim$9, e.g., @Remy2014). This high metal enrichment in strong starbursts compared to main-sequence galaxies could be explained by several mechanisms: - the transformation of gas into stars is quicker and the metals are not diluted by the accretion of pristine gas; - a fraction of the external layers of low-metallicity gas far from the regions of star formation could be ejected by the strong outflows caused by these extreme starbursts; - a top-heavy IMF could produce quickly lots of metals through massive stars without increasing too rapidly the total stellar mass because of mass losses. This high ratio in strong starbursts is discussed in details in @Tan2014.\ When it comes to the comparison with the models, one can understand the lower dust-to-stellar mass ratios predicted by the model as resulting from the predicted gas metallicities. @Lagos2012 predict that the average gas metallicity in strong starbursts is close to 0.4 solar metallicities (12+log(O/H)$\sim$8.3), which is about 4 times lower than we can infer from a gas-to-dust mass ratio of $\approx 50$ (see previous paragraph). While the @Lacey2014 model predicts gas metallicities for starbursts that are on average close to solar metallicity (12+log(O/H)$\sim$8.7), 2 times too low for the inferred metallicity of the strong starbursts we observe. We note that both models predict main sequence galaxies having higher metallicities than bright starbursts of the same stellar masses. This seems to contradict the observations and may be at the heart of why the models struggle to get the dust-to-stellar mass ratios of both the main sequence and starburst populations at the same time. ![\[fig:gasfrac\] Evolution of the mean molecular gas fraction in massive galaxies ($>3\times10^{10}$M$_\odot$). The starbursts are represented by red squares and the main-sequence galaxies by blue triangles or light blue diamonds depending on wether the gas fraction is estimated using a broken or an universal FMR, respectively. These results are compared with previous estimate using dust masses of @Magdis2012b [black plus] and @Santini2014 [gray area], using CO for two z$>$3 galaxies [@Magdis2012a black crosses], and the compilation of CO measurements of @Saintonge2013 [black asterisks]. The predictions of the models of @Lagos2012 and @Lacey2014 for the same mass cut are overplotted with a three-dot-dash line and a long-dash line, respectively.](fgas_z.eps) Evolution of the fraction of molecular gas {#sect:fgas} ------------------------------------------ Finally, we deduced the mean mass of molecular gas from the dust mass using the same method following @Magdis2011 and @Magdis2012b. They assumed that the gas-to-dust ratio depends only on gas metallicity and used the local relation of @Leroy2011[^7]: $$\textrm{log} \left ( \frac{M_{\rm dust}}{M_{\rm mol}} \right ) = (10.54 \pm 1.0) - (0.99\pm0.12) \times {12 + \textrm{log(O/H)}}. \label{eq:gdr}$$ Given the relatively high stellar mass of our samples, and the rising gas masses and ISM pressures to high redshifts [@Obreschkow2009], we expect the contribution of atomic hydrogen to the total gas mass to be negligible and we neglect it in the rest of the paper, considering total gas mass or molecular gas mass to be equivalent. For main-sequence galaxies, the gas metallicity is estimated using the FMR as explained in Sect.\[sect:U\]. We converted the values provided by the FMR from the KD02 to the PP04 metallicity scale using the prescriptions of @Kewley2008 before using it in Eq.\[eq:gdr\].\ The gas metallicity in strong starbursts cannot be estimated using the FMR. Indeed, this relation predicts that, at fixed stellar mass, objects forming more stars are less metallic. This effect is expected in gas regulated systems, because a higher accretion of pristine gas involves a stronger SFR, but also a dilution of metals [e.g., @Lilly2013]. This phenomenon is not expected to happen in starbursts, since their high SFRs are not caused by an excess of accretion, but more likely by a major merger. These high-redshift starbursts are probably progenitors of current, massive, elliptical galaxies [e.g., @Toft2014]. We thus assumed that their gas metallicity is similar and used a value of 12+log(O/H) = 9.1$\pm$0.2 (see a detailed discussion in @Magdis2011 and @Magdis2012b).\ We then derived the molecular gas fraction in main-sequence galaxies, defined in this paper as $\rm M_{mol} / (M_\star + M_{mol})$. The results are presented in Fig.\[fig:gasfrac\]. We found a quick rise up to z$\sim$2. At higher redshifts, the recovered trend depends on the assumptions on the gas metallicity. The rise of the gas fraction in main-sequence galaxies continues at higher redshift if we assume the broken FMR favored by the recent studies, but flattens with a universal FMR. If the broken FMR scenario is confirmed, there could thus be no flattening or reversal of the molecular gas fraction at z$>$2 contrary to what is claimed in @Magdis2012a, @Saintonge2013, and @Tan2013. Our estimations agree with the previous estimates of @Magdis2012b at z=1, but are 1$\sigma$ lower at z=2, because the bias introduced by clustering was corrected in our study. Our results also agree at 1$\sigma$ with the analysis of @Santini2014 at the same stellar mass up to z=2.5 after converting the stellar mass from a Salpeter to a Chabrier IMF convention. However, our estimates are systematically higher than theirs and agree better with the CO data. Our measurements also agree with the compilation of CO measurements of @Saintonge2013 and the two galaxies studied at z$\sim$3 by @Magdis2012a. These measurements are dependent on the assumed $\alpha_{\rm CO}$ conversion factor, and on the completeness corrections. The good agreement with this independent method is thus an interesting clue to the reliability of these two techniques.\ Strong starbursts have molecular gas fractions 1$\sigma$ higher than main-sequence galaxies, but follows the same trend. @Sargent2014 predicted that starbursts on average should have a deficit of gas compared to the main sequence (but that gas fraction are expected to rise continuously as the sSFR-excess with respect to the MS increases). Here we selected only the most extreme starbursts with an excess of sSFR of a factor of 10 instead of the average value of $\sim$4. These extreme starbursts may only be possible by the mergers of two gas-rich galaxies galaxies already above the main-sequence before the merger. This could explain this small positive offset compared to the main-sequence sample.\ We also compared our results with the models of @Lagos2012 and @Lacey2014 presented in Sect.\[sect:mdms\]. Both models agree well with our measurements of the gas fraction for starburst galaxies at all redshifts and main-sequence galaxies at 1.5$<$z$<$3. Both the @Lagos2012 and @Lacey2014 models overpredict the molecular gas fraction at z$<$0.5 at a 1-2$\sigma$ level. At reshifts $z>3$, the @Lacey2014 model agrees better with the universal FMR scenario at z$>$3, while the @Lagos2012 model is more compatible with the broken FMR. The fact that both models predict molecular gas fractions that in overall agree with the observations supports our interpretation in Sect. \[sect:mdms\], which points to the model of metal enrichment as the source of discrepancy in the dust-to-stellar mass ratios.\ Evolution of the depletion time ------------------------------- We estimated the mean depletion time of the molecular gas, defined in our analysis as the ratio between the mass of molecular gas and the SFR. Figure\[fig:tdep\] shows our results. The depletion time in strong starbursts does not evolve with redshift and is compatible with 100Myr, the typical timescale of the strong boost of star formation induced by major mergers [e.g., @Di_Matteo2008]. This timescale is longer in main-sequence galaxies and slightly (1$\sigma$) evolves with redshift at z$<$1. It decreases from 1.3$_{-0.5}^{+0.7}$Gyr at z$\sim$0.375 to $\sim$500Myr around z$\sim$1.5 and is stable at higher redshift in the case of a broken FMR (but continues to decrease with redshift for a universal FMR). This timescale is similar to the maximum duration high-redshift massive galaxies can stay on the main-sequence before reaching the quenching mass around 10$^{11}$M$_\odot$ [@Heinis2014]. The mass of molecular gas and stars contained in these high-redshift objects is already sufficient to reach this quenching mass without any additional accretion of gas.\ ![\[fig:tdep\] Evolution of the mean molecular gas depletion time. The symbols are the same as in Fig.\[fig:gasfrac\].](tdep_z.eps) Discussion ========== What is the main driver of the strong evolution of the specific star formation rate? ------------------------------------------------------------------------------------ ![\[fig:iKS\] Relation between the mean SFR rate and the mean molecular gas mass in our galaxy samples, i.e., integrated Kennicutt-Schmidt law. The solid line and the dashed line are the center of the relation fitted by @Sargent2014 on a compilation of data for main-sequence galaxies and starbursts, respectively. The dotted lines represent the 1$\sigma$ uncertainties on these relations.](iKS.eps) The triangles and diamonds represent the average position of massive, main-sequence galaxies in this diagram assuming a broken FMR and an universal FMR, respectively. The squares indicates the average position of strong starbursts. We checked the average position of our selection of massive galaxies in the integrated Kennicutt-Schmidt diagram (SFR versus mass of molecular gas) to gain insight on their mode of star formation. In this diagram, normal star-forming galaxies and starbursts follow two distinct sequences. For comparison, we used the fit of a recent data compilation performed by @Sargent2014. The results are presented in Fig.\[fig:iKS\].\ The average position of our sample of strong starbursts is in the 1$\sigma$ confidence region of @Sargent2014 for starbursts. They are systematically below the central relation, but the uncertainty is dominated by the systematic uncertainties on their gas metallicity. In addition, @Sargent2014 suggested that the SFEs of starbursts follow a continuum of values depending on their boost of sSFR. Our objects are thus not expected to be exactly on the central relation. The interpretation of the results for main-sequence galaxies is dependent on the hypothesis on the gas metallicity. In the scenario of a broken FMR favored by recent observations, the average position of main-sequence galaxies at all redshifts falls on the relation of normal star-forming galaxies. This suggests that the star formation is dominated by galaxies forming their stars through a normal mode at all redshifts below z=4. In the case of a non-evolving FMR, the massive high-redshift galaxies do not stay on the normal star-forming sequence and have higher SFEs.\ If the scenario of a broken FMR favored by the most recent observations is consolidated, the strong star-formation observed in massive high-redshift galaxies would thus be caused by huge gas reservoirs probably fed by an intense cosmological accretion. This strong accretion of primordial gas dilute the metals produced by the massive stars [e.g., @Bouche2010; @Lilly2013]. Consequently, the gas-to-dust ratio is much lower at high redshift than at low redshift. Since the star-formation efficiency is only slowly evolving (SFR$\propto$M$_{\rm mol}^{1.2}$), the number of UV photons absorbed per mass of dust is thus higher and the dust temperature is warmer as observed in our analysis (see Sect.\[sect:U\]). This scenario provides thus a consistent interpretation of evolution of both the sSFR and the dust temperature of massive galaxies with redshift.\ Limitations of our approach --------------------------- Our analysis provided suggestive results. However, it relies on several hypotheses, which cannot be extensively tested yet. In this section, we discuss the potential limitation of our analysis.\ The evolution of the metallicity relations at z$>$2.5 was measured only by a few pioneering works, which found that the normalization of the FMR evolves at z$>$2.5. We used a simple renormalization depending on redshift to take this evolution into account. The redshift sampling of these studies is relatively coarse and we used a simple linear evolution with redshift. Future studies based on larger samples will allow a finer sampling of the evolution of the gas metallicity in massive galaxies at high redshift. However, the current results are very encouraging. The current assumption of a broken FMR allows us to recover naturally both the evolution of the $\langle U \rangle$ parameter and the integrated Schmidt-Kennicutt relation at high redshift.\ The gas metallicity of strong starbursts was more problematic to set. We can reasonably guess it assuming they are progenitors of the most massive galaxies. However, direct measurements of their gas metallicity are difficult to perform using optical/near-IR spectroscopy because of their strong dust attenuation. The millimeter spectroscopy of fine-structure lines with ALMA will be certainly an interesting way to determine the distribution of gas metallicity of strong starbursts in the future [e.g., @Nagao2011].\ The validity of the calibration of the gas-to-dust ratio versus gas metallicity relation in most extreme environment is also uncertain and difficult to test with the current data sets. @Saintonge2013 found an offset of a factor 1.7 for a population of lensed galaxies and discussed the possible origins of the tension between the gas content estimated from CO and from dust. However, we found no offset with the integrated Kennicutt-Schmidt relation in our analysis and a good agreement with the compilation of CO measurements of gas fractions. The lensed galaxies of @Saintonge2013 could be a peculiar population because they are UV-selected and then biased toward dust-poor systems. They could also be affected differential magnification effects or *Herschel*-selection biases. The hypotheses performed to estimate the gas metallicity are also different between their and our analysis (standard mass-metallicity relation versus broken FMR).\ Finally, the stacking analysis only provides an average measurement of a full population. Thus it is difficult to estimate the heterogeneity of the stacked populations. Bootstrap techniques can be applied to estimate the scatter on the flux density at a given wavelength [@Bethermin2012b]. However, because of the correlation between $\langle U \rangle$ and M$_d$, this technique cannot be applied to measure the scatter on each of these parameters.\ Conclusion ========== We used a stacking analysis to measure the evolution of the average mid-infrared to millimeter emission of massive star-forming galaxies up to z=4. We then derived the evolution of the mean physical parameters of these objects. Our main findings are the following. - The mean intensity of the radiation field $\langle U \rangle$ in main-sequence galaxies, which is strongly correlated with their dust temperature, rises rapidly with redshift: $\langle U \rangle = (3.0\pm1.1) \times (1+z)^{1.8 \pm 0.4}$. This evolution can be interpreted considering the decrease in the gas metallicity of galaxies at constant stellar mass with increasing redshift. We found no evidence for an evolution of $\langle U \rangle$ in strong starbursts up to z=3.\ - The mean ratio between the dust mass and the stellar mass in main-sequence galaxies rises between z=0 and z=1 and exhibit a plateau at higher redshift. The strong starbursts have a higher ratio by a factor of 5.\ - The average fraction of molecular gas ($\rm M_{mol} / (M_\star + M_{mol})$) rises rapidly with redshift and reaches $\sim$60% at z=4. A similar evolution is found in strong starbursts, but with slightly higher values. These results agree with the pilot CO surveys performed at high redshift.\ - We compare with two state-of-the-art semi-analytic models that adopt either a universal IMF or a top-heavy IMF in starbursts and find that the models predict molecular gas fractions that agree well with the observations but the predicted dust-to-stellar mass ratios are either too high or too low. We interpret this as being due to the way metal enrichment is dealt with in the simulations. We suggest different mechanisms that can help overcome this issue. For instance, outflows affecting more metal depleted gas that is in the outer parts of galaxies.\ - The average position of the massive main-sequence galaxies in the integrated Kennicutt-Schmidt diagram corresponds to the sequence of normal star-forming galaxies. This suggests that the bulk of the star-formation up to z$\sim$4 is dominated by the normal mode of star-formation and that the extreme SFR observed are caused by huge gas reservoirs probably induced by the very intense cosmological accretion. The strong starbursts follow another sequence with a 5–10 times higher star-formation efficiency.\ We thank the anonymous referee for providing constructive comments. We acknowledge Morgane Cousin, Nick Lee, Nick Scoville, and Christian Maier for their interesting discussions/suggestions, Laure Ciesla for providing an electronic table of the physical properties of the HRS sample, and Amélie Saintonge for providing her compilation of data. We gratefully acknowledge the contributions of the entire COSMOS collaboration consisting of more than 100 scientists. The HST COSMOS program was supported through NASA grant HST-GO-09822. More information on the COSMOS survey is available at <http://www.astro.caltech.edu/cosmos>. ased on data obtained from the ESO Science Archive Facility. Based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under ESO programme ID 179.A-2005 and on data products produced by TERAPIX and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium. MB, ED, and MS acknowledge the support of the ERC-StG UPGAL 240039 and ANR-08-JCJC-0008 grants. AK acknowledges support by the Collaborative Research Council 956, sub-project A1, funded by the Deutsche Forschungsgemeinschaft (DFG). Estimation and correction on the bias caused by the galaxy clustering on the stacking results {#Annexestacking} ============================================================================================= As explained in Sect.\[sect:stacking\], the standard stacking technique can be strongly affected by the bias caused by the clustering of the galaxies. We use two independent methods to estimate and correct it. Estimation of the bias using a simulation based on the real catalog {#sect:simu} ------------------------------------------------------------------- We performed an estimate of the bias induced by the clustering using a realistic simulation of the COSMOS field based on the positions and stellar masses of the real sources. The flux of each source in this simulation is estimated using the ratio between the mean far-IR/(sub-)mm fluxes and the stellar mass found by a first stacking analysis. The galaxies classified as passive are not taken into account in this simulation. This technique assumes implicitly a flat sSFR-M$_\star$ relation, since we use a constant SFR/M$_\star$ ratio versus stellar mass at fixed redshift. However, we checked that using a more standard sSFR$\propto$M$_\star^{-0.2}$ relation [e.g., @Rodighiero2011] has a negligible impact on the results. We applied no scatter around this relation in our simulation for simplicity. As mean stacking is a linear operation, the presence or not of a scatter has no impact on the results [@Bethermin2012b].\ A simulated map is thus produced using all the star-forming galaxies of the @Ilbert2013 catalog. In order to avoid edge effects (absence of sources and thus a lower background caused by the faint unresolved sources in the region covered by the optical/near-IR data), we fill the uncovered regions drawing with replacement sources from the UltraVISTA field and putting them at a random position. The number of drawn sources is chosen to have exactly the same number density inside and outside the UltraVISTA field.\ Finally, we measured the mean fluxes of the M$_\star>$3$\times$10$^{10}$M$_\odot$ sources by stacking in the simulated maps, using exactly the same photometric method as for the real data. We finally computed the relative bias between the recovered flux and the input flux ($S_{\rm out}/S_{\rm in}-1$). The results are shown Fig.\[fig:clusbias\] (blue triangles). The uncertainties are computed a bootstrap method. As expected, the bias increases with the size of the beam. We can see a rise of the bias with redshift up to z$\sim$2. This trend can be understood considering the rise of the clustering of the galaxy responsible for the cosmic infrared background [@Planck_CIB_2013] and a rather stable number density of emitters especially below z=1 [@Bethermin2011; @Magnelli2013; @Gruppioni2013]. At higher redshift, we found a slow decrease. This trend is probably driven by the decrease in the infrared luminosity density at high redshift [@Planck_CIB_2013; @Burgarella2013] combined with the decrease in the number density of infrared emitters [@Gruppioni2013].\ ![\[fig:clusbias\] Relative bias induced by the clustering as a function of redshift at the various wavelengths we used in our analysis. The FWHM of the beam is provided in brackets. The blue triangles are the estimations from the simulation (Sect.\[sect:simu\]) and the red diamonds are provided by the fit of the clustering component in map space (Sect.\[sect:fitclus\]). These numbers are only valid for a complete sample of M$_\star > 3 \times 10^{10}$M$_\odot$ galaxies.](Stacking_bias_Mcut310.eps) Estimation of the bias fitting the clustering contribution in the stacked images {#sect:fitclus} -------------------------------------------------------------------------------- The method presented in the previous section only takes into account the contamination of the stacks by known sources. However, faint galaxy populations could have a non negligible contribution, despite their total contribution to the infrared luminosity and their clustering are expected to be small. We thus used a second method to estimate the bias caused by the clustering which takes into account a potential contamination by these low-mass galaxies. This method is based on a simultaneous fit in the stacked images of three components: a point source at the center of the image, a clustering contamination, and a background. This technique was already successfully used by several previous works based on *Herschel* and *Planck* data [@Bethermin2012b; @Heinis2013; @Heinis2014; @Welikala2014].\ In presence of clustering, the outcome of a stacking is not only a PSF with the mean flux of the population and a constant background (corresponding to the surface brightness of all galaxy populations i.e., the cosmic infrared background). There is in addition a signal coming from the greater probability of finding another neighboring infrared galaxy compared to the field because of galaxy clustering. The signal in the stacked image can thus be modeled by [@Bavouzet2008; @Bethermin2010b] $$m(x,y) = \alpha \times \textrm{PSF}(x,y) + \beta \times (\textrm{PSF} \ast w)(x,y) +\gamma,$$ where $m$ is the stacked image, PSF the point spread function, and $w$ the auto-correlation function. The symbol $\ast$ represents the convolution. $\alpha$, $\beta$, and $\gamma$ are free parameters corresponding to the intensity of the mean flux of the population, the clustering signal, and the background, respectively. This method works only if the PSF is well-known, the extension of the sources is negligible compared to the PSF, and the effects of the filtering are small at the scale of the stacked image. Consequently, we applied this method only to the SPIRE data for which these hypotheses are the most solid. The uncertainties on the clustering bias ($\beta / \alpha$ for the photometry we chose to use for SPIRE data) are estimated fitting the model described previously on a set of stacked images produced from 1000 bootstrap samples. The results are shown in Fig.\[fig:clusbias\] (red diamonds).\ Corrections of the measurements ------------------------------- In Fig.\[fig:clusbias\], we can see that the two methods provide globally consistent estimates. This confirms that the low-mass galaxies not included in the UltraVISTA catalog have a minor impact. We found few outliers for which the two methods disagree. In particular, in the 1.5$<$z$<$1.75 bin, the estimation from the simulation is higher than the trend of the redshift evolution at all wavelengths, and the results from the profile fitting are lower. This could be caused, as instance, by a structures in the field or a systematic effect in the photometric redshift. Because of these few catastrophic outliers, we chose to use a correction computed from a fit of the redshift evolution of the bias instead of an individual estimate in each redshift slice.\ The evolution of the bias with redshift is fitted independently at each wavelength. We chose to use a simple, second-order, polynomial model ($a z^2 + bz + c$). We used only the results from the simulation to have a consistent treatment of the various wavelengths. The scatter of the residuals is larger than the residuals, probably because bootstrap does not take into account the variance coming from the large-scale structures. We thus used the scatter of the residuals to obtain a conservative estimate of the uncertainties on the bias. In Fig.\[fig:clusbias\], the best fit is represented by a solid line and the 1$\sigma$ confidence region by a dashed line.\ In a few case, the bias at z$>$3 can converge to unphysical negative values. We then apply no corrections, but combine the typical uncertainty on the bias to the error bars. A special treatment is also applied to the samples of strong starbursts. Their flux is typically 10 times brighter in infrared by construction (their sSFR is 10 times larger than the main sequence). In contrast, the clustering signal is not expected to be significantly stronger, because the clustering of massive starbursts and main-sequence galaxies is relatively similar [@Bethermin2014]. We thus divide the bias found for the full population of galaxy by a factor of 10 to estimate the one of the starbursts for simplicity.\ Testing another method ---------------------- We also tried to apply the <span style="font-variant:small-caps;">simstack</span> algorithm [@Viero2013b] to our data. This algorithm is adapted from @Kurczynski2010 and uses the position of the known sources to deblend their contamination. Contrary to @Kurczynski2010, <span style="font-variant:small-caps;">simstack</span> can consider a large set of distinct galaxy populations. The mean flux of the each population is used to estimate how sources contaminate their neighbors. All populations are treated simultaneously. This is the equivalent of PSF-fitting codes but applied to a full population instead of each source individually. Unfortunately, this method is not totally unbiased in our case. We found biases up to 15% running <span style="font-variant:small-caps;">simstack</span> on the simulation presented in Sect.\[sect:simu\], probably because the catalog of mass-selected sources is not available around bright sources. At the edge of the optical/near-IR-covered region, the flux coming from the sources outside the covered area is not corrected, when the flux from all neighbors is taken into account at the middle of zone where the mass catalog is extracted. Indeed, the algorithm works correctly if we put on the simulation only sources present in the input catalog.\ Fit residuals {#sect:residuals} ============= Figures\[fig:res\] and \[fig:res\_sb10\] shows the residuals of the fits of our mean SEDs derived by stacking. We did not find any systematic trend, except a 2$\sigma$ underestimation of the millimeter data in main-sequence galaxies at z$>$3.\   [^1]: [*Herschel*]{} is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. [^2]: The minimum expected flux for our mass-selected sample of strong starbursts is computed using the three-dot-dash curve in Fig.\[SBcomp\] and the @Magdis2012b starburst template. [^3]: APEX project IDs: 080.A–3056(A), 082.A–0815(A) and 086.A–0749(A). [^4]: More details on the CRUSH settings can be found at: [http://www.submm.caltech.edu/\$\\sim\$sharc/crush/v2/README](http://www.submm.caltech.edu/$\sim$sharc/crush/v2/README) [^5]: The bias $b$ is defined by $w_{\rm gal} = b^2 w_{\rm DM}$, where $w_{\rm gal}$ and $w_{\rm DM}$ are the projected two-point correlation function of galaxies and dark matter, respectively. The higher the bias is, the stronger is the clustering density of galaxies compared to dark matter. [^6]: We could have used the mean stellar masses in each redshift bin provided in Table\[tab:physpar\]. However, assuming a single value of the stellar mass at all redshift has a negligible impact on the results and the tracks are smoother. [^7]: converted to PP04 convention

--- abstract: 'Treewidth is a graph parameter of fundamental importance to algorithmic and structural graph theory. This paper describes several graph parameters tied to treewidth, including separation number, tangle number, well-linked number and cartesian tree product number. We prove that these parameters are tied to treewidth. In a number of cases we also improve known bounds, provide simpler proofs and show that the inequalities presented are tight.' address: - 'Department of Mathematics and Statistics The University of Melbourne Melbourne, Australia' - 'School of Mathematical Sciences Monash University Melbourne, Australia' author: - 'Daniel J. Harvey' - 'David R. Wood' bibliography: - 'myBibliography.bib' - 'extra.bib' title: Parameters Tied to Treewidth --- [^1] [^2] Introduction ============ Treewidth is an important graph parameter for two key reasons. Firstly, treewidth has many algorithmic applications; for example, there are many results showing that NP-Hard problems can be solved in polynomial time on classes of graphs with bounded treewidth (see @Bodlaender-AC93 for a survey). Treewidth is inherently related to graph separators, which are “small" sets of vertices whose removal leaves no component with more than half the vertices (or thereabouts). Separators are particularly useful when using dynamic programming to solve graph problems; find and delete a separator, recursively solve the problem on the remaining components, and then combine these solutions to obtain a solution for the original problem. Secondly, treewidth is a key parameter in graph structure theory, especially in Robertson and Seymour’s seminal series of papers on graph minors [@RS-GraphMinors]. Ultimately, the purpose of these papers was to prove what it now known as the Graph Minor Theorem (often referred to as Wagner’s Conjecture), which states that any class of minor-closed graphs (other than the class of all graphs) has a finite set of forbidden minors. In order to prove this, Robertson and Seymour separately considered classes with bounded treewidth and classes with unbounded treewidth. The Graph Minor Theorem is (comparatively) easy to prove for classes with bounded treewidth [@RS-GraphMinorsIV-JCTB90]. In order to prove the Graph Minor Theorem for classes with unbounded treewidth, Robertson and Seymour showed that graphs with large treewidth contain large grid minors. This Grid Minor Theorem has been reproved by many researchers; we will discuss it more thoroughly in Section \[section:gridminors\]. In proving these results, the parameters *linkedness* and *well-linked number* were used. At the heart of the Graph Minor Theorem is the Graph Minor Structure Theorem, which describes how to construct a graph in a minor-closed class; see @KM-GC07 for a survey of several versions of the Graph Minor Structure Theorem. The most complex version, and the one used in the proof of the Graph Minor Theorem, describes the structure of graphs in a minor-closed class with unbounded treewidth in terms of *tangles*. Robertson and Seymour combined all these ingredients in their proof of the Graph Minor Theorem. The purpose of this paper is to present a number of known graph parameters that are closely related to treewidth, including those mentioned above such as separation number, linkedness, well-linked number and tangle number. Formally, a *graph parameter* is a real-valued function $\alpha$ defined on all graphs such that $\alpha(G_1)=\alpha(G_2)$ whenever $G_1$ and $G_2$ are isomorphic. Two graph parameters $\alpha(G)$ and $\beta(G)$ are *tied*[^3] if there exists a function $f$ such that for every graph $G$, $$\alpha(G) \leq f(\beta(G)) \text{ and } \beta(G) \leq f(\alpha(G)).$$ Moreover, say that $\alpha$ and $\beta$ are *polynomially tied* if $f$ is a polynomial. Drawing on results in the literature, we prove the following: \[theorem:pttt\] The following graph parameters are polynomially tied: - treewidth, - bramble number, - minimum integer $k$ such that $G$ is a spanning subgraph of a $k$-tree, - minimum integer $k$ such that $G$ is a spanning subgraph of a chordal graph with no $(k+2)$-clique, - separation number, - branchwidth, - tangle number, - lexicographic tree product number, - cartesian tree product number, - linkedness, - well-linked number, - maximum order of a grid minor, - maximum order of a grid-like-minor, - Hadwiger number of the Cartesian product $G\square K_2$ (viewed as a function of $G$), - fractional Hadwiger number, - $r$-integral Hadwiger number for each $r\geq 2$. @Fox11 states (without proof) a theorem similar to Theorem \[theorem:pttt\] with the parameters treewidth, bramble number, separation number, maximum order of a grid minor, fractional Hadwiger number, and $r$-integral Hadwiger number for each $r\geq 2$. Indeed, this statement of Fox motivated the present paper. This paper investigates the parameters in Theorem \[theorem:pttt\], showing where these parameters have been useful, and provides proofs that each parameter is tied to treewidth (except in a few cases). In a number of cases we improve known bounds, provide simpler proofs and show that the inequalities presented are tight. The following graph is a key example. Say $n,k$ are integers. Let $\psi_{n,k}$ be the graph with vertex set $A \cup B$, where $A$ is a clique on $n$ vertices, $B$ is an independent set on $kn$ vertices, and $A \cap B = \emptyset$, such that each vertex of $A$ is adjacent to exactly $k(n-1)$ vertices of $B$ and each vertex of $B$ is adjacent to exactly $n-1$ vertices of $A$. (Note it always possible to add edges in this fashion; pair up each vertex in $A$ with $k$ vertices in $B$ such that all pairs are disjoint, and then add all edges from $A$ to $B$ except those between paired vertices.)  Treewidth and Basics ==================== Let $G$ be a graph. A *tree decomposition* of $G$ is a pair $(T, (B_x \subseteq V(G))_{x \in V(T)})$ consisting of - a tree $T$, - a collection of *bags* $B_x$ containing vertices of $G$, indexed by the nodes of $T$. The following conditions must also hold: - For all $v \in V(G)$, the set $\{ x \in V(T) : v \in B_x\}$ induces a non-empty subtree of $T$. - For all $vw \in E(G)$, there is some bag $B_x$ containing both $v$ and $w$. The *width* of a tree decomposition is defined as the size of the largest bag minus 1. The treewidth ${\textsf{\textup{tw}}}(G)$ is the minimum width over all tree decompositions of $G$. Often, for the sake of simplicity, we will refer to a tree decomposition simply as $T$, leaving the set of bags implied whenever this is unambiguous. For similar reasons, often we say that bags $X$ and $Y$ are adjacent (or we refer to an edge $XY$), instead of the more accurate statement that the nodes of $T$ indexing $X$ and $Y$ are adjacent. Treewidth was defined by @Halin76 (in an equivalent form which Halin called $S$-functions) and independently by @RS-GraphMinorsII-JAlg86. Intuitively, a graph with low treewidth is simple and treelike — note that a tree itself has treewidth 1. (In fact, ensuring this fact is the reason that 1 was subtracted in the definition of width.) On the other hand, a complete graph $K_{n}$ has treewidth $n-1$. Say a tree decomposition is *normalised* if each bag has the same size, and if $|X-Y|=|Y-X|=1$ whenever $XY$ is an edge. \[lemma:normal\] If $G$ has a tree decomposition of width $k$, then $G$ has a normalised tree decomposition of width $k$. Let $T$ be the tree decomposition of $G$ with width $k$. Thus $T$ contains a bag of size $k+1$. If some bag of $T$ does not contain $k+1$ vertices, then as $T$ is connected, there exist adjacent bags $X$ and $Y$ such that $|X|=k+1$ and $|Y|<k+1$. Then $X-Y$ is non-empty; take some vertex of $X-Y$ and add it to $Y$. This increases $|Y|$, so repeat this process until all bags have size $k+1$. Now, consider an edge $XY$. Since $|X|=|Y|$, it follows $|X-Y|=|Y-X|$. If $|X-Y| > 1$, then let $v \in X-Y$ and $u \in Y-X$. Subdivide the edge $XY$ of $T$ and call the new bag $Z$. Let $Z = X - \{v\} + \{u\}$. Now $|X-Z| = 1$ and $|Y-Z|=|Y-X|-1$, so repeat this process until $|X-Y|=|Y-X| \leq 1$ for each pair of adjacent bags. Finally, if $XY$ is an edge and $|X-Y|=0$, then contract the edge $XY$, and let the bag at the contracted node be $X$. Repeat this process so that if $X$ and $Y$ are a pair of adjacent bags, then $|X-Y|=|Y-X|=1$. All of these operations preserve tree decomposition properties and width. Hence this modified $T$ is our desired normalised tree decomposition. A *$k$-colouring* of a graph $G$ is a function that assigns one of $k$ colours to each vertex of $G$ such that no pair of adjacent vertices are assigned the same colour. The *chromatic number* $\chi(G)$ is the minimum number $k$ such that $G$ has a $k$-colouring. A graph $H$ is a *minor* of a graph $G$ if a graph isomorphic to $H$ can be constructed from $G$ by vertex deletion, edge deletion and edge contraction. *Edge contraction* means to take an edge $vw$ and replace $v$ and $w$ with a new vertex $x$ adjacent to all vertices originally adjacent to $v$ or $w$. If $H$ is a minor of $G$, say that $G$ has an $H$-minor. The *Hadwiger number* ${\textsf{\textup{had}}}(G)$ is the order of the largest complete minor of $G$. The Hadwiger number is most relevant to *Hadwiger’s Conjecture* [@Hadwiger43], often considered one of the most important unsolved conjectures in graph theory, which states that $\chi(G) \leq {\textsf{\textup{had}}}(G)$. Hadwiger’s Conjecture can be seen as an extension of the Four Colour Theorem, since every planar graph has ${\textsf{\textup{had}}}(G) \leq 4$. While the conjecture remains unsolved in general, it has been proved for ${\textsf{\textup{had}}}(G) \leq 5$ [@RST-Comb93]. Given a graph $H$, an *$H$-model* of $G$ is a set of pairwise vertex-disjoint connected subgraphs of $G$, called *branches*, indexed by the vertices of $H$, such that if $vw \in E(H)$, then there exists an edge between the branches indexed by $v$ and $w$. If $G$ has an $H$-model, then repeatedly contract the edges inside each branch and delete extra vertices and edges to obtain a copy of $H$. Thus if $G$ has an $H$-model, then $H$ is a minor of $G$. Similarly, if $H$ is a minor of $G$, “uncontract" each vertex in the minor to obtain an $H$-model of $G$. Models are helpful when dealing with questions relating to minors, as they describe how the $H$-minor “sits” in $G$. Brambles ======== Two subgraphs $A$ and $B$ of a graph $G$ *touch* if $V(A)\cap V(B)\neq\emptyset$, or some edge of $G$ has one endpoint in $A$ and the other endpoint in $B$. A *bramble* in $G$ is a set of connected subgraphs of $G$ that pairwise touch. A set $S$ of vertices in $G$ is a *hitting set* of a bramble ${\ensuremath{\mathcal{B}}}$ if $S$ intersects every element of ${\ensuremath{\mathcal{B}}}$. The *order* of ${\ensuremath{\mathcal{B}}}$ is the minimum size of a hitting set. The *bramble number* of $G$ is the maximum order of a bramble in $G$. Brambles were first defined by @SeymourThomas-JCTB93, where they were called *screens of thickness $k$*. Seymour and Thomas proved the following result. \[theorem:twdual\] For any graph $G$, $${\textsf{\textup{tw}}}(G) = {\textsf{\textup{bn}}}(G) - 1.$$ Here, we present a short proof showing one direction of this result. The other (more difficult) direction can be found in [@SeymourThomas-JCTB93]; see @ShortDiestel for a shorter proof. Let $\beta$ be a bramble in $G$ of maximum order, and let $T$ be the underlying tree in a tree decomposition of $G$. For a subgraph $A \in \beta$, let $T_{A}$ be the subgraph of $T$ induced by the nodes of $T$ whose bags contain vertices of $A$. Since $A$ is connected, $T_A$ is also connected. Similarly, if $A,B \in \beta$, then since these subgraphs touch, there is a node of $T$ in both $T_A$ and $T_B$. So the set of subtrees $\{ T_A : A \in \beta \}$ pairwise intersect. By the Helly Property of trees, there is some node $x$ that is in all such $T_A$. The bag indexed by $x$ contains a vertex from each $A \in \beta$, so it is a hitting set of $\beta$. Hence that bag has order at least ${\textsf{\textup{bn}}}(G)$, and so ${\textsf{\textup{tw}}}(G) \geq {\textsf{\textup{bn}}}(G) - 1$. Note that this means that the bramble number is equal to the size of the largest bag in a minimum width tree decomposition. Brambles are useful for finding a lower bound on the treewidth of a graph. Consider the following: given a valid tree decomposition $T$ for a graph $G$, then ${\textsf{\textup{tw}}}(G)$ is at most the width of $T$. Brambles provide the equivalent functionality for the lower bound — given a valid bramble of a graph $G$, it follows that the bramble number is at least the order of that bramble, giving us a lower bound on the treewidth. (For examples of this, see @bodlaenderbramble, @lucenabramble and Lemma \[lemma:gridbramble\].) $k$-Trees and Chordal Graphs ============================ In certain applications, such as graph drawing [@DMW05; @DiGiacomo] or graph colouring [@KP-DM08; @Albertson-EJC04], it often suffices to consider only the edge-maximal graphs of a given family to obtain a result. The language of $k$-trees and chordal graphs provides an elegant description of the edge-maximal graphs with treewidth at most $k$. A vertex $v$ in a graph $G$ is *$k$-simplicial* if it has degree $k$ and its neighbours induce a clique. A graph $G$ is a *$k$-tree* if either: - $G = K_{k+1}$ - $G$ has a $k$-simplicial vertex $v$ and $G-v$ is also a $k$-tree. Note that there is some discrepancy over this definition; certain authors use $K_{k}$ in the base case. This means that $K_{k}$ is a $k$-tree, but creates no other changes. $k$-trees have a strong tie to treewidth; see Lemma \[lemma:ktree\]. A graph is *chordal* if it contains no induced cycle of length at least four. That is, every cycle that is not a triangle contains a chord. @gavril74 showed that the chordal graphs are exactly the intersection graphs of subtrees of a tree $T$. So construct a tree decomposition with underlying tree $T$ as follows. Think of each $v \in V(G)$ as a subtree of $T$; place $v$ in the bags indexed by the nodes of that subtree. It can easily be seen that this is a tree decomposition of $G$ in which every bag is a clique (that is, every possible edge exists), since should two vertices share a bag, then their subtrees intersect and the vertices are adjacent. It also follows that the graph arising from a tree decomposition with all possible edges (that is, two vertices are adjacent if and only if they share a bag) is a chordal graph. Chordal graphs are therefore interesting by being the edge-maximal graphs for a fixed tree-width. \[lemma:ktree\] $$\begin{aligned} {\textsf{\textup{tw}}}(G) &= \min\{ k: G \text{ is a spanning subgraph of a $k$-tree }\}. \\ &= \min\{ k: G \text{ is a spanning subgraph of a chordal graph with no $(k+2)$-clique }\}.\end{aligned}$$ For simplicity, let $a(G) = \min\{ k: G$ is a spanning subgraph of a $k$-tree$\}$ and $b(G) = \min\{ k: G$ is a spanning subgraph of a chordal graph with no $(k+2)$-clique$\}$. First, show $b(G) \leq a(G)$. @FG65 showed that a graph $H$ is chordal if and only if it has a *perfect elimination ordering*; that is, an ordering of the vertex set such that for each $v \in V(H)$, $v$ and all vertices adjacent to $v$ which are after $v$ in the ordering form a clique. If $H$ be an $a(G)$-tree such that $G$ is a spanning subgraph of $H$, then there is a simple perfect elimination ordering for $H$. (Repeatedly delete the $a(G)$-simplicial vertices to obtain $K_{a(G)+1}$, and consider the order of deletion.) So $H$ is chordal. It is clear that each $v$ has only $a(G)$ neighbours after it in this ordering, so $H$ has no $(a(G)+2)$-clique. (For any clique, consider the first vertex of the clique in the ordering, and note at most $a(G)$ other vertices are in the clique.) Thus $b(G) \leq a(G)$. Second, show $a(G) \leq {\textsf{\textup{tw}}}(G)$. Assume for the sake of a contradiction that $G$ is a vertex-minimal counterexample, and say $G$ has treewidth $k$. It is easy to see $a(G) \leq {\textsf{\textup{tw}}}(G)$ when $G$ is complete, so assume otherwise. Let $T$ be a tree decomposition of $G$ with minimum width. By Lemma \[lemma:normal\], assume $T$ is normalised. Note since $G$ is not complete, $T$ contains more than one bag. Let $G'$ be the graph created by taking $G$ and adding all edges $vw$, where $v$ and $w$ share some bag of $T$. So $G$ is a spanning subgraph of $G'$ and $T$ is a tree decomposition of $G'$ as well as $G$. By the normalisation, there is a vertex $v \in V(G')$ such that $v$ appears in a leaf bag $B$ of $T$ and nowhere else. Hence $v$ has exactly $k$ neighbours in $G'$, which form a clique as they are all in $B$. Since it is smaller than the minimal counterexample, $a(G'-v) \leq {\textsf{\textup{tw}}}(G'-v) \leq k$. Since $G'-v$ contains a $(k+1)$-clique (consider a bag of $T$ other than $B$), it follows $a(G'-v) \geq k$. Thus $a(G'-v)=k$, and $G'-v$ is a spanning subtree of a $k$-tree $H$. As $v$ is $k$-simplicial in $G'$, it follows $G'$ (and thus $G$) is a spanning subgraph of a $k$-tree, which contradicts our assumption. Finally, show ${\textsf{\textup{tw}}}(G) \leq b(G)$. $G$ is a spanning subgraph of a $H$, chordal graph with no $(b(G)+2)$-clique. There is a tree decomposition of $H$ where every bag is a clique; this means it has width at most $b(G)$. This tree decomposition is also a tree decomposition for $G$, so ${\textsf{\textup{tw}}}(G) \leq b(G)$. Hence, it follows that $b(G) \leq a(G) \leq {\textsf{\textup{tw}}}(G) \leq b(G)$, which is sufficient to prove our desired result. Separators {#section:separators} ========== For a graph $G$, a set $S \subseteq V(G)$, and some $c \in [\frac{1}{2},1)$, a $(k,S,c)$-*separator* is a set $X \subseteq V(G)$ with $|X| \leq k$, such that no component of $G-X$ contains more than $c|S-X|$ vertices of $S$. Note that a $(k,S,c)$-separator is also a $(k,S,c')$-separator for all $c' \geq c$. Define the *separation number* ${\textsf{\textup{sep}}}_c(G)$ to be the minimum integer $k$ such that there is a $(k,S,c)$-separator for all $S \subseteq V(G)$. We also consider the following variant: a $(k,S,c)^*$-*separator* is a set $X \subseteq V(G)$ with $|X| \leq k$ such that no component of $G-X$ contains more than $c|S|$ vertices of $S-X$. Define ${\textsf{\textup{sep}}}_c^*(G)$ analogously to ${\textsf{\textup{sep}}}_c(G)$, but with respect to these variant separators. It follows from the definition that ${\textsf{\textup{sep}}}_c^*(G) \leq {\textsf{\textup{sep}}}_c(G)$. Separators can be seen as a generalisation of the ideas presented in the famous planar separator theorem [@LT79], which essentially states that a planar graph $G$ with $n$ vertices has a $(O(\sqrt{n}),V(G),\frac{2}{3})^*$-separator. Unfortunately, the precise definition of a separator and the separation number is inconsistent across the literature. The above definition is an attempt to unify all existing definitions. @RS-GraphMinorsII-JAlg86 gave the first lower bound on ${\textsf{\textup{tw}}}(G)$ in terms of separators, though they do not use the term, nor do they give an explicit definition of separation number. This definition is equivalent to our standard definition but with $c$ fixed at $\frac{1}{2}$. @GM-JCTB, give the above variant definition, with $c$ fixed at $\frac{1}{2}$, and instead call it a *balanced separator*. @Reed97 defines separators using our standard definition, with $c=\frac{2}{3}$. @Bodlaender-TCS98 defines type-1 and type-2 separators, which have variable proportion (i.e. allow for different values of $c$), but are not defined on sets other than $V(G)$. Sometimes [@Fox11; @GM-JCTB; @Bodlaender-TCS98] instead of considering components in $G-X$, separators are defined as partitioning the vertex set of $G-X$ into exactly two parts $A$ and $B$, such that no edge has an endpoint in both parts and $|A \cap S|,|B \cap S| \leq c|S|$. (In fact, @Bodlaender-TCS98 uses both this definition and the standard “components of $G-X$" definition as the difference between type-1 and type-2 separators.) As long as $c \geq \frac{2}{3}$, this is equivalent to considering the components, since Lemma \[lemma:comptrick\] and Corollary \[corollary:comptrick\] allow partitioning of the components into parts $A$ and $B$. However, for lower values of $c$ this no longer holds, for example, if $c=\frac{1}{2}$, it is possible that each component contains exactly $\frac{1}{3}$ of the vertices of $S$, so there is no acceptable partition into $A$ and $B$. As a result, $c=\frac{2}{3}$ and $c=\frac{1}{2}$ are the most “natural" choices for $c$. Fortunately, ${\textsf{\textup{sep}}}_c(G),{\textsf{\textup{sep}}}_c^*(G),{\textsf{\textup{sep}}}_{c'}(G)$ and ${\textsf{\textup{sep}}}_{c'}^*(G)$ are all tied for all $c,c' \in [\frac{1}{2},1)$. [^4] @RS-GraphMinorsII-JAlg86 proved that $${\textsf{\textup{sep}}}_{\frac{1}{2}}(G)\leq {\textsf{\textup{tw}}}(G)+1\enspace.$$ (Of course, they did not use our notation.) @RS-GraphMinorsII-JAlg86 [@RS-GraphMinorsXIII-JCTB95] also proved that $${\textsf{\textup{tw}}}(G)+1\leq 4\,{\textsf{\textup{sep}}}_{\frac{2}{3}}(G) - 2\enspace.$$ (@Reed97 [@Reedktreefinder] gives a more accessible proof of this upper bound.) We now provide a series of lemmas to prove a slightly stronger result, that replaces the multiplicative constant “4" by “3". First, we prove a useful lemma for dealing with components of a graph. \[lemma:comptrick\] For every graph $G$ and for all sets $X, S \subseteq V(G)$ such that no component of $G-X$ contains more than half of the vertices of $S-X$, it is possible to partition the components of $G-X$ into at most three parts such that no part contains more than half the vertices of $S-X$. If $G-X$ has at most three components, the claim follows immediately. Hence assume $G-X$ has at least four components. Initially, let each part simply contain a single component. Merge parts as long as the merge does not cause the new part to contain more than half the vertices of $S-X$. Now if two parts contain more than $\frac{1}{4}$ of the vertices of $S-X$ each, then all other parts (of which there must be at least two) contain, in total, less than $\frac{1}{2}$ of the vertices of $S-X$. Then merge all other parts together, leaving the partition with exactly three parts. Alternatively only one part (at most) contains more than $\frac{1}{4}$ of the vertices of $S-X$. So at least three parts contain at most $\frac{1}{4}$ of the vertices of $S-X$, and so merge two of them. This lowers the number of parts in the partition. As long as there are four or more parts, one of these operations can be performed, so repeat until at most three parts remain. \[corollary:comptrick\] For every graph $G$ and for all sets $X, S \subseteq V(G)$ such that no component of $G-X$ contains more than two-thirds of the vertices of $S-X$, it is possible to partition the components of $G-X$ into at most two parts such that no part contains more than two-thirds the vertices of $S-X$. This corollary follows by a very similar argument to Lemma \[lemma:comptrick\]. The following argument is similar to that provided in [@RS-GraphMinorsII-JAlg86]. \[lemma:sepleqtw\] For any graph $G$ and for all $c \in [\frac{1}{2},1)$, $${\textsf{\textup{sep}}}_c(G) \leq {\textsf{\textup{tw}}}(G)+1.$$ Fix $S \subseteq V(G)$ and let $k := {\textsf{\textup{tw}}}(G)+1$. It is sufficient to construct a $(k,S,\frac{1}{2})$-separator for $G$. $G$ has a normalised tree decomposition $T$ with maximum bag size $k$, by Lemma \[lemma:normal\]. Consider a pair of adjacent bags $X,Y$. Let $T_X$ and $T_Y$ be the subtrees of $T-XY$ containing bags $X$ and $Y$ respectively. Let $U_X \subseteq V(G)$ be the set of vertices only appearing in bags of $T_X$, and $U_Y$ the set of vertices only appearing in bags of $T_Y$. Then $U_X, X \cap Y, U_Y$ is a partition of $V(G)$ such that no edge has an endpoint in $U_X$ and $U_Y$. Each component of $G-(X \cap Y)$ is contained entirely within $U_X$ or $U_Y$. Say $Q \subseteq V(G)$ is *large* if $|Q \cap S| > \frac{1}{2}|S-(X \cap Y)|$. If neither $U_X$ or $U_Y$ is large, then no component of $G-(X \cap Y)$ is large. Hence $X \cap Y$ is a $(|X \cap Y|, S,\frac{1}{2})$-separator. Since $|X \cap Y| \leq |Y| \leq k$, this is sufficient. Alternatively, for all edges $XY \in E(T)$, exactly one of $U_X$ and $U_Y$ is large. (If both sets are large, then $|S-(X \cap Y)| = |U_X \cap S| + |U_Y \cap S| > |S-(X \cap Y)|$, which is a contradiction.) Orient the edge $XY \in E(T)$ towards $X$ if $U_X$ is large, or towards $Y$ if $U_Y$ is large. Now there must be a bag $B$ with outdegree 0. If $B$ is a $(|B|, S,\frac{1}{2})$-separator, then as $|B|=k$, the result is achieved. Otherwise, exactly one component $C$ of $G-B$ is large. The vertices of $C$ only appear in the bags of a single subtree of $T-B$. Label that subtree as $T'$, and let $A$ denote the bag of $T'$ adjacent to $B$. Recall there is a partition $V(G)$ into $U_A, A \cap B, U_B$ where $|U_B \cap S| > \frac{1}{2}|S-(A \cap B)|$, since the edge $AB$ is oriented towards $B$. Hence $|U_A \cap S| < \frac{1}{2}|S-(A \cap B)|$. Also note the vertices of $G-B$ that only appear in the bags of $T'$ are exactly the vertices of $U_A$. Hence $C \subseteq U_A$, and $|U_A \cap S| > \frac{1}{2}|S-B|$. So $\frac{1}{2}|S-B| < |U_A \cap S| < \frac{1}{2}|S-(A \cap B)|$. By our normalisation, $|A \cap B| = |B|-1$. So $|S-B| \geq |S-(A \cap B)|-1$. Thus $|S-(A \cap B)|-1 < 2|U_A \cap S| < |S-(A \cap B)|$, which is a contradiction as $|S-(A \cap B)|-1$, $2|U_A \cap S|$ and $|S-(A \cap B)|$ are all integers. Now we provide a proof of the upper bound. For any graph $G$, for all $c \in [\frac{1}{2},1)$, $${\textsf{\textup{bn}}}(G) \leq \frac{1}{1-c}{\textsf{\textup{sep}}}_c^*(G).$$ Say $\beta$ is an optimal bramble of $G$ with a minimum hitting set $H$. That is, $|H|={\textsf{\textup{bn}}}(G)$. For the sake of a contradiction, assume that $(1-c){\textsf{\textup{bn}}}(G) > {\textsf{\textup{sep}}}_c^*(G)$. So there is a $({\textsf{\textup{sep}}}_c^*(G),H,c)^*$-separator $X$. If $X$ is a hitting set for $\beta$ then ${\textsf{\textup{bn}}}(G) \leq |X| \leq sep_c^*(G) < (1-c){\textsf{\textup{bn}}}(G)$, which is a contradiction. So $X$ is not a hitting set for $\beta$. Thus some bramble element of $\beta$ is entirely within a component of $G-X$. Only one such component can contain bramble elements. Call this component $C$. Then we can hit every bramble element of $\beta$ with the vertices of $X$ or the vertices of $H$ inside $C$, that is, $X \cup (H \cap V(C))$ is a hitting set. Since $X$ is a $({\textsf{\textup{sep}}}_c^*(G),H,c)^*$-separator, $|H \cap V(C)| \leq c|H|$. Thus $|X \cup (H \cap V(C))| = |X| + |H \cap V(C)| \leq |X| + c|H| \leq {\textsf{\textup{sep}}}_c^*(G) + c|H| < (1-c)|H| + c|H| = |H|$. Thus $X \cup (H \cap V(C))$ is a hitting set smaller than the minimum hitting set, a contradiction. Hence, from the above it follows that for $c \in [\frac{1}{2},1)$, $${\textsf{\textup{sep}}}_c^*(G) \leq {\textsf{\textup{sep}}}_c(G) \leq {\textsf{\textup{tw}}}(G) + 1 = {\textsf{\textup{bn}}}(G) \leq \frac{1}{1-c}{\textsf{\textup{sep}}}_c^*(G) \leq \frac{1}{1-c}{\textsf{\textup{sep}}}_c(G).$$ Each of the above inequalities is tight. For a fixed $c \in [\frac{1}{2},1)$, let $k,n$ be integers such that $k > \frac{c}{1-c}+1$ and $n \geq \frac{k-1}{1-c}$. Then ${\textsf{\textup{sep}}}_c^*(\psi_{n,k}) = {\textsf{\textup{sep}}}_c(\psi_{n,k}) = n$, which proves that the first and last inequalities are tight. (See @thesis for a proof of this result.) The remaining two inequalities are tight due to the complete graph $K_n$. Branchwidth and Tangles ======================= A *branch decomposition* of a graph $G$ is a pair $(T, \theta)$ where $T$ is a tree with each node having degree 3 or 1, and $\theta$ is a bijective mapping from the edges of $G$ to the leaves of $T$. A vertex $x$ of $G$ is *across* an edge $e$ of $T$ if there are edges $xy$ and $xz$ of $G$ mapped to leaves in different subtrees of $T-e$. The *order* of an edge $e$ of $T$ is the number of edges of $G$ across $e$. The *width* of a branch decomposition is the maximum order of an edge. Finally, the *branchwidth* ${\textsf{\textup{bw}}}(G)$ of a graph $G$ is the minimum width over all branch decompositions of $G$. Note that if $|E(G)| \leq 1$, there are no branch decompositions of $G$, in which case we define ${\textsf{\textup{bw}}}(G)=0$. @RS-GraphMinorsX-JCTB91 first defined branchwidth, where it was defined more generally for hypergraphs; here we just consider the case of simple graphs. Tangles were first defined by @RS-GraphMinorsX-JCTB91. Their definition is in terms of sets of separations of graphs. (Note, importantly, that a *separation* is not the same as a *separator* as defined in Section \[section:separators\].) We omit their definition and instead present the following, initially given by @Reed97. A set $\tau$ of connected subgraphs of a graph $G$ is a *tangle* if for all sets of three subgraphs $A,B,C \in \tau$, there exists either a vertex $v$ of $G$ in $V(A \cap B \cap C)$, or an edge $e$ of $G$ such that each of $A,B$ and $C$ contain at least one endpoint of $e$. Clearly a tangle is also a bramble—this is the main advantage of this definition. The *order* of a tangle is equal to its order when viewed as a bramble. The *tangle number* ${\textsf{\textup{tn}}}(G)$ is the maximum order of a tangle in $G$. When defined with respect to hypergraphs, treewidth and tangle number are tied to the maximum of branchwidth and the size of the largest edge. So for simple graphs, there are a few exceptional cases when ${\textsf{\textup{bw}}}(G) < 2$, which we shall deal with briefly. If $G$ is connected and ${\textsf{\textup{bw}}}(G) \leq 1$, then $G$ has at most one vertex with degree greater than 1 (that is, $G$ is a star), and ${\textsf{\textup{bn}}}(G)={\textsf{\textup{tn}}}(G) \leq 2$. Henceforth, assume ${\textsf{\textup{bw}}}(G) \geq 2$. @RS-GraphMinorsX-JCTB91 prove the following relation between tangle number and branchwidth; we omit the proof. Instead we show that ${\textsf{\textup{tn}}}(G),{\textsf{\textup{bw}}}(G),{\textsf{\textup{bn}}}(G)$ and ${\textsf{\textup{tw}}}(G)$ are all tied by small constant factors. \[theorem:tangbw\] For a graph $G$, if ${\textsf{\textup{bw}}}(G) \geq 2$, then $${\textsf{\textup{bw}}}(G) = {\textsf{\textup{tn}}}(G).$$ @RS-GraphMinorsX-JCTB91 proved that ${\textsf{\textup{bn}}}(G) \leq \frac{3}{2}{\textsf{\textup{tn}}}(G)$. @Reed97 provided a short proof that ${\textsf{\textup{bn}}}(G) \leq 3\,{\textsf{\textup{tn}}}(G)$. Here, we modify Reed’s proof to show that ${\textsf{\textup{bn}}}(G) \leq 2\,{\textsf{\textup{tn}}}(G)$. For every graph $G$, $${\textsf{\textup{tn}}}(G)\leq {\textsf{\textup{bn}}}(G)\leq 2\,{\textsf{\textup{tn}}}(G).$$ Since every tangle is also a bramble, ${\textsf{\textup{tn}}}(G) \leq {\textsf{\textup{bn}}}(G)$. To prove that ${\textsf{\textup{bn}}}(G) \leq 2\,{\textsf{\textup{tn}}}(G)$, let $k:={\textsf{\textup{bn}}}(G)$, and say $\beta$ is a bramble of $G$ of order $k$. Consider a set $S \subseteq V(G)$ with $|S| < k$. If two components of $G-S$ entirely contain a bramble element of $\beta$, then those two bramble elements do not touch. Alternatively, if no component of $G-S$ entirely contains a bramble element, then all bramble elements use a vertex in $S$, and $S$ is a hitting set of smaller order than the minimum hitting set. Thus exactly one component $S'$ of $G-S$ entirely contains a bramble element of $\beta$. Clearly, $V(S') \cap S = \emptyset$. Define $\tau := \{ S' : S \subseteq V(G), |S| < \frac{k}{2} \}$. To prove that $\tau$ is a tangle, let $T_1, T_2, T_3$ be three elements of $\tau$. Say $T_i = S'_i$ for each $i$. Since $|S_1 \cup S_2| < k$, some bramble element $B_1$ of $\beta$ does not intersect $S_1 \cup S_2$. Similarly, some bramble element $B_2$ does not intersect $S_2 \cup S_3$. Since $B_1$ does not intersect $S_1$, it is entirely within one component of $G-S_1$, that is, $B_1 \subseteq T_1$. Similarly, $B_1 \subseteq T_2$ and $B_2 \subseteq T_2 \cap T_3$. Since $B_1,B_2 \in \beta$, they either share a vertex $v$, or there is an edge $e$ with one endpoint in $B_1$ and the other in $B_2$. In the first case, $v \in V(T_1 \cap T_2 \cap T_3)$. In the second case, one endpoint of $e$ is in $T_1 \cap T_2$, the other in $T_2 \cap T_3$. It follows that $\tau$ is a tangle. The order of $\tau$ is at least $\frac{k}{2}$, since a set $X$ of size less than $\frac{k}{2}$ has a defined $X' \in \tau$, and so $X$ does not intersect all subgraphs of $\tau$. Then ${\textsf{\textup{tn}}}(G) \geq \frac{k}{2}$. We now provide a proof for a direct relationship between branchwidth and treewidth. Note again these proofs are modified versions of those in [@RS-GraphMinorsX-JCTB91]. \[lemma:twbranch\] For a graph $G$, if ${\textsf{\textup{bw}}}(G) \geq 2$ then $${\textsf{\textup{bw}}}(G) \leq {\textsf{\textup{tw}}}(G)+1 \leq \frac{3}{2}\,{\textsf{\textup{bw}}}(G).$$ We prove the second inequality first. Assume no vertex is isolated. Let $k:= {\textsf{\textup{bw}}}(G)$, and let $(T,\theta)$ be a branch decomposition of order $k$. We construct a tree decomposition with $T$ as the underlying tree, and where $B_x$ will denote the bag indexed by each node $x$ of $T$. A node $x$ in $T$ has degree 3 or 1. If $x$ has degree 1, then let $B_{x}$ contain the two endpoints of $e = \theta^{-1}(x)$. If $x$ has degree 3, then let $B_{x}$ be the set of vertices that are across at least one edge incident to $x$. We now show that this is a tree decomposition. Every vertex appears at least once in the tree decomposition. Also, for every edge $vw \in E(G)$, the bag of the leaf node $\theta(vw)$ contains both $v$ and $w$. If we consider vertex $v \in V(G)$ incident with $vw$ and $vu$, then $v$ is across every edge in $T$ on the path from $\theta(vw)$ to $\theta(vu)$. Thus, $v$ is in every bag indexed by a node on that path. Such a path exists for all neighbours $w,u$ of $v$. It follows that the subtree of nodes indexing bags containing $v$ form a subtree of $T$. Thus $(T, (B_x)_{x \in V(T)})$ is a tree decomposition of $G$. A bag indexed by a leaf node has size 2. If $x$ is not a leaf, then $B_x$ contains the vertices that are across at least one edge incident to $x$. Suppose $v$ is across exactly one such edge $e$. Then there exists $\theta(vw)$ and $\theta(vu)$ in different subtrees of $T-e$. Without loss of generality, $\theta(vw)$ is in the subtree containing $x$. But then the path from $x$ to $\theta(vw)$ uses one of the other two edges incident to $x$. Hence if $v$ is in $B_{x}$ then $v$ is across at least two edges incident to $x$. If the sets of vertices across the three edges incident to $x$ are $A,B$ and $C$ respectively, then $|A| + |B| + |C| \geq 2|B_{x}|$. But $|A|+|B|+|C| \leq 3k$. Therefore, regardless of whether $x$ is a leaf, $|B_{x}| \leq \max\{2,\frac{3}{2}k\} =\frac{3}{2}k$ (since $k \geq 2$). Therefore ${\textsf{\textup{tw}}}(G)+1 \leq \frac{3}{2}k$. Now we prove the first inequality. Let $k:={\textsf{\textup{tw}}}(G)+1$. Hence there exists a tree decomposition $(T, (B_x)_{x \in V(T)})$ with maximum bag size $k$; choose this tree decomposition such that $T$ is node-minimal, and such that the subtree induced by $\{ x \in V(T) : v \in B_x\}$ is also node-minimal for each $v \in V(G)$. If $k < 2$, then $G$ contains no edge, and ${\textsf{\textup{bw}}}(G) = 0$. Now assume $k \geq 2$ and $E(G) \neq \emptyset$. As this result is trivial when $G$ is complete, we assume otherwise, and thus $T$ is not a single node. Note the following facts: if $x$ is a node of $T$ with degree 2, then there exists some pair of adjacent vertices $v,w$ such that $B_x$ is the only bag containing $v$ and $w$. (Otherwise, $T$ would violate the minimality properties.) Similarly, if $x$ is a leaf node, then there exists some $v \in V(G)$ such that $B_x$ is the only bag containing $v$. $B_x$ also contains the neighbours of $v$, but nothing else. Now, for every edge $vw \in E(G)$, choose some bag $B_x$ containing $v$ and $w$. Unless $x$ is a leaf with $B_x = \{v,w\}$, add to $T$ a new node $y$ adjacent to $x$, such that $B_y = \{v,w\}$. Clearly $(T, (B_x)_{x \in V(T)})$ is still a tree decomposition of the same width. From our above facts, every leaf node is either newly constructed or was already of the form $B_x = \{v,w\}$. Also, every node that previously had degree 2 now has higher degree. A node that was previously a leaf either remains a leaf, or now has degree at least 3. So no node of the new $T$ has degree 2. If a node $x$ has degree greater than 3, then delete the edges from $x$ to two of its neighbours (denoted $y,z$), and add to $T$ a new node $s$ adjacent to $x,y$ and $z$. Let $B_s := B_x \cap (B_y \cup B_z)$. Clearly this is still a tree decomposition of the same width. Now the degree of $x$ has been reduced by 1, and the new node has degree 3. Repeat this process until all nodes have either degree 3 or 1. Since each leaf bag contains exactly the endpoints of an edge (and no edge has both endpoints in more than one leaf), there is a bijective mapping $\theta$ that takes $vw \in E(G)$ to the leaf node containing $v$ and $w$. Together with $T$, this gives a branch decomposition of $G$. If $xy \in E(T)$, then all edges of $G$ across $xy$ are in $B_x \cap B_y$. So the order of this branch decomposition is at most $k$. Thus ${\textsf{\textup{bw}}}(G) \leq {\textsf{\textup{tw}}}(G)+1$. (Note that our minimality properties would imply that $|B_x \cap B_y| < k$, however converting the tree to ensure that all nodes have degree 3 or 1 does not necessarily maintain this.) @RS-GraphMinorsX-JCTB91 showed the bounds in Lemma \[lemma:twbranch\] are tight. To show this, we shall use Theorem \[theorem:tangbw\] and consider the bounds in terms of ${\textsf{\textup{tn}}}(G)$. $K_n$ exhibits the upper bound on ${\textsf{\textup{tw}}}(G)$ when $n$ is divisible by 3; let the tangle contain all subgraphs with more than $\frac{2}{3}n$ vertices. The graph $K_{n,n}$ minus a perfect matching exhibits the lower bound on ${\textsf{\textup{tw}}}(G)$ when $n \geq 4$; let the tangle contain all connected subgraphs with at least $n$ vertices. Tree Products ============= Let the *lexicographic tree product number* of $G$, denoted by ${\textsf{\textup{ltp}}}(G)$, be the minimum integer $k$ such that $G$ is a minor of $T[K_k]$ for some tree $T$. Here $T[K_k]$ is the lexicographic product, which is the graph obtained from $T$ by replacing each vertex by a copy of $K_k$ and each edge by a copy of $K_{k,k}$. We now show that ${\textsf{\textup{tw}}}$ and ${\textsf{\textup{ltp}}}$ are within constant factors of each other. \[lemma:ltp\] For every graph $G$, $${\textsf{\textup{ltp}}}(G)-1\leq {\textsf{\textup{tw}}}(G)\leq2{\textsf{\textup{ltp}}}(G)-1\enspace.$$ First we prove that ${\textsf{\textup{ltp}}}(G)\leq{\textsf{\textup{tw}}}(G)+1$. Consider a tree decomposition of $G$ with width $k:={\textsf{\textup{tw}}}(G)$ whose underlying tree is $T$. Clearly, $G$ is a minor of $T[K_{k+1}]$. Thus ${\textsf{\textup{ltp}}}(G)\leq k+1$. Now we prove that ${\textsf{\textup{tw}}}(G)\leq2{\textsf{\textup{ltp}}}(G)-1$. Let $T$ be a tree such that $G$ is a minor of $T[K_k]$ where $k:={\textsf{\textup{ltp}}}(G)$. For each vertex $v$ of $T$ let $K_v$ be the copy of $K_k$ that replaces $v$ in the construction of $T[K_k]$. Let $T'$ be the tree obtained from $T$ by subdividing each edge. Now we construct a tree decomposition of $T[K_k]$ whose underlying tree is $T'$. For each vertex $v$ of $T$, let the bag at $v$ consist of $K_v$. For each edge $vw$ of $T$ subdivided by vertex $x$, let the bag at $x$ consist of $K_v\cup K_w$. Thus each edge of $T[K_k]$ is in some bag, and the set of bags that contain each vertex of $T[K_k]$ form a connected subtree of $T'$. Hence we have a tree decomposition of $T'$. Each bag has size at most $2k$. Hence ${\textsf{\textup{tw}}}(T[K_k])\leq 2k-1$. (In fact, ${\textsf{\textup{tw}}}(T[K_k])= 2k-1$ since $T[K_k]$ contains $K_{2k}$.) Thus every minor of $T'$, including $G$, has treewidth at most $2k-1$. If $T$ is a tree, let $T^{(k)}$ denote the cartesian product of $T$ with $K_k$. That is, the graph with vertex set $\{(x,i): x \in T, i \in \{1, \dots, k\}\}$ and with an edge between $(x,i)$ and $(y,j)$ when $x=y$, or when $xy \in E(T)$ and $i=j$. Then define the *cartesian tree product number* of $G$, ${\textsf{\textup{ctp}}}(G)$, to be the minimum integer $k$ such that $G$ is a minor of $T^{(k)}$. ${\textsf{\textup{ctp}}}(G)$ was first defined by @holst and @yves, however they did not use that name or notation. They also proved the following result. We provide a different proof. \[lemma:ctp\] For every graph $G$, $${\textsf{\textup{ctp}}}(G)-1 \leq {\textsf{\textup{tw}}}(G) \leq {\textsf{\textup{ctp}}}(G).$$ Let $k:= {\textsf{\textup{tw}}}(G)$. By Lemma \[lemma:ktree\], $G$ is the spanning subgraph of a chordal graph $G'$ that has a $(k+1)$-clique but no $(k+2)$-clique. Let $(T, (B_x \subseteq V(G))_{x \in V(T)})$ be a minimum width tree decomposition of $G'$. This has width $k$ and is also a tree decomposition of $G$. To prove the first inequality, it is sufficient to show that $G$ is a minor of $T^{(k+1)}$. Let $c$ be a $(k+1)$-colouring of $G'$. (It is well-known that chordal graphs are perfect.) For each $v \in V(G)$, define the connected subgraph $R_v$ of $T^{(k+1)}$ such that $R_v := \{(x,c(v)): v \in B_x\}$. If $(x,i) \in V(R_v) \cap V(R_w)$ then both $v$ and $w$ are in $B_x$ and $c(v)=c(w)=i$. But if $v$ and $w$ share a bag then $vw \in E(G')$, which contradicts the vertex colouring $c$. So the subgraphs $R_v$ are pairwise disjoint, for all $v \in V(G)$. If $vw \in E(G)$, then $v$ and $w$ share a bag $B_x$. Hence there is an edge $(x,c(v))(x,c(w))$ between the subgraphs $R_v$ and $R_w$. Hence the $R_v$ subgraphs form a $G$-model of $T^{(k+1)}$. Now we prove the second inequality. Let $k:= {\textsf{\textup{ctp}}}(G)$, and choose tree $T$ such that $G$ is a minor of $T^{(k)}$. As ${\textsf{\textup{tw}}}(G) \leq {\textsf{\textup{tw}}}(T^{(k)})$, it is sufficient to show that ${\textsf{\textup{tw}}}(T^{(k)}) \leq k$. Let $T'$ be the tree $T$ with each edge subdivided $k$ times. Label the vertices created by subdividing $xy \in E(T)$ as $xy(1), \dots, xy(k)$, such that $xy(1)$ is adjacent to $x$ and $xy(k)$ is adjacent to $y$. Construct $(T', (B_x \subseteq V(G))_{x \in V(T')})$ as follows. For a vertex $x \in T$, let $B_x = \{(x,i) | i \in \{1, \dots, k\}\}$. For a subdivision vertex $xy(j)$, let $B_{xy(j)} = \{(x,i),(y,i') | 1 \leq i' \leq j \leq i \leq k\}$. This is a valid tree decomposition with maximum bag size $k+1$. Hence ${\textsf{\textup{tw}}}(T^{(k)}) \leq k$ as required. All these bounds are tight. Let $k,n$ be integers such that $n \geq 3$. Then the first inequalities in Lemma \[lemma:ltp\] and Lemma \[lemma:ctp\] are tight for $\psi_{n,k}$ [@thesis]. (Also see @MS for a similar result.) The second inequalities are tight for the complete graph $K_n$ (for Lemma \[lemma:ltp\], ensure that $n$ is even). Linkedness ========== @Reed97 introduced the following definition. For a positive integer $k$, a set $S$ of vertices in a graph $G$ is *$k$-linked* if for every set $X$ of fewer than $k$ vertices in $G$ there is a component of $G-X$ that contains more than half of the vertices in $S$. The *linkedness* of $G$, denoted by ${\textsf{\textup{link}}}(G)$, is the maximum integer $k$ for which $G$ contains a $k$-linked set. Linkedness is used by @Reed97 in his proof of the Grid Minor Theorem. \[lemma:lemmalink\] For every graph $G$, $${\textsf{\textup{link}}}(G)\leq {\textsf{\textup{bn}}}(G)\leq2\,{\textsf{\textup{link}}}(G)\enspace.$$ First we prove that ${\textsf{\textup{link}}}(G)\leq {\textsf{\textup{bn}}}(G)$. Let $k:={\textsf{\textup{link}}}(G)$. Let $S$ be a $k$-linked set of vertices in $G$. Thus, for every set $X$ of fewer than $k$ vertices there is a component of $G-X$ that contains more than half of the vertices in $S$. This component is unique. Call it the *big* component. Let $\beta$ be the set of big components (taken over all such sets $X$). Clearly, any two elements of $\beta$ intersect at a vertex in $S$. Hence $\beta$ is a bramble. Let $H$ be a hitting set for $\beta$. If $|H|<k$ then (by the definition of $k$-linked) there is a component of $G-H$ that contains more than half of the vertices in $S$, implying $H$ does not hit some big component. This contradiction proves that $|H|\geq k$. Hence $\beta$ is a bramble of order at least $k$. Therefore ${\textsf{\textup{bn}}}(G)\geq k={\textsf{\textup{link}}}(G)$. Now we prove that ${\textsf{\textup{bn}}}(G)\leq 2\,{\textsf{\textup{link}}}(G)$. Assume for the sake of a contradiction that ${\textsf{\textup{bn}}}(G) > 2\,{\textsf{\textup{link}}}(G)$. Let $k := {\textsf{\textup{link}}}(G)$, so $G$ is not $(k+1)$-linked. Let $H$ be a minimum hitting set for a bramble $\beta$ of $G$ of largest order. Since $H$ is not $(k+1)$-linked, there exists a set $X$ of order at most $k$ such that no component of $G-X$ contains more than half of the vertices in $H$. Note that at most one component of $G-X$ can entirely contain a bramble element of $\beta$ (otherwise two bramble elements do not touch). If no component of $G-X$ entirely contains a bramble element of $\beta$, then $X$ is a hitting set for $\beta$ of order $|X| \leq k < \frac{1}{2}{\textsf{\textup{bn}}}(G)$, which contradicts the order of the minimum hitting set. Finally, if a component of $G-X$ entirely contains some bramble element of $\beta$, then let $H' \subset H$ be the set of vertices of $H$ in that component. Now $H'$ intersects all of the bramble elements contained in the component (since those bramble elements do not intersect any other vertices of $H$), and $X$ intersects all remaining bramble elements, as in the previous case. Thus, $H' \cup X$ is a hitting set for $\beta$. However, $|X| \leq k < \frac{1}{2}{\textsf{\textup{bn}}}(G)$, and by the choice of $X$, $|H'| \leq \frac{1}{2}|H| = \frac{1}{2}{\textsf{\textup{bn}}}(G)$. So $|H' \cup X| = |H'| + |X| < {\textsf{\textup{bn}}}(G)$, again contradicting the order of the minimum hitting set. When $n$ is even ${\textsf{\textup{link}}}(K_n) = \frac{n}{2}$, so the second inequality is tight. The first inequality is tight since ${\textsf{\textup{link}}}(\psi_{n,k}) = {\textsf{\textup{bn}}}(\psi_{n,k}) = n$ when $k \geq 2$ and $n \geq 3$ [@thesis]. Well-linked and $k$-Connected Sets ================================== For a graph $G$, a set $S \subseteq V(G)$ is *well-linked* if for every pair $A,B \subseteq S$ such that $|A|=|B|$, there exists a set of $|A|$ vertex-disjoint paths from $A$ to $B$. If we can ensure these vertex-disjoint paths also have no internal vertices in $S$, then $S$ is *externally-well-linked*. The notion of a well-linked set was first defined by @Reed97, while a similar definition was used by @RST-JCTB94. Reed also described externally-well-linked sets in the same paper (but did not define it explicitly) and stated but did not prove that $S$ is well-linked iff $S$ is externally-well-linked. We provide a proof below. Let ${\textsf{\textup{wl}}}(G) := \max\{|S| : S \subseteq V(G), S$ is well-linked$\}$ denote the *well-linked number* of $G$. \[lemma:extwl\] $S$ is well-linked iff $S$ is externally-well-linked. It should be clear that if $S$ is externally-well-linked that $S$ is well-linked, so we prove the forward direction. Let $S \subseteq V(G)$ be well-linked. It is sufficient to show that for all $A,B \subseteq S$ with $|A|=|B|$ there are $|A|$ vertex-disjoint paths from $A$ to $B$ that are internally disjoint from $S$. Define $C := S - (A \cup B)$ and $A' := A \cup C$ and $B' := B \cup C$. Now $S = A' \cup B'$. Since $S$ is well-linked and $|A'| = |B'|$, there are $|A'|$ vertex-disjoint paths between $A'$ and $B'$. Each such path uses exactly one vertex from $A'$ and one vertex from $B'$. Thus, if $v \in C \subseteq A \cap B$, then the path containing $v$ must simply be the singleton path $\{v\}$. Thus this path set contains a set of singleton paths for each vertex of $C$ and, more importantly, a set of paths starting in $A'-C = A$ and ending at $B'-C = B$. Since every vertex of $S$ is in either $A'$ or $B'$, and each path starts at a vertex in $A'$ and ends at one in $B'$, no internal vertex of these paths is in $S$. This is the required set of disjoint paths from $A$ to $B$ that are internally disjoint from $S$. @Reed97 proved that ${\textsf{\textup{bn}}}(G) \leq {\textsf{\textup{wl}}}(G) \leq 4\,{\textsf{\textup{bn}}}(G)$. We provide that proof of the first inequality and modify the proof of the second to give: \[lemma:welllink\] For every graph $G$, $${\textsf{\textup{bn}}}(G) \leq {\textsf{\textup{wl}}}(G) \leq 3\,{\textsf{\textup{link}}}(G) \leq 3\,{\textsf{\textup{bn}}}(G).$$ We first prove that ${\textsf{\textup{bn}}}(G) \leq {\textsf{\textup{wl}}}(G)$. Assume for the sake of a contradiction that ${\textsf{\textup{wl}}}(G) < {\textsf{\textup{bn}}}(G)$. Let $\beta$ be a bramble of largest order, and $H$ a minimal hitting set of $\beta$. Thus $H$ is not well-linked (since $|H| = {\textsf{\textup{bn}}}(G) > {\textsf{\textup{wl}}}(G)$). Choose $A,B \subseteq H$ such that $|A|=|B|$ but there are not $|A|$ vertex-disjoint paths from $A$ to $B$. By Menger’s Theorem, there exists a set of vertices $C$ with $|C| < |A|$ such that after deleting $C$, there is no $A$-$B$ path in $G$. Now consider a bramble element of $\beta$. If two components of $G-C$ entirely contain bramble elements, then those bramble elements cannot touch. Thus, it follows that at most one component of $G-C$ entirely contains some bramble element. Label this component $C'$; if no such component exists label $C'$ arbitrarily. $C'$ does not contain vertices from both $A$ and $B$, so without loss of generality assume $A \cap C' = \emptyset$. Thus all bramble elements entirely within $C'$ are hit by vertices of $H-A$, and all others are hit by $C$. So $(H-A) \cup C$ is a hitting set for $\beta$, but $|(H-A) \cup C| = |H|-|A|+|C| < |H|$, contradicting the minimality of $H$. Hence ${\textsf{\textup{bn}}}(G) \leq {\textsf{\textup{wl}}}(G)$. Now we show that ${\textsf{\textup{wl}}}(G) \leq 3\,{\textsf{\textup{link}}}(G)$. For the sake of a contradiction, say $3\,{\textsf{\textup{link}}}(G)< {\textsf{\textup{wl}}}(G)$. Define $k := \frac{1}{3}{\textsf{\textup{wl}}}(G)$. Let $S$ be the largest well-linked set. That is, $|S| = {\textsf{\textup{wl}}}(G)$. By Lemma \[lemma:extwl\] $S$ is externally-well-linked. $S$ is not ${\ensuremath{\protect\lceilk\rceil}}$-linked as ${\textsf{\textup{link}}}(G) < {\ensuremath{\protect\lceilk\rceil}}$. Hence there exists a set $X \subseteq V(G)$ with $|X| < {\ensuremath{\protect\lceilk\rceil}}$ such that $G-X$ has no component containing more than $\frac{1}{2}|S|$ vertices of $S$. Since $|X|$ is an integer, $|X| < k$. Let $a := |X \cap S|$. Using an argument similar to Lemma \[lemma:comptrick\], the components of $G-X$ can be partitioned into two or three parts, each with at most $\frac{1}{2}|S|$ vertices of $S$. Some part contains at least a third of the vertices of $S-X$. Let $A$ be the set of vertices in $S$ contained in that part, and let $B$ be the set of vertices in $S$ in the other parts of $G-X$. Now $\frac{1}{2}|S| \geq |A| \geq \frac{1}{3}|S-X| = \frac{1}{3}(|S| - a)$, and so $|B| \geq |S| - |S \cap X| - |A| \geq |S| - a - \frac{1}{2}|S|$. Remove vertices arbitrarily from the largest of $A$ and $B$ until these sets have the same order. Hence $|A|=|B|$ and $|A| \geq \min\{\frac{1}{3}(|S| - a), \frac{1}{2}|S| - a\}$. Since $A,B \subseteq S$ and $S$ is externally-well-linked, there are $|A|$ vertex-disjoint paths from $A$ to $B$ with no internal vertices in $S$. Since $A$ and $B$ are in different components of $G-X$, these paths must use vertices of $X$, but more specifically, vertices of $X-S$. Thus there are at most $|X-S|$ such paths. Thus $|A| \leq |X-S| < k-a$. Either $\frac{1}{3}(|S|-a) \leq |A| < k-a$ or $\frac{1}{2}|S| - a \leq |A| < k-a$, so $|S| < 3k$. However, $|S|={\textsf{\textup{wl}}}(G)=3k$, which is a contradiction. The final inequality follows from Lemma \[lemma:lemmalink\]. The first inequality in Lemma \[lemma:welllink\] is tight since ${\textsf{\textup{bn}}}(K_n) = {\textsf{\textup{wl}}}(K_n) = n$. We do not know if the second inequality is tight, but ${\textsf{\textup{wl}}}(G) \leq 2\,{\textsf{\textup{bn}}}(G)-2$ would be best possible since ${\textsf{\textup{bn}}}(K_{2n,n}) = n+1$ and ${\textsf{\textup{wl}}}(K_{2n,n}) = 2n$ (the larger part is the largest well-linked set). @diestelHC defined the following: $S \subseteq V(G)$ is *$k$-connected* in $G$ if $|S| \geq k$ and for all subsets $A,B \subseteq S$ with $|A| = |B| \leq k$, there are $|A|$ vertex-disjoint paths from $A$ to $B$. If we can ensure these vertex-disjoint paths have no internal vertex or edge in $G[S]$, then $S$ is *externally $k$-connected*. This construction was used in [@diestelHC] to prove a short version of the grid minor theorem. Note the obvious connection to well-linked sets; $X$ is well-linked iff $X$ is $|X|$-connected. Also note that @Diestel00a, in his treatment of the grid minor theorem, provides a slightly different formulation of externally $k$-connected sets, which only requires vertex-disjoint paths between $A$ and $B$ when they are disjoint subsets of $S$. These definitions are equivalent, which can be proven using a similar argument as in Lemma \[lemma:extwl\]. @Diestel00a also does not use the concept of $k$-connected sets, just externally $k$-connected. @diestelHC prove the following, but due to its similarity between $k$-connected sets and well-linked sets, we omit the proof. If $G$ has ${\textsf{\textup{tw}}}(G) < k$ then $G$ contains $(k+1)$-connected set of size $\geq 3k$. If $G$ contains no externally $(k+1)$-connected set of size $\geq 3k$, then ${\textsf{\textup{tw}}}(G) < 4k$. Grid Minors {#section:gridminors} =========== A key part of the previously mentioned Graph Minor Structure Theorem is as follows: given a fixed planar graph $H$, there exists some integer $r_{H}$ such that every graph with no $H$-minor has treewidth at most $r_{H}$. This cannot be generalised to when $H$ is non-planar, since there exist planar graphs, the grids, with unbounded treewidth. (By virtue of being planar, the grids do not have a non-planar $H$ as a minor.) In fact, since every planar graph is the minor of some grid, it is sufficient to just consider the grids, which leads to the Grid Minor Theorem: For each integer $k$ there is a minimum integer $f(k)$ such that every graph with treewidth at least $f(k)$ contains the $k \times k$ grid as a minor. All of our previous sections have provided parameters with linear ties to treewidth. However, the order of the largest grid minor is not linearly tied to treewidth. The initial bound on $f(k)$ by @RS-GraphMinorsV-JCTB86 was an iterated exponential tower. Later, @RST-JCTB94 improved this to $f(k) \leq 20^{2k^5}$. They also note, by use of a probabilistic argument, that $f(k) \geq \Omega(k^2\log k)$. @diestelHC obtained an upper bound of $2^{5k^{5}\log k}$, which is actually slightly worse than the bound provided by Robertson, Seymour and Thomas, but with a more succinct proof. @KenandYusuke proved that $f(k) \leq 2^{O(k^2 \log k)}$, and @SeymourLeaf proved that $f(k) \leq 2^{O(k\log k)}$. Very recently, @CCpoly proved a polynomial bound of $f(k) \leq O(k^{228})$. Together with the following lower bound, this implies that treewidth and the order of the largest grid-minor are polynomially tied. \[lemma:gridbramble\] If $G$ contains a $k \times k$ grid minor, then ${\textsf{\textup{tw}}}(G) \geq k$. If $H$ is a minor of $G$ then ${\textsf{\textup{tw}}}(H) \leq {\textsf{\textup{tw}}}(G)$. Thus it suffices to prove that the $k \times k$ grid $H$ has treewidth at least $k$, which is implied if ${\textsf{\textup{bn}}}(H) \geq k+1$. Consider $H$ drawn in the plane. For a subgraph $S$ of $H$, define a *top vertex* of $S$ in the obvious way. (Note it is not necessarily unique.) Similarly define *bottom vertex*, *left vertex* and *right vertex*. Let subgraph $H'$ of $H$ be the top-left $(k-1) \times (k-1)$ grid in $H$. A *cross* is a subgraph containing exactly one row and column from $H'$, and no vertices outside $H'$. Let $X$ denote the bottom row of $H$, and $Y$ the right column without its bottom vertex. Let $\beta:= \{X,Y, \text{ all crosses}\}$. A pair of crosses intersect in two places. There is an edge from a bottom vertex of a cross to $X$ and a right vertex of a cross to $Y$. There is also an edge from the right vertex of $X$ to the bottom vertex of $Y$. Hence $\beta$ is a bramble. If $Z$ is a hitting set for $\beta$, it must contain $k-1$ vertices of $V(H')$, for otherwise a row and column are not hit, and so a cross is not hit. $Z$ must also contain two other vertices to hit $X$ and $Y$. So $|Z| \geq k+1$, as required. Grid-like Minors ================ A *grid-like-minor of order $t$* of a graph $G$ is a set of paths ${\ensuremath{\mathcal{P}}}$ with a bipartite intersection graph that contains a $K_t$-minor. Note that if the intersection graph of ${\ensuremath{\mathcal{P}}}$ is partitioned $A$ and $B$, then we can think of the set of paths $A$ as being the “rows" of the “grid", and the set $B$ being the “columns". Also note that an actual $k$-by-$k$ grid gives rise to a set ${\ensuremath{\mathcal{P}}}$ with an intersection graph that is complete bipartite and as such contains a complete minor of order $k$+1. Let ${\textsf{\textup{glm}}}(G)$ be the maximum order of a grid-like-minor of $G$. Grid-like-minors were first defined by @ReedWood-EuJC as a weakening of a grid minor; see Section \[section:gridminors\]. As a result of this weakening, it is easier to tie ${\textsf{\textup{glm}}}(G)$ to ${\textsf{\textup{tw}}}(G)$. This notion has also been applied to prove computational intractability results in monadic second order logic; see @MR2904950 [@Ganian14] and @KT10 [@MR2809681]. The *fractional Hadwiger number* ${\textsf{\textup{had}}}_f(G)$ of $G$ is the maximum $h$ for which there is a bramble ${\ensuremath{\mathcal{B}}}$ in $G$, and a weight function $w :{\ensuremath{\mathcal{B}}}\rightarrow\mathbb{R}_{\geq 0}$, such that $h =\sum_{X\in{\ensuremath{\mathcal{B}}}}w(X)$ and for each vertex $v$, the sum of the weights of the subgraphs in ${\ensuremath{\mathcal{B}}}$ that contain $v$ is at most 1. For example, the branch sets of a $K_{{\textsf{\textup{had}}}(G)}$-minor form a bramble, for which we may weight each vertex by 1. Thus ${\textsf{\textup{had}}}_f(G) \geq {\textsf{\textup{had}}}(G)$. For a positive integer $r$, the *$r$-integral Hadwiger number* ${\textsf{\textup{had}}}_r(G)$ is defined the same as the fractional Hadwiger number, except that all the weights must be multiples of $\frac{1}{r}$. The graph $G\square K_2$ (that is, the Cartesian product of $G$ with $K_2$) consists of two disjoint copies of $G$ with an edge between corresponding vertices in the two copies. \[lemma:glmhad\] For every graph $G$ and integer $r\geq2$, $${\textsf{\textup{glm}}}(G) \leq {\textsf{\textup{had}}}(G\square K_2) \leq 3\,{\textsf{\textup{had}}}_r(G) \enspace,$$ and if $r$ is even then $${\textsf{\textup{glm}}}(G) \leq {\textsf{\textup{had}}}(G\square K_2) \leq 2\,{\textsf{\textup{had}}}_r(G) \leq 2\,{\textsf{\textup{had}}}_f(G)\enspace.$$ @ReedWood-EuJC proved that ${\textsf{\textup{glm}}}(G) \leq {\textsf{\textup{had}}}(G\square K_2)$. Here we provide a proof. Let $t:={\textsf{\textup{glm}}}(G)$. It suffices to show there exists a $K_t$-model in $G\square K_2$. Label the vertices of $K_2$ as $1$ and $2$, so a vertex of $G \square K_2$ has the form $(v,i)$ where $v \in V(G)$ and $i \in \{1,2\}$. If $S$ is a subgraph of $G$, define $(S,i)$ to be the subgraph of $G \square K_2$ induced by $\{(v,i) | v \in S\}$. Let $H$ be the intersection graph of a set of paths ${\ensuremath{\mathcal{P}}}$ with bipartition $A,B$, such that $H$ has a $K_t$-minor. For each $P \in {\ensuremath{\mathcal{P}}}$, let $P' := (P,i)$ where $i=1$ if $P \in A$, and $i=2$ if $P \in B$. If $PQ \in E(H)$, then without loss of generality $P \in A$ and $Q \in B$, and there exists a vertex $v$ such that $v \in V(P) \cap V(Q)$. Then the edge $(v,1)(v,2) \in E(G \square K_2)$ has one endpoint in $P'$ and the other in $Q'$. So $P' \cup Q'$ is connected. Let $X_1, \dots, X_t$ be the branch sets of a $K_t$-model in $H$. Define $X_i':= \bigcup_{P \in X_i} P'$. Now each $X_i'$ is connected. It is sufficient to show, for $i \neq j$, that $V(X_i' \cap X_j') = \emptyset$ and there exists an edge of $G \square K_2$ with one endpoint in $X_i'$ and the other in $X_j'$. If there exists $v \in V(X_i' \cap X_j')$ then there exists $P'$ such that $v \in P'$ and $P' \in X_i' \cap X_j'$. But then $P \in X_i \cap X_j$, which is a contradiction. So $V(X_i' \cap X_j') = \emptyset$. Also, since $X_1, \dots, X_t$ is a $K_t$-model of $H$, there exists some $PQ \in E(H)$ such that $P \in X_i$ and $Q \in X_j$. From above, there exists an edge between $P'$ and $Q'$ in $G \square K_2$, which is sufficient. For the other inequalities, let $X_1,\dots,X_t$ be the branch sets of a $K_t$-minor in $G\square K_2$, where $t:={\textsf{\textup{had}}}(G\square K_2)$. Let $X'_i$ be the projection of $X_i$ into the first copy of $G$. Thus $X'_i$ is a connected subgraph of $G$. If $X_i$ and $X_j$ are joined by an edge between the two copies of $G$, then $X'_i$ and $X'_j$ intersect. Otherwise, $X_i$ and $X_j$ are joined by an edge within one of the copies $G$, in which case, $X'_i$ and $X'_j$ are joined by an edge in $G$. Thus $X'_1,\dots,X'_t$ is a bramble in $G$. Weight each $X'_i$ by ${\ensuremath{\protect\lfloor\frac{r}{2}\rfloor}}/r$, which is at least $\frac{1}{3}$ and at most $\frac{1}{2}$. Since $X_1,\dots,X_t$ are pairwise disjoint, each vertex of $G$ is in at most two of $X'_1,\dots,X'_t$. Hence the sum of the weights of $X'_i$ that contain a vertex $v$ is at most $1$. Hence ${\textsf{\textup{had}}}_r(G)$ is at least the total weight, which is at least $\frac{t}{3}$. That is, ${\textsf{\textup{had}}}(G\square K_2)\leq 3{\textsf{\textup{had}}}_r(G)$. If $r$ is even then the total weight equals $\frac{t}{2}$ and ${\textsf{\textup{had}}}(G\square K_2)\leq 2\,{\textsf{\textup{had}}}_r(G)$, which is at most $2\,{\textsf{\textup{had}}}_f(G)$ by definition. \[lemma:hadbn\] For every graph $G$, $${\textsf{\textup{had}}}_f(G) \leq {\textsf{\textup{bn}}}(G)\enspace.$$ Let ${\ensuremath{\mathcal{B}}}$ be a bramble in $G$ and let $w:{\ensuremath{\mathcal{B}}}\rightarrow\mathbb{R}_{\geq 0}$ be a weight function, such that ${\textsf{\textup{had}}}_f(G) =\sum_{X\in{\ensuremath{\mathcal{B}}}}w(X)$ and for each vertex $v$, the sum of the weights of the subgraphs in ${\ensuremath{\mathcal{B}}}$ that contain $v$ is at most 1. Let $S$ be a hitting set for ${\ensuremath{\mathcal{B}}}$. Thus $$|S|=\sum_{v\in S}1 \geq \sum_{v\in S}\sum_{X\in{\ensuremath{\mathcal{B}}}:v\in X}w(X) = \sum_{X\in{\ensuremath{\mathcal{B}}}}|X\cap S|w(X) \geq \sum_{X\in{\ensuremath{\mathcal{B}}}}w(X) ={\textsf{\textup{had}}}_f(G)\enspace.$$ That is, the order of ${\ensuremath{\mathcal{B}}}$ is at least ${\textsf{\textup{had}}}_f(G)$. Hence ${\textsf{\textup{bn}}}(G) \geq {\textsf{\textup{had}}}_f(G)$. Note this lemma is tight; consider $G = K_n$. @Wood-ProductMinor proved that ${\textsf{\textup{had}}}(G\square K_2)\leq 2{\textsf{\textup{tw}}}(G)+2$ and @ReedWood-EuJC proved that ${\textsf{\textup{glm}}}(G) \leq 2{\textsf{\textup{tw}}}(G)+2$. More precisely, Lemma \[lemma:glmhad\] and Lemma \[lemma:hadbn\] imply that $${\textsf{\textup{glm}}}(G) \leq {\textsf{\textup{had}}}(G\square K_2)\leq 2 {\textsf{\textup{had}}}_f(G) \leq 2{\textsf{\textup{bn}}}(G) = 2{\textsf{\textup{tw}}}(G)+2\enspace,$$ and for every integer $r\geq2$, $${\textsf{\textup{glm}}}(G) \leq 3 {\textsf{\textup{had}}}_r(G) \leq 3 {\textsf{\textup{had}}}_f(G) \leq 3{\textsf{\textup{bn}}}(G) =3{\textsf{\textup{tw}}}(G)+3\enspace.$$ Conversely, @ReedWood-EuJC proved that $${\textsf{\textup{tw}}}(G) \leq c\, {\textsf{\textup{glm}}}(G)^4 \sqrt{ \log {\textsf{\textup{glm}}}(G) }\enspace,$$ for some constant $c$. Thus ${\textsf{\textup{glm}}}$, ${\textsf{\textup{had}}}(G\square K_2)$, ${\textsf{\textup{had}}}_f$, ${\textsf{\textup{had}}}_r$ for each $r\geq2$, and ${\textsf{\textup{tw}}}$ are tied by polynomial functions. Fractional Open Problems ======================== Given a graph $G$ define a *$b$-fold colouring* for $G$ to be an assignment of $b$ colours to each vertex of $G$ such that if two vertices are adjacent, they have no colours assigned in common. We can consider this a generalisation of standard graph colouring, which is equivalent when $b=1$. A graph $G$ is *$a\!:\!b$-colourable* when there is a $b$-fold colouring of $G$ with $a$ colours in total. Then define the *$b$-fold chromatic number* $\chi_b(G) := \min\{a| G$ is $a\!:\!b$-colourable$\}$. So $\chi_1(G) = \chi(G)$. Then, define the *fractional chromatic number* $\chi_f(G) = \lim_{b \rightarrow \infty} \frac{\chi_b(G)}{b}$. See @SU-FGT97 for an overview of the topic. @ReedSeymour-JCTB98 proved that $\chi_f(G) \leq 2\,{\textsf{\textup{had}}}(G)$. Hence there is a relationship between the fractional chromatic number and Hadwiger’s number. We have $$\chi_f(G)\leq \chi(G)\quad\text{and}\quad{\textsf{\textup{had}}}(G)\leq{\textsf{\textup{had}}}_f(G)\leq{\textsf{\textup{tw}}}(G)+1\enspace.$$ Hadwiger’s Conjecture asserts that $\chi(G)\leq{\textsf{\textup{had}}}(G)$, thus bridging the gap in the above inequalities. Note that $\chi(G)\leq{\textsf{\textup{tw}}}(G)+1$. (Since $G$ has minimum degree at most ${\textsf{\textup{tw}}}(G)$, a minimum-degree-greedy algorithm uses at most ${\textsf{\textup{tw}}}(G)+1$ colours.) Thus the following two questions provide interesting weakenings of Hadwiger’s Conjecture: $\chi(G)\leq {\textsf{\textup{had}}}_f(G)$. $\chi_f(G)\leq {\textsf{\textup{had}}}_f(G)$. Finally, note that the above results prove that ${\textsf{\textup{had}}}_3$ is bounded by a polynomial function of ${\textsf{\textup{had}}}_2$. Is ${\textsf{\textup{had}}}_3(G) \leq c\,{\textsf{\textup{had}}}_2(G)$ for some constant $c$? Acknowledgements {#acknowledgements .unnumbered} ================ Many thanks to Jacob Fox and Bruce Reed for instructive conversations. [^1]: Research of D.R.W. is supported by the Australian Research Council. [^2]: Research of D.J.H. is supported by an Australian Postgraduate Award. [^3]: Occasionally, other authors use the term *comparable* [@Fox11] [^4]: @Fox11 defines a separator to be a set $X \subseteq V(G)$ that partitions $V(G)$ into $X \cup A \cup B$ with no $A-B$ edge and $|A|,|B| \leq \frac{2}{3}|V(G)|$. Fox then defines the separation number to be the minimum integer $k$ such that each subgraph of $G$ has a separator of size $k$. However, we will not consider this definition in this paper.

--- abstract: 'Conformally related metrics and Lagrangians are considered in the context of scalar-tensor gravity cosmology. After the discussion of the problem, we pose a lemma in which we show that the field equations of two conformally related Lagrangians are also conformally related if and only if the corresponding Hamiltonian vanishes. Then we prove that to every non-minimally coupled scalar field, we may associate a unique minimally coupled scalar field in a conformally related space with an appropriate potential. The latter result implies that the field equations of a non-minimally coupled scalar field are the same at the conformal level with the field equations of the minimally coupled scalar field. This fact is relevant in order to select physical variables among conformally equivalent systems. Finally, we find that the above propositions can be extended to a general Riemannian space of $n$-dimensions.' author: - Michael Tsamparlis - Andronikos Paliathanasis - Spyros Basilakos - Salvatore Capozziello title: Conformally related metrics and Lagrangians and their physical interpretation in cosmology --- Introduction ============ The detailed analysis of the current high quality cosmological data (Type Ia supernovae, cosmic microwave background, baryonic acoustic oscillations, etc), converge towards a new emerging “Cosmological Standard Model”. This cosmological model is spatially flat with a cosmic dark sector constituted by cold dark matter and some sort of dark energy, associated with large negative pressure, in order to explain the observed accelerating expansion of the universe (see [@Teg04; @Spergel07; @essence; @Kowal08; @Hic09; @komatsu08; @LJC09; @BasPli10] and references therein). Despite the mounting observational evidences on the existence of the dark energy component in the universe, its nature and fundamental origin remains an intriguing enigma challenging the very foundations of theoretical physics. Indeed, during the last decade there has been an intense theoretical debate among cosmologists regarding the nature of this exotic “dark energy”. The absence of a fundamental physical theory, concerning the mechanism inducing the cosmic acceleration, has opened a window to a plethora of alternative cosmological scenarios. Most are based either on the existence of new fields in nature (dark energy) or in some modification of Einstein’s general relativity, with the present accelerating stage appearing as a sort of geometric effect (see [@Ratra88; @curvature; @mauro; @report; @repsergei; @Oze87; @Weinberg89; @Lambdat; @Bas09c; @Wetterich:1994bg; @Caldwell98; @Brax:1999gp; @KAM; @fein02; @Caldwell; @Bento03; @chime04; @Linder2004; @LSS08; @Brookfield:2005td; @Boehmer:2007qa; @Starobinsky-2007; @Ame10] and references therein). In order to investigate the dynamical properties of a particular dark energy model, we need to specify the covariant Einstein-Hilbert action of the model and find out the corresponding energy-momentum tensor. This methodology provides a mathematically consistent way to incorporate dark energy in cosmology. However, in the literature, there are many Lagrangians [@Ame10] which describe differently the physical features of the scalar field or the modified gravity [@report]. Because of the large amount of dark energy models, it is essential to study them in a unified context in order to discriminate the true physical variables. From our viewpoint, this framework has to be at the level of geometry since the various Lagrangians which describe the nature of dark energy are embedded in the space-time. From a theoretical point of view, an easy way to study dynamics in a unified manner is to look for conformally related Lagrangians. In fact, the idea to use conformally related metrics and Lagrangians as a cosmological tool is not new. In particular, it has been proposed that the existence of conformally equivalent Lagrangians can be used in order to select viable cosmological models [@allemandi]. In general, the presence of scalar fields into the gravitational action can give rise to two classes of theories: minimally and non-minimally coupled Lagrangians. In the first case, the gravitational coupling is the standard Newton constant and the scalar field Lagrangian is simply added to the Ricci scalar. It can consist of $i)$ a kinetic term, $ii)$ a kinetic term and a self-interaction potential, or $iii)$ just an interaction potential. In the first case, the scalar field is nothing else but a cyclic variable and then it is related to a conserved quantity. The second case is the relevant one since, by a variational principle, it is possible to obtain a Klein-Gordon equation where the self-interacting potential $V(\phi)$ leads the dynamics. The third case means that the scalar field has no dynamics. When the coupling is the Newton constant, we are in the so-called [*Einstein frame*]{}. In fact, as firstly pointed out by Brans and Dicke [@brans], a gravitational theory can be made more “Machian” by relaxing the hypothesis that the coupling is constant. They introduced a scalar field $\phi$ non-minimally coupled to the Ricci scalar $R$ and a kinetic term for such a scalar field into the gravitational action. The result was that the coupling was non-minimal and coordinate dependent. In other words, the gravitational interaction is assumed to change with distance and time. This approach can be generalized considering scalar-tensor theories of gravity where also a self-interaction potential comes into the game or more than one scalar field are taken into account. In general, any gravitational theory not simply linear in the Ricci scalar can be reduced to a scalar-tensor one. In the case of $f(R)$-gravity, it is straightforward to show that an [*O’Hanlon representation*]{} by scalar fields is possible [@ohanlon]. In this case, a scalar field is non-minimally coupled to the Ricci scalar and a self-interacting potential is present while there is no kinetic term. Scalar field dynamics is guaranteed by the non-minimal coupling and the potential (see [@report] and references therein). In other words, the further gravitational degrees of freedom, coming from the fact that $f(R)\neq R$, can be figured out as a scalar field. As soon as we are considering non-minimal couplings or higher-order terms in the Lagrangian, we are in the so-called [*Jordan frame*]{}, after Jordan who first introduced this notion [@Jordan38; @Jordan52]. In this article we would like to address the following basic question: *In the framework of the scalar field cosmology, is it possible to relate the available Lagrangians in a conformal way?* The structure of the paper is as follows. In section II, we discuss the issue of conformal transformations considering its historical development and connection with physical theories. From our point of view, this section is essential in order to fix the problem showing the urgency to discriminate physical variables among the conformally related models. The basic theoretical elements of the conformally related metric are presented in section III, where we also introduce the concept of conformally related Lagrangians and prove a lemma which shows that the field equations for two conformally related Lagrangian are also conformally related if the corresponding Hamiltonian vanishes. In section IV, we discuss the conformal equivalence of Lagrangians for scalar fields in a Riemannian space of dimension $4$ (for an extension see appendix A). In particular we enunciate a theorem which proves that the field equations for a non-minimally coupled scalar field are the same at the conformal level with the field equations of the minimally coupled scalar field. The necessity to preserve Einstein’s equations in the context of Friedmann-Lemaître-Robertson-Walker (FRLW) space-time leads us to apply, in section V, the current general analysis to the scalar field (quintessence or phantom) flat FLRW cosmologies. Finally, we draw our main conclusions and discuss results in section VI. What is the physical frame? =========================== Some considerations are in order at this point. The conformal transformations from the Jordan to the Einstein frame are geometrical maps allowing to set many features of scalar-tensor gravity, $f(R)$ gravity and, in general, any modified theory of gravity. Taking into account both the Jordan and the Einstein conformal frames (infinite conformal frames can be chosen assuming a suitable conformal factor), the question is whether the frames are infinitely many *physically* equivalent or only mathematically related. In other words, the problem is whether the physical information contained in the theory is “preserved” or not by the conformal transformations. In other words, one has the metric $g_{ij} $ and its conformally related one $\bar{g}_{ij}$: the question is what is the “physical metric”, *i.e.*, the metric from which curvature, geometry, and physical effects have to be calculated and compared with experiments and observations [@mercadante]. More precisely, every Killing and homothetic vector is also a symmetry of the energy-momentum tensor. The latter is also a [*metric*]{} of spacetime because a “metric” can be considered any second order tensor. Therefore, up to the homothetic group, metric, energy-momentum tensor, and Ricci tensor have the same symmetries. This is not the case for the conformal group and then the requirement that the theory is invariant under conformal transformations is an additional assumption not related to the gauge invariance. Furthermore, we are concerned with Lagrangians which means that we assume the same kinematics on which these Lagrangians set up a dynamics. In general, kinematics has other symmetries with respect to dynamics and these symmetries are not related to those of the metric because no field equation relates kinematic quantities to metric. Kinematic symmetries are constrained only by the Ricci identity which gives the constraint and propagation equations. The issue of “which is the physical frame” has been debated for a long time and it emerges as soon as some authors argue in favor of one frame against the other, and others support the idea that the two frames are physically equivalent. In the latter case, authors claim that the issue is a pseudo-problem. The final result is that there is a good deal of confusion in the literature. Fierz was the first to pose the problem [@GrumillerKummerVassilevich03], but the main argument is due to Dicke, who discussed the conformal transformation for Brans-Dicke theory [@Dicke62]. The point was that physics must be invariant under a rescaling of units and the conformal transformation is a local rescaling: units do not change rigidly over the entire spacetime manifold, but by amounts which are different at different spacetime points. From Dicke’s point of view, the two frames are equivalent provided that mass, length, time, and quantities derived from them scale with appropriate powers of the conformal factor in the Einstein frame [@Dicke62]. From this point of view, it is not difficult to see why many authors consider the issue of which is the physical frame a pseudo-problem. In principle, it is difficult to object to this argument, but there are some difficulties. Even though the above argument is clear in principle, its application to practical situations gives rise to problems. The assumption that the two conformal frames are just different *representations* of the same theory, similar to different gauges of a gauge theory, has to be checked explicitly by using the field equations of a given system. Physical equivalence is a vague concept because one can consider many different matter (or test) fields in curved spacetime and different types of physics, or different physical aspects of a problem. When checking explicitly the physical equivalence between the two frames, one has to specify which physical field, or physical process is considered and the equations describing it. The equivalence has to be shown explicitly, but there is no proof that holds in any situation (e.g. scalar fields, spinors, cosmology, black holes derived from the same theory). While physical equivalence can be proven for various physical aspects, no proof comprehensive of all physical fields and different physical applications exists. It is important to stress that Dicke seems to mix the concept of dimensional units and the concept of measuring units. For example, spatial distance has the dimensional unit length $L$ but the measuring unit length, as we already know from Special Relativity, has to be defined in a relativistic inertial frame (RIF). After, by some rule, spatial lengths are compared in different RIFs. One of such rules is the Einstein one. It can be defined by means of light signals which are simultaneous in the RIF in which the measurement is taking place. This is the so called [*rest length*]{} (which, by assumption, coincides with the corresponding Euclidian length measured by a Newtonian inertial observer). This type of measurement has nothing to do with the concept of dimension which, for all RIFs, is the same i.e. the time $T$ (the second, in units, where $c=1$, see e.g. [@mikebook]). Dicke is aware of that and then says [*“ Generally there may be more than one feasible way of establishing the equality of units at different spacetime points”*]{} [@Dicke62]. It is essential to stress that such an approach is based on conventions which may lead to absurd results in real measurements. Geometrically it is a 1:1 map of two points defining a spacelike interval from the rest space of one observer at a spacetime point to a spacelike plane of another observer at another spacetime point. This definition it is not a coordinate transformation and there is no point or meaning to consider symmetry invariance with this transformation. Dicke says [*“It is evident that the equations of motion of matter must be invariant under a general coordinate dependent transformation of units”*]{} [@Dicke62]. This is a misunderstanding that can give rise to confusion. The method of comparing lengths at different spacetime points is a kinematic assumption while equations of motion give dynamics. Furthermore, Dicke’s argument is purely classical. In cosmology, black hole physics, and quantum fields in curved space, the equivalence of conformal frames is not clear. At quantum level, this equivalence is not proven due to the lack of a definitive quantum gravity theory: in fact, when the metric $g_{ij}$ is quantized, inequivalent quantum theories can be found [@GrumillerKummerVassilevich03; @AshtekarCorichi03]. One can consider the semiclassical regime in which gravity is classical and matter fields are quantized: again, one would expect the conformal frames to be inequivalent because the conformal transformations can be seen as Legendre transformations [@MagnanoSokolowski94], similar to the Legendre transformations of classical mechanics of point particles which switches from the canonical Lagrangian coordinates $q$ to the variables $\left\{ q,p\right\} $ of the Hamiltonian formalism. Now, it is well known that Hamiltonians that are classically equivalent become inequivalent when quantized, producing different energy spectra and scattering amplitudes [@CalogeroDegasperis04; @GlassScanio77]. However, the conformal equivalence between Jordan and Einstein frame seems to hold to some extent at the semi-classical level [@Flanagan04b]. Again, only a particular kind of physics has been considered and one cannot make statements about all possible physical situations. It is important to point out a very basic argument among particle physicists that relies on the equivalence theorem of Lagrangian field theory. It states that the $S$-matrix is invariant under local (nonlinear) field redefinitions [@Dyson48; @Blasietal99]. Since the conformal transformation is, essentially, a field redefinition, it would seem that quantum physics is invariant under the change of the conformal frame. However, the field theory in which the equivalence theorem is derived applies to gravity only in the perturbative regime in which the fields deviate slightly from the Minkowski space-time. In this regime, tree level quantities can be calculated in any conformal frame with same result, but in the non-perturbative regime field theory and the equivalence theorem do not apply. Unfortunately, the scaling of units in the Einstein frame often produces results that either do not make sense or are incorrect in the Jordan frame, reinforcing the opposite view that the two frames are not equivalent. While Dicke’s explanation is very appealing and several claims supporting the view that the two frames are inequivalent turned out to be incorrect because they simply neglected the scaling of units in the Einstein frame, one should not forget that Dicke’s argument is not inclusive of all areas of physics and it is better to check explicitly that the physics of a certain field does not depend on the conformal representation and not make sweeping statements. Certain points have been raised in the literature which either constitute a problem for Dicke’s view, or, at least, indicate that this viewpoint cannot be applied blindly, including the following ones. For example, massive particles follow time-like geodesics in the Jordan frame, while they deviate from geodesic motion in the Einstein frame due to a force proportional to the gradient of the conformal scalar field [@FaraoniNadeauconfo]. Hence, the Weak Equivalence Principle is satisfied only in the Jordan frame but not in the Einstein frame due to the coupling of the scalar field to the ordinary matter. Since the Equivalence Principle is the foundation of any relativistic theory of gravity, this aspect is important and there are two ways to consider it. One can ask for the two conformal frames are equivalent also with respect to the Equivalence Principle. This means that Equivalence Principle is formulated in a way that depends on the conformal frame representation. Then, a representation-independent formulation must be sought for. However, up to now, no definite result exists in this direction. On the other hand, we can ask for the violation of the Weak Equivalence Principle in the Einstein frame by saying that the “physical equivalence” of the two frames must be precisely defined and this concept cannot be used blindly. In fact the Equivalence Principle holds only in one frame but not in the other. This fact could be used as an argument against the physical equivalence of the frames. However, the fact that the Equivalence Principle holds in a given frame and not in [*all frames*]{} means that it is not a covariant requirement but a kinematic one. In other words, the Equivalence Principle could not be sufficient to discriminate among conformally related frames. In the scalar-tensor theories of gravity, the energy conditions are easily violated in the Jordan frame, but they are satisfied in the Einstein frame [@MagnanoSokolowski94]. This fact does not eliminate singularities and then the two frames remain equivalent with respect to the singularities and not with respect to the energy conditions. This difficulty arises because part of the matter sector of the theory, in the Einstein frame, comes from the conformal factor; in other words, the conformal transformation mixes matter and geometric degrees of freedom, which is the source of many interpretational problems [@Cap01; @Cap02]. Thus, even if the theory turns out to be independent of the conformal representation, its interpretation is not. There are results in cosmology in which the universe accelerates in one frame but not in the other. From the pragmatic point of view of an astronomer attempting to fit observational data (for example, type Ia supernovae data to a model of the present acceleration of the universe), the two frames certainly do not appear to be “physically equivalent” [@nodicap2; @CapozzielloPrado]. To approach correctly the problem of physical equivalence under conformal transformations, one has to compare physics in different conformal frames at the level of the Lagrangian, of the field equations, and of their solutions [@allemandi]. This comparison may not always be easy but, in certain cases, it is extremely useful to discriminate between frames. It has been adopted, for example, in Ref. [@cno], to compare cosmological models in the Einstein and the Jordan frame. Specifically, it has been shown that solutions of $f(R)$ and scalar-tensor gravity cannot be assumed to be physically equivalent to those in the Einstein frame when matter fields are given by generalized Equations of State. In these and in other situations, one must specify precisely what “physical equivalence” means. In certain situations physical equivalence is demonstrated simply by taking into account the coupling of the scalar field to matter and the varying units in the Einstein frame, but in other cases the physical equivalence is not obvious and it does not seem to hold. At the very least, this equivalence, if it is valid at all, must be defined in precise terms and discussed in ways that are far from obvious. For this reason, it would be too simplistic to dismiss the issue of the conformal frame as a pseudo-problem that has been solved for all physical situations of interest. Conformally related metrics and Lagrangians =========================================== Taking in mind the above discussion, we want to seek for geometrical structures that are conformally invariant. Our aim is to compare cosmological models coming from scalar-tensor gravity in order to select conformal quantities in view of a possible physical meaning. This is a very delicate issue that has to be discussed in details. On one hand, an invariant quantity, (i.e. a quantity that remains the same under conformal transformation) should have a physical meaning. However, in cosmology, this statement does not necessarily hold since the problem of equivalence between the two frames is not well posed and it may happen that one of them has to be taken as the physical one in a particular case and then such invariant quantity would not have a physical meaning. This situation often happen if cosmological solutions fit data and then related quantities are assumed as “physical” [@CapozzielloPrado]. On the other hand, if a scalar field describes an actual particle in a given frame, then its properties (e.g, its mass and couplings to other fields) would change in the conformally-related metric. This fact does not mean that its properties have no physical meaning. In other words, the identification of conformally invariant physical quantities is a very difficult task if it is not based on first principles. With these considerations in mind, let us start with defining some geometrical structures that will be useful in the discussion. A vector field $X^{a}$ is a Conformal Killing Vector [@Yano] (hereafter CKV) of the metric $g_{ij}$ if there exists a function $\psi\left( x^{k}\right) $ so that: $$\mathcal{L}_{X}g_{ij}=2\psi\left( x^{k}\right) g_{ij}\label{CLN.01}$$ where $\mathcal{L}_{X}$ is the Lie derivative. In case that $\psi_{;a}=0$ i.e. $\psi=$constant, the vector $X$ is called homothetic (hereafter HV) while if $\psi=0$ then the vector $X$ is a Killing vector (hereafter KV). In this context, two metrics $g_{ij},\bar{g}_{ij}$ are said to be conformal or conformally related if there exists a function $N^{2}\left( x^{k}\right) $ so that $\bar{g}_{ij}=N^{2}\left( x^{k}\right) g_{ij}$. From the mathematical point of view the CKVs form the so called *conformal algebra* of the metric. The conformal algebra contains two closed sub-algebras the *Homothetic algebra* and the *Killing algebra*. Interestingly the above algebras are related as follows: $$KV_{s}\subseteq HV_{s}\subseteq CKV_{s}\;.\label{CLN.02}$$ The dimension of the conformal algebra of an $n-$ dimensional metric $(n>2)$ of constant curvature equals $\frac{(n+1)(n+2)}{2}$, the dimension of the Killing algebra $\frac{n(n+1)}{2}$ and that of the homothetic algebra $\frac{n(n+1)}{2}+1$. Note that two conformally related metrics have the same conformal algebra [@TsampC], however not the same subalgebras. Indeed if $X$ is a CKV for the metric $g_{ij}$ i.e. $\mathcal{L}_{X}g_{ij}=2\psi\left( x^{k}\right) g_{ij}$ then for the metric $\bar{g}_{ij}$ the vector $X$ is again a CKV with conformal factor $\bar{\psi}\left( x^{k}\right) ,$ that is: $$\mathcal{L}_{X}\bar{g}_{ij}=2\bar{\psi}\left( x^{k}\right) \bar{g}_{ij}\label{CLN.03}$$ where the conformal factors $\psi\left( x^{k}\right) $, $\bar{\psi}\left( x^{k}\right) $ are related as follows: $$\bar{\psi}\left( x^{k}\right) =\psi\left( x^{k}\right) +\mathcal{L}_{X}\left( \ln N\right) .\label{CLN.04}$$ The Ricci scalars of the conformally related metrics $g_{ij},\bar{g}_{ij}$ are related as follows [@HawkingB]:$$\bar{R}=N^{-2}R-2(n-1)N^{-3}\Delta_{2}N-(n-1)(n-4)\Delta_{1}N\label{CLN.04.1}$$ where (note that $\Delta_{2}N$ contains the covariant derivative whereas $\Delta_{1}N$ the partial derivative): $$\begin{aligned} \Delta_{1}N & =g_{ij}N^{,i}N^{,j}\label{CLN.04.2}\\ \Delta_{2}N & =g_{ij}N^{;ij}.\label{CLN.04.3}$$ From the above discussion it becomes clear that all two dimensional spaces are Einstein Spaces $\left( \text{i.e. }R_{ab}=\frac{R}{2}g_{ab}\right) $ and conformally flat. The metric of a two dimensional space can be written in the generic form: $$ds^{2}=N^{2}\left( x,y\right) \left( \varepsilon dx^{2}+dy^{2}\right) ~~,~\varepsilon=\pm1\;.\label{CLN.10}$$ Conformal Lagrangians --------------------- Due to the fact that almost every dynamical system is described by a corresponding Lagrangian, below we study generically, as much as possible, the problem of the conformal Lagrangians and then we apply the current ideas to the scalar field cosmology. To begin with, consider the Lagrangian of a particle moving under the action of a potential $V(x^{k})$ in a Riemannian space with metric $g_{ij}$ $$L=\frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}-V\left( x^{k}\right) ~,~\dot {x}^{i}=\frac{dx^{i}}{dt}\label{CLN.05}$$ where $t$ is a path parameter. The equations of motion follow from the action $$S=\int dxdtL\left( x^{k},\dot{x}^{k}\right) =\int dxdt\left[ \frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}-V\left( x^{k}\right) \right] .\label{CLN.06}$$ Changing the variables in Eq.(\[CLN.06\]) from $t$ to $\tau$ via the relation: $$d\tau=N^{2}\left( x^{i}\right) dt \label{tran1A}$$ the action is given by $$S=\int dx\frac{d\tau}{N^{2}\left( x^{k}\right) }\left[ \frac{1}{2}g_{ij}N^{4}\left( x^{k}\right) x^{\prime i}x^{\prime j}-V\left( x^{k}\right) \right] ~~\label{CLN.07}$$ where $x^{\prime i}=\frac{dx^{i}}{d\tau}$. Obviously, the Lagrangian in the new coordinate system $(\tau,x^{i})$ becomes:$$\bar{L}\left( x^{k},x^{\prime k}\right) =\frac{1}{2}N^{2}\left( x^{k}\right) g_{ij}x^{\prime i}x^{\prime j}-\frac{V\left( x^{k}\right) }{N^{2}\left( x^{k}\right) }.\label{CLN.08}$$ Now if we consider a conformal transformation of the metric $\bar{g}_{ij}=N^{2}\left( x^{k}\right) g_{ij}$ and a new potential function $\bar {V}\left( x^{k}\right) =\frac{V\left( x^{k}\right) }{N^{2}\left( x^{k}\right) }$ then the new Lagrangian $\bar{L}\left( x^{k},x^{\prime k}\right) $ takes the following form: $$\bar{L}\left( x^{k},x^{\prime k}\right) =\frac{1}{2}\bar{g}_{ij}x^{\prime i}x^{\prime j}-\bar{V}\left( x^{k}\right) \label{CLN.09}$$ implying that Eq.(\[CLN.09\]) is of the same form as the Lagrangian $L$ in Eq.(\[CLN.05\]). From now on *the Lagrangian $L\left( x^{k},\dot {x}^{k}\right) ~$ of Eq.(\[CLN.05\]) and the Lagrangian $\bar{L}\left( x^{k},x^{\prime k}\right) $ of Eq.(\[CLN.09\]) will be called conformal*. In this framework, the action remains the same i.e. it is invariant under the change of parameter, the equations of motion in the new variables $(\tau ,x^{i})$ will be the same with the equations of motion for the Lagrangian $L$ in the original coordinates $(t,x^{k})$ . It has been shown [@TsamGE] that the Noether symmetries of a Lagrangian of the form (\[CLN.05\]) follow the homothetic algebra of the metric $g_{ij}.$ The same applies to the Lagrangian $\bar{L}\left( x^{k},x^{\prime k}\right) $ and the metric $\bar{g}_{ij}.$ As we have remarked the conformal algebra of the metrics $g_{ij},\bar{g}_{ij}$ (as a set) is the same however their closed subgroups of HVs and KVs are in general different[^1]. Now, we formulate and prove the following proposition: ***Lemma:*** *The Euler-Lagrange equations for two conformal Lagrangians transform covariantly under the conformal transformation relating the Lagrangians iff the Hamiltonian vanishes.* ***Proof:*** Consider the Lagrangian $L=\frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}-V\left( x^{k}\right) $ whose Euler-Lagrange equations are: $$\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{i}\dot{x}^{j}+V^{,i}=0 \label{CLN1.01}$$ where $\Gamma_{jk}^{i}$ are the Christofell symbols. The corresponding Hamiltonian is given by $$E=\frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}+V\left( x^{k}\right) \;. \label{CLN1.02}$$ For the conformally related Lagrangian $\bar{L}\left( x^{k},x^{\prime k}\right) =\left( \frac{1}{2}N^{2}\left( x^{k}\right) g_{ij}x^{\prime i}x^{\prime j}-\frac{V\left( x^{k}\right) }{N^{2}\left( x^{k}\right) }\right) $ where $N_{,j}\neq0$ the resulting Euler Lagrange equations are$$x^{\prime\prime i}+\hat{\Gamma}_{jk}^{i}x^{\prime j}x^{\prime k}+\frac {1}{N^{4}}V^{,i}-\frac{2V}{N^{5}}N^{,i}=0 \label{CLN1.03}$$ where $$\hat{\Gamma}_{jk}^{i}=\Gamma_{jk}^{i}+(\ln N)_{,k}\delta_{j}^{i}+(\ln N)_{,j}\delta_{k}^{i}-(\ln N)^{,i}g_{jk} \label{CLN1.04}$$ and the corresponding Hamiltonian is$$\bar{E}=\frac{1}{2}N^{2}\left( x^{k}\right) g_{ij}\dot{x}^{i}\dot{x}^{j}+\frac{V\left( x^{k}\right) }{N^{2}\left( x^{k}\right) }. \label{CLN1.05}$$ In order to show that the two equations of motion are conformally related we start from Eq.(\[CLN1.03\]) and apply the conformal transformation $$\begin{aligned} x^{\prime i} & =\frac{dx^{i}}{d\tau}=\frac{dx^{i}}{dt}\frac{dt}{d\tau}=\dot{x}^{i}\frac{1}{N^{2}}\\ x^{^{\prime\prime}i} & =\ddot{x}^{i}\frac{1}{N^{4}}-2\dot{x}^{i}\dot{x}^{j}\left( \ln N\right) _{,j}\frac{1}{N^{4}}.\end{aligned}$$ Replacing in Eq.(\[CLN1.03\]) we find:$$\ddot{x}^{i}\frac{1}{N^{4}}-2\dot{x}^{i}\dot{x}^{j}\left( \ln N\right) _{,j}\frac{1}{N^{4}}+\frac{1}{N^{4}}\hat{\Gamma}_{jk}^{i}\dot{x}^{j}\dot {x}^{k}+\frac{1}{N^{4}}V^{,i}-\frac{2V}{N^{5}}N^{,i}=0\,.$$ Replacing $\hat{\Gamma}_{jk}^{i}$ from Eq.(\[CLN1.04\]) we have $$\begin{aligned} \ddot{x}^{i}-2\dot{x}^{i}\dot{x}^{j}\left( \ln N\right) _{,j}+\Gamma _{jk}^{i}\dot{x}^{j}\dot{x}^{k}+2(\ln N)_{,j}\dot{x}^{j}\dot{x}^{i}\\ -(\ln N)^{,i}g_{jk}\dot{x}^{j}\dot{x}^{k}+V^{,i}-2V(\ln N)^{,i}=0\end{aligned}$$ from which follows$$\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x}^{k}+V^{,i}-(\ln N)^{,i}\left( g_{jk}\dot{x}^{j}\dot{x}^{k}+2V\right) =0.$$ Obviously, the above Euler-Lagrange equations coincide with Eqs.(\[CLN1.01\]) if and only if $\left( g_{jk}\dot{x}^{j}\dot{x}^{k}+2V\right) =0$, which implies that the Hamiltonian of Eq.(\[CLN1.02\]) vanishes. The steps are reversible hence the inverse is also true. The physical meaning of such a result is that systems with vanishing energy are conformally invariant at the level of equations of motion. Conformally equivalent Lagrangians in the scalar field Cosmology ================================================================ Let us discuss now the conformal equivalence of Lagrangians for scalar fields in a general Riemannian space of four dimensions (see also [@allemandi]). The field equations in the scalar-tensor cosmology can be derived from two different variational principles. In the first case we consider a scalar field $\phi$ which is minimally coupled to gravity and the equations of motion follow from the action $$S_{M}=\int d\tau dx^{3}\sqrt{-g}\left[ R+\frac{\varepsilon}{2}g_{ij}\phi ^{;i}\phi^{;j}-V\left( \phi\right) \right] \label{CLN.11}$$ where $\varepsilon=\pm 1$ defines quintessence or phantom field cosmology respectively. In the second case we assume a scalar field $\psi$ (different from the minimally coupled scalar field $\phi$) which interacts with the gravitational field (non minimal coupling) and the corresponding action is given by $$S_{NM}=\int d\tau dx^{3}\sqrt{-\bar{g}}\left[ F\left( \psi\right) \bar {R}+\frac{\varepsilon}{2}\bar{g}^{ij}\psi_{;i}\psi_{;j}-\bar{V}\left( \psi\right) \right] \label{CL.12.0}$$ where $F(\psi)$ is the coupling function between the gravitational and the scalar field $\psi$ respectively. Below we pose the following proposition. ***Theorem:*** *The field equations for a non minimally coupled scalar field $\psi$ with Lagrangian $\bar{L}\left( \tau,x^{k},\dot {x}^{k}\right) $ and coupling function $F(\psi)$ in the gravitational field $\bar{g}_{ij}$ are the same with the field equations of the minimally coupled scalar field $\Psi$ for a conformal Lagrangian $L\left( \tau,x^{k},\dot {x}^{k}\right) $ in the conformal metric $g_{ij}$ $=N^{-2}\bar{g}_{ij}$ where the conformal function is $N=\frac{1}{\sqrt{-2F\left( \psi\right) }}$ with $F\left( \psi\right) <0.$ The inverse is also true, that is, to a minimally coupled scalar field it can be associated a unique non minimally coupled scalar field in a conformal metric and with a different potential function.* ***Proof:*** We first start with the action of Eq.(\[CL.12.0\]). Let $g_{ij}$ be the conformally related metric (this is not a coordinate transformation): $$g_{ij}=N^{-2}\bar{g}_{ij}.$$ Then the action provided by Eq.(\[CL.12.0\]) becomes [^2]:$$S_{NM}=\int d\tau dx^{3}N^{4}\sqrt{-g}\left[ F\left( \psi\right) \bar {R}+\frac{\varepsilon}{2}N^{-2}g^{ij}\psi_{;i}\psi_{;j}-\bar{V}\left( \psi\right) \right] .$$ Inserting the Ricci scalar $\bar{R}$ (using $n=4)$ from Eq.(\[CLN.04.1\]) into the latter equation we find: $$\begin{aligned} \label{ICLN} S_{NM} &=&\int d\tau dx^{3}N^{4}\sqrt{-g}\left[ F\left( \psi \right) N^{-2}R-6F\left( \psi \right) N^{-3}\Delta _{2}N +\frac{\varepsilon }{2}N^{-2}\Delta _{1}\psi -\bar{V}\left( \psi \right) \right]\,. $$ Now we can define the conformal function $N$ in terms of the coupling function $F(\psi)$ \[where $F\left( \psi\right) <0)$\]: $$\label{CLN122}N=\frac{1}{\sqrt{-2F\left( \psi\right) }}~.$$ with $$\label{CLN123}N_{;i}=\frac{F_{\psi}\psi_{;i}}{\left( -2F\right) ^{\frac {3}{2}}}\;.$$ Using Eqs.(\[CLN122\]) and (\[CLN123\]) the first term of the integral in Eq.(\[ICLN\]) becomes: $$\int d\tau dx^{3}\sqrt{-g}F\left( \psi\right) N^{2}R=\int d\tau dx^{3}\sqrt{-g}\left( -\frac{R}{2}\right) .$$ On the other hand the second term in Eq.(\[ICLN\]) gives, after integration by parts: $$\begin{aligned} \label{ICLN1} \int d\tau dx^{3}\sqrt{-g}\left[ -6F\left( \psi \right) N\Delta _{2}N\right] &=&\int d\tau dx^{3}\sqrt{-g}\left( -6\frac{F}{\sqrt{-2F}} N_{;ij}g^{ij}\right) \nonumber\\ &=&\int d\tau dx^{3}\sqrt{-g}\left[ -6\frac{F}{\sqrt{-2F}}\frac{1}{\sqrt{-g}}\left( \sqrt{-g}g^{ij}N_{,k}\right) _{,j}\right] \nonumber\\ &=&\int d\tau dx^{3}\left[ -6\frac{F}{\sqrt{-2F}}\left( \sqrt{-g} g^{ij}N_{,k}\right) _{,j}\right] \nonumber\\ &=&\int d\tau dx^{3}\sqrt{-g}\left( 3\frac{F_{\psi }}{\sqrt{-2F}}\psi _{;j}N_{;i}g^{ij}\right) \nonumber \\ &=& \int d\tau dx^{3}\sqrt{-g}\left[ 3\frac{F_{\psi }^{2}}{\left( -2F\right) ^{2}}\psi _{;i}\psi _{;j}g^{ij}\right] \;.\end{aligned}$$ The third term provides: $$\frac{\varepsilon}{2}N^{2}\Delta_{1}\psi=\frac{\varepsilon}{4F}\psi_{;i}\psi_{;j}g^{ij}\,.$$ Finally, collecting all terms and inserting them into Eq.(\[ICLN\]), the action is written as $$\begin{aligned} S_{NM} &=& \int d\tau dx^{3}\sqrt{-g}\left[ -\frac{R}{2}+3\frac{F_{\psi }^{2}}{4F^{2} }\psi _{;i}\psi _{;j}g^{ij}-\frac{\varepsilon }{4F}\psi _{;i}\psi _{;j}g^{ij}-\frac{\bar{V}\left( \psi \right) }{4F^{2}}\right] \nonumber\\ &=& \int d\tau dx^{3}\sqrt{-g}\left[ -\frac{R}{2}+\frac{\varepsilon }{2}\left( \frac{3\varepsilon F_{\psi }^{2}-F}{2F^{2}}\right) \psi _{;i}\psi _{;j}g^{ij}-\frac{\bar{V}\left( \psi \right) }{4F^{2}}\right] \;. \label{CLN.12.5}\end{aligned}$$ Interestingly, introducing the scalar field $\Psi$ with the requirement: $$d\Psi=\sqrt{\left( \frac{3\varepsilon F_{\psi}^{2}-F}{2F^{2}}\right) }d\psi\label{CLN.12.6}$$ the action of Eq.(\[CLN.12.5\]) can be written as follows $$S_{NM}=\int d\tau dx^{3}\sqrt{-g}\left[ -\frac{R}{2}+\frac{\varepsilon}{2}\Psi_{;i}\Psi_{;j}g^{ij}-\frac{\bar{V}\left( \Psi\right) }{4F\left( \Psi\right) ^{2}}\right] \;.\label{CLN.12.7}$$ We conclude that the scalar field $\Psi$ is minimally coupled (modulus a constant) to the gravitational field. In other words, we find that to every non-minimally coupled scalar field, we may associate a unique minimally coupled scalar field in a conformally related space with an appropriate potential. All considerations are reversible, hence the result is reversible. Finally, we would like to remark that the above theorem can be extended to general Riemannian spaces of $n-$dimensions (see appendix A). Conformal Lagrangians in FRLW cosmology ======================================= In this section we consider a spatially flat $\left( K=0\right) $ FRLW spacetime[^3] whose metric is$$ds^{2}=-dt^{2}+a^{2}\left( t\right) \delta_{ij}dx^{i}dx^{j}\label{CLN.13}$$ where $\delta_{ij}$ is the 3-space metric in Cartesian coordinates. The Lagrangian of a scalar field $\phi$ minimally coupled to gravity in this coordinate system $(a,\phi)$ is $$L_{M}=-3a\dot{a}^{2}+\frac{\varepsilon}{2}a^{2}\dot{\phi}^{2}-a^{3}V\left( \phi\right) .\label{CLN.14}$$ On the other hand, the Lagrangian of the non minimally coupled scalar field $\psi$ in the coordinate system $(a,\psi)$ is given by $$L_{NM}=6F\left( \psi\right) a\dot{a}^{2}+6F_{\psi}\left( \psi\right) a^{2}\dot{a}\dot{\psi}+\frac{\varepsilon}{2}a^{3}\dot{\psi}^{2}-a^{3}V\left( \psi\right) \label{CLN.15}$$ where $F(\psi)<0$ is the coupling function. The Hamiltonian of the above Lagrangian is $$E=6F\left( \psi\right) a\dot{a}^{2}+6F_{\psi}\left( \psi\right) a^{2}\dot{a}\dot{\psi}+\frac{\varepsilon}{2}a^{3}\dot{\psi}^{2}+a^{3}V\left( \psi\right) .\label{CLN.15e}$$ We construct a conformal Lagrangian which corresponds to a minimally coupled scalar field. To do that we introduce the following transformation (see [@Cap01; @Cap02]): $$A\left( t\right) =\sqrt{-2F(t)}a(t)\;.\label{CLN.15a}$$ Then the Lagrangian (\[CLN.15\]) takes the form:$$\begin{aligned} L_{NM} & =\frac{1}{\sqrt{-2F}}\left[ -3A\dot{A}^{2}+\frac{\varepsilon}{2}\left( \frac{3\varepsilon F_{\psi}^{2}-F}{2F^{2}}\right) A^{3}~\dot{\psi }^{2}\right] \nonumber\\ & -\frac{A^{3}}{\left( -2F\right) ^{\frac{3}{2}}}V\left( \psi\right) \;.\label{CLN.16}$$ It is interesting to mention here that the cross term $\dot{a}\dot{\psi}$ disappears from Eq.(\[CLN.16\]). Utilizing simultaneously Eq.(\[CLN.12.6\]) and the conformal transformation $$d\tau=\sqrt{-2F\left( \psi\right) }dt\label{CLN.17a}$$ we find, after some algebra, that Eq.(\[CLN.16\]) can be written as $$L_{M}\left( A,A^{\prime},\Psi,\Psi^{\prime}\right) =-3AA^{\prime2}+\frac{\varepsilon}{2}A^{3}\Psi^{\prime2}-A^{3}\bar{V}(\Psi)\label{LLC}$$ where $$\bar{V}(\Psi)=\frac{A^{3}}{\left( -2F\right) ^{\frac{3}{2}}}V\left( \Psi\right) \;.\label{CLN.17}$$ Notice that the prime denotes derivative with respect to the conformal time $\tau$. Evidently, the functional form of the Lagrangian (\[LLC\]) has the general form of Eq.(\[CLN.14\]) proving our assessment. Furthermore, considering in the new coordinates $(\tau,x^{i})$ the metric$$d\bar{s}^{2}=-d\tau^{2}+A^{2}\left( \tau\right) \delta_{ij}dx^{i}dx^{j}\label{CLN.19}$$ we find that the term $3AA^{\prime2}$ equals the Ricci scalar $\bar{R}$ of the conformally flat metric $d\bar{s}^{2}.$ In other words, the Lagrangian (\[LLC\]) can be seen as the Lagrangian of a scalar field $\Psi$ of potential $\bar{V}\left( \Psi\right) $ which is minimally coupled to the gravitational field $\bar{g}_{ij}$ in the space with metric $d\bar{s}^{2}$. Replacing the quantity $A\left( \tau\right) $ and the coordinate $\tau$ from Eq.(\[CLN.15a\]) and Eq.(\[CLN.17a\]) respectively, we obtain: $$d\bar{s}^{2}=\sqrt{-2F}\left[ -dt^{2}+a^{2}\left( t\right) \delta _{ij}dx^{i}dx^{j}\right] =\sqrt{-2F}ds^{2}\label{CLN.20}$$ that is, the metric $d\bar{s}^{2}$ is conformally related to the metric $ds^{2}$ with conformal function $\sqrt{-2F}.$ This means that the non-minimally coupled scalar field in the gravitational field $ds^{2}$ is equivalent to a minimally coupled scalar field - with appropriate potential defined in terms of the coupling function - in the gravitational field $d\bar{s}^{2}.$ For the benefit of the reader, we would like to stress that the above geometrical/dynamical result is reversible in the sense that a minimally coupled scalar field $\phi$ in a metric $ds^{2}$ can be seen as a non-minimally coupled scalar field $\psi$ in the flat FRLW space in which the Eq.(\[CLN.15\]) is equivalent to the minimally coupled scalar field $\Psi=\Psi(\psi)$ in the conformally related metric $d\bar{s}^{2},$ where the conformal function is defined in terms of the coupling function. Equivalently the Lagrangians $L_{M}$ and $L_{NM}$ are conformally related. Finally, we want to stress that the result of the previous lemma is automatically recovered since the Hamiltonian (\[CLN.15e\]) is equal to zero being the $\{0,0\}$ Einstein equation of the system (see also [@CapRev]). Discussion and Conclusions ========================== In this article we have investigated conformally related metrics and Lagrangians in the context of scalar-tensor cosmology. The aim is to select which is the frame where conformally related solutions have an immediate physical meaning. As discussed in section II, no final statement is available for the problem if solutions have to be interpreted either in the Jordan frame or in the Einstein frame since the physical equivalence can be questioned according to several issues (quantum vs classical measurements, energy conditions, choice of physical units, etc.). Due to this situation, it is is too simplistic to consider the problem of conformal frames just a pseudo-problem since we are facing only a mathematical equivalence. Clearly, it has to be addressed at three levels: $i)$ Lagrangians (or in general, effective actions); $ii)$ field equations; $iii)$ solutions. Actually, the last issue means also the choice of a set of observables where the interpretation of solutions is evident. To this goal, seeking for dynamical quantities invariant under conformal transformations is a fundamental issue. However, such quantities have to be related to geometry and possibly to be conserved like Noether symmetries. With this target in mind, we have firstly proved a lemma which shows that the field equations of two conformally related Lagrangians are also conformally related if the corresponding Hamiltonian vanishes. This fact is extremely relevant being the Hamiltonian the energy constraint of a given mechanical system and, in particular, it constitutes a non-holonomic constraint for dynamical systems describing cosmological models. It is the $\{0,0\}$ Einstein equation of the system. Secondly, we have found that to every non-minimally coupled scalar field, we can associate a unique minimally coupled scalar field in a conformally related space with an appropriate potential. The existence of such a connection can be used in order to study the dynamical properties of the various cosmological models, since the field equations of a non-minimally coupled scalar field are the same, at the conformal level, of the field equations of the minimally coupled scalar field. The above propositions can be extended to general Riemannian spaces in $n$-dimensions. It is worth stressing that the above results are in agreement with the so called [*Bicknell’s Theorem*]{} which states that a general non-linear $f(R)$ Lagrangian is equivalent to a minimally coupled scalar field with a general potential in the Einstein frame. In Ref. [@Bick74], this result is achieved in the case of $R^2$-gravity. In [@schmidt], the result is generalized to any analytic $f(R)$-gravity. We’d like to point out that in a recent paper [@BB], based on the Noether symmetry approach, we have studied the issue of physical solutions in $f(R)$ gravity models and scalar field dark energy models. Starting from these results, it is possible to identify the Noether symmetries, the physical solutions and the corresponding conformal properties of the scalar tensor theories (including $f(T)$ gravity). Such an analysis is in progress. In general, the Noether symmetries play an important role in physics because they can be used to simplify a given system of differential equations as well as to determine the integrability of the system. The latter will provide the necessary platform in order to solve the equations of motion analytically and thus to obtain the evolution of the physical quantities. In cosmology, such a method is extremely relevant in order to compare cosmographic parameters, such as scale factor, Hubble expansion rate, deceleration parameter, density parameters with observations [@CapRev; @BB; @felice; @nesseris]. In this appendix we generalize the theorem of section III to a Riemannian space of dimension $n$. Briefly, we consider the non minimally coupled scalar field $\psi $ whose field equations are obtained from the action: $$\begin{aligned} S_{NM} &=&\int dx^{n}N^{n}\sqrt{-g}\left[ F\left( \psi \right) \bar{R}+\frac{\varepsilon }{2}N^{-2}g^{ij}\psi _{,i}\psi _{,j}-\bar{V}\left( \psi \right) \right] \nonumber \\ &=&\int dx^{n}\sqrt{-g}\left[ \begin{array}{c} F\left( \psi \right) N^{n-2}R-2(n-1)F\left( \psi \right) N^{n-3}\Delta _{2}N+ \\ -F\left( \psi \right) N^{n}(n-1)(n-4)\Delta _{1}N+\frac{\varepsilon }{2}N^{n-2}g^{ij}\psi _{,i}\psi _{,j}-N^{n}\bar{V}\left( \psi \right)\end{array}\right] \label{GENA}\end{aligned}$$ where in order to derive the last equality we have used Eq.(\[CLN.04.1\]). Note that we can define the function $N(x^{i})$ in terms of the coupling function $F(\psi)$ by the requirement: $$N^{n-2}=-\frac{1}{2F}\;\;\;\;F=-\frac{N^{2-n}}{2}$$ which also implies that $$N=\frac{1}{\left( -2F\right) ^{\frac{1}{n-2}}}\rightarrow N^{-1}=\left( -2F\right) ^{\frac{1}{n-2}} \label{N11}$$ $$N_{;i}N_{;j}^{-1}=-\frac{F_{\psi}^{2}}{\left( n-2\right) ^{2}F^{2}}\psi _{;i}\psi_{;j}\;. \label{N12}$$ We start now to treat the terms of the action in Eq.(\[GENA\]). In particular, the first term gives: $$\int dx^{n}\sqrt{-g}\left( F\left( \psi\right) N^{n-2}R\right) =\int dx^{n}\sqrt{-g}\left( -\frac{R}{2}\right) \;.\label{ICLN01}$$ If we utilize Eqs.(\[N11\]) and (\[N12\]) then the second (integrating by parts) and the third terms of the general action are $$\begin{aligned} \label{ICLN11} \int dx^{n}\sqrt{-g}\left[ -2(n-1)F\left( \psi \right) N^{n-3}N_{;ij}g^{ij}\right] &=&\int dx^{n}\sqrt{-g}\left[ (n-1)N^{2-n}N^{n-3}N_{;ij}g^{ij}\right] \nonumber \\ &=&\int dx^{n}\sqrt{-g}\left[ (n-1)N^{-1}\frac{1}{\sqrt{-g}}\left( \sqrt{-g}g^{ij}N_{,k}\right) _{,j}\right] \nonumber \\ &=&\int dx^{n}\sqrt{-g}\left[ (n-1)N^{-1}N_{;ij}g^{ij}\right] \nonumber \\ &=&\int dx^{n}\sqrt{-g}\left[ -(n-1)\left( N^{-1}\right) _{;j}N_{;i}g^{ij}\right] \nonumber \\ &=&\int dx^{n}\sqrt{-g}\left[ \frac{(n-1)}{\left( n-2\right) ^{2}}\frac{F_{\psi }^{2}}{F^{2}}\psi _{;i}\psi _{;j}g^{ij}\right]\;.\end{aligned}$$ $$\begin{aligned} \label{ICLN22} \int dx^{n}\sqrt{-g}\left[ F\left( \psi \right) N^{n}(n-1)(n-4)\Delta _{1}N\right) &=&\int dx^{n}\sqrt{-g}\left( -\frac{N^{2-n}}{2}N^{n}(n-1)(n-4)\Delta _{1}N\right] \nonumber \\ &=&\int dx^{n}\sqrt{-g}\left[ -\frac{1}{2}N^{2}(n-1)(n-4)\Delta _{1}N\right] \nonumber \\ &=&\int dx^{n}\sqrt{-g}\left[ -\frac{1}{2}\frac{(n-1)(n-4)}{\left( n-2\right) ^{2}}\frac{F_{\psi }^{2}}{\left( -2F\right) ^{\frac{4}{2-n}}F^{2}}\psi _{;i}\psi _{;j}g^{ij}\right] \;.\end{aligned}$$ To this end the final term gives: $$\label{ICLN33}\int dx^{n}\sqrt{-g}\left( \frac{\varepsilon}{2}N^{n-2}g^{ij}\psi_{,i}\psi_{,j}\right) =\int dx^{n}\sqrt{-g}\left( -\frac {\varepsilon}{4}\frac{1}{F}g^{ij}\psi_{;i}\psi_{;j}\right).$$ Now we change the variable $\psi$ to $\Psi$ as follows $$\label{ICLN44} d\Psi =\left[ \frac{2\varepsilon (n-1)}{\left( n-2\right) ^{2}}\frac{F_{\psi }^{2}}{F^{2}}-\varepsilon \frac{(n-1)(n-4)}{\left( n-2\right) ^{2}}\frac{F_{\psi }^{2}}{\left( -2F\right) ^{\frac{4}{2-n}}F^{2}}-\frac{1}{2F}\right] ^{\frac{1}{2}}d\psi \;.$$ Collecting the results of the above terms namely Eqs.(\[ICLN01\]),(\[ICLN11\]),(\[ICLN22\]),(\[ICLN33\]) and (\[ICLN44\]) we find after some non-trivial algebra that the general action of Eq.(\[GENA\]) is written is terms of $\psi$ as follows $$S_{NM}=\int dx^{n}\sqrt{-g}\left[ -\frac{R}{2}+\frac{\varepsilon}{2}\Psi _{;i}\Psi_{;j}g^{ij}-\frac{\bar{V}\left( \Psi\right) }{\left( -2F\right) ^{\frac{n}{n-2}}}\right] .$$ We would like to remind the reader that the new scalar field $\Psi$ is minimally coupled to the gravitational field $g_{ij}$ and that the potential of $\Psi$ is given by $\frac{\bar{V}\left( \Psi\right) }{\left( -2F\right) ^{\frac{n}{n-2}}}$. 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--- abstract: 'We investigate the morphology dependence of the Tully-Fisher (TF) relation, and the expansion of the relation into a three-dimensional manifold defined by luminosity, total circular velocity and a third dynamical parameter, to fully characterise spiral galaxies across all morphological types. We use a full semi-analytic hierarchical model (based on Croton et al. 2006), built on cosmological simulations of structure formation, to model galaxy evolution and build the theoretical Tully-Fisher relation. With this tool, we analyse a unique dataset of galaxies for which we cross-match luminosity with total circular velocity and central velocity dispersion. We provide a theoretical framework to calculate such measurable quantities from hierarchical semi-analytic models. We establish the morphology dependence of the TF relation in both model and data. We analyse the dynamical properties of the model galaxies and determine that the parameter $\sigma/V_{\rm C}$, i.e. the ratio between random and total motions defined by velocity dispersion and circular velocity, accurately characterises the varying slope of the TF relation for different model galaxy types. We apply these dynamical cuts to the observed galaxies and find indeed that such selection produces a differential slope of the TF relation. The TF slope in different ranges of $\sigma/V_{\rm C}$ is consistent with that for the traditional photometric classification in Sa, Sb, Sc. We conclude that $\sigma/V_{\rm C}$ is a good parameter to classify galaxy type, and we argue that such classification based on dynamics more closely mirrors the physical properties of the observed galaxies, compared to visual (photometric) classification. We also argue that dynamical classification is useful for samples where eye inspection is not reliable or impractical. We conclude that $\sigma/V_{\rm C}$ is a suitable parameter to characterise the hierarchical assembly history that determines the disk-to-bulge ratio, and to expand the TF relation into a three-dimensional manifold, defined by luminosity, circular velocity and $\sigma/V_{\rm C}$.' title: 'The Fundamental Manifold of spiral galaxies: ordered versus random motions and the morphology dependence of the Tully-Fisher relation.' --- galaxies: formation galaxies: evolution galaxies: kinematics and dynamics galaxies: structure galaxies: fundamental parameters galaxies: spiral Introduction ============ Galaxy scaling relations highlight the regularities that characterise galaxy formation. Of particular interest is the physical connection between the dynamical state of a galaxy, determined by its assembly history, and the galaxy photometric properties, tracing its stellar populations and its star formation history. The main dynamical quantity in such an analysis is the galaxy total gravitational potential, which determines the galaxy rotation curve. In the case of spiral galaxies, this is usually parameterised with the circular velocity at a given radius, and in the case of ellipticals, with the central velocity dispersion. Both these quantities have a monothonic dependence on luminosity, thus giving rise to the Tully-Fisher relation (TF; Tully $\&$ Fisher, 1977) for spirals, and the Faber-Jackson relation (FJ; Faber $\&$ Jackson 1976) for ellipticals. However galaxy structure cannot be completely captured by such simple scalings, and the underlying complexity emerges with additional parameters. In the case of ellipticals, the triaxial mass distribution causes an expansion of the FJ relation into a third dimension, recasting it as a scaling between velocity dispersion $\sigma$, surface brightness $I_e$ and effective radius $r_e$, known as the Fundamental Plane (FP; Djorgovski $\&$ Davis 1987, Dressler et al. 1987), a manifestation of the virial theorem in observational quantities (see for instance Cappellari et al. 2006). The slope of the FP indicates that the structure of elliptical galaxies is not a self-similar scaled version of a single object as a function of mass. In the case of spirals, the apparent simplicity of these systems is broken by the observational evidence of a morphology dependence of the TF slope (Springob et al. 2007, Masters et al. 2008). Such feature is also predicted by hierarchical galaxy formation models (Tonini et al. 2011), which naturally produce a varying TF slope that depends on the galaxy internal dynamics, a signature of its assembly history. Indeed, there is an extensive body of work in the literature supporting the observational evidence of a dependence of the TF on galaxy type; the TF for late-type spirals (Masters et al. 2006, Courteau 1997, Giovanelli et al. 1997) is not followed by early-type spirals and S0 types (Williams et al. 2010, Bedregal et al. 2006, Neistein et al. 1999), dwarf galaxies (McGaugh et al. 2000, Begum et al. 2008), barred spirals (Courteau et al. 2003), and polar ring galaxies (Iodice et al. 2003). The evidence suggests that a third parameter beside circular velocity and luminosity enters the TF relation. This parameter depends on galaxy type, and expands the TF into a three-dimensional manifold that describes the structure of spirals or alternatively, of all galaxies. A general three-dimensional manifold for galaxies regardless of type has been investigated (see Zaritsky et al. 2008), with the set of observables ($V_c, I_e, r_e$) in analogy with the ellipticals FP. The approach of this work is the use of analytic toy models, which cannot capture the intrinsic complexity of galaxies due to their hierarchical mass assembly, but resort to absorb it into a single parameter, $\Gamma_e$, the mass-to-light ratio inside the half-light radius $r_e$, and to fit this parameter assuming that galaxies lie on the manifold. Catinella et al. (2012) make use instead of dynamical indicators such as rotational velocity and dispersion to obtain a generalised baryonic FJ relation that holds for all galaxy types. Courteau et al. (2007) investigate galaxy size, and analyse the spiral size-luminosity and luminosity-velocity relations and their dependence on morphology and stellar population content. In this work we consider two questions: 1) is there a third dynamical parameter that characterises the morphology-dependence of the Tully-Fisher relation, and can be used to expand such relation into a three-dimensional manifold, describing the structure of spiral galaxies?; and 2) can we use this parameter to classify galaxy morphology, when visualisation is not available (for instance at high redshift) or impractical (for instance for large surveys)? The novelty of our analysis is that we take advantage of a full semi-analytic hierarchical model (based from Croton et al. 2006) to define a dynamical parameter that characterises galaxy morphology, predicts the TF relation for different galaxy types and defines the spiral galaxy manifold. The model is built on cosmological simulations of structure formation, to model galaxy formation and evolution and build the theoretical TF relation. This tool is ideal for accounting for the merger history of galaxies and their complex star formation history, which is recorded in their rotation curve and stellar populations. We take particular care in producing dynamical and photometric quantities that can be directly compared with observations. With this tool, we analyse a unique dataset of galaxies: we build a sample of observed galaxies carrying the information on luminosity, circular velocity and central velocity dispersion, derived from the GALEX Arecibo SDSS Survey (Catinella et al. 2013). To this sample we apply our theoretical predictions. In Section 2 we describe the model. In Section 3 we introduce our observational sample and present our main results. In Section 4 we discuss our findings and present our conclusions. The model ========= Semi-analytic models are a powerful tool for investigating the TF relation in a cosmological framework. They naturally interlink the dynamics of structure formation with the galaxy emission; the galaxy assembly and star formation histories derive directly from the hierarchical growth of structures. In addition, such models allow for a thorough statistical analysis. The model galaxies are obtained with the semi-analytic model by Croton et al. (2006), with the spectro-photometric model (including dust absorption and emission) described in Tonini et al. (2012). We implement the prescription for the galaxy rotation curves by Tonini et al. (2011), where the velocity profile is determined by the mass distribution of all galaxy components (dark matter, stellar disk and bulge, and gas), and the total circular velocity is $$V_{\rm C}^2(r)=V^2_{\rm DM}(r)+V^2_{\rm disk}(r)+V^2_{\rm bulge}(r) \label{Vc}$$ Each of the velocity terms in the equation is of the type $V^2 \propto G \ M(r)/r$ where $r$ is the galactocentric radius and $M(r)$ is the mass profile: a truncated isothermal sphere for the dark matter halo, a flat exponential disk, and a Hernquist (1990) profile for the bulge (see Tonini et al. 2011 for a complete description; see also Tonini et al. 2006a, Salucci et al. 2007). The slope of the TF relation in models, and in particular its tilting with galaxy type observed in the data, is the manifestation of the connection between galaxy dynamics and star formation and assembly history. For spiral galaxies in general, morphology can be understood in terms of bulge-to-disk ratios. Dynamically, the growth of a bulge in the center of a disk is the result of secular (angular momentum redistribution and stellar migration, bars) and violent (mergers) processes, all of which leave a trace in the galaxy rotation curve. At the same time, the star formation history of the galaxy, which is affected by the dynamical evolution, imprints the bulge-to-disk luminosity. A theoretical determination of the TF relation needs to incorporate each of these effects to be successful, a job that hierarchical semi-analytic models are best suited to accomplish. Which radius? An angular momentum problem ----------------------------------------- When studying the morphology dependence of the TF relation, it becomes especially important to measure the TF at a meaningful, physically motivated radius. Because of the different radial profiles of the galaxy dynamical components (disk, bulge, dark matter halo, gas), *1)* the slope of the TF relation varies with the radius at which the rotation velocity is calculated (Yegorova et al. 2007) and *2)* the same radius in galaxies of different morphology probes different dynamical regions, thus introducing an artificial scatter in the TF. Traditionally, the velocity at a galactocentric radius $r=2.2R_{\rm D}$, where $R_{\rm D}$ is the exponential stellar disk scale-length, has often been used to build the TF relation from observations. This velocity roughly corresponds to the peak velocity of a bulgeless disk; in the case of a galaxy with a substantial bulge however, the region around $r=2.2R_{\rm D}$ contains a different mix of dark matter, gas and stars, and the peak of the rotation curve is actually at a different radius. Following Tonini et al. (2011), we adopt a *dynamical* definition of the disk scale-length, that corresponds to a *fixed angular momentum* rather than a fixed galactocentric distance. With this definition the value of $R_{\rm D}$ self-regulates in the presence of a bulge. The formation of the bulge, both from secular evolution and mergers, implies that stars migrate radially or with inspiralling orbits to the centre of the galaxy, losing all their angular momentum and settling into a pressure-supported configuration. This lost angular momentum is transferred to the disk (Dutton et al. 2007; see also Tonini et al. 2006), with the net effect of increasing the disk size. For a bulge of mass $M_{\rm bulge}$ forming in a disk of mass $M_{\rm disk}$ with initial scale-length $R_{\rm D_{\rm old}}$, the disk scale-length after angular momentum transfer is $$R_{\rm D}=R_{\rm D_{\rm old}} \left( 1+(1-f_{\rm x}) \frac{M_{\rm bulge}}{M_{\rm disk}} \right)~, \label{rdafter}$$ with the fiducial value $f_{\rm x}=0.25$ indicated by Dutton et al. (2007). The new $R_{\rm D}$ represents a ’corrected’ disk scale-length, that takes into account the additional gravitational potential of the bulge. After the correction, all galaxies move onto the disk mass-disk scale length relation that holds for Sc galaxies (see Tonini et al. 2011). After definining a $R_{\rm D}$ that evolves with morphology, it then makes physical sense to adopt $r=2.2R_{\rm D}$ as our radius of choice to build the TF relation. Integrated galaxy velocity dispersion ------------------------------------- From the mass distribution we can build velocity profiles for all galaxy components. The theoretical velocity dispersion is the sum of the contribution to the rotation curve by all components that are pressure-supported, namely the dark matter halo and the bulge: $$\sigma(r)=\sqrt{V^2_{\rm DM}(r)+V^2_{\rm bulge}(r)}~, \label{sigma}$$ where each velocity term in the equation is determined by the mass profile of the dynamical component: $V_i^2(r) \propto GM_i(r)/r$ (see Tonini et al. 2011 for a detailed description). However this particular kind of output is not readily comparable with observations. In fact, in the literature the observed samples usually provide a single-value galaxy velocity dispersion for each object, which is obtained from the broadening of distinct spectral features due to the internal motions of the stars. This measure is an integrated quantity over a galactocentric radius generally determined by telescope aperture or detection limit. The model on the other hand outputs intrinsic, physical galaxy properties, thus the comparison with the observed spectral line dispersion requires *1)* the definition of a ’model aperture radius’ inside which to compute the velocity dispersion, and *2)* the definition of an integrated velocity dispersion inside this radius. In defining such a radius, there are two factors to consider: *1)* the galaxy dynamics becomes increasingly dark matter-dominated at larger galactocentric distances, but *2)* the only visible tracers of the galaxy velocity dispersion are the stars in the bulge, since the disk is modeled as completely rotation-supported. Therefore, if we were to ’observe’ a model galaxy, the entirety of the velocity dispersion signal in the spectral features would come from the stars in the bulge. For this reason, we assume that the ’model aperture radius’ corresponds to the bulge outer limit. The latter is calculated as $R_{out}=3.5 R_S$, where $R_S$ is the characteristic scale-length of the Hernquist density profile (following Tonini et al. 2011; this is consistent with other semi-analytic models, for instance GalICS, Hatton et al. 2003). $R_S$ is not well constrained from observations and its relation to other physical parameters is quite uncertain, depending on the formation mechanisms of bulges (for instance, are bulges formed in merger events, or are they formed through secular evolution from the disk?). For this reason, we introduce a random scatter in the value $R_S$, only costrained to be at most half of the disk scale-length $R_D$, which is reasonable for spiral galaxies. The width of a spectral line is in principle obtained by averaging the velocity dispersion over all the stars; in models, that calculate theoretical velocity dispersion profiles, the equivalent quantity is the mass-averaged velocity dispersion, calculated over radial bins out to the bulge outer limit: $$\sigma = \frac{\sum_n M_n \ \sigma_n}{\sum_n M_n}~, \label{massweight}$$ where $M_n$ is the bulge mass in the $nth$ shell (out to the bulge outer limit), and $\sigma_n$ is the total velocity dispersion in the centre of the bin, calculated from Eq. (\[sigma\]). Selection of the model galaxies ------------------------------- In the model, morphology can be defined in terms of the galaxy physical parameters, like the mass of the bulge and the disk (see Tonini et al. 2011). This method has the advantage of grouping together objects that share a similar formation history, thus favouring a more detailed study of the physics involved in their evolution. On the other hand, this type of selection is hard to apply to observations, and it involves some model-dependencies in the conversion between colors and luminosities to masses and ages, thus confusing the comparison between models and data. ![The distribution of the mass ratio $M_{\rm bulge}/M_{\rm disk}$ for the model Sc, Sb and Sa galaxies (*blue, orange and red respectively*).[]{data-label="histo"}](f1.eps) In most observations the determination of the Hubble type employs the use of some photometric criterion, based on colors or on the relative luminosities of bulge and disk when available. To facilitate the comparison, we classify our model galaxies with one of such methods, based on the bulge-to-total luminosity ratio in the B band, following Simien $\&$ De Vaucouleurs (1986): after defining $\mu_B=M_B(bulge)-M_B(total)$, Sc galaxies are characterised by $2.3 < \mu_B < 4.15$, Sb galaxies by $1.23 < \mu_B < 2.01$ and Sa galaxies by $0.8 < \mu_B < 1.23$. In Fig. (\[histo\]) we compare this classification with the actual mass ratio of bulge and disk, $M_{bulge}/M_{disk}$. With this classification, we find that the Sc types are well represented by galaxies with small bulges (less than $\sim 20 \% $ of the disk mass). On the other hand, Sb types show a wide variety of bulge-to-disk ratios (peaking between 0.2 and 0.7), and Sa galaxies are rare below 1, where they show a very flat distribution. Both Sb and Sa types leak into the S0 and elliptical regimes, defined in the model for values $M_{bulge}/M_{disk}>1$. It is not difficult to imagine that such a scenario is present in observed galaxies too, especially if a band-pass is used where the luminosity evolves rapidly with the age of the stellar populations, and is moreover subject to dust extinction, like the B band. The photometric definition of Sb and Sa model galaxies select objects that do not belong to a uniform population in terms of physical parameters and formation history. In general photometric classifications may rely on more than one band, and the accuracy of the classification increases with the number of bands available. However, the photometric classification relies on the fact that the stellar populations of bulge and disk are *visibly* different, and that might not be the case, especially for more massive and evolved systems. Even the most sophisticated photometric schemes in fact cannot easily disentangle the effects of age, metallicity, and reddening (see Pforr et al. 2012, 2013). The degeneracies in luminosity and colours of different stellar populations confuse the mass-to-light ratio and the decomposition of bulge and disk, and we believe this is one factor at the origin of the increased observed scatter in the TF relation for these galaxy types. In fact, the mass-to-light ratio, by mirroring the balance between different dynamical components, is the main factor that shapes the rotation curve, which in turn determines the slope of the TF relation. Comparison with observations ---------------------------- To compare the model with observed data, we employ the sample of massive galaxies of GASS (GALEX Arecibo SDSS) survey, Data Release 3 (Catinella et al. (2013; see also Catinella et al. 2010 for a complete description of the survey), which provides circular velocities obtained from HI linewidths. To this sample we add measurements of the central velocity dispersions measurements from the Sloan Digital Sky Survey, Data Release 9 (Ahn et al. 2012), and K-band magnitudes from the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006). Distances are estimated from the 21 cm redshifts (see Mould et al 2000). To minimise the scatter we consider only galaxies with inclinations $> 45^{\circ}$. The final sample after cutting away non-detections and poor-quality detections (see Catinella et al. 2013) consists of $\sim 340$ galaxies. The comparison between model velocities and observed quantities is complicated by some unavoidable caveats, that we address below. $\bullet$ In the GASS sample the circular velocity is obtained from HI linewidths. The HI linewidth is an integrated quantity, that we compare with the amplitude of the theoretical rotation curve at one radius. However $r=2.2 R_D$ is large enough for the rotation curve to have already peaked; following Catinella et al. (2006) and Verheijen (2001), we know that in massive galaxies (such as those in the GASS sample) the rotation curve rises rapidly and it remains flat for most of the radial range. Therefore, a measurement of the HI linewidth will be largely dominated by the signal coming from gas that would contribute to the flat region of the rotation curve. Being this the case, the HI linewidth will yield a value of the circular velocity consistent with that which would be measured from the rotation curve itself. $\bullet$ A source of scatter in the Tully-Fisher, whether theoretical or observational, comes from the choice of a radius at which to measure it, or the choice of an aperture within which to collect the signal and determine a linewidth. As described in Yegorova et al. (2007) and Salucci et al. (2007), rotation curves are actually rarely really flat, and measurements encompassing different radii will give rise to different slopes of the TF relation. This is due to the different density profiles of the galaxy components (disk, bulge, dark matter halo), each of which dominate the curve at different radii. As different radii map different dynamical regimes in the galaxy, so the signal in the HI linewidth, or the amplitude of the rotation curve, will be dominated by different components depending on aperture or radius. In the model we choose a radius large enough to be well away from the bulge, and moreover apply our correction of $R_D$ in Eq. (\[rdafter\]) that ’resets’ the curve in the presence of the bulge. However the farther out we go, the more the curve is dominated by the dark matter halo. As already pointed out, for massive spiral galaxies such as those in the GASS sample the shape of the rotation curve is well behaved outside the bulge (Catinella et al. 2006, Verheijen 2001). However in general, as described by Verheijen (2001), the shape of the curve depends on the galaxy mass and type, and in particular the radius at which the dark matter halo starts to dominate the curve is smaller and smaller for less massive galaxies, with dwarfs having a constantly rising curve out to the last measured point. Therefore, an important source of scatter in the TF relation is driven by the shape of the curve, $dV/dr$, which in turn is determined by the baryonic/dark matter mass ratio or the total $M/L$ ratio (as has been highlighted for the B band in Fig. \[histo\]). $\bullet$ In the model the velocity dispersion is traced by the bulge stars, since the disk is fully rotationally supported, and therefore we set our "aperture” as encompassing the radial extension of the bulge. Note that this implies a different radius for each galaxy, and corresponds to the assumption that the aperture is always larger than the target. The velocity dispersion for the observed sample on the contrary is determined by the SDSS fiber line-width, which has a fixed aperture centered on the centre of each galaxy. We argue that this difference is however smaller than the scatter we obtain in the theoretical values of the velocity dispersion $\sigma$, dominated by the intrinsic scatter in the galaxy assembly histories. $\bullet$ Regarding the theoretical velocity dispersion, a quantity more directly comparable with data would be a luminosity-averaged (rather than mass-averaged) dispersion, but this would require an additional layer of modeling, in particular of the radial dependence of the mass-to-light ratio, that is not very well constrained observationally and represents an important source of scatter. In effect our implementation corresponds to assuming a constand mass-to-light ratio for the bulge stars; although this is not strictly true in general, it is nonetheless a fairly reasonable assumtpion for the K band, given that the bulge stars are mostly old (a few Gyrs) and that the spatial extension of the bulge is rather small compared to the disk. In addition, any differences between the mass profile and the K-band luminosity profile are smoothed out by the averaging operation (checked for convergence over different binning grids), and only very large differences, which are unlikely in the K band, would produce significantly different values of $\sigma$. $\bullet$ We do not include the disk contribution to $\sigma$. The disk is modeled as fully rotationally supported, so the disk stars are dispersion-free. In addition, we find that the disk contribution to the total gravitational potential inside the bulge radius is negligible and its effect on the calculated $\sigma$ is marginal. Observationally the inner disk would add to the signal coming from the bulge, depending on inclination. The model galaxies do not suffer from inclination effects, and rather than attempting to model them, we choose to use data that have been corrected for inclination. In addition we argue that, since we are restricting ourselves to the K band, the disk contribution to the estimate of $\sigma$ in observed galaxies is not likely to be significant. Results ======= The observational sample we use is not a Tully-Fisher dedicated sample, and galaxies were not selected based on the quality of their rotation curves or their morphological type. The goal in this work is to characterise the physical properties of the galaxies in this sample so that they would naturally produce a morphology-dependent TF relation. In other words, we want to find the physical parameters that uniquely characterise the Tully-Fisher manifold of spiral galaxies, by which a galaxy type is defined based on dynamics and physical parameters linked to its assembly history, rather than visual inspection. ![The Tully-Fisher relation for the semi-analytic model galaxies, divided according to morphology (*light blue, orange and red points* for Sc, Sb and Sa galaxies respectively), compared with the observed sample (*black squares*).[]{data-label="tfmodel"}](f2.eps) We start by building the TF relation for the observed sample, plotted in Fig. (\[tfmodel\]) in terms of $W=2 V_{\rm C}$ and represented by the *black squares*. Note that the majority of the galaxies in the sample follow a reasonably-defined relation, but with a rather large scatter. In addition, there are numerous objects that significantly deviate from such behaviour. Fig. (\[tfmodel\]) also shows the Tully-Fisher relation for the model galaxies, divided according to morphology: *light blue, orange and red points* represent Sc, Sb and Sa galaxies respectively. As the bulge-to-disk mass ratio increases, the slope of the TF relation flattens, due to the combination of two effects: 1) the stellar populations in the bulge are on average older and therefore their $M/L$ is higher (for instance in the K band, the very luminous post-main sequence phases have faded away; Maraston 2005), while at the same time the velocity is higher due to the boost the bulge inflicts on the rotation curve, and 2) the scatter in the bulge-to-disk mass ratio increases with mass, because larger galaxies have a wider variety of merger histories. Therefore an Sa galaxy that by chance has a smaller than average bulge will scatter back towards the main TF relation, while an Sa galaxy with a larger than average bulge will scatter further away from the TF, and more so with increasind galaxy mass. The overall effect is that of flattening the TF slope. In addition, at the very high-mass end of the mass distribution AGN feedback causes a further increase of the $M/L$ ratio. Model and data roughly occupy the same locus in the plot (ignoring the data outliers for the moment), so we use the model to provide insight into the observed galaxies. In the model, the galaxy type and its evolutionary history (short $vs$ prolongued star formation history, merger-rich $vs$ merger-poor assembly) is at first order characterised by the mass ratio between the spherical and disky components $M_{\rm bukge}/M_{\rm disk}$. A more observationally-friendly quantity to express this is the ratio between random and total motions. In the model this corresponds to $\sigma/V_{\rm C}$, where $\sigma$ is defined by Eq. (\[sigma\]), and $V_{\rm C}^2=\sigma^2+V_{\rm disk}^2$ (Eq. (\[Vc\])). The bulge and dark matter halo mostly contribute with velocity dispersion, and the disk mostly with rotation.  After splitting the model galaxies in photometric classes Sa, Sb and Sc, we consider the ratio $\sigma/V_{\rm C}$ for each type, and plot it in Fig. (\[sigmamodel\]), as a function of the total circular velocity $V_{\rm C}$, calculated at $2.2 \ R_{\rm D}$ (where $R_{\rm D}$ is defined by Eq. (\[rdafter\])); these are the *circles, colour-coded as in Fig. (\[tfmodel\])*. The model galaxies show a differentiation in $\sigma/V_{\rm C}$ depending on the B-band selected morphology, with later-type spirals showing a smaller total velocity dispersion than the earlier types, for a given total circular velocity, and a smaller scatter. This shows how dynamics and star formation history are interlinked in the model: the hierarchical build-up of galaxies grows bulges through mergers and evolves the star formation rates to produce redder early-type objects. Moreover, following the nature of hierarchical mass assembly, earlier-type galaxies live in halos with a richer and more varied merger history, that affects both the bulge and the halo mass; this increases the scatter in $\sigma/V_{\rm C}$. Notice, however, how $\sigma/V_{\rm C}$ does not depend on the total circular velocity $W$, for a given morphological type. We then compare the observed sample in the same space (*squares*). The morphology of the galaxies in the sample is not known, and we use the relation between photometric type and $\sigma/V_{\rm C}$ of the model to classify them: we colour-code the data based on where they lay with respect to the model, in bands of $\sigma/V_{\rm C}$. We assign *blue* colour to the galaxies with very low velocity dispersion, with values of $\sigma/V_{\rm C}$ typical of model Sc galaxies, *red* to the more dispersion-dominated, with values of $\sigma/V_{\rm C}$ typical of model Sa, and *orange* to the intermediate class, typical of model Sb objects. There is a certain degree of overlap in the model, i.e. there are ranges of $\sigma/V_{\rm C}$ that are occupied by both model Sb and Sa galaxies and, to a lesser degree, by both model Sc and Sb (in analogy with Fig. (\[histo\])). We consider the $\sigma/V_{\rm C}$ intervals with a clear predominance of one type above the others, and colour-code the data accordingly, keeping in mind that the overlap between classes is going to be a source of scatter (particularly between *orange* and *red* data points). In addition, we colour in *green* the outliers, i.e. galaxies that do not fall in the locus of the model (those with $\sigma/V_{\rm C} > 0.8$ and $\sigma/V_{\rm C} < 0.35$). We also consider as outliers all objects with $V_{\rm C}<100 \ km/s$, which fall way off both the bulk of the data and the model in the TF plot (Fig. (\[tfmodel\]); moreover, the model starts to suffer from resolution effects at those masses, given the mass resolution of the Millennium simulation). The outliers are mostly dispersion-dominated (in accord with the analysis by Catinella et al. 2012), a feature that in the model is a signature of an early-type/elliptical galaxy. A few outliers show instead a very low velocity dispersion. However, in the range of circular velocities considered here, ratios $\sigma/V_{\rm C}<0.35$ yield velocity dispersions $\sigma < 70 \ km/s$, which is the resolution limit of the Sloan spectrograph. For this reason, such values of the velocity dispersion cannot be considered reliable (Bernardi et al. 2003). Is $\sigma/V_{\rm C}$ a good parameter to characterise observed galaxies? In other words, have we selected a spiral sample, based entirely on theoretical expectations of the ratio between random and total motions? And is this classification a good proxy for galaxy morphology, i.e. will the sub-classes produce different TF relations, in accord with observations?   Fig. (\[tfmodel2\]) shows again the TF relation for the observed sample, after we have divided the galaxies in bands of $\sigma/V_{\rm C}$, following the prediction of the semi-analytic model described in Fig. (\[sigmamodel\]). In the same plot we show the model galaxies (*circles*). In addition in Fig. (\[tfmodel3\]) we show the same plot, split into 3 panels for clarity. Indeed, the three subsamples follow three distinct TF relations. As bulges grow larger in galaxies, the TF slope flattens as expected, mostly due to the increased mass-to-light ratio due to the spherical component. At the same time, the scatter increases; as this happens both in the data and in the model, this must be intrinsic rather than due to increased uncertainties in the measurements of $W$ and $M_K$. In models, the increased scatter is due to the hierarchical nature of galaxy assembly, more prominent in bulge-heavy galaxies because the bulge is a signature of a significant merger history (Tonini et al. 2011; see also Tonini, 2013). A linear regression[^1] of the TF relation for the three subsamples yields the values $[-5.35 \pm 0.40, -6.50 \pm 0.42, -7.34 \pm 0.72]$ for the TF slope of Sa, Sb and Sc-types respectively (also consistent with the theoretical values obtained in Tonini et al. 2011, and the observations of Masters et al. 2008). The best fit lines are shown in Figs. (\[tfmodel2\], \[tfmodel3\]) in *red, orange and light blue* colours respectively. The model galaxies are shown in the background, colour-coded as usual. The slopes and zero-points for the more rotationally-supported objects (*light blue and orange lines*) are well matched with the model Sc and Sb TF relations. The TF slope for the observed more dispersion-dominated objects is consistent with that of model Sa galaxies, although the zero-point is offset by about $~0.5 \ mag$ or by $~50 \%$ in the velocity. This might be due to selection effects. In fact, the GASS sample was selected based on the HI signature; while in the model Sc and Sb galaxies are pretty uniformly gas-rich, the earlier types present a larger scatter in gas content, depending on the assembly history, and therefore this class of model objects cannot be matched in its entirety by the GASS sample. As a sanity check, we also visually inspect the observed galaxy sample, to verify to what degree the morphology selection based on $\sigma/V_{\rm C}$ mirrors the traditional Sa, Sb and Sc classification. We find that for about $70 \%$ of the sample the two selections match exactly, with an expecially good match for later types, while for the remaining galaxies there is an offset of one type, predominantly involving Sb-type objects. Notice how this is in agreement with the predicted uncertainty regarding Sb galaxies pointed out in Fig. (\[histo\]). Fig. (\[tfmodel2\]) shows that $\sigma/V_{\rm C}$ is a good dynamical parameter to characterise spiral galaxies, and it is a good proxy for morphology. The data selected based on $\sigma/V_{\rm C}$ produces morpholgy-dependent TF relations, consistent with the theoretical expectations and previous observational results. The spiral galaxy scaling relation between luminosity and dynamics seems fully characterised in the three-dimensional space $[M_{\rm K}, V_{\rm C}, \sigma/V_{\rm C}]$ across all spiral types. Discussion and conclusions ========================== The Tully-Fisher relation is the product of virial equilibrium combined with the star formation history. It links a measure of the total gravitational potential, $V_{\rm C}$, with the luminosity in various bands, which maps the stellar mass as a function of age. The fact that the TF relation holds for all spirals shows that there is one principal parameter that largely governs galaxy evolution, i.e. the total mass. On the other hand, the perturbations in the TF, such as the observed morphology dependence, indicate that at least a second parameter plays a detectable role. Such parameter is linked to the mass *distribution* inside the galaxy, a product of both secular evolution and hierarchical mass assembly (an ideal scenario to study with a semi-analytic model). The mass distribution might be parameterised with some definition of effective radius, as in the case or ellipticals, but this method relies on a photometric classification that in both models and observations is affected by systematics. Another route is to consider that mass distribution is interlinked with the distribution of the internal motions. In these terms, a dynamical parameter such as $\sigma/V_{\rm C}$ represents as well the mass concentration as the fraction of random over total motions inside the galaxy. In the formalism of virial equilibrium phase-space analysis, it represents an angular momentum parameter. This makes it a very clear-cut, physically well defined quantity to determine in models, but it also has observational advantages. In fact, while a morphology analysis or the determination of effective radii is very uncertain for distant galaxies, central velocity dispersion and circular velocity are relatively easy to determine from galaxy spectra, pushing the analysis of the TF relation to higher redshifts. In addition, this method is ideal in the case of large galaxy surveys, where a classification based on visual inspection is impractical. The parameter $\sigma/V_{\rm C}$ is tracked by the emission of a fraction of the stellar populations, those that are not rotationally supported. These are mostly the bulge stars, a component of intermediate to old age, with a higher mass-to-light ratio than the disk. The prominence of this component causes the velocity-luminosity relation to shift from that of the Sc types (close to pure disks). Thus $\sigma/V_{\rm C}$ is a good proxy for galaxy morphology, it directly relates to the bulge-to-total mass ratio, which in turn can be linked to the bulge-to-disk luminosity ratios traditionally used in the morphology classifications. For this reason the varying slope of the TF relation according to $\sigma/V_{\rm C}$ corresponds to that seen for varying morphology types classified according to luminosity ratios. The parameter $\sigma/V_{\rm C}$ has the additional advantage that it is well defined in all galaxies, and therefore it can be used to expand the present analysis to S0 and elliptical galaxies, where $M_{\rm bulge}/M_{\rm disk}>1$. A future work, based on a larger observed galaxy sample that includes ellipticals and S0s, will address the determination of a generalised galaxy manifold, defined by the circular velocity $[V_{\rm C}]$, the ratio of random-over-total motions$ \sigma/V_{\rm C}$, and the luminosity $L$ (or alternatively the total mass-to-light ratio). In this work we characterised the morphology dependence of the Tully-Fisher relation with a physical parameter, and employed it along with circular velocity and luminosity to define a three-dimensional manifold that determines the structure of spiral galaxies. We built and analysed a sample of observed galaxies and compared the observed Tully-Fisher relation and the central galaxy velocity dispersion with the predictions by a hierarchical semi-analytic model based on Croton et al. (2006). Our results are the following: $\bullet$ the model predicted K-band TF relation is a good match to the data; the hierarchical galaxy formation model fully captures the velocity-luminosity relation for spirals. The model galaxies, classified as Sa, Sb and Sc galaxies with a photometric criterion, show a differentiation of the TF slope, zero-point and scatter with the galaxy type; $\bullet$ we define a theoretical galaxy velocity dispersion as the component of the rotation curve generated by the spherical, pressure supported mass components, i.e. bulge and dark matter halo, and traced by the stars in the bulge; to compare it with the central (aperture-defined) velocity dispersion measured from galaxy spectra, we compute the mass-average of such component over its density profile; $\bullet$ we compare the observed ratio of the velocity dispersion over total circular velocity $\sigma/V_{\rm C}$ as a function of $V_{\rm C}$, with the predictions of the semi-analytic model, finding a good match. The model predicts a correspondence between $\sigma/V_{\rm C}$ a and the photometrically-determined galaxy type, with the earlier-types exhibiting a higher $\sigma/V_{\rm C}$ and a larger scatter; $\bullet$ we divide the observed galaxies in 3 subsamples of different average $\sigma/V_{\rm C}$ following the model trend, and recalculate the TF relation separately for the 3 subsamples; we find that they follow 3 distinct TF relation, with decreasing slope for increasing $\sigma/V_{\rm C}$. The slope of the TF relation for each class of galaxies characterised by $\sigma/V_{\rm C}$ is in agreement with previous results in the literature for Sa, Sb and Sc galaxies. In addition, this method naturally exclude the TF outliers, thus reducing the scatter on the TF relation; $\bullet$ we find that $\sigma/V_{\rm C}$ is a good dynamical parameter to characterise galaxy morphology, yielding a classification consistent with the photometrically defined Sa, Sb and Sc types. We conclude that $\sigma/V_{\rm C}$ is a good, physically motivated third parameter to characterise the TF across the spiral galaxy population. Along with the total velocity $V_{\rm C}$ and the luminosity $M_{\rm K}$, $\sigma/V_{\rm C}$ it thus defines a three-dimensional spiral galaxy manifold that fully characterise the spiral galaxy population. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank the anonymous Referee for her/his comments and suggestions, very beneficial to this work. We would like to thank Barbara Catinella, Simon Mutch and Darren Croton for their insight and the useful discussions. JM is funded by the Australian Research Council Discovery Projects. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. 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--- abstract: 'We use the POWHEG formalism in the  event generator to match QCD real-emission matrix elements with the parton shower for a range of decays relevant to Beyond the Standard Model physics searches. Applying this correction affects the shapes of experimental observables and so changes the number of events passing selection criteria. To validate this approach, we study the impact of the correction on Standard Model top quark decays. We then illustrate the effect of the correction on Beyond the Standard Model scenarios by considering the invariant-mass distribution of dijets produced in the decay of the lightest Randall-Sundrum graviton and transverse momentum distributions for decays in Supersymmetry. We consider only the effect of the POWHEG correction on the simulation of the hardest emission in the shower and ignore the normalisation factor required to correct the total widths and branching ratios to next-to-leading order accuracy.' bibliography: - 'Herwig.bib' --- IPPP/13/17\ DCPT/13/34\ MCNET-13-03\ \ \ [**]{}\ [**]{}\ [ ** ]{} ——————————————————————————————————— ——————————————————————————————————— Introduction {#sec:Intro} ============ For Beyond the Standard Model (BSM) scenarios with additional new particles, the decays of these particles determine the experimental signals we would observe at collider experiments. If the new particles have a well separated mass spectrum, long decay chains will occur when a heavy new particle is produced. For decays involving coloured particles, hard quantum chromodynamic (QCD) radiation at each step in the decay chain will alter the structure of the event and therefore the number of events passing experimental selection criteria. The effects of radiation are also important in models with degenerate new particle mass spectra, where decay chains are typically limited to one step. Searches for these compressed spectra scenarios look for events in which hard radiation in the initial-state shower recoils against missing transverse energy in the final state to give an observable signal[^1]. The emission of hard QCD radiation in the decay of new particles could either enhance or reduce this effect and so must be taken into account. Therefore, accurate simulation of hard radiation in the decays of BSM particles is necessary in order to optimise searches for new physics. Monte Carlo event generators use fixed-order matrix elements combined with parton showers and hadronization models to simulate particle collisions. In the  event generator [@Bahr:2008pv; @Arnold:2012fq], the decays of unstable fundamental particles are treated separately from the hard process which produced them, prior to the parton shower phase, using the narrow width approximation. Decays are generated using the algorithm described in [@Richardson:2001df], which ensures spin correlations are correctly treated. The parton shower utilises an approximation that resums the leading collinear and leading-colour soft logarithms [@Buckley:2011ms] and so does not accurately describe QCD radiation in the regions of phase space where the transverse momenta of the emitted partons are high. The Positive Weight Hardest Emission Generator (POWHEG) formalism [@Nason:2004rx] is one method that allows the simulation of high transverse momentum (hard) radiation to be improved upon by using the real-emission matrix element to produce the hardest emission in the shower. This approach affects both the overall cross sections for inclusive processes and results in local changes to the shapes of distributions sensitive to the hardest emission. In particular, local changes to observables such as jet transverse momenta are important since they can impact on the proportion of events passing selection criteria in new physics searches. Since BSM signals often consist of only a few events, this can in turn result in significant changes to the exclusion bounds that can be set. The POWHEG formalism has been successfully applied to a wide range of hard production processes, for example [@Frixione:2007nw; @Alioli:2008gx; @Hamilton:2008pd; @Alioli:2008tz; @Nason:2009ai; @Alioli:2009je; @Hamilton:2009za; @Re:2010bp; @Hamilton:2010mb; @Alioli:2010qp; @Alioli:2010xa; @Platzer:2011bc; @Oleari:2011ey; @Melia:2011gk; @Jager:2011ms; @Melia:2011tj; @D'Errico:2011sd; @D'Errico:2011um; @Jager:2012xk; @Re:2012zi; @Jager:2013mu], and particle decays [@LatundeDada:2008bv; @Richardson:2012bn] in the Standard Model (SM) as well as selected BSM processes [@Papaefstathiou:2009sr; @Bagnaschi:2011tu; @Klasen:2012wq; @FridmanRojas:2012yh; @Jager:2012hd]. Next-to-leading order (NLO) corrections to BSM particle decays have also previously been studied, for example in [@Horsky:2008yi] where the Supersymmetric-QCD correction to the decay $\tilde{q} \rightarrow q \tilde{\chi} $ was calculated. In this work, we present results from the implementation of the POWHEG method in  for a range of decays relevant for new physics searches. A similar approach based on generic spin structures is used to apply a matrix-element correction to hard radiation in particle decays in  [@Norrbin:2000uu]. The POWHEG formalism will be reviewed in Sect. \[sec:POWHEG\] and in Sect. \[sec:TopQuark\] our implementation of the POWHEG correction will be described in full for the example of top quark decay. In Sect. \[sec:BSMDecays\], details of the decay modes implemented will be given. The impact of the correction on the decay of the lightest graviton in the Randall-Sundrum (RS) model [@Randall:1999ee] will be studied in Sect. \[sec:RSResults\]. Finally, results from a selection of decays in the Constrained Minimal Supersymmetric Standard Model (CMSSM) will be presented in Sect. \[sec:MSSMResults\]. POWHEG Method {#sec:POWHEG} ============= In this section, a brief outline of the POWHEG method is given. Further details can be found in [@Frixione:2007vw]. In the conventional parton shower approach, the inclusive differential cross section for the highest transverse momentum emission from an $N$-body process is given by d \^[[PS]{}]{}=B(\_N) d\_N . \[eq:PS\] Here we are considering a parton shower ordered in terms of the transverse momentum of the emitted parton, $p_T$. $ \Phi_N$ are the phase space variables of the $N$-body leading-order (LO) process and $B$ is the Born-level matrix element squared, including the relevant flux factor, such that the total LO cross section is $ \sigma^{\rm{LO}} = \int B(\Phi_N) d\Phi_N$. $ \mathcal{P}$ is the unregularized Altarelli-Parisi splitting kernel and $ \Phi_R$ is a set of variables parameterizing the phase space of the additional radiated parton. The radiative phase space is limited to the region $p_T \left(\Phi_R \right)>p_{T \rm{min}}$, where $p_{T \rm{min}} $ is a transverse momentum cut-off introduced to regularize the infra-red (IR) divergences in the splitting kernel. The Sudakov form factor for the parton shower is (p\_T) = ( - d\_R (p\_T (\_R) -p\_T ) ). \[eq:PSSudakov\] The square bracket in Eq. \[eq:PS\] integrates to unity which ensures that the total cross section is given by the LO result. In the POWHEG approach, the inclusive differential cross section for the hardest emission is given by the QCD NLO differential cross section, that is d \^[[PO]{}]{}=|[B]{}(\_N) d\_N , \[eq:PO\] where $\bar{B}(\Phi_N) $ is defined by $$\begin{aligned} \bar{B}(\Phi_N) = B(\Phi_N) &+ \left[V(\Phi_N) + \int C(\Phi_N, \Phi_R) d\Phi_R\right] \nonumber \\ &+ \int\left[R(\Phi_N, \Phi_R) d\Phi_R- C(\Phi_N, \Phi_R) d\Phi_R\right]. \label{eq:Bbar}\end{aligned}$$ The real-emission contribution, $R(\Phi_N, \Phi_R)$, corresponds to the radiation of an additional parton from the LO interaction and the virtual contribution, $V(\Phi_N)$, comes from the 1-loop correction to the LO process. $C(\Phi_N, \Phi_R)$ is a counter term with the same singular behaviour as the real and virtual contributions and is introduced to ensure the two square brackets in Eq. \[eq:Bbar\] are separately finite. The Sudakov form factor appearing in Eq. \[eq:PO\] is \^[PO]{}(p\_T) = ( - d\_R (p\_T (\_R ) -p\_T ) ). \[eq:POSudakov\] As with the conventional parton shower approach, the square bracket in Eq. \[eq:PO\] will integrate to unity and hence the total inclusive cross section will be given by the NLO result. Typically, the counter term, $C(\Phi_N, \Phi_R)$, can be rewritten as a sum of dipole functions, $\mathcal{D}_i $, each of which describes the behaviour of the real-emission matrix element in a singular region of phase space, i.e. when the emitted parton becomes soft or collinear to one of the legs in the Born process. By doing so, the different singular regions can be separated such that Eq. \[eq:POSudakov\] becomes a product of Sudakov form factors \^[PO]{}(p\_T) = \_i ( - d\_R (p\_T (\_R ) -p\_T ) ), \[eq:POSudakov2\] each of which describes the non-emission probability in a particular region of phase space specified by the dipole function $\mathcal{D}_i$. When applying the POWHEG method to a parton shower ordered in transverse momentum, the hardest emission is generated first using the POWHEG Sudakov form factor in Eq. \[eq:POSudakov2\]. Subsequent emissions are generated with the normal parton shower Sudakov given in Eq. \[eq:PSSudakov\], with the requirement that no parton shower emission has higher transverse momentum than the emission described by $R(\Phi_N, \Phi_R)$. However, to allow QCD coherence effects to be included, an angularly ordered parton shower is used in . Ordering the parton shower in terms of an angular variable means the first emission in the shower may not be the hardest. The POWHEG approach can be reconciled with angularly ordered parton showers by dividing the shower into several steps [@Nason:2004rx]. The hardest emission in the shower is generated first using the POWHEG Sudakov form factor and the value of the angular evolution variable corresponding to this emission is determined. An angularly ordered shower, running from the shower starting scale down to the scale of the hardest emission, is then generated. This *truncated* parton shower simulates coherent soft wide-angle radiation. The hardest emission is then inserted and the shower continues until the IR cut-off of the evolution variable is reached. Finally, in both stages of the parton shower, emissions generated by the shower are discarded if they have higher transverse momentum than the emission generated using the POWHEG Sudakov form factor. Top Quark Decays {#sec:TopQuark} ================ In this section, we describe our implementation of the POWHEG formalism for the example of a top quark decaying to a $W$ boson and a bottom quark. To implement the full POWHEG correction to this decay, the Born configuration must be generated according to Eq. \[eq:Bbar\] and the hardest emission in the parton shower simulated using Eq. \[eq:POSudakov2\]. However, in this work we consider only the effect of the POWHEG correction on the simulation of the hardest emission in the shower and hence generate the Born configuration using only $B \left(\Phi_N \right)$, the leading order contribution in Eq. \[eq:Bbar\]. As such, we use the existing LO  implementation of top quark decay and modify the shower such that the hardest emission is generated according to Eq. \[eq:POSudakov2\]. Justification for excluding the normalisation factor of the POWHEG correction will be given in Sect. \[sec:BSMDecays\]. Application of the POWHEG correction to top quark decays, along with top quark pair production in $e^+ e^-$ collisions, has been previously studied in [@LatundeDada:2008bv] for massless bottom quarks. In this work, we retain the mass of the bottom quark throughout. Implementation in   {#sec:TopQuarkImp} ------------------- In , the decays of fundamental particles are performed in the rest frame of the decaying particle. In this frame, we are free to choose the orientation of the $W$ boson to be along the negative $z$-direction and so, at LO, the bottom quark is orientated along positive $z$-direction. The squared, spin and colour averaged matrix element for the LO process is given by |\_B|\^2 = (m\_t\^4+m\_b\^4-2m\_w\^4+m\_t\^2m\_w\^2+m\_b\^2m\_w\^2-2m\_t\^2m\_b\^2 ), where $m_t$, $m_b$ and $m_W$ are the masses of the top quark, bottom quark and $W$ boson respectively and $g$ is the weak interaction coupling constant. The relevant CKM factor has been set equal to 1. The squared, spin and colour averaged matrix element for the $\mathcal{O} \left(\alpha _s \right)$ real-emission correction to the decay $t \rightarrow W b$ is $$\begin{gathered} |\mathcal{M}_R|^2 = g^2 g_s^2 C_F \left\{ - \frac{|\mathcal{M}_B|^2} {g^2} \left(\frac{p_b}{p_b.p_g} - \frac{p_t} {p_t.p_g} \right)^2 + \right. \\ \left. \left(\frac{p_g.p_t}{p_b.p_g} + \frac{p_b.p_g}{p_t.p_g} \right) \left(1 + \frac{m_t^2}{2m_w^2} + \frac{m_b^2}{2m_w^2} \right) - \frac{1}{m_w^2} \left(m_t^2+m_b^2 \right)\right\},\end{gathered}$$ where $g_s$ is the strong coupling constant, $C_F= \frac{4}{3}$ and $p_t$, $p_b$, $p_W$ and $p_g$ are the four-momenta of the top quark, bottom quark, $W$ boson and gluon. In general, the orientation of the decay products in the three-body final state is such that the emitting parton absorbs the transverse recoil coming from the emission of the gluon and the spectator particle continues to lie along the negative $z$-direction. When the radiation originates from the top quark, the bottom quark effectively acts as the emitting particle so that we remain in the rest frame of the top quark. Therefore, for emission from both the top and the bottom quarks, the momenta of the decay products are p\_W = (E\_W,0,0,- ), p\_b = (E\_b, -p\_T ( ), -p\_T ( ), ), p\_g = (E\_g, p\_T ( ), p\_T ( ), ), where $E_x$ is the energy of particle $x$, and $p_T$ and $\phi$ are the transverse momentum and azimuthal angle of the gluon. The Lorentz invariant phase space element, ${\mathrm{d}}\Phi_R$, describing the emission of the additional gluon is obtained from the relation \_3 = \_2 \_R , where \_N = (2 )\^4 \^4 (p\_t - \_[i=1]{}\^N p\_i ) \_[i=1]{}\^[N]{} and $\mathbf{p_i}$ is the three-momentum of particle $i$. We choose to parameterize the radiative phase space in terms of the transverse momentum, $p_T$, rapidity, $y$, and azimuthal angle, $\phi$, of the gluon and so find \_R = J p\_T dy , where the Jacobian factor, $J$, is [^2] J = . This parametrization has the advantage of simplifying the Heaviside function in the POWHEG Sudakov form factor to a lower limit in the integration over $p_T$. The final components required for the implementation of the POWHEG Sudakov form factor in Eq. \[eq:POSudakov2\] are the dipole functions, $\mathcal{D}_i$, which describe the singular behaviour of the real-emission matrix element. We use the dipole functions defined in the Catani-Seymour subtraction scheme, details of which can be found in [@Catani:1996vz; @Catani:2002hc], to describe the singular behaviour resulting from emissions from the decay products. The dipole used to describe radiation from the top quark is as follows \_i = |\_B|\^2. \[eq:initialDipole\] It contains only soft enhancements since, in the top quark rest frame, collinear enhancements are suppressed. Using the above information, the hardest emission in the shower can then be generated according to Eq. \[eq:POSudakov2\] using the veto algorithm[^3], which proceeds as follows: 1. Trial values of the radiative phase space variables are generated. The transverse momentum of the emission is generated by solving \^[[over]{}]{}(p\_T ) = (- \^[p\^[[max]{}]{}\_T]{}\_[p\_T]{} p\_T( \_R) ) = , \[eq:SudakovOver\] where $p^{\rm{max}}_T = \frac{\left(m_t -m_W \right)^2 -m_b^2} {2 \left(m_t - m_W \right)}$ is the maximum possible $p_T$ of the gluon. $y_{\rm{max}}$ and $y_{\rm{min}}$ are the upper and lower bounds on the gluon rapidity, chosen to overestimate the true rapidity range. $C$ is a constant chosen such that the integrand in Eq. \[eq:SudakovOver\] always exceeds the integrand in Eq. \[eq:POSudakov2\] and $\mathcal{R}$ is a random number distributed uniformly in the range $[0,1]$. Values of $y$ and $\phi$ are generated uniformly in the ranges $[y_{\rm{min}}, y_{\rm{max}}]$ and $[0, 2 \pi]$ respectively; 2. If $p_T < p^{\rm{min}}_T$, no radiation is generated and the event is hadronized directly. We set $p^{\rm{min}}_T=1{\,\mathrm{GeV}}$ throughout this work; 3. If $p_T \geq p^{\rm{min}}_T$, the momenta of the $W$ boson, bottom quark and gluon are calculated using the generated values of the radiative variables. Doing so, yields two possible values of $E_W$ that must both be retained and used in the remainder of the calculation. If the resulting momenta do not lie within the physically allowed region of phase space, we veto this configuration, set $p^{\rm{max}}_T = p_T$ and return to step 1; 4. Events within the physical phase space are accepted with a probability given by the ratio of the true to overestimated integrands in Eqs. \[eq:POSudakov2\] and \[eq:SudakovOver\] respectively. If the event is rejected, we set $p^{\rm{max}}_T = p_T$ and return to step 1; Using this procedure, a trial emission is generated for each dipole, $\mathcal{D}_i$, in Eq. \[eq:POSudakov2\]. The configuration which gives the highest $p_T$ emission is selected. The existing  framework, detailed in [@Hamilton:2008pd], is then used to generate the remainder of the parton shower. Parton Level Results {#sec:TopQuarkRes} -------------------- To validate our implementation of the algorithm described in Sec. \[sec:TopQuarkImp\], Dalitz style plots were generated for the decay $t \rightarrow W b$ and are shown in Fig. \[fig:top\_Dalitz\]. The Dalitz variables, $x_W$ and $x_g$, were defined by the relation, $x_i = \frac{2E_i}{m_t}$, where $E_i$ is the energy of particle $i$ in the rest frame of the top quark. The left-hand plot in Fig. \[fig:top\_Dalitz\] shows the distribution obtained using the POWHEG style correction. In this case, $x_g$ is the energy fraction of the gluon generated using the full real-emission matrix element. The distribution on the right-hand side of Fig. \[fig:top\_Dalitz\] was generated using the conventional parton shower, limited to one emission in the final state, and so here $x_g$ is the energy fraction of a gluon produced using the parton shower splitting kernels. On both distributions, the black outline indicates the physical phase space boundaries. The enclosed area is divided into a section populated by radiation from the bottom quark (above the green dashed line), sections populated by radiation from the top quark (below the blue dotted lines) and a *dead region* (between the blue dotted and green dashed lines) that corresponds to hard gluon radiation and is not populated by the conventional parton shower. These boundaries correspond to the theoretical limits of the  parton shower with symmetric phase space partitioning, described in [@Gieseke:2003rz], in which the starting values of the shower evolution variables for the top and bottom quarks are chosen such that the volumes of phase space accessible to emissions from each quark are approximately equal. As expected, in both plots we see a high density of points in the limit $x_g \rightarrow 0$, corresponding to soft gluon emission. The POWHEG corrected distribution also has a concentration of points along the upper physical phase space boundary where $x_W$ is maximal and emissions are collinear to the bottom quark. The density of points along the upper boundary is reduced in the parton shower distribution and points are instead concentrated along the lower boundary of the bottom quark emission region. As discussed in [@Gieseke:2003rz], the parton shower approximation agrees with the exact matrix element in the case of collinear radiation from the bottom quark but overestimates it elsewhere in the bottom quark emission region. The factor by which the parton shower approximation exceeds the exact matrix element, increases towards the lower boundary of the region and therefore we see an excess of points near the boundary. The parton shower distribution also has a high density of points in the top quark emission region for $x_g \lesssim 0.53$. This enhancement is again the result of the parton shower approximation overestimating the exact matrix element in this area [@Gieseke:2003rz]. In general, we see that the parton shower in  produces areas of high emission density which do not correspond to physically enhanced areas of phase space and therefore has a tendency to overpopulate hard regions of phase space. On the other hand, the POWHEG emission is distributed according to the exact real-emission matrix element and so correctly populates the physically enhanced regions of phase space with no additional spurious high density regions. Finally, we also see that the POWHEG corrected distribution fills the dead region of phase space that is not populated by the standalone parton shower. ![Dalitz distributions for the decay $t \rightarrow W b$ with (left) and without (right) the POWHEG style correction. The black outline indicates the physically allowed region of phase space. In the conventional parton shower approach, the region above the green dashed line is populated with radiation from the bottom quark and the regions below the blue dotted lines with radiation from the top quark. These boundaries correspond to the limits of the parton shower with symmetric phase space partitioning.[]{data-label="fig:top_Dalitz"}](figure_1a) ![Dalitz distributions for the decay $t \rightarrow W b$ with (left) and without (right) the POWHEG style correction. The black outline indicates the physically allowed region of phase space. In the conventional parton shower approach, the region above the green dashed line is populated with radiation from the bottom quark and the regions below the blue dotted lines with radiation from the top quark. These boundaries correspond to the limits of the parton shower with symmetric phase space partitioning.[]{data-label="fig:top_Dalitz"}](figure_1b) ![Comparison of distributions generated using the standalone parton shower with those generated using a matrix element or POWHEG style correction to the decay $t \rightarrow W b$. Parton level $e^+e^- \rightarrow t \bar{t}$ events were generated at $\sqrt{s}=360 {\,\mathrm{GeV}}$. The left-hand plot shows the distribution of the minimum jet separation, $\Delta R$, and the right-hand plot the logarithm of the jet measure, $y_{32}$. []{data-label="fig:top_parton"}](figure_2a) ![Comparison of distributions generated using the standalone parton shower with those generated using a matrix element or POWHEG style correction to the decay $t \rightarrow W b$. Parton level $e^+e^- \rightarrow t \bar{t}$ events were generated at $\sqrt{s}=360 {\,\mathrm{GeV}}$. The left-hand plot shows the distribution of the minimum jet separation, $\Delta R$, and the right-hand plot the logarithm of the jet measure, $y_{32}$. []{data-label="fig:top_parton"}](figure_2b) To study the impact of the POWHEG style correction on top quark decays, parton level $e^+ e^- \rightarrow t \bar{t}$ events were simulated and analysed as in [@Hamilton:2006ms]. Events were generated at a centre-of-mass energy close to the $t \bar{t}$ threshold, $\sqrt{s} =360 {\,\mathrm{GeV}}$, to minimize the effects of radiation from the initial-state shower. Unless otherwise stated, in this study we use the default set of tuned perturbative and non-perturbative parameters, or *event tune*, in  version 2.6 [@Arnold:2012fq]. Final-state partons were clustered into three jets using the [@Cacciari:2011ma] implementation of the $k_T$ algorithm. The $W$ bosons were decayed leptonically and their decay products excluded from the jet clustering. Events were discarded if they contained a jet with $p_{T}<10 {\,\mathrm{GeV}}$ or the minimum jet separation[^4], $\Delta R$ , did not satisfy $ \Delta R \geq 0.7$. Using events that passed these selection criteria, differential distributions were plotted of $ \Delta R $ and $\log \left(y_{32}\right)$, where $y_{32}$ is the value of the jet resolution parameter[^5] at which a three jet event is classified as a two jet event. The resulting distributions are shown in the left and right-hand plots in Fig. \[fig:top\_parton\]. Distributions generated using the normal parton shower and the parton shower including the POWHEG style correction, are shown by the blue dashed and black solid lines respectively. The red dotted lines in Fig. \[fig:top\_parton\] show the distributions obtained when the existing implementation of hard and soft matrix element corrections (MEC) [@Hamilton:2006ms] are applied to the normal  parton shower. Hard matrix element corrections use the full $t \rightarrow W b g$ matrix element to distribute emissions in the dead regions of phase space that are not populated by the parton shower. Soft matrix element corrections use the full real-emission matrix element to correct emissions generated by the parton shower that lie outside the areas of phase space where the parton shower approximation is valid, i.e. away from the soft and collinear limits. Applying these corrections ensures that the hardest emission in the shower is generated according to the exact matrix element, therefore, we expect a high level of agreement between the POWHEG and matrix element corrected distributions. The bottom panel in each plot shows the ratio of the parton shower and matrix element corrected distributions to the POWHEG corrected distribution. In both plots, we include error bars indicating the statistical uncertainty. As discussed in [@Hamilton:2006ms], applying the matrix element corrections has the effect of softening both the $\Delta R $ and $\log \left(y_{32} \right) $ distributions. This is due to the soft matrix element correction rejecting a portion of the high $p_T$ emissions generated by the parton shower. The magnitude of the observed effect illustrates the importance of matching the parton shower to the exact matrix element in high $p_T$ regions. As expected, the distributions generated using the POWHEG style and matrix element corrections are very similar although, for both variables, the POWHEG style correction yields slightly harder distributions. The discrepancies between the distributions are the result of a number of subtle differences between the POWHEG and matrix element correction schemes. Firstly in the matrix element correction approach, events in the dead region are generated using the fixed-order real-emission matrix element only, without any Sudakov suppression, and subsequent showering of the resulting configuration is simulated starting from the $1 \rightarrow 3 $ process. However, in the POWHEG approach the hardest emission in the shower is reinterpreted such that the conventional parton shower instead begins from the Born hard configuration. The scale of the hardest emission is generated, and then the shower proceeds as normal except that the hardest emission is fixed at the generated scale. In addition to this, the soft matrix element correction is applied to all emissions in the parton shower which are the hardest so far. Normally this leads to the correction of both the hardest emission and a number of other emissions with large values of the evolution parameter, but smaller transverse momentum. These differences all contribute to the discrepancies between the POWHEG style and matrix element corrected distributions although it is unclear which would have the largest effect. However, the difference between the POWHEG style and matrix element corrected results is comparatively small. The agreement between the two approaches serves to further validate our implementation of the POWHEG formalism. Finally, we note that the POWHEG style approach is preferable to the original matrix element correction scheme since it is significantly simpler to implement in . Decays of BSM Particles {#sec:BSMDecays} ======================= As discussed in Sect. \[sec:Intro\], it is important that the simulation of QCD radiation in the decays of BSM particles is done in the most accurate way possible. In this work, we present results illustrating the effect of consistently matching the QCD real-emission matrix element with the parton shower in  through the POWHEG formalism. This technique has been applied to a range of decays that occur in most of the well studied BSM scenarios. Tab. \[tab:spins\] shows the combinations of incoming and outgoing spins for which this method is used and each spin structure is implemented for the colour flows given in Tab. \[tab:colour\]. However, models with coloured tensor particles are beyond the scope of this work and therefore decays involving incoming tensor particles were limited to colour flows in which the tensor is a colour singlet. The LO and real-emission matrix elements appearing in the POWHEG Sudakov form factor in Eq. \[eq:POSudakov\] are calculated using helicity amplitude methods to correctly incorporate spin correlations [@Richardson:2001df]. The dipole functions, $\mathcal{D}_i $, are defined as in the Catani-Seymour dipole subtraction method[@Catani:1996vz; @Catani:2002hc] when describing radiation from the decay products. In this approach, dipoles describing quasi-collinear radiation from massive vector bosons are not well defined. Therefore, the Fermion-Fermion-Vector, Scalar-Scalar-Vector and Tensor-Vector-Vector decays are limited to the situation where any final-state coloured vector particles are massless. The Vector-Fermion-Fermion and Vector-Scalar-Scalar decays do, however, include radiation from massive incoming vector particles. Decays are performed in the rest frame of the decaying particle [@Bahr:2008pv] and therefore the dipole describing the singular behaviour of this particle will only contain a universal soft contribution. This is a well defined, spin-independent function given, for the example colour flow $3 \rightarrow 3\,0$, by Eq. \[eq:initialDipole\]. Finally, in this work we focus solely on the effect of the POWHEG correction on the simulation of the hardest emission in the shower and have not implemented the normalisation factor coming from the presence of $\bar{B}$ rather than $B$ in Eq. \[eq:PO\]. In many cases, the partial widths and branching ratios used in the simulation are calculated by an external program, for example SDECAY [@Muhlleitner:2003vg], and so already include NLO corrections. These values are then passed to  by means of a spectrum file in SUSY Les Houches Accord format [@Allanach:2004ub; @Allanach:2008qq]. In cases where the calculation of the widths and branching ratios is performed in , generated distributions can be rescaled by a global normalisation factor to achieve NLO accuracy for suitably inclusive observables when the necessary calculations exist. Results ======= Randall-Sundrum Graviton {#sec:RSResults} ------------------------ The effect of applying the POWHEG correction to the decay of the lightest RS graviton was investigated using the  implementation of the RS model. LHC proton-proton collisions with a centre-of-mass (CM) energy of $\sqrt{s}=8 {\,\mathrm{TeV}}$ were simulated. The lightest graviton, $G$, was produced as a resonance and allowed to decay via $G \rightarrow gg $ and $G \rightarrow q \bar{q}$ for $q = u,d,s,c,b $. The mass of the graviton was chosen to be $m_G = 2.23 {\,\mathrm{TeV}}$ which corresponds to the lower bound on the allowed graviton mass for the coupling $k/ \bar{M}_{pl} = 0.1 $ in [@Aad:2012cy]. An analysis based on the ATLAS experiment’s search for new phenomena in dijet distributions [@ATLAS:2012pu] was then carried out. Jets were constructed using the [@Cacciari:2011ma] implementation of the anti-$k_t$ algorithm [@Cacciari:2008gp] with the energy recombination scheme and a distance parameter $R=0.6$. Jets with $|y| \geq 4.4 $ were discarded, where $y$ is the rapidity of the jet in the $pp$ CM frame. Events with less than two jets passing this constraint were vetoed. The rapidities of the two highest $p_T$ jets in the $pp$ CM frame are given by $y_{1}$ and $y_{2}$. In the dijet CM frame formed by the two hardest jets, their corresponding rapidities are $y_{*}$ and $-y_{*}$ where $y_{*}= \frac{1}{2} (y_1 - y_2)$. Events not satisfying $|y_{*}| < 0.6$ and $|y_{1,2}| < 2.8$ were discarded. The dijet invariant mass, $m_{jj}$, was formed from the vector sum of the two hardest jet momenta and events were vetoed if $m_{jj} \leq 1.0 {\,\mathrm{TeV}}$. ![Dijet invariant mass distribution for the lightest RS graviton decaying to jets. The left-hand plot shows the distribution in the full range while the right-hand plot emphasises the effect on the peak region $2.1 {\,\mathrm{TeV}}\leq m_{jj} \leq 2.3 {\,\mathrm{TeV}}$. The mass of the graviton was $m_G = 2.23 {\,\mathrm{TeV}}$ and the coupling $k/ \bar{M}_{pl} = 0.1 $. LHC events were simulated with $\sqrt{s}=8 {\,\mathrm{TeV}}$. The yellow and orange bands were generated by varying the event tune parameters in the POWHEG corrected and conventional parton shower distributions respectively.[]{data-label="fig:mass distribution"}](figure_3a) ![Dijet invariant mass distribution for the lightest RS graviton decaying to jets. The left-hand plot shows the distribution in the full range while the right-hand plot emphasises the effect on the peak region $2.1 {\,\mathrm{TeV}}\leq m_{jj} \leq 2.3 {\,\mathrm{TeV}}$. The mass of the graviton was $m_G = 2.23 {\,\mathrm{TeV}}$ and the coupling $k/ \bar{M}_{pl} = 0.1 $. LHC events were simulated with $\sqrt{s}=8 {\,\mathrm{TeV}}$. The yellow and orange bands were generated by varying the event tune parameters in the POWHEG corrected and conventional parton shower distributions respectively.[]{data-label="fig:mass distribution"}](figure_3b) The dijet mass distribution after the above selection criteria were applied, is shown in the left-hand plot in Fig. \[fig:mass distribution\]. The blue dashed line shows the invariant mass distribution for the LO matrix element combined with the parton shower while the black solid line shows the result including the POWHEG correction to the graviton decay. Both distributions were generated using the optimum set of tuned perturbative and non-perturbative parameters found in[@Richardson:2012bn]. From Fig. \[fig:mass distribution\], we see that including the POWHEG correction causes a decrease of $\mathcal{O} \left( 40 \% \right) $ in the number of events in the region $2.1 {\,\mathrm{TeV}}\leq m_{jj} \leq 2.3 {\,\mathrm{TeV}}$. This effect is highlighted in the right-hand plot in Fig. \[fig:mass distribution\], which shows the dijet mass distribution in this range. In the conventional parton shower approach, the majority of the graviton’s momentum will be carried by the two partonic decay products. When the POWHEG correction is applied, the highest $p_T$ emission in the shower will typically be quite hard and so a significant fraction of the the graviton’s momentum will be missed by considering the invariant mass of only the hardest two jets, therefore shifting the distribution to lower values of $m_{jj}$. To give an estimate of the uncertainty arising from our choice of event tune, the dijet mass distributions were generated at ten points in the event tune parameter space and error bands were created showing the maximum and minimum values from the resulting set of distributions. A description of the varied parameters can be found in[@Richardson:2012bn] and their values at each of the ten points are given in Tab.$\,$2 of[@Richardson:2012bn]. The error bands are shown in yellow and orange for the distributions with and without the POWHEG correction respectively. The impact of the POWHEG correction is still clearly evident once this uncertainty has been taken into account. Constrained Minimal Supersymmetric Standard Model {#sec:MSSMResults} ------------------------------------------------- In addition to the results presented in Sect. \[sec:RSResults\], the effect of the POWHEG correction was also studied in the context of the CMSSM model. The high scale parameters of the model were chosen to be $m_0 = 1220 {\,\mathrm{GeV}}$, $m_{1/2} = 630 {\,\mathrm{GeV}}$, $\tan{\beta} = 10$, $A_0 = 0$ and $\mu >0$. This point lies just outside the exclusion limits set by the ATLAS experiment in [@ATLAS:2012ona]. The corresponding weak scale parameters and decay modes were calculated using ISAJET 7.80 [@Paige:2003mg] and the resulting masses of the Supersymmetric (SUSY) particles relevant to this study are given in Tab. \[tab:mass\]. The  implementation of the MSSM model was used to generate LHC $pp$ collisions at a centre-of-mass energy of $\sqrt{s}=8 {\,\mathrm{TeV}}$. Here we focus on the effect of the correction to the parton shower and so hadronization and the underlying event are not simulated. In the following sections, the impact of the POWHEG correction on two archetypal decays is presented. In both cases, the decaying SUSY particle is pair produced in the hard process and the two subsequent decays are then analysed separately in the rest frame of the decaying particle. Dalitz style distributions were produced, as described in Sec. \[sec:TopQuarkRes\], for both the POWHEG corrected emission and the normal parton shower limited to one final-state emission. In addition, transverse momentum distributions of the hardest jet not coming from a visible decay product were also studied. To do so, the full parton shower was generated, with and without the POWHEG style correction, and events were analysed by clustering all visible final-state particles into jets using the  implementation of the anti-$k_T$ algorithm with the energy recombination scheme and $R=0.4$. Jets with $p_T \leq 20 {\,\mathrm{GeV}}$ or $|\eta|>4.0 $ were discarded. Events were required to have at least $n+1$ jets passing the selection criteria, where $n$ is the number of visible decay products. $m_{\tilde{u}_L}$ $m_{\tilde{g}}$ $m_{\tilde{t}_1}$ $m_{\tilde{\chi}_1^0}$ ------------------------------ ------------------------------ ------------------------------ ----------------------------- $1812.91 {\,\mathrm{GeV}}\ $ $1546.56 {\,\mathrm{GeV}}\ $ $1278.14 {\,\mathrm{GeV}}\ $ $279.22 {\,\mathrm{GeV}}\ $ : Masses of the SUSY particles relevant to the decays studied in Secs. \[sec:chiResults\] and \[sec:gluinoResults\]. Values were obtained using ISAJET 7.80 with the high scale parameters $m_0 = 1220 {\,\mathrm{GeV}}$, $m_{1/2} = 630 {\,\mathrm{GeV}}$, $\tan{\beta} = 10$, $A_0 = 0$ and $\mu >0$. []{data-label="tab:mass"} ### $\tilde{u}_L \rightarrow u \, \tilde{\chi}_1^0$ {#sec:chiResults} Events were generated in which $\tilde{u}_L$ and its associated anti-particle were produced and then decayed via the mode $\tilde{u}_L \rightarrow u \, \tilde{\chi}_1^0$. Dalitz style distributions with and without the POWHEG correction were produced and are shown in the left and right-hand plots in Fig. \[fig:squark\_Dalitz\]. The black outline indicates the kinematic limits of phase space and the green dashed and blue dotted lines are the boundaries of the emission regions of the conventional parton shower with the most symmetric choice of shower phase space partitioning. Emissions from the up quark populate the area above the green dashed line, while the regions below the blue dotted lines are filled by emissions from the $\tilde{u}_L$. The area between the green and blue lines is the dead zone, unpopulated by the normal parton shower. In the POWHEG corrected distribution, points are concentrated in the soft region as $x_g \rightarrow 0$ and along the upper boundary of physical phase space where the gluon in collinear to the up quark. However, in the normal parton shower distribution fewer points lie along the upper physical phase space boundary and instead there is an concentration of points in the $\tilde{u}_L$ emission region with $ x_g \lesssim 0.85$ and along the lower boundary of the up quark emission region. In analogy to the case of top quark decay, it is likely that these unphysical high density regions are due to the parton shower kernels overestimating the exact real-emission matrix element. Finally, we see that including the POWHEG correction ensures that the region of phase space inaccessible to the normal parton shower is populated. ![Dalitz distributions for the decay $\tilde{u}_L \rightarrow u \, \tilde{\chi}_1^0$ with (left) and without (right) the POWHEG style correction. The black outline indicates the physically allowed region phase space. In the conventional parton shower approach, the region above the green dashed line is populated with radiation from the up quark and the regions below the blue dotted lines with radiation from the $\tilde{u}_L$. These boundaries correspond to the limits of the parton shower with symmetric phase space partitioning.[]{data-label="fig:squark_Dalitz"}](figure_4a) ![Dalitz distributions for the decay $\tilde{u}_L \rightarrow u \, \tilde{\chi}_1^0$ with (left) and without (right) the POWHEG style correction. The black outline indicates the physically allowed region phase space. In the conventional parton shower approach, the region above the green dashed line is populated with radiation from the up quark and the regions below the blue dotted lines with radiation from the $\tilde{u}_L$. These boundaries correspond to the limits of the parton shower with symmetric phase space partitioning.[]{data-label="fig:squark_Dalitz"}](figure_4b) ![Transverse momentum distributions of the the second hardest jet in the decay $\tilde{u}_L \rightarrow u \, \tilde{\chi}_1^0$ in the rest frame of the $\tilde{u}_L$. Events were generated with and without the POWHEG correction using the CMSSM model with , $m_{1/2} = 660 {\,\mathrm{GeV}}$, $\tan{\beta} = 10$, $A_0 = 0$ and $\mu >0$ at the LHC with $\sqrt{s}=8 {\,\mathrm{TeV}}$. []{data-label="fig:squark distribution"}](figure_5) Differential distributions of the transverse momentum of the subleading jet[^6], $p_{T,2}$, in each decay were also generated and are shown in Fig. \[fig:squark distribution\]. The blue dashed line corresponds to the distribution generated using the LO matrix element combined with the parton shower while the black solid line shows the result with the POWHEG correction to the decay applied. The bottom panel in Fig. \[fig:squark distribution\] shows the ratio of the parton shower and POWHEG corrected results and in both distributions error bars are included to indicate statistical uncertainty. As demonstrated in Fig. \[fig:squark\_Dalitz\], the parton shower has a tendency to over-populate the hard regions of phase space. Hence, including the POWHEG correction reduces the $p_T$ of the hardest emission in the decay. This phenomenon is reflected in the $p_{T,2}$ distributions. When the POWHEG correction is applied, the $p_{T,2}$ distribution is softened such that there is a reduction in the number of events passing the jet $p_T$ selection criteria of $\mathcal{O} \left(20 \% \right)$. The softening is less pronounced at low values of $p_{T,2}$ where the parton shower splitting kernels give a good approximation to the exact matrix element. Here the standalone parton shower and POWHEG corrected distributions are similar. At larger values of $p_{T,2}$, the impact of the POWHEG correction is again reduced as, in this region, the subleading jet in the POWHEG corrected distribution typically has a significant contribution from partons generated by the normal parton shower in addition to the hardest emission coming from the POWHEG correction. ### $\tilde{g} \rightarrow \tilde{t}_1 \, \bar{t}$ {#sec:gluinoResults} Finally, we investigate the impact of the POWHEG style correction on the decay mode $\tilde{g} \rightarrow \tilde{t}_1 \, \bar{t}$. The left and right-hand plots in Fig. \[fig:gluino\_Dalitz\] show Dalitz distributions for this decay with and without the POWHEG correction respectively. In both plots, the black outline indicates the kinematically allowed region phase space. The solid coloured lines show the boundaries of the parton shower emission regions in the scenario where the $\bar{t}$ absorbs the $p_T$ of the gluon and the $\tilde{t}_1$ is orientated along the negative $z$-axis in the $\tilde{g}$ rest frame. The region above the pale green line is populated by emissions from the $\bar{t}$ and the areas below the dark blue lines are filled by emissions from the $\tilde{g}$. In this scenario, the two emission regions overlap and there is no region of phase space left unpopulated by the parton shower. The dashed coloured lines indicate the emission boundaries of the parton shower when the $\tilde{t}_1$ absorbs the transverse recoil of the emission and the $\bar{t}$ is aligned with the negative $z$-axis. The pale green dashed line is the upper limit for emissions coming from the $\tilde{t}_1$ and the dark blue dashed lines are the lower boundaries from emissions from the $\tilde{g}$. From the left-hand plot of Fig. \[fig:gluino\_Dalitz\], we see that the majority of points in the POWHEG corrected distribution are concentrated in the soft region of phase space. High density regions corresponding to emissions collinear to the $\bar{t}$ or $\tilde{t}_1$ are suppressed due to the large masses of the decay products. In the parton shower distribution, points are concentrated in the soft region and along the lower boundary of the $\bar{t}$ and dashed $\tilde{g}$ emission regions. The latter two unphysical regions of over-population again highlight the importance of correcting hard emissions in the parton shower using the exact real-emission matrix element. The transverse momentum distribution of the third hardest jet in the rest frame of the $\tilde{g}$ were also plotted and are shown in Fig. \[fig:gluino stable\]. To focus on the effect of the POWHEG correction, the decay products, $\bar{t}$ and $\tilde{t}_1 $, were not allowed to decay further. The blue dashed and black solid lines in Fig. \[fig:gluino stable\] correspond to the parton shower and POWHEG correction distributions respectively. The bottom panel of the plot shows the ratio of the parton shower and POWHEG corrected results and in both distributions error bars are included to indicate statistical uncertainty. As in Sect. \[sec:chiResults\], we find that the POWHEG correction decreases the total number of events passing the jet $p_T$ selection criterion. The effect is more pronounced in this case, with an $\mathcal{O} \left(40 \% \right )$ reduction. The parton shower distribution significantly exceeds the POWHEG corrected distribution at small $p_{T,3}$, however, at higher values of $p_{T,3}$ the two distributions are similar. At lower values of $p_{T,3}$, the main contribution to the third hardest jet in the POWHEG corrected distribution is from the hardest emission in the decay, generated using the real-emission matrix element. Therefore, we expect the uncorrected distribution to exceed the corrected one in this region. However, the maximum possible $p_T$ of the gluon generated ![Dalitz distributions for the decay $\tilde{g} \rightarrow \tilde{t}_1 \, \bar{t}$ with (left) and without (right) the POWHEG style correction applied. The solid (dashed) coloured lines indicate the parton shower emission regions when the $\bar{t}$ $\left(\tilde{t}_1\right)$ absorbs the transverse recoil of the emission. The solid (dashed) pale green line shows the lower (upper) boundary for radiation from the $\bar{t}$ $\left(\tilde{t}_1 \right) $. The dark blue solid (dashed) lines are the equivalent upper (lower) boundaries for radiation from the $\tilde{g}$. All boundaries correspond to the case of symmetric phase space partitioning and the black outline shows the kinematically allowed region of phase space.[]{data-label="fig:gluino_Dalitz"}](figure_6a) ![Dalitz distributions for the decay $\tilde{g} \rightarrow \tilde{t}_1 \, \bar{t}$ with (left) and without (right) the POWHEG style correction applied. The solid (dashed) coloured lines indicate the parton shower emission regions when the $\bar{t}$ $\left(\tilde{t}_1\right)$ absorbs the transverse recoil of the emission. The solid (dashed) pale green line shows the lower (upper) boundary for radiation from the $\bar{t}$ $\left(\tilde{t}_1 \right) $. The dark blue solid (dashed) lines are the equivalent upper (lower) boundaries for radiation from the $\tilde{g}$. All boundaries correspond to the case of symmetric phase space partitioning and the black outline shows the kinematically allowed region of phase space.[]{data-label="fig:gluino_Dalitz"}](figure_6b) ![Comparison of parton level distributions generated with and without the POWHEG correction for the decay $\tilde{g} \rightarrow \tilde{t}_1 \, \bar{t}$ with stable decay products. Results are for the CMSSM model with $m_0 = 1220 {\,\mathrm{GeV}}$, , $\tan{\beta} = 10$, $A_0 = 0$ and $\mu >0$ and LHC events with $\sqrt{s}=8 {\,\mathrm{TeV}}$. Shown are the $p_T$ distributions of the the third hardest jet in the rest frame of the $\tilde{g}$. []{data-label="fig:gluino stable"}](figure_7) by the POWHEG correction is[^7] $p^{\rm{max}}_T \approx 75{\,\mathrm{GeV}}$. Jets contributing to the POWHEG corrected distribution above this limit include a number of other partons generated by the normal parton shower in addition to the hardest emission. This reduces the effect of the correction at higher values of $p_{T,3}$. Therefore, we find that applying the POWHEG correction has a more significant impact on the number of events passing selection criteria when the value of the $p_{T,3}$ selection criterion lies below $p^{\rm{max}}_T$ of the gluon produced in the POWHEG correction. Conclusions =========== In this work, we used the real-emission matrix element to generate hard QCD radiation in a range of particle decays in the  event generator. This method is particularly relevant to new physics searches based on the decays of heavy new particles. The POWHEG corrections to these decays can change the shapes of certain experimental observables, thus altering the number of signal events passing selection criteria and modifying the exclusion bounds that can be set on the masses of the new particles. This correction will be available in  version 2.7. The algorithm used to implement the POWHEG style correction in  was described in detail for the decay $t \rightarrow W b$. Dalitz style distributions of the first emission in the conventional parton shower and POWHEG corrected approach were produced and showed that, while the POWHEG style correction ensures the majority of emissions lie in the soft and collinear limits, the parton shower has erroneous, unphysical regions of high emission density. This causes the parton shower to overpopulate the high $p_T$ regions of phase space. Differential distributions of the minimum jet separation and logarithm of the jet measure were also generated with the POWHEG style correction and compared to those generated with the existing  implementation of hard and soft matrix element corrections. The two techniques exhibit a high level of agreement therefore demonstrating the validity of our approach. In addition to this, distributions were generated using the normal parton shower. In agreement with the results from the Dalitz plots, these distributions were found to be considerably harder than those generated with the matrix element or POWHEG style corrections. The impact of applying the POWHEG style correction to a BSM decay was studied by plotting the invariant mass distribution of dijets produced in the decay of the lightest RS graviton, $G \rightarrow gg$ and $G \rightarrow q \bar{q}$. Applying the POWHEG correction was found to have a considerable impact on the height of the distribution in the dijet mass peak. The number of events passing selection criteria in the mass range $2.1 {\,\mathrm{TeV}}\leq m_{jj} \leq 2.3 {\,\mathrm{TeV}}$ dropped by $\mathcal{O} \left(40 \% \right) $ when the correction was applied. This is a consequence of the dijet invariant mass not including the hardest emission in the shower that carries a significant fraction of the graviton’s momentum when it is simulated using the real-emission matrix element. The sizable impact of the correction in this scenario illustrates the importance of including higher order corrections when optimising experimental searches. The impact of the POWHEG correction was also investigate for two decays in the CMSSM model by studying the transverse momentum distributions of the hardest jet generated by the shower. At values of the transverse momentum less than the upper limit of the POWHEG correction, it was found that the POWHEG corrected distributions were significantly reduced with respect to those generated with the conventional parton shower. Above this cutoff, the normal parton shower and POWHEG corrected distributions were found to be similar. In this work, we have used the POWHEG formalism to improve the simulation of hard radiation in particle decays and studied the resulting effect on a number of distributions. However, hard radiation in the initial-state parton shower can also have a significant impact on these distributions. Hence, in order to achieve accurate simulation of hard radiation in BSM processes we must also include effects from the initial-state shower. Using the POWHEG formalism to improve the simulation of the hardest initial-state emission in the shower will be the subject of future work. Acknowledgements ================ We are grateful for help from the other members of the  collaboration. This work was supported by the Science and Technology Facilities Council. We also acknowledge the support of the European Union via MCNet. [^1]: See [@LeCompte:2011cn] for a recent study. [^2]: $\lambda(x,y,z) = \sqrt{x^2+y^2+z^2-2xy-2xy-2yz} $. [^3]: A good description of the veto algorithm can be found in [@Sjostrand:2006za]. [^4]: $\Delta R = \min_{ij}\sqrt{\Delta \eta_{ij} ^2 + \Delta \phi_{ij} ^2}$ where the indices $i,j$ run over the three hardest jets and . $\Delta \eta_{ij}$ and $\Delta \phi_{ij}$ are the differences in pseudorapidity and azimuthal angle of jets $i$ and $j$ respectively. [^5]: $y_{32} = \frac{2} {s} \min_{ij} \left(\min \left(E_i^2, E_j^2 \right) \left(1-\cos \theta_{ij} \right)\right)$ where again the indices $i,j$ run over the three hardest jets with $i \neq j$. $E_i$ is the energy of jet $i$ and $\theta_{ij}$ the polar angle between jets $i$ and $j$. [^6]: Jets are ordered in terms of their transverse momentum such that $p_{T,1} > p_{T,2} > p_{T,3} $ etc. [^7]: The value of $p^{\rm{max}}_T$ was calculated using the formula for $p^{\rm{max}}_T$ in top quark decay, given on page , with the replacements $m_t \rightarrow m_{\tilde{g}}$, $m_W \rightarrow m_{\tilde{t}_1}$ and $m_b \rightarrow m_t$

--- abstract: | We consider the simple exclusion process with $k$ particles on a segment of length $N$ performing random walks with transition $p>1/2$ to the right and $q=1-p$ to the left. We focus on the case where the asymmetry in the jump rates $b=p-q>0$ vanishes in the limit when $N$ and $k$ tend to infinity, and obtain sharp asymptotics for the mixing times of this sequence of Markov chains in the two cases where the asymmetry is either much larger or much smaller than $(\log k)/N$. We show that in the former case ($b \gg (\log k)/N$), the mixing time corresponds to the time needed to reach macroscopic equilibrium, like for the strongly asymmetric (i.e. constant $b$) case studied in [@LabLac16], while the latter case ($b \ll (\log k)/N$) macroscopic equilibrium is not sufficient for mixing and one must wait till local fluctuations equilibrate, similarly to what happens in the symmetric case worked out in [@Lac16]. In both cases, convergence to equilibrium is abrupt: we have a cutoff phenomenon for the total-variation distance. We present a conjecture for the remaining regime when the asymmetry is of order $(\log k) / N$. [**MSC 2010 subject classifications**]{}: Primary 60J27; Secondary 37A25, 82C22.\ [**Keywords**]{}: [*Exclusion process; WASEP; Mixing time; Cutoff.*]{} address: - 'Université Paris-Dauphine, PSL Research University, Ceremade, CNRS, 75775 Paris Cedex 16, France.' - 'IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil.' author: - Cyril Labbé - Hubert Lacoin bibliography: - 'library.bib' title: | Mixing time and cutoff for the weakly asymmetric\ simple exclusion process --- Introduction ============ The simple exclusion process is a model of statistical mechanics that provides a simplified picture for a gas of interacting particles. Particles move on a lattice, each of them performing a nearest neighbor random walk independently of the others, and interact only via the exclusion rule that prevents any two particles from sharing the same site (when a particle tries to jump on a site which is already occupied, this jump is cancelled). In spite of its simplicity, this model displays a very rich behavior and has given rise to a rich literature both in theoretical physics and mathematics, see for instance [@KipLan; @Liggett] and references therein. In the present paper, we study relaxation to equilibrium for a particular instance of the simple exclusion process in which the lattice is a segment of length $N$ and particles feel a bias towards the right that vanishes when $N$ tends to infinity. This setup is often referred to as the Weakly Asymmetric Simple Exclusion Process (WASEP): it interpolates between the symmetric case (SSEP) and the one with a positive constant bias (ASEP). While convergence to equilibrium for a particle system can be considered on a macroscopic scale via the evolution of the particle density or hydrodynamic profile (see e.g. [@KipLan] and references therein), an alternative and complementary viewpoint (when the system is of finite size) consists in measuring the so-called ${\varepsilon}$-Total Variation Mixing Time [@LevPerWil]. It is defined as the first time at which the total variation distance to the stationary state, starting from the “worst" initial condition, falls below a given threshold $\epsilon$. Compared to the hydrodynamic profile, this provides a much more microscopic information on the particle system. The problem of mixing time of the simple exclusion process on the segment has been extensively studied both in the symmetric [@Wil04; @Lac16; @Lac162] and the asymmetric setup [@Benjamini; @LabLac16] and it has been proved in [@Lac16] and [@LabLac16] respectively that in both cases, the worst case total variation distance drops abruptly from its maximal value $1$ to $0$, so that the mixing time does not depend at first order on the choice of the threshold ${\varepsilon}$ - a phenomenon known as cutoff and conjectured to hold for a large class of Markov chains as soon as the mixing time is of a larger order than the relaxation time (which is defined as the inverse of the spectral gap of the generator). However the patterns of convergence to equilibrium in the symmetric and asymmetric cases are very different. Let us for simplicity focus on the case with a density of particles $k=\alpha N$, $\alpha\in (0,1)$. In the symmetric case, the time scale associated with the hydrodynamic profile is $N^2$ and the limit is given by the heat equation [@KOV] (which takes an infinite time to relax to its equilibrium profile which is flat) and microscopic mixing occurs on a larger time scale $N^2\log N$. In the asymmetric setup the hydrodynamic limit is given by the inviscid Burgers’ equation with a shorter time scale $N$ [@Reza] (see also [@LabbeKPZ; @LabLac16] for adaptations of this result to the segment). The equilibrium profile for this equation is reached after a finite time and in this case, the mixing time is of order $N$ and corresponds exactly to the time at which macroscopic equilibrium is attained. The aim of this paper is to understand better the role of the asymmetry in mixing and how one interpolates between the symmetric and asymmetric regimes. This leads us to consider a model with an asymmetry that vanishes with the scale of observation, usually referred to as Weakly Asymmetric Exclusion Process (WASEP). While hydrodynamic limit [@Demasi89; @Gartner88; @KipLan] and fluctuations scaling limits [@DG91; @BG97; @LabbeKPZ] for WASEP are now well understood, much less is known about how a weak asymmetry affects the mixing time of the system. A first step in this direction was made in [@LevPer16]. Therein the order of magnitude for the mixing time was identified for all possible intensities of vanishing bias, but with different constant for the upper and the lower bounds. Three regimes where distinguished (in the case where there is a density of particles): - When $b_N\le 1/N$, the mixing time remains of the same order as that of the symetric case $N^2\log N$. - When $1/N\le b_N\le (\log N)/N$, the mixing time is of order $(b_N)^{-2}\log N$. - When $(\log N)/N\le b_N\le 1$, the mixing time is of order $(b_N)^{-1} N$. The transition occurring around $b_N\approx N^{-1}$ is the one observed for the hydrodynamic limit: It corresponds to a crossover regime where the limit is given by a viscous Burger’s equation [@Demasi89; @Gartner88; @KipLan] which interpolates between the heat and the inviscid Burgers’ equations. The one occurring for $b_N\approx N^{-1}\log N$ is however not observed in the macrospic profile and is specific to mixing times. In the present work, we identify the full asymptotic of the mixing time (with the right constant) when the bias is either negligible compared to, or much larger than $\log N / N$ (or $\log k/N$ when the total number of particle is not of order $N$). This implies cutoff in these two regimes. Our result and its proof provide a better understanding of the effect of asymmetry on microscopic mixing: When $b_N\gg N^{-1}\log N$, the pattern of relaxation is identical to that of the fully asymmetric case and microscopic equilibrium is reached exactly when the macroscopic profile hits its equilibrium state. When $b_N\ll N^{-1}\log N$ the pattern of relaxation resembles that of the symmetric case, the mixing time corresponds to the time needed to equilibrate local fluctuations, in particular in the case $(B)$ described above (or more precisely when $1/N\ll b_N\ll (\log N)/N$) this time does not correspond to the time needed to reach macroscopic equilibrium. We could not prove such a result in the crossover regime $b_N\approx N^{-1}\log N$: In this case the time to reach macroscopic equilibrium and that to equilibrate local fluctuations are of the same order and the two phenomena are difficult to separate. In Section \[conjectos\] we provide a conjecture for the mixing time in this regime in the case of vanishing density. However the techniques developed here are not sufficient to obtain sharp results in this case. Model and results ================= Mixing time for the WASEP ------------------------- Given $N\in \mathbb N$, $k\in {\llbracket}1,N-1 {\rrbracket}$ (we use the notation ${\llbracket}a,b{\rrbracket}=[a,b]\cap {\mathbb{Z}}$) and $p\in(1/2,1]$, the Asymmetric Simple Exclusion Process on ${\llbracket}1, N {\rrbracket}$ with $k$ particles and parameter $p$ is the random process on the state space $${\Omega}_{N,k}^0:=\Big\{ \xi\in\{0,1\}^{N} \ : \ \sum_{x=1}^N \xi(x)=k\Big\},$$ associated with the generator $$\label{defgen} {\mathcal{L}}_{N,k} f(\xi):= \sum_{y=1}^{N-1} \left( q{\mathbf{1}}_{\{\xi(y)< \xi(y+1)\}}+p{\mathbf{1}}_{\{\xi(y)> \xi(y+1)\}}\right)(f(\xi^y)-f(\xi)),$$ where $q=1-p$ and $$\label{flipz} \xi^y(x):=\begin{cases} \xi(y+1) \quad &\text{ if } x=y,\\ \xi(y) \quad &\text{ if } x=y+1,\\ \xi(x) \quad& \text{ if } x\notin \{y,y+1\}. \end{cases}$$ In a more intuitive manner we can materialize the positions of $1$ by particles, and say that the particles perform random walks with jump rates $p$ to the right and $q=1-p$ to the left: These random walks are independent from one another except that any jump that would put a particle at a location already occupied by another particle is cancelled. Having in mind this particle representation, we let for $i\in {\llbracket}1, k{\rrbracket}$, $\xi_i$ denote the position of the $i$-th leftmost particle $$\xi_i:= \min\left\{ y\in {\llbracket}1,N {\rrbracket}\ : \ \sum_{x=1}^y \xi(x)=i \right\}.$$ We let $P^{N,k}_t$ denote the associated semi-group and $(\eta^\xi(t,\cdot))_{t\ge 0}$ denote the trajectory of the Markov chain starting from initial condition $\xi \in {\Omega}_{N,k}^0$. This Markov chain is irreducible, and admits a unique invariant (and reversible) probability measure $\pi_{N,k}$ given by $$\pi_{N,k}(\xi):= \frac{1}{Z_{N,k}}{\lambda}^{-A(\xi)}.$$ where $\lambda = p/q$, $Z_{N,k}:= \sum_{\xi\in {\Omega}_{N,k}^0} {\lambda}^{-A(\xi)}$, and $$\label{def:A} A(\xi):= \sum_{i=1}^k (N-k+i-\xi_i)\ge 0$$ denotes the minimal number of moves that are necessary to go from $\xi$ to the configuration $\xi^{\min}$ where all the particles are on the right $\xi^{\min}(x):={\mathbf{1}}_{[N-k+1,N]}(x)$ (this terminology is justified by the fact that $\xi^{\min}$ is minimal for the order introduced in Section \[Sec:Prelim\]). Recall that the total-variation distance between two probability measures defined on the same state-space ${\Omega}$ is defined by $$\|\alpha-{\beta}\|_{TV}=\sup_{A\subset {\Omega}} \alpha(A)-{\beta}(A),$$ where the $\sup$ is taken over all measurable sets $A$. The mixing time associated to the threshold ${\varepsilon}\in (0,1)$ is defined by $${T_{\rm mix}}^{N,k}({\varepsilon}):=\inf\{ t \ge 0 \ : \ d^{N,k}(t)\le {\varepsilon}\},$$ where $d^{N,k}(t)$ denotes the total-variation distance to equilibrium at time $t$ starting from the worst possible initial condition $$\label{tvdis} d^{N,k}(t):= \max_{\xi\in {\Omega}_{N,k}^0} \| P^{N,k}_t(\xi, \cdot)-\pi_{N,k}\|_{TV}.$$ We want to study the asymptotic behavior of the mixing time for this system when both the size of the system and the number of particles tend to infinity. A natural case to consider is when there is a non-trivial density of particles, that is $k/N\to \alpha \in (0,1)$, but we decide to also treat the boundary cases of vanishing density ($\alpha=0$) and full density ($\alpha=1$). By symmetry we can restrict to the case when $k=k_N\le N/2$: indeed, permuting the roles played by particles and empty sites boils down to reversing the direction of the asymmetry of the jump rates. Note that we will always impose $k\ge 1$ since when $k=0$ the process is trivial. The asymptotic behavior of ${T_{\rm mix}}^{N,k}({\varepsilon})$ in the case of constant bias ($p>1/2$ is fixed when $N$ goes to infinity) has been obtained in a previous work. We have for every ${\varepsilon}>0$, every $\alpha \in [0,1]$ and every sequence $k_N$ such that $k_N/N \to \alpha$ $$\lim_{N\to \infty}\frac{{T_{\rm mix}}^{N,k_N}({\varepsilon})}{N}= \frac{(\sqrt{\alpha}+\sqrt{1-\alpha})^2}{p-q}\;.$$ The result implies in particular that at first order, the mixing time does not depend on ${\varepsilon}\in(0,1)$, meaning that on the appropriate time-scale, for large values of $N$ the distance to equilibrium drops abruptly from $1$ to $0$. This phenomenon is referred to as *cutoff* and was first observed in the context of card shuffling [@AldDia; @DiaSha]. It is known to occur for a large variety of Markov chains, see for instance [@LevPerWil]. In the context of the exclusion process, it has been proved in [@Lac16] that cutoff holds for the Symmetric Simple Exclusion Process (SSEP) which is obtained by setting $p=1/2$ in the generator . When $p=1/2$, for any sequence $k_N$ that tends to infinity and satisfies $k_N\le N/2$ for all $N$, we have $$\lim_{N\to \infty}\frac{{T_{\rm mix}}^{N,k_N}({\varepsilon})}{N^2\log k_N}= \frac{1}{\pi^2}.$$ While cutoff occurs in the two cases, it appears to be triggered by different mechanisms. When $p>1/2$, the mixing time is determined by the time needed for the particle density profile to reach its macroscopic equilibrium: After rescaling time and space by $N$, the evolution of the particle density has a non-trivial scaling limit (the inviscid Burgers’ equation with zero-flux boundary conditions), which fixates at time $\frac{(\sqrt{\alpha}+\sqrt{1-\alpha})^2}{p-q}$. The first order asymptotic for the mixing time is thus determined by the time the density profile needs to reach equilibrium. When $p=1/2$, the right-time scale to observe a macroscopic motion for the particles is $N^2$, and it is worth mentioning that the scaling limit obtained for the particle density (the heat equation) does not fixate in finite time. To reach equilibrium, however, we must wait for a longer time, of order $N^2\log N$, which is the time needed for local fluctuations in the particle density to come to equilibrium. We are interested in studying the process when the drift tends to zero: this requires to understand the transition between these two patterns of relaxation to equilibrium. Hence we consider $p$ to be a function of $N$ which is such that the bias towards the right $b_N:=p_N-q_N=2p_N-1$ vanishes $$\label{vanishing} \lim_{N\to \infty} b_N=0.$$ In this regime, the model is sometimes called WASEP for Weakly Asymetric Simple Exclusion Process. Its convergence to equilibrium has already been studied in[@LabbeKPZ; @LevPer16]. In [@LevPer16] the authors identify the order of magnitude of the mixing time as a function of $b_N$ in full generality. However the approach used in [@LevPer16] does not allow to find the exact asymptotic for the mixing time nor to prove cutoff, and does not answer our question concerning the pattern of relaxation to equilibrium. Results ------- We identify two main regimes for the pattern of relaxation to equilibrium. The *large bias* regime where $$\label{largebias} \lim_{N\to \infty} \frac{N b_N}{(\log k_N)\vee 1}=\infty.$$ and the *small bias* regime where $$\label{smallbias} \begin{cases} \lim\limits_{N\to \infty} \frac{Nb_N}{\log k_N}=0,\\ \lim\limits_{N\to \infty} k_N=\infty. \end{cases}$$ We identify the asymptotic expression for the mixing time in both regimes. In the large bias regime we show that the mixing time coincides with the time needed by the particle density to reach equilibrium like in the constant bias case. \[Th:largebias\] When holds, and $\lim_{N\to \infty} k_N/N=\alpha\in [0,1]$, we have for every ${\varepsilon}\in (0,1)$ $$\lim_{N\to \infty} \frac{b_N {T_{\rm mix}}^{N,k_N}({\varepsilon})}{N}= \left(\sqrt{\alpha}+\sqrt{1-\alpha}\right)^2.$$ To state our result in the small bias regime, let us introduce the quantity $$\label{thegap} \operatorname{\mathrm{gap}}_{N}:= (\sqrt{p_N}-\sqrt{q_N})^2+ 4 \sqrt{p_Nq_N} \sin\left( \frac{\pi}{2N}\right)^2,$$ which corresponds to the spectral gap associated with the generator . Notice that it does not depend on the number $k$ of particles in the system. The pattern of relaxation is similar to the one observed in the symmetric case. \[Th:smallbias\] When holds, we have $$\lim_{N\to \infty} \frac{\operatorname{\mathrm{gap}}_{N} {T_{\rm mix}}^{N,k_N}({\varepsilon})}{\log k_N}= \frac12.$$ Using Taylor expansion for $\operatorname{\mathrm{gap}}_N$ we have, whenever $b_N$ tends to zero $$\label{eq:taylorgap} \operatorname{\mathrm{gap}}_{N}\stackrel{N\to \infty}{\sim} \frac{1}{2}\left( b^2_N+ \left(\frac{\pi}{N}\right)^2 \right).$$ Thus in particular we have $${T_{\rm mix}}^{N,k_N}({\varepsilon})\sim \begin{cases} \frac{\log k_N}{b_N^2} & \text{if } 1/N \ll b_N\ll \log k_N/N, \\ \frac{1}{\pi^2} N^2 \log k_N &\text{ if } 0< b_N \ll 1/N, \\ \frac{1}{\pi^2+\beta^2} N^2 \log k_N &\text{ if } b_N\sim \beta/N. \end{cases}$$ Note that our classification of regimes - does not cover all possible choices of $b_N$. Two cases have been excluded for very different reasons: - When $b_N=O(N^{-1})$ and $k_N$ is bounded, then we have a system of $k$ diffusive interacting random walks. This system does not exhibit cutoff and has a mixing time of order $N^2$ (The upper bound can actually be deduced from argument presented in Section \[preuv2\] and the lower bound is achieved e.g. by looking at the expectation and variance of the number of particles on right half of the segment, like what is done in [@Morris06 Section 6] ). - When $b_N$ is of order $\log k_N/N$ the time at which the density profile reaches its equilibrium and the time needed for local fluctuations to reach their equilibrium values are of the same order and we believe that there is an interaction between the two phenomena. We provide a more detailed conjecture in Section \[conjectos\] We have not included here results concerning the biased card shuffling considered in [@Benjamini; @LabLac16]. Let us mention that while our analysis should also yield optimal bound for the mixing time of this process when holds (i.e. ${T_{\rm mix}}^{k,N}({\varepsilon})\sim 2 N /(p_N-q_N)$), it seems much more difficult to prove the equivalent of Theorem \[Th:smallbias\]. The main reason is that the coupling presented in Section \[Appendix:Coupling\] cannot be extended to a coupling on the permutation process. Building on and adapting the techniques presented in [@Lac16 Section 5] it should a priori be possible to obtain a result concerning the mixing time starting from an extremal condition (the identity or its symmetric), but this is out of the scope of the present paper. Conjecture in the regime $b_N\asymp \log k_N/N$ {#conjectos} ----------------------------------------------- Let us here formulate, and heuristically support a conjecture concerning the mixing-time in the crossover regime where $$\label{limbeta} \lim_{N\to \infty} \frac{b_N N}{\log k_N}={\beta}.$$ for some ${\beta}\in(0,\infty)$. For the ease of exposition, while it should be in principle possible to extend the heuristic to the case of positive density (see Remark \[pluscomplique\] below) we restrict ourselves to the case $\lim_{N\to \infty }k_N/N=0$. The justification we provide for the conjecture might be better understood after a first reading of the entire paper. \[jecture\] When $b_N$ and $k_N$ display the asymptotic behavior given by , we have for every ${\varepsilon}>0$ $$\lim_{N \to \infty} \frac{{T_{\rm mix}}^{N,k_N}({\varepsilon}) \log k_N}{N^2}= \begin{cases} \frac{2}{{\beta}}+\frac{1}{{\beta}^2}, \quad &\text{ if } {\beta}\le 1/2,\\ \left( \frac{\sqrt{2}+2\sqrt{{\beta}}}{2{\beta}} \right)^2, \quad &\text{ if } {\beta}\ge 1/2. \end{cases}$$ To motivate this conjecture let us first describe the equilibrium measure and its dependence on ${\beta}$. As we are in the low-density regime, the equilibrium measure is quite close to the product measure one would obtain for the system without exclusion rules: the $k$ particles are approximately IID distributed with the distance from the right extremity being a geometric of parameter $q_N/p_N\approx e^{- \frac{2{\beta}\log k}{N}}$. Hence the probability of having a particle at site $\lfloor zN \rfloor$ for $z\in [0,1]$ is roughly of order $k^{[1-2{\beta}(1-z)]+o(1)}/N$. Thus, while particles are concentrated near the right extremity at equilibrium, the equilibrium “logarithmic density” of particles exhibit a non trivial profile in the sense that for any $z> 1-(2{\beta})^{-1}$ we have $$\label{liquib} \lim_{{\varepsilon}\to 0}\lim_{N\to \infty}\frac{\log \left( \sum_{i= (z-{\varepsilon})N}^{(z+{\varepsilon})N} \xi(i) \right) }{\log k_N} \Rightarrow 1-2{\beta}(1-z),$$ where the convergence holds in probability under the equilibrium measure $\pi_{N,k}$ when $N$ tends to infinity and ${\varepsilon}$ tends to zero in that order. The typical distance to zero of the left-most particle at equilibrium is also given by this profile in the sense that it is typically $o(N)$ when ${\beta}\le 1/2$ and of order $N(1-\frac 1 {2{\beta}})$ when ${\beta}\ge 1/2$. While we only give heuristic justification for these statements concerning equilibrium, it is worth mentioning that they can be made rigorous by using the techniques exposed in Section \[Sec:Prelim\]. To estimate the mixing time, we assume that the system gets close to equilibrium once the number of particles on any “mesoscopic” interval of the form $[ (z-{\varepsilon})N, (z+{\varepsilon})N ]$ is close to its equilibrium value. While the mean number of particle is of order $k^{1-2{\beta}(1-z)}$ (cf. ), the typical equilibrium fluctuation around this number should be the given by the square root due to near-independence of different particles and thus be equal to $k^{\frac{1}{2}-{\beta}(1-z)}$. To estimate the surplus of particles in this interval at time $t N^2 (\log k)^{-1}$ for $t> 1/{\beta}$ (note that $t=1/{\beta}$ is the time of macroscopic equilibrium where most particles are packed on the right), we consider the number of particles that end up there after keeping a constant drift of order $z (\log k) /(N t)$, which is smaller than $b_N$. Neglecting interaction between particles and making a Brownian approximation for the random walk with drift, we obtain that the expected number of particles following this strategy is given by $$k \exp\left(- \log k \frac{({\beta}t-z)^2}{2 t} \right)=k^{1-\frac{({\beta}t-z)^2}{2 t}}.$$ Hence equilibrium should be attained when this becomes negligible with respect to the typical fluctuation $k^{\frac{1}{2}-{\beta}(1-z)}$ for all values of $z$ where we find particles at equilibrium. That is, when the inequality $$\label{fluqueton} 1-\frac{({\beta}t-z)^2}{2 t}< \frac{1}{2}-{\beta}(1-z),$$ is valid for all $z\in[0,1]$ if ${\beta}\le 1/2$ or for all $z\ge 1-\frac{1}{2{\beta}}$ if $\beta\ge 1/2$. A rapid computation show that one only needs to satisfy the condition for the smallest value of $z$ (either $1$ or $1-\frac{1}{2{\beta}}$), which boils down to finding the roots of a degree two polynomial. This yields that we must have $t>t_0$ where $$t_0:=\begin{cases} \frac{2}{{\beta}}+\frac{1}{{\beta}^2} &\text{ if } {\beta}\le 1/2,\\ \left( \frac{\sqrt{2}+2\sqrt{{\beta}}}{2{\beta}} \right)^2 &\text{ if } {\beta}\ge 1/2. \end{cases}$$ \[pluscomplique\] Describing the equilibrium “logarithmic profile” of particles when the system has positive density is also possible (note that on the right of $(1-\alpha) N$ it is the density of empty-sites that becomes the quantity of interest). It is thus reasonable to extend the heuristic to that case. However the best strategy to produce a surplus of particle in that case becomes more involved, as the zones with positive density of particles, which are described by the hydrodynamic evolution given in Proposition \[prop:lidro\], play a role in the optimization procedure. For this reason we did not wish to bring the speculation one step further. Organization of the paper ------------------------- In the remainder of the paper we drop the subscript $N$ in $k_N$ in order to simplify the notation. The article is organised as follows. In Section \[Sec:Prelim\], we introduce the representation through height functions and collect a few results on the invariant measure, the spectral gap and the hydrodynamic limit of the process. In Sections \[Sec:LBLB\] and \[Sec:UBLB\], we consider the large bias case and prove respectively the lower and upper bounds of Theorem \[Th:largebias\]: While the lower bound essentially follows from the hydrodynamic limit, the upper bound is more involved and is one of the main achievement of this paper. In Sections \[Sec:LBSB\] and \[Sec:UBSB\], we deal with the small bias case and prove respectively the lower and upper bounds of Theorem \[Th:smallbias\]. Here again, the lower bound is relatively short and follows from similar argument as those presented by Wilson [@Wil04] in the symmetric case, while the upper bound relies on a careful analysis of the area between the processes starting from the maximal and minimal configurations and under some grand coupling. Preliminaries and technical estimates {#Sec:Prelim} ===================================== Height function ordering and grand coupling ------------------------------------------- To any configuration of particles $\xi\in\Omega_{N,k}^0$, we can associate a so-called height function $h=h(\xi)$ defined by $h(\xi)(0) = 0$ and $$h(\xi)(x) = \sum_{y=1}^x \big(2\xi(y) - 1\big)\;,\quad x\in {\llbracket}1,N{\rrbracket}\;.$$ For simplicity, we often abbreviate this in $h(x)$. The height function is a lattice path that increases by $1$ from $\ell-1$ to $\ell$ if there is a particle at site $\ell$, and decreases by $1$ otherwise. Its terminal value therefore only depends on $k$ and $N$. The set of height functions obtained from $\Omega_{N,k}^0$ through the above map is denoted $\Omega_{N,k}$.\ The particle dynamics can easily be rephrased in terms of height functions: every upward corner ($h(x)=h(x-1)+1=h(x+1)+1$) flips into a downward corner ($h(x)=h(x-1)-1=h(x+1)-1$) at rate $p$, while the opposite occurs at rate $q$. We denote by $(h^\zeta(t,\cdot),t\ge 0)$ the associated Markov process starting from some initial configuration $\zeta \in \Omega_{N,k}$.\ It will be convenient to denote by $\wedge$ the maximal height function: $$\wedge(x) = x \wedge (2k-x)\;,\quad x\in{\llbracket}1,N{\rrbracket}\;,$$ and by $\vee$ the minimal height function: $$\vee(x) = (-x)\vee(x-2N+2k)\;,\quad x\in{\llbracket}1,N{\rrbracket}\;.$$ Though the dependence on $k$ is implicit in the notations $\wedge,\vee$, this will never raise any confusion as the value $k$ will be clear from the context. It is possible to construct simultaneously on a same probability space and in a Markovian fashion, the height function processes $(h_t^\zeta,t\ge 0)$ starting from all initial conditions $\zeta\in\cup_k \Omega_{N,k}$ and such that the following monotonicity property is satisfied for all $k$ and all $\zeta,\zeta' \in \Omega_{N,k}$: $$\label{eq:grand} \zeta \le \zeta' \Rightarrow h_t^\zeta \le h_t^{\zeta'}\;,\quad \forall t\ge 0\;.$$ Here, $\zeta\le \zeta'$ simply means $\zeta(x) \le \zeta'(x)$ for all $x\in{\llbracket}0,N{\rrbracket}$. We call such a construction a monotone Markovian grand coupling, and we denote by ${\mathbb{P}}$ the corresponding probability distribution. The existence of such a grand coupling is classical, see for instance [@LabLac16 Proposition 4]. In a portion of our proof, we require to use a specific grand coupling which is not the one displayed in [@LabLac16] and for this reason we provide an explicit construction in Appendix \[Appendix:Coupling\]. Once a coupling is specified, by enlarging our probability space, one can also define the process $h^{\pi}_t$ which is started from an initial condition sampled from the equilibrium measure $\pi_{N,k}$, independently of $h_t^{\zeta}, \zeta\in {\Omega}_{N,k}$. Let us end up this section introducing the (less canonical) notation $$\label{eq:strict} \zeta<\zeta' \quad \Leftrightarrow \quad \left( \zeta\le \zeta' \text{ and } \zeta\ne \zeta' \right).$$ We say that a function $f$ on $\Omega_{N,k}$ is increasing (strictly) if $f(\zeta)\le f(\zeta')$ ($f(\zeta)< f(\zeta')$) whenever $\zeta<\zeta'$. The minimal increment of an increasing function is defined by $$\label{eq:minincr} \delta_{\min}(f)=\min_{\zeta,\zeta'\in {\Omega}_{N,k}, \zeta<\zeta'} f(\zeta')-f(\zeta).$$ The equilibrium measure in the large-bias case ---------------------------------------------- For $\xi\in {\Omega}_{N,k}^0$ we set $$\label{def1:lknrkn} \begin{split} \ell_{N}(\xi)&=\min\{ x\in {\llbracket}1, N{\rrbracket}\ : \ \xi(x)= 1 \},\\ r_{N}(\xi)&=\max\{ x\in {\llbracket}1, N{\rrbracket}\ : \ \xi(x)= 0 \}. \end{split}$$ A useful observation on the invariant measure is the following. Given $\xi$, we define $\chi(\xi)$ as the sequence of particle spacings: $$\chi_i:= \xi_{i+1}-\xi_i\;,\; \text{ for } i\in {\llbracket}1,k-1{\rrbracket}, \quad \chi_k=N+1-\xi_k.$$ From , under $\pi_{N,k}$ the probability of a given configuration is proportional to $$\label{eq:weights} {\lambda}^{-\chi_1} {\lambda}^{-2\chi_2} \ldots {\lambda}^{-k \chi_k},$$ In other terms, under the invariant measure the particle spacings $(\chi_i)_{1\le i \le k}$ are distributed like independent geometric variables, with respective parameters ${\lambda}^{-i}$, conditioned to the event $\sum_{i=1}^k \chi_k\le N$. \[lem:lbeq\] When holds we have for any ${\varepsilon}>0$ $$\begin{split} \lim_{N\to \infty}& \pi_{N,k}( \ell_N \le (N-k)-{\varepsilon}N)=0,\\ \lim_{N\to \infty}& \pi_{N,k}( r_N \ge (N-k)+{\varepsilon}N)=0, \end{split}$$ By symmetry it is sufficient to prove the result for $\ell_N$ only, but for all $k\in{\llbracket}1,N-1{\rrbracket}$. Note that there is nothing to prove regarding $\ell_N$ if $\alpha:=\lim_{N\to \infty} k/N =1$, so we assume that $\alpha\in[0,1)$.\ Let $(X_i)_{1\le i \le k}$ be independent geometric variables, with respective parameters ${\lambda}^{-i}$. The sum of their means satisfies (recall that ${\lambda}-1$ is of order $b_N$) $$\begin{aligned} \sum_{i=1}^k \frac{1}{1-{\lambda}^{-i}} = k+ \sum_{i=1}^k \frac{{\lambda}^{-i}}{1-{\lambda}^{-i}}\le k+ C b_N^{-1} \log \min( k, b_N^{-1})\;,\end{aligned}$$ for some constant $C>0$. The large bias assumption ensures that $b_N^{-1} \log \min( k_N, b_N^{-1})=o(N)$. Hence using the Markov inequality, we obtain that if $(X_i)_{1\le i \le k}$ is a sequence of such geometric variables, and if is satisfied, then for any ${\varepsilon}>0$ $${\mathbb{P}}\Big(\sum_{i=1}^k X_i\ge k+{\varepsilon}N\Big) \le \frac{{\mathbb{E}}\big[\big|\sum_{i=1}^k X_i - k\big|\big]}{{\varepsilon}N}=\frac{{\mathbb{E}}\big[\sum_{i=1}^k X_i - k\big]}{{\varepsilon}N}\;,$$ so that $$\lim_{N\to\infty} {\mathbb{P}}\Big(\sum_{i=1}^k X_i\ge k+{\varepsilon}N\Big) = 0\;.$$ The above inequality for ${\varepsilon}<1-\alpha$ implies that ${\mathbb{P}}(\sum_{i=1}^k X_i\le N)\ge 1/2$ for all $N$ large enough meaning that the conditioning only changes the probability by a factor at most $2$. Then, we can conclude by noticing that $\ell_N=\xi_1= N-\sum_{i=1}^k \chi_k$. The equilibrium measure in the small-bias case {#Subsec:eqSmallBias} ---------------------------------------------- We aim at showing that with large probability the density of particles everywhere is of order $k^{1+o(1)}/N$. Given $\xi\in {\Omega}^0_{N,k}$ we let $Q_1(\xi)$, resp. $Q_2(\xi)$, denote the largest gap between two consecutive particles, resp. between two consecutive empty sites. $$\begin{split} Q_1(\xi)&:=\max\{ n\ge 1 \ : \ \exists i\in {\llbracket}0,N-n{\rrbracket}, \ \forall x\in {\llbracket}i+1,i+n{\rrbracket}, \xi(x)=0 \},\\ Q_2(\xi)&:=\max\{ n\ge 1 \ : \ \exists i\in {\llbracket}0,N-n{\rrbracket}, \ \forall x\in {\llbracket}i+1,i+n{\rrbracket}, \xi(x)=1 \}, \end{split}$$ and $Q(\xi)=\max(Q_1(\xi),Q_2(\xi))$. \[lem:dens\] For all $x\in {\llbracket}1,N{\rrbracket}$, we have $$\label{encadr} \frac{k}{N} {\lambda}^{x-N} \le \pi_{N,k}(\xi(x) = 1) \le \frac{k}{N} {\lambda}^{x-1}\;.$$ Furthermore, there exists a constant $c>0$ such that for all choices of $N\ge 1$, $u>1$ and $p_N\in (1/2,1]$ and all $k\le N/2$ $$\label{splam} \pi_{N,k}\left(Q(\xi)\ge \frac{{\lambda}^{N} N u}{k} \right) \le 2k e^{-cu}\;.$$ We set $A_x:= \{\xi(x) = 1\}$, we first prove that for all $y\in {\llbracket}1,N-1{\rrbracket}$ we have $$\label{swiz} \pi_{N,k}(A_y) \le \pi_{N,k}(A_{y+1}) \le {\lambda}\pi_{N,k}(A_y).$$ We observe that the map $\xi \mapsto \xi^y$ defined in induces a bijection from $A_y$ to $A_{y+1}$ and that for every $\xi\in A^y$, $$\pi_{N,k}(\xi) \le \pi_{N,k}(\xi^y)\le {\lambda}\pi_{N,k}(\xi).$$ The reader can check indeed that $\pi_{N,k}(\xi^y)= {\lambda}\pi_{N,k}(\xi)$ if $\xi\in A_y \setminus A_{y+1}$ and that $\xi^y=\xi$ if $\xi\in A_y \cap A_{y+1}$. The desired inequality is then obtained by summing over $\xi\in A_y$. By iterating we obtain $${\lambda}^{x-N}\pi_{N,k}(A_1) \le \pi_{N,k}(A_{x}) \le {\lambda}^{x-1} \pi_{N,k}(A_1).$$ By monotony of $\pi_{N,k}(A_y)$ in $y$ and the fact that there are $k$ particles $$N\pi_{N,k}(A_1) \le \sum_{y=1}^N \pi_{N,k}(A_y)=k \le N \pi_{N,k}(A_N),$$ and thus can be deduced. We pass to the proof of . We can perform the same reasoning as above but limiting ourselves to configurations with no particles in some set $I\subset {\llbracket}1, N{\rrbracket}$. Setting $B_I:=\{ \forall y\in I, \xi(y)=0 \}$ we obtain similarly to (exchanging directly the content of $x$ and $y$ instead of nearest neighbors) that for every $x,y \in {\llbracket}1, N{\rrbracket}\setminus I$ with $x< y$ $$\pi_{N,k}(A_x \cap B_I ) \le \pi_{N,k}(A_y \cap B_I ) \le {\lambda}^{y-x} \pi_{N,k}(A_x \cap B_I ).$$ This allows to deduce that $$\pi_{N,k}(\xi(x)=1 \ | \ \forall y\in I, \xi(y)=0) \ge \frac{k}{N-|I|}{\lambda}^{x-N},$$ and yields by induction $$\pi_{N,k}(\forall x\in I, \ \xi(x)=0)\le \left(1-{\lambda}^{-N} \frac{k}{N}\right)^{|I|} \le \exp\left(-|I| {\lambda}^{-N} \frac{k}{N}\right).$$ Then noticing that $\{Q_1(\xi)\ge 2m\}$ implies that an interval of the type ${\llbracket}mi+1, m(i+1){\rrbracket}$ is empty, a union bound yields that $$\pi_{N,k}(Q_1(\xi)\ge 2m)\le \left\lfloor \frac{N}{m}\right\rfloor \exp\left(-m{\lambda}^{-N} \frac{k}{N}\right).$$ This remains true for $Q_2(\xi)$ upon replacing $k$ by $N-k$, and this concludes the proof of if one choses $m=\frac{{\lambda}^{N} N u}{2k}$ . Eigenfunctions and contractions {#sec:eigen} ------------------------------- The exact expression of the principal eigenfunction / eigenvalue has been derived in previous works [@LevPer16; @LabLac16]. It turns out that it can be obtained by applying a discrete Hopf-Cole transform to the generator of our Markov chain. Let us recall some identities in that direction as they will be needed later on; the details can be found in [@LabLac16 Section 3.3]. We set $$\label{Eq:rho} \varrho := \big(\sqrt p - \sqrt q\big)^2 \sim \frac{b_N^2}{2}\;,$$ and we let $a_{N,k}$ be the unique solution of $$\begin{cases} (\sqrt{pq}\, {\Delta}-\varrho)a(x)=0\;,\quad x\in {\llbracket}1,N-1 {\rrbracket}\;,\\ a(0)=1\;,\quad a(N)= {\lambda}^{\frac{2k-N}{2}} \;, \end{cases}$$ where ${\Delta}$ denotes the discrete Laplace operator $$\label{laplace} {\Delta}(f)(x)=f(x+1)+f(x-1)-2f(x), \quad x\in {\llbracket}1,N-1 {\rrbracket}.$$ If $(h^\zeta_t,t\ge 0)$ denotes the height function process starting from some arbitrary initial condition $\zeta\in\Omega_{N,k}$, then the map $$V(t,x) := {\mathbb{E}}[\lambda^{\frac12 h^{\zeta}_t(x)} - a_{N,k}(x)]\;,\quad t\ge 0\;,\quad x\in {\llbracket}0,N{\rrbracket}\;,$$ solves $$\label{Eq:V} \begin{cases} \partial_t V(t,x)= (\sqrt{pq}\,{\Delta}- \varrho) V(t,x)\;,\quad x\in{\llbracket}1,N-1 {\rrbracket}\;.\\ V(t,0) = V(t,N) = 0\;. \end{cases}$$ This allows to identify $N-1$ eigenvalues and eigenfunctions of the generator ${\mathcal{L}}_{N,k}$ of the Markov chain: for every $j\in\{1,\ldots,N-1\}$, the map $$\label{eq:fj} f^{(j)}_{N,k}(\zeta) = \sum_{x=1}^{N-1} \sin \left(\frac{j x\pi}{N}\right) \left( \frac{\lambda^{\frac12 \zeta(x)} - a_{N,k}(x)}{\lambda - 1} \right)\;,$$ defines an eigenfunction with eigenvalue $$-\gamma_j = -\varrho - 4\sqrt{p_Nq_N} \sin\left( \frac{j \pi}{2N}\right)^2\;.$$ The eigenvalue $\gamma_1$ corresponds to the spectral gap of the generator (this is related to the fact that the corresponding eigenfunction is monotone, see [@LabLac16 Section 3.3] for more details), and for this reason we adopt the notation $$\operatorname{\mathrm{gap}}_N :=\gamma_1= \varrho + 4\sqrt{p_Nq_N} \sin\left( \frac{\pi}{2N}\right)^2.$$ We also set $f_{N,k}:=f^{(1)}_{N,k}(\zeta)$ for the corresponding eigenfunction. Notice that this is a *strictly* increasing function (recall ). An immediate useful consequence of the eigenvalue equation is that $$\label{eq:contract1} {\mathbb{E}}[f_{N,k}(h^{\zeta'}_t)-f_{N,k}(h^\zeta_t)]= e^{-\operatorname{\mathrm{gap}}_N t} \left( f_{N,k}(\zeta')-f_{N,k}(\zeta) \right).$$ To close this section, let us introduce another function which is not an eigenfunction, but is also strictly increasing and enjoys a similar contraction property $$f^{(0)}_{N,k}(\zeta):=\sum_{x=1}^{N-1} \frac{{\lambda}^{\zeta(x)/2}-a_{N,k}(x)}{{\lambda}-1}.$$ As a direct consequence of at time zero, we have (using the notation introduced in ) $$\begin{gathered} ({\mathcal{L}}_{N,k}f^{(0)}_{N,k})(\zeta)= -\varrho f^{(0)}_{N,k}(\zeta)+\sqrt{pq}\sum_{x=1}^{N-1} \frac{{\Delta}({\lambda}^{\zeta/2}-a_{N,k})(x)}{{\lambda}-1} \\=-\varrho f^{(0)}_{N,k}(\zeta)-\frac{\sqrt{pq}}{{\lambda}-1}\left[ {\lambda}^{\frac{\zeta(N-1)}2}+{\lambda}^{\frac{\zeta(1)}{2}}-a_{N,k}(N-1)-a_{N,k}(1) \right].\end{gathered}$$ In particular, we obtain for $\zeta \le \zeta'$ $$\begin{aligned} &({\mathcal{L}}_{N,k}f^{(0)}_{N,k})(\zeta')- ({\mathcal{L}}_{N,k}f^{(0)}_{N,k})(\zeta)\\&= -\varrho \left( f^{(0)}_{N,k}(\zeta')-f^{(0)}_{N,k}(\zeta) \right)- \frac{\sqrt{pq}}{{\lambda}-1} \left[{\lambda}^{\frac{\zeta'(N-1)}2}+{\lambda}^{\frac{\zeta'(1)}2}- {\lambda}^{\frac{\zeta(N-1)}2}- {\lambda}^{\frac{\zeta(1)}2}\right]\\ &\le -\varrho \left( f^{(0)}_{N,k}(\zeta')-f^{(0)}_{N,k}(\zeta) \right).\end{aligned}$$ Considering a monotone coupling between $(h^{\zeta'}_t)_{t\ge 0}$ and $(h^{\zeta}_t)_{t\ge 0}$, we obtain that $$\begin{gathered} \partial_t {\mathbb{E}}[f^{(0)}_{N,k}(h^{\zeta'}_t)-f^{(0)}_{N,k}(h^\zeta_t)]= {\mathbb{E}}[({\mathcal{L}}_{N,k}f^{(0)}_{N,k})(h_t^{\zeta'})- ({\mathcal{L}}_{N,k}f^{(0)}_{N,k})(h_t^\zeta)]\\ \le -\varrho {\mathbb{E}}[f^{(0)}_{N,k}(h^{\zeta'}_t)-f^{(0)}_{N,k}(h^\zeta_t)],\end{gathered}$$ and thus $$\label{eq:contract0} {\mathbb{E}}[f^{(0)}_{N,k}(h^{\zeta'}_t)-f^{(0)}_{N,k}(h^\zeta_t)]\le e^{-\varrho t} \left( f^{(0)}_{N,k}(\zeta')-f^{(0)}_{N,k}(\zeta) \right).$$ The hydrodynamic limit ---------------------- We are interested in the macroscopic evolution of the height function. For $\alpha\in [0,1]$, we define $\vee_{\alpha} : [0,1] \to {\mathbb{R}}$, $\wedge_{\alpha} : [0,1] \to {\mathbb{R}}$ as $$\vee_{\alpha}(x):= \max(-x,x-2(1-\alpha))\;,\qquad \wedge_{\alpha}(x):= \min(x,2\alpha-x)\;,$$ and we let $g_{\alpha}: {\mathbb{R}}_+\times [0,1]\to {\mathbb{R}}$ be defined as follows $$\begin{split} g^0_{\alpha}(t,x)&:= \begin{cases} \alpha-\frac{t}{2}-\frac{(x-\alpha)^2}{2t}, \quad &\text{ if } |x-\alpha| \le t, \\ \wedge^{\alpha}(x), \quad & \text{ if } |x-\alpha| \ge t, \end{cases}\\ g_{\alpha}(t,x)&:= \max(\vee_{\alpha}(x), g^0_{\alpha}(t,x)). \end{split}$$ \[prop:lidro\] Assume that $Nb_N = N(p_N-q_N) \to\infty$ and that $k_N/N \to \alpha \in (0,1)$. Then, after an appropriate space-time scaling, $h^{\wedge}(\cdot,\cdot)$ converges to $g_\alpha$ in probability as $N\to\infty$. More precisely we have for any ${\varepsilon}>0$, $T>0$, $$\lim_{N\to \infty} {\mathbb{P}}\left[ \sup_{t\le T}\sup_{x\in[0,1]}\left| \frac{1}{N} h\Big( \frac{Nt}{b_N}, Nx \Big)-g_{\alpha}(t,x) \right|\ge {\varepsilon}\right]=0.$$ This is essentially the content of [@LabbeKPZ Th 1.3] where the hydrodynamic limit of the density of particles is shown to be given by the inviscid Burgers’ equation with zero-flux boundary conditions: when starting from the maximal initial condition, this yields (after integrating the density in space) the explicit solution $g_\alpha$.\ Actually the setting of [@LabbeKPZ Th 1.3] is more restrictive as the number of particles is taken to be $k=N/2$ and $p_N-q_N = 1/N^{\alpha}$ with $\alpha \in (0,1)$. However, a careful inspection of the proof shows that we only require $N^{1-\alpha}$ to go to infinity: this corresponds to the assumption $N(p_N-q_N) \to \infty$ which is in force in the statement of the proposition so that the proof carries through *mutatis mutandis*. Lower bound on the mixing time for large biases {#Sec:LBLB} =============================================== In the large bias case, the last observable that equilibrates is the position of the leftmost particle. Obtaining a lower bound on the mixing time is thus relatively simple: we have to show that for arbitrary $\delta>0$ at time $$s_{\delta}(N):=[(\sqrt{\alpha}+\sqrt{1-\alpha})^2-\delta]N b_N^{-1},$$ the leftmost particle has not reached its equilibrium position given by Lemma \[lem:lbeq\]. This is achieved by using the hydrodynamic limit for $\alpha>0$, and a simple comparison argument for $\alpha=0$. When is satisfied, for every $\delta>0$ we have $$\lim_{N\to \infty} \left\| {\mathbb{P}}\left( \ell_N(\eta^{\wedge}_{s_{\delta}(N)}) \in \cdot \right)- \pi_N(\ell_N \in \cdot)\right\|_{TV}=1.$$ As a consequence for all ${\varepsilon}>0$ and $N$ sufficiently large $${T_{\rm mix}}^{N,k}(1-{\varepsilon})\ge s_{\delta}(N).$$ The case $\alpha=0$ ------------------- Given ${\delta}>0$ we want to prove that the system is not mixed at time $s_{\delta}(N):=(1-{\delta})N b_N^{-1}$. We know from Lemma \[lem:lbeq\], that when $\alpha=0$ and holds, at equilibrium we have $$\lim_{N\to \infty}\pi_{N,k}(\ell_N\le (1-{\delta}/2)N)=0.$$ On the other hand observing that the position of the first particle is dominated by a random walk on ${\mathbb{N}}$ with jump rates $p_N$ to the right and $q_N$ to the left, it is standard to check that whenever $\lim_{N\to \infty} b_N N=\infty$ $$\lim_{N\to \infty} {\mathbb{P}}\left( \ell_N(\eta^{\wedge}_{s_\delta})\le (1-{\delta}/2)N \right)=1.$$ The case $\alpha\in (0,1/2]$ ---------------------------- Setting $x_\delta:= 1-\alpha-c_\alpha \delta$, for some positive constant $c_\alpha$ sufficiently small, we observe that the hydrodynamic profile at the rescaled time corresponding to $s_\delta$ is above the minimum at $x_{\delta}$ $$g_{\alpha}(x_{\delta}, (\sqrt{\alpha}+\sqrt{1-\alpha})^2-\delta)>\vee_{\alpha}(x_{\delta}).$$ The reader can check that $c_{\alpha}=1/3$ works for all $\alpha\in(0,1/2)$. Thus whenever $\lim_{N\to \infty} b_N N=\infty$ Proposition \[prop:lidro\] yields that $$\lim_{N\to \infty} {\mathbb{P}}\left( \ell_N(\eta^{\wedge}_{s_\delta}) \le (1-\alpha-c_{\alpha}\delta)N \right)=1.$$ On the other hand we know from Lemma \[lem:lbeq\] that when holds, at equilibrium we have for any $\delta>0$ $$\lim_{N\to \infty}\pi_{N,k}(\ell_N\le (1-\alpha-c_{\alpha}\delta)N)=0.$$ Upper bound on the mixing time for large biases {#Sec:UBLB} =============================================== Let us recall how a grand coupling satisfying the order preservation property is of help to establish an upper bound on the mixing time. Recalling , we have by the triangle inequality $$\label{eq:zone} d^{N,k}(t)\le \max_{\zeta,\zeta' \in {\Omega}_{N,k}}\|P^{N,k}_{t}(\zeta',\cdot)-P^{N,k}_{t}(\zeta,\cdot)\|_{TV}.$$ On the other hand if ${\mathbb{P}}$ is a monotone grand coupling, one observes that the extremal initial conditions are the last to couple so that one has $$\label{eq:ztwo} \|P^{N,k}_{t}(\zeta',\cdot)-P^{N,k}_{t}(\zeta,\cdot)\|_{TV}\le {\mathbb{P}}[ h^{\zeta}_t\ne h^{\zeta'}_t] \le {\mathbb{P}}[ h^{\vee}_t\ne h^{\wedge}_t].$$ Hence to establish an upper bound on the mixing time, it suffices to obtain a good control on the merging time $$\label{eq:mergin} \tau:=\inf\{ t>0 \ : \ h^{\vee}_t= h^{\wedge}_t\}.$$ Let us set for this section $$\label{deltaz} t_{\delta}(N):=[(\sqrt{\alpha}+\sqrt{1-\alpha})^2+\delta]N b_N^{-1}.$$ \[lapropo\] When is satisfied, for every $\delta>0$, and any monotone grand coupling we have $$\lim_{N\to \infty} {\mathbb{P}}\left( \tau \le t_{\delta}(N)\right)=1.$$ As a consequence for all ${\varepsilon}>0$ and $N$ sufficiently large ${T_{\rm mix}}^{N,k}({\varepsilon})\le t_{\delta}(N)$. When $\alpha=1/2$, a sharp estimate on $\tau$ can be obtained directly from spectral considerations (Section \[specz\]), but when $\alpha\in [0,1/2)$ we need a refinement of the strategy used in [@LabLac16]: The first step (Proposition \[lerimo\]) is to obtain a control on the position of the leftmost particle which matches the lower bound provided by the hydrodynamic limit. This requires a new proof since the argument used in [@LabLac16] is not sharp enough to cover all biases. The second step is to use contractive functions once the system is at macroscopic equilibrium, this is sufficient to treat most cases. A third and new step is required to treat the case when the bias $b_N$ of order $\log N/ N$ or smaller: as we are working under the assumption we only need to treat this case when $k=N^{o(1)}$. In this third step we use diffusive estimates to control the hitting time of zero for the function $f^{(0)}_{N,k}(h^{\wedge}_t)-f^{(0)}_{N,k}(h^{\vee}_t)$ where $f^{(0)}$ was introduced in Subsection \[sec:eigen\]. The special case $\alpha=1/2$ {#specz} ----------------------------- In the special case $\alpha=1/2$, a sharp upper-bound can be deduced in a rather direct fashion from spectral estimates repeating the computation performed in [@Wil04 Section 3.2] for the symmetric case. This fact is itself a bit surprising since this method does not yield the correct upper bound in the symetric case nor in the constant bias case. Recall the definition of $f_{N,k}$ in Equation and below. It being a strictly monotone function and ${\mathbb{P}}$ being a monotone coupling, we obtain using Markov’s inequality (recall ) $$\label{eq:mark} {\mathbb{P}}(\tau>t)= {\mathbb{P}}\left[ f_{N,k}(h^{\wedge}_t) > f_{N,k}(h^{\vee}_t)\right] \le \frac{{\mathbb{E}}\left[ f_{N,k}(h^{\wedge}_t) - f_{N,k}(h^{\vee}_t)\right]}{\delta_{\min}(f_{N,k})}.$$ The expectation decays exponentially with rate $\operatorname{\mathrm{gap}}_N$ and it is not difficult to check that $$\label{delmin} \delta_{\min}(f_{N,k})\ge {\lambda}^{(k-N)/2} \sin\left(\frac{\pi}{N}\right)\ge N^{-1}{\lambda}^{(k-N)/2}.$$ Hence Equation becomes $$\label{eq:zthree} {\mathbb{P}}(\tau>t) \le N {\lambda}^{\frac{N-k}{2}} e^{-\operatorname{\mathrm{gap}}_N t}\left( f_{N,k}(\wedge)-f_{N,k}(\vee) \right) \le \frac{N^2 {\lambda}^{N/2}}{{\lambda}-1}e^{-\operatorname{\mathrm{gap}}_N t},$$ where in the last inequality we used that $$\sin\left(\frac{x\pi}{N}\right)\left({\lambda}^{\wedge(x)/2}-{\lambda}^{\vee(x)/2}\right)\le {\lambda}^{\wedge(x)/2}\le {\lambda}^{k/2}.$$ Recall that we assume that holds and $b_N$ tends to zero. Recalling , we obtain the following asymptotic equivalent $$\label{eq:lezequiv} \operatorname{\mathrm{gap}}_N \stackrel{N\to \infty}{\sim} b_N^2/2 \quad \text{ and } \quad \log {\lambda}_N \stackrel{N\to \infty}{\sim} 2b_N$$ Furthermore for $N$ sufficiently large we have $({\lambda}-1)^{-1}\le N$. Hence recalling that $t_\delta=(2+\delta)b^{-1}_N N$ and using in we obtain for all $N$ sufficiently large $${\mathbb{P}}(\tau>t_\delta)\le \frac{N^2}{{\lambda}-1} \exp\left(\frac{N}{2}\log {\lambda}-\operatorname{\mathrm{gap}}_N t_{{\delta}} \right) \le N^3 e^{-\frac{\delta}{4}b_N N}.$$ and the left-hand side vanishes when $N$ tends to infinity as a consequence of (recall that as $\alpha=1/2$, $k$ is of order $N$) The reason why the computation above yields a sharp upper bound is because: (A) The difference of order between $\delta_{\min}(f_{N,k})$ and the typical fluctuation of $f_{N,k}$ at equilibrium is negligible in the computation. (B) Until shortly before the mixing time the quantity $\log \left[f_{N,k}(h^{\wedge}_t) - f_{N,k}(h^{\vee}_t)\right]$ has the same order of magnitude as $\log {\mathbb{E}}\left[f_{N,k}(h^{\wedge}_t) - f_{N,k}(h^{\vee}_t)\right].$ In the case of symmetric exclusion, $(A)$ does not hold, while when the bias is constant, (B) fails to hold. In the weakly asymmetric case, when $\alpha\ne 1/2$, the reader can check by combining Propositions \[prop:lidro\] and \[lerimo\] that (B) holds until time $4\alpha$ (in the macroscopic time-scale) after which $g_{\alpha}(t,\cdot)$ stops to display a local maximum in the interval $(l_{\alpha}(t),r_{\alpha}(t))$ and $f_{N,k}(h^{\wedge}_t) - f_{N,k}(h^{\vee}_t)$ starts to decay much faster than its average. The case $\alpha \ne1/2$: scaling limit for the boundary processes ------------------------------------------------------------------ In order to obtain a sharp upper-bound for $\alpha \ne 1/2$, we rely on a scaling limit result in order to control the value of $f_{N,k}(h^{\wedge}_t)-f_{N,k}(h^{\vee}_t)$ up to a time close to the mixing time, and then we use the contractive estimate to couple $h^{\wedge}_t$ with $h^{\vee}_t$. Note that Proposition \[prop:lidro\] is not sufficient to estimate $f_{N,k}(h^{\wedge}_t)$: we also need a control on the positions of the left-most particle and right-most empty site in our particle configuration. In the case when $\alpha=0$ and the bias is of order $\log N/N$ or smaller (this is possible when is satisfied and $k$ grows slower than any power of $N$), we need an additional step, based on diffusion estimates, to couple the two processes. In this last case also, the factor $N^{-1}$ in causes some difficulty. For that reason we use $f^{(0)}_{N,k}$ and instead of $f_{N,k}$ and : observe that $\delta_{\min}(f^{(0)}_{N,k})={\lambda}^{\frac{k-N}{2}}$. Let us define $[L_N(t),R_N(t)]$ to be the interval on which $h^{\wedge}_t$ and $\vee$ differ. More explicitly, we set $$\begin{split}\label{def:lknrkn} L_N(t)&:=\max\{x \ : \ h^{\wedge}(t,x)=-x\},\\ R_N(t)&:=\min\{x \ : \ h^{\wedge}(t,x)= x-2(N-k)\}, \end{split}$$ or equivalently $L_N(t):=\ell_N(\eta^{\wedge}_t)-1$ and $R_N(t):=r_N(\eta^{\wedge}_t)$. We let $\ell_\alpha$ and $r_{\alpha}$ denote the most likely candidates for the scaling limits of $L_N$ and $R_N$ that can be inferred from the hydrodynamic behavior of the system (cf. Proposition \[prop:lidro\]): $$\begin{split} \ell_\alpha(t)&=\begin{cases} 0 \quad &\text{ if } t\le \alpha\;,\\ (\sqrt{t}-\sqrt{\alpha})^2 \quad &\text{ if } t\in \left(\alpha, (\sqrt{\alpha}+\sqrt{1-\alpha})^2 \right)\;,\\ 1-\alpha \quad &\text{ if } t\ge (\sqrt{\alpha}+\sqrt{1-\alpha})^2\;, \end{cases} \\ r_{\alpha}(t)&=\begin{cases} 1 \quad &\text{ if } t\le 1-\alpha\;,\\ 1-(\sqrt{t}-\sqrt{1-\alpha})^2 \quad &\text{ if } t\in \left(1-\alpha, (\sqrt{\alpha}+\sqrt{1-\alpha})^2 \right)\;,\\ 1-\alpha \quad &\text{ if } t\ge (\sqrt{\alpha}+\sqrt{1-\alpha})^2\;. \end{cases}\end{split}$$ We prove that $\ell_{\alpha}$ and $r_{\alpha}$ are indeed the scaling limits of $L_N$ and $R_N$. \[lerimo\] If holds and $k_N/N\to \alpha$ then for every $t>0$ we have the following convergences in probability $$\lim_{N\to \infty} \frac{1}{N} L_N\left( b^{-1}_NNt \right)= \ell_\alpha(t),\qquad\lim_{N\to \infty} \frac{1}{N} R_N\left( b_N^{-1} Nt \right)= r_\alpha(t).$$ The assumption is optimal for the above result to hold. To see this, the reader can check that when fails, at equilibrium $\ell_N$ and $r_N$ are typically at a macroscopic distance from $(1-\alpha)N$. The proof of Proposition \[lerimo\] is presented in the next subsections. Let us now check that it yields the right bound on mixing time. First, notice that the inequalities still hold with $f_{N,k}$ replaced by $f_{N,k}^{(0)}$ since the latter is also a strictly increasing function in the sense of . Next observe that Proposition \[lerimo\] allows an acute control on the quantity $$\frac{f^{(0)}_{N,k}(h^{\wedge}_t)-f^{(0)}_{N,k}(h^{\vee}_t)}{\delta_{\min} (f^{(0)}_{N,k})}\;.$$ We summarize the argument in a lemma. \[lem:difrence\] Set $D_N(\zeta):= \max\left(|L_N(\zeta)-N+k|, |R_N(\zeta)-N+k|\right)$. We have for $\zeta' \ge \zeta$ $$\frac{f^{(0)}_{N,k}(\zeta')-f^{(0)}_{N,k}(\zeta)}{\delta_{\min} (f^{(0)}_{N,k})}\le Nk {\lambda}^{D_N(\zeta')}.$$ We assume that $\zeta' \ne \zeta$. Then, $$\label{linex} \frac{\lambda^{\frac12 \zeta'(x)} - \lambda^{\frac12 \zeta(x)}}{{\lambda}-1} = \sum_{n=0}^{\frac{\zeta'(x)-\zeta(x)}{2}-1} \lambda^{\frac12\zeta(x)+n}\\ \le \lambda^{\frac{\zeta'(x)}{2}} \frac{\left(\zeta'(x)-\zeta(x)\right)}{2}.$$ Now for $x\le L_N(\zeta')$ or $x\ge R_N(\zeta')$ we necessarily have $\zeta(x)=\zeta'(x)=\vee(x)$. For $x\in {\llbracket}L_N(\zeta')+1,R_N(\zeta')-1{\rrbracket}$, the fact that $\zeta'$ is $1$-Lipschitz yields $$\zeta'(x)\le k-N+2 D_N(\zeta').$$ Recall that $\delta_{\min} (f^{(0)}_{N,k}) = {\lambda}^{(k-N)/2}$. Hence one obtains from $$\begin{aligned} \frac{f^{(0)}_{N,k}(\zeta')-f^{(0)}_{N,k}(\zeta)}{\delta_{\min} (f^{(0)}_{N,k})} &\le \sum_{x=1}^N \lambda^{\frac{\zeta'(x)-(k-N)}{2}} \frac{\left(\zeta'(x)-\zeta(x)\right)}{2} \\ &\le \lambda^{D_N(\zeta')} \sum_{x=1}^N \frac{\left(\zeta'(x)-\zeta(x)\right)}{2}\le \lambda^{D_N(\zeta')} Nk.\end{aligned}$$ In the last inequality we simply used that $\zeta'(x)-\zeta(x)\le 2k$ (there are at most $k$ sites where the increment of $\zeta'$ is larger than that of $\zeta$). We can now apply Proposition \[lerimo\] to obtain an estimate on the mixing time. For convenience we treat the case of smaller bias separately. ### Proof of Proposition \[lapropo\] when $b_N\gg (\log N)/ N$ {#preuv1} We assume that $$\label{hypo} \lim_{N\to \infty}(b_N N)/\log N=\infty.$$ We consider first the system at time $t_0(N):= (\sqrt{\alpha}+\sqrt{1-\alpha})^2 N b^{-1}_N$. From Proposition \[lerimo\], we know that at time $t_0$, $L_N$ and $R_N$ are close to their equilibrium positions: we have for $N$ sufficiently large and arbitrary $\delta, {\varepsilon}>0$ $$\begin{gathered} \label{eq:lax} {\mathbb{P}}[ L_N(t_0)\ge N-k-(\delta/20) N \ ; \ R_N(t_0)\le N-k+(\delta/20) N ]\\ =: {\mathbb{P}}[{\mathcal{A}}_N] \ge 1-({\varepsilon}/2).\end{gathered}$$ We let ${\mathcal{F}}_t$ denote the canonical filtration associated with the process. For $t\ge t_0$, repeating starting at time $t_0$ for $f^{(0)}_{N,k}$ and combining it with , we obtain that $$\label{eq:lux} {\mathbb{P}}[ \tau>t \ | \ {\mathcal{F}}_{t_0}]\le e^{-\varrho(t-t_0)}\frac{ f^{(0)}_{N,k}(h^{\wedge}_{t_0})-f^{(0)}_{N,k}(h^{\vee}_{t_0})}{\delta_{\min}(f^{(0)}_{N,k})}\;.$$ Note that on the event ${\mathcal{A}}_N$, we have $D_N(h^{\wedge}_{t_0})\le \delta N/20$. Thus using Lemma \[lem:difrence\] to bound the r.h.s. we obtain $$\label{eq:lox} {\mathbb{E}}\left[\frac{ f^{(0)}_{N,k}(h^{\wedge}_{t_0})-f^{(0)}_{N,k}(h^{\vee}_{t_0})}{\delta_{\min}(f^{(0)}_{N,k})} \ \Big| \ {\mathcal{A}}_N\right] \le kN{\lambda}^{\frac{\delta N}{20}}.$$ Hence averaging on the event ${\mathcal{A}}_N$ one obtains we obtain $${\mathbb{P}}( \tau>t )\le {\varepsilon}/2+ {\mathbb{P}}[ \tau>t \ | \ {\mathcal{A}}_N]\le {\varepsilon}/2 + e^{-\varrho(t-t_0)}kN{\lambda}^{\frac{\delta N}{20}}.$$ For $t=t_\delta=t_0+ \delta b_N^{-1} N$, replacing $\varrho$ and $\log {\lambda}$ by their equivalents given in and , one can check that for $N$ sufficiently large one has $$\label{oazt} N k {\lambda}^{\frac{\delta N}{20}} e^{-\varrho (t_{\delta}-t_0)} \le N k e^{-\frac{\delta N b_N}{20}}\le {\varepsilon}/2,$$ where the last inequality is valid for $N$ sufficiently large provided that holds. ### Proof of Proposition \[lapropo\]: the general case {#preuv2} If we no longer assume that holds, then an additional step is needed in order to conclude: this step relies on diffusion estimates proved in Appendix \[sec:diffu\]. From and , for any ${\varepsilon}, {\delta}>0$ we have for $N$ sufficiently large (recall ) $$\begin{gathered} {\mathbb{E}}\left[\frac{ f^{(0)}_{N,k}(h^\wedge_{t_{\delta/2}}) - f^{(0)}_{N,k}(h^\vee_{t_{\delta/2}})}{\delta_{\min}(f^{(0)}_{N,k})} \ \Big| \ {\mathcal{A}}_N \right]\\ \le e^{-\varrho(t_{\delta/2}-t_0)}{\lambda}^{ \frac{\delta N}{20}} k N \le e^{-\frac{\delta N b_N}{40}} k N\le e^{-\frac{\delta N b_N}{50}} N,\end{gathered}$$ where the second inequality relies on the the asymptotic equivalence in and the last one on .\ Now we can conclude using Proposition \[prop:solskjaer\]-(i) with $a:= 4{\varepsilon}^{-1} e^{-\delta N b_N/50} N$ and $$(M_{s})_{s\ge 0}:= \left(\frac{f^{(0)}_{N,k}(h^\wedge_{t_{\delta/2}+s})-f^{(0)}_{N,k}(h^\vee_{t_{\delta/2}+s})}{\delta_{\min}(f^{(0)}_{N,k})}\right)_{s\ge 0}.$$ Indeed $M_{s}$ is a non-negative supermartingale whose jumps are of size at least $1$ (recall that we have divided the weighted area by $\delta_{\min}(f^{(0)}_{N,k})$ in the definition of $M$). Furthermore, up to the merging time $\tau$, the two interfaces $h^\wedge$ and $h_\vee$ differ on some interval: on this interval $h^\wedge$ makes an upward corner ($\Delta h^\wedge < 0$) and $h^\vee$ makes a downward corner ($\Delta h^\vee > 0$). Consequently, the jump rate of $M$ is at least $1$ up to its hitting time of $0$. From Markov’s inequality we have (recall ) $${\mathbb{P}}[M_0 > a]\le {\mathbb{P}}[{\mathcal{A}}^{{\complement}}_N]+ a^{-1} {\mathbb{E}}[M_0 \ | \ {\mathcal{A}}_N ] \le 3{\varepsilon}/4.$$ Setting $r_{\delta}:= (\delta/2) N b^{-1}_N$ and applying , we have for all $N$ sufficiently large $${\mathbb{P}}[M_{r_{\delta}}>0 \ | \ M_0\le a ]\le 4 a (r_\delta)^{-1/2}\le \frac{16 {\varepsilon}^{-1}}{\sqrt{\delta/2}} \sqrt{N b_N} e^{-\frac{\delta N b_N}{50}}\le {\varepsilon}/4\;,$$ where the last inequality comes from the fact that $Nb_N$ diverges. Hence we conclude by observing that for $N$ sufficiently large $${\mathbb{P}}[\tau>t_{\delta}] ={\mathbb{P}}[M_{r_{\delta}}>0] \le {\mathbb{P}}[M_0 > a]+{\mathbb{P}}[M_{r_{\delta}}>0 \ | \ M_0\le a ]\le {\varepsilon}.$$ An auxiliary model to control the speed of the right-most particle {#sec:auxi} ------------------------------------------------------------------ Our strategy to prove Proposition \[lerimo\] is to compare our particle system with another one on the infinite line, for which a stationary probability exists. We consider $n$ particles performing the exclusion process on the infinite line with jump rate $p$ and $q$ and we add a “slower” $n+1$-th particle on the right to enforce existence of a stationary probability for the particle spacings. To make the system more tractable this extra particle is only allowed to jump to the right (so that it does not feel the influence of the $n$ others). Note that in our application, the number of particles $n$ does not necessarily coincide with $k$.\ The techniques developed in this section present some similarities to those used for the constant bias case in [@LabLac16 Section 6], but also present several improvements, the main conceptual change being the addition of a slow particle instead of modifying the biases in the process. This novelty presents two advantages: Firstly it considerably simplifies the computation since martingale concentration estimates are not needed any more. Secondly this allows to obtain control for the whole large bias regime , something that cannot be achieved even by optimizing all the parameters involved in [@LabLac16 Section 6]. More formally we consider a Markov process $(\hat \eta(t))_{t\ge 0}$ on the state space $$\Theta_n:= \{ \xi\in {\mathbb{Z}}^{n+1} \ : \ \xi_1<\xi_2<{\ifmmode\mathinner{\ldotp\kern-0.2em\ldotp\kern-0.2em\ldotp}\else.\kern-0.13em.\kern-0.13em.\fi}<\xi_{n+1}\}.$$ The coordinate $\hat \eta_i(t)$ denotes the position of the $i$-th leftmost particle at time $t$. The dynamics is defined as follows: the first $n$ particles, $\hat \eta_{i}(t)$, $i\in {\llbracket}1,n{\rrbracket}$ perform an exclusion dynamics with jump rates $p$ to the right and $q$ to the left while the last one $\hat \eta_{n+1}(t)$ can only jump to the right and does so with rate ${\beta}b={\beta}(p-q)$, for some $\beta<1$. We assume furthermore that initially we have $\hat \eta_{n+1}(0)=0$. The initial position of the other particles is chosen to be random in the following manner. We define $$\label{ladef} \mu_i:={\beta}+{\lambda}^{-i}(1-{\beta}).$$ and we assume that the spacings $\left( \hat \eta_{i+1}(0)-\hat \eta_{i}(0) \right)_{i=1}^n$ are independent with Geometric distribution $$\label{geom} {\mathbb{P}}\left[ \hat \eta_{i+1}(0)-\hat \eta_i(0)= m \right]=(1-\mu_i)\mu_i^{m-1}\;,\quad m\ge 1\;.$$ Our aim is to prove the following control on the position of the first particle in this system, uniformly in ${\beta}$ and $n$. In Subsections \[Subsec:alpha0\] and \[Subsec:alphaNon0\], we use this result in order to control the position of $L_N(t)$. \[deviats\] We have, $$\begin{gathered} \label{eq:woof} \lim_{A\to \infty}\sup_{t\ge 1, n\in {\mathbb{N}}, {\beta}\in(0,1)} {\mathbb{P}}\bigg[\hat \eta_1(t) \le t{\beta}b\\ - A \left( \sqrt{bt}+ \frac{1}{1-{\beta}}\left[n+ b^{-1} \log \min(n,b^{-1}) \right] \right) \bigg]= 0. \end{gathered}$$ The statement is not hard to prove, the key point is to observe that the distribution of particle spacings is stationary. \[station\] For all $t\ge 0$, $\left( \hat \eta_{i+1}(t)-\hat \eta_{i}(t) \right)_{i=1}^n$ are independent r.v. with distribution given by . We use the notation $(m_i)_{i=1}^n\in {\mathbb{N}}^n$ to denote a generic element in the configuration space for the process $\left( \hat \eta_{i+1}(t)-\hat \eta_{i}(t) \right)_{i=1}^n$. We need to show that the measure defined above is stationary. A measure $\pi$ is stationary if and only if we have $$\begin{aligned} & p \pi(m_1+1,{\ifmmode\mathinner{\ldotp\kern-0.2em\ldotp\kern-0.2em\ldotp}\else.\kern-0.13em.\kern-0.13em.\fi},m_n) \\ +\;&\sum_{i=1}^{n-1}\left[p\pi({\ifmmode\mathinner{\ldotp\kern-0.2em\ldotp\kern-0.2em\ldotp}\else.\kern-0.13em.\kern-0.13em.\fi}, m_i-1,m_{i+1}+1,{\ifmmode\mathinner{\ldotp\kern-0.2em\ldotp\kern-0.2em\ldotp}\else.\kern-0.13em.\kern-0.13em.\fi})+q\pi({\ifmmode\mathinner{\ldotp\kern-0.2em\ldotp\kern-0.2em\ldotp}\else.\kern-0.13em.\kern-0.13em.\fi},m_{i-1}+1,m_i-1,{\ifmmode\mathinner{\ldotp\kern-0.2em\ldotp\kern-0.2em\ldotp}\else.\kern-0.13em.\kern-0.13em.\fi})\right]{\mathbf{1}}_{\{m_i\ge 2\}}\\ +\;&q\pi({\ifmmode\mathinner{\ldotp\kern-0.2em\ldotp\kern-0.2em\ldotp}\else.\kern-0.13em.\kern-0.13em.\fi},m_{n-1}+1,m_n-1) {\mathbf{1}}_{\{m_n\ge 2\}} +{\beta}b \pi(m_1,{\ifmmode\mathinner{\ldotp\kern-0.2em\ldotp\kern-0.2em\ldotp}\else.\kern-0.13em.\kern-0.13em.\fi},m_n-1){\mathbf{1}}_{\{m_n\ge 2\}}\\ &= \pi(m_1,{\ifmmode\mathinner{\ldotp\kern-0.2em\ldotp\kern-0.2em\ldotp}\else.\kern-0.13em.\kern-0.13em.\fi},m_n)\left(q + \sum_{i=1}^{n-1} (p+q) {\mathbf{1}}_{\{m_i\ge 2\}}+ p{\mathbf{1}}_{\{m_n\ge 2\}} + {\beta}b \right),\end{aligned}$$ where in the sums, the dots stand for coordinates that are not modified (and $m_{i-1}$ simply has to be ignored when $i=1$). If we assume that $\pi$ is the product of geometric laws with respective parameters $\mu_i$ (not yet fixed) then the equation above is equivalent to the system $$\label{dasistem} \begin{cases} p\mu_1=q+ {\beta}(p-q), &\\ q \frac{\mu_{i-1}}{\mu_i}+p \frac{\mu_{i+1}}{\mu_i}=p+q, &\quad \forall i\in {\llbracket}1, n-1 {\rrbracket},\\ q\frac{\mu_{n-1}}{\mu_n}+\frac{{\beta}(p-q)}{\mu_n}=p.& \end{cases}$$ where we have taken the convention $\mu_0=1$. One can readily check that $\mu_i$ given by satisfies this equation. Note that the equations can be obtained directly simply by using the fact that the expected drifts of the particles starting from the geometric distributions are given by $p\mu_i-q \mu_{i-1}$ for the $i$-th particle $i\in {\llbracket}1,n {\rrbracket}$ and ${\beta}(p-q)$ for the $n+1$-th particle, and that stationarity implies that the drifts are all equal. However, the proof is necessary to show that this condition is also a sufficient one. Starting from stationarity allows us to control the distance between the first and last particle at all time. In particular we have $$\label{staz}\begin{split} {\mathbb{E}}\left[ \hat \eta_{n+1}(t)-\hat \eta_{1}(t)\right]&={\mathbb{E}}\left[ \hat \eta_{n+1} (0)-\hat \eta_{1}(0)\right]= \sum_{i=1}^n \frac{1}{1-\mu_i} =\frac{1}{1-{\beta}} \sum_{i=1}^n \frac{1}{1-{\lambda}^{-i}}\\ &\le \frac{1}{1-{\beta}} \left(n+ \frac{C}{{\lambda}-1}\log \left(\min( n, |{\lambda}-1|^{-1} ) \right)\right), \end{split}$$ for some universal constant $C$. By union bound, the probability in the l.h.s. of is smaller than $$\begin{gathered} {\mathbb{P}}\left[\hat \eta_{n+1}(t) \le t{\beta}b- A \sqrt{bt}\right]\\+ {\mathbb{P}}\left[\hat \eta_{n+1}(t) -\hat \eta_1(t)\ge \frac{A}{1-{\beta}}\left[n+b^{-1} \log \min(n,b^{-1})\right]\right].\end{gathered}$$ The first term is small because the expectation and the variance of $\hat \eta_{n+1}(t)$ are equal to $t{\beta}b$. The second can be shown to be going to zero with $A$ using and Markov’s inequality for $\hat \eta_{n+1}(t)-\hat\eta_1(t)$. Proof of Proposition \[lerimo\] in the case $\alpha=0$ {#Subsec:alpha0} ------------------------------------------------------ We restate and prove the result in this special case (observe that the result for $R_N$ is trivial for $\alpha=0$). Assume that $\alpha=0$ and holds. We have for any $C>0$ and any ${\varepsilon}>0$ $$\lim_{N\to \infty} \sup_{t\in [0,C b_N^{-1} N]} {\mathbb{P}}\left[ |L_N(t)- b_N t|\ge {\varepsilon}N \right]=0.$$ First let us remark that the convergence $$\lim_{N\to \infty} \sup_{t\in [0,C b_N^{-1} N]} {\mathbb{P}}\left[ L_N(t)\ge b_N t + {\varepsilon}N \right]=0$$ follows from the fact that the first particle is stochastically dominated by a simple random walk with bias $b_N \gg N^{-1}$ on the segment, starting from position $1$. It remains to prove that $$\label{srup} \lim_{N\to \infty} \sup_{t\in [0,C b_N^{-1} N]} {\mathbb{P}}\left[ L_N(t)\le b_N t- {\varepsilon}N \right]=0.$$ We provide the details for the most important case $C=1$, and then we briefly explain how to deal with the case $C>1$.\ We couple $\eta^{\wedge}(t)$ with the system $\hat \eta(t)$ of the previous section, choosing $n=k$ and ${\beta}=1-({\varepsilon}/2)$. The coupling is obtained by making the $i$-th particle in both processes try to jump at the same time (for $i\in {\llbracket}1, k {\rrbracket}$) with rate $p$ and $q$, and rejection of the moves occurs as consequences of the exclusion rule or boundary condition (for $\eta^{\wedge}$ only). Initially of course we have $$\label{initial} \forall i \in {\llbracket}1, n {\rrbracket}, \quad \eta^{\wedge}_{i}(0)\ge \hat \eta_i(0).$$ because of the choice of the initial condition for $\hat \eta$ (recall that by definition $\eta^{\wedge}_i(0)=i$). The boundary at zero, and the presence of one more particle on the right in $\hat \eta$ gives $\eta^\wedge$ only more pushes towards the right, so that the ordering is preserved at least until $\hat \eta_{n+1}$ reaches the right side of the segment and the effect of the other boundary condition starts to be felt: $$\eta^\wedge_{i}(t)\ge \hat \eta_i(t),\quad \forall i \in {\llbracket}1, n {\rrbracket}, \forall t\le {\mathcal{T}}$$ where ${\mathcal{T}}:= \inf \{t \ge 0 \ : \ \hat \eta _{n+1}(t)=N+1 \}$. Using the assumption , a second moment estimate and the fact that ${\beta}<1$, we have $$\lim_{N\to \infty} {\mathbb{P}}[ {\mathcal{T}}\le b^{-1}_N N] =0,$$ and hence $$\lim_{N\to \infty} \sup_{t\le b^{-1}_N N} {\mathbb{P}}\left[\eta^\wedge_1(t)\le \hat \eta_1(t)\right]=0.$$ Therefore, it suffices to control the probability of $\hat{\eta}_1(t) \le b_N t - {\varepsilon}N$. Observe that the assumptions $(k_N/N)\to 0$ and imply that for any given $A>0$, for all $N$ sufficiently large and for any $t\le b^{-1}_N N$ we have $$\left( \sqrt{b_N t}+ \frac{1}{1-{\beta}}\left[k_N+b_N^{-1}\log\min(k_N , b^{-1}_N) \right] \right)\le {\varepsilon}\frac{N}{2 A}.$$ Furthermore $(1-\beta)b_Nt \le {\varepsilon}N/ 2$ for all $t\in [0,b_N^{-1} N]$. Thus applying Proposition \[deviats\] we obtain that for $N$ sufficiently large $$\begin{gathered} \sup_{t\in [0,b_N^{-1} N]} {\mathbb{P}}\left[ \hat \eta_1(t)\le b_N t-{\varepsilon}N\right] \\ \le \sup_{t\in [0,b_N^{-1} N]} {\mathbb{P}}\left[ \hat \eta_1(t)\le {\beta}b_N t- A\left( \sqrt{b_N t}+ \frac{k_N+b_N^{-1}\log\min(k , b^{-1}_N)}{1-{\beta}}\right) \right]\le \delta.\end{gathered}$$ where $\delta$ can be made arbitrarily small by choosing $A$ large. This concludes the proof of for $C=1$. To treat the case $C>1$, it suffices to shift the particle system $\hat{\eta}(0)$ to the left by $\lfloor (C-1)b_N^{-1} N \rfloor$ and to apply the same arguments as before so we omit the details. Proof of Proposition \[lerimo\] in the case $\alpha\in(0,1)$ {#Subsec:alphaNon0} ------------------------------------------------------------ The roles of $L_N$ and $R_N$ being symmetric, we only need to prove the result for $L_N$ (but we do not assume here that $\alpha\le 1/2)$. A direct consequence of Proposition \[prop:lidro\] is that for all $s\in {\mathbb{R}}$ and ${\varepsilon}>0$ we have $$\lim_{N\to \infty}{\mathbb{P}}\left[ \eta^{\wedge}_1(b_N^{-1} N s ) \ge N\left(\ell_{\alpha}(s)+{\varepsilon}\right)\right] = 0.$$ Hence to conclude we want to prove that $$\label{toprove} \lim_{N\to \infty}{\mathbb{P}}\left[ \eta^{\wedge}_1(b_N^{-1} N s ) \le N\left(\ell_{\alpha}(s)-{\varepsilon}\right) \right]=0.$$ For the remainder of the proof $s$ and ${\varepsilon}$ are considered as fixed parameters. We set $\delta \in (0,\alpha)$, and $n=\lceil \delta N \rceil$. To prove , we are going to compare $(\eta^{\wedge}_i)_{i=1}^n$ to the particle system considered in Section \[sec:auxi\]. First we observe that as a consequence of Proposition \[prop:lidro\], we have, for any $T>0$ $$\label{camarch} \lim_{N \to \infty} {\mathbb{P}}\left[ \exists t\in [0, T], \ \eta^{\wedge}_{n+1}(b^{-1}_N Nt)\le N \ell_{\alpha}(t) \right]=0$$ We define the process $\hat \eta$ as in Section \[sec:auxi\] with ${\beta}=1-{\varepsilon}/(2s)$ but with a shifted initial condition. More precisely we set $$\hat \eta_{n+1}(0)= N\left(\ell_{\alpha}(s)-s \right)\le 0,$$ and choose the initial particle spacings to be independent and with geometric distributions given by . As is satisfied, we can couple the two processes in such a way that $$\forall i \in {\llbracket}1, n {\rrbracket}\;, \forall t\le {\mathcal{T}}', \quad \eta^{\wedge}_{i}(t)\ge \hat \eta_i(t),$$ where ${\mathcal{T}}':= \inf\{ t \ : \hat \eta_{n+1}(t)=\eta^{\wedge}_{n+1}(t)\}$. It is a simple exercise to show that for every $T>0$ the position of $\hat \eta_{n+1}$ satisfies the following law of large numbers $$\lim_{N\to \infty}{\mathbb{P}}\left[ \sup_{t\in [0,T]} \left|\frac{\hat \eta_{n+1}(b_N^{-1}N t)}{N}-\left(\ell_{\alpha}(s)-s \right)-{\beta}t\right|\ge \kappa \right]=0,\quad \forall \kappa >0\;,$$ which, combined with , yields $$\lim_{N\to \infty} {\mathbb{P}}[{\mathcal{T}}' \ge b^{-1}_N N s]=1.$$ and thus we only need to prove with $\eta^{\wedge}_1$ replaced by $\hat \eta_{1}$. More precisely we prove that given $\kappa>0$, one can find $\delta$ sufficiently small such that $$\label{provex} {\mathbb{P}}\left[ \hat\eta_1(b_N^{-1} N s ) \le N\left(\ell_{\alpha}(s)-{\varepsilon}\right) \right]\le \kappa.$$ Using Proposition \[deviats\] for $t=b^{-1}_N N s$ and $A= \delta^{-1/2}$ and taking into account the new initial condition, the probability of the event $$\left\{ \hat \eta_1(b^{-1}_N N s)\le N\left( \ell_{\alpha}(s)-\frac{{\varepsilon}}{2} \right)- \delta^{-1/2}\left( \sqrt{Ns}+ \frac{2s}{{\varepsilon}}[\delta N + b^{-1}_N \log b^{-1}_N] \right) \right\}$$ has a probability which can be made arbitrarily small if $\delta$ is chosen sufficiently small. We can then conclude that holds by observing that for $\delta$ sufficiently small and $N$ sufficiently large $$\delta^{-1/2}\left( \sqrt{Ns}+ \frac{2s}{{\varepsilon}}[\delta N + b^{-1}_N \log b^{-1}_N] \right)\le {\varepsilon}N/2.$$ Lower bound on the mixing time for small biases {#Sec:LBSB} =============================================== Until the end of the section, we assume that the small bias assumption holds. Let us set $s_{\delta}(N):= (1-\delta) \log k / (2\operatorname{\mathrm{gap}}_N)$. We show that at time $s_{\delta}$, equilibrium is not reached if one starts from one of the extremal conditions (some moderate efforts allow to replace $\max$ by $\min$ in the statement of the proposition). \[lbsb\] When assumption holds, we have $$\lim_{N\to \infty} \max_{\zeta\in \{\vee,\wedge\} } \| {\mathbb{P}}( h^{\zeta}_{s_{{\delta}(N)}}\in \cdot )-\pi_{N,k}\|_{TV}=1$$ As a consequence for every ${\varepsilon}\in (0,1)$, ${T_{\rm mix}}^{N,k}({\varepsilon})\ge s_{\delta}(N)$ for $N$ sufficiently large. The method to obtain a lower bound on the mixing time for small biases is similar to the one used in the symmetric case (see [@Wil04 Section 3.3]), and is based on the control of the two first moments of $f_{N,k}(h^{\wedge}_t)-f_{N,k}(\zeta)$ where $\zeta$ is independent of $h^\wedge_t$ and distributed according to $\pi_{N,k}$: if at time $t$ the mean of $f_{N,k}(h^\wedge_t)-f_{N,k}(\zeta)$ is much larger than its standard deviation, then the system is not at equilibrium (cf [@LevPerWil Proposition 7.12]). We present estimates for the first two moments that we prove at the end of the section. This first moment bound is elementary. \[lem:firstmom\] We have $$f_{N,k}(\wedge)-f_{N,k}(\vee)\ge \frac{1}{8} {\lambda}^{(k-N)/2} Nk\;,$$ and as a consequence, for every $t\ge 0$ $$\label{lunoulot} \max( f_{N,k}(\wedge), -f_{N,k}(\vee)) \ge \frac{1}{16} {\lambda}^{(k-N)/2} Nk\;.$$ The second moment estimates rely on the control of a martingale bracket. \[Lemma:BoundVar\] For all $t\ge 0$, $N\ge 1$ and all $k\in {\llbracket}1, N/2{\rrbracket}$ we have $${{\rm Var}}(f_{N,k}(h_t^\wedge)\le \frac{k \lambda^k}{2 \operatorname{\mathrm{gap}}_N},$$ The same bound holds for ${{\rm Var}}(f_{N,k}(h_t^\vee))$ and ${{\rm Var}}_{\pi_{N,k}}(f_{N,k})$. Let us assume for simplicity (recall ) that $$f_{N,k}(\wedge)\ge \frac{1}{16} {\lambda}^{(k-N)/2} Nk.$$ (if not we apply the same proof to $f_{N,k}(\vee)\le -\frac{1}{16} {\lambda}^{(k-N)/2} Nk$). By the material in Section \[sec:eigen\], we have $$\label{cleup} {\mathbb{E}}\left[ f_{N,k}(h^\wedge_t)\right] \ge \frac{1}{16} e^{-\operatorname{\mathrm{gap}}_N t} {\lambda}^{(k-N)/2} Nk.$$ Applying [@LevPerWil Proposition 7.12] for the probability measures $P_t^{N,k}(\wedge, \cdot)$ and $\pi_{N,k}$ and the function $f_{N,k}$ (recall that $E_{\pi_{N,k}}[f_{N,k}]=0$), we obtain that $$\|P_t^{N,k}(\wedge, \cdot)-\pi_{N,k} \|_{TV} \ge 1-\frac{2 \left({{\rm Var}}(f_{N,k}(h_t^\wedge))+{{\rm Var}}_{\pi_{N,k}}(f_{N,k}) \right)}{{\mathbb{E}}\left[f_{N,k}(h_t^\wedge)\right]^2}\;.$$ Using Lemma \[Lemma:BoundVar\] and , we obtain that $$\frac{{{\rm Var}}(f_{N,k}(h_t^\wedge))+{{\rm Var}}_{\pi_{N,k}}(f_{N,k})}{{\mathbb{E}}\left[f_{N,k}(h_t^\wedge)\right]^2}\le \frac{ 16^2 e^{2\operatorname{\mathrm{gap}}_N t} {\lambda}^N}{\operatorname{\mathrm{gap}}_N N^2 k}.$$ Now if we apply this inequality at time $s_{\delta} = (1-\delta) \log k / (2\operatorname{\mathrm{gap}}_N)$, then we obtain for any given ${\varepsilon}> 0$ and all $N$ sufficiently large $$d(t_1)\ge 1-2\frac{16^2 {\lambda}^N }{k^{\delta}\operatorname{\mathrm{gap}}_N N^2}\ge 1-{\varepsilon}.$$ where we used the small bias assumption . This yields ${T_{\rm mix}}^{N,k}({\varepsilon})\ge s_{\delta}$. We have $$\begin{aligned} f_{N,k}(\wedge) - f_{N,k}(\vee) &= \sum_{x=1}^{N-1} \sin \left(\frac{x\pi}{N}\right) \frac{\lambda^{\frac12 \wedge(x)} - \lambda^{\frac12 \vee(x)}}{\lambda-1}\\ &\ge \sum_{x=1}^{N-1} \sin \left(\frac{x\pi}{N}\right) \lambda^{\frac12 \vee(x)} \frac{\wedge(x)-\vee(x)}{2},\end{aligned}$$ where the last inequality is obtained similarly to . Since $\vee(x)\ge k-N$ for all $x$ and $\wedge(x)-\vee(x) \ge k$ for all $x\in \{N/4,\ldots,3N/4\}$, we conclude that $$\sum_{x=1}^{N-1} \sin \left(\frac{x\pi}{N}\right) \lambda^{\frac12 \vee(x)} \frac{\wedge(x)-\vee(x)}{2} \ge \frac{\sqrt 2}{2}{\lambda}^{\frac{k-N}{2}}\frac{N k}{4}.$$ By the material in Section \[sec:eigen\], we know $$M_t := f_{N,k}(h^\wedge_t) e^{\operatorname{\mathrm{gap}}_N t}$$ is a martingale. Its predictable bracket is given by $$\begin{aligned} \langle M_\cdot \rangle_t &= \int_0^t \sum_{x=1}^{N-1} \lambda^{h_s^\wedge(x)} \sin\Big(\frac{\pi x}{N}\Big)^2 e^{2\operatorname{\mathrm{gap}}_N s}\\ &\quad\times\Big(p_N{\mathbf{1}}_{\{\Delta h_s^\wedge(x) < 0\}} {\lambda}^{-2} + q_N {\mathbf{1}}_{\{\Delta h_s^\wedge(x) > 0\}}\Big) ds\;,\end{aligned}$$ and $M_t^2 - \langle M_\cdot \rangle_t$ is again a martingale. This yields the identity $${{\rm Var}}(f_{N,k}(h_t^\wedge)) = e^{-2\operatorname{\mathrm{gap}}_N t}{\mathbb{E}}\big[\langle M_\cdot \rangle_t\big].$$ To bound the predictable bracket of $M$, let us observe that the number of possible particle transitions to the right and to the left (the number of sites $x$ such that $\Delta h_s^\wedge(x)<0$, resp. $>0$) is bounded by $k$, and that for any $x$ and $\zeta\in {\Omega}_{N,k}$ we have $\lambda^{\zeta(x)} \le \lambda^{k}$. Therefore, we obtain the bound $$\begin{aligned} {\mathbb{E}}\big[\langle M_\cdot \rangle_t\big] &\le \int_0^t e^{2\operatorname{\mathrm{gap}}_N s} ds\, \lambda^k \sum_x {\mathbf{1}}_{\{\Delta h_s^\wedge(x) \ne 0\}}\le k\lambda^k \frac{e^{2\operatorname{\mathrm{gap}}_N t}}{2\operatorname{\mathrm{gap}}_N}\;,\end{aligned}$$ which yields the asserted bound. The case of $h^{\vee}_t$ is treated in the same manner by symmetry. Since the distribution of $h^\wedge_t$ converges to $\pi_{N,k}$ when $t$ tends to infinity we deduce that $${{\rm Var}}_{\pi_{N,k}}(f_{N,k}) = \lim_{t\rightarrow\infty} {{\rm Var}}(f_{N,k}(h_t^\wedge))\;,$$ which allows to conclude. Upper bound on the mixing time for small biases {#Sec:UBSB} =============================================== Until the end of the section we assume that the small bias assumption holds and that the different initial conditions are coupled using the monotone grand coupling ${\mathbb{P}}$ defined in Appendix \[Appendix:Coupling\]. We set for all $\delta > 0$ $$t_{\delta}(N):=(1+\delta)\frac{\log k}{2\operatorname{\mathrm{gap}}_N}.$$ Recall the definition of the merging time $\tau$ from . \[oldform\] Assume that holds. We have $$\label{laforme} \lim_{N\to \infty} {\mathbb{P}}[\tau < t_{\delta}(N)]=1.$$ As a consequence, for every ${\varepsilon}>0$ and all $N$ sufficiently large, ${T_{\rm mix}}^{N,k}({\varepsilon})\le t_{\delta}(N)$. Recall (see the paragraph after ) that $h^{\pi}_t$ denotes the chain with stationary initial condition. For practical reasons, it is simpler to couple two processes when at least one of them is at equilibrium. We thus prove by showing that $\lim_{N\to \infty}{\mathbb{P}}[\tau_i < t_{\delta}(N)]=1$ for $i\in \{1,2\}$ where $$\tau_1:=\inf\{ t>0 \ : \ h^{\wedge}_t=h^{\pi}_t \}\quad \text{ and }\quad \tau_2:=\inf\{ t>0 \ : \ h^{\vee}_t=h^{\pi}_t \}.$$ The argument being completely symmetric, we focus only on $\tau_1$. As in Sections \[preuv1\] and \[preuv2\], we interpret $\tau_1$ as the time at which the weighted area $A_t$ between the maximal and equilibrium interface vanishes $$\label{defAt} A_{t}:= \frac{f^{(0)}_{N,k}(h^{\wedge}_t)-f^{(0)}_{N,k}(h^{\pi}_t)}{\delta_{\min}(f^{(0)}_{N,k})} ={\lambda}^{\frac{N-k}{2}}\sum_{x=1}^{N-1} \frac{\lambda^{\frac12 h_t^\wedge(x)}-\lambda^{\frac12 h_t^\pi(x)}}{\lambda - 1} \;.$$ A simple computation based on the identity shows that $A$ is a supermartingale. While in the large bias case (Section \[Sec:UBLB\]) the choice of the grand coupling does not matter, here it is crucial to use a coupling which maximizes in a certain sense the fluctuation of the weighted area $A_t$, so that this process reaches zero as quickly as possible. The coupling defined in Appendix \[Appendix:Coupling\] makes the transitions for the two processes $h^{\wedge}$ and $h^{\pi}$ as independent as possible (some transitions must occur simultaneously for the two processes in order to preserve monotonicity). We consider $\eta >0$ small and introduce the successive stopping times ${\mathcal{T}}_i$ by setting $${\mathcal{T}}_0 := \inf\big\{t\ge t_{\delta/2}: A_t \le k^{\frac12 - \frac {\delta}5} N\big\}\;,$$ and $${\mathcal{T}}_{i} := \inf\big\{t\ge {\mathcal{T}}_{i-1}: A_t \le k^{\frac12 - i\eta - \frac {\delta}5} N\big\}\;,\quad i\ge 1\;.$$ We also set for coherence ${\mathcal{T}}_{\infty}:=\max(\tau_1,t_{\delta/2})$ the first time at which $A_t$ reaches $0$. Notice that some of these stopping times may be equal to $t_{\delta/2}$.\ Set $T_N:=\min( b^{-2}_N, N^2)$. To prove Proposition \[oldform\], we show first that $A_t$ shrinks to $k^{\frac12-\frac {\delta}5} N$ by time $t_{\delta/2}$ and then that it only needs an extra time $2T_N$ to reach $0$. The second step is performed by controlling each increment ${\Delta}{\mathcal{T}}_i:={\mathcal{T}}_i-{\mathcal{T}}_{i-1}$ separately for each $i$ smaller than some threshold $K:=\lceil 1/(2\eta)\rceil$. \[newform\] Given $\delta$, if $\eta$ is chosen small enough and $K:=\lceil 1/(2\eta)\rceil$, we have $$\lim_{N\to \infty}{\mathbb{P}}\left( \{ {\mathcal{T}}_{0} = t_{\delta/2}\}\cap \left(\bigcap_{i=1}^K \{{\Delta}{\mathcal{T}}_{i}\le 2^{-i} T_N\}\right)\cap\{{\mathcal{T}}_{\infty}-{\mathcal{T}}_{K}\le T_N \}\right)=1 \;.$$ Note that on the event defined in the lemma and for all $N$ large enough, we have $$\tau_1\le {\mathcal{T}}_{\infty}\le t_{\delta/2}+2T_N\le t_{\delta}.$$ Hence Proposition \[oldform\] follows as a direct consequence. The bound on ${\mathcal{T}}_0$ is proved in Section \[s:contract\], while that of on ${\mathcal{T}}_{\infty}-{\mathcal{T}}_{K}$ follows from Lemma \[Lemma:AN\] in Section \[s:dif\], the case of the other increments is more delicate and is detailed in Section \[s:inter\]. Contraction estimates {#s:contract} --------------------- The approach used in the first step bears some similarity with the one used in Section \[preuv1\], the notable difference being that is not sufficient here and we must work a bit more to show that ${\mathbb{E}}[A_t]$ decays with rate $\operatorname{\mathrm{gap}}_N$. \[Lemma:T0\] Given $\delta>0$ we have ${\mathbb{P}}\big( {\mathcal{T}}_0 > t_{\delta/2}\big) \to 0$ as $N\to\infty$. Note that $a(t,x):= {\mathbb{E}}\left[\frac{\lambda^{\frac12 h_t^\wedge(x)}-\lambda^{\frac12 h_t^\pi(x)}}{\lambda - 1} \right]$ is a solution of the equation $$\partial_t a = (\sqrt{pq}\, {\Delta}-\varrho)a\;,$$ with $a(t,0)=a(t,N)=0$. Diagonalising the operator on the right hand side, see Subsection \[sec:eigen\], we get the following bound on the $\ell^2$-norm of the solution: $$\sum_{x=1}^{N-1}a(t,x)^2 \le e^{-2\operatorname{\mathrm{gap}}_N t}\sum_{x=1}^{N-1}a(0,x)^2,$$ and using Cauchy-Schwartz inequality we obtain $${\lambda}^{\frac{k-N}{2}}{\mathbb{E}}[A_t] = \sum_{x=1}^{N-1}a(t,x)\le \sqrt{N} e^{-\operatorname{\mathrm{gap}}_N t}\sqrt{\sum_{x=1}^{N-1}a(0,x)^2}\le 2e^{-\operatorname{\mathrm{gap}}_N t} Nk {\lambda}^{k/2}\;.$$ Since $\lambda^{N/2}$ is, by the small bias assumption, asymptotically smaller than any power of $k$, Markov’s inequality concludes the proof. Diffusion estimate after time $t_{\delta/2}$ {#s:dif} -------------------------------------------- Now this part is much more delicate than Section \[preuv2\]. The reason being that since $T_N$ is extremely close to $t_{\delta}$, we need very accurate control on the derivative of the predictable bracket of $A_t$. Our first task is to use Proposition \[prop:solskjaer\] in order to control the increment of the bracket of $A$ in between the ${\mathcal{T}}_i$’s. Let us set $$\Delta_i \langle A\rangle := \langle A_\cdot\rangle_{{\mathcal{T}}_{i}} - \langle A_\cdot\rangle_{{\mathcal{T}}_{i-1}},\qquad\Delta_{\infty} \langle A\rangle := \langle A_\cdot\rangle_{{\mathcal{T}}_{\infty}} - \langle A_\cdot\rangle_{{\mathcal{T}}_{K}}\;,$$ and consider the event $${\mathcal{A}}_N:= \left\{ \forall i \in {\llbracket}1,K {\rrbracket}, \quad \Delta_i \langle A\rangle \le k^{1 - 2(i-1)\eta - \frac {\delta}4} N^2 \right\}\cap \left\{ \Delta_{\infty} \langle A\rangle \le T_N \right\}$$ \[Lemma:AN\] We have $\lim_{N\to \infty} {\mathbb{P}}[{\mathcal{A}}_N^{{\complement}}]=0$. We apply Proposition \[prop:solskjaer\]-(ii) to $(A_{t+{\mathcal{T}}_{i-1}})_{t\ge 0}$, with $a=k^{\frac12 - (i-1)\eta - \frac {\delta}5}N$, $b=k^{\frac12 - i\eta - \frac {\delta}5}N$. We obtain that for all $N$ sufficiently large and every $i\le K$ $${\mathbb{P}}[ \Delta_i \langle A\rangle\ge k^{1 - 2(i-1)\eta - \frac {\delta}4} N^2]\le k^{-\delta/100}.$$ Applying the same proposition to $(A_{t+{\mathcal{T}}_{K}})_{t\ge 0}$ with $a=k^{- \frac {\delta}5}N$ and $b=0$, we obtain $${\mathbb{P}}[ \Delta_\infty \langle A\rangle\ge T_N ]\le 8 N k^{-\frac {{\delta}}{5}}(T_N)^{-1/2},$$ and the r.h.s. tends to zero by assumption . The next step is to compare $\Delta_i \langle A\rangle$ with ${\mathcal{T}}_i-{\mathcal{T}}_{i-1}$. For the last increment this is easy: We have $\partial_t \langle A_\cdot\rangle\ge 1$ for any $t\le {\mathcal{T}}_{\infty}$ (from our construction $A$ changes its value at rate at least $1$, and its minimal increment in absolute value is $1$). We have thus ${\mathcal{T}}_{\infty}-{\mathcal{T}}_K \le {\Delta}_{\infty} \langle A \rangle$, and thus when ${\mathcal{A}}_N$ holds we have $$\label{Eq:Tinf} {\mathcal{T}}_{\infty}-{\mathcal{T}}_K \le T_N\;.$$ Control of intermediate increments {#s:inter} ---------------------------------- For all other increments we have to use a subtler control of the bracket. Let us set $${\mathbf{H}}(t):= {\lambda}^{\frac{N-k}{2}}\max_{x\in {\llbracket}0,N {\rrbracket}} \frac{{\lambda}^{\frac12 h^{\wedge}_t(x)}- {\lambda}^{\frac12 h^{\vee}_t(x)}}{{\lambda}-1}\;,$$ which corresponds roughly (up to a multiplicative factor ${\lambda}^{N}$) to the maximal height difference $\max h^{\wedge}_t(x)-h^{\vee}_t(x)$ and thus provides a bound for $\max_x h^{\wedge}_t(x)-h^{\pi}_t(x)$.\ Recall $Q(\cdot)$ from Subsection \[Subsec:eqSmallBias\], and set $Q(h_t^\pi) := Q(\eta_t^\pi)$ where $\eta_t^\pi$ is the particle configuration associated with $h_t^\pi$. \[Lemma:BrackDeriv\] We have $\partial_t \langle A_\cdot\rangle \ge \frac{ A_t }{6 {\mathbf{H}}(t) Q(h_t^{\pi})}$. As mentioned above, all the jumps of $A_t$ have amplitude larger than or equal to $1$. Moreover, $A_t$ performs a jump whenever $h^{\pi}_t$ performs a transition while $h^{\wedge}_t$ does not, or when the opposite occurs. As any such transition occurs at rate larger than $q_N\ge 1/3$, only considering the transitions for $h^{\pi}_t$ , we obtain the following lower bound for the drift of $\langle A_\cdot\rangle$ (recall ) $$\label{loopz} \partial_t \langle A_\cdot\rangle \ge \frac{1}{3} \#\{ x\in {\mathcal{C}}_t \ : \ {\Delta}(h^{\pi}_t)(x)\ne 0\}=:\frac{1}{3}\#{\mathcal{D}}_t$$ where $${\mathcal{C}}_t:= \big\{x\in{\llbracket}1,N-1{\rrbracket}\ : \ \exists y\in {\llbracket}x-1,x+1{\rrbracket},\ h^\wedge_t(y) > h^{\pi}_t(y) \big\}\;.$$ Now let ${\llbracket}a, b{\rrbracket}$ be a maximal connected component of ${\mathcal{C}}_t$, we claim that $$\label{zaap} \#( {\mathcal{D}}_t \cap {\llbracket}a,b {\rrbracket})\ge \max\left( \left\lfloor \frac{b-a}{Q(h^{\pi}_t)}\right\rfloor, 1 \right) \ge \frac{b-a}{2Q(h^{\pi}_t)}.$$ To check this inequality, notice that $\#( {\mathcal{D}}_t \cap {\llbracket}a,b {\rrbracket})\ge 1$ because $h^{\pi}_t$ cannot be linear on the whole segment ${\llbracket}a, b{\rrbracket}$. On the other hand, considering the particle configuration associated to $h^\pi_t$ and decomposing the segment ${\llbracket}a,b{\rrbracket}$ into maximal connected components containing either only particles or only holes, we see that any two consecutive components corresponds to a point in ${\mathcal{D}}_t$: since $Q(h^{\pi}_t)$ is an upper bound for the size of these components, we deduce that $\#( {\mathcal{D}}_t \cap {\llbracket}a,b {\rrbracket})\ge \left\lfloor \frac{b-a}{Q(h^{\pi}_t)}\right\rfloor$. Now we observe that $$\label{ziip} {\lambda}^{\frac{N-k}{2}}\sum_{x=a}^b \frac{{\lambda}^{\frac{h^{\wedge}_t(x)}{2}}-{\lambda}^{\frac{h^{\pi}_t(x)}{2}}}{{\lambda}-1} \le (b-a) {\mathbf{H}}(t).$$ Combining and and summing over all such intervals ${\llbracket}a,b{\rrbracket}$, we obtain $$A_t\le 2\,\#{\mathcal{D}}_t \, {\mathbf{H}}(t)\,Q(h^{\pi}_t),$$ and allows us to conclude. The last ingredient needed is then a bound on ${\mathbf{H}}$: The proof of this lemma is postponed to Subsection \[Subsec:Max\]. Recall that $t_0=\log k/(2\operatorname{\mathrm{gap}}_N)$. \[Prop:BoundMax0\] For any $c >0$ we have $$\lim_{N\to \infty} \sup_{t\ge t_0} {\mathbb{P}}\Big({\mathbf{H}}(t) > k^{\frac12 + c}\Big) = 0\;.$$ By Lemma \[Lemma:T0\], Lemma \[Lemma:AN\] and Equation , we already know that $$\lim_{N\to \infty}{\mathbb{P}}\left( \{ {\mathcal{T}}_{0} \le t_{\delta/2}\}\cap \{{\mathcal{T}}_{\infty}-{\mathcal{T}}_{K}\le T_N \}\right)=1 \;.$$ We define ${\mathcal{H}}_N$ to be the event on which particles are reasonably spread and ${\mathbf{H}}(t)$ is reasonably small for most of the times within the interval $[t_{\delta/2}, t_{\delta/2}+T_N]$, $${\mathcal{H}}_N:= \Big\{ \int^{t_{\delta/2}+T_N}_{t_{\delta/2}} {\mathbf{1}}_{\{\text{ ${\mathbf{H}}(t) \le k^{\frac12 + \frac{\delta}{80}}\} \cap \{Q(h_t^\pi) \le N k^{\frac{\delta}{80}-1}$ } \}} dt \ge T_N (1-2^{-(K+1)}) \Big\}\;.$$ By Markov’s inequality, Proposition \[lem:dens\] and Proposition \[Prop:BoundMax0\], we have $$\lim_{N\to \infty}{\mathbb{P}}({\mathcal{H}}_N) = 1\;.$$ We now work on the event ${\mathcal{H}}_N \cap {\mathcal{A}}_N \cap \{{\mathcal{T}}_0 \le t_{\delta/2}\}$ whose probability tends to $1$ according to Lemmas \[Lemma:T0\], \[Lemma:AN\]. We prove by induction that $\Delta{\mathcal{T}}_j \le 2^{-j} T_N$ for all $j\in {\llbracket}1,K{\rrbracket}$. Let us reason by contradiction and let $i$ be the smallest integer such that $\Delta{\mathcal{T}}_i > 2^{-i} T_N$. We have $$\label{indd} \Delta_i\langle A\rangle \ge \int_{{\mathcal{T}}_{i-1}}^{{\mathcal{T}}_{i-1}+2^{-i}T_N} \partial_t \langle A_\cdot \rangle{\mathbf{1}}_{\{\text{ ${\mathcal{H}}(t) \le k^{\frac12 + \frac{\delta}{80}}\}\cap\{Q(h_t^\pi) \le N k^{\frac{\delta}{80}-1}$ } \}} dt$$ Now, Lemma \[Lemma:BrackDeriv\] and the restriction with the indicator function provides a uniform lower bound on $\partial_t \langle A \cdot \rangle$. The assumption $\Delta{\mathcal{T}}_j \le 2^{-j} T_N$ for $j<i$ implies that ${\mathcal{T}}_{i-1}\le t_{\delta/2} + T_N(1-2^{-(i-1)})$, and thus the assumption that ${\mathcal{H}}_N$ holds implies that the indicator in is equal to one on a set of measure at least $2^{-i} - 2^{-(K+1)}\ge 2^{-(K+1)}$. All of this implies that $$\Delta_i\langle A\rangle \ge \frac16 T_N 2^{-(K+1)} k^{1 -i\eta -\frac{\delta}{40} - \frac{\delta}{5}}$$ On the other hand, since we work on ${\mathcal{A}}_N$ we have $\Delta_i\langle A\rangle \le k^{1-2(i-1)\eta-\frac{\delta}{4}}N^2$ so that we get a contradiction as soon as $\eta$ is small enough compared to $\delta$. Bounding the maximum {#Subsec:Max} -------------------- Recall the function $a_{N,k}$ defined in Subsection \[sec:eigen\]. Set $$H_1(t,x) := {\lambda}^{\frac{N-k}{2}}\frac{\lambda^{\frac12 h_t^\wedge(x)}-a_{N,k}(x)}{\lambda- 1}\;,\quad H_2(t,x) :={\lambda}^{\frac{N-k}{2}} \frac{\lambda^{\frac12 h_t^{\vee}(x)}-a_{N,k}(x)}{\lambda- 1}\;,$$ so that $$H_1(t,x)-H_2(t,x) = {\lambda}^{\frac{N-k}{2}}\frac{\lambda^{\frac12 h_t^\wedge(x)}-\lambda^{\frac12 h_t^{\vee}(x)}}{\lambda- 1}\;.$$ For every $i=1,2$, we define $${\mathbf{H}}_i(t):= \max_{x\in {\llbracket}0,N{\rrbracket}} |H_i(t,x)|\;.$$ Notice that ${\mathbf{H}}(t) \le {\mathbf{H}}_1(t) + {\mathbf{H}}_2(t)$ so that Proposition \[Prop:BoundMax0\] is a consequence of the following result. \[Prop:BoundMax\] For any $c >0$, there exists $c' >0$ such that for all $N$ large enough $$\sup_{t\ge t_0} \max_{i\in \{1,2\}} {\mathbb{P}}\Big({\mathbf{H}}_i(t) > k^{\frac12 + c}\Big) \le e^{- k^{c'}}\;.$$ The proof of this bound is split into two lemmas. First, we show that $H_i(t,\cdot)$ can not decrease too much. \[smsl\] We have for all $N$ sufficiently large, all $x\in {\llbracket}1,N-1{\rrbracket}$, all $t\ge 0$, every $i\in\{1,2\}$ and every $y\ge x$ $$\label{zladiff} H_i(t,y)-H_i(t,x)\ge -\frac{k^2(y-x)}{4N}.$$ It is of course sufficient to prove that $$H_i(t,x)-H_i(t,x-1)\ge -\frac{k^2}{4N}\;.$$ We have for any $\eta\in {\Omega}^0_{N,k}$, setting $h=h(\eta)$, $$\frac{\lambda^{\frac12 h(x)}-\lambda^{\frac12 h(x-1)}}{\lambda- 1} =\lambda^{\frac12 (h(x-1)-1)}\left(\eta(x)-\frac{1}{\sqrt{{\lambda}}+1}\right).$$ Note that $a_{N,k}(x)=E_{\pi_{N,k}}[ {\lambda}^{h(x)/2}]$ where $a_{N,k}$ was defined in Section \[sec:eigen\]. Hence using the fact that $\eta^{\wedge}_t(x)\ge 0$, we get (the same holds for $i=2$ and $h^\vee$): $$\begin{gathered} \label{Eq:H1H1} H_1(t,x)-H_1(t,x-1)\\ \ge {\lambda}^{\frac{N-k}{2}}\frac{ E_{\pi_{N,k}}\left[\lambda^{\frac{h(x-1)-1}{2}}\right]-\lambda^{\frac{h^{\wedge}_t(x-1)-1}{2}}}{\sqrt{{\lambda}}+1} -{\lambda}^{\frac{N-k}{2}} E_{\pi_{N,k}}\left[ \lambda^{\frac{h(x-1)-1}{2}}\eta(x)\right]. \end{gathered}$$ By Proposition \[lem:dens\] and the small bias assumption , we have for all $N$ large enough $${\lambda}^{\frac{N-k}{2}} E_{\pi_{N,k}}\left[ \lambda^{\frac{h(x-1)-1}{2}}\eta(x)\right]\le {\lambda}^{\frac{N}{2}}E_{\pi_{N,k}}[\eta(x)]\le {\lambda}^{2N} \frac{k}{N}\le \frac{k^2}{8N}.$$ Regarding the first term on the r.h.s. of , we simply notice that for $\zeta, \zeta'\in {\Omega}_{N,k}$, we have $\zeta(x)-\zeta'(x)\le 2k$ so that $${\lambda}^{\frac{N-k}{2}}|{\lambda}^{\frac{\zeta(x)}{2}}-{\lambda}^{\frac{\zeta'(x)}{2}}|\le ({\lambda}-1){\lambda}^{\frac{N}{2}} k \le \frac{k^{2}}{8N}\;.$$ This is sufficient to conclude. Let us introduce the average of $H_i(t,\cdot)$ over a box of size $\ell =\ell_{N,k}= \lceil \frac{N}{k^2}\rceil$ $$\bar{H}_i(t,y) := \frac1{\ell}\sum_{x=(\ell- 1)y+1}^{y\ell} H_i(t,x)\;.$$ As a consequence of Lemma \[smsl\] we have $${\mathbf{H}}_i(t)= \max_{x\in {\llbracket}0,N{\rrbracket}} |H_i(t,x)|\le \max_{y\in {\llbracket}1, N/\ell {\rrbracket}} \big| \bar{H}_i(t,y) \big|+1 \;.$$ The result is of course obvious when $\ell=1$. For $\ell \ge 2$, let us briefly explain why $\max H_i(t,x)\le \max |\bar{H}_i(t,y)|+1$ (the case for $-\min$ follows by symmetry). If $x_{\max}$ is the smallest $x$ at which the $\max$ is attained, we must distinguish two cases - $x_{\max}> \ell\left( \lfloor N/\ell \rfloor-1\right)+1\ge N-2\ell$, in which case applied for $x_{\max}$ and $N$ implies that $H_i(t,x_{\max})\le 1$, - $x_{\max}\le \ell\left( \lfloor N/\ell \rfloor-1\right)+1$ in which case one can compare $H_i(t,x_{\max})$ with $\bar{H}_i(t,y)$ for the smallest $y$ such that $x_{\max}\le y(\ell- 1)+1$ using again. Then, Proposition \[Prop:BoundMax\] is a direct consequence of the following bound on the averages of $H_i$. For any $a >0$, there exists $a' >0$ such that for all $N$ large enough $$\sup_{t\ge t_0} \max_{i\in\{1,2\}} {\mathbb{P}}\Big( \max_{y\in {\llbracket}1, N/\ell {\rrbracket}} |\bar{H}_i(t,y)| > k^{\frac12 + a}\Big) \le e^{- k^{a'}}\;.$$ We treat in details the bound of $\bar{H}_1$, since the bound of $\bar{H}_2$ follows from the same arguments. Using a decomposition of ${\lambda}^{\frac{k-N}{2}} H_1(t,\cdot)$, which is a solution of , on the basis of eigenfunction of the Laplacian formed by $\sin(i \pi \cdot)$, $i=1,\ldots,N-1$, we obtain the following expression for the mean $${\mathbb{E}}[H_1(t,x)]= {\lambda}^{\frac{N-k}{2}} \sum_{i=1}^{N-1} \frac2{N} e^{-\gamma_i t} f^{(i)}_{N,k}(\wedge) \sin\big( \frac{i \pi x}{N}\big),$$ and the fluctuation around it $$\begin{gathered} \label{2dterm} H_1(t,x)- {\mathbb{E}}[H_1(t,x)]\\ ={\lambda}^{\frac{N-k}{2}} \sum_{i=1}^{N-1} \frac{2} {N} \big(f^{(i)}_{N,k}(h^\wedge_t)-e^{-\gamma_i t} f^{(i)}_{N,k}(\wedge)\big) \sin\big(\frac{i \pi x}{N}\big).\end{gathered}$$ We bound separately the contributions to $\bar{H}_1$ coming from these two terms. We start with the mean. Since ${\lambda}^{\frac{1}{2}\wedge(y)} \ge a_{N,k}(y) \ge{\lambda}^{\frac{1}{2} \vee(y)}$ for every $y\in{\llbracket}0,N{\rrbracket}$, we have (recall ) $$\begin{aligned} |f^{(i)}_{N,k}(\wedge)| \le \sum_{y=1}^{N-1} \frac{{\lambda}^{\frac12 \wedge(y)} - a_{N,k}(y)}{{\lambda}-1} \le \sum_{y=1}^{N-1} \lambda^{\frac12 \wedge(y)}\,\frac{\wedge(y)-\vee(y)}{2} \le \lambda^{\frac{k}{2}} k N\;.\end{aligned}$$ Since $\lambda^{k/2}$ is negligible compared to any power of $k$, we deduce that for all $a>0$ and all $t\ge t_{0}$ we have for all $N$ large enough $$\sup_{x\in {\llbracket}0,N{\rrbracket}} \left|\sum_{i=1}^{N-1} \frac2{N} e^{-\gamma_i t} f^{(i)}_{N,k}(\wedge) \sin\big(i \pi \frac{x}{N}\big)\right| \le k^{(1+a)/2} \sum_{i=1}^{N-1} e^{(\gamma_1-\gamma_i) t_0}\;.$$ Notice that there exists $c>0$ such that for all $i\ge 2$ and all $N$ large enough $$\label{Eq:BoundGamma} \gamma_i - \gamma_1 \ge c \frac{i^2}{N^2}\;.$$ In addition, we have $N^2 \operatorname{\mathrm{gap}}_N \ll (\log k)^2$ by the small bias assumption , so that we get for $i\ge 2$ $$e^{(\gamma_1-\gamma_i) t_{0}} \le e^{- c \frac{i^2}{N^2} \frac{\log k}{2 \operatorname{\mathrm{gap}}_N}} \le e^{-c' \frac{i^2}{\log k}} \le e^{-c' \frac{i}{\log k}}\;,$$ so that for all $N$ large enough we have $$\sum_{i=2}^{N-1} e^{(\gamma_1-\gamma_i) t_{0}} \le \sum_{i=2}^{N-1} e^{-c' \frac{i}{\log k}} \le C \log k \;.$$ Recall that $\ell = \lceil N/k^2 \rceil $. Putting everything together and using assumption , we get that given $a>0$ for $N$ sufficiently large and all values of $y$ we have $$\label{stimean} {\mathbb{E}}[\bar H_1(t,y)] = \frac{ 2{\lambda}^{\frac{N-k}{2}} }{N\ell} \Big|\sum_{x=\ell(y-1)+1}^{\ell y}\sum_{i=1}^{N-1} e^{-\gamma_i t} f^{(i)}_{N,k}(\wedge) \sin\big(i \pi \frac{x}{N}\big)\Big| \le \frac{1}{2} k^{\frac12+a} \;.$$ We turn to the contribution coming from the second term . To that end, we rewrite it in the form $$\bar H_1(s,y)-{\mathbb{E}}\left[ \bar H_1(s,y) \right] = {\lambda}^{\frac{N-k}{2}}\sum_{i=1}^{N-1} \big(f^{(i)}_{N,k}(h_t^\wedge) - e^{-\gamma_i s}f^{(i)}_{N,k}(\wedge)\big) \Phi_{y,i}\;.$$ where (the second expression being obtained by summation by part) $$\begin{gathered} \Phi_{y,i} = \frac2{N \ell } \sum_{x=y(\ell-1)+1}^{y\ell} \sin\big( \frac{i\pi x}{N}\big)\\ = \frac1 {N \ell \sin \left(\frac{i\pi }{2N}\right)} \left[\cos\left( \frac{[2y(\ell-1)+1]i\pi }{2N}\right)- \cos\left( \frac{[2y\ell+1]i\pi }{2N}\right) \right]\;.\end{gathered}$$ Note that for all $N\ge1$, $y$ and $i$ we have $$\big|\Phi_{y,i}\big| \le 2\min\left( \frac 1 N, \frac{1}{ i \ell }\right) \;.$$ Now let us fix $t$ and $y$ and introduce the martingale $$N_s^{(t,y)} = {\lambda}^{\frac{N-k}{2}} \sum_{i=1}^{N-1} e^{\gamma_i (s-t)} \big(f^{(i)}_{N,k}(h_s^\wedge) - e^{-\gamma_i s}f^{(i)}_{N,k}(\wedge)\big) \Phi_{y,i}\;,\quad s\in [0,t]\;.$$ which satisfies $$N_0^{(t,y)}=0 \text{ and } N^{(t,y)}_t=\bar H_1(t,y)-{\mathbb{E}}\left[ \bar H_1(t,y) \right].$$ We wish to apply Lemma \[lem:expobd\] to the martingale $N^{(t,y)}$: the maximal jump rate of this process is bounded by $k$ and the maximal amplitude of the jump (cf. the notations introduced in Appendix \[lapC\]) satisfies $$\forall s \in [0,t], \quad S(s)\le {\lambda}^N \sum_{i=1}^{N-1} e^{\gamma_i(s-t)} \big|\Phi_{y,i}\big|\;.$$ Using that as a consequence of we have $\gamma_i\ge c i^2 N^{-2}$ for all $i\ge 1$ for some $c>0$, we deduce that there exist some constants $C, C'>0$ such that $$\begin{aligned} &\int_0^t S(s)^2 ds\le {\lambda}^{2N} \sum_{i,j=1}^{N-1} \frac{1}{\gamma_i+\gamma_j}|\Phi_{y,i} \Phi_{y,j}|\\ &\le C \lambda^{2N} \Big(\sum_{1\le i \le j \le \frac{N}{\ell}} \frac1{i^2+j^2} + \sum_{1\le i \le \frac{N}{\ell} < j} \frac{N}{j\ell} \frac1{i^2+j^2} + \sum_{\frac{N}{\ell} < i \le j} \frac{N^2}{i j\ell^2} \frac1{i^2+j^2}\Big)\\ &\le C' \lambda^{2N} \log k\;,\end{aligned}$$ Consequently, setting $\gamma= k^{-\frac{1}{2}-2a}$ and using the fact that from we have $\gamma \,C'\lambda^{2N} \log k < 1$ for $N$ sufficiently large, we apply and obtain $$\begin{gathered} {\mathbb{P}}(N_t^{(t,y)} > \frac 1 2 k^{\frac {1}{2} +a}) \le {\mathbb{E}}[e^{\gamma N_t^{(t,y)}}]e^{-\frac 1 2 \gamma k^{\frac{1}{2}+a}} \\ \le e^{C' e \gamma^2 k {\lambda}^{2N} \log k -\frac 1 2 \gamma k^{\frac{1}{2}+a}}\le e^{-\frac{1}{4}k^{-a}} \;.\end{gathered}$$ A similar same computation for $N_t^{(t,y)} <- \frac 1 2 k^{\frac {1}{2} +a}$ and a union bound yield $$\begin{gathered} {\mathbb{P}}(\sup_{y \in {\llbracket}1, N/\ell {\rrbracket}} |\bar H_i(t,y)-{\mathbb{E}}[ H_i(t,y)]| > \frac 1 2 k^{\frac{1}{2}+a})\\ ={\mathbb{P}}(\sup_{y \in {\llbracket}1, N/\ell {\rrbracket}} |N_t^{(t,y)}| > \frac 1 2 k^{\frac{1}{2}+a}) \le C k^2 e^{-\frac{1}{4}k^{-a}}, \end{gathered}$$ which combined with allows to conclude. A monotone grand coupling {#Appendix:Coupling} ========================= The construction below is similar to the one detailed in [@Lac16 Section 8.1] in the symmetric case. We consider a collection of independent Poisson clock processes ${\mathcal{P}}^{(i,\ell)}$ and ${\mathcal{Q}}^{(i,\ell)}$ with rate $p$ and $q$ respectively where $i\in{\llbracket}1 ,N{\rrbracket}$ and $\ell \in {\llbracket}-N,\ldots,N{\rrbracket}$: For each $(i,\ell)$, ${\mathcal{P}}^{(i,\ell)}$ resp. ${\mathcal{Q}}^{(i,\ell)}$ is a random increasing sequence of positive real numbers (or equivalently a random locally finite subset of $(0,\infty)$) whose first term and increments are independent geometric variables of mean $p^{-1}$ resp. $q^{-1}$. For every $k$ and every $\zeta\in \Omega_{N,k}$, we construct the process $(h_t^{\zeta})_{t\ge 0}$ as follows: The process is càd-làg and may only jump at the times specified by the clock process ${\mathcal{P}}$ and ${\mathcal{Q}}$. We enumerate these Poisson times in increasing order and if $t\in {\mathcal{P}}^{(i,\ell)}$ and if $h^\zeta_{t-}$ displays a local maximum at $i$ and height $\ell$, that is if $$h^\zeta_{t-}(i) = \ell = h^\zeta_{t-}(i-1)+1 = h^\zeta_{t-}(i+1)+1\;,$$ then we flip it downwards to a local minimum by setting , $h^\zeta_t(i) := h^\zeta_{t-}(i) - 2$, and $h^{\zeta}_t(j)=h^{\zeta}_{t-}(j)$ for $j\ne i$. A similar transition occurs if $Q(i,\ell)$ rings and if $h^\zeta_{t-}$ displays local minimum at $i$ and height $\ell$.\ It is simple to check that under this construction, $h^\zeta$ indeed evolves according to the right dynamics, and that monotonicity is preserved. Diffusion bounds for continuous-time supermartingales {#sec:diffu} ===================================================== In this section, we assume that $(M_t)_{t\ge 0}$ is a pure-jump supermartingale with bounded jump rate and jump amplitude. This implies in particular that, $M_t$ is square integrable for all $t>0$. With some abuse of notation, we use the notation $\langle M_\cdot \rangle_t$ for the predictable bracket associated with the martingale ${\widetilde}M_t=M_t-A_t$ where $A$ is the compensator of $M$. \[prop:solskjaer\] Let $(M_t)_{t\ge 0}$ be as above - Set $\tau= \inf\{ t\ge 0 \ : \ M_t=0\}$. Assume that $M_t$ is non-negative and that, until the absorption time $\tau$, its jump amplitude and jump rate are bounded below by $1$. Then we have for any $a\ge 1$ and all $u>0$ $$\label{lopes} {\mathbb{P}}[ \tau \ge a^2 u \ | \ M_0\le a ]\le 4 u^{-1/2}.$$ - Given $a \in{\mathbb{R}}$ and $b\le a$, we set $\tau_{b}:=\inf\{ t\ge 0 \ : \ M_t\le b\}$. If the amplitude of the jumps of $(M_t)_{t\ge 0}$ is bounded above by $a-b$, we have for any $u\ge 0$ $${\mathbb{P}}[ \langle M_\cdot \rangle_{\tau_b}\ge (a-b)^2 u \ | \ M_0\le a ]\le 8 u^{-1/2}.$$ The important building block for the proof of the above proposition is the following result. \[lem:supersub\] Let $(M_t)_{t\ge 0}$ be as above - If the amplitude of the jumps of $(M_t)_{t\ge 0}$ and the jump rate are bounded below by $1$ then for all ${\lambda}\in (0,1)$, $$\left(e^{-{\lambda}M_t-\frac{{\lambda}^2 t}{4}}\right)_{t\ge 0}$$ is a submartingale - If the amplitude of the jumps of $M_t$ is bounded above by $a$ then for any ${\lambda}\in (0,a^{-1})$ we have $$\left( \exp\left(-{\lambda}M_t- \frac{{\lambda}^2}{4} \langle M_\cdot \rangle_t\right)\right)_{t\ge 0}$$ is a submartingale. The result only needs to be proved for $u\ge 4$. Without loss of generality one can assume for the proof of both statements that $P[ M_0\le a ]=1$ and for the second one that $b=0$. We set ${\lambda}=2 a^{-1} u^{-1/2}$. For $(i)$, a direct application of the Martingale Stopping Theorem to the submartingale of Lemma \[lem:supersub\] yields: $${\mathbb{E}}\left[e^{-\frac{{\lambda}^2\tau}{4}}\right]\ge {\mathbb{E}}[e^{-{\lambda}M_0 }]\ge e^{-{\lambda}a}=e^{-2u^{-1/2}}.$$ On the other hand one has $${\mathbb{E}}\left[e^{-\frac{{\lambda}^2\tau}{4}}\right]\le 1-(1-e^{-\frac{{\lambda}^2}{4} a^2u}){\mathbb{P}}[\tau \ge a^2 u ] \le 1-\frac{1}{2}{\mathbb{P}}[\tau \ge a^2 u].$$ The combination of the two yields $${\mathbb{P}}[\tau \ge a^2 u]\le 2(1-e^{-2u^{-1/2}})\le 4 u^{-1/2}.$$ For $(ii)$, the arguments of the previous case apply almost verbatim if one replaces $\tau$ by $T:=\langle M_\cdot \rangle_{\tau}$. The only thing one has to take into account is that $M_{\tau}$ is not necessarily equal to $0$, but the assumption on the amplitude of jumps yields $M_{\tau}\ge -a$. The Martingale Stopping Theorem gives us $${\mathbb{E}}\left[e^{-\frac{{\lambda}^2 T}{4}}\right]\ge e^{-{\lambda}a } {\mathbb{E}}\left[e^{-{\lambda}M_{\tau}-\frac{{\lambda}^2 T}{4}}\right]\ge e^{-{\lambda}a} {\mathbb{E}}[e^{-{\lambda}M_0}] \ge e^{-4u^{-1/2}}\;.$$ Repeating the rest of the computation yields $${\mathbb{P}}[T \ge a^2 u]\le 2(1-e^{-4u^{-1/2}})\le 8 u^{-1/2}\;.$$ Until the end of the proof, we write ${\mathbb{E}}_t$ for the conditional expectation given $(M_s)_{s\le t}$.\ Case (i). Take ${\lambda}\in (0,1)$. The submartingale identity we need to prove can be written as follows $$\forall s, t\ge 0, \quad \log {\mathbb{E}}_t\Big[e^{-{\lambda}(M_{t+s}-M_t)}\Big]\ge s\frac{{\lambda}^2}{4},$$ Taking derivative, we deduce that it suffices to prove that for all $t,s\ge 0$ we have $$\label{toprov} \lim_{h\downarrow 0} \frac1{h}{\mathbb{E}}_t\Big[e^{-{\lambda}( M_{t+s+h}-M_t)}-e^{-{\lambda}(M_{t+s}-M_t)}\Big] \ge \frac{{\lambda}^2}{4} {\mathbb{E}}_t\Big[e^{-{\lambda}(M_{t+s}-M_t)}\Big]\;.$$ Notice that for all $x\in {\mathbb{R}}$ $$\label{Eq:ExpoTaylor} e^{-x}+x-1\ge \frac{\min(1,x^{2})}{4}\;.$$ Thus, using the supermartingale property of $M$ we have for all ${\lambda}\in(0,1)$ $$\begin{aligned} &{\mathbb{E}}_t\Big[e^{-{\lambda}(M_{t+s+h}-M_t)}-e^{-{\lambda}(M_{t+s}-M_t)}\Big]\\ &= {\mathbb{E}}_t\Big[e^{-{\lambda}(M_{t+s}-M_t)}{\mathbb{E}}_{t+s}\big[e^{-{\lambda}(M_{t+s+h}-M_{t+s})}-1\big]\Big]\\ &\ge {\mathbb{E}}_t\Big[e^{-{\lambda}(M_{t+s}-M_t)}{\mathbb{E}}_{t+s}\big[e^{-{\lambda}(M_{t+s+h}-M_{t+s})}+{\lambda}(M_{t+s+h}-M_{t+s})-1\big]\Big]\\ &\ge \frac{{\lambda}^2}{4}{\mathbb{E}}_t\Big[e^{-{\lambda}(M_{t+s}-M_t)}{\mathbb{E}}_{t+s}\big[\min(1,(M_{t+s+h}-M_{t+s})^2)\big]\Big]\;.\end{aligned}$$ The assumption on the jump rates and the jump amplitudes yield $$\liminf_{h\to 0} \frac1{h}{\mathbb{E}}_{t+s}\big[\min(1,(M_{t+s+h}-M_{t+s})^2)\big]\ge 1,$$ so that Fatou’s Lemma concludes the proof. Case (ii). We can assume without loss of generality that $a=1$. Here again, taking the derivative of the submartingale identity that we want to establish, it suffices to prove that for all $t,s \ge 0$ we have $$\begin{aligned} \liminf_{h\downarrow 0} &\frac1{h}{\mathbb{E}}_t\Big[e^{-{\lambda}M_{t+s+h} - \frac{{\lambda}^2}{4}\langle M_\cdot\rangle_{t+s+h}} -e^{-{\lambda}M_{t+s}- \frac{{\lambda}^2}{4}\langle M_\cdot\rangle_{t+s}} \Big] \ge 0\;.\end{aligned}$$ Taking the conditional expectation w.r.t. $M_{t+s}$, we see that it suffices to prove the existence of some deterministic constant $C>0$ such that $$\label{Eq:subderiv} {\mathbb{E}}_{t+s}\Big[e^{-{\lambda}(M_{t+s+h}-M_{t+s}) - \frac{{\lambda}^2}{4}(\langle M_\cdot\rangle_{t+s+h}-\langle M_\cdot\rangle_{t+s})} -1 \Big] \ge -C h^2\;,$$ for all $h$ small enough.\ Without loss of generality, we can assume that $t+s = 0$ and $M_0=0$. Recall that ${\widetilde}M\ge M$ is the martingale which is obtained by subtracting the (negative) compensator. Thus $$\begin{aligned} e^{-{\lambda}M_{h} - \frac{{\lambda}^2}{4} \langle M_\cdot\rangle_{h}} -1 &\ge e^{-{\lambda}{\widetilde}{M}_{h} - \frac{{\lambda}^2}{4} \langle M_\cdot\rangle_{h}} -1\\ &\ge \big(1-{\lambda}{\widetilde}M_{h}+ \frac{1}{4}\min(1, {\lambda}^{2} {\widetilde}M_{h}^2)\big) \big(1- \frac{{\lambda}^2}{4} \langle M_\cdot\rangle_{h}\big)-1 \\ &\ge -{\lambda}{\widetilde}M_{h}+ \frac{{\lambda}^2}{4}\big( {\widetilde}M_{h}^2- \langle M_\cdot\rangle_{h}\big) - \frac{{\lambda}^2}{4}\big({\widetilde}M_{h}^2-{\lambda}^{-2}\big)_+ \\ &\quad+ \frac{{\lambda}^2}{4}\langle M_\cdot\rangle_{h} \big({\lambda}{\widetilde}M_{h} - \frac{1}{4}\min(1, {\lambda}^2 {\widetilde}M_{h}^2) \big)\;,\end{aligned}$$ so that $$\begin{aligned} {\mathbb{E}}\Big[e^{-{\lambda}M_{h} - \frac{{\lambda}^2}{4} \langle M_\cdot\rangle_{h}} -1\Big] &\ge {\mathbb{E}}\Big[- \frac{{\lambda}^2}{4}\big({\widetilde}M_{h}^2-{\lambda}^{-2}\big)_+\\ &\quad+ \frac{{\lambda}^2}{4}\langle M_\cdot\rangle_{h} \big({\lambda}{\widetilde}M_{h} - \frac{1}{4}\min(1, {\lambda}^2 {\widetilde}M_{h}^2) \big)\Big]\;.\end{aligned}$$ Take ${\lambda}\in (0,1)$. Our assumptions on the increments and jump rates imply that for some constant $C>0$ we have $$\begin{split} {\mathbb{E}}\left [ \big({\widetilde}M_{h}^2-{\lambda}^{-2}\big)_+ \right]&\le C h^2,\\ \langle M_\cdot\rangle_{h} &\le C h,\\ \max\left( {\mathbb{E}}\left[ |{\widetilde}M_{h}| \right], {\mathbb{E}}\left[ \min(1, {\lambda}^2 {\widetilde}M_{h}^2) \right]\right)&\le Ch, \end{split}$$ (the compensator being of order $h$ the estimates for ${\widetilde}M$ can be deduced from that for $M$), which allows to conclude that holds. Exponential moments of continuous-time martingales {#lapC} ================================================== Let $(M_t)_{t\ge 0}$ be a martingale defined as a function of a continuous time Markov chain on a finite state space $$M_t=f(t,X_t)\;,$$ where $f$ is differentiable in time. We let $B$ denote the maximal jump rate for $X$ and let $S(t)$ denote the maximal amplitude for a jump of $M$ at time $t$: $$S(t):=\max_{\xi\sim \xi'} |f(t,\xi)-f(t,\xi')|\;.$$ \[lem:expobd\] For any ${\lambda}> 0$ we have $${\mathbb{E}}\left[ e^{{\lambda}M_t} \right] \le \exp \left( B\int^t_0 \left[e^{{\lambda}S(s)}-{\lambda}S(s)-1\right]{\,\text{\rm d}}s \right).$$ In particular if ${\lambda}S(t)\le 1$ for all $t\ge 0$ then we have $$\label{Eq:ExpoBdMgale} {\mathbb{E}}\left[ e^{{\lambda}M_t} \right] \le \exp \left( B e {\lambda}^2 \int^t_0 S^2(s) {\,\text{\rm d}}s \right).$$ We are going to show that for all $t\ge 0$ $$\partial_t \log {\mathbb{E}}\left[ e^{{\lambda}M_t} \right]= \frac{\partial_t {\mathbb{E}}\left[ e^{{\lambda}M_t} \right]}{{\mathbb{E}}\left[ e^{{\lambda}M_t} \right]}\le B [e^{{\lambda}S(t)}-{\lambda}S(t)-1].$$ To that end, it is sufficient to show that almost surely $$\partial_s {\mathbb{E}}\left[ e^{{\lambda}(M_{t+s}-M_t)} - 1 \ | \ {\mathcal{F}}_t \right] |_{s=0} \le B [e^{{\lambda}S(t)}-{\lambda}S(t)-1].$$ We let $\Delta_s M=M_{t+s}-M_t$ denote the martingale increment and as in the previous section write ${\mathbb{E}}_t$ for the conditional expectation w.r.t. $M_t$ . By the martingale property, we have $$\begin{aligned} {\mathbb{E}}_t \left[ e^{{\lambda}\Delta_s M}-1 \right]&={\mathbb{E}}_t \left[ e^{{\lambda}\Delta_s M}-{\lambda}\Delta_s M-1 \right] \le {\mathbb{E}}_t \left[ e^{{\lambda}|\Delta_s M|}-{\lambda}|\Delta_s M|-1 \right].\end{aligned}$$ Note that $|\Delta_s M|$ is stochastically dominated by $$\left[\max_{u\in[t,t+s]} S(u)\right] {\mathcal{W}}+ s\times \max_{u\in[t,t+s]} \| \partial_u f(u, \cdot)\|_{\infty}$$ where ${\mathcal{W}}$ is a Poisson variable of parameter $B s$. As $S$ is Lipshitz we conclude that $${\mathbb{E}}_t \left[ \frac{e^{{\lambda}|\Delta_s M|}-{\lambda}|\Delta_s M|-1}{s} \right]\le B[e^{{\lambda}S(t)}-{\lambda}S(t)-1]+ c s.$$

--- abstract: 'We study the evolution of cosmological perturbations in a non-singular bouncing cosmology with a bounce phase which has superimposed oscillations of the scale factor. We identify length scales for which the final spectrum of fluctuations obtains imprints of the non-trivial bounce dynamics. These imprints in the spectrum are manifested in the form of damped oscillation features at scales smaller than a characteristic value and an increased reddening of the spectrum at all the scales as the number of small bounces increases.' author: - Robert Brandenberger - Qiuyue Liang - 'Rudnei O. Ramos' - Siyi Zhou title: Fluctuations through a Vibrating Bounce --- Introduction ============ Some bouncing cosmologies provide an alternative to cosmological inflation as a way to obtain primordial cosmological fluctuations (see, e.g., Ref. [@BP] for a recent review). Specifically, in a model which contains a matter-dominated phase of contraction, initial vacuum fluctuations in the far past which exit the Hubble radius during the matter-dominated contracting phase evolve into a scale-invariant spectrum of cosmological perturbations [@FB; @Wands]. For example, in the case of an Ekpyrotic contracting universe [@Ekp] entropy fields can source scale-invariant curvature fluctuations [@newEkp]. In all bouncing cosmologies, new physics is required in order to obtain a non-singular cosmological bounce. Such new physics could come from the matter sector (see, e.g., Refs. [@matterBounce-1; @matterBounce-2]), from modifications of the classical gravitational action (as for example in Horava-Lifshitz gravity [@HLbounce] or in the non-local gravity construction of Ref. [@Biswas]), or from quantum gravity effects. Examples of the latter are the bounce in loop quantum cosmology (see, e.g., Ref. [@LQCbounce] for reviews), in deformed AdS/CFT cosmology [@Elisa], the S-brane bounce of Ref. [@Costas] and the temperature bounce in String Gas Cosmology [@BV]. Concerning the robustness of the computations of the spectrum of cosmological fluctuations, an advantage of bouncing cosmologies (without an inflationary phase after the bounce) is that the physical length of modes which are probed in current observations remain in the far infrared throughout the cosmological evolution as long as the energy density at the bounce point is smaller than the Planck density. Hence, the computations can be done in the realm where effective field theory is well justified. This is in contrast to the situation in inflationary cosmology [@MB] where the physical wavelengths of even the largest scales which are currently observed are smaller than the Planck length at the beginning of inflation (provided that the inflationary phase lasts slightly longer than the minimal period it has to last in order to solve the horizon and flatness problems of Standard Big Bang Cosmology). A key question is to whether the predictions for cosmological perturbations at late times in the expanding phase are sensitive to the details of the bounce phase. For simple parametrizations of the bounce phase, detailed studies have shown that the spectral shape does not change during the bounce phase provided that the duration of the bounce phase is shorter than the length scale of the fluctuations at the bounce point (see, e.g., Ref. [@matterBounce-2] in the case of matter-driven bounces, Ref. [@HLflucts] in the case of the Horava-Lifshitz bounce, Ref. [@Elisa] in the case of the AdS/CFT bounce, and Ref. [@Subodh] for the S-brane bounce). On the other hand, there are examples where the bounce phase yields dramatic changes in the spectrum [@BXue]. The reason why such dramatic changes are possible is that the Hubble radius at the bounce point is infinite, and we cannot invoke the freezing of cosmological perturbations on super-Hubble scales to argue for a constancy of the spectrum[^1]. To further analyze the sensitivity of the spectrum of cosmological fluctuations on the details of the bounce phase, we here consider a toy model where the scale factor undergoes small amplitude oscillations during the bounce phase. Such a behavior may emerge from certain models motivated by ideas from loop quantum gravity [@Alesci]. Heuristically, one would argue that those small bounces will not influence the large scale modes provided that the wavelengths of these modes are so large scale that they would not feel the small scale fluctuations of the scale factor. On the other hand, smaller scale modes whose wavelength is comparable or smaller to the total duration of the bounce phase should be sensitive to the details of the dynamics during the bounce. In this work, we would like to give a careful treatment to see if this is really the case. In the case of a cyclic cosmology, when the time interval between cycles is larger than the wavelength of the modes being considered, it is generally sufficient [@RHBcyclic] to consider only the dominant modes in each phase (except during the bounce phase and when mode matching conditions are applied [@Durrer]). In our case however, the time scale between the small bounces is very small compared to the length scales of interest, and hence we cannot just focus on the dominant modes because for those small time durations the subdominant modes can also have an effect on the primordial power spectra. We need to keep all the contributions and give a comprehensive analysis. This paper is organized as follows. In Sec. \[sec2\], we specify our setup for the intermediate small bounce feature and discuss about the relevant scales involved. In Sec. \[sec3\], we present the calculation of power spectrum. A detailed presentation of the required matching conditions is given and the specific results are given for the two specific small inter bounce features we considered. An analysis of these results for the power spectrum is then given in Sec. \[sec4\]. In Sec. \[sec5\], we give a generalization for the case of a large number of small bounces. In Sec. \[sec6\] we present our conclusions. An appendix is included to discuss some of the technical details. Setup {#sec2} ===== We will consider a spatially flat Friedmann-Lemaitre-Robertson-Walker space-time in which the metric is given by $$ds^2 \, = \, dt^2 - a(t)^2 d{\bf x}^2 \, ,$$ where $t$ is physical time, ${\bf x}$ are the comoving spatial coordinates, and $a(t)$ is the cosmological scale factor. It will be convenient to use conformal time $\tau$ related to the physical time via $dt = a(t) d\tau$. We will consider only linear cosmological perturbations (see, e.g., Ref. [@MFB] for a comprehensive review). In this case, fluctuations evolve independently in Fourier space. We will label the fluctuation modes in terms of their comoving wave number $k$. We consider a non-singular symmetric bouncing cosmology in which the cosmological scale factor $a(\tau)$ has the form shown in Fig. \[bounceplot\], i.e., for which $a(\tau)$ has one “oscillation” between the onset and end of the bounce phase. We consider the two forms depicted in Fig. \[bounceplot\], with one peak between the time $- \tau_B$ when $a(\tau)$ reaches its first minimum, and the time $+ \tau_B$ when the second minimum of $a(\tau)$ is taken on. Specifically, we consider two specific models: Model 1, with a flat plateau about $\tau = 0$; and the Model 2, with a kink of $a(\tau)$ at $\tau = 0$, which is the limiting case of the first model when the duration of the flat plateau equals to zero. The time interval of the plateau region is $-\Delta < \tau < \Delta$ given by some conformal time $\Delta$ with $\Delta < \tau_B$. The forms shown in Fig. \[bounceplot\] are simpler enough such as to allow an analytical study, but already of sufficient complexity such as to provide the main relevant features in the power spectrum that we might also observe in some more complex setup for the inter-bounce features. In particular, our results can be easily generalized to the case of many oscillations, as we will later discuss in Sec. \[sec5\]. As seen in Fig. \[bounceplot\], for the two forms shown the time interval can be divided into five intervals. The first is the initial contracting phase (Phase I) $\tau < - \tau_B$. The second is the intermediate expanding phase (Phase II). The third is Phase III with static scale factor, the fourth (Phase IV) is the intermediate contracting phase and Phase V is the final expanding phase. Fluctuation modes exit the Hubble radius in the initial contracting phase. The top panels in Fig. \[bounceplot\] give a sketch of the scale factor, the lower panels show the corresponding time evolution of the comoving Hubble radius. The vertical axis of the lower panels can also be viewed as a label for comoving wavelength. In this way, it is easy to read off when various modes enter and exit the Hubble radius. All scales re-enter and re-exit the Hubble radius several times since at the extrema of $a(t)$ the Hubble radius is infinite. We will treat the transitions at $-\tau_B$ and $\tau_B$ as instantaneous[^2]. More specifically, we will cut out a time interval $$-\tau_B - \epsilon < \tau < - \tau_B + \epsilon,$$ (with $\epsilon \ll \tau_B$) and correspondingly another time interval of the same length about $\tau_B$ and we will match the solutions between the neighboring phases making use of the matching conditions given in Refs. [@HV; @DM], which are the cosmological version of the Israel [@Israel] ones[^3]. Hence, the only Hubble re-entry which is important to us is the one which occurs between $- \tau_B$ and $+ \tau_B$. In the first model, given by the plots on the left shown in Fig. \[bounceplot\], there are two characteristic comoving length scales. The first is $k_*^{-1}$ which is defined as the length which re-enters the Hubble radius at time $- \Delta$. The second one, $k_{\Delta}^{-1}$, is the mode which undergoes one oscillation between $\tau = - \Delta$ and $\tau = + \Delta$ (we are assuming $k_{\Delta}^{-1} > k_*^{-1}$ if this is not satisfied then we recover the results for the second model, given by the plots on the right shown in Fig. \[bounceplot\]). Modes with wavelength smaller than $k_*^{-1}$ enter the Hubble radius during Phase II and exit again during Phase IV. Modes with $k_*^{-1} < k^{-1} < k_{\Delta}^{-1}$ are inside the Hubble radius only during Phase III. For these modes the matching occurs at times $- \Delta$ and $+ \Delta$. This is also true for modes with $k^{-1} > k_{\Delta}^{-1}$. These modes, however, undergo a negligible amount of oscillations in Phase III. In the case that $k_\Delta^{-1}>k_*^{-1}$, we have three different behaviors of the power spectrum. For the very large scale modes $k^{-1}\gg k_\Delta^{-1}$, the power spectrum does not feel the influence of the small bump of the scale factor. For the modes $k_*^{-1}<k^{-1}<k_\Delta^{-1}$, there is a complicated change of the power spectrum induced by the flat plateau. For the modes $k^{-1}<k_*^{-1}$, the change of the power spectrum approaches the well known result for cyclic cosmologies [@RHBcyclic], as we will explicitly verify later on below. In the case that $k_\Delta^{-1}<k_*^{-1}$, there is only one characteristic scale $k_*^{-1}$. The mode with $k^{-1}>k_*^{-1}$ will not feel the influence of the bump, while the mode with $k^{-1}<k_*^{-1}$ will be changed by the bump according to the well known result for cyclic cosmologies. A special situation belonging to this case is the limiting case $\Delta \rightarrow 0$. We divide the evolution of fluctuation modes into five regions as shown on Fig. \[bounceplot\] [^4]. The five regions are denoted by Region I, Region II, Region III, Region IV and Region V, respectively. Region I and Region II are separated by the time $-\tau_B$. Region IV and Region V are separated by $\tau_B$. The separation between Regions II and III, and between Regions III and IV are more complicated. Because of the existence of the flat plateau (or of the local maximum of the comoving Hubble radius in the case of Model 2), we can see that there is a clear distinction between the large scale and small scale modes separated by a characteristic scale $k_*^{-1}$. Small scale modes (i.e. $k^{-1}<k_*^{-1}$) enter the Hubble radius at time $-\tau_H(k) \leq - \Delta$ and exit the Hubble radius at $\tau_H(k) > \Delta$, and the separation between Regions II and III and between Regions III and IV are given by $-\tau_H(k)$ and $\tau_H(k)$, respectively. For large scale modes, the separations between Regions II and III, and between Regions III and IV are given by the times $-\Delta$ and $\Delta$, respectively, because the modes enter and exit the Hubble radius at these two times. The situation in the case of Model 2 is simpler. The evolution of small scale modes is the same as the case with a flat plateau, while the large scale modes have only four regions which we denote by Regions I, II, IV and V, respectively. The separation between Regions II and IV, in this case, is the time $\tau= 0$. The Computation of the Power Spectrum {#sec3} ===================================== We are interested in the power spectrum of the primordial curvature perturbation $\zeta$ (see, e.g., Ref. [@MFB] for a review of the theory of cosmological perturbations). We quantize the linear fluctuations and write them in terms of the more convenient Mukhanov-Sasaki variable $v$. In the case of a constant equation of state, the relation between $\zeta$ and $v$ is $$\begin{aligned} v = C a \zeta \, ,\end{aligned}$$ where $C$ is a constant. Thus, the equation of motion for the mode function $v$, in momentum space, is given by $$\begin{aligned} v'' + \bigg(k^2 - \frac{a''}{a} \bigg) v = 0 \, .\end{aligned}$$ For the scale factor $a\sim \tau^q$, the solution of the Mukhanov-Sasaki equation is given by $$\begin{aligned} v (\tau) = c_1(k) \sqrt{\tau} J_{\alpha} (k\tau) + c_2(k) \sqrt{\tau} Y_{\alpha} (k\tau),\quad \alpha\equiv\ q-\frac{1}{2} ,\end{aligned}$$ where $J(x)$ and $Y(x)$ are Bessel functions of the first and second kind, respectively. On sub-Hubble scales, the solutions are oscillatory, on super-Hubble scales they can be approximated by a power law. To see this, we note that the expansion of the Bessel function solutions for small argument, $x\ll 1$, is $$\begin{aligned} J_\alpha (x) &= \sum_{m = 0}^\infty \frac{(-1)^m}{m!\Gamma(m+\alpha+1)} \left( \frac{x}{2} \right)^{2m+\alpha}, \\ Y_\alpha (x) &= \frac{\cos(\alpha\pi)}{\sin(\alpha\pi)} J_\alpha(x) - \frac{1}{\sin(\alpha\pi)}J_{-\alpha}(x) \nonumber \\ &= \frac{\cos(\alpha \pi)}{\sin(\alpha \pi)} \sum_{m = 0}^\infty \frac{(-1)^m}{m!\Gamma(m+\alpha+1)} \left( \frac{x}{2} \right)^{2m+\alpha} - \frac{1}{\sin(\alpha\pi)}\sum_{m = 0}^\infty \frac{(-1)^m}{m!\Gamma(m-\alpha+1)} \left( \frac{x}{2} \right)^{2m-\alpha},\end{aligned}$$ we can express the mode function in terms of a series expansion $$\begin{aligned} v(\tau) = \sum_{m=0}^{\infty} d_{1m} (k) \tau^{q+2m} + \sum_{m=0}^{\infty} d_{2m} (k) \tau^{1-q+2m} , \label{vtaum}\end{aligned}$$ where $d_{1m}(k)$ and $d_{2m}(k)$ are given, respectively, by $$\begin{aligned} d_{1m}(k) & = \left[ c_1(k) + \frac{\cos(\alpha\pi)}{\sin(\alpha\pi)} c_2(k) \right] \frac{(-1)^m}{m!\Gamma(m+\alpha+1)} \left( \frac{k}{2} \right)^{2m+\alpha}, \\ d_{2m}(k) & = -\frac{c_2(k)}{\sin(\alpha\pi)} \frac{(-1)^m}{m!\Gamma(m-\alpha+1)} \left( \frac{k}{2} \right)^{2m-\alpha} .\end{aligned}$$ Since we are interested in those modes that went classical (crossed the Hubble radius), such that $k \tau \ll 1$, the higher order terms of $v(\tau)$ given by $m> 0$ are subleading. Thus, in the following, we can just focus on the $m=0$ terms in Eq. (\[vtaum\]). The scale factors and the solutions to the mode functions of the five regions can be obtained by shifting the time coordinate. Thus, they are given, respectively, by - Region I (contracting), where $\tau<-\tau_B$, we have that $$\begin{aligned} a \sim (-\tau-\tau_B)^{q_2} \sim(-t-t_B)^{p_2}, \quad v_{1} = c_{11} (-\tau_B-\tau)^{1-q_2} + c_{12} (-\tau_B-\tau)^{q_2},\end{aligned}$$ - Region II (expanding), where $ -\tau_B<\tau<-\Delta\,\,\, {\rm for}\,\,\, k^{-1}>k_*^{-1}, \,\,\, {\rm and}\,\,\,-\tau_B<\tau<-\tau_H \,\,\, {\rm for}\,\,\, k^{-1}<k_*^{-1}$, we have that $$\begin{aligned} a\sim (\tau+\tau_B)^{q_1}\sim(t+t_B)^{p_1},\quad v_2 = c_{21}(\tau+\tau_B)^{1-q_1} + c_{22} (\tau+\tau_B)^{q_1},\end{aligned}$$ - Region III (intermediate), where $-\Delta<\tau<\Delta\,\,\, {\rm for}\,\,\, k^{-1}>k_*^{-1}, \,\,\, {\rm and}\,\,\, -\tau_H<\tau<\tau_H \,\,\, {\rm for}\,\,\, k^{-1}<k_*^{-1}$, we have that $$\begin{aligned} a\sim {\rm constant\,\,\,in\,\,\,-\Delta<\tau<\Delta} , \quad v_3 = c_{31} e^{i k \tau} + c_{32} e^{-i k \tau},\end{aligned}$$ - Region IV (contracting), where $\Delta<\tau<\tau_B\,\,\, {\rm for}\,\,\, k^{-1}>k_*^{-1}, \,\,\, {\rm and}\,\,\, \tau_H<\tau<\tau_B \,\,\, {\rm for}\,\,\, k^{-1}<k_*^{-1}$, we have that $$\begin{aligned} a \sim (-\tau+\tau_B)^{q_1} \sim (-t+t_B)^{p_1},\quad v_4 = c_{41} (-\tau+\tau_B)^{1-q_1} +c_{42} (-\tau+\tau_B)^{q_1},\end{aligned}$$ - Region V (expanding), where $\tau>\tau_B$, we have that $$\begin{aligned} a\sim (\tau-\tau_B)^{q_2} \sim (t-t_B)^{p_2},\quad v_5 = c_{51} (\tau - \tau_B)^{1-q_2} + c_{52} (\tau-\tau_B)^{q_2} .\end{aligned}$$ Note that in the above, the time $\tau_H$ depends on $k$, $\tau_H\equiv \tau_H(k)$. To simplify the notation we do not write the k-dependence explicitly. Note also that $q_2$ (the power of the scale factor in conformal time) and $p_2$ (the power of the scale factor in cosmological time) are the indices of the scale factor during the initial contracting and final expanding phase, and that $q_1$ and $p_1$ are the indices in the intervening periods. In particular, the index 1 in these quantities should not be confused with the index in Region I. Note also that the $q_i$ are related to the $p_i$ through the relation $$q_i = \frac{p_i}{1-p_i},\;\;\;i=1,2. \label{qipi}$$ Matching conditions ------------------- Let us now discuss the process of matching between the different regions discussed above. The matching conditions for the metric across a space-like hypersurface were derived in Hwang-Vishniac [@HV] and Deruelle-Mukhanov [@DM] and are the generalization of the Israel matching conditions [@Israel]. For cosmological fluctuations, the matching conditions say the solutions in different regions can be connected by enforcing two conditions, namely the continuity of both $v$ and its derivative across the boundary surface. As mentioned earlier, for the two non-singular bouncing points $- \tau_B$ and $\tau_B$ we match the solutions at times $\mp\tau_B-\epsilon$ and $\mp\tau_B+\epsilon$, neglecting any evolution in the intervening time period. This is similar to what was done in Refs. [@Elisa; @Subodh]. A second justification of this method (in addition to the one given earlier) is that for modes we are interested in, the time interval $2\epsilon$ is so small, thus the mode functions do not have enough time to oscillate inside the Hubble radius. On the other hand, in our first model (with a flat plateau for $a(t)$, model 1) we consider the case that the interval $2\Delta$ is sufficiently long such that some of the modes we are interested in have time to oscillate while the mode is inside the Hubble radius. Very large scale modes, on the other hand, still do not oscillate inside the Hubble radius. ### Between region I and region II The matching conditions are $$\begin{aligned} v_1(-\tau_B-\epsilon) = v_2 (-\tau_B+\epsilon), \quad v'_1(-\tau_B-\epsilon) = v'_2 (-\tau_B+\epsilon) \, .\end{aligned}$$ We write down the equations explicitly in terms of coefficients $c_{ij}$ of the fundamental solutions of the equation of motion. The index $i$ stands for the region, the index $j$ (either $1$ or $2$) running over the two different modes: $$\begin{aligned} \begin{pmatrix} \epsilon^{1-q_2} & \epsilon^{q_2} \\ -(1-q_2) \epsilon^{-q_2} & -q_2\epsilon^{q_2-1} \end{pmatrix} \begin{pmatrix} c_{11} \\ c_{12} \end{pmatrix} = \begin{pmatrix} \epsilon^{1-q_1} & \epsilon^{q_1} \\ (1-q_1) \epsilon^{-q_1} & q_1 \epsilon^{q_1-1} \end{pmatrix} \begin{pmatrix} c_{21} \\ c_{22} \end{pmatrix} .\end{aligned}$$ ### Between region II and region III To be explicit, we focus in this case on the large scale modes for which the time duration of Region III is $-\Delta<\tau<\Delta$. For the small scale modes which enter the Hubble radius before $- \Delta$, we just make the substitution $\Delta\rightarrow\tau_H(k)$. Apart from that the discussion is the same. For the next subsection the convention will be the same. The matching conditions in this case are $$\begin{aligned} v_2(-\Delta) = v_3 (-\Delta),\quad v'_2(-\Delta) = v'_3 (-\Delta),\end{aligned}$$ which in matrix form can be expressed as $$\begin{aligned} \begin{pmatrix} \delta^{1-q_1} & \delta^{q_1} \\ (1-q_1) \delta^{-q_1} & q_1 \delta^{q_1-1} \end{pmatrix} \begin{pmatrix} c_{21} \\ c_{22} \end{pmatrix} = \begin{pmatrix} e^{-ik \Delta} & e^{i k \Delta} \\ i k e^{-ik\Delta} & - i k e^{i k \Delta} \end{pmatrix} \begin{pmatrix} c_{31} \\ c_{32} \end{pmatrix},\end{aligned}$$ and where in the above equation we have defined $\delta$ as $$\delta = \tau_B - \Delta . \label{delta}$$ Note that for the small scale modes the definition of $\delta$ should be changed to $\delta\to \delta(k) \equiv \tau_B-\tau_H(k)$. ### Between region III and region IV The matching conditions are $$\begin{aligned} v_3(\Delta) = v_4 (\Delta), \quad v'_3(\Delta) = v'_4 (\Delta) \, .\end{aligned}$$ In matrix form this yields $$\begin{aligned} \begin{pmatrix} e^{ik\Delta} & e^{-ik\Delta} \\ i k e^{ik\Delta} & -i k e^{-ik\Delta} \end{pmatrix} \begin{pmatrix} c_{31} \\ c_{32} \end{pmatrix} = \begin{pmatrix} \delta^{1-q_1} & \delta^{q_1} \\ -(1-q_1) \delta^{-q_1} & -q_1\delta^{q_1-1} \end{pmatrix} \begin{pmatrix} c_{41} \\ c_{42} \end{pmatrix}.\end{aligned}$$ ### Between region IV and region V The matching conditions in this case are $$\begin{aligned} v_4(\tau_B-\epsilon) = v_5 (\tau_B+\epsilon),\quad v'_4(\tau_B-\epsilon) = v'_5 (\tau_B+\epsilon) \, .\end{aligned}$$ In matrix form this yields $$\begin{aligned} \begin{pmatrix} \epsilon^{1-q_1} & \epsilon^{q_1} \\ -(1-q_1) \epsilon^{-q_1} & -q_1 \epsilon^{q_1-1} \end{pmatrix} \begin{pmatrix} c_{41} \\ c_{42} \end{pmatrix} = \begin{pmatrix} \epsilon^{1-q_2} & \epsilon^{q_2} \\ (1-q_2) \epsilon^{-q_2} & q_2 \epsilon^{q_2-1} \end{pmatrix} \begin{pmatrix} c_{51} \\ c_{52} \end{pmatrix}.\end{aligned}$$ Analysis and Results for the Power Spectrum {#sec4} =========================================== Combining the results of the previous section we find that the final mode coefficients can be written in terms of the initial ones via $$\begin{aligned} \nonumber & \mathcal C_5 = \begin{pmatrix} c_{51} \\ c_{52} \end{pmatrix},\end{aligned}$$ where[^5] $$\begin{aligned} c_{51}&=&\frac{c_{11}}{(1-2q_2)(1-2q_1)2k(1-2q_1)} \left[ a_{11} (1 - 2 (q_1-1) q_1 - 2 (q_2 -1 ) q_2) \right. \nonumber \\ &-& \left. a_{12} ( (q_1 - q_2 - 1) (q_1 + q_2 - 2) \epsilon^{1 - 2 q_1} ) + a_{21} (q_1 - q_2 + 1) (q_1 + q_2) \epsilon^{2 q_1 - 1} \right], \label{c51} \\ c_{52}&=&\frac{c_{11}}{(1-2q_2)(1-2q_1)2k(1-2q_1)} \left[ 2 a_{11} (1 + q_1 - q_2) (-2 + q_1 + q_2) \epsilon^{1 - 2 q_2} \right. \nonumber \\ &+& \left. a_{12} (-2 + q_1 + q_2)^2 \epsilon^{-2 (-1 + q_1 + q_2)} + a_{21} (1 + q_1 - q_2)^2 \epsilon^{2 (q_1 - q_2)}\right], \label{c52}\end{aligned}$$ and $$\begin{aligned} \label{a11} a_{11} & = \frac{2\left[k^2\delta^2+(q_1-1)q_1\right]\sin (2k\Delta)}{\delta} - 2 k \cos (2 k \Delta), \\ \label{a12} a_{12} & = \delta^{2q_1-2} \left[-4kq_1\delta\cos(2k\Delta)-2(q_1-k\delta)(q_1+k\delta) \sin(2k\Delta)\right], \\ \label{a21} a_{21} & = \delta^{-2q_1}\left[2(q_1-k\delta-1)(q_1+k\delta-1)\sin(2k\Delta)- 4k(q_1-1)\delta\cos(2k\Delta)\right] .\end{aligned}$$ Note that these coefficients oscillate as a function of $k$. These oscillations are important, however, only for small wavelength fluctuations. For these we will obtain oscillations in the power spectrum. The final general result for the power spectrum is given by $$\begin{aligned} P_{\zeta} & = \zeta^2 k^{3} \sim \left( \frac{v}{z} \right)^2 k^{3} \nonumber \\ & = \left[ \frac{c_{51}(\tau-\tau_B)^{1-q_2}+c_{52}(\tau-\tau_B)^{q_2}}{(\tau-\tau_B)^{q_2}} \right]^2 k^{3} \nonumber \\ &= \left[ c_{51} (\tau-\tau_B)^{1-2q_2} + c_{52} \right]^2 k^{3} \, . \label{finalPzeta}\end{aligned}$$ Below we will analyze some of the specific cases given by our two models when applying the result given by Eq. (\[finalPzeta\]). Limiting case of Instantaneous matching --------------------------------------- We first consider the limit as the duration of the plateau region of $a(t)$ goes to zero, corresponding to what we have denoted by Model 2 in Sec. \[sec2\]. This is the limit $\Delta\rightarrow 0$. In this case, large scale modes $k^{-1} > k_*^{-1}$ do not enter the Hubble radius in the region near $t = 0$, and we can set $\Delta = 0$ in the matching condition equations, i.e., $$\begin{aligned} \sin(2k\Delta) \rightarrow 0, \quad \cos(2 k \Delta) \rightarrow 1 \, .\end{aligned}$$ On the other hand, small scale modes $k^{-1} < k_*^{-1}$ will enter the Hubble radius at a time given by $-\tau_H(k)$, and in the matching condition equations we must replace $\Delta$ by $\tau_H(k)$. ### Large scale modes $k^{-1} > k_*^{-1}$ Let us first consider the case for large scale modes $k^{-1} > k_*^{-1}$. In this case we have that $$\begin{aligned} a_{11}\rightarrow -2k, \quad a_{12}\rightarrow -4k q_1\delta^{2q_1-1}, \quad a_{21}\rightarrow -4k (q_1-1)\delta^{-2q_1+1}.\end{aligned}$$ Because we are interested in the parameter region $1/3<p<1$, then, written in terms of $q$, we have $q>1/2$. So the $c_{51}$ mode in the expression for the power spectrum Eq. (\[finalPzeta\]) is a decaying solution. Hence, we can focus on the constant mode $c_{52}$, and thus the power spectrum in this case becomes $$\begin{aligned} \nonumber &P_{\zeta} \sim c_{52}^2 k^{3} =\left\{ \frac{c_{11}}{(1-2q_2)(1-2q_1)2(1-2q_1)} \right. \\ \nonumber & \left. \times \left[ -4 (1 + q_1 - q_2) (-2 + q_1 + q_2) \epsilon^{1 - 2 q_2}-4q_1\delta^{2q_1-1} (-2 + q_1 + q_2)^2 \epsilon^{-2 (-1 + q_1 + q_2)} \right. \right. \nonumber \\ & \left. \left. -4(q_1-1)\delta^{-2q_1+1} (1 + q_1 - q_2)^2 \epsilon^{2 (q_1 - q_2) } \right] \right\}^2 k^{3} .\end{aligned}$$ The initial power spectrum is $$\begin{aligned} P_i = P_{\zeta} (-\tau_B-\epsilon) = \zeta^2 k^{3} = c_{11}^2 \epsilon^{2-4q_2} k^{3}\end{aligned}$$ and, thus, we can relate the final to the initial power spectrum as $$\begin{aligned} \label{powerspectrum} P_{\zeta} = \left(A_1 + A_2 \delta^{2q_1-1} \epsilon^{1-2 q_1 } + A_3\delta^{-2q_1+1} \epsilon^{2q_1-1} \right)^2 P_i \, ,\end{aligned}$$ where $A_1$, $A_2$ and $A_3$ are constants that do not depend on $k$. Their explicit forms are $$\begin{aligned} &A_1 = \frac{-2(1+q_1-q_2)(-2+q_1+q_2)}{(1-2q_2)(1-2q_1)^2},\\ &A_2 = \frac{-2q_1(-2+q_1+q_2)^2}{(1-2q_2)(1-2q_1)^2}, \\ &A_3 = \frac{-2(q_1-1)(1+q_1-q_2)^2}{(1-2q_2)(1-2q_1)^2}.\end{aligned}$$ For very large scale modes $k^{-1} \gg k_*^{-1}$, $\delta \to \tau_B$ and $\delta$ can be approximated as a constant time interval. Thus, the power spectrum in this case becomes $$\begin{aligned} \label{powerspectrum1} (A_1 + A_2 \tau_B^{2q_1-1} \epsilon^{1-2 q_1 } + A_3 \tau_B^{-2q_1+1} \epsilon^{2q_1-1} )^2 P_i .\end{aligned}$$ The first conclusion we draw from this result is that the shape of the spectrum for large scale modes does not change during the bounce. This agrees with the conclusions of previous work on simple bounce models [@matterBounce-2]. The amplitude, on the other hand, is amplified. Recall that $2q_i - 1 > 0$, and that $\epsilon \ll \tau_B$. Hence, it is the second term in Eq. (\[powerspectrum1\]) which dominates, and we conclude that the amplitude of the spectrum is amplified by a factor of $${\cal A} \, = \, A_2^2 \bigg(\frac{\tau_B}{\epsilon}\bigg)^{4q_1 - 2} . \label{Amplitude}$$ This result can also be understood easily: Fluctuations grow both in the contracting and in the expanding phase. In fact, the fluctuations diverge in the limit when the scale factor becomes zero. Hence, without an effective cutoff $\epsilon$ we would get a divergence in the spectrum. With a cutoff, the enhancement factor of the amplitude of the power spectrum will be determined by the dimensionless ratio between $\tau_B$ and $\epsilon$ to a power which depends on the growth rate of the fluctuations on super-Hubble scales, i.e., on $q_1$ (see the discussion of these issues in a more general context in the review article Ref. [@RHBbounceReview]). ### Small scale modes $k^{-1} < k_*^{-1}$ For small scale modes $k^{-1}<k_{*}^{-1}$, we set $\delta$ equal to the Hubble crossing time. Thus, we can use the Hubble crossing condition $a H= k$, which from $a\sim \tau^q$ gives $$\begin{aligned} \delta =q_1 k^{-1} .\end{aligned}$$ But we need to have $k \tau_H = k\tau_B - k\delta= k\tau_B - q_1 $. As a consequence of the oscillations in the coefficients $a_{ij}$, Eqs. (\[a11\]), (\[a12\]) and (\[a21\]), the final power spectrum of fluctuations will oscillate for small wavelengths. This is explicitly manifested when we show a numerical example for the power spectrum in Fig. \[bounceplot1\], where we chose an initial pre-bounce spectrum which is scale-invariant. We see that the scale-invariance of the spectrum is maintained on large scales, but that on small length scales there is both a change in the slope of the spectrum, and superimposed oscillations. In the following we discuss in what range we can reproduce the results of Ref. [@RHBcyclic], which hold for a cyclic cosmology. In that work, it was found that for modes which re-enter the Hubble radius during the bounce phase, the index of the spectrum of cosmological perturbations changes during each cycle. For a matter-dominated contracting phase the change in the index $n_s$ of the power spectrum was determined to be $\Delta n_s = - 2$. The results of Ref. [@RHBcyclic] are applicable when $k^{-1}<k_*^{-1}$, but for quite large scales such that $k^{-1}\rightarrow k_*^{-1}$. In this range we have $k\tau_H\rightarrow 0$. Using this in Eq. (\[powerspectrum\]), we obtain that $$\begin{aligned} \label{powerspectrum_cyclic} P_f \, = \, \left[ A_1 + A_2 (k \epsilon/q_1)^{1-2q_1} + A_3 (k\epsilon/q_1)^{2q_1-1} \right]^2 P_i.\end{aligned}$$ Since $2q_1 - 1 > 0$ it is the second term in Eq. (\[powerspectrum\_cyclic\]) which dominates. Hence, we conclude that there is a change in the index of the power spectrum by $$\Delta n_s = 2-4 q_1= - 2 \frac{3 p_1 - 1}{1 - p_1} , \label{tiltns}$$ which coincides with the results of Ref. [@RHBcyclic]. This is as expected because the case studied in Ref. [@RHBcyclic] corresponds to a big bounce where $\delta$ is (cosmologically) large. Case with a flat plateau ------------------------ In the case with a flat plateau and when $\Delta$ is very small, we have just one characteristic comoving mometum. However, when $\Delta$ is big, we have two key comoving momenta which are characterized by the mode which cross the Hubble radius at $\Delta$ and $\tau_B- \Delta$, respectively. In this subsection, we would like to analyze in detail these two cases. First we would like to calculate the critical comoving momentum $k_*^{-1}$. We start by analyzing the Hubble parameter $H$. The corresponding comoving Hubble parameter in region II is $$\begin{aligned} aH =q_1 (\tau+\tau_B)^{-1}.\end{aligned}$$ The critical scale $k_{*}$, which is obtained by $k_*=aH(\tau=-\Delta)$, is therefore $$\begin{aligned} k_{*} = q_1 (\tau_B - \Delta)^{-1}.\end{aligned}$$ The analysis here is similar to the instantaneous matching case of the previous subsection and we can obtain the power spectrum as $$\begin{aligned} &P_{\zeta} \sim c_{52}^2 k^{3} = \left\{ \frac{ 1 }{(1-2q_2)(1-2q_1)2k(1-2q_1)} \right. \nonumber \\ & \left. \times \left[ 2 a_{11} (1 + q_1 - q_2) (-2 + q_1 + q_2) + a_{12} (-2 + q_1 + q_2)^2 \epsilon^{1 -2 q_1 } \right. \right. \nonumber \\ & \left. \left. + a_{21} (1 + q_1 - q_2)^2 \epsilon^{2 q_1 -1} \right] \right\}^2 P_i , \label{powerspectrumforinstantaneous}\end{aligned}$$ where $a_{11}$, $a_{12}$ and $a_{21}$ were already defined by Eqs. , and , respectively. Model with no Region III ------------------------ Let us here consider the model with no Region III. We expect that the result we obtained in the previous Subsection will approach the result derived here in the limit when $\Delta\rightarrow 0$. The matching condition of Region I and II, Region IV and V are completely the same as in the flat plateau case, so here we only write down the matching condition between Region II and IV: $$\begin{aligned} v_2(0) = v_4 (0),\quad v'_2(0) = v'_4 (0) ,\end{aligned}$$ which can be written in terms of the more convenient matrix form $$\begin{aligned} \begin{pmatrix} \tau_B^{1-q_1} & \tau_B^{q_1} \\ (1-q_1) \tau_B^{-q_1} & q_1 \tau_B^{q_1-1} \end{pmatrix} \begin{pmatrix} c_{21} \\ c_{22} \end{pmatrix} = \begin{pmatrix} \tau_B^{1-q_1} & \tau_B^{q_1} \\ -(1-q_1) \tau_B^{-q_1} & -q_1 \tau_B^{q_1-1} \end{pmatrix} \begin{pmatrix} c_{41} \\ c_{42} \end{pmatrix}.\end{aligned}$$ Combining these matching results we obtain the power spectrum completely the same as that of the instantaneous matching of the previous section. Numerical Examples ------------------ In Fig. \[bounceplot1\] we show the form of the spectrum of cosmological perturbations and its tilt as a function of the comoving wavenumber $k$ for the two models we have considered in the previous sections. We can clearly identify in Fig. \[bounceplot1\](a) the characteristic scales for each of the two models we have defined in Sec. \[sec2\]. For the flat plateau model (Model 1), there are two relevant comoving scales, $$\begin{aligned} &&k_{*,{\rm flat}}= \frac{q_1}{\tau_B-\Delta}, \label{kflat} \\ &&k_{*,{\rm osc}}= \frac{q_1}{\tau_B}. \label{kosc}\end{aligned}$$ In model 1 the spectrum is always evolving. On large scales $k < k_{*,{\rm osc}}$ there is both an amplification of the spectrum and a damping evolution. On small scales $k > k_{*,{\rm osc}}$ the power spectrum shows superimposed damped oscillations. In the instantaneous case $\Delta=0$ (Model 2), the characteristic comoving scale is $k_{*,{\rm inst}} \equiv k_{*,{\rm osc}}$, the same as Eq. (\[kosc\]). In the model 2, on large scales $k < k_{*,{\rm inst}}$ the spectral shape is unchanged during the bounce and only the amplitude increases, as identified in Eq. (\[Amplitude\]). On smaller scales $k > k_{*,{\rm inst}}$ there is a change in the spectral index and the power spectrum, as in the case of model 1, shows superimposed damped oscillations. The results Fig. \[bounceplot1\](b) show that for Model 1 (black dashed line) the spectral tilt always decreases with the momentum. The discontinuity at $k_{*,{\rm flat}}$ (denoted by the black dashed vertical line) is an unphysical feature that appears as a consequence of the shape we have considered and should not appear in realistic smooth shapes. The same is true for the Model 2 case (red solid line), where the discontinuity happens at the characteristic scale $k_{*,{\rm inst}}$ in this case and comes from the kink like shape considered in this model. Other than that, the spectral index is unchanged (and null) for large scales modes $k< k_{*,{\rm inst}}$ and then decreases for small scale modes $k> k_{*,{\rm inst}}$ and agrees with that of model 1 from this point on, where the index of the power spectrum for both models acquires a large red tilt, and there are superimposed oscillations. ![\[bounceplot2\] The power spectrum for different parameters, namely $q_1= q_2 = 2/3$, $\epsilon = 0.01 \tau_B$, for the cases of $\Delta=0.7 \tau_B$ (blue dotted line), and $\Delta=0.9 \tau_B$ (red solid line) for the case of the flat plateau (Model 1) and for the instantaneous case (Model 2), where $\Delta=0$ (black dashed line). Again, the vertical lines denote the positions of the characteristic scales $k_*$ for each model.](bouncefig3.pdf){width="45.00000%"} Figure \[bounceplot2\] shows the results for different parameter values, parameters for which the two scales $k_*$ for the models with and without a plateau for $a(t)$ are more widely separated than they are for the earlier parameter values. For both models there are oscillations of the power spectrum for $k$ values between the two critical $k_*$ values. These results in particular show that as $\Delta \to \tau_B$, the scale $k_{*,{\rm flat}}$ can occur deeper in the oscillating regime $k>k_{*,{\rm osc}}$ for the spectrum. A study of the oscillating regime for small scales $k>k_{*,{\rm osc}}$, and which is common for both models considered here, is given in the Appendix \[appA\]. In particular, it is shown that the envelope function of the power spectrum for the small scale modes keeps the spectral tilt $n_s = -4q_1+2$, as also seen in the previous Eq. (\[tiltns\]). Generalization to $n$ small bounces {#sec5} =================================== In this section, we would like to analyze the case where there are $n$ small bounces see Fig \[nbounceplot\]. Since the transfer matrix of the flat plateau case is quite involved, we would like to first consider Model 2 (no plateau interval) for illustrative purposes. ![\[nbounceplot\] An illustration of the generalization to n-vibrations (or small bounces). ](bouncefig4.pdf){width="45.00000%"} Based on our previous calculations, we can easily write down the transfer matrices for the coefficient vector. We define the following useful matrices $$\begin{aligned} M_1(\tau,q) = \begin{pmatrix} \tau^{1-q} & \tau^q \\ -(1-q) \tau^{-q} & -q \tau^{q-1} \end{pmatrix},\quad M_2(\tau,q) = \begin{pmatrix} \tau^{1-q} & \tau^q \\ (1-q) \tau^{-q} & q \tau^{q-1} \end{pmatrix} .\end{aligned}$$ For large scale modes, we define the combination of matrices $$\begin{aligned} N = M^{-1}_2(\tau_B,q_1)M_1(\tau_B,q_1)M^{-1}_2(\epsilon,q_1)M_1(\epsilon,q_1) ,\end{aligned}$$ which becomes $$\begin{aligned} N = \frac{1}{(1-2q_1)^2} \begin{pmatrix} 4(q_1-1)q_1 \tau_B^{2q_1-1}\epsilon^{1-2q_1} + 1 & 2q_1(\epsilon^{2q_1-1}-\tau_B^{2q_1-1}) \\ 2(q_1-1)(\tau_B^{1-2q_1}-\epsilon^{1-21_1}) & 4(q_1-1)q_1\epsilon^{2q_1-1}\tau_B^{1-2q_1}+1 \end{pmatrix}.\end{aligned}$$ Taking the two bump model as an example, we obtain the final coefficient vector to be $$\begin{aligned} \mathcal C_F = M_2^{-1} (\epsilon ,q_2) M_1(\epsilon ,q_1) N M_2^{-1} (\tau_B, q_1) M_1(\tau_B, q_1) M_2^{-1} (\epsilon,q_1) M_1(\epsilon, q_2) \mathcal C_I .\end{aligned}$$ We set the initial coefficient matrix $\mathcal C_I$ to be $$\begin{aligned} \mathcal C_I = \begin{pmatrix} c_{11} \\ 0 \end{pmatrix},\end{aligned}$$ and then we get $$\begin{aligned} \nonumber c_{52} =&\frac{1}{(1-2q_1)(1-2q_2^3)} \left[ 8 c_{11} \left(q_1-1\right){}^2 q_1 \left(q_1-q_2+1\right){}^2 \epsilon ^{4 q_1-2 q_2-1} \tau _B^{2-4 q_1} \right. \\ \nonumber & \left. -4 c_{11} \left(q_1-1\right) \left(q_1-q_2+1\right) \left(2 q_1^2+2 q_2 q_1-5 q_1+q_2-1\right) \epsilon ^{2 q_1-2 q_2} \tau _B^{1-2 q_1} \right. \\ \nonumber & \left. -4 c_{11} q_1 \left(q_1+q_2-2\right) \left(2 q_1^2-2 q_2 q_1+q_1+3 q_2-4\right) \epsilon ^{-2 q_1-2 q_2+2} \tau _B^{2 q_1-1} \right. \\ \nonumber & \left. -8 c_{11} \left(q_1-1\right) q_1^2 \left(q_1+q_2-2\right){}^2 \epsilon ^{-4 q_1-2 q_2+3} \tau _B^{4 q_1-2} \right. \\ \nonumber & \left. +2 c_{11} \left(4 q_1^4-8 q_1^3-4 q_2^2 q_1^2+16 q_2 q_1^2-10 q_1^2+4 q_2^2 q_1-16 q_2 q_1+14 q_1+2 q_2^2-5 q_2+3\right) \epsilon ^{1-2 q_2} \right] .\end{aligned}$$ On very large scale there is no change in the spectral slope, as expected. Now we want to deal with the small scale case. We need to define two more matrices $$\begin{aligned} L_1 = \begin{pmatrix} e^{-ik\tau_H} & e^{i k \tau_H} \\ ik e^{-i k \tau_H} & -i k e^{i k \tau_H} \end{pmatrix}, \quad L_2 = \begin{pmatrix} e^{ik\tau_H} & e^{-i k \tau_H} \\ ik e^{i k \tau_H} & -i k e^{-i k \tau_H} \end{pmatrix}.\end{aligned}$$ Then we have $$\begin{aligned} \nonumber \mathcal C_F = & M_2^{-1} (\epsilon,q_2) M_1(\epsilon,q_1) M_1^{-1} (\delta,q_1) L_2 L_1^{-1} M_2(\delta,q_1) M_2^{-1} (\epsilon,q_1) \\ &\times M_1(\epsilon,q_1) M_1^{-1} (\delta,q_1) L_2 L_1^{-1} M_2(\delta,q_1) M_2^{-1} (\epsilon,q_1) M_1(\epsilon,q_2) \mathcal C_I,\end{aligned}$$ and the general result has the form $$\begin{aligned} P_{\zeta} = \left[\# + \# (\epsilon k)^{2-4q_1} + \# (\epsilon k)^{4q_1-2} + \# (\epsilon k)^{1-2q_1} + \# (\epsilon k)^{2q_1-1}\right]^2 P_i \, .\end{aligned}$$ Since we are interested in modes which exit the Hubble radius before the time $-(\tau_B + \epsilon)$, we consider values of $k$ for which $\epsilon k \ll 1$. Hence, in this range of $k$ values it is the second term above which dominates and we find the scaling $$P_{\zeta} \, \sim \, (\epsilon k)^{4 - 8q_1} P_i ~.$$ Thus, small scale modes acquire a red tilt compared to the initial spectrum. If the initial spectrum is scale invariant, then the resulting spectral index for small scale modes is $$n_s - 1 = 4 - 8q_1 ~.$$ Similarly, we can obtain the spectral index change for $n$ small bounces, which is $$n_s - 1 = (2 - 4q_1)n ~.$$ Summary and Conclusions {#sec6} ======================= In this paper we have analyzed in detail the power spectrum of curvature fluctuations in a bouncing cosmology in which the bounce phase has small vibrations, i.e., small bounces. To be specific we have mostly considered the case of one small bounce with characteristic time scales $\tau_B$ and $\Delta < \tau_B$ which are much smaller than cosmological times. We have given a detailed study of the necessary matching conditions required to obtain the complete form for the power spectrum. The matchings connect at least five different phases for a given momentum scale which need to be treated with care. In our study, we have adopted two simplified models for the shape of the vibrations, allowing a complete analytical study. Despite the apparent simplicity of these models, they are already of sufficient complexity to allow to extract similar features that can emerge in more realistic models. In particular, similar structures that we have considered here can appear in bounce models coming from quantum gravity, as those recently proposed in Ref. [@Alesci], which makes this study of particular importance. Our results for the power spectrum shows that there is an amplification of its amplitude and it also tends to get redder at large scales as the number of vibrations increase. At small scales the power spectrum features superimposed damped oscillations. The reddening of the spectrum for scales which enter the small bounce agrees with the results found in Ref. [@RHBcyclic]. The oscillations in the power spectrum which are seen on small scales are reminiscent of oscillations which are obtained in some other approaches to the [*Trans-Planckian problem*]{} for cosmological fluctuations. For example, if initial conditions are set on a time-like [*new physics hypersurface*]{} [@newphysics] such that modes $k$ are initiated when the physical wavelength associated with $k$ equals a fixed physical length (e.g. the Planck length), and they are initiated in the same state (e.g. the state which locally looks like the Bunch-Davies vacuum [@BD]), then oscillations in the spectrum result. Both the qualitative and quantitative changes in the power spectrum that we have obtained can produce observed effects in spectrum of cosmological perturbations accessible through the measurements of the cosmic microwave background radiation. These effects can manifest themselves both in pure bouncing cosmologies (no subsequent inflationary period) and in scenarios where there is a post-bounce inflationary phase. For instance, those bounce vibrations can induce particle production, changing the vacuum state such as to be different from the usual Bunch-Davis one, similar to recent pre-inflationary studies in Loop Quantum Cosmology [@Zhu:2017jew]. The modifications we have obtained in this work could then be used to put constraints on these possible features that can appear in these bounce models and which deserve further study. The results we have presented here provides then an important first step in understanding these effects and which we hope to address elsewhere. Envelope of the Power Spectrum for Small Scale Modes {#appA} ==================================================== In this section, we would like to calculate the envelope of the power spectrum for small scale modes. Since the model without plateau is a special limit of the model with a non-vanishing flat plateau, we just focus on the latter. We can simply set $\Delta \rightarrow 0$ to get the answer for the model without a plateau. By collecting the relevant terms in the power spectrum, we can write it in the form $$\begin{aligned} P_{\zeta } = \left[C_1\sin(2k\Delta) + C_2 \cos(2k\Delta)\right]^2 P_i \, .\end{aligned}$$ The envelope of the power spectrum is thus $$\begin{aligned} P_{\zeta {\rm (env)}} = (C_1^2 + C_2^2 )P_i \, ,\end{aligned}$$ where the coefficients $C_1$ and $C_2$ are given by $$\begin{aligned} \nonumber C_1& = \frac{k^{-1}}{(1-2q_2)(1-2q_1)^2} \left\{ 2 \delta^{-1} \left[k^2\delta^2+(q_1-1)q_1 \right](1+q_1-q_2)(-2+q_1+q_2) \right. \\ & \left. - \delta^{2q_1-2} (q_1-k\delta) (q_1+k\delta) (-2+q_1+q_2)^2 \epsilon^{1-2q_1} + \delta^{-2q_1} (q_1-k\delta-1)(q_1+k\delta-1)(1+q_1-q_2)^2\epsilon^{2q_1-1} \right\}, \label{C1env} \\\nonumber C_2 & = \frac{1}{(1-2q_2)(1-2q_1)^2} \left[ -2(1+q_1-q_2)(-2+q_1+q_2) \right. \\ & \left. -\delta^{2q_1-1} 2 q_1 (-2+q_1+q_2)^2 \epsilon^{1-2q_1} - \delta^{-2q_1+1} 2 (q_1-1) (1+q_1-q_2)^2 \epsilon^{2q_1-1} \right] \, . \label{C2env}\end{aligned}$$ We are interested in the parameter region $\epsilon / \delta \ll1$ (recall that the time scale $\epsilon$ is expected to be of the order of the Planck scale, whereas $\delta$ will be parametrically larger since it is associated with the time scale of the bounce). We are also interested in the range of values $1/3<p<1$, or equivalently, $1/2<q<+\infty$. We can then determine which are the dominant terms in $C_1$ and $C_2$, which from Eqs. (\[C1env\]) and (\[C1env\]), they are given by $$\begin{aligned} C_1& \simeq \frac{k^{-1}}{(1-2q_2)(1-2q_1)^2} \left[ - \delta^{2q_1-2} (q_1-k\delta) (q_1+k\delta) (-2+q_1+q_2)^2 \epsilon^{1-2q_1} \right], \\ C_2 & \simeq \frac{1}{(1-2q_2)(1-2q_1)^2} \left[ -\delta^{2q_1-1} 2 q_1 (-2+q_1+q_2)^2 \epsilon^{1-2q_1} \right] \, .\end{aligned}$$ When $k$ is close to the $k_*$, then $\delta \rightarrow \tau_B-\Delta$ which is a constant. In this case, taking the square of $C_1$, we get terms with different with spectral indices, but the dominant contribution is the term with the lowest power of $k$, which the gives that the slope of the envelope (for an initial spectrum which is scale-invariant) will be $$\begin{aligned} n_s -1 = -2 ,\end{aligned}$$ because we have $k\delta<q_1$ in this range. This can be seen from the numerical results shown in Fig. \[bounceplot3\] for the two models we have considered. The change in the spectral slope is due to the matching conditions. Each time, we can get factors of $1/k$ or $k$ when we match the solution across the boundaries of Regions II and III, and of Regions III and IV. Note that in a generic case when we have a smooth evolution of the scale factor, we expect that there will be no discontinuities in the power spectrum. Thus, in a generic case, we do not expect that we always get an interval of wavenumber with a spectrum of slope $n_s=-2$. What we expect in the case of a smoothly evolving scale factor is that on very large scales, we get a scale invariant spectrum (the actual spectrum, not just the envelope), and then it will smoothly transit to a spectrum with tilt $n_s=-4q_1+2$ when we look at the envelope only. We see oscillations with amplitude given by the envelope function on intermediate and small scales. The coefficient $C_2^2$ gives a scale invariant power spectrum $$\begin{aligned} n_s -1 = 0,\end{aligned}$$ but its amplitude is suppressed by $k\delta$ compared to the amplitude of $C_1$. To be a bit more precise (still in the case of constant $\delta$), we can write $$\begin{aligned} C_1 = A_1 k^{-1} + A_2 k \, ,\end{aligned}$$ where the constants $A_1$ and $A_2$ are $$\begin{aligned} \nonumber A_1 & = \frac{1}{(1-2q_2)(1-2q_1)^2} \left[ 2 \delta^{-1} (q_1-1)q_1 (1+q_1-q_2) (-2+q_1+q_2) - \delta^{2q_1-2} q_1^2 (-2+q_1+q_2)^2 \epsilon^{1-2q_1} \right. \\ &\left. -\delta^{-2q_1} (q_1-1)^2 (1+q_1-q_2)^2 \epsilon^{2q_1-1} \right], \\ \nonumber A_2 & = \frac{1}{(1-2q_2)(1-2q_1)^2} \left[ 2\delta(1+q_1-q_2)(-2+q_1+q_2) + \delta^{2q_1} (-2+q_1+q_2)^2 \epsilon^{1-2q_1} \right. \\ &- \left. \delta^{-2q_1+2} (1+q_1-q_2)^2 \epsilon^{2q_1-1} \right] .\end{aligned}$$ The spectral index is computed as $$\begin{aligned} n_s - 1 = \frac{d\ln P_{\zeta {\rm (env)}}}{d\ln k} = \frac{2 P_i}{P_{\zeta {\rm (env)}}} (A_2^2 k^2 -A_1^2 k^{-2}).\end{aligned}$$ The power spectrum is hence comprised of several terms with different spectral tilts $n_s$ $$\begin{aligned} n_s-1 = 2, 1, 0, -1, -2,\end{aligned}$$ More generally (for larger values of $k$ when $\delta$ is not constant), we have $$\begin{aligned} P_{\zeta } = \left\{D_1\sin[2(k\tau_B-q_1)] + D_2 \cos[2k(k\tau_B-q_1)]\right\}^2 P_i \, .\end{aligned}$$ The envelope of the power spectrum is thus $$\begin{aligned} P_{\zeta {\rm (env)}} = (D_1^2 + D_2^2 )P_i \, ,\end{aligned}$$ where the coefficients $D_1$ and $D_2$ are given by $$\begin{aligned} D_1 & = \frac{1}{(1-2q_2)(1-2q_1)} \left[ -2 (1+q_1-q_2) (-2+q_1+q_2) + k^{2q_1-1} q_1^{-2q_1} (1+q_1-q_2)^2 \epsilon^{2q_1-1} \right] , \\ D_2 & = \frac{1}{(1-2q_2)(1-2q_1)^2} \left[ -2(1+q_1-q_2)(-2+q_1+q_2) + k^{1-2q_1} q_1^{2q_1-1} (-2q_1)(-2+q_1+q_2)^2\epsilon^{1-2q_1} \right. \nonumber \\ & \left. -2 k^{2q_1-1} q_1^{-2q_1+1} (q_1-1) (1+q_1-q_2)^2 \epsilon^{2q_1-1} \right] \, .\end{aligned}$$ We now can see that this envelope function reproduces the result of Ref. [@RHBcyclic]. We have $$\begin{aligned} D_1 &= B_1 + B_2 k^{2q_1-1},\\ D_2 &= E_1 + E_2 k^{1-2q_1} + E_3 k^{2q_1-1} \, ,\end{aligned}$$ where the constants $B_1$, $B_2$, $E_1$, $E_2$ and $E_3$ are given by $$\begin{aligned} B_1 &= \frac{-2 (1+q_1-q_2) (-2+q_1+q_2)}{(1-2q_2)(1-2q_1)},\\ B_2 &= \frac{ q_1^{-2q_1} (1+q_1-q_2)^2 \epsilon^{2q_1-1} }{(1-2q_2)(1-2q_1)},\\ E_1 &= \frac{-2(1+q_1-q_2)(-2+q_1+q_2)}{(1-2q_2)(1-2q_1)^2},\\ E_2 &= \frac{q_1^{2q_1-1} (-2q_1)(-2+q_1+q_2)^2\epsilon^{1-2q_1}}{(1-2q_2)(1-2q_1)^2}, \\ E_3 &= \frac{-2q_1^{-2q_1+1} (q_1-1) (1+q_1-q_2)^2 \epsilon^{2q_1-1}}{(1-2q_2)(1-2q_1)^2} .\end{aligned}$$ The spectral tilt is then given by $$\begin{aligned} n_s - 1 & = \frac{P_i}{P_{\zeta {\rm (env)}}} \left\{ 2 D_1 B_2 (2q_1-1) k^{2q_1-1} + 2 D_2 \left[E_2(1-2q_1) k^{-2q_1+1} + E_3 (2q_1-1) k^{2q_1-1} \right] \right\} \, . \label{nsenv}\end{aligned}$$ The expression (\[nsenv\]) is comprised of several terms with spectral tilts $n_s$ given by $$\begin{aligned} n_s-1 = 4q_1-2, 2q_1-1, 0, -4q_1+2, -2q_1+1 \, .\end{aligned}$$ Since are interested in modes with $k\epsilon<1$ and parameter values $1/3<p<1$ (or, equivalently, $1/2<q<+\infty$) we can determine the dominant terms in $D_1$ and $D_2$ and find them to be $$\begin{aligned} D_1 \rightarrow 0,\quad D_2 \rightarrow E_2 k^{1-2q_1} \, .\end{aligned}$$ Thus, the dominant contribution to the power spectrum is $$\begin{aligned} P_{\zeta {\rm (env)} } = E_2^2 k^{2-4q_1} P_i \, ,\end{aligned}$$ which corresponds to a spectral tilt of $$\begin{aligned} n_s = -4q_1+2 \, .\end{aligned}$$ Acknowledgments {#acknowledgments .unnumbered} =============== One of us (R.B.) is grateful to Emanuele Alesci and Stefano Liberati for discussions about the model of [@Alesci] which led to this project. He also thanks Stefano Liberati and the other organizers of the [*Probing the Spacetime Fabric: from Concepts to Phenomenology*]{} workshop help in July 2017 at SISSA for inviting him to participate and speak. The research at McGill was supported in part by an NSERC Discovery grant and by the Canada Research Chair program. Q.L acknowledge financial support from the University of Science and Technology of China, and from the CAST Young Elite Scientists Sponsorship Program (2016QNRC001), and by the NSFC (grant Nos. 11421303, 11653002). SZ is supported by the Hong Kong PhD Fellowship Scheme (HKPFS) issued by the Research Grants Council (RGC) of Hong Kong. R.O.R is partially supported by research grants from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), grant No. 303377/2013-5 and Fundação Carlos Chagas Filho de Amparo á Pesquisa do Estado do Rio de Janeiro (FAPERJ), grant No. E - 26/201.424/2014. 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--- abstract: 'The Fuchs-Peres-Brandt (FPB) probe realizes the most powerful individual attack on Bennett-Brassard 1984 quantum key distribution (BB84 QKD) by means of a single controlled-NOT (CNOT) gate. This paper describes a complete physical simulation of the FPB-probe attack on polarization-based BB84 QKD using a deterministic CNOT constructed from single-photon two-qubit quantum logic. Adding polarization-preserving quantum nondemolition measurements of photon number to this configuration converts the physical simulation into a true deterministic realization of the FPB attack.' author: - 'Jeffrey H. Shapiro' - 'Franco N. C. Wong' title: 'Attacking quantum key distribution with single-photon two-qubit quantum logic' --- Introduction ============ Bennett-Brassard 1984 quantum key distribution (BB84 QKD) using single-photon polarization states works as follows [@BB84]. In each time interval allotted for a bit, Alice transmits a single photon in a randomly selected polarization, chosen from horizontal ($H$), vertical ($V$), $+$45$^\circ$, or $-$45$^\circ$, while Bob randomly chooses to detect photons in either the $H$/$V$ or $\pm$45$^\circ$ bases. Bob discloses to Alice the sequence of bit intervals and associated measurement bases for which he has detections. Alice then informs Bob which detections occurred in bases coincident with the ones that she used. These are the *sift events, i.e., bit intervals in which Bob has a detection *andhis count has occurred in the same basis that Alice used. An *error  event is a sift event in which Bob decodes the incorrect bit value. Alice and Bob employ a prescribed set of operations to identify errors in their sifted bits, correct these errors, and apply sufficient privacy amplification to deny useful key information to any potential eavesdropper (Eve). At the end of the full QKD procedure, Alice and Bob have a shared one-time pad with which they can communicate in complete security.*** In long-distance QKD systems, most of Alice’s photons will go undetected, owing to propagation loss and detector inefficiencies. Dark counts and, for atmospheric QKD systems, background counts can cause error events in these systems, as can intrusion by Eve. Employing an attenuated laser source, in lieu of a true single-photon source, further reduces QKD performance as such sources are typically run at less than one photon on average per bit interval, and the occurrence of multi-photon events, although rare at low average photon number, opens up additional vulnerability. Security proofs have been published for ideal BB84 [@security1], as have security analyses that incorporate a variety of non-idealities [@security2]. Our attention, however, will be directed toward attacking BB84 QKD, as to our knowledge no such experiments have been performed, although a variety of potentially practical approaches have been discussed [@attacks]. Our particular objective will be to show that current technology permits physical simulation of the Fuchs-Peres-Brandt (FPB) probe [@FPB], i.e., the most powerful individual attack on single-photon BB84, and that developments underway in quantum nondemolition (QND) detection may soon turn this physical simulation into a full implementation of the attack. Thus we believe it is of interest to construct the physical simulation and put BB84’s security to the test: how much information can Eve really derive about the key that Alice and Bob have distilled while keeping Alice and Bob oblivious to her presence. The remainder of this paper is organized as follows. In Sec. II we review the FPB probe and its theoretical performance. In Sec. III we describe a complete physical simulation of this probe constructed from single-photon two-qubit (SPTQ) quantum logic. We conclude, in Sec. IV, by showing how the addition of polarization-preserving QND measurements of photon number can convert this physical simulation into a true deterministic realization of the FPB attack on polarization-based BB84. The Fuchs-Peres-Brandt Probe ============================ In an individual attack on single-photon BB84 QKD, Eve probes Alice’s photons one at a time. In a collective attack, Eve’s measurements probe groups of Alice’s photons. Less is known about collective attacks [@collective], so we will limit our consideration to individual attacks. Fuchs and Peres [@FP] described the most general way in which an individual attack could be performed. Eve supplies a probe photon and lets it interact with Alice’s photon in a unitary manner. Eve then sends Alice’s photon to Bob, and performs a probability operator-valued measurement (POVM) on the probe photon she has retained. Slutsky [*et al.*]{} [@Slutsky] demonstrated that the Fuchs-Peres construct—with the appropriate choice of probe state, interaction, and measurement—affords Eve the maximum amount of Rényi information about the error-free sifted bits that Bob receives for a given level of disturbance, i.e., for a given probability that a sifted bit will be received in error. Brandt [@FPB] extended the Slutsky [*et al.*]{} treatment by showing that the optimal probe could be realized with a single CNOT gate. Figure 1 shows an abstract diagram of the resulting Fuchs-Peres-Brandt probe. In what follows we give a brief review of its structure and performance—see [@FPB] for a more detailed treatment—where, for simplicity, we assume ideal conditions in which Alice transmits a single photon per bit interval, there is no propagation loss and no extraneous (background) light collection, and both Eve and Bob have unity quantum efficiency photodetectors with no dark counts. These ideal conditions imply there will not be any errors on sifted bits in the absence of eavesdropping; the case of more realistic conditions will be discussed briefly in Sec. IV. In each bit interval Alice transmits, at random, a single photon in one of the four BB84 polarization states. Eve uses this photon as the control-qubit input to a CNOT gate whose computational basis—relative to the BB84 polarization states—is shown in Fig. 2, namely $$\begin{aligned} |0\rangle &\equiv& \cos(\pi/8)|H\rangle + \sin(\pi/8)|V\rangle \\ %[.05in] |1\rangle &\equiv& -\sin(\pi/8)|H\rangle + \cos(\pi/8)|V\rangle,\end{aligned}$$ in terms of the $H/V$ basis. Eve supplies her own probe photon, as the target-qubit input to this CNOT gate, in the state $$|T_{\rm in}\rangle \equiv C|+\rangle + S|-\rangle, \label{probeinput}$$ where $C = \sqrt{1-2P_E}$, $S = \sqrt{2P_E}$, $|\pm\rangle = (|0\rangle \pm |1\rangle)/\sqrt{2}$, and $0\le P_E\le 1/2$ will turn out to be the error probability that Eve’s probe creates on Bob’s sifted bits [@footnote1]. So, as $P_E$ increases from 0 to 1/2, $|T_{\rm in}\rangle$ goes from $|+\rangle$ to $|-\rangle$. The (unnormalized) output states that may occur for this target qubit are $$\begin{aligned} |T_\pm\rangle &\equiv& C|+\rangle \pm \frac{S}{\sqrt{2}}|-\rangle \\ %[.05in] |T_E\rangle &\equiv& \frac{S}{\sqrt{2}}|-\rangle.\end{aligned}$$ Here is how the FPB probe works. When Alice uses the $H/V$ basis for her photon transmission, Eve’s CNOT gate effects the following transformation, $$\begin{aligned} |H\rangle|T_{\rm in}\rangle &\longrightarrow& |H\rangle|T_-\rangle + |V\rangle|T_E\rangle \label{Hin_out} \\ %[.05in] |V\rangle|T_{\rm in}\rangle &\longrightarrow& |V\rangle|T_+\rangle +|H\rangle|T_E\rangle,\label{Vin_out} %\\[-.075in] \nonumber\end{aligned}$$ where the kets on the left-hand side denote the Alice$\otimes$Eve state of the control and target qubits at the CNOT’s input and the kets on the right-hand side denote the Bob$\otimes$Eve state of the control and target qubits at the CNOT’s output. Similarly, when Alice uses the $\pm 45^\circ$ basis, Eve’s CNOT gate has the following behavior, $$\begin{aligned} |\mbox{$+$}45^\circ\rangle|T_{\rm in}\rangle &\longrightarrow& |\mbox{$+$}45^\circ\rangle|T_+\rangle + |\mbox{$-$}45^\circ\rangle|T_E\rangle \label{plus_in_out}\\ %[.05in] |\mbox{$-$}45^\circ\rangle|T_{\rm in}\rangle &\longrightarrow& |\mbox{$-$}45^\circ\rangle|T_-\rangle +|\mbox{$+$}45^\circ\rangle|T_E\rangle. \label{minus_in_out}\end{aligned}$$ Suppose that Bob measures in the basis that Alice has employed *and his outcome matches what Alice sent. Then Eve can learn their shared bit value, once Bob discloses his measurement basis, by distinguishing between the $|T_+\rangle$ and $|T_-\rangle$ output states for her target qubit. Of course, this knowledge comes at a cost: Eve has caused an error event whenever Alice and Bob choose a common basis and her target qubit’s output state is $|T_E\rangle$. To maximize the information she derives from this intrusion, Eve applies the minimum error probability receiver for distinguishing between the single-photon polarization states $|T_+\rangle$ and $|T_-\rangle$. This is a projective measurement onto the polarization basis $\{|d_+\rangle,|d_-\rangle\}$, shown in Fig. 3 and given by $$\begin{aligned} |d_+\rangle &=& \frac{|+\rangle + |-\rangle}{\sqrt{2}} = |0\rangle \\ |d_-\rangle &=& \frac{|+\rangle - |-\rangle}{\sqrt{2}} = |1\rangle. \end{aligned}$$* Two straightforward calculations will now complete our review of the FPB probe. First, we find the error probability that is created by Eve’s presence. Suppose Alice and Bob use the $H/V$ basis and Alice has sent $|H\rangle$. Alice and Bob will incur an error if the control$\otimes$target output from Eve’s CNOT gate is $|V\rangle|T_E\rangle$. The probability that this occurs is $\langle T_E| T_E\rangle = S^2/2 = P_E$. The same conditional error probability ensues for the other three error events, e.g., when Alice and Bob use the $\pm 45^\circ$ basis, Alice sends $|$$+45^\circ\rangle$, and the CNOT output is $|$$-45^\circ\rangle|T_E\rangle$. It follows that the unconditional error probability incurred by Alice and Bob on their sift events is $P_E$. Now we shall determine the Rényi information that Eve derives about the sift events for which Alice and Bob do not suffer errors. Let $B = \{0,1\}$ and $E = \{0,1\}$ denote the ensembles of possible bit values that Bob and Eve receive on a sift event in which Bob’s bit value agrees with Alice’s. The Rényi information (in bits) that Eve learns about each Alice/Bob error-free sift event is $$\begin{aligned} I_R &\equiv& -\log_2\!\left(\sum_{b= 0}^1P^2(b)\right) \nonumber \\ &+&\sum_{e = 0}^1P(e)\log_2\!\left(\sum_{b = 0}^1 P^2(b\mid e)\right),\end{aligned}$$ where $\{P(b), P(e)\}$ are the prior probabilities for Bob’s and Eve’s bit values, and $P(b\mid e)$ is the conditional probability for Bob’s bit value to be $b$ given that Eve’s is $e$. Alice’s bits are equally likely to be 0 or 1, and Eve’s conditional error probabilities satisfy [@Helstrom] $$\begin{aligned} \lefteqn{P(e = 1\mid b = 0) = P(e = 0\mid b = 1)} \\ &=& \frac{1}{2}\!\left(1 - \sqrt{1 - \frac{|\langle T_+|T_-\rangle|^2}{\langle T_+|T_+\rangle \langle T_-|T_-\rangle}}\right) \\ &=& \frac{1}{2}\!\left(1- \frac{\sqrt{4P_E(1-2P_E)}}{1-P_E}\right).\end{aligned}$$ These results imply that $b$ is also equally likely to be 0 or 1, and that $P(b\mid e) = P(e\mid b)$, whence $$I_R = \log_2\!\left(1 + \frac{4P_E(1-2P_E)}{(1-P_E)^2}\right),$$ which we have plotted in Fig. 4. Figure 4 reveals several noteworthy performance points for the FPB probe. The $I_R = 0, P_E = 0$ point in this figure corresponds to Eve’s operating her CNOT gate with $|T_{\rm in}\rangle = |+\rangle$ for its target qubit input. It is well known that such an input is unaffected by and does not affect the control qubit. Thus Bob suffers no errors but Eve gets no Rényi information. The $I_R = 1, P_E = 1/3$ point in this figure corresponds to Eve’s operating her CNOT gate with $|T_{\rm in}\rangle = \sqrt{1/3}|+\rangle + \sqrt{2/3}|-\rangle$, which leads to $|T_\pm\rangle \propto |d_\pm\rangle$. In this case Eve’s Fig. 3 receiver makes no errors, so she obtains the maximum (1 bit) Rényi information about each of Bob’s error-free bits. The $I_R = 0, P_E = 1/2$ point in this figure corresponds to Eve’s operating her CNOT gate with $|T_{\rm in}\rangle = |-\rangle$, which gives $|T_+\rangle = |T_-\rangle = |T_E\rangle = \sqrt{1/2}|-\rangle$. Here it is clear that Eve gains no information about Bob’s error-free bits, but his error probability is 1/2 because of the action of the $|-\rangle$ target qubit on the control qubit. Physical Simulation in SPTQ Logic ================================= In single-photon two-qubit quantum logic, each photon encodes two independently controllable qubits [@SPTQ1]. One of these is the familiar polarization qubit, with basis $\{|H\rangle,|V\rangle\}$. The other we shall term the momentum qubit—because our physical simulation of the FPB probe will rely on the polarization-momentum hyperentangled photon pairs produced by type-II phase matched spontaneous parametric downconversion (SPDC)—although in the collimated configuration in which SPTQ is implemented its basis states are single-photon kets for right and left beam positions (spatial modes), denoted $\{|R\rangle, |L\rangle\}$. Unlike the gates proposed for linear optics quantum computing [@KLM], which are scalable but non-deterministic, SPTQ quantum logic is deterministic but not scalable. Nevertheless, SPTQ quantum logic suffices for a complete physical simulation of polarization-based BB84 being attacked with the FPB probe, as we shall show. Before doing so, however, we need to comment on the gates that have been demonstrated in SPTQ logic. It is well known that single qubit rotations and CNOT gates form a universal set for quantum computation. In SPTQ quantum logic, polarization-qubit rotations are easily accomplished with wave plates, just as is done in linear optics quantum computing. Momentum-qubit rotations are realized by first performing a SWAP operation, to exchange the polarization and momentum qubits, then rotating the polarization qubit, and finally performing another SWAP. The SWAP operation is a cascade of three CNOTs, as shown in Fig. 5. For its implementation in SPTQ quantum logic the left and right CNOTs in Fig. 5 are momentum-controlled NOT gates (M-CNOTs) and the middle CNOT is a polarization-controlled NOT gate (P-CNOT). (An M-CNOT uses the momentum qubit of a single photon to perform the controlled-NOT operation on the polarization qubit of that same photon, and vice versa for the P-CNOT gate.) Experimental demonstrations of deterministic M-CNOT, P-CNOT, and SWAP gates are reported in [@SPTQ1; @SPTQ2]. Figure 6 shows a physical simulation of polarization-based BB84 under FPB attack when Alice has a single-photon source and Bob employs active basis selection; Fig. 7 shows the modification needed to accommodate Bob’s using passive basis selection. In either case, Alice uses a polarizing beam splitter and an electro-optic modulator, as a controllable half-wave plate (HWP), to set the randomly-selected BB84 polarization state for each photon she transmits. Moreover, she employs a single spatial mode, which we assume coincides with the $R$ beam position in Eve’s apparatus. Eve then begins her attack by imposing the probe state $|T_{\rm in}\rangle$ on the momentum qubit. She does this by applying a SWAP gate, to exchange the momentum and polarization qubits of Alice’s photon, rotating the resulting polarization qubit (with the HWP in Fig. 6) to the $|T_{\rm in}\rangle$ state, and then using another SWAP to switch this state into the momentum qubit. This procedure leaves Alice’s BB84 polarization state unaffected, although her photon, which will ultimately propagate on to Bob, is no longer in a single spatial mode. Eve completes the first stage of her attack by sending Alice’s photon through a P-CNOT gate, which will accomplish the state transformations given in Eqs. (\[Hin\_out\])–(\[minus\_in\_out\]), and then routing it to Bob. If Bob employs active basis selection (Fig. 6), then in each bit interval he will use an electro-optic modulator—as a controllable HWP—plus a polarizing beam splitter to set the randomly-selected polarization basis for his measurement. The functioning of this basis-selection setup is unaffected by Alice’s photon no longer being in a single spatial mode. The reason that we call Fig. 6 a physical simulation, rather than a true attack, lies in the measurement box. Here, Eve has invaded Bob’s turf, and inserted SWAP gates, half-wave plates, polarizing beam splitters, and additional photodetectors, so that she can forward to Bob measurement results corresponding to photon counting on the polarization basis that he has selected while she retains the photon counting results corresponding to her $\{|d_+\rangle, |d_-\rangle\}$ measurement. Clearly Bob would never knowingly permit Eve to intrude into his receiver box in this manner. Moreover, if Eve could do so, she would not bother with an FPB probe as she could directly observe Bob’s bit values. If Bob employs passive basis selection (Fig. 7), then he uses a 50/50 beam splitter followed by static-HWP analysis in the $H$-$V$ and $\pm 45^\circ$ bases, with only the former being explicitly shown in Fig. 7. The rest of Eve’s attack mimics what was seen in Fig. 6, i.e., she gets inside Bob’s measurement boxes with SWAP gates, half-wave plates, and additional detectors so that she can perform her probe measurement while providing Bob with his BB84 polarization-measurement data. Because the Fig. 7 arrangement requires that twice as many SWAP gates, twice as many half-wave plates, and twice as many single-photon detectors be inserted into Bob’s receiver system, as compared to what is needed in the Fig. 6 setup, we shall limit the rest of our discussion to the case of active basis selection as it leads to a more parsimonious physical simulation of the Fuchs-Peres-Brandt attack. We recognize, of course, that the decision to use active basis selection is Bob’s to make, not Eve’s. More importantly, however, in Sec. IV we will show how the availability of polarization-preserving QND photon-number measurements can be used to turn Fig. 6 into a true, deterministic implementation of the FPB attack. The same conversion can be accomplished for passive basis selection. Before turning to the true-attack implementation, let us flesh out some details of the measurement box in Fig. 6 and show how SPDC can be used, in lieu of the single-photon source, to perform this physical simulation. Let $|\psi_{\rm out}\rangle$ denote the polarization$\otimes$momentum state at the output of Eve’s P-CNOT gate in Fig. 6. Bob’s polarization analysis box splits this state, according to the basis he has chosen, so that one basis state goes to the upper branch of the measurement box while the other goes to the lower branch of that box. This polarization sorting does nothing to the momentum qubit, so the SWAP gates, half-wave plates, and polarizing beam splitters that Eve has inserted into the measurement box accomplish her $\{|d_+\rangle, |d_-\rangle\}$ projective measurement, i.e., the horizontal paths into photodetectors in Fig. 6 are projecting the momentum qubit of $|\psi_{\rm out}\rangle$ onto $|d_-\rangle$ and the vertical paths into photodetectors in Fig. 6 are projecting this state onto $|d_+\rangle$. Eve records the combined results of the two $|d_+\rangle$ versus $|d_-\rangle$ detections, whereas Bob, who only sees the combined photodetections for the upper and lower branches entering the measurement box, gets his BB84 polarization data. Bob’s data is impaired, of course, by the effect of Eve’s P-CNOT. Single-photon on-demand sources are now under development at several institutions [@single], and their use in BB84 QKD has been demonstrated [@singleBB84]. At present, however, it is much more practical to use SPDC as a heralded source of single photons [@herald]. In SPDC, signal and idler photons are emitted in pairs, thus detection of the signal photon heralds the presence of the idler photon. Moreover, with appropriate configurations [@bidirectional], SPDC will produce photons that are simultaneously entangled in polarization and in momentum. This hyperentanglement leads us to propose the Fig. 8 configuration for physically simulating the FPB-probe attack on BB84. Here, a pump laser drives SPDC in a type-II phase matched $\chi^{(2)}$ crystal, such as periodically-poled potassium titanyl phosphate (PPKTP), producing pairs of orthogonally-polarized, frequency-degenerate photons that are entangled in both polarization and momentum. The first polarizing beam splitter transmits a horizontally-polarized photon and reflects a vertically-polarized photon while preserving their momentum entanglement. Eve uses a SWAP gate and (half-wave plate plus polarizing beam splitter) polarization rotation so that her photodetector’s clicking will, by virtue of the momentum entanglement, herald the setting of the desired $|T_{\rm in}\rangle$ momentum-qubit state on the horizontally-polarized photon emerging from the first polarizing beam splitter. Alice’s electronically controllable half-wave plate sets the BB84 polarization qubit on this photon, and the rest of the Fig. 8 configuration is identical to that shown and explained in Fig. 6. Inasmuch as the SPDC source and SPTQ gates needed to realize the Fig. 8 setup have been demonstrated, we propose that such an experiment be performed. Simultaneous recording of Alice’s polarization choices, Bob’s polarization measurements and Eve’s $|d_+\rangle$ versus $|d_-\rangle$ results can then be processed through the BB84 protocol stack to study the degree to which the security proofs and eavesdropping analyses stand up to experimental scrutiny. The Complete Attack =================== Although the FPB attack’s physical simulation, as described in the preceding section, is both experimentally feasible and technically informative, any vulnerabilities it might reveal would only be of academic interest were there no practical means to turn it into a true deterministic implementation in which Eve did *not need to invade Bob’s receiver. Quantum nondemolition measurement technology provides the key to creating this complete attack. As shown in the appendix, it is possible, in principle, to use cross-phase modulation between a strong coherent-state probe beam and an arbitrarily polarized signal beam to make a QND measurement of the signal beam’s total photon number while preserving its polarization state. Cross-phase modulation QND measurement of photon number has long been a topic of interest in quantum optics [@Imoto], and recent theory has shown that it provides an excellent new route to photonic quantum computation [@Nemoto]. Thus it is not unwarranted to presume that polarization-preserving QND measurement of total photon number may be developed. With such technology in hand, the FPB-probe attack shown in Fig. 9 becomes viable. Here, Eve imposes a momentum qubit on Alice’s polarization-encoded photon and performs a P-CNOT operation exactly as discussed in conjunction with Figs. 6 and 8. Now, however, Eve uses a SWAP-gate half-wave plate combination so that the $|d_+\rangle$ and $|d_-\rangle$ momentum qubit states emerging from her P-CNOT become $|V\rangle$ and $|H\rangle$ states entering the polarizing beam splitter that follows the half-wave plate. This beam splitter routes these polarizations into its transmitted and reflected output ports, respectively, where, in each arm, Eve employs a SWAP gate, a polarization-preserving QND measurement of total photon number, and another SWAP gate. The first of these SWAPs returns Alice’s BB84 qubit to polarization, so that a click on Eve’s polarization-preserving QND apparatus completes her $\{|d_+\rangle, |d_-\rangle\}$ measurement without further scrambling Alice’s BB84 qubit beyond what has already occurred in Eve’s P-CNOT gate. The SWAP gates that follow the QND boxes then restore definite ($V$ and $H$) polarizations to the light in the upper and lower branches so that they may be recombined on a polarizing beam splitter. The SWAP gate that follows this recombination then returns the BB84 qubit riding on Alice’s photon to polarization for transmission to and measurement by Bob. This photon is no longer in the single spatial mode emitted by Alice’s transmitter, hence Bob could use spatial-mode discrimination to infer the presence of Eve, regardless of the $P_E$ value she had chosen to impose. Eve, however, can preclude that possibility. Because the result of her $\{|d_+\rangle,|d_-\rangle\}$ measurement tells her the value of the momentum qubit on the photon being sent to Bob, she can employ an additional stage of qubit rotation to restore this momentum qubit to the $|R\rangle$ state corresponding to Alice’s transmission. Also, should Alice try to defeat Eve’s FPB probe by augmenting her BB84 polarization qubit with a randomly-chosen momentum qubit, Eve can use a QND measurement setup like that shown in Fig. 9 to collapse the value of that momentum qubit to $|R\rangle$ or $|L\rangle$, and then rotate that momentum qubit into the $|R\rangle$-state spatial mode before applying the FPB-probe attack. At the conclusion of her attack, she can then randomize the momentum qubit on the photon that will be routed on to Bob without further impact—beyond that imposed by her P-CNOT gate—on that photon’s polarization qubit. So, unless Alice and Bob generalize their polarization-based BB84 protocol to include cooperative examination of the momentum qubit, Alice’s randomization of that qubit will neither affect Eve’s FPB attack, nor provide Alice and Bob with any additional evidence, beyond that obtained from the occurrence of errors on sifted bits, of Eve’s presence.* Some concluding remarks are now in order. We have shown that a physical simulation of the Fuchs-Peres-Brandt attack on polarization-based BB84 is feasible with currently available technology, and we have argued that the development of polarization-preserving QND technology for measuring total photon number will permit mounting of a true deterministic FBP-probe attack. Our analysis has presumed ideal conditions in which Alice employs a single-photon source, there is no propagation loss and no extraneous (background) light collection, and both Eve and Bob have unity quantum efficiency photodetectors with no dark counts. Because current QKD systems typically employ attenuated laser sources, and suffer from propagation loss, photodetector inefficiencies, and extraneous counts, it behooves us to at least comment on how such non-idealities could impact the FPB probe we have described. The use of an attenuated laser source poses no problem for the configurations shown in Figs. 6–9. This is because the single-qubit rotations and the CNOT gates of SPTQ quantum logic effect the same transformations on coherent states as they do on single-photon states. For example, the same half-wave plate setting that rotates the single-photon $|H\rangle$ qubit into the single-photon $|V\rangle$ qubit will transform the horizontally-polarized coherent state $|\alpha\rangle_H$ into the vertically-polarized coherent state $|\alpha\rangle_V$. Likewise, the SPTQ P-CNOT gate that transforms a single photon carrying polarization ($|H\rangle = |0\rangle, |V\rangle = |1\rangle$) and momentum ($|R\rangle = |0\rangle, |L\rangle = |1\rangle$) qubits according to $$\begin{aligned} \lefteqn{c_{HR}|HR\rangle + c_{HL}|HL\rangle + c_{VR}|VR\rangle + c_{VL}|VL\rangle \longrightarrow } \nonumber \\ &&c_{HR}|HR\rangle + c_{HL}|HL\rangle + c_{VR}|VL\rangle + c_{VL}|VR\rangle,\end{aligned}$$ will transform the four-mode coherent-state input with eigenvalues $$\begin{aligned} \lefteqn{\hspace*{-.5in} \left[\begin{array}{cccc} \langle\hat{a}_{HR}\rangle & \langle\hat{a}_{HL}\rangle & \langle\hat{a}_{VR}\rangle & \langle\hat{a}_{VL}\rangle \end{array}\right] = }\nonumber \\ &&\hspace*{.25in}\left[\begin{array}{cccc} \alpha_{HR} & \alpha_{HL} & \alpha_{VR} & \alpha_{VL}\end{array}\right],\end{aligned}$$ into a four-mode coherent-state output with eigenvalues $$\begin{aligned} \lefteqn{\hspace*{-.5in}\left[\begin{array}{cccc} \langle\hat{a}_{HR}\rangle & \langle\hat{a}_{HL}\rangle & \langle\hat{a}_{VR}\rangle & \langle\hat{a}_{VL}\rangle \end{array}\right] = }\nonumber \\ &&\hspace*{.25in} \left[\begin{array}{cccc} \alpha_{HR} & \alpha_{HL} & \alpha_{VL} & \alpha_{VR}\end{array}\right],\end{aligned}$$ where the $\hat{a}$’s are annihilation operators for modes labeled by their polarization and beam positions. It follows that the coherent-state $P_E$ and $I_B$ calculations mimic the qubit derivations that we presented in Sec. III, with coherent-state inner products taking the place of qubit-state inner products. At low average photon number, these coherent-state results reduce to the qubit expressions for events which give rise to clicks in the photodetectors shown in Figs. 6–9. Finally, a word about propagation loss, detector inefficiencies, and extraneous counts from dark current or background light is in order. All of these non-idealities actually help our Eve, in that they lead to a non-zero quantum bit error rate between Alice and Bob in the absence of the FPB attack. If Eve’s $P_E$ value is set below that baseline error rate, then her presence should be undetectable. The authors acknowledge useful technical discussions with Howard Brandt, Jonathan Smith and Stewart Personick. This work was supported by the Department of Defense Multidisciplinary University Research Initiative program under Army Research Office grant DAAD-19-00-1-0177 and by MIT Lincoln Laboratory. QND Measurement =============== Here we show that it is possible, in principle, to use cross-phase modulation between a strong coherent-state probe beam and an arbitrarily-polarized signal beam to make a QND measurement of the signal beam’s total photon number. Let $\{\hat{a}_H, \hat{a}_V, \hat{a}_P\}$ be the annihilation operators of the horizontal and vertical polarizations of the signal beam and the (single-polarization) probe beam, respectively at the input to a cross-phase modulation interaction. We shall take that interaction to transform these annihilation operators according to the following commutator-preserving unitary operation, $$\begin{aligned} \hat{a}_H &\longrightarrow& \hat{a}'_H \equiv \exp(i\kappa \hat{a}_P^\dagger\hat{a}_P)\hat{a}_H \\ \hat{a}_V &\longrightarrow& \hat{a}'_V \equiv \exp(i\kappa \hat{a}_P^\dagger\hat{a}_P)\hat{a}_V\\ \hat{a}_P &\longrightarrow&\hat{a}'_P \equiv \exp[i\kappa(\hat{a}_H^\dagger\hat{a}_H + \hat{a}_V^\dagger\hat{a}_V)]\hat{a}_P, \end{aligned}$$ where $0 < \kappa \ll 1$ is the cross-phase modulation coupling coefficient. When the probe beam is in a strong coherent state, $|\sqrt{N}_P\rangle$ with $N_P\gg 1/\kappa^2$, the total photon number in the signal beam can be inferred from a homodyne-detection measurement of the appropriate probe quadrature. In particular, the state of $\hat{a}'_P$ will be $|\sqrt{N}_P\rangle$ when the signal beam’s total photon number is zero, and its state will be $|(1+i\kappa)\sqrt{N}_P\rangle$ when the signal beam’s total photon number is one, where $\kappa \ll 1$ has been employed. Homodyne detection of the $\hat{a}'_{P2} \equiv {\rm Im}(\hat{a}'_P)$ quadrature thus yields a classical random-variable outcome $\alpha'_{P2}$ that is Gaussian distributed with mean zero and variance 1/4, in the absence of a signal-beam photon, and Gaussian distributed with mean $\kappa\sqrt{N}_P$ and variance 1/4 in the presence of a signal-beam photon. Note that these conditional distributions are independent of the polarization state of the signal-beam photon when it is present. Using the decision rule, “declare signal-beam photon present if and only if $\alpha'_{P2} > \kappa\sqrt{N}_P/2$,” it is easily shown that the QND error probability is bounded above by $\exp(-\kappa^2 N_P/2)/2 \ll 1$. The preceding polarization independent, low error probability QND detection of the signal beam’s total photon number does *not disturb the polarization state of that beam. This is so because the probe imposes the same nonlinear phase shift on both the $H$ and $V$ polarizations of the signal beam. Hence, if the signal-beam input is in the arbitrarily-polarized single-photon state, $$|\psi_S\rangle = c_H |1\rangle_{H}|0\rangle_{V} + c_V |0\rangle_{H}|1\rangle_{V}, \vspace*{.075in}$$ where $|c_H|^2 + |c_V|^2 = 1$, then, except for a physically unimportant absolute phase, the signal-beam output will also be in the state $|\psi_S\rangle$.* [2]{} C. H. Bennett and G. Brassard, *Proc. of IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India, 1984, p. 175 (IEEE, New York, 1984); see, e.g., N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. [**74,**]{} 145 (2002) for a review of progress in both theory and experiment. P. W. Shor and J. 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Fuchs and A. Peres, Phys. Rev. A [**53,**]{} 2038 (1996). B. A Slutsky, R. Rao, P,-C. Sun, and Y. Fainman, Phys. Rev. A [**57,**]{} 2383 (1998). Equation (\[probeinput\]) corrects an error in [@FPB]. Because of an extraneous root problem, the expression for Eve’s target-qubit input state in that paper is only correct for $0\le P_E\le 1/4$. This can be seen by comparing the input probe state $|A_2\rangle$ from Eq. (207) of [@FPB] with our target-qubit input state $|T_{\rm in}\rangle$ from Eq. (\[probeinput\]). The former coincides with the latter for $0\le P_E\le 1/4$, but not for $1/4 < P_E \le 1/2$. Indeed, because Brandt’s $|A_2\rangle$ states for $P_E = 1/4 \pm x$ are identical, for all $0\le x\le 1/4$, it is clear that his two-state FPB probe traces out the *same Rényi information trajectory when $P_E$ increases from 1/4 to 1/2 as it does when $P_E$ decreases from 1/4 to 0. In subsequent work \[H.E. 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--- abstract: | This paper contains a rich collection of results related to weight structures and $t$-structures. For any weight structure $w$ we study [*pure*]{} (co)homological functors; these “ignore all weights except weight zero” and have already found several applications (in particular, to Picard groups of triangulated categories). Moreover, we study [*virtual $t$-truncations*]{} of cohomological functors. The resulting functors are defined in terms of $w$ but are closely related to $t$-structures; so we prove in several cases that a weight structure $w$ “gives” a $t$-structure (that is [*adjacent*]{} or [*$\Phi$-orthogonal*]{} to it). We also study in detail [*well generated*]{} weight structures (and prove that any [ *perfect*]{} set of objects generates a certain weight structure). We prove the existence of weight structures right adjacent to [*compactly generated*]{} $t$-structures (using Brown-Comenetz duality); this implies that the hearts of the latter have injective cogenerators and satisfy the AB3\* axiom. Actually, “most of” these hearts are Grothendieck abelian (due to the existence of “bicontinuously orthogonal” weight structures). It is convenient for us to use the notion of [*torsion pairs*]{}; these essentially generalize both weight structures and $t$-structures. We prove several new properties of torsion pairs; in particular, we generalize a theorem of D. Pospisil and J. Šťovíček to obtain a complete classification of compactly generated torsion pairs. author: - 'Mikhail V. Bondarko [^1]' title: 'On torsion pairs, (well generated) weight structures, adjacent $t$-structures, and related (co)homological functors' --- [***2010 Mathematics Subject Classification.*** Primary 18E30 18E40 18G25; Secondary 14F05 55P42.]{} [***Key words and phrases*** Triangulated category, torsion pair, weight structure, $t$-structure, adjacent structure, cohomological functor, pure functor, virtual $t$-truncation, compact object, perfect class, symmetric classes, Brown-Comenetz duality, well generated category, derived category of coherent sheaves, duality, pro-objects, projective class.]{} Introduction {#introduction .unnumbered} ============ The main goal of the current paper is to demonstrate the utility of weight structures to the construction and study of $t$-structures and of (co)homological functors from triangulated categories (into abelian ones). In particular, for a weight structure $w$ we study [*$w$-pure functors*]{} (i.e., those that “only see $w$-weight zero”).[^2]  Functors of this type have already found interesting applications in several papers (note in particular that the results of our §\[sdetect\] are important for the study of Picard groups of triangulated categories in [@bontabu]). So the author believes that the reader not interested in the construction weight structures and $t$-structures (that we will start discussing very soon) may still benefit from §\[sws\] of the paper where a rich collection of properties of pure functors and [*virtual $t$-truncations*]{} of (co)homological functors (with respect to $w$) is proved. Now, virtual $t$-truncations are defined in terms of weight structures; still they are closely related to $t$-structures (whence the name). Respectively, our results yield the existence of some vast new families of $t$-structures. To describe one of the main results of this sort here we recall that a $t$-structure $t=({\underline{C}}^{t\le 0}, {\underline{C}}^{t\ge 0})$ (for a triangulated category ${\underline{C}}$) is said to be (right) adjacent to $w$ if ${\underline{C}}^{t\le 0}={\underline{C}}_{w\ge 0}$.[^3] For a triangulated category ${\underline{C}}$ that is closed with respect to (small) coproducts and a weight structure $w$ on it we will say that $w$ is [*smashing*]{} whenever ${\underline{C}}_{w\ge 0}$ is closed with respect to ${\underline{C}}$-coproducts (note that ${\underline{C}}_{w\le 0}$ is $\coprod$-closed automatically). \[tadjti\] Let ${\underline{C}}$ be a triangulated category that is closed with respect to coproducts and satisfies the following Brown representability property: any functor ${\underline{C}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ that respects (${\underline{C}}{{^{op}}}$)-coproducts is representable.[^4] Then for a weight structure $w$ on ${\underline{C}}$ there exists a $t$-structure right adjacent to it if and only if $w$ is smashing. Moreover, the heart of $t$ (if $t$ exists) is equivalent to the category of all those additive functors ${{\underline{Hw}}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ that respect products.[^5] Note here that triangulated categories closed with respect to coproducts have recently become very popular in homological algebra and found applications in various areas of mathematics; so a significant part of this paper is dedicated to categories of this sort. Still we prove certain alternative versions of Theorem \[tadjti\]; some of them can be applied to “quite small” triangulated categories. So, if instead of the Brown representability condition for ${\underline{C}}$ we demand it to satisfy the [*$R$-saturatedness one*]{} (see Definition \[dsatur\] below; this is an “$R$-linear finite” version of the Brown representability) then for any [*bounded*]{} $w$ on ${\underline{C}}$ there will exist a $t$-structure right adjacent to it. Note that this version of the result can be applied to the bounded derived category $D^b(X)$ of coherent sheaves on regular separated finite-dimensional scheme that is proper over the spectrum of a Noetherian ring $R$ (see Proposition \[psatur\](II) and Remark \[rsatur\](1)). We also prove two generalizations of this existence result (see Proposition \[psaturdu\] and Remark \[roq\]); they “produce” certain $t$-structures from weight structures on the bounded derived category of coherent sheaves on any proper $R$-scheme $X$ and also on the triangulated category of perfect complexes on $X$.[^6] So, one may (roughly) say that any “reasonable” weight structure (on a triangulated category satisfying some Brown representability condition) can be used to construct certain $t$-structures. This demonstrates the importance of constructing weight structures. However, the $t$-structures constructed using Theorem \[tadjti\] appear to be somewhat “exotic” (yet cf. Theorem 1 of [@zvon]) and possibly the $t$-structures constructed via the aforementioned “$R$-saturated” versions of the theorem are “more useful”. Still we also prove (using adjacent and [*$\Phi$-orthogonal*]{} weight structures in a crucial way) several properties of [compactly generated]{} $t$-structures.[^7] To formulate the following theorem we need some definitions. For a class of objects $S$ of a triangulated category ${\underline{C}}$ we will write $S{{}^{\perp}}$ (resp. ${{}^{\perp}}S$) for the class of those $M\in {\operatorname{Obj}}{\underline{C}}$ such that the morphism group ${\underline{C}}(N,M)$ (resp. ${\underline{C}}(M,N)$) is zero for all $N\in S$. We will say that a $t$-structure $t=({\underline{C}}^{t\le 0},{\underline{C}}^{t\ge 0})$ on ${\underline{C}}$ is [*generated*]{} by a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ whenever ${\underline{C}}^{t\ge 0}=\cap_{i\ge 1}({\mathcal{P}}{{}^{\perp}}[i])$. \[tgroth\] Let ${\underline{C}}$ be a triangulated category that is closed with respect to coproducts; let $t$ be a $t$-structure on it generated by a set of compact objects (we will say that $t$ is compactly generated if such a generating set ${\mathcal{P}}$ exists).[^8] 1. Then the heart ${{\underline{Ht}}}$ of $t$ has an injective cogenerator and satisfies the AB3\* axiom. 2\. Assume in addition either that ${\underline{C}}$ is the homotopy category of a proper simplicial stable model category or that $t$ is non-degenerate. Then ${{\underline{Ht}}}$ is a Grothendieck abelian category. Past 2 of this theorem answers Question 3.8 of [@parrasao] “in all reasonable cases”;[^9] part 1 appears to be new also. Now we describe a result that was important for the proof of part 1 of Theorem \[tgroth\]. One of the main topics of this paper is the study of the relation of (smashing) weight structures to perfect sets of objects (that we will now define; recall that these are also closely related to the Brown representability condition).[^10] \[textw\] Assume that ${\underline{C}}$ is closed with respect to small coproducts. Let ${\mathcal{P}}$ be a [*perfect*]{} set of objects of ${\underline{C}}$ (i.e., assume that the class [*${\mathcal{P}}$-null*]{} of those morphisms that are annihilated by corepresentable functors of the type ${\underline{C}}(P,-)$ for $P\in {\mathcal{P}}$ is closed with respect to coproducts).[^11] Then the couple $w=(L,R)$ is a smashing weight structure, where $R=\cap_{i<0}({\mathcal{P}}{{}^{\perp}}[i])$ and $L=({{}^{\perp}}R)[1]$. Moreover, the class $L$ may be described “more explicitly” in terms of ${\mathcal{P}}$; cf. Corollary \[cwftw\] below. Thus if we assume in addition that the Brown representability condition is fulfilled for ${\underline{C}}$[^12]then we also obtain the (right) adjacent $t$-structure $t=(R, (R{{}^{\perp}})[1])$. This $t$-structure is [*cogenerated*]{} by any class ${\mathcal{P}}'$ that is [*weakly symmetric*]{} to ${\mathcal{P}}$ (see Definition \[dsym\](\[iwsym\]) below; if the elements of ${\mathcal{P}}$ are compact then we can construct ${\mathcal{P}}'$ using [*Brown-Comenetz duality*]{}). Somewhat surprisingly to the author, a similar chain of arguments gives the existence of a weight structure $w$ that is [*right adjacent*]{} to a given compactly generated $t$-structure (i.e., ${\underline{C}}^{t\ge 0}={\underline{C}}_{w\le 0}$); this yields the proof of Theorem \[tgroth\](1).[^13] Moreover, the “opposite” weight structure $w{{^{op}}}$ in the category ${\underline{C}}{{^{op}}}$ is perfectly generated but not compactly generated (so, there exist plenty of perfectly generated weight structures that are not compactly generated; recall that the latter class of weight structures was introduced in [@paucomp]). We also prove the following “well generatedness” result for weight structures (obtaining in particular that all smashing weight structures on well generated categories can be obtained from Theorem \[textw\]). \[pwgwstr\] Assume that ${\underline{C}}$ is a well generated triangulated category (i.e., there exists a [*regular*]{} cardinal ${\alpha}$ and a perfect set $S$ of [*${\alpha}$-small*]{} objects such that $S{{}^{\perp}}={\{0\}}$; see Definition \[dwg\](\[idpg\])). Then for any smashing weight structure $w$ on ${\underline{C}}$ there exists a cardinal ${\alpha}'$ such that for any regular ${\beta}\ge {\alpha}'$ the weight structure $w$ is strongly ${\beta}$-well generated in the following sense: the couple $({\underline{C}}_{w\le 0}\cap {\operatorname{Obj}}{\underline{C}}^{\beta}, {\underline{C}}_{w\ge 0}\cap {\operatorname{Obj}}{\underline{C}}^{\beta})$ is a weight structure on the triangulated subcategory ${\underline{C}}^{\beta}$ of ${\underline{C}}$ consisting of ${\beta}$-compact objects (see Definition \[dbecomp\](\[idcomp\])), the class ${\mathcal{P}}={\underline{C}}_{w\le 0}\cap {\operatorname{Obj}}{\underline{C}}^{\beta}$ is essentially small and perfect, and $w=(L,R)$ for $L$ and $R$ being the classes described in Theorem \[textw\] (if these conditions are fulfilled for $w$ and some ${\mathcal{P}}$ then we also say that $w$ is perfectly generated by ${\mathcal{P}}$). A significant part of the “easier” results of the current paper is stated in terms of [*torsion pairs*]{} (as defined in [@aiya]; cf. Remark \[rgen\](4) below); these essentially generalize both weight structures and $t$-structures.[^14]  This certainly makes the corresponding results more general; note also that (the main subject of) [@bkw] yields an interesting family of examples of torsion pairs that do not come either from weight structures or from $t$-structures. Probably the most interesting result about general torsion pairs proved in this paper is the classification of compactly generated ones (in Theorem \[tclass\]); we drop the assumption that ${\underline{C}}$ is a “stable derivator” category that was necessary for the proof of the closely related Corollary 3.8 of [@postov]. We also relate [*adjacent*]{} torsion pairs to “Brown-Comenetz-type symmetry”. We also study [*dualities*]{} between triangulated categories and their relation to torsion pairs. We demonstrate that for a $t$-structure $t$ on ${\underline{C}'}$ that does not possess a left adjacent weight structure there still may exist $w$ on a category ${\underline{C}}$ that is [$\Phi$-orthogonal ]{} to $t$, where $\Phi:{\underline{C}}{{^{op}}}\times {\underline{C}}'\to {\underline{\operatorname{Ab}}}$ is a (“bicontinuous”) duality of triangulated categories. Moreover, for an interesting sort of dualities that we study in §\[sdual2\] and for any compactly generated $t$ on ${\underline{C}'}$ the objects of the heart of the corresponding orthogonal $w$ give faithful exact “stalk” functors ${{\underline{Ht}}}\to {\underline{\operatorname{Ab}}}$ that respect coproducts; taking the functor $\Phi(P,-)$ for $P$ being a [*cogenerator*]{} of ${{\underline{Hw}}}$ we easily obtain that ${{\underline{Ht}}}$ is an AB5 abelian category (cf. Theorem \[tgroth\]). So we demonstrate once again that weight structures “shed some light on $t$-structures”. Moreover, dualities are important for [@bgn] where they are applied to the study of coniveau spectral sequences and the homotopy $t$-structures on various motivic stable homotopy categories. This matter is also related to the study of the stable homotopy category in [@prospect]. Let us now describe the contents of the paper. Some more information of this sort may be found in the beginnings of sections. In §\[sold\] we we define torsion pairs and prove several of their properties; these are mostly simple but new. We also relate torsion pairs to $t$-structures and study $t$-projective objects (essentially following [@zvon]); they are related to adjacent weight structures that we will study later. We start §\[sws\] from recalling some basics on weight structures (among those are some properties of weight complexes; though these are not really new, we treat this subject more accurately than in [@bws] where weight complexes were originally defined). Next we introduce pure functors (in §\[sdetect\]; their “construction” in Proposition \[ppure\] appears to be a very useful statement). An “intrinsic” definition of pure functors is given by Proposition \[pwrange\]. We relate the weight range of functors to virtual $t$-truncations. We also study the properties of virtual $t$-truncations and weight complexes under the assumption that $w$ is smashing. As an application, we treat the representability for virtual $t$-truncations of representable functors; this result has important applications to the construction of (adjacent) $t$-structures below. In §\[sadjbrown\] we investigate the question when weight structures and $t$-structures admit (right or left) adjacent $t$-structures or weight structures (respectively); we also study adjacent “structures” if they exist. To prove the existence of these adjacent structures on a triangulated category ${\underline{C}}$ we usually assume a certain Brown representability-type condition for ${\underline{C}}$. We also recall the definition of perfect and symmetric classes and relate them to Brown-Comenetz duality and torsion pairs. This gives a funny (general) criterion for the existence of an adjacent torsion pair in terms of “symmetry” along with a “new” description of a $t$-structure that is right adjacent to a given compactly generated weight structure. In §\[spgtp\] we study compactly generated torsion pairs and perfectly generated weight structures; we prove several new and interesting results about them (and some of these statements were formulated above). In particular, we prove that for any “symmetric” ${\mathcal{P}},{\mathcal{P}}'\subset {\operatorname{Obj}}{\underline{C}}$ there exist adjacent $t$ and $w$ (co)generated by them. This implies the existence of a weight structure right adjacent to a given compactly generated $t$; hence the category ${{\underline{Ht}}}$ has an injective cogenerator and satisfies the AB3\* axiom. Combining this fact with the results of [@humavit] we deduce that ${{\underline{Ht}}}$ is Grothendieck abelian whenever $t$ is non-degenerate. The results and arguments of the section are closely related to the properties of localizing subcategories of triangulated categories as studied by A. Neeman, H. Krause and other authors; see Remark \[rtst2\](\[it6\]) below for an “explanation” of this similarity. In §\[skan\] we study dualities between triangulated categories and torsion pairs orthogonal with respect to them. Considering a “very simple” duality we prove that for certain weight structures inside the bounded derived category of coherent sheaves on a scheme $X$ that is proper over ${{\operatorname{Spec}\,}}R$ (where $R$ is a noetherian ring) there necessarily exist (right or left) orthogonal $t$-structures. Our main tools for constructing “more complicated” dualities are Kan extensions of (co)homological functors from a triangulated subcategory ${\underline{C}_0}$ to ${\underline{C}}$; their properties are rather interesting for themselves. We describe a duality between the homotopy category of filtered pro-objects for a stable proper Quillen model category ${\mathcal{M}}$ with ${\operatorname{Ho}}({\mathcal{M}})$. The properties of this duality imply that for any compactly generated $t$-structure on ${\operatorname{Ho}}({\mathcal{M}})$ its heart is a Grothendieck abelian category; they are also applied in [@bgn] to the study of [*motivic pro-spectra* ]{} and [*generalized coniveau spectral sequences*]{}. For the convenience of the reader we also make a list of the main definitions and notation used in this paper. Regular cardinals, Karoubi-closures and related matters, $D\perp E$, ${{}^{\perp}}D$, and $D{{}^{\perp}}$, suspended, cosuspended, extension-closed, and strict classes of objects, extension-closures, envelopes, subcategories generated by classes of objects of triangulated categories, homological and cohomological functors, compact objects, localizing and colocalizing subcategories, cogenerators, Hom-generators, compactly generated categories, cc, cp, and pp functors, and the Brown representability condition (along with its dual) are defined in §\[snotata\]; (smashing, cosmashing, countably smashing, adjacent, and compactly generated) torsion pairs (often denoted by $s=({\mathcal{LO}},{\mathcal{RO}})$; sometimes we also use the notation $(L_sM,R_sM)$), generators for them, and ${\mathcal{P}}$-null morphisms are introduced in §\[shop\]; $t$-structures ($t=({\underline{C}}^{t\le 0},{\underline{C}}^{t\ge 0})$) and several of their “types” (including smashing and cosmashing $t$-structures), their hearts (${{\underline{Ht}}}$), associated torsion pairs, $t$-homology ($H_0^t$), and $t$-projective objects are defined in §\[sts\]; weight structures ($w=({\underline{C}}_{w\le 0},{\underline{C}}_{w\ge 0})$; we also define ${\underline{C}}_{w=i}$ and ${\underline{C}}_{[m,n]}$), their “types”, hearts (${{\underline{Hw}}}$), and associated (weighty) torsion pairs, adjacent weight and $t$-structures, $m$-weight decompositions, negative subcategories, and the weight structure ${w^{st}}$ are introduced in §\[ssws\]; (weight) Postnikov towers along with the corresponding filtrations and (weight) complexes, weakly homotopic morphisms of complexes, and the category ${K_{\mathfrak{w}}}({{\underline{Hw}}})$ are defined in §\[sswc\]; pure homological functors ($H^{{\mathcal{A}}}$) and purely $R$-representable homology (with values in ${\operatorname{PShv}^R}({\underline{B}})$) is introduced in §\[sdetect\], virtual $t$-truncations ($\tau^{\ge m}H$ and $\tau^{\le m}H$), weight range of functors, and pure cohomological functors ($H_{{\mathcal{A}}}$) are defined in §\[svtt\]; functors of $R$-finite type and $R$-saturated categories are introduced in §\[sadjt\]; ${\alpha}$-small objects, countably perfect and perfect classes of objects, perfectly generated and well generated categories, weakly symmetric and symmetric classes, and Brown-Comenetz duals of functors and objects are defined in §\[scomp\]; countable homotopy colimits $\operatorname{\varinjlim}Y_i$, strongly extension-closed classes and strong extension-closures, (naive) big hulls, and zero classes of (collections of) functors are introduced in §\[scoulim\]; perfectly generated weight structures, ${\mathcal{P}}$-approximations, and contravariantly finite classes are defined in §\[sperfws\]; weakly and strongly ${\beta}$-well generated weight structures (and torsion pairs) are studied in §\[swgws\]; coextended functors and coextensions are defined in §\[scoext\]; (nice) dualities and orthogonal structures are introduced in §\[sdual1\]; biextensions are defined in §\[sdual1\]; stalk functors are introduced in §\[sgengroth\]; certain model structures on pro-objects are recalled in §\[sprospectra\]. The author is deeply grateful to prof. F. Déglise, prof. G.C. Modoi, prof. Salorio M.J. Souto, and prof. J. Šťovíček for their very useful comments. On torsion pairs and $t$-structures (“simple properties”) {#sold} ========================================================= This section is dedicated to the basics on torsion pairs and $t$-structures. In §\[snotata\] we introduce some notation and recall several important properties of triangulated categories (mostly from [@neebook]). In §\[shop\] we define and study torsion pairs (in the terminology of [@aiya]). Our results are rather easy; yet the author does not know any references for most of them. In §\[sts\] we recall some basics on $t$-structures and relate them to torsion pairs. We also study the notions of $t$-projective objects for the purpose of using them in §\[sadjw\] and later. Some categorical preliminaries {#snotata} ------------------------------ When we will write $i\ge c$ or $i\le c$ (for some $c\in {{\mathbb{Z}}}$) we will mean that $i$ is an integer satisfying this inequality. A cardinal ${\alpha}$ is said to be [*regular*]{} if it cannot be presented as a sum of less then ${\alpha}$ cardinals that are less than ${\alpha}$. Most of the categories of this paper will be locally small. When considering a category that is not locally small we will usually say that it is (possibly) big; we will not need much of these categories. For categories $C,D$ we write $D\subset C$ if $D$ is a full subcategory of $C$. Given a category $C$ and $X,Y\in{\operatorname{Obj}}C$ we will write $C(X,Y)$ for the set of morphisms from $X$ to $Y$ in $C$. We will say that $X$ is a [*retract*]{} of $Y$ if ${\operatorname{id}}_X$ can be factored through $Y$.[^15]  For a category $C$ the symbol $C^{op}$ will denote its opposite category. For a subcategory $D\subset C$ we will say that $D$ is [*Karoubi-closed*]{} in $C$ if it contains all retracts of its objects in $C$. We call the smallest Karoubi-closed subcategory $\operatorname{\operatorname{Kar}}_C(D)$ of $C$ containing $D$ the [*Karoubi-closure*]{} of $D$ in $C$. The [*Karoubi envelope*]{} $\operatorname{\operatorname{Kar}}({\underline{B}})$ (no lower index) of an additive category ${\underline{B}}$ is the category of “formal images” of idempotents in ${\underline{B}}$ (so ${\underline{B}}$ is embedded into an idempotent complete category). ${\underline{\operatorname{Ab}}}$ is the category of abelian groups. ${\underline{C}}$, ${\underline{C}}'$, ${\underline{C}_0}$, ${\underline{D}}$, ${{\underline{D}}_0}$, and ${\underline{E}}$ will always denote certain triangulated categories. ${\underline{C}}$ will often be endowed with a weight structure $w$; we always assume that this is the case in those formulations where $w$ is mentioned without any explanations. For $f\in{\underline{C}}(X,Y)$, where $X,Y\in{\operatorname{Obj}}{\underline{C}}$, we call the third vertex of (any) distinguished triangle $X\stackrel{f}{\to}Y\to Z$ a cone of $f$. We will often consider some representable and corepresentable functors and their restrictions. So for ${\underline{D}}'$ being a full triangulated subcategory of a triangulated category ${\underline{D}}$ and $M\in {\operatorname{Obj}}{\underline{D}}$ we will often write $H^M: {\underline{D}}'\to {\underline{\operatorname{Ab}}}$ for the restriction of the corepresentable (homological) functor ${\underline{D}}(M,-)$ to ${\underline{D}}'$ (yet $H^P$ in Proposition \[porthop\] will denote a certain [*coextension*]{}); $H_M: {\underline{D}}'{{^{op}}}\to {\underline{\operatorname{Ab}}}$ is the restriction of the functor ${\underline{D}}(-,M)$ to ${\underline{D}}'$. We will often be interested in the case ${\underline{D}}'={\underline{D}}$ in this notation; we assume that the domain of the functors $H^M$ and $H_M$ is the category ${\underline{C}}$ if not specified otherwise. For $X,Y\in {\operatorname{Obj}}{\underline{C}}$ we will write $X\perp Y$ if ${\underline{C}}(X,Y)={\{0\}}$. For $D,E\subset {\operatorname{Obj}}{\underline{C}}$ we will write $D\perp E$ if $X\perp Y$ for all $X\in D,\ Y\in E$. For $D\subset{\operatorname{Obj}}{\underline{C}}$ we will write $D^\perp$ for the class $$\{Y\in {\operatorname{Obj}}{\underline{C}}:\ X\perp Y\ \forall X\in D\};$$ sometimes we will write $\perp_{{\underline{C}}}$ instead to indicate the category that we are considering. Dually, ${}^\perp{}D$ is the class $\{Y\in {\operatorname{Obj}}{\underline{C}}:\ Y\perp X\ \forall X\in D\}$. In this paper all complexes will be cohomological, i.e., the degree of all differentials is $+1$; respectively, we use cohomological notation for their terms. We will use the term [*exact functor*]{} for a functor of triangulated categories (i.e., for a functor that preserves the structures of triangulated categories). We will say that a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ is [*suspended*]{} if ${\mathcal{P}}[1]\subset {\mathcal{P}}$; ${\mathcal{P}}$ is [*cosuspended*]{} if ${\mathcal{P}}[-1]\subset {\mathcal{P}}$. A class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ is said to be [*extension-closed*]{} if $0\in {\mathcal{P}}$ and for any distinguished triangle $A\to B\to C$ in ${\underline{C}}$ we have the implication $A,C\in {\mathcal{P}}\implies B\in {\mathcal{P}}$. In particular, an extension-closed ${\mathcal{P}}$ is [*strict*]{} (i.e., contains all objects of ${\underline{C}}$ isomorphic to its elements). The smallest extension-closed class ${\mathcal{P}}'$ containing a given ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ will be called the [*extension-closure*]{} of ${\mathcal{P}}$. The smallest extension-closed Karoubi-closed ${\mathcal{P}}'\subset {\operatorname{Obj}}{\underline{C}}$ containing ${\mathcal{P}}$ will be called the [*envelope*]{} of ${\mathcal{P}}$. We call the smallest Karoubi-closed triangulated subcategory ${\underline{D}}$ of ${\underline{C}}$ such that ${\operatorname{Obj}}{\underline{D}}$ contains ${\mathcal{P}}$ the [*triangulated subcategory densely generated by*]{} ${\mathcal{P}}$; we will write ${\underline{D}}={\langle}{\mathcal{P}}{\rangle}_{{\underline{C}}}$. We will say that ${\underline{C}}$ [*has coproducts*]{} (resp. products, resp. countable coproducts) whenever it contains arbitrary small coproducts (resp. products, resp. countable coproducts) of families of its objects. ${\underline{A}}$ will usually denote some abelian category; the case ${\underline{A}}={\underline{\operatorname{Ab}}}$ is the most important one for the purposes of this paper. We will call a covariant additive functor ${\underline{C}}\to {\underline{A}}$ for an abelian ${\underline{A}}$ [*homological*]{} if it converts distinguished triangles into long exact sequences; homological functors ${\underline{C}}^{op}\to {\underline{A}}$ will be called [*cohomological*]{} when considered as contravariant functors from ${\underline{C}}$ into $ {\underline{A}}$. For additive categories $C,D$ the symbol $\operatorname{\operatorname{AddFun}}(C,D)$ will denote the (possibly, big) category of additive functors from $C$ to $D$. Certainly, if $D$ is abelian then an $\operatorname{\operatorname{AddFun}}(C,D)$-complex $X\to Y\to Z$ is exact in $Y$ whenever $X(P)\to Y(P)\to Y(P)$ is exact (in $D$) for any $P\in {\operatorname{Obj}}C$. Below we will sometimes need some properties of the Bousfield localization setting (cf. §9.1 of [@neebook]; most of these statements are contained in Propositions 1.5 and 1.6 of [@bondkaprserr]). \[pbouloc\] Let $F:{\underline{E}}\to {\underline{C}}$ be a full embedding of triangulated categories; assume that ${\underline{E}}$ is Karoubi-closed in ${\underline{C}}$. Denote by ${\underline{D}}$ the full subcategory of ${\underline{C}}$ whose object class is ${\operatorname{Obj}}{\underline{E}}{{}^{\perp}}$; denote the embedding ${\underline{D}}\to {\underline{C}}$ by $i$. Then the following statements are valid. I. A (left or right) adjoint to an exact functor is exact. II\. ${\underline{D}}$ is a Karoubi-closed triangulated subcategory of ${\underline{C}}$. III\. Assume that $F$ possesses a right adjoint $G$. 1. \[ibou1\] Then for any $N\in {\operatorname{Obj}}{\underline{C}}$ there exists a distinguished triangle $$\label{ebou} N'\stackrel{f}{\to} N\stackrel{g}{\to} N''\to N'[1]$$ with $N'\in {\operatorname{Obj}}{\underline{E}}$ and $ N'' \in{\operatorname{Obj}}{\underline{D}}$ (here we consider ${\underline{E}}$ as a subcategory of ${\underline{C}}$ via $F$), and the triangle (\[ebou\]) is unique up to a canonical isomorphism. 2. \[ibouort\] ${{}^{\perp}}{\operatorname{Obj}}{\underline{D}}={\operatorname{Obj}}{\underline{E}}$. 3. \[ibou2\] The functor $i$ possesses an (exact) left adjoint $A$ and the morphism $g$ in (\[ebou\]) is given by the unit of this adjunction. Moreover, this unit transformation yields an equivalence of the Verdier localization of ${\underline{C}}$ by ${\underline{E}}$ (that is locally small in this case) to ${\underline{D}}$. 4. \[ibou3\]The morphism $f$ in (\[ebou\]) is given by the counit of the adjunction $F\dashv G$, and this counit gives an equivalence ${\underline{E}}\cong {\underline{C}}/{\underline{D}}$. IV\. A full embedding $F$ as above possesses a right adjoint if and only if for any $N\in {\operatorname{Obj}}{\underline{C}}$ there exists a distinguished triangle (\[ebou\]). I. This is Lemma 5.3.6 of [@neebook]. II. Obvious. III.\[ibou1\]. The existence of (\[ebou\]) is immediate from Proposition 9.1.18 (that says that the Verdier localization ${\underline{C}}\to {\underline{C}}/{\underline{E}}$ is a [*Bousfield localization*]{} functor in the sense of Definition 9.1.1 of ibid.) and Proposition 9.1.8 of ibid. (cf. also Proposition 1.5 of [@bondkaprserr]). Lastly, the essential uniqueness of (\[ebou\]) easily follows from [@bbd Proposition 1.1.9] (cf. also Proposition \[pbw\](\[icompl\]) below). \[ibouort\]. According to the aforementioned Proposition 9.1.18 of [@neebook], we can deduce the assertion from Corollary 9.1.14 of ibid. \[ibou2\]. The “calculation” of $g$ is given by Proposition 9.1.8 of ibid. also. It remains to apply Theorem 9.1.16 of ibid. \[ibou3\]. Corollary 9.1.14 of ibid. allows to deduce the assertion from the previous one. IV. The “only if” part of the assertion is given by assertion III.\[ibou1\]. The converse implication is given by Proposition 9.1.18 of ibid. Now we recall some terminology, notation, and statements related to infinite coproducts and products in triangulated categories. Some of these definitions and results may will be generalized in §\[scomp\] below. All the coproducts and products in this paper will be small. We will say that a subclass ${\mathcal{P}}$ of ${\operatorname{Obj}}{\underline{C}}$ is [*coproductive*]{} (resp. [*productive*]{}) if it is closed with respect to all (small) coproducts (resp. products) that exist in ${\underline{C}}$. For triangulated categories closed with respect to (all) coproducts or products we will just say that these categories have coproducts (resp. products).[^16] We recall the following very useful statement. \[pcoprtriang\] Assume that ${\underline{C}}$ has coproducts (resp. products, resp. countable coproducts). Then ${\underline{C}}$ is Karoubian and all (small) coproducts (resp. products, resp. countable coproducts) of distinguished triangles in ${\underline{C}}$ are distinguished. The first of the assertions is given by Proposition 1.6.8 of [@neebook], and the second one is is given by Proposition 1.2.1 and Remark 1.2.2 of ibid. \[dcomp\] Assume that ${\underline{C}}$ has coproducts; ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$. 1. \[idcompa\] An object $M$ of ${\underline{C}}$ is said to be [*compact*]{} if the functor $H^M={\underline{C}}(M,-):{\underline{C}}\to {\underline{\operatorname{Ab}}}$ respects coproducts. 2. \[idloc\] For ${\underline{D}}\subset {\underline{C}}$ (${\underline{D}}$ is a triangulated category that may be equal to ${\underline{C}}$) one says that ${\mathcal{P}}$ generates ${\underline{D}}$ [*as a localizing subcategory*]{} of ${\underline{C}}$ if ${\underline{D}}$ is the smallest full strict triangulated subcategory of ${\underline{C}}$ that contains ${\mathcal{P}}$ and is closed with respect to ${\underline{C}}$-coproducts. If this is the case then we will also say that ${\mathcal{P}}$ [*cogenerates*]{} ${\underline{D}}{{^{op}}}$ as a [*colocalizing*]{} subcategory of ${\underline{C}}{{^{op}}}$. 3. \[idhg\] We will say that ${\mathcal{P}}$ [*Hom-generates*]{} a full triangulated category ${\underline{D}}$ of ${\underline{C}}$ containing ${\mathcal{P}}$ if ${\operatorname{Obj}}{\underline{D}}\cap(\cup_{i\in {{\mathbb{Z}}}}{\mathcal{P}}[i]){{}^{\perp}}={\{0\}}$. 4. \[idcg\] We will say that ${\underline{C}}$ is [*compactly generated*]{} if it is Hom-generated by a set of compact objects. 5. \[idcc\] It will be convenient for us to use the following somewhat clumsy terminology: a homological functor $H:{\underline{C}}\to {\underline{A}}$ (where ${\underline{A}}$ is an abelian category) will be called a [*cc*]{} functor if it respects coproducts (i.e., the image of any coproduct in ${\underline{C}}$ is the corresponding coproduct in ${\underline{A}}$); $H$ will be called a [*wcc*]{} functor if it respects countable coproducts. A cohomological functor $H$ from ${\underline{C}}$ into ${\underline{A}}$ will be called a [*cp*]{} functor if it converts all (small) coproducts into ${\underline{A}}$-products. Dually, for a triangulated category ${\underline{D}}$ that has products we will call a homological functor $H:{\underline{D}}\to {\underline{A}}$ a [*pp*]{} functor if its respects products. 6. \[idbrown\] We will say that ${\underline{C}}$ satisfies the [*Brown representability*]{} property whenever any cp functor from ${\underline{C}}$ into ${\underline{\operatorname{Ab}}}$ is representable. Dually, we will say that a triangulated category ${\underline{D}}$ satisfies the dual Brown representability property if it has products and any pp functor from ${\underline{C}}$ into ${\underline{\operatorname{Ab}}}$ is corepresentable (i.e., if ${\underline{D}}{{^{op}}}$ satisfies the Brown representability property). \[pcomp\] Assume that ${\underline{C}}$ has coproducts. I. Let ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$. 1\. If ${\mathcal{P}}$ generates ${\underline{C}}$ as its own localizing subcategory then it also Hom-generates ${\underline{C}}$. Conversely, if ${\mathcal{P}}$ Hom-generates ${\underline{C}}$ and the embedding into ${\underline{C}}$ of its localizing subcategory ${\underline{C}}'$ generated by ${\mathcal{P}}$ possesses a right adjoint then ${\underline{C}}'={\underline{C}}$. 2\. More generally, denote by $C$ the smallest coproductive extension-closed subclass of ${\operatorname{Obj}}{\underline{C}}$ containing ${\mathcal{P}}$. Then ${\mathcal{P}}{{}^{\perp}}\cap C={\{0\}}$. II.1. Assume that ${\underline{C}}$ is compactly generated. Then both ${\underline{C}}$ and ${\underline{C}}{{^{op}}}$ satisfy the Brown representability property. 2\. Assume that ${\underline{C}}$ satisfies the Brown representability property. Then it has products and any exact functor $F$ from ${\underline{C}}$ (into a triangulated category ${\underline{D}}$) that respects coproducts possesses an exact right adjoint $G$. I.1. The first part of the assertion is essentially a part of [@neebook Proposition 8.4.1] (note that the simple argument used for the proof of this implication does not require ${\mathcal{P}}$ to be a set); it is also a particular case of assertion I.2. Now assume that the embedding ${\underline{C}}'\to {\underline{C}}$ possesses a right adjoint. Then Proposition \[pbouloc\](III.\[ibou1\]) implies that ${\underline{C}}'={\underline{C}}$ whenever ${\operatorname{Obj}}{\underline{C}}'{{}^{\perp}}={\{0\}}$ (note here that ${\underline{C}}'$ is a strict subcategory of ${\underline{C}}$). Lastly, ${\operatorname{Obj}}{\underline{C}}'{{}^{\perp}}$ is certainly zero if ${\mathcal{P}}$ Hom-generates ${\underline{C}}$ (since ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}'$). 2\. Similarly to the proof of loc. cit., if $N$ belongs to ${\mathcal{P}}^{\perp}\cap C$ then the class $C\cap {}^{\perp}N$ contains ${\mathcal{P}}$, coproductive and extension-closed. Hence $N\perp N$ and we obtain $N=0$. II.1. The Brown representability property for ${\underline{C}}$ is given by Proposition 8.4.2 of [@neebook]. The Brown representability for ${\underline{C}}{{^{op}}}$ is immediate from the combination of Theorem 8.6.1 with Remark 6.4.5 of ibid. 2\. The first part of the assertion is given by Proposition 8.4.6 of [@neebook]. The second part is immediate from Theorem 8.4.4 of ibid. (combined with Proposition \[pbouloc\](I)). On torsion pairs {#shop} ---------------- As we have already said, this paper is mostly dedicated to the study of weight structures and $t$-structures. Now, these notions have much in common; so we start from recalling an (essentially) more general definition of a torsion pair (in the terminology of [@aiya Definition 1.4]; in [@postov Definition 3.2] torsion pairs were called complete Hom-orthogonal pairs).[^17] \[dhop\] A couple $s$ of classes ${\mathcal{LO}},{\mathcal{RO}}\subset{\operatorname{Obj}}{\underline{C}}$ (of $s$-left orthogonal and $s$-right orthogonal objects, respectively) will be said to be a [*torsion pair*]{} (for ${\underline{C}}$) if ${\mathcal{LO}}^{\perp}={\mathcal{RO}}$, ${\mathcal{LO}}={}^{\perp}{\mathcal{RO}}$, and for any $M\in{\operatorname{Obj}}{\underline{C}}$ there exists a distinguished triangle $$\label{swd} L_sM\stackrel{a_M}{\to} M\stackrel{n_M}{\to} R_sM{\to} L_sM[1]$$ such that $L_sM\in {\mathcal{LO}}$ and $ R_sM\in {\mathcal{RO}}$. We will call any triangle of this form an [*$s$-decomposition*]{} of $M$; $a_M$ will be called an [*$s$-decomposition morphism*]{}. We will also need the following auxiliary definitions. \[dhopo\] Let $s=({\mathcal{LO}},{\mathcal{RO}})$ be a torsion pair. 1\. We will say that $s$ is [*coproductive*]{} (resp. [*productive*]{}) if ${\mathcal{RO}}$ is coproductive (resp. ${\mathcal{LO}}$ is productive). We will also say that $s$ is smashing (resp. cosmashing) if ${\underline{C}}$ in addition has coproducts (resp. products). We will also use the following modification of the smashing condition: we will say that $s$ is [*countably smashing*]{} whenever ${\underline{C}}$ has coproducts and ${\mathcal{RO}}$ is closed with respect to countable ${\underline{C}}$-coproducts.[^18] 2. For another torsion pair $s'=({\mathcal{LO}}',{\mathcal{RO}}')$ for ${\underline{C}}$ we will say that $s$ is left adjacent to $s'$ or that $s'$ is right adjacent to $s$ if ${\mathcal{RO}}={\mathcal{LO}}'$. 3\. We will say (following [@postov Definition 3.1]) that $s$ is [*generated by ${\mathcal{P}}\subset {\mathcal{LO}}$*]{} if ${\mathcal{P}}^\perp={\mathcal{RO}}$.[^19] We will say that $s$ is [*compactly generated*]{} if it is generated by some set of compact objects. 4\. For ${\underline{C}'}$ being a full triangulated subcategory of ${\underline{C}}$ we will say that $s$ restricts to it whenever $({\mathcal{LO}}\cap {\operatorname{Obj}}{\underline{C}'},{\mathcal{RO}}\cap {\operatorname{Obj}}{\underline{C}'})$ is a torsion pair for ${\underline{C}}'$. 5\. For a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ we will say that a ${\underline{C}}$-morphism $h$ is [*${\mathcal{P}}$-null*]{} whenever for all $M\in {\mathcal{P}}$ we have $H^M(h)=0$ (where $H^M={\underline{C}}(M,-):{\underline{C}}\to {\underline{\operatorname{Ab}}}$).[^20] \[rgen\] 1. If ${\mathcal{P}}$ generates a torsion pair $s$ then ${\mathcal{RO}}= {\mathcal{P}}^\perp$ and ${\mathcal{LO}}={}^{\perp}{\mathcal{RO}}$; thus ${\mathcal{P}}$ determines $s$ uniquely. So we will say that $s$ is [**the**]{} torsion pair generated by ${\mathcal{P}}$. 2\. On the other hand, for a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ the corresponding couple $s=({\mathcal{LO}},{\mathcal{RO}})$ (where ${\mathcal{RO}}= {\mathcal{P}}^\perp$ and ${\mathcal{LO}}={}^{\perp}{\mathcal{RO}}$) certainly satisfies the orthogonality properties prescribed by Definition \[dhop\]. Yet simple examples demonstrate that the existence of $s$-decompositions can fail in general. In [@postov Definition 3.1] a couple that satisfies only the orthogonality properties in Definition \[dhop\] was called a [*Hom-orthogonal pair*]{} (in contrast to complete Hom-orthogonal pairs). The reader may easily note that several of our arguments below work for “general” Hom-orthogonal pairs; yet the author has chosen not to treat this more general definition in the current paper. 3. The object $M$ “rarely” determines its $s$-decomposition triangle (\[swd\]) canonically (cf. Remark \[rstws\](1) below). Yet we will often need some choices of its ingredients; so we will use the notation of (\[swd\]). 4. Our definition of torsion pair actually follows [@postov Definition 3.2] and differs from Definition 1.4(i) of [@aiya]. However, Proposition \[phop\](9) below yields immediately that these two definitions are equivalent. As noted in [@postov], some other authors use the term “torsion pair” to denote the couple $s$ associated with a $t$-structure (see Remark \[rtst1\](\[it1\]) below). So, the term “complete Hom-orthogonal pair” would be less ambiguous; yet it does not fit well (linguistically) with the notion of $\Phi$-orthogonality that we will introduce below. We make some simple observations. \[phop\] Let $s=({\mathcal{LO}},{\mathcal{RO}})$ be a torsion pair for ${\underline{C}}$, $i, j\in {{\mathbb{Z}}}$. Then the following statements are valid. 1\. Both ${\mathcal{LO}}$ and ${\mathcal{RO}}$ are Karoubi-closed and extension-closed in ${\underline{C}}$. 2\. ${\mathcal{LO}}$ is coproductive and ${\mathcal{RO}}$ is productive. 3\. If $s$ is coproductive (resp. productive) then the class ${\mathcal{RO}}[i]\cap {\mathcal{LO}}[j]$ is coproductive (resp. productive) also. 4\. Assume that ${\underline{C}}$ has coproducts. Then $s$ is (countably) smashing if and only if the coproduct of any $s$-decompositions of $M_i\in {\operatorname{Obj}}{\underline{C}}$ gives an $s$-decomposition of $\coprod M_i$; here $i$ runs through any (countable) index set. Dually, if ${\underline{C}}$ has products then $s$ is cosmashing if and only if the product of any $s$-decompositions of any small family of $M_i\in {\operatorname{Obj}}{\underline{C}}$ gives an $s$-decomposition of $\prod M_i$. 5. If $s$ is left adjacent to a torsion pair $s'$ (for ${\underline{C}}$) then $s$ is coproductive and $s'$ is productive. 6\. $s^{op}=({\mathcal{RO}},{\mathcal{LO}})$ is a torsion pair for ${\underline{C}}{{^{op}}}$. 7\. $s$-decompositions are “weakly functorial” in the following sense: any ${\underline{C}}$-morphism $g:M\to M'$ can be completed to a morphism between any choices of $s$-decompositions of $M$ and $M'$, respectively. In particular, if $M\in {\mathcal{LO}}$ then it is a retract of any choice of $L_sM$ (see Remark \[rgen\](3)). 8\. A morphism $h\in {\underline{C}}(M,N)$ is ${\mathcal{LO}}$-null if and only if factors through an element of ${\mathcal{RO}}$. Moreover, the couple $({\mathcal{LO}},{\mathcal{LO}}-{\text{null}})$ (the latter is the class of ${\mathcal{LO}}$-null morphisms) is a projective class in the sense of [@christ] (see Remark \[rwsts\](3) below). 9\. For $L,R\subset {\operatorname{Obj}}{\underline{C}}$ assume that $L\perp R$ and that for any $M\in {\operatorname{Obj}}{\underline{C}}$ there exists a distinguished triangle $l\to M\to r\to l[1]$ for $l\in L$ and $r\in R$. Then $(\operatorname{\operatorname{Kar}}_{{\underline{C}}}(L),\operatorname{\operatorname{Kar}}_{{\underline{C}}}(R))$ is a torsion pair for ${\underline{C}}$. 10. Assume that $({\mathcal{LO}}',{\mathcal{RO}}')$ is a torsion pair in a triangulated category ${\underline{C}}'$; assume that $F:{\underline{C}}\to {\underline{C}}'$ is an exact functor that is essentially surjective on objects, and such that $F({\mathcal{LO}})\subset {\mathcal{LO}}'$ and $F({\mathcal{RO}})\subset {\mathcal{RO}}'$. Then $({\mathcal{LO}}',{\mathcal{RO}}')= (\operatorname{\operatorname{Kar}}_{{\underline{C}}'}(F({\mathcal{LO}})),\operatorname{\operatorname{Kar}}_{{\underline{C}}'}(F({\mathcal{RO}})))$. 1–3, 5, 6. Obvious. 4. The “only if” implication follows immediately from Proposition \[pcoprtriang\] (along with its dual). Conversely, assume that (countable) coproducts of $s$-decompositions are $s$-decompositions also. Since for any (countable) set of $R_i\in {\mathcal{RO}}$ the distinguished triangles $0\to R_i\to R_i\to 0$ are $s$-decompositions of $R_i$, we obtain $\coprod R_i\in{\mathcal{RO}}$; hence $s$ is (countably) smashing. To prove the remaining (“cosmashing”) part of the assertion one can apply duality (along with assertion 6). 7\. According to [@bbd Proposition 1.1.9], to prove the first part of the assertion it suffices to verify the following: for any $s$-decomposition triangles (\[swd\]) and $L_sM'\to M'\to R_sM' {\to} L_sM'[1]$ the composition $L_sM\to M\to M'\to R_sM'$ vanishes. This certainly follows from ${\mathcal{LO}}\perp{\mathcal{RO}}$. The “in particular” part of the assertion follows if we take $g={\operatorname{id}}_M$, $M'=M$, and take the triangle $M\stackrel{{\operatorname{id}}_M}{\to} M\to 0\to M[1]$ as (“the first”) $s$-decomposition of $M$. 8\. Since ${\mathcal{LO}}\perp{\mathcal{RO}}$, any morphism that factors through ${\mathcal{RO}}$ is ${\mathcal{LO}}$-null. Conversely, if $h:M\to N$ is ${\mathcal{LO}}$-null then for any choice of an $s$-decomposition of $M$ the composition $h\circ a_M$ is zero (see (\[swd\]) for the notation). It certainly follows that $h$ factors through $R_sM\in {\mathcal{RO}}$. Next, for any $L\in {\mathcal{LO}}$ and $h\in {\mathcal{LO}}$-null we have $H^L(h)=0$ by definition (see §\[snotata\] or Definition \[dhopo\](5) for this notation). Arguing as in the proof [@christ Lemma 3.2], we easily obtain that (to prove the second part of the assertion) it remains to construct for $X\in {\operatorname{Obj}}{\underline{C}}$ a morphism $a:L\to X$ such that the cone morphism for it is a ${\mathcal{LO}}$-null one. According to the first part of the assertion, for this purpose we can take $a$ being any choice of $a_X$ (in the notation of (\[swd\])). 9\. Certainly, $\operatorname{\operatorname{Kar}}_{{\underline{C}}}(L)\perp \operatorname{\operatorname{Kar}}_{{\underline{C}}}(R)$. Assume that $M\in {{}^{\perp}}R$. Then in the corresponding distinguished triangle $l\to M\stackrel{f}{\to} r\to l[1]$ we have $f=0$; hence $l\in \operatorname{\operatorname{Kar}}_{{\underline{C}}}(L)$ and ${{}^{\perp}}R \subset \operatorname{\operatorname{Kar}}_{{\underline{C}}}(L)$. Dually (cf. assertion 6) if $M \in \operatorname{\operatorname{Kar}}_{{\underline{C}}}(R)$ then it is a retract of the corresponding $r$; thus $L^\perp\subset \operatorname{\operatorname{Kar}}_{{\underline{C}}}(R)$. This concludes the proof. 10\. For any object $M'$ of ${\underline{C}}'$ choose $M\in {\operatorname{Obj}}{\underline{C}}$ such that $F(M)\cong M'$; choose an $s$-decomposition $L_sM\to M \to R_sM\to L_sM[1]$ of $M$. Then we obtain a distinguished triangle $F(L_sM)\to M' \to F(R_sM)\to F(L_sM)[1]$. Next, the relation of $s$ to $s'$ implies that $F({\mathcal{LO}})\perp F({\mathcal{RO}})$. Hence $(\operatorname{\operatorname{Kar}}_{{\underline{C}}'}(F({\mathcal{LO}})),\operatorname{\operatorname{Kar}}_{{\underline{C}}'}(F({\mathcal{RO}}))$ is a torsion pair in ${\underline{C}}'$. Next, ${\mathcal{LO}}'$ and ${\mathcal{RO}}'$ are Karoubi-closed in ${\underline{C}}'$; hence $\operatorname{\operatorname{Kar}}_{{\underline{C}}'}(F({\mathcal{LO}}))\subset {\mathcal{LO}}'$ and $\operatorname{\operatorname{Kar}}_{{\underline{C}}'}(F({\mathcal{RO}}))\subset {\mathcal{RO}}'$. On the other hand, $$\operatorname{\operatorname{Kar}}_{{\underline{C}}'}(F({\mathcal{LO}}))={{}^{\perp}}(\operatorname{\operatorname{Kar}}_{{\underline{C}}'}(F({\mathcal{RO}})))\supset {{}^{\perp}}{\mathcal{RO}}'={\mathcal{LO}}';$$ dualizing we obtain the remaining inclusion $\operatorname{\operatorname{Kar}}_{{\underline{C}}'}(F({\mathcal{RO}}))\supset {\mathcal{RO}}$. \[rwsts\] 1. We have to pay a certain price for uniting weight structures and $t$-structures in a single definition. The problem is that we have ${\mathcal{LO}}[1]\subset {\mathcal{LO}}$ for $t$-structures, whereas for weight structures the opposite inclusion is fulfilled (and we will see below that these inclusions actually characterize $t$-structures and weight structures). So, the left orthogonal class for a weight structure is actually “a right one with respect to shifts”. Also, the definition of left and right adjacent (weight and $t$-) structures in [@bws] was “symmetric”, i.e., $w$ being left adjacent to $t$ and $t$ being left adjacent to $w$ were synonyms; in contrast, our current convention follows Definition 3.10 of [@postov]. So, $w$ and $t$ being left adjacent in the sense introduced in the previous papers is equivalent to the torsion pair associated with $w$ (see Remark \[rwhop\](1) below) being left adjacent for the torsion pair associated with $t$ “up to a shift” (see Remark \[rstws\](4)).[^21] Lastly, we will study the [*hearts*]{} both for weight structures and $t$-structures. The corresponding definitions are very much similar; yet we are not able to give a single definition in terms of torsion pairs. 2\. Certainly, part 6 of the proposition essentially (see part 1 of this remark) generalizes Proposition \[pbw\](\[idual\]) below, whereas Proposition \[phop\](7) is closely related to Proposition \[pbw\](\[icompl\]). In particular, Proposition \[phop\](7) easily implies that for any functor $H:{\underline{C}}\to {\underline{A}}$ (for any abelian category ${\underline{A}}$) the correspondence $M\mapsto \operatorname{\operatorname{Im}}(H(L_sM)\to H(M))$ for $M\in {\operatorname{Obj}}{\underline{C}}$ gives a well-defined subfunctor of $H$; cf. Proposition 2.1.2(1) of [@bws]. Thus $s$ yields a certain (two-step) filtration on any (co)homology theory defined on ${\underline{C}}$. This is a certain generalization of the weight filtration defined in ibid. (cf. Remark \[rwhop\] below). The main distinction is that we don’t know (in general) the relation between $s$ and the “shifted” torsion pairs $s[i]=({\mathcal{LO}}[i],{\mathcal{RO}}[i])$; thus there appears to be no reasonable way to obtain “longer” filtrations using this observation. 3\. Recall (see Proposition 2.6 of [@christ]; cf. also [@modoi]) that a projective class in a triangulated category ${\underline{C}}$ is a couple $({\mathcal{P}},I)$ for ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ and $I$ being the class of ${\mathcal{P}}$-null morphisms that satisfy the following additional conditions: ${\mathcal{P}}$ is the largest class such that all elements of $I$ are ${\mathcal{P}}$-null and for any $M\in {\operatorname{Obj}}{\underline{C}}$ there exists a distinguished triangle $L\to M\stackrel{n}{\to} R\to L[1]$ such that $L\in {\mathcal{P}}$ and $n\in I$. The author has proved the relation of torsion pairs to projective classes for the purposes of applying it in §\[swgws\] below. He was not able to get anything useful from this relation yet; so the reader may ignore projective classes in this text. Note however that knowing the notion of a projective class is necessary to trace the (close) relation of our proof of Theorem \[tpgws\] to Lemma 2.2 of [@modoi]. We also prove some simple statements on torsion pairs in categories that have coproducts. \[phopft\] Let $s=({\mathcal{LO}},{\mathcal{RO}})$ be a torsion pair generated by some ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ in a triangulated category ${\underline{C}}$. Then the following statements are valid. I. Assume that ${\underline{C}}$ has coproducts and ${\mathcal{P}}$ is a set of compact objects. Denote by ${\underline{D}}$ the localizing subcategory of ${\underline{C}}$ generated by ${\mathcal{P}}$; $E=(\cup_{i\in {{\mathbb{Z}}}}{\mathcal{P}}[i])^\perp$. 1\. ${\mathcal{LO}}\subset {\operatorname{Obj}}{\underline{D}}$, whereas ${\mathcal{RO}}$ is precisely the class of extensions of elements of $E$ by that of $D={\operatorname{Obj}}{\underline{D}}\cap {\mathcal{RO}}$. 2\. $({\mathcal{LO}},D)$ is a torsion pair for ${\underline{D}}$. II\. Let $F:{\underline{C}}\to {\underline{C}}'$ be an exact functor that possesses a right adjoint $G$. 1. Assume in addition that $F$ is a full embedding. Then $s'=({\mathcal{LO}}', {\mathcal{RO}}')$ is the torsion pair generated by $F({\mathcal{P}})$ in ${\underline{C}}'$, where ${\mathcal{RO}}'$ is the class of extensions of elements of $F({\operatorname{Obj}}{\underline{C}})^{\perp_{{\underline{C}}'}}$ by that of $F({\mathcal{RO}})$, and ${\mathcal{LO}}'$ is the closure of $F({\mathcal{LO}})$ with respect to ${\underline{C}}'$-isomorphisms. 2\. Let $s'=({\mathcal{LO}}', {\mathcal{RO}}')$ be an arbitrary torsion pair in ${\underline{C}}'$. Then the following conditions are equivalent: a). $F({\mathcal{LO}})\subset {\mathcal{LO}}'$; b.) $F({\mathcal{P}})\subset {\mathcal{LO}}'$; c). $G({\mathcal{RO}}')\subset {\mathcal{RO}}$. 3\. For $s'$ as in the previous assertion assume in addition that $F({\mathcal{LO}})\subset {\mathcal{LO}}'$ and $F({\mathcal{RO}})\subset {\mathcal{RO}}'$, and $F$ is essentially surjective on objects. Then $s'$ is generated by $F({\mathcal{P}})$. III\. Assume that ${\underline{C}}$ has coproducts and all elements of ${\mathcal{P}}$ are compact in it. Then $s$ is smashing. I.1. Certainly, any extension of an element of $E$ by an object of ${\underline{D}}$ belongs to ${\mathcal{P}}^\perp={\mathcal{RO}}$. Next, Proposition \[pcomp\](II.2) gives the existence of an exact right adjoint $G$ to the embedding of ${\underline{C}'}$ into ${\underline{C}}$. Moreover, any object of ${\underline{C}}$ is an extension of an element of ${\underline{C}}$ by an object of ${\underline{D}}$, and $G$ is equivalent to the localization of ${\underline{C}}$ by the full triangulated subcategory ${\underline{E}}$ whose object class is $E$ according to Proposition \[pbouloc\](III.\[ibou1\],\[ibou2\]). Hence ${\mathcal{LO}}\subset {\operatorname{Obj}}{\underline{D}}$ and we obtain that any element of ${\mathcal{RO}}$ can be presented as an extension of the aforementioned form. 2\. Obviously, $D={\mathcal{LO}}^{\perp_{{\underline{C}'}}}$. Next, the presentation of elements of ${\mathcal{RO}}$ by extensions as above yields that ${\mathcal{LO}}={}^{\perp_{{\underline{C}}'}} D$. It remains to verify the existence of the corresponding decompositions. For $M\in {\operatorname{Obj}}{\underline{C}'}$ we apply $G$ to (any) its $s$-decomposition; this is easily seen to yield a decomposition of $G(M)\cong M$ with respect to the couple $({\mathcal{LO}},D)$. II.1. We can certainly assume that ${\underline{C}}$ is a (full) strict subcategory of ${\underline{C}}'$. We apply Proposition \[phop\](9). Obviously, ${\mathcal{LO}}'\perp{\mathcal{RO}}'$. Since ${\underline{C}}$ is Karoubi-closed in ${\underline{C}}'$, ${\mathcal{LO}}'$ is Karoubi-closed in ${\underline{C}}'$. Next, for any $N\in {\operatorname{Obj}}{\underline{C}}'$ there exists a essentially unique distinguished triangle $$\label{eglud} N'\to N \to N''\to N'[1]$$ with $N'=G(N)\in{\operatorname{Obj}}{\underline{C}}$ and $N''\in {\underline{C}}^{\perp_{{\underline{C}}'}}$; see Proposition \[pbouloc\](III,\[ibou1\]). Since ${\mathcal{LO}}\perp N''$, we obtain ${\mathcal{LO}}^\perp={\mathcal{RO}}'$. So it remains to verify that any $N$ (as above) possesses an $s'$-decomposition. We choose an $s$-decomposition $L_sN'\to N'\to R_sN'\to L_sN'[1]$ of $N'$. Applying the octahedral axiom to this distinguished triangle along with (\[eglud\]) we obtain a presentation of $N$ as an extension of $R$ by $L_sN'$, where $R$ is some extension of $N''$ by $R_sN'$. Thus we obtain an $s'$-decomposition of $N$. 2. Certainly, c) is equivalent both to ${\mathcal{P}}\perp G({\mathcal{RO}}')$ and to ${\mathcal{LO}}\perp G({\mathcal{RO}}')$. Applying the adjunction $F \dashv G$ we obtain the equivalences in question. 3\. For $M'\in {\operatorname{Obj}}{\underline{C}}$ we obviously have the following chain of equivalences: $$F({\mathcal{P}})\perp M'\iff {\mathcal{P}}\perp G(M') \iff G(M')\in {\mathcal{RO}}\iff {\mathcal{LO}}\perp G(M')\iff F({\mathcal{LO}}')\perp M'.$$ Now, Proposition \[phop\](10) implies immediately that $F({\mathcal{LO}}){{}^{\perp}}={\mathcal{RO}}'$; hence $F({\mathcal{P}}){{}^{\perp}}={\mathcal{RO}}'$ indeed. III\. Obvious; cf. also Proposition \[psym\](\[isymcomp\]) below. $t$-structures: recollection, relation to torsion pairs, and $t$-projectives {#sts} ---------------------------------------------------------------------------- Now we pass to $t$-structures. Certainly, one can easily define them in terms of torsion pairs; still we give the “classical” definition of a $t$-structure here for fixing the notation and for recalling its relation to Definition \[dhop\] (explicitly). \[dtstr\] A couple of subclasses ${\underline{C}}^{t\ge 0},{\underline{C}}^{t\le 0}\subset{\operatorname{Obj}}{\underline{C}}$ will be said to be a $t$-structure $t$ on ${\underline{C}}$ if they satisfy the following conditions: \(i) ${\underline{C}}^{t\ge 0},{\underline{C}}^{t\le 0}$ are strict, i.e., contain all objects of ${\underline{C}}$ isomorphic to their elements. \(ii) ${\underline{C}}^{t\ge 0}\subset {\underline{C}}^{t\ge 0}[1]$, ${\underline{C}}^{t\le 0}[1]\subset {\underline{C}}^{t\le 0}$. \(iii) ${\underline{C}}^{t\le 0}[1]\perp {\underline{C}}^{t\ge 0}$. \(iv) For any $M\in{\operatorname{Obj}}{\underline{C}}$ there exists a [*$t$-decomposition*]{} distinguished triangle $$\label{tdec} L_tM\to M\to R_tM{\to} L_tM[1]$$ such that $L_tM\in {\underline{C}}^{t\le 0}, R_tM\in {\underline{C}}^{t\ge 0}[-1]$. We also need the following auxiliary definitions. \[dtsto\] Let $n\in {{\mathbb{Z}}}$; let $t$ be a $t$-structure on ${\underline{C}}$. 1. \[ito1\] ${\underline{C}}^{t\ge n}$ (resp. ${\underline{C}}^{t\le n}$) will denote ${\underline{C}}^{t\ge 0}[-n]$ (resp. ${\underline{C}}^{t\le 0}[-n]$); ${\underline{C}}^{t=0}={\underline{C}}^{t\ge 0}\cap {\underline{C}}^{t\le 0}$. 2. \[ito2\] ${{\underline{Ht}}}$ will be the full subcategory of ${\underline{C}}$ whose object class is ${\underline{C}}^{t=0}$. 3. \[ito3\] We will say that $t$ is [*right non-degenerate*]{} if $\cap_{i\in {{\mathbb{Z}}}}{\underline{C}}^{t\ge i} ={\{0\}}$. We will say that $t$ is (just) [*non-degenerate*]{} if we also have $\cap_{i\in {{\mathbb{Z}}}}{\underline{C}}^{t\le i} ={\{0\}}$. 4. \[ito4\] We will say that $t$ is [*bounded above*]{} if ${\underline{C}}=\cup_{i\in {{\mathbb{Z}}}}{\underline{C}}^{t\le i}$. 5. \[itoe\] Let ${\underline{C}}'$ be a triangulated category endowed with a $t$-structure $t'$. We will say that an exact functor $F:{\underline{C}}\to {\underline{C}}'$ is [*$t$-exact*]{} whenever $F({\underline{C}}^{t\le 0})\subset {\underline{C}}'^{t'\le 0}$ and $F({\underline{C}}^{t\ge 0})\subset {\underline{C}}'^{t'\ge 0}$. \[rtst1\] 1. \[it1\] Recall that ${\underline{C}}^{t\le n}={}^{\perp}({\underline{C}}^{t\ge n+1})$ and ${\underline{C}}^{t\ge n}={\underline{C}}^{t\le n-1}{}^{\perp}$ (for $t$ and $n$ as above). Thus for ${\mathcal{LO}}={\underline{C}}^{t\le 0}$ and ${\mathcal{RO}}={\underline{C}}^{t\ge 1}$ the couple $s=({\mathcal{LO}},{\mathcal{RO}})$ is a torsion pair for ${\underline{C}}$ that we will call the torsion pair associated with $t$. Conversely, if for a torsion pair $s$ we have ${\mathcal{LO}}[1]\subset {\mathcal{LO}}$ then $({\mathcal{LO}},{\mathcal{RO}}[1])$ is a $t$-structure (that we will say to be associated with $s$). For a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ we will say that ${\mathcal{P}}$ generates $t$ whenever it generates the associated $s$ (certainly, then we have ${\mathcal{P}}\subset {\mathcal{LO}}={\underline{C}}^{t\le 0}$); we will say that $t$ is compactly generated whenever $s$ is. We will say that $t$ is coproductive (resp., productive; resp., smashing or cosmashing) whenever $s$ is; we will say that $t$ restricts to ${\underline{C}'}$ (see Definition \[dhopo\](4)) whenever $s$ does. 2. \[itd\] It is well known (and also follows from the previous part of this remark along with Proposition \[phop\](6)) that $({\mathcal{LO}},{\mathcal{RO}}[1])$ is a $t$-structure on ${\underline{C}}$ if and only if $({\mathcal{RO}}, {\mathcal{LO}}[-1])$ gives a $t$-structure on ${\underline{C}}{{^{op}}}$. 3. \[it3\] Recall that the triangle (\[tdec\]) is canonically (and functorially) determined by $M$. So for $M'=M[n]$ and $L_tM'\to M'\to R_tM'{\to} L_tM'[1]$ being its $t$-decomposition we will write $t^{\le n}M$ for $L_tM'[-n]$ (and this notation is ${\underline{C}}$-functorial). Moreover, the functor $t^{\le n}: {\underline{C}}\to {\underline{C}}^{t\le n}$ (considered as a full subcategory of ${\underline{C}}$) is right adjoint to the embedding $ {\underline{C}}^{t\le n}\to {\underline{C}}$. Dually, we obtain a functor $t^{\ge n+1}:M\mapsto R_tM'[-n-1]$, and $t^{\ge n+1}$ is left adjoint to the embedding $ {\underline{C}}^{t\ge n+1}\to {\underline{C}}$. It certainly follows that any $t$-exact functor “commutes with these adjoints”. In particular, this fact may be applied in the case where ${\underline{C}}^{t\le 0}$ is the class of objects of a full triangulated subcategory ${\underline{E}}$ of ${\underline{C}}$ (cf. Proposition \[prtst\](\[itp4\]) below); in this case the triangle (\[tdec\]) is just the triangle (\[ebou\]). 4. \[it4\] Recall that ${{\underline{Ht}}}$ is necessarily an abelian category with short exact sequences corresponding to distinguished triangles in ${\underline{C}}$. Moreover, have a canonical isomorphism of functors $t^{\le 0}\circ t^{\ge 0}=t^{\le 0}\circ t^{\le 0}$ (if we consider these functors as endofunctors of ${\underline{C}}$). This composite functor $H_0^t$ actually takes values in ${{\underline{Ht}}}\subset {\underline{C}}$, and it is homological if considered this way. We prove a few properties of $t$-structures related to these observations. \[prtst\] Let $t$ be a $t$-structure on ${\underline{C}}$. Then the following statements are valid. 1. \[it2\] Assume that $t$ is smashing. Then ${{\underline{Ht}}}$ is an AB4 category that is closed with respect to ${\underline{C}}$-coproducts, and $H_0^t$ is a cc functor (see Definition \[dcomp\](\[idcc\])). 2. \[it2d\] Dually, if $t$ is cosmashing then ${{\underline{Ht}}}$ is an AB4\* category, the embedding ${{\underline{Ht}}}\to {\underline{C}}$ respects products, and $H_0^t$ is a pp functor. 3. \[itp4\] Triangulated subcategories $L\subset {\underline{C}}$ possessing a right adjoint $G$ to the embedding functor $L\to {\underline{C}}$ are in one-to-one correspondence with torsion pairs $({\mathcal{LO}},{\mathcal{RO}})$ such that ${\mathcal{LO}}[1]={\mathcal{LO}}$; the correspondence sends $L$ into $({\operatorname{Obj}}L,{\operatorname{Obj}}L{{}^{\perp}})$. Moreover, any $({\mathcal{LO}},{\mathcal{RO}})$ of this sort is a $t$-structure.[^22] 4. \[it4sm\] Assume that ${\underline{C}}$ has coproducts. Then smashing torsion pairs $({\mathcal{LO}},{\mathcal{RO}})$ such that ${\mathcal{LO}}[1]={\mathcal{LO}}$ are in one-to-one correspondence with those exact embeddings $L\to {\underline{C}}$ such that the corresponding $G$ (exists and) respects coproducts. 5. \[ittadj\] Assume that ${\underline{C}}$ is compactly generated; let $F:{\underline{C}}\to {\underline{C}}'$ be a full exact embedding respecting coproducts. Then $({\underline{C}}'^{t'\le 0}, {\underline{C}}'^{t'\ge 0})$ is the $t$-structure generated by $F({\underline{C}}^{t\le 0})$ in ${\underline{C}}'$, where ${\underline{C}}'^{t'\ge 0}$ is the class of extensions of elements of $F({\operatorname{Obj}}{\underline{C}})^{\perp_{{\underline{C}}'}}$ by that of $F({\underline{C}}^{t\ge 0})$, and ${\underline{C}}'^{t'\le 0}$ is the closure of $F({\underline{C}}^{t\le 0})$ with respect to ${\underline{C}}'$-isomorphisms. \[it2\]. ${{\underline{Ht}}}$ is closed with respect to ${\underline{C}}$-coproducts according to Proposition \[phop\](3). Thus ${{\underline{Ht}}}$ is an AB4 category according to Proposition \[pcoprtriang\]. Next, the endofunctors $t^{\le 0} $ and $ t^{\ge 0}$ of ${\underline{C}}$ respect coproducts according to Proposition \[phop\](4); hence their composition also does. \[it2d\]. This is just the categorical dual to the previous assertion (see Remark \[rtst1\](\[itd\]). \[itp4\]. If $G$ exists then $({\operatorname{Obj}}L,{\operatorname{Obj}}L{{}^{\perp}})$ is easily seen to be a torsion pair; combine Proposition \[pbouloc\](III.\[ibou1\], II) with Proposition \[phop\](9). Moreover, we certainly have ${\operatorname{Obj}}L[1]={\operatorname{Obj}}L$. Conversely, if ${\mathcal{LO}}={\mathcal{LO}}[1]$ then the corresponding $L$ is triangulated since it is extension-closed (see Proposition \[phop\](1)). Next, $({\mathcal{LO}},{\mathcal{RO}})$ is a $t$-structure according to Remark \[rtst1\](\[it1\]). Hence $G$ exists according to part \[it3\] of the remark. \[it4sm\]. We should check which torsion pairs $s=({\mathcal{LO}},{\mathcal{RO}})$ with ${\mathcal{LO}}={\mathcal{LO}}[1]$ are smashing. According to Proposition \[phop\](4) we should check whether coproducts of $s$-decompositions are $s$-decompositions. Since $s$ is a $t$-structure, $t$-decompositions are canonical (see Remark \[rtst1\](\[it1\]) hence we should check when the endofunctor $t^{\le 0}$ respects coproducts. Now, the embedding $i:L\to {\underline{C}}$ respects coproducts since it possesses a right adjoint; thus $ G$ respects coproducts if and only if $i\circ G=-^{t\le 0}$ does. \[ittadj\]. Since ${\underline{C}}$ is compactly generated and $F$ respects coproducts, it possesses a right adjoint (see Proposition \[pcomp\](II)). Hence it remains to apply Proposition \[phopft\](II.1) (we take ${\mathcal{P}}={\underline{C}}^{t\le 0}$ in it and invoke Remark \[rtst1\](\[itd\])). \[rtst2\] 1. \[ismashs\] If ${\underline{C}}$ has coproducts and the embedding $i:L\to {\underline{C}}$ possesses a right adjoint respecting coproducts then $L$ is called a [*smashing subcategory*]{} of ${\underline{C}}$; see [@kellerema]. Moreover, these conditions are equivalent to the perfectness of the class ${\operatorname{Obj}}L$ in ${\underline{C}}$ (see Definition \[dwg\](\[idpc\]) below) according to Proposition \[psym\](\[iperftp\]). 2. \[it5s1\] Both $t$-structures and weight structures are essentially particular cases of torsion pairs corresponding to the cases ${\mathcal{LO}}[1]\subset {\mathcal{LO}}$ and ${\mathcal{LO}}\subset {\mathcal{LO}}[1]$, respectively (see Remark \[rtst1\](\[it1\]) and Remark \[rwhop\](1) below). So, the “shift-stable” torsion pairs described in Proposition \[prtst\](\[itp4\]) yield the “intersection” of these cases. 3. \[it6\] So it is no wonder that the results and arguments of §\[sperfws\] below are closely related to the properties of localizing subcategories of triangulated categories as studied by A. Neeman, H. Krause and others. 4. \[ialsmash\] Proposition \[prtst\](\[it4sm\]) can easily be generalized as follows: if ${\alpha}$ is a regular cardinal (see §\[snotata\]) and ${\underline{C}}$ is closed with respect to coproducts of less than ${\alpha}$ objects then for an exact embedding $L\to {\underline{C}}$ possessing a right adjoint $G$ the functor $G$ respects coproducts of less than ${\alpha}$ objects if and only if the class ${\operatorname{Obj}}L^{\perp_{{\underline{C}}}}$ is closed with respect to ${\underline{C}}$-coproducts of less than ${\alpha}$ elements. Now we introduce $t$-projective objects (essentially following [@zvon]). \[dpt\] Let $t$ be a $t$-structure on ${\underline{C}}$. Then we will write $P_t$ for the class ${}^\perp ({\underline{D}}^{t\ge 1}\cup {\underline{D}}^{t\le -1})$; we will say that its elements are [*$t$-projective*]{}.[^23] We prove some simple statements relating $t$-projectives to exact functors from ${{\underline{Ht}}}$ into abelian groups. \[pgen\] Let $t$ be a $t$-structure for ${\underline{C}}$. Then the following statements are valid. 1. \[ipgen1\] Let $N\in {\underline{C}}^{t\le 0}$. Then we have $t^{\ge 0}N\cong H_0^t(N)$, and the object $H_0^t(N)$ corepresents the restriction of the functor $H^N={\underline{C}}(N,-)$ to ${{\underline{Ht}}}$. 2. \[ipgen2\] $P_t\subset {\underline{C}}^{t\le 0}$, and for any $P\in P_t$ we have natural isomorphisms of functors $$\label{ept} {\underline{C}}(P,-)\cong {\underline{C}}(P,H_0^t(-))\cong {{\underline{Ht}}}(H_0^t(P), H_0^t(-));$$ the first of them is induced by the transformations $ {\operatorname{id}}_{{\underline{C}}}\to t^{\ge 0}$ and $H_0^t\to t^{\ge 0}$. 3. \[ipgen25\] For any $P\in P_t$ the object $H_0^t(P)$ is projective in ${{\underline{Ht}}}$. 4. \[ipgen4\] Assume that ${\underline{C}}$ (has products and) satisfies the dual Brown representability condition; assume that $t$ is cosmashing. Then $H_0^t$ gives an equivalence of (the full subcategory of ${\underline{C}}$ given by) $P_t$ with the subcategory of projective objects of ${{\underline{Ht}}}$. \[ipgen1\]. The first part of the assertion is just a particular case of the definition of $H_0^t(-)$ (see Remark \[rtst1\](\[it4\])). The second part is very easy also (cf. the proof of [@zvon Lemma 2(1)]); just apply the fact that the functor $t^{\ge 0}$ is left adjoint to the embedding $ {\underline{C}}^{t\ge 0}\to {\underline{C}}$ (see Remark \[rtst1\](\[it3\])). \[ipgen1\]. The first part of the assertion is immediate from Remark \[rtst1\](\[it1\]) (since it gives ${\underline{C}}^{t\le 0}= {{}^{\perp}}{\underline{C}}^{t\ge 1}$). The first isomorphism in (\[ept\]) follows easily from the definitions of $P_t$ and $H^t_0$, whereas the second one is given by assertion \[ipgen1\]. \[ipgen25\]. The statement is given by Lemma 2(1) of [@zvon] (and also easily follows from assertion \[ipgen1\]). \[ipgen4\]. We should prove that any projective object $P_0$ of ${{\underline{Ht}}}$ “lifts” to $P_t$. Consider the functor $H^{P_0}={{\underline{Ht}}}(P_0,-)\circ H_0^t:{\underline{C}}\to {\underline{\operatorname{Ab}}}$. Since $H^t_0$ is a (homological) pp functor according to (the dual to) Proposition \[prtst\](\[it2\]), $H^{P_0}$ is a pp functor also. Hence it is corepresentable by some $P\in {\operatorname{Obj}}{\underline{C}}$ that obviously belongs to $P_t$. It remains to apply assertions \[ipgen1\] and \[ipgen2\] to prove that $H_0^t(P)\cong P_0$. \[rcompgen\] Now assume that $t$ is generated by the class $\cup_{i\ge 0}{\mathcal{P}}[i]$ for some ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ (i.e., ${\underline{C}}^{t\ge 0}=\cap_{i\ge 1} {\mathcal{P}}{{}^{\perp}}[i]$; it certainly follows that ${\mathcal{P}}\subset {\underline{C}}^{t\le 0}$). Then part \[ipgen1\] of our proposition implies that ${\underline{C}}^{t=0}\cap (\{H_0^t(P):\ P\in {\mathcal{P}}\}{{}^{\perp}})={\{0\}}$, i.e., this class Hom-generates ${{\underline{Ht}}}$. Indeed, if $M$ is a non-zero element of ${\underline{C}}^{t=0}$ then $M\notin {\underline{C}}^{t\ge 1}$. Hence there exists $P\in {\mathcal{P}}$ such that $P\not\perp M$; thus $H_0^t(P) \not\perp M$ also. Moreover, if ${\mathcal{P}}$ is a set and $t$ is smashing then the ${\underline{C}}$-coproduct $\coprod_{P\in {\mathcal{P}}} H_0^t(P)$ Hom-generates ${{\underline{Ht}}}$ also (see Proposition \[phop\](3)). Weight structures: reminder and pure functors {#sws} ============================================= In §\[ssws\] we recall some basics on weight structures (so, the only new result of the subsection is the remark on the relation of weight structures to torsion pairs). In §\[sswc\] we recall some properties of the weight complex functors. Our treatment of this subject is “more accurate” than the original one in [@bws]. In §\[sdetect\] we construct [*pure*]{} (cf. Remark \[rwrange\](5)) homological functors from ${\underline{C}}$ starting from additive functors from ${{\underline{Hw}}}$ into abelian categories. We also study conditions ensuring that a functor of this sort “detects weights of objects”. The results of this section are important for the study of Picard groups of triangulated categories (endowed with weight structures) carried over in [@bontabu]. In §\[svtt\] we recall the notion of virtual $t$-truncations of (co)homological functors from ${\underline{C}}$ and relate them to functors of limited weight range (that are important for [@bkw]). In §\[sprcoprod\] we prove that weight decompositions and weight complexes “respect coproducts” whenever $w$ is smashing; it follows that virtual $t$-truncations of cc and cp functors are cc and cp functors, respectively (in this case). We also study generators (see Remark \[rgenw\]) for ${{\underline{Hw}}}$. Weight structures: basics {#ssws} -------------------------- Let us recall the definition of one of the main notions of this paper. \[dwstr\] A couple of subclasses ${\underline{C}}_{w\le 0},{\underline{C}}_{w\ge 0}\subset{\operatorname{Obj}}{\underline{C}}$ will be said to define a [*weight structure*]{} $w$ for a triangulated category ${\underline{C}}$ if they satisfy the following conditions. \(i) ${\underline{C}}_{w\le 0}$ and ${\underline{C}}_{w\ge 0}$ are Karoubi-closed in ${\underline{C}}$ (i.e., contain all ${\underline{C}}$-retracts of their elements).[^24] (ii) [**Semi-invariance with respect to translations.**]{} ${\underline{C}}_{w\le 0}\subset {\underline{C}}_{w\le 0}[1]$ and ${\underline{C}}_{w\ge 0}[1]\subset {\underline{C}}_{w\ge 0}$. \(iii) [**Orthogonality.**]{} ${\underline{C}}_{w\le 0}\perp {\underline{C}}_{w\ge 0}[1]$. \(iv) [**Weight decompositions**]{}. For any $M\in{\operatorname{Obj}}{\underline{C}}$ there exists a distinguished triangle $LM\to M\to RM {\to} LM[1]$ such that $LM\in {\underline{C}}_{w\le 0} $ and $ RM\in {\underline{C}}_{w\ge 0}[1]$. We will also need the following definitions. \[dwso\] Let $i,j\in {{\mathbb{Z}}}$. 1. \[id1\] The full subcategory ${{\underline{Hw}}}\subset {\underline{C}}$ whose object class is ${\underline{C}}_{w=0}={\underline{C}}_{w\ge 0}\cap {\underline{C}}_{w\le 0}$ is called the [*heart*]{} of $w$. 2. \[id2\] ${\underline{C}}_{w\ge i}$ (resp. ${\underline{C}}_{w\le i}$, ${\underline{C}}_{w= i}$) will denote ${\underline{C}}_{w\ge 0}[i]$ (resp. ${\underline{C}}_{w\le 0}[i]$, ${\underline{C}}_{w= 0}[i]$). 3. \[id3\] ${\underline{C}}_{[i,j]}$ denotes ${\underline{C}}_{w\ge i}\cap {\underline{C}}_{w\le j}$; so, this class equals ${\{0\}}$ if $i>j$. ${\underline{C}}^b\subset {\underline{C}}$ will be the category whose object class is $\cup_{i,j\in {{\mathbb{Z}}}}{\underline{C}}_{[i,j]}$. 4. \[id4\] We will say that $({\underline{C}},w)$ is [*bounded*]{} if ${\underline{C}}^b={\underline{C}}$ (i.e., if $\cup_{i\in {{\mathbb{Z}}}} {\underline{C}}_{w\le i}={\operatorname{Obj}}{\underline{C}}=\cup_{i\in {{\mathbb{Z}}}} {\underline{C}}_{w\ge i}$). Respectively, we will call $\cup_{i\in {{\mathbb{Z}}}} {\underline{C}}_{w\le i}$ (resp. $\cup_{i\in {{\mathbb{Z}}}}{\underline{C}}_{w\ge i}$) the class of $w$-[*bounded above*]{} (resp. $w$-[*bounded below*]{}) objects; we will say that $w$ is bounded above (resp. bounded below) if all the objects of ${\underline{C}}$ satisfy this property. 5. \[ideg\] We will call the elements of $\cap_{i\in {{\mathbb{Z}}}}{\underline{C}}_{w\le i}$ (resp. of $\cap_{i\in {{\mathbb{Z}}}}{\underline{C}}_{w\ge i}$) [*right degenerate*]{} (resp. [*left degenerate*]{}). Respectively, we will say that $w$ is [*non-degenerate*]{} if $\cap_{i\in {{\mathbb{Z}}}}{\underline{C}}_{w\le i}=\cap_{i\in {{\mathbb{Z}}}}{\underline{C}}_{w\ge i}={\{0\}}$ (i.e., if all degenerate objects of ${\underline{C}}$ are trivial). We will say that $w$ is [*right non-degenerate*]{} (resp. [*left non-degenerate*]{}) if $\bigcap\limits_{i\in {{\mathbb{Z}}}} {\underline{C}}_{w \le i} = \{0\}$ (resp. $\bigcap\limits_{i\in {{\mathbb{Z}}}} {\underline{C}}_{w \ge i} = \{0\}$). 6. \[idadj\] If $t$ is a $t$-structure on ${\underline{C}}$ then we will say that $w$ is left adjacent to $t$ or that $t$ is right adjacent to $w$ if ${\underline{C}}_{w\ge 0}={\underline{C}}^{t\le 0}$. Dually, $w$ is right adjacent to $t$ whenever ${\underline{C}}_{w\le 0}={\underline{C}}^{t\ge 0}$. 7. \[ineg\] An additive subcategory $D\subset {\operatorname{Obj}}{\underline{C}}$ will be called [*negative*]{} if for any $i>0$ we have ${\operatorname{Obj}}D\perp {\operatorname{Obj}}D[i]$. 8. \[idwe\] Let ${\underline{C}}'$ be a triangulated category endowed with a weight structure $w'$; let $F:{\underline{C}}\to {\underline{C}}'$ be an exact functor. We will say that $F$ is [*left weight-exact*]{} (with respect to $w,w'$) if it maps ${\underline{C}}_{w\le 0}$ to ${\underline{C}}'_{w'\le 0}$; it will be called [*right weight-exact*]{} if it maps ${\underline{C}}_{w\ge 0}$ to ${\underline{C}}'_{w'\ge 0}$. $F$ is called [*weight-exact*]{} if it is both left and right weight-exact. \[rstws\] 1\. Similarly to Remark \[rgen\](3), we will sometimes need a choice of a weight decomposition of $M[-m]$ shifted by $[m]$. So we take a distinguished triangle $$\label{ewd} w_{\le m}M\to M\to w_{\ge m+1}M$$ with some $ w_{\ge m+1}M\in {\underline{C}}_{w\ge m+1}$, $ w_{\le m}M\in {\underline{C}}_{w\le m}$; we will call it an [*$m$-weight decomposition*]{} of $M$, and call arbitrary choices of $w_{\ge m+1}M$ and $ w_{\le m}M$ [*weight truncations of $M$*]{} (for all $m\in {{\mathbb{Z}}}$). We will use this notation below (though $w_{\ge m+1}M$ and $ w_{\le m}M$ are not canonically determined by $M$). Moreover, when we will write arrows of the type $w_{\le m}M\to M$ or $M\to w_{\ge m+1}M$ we will always assume that they come from some $m$-weight decomposition of $M$. 2\. A simple (and still useful) example of a weight structure comes from the stupid filtration on the homotopy categories of cohomological complexes $K(B)$ for an arbitrary additive $B$. In this case $K(B)_{{w^{st}}\le 0}$ (resp. $K(B)_{{w^{st}}\ge 0}$) will be the class of complexes that are homotopy equivalent to complexes concentrated in degrees $\ge 0$ (resp. $\le 0$); see Remark 1.2.3(1) of [@bonspkar] for more detail. The heart of this weight structure is the Karoubi-closure of $B$ in $K(B)$ (that is actually equivalent to $\operatorname{\operatorname{Kar}}(B)$). 3\. In the current paper we use the “homological convention” for weight structures; it was previously used in [@wildcons], [@wildab], [@brelmot], [@bkw], [@bonspkar], and [@bgn], whereas in [@bws], [@bger], [@bach], and [@bontabu] the “cohomological convention” was used.[^25] In the latter convention the roles of ${\underline{C}}_{w\le 0}$ and ${\underline{C}}_{w\ge 0}$ are interchanged, i.e., one considers ${\underline{C}}^{w\le 0}={\underline{C}}_{w\ge 0}$ and ${\underline{C}}^{w\ge 0}={\underline{C}}_{w\le 0}$. Note also that in this paper we will (following [@bbd] and coherently with all the papers of the author cited here) use the “cohomological” convention for $t$-structures. This “discrepancy between conventions” will force us to put somewhat weird “$-$” signs in some of the formulas (cf. also Definition \[dwso\](\[idadj\])); however, it is coherent with Definition \[dhopo\](2) (of adjacent torsion pairs). Lastly, in [@bws] both “halves” of $w$ were required to be additive. Yet this additional restriction is easily seen to follow from the remaining axioms; see Remark 1.2.3(4) of [@bonspkar]. 4\. As we had already noted in Remark \[rwsts\], the current definition of right and left adjacent “structures” is somewhat different from the one used in previous papers of the author. Also, if $w$ is left or right adjacent to $t$ then the associated torsion pairs are only “adjacent up to a shift”; yet this is easily seen to make no difference in the proofs. Now we recall some basic properties of weight structures. \[pbw\] Let ${\underline{C}}$ be a triangulated category endowed with a weight structure $w$, $M, M',M''\in {\operatorname{Obj}}{\underline{C}}$, $i,m,n\in {{\mathbb{Z}}}$. Then the following statements are valid. 1. \[idual\] The axiomatics of weight structures is self-dual, i.e., for ${\underline{D}}={\underline{C}}^{op}$ (so ${\operatorname{Obj}}{\underline{C}}={\operatorname{Obj}}{\underline{D}}$) there exists the (opposite) weight structure $w'$ for which ${\underline{D}}_{w'\le 0}={\underline{C}}_{w\ge 0}$ and ${\underline{D}}_{w'\ge 0}={\underline{C}}_{w\le 0}$. 2. \[iextw\] ${\underline{C}}_{w\le i}$, ${\underline{C}}_{w\ge i}$, and ${\underline{C}}_{w=i}$ are Karoubi-closed and extension-closed in ${\underline{C}}$ (and so, additive). 3. \[iort\] ${\underline{C}}_{w\ge i}=({\underline{C}}_{w\le i-1})^{\perp}$ and ${\underline{C}}_{w\le i}={}^{\perp} ({\underline{C}}_{w\ge i+1})$. 4. \[igenlm\] The class ${\underline{C}}_{[m,n]}$ is the extension-closure of $\cup_{m\le j\le n}{\underline{C}}_{w=j}$. 5. \[iwd0\] If $M\in {\underline{C}}_{w\ge m}$ then $w_{\le n}M\in {\underline{C}}_{[m,n]}$ (for any $n$-weight decomposition of $M$). Dually, if $M\in {\underline{C}}_{w\le n}$ then $w_{\ge m}M\in {\underline{C}}_{[m,n]}$. 6. \[icompl\] Assume that $ m\le n$. The for any (fixed) $m$-weight decomposition of $M$ and an $n$-weight decomposition of $M'$ (see Remark \[rstws\](1)) any morphism $g\in {\underline{C}}(M,M')$ can be extended to a morphism of the corresponding distinguished triangles: $$\label{ecompl} \begin{CD} w_{\le m} M@>{c}>> M@>{}>> w_{\ge m+1}M\\ @VV{h}V@VV{g}V@ VV{j}V \\ w_{\le n} M'@>{}>> M'@>{}>> w_{\ge n+1}M' \end{CD}$$ Moreover, if $m<n$ then this extension is unique (provided that the rows are fixed). 7. \[iwdext\] For any distinguished triangle $M\to M'\to M''\to M[1]$ and any weight decompositions $LM\stackrel{a_{M}}{\to} M\stackrel{n_{M}}{\to} R_M\to LM[1]$ and $LM''\stackrel{a_{M''}}{\to} M''\stackrel{n_{M''}}{\to} R_M''\to LM''[1]$ there exists a commutative diagram $$\begin{CD} LM @>{}>>LM'@>f>> LM''@>{}>>LM[1]\\ @VV{a_M}V@VV{a_{M'}}V @VV{a_{M''}}V@VV{a_{M}[1]}V\\ M@>{}>>M'@>{}>>M''@>{}>>M[1]\\ @VV{n_M}V@VV{n_{M'}}V @VV{n_{M''}}V@VV{n_{M}[1]}V\\ RM@>{}>>RM'@>{}>>RM''@>{}>>M[1]\end{CD}$$ in ${\underline{C}}$ whose rows are distinguished triangles and the second column is a weight decomposition (along with the first and the third one). 8. \[isplit\] If an ${{\underline{Hw}}}$-morphism $f:A\to B$ is split surjective then there exists an object $C\in {\underline{C}}_{w=0}$ such that $f$ is isomorphic to the canonical epimorphism $B \bigoplus C\to C$. 9. \[iwdmod\] If $M$ belongs to $ {\underline{C}}_{w\le 0}$ (resp. to ${\underline{C}}_{w\ge 0}$) then it is a retract of any choice of $w_{\le 0}M$ (resp. of $w_{\ge 0}M$). All of the assertions except the two last ones were essentially established in [@bws] (pay attention to Remark \[rstws\](3)!). Assertion \[isplit\] is immediate from assertion \[iextw\] (that says that ${\underline{C}}_{w=0}$ is Karoubi-closed in ${\underline{C}}$). \[iwdmod\]. The case $M\in {\underline{C}}_{w\le 0}$ of the assertion follows immediately from Proposition \[phop\](7) (see Remark \[rwhop\](1) below). The case $M\in {\underline{C}}_{w\ge 0}$ is just the categorical dual of the first case; see assertion \[idual\]. \[rwhop\] 1\. Similarly to Remark \[rtst1\](\[it1\]), part \[iort\] of our proposition yields that for ${\mathcal{LO}}={\underline{C}}_{w\le 0}$ and ${\mathcal{RO}}={\underline{C}}_{w\ge 1}$ the couple $s=({\mathcal{LO}},{\mathcal{RO}})$ is a torsion pair; we will call it the torsion pair associated with $w$. Conversely, if for a torsion pair $s$ we have ${\mathcal{LO}}\subset {\mathcal{LO}}[1]$ then $({\mathcal{LO}},{\mathcal{RO}}[-1])$ is a weight structure that we will say to be associated with $s$. In this case we will say that $s$ is [*weighty*]{}; for a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ we will say that ${\mathcal{P}}$ generates $w$ whenever it generates $s$ (certainly, then we have ${\mathcal{P}}\subset {\mathcal{LO}}={\underline{C}}_{w\le 0}$, ${\mathcal{P}}{{}^{\perp}}={\underline{C}}_{w\ge 1}$, and $w$ is determined by ${\mathcal{P}}$); $w$ is compactly generated whenever $s$ is. Respectively, we will say that $w$ is smashing (resp. cosmashing, countably smashing; resp. generated by ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$) whenever the corresponding $s$ is. Lastly, we will say that $w$ restricts to a triangulated subcategory ${\underline{C}'}$ of ${\underline{C}}$ whenever the associated $s$ does; cf. Definition \[dhopo\](4). 2. Certainly, the “most interesting” weight and $t$-structures are the non-degenerate ones. However, this non-degeneracy condition can be quite difficult to check (and it actually fails for certain “important” weight structures; see Remark 2.4.4(1) of [@bkw]). So we prefer not to avoid degenerate weight and $t$-structures in this paper; this makes Proposition \[phopft\](II.1) an important tool. Now we describe (some) consequences of this proposition for a torsion pair associated with a weight structure $w$ (resp. with a $t$-structure $t$; see Remark \[rtst1\](\[it1\])). So, assume that ${\underline{C}}$ is a full strict triangulated subcategory of a triangulated category ${\underline{C}}'$, and assume that the embedding ${\underline{C}}\to {\underline{C}}'$ possesses a right adjoint $G$. Then there exists a weight structure $w'$ (resp. a $t$-structure $t'$) on ${\underline{C}}'$ such that ${\underline{C}}'_{w'\le 0}={\underline{C}}_{w\le 0}$ (resp. ${\underline{C}}'^{t'\le 0}={\underline{C}}^{t\le 0}$). It certainly follows that ${\underline{C}}_{w'\ge 0}\cap {\operatorname{Obj}}{\underline{C}}={\underline{C}}_{w\ge 0}$ (resp. ${\underline{C}}^{t'\ge 0}\cap {\operatorname{Obj}}{\underline{C}}={\underline{C}}^{t\ge 0}$); hence ${{\underline{Hw}}}'={{\underline{Hw}}}$ (resp. ${{\underline{Ht}}}'={{\underline{Ht}}}$). Moreover, if $w$ (resp. $t$) is generated by a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ in ${\underline{C}}$ then $w'$ (resp. $t'$) is generated by ${\mathcal{P}}$ in ${\underline{C}}'$. 3\. Now we apply to weight structures Proposition \[phopft\](II.2). Assume that $F:{\underline{C}}\to {\underline{C}}'$ is an exact functor, $G$ is its (exact) right adjoint, and $w$ is generated by some ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$. Then Proposition \[phopft\](II.2) (combined with the first part of this remark) yields the following: $F$ is left weight-exact (with respect to $w$ and some weight structure $w'$ on ${\underline{C}}'$) if and only if $F({\mathcal{P}})\subset {\underline{C}}'_{w'\le 0}$; these conditions are equivalent to the right weight-exactness of $G$. Note that this statement can certainly be applied for ${\mathcal{P}}={\underline{C}}_{w\le 0}$; hence $F$ is left weight-exact if and only if $G$ is right weight-exact. 4\. Recall also that for a compactly generated triangulated category ${\underline{C}}$ and $F:{\underline{C}}\to {\underline{C}}'$ being an exact functor respecting coproducts Proposition \[pcomp\](II) gives the existence of $G$. Hence these assumptions imply that $F$ is left weight-exact if and only if $F({\mathcal{P}})\subset {\underline{C}}'_{w'\le 0}$. 5\. We will also need the categorical dual of the latter observation. Certainly, it is formulated is follows: if ${\underline{C}}$ is cocompactly cogenerated (i.e., there exists a set of objects of ${\underline{C}}$ that compactly generates ${\underline{C}}{{^{op}}}$), $F$ is an exact functor ${\underline{C}}\to {\underline{C}}'$ that respects products, $w$ and $w'$ are are weight structures on ${\underline{C}}$ and ${\underline{C}}'$, respectively, and $w$ is cogenerated by some ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ (i.e., ${\underline{C}}_{w\le -1}={{}^{\perp}}{\mathcal{P}}$) then $F$ is right weight-exact if and only if $F({\mathcal{P}})\subset {\underline{C}}'_{w'\ge 0}$. 6\. Applying Proposition \[phop\](10) to weighty torsion pairs we certainly obtain the following: if $F:{\underline{C}}\to {\underline{C}}'$ is a weight-exact functor that is surjective on objects then ${\underline{C}}'_{w'\ge 0} = \operatorname{\operatorname{Kar}}_{{\underline{C}}'}(F({\underline{C}}_{w\ge 0}))$ and ${\underline{C}}'_{w'\le 0} = \operatorname{\operatorname{Kar}}_{{\underline{C}}'}(F({\underline{C}}_{w\le 0}))$. 7\. Lastly, assume that $F$ as above possesses a right adjoint. Then Proposition \[phopft\](II.3) implies that the weight structure $w'$ is generated by $F({\mathcal{P}})$. Once again, the existence of right adjoint is automatic whenever ${\underline{C}}$ is compactly generated and $F$ respects coproducts. 8\. Assume that $w$ is generated by a class ${\mathcal{P}}$. Then $\cap_{i\in {{\mathbb{Z}}}}{\underline{C}}_{w\ge i}=\cap_{i\in {{\mathbb{Z}}}}{\mathcal{P}}{{}^{\perp}}$. Thus ${\mathcal{P}}$ is left non-degenerate if and only if ${\mathcal{P}}$ Hom-generates ${\underline{C}}$. On weight Postnikov towers and weight complexes {#sswc} ----------------------------------------------- To define the weight complex functor we will need the following definitions. \[dfilt\] Let $M\in {\operatorname{Obj}}{\underline{C}}$. 1\. A datum consisting of $M_{\le i}\in {\operatorname{Obj}}{\underline{C}}$, $h_i\in {\underline{C}}(M_{\le i},M)$, $j_i\in {\underline{C}}(M_{\le i},M_{\le i+1})$ for $i$ running through integers will be called a [*filtration on $M$*]{} if we have $h_{i+1}\circ j_i=h_i$ for all $i\in {{\mathbb{Z}}}$; we will write ${{\operatorname{Fil}}}_*M$ for this filtration. A filtration will be called [*bounded*]{} if there exist $l\le m\in {{\mathbb{Z}}}$ such that $M_{\le i}=0$ for all $i<l$ and $h_i$ are isomorphisms for all $i\ge m$. 2\. A filtration as above equipped with distinguished triangles $M_{\le i-1}\stackrel{j_{i-1}}{\to}M_{\le i}\to M_i$ for all $i\in {{\mathbb{Z}}}$ will be called a [*Postnikov tower*]{} for $M$ or for ${{\operatorname{Fil}}}_*M$; this tower will be denoted by $Po_{{\operatorname{Fil}}}$. We will use the symbol $M^p$ to denote $M_{-p}[p]$; we will call $M^p$ the [*factors*]{} of $Po_{{\operatorname{Fil}}}$. 3. If ${{\operatorname{Fil}}}_*M'=(M'_{\le i}, h'_i, j_i)$ is a filtration of $M'\in {\operatorname{Obj}}{\underline{C}}$ and $g\in {\underline{C}}(M,M')$ then we will call $g$ along with a collection of $g_{\le i}\in {\underline{C}}(M_{\le i}, M'_{\le i})$ a [*morphism of filtrations compatible with $g$*]{} if $g\circ h_i=h'_i\circ g_{\le i}$ and $j'_i\circ g_{\le i} =g_{\le i+1}\circ j_i$ for all $i\in {{\mathbb{Z}}}$. \[rwcomp\] 1. Composing (and shifting) arrows from triangles in $Po_{{\operatorname{Fil}}}$ for all pairs of two subsequent $i$s one can construct a complex whose $i$th term equals $M^i$ (it is easily seen that this is a complex indeed; cf. Proposition 2.2.2 of [@bws]). We will call it a complex [*associated with*]{} $Po_{{\operatorname{Fil}}}$. 2\. Certainly, any filtration yields a Postnikov tower (uniquely up to a non-unique isomorphism). Furthermore, it is easily seen that any morphism of filtrations extends to a morphism of the corresponding Postnikov towers (defined in the obvious way). Besides, any morphism of Postnikov towers yields a morphism of the associated complexes. Lastly, note that morphisms of filtrations and Postnikov towers can certainly be added and composed. 3\. The triangles in $Po_{{\operatorname{Fil}}}$ also give the following statement immediately: if a filtration of $M$ is bounded then $M$ belongs to the extension-closure of $\{M_i\}$. \[dwpt\] Assume that ${\underline{C}}$ is endowed with a weight structure $w$. 1\. We will call a filtration (see Definition \[dfilt\]) ${{\operatorname{Fil}}}_*M$ of $M\in {\operatorname{Obj}}{\underline{C}}$ a [*weight filtration*]{} (of $M$) if the morphisms $h_i:M_{\le i}\to M$ yield $i$-weight decompositions for all $i\in {{\mathbb{Z}}}$ (in particular, $M_{\le i}=w_{\le i}M$). We will call the corresponding $Po_{{\operatorname{Fil}}}$ a [*weight Postnikov tower*]{} for $M$. 2\. ${\operatorname{Post}_w({\underline{C}})}$ will denote the category whose objects are objects of ${\underline{C}}$ endowed with arbitrary weight Postnikov towers and whose morphisms are morphisms of Postnikov towers. ${{\underline{C}}_w}$ will be the category whose objects are the same as for ${\operatorname{Post}_w({\underline{C}})}$ and such that ${{\underline{C}}_w}(Po_{{{\operatorname{Fil}}}_M},Po_{{{\operatorname{Fil}}}_{M'}})=\operatorname{\operatorname{Im}}({\operatorname{Post}_w({\underline{C}})}(Po_{{{\operatorname{Fil}}}_M},Po_{{{\operatorname{Fil}}}_{M'}})\to {\underline{C}}(M,M'))$ (i.e., we kill those morphisms of towers that are zero on the underlying objects). 3\. For an additive category ${\underline{B}}$, complexes $A,B\in {\operatorname{Obj}}K({\underline{B}})$, and morphisms $m_1,m_2\in C({{\underline{Hw}}})(A,B)$ we will write $m_1\backsim m_2$ if $m_1-m_2=d_Bh+jd_A$ for some collections of arrows $j^*,h^*:A^*\to B^{*-1}$. We will call this relation the [*weak homotopy one*]{}.[^26]  The following statements were essentially proved in [@bws]. Moreover, the first two of them easily follow from Proposition \[pbw\](\[icompl\]) (along with the corresponding definitions). \[pwt\] In addition to the notation introduced above assume that ${\underline{B}}$ is an additive category. 1. \[iwpt1\] Any choice of $i$-weight decompositions of $M$ for $i$ running through integers naturally yields a canonical weight filtration for $M$ (with $M_{\le i}=w_{\le i}M$). Moreover, we have $\operatorname{\operatorname{Cone}}(Y_i\to M)\in {\underline{C}}_{w\ge i+1}$ and $M^i\in {\underline{C}}_{w=0}$. 2. \[iwpt2\] Any $g\in {\underline{C}}(M,M')$ can be extended to a morphism of (any choice of) weight filtrations for $M$ and $M'$, respectively; hence it also extends to a morphism of weight Postnikov towers. 3. \[iwpt3\] The natural functor ${{\underline{C}}_w}\to {\underline{C}}$ is an equivalence of categories. 4. \[iwhecat\] Factoring morphisms in $K({\underline{B}})$ by the weak homotopy relation yields an additive category ${K_{\mathfrak{w}}}({\underline{B}})$. Moreover, the corresponding full functor $K({\underline{B}})\to {K_{\mathfrak{w}}}({\underline{B}})$ is (additive and) conservative. 5. \[iwhefu\] Let ${{\mathcal{A}}}:{\underline{B}}\to {\underline{A}}$ be an additive functor, where ${\underline{A}}$ is any abelian category. Then for any $B,B'\in {\operatorname{Obj}}K({\underline{B}})$ any pair of weakly homotopic morphisms $m_1,m_2\in C({{\underline{Hw}}})(B,B')$ induce equal morphisms of the homology $H_*({{\mathcal{A}}}(B^i))\to H_*({{\mathcal{A}}}(B'^i))$. 6. \[iwhefun\] Sending an object of ${{\underline{C}}_w}$ into the complex described in Remark \[rwcomp\](1) yields a well-defined additive functor $t=t_w:{{\underline{C}}_w}\to {K_{\mathfrak{w}}}({{\underline{Hw}}})$. We will call this functor the [*weight complex*]{} one.[^27] We will often write $t(M)$ for $M\in {\operatorname{Obj}}{\underline{C}}$ assuming that some weight Postnikov tower for $M$ is chosen; we will say that $t(M)$ is [*a choice of a weight complex*]{} for $M$. 7. \[iwcex\] If $M_0\stackrel{f}{\to} M_1\to M_2$ is a distinguished triangle in ${\underline{C}}$ then any possible “lift” of $f$ (along with $M_0$ and $M_1$) to ${{\underline{C}}_w}$ can be completed to a lift of the couple of morphisms $M_0\stackrel{f}{\to} M_1\to M_2$ to ${{\underline{C}}_w}$ such that corresponding morphisms $t(M_0)\to t(M_1)\to t(M_2)$ yield a distinguished triangle in $K({{\underline{Hw}}})$. \[rwc\] So, for an object $M$ of ${\underline{C}}$ its weight complex $t(M)$ is well-defined up to a $K({{\underline{Hw}}})$-endomorphism that is weakly homotopic to zero; thus it is defined in $K({{\underline{Hw}}})$ up to a (not necessarily unique) isomorphism. In particular, if $M\in {\underline{C}}_{w\ge -n}$ for some $n\in {{\mathbb{Z}}}$ then we can take $M_{\le i}=w_{\le i}M=0$ for all $i<-n$; hence any choice of $t(M)$ is homotopy equivalent to a complex concentrated in degrees at most $n$. Similarly, if $M\in {\underline{C}}_{w\le -n}$ then we can take $M_{\le i}=w_{\le i}M=M$ for all $i\le -n$; thus $t(M)$ is homotopy equivalent to a complex concentrated in degrees at least $n$. Hence $t(M)\cong 0$ whenever $M$ is left or right degenerate. On pure functors and detecting weights {#sdetect} -------------------------------------- \[ppure\] Assume that ${\underline{C}}$ is endowed with a weight structure $w$. Let ${{\mathcal{A}}}:{{\underline{Hw}}}\to {\underline{A}}$ be an additive functor, where ${\underline{A}}$ is any abelian category. Choose a weight complex $t(M)=(M^j)$ for each $M\in {\operatorname{Obj}}{\underline{C}}$, and denote by $H(M)=H^{{{\mathcal{A}}}}(M)$ the zeroth homology of the complex ${{\mathcal{A}}}(M^{j})$. Then $H(-)$ yields a homological functor that does not depend on the choices of weight complexes. Moreover, the assignment ${{\mathcal{A}}}\mapsto H^{{\mathcal{A}}}$ is natural in ${{\mathcal{A}}}$. Immediate from Proposition \[pwt\] (\[iwpt3\], \[iwhefu\],\[iwcex\]). \[rpure\] 1. (Co)homological functors of this type have already found interesting applications in [@kellyweighomol], [@bach], [@bscwh], [@bontabu], and [@bgn]. We will prove some statements relevant for the latter paper just now. We will not apply all the remaining results of this subsection elsewhere in the paper. 2. We will call a functor $H:{\underline{C}}\to {\underline{A}}$ [*pure*]{} (or $w$-pure) if it equals $H^{{\mathcal{A}}}$ for a certain ${{\mathcal{A}}}:{{\underline{Hw}}}\to {\underline{A}}$. In the next subsection we will prove that this definition of purity is equivalent to another (“intrinsic”) one. So we prove that pure functors can be used to “detect weights”; these results are crucial for [@bontabu]. This notion of detecting weights is closely related to the one of [*weight-conservativity*]{} that was introduced in [@bach]. To prove the most general case of we have to recall a result from [@bkw].[^28]  \[lbkw\] Let $m\in {{\mathbb{Z}}}$, $M\in {\operatorname{Obj}}{\underline{C}}$, where ${\underline{C}}$ is endowed with a weight structure $w$. Then $t(M)$ belongs to $K({{\underline{Hw}}})_{{w^{st}}\ge -m }$ (resp. to $K({{\underline{Hw}}})_{{w^{st}}\le -m }$; see Remark \[rstws\](2)) if and only if $M\bigoplus M[1]$ is an extension of an element of ${\underline{C}}_{w\ge -m}$ by a right degenerate object (resp. $M\bigoplus M[-1]$ is an extension of a left degenerate object by an element of ${\underline{C}}_{w\le -m}$). This statement is contained Corollary 3.1.5 of ibid. For a homological functor $H$ the symbol $H_i$ will be used to denote the composite functor $H\circ [i]$. \[pdetect\] Adopt the notation of Proposition \[ppure\] and assume that the following conditions are fulfilled: \(i) the image of ${{\mathcal{A}}}$ consists of ${\underline{A}}$-projective objects only; \(ii) if an ${{\underline{Hw}}}$-morphism $h$ does not split (i.e., it is not a retraction) then ${{\mathcal{A}}}(h)$ does not split also. Then for any $M\in {\operatorname{Obj}}{\underline{C}}$, $m\in {{\mathbb{Z}}}$, and $H=H^{{\mathcal{A}}}$ we have the following: $M$ is $w$-bounded below and $H_i(M)=0$ for all $i> m$ if and only if $M\in {\underline{C}}_{w\ge -m}$. The “if” part of the statement is very easy (for any ${{\mathcal{A}}}$); just combine the definition of $H^{{\mathcal{A}}}$ with Remark \[rwc\]. Now we prove the converse application. If $M$ is $w$-bounded below then the object $M'=M\bigoplus M[1]$ is $w$-bounded below also. We also have $H_i(M')=0$ for all $i> m$, and it suffices to prove that $M'\in {\underline{C}}_{w\ge -m}$ (by the axiom (i) of weight structures). Now chose the minimal integer $n\ge m$ such that $t(M')\in K(B)_{{w^{st}}\ge -n}$ (see Remark \[rstws\](2)). Then Lemma \[lbkw\] yields the existence of a distinguished triangle $M_1'\stackrel{g}{\to} M'{\to} M'_2\to M'_1[1]$ such that $M'_1$ is right $w$-degenerate and $M'_2\in {\underline{C}}_{w\ge -n}$. Since $M'$ is bounded, $g=0$; hence $M'$ is a retract of $M'_2$ and so belongs to ${\underline{C}}_{w\ge -n}$ itself. It remains to prove that $n=m$. Assume the converse. Then $t(M')$ is homotopy equivalent to an ${{\underline{Hw}}}$-complex $(N^i)$ concentrated in degrees $\le n$, and Proposition \[pbw\](\[isplit\]) yields that the boundary morphism $d^{n-1}_N:N^{n-1}\to N^n$ does not split. Thus ${{\mathcal{A}}}(d^{n-1}_N)$ does not split also. Since ${{\mathcal{A}}}(N^n)$ is projective, this non-splitting implies that $H_n(M')\neq 0$. Thus $n\le m$. Now we formulate a simple corollary from the proposition that will be applied in [@bontabu]. \[cbontabu\] Let ${{\mathcal{A}}}:{{\underline{Hw}}}\to {\underline{A}}$ be a full additive conservative functor whose target is semi-simple. Then for a $w$-bounded object $M$ of ${\underline{C}}$ we have $M\in {\underline{C}}_{w=0}$ if and only if $H^{{\mathcal{A}}}_i(M)=0$ for all $i\neq 0$. Once again, the “if” part of the statement is simple (and can easily be deduced from the previous proposition). So we prove the converse implication. Now, Lemma \[lbkw\] allows us to assume that ${\underline{C}}=K({{\underline{Hw}}})$ and $w={w^{st}}$ (since it enables us to treat $t(M)$ instead of $M$). Hence it suffices to prove that $M\in {\underline{C}}_{w\ge 0}$ since then we will also have $M\in {\underline{C}}_{w\le 0}$ by duality (see Proposition \[pbw\](\[idual\]; note that our assumptions on ${{\mathcal{A}}}$ and ${\underline{A}}$ are self-dual and the construction of $H^{{\mathcal{A}}}$ is also so in this case).[^29] Hence it suffices to verify that the functor ${{\mathcal{A}}}$ satisfies the assumptions of Proposition \[pdetect\]. The latter is very easy: all elements of ${{\mathcal{A}}}({{\underline{Hw}}})$ are projective in ${\underline{A}}$ since all objects of ${\underline{A}}$ are (recall that ${\underline{A}}$ is semi-simple), and a full conservative functor obviously does not send non-split morphisms into split ones. \[rdetect\] 1. As we have (essentially) just noted, condition (ii) of Proposition \[pdetect\] is obviously fulfilled both for ${{\mathcal{A}}}$ and for the opposite functor ${{\mathcal{A}}}^{op}:{{\underline{Hw}}}^{op}\to {\underline{A}}^{op}$ whenever ${{\mathcal{A}}}$ is a full conservative functor. In particular, it suffices to assume that ${{\mathcal{A}}}$ is a full embedding. Hence it may be useful to assume (in addition to assumption (i) of the proposition) that the image of ${{\mathcal{A}}}$ consists of injective objects only. 2. Now we describe a general method for constructing a full embedding ${{\mathcal{A}}}$ whose image consists of ${\underline{A}}$-projective objects. Assume that ${{\underline{Hw}}}$ is an essentially small (additive) $R$-linear category, where $R$ is a commutative unital ring (certainly, one may take $R={{\mathbb{Z}}}$ here). Denote some small skeleton of ${{\underline{Hw}}}$ by ${\underline{B}}$ (to avoid set-theoretical difficulties). Consider the abelian category ${\operatorname{PShv}^R}({\underline{B}})$ of $R$-linear contravariant functors from ${\underline{B}}$ into the category of $R$-modules (cf. §\[snotata\]). Then ${\underline{B}}$ (and so, also ${{\underline{Hw}}}$) embeds into the full subcategory of projective objects of ${\operatorname{PShv}^R}({\underline{B}})$ (by the Yoneda lemma; see Lemma 5.1.2 of [@neebook]). Hence this functor “detects weights” (in the sense of Proposition \[pdetect\]). 3\. The objects in the essential image of this functor may be called [*purely $R$-representable homology*]{}. Since they are usually not injective in ${\operatorname{PShv}^R}({\underline{B}})$, a dual construction may be useful for checking whether $M\in {\underline{C}}_{w\le -m}$. 4\. It is easily seen that in the proof of our corollary (and so, also of the bounded case of Proposition \[pdetect\]) one can replace the usage of Lemma \[lbkw\] by that of [@bws Theorem 3.3.1(IV)]. 5\. Condition (ii) of Proposition \[pdetect\] is certainly necessary. Indeed, if $h\in {\operatorname{Mor}}({{\underline{Hw}}})$ does not split whereas ${{\mathcal{A}}}(h)$ does, then one can easily check that $\operatorname{\operatorname{Cone}}(h)\in {\underline{C}}_{w\ge 0}\setminus {\underline{C}}_{w\ge 1}$ and $H_i(\operatorname{\operatorname{Cone}}(h))=0$ for $i\neq -1$. Now we prove a certain unbounded version of Proposition \[pdetect\]. Note that its proof may be (slightly) simplified if we assume that ${{\underline{Hw}}}$ is Karoubian (cf. [@bkw §3.1] that demonstrates that the general case can be “reduced” to this one). \[pdetectsse\] Assume that ${{\mathcal{A}}}:{{\underline{Hw}}}\to {\underline{A}}$ (as in the previous proposition) is a full functor, for any $N\in {\underline{C}}_{w=0}$ the ideal $\operatorname{\operatorname{Ker}}({{\underline{Hw}}}(N,N )\to {\underline{A}}({{\mathcal{A}}}(N),{{\mathcal{A}}}(N))$ is a nilpotent ideal of the endomorphism ring ${{\underline{Hw}}}(N,N)$, and ${\underline{A}}$ is abelian semi-simple. 1\. Then $H_i(M)=0$ for all $i> m$ (resp. for all $i<m$) if and only if $M\bigoplus M[1]$ is an extension of an element of ${\underline{C}}_{w\ge -m}$ by a right degenerate object (resp. $M\bigoplus M[-1]$ is an extension of an element of ${\underline{C}}_{w\le -m}$ by a left degenerate object). 2\. Assume in addition that $w$ is non-degenerate. Then these two conditions are equivalent to $M\in {\underline{C}}_{w\ge -m}$ (resp. $M\in {\underline{C}}_{w\le -m}$). In particular, if $H_i(M)=0$ for all $i \neq m$ then $M\in {\underline{C}}_{w= -m}$. 1\. It suffices to verify that $t(M)$ belongs to $K({{\underline{Hw}}})_{{w^{st}}\le -m }$ (resp. to $K({{\underline{Hw}}})_{{w^{st}}\ge -m }$). Indeed, then applying Lemma \[lbkw\] once again we will certainly obtain the result. Note also that it suffices to verify the “main” version of this assertion since the “resp.” one is its dual. Now, by Proposition \[pbw\](\[iort\]) it suffices to check that $t(M)$ is $K(\operatorname{\operatorname{Kar}}({{\underline{Hw}}}))$-isomorphic to a complex concentrated in degrees $\le m$. To construct the latter we extend ${{\mathcal{A}}}$ to an additive functor ${{\mathcal{A}}}': \operatorname{\operatorname{Kar}}({{\underline{Hw}}})\to {\underline{A}}$ and find a complex $T=(t^i)\in {\operatorname{Obj}}C(\operatorname{\operatorname{Kar}}({{\underline{Hw}}}))$ that is $K(\operatorname{\operatorname{Kar}}({{\underline{Hw}}}))$-isomorphic to $t(M)$ such that the boundary $d_t^{m}:t^{m}\to t^{m+1}$ is killed by ${{\mathcal{A}}}'$. This is easily seen to be possible according to Theorem 1.3 of [@wildab] (cf. also Theorem 2.2 of ibid.). Indeed, our assumptions on ${{\mathcal{A}}}$ imply immediately that ${{\underline{Hw}}}$ is [*semi-primary*]{} in the sense of . Then $\operatorname{\operatorname{Kar}}({{\underline{Hw}}})$ is semi-primary also according to Proposition 2.3.4(c) of ibid.; hence ${{\mathcal{A}}}': \operatorname{\operatorname{Kar}}({{\underline{Hw}}})\to {\underline{A}}$ satisfies the “kernel nilpotence assumption” similar to that for ${{\mathcal{A}}}$. Thus Theorem 1.3 of [@wildab] says that the cone of any $\operatorname{\operatorname{Kar}}({{\underline{Hw}}})$-morphism is $K(\operatorname{\operatorname{Kar}}({{\underline{Hw}}}))$-isomorphic to a cone of a morphism killed by ${{\mathcal{A}}}'$, and so we can “replace” the $m$th boundary of $t(M)$ by a morphism satisfying this condition. For this complex $T$ the result of the applying ${{\mathcal{A}}}'$ termwisely to its stupid truncation complex ${w^{st}}_{\le -m-1}T$ is certainly zero. Hence the identity of ${w^{st}}_{\le -m-1}T$ is homotopy equivalent to an endomorphism killed by the termwise application of ${{\mathcal{A}}}'$. Applying our nilpotence assumption to a sufficiently high power of this ($K(\operatorname{\operatorname{Kar}}({{\underline{Hw}}}))$-invertible) endomorphism we obtain that for any $j\in {{\mathbb{Z}}}$ the complex ${w^{st}}_{\le -m-1}T$ is homotopy equivalent to a complex concentrated in degrees $\ge j$. Hence ${w^{st}}_{\le -m-1}T$ is contractible and we obtain $t(M)$ is ${K(\operatorname{\operatorname{Kar}}({{\underline{Hw}}}))}$-isomorphic to its stupid truncation ${w^{st}}_{\ge -m}T$. 2. Assertion 1 implies that $M\bigoplus M[1]$ belongs to ${\underline{C}}_{w\ge -m}$ (resp. $M\bigoplus M[-1]$ belongs to ${\underline{C}}_{w\le -m}$) itself. Hence $M$ belongs to ${\underline{C}}_{w\ge -m}$ (resp. to ${\underline{C}}_{w\le -m}$) also. The “in particular” part of the assertion follows immediately. \[rpsh\] 1. Both Propositions \[pdetect\] and \[pdetectsse\] are easily seen to imply certain generalizations of [@wildcons Theorem 1.5]. 2\. Proposition \[pdetectsse\] can probably generalized. In particular, it appears to be sufficient to assume for any $N\in {\underline{C}}_{w=0}$ that all the endomorphisms in $\operatorname{\operatorname{Ker}}({{\underline{Hw}}}(N,N )\to {\underline{A}}({{\mathcal{A}}}(N),{{\mathcal{A}}}(N)))$ are nilpotent. However, it is demonstrated in [@bontabu] that the case where ${{\mathcal{A}}}$ is a full embedding (into a semi-simple category) is quite interesting (already). In this case the proof can be simplified since then $t(M)$ is obviously a retract of ${w^{st}}_{\ge -m}t(M)$ (resp. of ${w^{st}}_{\le -m}t(M)$). On virtual $t$-truncations and cohomology of bounded weight range {#svtt} ----------------------------------------------------------------- We recall the notion of virtual $t$-truncations for a cohomological functor $H:{\underline{C}}\to {\underline{A}}$ (as defined in §2.5 of [@bws] and studied in more detail in §2 of [@bger]). These truncations allow us to “slice” $H$ into $w$-pure pieces. These truncations behave as if they were given by truncations of $H$ in some triangulated “category of functors” ${\underline{D}}$ with respect to some $t$-structure (whence the name). Moreover, this is often actually the case (and we will discuss this matter below); yet the definition does not require the existence of ${\underline{D}}$ (and so, does not depend on its choice). Our choice of the numbering for them is motivated by the cohomological convention for $t$-structures; this convention combined with the homological numbering for weight structures causes the (somewhat weird) “$-$” signs in the definitions and formulations of this section. \[dvtt\] Let $H$ be a cohomological functor from ${\underline{C}}$ into an abelian category ${\underline{A}}$; assume that ${\underline{C}}$ is endowed with a weight structure $w$ and $ n\in {{\mathbb{Z}}}$. We define the [*virtual $t$-truncation*]{} functors $\tau^{\ge -n }(H)$ (resp. $\tau^{\le -n }(H)$) by the correspondence $$M\mapsto\operatorname{\operatorname{Im}}(H(w_{\le n+1}M)\to H(w_{\le n}M)) ;$$ (resp. $M\mapsto\operatorname{\operatorname{Im}}(H(w_{\ge n}M)\to H(w_{\ge n-1}M)) $); here we take arbitrary choices of the corresponding weight truncations of $M$ and connect them using Proposition \[pbw\](\[icompl\]). We recall the main properties of these constructions that were established in [@bws §2.5 and Theorem 4.4.2(7,8)]. \[pwfil\] In the notation of the previous definition the following statements are valid. 1. The objects $\tau^{\ge -n}(H)(M)$ and $\tau^{\le -n}(H)(M)$ are ${\underline{C}}$-functorial in $M$ (and so, the virtual $t$-truncations of $H$ are well-defined functors). 2\. The functors $\tau^{\ge -n }(H)$ and $\tau^{\le -n }(H)$ are cohomological. 3\. There exist natural transformations that yield a long exact sequence $$\label{evtt} \begin{gathered} \dots \to \tau^{\ge -n +1}(H)\circ [-1] \to \tau^{\le -n }(H)\to H \\ \to \tau^{\ge -n +1}(H)\to \tau^{\le -n }(H)\circ [1]\to \dots\end{gathered}$$ (i.e., the result of applying this sequence to any object of ${\underline{C}}$ is a long exact sequence); the shift of this exact sequence by $3$ positions is given by composing the functors with $-[1]$. 4. Assume that there exists a $t$-structure $t$ that is right adjacent to $w$. Then for any $M\in {\operatorname{Obj}}{\underline{C}}$ and $H_M={\underline{C}}(-,M):{\underline{C}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ the functors $ \tau^{\le -n }(H_M)$ and $\tau^{\ge -n}(H_M)$ are represented by $t^{\le -n}M$ and $t^{\ge -n}M$, respectively. Moreover, a stronger and more general statement than part 4 of this proposition is given by [@bger Proposition 2.5.4(1)]; it will be applied in the proof of Proposition \[psaturdu\](2) below. Now we define weight range and introduce notation for pure cohomological functors. Some of these statements will be applied below, whereas other ones are proved here for the purpose of applying them in [@bkw]. \[drange\] 1. Let $m,n\in {{\mathbb{Z}}}$; let $H$ be as above. Then we will say that $H$ is [*of weight range*]{} $\ge m$ (resp. $\le n$, resp. $[m,n]$) if it annihilates ${\underline{C}}_{w\le m-1}$ (resp. ${\underline{C}}_{w\ge n+1}$, resp. both of these classes). 2\. Let ${{\mathcal{A}}}:{{\underline{Hw}}}^{op}\to {\underline{A}}$ be an additive functor. Then for ${{\mathcal{A}}}^{op}$ being the opposite functor ${{\underline{Hw}}}\to {\underline{A}}^{op}$ we will write $H_{{\mathcal{A}}}$ for the cohomological functor from ${\underline{C}}$ into ${\underline{A}}$ obtained from $H^{{{\mathcal{A}}}^{op}}$ (see Proposition \[ppure\]) by means of reversion of arrows. We will functors obtained using this construction [*pure cohomological*]{} ones. \[pwrange\] In the notation of the previous definition the following statements are valid. 1. \[iwrvt\] The functor $\tau^{\ge -n}(H)$ is of weight range $\le n$, and $\tau^{\le -m}(H)$ is of weight range $\ge m$. 2. \[iwrcrit\] Assume that $w$ is bounded. Then $H$ is of weight range $\le n$ (resp. of weight range $\ge m$) if and only if it kills ${\underline{C}}_{w=i}$ for all $i>n$ (resp. for $i>m$). 3. \[iwridemp\] We have $\tau^{\ge -n}(H)\cong H$ (resp. $\tau^{\le -m}(H)\cong H$) if and only if $H$ is of weight range $\le n$ (resp. of weight range $\ge m$). 4. \[iwrcomm\] We have $\tau^{\ge -n}(\tau^{\le -m})(H)\cong \tau^{\le -m}(\tau^{\ge -n})(H)$. 5. \[iwfil4\] If $H$ is of weight range $\ge m$ then $\tau^{\ge -n}(H)$ is of weight range $[m,n]$. 6. \[iwrd\] Dually, if $H$ is of weight range $\le n$ then $\tau^{\le -m}(H)$ is of weight range $[m,n]$. 7. \[iwrvan\] If $m>n$ then the only functors of weight range $[m,n]$ are zero ones. 8. \[iwrpure\] The functors of weight range $[m,m]$ are exactly those of the form $H_{{\mathcal{A}}}\circ [-m]$ (see Definition \[drange\](2)), where ${{\mathcal{A}}}:{{\underline{Hw}}}\to {\underline{A}}^{op}$ is an additive functor. 9. \[iwfil3\] The (representable) functor $H_M:{\underline{C}}\to {\underline{\operatorname{Ab}}}$ if of weight range $\ge m$ if and only if $M\in {\underline{C}}_{w\ge m}$. 10. \[iwfil5\] If $H$ is of weight range $[m,n]$ then the morphism $H(w_{\ge m}M)\to H(M)$ is surjective and the morphism $H(M)\to H(w_{\le n}M)$ is injective (here we take arbitrary choice of the corresponding weight decompositions of $M$ and apply $H$ to their connecting morphisms). Let $M\in {\underline{C}}_{w\ge n+1}$. Then we can take $w_{\le n}(M)=0$. Thus $\tau^{\ge -n}(H)(M)=0$, and we obtain the first part of assertion \[iwrvt\]. It second part is easily seen to be dual to the first part. The “only if” part of assertion \[iwrcrit\] is immediate from the definition of weight range. The converse implication easily follows from Proposition \[pbw\](\[igenlm\]). Assertion \[iwridemp\] is precisely Theorem 2.3.1(III2,3) of [@bger]; assertion \[iwrcomm\] is given by part II.3 of that theorem. Now let $H$ be of weight range $\ge m$. Then $\tau^{\ge -n}(H)\cong \tau^{\ge -n}(\tau^{\le -m})(H)\cong \tau^{\le -m}(\tau^{\ge -n})(H)$ (according to the two previous assertions). It remains to apply assertion \[iwrvt\] to obtain assertion \[iwfil4\]. Assertion \[iwrd\] can be proved similarly; it is also easily seen to be dual to assertion \[iwfil4\]. Next, for any $l\in {{\mathbb{Z}}}$ and any cohomological $H$ any choice of an $l$-weight decomposition triangle (cf. (\[ewd\])) for $M$ gives the long exact sequence $$\label{eles} \begin{gathered} \dots \to H((w_{\le l}M)[1])\to H(w_{\ge l+1}M)\to H(M)\\ \to H(w_{\le l}M)\to H((w_{\ge l+1}M)[-1])\to\dots \end{gathered}$$ The exactness of this sequence in $H(M)$ for $l=n$ immediately gives assertion \[iwrvan\]. \[iwrpure\]. It certainly suffices to verify the statement for $m=0$. Now, the functor $H^{{{\mathcal{A}}},op}$ is easily seen to be of weight range $[0,0]$ for any additive functor ${{\mathcal{A}}}:{\underline{C}}\to {\underline{A}}^{op}$; see Remark \[rwc\]. Conversely, let $H$ be of weight range $[0,0]$. We take the functor ${{\mathcal{A}}}$ being the restriction of $H^{op}:{\underline{C}}\to {\underline{A}}^{op}$ to ${{\underline{Hw}}}$. Then we should check that $H$ sends $M\in {\operatorname{Obj}}{\underline{C}}$ into the homology in $H(M^0)$ of the complex $H(M^{-*})$ (where $t(M)=(M^*)$). This is an immediate consequence of the properties of the [*weight spectral sequence*]{} converging to $H^*(M)$; see Theorem 2.4.2 of [@bws]. Indeed, this spectral sequence converges according to part II(ii) of this theorem, and it remains to apply the vanishing for $H(M^i[j])$ for $j\neq 0$. Assertion \[iwfil3\] is immediate from Proposition \[pbw\](\[iort\]). Assertion \[iwfil5\] is an immediate consequence of assertion \[iwfil3\]; just apply (\[eles\]) for $l=m$ and for $l=m-1$, respectively. \[rwrange\] 1. So, we call cohomological functors of weight range $[0,0]$ and their opposite homological functors $w$-pure ones; this terminology is compatible with Remark \[rpure\](2) according to part \[iwrpure\] of our proposition. 2\. Actually, the arguments used in the proof of this statement are easily seen to be functorial enough to yield an equivalence of the (possibly) big category of pure (cohomological) functors ${\underline{C}}\to {\underline{A}}$ with the one of additive contravariant functors ${{\underline{Hw}}}\to {\underline{A}}$. 3\. Sending $H$ into the pure functor $\tau^{\ge -m}(\tau^{\le -m})(H\circ [m])\cong \tau^{\le -m}(\tau^{\ge -m})(H\circ [m])$ (for $m\in {{\mathbb{Z}}}$) yields a sort of “pure homology” for $H$. It will correspond to the homology of an object representing (or, more generally, [*$\Phi$-representing*]{}) $H$ in the settings that we will consider below. 4. Certainly, a homological functor $H$ from ${\underline{C}}$ into ${\underline{A}}$ may be considered as a cohomological functor from ${\underline{C}}$ into ${\underline{A}}^{op}$. Thus one can easily dualize the aforementioned results; we give some more detail for this here to refer to them later. Obviously, the virtual $t$-truncation functor $\tau^{\ge -n }(H)$ (resp. $\tau^{\le -n }(H)$) will be defined by the correspondence $M\mapsto\operatorname{\operatorname{Im}}(H(w_{\le n}M)\to H(w_{\le n+1}M)) $ (resp. $M\mapsto\operatorname{\operatorname{Im}}(H(w_{\ge n}M) \to H(w_{\ge n-1}M)$), whereas the arrows in (\[evtt\]) should be reversed. 5\. The author is using the term “pure” due to the relation of pure functors to Deligne’s purity of cohomology. To explain it we recall that various categories of Voevodsky motives are endowed with so-called Chow weight structures; the first of these weight structures was constructed in [@bws] where it was proved that the category ${\operatorname{DM_{gm}}}(k)$ of geometric motives over a characteristic zero field $k$ is endowed with a weight structure $w_{{\operatorname{Chow}}}(k)$ whose heart is the category of Chow motives over $k$.[^30] Now, for any $r\in {{\mathbb{Z}}}$ the $r$th level of the Deligne’s weight filtration of either of singular of étale cohomology of motives certainly kills ${\operatorname{Chow}}[i]$ for all values of $i$ except one (and the remaining value of $i$ is either $r$ or $-r$ depending on the choice of the convention for Deligne’s weights).[^31]  Thus (the corresponding shifts of) Deligne’s pure factors of (singular and étale) cohomology are pure with respect to $w_{{\operatorname{Chow}}(k)}$. We also formulate a simple statement that will be applied in a succeeding paper. \[pcrivtt\] For $M\in {\operatorname{Obj}}{\underline{C}}$ the following conditions are equivalent. \(i) $M\in {\underline{C}}_{w\ge 0}$. \(ii) $H(M)=0$ for any cohomological $H$ from ${\underline{C}}$ into (an abelian category) ${\underline{A}}$ that is of weight range $\le -1$. \(iii) $(\tau^{\ge 1}H_N)(M) ={\{0\}}$ for any $N\in {\operatorname{Obj}}{\underline{C}}$. \(iv) $(\tau^{\ge 1}H_M)(M) ={\{0\}}$. Condition (i) implies condition (ii) by definition; certainly, (iii) $\implies$ (iv). Next, condition (ii) implies condition (iii) according to Proposition \[pwrange\](\[iwrvt\]). Now we prove that condition (iv) implies that $M\in {\underline{C}}_{w\ge 0}$. We consider the commutative diagram $$\label{epr} \begin{CD} w_{\le -1} M@>{c_{-1}}>> M@>{}>> w_{\ge 0}M\\ @VV{h}V@VV{{\operatorname{id}}_M}V@ VV{}V \\ w_{\le 0} M@>{c_0}>> M@>{}>> w_{\ge 1}M \end{CD}$$ given by Proposition \[pbw\](\[icompl\])) (we take $g={\operatorname{id}}_M$ in the proposition). Since $(\tau^{\ge 1}H_M)(M) ={\{0\}}$, $c_0\circ h=0$; hence $c_{-1}=0$ also. Hence the upper distinguished triangle in (\[epr\]) yields that $M$ is a retract of $w_{\ge 0}M\in {\underline{C}}_{w\ge 0}$. On the relation of smashing weight structures to cc and cp functors {#sprcoprod} ------------------------------------------------------------------- We prove a collection of properties of smashing weight structures (see Definition \[dhop\]). \[ppcoprws\] Let $w$ be a smashing weight structure (on ${\underline{C}}$), $i,j\in {{\mathbb{Z}}}$; let ${\underline{A}}$ be an AB4 abelian category, and ${\underline{A}}'$ be an AB4\* abelian category. Then the following statements are valid. 1. \[icopr1\] The classes ${\underline{C}}_{w\le j}$, ${\underline{C}}_{w\ge i}$, and ${\underline{C}}_{[i,j]}$ are closed with respect to ${\underline{C}}$-coproducts. 2. \[icoprhw\] In particular, the category ${{\underline{Hw}}}$ is closed with respect to ${\underline{C}}$-coproducts, and the embedding ${{\underline{Hw}}}\to {\underline{C}}$ respects coproducts. 3. \[icopr2\] Coproducts of weight decompositions are weight decompositions. 4. \[icopr3\] Coproducts of weight Postnikov towers are weight Postnikov towers. 5. \[icopr4\] The categories ${{\underline{C}}_w}$ and ${K_{\mathfrak{w}}}({{\underline{Hw}}})$ are closed with respect to coproducts, and the functor $t$ respects coproducts. 6. \[icopr5\] Pure functors ${\underline{C}}\to {\underline{A}}$ respecting coproducts are exactly the functors of the form $H^{{\mathcal{A}}}$ (see Proposition \[ppure\]), where ${{\mathcal{A}}}:{{\underline{Hw}}}\to {\underline{A}}$ is an additive functor respecting coproducts. Moreover, this correspondence is an equivalence of (possibly, big) categories. 7. \[icopr6\] If $H:{\underline{C}}\to {\underline{A}}$ is a cc functor (see Definition \[dcomp\](\[idcc\])) then $\tau^{\ge i }(H)$ and $\tau^{\le i }(H)$ are cc functors also. 8. \[icopr6p\] $H':{\underline{C}}^{op}\to {\underline{A}}'$ is a cp functor then $\tau^{\ge i }(H')$ and $\tau^{\le i }(H')$ are cp functors also. 9. \[icopr5p\] Pure cohomological (see Definition \[drange\](2)) cp functors from ${\underline{C}}$ into ${\underline{A}}'$ are exactly those of the form $H_{{\mathcal{A}}}$ for ${{\mathcal{A}}}:{{\underline{Hw}}}^{op}\to {\underline{A}}'$ being an additive functor that sends ${{\underline{Hw}}}$-coproducts into products. 10. \[icopr7\] Let ${\underline{D}}$ be the localizing subcategory of ${\underline{C}}$ generated by a class of objects $\{D_l\}$, and assume that for any of the $D_l$ a choice of (the terms of) its weight complex $t(D_l)=(D_l^k)$ is fixed. Then any element of ${\underline{C}}_{w=0}\cap {\operatorname{Obj}}{\underline{D}}$ is a retract of a coproduct of a family of $D_l^k$. 11. \[icopr7p\] For $\{D_l\}$ and ${\underline{D}}$ as in the previous assertions assume that for any of the $D_l$ a choice of $w_{\le k}D_l$ and of $w_{\ge k}D_l$ for $k\in {{\mathbb{Z}}}$ is fixed (we do not assume any relation between these choices). Then for any $D\in{\operatorname{Obj}}{\underline{D}}$ and any $m\in {{\mathbb{Z}}}$ there exists a choice of $(w_{\le m}D)[-m]$ (resp. of $(w_{\ge m}D)[-m]$) belonging to the smallest coproductive extension-closed subclass $D_1$ (resp. $D_2$) of ${\operatorname{Obj}}{\underline{C}}$ containing $(w_{\le k}D_l)[-k]$ (resp. $(w_{\ge k}D_l)[-k]$) for all $l$ and all $k\in{{\mathbb{Z}}}$. Moreover, ${\underline{C}}_{w\le 0}\cap {\operatorname{Obj}}{\underline{D}}$ and ${\underline{C}}_{w\ge 0}\cap {\operatorname{Obj}}{\underline{D}}$ lie in $D_1$ and $D_2$, respectively. Furthermore, if ${\alpha}$ is a regular cardinal and $D$ belongs to the smallest triangulated category of ${\underline{C}}$ that contains $D_l$ and closed under ${\underline{C}}$-coproducts of less than ${\alpha}$ objects then all $(w_{\le m}D)[-m]$ can be chosen to belong to the smallest extension-closed subclass of ${\operatorname{Obj}}{\underline{C}}$ containing $(w_{\le k}D_l)[-k]$ and closed under coproducts of less than ${\alpha}$ objects. <!-- --> 1. This is essentially a particular case of Proposition \[phop\](2,3); see Remark \[rwhop\](1). 2. Immediate from the previous assertion. 3. Recalling Remark \[rwhop\](1) once again, we reduce the statement to Proposition \[phop\](4). 4. Immediate from the previous assertion. 5. Immediate from assertions \[icopr1\] and \[icopr3\]. 6. It certainly follows from Proposition \[pwrange\](\[iwrpure\]) that pure functors are exactly those of the type $H^{{\mathcal{A}}}$. Since ${{\mathcal{A}}}$ is the restriction of $H^{{\mathcal{A}}}$ to ${{\underline{Hw}}}$, this (restriction) correspondence is functorial. Next, if $H$ respects coproducts then its restriction to ${{\underline{Hw}}}$ also does according to assertion \[icoprhw\]. Conversely, if ${{\mathcal{A}}}$ respects coproducts then $H^{{\mathcal{A}}}$ also does according to assertion \[icopr4\]. \[icopr6\], \[icopr6p\]. Immediate from assertion \[icopr2\]. \[icopr5p\]. Similarly to assertion \[icopr5\], ${{\mathcal{A}}}$ is the restriction of $H_{{\mathcal{A}}}$ to ${{\underline{Hw}}}$; hence it sends coproducts into products according to assertion \[icoprhw\]. Conversely, if ${{\mathcal{A}}}$ sends coproducts into products then $H^{{\mathcal{A}}}$ also does according to assertion \[icopr4\]. \[icopr7\]. Combining assertion \[icopr4\] with Proposition \[pwt\](\[iwcex\]) we obtain that for any $M\in {\operatorname{Obj}}{\underline{D}}$ there exists a choice of $t(M)$ all of whose terms are coproducts of $D_l^k$. If $M$ also belongs to ${\underline{C}}_{w=0}$ then we obtain that $M$ is a retract of an element of this form according to Remark \[rwc\]. \[icopr7p\]. Combining assertion \[icopr2\] with Proposition \[pbw\](\[iwdext\]) we obtain that the class $C$ of those $D\in {\operatorname{Obj}}{\underline{C}}$ such that for any $m\in {{\mathbb{Z}}}$ there exist a choice of $(w_{\le m}D)[-m]$ and of $(w_{\ge m}D)[-m]$ belonging to $D_1$ (resp. to $D_2$) is a coproductive class of objects of a full triangulated subcategory of ${\underline{C}}$ (i.e., there exists a triangulated ${\underline{C}'}\subset {\underline{C}}$ such that ${\operatorname{Obj}}{\underline{C}'}=C$ and $C$ is coproductive). Thus $C$ contains ${\operatorname{Obj}}{\underline{D}}$. Next, if $d$ belongs to $ {\operatorname{Obj}}{\underline{D}}\cap {\underline{C}}_{w\le 0}$ (resp. to $ {\operatorname{Obj}}{\underline{D}}\cap {\underline{C}}_{w\ge 0}$) then the existence of $w_{\le 0}M$ belonging to $D_1$ (resp. of $w_{\ge 0}M$ belonging to $D_2$) implies that $d$ belongs to the Karoubi-closure of $D_1$ (resp. of $D_2$) according to Proposition \[pbw\](\[iwdmod\]). Thus to prove the “moreover” part of the assertion it remains to note that $D_1$ and $D_2$ are Karoubian according to Remark \[rcoulim\](4) below. The proof of the “furthermore” part of the assertion is similar. \[ral\] In all the part of our proposition one can replace arbitrary small coproducts by coproducts of less than ${\alpha}$ objects in all occurences (where ${\alpha}$ is any regular infinite cardinal). In particular, in assertions \[icopr4\] and \[icopr6p\] one can assume that ${\underline{C}}$ and ${\underline{C}}_{w\le 0}$ are closed with respect to ${\underline{C}}$-coproducts of less than ${\alpha}$ of their objects; then $t$ respects these coproducts also, and virtual $t$-truncations of cohomological functors that convert coproducts of less than ${\alpha}$ objects into the corresponding products fulfil this condition as well.[^32] Part \[icopr6p\] of our proposition immediately implies the following corollary that will be important for us below. \[cvttbrown\] Assume that $w$ is smashing. I.1. If ${\underline{C}}$ satisfies the Brown representability condition (see Definition \[dcomp\](\[idbrown\])) then virtual $t$-truncations of representable functors are representable. 2\. If ${\underline{C}}$ is generated by a set of objects as its own localizing subcategory[^33] then ${{\underline{Hw}}}$ has a generator, i.e., there exists $P\in {\underline{C}}_{w=0}$ such that any object of ${{\underline{Hw}}}$ is a retract of a coproduct of (copies of) $P$. II\. Let $F:{\underline{C}}\to {\underline{C}}'$ be an exact functor respecting coproducts. Adopt the notation of Proposition \[ppcoprws\](\[icopr7p\]) and let $w'$ be a weight structure for ${\underline{C}}'$; 1\. Then $F$ is left (resp. right) weight-exact if and only if $F(w_{\le k}D_l)[-k]\in {\underline{C}}'_{w'\le 0}$ (resp. $F(w_{\ge k}D_l)[-k]\in {\underline{C}}'_{w'\ge 0}$) for all $l$ and all $k\in {{\mathbb{Z}}}$. 2\. Assume that all $D_l$ are $w$-bounded; let ${\mathcal{P}}\subset {\underline{C}}_{w=0}$ be a generating class for ${{\underline{Hw}}}$ (i.e., any object of ${{\underline{Hw}}}$ is a retract of a coproduct of elements of ${\mathcal{P}}$). Then $F$ is left (resp. right) weight-exact if and only if $F({\mathcal{P}})\subset {\underline{C}}'_{w'\le 0}$ (resp. $F({\mathcal{P}})\subset {\underline{C}}'_{w'\ge 0}$). I.1. The Brown representability condition says that ${\underline{C}}$-representable functors are precisely all cp functors from ${\underline{C}}$ into ${\underline{\operatorname{Ab}}}$. Hence the statement follows from Proposition \[ppcoprws\](\[icopr6p\]) indeed. 2\. Just take ${\underline{D}}={\underline{C}}$ in part \[icopr7\] of the proposition; then we can take $P$ being the coproduct of the corresponding $D_l^k$ (recall that ${{\underline{Hw}}}$ has coproducts!). II\. If $F$ is left (resp. right) weight-exact then the images of all elements of ${\underline{C}}_{w\le 0}$ (resp. of ${\underline{C}}_{w\ge 0}$) belong to ${\underline{C}}'_{w'\le 0}$ (resp. to ${\underline{C}}'_{w'\ge 0}$), and we obtain “one half” of the implications in question. The converse implication for assertion II.1 follows from Proposition \[ppcoprws\](\[icopr7p\]) immediately. Now we check the converse implication for assertion II.2. Since $F$ respects coproducts and $F({\mathcal{P}})\subset {\underline{C}}'_{w'\le 0}$ (resp. $F({\mathcal{P}})\subset {\underline{C}}'_{w'\ge 0}$), we obtain $F({\underline{C}}_{w=0})\subset {\underline{C}}'_{w'\le 0}$ (resp. $F({\underline{C}}_{w=0})\subset {\underline{C}}'_{w'\ge 0}$). Thus Proposition \[pbw\](\[igenlm\]) yields the result easily. \[rgenw\] 1. Note that our definition of a generator for ${{\underline{Hw}}}$ is much more “restrictive” than the assumption that $\{P\}$ Hom-generates ${{\underline{Hw}}}$ (cf. Remark \[rcompgen\]) and even than the (usual) “abelian version” of the definition of generators (cf. Theorem \[tab5\](1)). Moreover, it is much easier to specify generators in ${{\underline{Hw}}}$ (if $w$ is smashing) than in ${{\underline{Ht}}}$. 2. A certain “finite dimensional” analogue of part I.1 of this corollary is given by Proposition \[psatur\](1) below. On adjacent weight and $t$-structures and Brown representability-type conditions {#sadjbrown} ================================================================================ In this section we study some general conditions ensuring that a torsion pair admits a (right or left) torsion pair. In §\[sadjt\] we prove that a weight structure admits a right adjacent $t$-structure if and only if the virtual $t$-truncations of representable functors are representable. It easily follows that in a triangulated categories satisfying the Brown representability condition a weight structure admits a right adjacent $t$-structure if and only if it is smashing. Certainly, the dual to this statement is also true. Moreover, a similar argument demonstrates that if the representable functors from an $R$-linear category ${\underline{C}}$ are precisely the [*$R$-finite type*]{} ones (i.e., if ${\underline{C}}$ is [*$R$-saturated*]{}) then all bounded weight structures on ${\underline{C}}$ admit right adjacent $t$-ones. Note here that if $R$ is Noetherian then for $X$ being a regular separated finite-dimensional scheme that is proper over ${{\operatorname{Spec}\,}}R$ its bounded derived category of coherent sheaves (as well as its dual) is $R$-saturated according to a recent result of Neeman. In §\[sadjw\] we study when a $t$-structure $t$ admits a (left or right) adjacent weight structure $w$; however, the results of this section are “not as nice” as their “mirror” ones in §\[sadjt\] (at least, in the case where ${\underline{C}}$ has coproducts; cf. Remark \[rnondeg\]). In §\[scomp\] we recall the notions of perfectly generated and well generated triangulated categories along with their (Brown representability) properties; we also relate perfectness to smashing torsion pairs. Next we define symmetric classes and study their relation to perfect classes, Brown-Comenetz duality, and adjacent torsion pairs (obtaining a new criterion for the existence of the latter). This gives one more “description” of a $t$-structure that is right adjacent to a given compactly generated weight structure. On the existence of adjacent $t$-structures {#sadjt} ------------------------------------------- \[phadj\] Let $w$ be left adjacent to a $t$-structure $t$ on ${\underline{C}}$. Then ${{\underline{Ht}}}$ is a full exact subcategory of the (possibly, big) abelian category $\operatorname{\operatorname{AddFun}}({{\underline{Hw}}}^{op},{\underline{\operatorname{Ab}}})$ (see §\[snotata\]). Moreover, ${\underline{C}}_{w=0}=P_t$ and the functor $H^P={\underline{C}}(P,-)$ is isomorphic to ${{\underline{Ht}}}(H_0^t(P), H_0^t(-))$ for any $P\in {\underline{C}}_{w=0}$. The first part of the assertion is given by part 4 of [@bws Theorem 4.4.2]. The equality ${\underline{C}}_{w=0}=P_t$ is immediate from Remark \[rtst1\](\[it1\]) (recall Definition \[dwso\](\[idadj\])). The last of the assertions is given by Proposition \[pgen\](\[ipgen2\]). As we have essentially already noted (see Proposition \[phop\](5)), if a weight structure possesses a left adjacent $t$-structure then it is coproductive (i.e., the associated torsion pair is coproductive). Now we prove that the converse implication is valid also if ${\underline{C}}$ satisfies the Brown representability property (in particular, if it is compactly generated or [*perfectly generated*]{} in the sense of Definition \[dwg\](\[idpc\]) below). \[tadjt\] Assume that ${\underline{C}}$ satisfies the Brown representability condition. 1. Then for a weight structure $w$ on ${\underline{C}}$ there exists a $t$-structure right adjacent to it if and only if $w$ is smashing. 2. If a right adjacent $t$ exists then ${{\underline{Ht}}}$ is equivalent to the full subcategory of $\operatorname{\operatorname{AddFun}}({{\underline{Hw}}}^{op},{\underline{\operatorname{Ab}}})$ consisting of those functors that sends ${{\underline{Hw}}}$-coproducts into products. 1\. The “only if” assertion is immediate from Proposition \[phop\](5) (and very easy for itself). Conversely, assume that $w$ is smashing. According to Proposition \[padjt\](\[ile4\]) below it suffices to verify that for any $M\in {\operatorname{Obj}}{\underline{C}}$ the functor $\tau^{\le 0}H_M$ is representable (in ${\underline{C}}$; recall that $H_M$ denotes the functor ${\underline{C}}(-,M)$). The latter statement is given by Corollary \[cvttbrown\](I.1). 2\. Applying Proposition \[phadj\] we obtain that is suffices to find out which functors ${{\underline{Hw}}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ are represented by objects of ${{\underline{Ht}}}$. Since the embedding ${{\underline{Hw}}}\to {\underline{C}}$ respects coproducts (see Proposition \[ppcoprws\](\[icopr1\])), all these functors send ${{\underline{Hw}}}$-coproducts into ${\underline{\operatorname{Ab}}}$-products. Conversely, let ${{\mathcal{A}}}:{{\underline{Hw}}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ be an additive functor converting coproducts into products. Then the corresponding $H_{{\mathcal{A}}}$ is a cp functor (see Definition \[dcomp\](\[idcc\])) according to Proposition \[ppcoprws\](\[icopr5p\]). Hence it is representable by some $M\in {\operatorname{Obj}}{\underline{C}}$. Since $H_{{\mathcal{A}}}$ is also of weight range $[0,0]$ (see Proposition \[pwrange\](\[iwrpure\])), we have $M\in {\underline{C}}^{t=0}$. \[rstable\] 1\. Proposition \[pwsym\](\[iwsymcgwt\]) below gives some more information on the adjacent weight structure $t$ whenever $w$ is compactly generated (see Remark \[rwhop\](1)). 2. Recall from Proposition \[prtst\](\[itp4\], \[it4sm\]) that “shift-stable” weight structures are in one-to-one correspondence with exact embeddings $i:L\to {\underline{C}}$ possessing right adjoints. Hence applying our theorem in this case we obtain the following: if $i$ possesses a right adjoint respecting coproducts and ${\underline{C}}$ satisfies the Brown representability condition then for the full triangulated subcategory $R$ of ${\underline{C}}$ with ${\operatorname{Obj}}R=L{{}^{\perp}}$ the embedding $R\to {\underline{C}}$ possesses a right adjoint also. Thus $R$ is [*admissible*]{} in ${\underline{C}}$ in the sense of [@bondkaprserr] and the embedding $R\to{\underline{C}}$ may be completed to a [*gluing datum*]{} (cf. [@bbd §1.4] or [@neebook §9.2]). So we re-prove Corollary 2.4 of [@nisao]. \[cdualt\] Let ${\underline{C}}$ be a category satisfying the dual Brown representability property (recall that this is the case if ${\underline{C}}$ is compactly generated; see Proposition \[pcomp\](II.1)). 1. Then for a weight structure $w$ on ${\underline{C}}$ there exists a $t$-structure left adjacent to it if and only if $w$ is cosmashing. 2. If these equivalent conditions are fulfilled then ${{\underline{Ht}}}$ is anti-equivalent to the subcategory of $\operatorname{\operatorname{AddFun}}({{\underline{Hw}}},{\underline{\operatorname{Ab}}})$ consisting of those functors that respect products. This is just the categorical dual to Theorem \[tadjt\]. Now we re-formulate the existence of a $t$-structure right adjacent to $w$ in terms of virtual $t$-truncations; this finishes the proof of Theorem \[tadjt\]. \[padjt\] Let $w$ be a weight structure for ${\underline{C}}$, $M\in {\operatorname{Obj}}{\underline{C}}$, and assume that for the functor $H_M={\underline{C}}(-,M)$ its virtual $t$-truncation $\tau^{\le 0}H_M$ is represented by some object $M^{\le 0}$ of ${\underline{C}}$. Then the following statements are valid. 1. \[ile1\] $M^{\le 0}$ belongs to ${\underline{C}}_{w\ge 0}$. 2. \[ile2\] The natural transformation $\tau^{\le -n }(H_M)\to H_M$ mentioned in (\[evtt\]) is induced by some $f\in {\underline{C}}(M^{\le 0},M)$. 3. \[ile3\] The object $\operatorname{\operatorname{Cone}}(f)$ belongs to ${\underline{C}}_{w\ge 0}^{\perp}$. 4. \[ile4\] There exists a $t$-structure $t$ (on ${\underline{C}}$) right adjacent to $w$ if and only if the functor $\tau^{\le 0}H_{M'}$ is ${\underline{C}}$-representable for any object $M'$ of ${\underline{C}}$. 5. \[ile5\] For $M'\in {\operatorname{Obj}}{\underline{C}}$ the representability of the functor $\tau^{\le 0}H_{M'}$ is equivalent to that of $\tau^{\ge 1}H_{M'}$. 1\. $\tau^{\le 0}H_M$ is of weight range $\ge 0$ (see Proposition \[pwrange\](\[iwrvt\])). Hence the assertion follows from part \[iwfil3\] of the same proposition. 2\. Immediate from the Yoneda lemma. 3\. For any $N\in {\operatorname{Obj}}{\underline{C}}$ applying the functor $H^N={\underline{C}}(N,-)$ to the distinguished triangle $M^{\le 0}\to M \to M^{\ge 1} \to M^{\le 0}[1]$ one obtains a long exact sequence that yields the following short one: $$\label{eshort} \begin{gathered} 0\to \operatorname{\operatorname{Coker}}(H^N(M^{\le 0})\stackrel{h^1_N}{\to} H^N(M))\to H^N(\operatorname{\operatorname{Cone}}(f)) \\ \to \operatorname{\operatorname{Ker}}(H^N(M^{\le 0}[1])\stackrel{h^2_N}{\to} H^N(M[1]))\to 0. \end{gathered}$$ So, for any $N\in {\underline{C}}_{w\ge 0}$ we should check that $h^1_N$ is surjective and $h^2_N$ is injective. Applying (\[evtt\]) to the functor $H_M$ (in the case $n=0$) we obtain a long exact sequence of functors $$\label{evttp} \dots\to \tau^{\le 0 }(H_M)\to H_M\to \tau^{\ge 1}(H_M)\to \tau^{\le 0}(H_M)\circ [1]\to H_M\to \dots$$ Applying this sequence of functors to $N$ we obtain that the surjectivity of $h^1_N$ along with the injectivity of $h^2_N$ is equivalent to $\tau^{\ge 1}(H_M)(N)={\{0\}}$. So, recalling Proposition \[pwrange\](\[iwrvt\]) once again (to obtain that $\tau^{\ge 1}(H_M)$ is of weight range $\le -1$) we conclude the proof. 4\. The “only if” part of the assertion is immediate from Proposition \[pwfil\](4). To prove the converse implication we should check that the couple $({\underline{C}}_{w\ge 0}, {\underline{C}}_{w\ge 0}^{\perp}[1])$ is a $t$-structure if our representability assumption is fulfilled. It is easily seen that the only non-trivial axiom check here is the existence of $t$-decompositions (see Definition \[dtstr\]), which is given by the previous assertion. 5\. If $\tau^{\le 0}H_{M'}$ is representable then the previous assertions imply the existence of a distinguished triangle $M'^{\le 0}\to M' \to M'^{\ge 1} \to M'^{\le 0}[1]$ with $M'\in {\underline{C}}_{w\ge 0}^{\perp}$. Then the object $M'^{\ge 1} $ represents the functor $\tau^{\ge 1}H_{M'}$ according to Theorem 2.3.1(III.4) of [@bger] (and so, $\tau^{\ge 1}H_{M'}$ is representable). The proof of the converse implication is similar. Now we describe one more application of Proposition \[padjt\]. It relies on a modified version of the Brown representability property that we will now define. \[dsatur\] Let $R$ be an associative commutative unital ring, and assume that ${\underline{C}}$ is $R$-linear. 1\. We will say that an $R$-linear cohomological functor $H$ from ${\underline{C}}$ into $R-{\operatorname{Mod}}$ is [*of $R$-finite type*]{} whenever for any $M\in {\operatorname{Obj}}{\underline{C}}$ the $R$-module $H(M)$ is finitely generated and $H(M[i])={\{0\}}$ for almost all $i\in {{\mathbb{Z}}}$. 2\. We will say that ${\underline{C}}$ is [*$R$-saturated*]{} if the representable functors from ${\underline{C}}$ are exactly all the $R$-finite type ones. 3\. The symbol $\operatorname{\operatorname{AddFun}_R}(C,D)$ will denote the (possibly, big) category of $R$-linear (additive) functors from $C$ into $D$ whenever $C$ and $D$ are $R$-linear categories. 4\. We will write $R-{\operatorname{mod}}$ for the category of finitely generated $R$-modules. \[psatur\] Assume that ${\underline{C}}$ is $R$-linear and endowed with a bounded weight structure $w$. I. Then all virtual $t$-truncations of functors of $R$-finite type are of $R$-finite type also. II\. Assume in addition that ${\underline{C}}$ is $R$-saturated. Then the following statements are valid. 1\. For any $i\in {{\mathbb{Z}}}$ and $M\in {\operatorname{Obj}}{\underline{C}}$ the functors $\tau^{\le -i }(H_M)$ and $\tau^{\ge - i }(H_M)$ are representable. 2\. There exists a $t$-structure right adjacent to $w$. 3\. Its heart ${{\underline{Ht}}}$ naturally embeds into the category $\operatorname{\operatorname{AddFun}_R}({{\underline{Hw}}}{{^{op}}},R-{\operatorname{mod}})$. This embedding is essentially surjective whenever $R$ is noetherian. I. Recall that virtual $t$-truncations (of cohomological functors) are cohomological. Moreover, virtual $t$-truncations of $R$-linear functors are obviously $R$-linear. Now, let $H$ be a functor of $R$-finite type. It obviously follows that the values of $\tau^{\ge 0}(H)$ are finitely generated $R$-modules. Next, for any $N\in {\operatorname{Obj}}{\underline{C}}$ we can take $w_{\ge 0}(N[j])$ being $0$ for $j$ small enough and being equal to $N[j]$ for $j$ large enough (recall that $w$ is bounded); hence the functor $\tau^{\ge 0}(H)$ is of $R$-finite type. Applying this argument to $H\circ [-i]$ (for $i\in {{\mathbb{Z}}}$) we obtain that the functor $\tau^{\ge - i}(H)$ is of $R$-finite type also. The proof for $\tau^{\le - i }(H)$ is similar. II.1. Immediate from assertion I combined with the definition of saturatedness. 2. According to Proposition \[padjt\](\[ile4\]), the assertion follows from the previous one. 3\. Certainly, restricting functors of $R$-finite type from ${\underline{C}}$ to ${{\underline{Hw}}}$ gives functors of the type described. This restriction gives an embedding of ${{\underline{Ht}}}$ into $\operatorname{\operatorname{AddFun}_R}({{\underline{Hw}}}{{^{op}}},R-{\operatorname{mod}})$ according to Proposition \[phadj\]. Lastly, if a functor $A$ belongs to $\operatorname{\operatorname{AddFun}_R}({{\underline{Hw}}}{{^{op}}},R-{\operatorname{mod}})$ and $R$ is noetherian then the pure functor $H_{{\mathcal{A}}}$ (see Definition \[drange\](2)) is easily seen to be of $R$-finite type (as a cohomological functor from ${\underline{C}}$ into $R-{\operatorname{mod}}$), whereas the object representing it belongs to ${\underline{C}}^{t=0}$ according to Proposition \[pwrange\](\[iwrpure\]). \[rsatur\] 1. Note now that Corollary 0.5 of [@neesat] easily implies that ${\underline{C}}$ is $R$-saturated whenever $R$ is a noetherian ring, $X$ is a regular separated finite-dimensional scheme that is proper over ${{\operatorname{Spec}\,}}R$, and ${\underline{C}}=D^b(X)$ (the bounded derived category of coherent sheaves). Indeed, the assumptions on $X$ imply that in this case the derived category of perfect complexes of sheaves equals ${\underline{C}}$, and then loc. cit. gives the result immediately (cf. Remark \[roq\](1) below). Moreover, in this case ${\underline{C}}\cong {\underline{C}}{{^{op}}}$ (since there exists a dualizing complex on $X$). Thus $t$-structures left adjacent to bounded weight structures on $D^b(X)$ exist also. We recall also that in the case where $R$ is a field the saturatedness is question is given by Corollary 3.1.5 of [@bvdb]. Moreover, Theorem 4.3.4 of ibid. is a certain a “non-commutative geometric” analogue of this statement. 2\. Now we discuss possible weight structures on the category ${\underline{C}}=D^b(X)$ (as above). We recall that bounded weight structures for ${\underline{C}}$ are determined by their hearts (see Proposition \[pbw\](\[igenlm\])), whereas the latter are precisely all the negative additive Karoubi-closed subcategories of ${\underline{C}}$ that densely generate it (see Corollary 2.1.2 of [@bonspkar]). Moreover, the author suspects that in the “geometric” examples mentioned above there exists a single “dense” generator $P$ of ${{\underline{Hw}}}$, i.e., all objects of ${{\underline{Hw}}}$ are retracts of (finite) powers of $P$. One can construct a rich family of bounded weight structures on $D^b(X)$ at least in the case where $X={\mathbb{P}}^n$ (for some $n>0$; the author does not know whether it is necessary to assume that $R$ is a field here) since one can use gluing for constructing weight structures (cf. [@bws §8.2]). More generally, it suffices to assume that $D^b(X)$ possesses a [*full exceptional collection*]{} of objects. Note however that degenerate weight structures are certainly possible in triangulated categories of this type (cf. Proposition \[prtst\](\[itp4\])); so, these weight structures are not bounded. Still the author conjectures that the boundedness restriction is not actually necessary (the proof may rely on the existence of a [*strong generator*]{} in the sense of [@bvdb] for ${\underline{C}}$). 3\. The only examples of $R$-saturated triangulated categories for a non-noetherian $R$ known to the author are direct sums of $R/J_i$-saturated categories, where $\{J_i\}$ is a finite collection of ideals of $R$ and all $R/J_i$ are noetherian. So, a general $R$ was mentioned in the proposition just for the sake of generality. Our definition of $R$-saturatedness may be “not optimal”; it could make sense to put finitely presented modules instead of finitely generated ones into the definition. Note however that any length two ${{\underline{Hw}}}$-complex is a weight complex of some object of ${\underline{C}}$. This allows to describe ${{\underline{Ht}}}$ completely (cf. Proposition \[psatur\](II.3)) for any possible definition of $R$-saturatedness (and can possibly help choosing among the definitions). 4\. Actually, all of the statements of this paper may easily be proved in the $R$-linear context (cf. [@zvon]). This is formally a generalization (since one can take $R={{\mathbb{Z}}}$); yet Propositions \[psatur\] and \[psaturdu\] (along with the examples to them) appear to be the only statements for which this setting is really actual. Note also that below we define Brown-Comenetz duals (of objects and functors) “using” the group ${{\mathbb{Q}}}/{{\mathbb{Z}}}$. However, the only property of this group that we will actually apply is that it is an injective cogenerator of the category ${\underline{\operatorname{Ab}}}$. Hence for an $R$-linear category ${\underline{C}}$ one can replace our Definition \[dsym\](\[ibcomf\],\[ibcomo\]) below by any its “$R$-linear analogue”; this may make sense if $R$ is a field. On adjacent weight structures {#sadjw} ----------------------------- In this subsection we will assume that ${\underline{C}}$ is endowed with a $t$-structure $t$. For the construction of certain weight structures we will need the following statement. \[lconstws\] Assume that certain extension-closed classes ${\underline{C}}_-$ and ${\underline{C}}_+$ of objects of ${\underline{C}}$ satisfy the axioms (i)—(iii) of Definition \[dwstr\] (for ${\underline{C}}_{w\le 0}$ and $ {\underline{C}}_{w\ge 0}$, respectively). Let us call a ${\underline{C}}$-distinguished triangle $X\to M\to Y[1]$ a [*pre-weight decomposition*]{} of $M$ if $X$ belongs to ${\underline{C}}_-$ and $Y$ belongs to ${\underline{C}}'_+$. Then the following statements are valid. 1\. The class $C$ of objects possessing pre-weight decompositions is extension-closed (in ${\underline{C}}$). 2\. Assume that $C$ contains a subclass ${\mathcal{P}}$ such that ${\mathcal{P}}={\mathcal{P}}[1]$. Then $C$ also contains the object class of the smallest strict subcategory of ${\underline{C}}$ containing ${\mathcal{P}}$. 3. Assume that ${\underline{C}}$ has coproducts, and ${\underline{C}}_-$ and ${\underline{C}}_+$ are coproductive. Then $C$ is coproductive also. 1\. Immediate from Theorem 2.1.1(I.1) of [@bonspkar] (cf. also Remark 1.5.5(1) of [@bws]). 2\. Immediate from assertion 1. 3\. Once again, it suffices to recall Proposition \[pcoprtriang\]. Now we prove a simple statement on the existence of $w$ that is left adjacent to $t$. \[pconstrwfromt\] I. Assume that there exists a weight structure $w$ left adjacent to $t$. Then for any $M\in {\underline{C}}^{t=0}$ there exists an ${{\underline{Ht}}}$-epimorphism from the ${{\underline{Ht}}}$-projective object $H_0^t(P)$ into $ M$ for some $P\in P_t$, and the functor $H_0^t$ induces an equivalence of $\operatorname{\operatorname{Kar}}({{\underline{Hw}}})$ with the category of projective objects of ${{\underline{Ht}}}$. II. The converse implication is valid under any of the following additional assumptions. 1\. $t$ is bounded above (see Definition \[dtstr\]). 2\. There exists an integer $n$ such that ${\underline{C}}^{t\ge n}\perp {\underline{C}}^{t\le 0}$. I. Fix $M\in {\underline{C}}^{t=0}$ and consider its weight decomposition $P\stackrel{p}{\to} M\to M'\to P[1]$. Since $M\in {\underline{C}}^{t\le 0}={\underline{C}}_{w\ge 0}$, we have $P\in {\underline{C}}_{w=0}$ according to Proposition \[pbw\](\[iwd0\])). Next, since $P\in {\underline{C}}_{w\ge 0}={\underline{C}}^{t\le 0}$, the object $P_0=t^{\ge 0}P$ equals $H_0^t(P)$; hence $P_0$ is projective in ${{\underline{Ht}}}$ according to Proposition \[pgen\](\[ipgen25\]). The adjunction property for the functor $t^{\ge 0}$ (see Remark \[rtst1\](\[it3\])) implies that $p$ factors through the $t$-decomposition morphism $P\to P_0$. Now we check that the corresponding morphism $P_0\to M$ is an ${{\underline{Ht}}}$-epimorphism. This is certainly equivalent to its cone $C$ belonging to ${\underline{C}}^{t\le -1}$. The octahedral axiom of triangulated categories gives a distinguished triangle $(t^{\le -1}P)[1]\to M'\to C\to (t^{\le -1}P)[2]$; it yields the assertion in question since $M'\in {\underline{C}}_{w\ge 1}={\underline{C}}^{t\le -1}$ and the class ${\underline{C}}^{t\le -1}$ is extension-closed. Next, the category of projective objects of ${{\underline{Ht}}}$ is certainly Karoubian. According to Proposition \[phadj\] it remains to verify that for any projective object $Q$ of ${{\underline{Ht}}}$ there exists $R\in P_t$ such that $Q $ is a retract of $ H_0^t(R)$. Now, the first part of the assertion implies the existence of an ${{\underline{Ht}}}$-epimorphism $H_0^t(R')\to Q$ for some $R'\in P_t$. Since $H_0^t(R')$ is projective in ${{\underline{Ht}}}$, this epimorphism splits, i.e., $Q$ equals the image of some idempotent isomorphism of $H_0^t(R')$. Lifting this endomorphism to ${{\underline{Hw}}}$ we obtain the result. II.1. We set ${\underline{C}}_{w\ge 0}={\underline{C}}^{t\le 0}$ and take ${\underline{C}}_{w\le 0}$ to be the envelope of $\cup_{i<0} P_t[i]$.[^34]We should prove that this couple yields a weight structure for ${\underline{C}}$, since this weight structure would certainly be adjacent to $t$. Now, this “candidate weight structure” obviously satisfies axioms (i) and (ii) in Definition \[dwstr\]. Next, since $P_t[i]\perp {\underline{C}}^{t\le 0}$ for any $i<0$; hence the orthogonality axiom (iii) is fulfilled also. It remains to verify the existence of a weight decomposition for any $M\in {\underline{C}}^{t\le i}$ by induction on $i$. The statement is obvious for $i< 0$ since $M\in {\underline{C}}_{w\ge 1}={\underline{C}}^{t\le -1}$ and we can take a “trivial” weight decomposition $0\to M\to M\to 0$. Now assume that existence of $w$-decompositions is known for any $M\in {\underline{C}}^{t\le j}$ for some $j\in {{\mathbb{Z}}}$. We should verify the existence of weight decomposition of an element $N$ of ${\underline{C}}^{t\le j+1}$. Certainly, $N$ is an extension of $N'[-j-1]=H_0^t(N[j+1])[-j-1]$ by $t^{\le j}N$ (see Remark \[rtst1\](\[it3\]) for the notation). Since the latter object possesses a weight decomposition, Lemma \[lconstws\](1) allows us to verify the existence of a weight decomposition of $N'[-j-1]$ (instead of $N$). Now we choose a surjection $t^{=0}P\to N'$ whose existence is given by our assumptions. Then a cone $C $ of the corresponding composed morphism $P\to N'$ is easily seen to belong to $t^{\le -1}{\underline{C}}$. Since both $P$ and $C$ possess weight decompositions, applying loc. cit. once again we obtain the assertion in questions. 2\. We take ${\underline{C}}_{w\ge 0}={\underline{C}}^{t\le 0}$ and ${\underline{C}}_{w\le 0}={}^{\perp}{\underline{C}}^{t\le -1}$. Once again it suffices to verify the existence of a $w$-decomposition for an object $M$ of ${\underline{C}}$. We consider the (full) triangulated subcategory ${\underline{C}}'$ consisting of $t$-bounded below objects, i.e., ${\operatorname{Obj}}{\underline{C}}'=\cup_{i\in {{\mathbb{Z}}}}{\underline{C}}^{t\le i}$. According to the previous assertion, any object of ${\underline{C}}'$ possesses a weight decomposition with respect to the corresponding weight structure; thus it also possesses a $w$-decomposition. Now, the $t$-decomposition of the object $M[n-2]$ yields a presentation of $M$ as extension of an element $M'$ of ${\underline{C}}^{t\ge n-1}$ by an element $M''$ of ${\underline{C}}^{t\le n-2}$. Since $M''\in {\operatorname{Obj}}{\underline{C}}'$, it possesses a $w$-decomposition. Next, our “extra” orthogonality assumption on $t$ yields that $M''\in {\underline{C}}_{w\le 0}$; hence one take the triangle $M''\to M''\to 0\to M''[1]$ as a $w$-decomposition of $M''$. Lastly, applying Lemma \[lconstws\](1) once again we obtain that $M$ possesses a $w$-decomposition also. \[rexenproj\] 1. Assume that the category ${\underline{C}}{{^{op}}}$ is $R$-saturated (see Definition \[dsatur\]; in particular, ${\underline{C}}$ may equal the category $D^b(X)$ or $D^b(X){{^{op}}}$ for $X$ being regular separated finite-dimensional scheme that is proper over ${{\operatorname{Spec}\,}}R$ for a Noetherian $R$) and $t$ is a bounded above $t$-structure on ${\underline{C}}$. Then our proposition (combined with Proposition \[phadj\]) easily implies that there exists a weight structure left adjacent to $t$ if and only if ${{\underline{Ht}}}$ has enough projectives (since the corresponding pure functors are corepresented by elements of $P_t$). Moreover, in this case ${{\underline{Hw}}}$ is equivalent to ${\operatorname{Proj}}{\underline{A}}$. 2\. The assumption of the existence of an ${{\underline{Ht}}}$-epimorphism $t^{=0}P\to M$ with $P\in P_t$ for any $M\in {\underline{C}}^{t=0}$ naturally generalizes the condition of the existence of enough projectives that allows to relate the derived category of ${\underline{A}}$ to $K({\operatorname{Proj}}{\underline{A}})$. Note however that in this setting we have ${{\underline{Hw}}}\subset {{\underline{Ht}}}$; this is not the case in general. Moreover, the condition ${\underline{C}}^{t\ge n}\perp {\underline{C}}^{t\le 0}$ for $n\gg 0$ is a natural generalization of the finiteness of the cohomological dimension condition (for an abelian category). 3\. One can easily see that $P_t$ is [*negative*]{} for any $t$. So our existence of $w$ results are closely related to the statements on “constructing $w$ from a negative subcategory”; see §2.2 of [@bsnew], [@bws §4.3,4.5], [@bonspkar Corollary 2.1.2], and Remark \[rsatur\](2) above. \[tadjw\] Assume that ${\underline{C}}$ has coproducts and is endowed with a $t$-structure $t$ such that its localizing subcategory ${\underline{C}}'$ generated by ${\underline{C}}^{t\le 0}$ satisfies the dual Brown representability condition. 1. Then there exists a weight structure left adjacent to $t$ if and only if $t$ is productive and the category ${{\underline{Ht}}}$ has enough projectives. 2. If such a left adjacent $w$ exists then ${{\underline{Hw}}}$ is equivalent to the subcategory of projective objects of ${{\underline{Ht}}}$. 1\. If $w$ exists then $t$ is productive according to Proposition \[phop\](5). Next, the existence of enough projectives in ${{\underline{Ht}}}$ follows from Proposition \[pconstrwfromt\](I). Conversely, assume that $t$ is productive and the category ${{\underline{Ht}}}$ has enough projectives. Once again, for the “candidates” ${\underline{C}}_{w\ge 0}={\underline{C}}^{t\le 0}$ and ${\underline{C}}_{w\le 0}={{}^{\perp}}({\underline{C}}_{w\ge 0}[1])$ it suffices to verify the existence of a weight decomposition for any $Y\in {\operatorname{Obj}}{\underline{C}}$. For each projective object $P_0$ of ${{\underline{Ht}}}$ Proposition \[pgen\](\[ipgen4\]) gives the existence of $P\in P_t$ such that $H_0^t(P)\cong P_0$ (here $H^t_0$ is the $t$-homology on ${\underline{C}}$; see Remark \[rtst1\](\[it4\])). Thus the existence of enough projectives in ${{\underline{Ht}}}$ is equivalent to the fact that for any $M\in {\underline{C}}^{t=0}$ there exists an ${{\underline{Ht}}}$-epimorphism $H_0^t(P)\to M$ for $P\in P_t$. We take ${\underline{C}_0}\subset {\underline{C}}$ being the triangulated category of $t$-bounded below objects (i.e., ${\operatorname{Obj}}{\underline{C}_0}=\cup_{i\in{{\mathbb{Z}}}}{\underline{C}}^{t\le i}$). According to Proposition \[pconstrwfromt\](II.1), there exists a weight structure $w_0$ for ${\underline{C}_0}$ with ${\underline{C}}_{{w_0}\ge 0}={\underline{C}}^{t\le 0}$. Now we study the class $C$ of objects possessing pre-weight decompositions with respect to $w$ in the terms of Lemma \[lconstws\]. The existence of ${w_0}$ certainly implies that $C$ contains ${\operatorname{Obj}}{\underline{C}_0}$. Applying parts 2 and 3 of the lemma we obtain that $C$ actually contains ${\operatorname{Obj}}{\underline{C}'}$. On the other hand, since the class $C'={{}^{\perp}}{\operatorname{Obj}}{\underline{C}'}$ is contained in ${\underline{C}}_{w\le 0}$, $C$ also contains $C'$. According to Lemma \[lconstws\](1), it remains to verify that any object of ${\underline{C}}$ can be presented as an extension of an object of ${\underline{C}}'$ by an element of $C'$. The latter is immediate from Proposition \[pcomp\](II) combined with Proposition \[pbouloc\](III.\[ibou1\]). 2\. Similarly to the proof of Theorem \[tadjt\](II), Proposition \[phadj\] gives an embedding of ${{\underline{Hw}}}$ into the category of projective objects of ${{\underline{Ht}}}$. Hence the arguments used in the proof of assertion 1 (when $P$ was constructed from $P_0$) allow us to conclude the proof. \[restrt\] 1\. It appears that one is “usually” interested in the case where ${\underline{C}'}={\underline{C}}$. 2. It can be easily seen that in Theorem \[tadjt\] we could have replaced the Brown representability assumption for ${\underline{C}}$ by that for the category ${\underline{C}}'$ being its localizing category generated by ${\underline{C}}_{w\ge 0}$ (and so also by ${\underline{C}_0}=\cup_{i\in {{\mathbb{Z}}}} {\underline{C}}_{w\le i}$). 3\. It is actually not necessary to assume that the whole ${\underline{C}}$ has coproducts when defining ${\underline{C}}'$ (in both of these settings). Indeed, it suffices to assume that ${\underline{C}}^{t\le 0}$ is contained in some triangulated category ${\underline{C}}''\subset {\underline{C}}$ that has coproducts such that the embedding ${\underline{C}}''\to {\underline{C}}$ respects them. Certainly the dual to Theorem \[tadjw\] is also valid; it is formulated as follows. \[cadjw\] Assume that ${\underline{C}}$ has products and is endowed with a $t$-structure $t$ such that its colocalizing subcategory ${\underline{C}}'$ cogenerated by ${\underline{C}}^{t\ge 0}$ (see Definition \[dcomp\](\[idloc\])) satisfies the Brown representability condition. 1. Then there exists a weight structure right adjacent to $t$ if and only if $t$ is coproductive and the category ${{\underline{Ht}}}$ has enough injectives. 2. If such a left adjacent $w$ exists then ${{\underline{Hw}}}$ is equivalent to the subcategory of injective objects of ${{\underline{Ht}}}$. \[rnondeg\] 1. As it often happens when dealing with “large” triangulated categories, the “roles” of Theorem \[tadjw\] and Corollary \[cadjw\] seem to be somewhat different. This “asymmetry” occurs when one tries to apply these statements to a compactly generated (or more generally, [*well generated*]{}; see Definition \[dwg\](\[idpg\]) below) ${\underline{C}}$; recall that these condition are far from being self-dual (see Corollary E.1.3 and Remark 6.4.5 of [@neebook]). Now, if ${\underline{C}}$ is well generated then it appears to be “quite reasonable” to consider smashing $t$-structures (only); moreover, the existence of enough injectives seems to be a rather “reasonable” restriction on $t$ (see Corollary \[csymt\] below; note however that is not clear how to prove it in general without constructing a right adjacent weight structure [**first**]{}). Yet it may be difficult to check the Brown representability condition for ${\underline{C}}'$ (unless ${\underline{C}'}={\underline{C}}$;[^35] still checking the latter could be difficult also). On the other hand, it appears that for all “known” cosmashing $t$-structures the left adjacent weight structures can be easily described without using the results of this subsection. So, both Theorem \[tadjw\] and Corollary \[cadjw\] do not appear to be really “practical”. 2\. One results on the existence of adjacent weight and $t$-structures are certainly “not symmetric”: constructing a (right or left) adjacent $t$-structure is “much easier”. On the other hand, Theorem \[tpgws\] below is a tool of constructing weight structures that appears not to possess a $t$-structure analogue (see Remark \[rigid\](1)). Rather funnily, these two “asymmetries” appear to “compensate” each other. So, the “main general” sorts of weight structures for compactly generated categories that are “easy to construct” are the compactly generated ones (that are smashing) and the (cosmashing) weight structures right adjacent to compactly generated $t$-structures, whereas the main sorts of $t$-structures are the compactly generated ones and the ones right adjacent to compactly generated weight structures (these are smashing and cosmashing, respectively). 3\. Note also that there do exist smashing examples that are not compactly generated (inside a compactly generated ${\underline{C}}$). In [@kellerema] for a ring $R$ along with its two-sided ideal $I$ satisfying certain conditions possesses a right adjoint respecting coproducts but $L$ is not generated by compact objects of ${\underline{C}}$; a family of couples $(R,I)$ satisfying the conditions the following was proved: the embedding $i$ of the localizing subcategory $L$ generated by $I$ into the derived category ${\underline{C}}$ of right $R$-modules proved in Proposition \[prtst\](\[it4sm\]), the corresponding couple $s_L=({\operatorname{Obj}}L,{\operatorname{Obj}}L^{\perp_{{\underline{C}}}})$ is a “shift-stable” torsion pair for ${\underline{C}}$ (cf. Remark \[rtst2\](\[it5s1\]); so, it is a weight structure and a $t$-structure simultaneously); still it is certainly not compactly generated. One can probably obtain other examples of non-compactly generated weight structures for ${\underline{C}}$ of this type “starting from” $s_L$ since one can (considering it as a weight structure and) “join” it with any compactly generated weight structure for ${\underline{C}}$ (see Remark \[revenmorews\](1) below). Moreover, in Remark \[rsymt\](\[irsymt1\],\[irsymt4\]) below a rich family of smashing weight structures on categories dual to compactly generated ones is described; these weight structures are not compactly generated. On (countably) perfect and symmetric classes; their relation to Brown-Comenetz duality and adjacent torsion pairs {#scomp} ----------------------------------------------------------------------------------------------------------------- Now we recall the notion of perfectly generated triangulated categories; categories of this type satisfy the Brown representability property according to results of A. Neeman and H. Krause. \[dwg\] Let ${\alpha}$ be a regular infinite cardinal.[^36] 1. \[idsmall\] An object $M$ of ${\underline{C}}$ is said to be [*${\alpha}$-small*]{} if for any set of $N_i\in {\operatorname{Obj}}{\underline{C}}$ any morphism $M\to \coprod N_i$ factors through the coproduct of a subset of $\{N_i\}$ of cardinality less then ${\alpha}$. 2. \[idpc\] We will say that a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ is [*countably perfect*]{} if the class of ${\mathcal{P}}$-null morphisms is closed with respect to countable coproducts (recall that $h$ is ${\mathcal{P}}$-null if for all $M\in {\mathcal{P}}$ we have $H^M(h)=0$; see Definition \[dhopo\](5)). We will say that ${\mathcal{P}}$ is [*perfect*]{} if the class of ${\mathcal{P}}$-null morphisms is closed with respect to arbitrary coproducts. 3. \[idpg\] We will say that ${\underline{C}}$ is [*perfectly generated*]{} if there exists a countably perfect [**set**]{} ${\mathcal{P}}_0\subset {\operatorname{Obj}}{\underline{C}}$ that Hom-generates it. We will also say that ${\underline{C}}$ is ${\alpha}$-[*well generated*]{} (or just well generated) if (in addition) all elements of ${\mathcal{P}}_0$ are ${\alpha}$-small and ${\mathcal{P}}_0$ is perfect. Now we prove a few properties of these definitions. \[psym\] Let ${\mathcal{P}}$ be a class of objects of a triangulated category ${\underline{C}}$ that has coproducts. Denote by ${\underline{C}}'$ the localizing subcategory of ${\underline{C}}$ generated by ${\mathcal{P}}$ and denote by ${\underline{D}}$ the full subcategory of ${\underline{C}}$ whose object class equals ${\operatorname{Obj}}{\underline{C}'}{{}^{\perp}}$; denote the embedding ${\underline{C}}'\to {\underline{C}}$ by $i$. Then the following statements are valid. 1. \[isymcomp\] If a set ${\mathcal{Q}}$ Hom-generates ${\underline{C}}$ then it also ${{\aleph_0}}$-well generates it if and only if all elements of ${\mathcal{Q}}$ are compact. 2. \[isymeu\] ${\underline{D}}$ is triangulated and ${\operatorname{Obj}}{\underline{D}}=\cap_{i\in {{\mathbb{Z}}}}({\mathcal{P}}{{}^{\perp}}[i])$. Moreover, if ${\mathcal{P}}$ is (countably) perfect then ${\underline{D}}$ is closed with respect to (countable) ${\underline{C}}$-coproducts. 3. \[isymbr\] Assume that ${\mathcal{P}}$ is a (countably) perfect set. Then ${\underline{C}}'$ is perfectly generated by ${\mathcal{P}}$ and has the Brown representability property. Moreover, the embedding $i$ possesses an exact right adjoint $G$ that gives an equivalence ${\underline{C}}/{\underline{D}}\to {\underline{C}}'$ and respects (countable) coproducts.[^37] 4. \[isymuni\] If ${\mathcal{P}}_i$ is a collection of (countably) perfect subclasses of ${\operatorname{Obj}}{\underline{C}}$ then $\cup {\mathcal{P}}_i$ is (countably) perfect also. 5. \[iperftp\] Assume that $s=({\mathcal{LO}},{\mathcal{RO}})$ is a torsion pair for ${\underline{C}}$. Then $s$ is (countably) smashing if and only if ${\mathcal{LO}}$ is (countably) perfect. \[isymcomp\]. Certainly, ${{\aleph_0}}$-small objects are precisely the compact ones. Hence any ${{\aleph_0}}$-well generating set consists of compact objects. To get the converse implication it suffices to note that any class of compact objects is perfect (see assertion \[isymuni\]). \[isymeu\]. The first part of the assertion is obvious; the second one follows immediately from Proposition \[pwsym\](\[iwsmor\]). \[isymbr\]. The set ${\mathcal{P}}$ Hom-generates the category ${\underline{C}'}$ according to Proposition \[pcomp\](I.1); certainly it is perfect in it. Hence Lemma \[lperf\](\[ipereq\]) below (see Remark \[requivdef\]) implies that ${\mathcal{P}}$ perfectly generates ${\underline{C}'}$ (also) in the sense of [@kraucoh Definition 1]. Hence the Brown representability condition for ${\underline{C}'}$ is given by Theorem A of [@kraucoh]. Given this condition the existence of $G$ follows from Proposition \[pcomp\](II.2). Lastly, Proposition \[prtst\](\[it4sm\]) says that $G$ respects (countable) coproducts. \[isymuni\]. It suffices to note that uniting ${\mathcal{P}}_i$ corresponds to intersecting the corresponding classes of null morphisms. \[iperftp\]. Immediate from Proposition \[phop\](8). \[rwg\] 1. Recall also that any well generated triangulated category possessing a combinatorial model satisfies the dual Brown representability property; see §0 of [@neefrosi] (the statement is given by the combination of Theorems 0.17 and 0.14 of ibid.). 2\. It is well known that the class of well generated triangulated categories is “much bigger” than that of compactly generated ones; the class of perfectly generated categories is even bigger.[^38] Moreover, if ${\underline{C}}$ is well generated then any its subcategory ${\underline{C}}'$ generated by a set of objects as a localizing subcategory is well generated also; the Verdier localization ${\underline{C}}/{\underline{C}'}$ exists and is well generated (see Theorem 4.4.9 of [@neebook]). Note that the obvious analogue of this result for compactly generated categories is wrong (so, if ${\underline{C}}$ is compactly generated then its set-generated localizing subcategory ${\underline{C}}'$ along with the quotient ${\underline{C}}/{\underline{C}}'$ is only well generated in general). In particular, in [@neeshman] it was proved that the derived category of sheaves on a non-compact manifold is well generated but not compactly generated, whereas it is a localization of the compactly generated derived category of presheaves. However, the reader may assume that all the perfectly generated triangulated categories we study are actually compactly generated since most of our results are quite interesting in this case also. 3. Certainly, a set ${\mathcal{P}}$ is (countably) perfect if and only if the coproduct of its elements forms a (countably) perfect set. Now, constructing perfect sets (and classes) “without using compact objects” is rather difficult. The main source of “non-compact” perfect classes in our paper is a certain Brown-Comenetz-type “symmetry”; the idea originates from [@kraucoh]. We start with some definitions. \[dsym\] Let ${\mathcal{P}}$ and ${\mathcal{P}}'$ be subclasses of ${\operatorname{Obj}}{\underline{C}}$, $P\in {\operatorname{Obj}}{\underline{C}}$. 1. \[iwsym\] We will say that ${\mathcal{P}}$ is [*weakly symmetric*]{} to ${\mathcal{P}}'$ if ${\mathcal{P}}{{}^{\perp}}={{}^{\perp}}{\mathcal{P}}'$. 2. \[iconull\] We will say that a ${\underline{C}}$-morphism $h$ is [*${\mathcal{P}}$-conull*]{} whenever for all $M\in {\mathcal{P}}$ we have $H_M(h)=0$ (where $H_M={\underline{C}}(-,M):{\underline{C}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$). 3. \[isym\] We will say that ${\mathcal{P}}$ is [*symmetric*]{} to ${\mathcal{P}}'$ if the class ${\mathcal{P}}$-null (see Definition \[dhopo\](5)) coincides with the class of ${\mathcal{P}}'$-conull morphisms. 4. \[ibcomf\] Let $H:{\underline{C}}\to {\underline{\operatorname{Ab}}}$ be a homological functor. Then we will call the functor ${\widehat{H}}:M\mapsto {\underline{\operatorname{Ab}}}(H(M),{{\mathbb{Q}}}/{{\mathbb{Z}}}):{\underline{C}}\to {\underline{\operatorname{Ab}}}$ the [*Brown-Comenetz dual*]{} of $H$. 5. \[ibcomo\] We will call an object of ${\underline{C}}$ the [*Brown-Comenetz dual*]{} of $P$ and denote it by $\hat{P}$ if it represents the Brown-Comenetz dual of the functor $H^M={\underline{C}}(M,-):{\underline{C}}\to {\underline{\operatorname{Ab}}}$. \[psymb\] Assume that ${\mathcal{P}}$, ${\mathcal{P}}'$, along with certain ${\mathcal{P}}_i$ and ${\mathcal{P}}'_i$ for $i$ running through some $I$, are subclasses of ${\operatorname{Obj}}{\underline{C}}$; let $P\in {\operatorname{Obj}}{\underline{C}}$ and $h$ be a ${\underline{C}}$-morphism. Assume that $H:{\underline{C}}\to {\underline{\operatorname{Ab}}}$ is a homological functor respecting coproducts. I. Then the following statements are valid. 1. \[iwsmor\] $P$ belongs to ${\mathcal{P}}{{}^{\perp}}$ if and only if the morphism ${\operatorname{id}}_P$ is ${\mathcal{P}}$-null; dually, $P\in {{}^{\perp}}{\mathcal{P}}$ if and only if ${\operatorname{id}}_M$ is ${\mathcal{P}}$-conull. 2. \[iwsym1\] If ${\mathcal{P}}$ is symmetric to ${\mathcal{P}}'$ then it is also weakly symmetric to ${\mathcal{P}}'$. If we assume in addition that ${\underline{C}}$ has coproducts then ${\mathcal{P}}$ is perfect. 3. \[iws2\] ${\mathcal{P}}$ is (weakly) symmetric to ${\mathcal{P}}'$ if and only if ${\mathcal{P}}'$ is (weakly) symmetric to ${\mathcal{P}}$ in the category ${\underline{C}}{{^{op}}}$.[^39] 4. \[iws1\] If ${\mathcal{P}}_i$ is (weakly) symmetric to ${\mathcal{P}}'_i$ for any $i\in I$ then $\cup{\mathcal{P}}_i$ is (weakly) symmetric to $\cup{\mathcal{P}}'_i$. 5. \[isbcd\] The Brown-Comenetz dual functor ${\widehat{H}}$ is a cohomological functor that converts ${\underline{C}}$-coproducts into ${\underline{\operatorname{Ab}}}$-products. Moreover, if ${\widehat{H}}$ is represented by some $N\in {\operatorname{Obj}}{\underline{C}}$ then $h$ is $\{N\}$-conull if and only if $H(h)=0$. 6. \[isym4\] Assume that ${\underline{C}}$ has coproducts, ${\mathcal{P}}$ is symmetric to ${\mathcal{P}}'$, and both of them are sets. Then the localizing subcategory ${\underline{C}'}$ generated by ${\mathcal{P}}$ satisfies both the Brown representability condition and its dual, the embedding $i:{\underline{C}}'\to {\underline{C}}$ has an (exact) right adjoint $G$, and ${\mathcal{P}}$ is symmetric to the class $G({\mathcal{P}}')$ in ${\underline{C}}'$. Moreover, for ${\underline{D}}$ being the full subcategory of ${\underline{C}}$ whose object class equals ${\operatorname{Obj}}{\underline{C}'}{{}^{\perp}}$, the functor $G$ gives an equivalence to ${\underline{C}}'$ of the full subcategory ${\underline{D}}'$ of ${\underline{C}}$ whose object class equals ${\operatorname{Obj}}{\underline{D}}{{}^{\perp}}$. Furthermore, the embedding ${\underline{D}}'\to {\underline{C}}$ respects products and possesses a left adjoint, and ${\underline{D}}'^{op}$ (has coproducts and) is perfectly generated by ${\mathcal{P}}'$. II\. Assume in addition that ${\underline{C}}$ (has coproducts) and satisfies the Brown representability condition. 1. \[iws21\] Then ${\widehat{H}}$ is representable by some $N\in {\operatorname{Obj}}{\underline{C}}$. 2. \[iws22\] Assume that $P$ is compact in ${\underline{C}}$. Then its Brown-Comenetz dual object $\hat{P}$ exists. 3. \[iws23\] Assume that all objects of ${\mathcal{P}}$ are compact. Then ${\mathcal{P}}$ is symmetric to the set of the Brown-Comenetz duals of elements of ${\underline{C}}$. I.\[iwsmor\]. Obvious. \[iwsym1\]. The first part of the assertion is immediate from the previous one; the second one is obvious. \[iws2\], \[iws1\]. Obvious. \[isbcd\]. Certainly, ${\widehat{H}}$ converts ${\underline{C}}$-coproducts into products of abelian groups. It is cohomological since ${{\mathbb{Q}}}/{{\mathbb{Z}}}$ is an injective object of ${\underline{\operatorname{Ab}}}$. Since it also cogenerates ${\underline{\operatorname{Ab}}}$, we obtain that $H(h)=0$ if ${\widehat{H}}(h)=0$, whereas the converse implication is automatic. \[isym4\]. The set ${\mathcal{P}}$ is perfect according to assertion I.\[iwsym1\]; hence ${\underline{C}}'$ is perfectly generated by ${\mathcal{P}}$. Thus ${\underline{C}'}$ satisfies Brown representability and $i$ possesses an exact right adjoint $G$ that respects coproducts and gives an equivalence ${\underline{C}}/{\underline{D}}\to {\underline{C}}'$ (see Proposition \[psym\](\[isymbr\])). Thus $G$ restricts to a fully faithful functor $j$ from ${\underline{D}}'$ into ${\underline{C}'}$. Next, the adjunction of $i$ to $G$ immediately yields that ${\mathcal{P}}$ is symmetric to the class $G({\mathcal{P}}')$ in ${\underline{C}}'$ indeed. Hence ${\underline{C}'}$ has the Brown representability property; thus it has products according to Proposition \[pcomp\](II.2). Thus $G({\mathcal{P}}')$ perfectly generates the category ${\underline{C}}'^{op}$; therefore ${\underline{C}'}$ satisfies the dual Brown representability condition.[^40] Now, ${\underline{D}}'$ is certainly a triangulated subcategory of ${\underline{C}}$ that is closed with respect to ${\underline{C}}$-products. We have ${\operatorname{Obj}}{\underline{D}}\perp {\mathcal{P}}'$ since ${\mathcal{P}}\perp {\operatorname{Obj}}{\underline{D}}$; hence ${\mathcal{P}}'$ perfectly generates ${\underline{D}}'^{op}$ (since this statement becomes true after we apply $j$) and $j$ is an equivalence. It remains to apply (the dual to) Proposition \[pcomp\](II.2). II.\[iws21\],\[iws22\]. By the definition of Brown representability, it suffices to note that both ${\widehat{H}}$ and $\widehat{H^P}$ are cp functors. \[iws23\]. Easy; combine assertions II.\[iws22\], I.\[isbcd\], and I.\[iws1\]. Now we establish a (somewhat funny) general criterion for a torsion pair $s'$ to be adjacent to $s$. \[pwsym\] Let ${\mathcal{P}}$, ${\mathcal{P}}'$ be subclasses of ${\operatorname{Obj}}{\underline{C}}$; for torsion pairs $s=({\mathcal{LO}},{\mathcal{RO}})$ and $s'=({\mathcal{LO}},{\mathcal{RO}})$ assume that a class ${\mathcal{P}}_s\subset {\operatorname{Obj}}{\underline{C}}$ generates $s$ and some ${\mathcal{P}}_{s'}$ [*cogenerates*]{} $s'$ (i.e., ${\mathcal{LO}}={{}^{\perp}}{\mathcal{P}}_{s'}$ and so $s'\subset {\mathcal{RO}}'$). Then the following statements are valid. 1. \[iws3\] ${\mathcal{P}}_s$ is weakly symmetric to ${\mathcal{P}}'$ if and only if ${\mathcal{LO}}$ is weakly symmetric to ${\mathcal{P}}'$. 2. \[iws3p\] ${\mathcal{P}}$ is weakly symmetric to ${\mathcal{P}}_{s'}$ if and only if ${\mathcal{P}}$ is weakly symmetric to ${\mathcal{RO}}'$. 3. \[iws4\] The following conditions are equivalent: \(i) ${\mathcal{P}}_s$ is weakly symmetric to ${\mathcal{P}}_{s'}$. \(ii) ${\mathcal{LO}}$ is symmetric to ${\mathcal{RO}}'$. \(iii) There exist a class ${\mathcal{Q}}$ that generates $s$ and ${\mathcal{Q}}'$ that cogenerates $s'$ such that ${\mathcal{Q}}$ is symmetric to ${\mathcal{Q}}'$. \(iv) $s$ is left adjacent to $s'$. 4. \[iwsymcgatp\] Assume that ${\underline{C}}$ (has coproducts) and satisfies the Brown representability property, ${\mathcal{P}}_s$ is a class of compact objects,[^41]  and $s'$ is right adjacent to $s$. Then $s$ is smashing, $s'$ is cosmashing, and $s'$ is cogenerated by the Brown-Comenetz duals of elements of ${\mathcal{P}}_s$ (i.e., ${\mathcal{LO}}'={{}^{\perp}}\widehat{{\mathcal{P}}_s}$ where $\widehat{{\mathcal{P}}_s}=\{\hat{P}:\ P\in {\mathcal{P}}_s\}$). 5. \[iwsymcgwt\] For ${\underline{C}}$ as in the previous assertion assume that ${\mathcal{P}}$ is a class of compact objects and it generates a weight structure $w$. Then $w$ is smashing, there exists a cosmashing $t$-structure $t$ right adjacent to $w$, and $t$ is cogenerated by the Brown-Comenetz duals of elements of ${\mathcal{P}}$ (i.e., ${\underline{C}}^{t\le -1}={{}^{\perp}}\widehat{{\mathcal{P}}}$, where $\widehat{{\mathcal{P}}}=\{\hat{P}:\ P\in {\mathcal{P}}\}$). \[iws3\], \[iws3p\]: obvious (recall the corresponding definitions). \[iws4\]. By definition, $s$ is left adjacent to $s'$ if and only if ${\mathcal{LO}}{{}^{\perp}}={{}^{\perp}}{\mathcal{RO}}'$; hence (i) is equivalent to (iv). Applying assertion \[iwsmor\] we also obtain that (iii) implies (i). Next, the equivalence of (ii) to (iv) is immediate from Proposition \[phop\](8). Since (ii) implies (iii), we obtain the result. \[iwsymcgatp\]. $s$ is smashing and $s'$ is cosmashing according to Proposition \[phop\](5) (recall that ${\underline{C}}$ has products according to Proposition \[pcomp\](II.2)). Next, ${\mathcal{P}}_s$ is symmetric to $\widehat{{\mathcal{P}}_s}$ according to Proposition \[psymb\](II.\[iws23\]). Hence it remains to apply the previous assertion. \[iwsymcgwt\]. Certainly, $w$ is smashing (see Proposition \[phopft\](III)). Hence $t$ exists according to Theorem \[tadjt\], and it remains to apply the previous assertion. \[rwsym\] 1\. Part \[iwsymcgwt\] of corollary is the first application of part \[iwsymcgatp\]; another application is (essentially) Corollary \[csymt\] below. Note also that Theorem 3.11 of [@postov] states that in a compactly generated [*algebraic*]{} triangulated category ${\underline{C}}$ for any compactly generated torsion pair $s$ (i.e., we assume ${\mathcal{P}}_s$ to be a set in assertion \[iwsymcgatp\]) there exists a torsion pair right adjacent to it. Hence this torsion pair $s'$ is cogenerated by the corresponding $\widehat{{\mathcal{P}}_s}$ (also). 2. In theory, (part \[iws4\] of) our proposition gives a complete description of all couples of adjacent torsion pairs: one can start with $s$, take a generating class ${\mathcal{P}}_s$ for it (that may be equal to ${\mathcal{LO}}$), find a class ${\mathcal{P}}_{s'}$ that is weakly symmetric to ${\mathcal{P}}_s$ (if any), and “cogenerate” $s'$ (if ${\mathcal{P}}_{s'}$ does cogenerate some torsion pair). Still constructing (weakly) symmetric classes and cogenerating torsion pairs by them appears to be rather difficult in general; the author can only “do” this by applying the Brown-Comenetz duality. So, one may say that we construct (weakly) symmetric classes “elementwisely”; the author wonders whether a “more involved” method exists. However, our proposition demonstrates the relation of adjacent torsion pairs to “Brown-Comenetz-type symmetry”; this point of view appears to be new. 3\. The problem with the symmetry condition is that the class ${\mathcal{P}}_s$-null is not determined by $s$; so even if $s$ and $s'$ are adjacent, it may be difficult to find “small” symmetric ${\mathcal{Q}}$ and ${\mathcal{Q}}'$ as in condition \[iws4\](iii). So, we only have two (rather) “extreme” (“basic”) types of symmetric classes: the ones coming from classes of compact objects (that “usually” have bounded cardinality) via Brown-Comenetz duality and the “big” ones of the “type” $({\mathcal{LO}},{\mathcal{RO}}')$ (see condition \[iws4\](ii))).[^42] Note also that in the proof of Theorem \[tsymt\] we do not actually need ${\mathcal{P}}$ to be symmetric; it suffices to assume that ${\mathcal{P}}$ is weakly symmetric to ${\mathcal{P}}'$ and ${\mathcal{P}}'$ is a set that is perfect in the category ${\underline{C}}{{^{op}}}$. On perfectly generated weight structures and torsion pairs {#spgtp} ========================================================== In this section we will always assume that ${\underline{C}}$ has coproducts and ${\mathcal{P}}$ is a subclass of ${\operatorname{Obj}}{\underline{C}}$. Our goal is to study the case when ${\mathcal{P}}$ is a (countably) perfect class and it generates a torsion pair $s$ for ${\underline{C}}$. Our results are more satisfactory when $s$ is weighty; in particular, we prove that all weight structures on well generated triangulated categories are ([*perfectly generated*]{} and) [*strongly well generated*]{}. In §\[scoulim\] we recall the notion of countable homotopy colimit in ${\underline{C}}$ (that is one of the main tools of this section) and introduce several related notions and facts. In §\[scghop\] we prove that compactly generated torsion pairs are in one-to-one correspondence with extension-closed Karoubi-closed essentially small classes of compact objects (in ${\underline{C}}$). This result (slightly) generalizes Theorem 3.7 of [@postov] (along with Corollary 3.8 of ibid.). In §\[sperfws\] we study the (naturally defined) perfectly generated weight structures. The existence Theorem \[tpgws\] is absolutely new; still its proof has some “predecessors”. It appears to be rather difficult to construct examples of perfectly generated weight structures that are not compactly generated; still we construct a curious family of those using suspended symmetric sets $({\mathcal{P}},{\mathcal{P}}')$ (see Definition \[dsym\](\[isym\])). For any couple of this sort the set ${\mathcal{P}}'$ is perfect in the category ${\underline{C}}{{^{op}}}$; so we obtain a certain weight structure $w$ on ${\underline{C}}$ whereas its left adjacent $t$-structure (whose existence is essentially given by Corollary \[cdualt\]) is generated by ${\mathcal{P}}$. The author does not know how to construct “new” $t$-structures using this result; it implies however that for any compactly generated $t$-structure there exists a right adjacent weight structure.[^43] Now, the opposite to this weight structure (in ${\underline{C}}{{^{op}}}$) is perfectly generated but (“almost never”) compactly generated, whereas the existence of $w$ implies that the category ${{\underline{Ht}}}$ has an injective cogenerator and satisfies the AB3\* axiom. Moreover, we deduce (using the results of [@humavit]) that ${{\underline{Ht}}}$ is Grothendieck abelian whenever $t$ is non-degenerate. In §\[swgws\] we develop a certain theory of well generated torsion pairs. We prove several relations between torsion pairs and (countably) perfect classes. Probably, the most interesting result in this section is the fact that a smashing weight structure on a well generated triangulated category is [*strongly well generated*]{} (i.e., it restricts to the subcategory of [*${\beta}$-compact*]{} objects for any large enough regular ${\beta}$ and it can be “recovered” from this restriction); in particular, it is perfectly generated (i.e., it may be constructed using Theorem \[tpgws\]). On countable homotopy colimits {#scoulim} ------------------------------ We recall the basics of the theory of countable (filtered) homotopy colimits in triangulated categories (as introduced in [@bokne]; some more detail can be found in [@neebook]; cf. also §4.2 of [@bws]). We will only apply the results of this subsection to triangulated categories that have coproducts; so we will not mention this restriction below. \[dcoulim\] For a sequence of objects $Y_i$ of ${\underline{C}}$ for $i\ge 0$ and maps $\phi_i:Y_{i}\to Y_{i+1}$ we consider the morphism $a:\oplus {\operatorname{id}}_{Y_i}\bigoplus \oplus (-\phi_i): D\to D$ (we can define it since its $i$-th component can be easily factorized as the composition $Y_i\to Y_i\bigoplus Y_{i+1}\to D$). Denote a cone of $a$ by $Y$. We will write $Y=\operatorname{\varinjlim}Y_i$ and call $Y$ a [*homotopy colimit*]{} of $Y_i$ (we will not consider any other homotopy colimits in this paper). Moreover, $\operatorname{\operatorname{Cone}}(\phi_i)$ will be denoted by $Z_{i+1}$ and we set $Z_0=Y_0$. \[rcoulim\] 1. Note that these homotopy colimits are not really canonical and functorial in $Y_i$ since the choice of a cone is not canonical. They are only defined up to non-canonical isomorphisms; still this is satisfactory for our purposes. Note also that the definition of $Y$ gives a canonical morphism $D\to Y$. 2\. By Lemma 1.7.1 of [@neebook], a homotopy colimit of $Y_{i_j}$ is the same (up to an isomorphism) for any subsequence of $Y_i$. In particular, we can discard any (finite) number of first terms in $(Y_i)$. 3\. By Lemma 1.6.6 of [@neebook], $M$ is a homotopy colimit of $M\stackrel{{\operatorname{id}}_M}{\to}M\stackrel{{\operatorname{id}}_M}{\to} M\stackrel{{\operatorname{id}}_M}{\to} M\stackrel{{\operatorname{id}}_M}{\to}\dots$. 4. More generally, if $p$ is an idempotent endomorphism of $M$ then $p$ is isomorphic to a retraction of $M$ onto a homotopy colimit $N$ of $M\stackrel{p}{\to}M\stackrel{p}{\to} M\stackrel{p}{\to} M\stackrel{p}{\to}\dots$ (see the proof of Proposition 1.6.8 of ibid). It easily follows that any extension-closed coproductive subclass of ${\operatorname{Obj}}{\underline{C}}$ that is closed either with respect $[1]$ or $[-1]$ is Karoubian; see also Corollary 2.1.3(2) of [@bsnew] for an alternative proof of this fact. 5\. Below we will often want to say something on some (co)homology of $Y$ along with morphisms from it. We start from treating representable cohomology. Let $T$ be an object of ${\underline{C}}$ and consider the cp functor $H_T={\underline{C}}(-,T)$ from ${\underline{C}}$ into ${\underline{\operatorname{Ab}}}$. Then the distinguished triangle defining $Y$ certainly yields a long exact sequence $$\begin{aligned} \dots\to H_T(D[1])\stackrel{H_T(a[1])} \to H_T(D[1])(\cong \prod H_T(Y_i[1]))\to H_T(Y) \\ \to H_T(D)(\cong \prod H_T(Y_i)) \stackrel{H_T(a)}{\to} H_T(D)\to\dots \end{aligned}$$ According to Remark A.3.6 of [@neebook] this gives a short exact sequence $$0\to \operatorname{\varprojlim}^1 H_T(Y_i[1])\to H_T(Y)\to \operatorname{\varprojlim}H_T(Y_i) \to 0.$$ Thus any morphism $f\in {\underline{C}}(Y,T)$ gives a “coherent” system of morphisms $Y_i\to T$. Conversely, for any coherent system $(f_i)$ of this sort there exists its “lift” to some $f\in {\underline{C}}(Y,T)$ that we will say to be [*compatible*]{} with $(f_i)$. Certainly, the compatibility of $f$ with $(f_i)$ is fulfilled if and only if the composition $Y_i\to \coprod Y_i\to Y\stackrel{f}{\to} T$ equals $f_i$ for any $i\ge 0$. Hence for any functor $F$ from ${\underline{C}}$ the composition $F(Y_i)\to F(\coprod Y_i)\to F(Y)\stackrel{F(f)}{\to} F(T)$ equals $F(f_i)$. We study the behaviour of homotopy colimits under cp and wcc functors. \[lcoulim\] Assume that $Y=\operatorname{\varinjlim}Y_i$ (in ${\underline{C}}$), ${\underline{A}}$ is an abelian category; let $H$ (resp. $H'$) be a cp (resp. a wcc; see Definition \[dcomp\](\[idcc\])) functor from ${\underline{C}}$ into ${\underline{A}}$. Then the following statements are valid. 1\. The obvious connecting morphisms $Y_i\to Y$ give an epimorphism $H(Y)\to \operatorname{\varprojlim}H(Y_i)$. 2\. This epimorphism $H(Y)\to \operatorname{\varprojlim}H(Y_i)$ is an isomorphism whenever ${\underline{A}}$ is an AB4\* category and all the morphisms $H(\phi_i[1])$ are epimorphic for $i\gg 0$. 3\. $H'(Y) $ naturally surjects onto $ \operatorname{\varinjlim}H'(Y_i)$. This surjection is an isomorphism if either (i) there exist objects $A$ and $A_i\in {\operatorname{Obj}}{\underline{A}}$, along with compatible isomorphisms $H'(Y_i[1])\cong A_i\bigoplus A$ and $H(\phi_i[1])\cong (0:A_i\to A_{i+1}) \bigoplus {\operatorname{id}}_A$, for $i\gg 0$ or \(ii) ${\underline{A}}$ is an AB5 category. In particular, if $C$ is compact then ${\underline{C}}(C,Y)\cong \operatorname{\varinjlim}{\underline{C}}(C,Y_i)$. 1\. We argue as in Remark \[rcoulim\](5). We have a long exact sequence $$\dots\to H(D[1])\stackrel{H(a[1])} \to H(D[1])\to H(Y)\to H(D)\stackrel{H(a)}{\to} H(D)\to\dots.$$ Since $H(D)\cong \prod H(Y_i)$, the kernel of $H(a)$ equals $\operatorname{\varprojlim}H(Y_i)$ (and this inverse limit exists in ${\underline{A}}$). This yields the result. 2\. Remark A.3.6 of [@neebook] yields that (in this case) the cokernel of $H(a[1])$ equals the $1$-limit of the objects $H(Y_i[1])$. Next, by Remark \[rcoulim\](1) we can assume that the homomorphisms $\phi[1]^*$ are surjective for all $i$. Hence the statement is given by Lemma A.3.9 of ibid. 3\. Similarly to the proof of assertion 1, we consider the long exact sequence $$\dots\to H'(D)\stackrel{H'(a)}{\to} H'(D)\to H'(Y)\to H'(D[1]) \stackrel{H'(a[1])} \to H'(D[1]) \to\dots.$$ Since $H'(D)\cong \coprod H'(Y_i)$, it easily follows that the cokernel of $H'(a)$ is $\operatorname{\varinjlim}H'(Y_i)$, this gives the first part of the assertion. To prove its second part we should verify that $H'(a[1])$ is monomorphic (if either of the two additional assumption is fulfilled). We will write $B_i$ and $f_i$ for $H'(Y_i[1])$ and $H(\phi_i[1])$, respectively, whereas $H'(a[1])$ (that can certainly be expressed in terms of ${\operatorname{id}}_{B_i}$ and $f_i$) will be denoted by $h$. If (i) is valid then we can certainly assume that $B_i\cong A_i\bigoplus A$ and $f_i\cong {\operatorname{id}}_A\bigoplus 0$ for all $i\ge 0$. Moreover, the additivity of the object $\operatorname{\operatorname{Ker}}(h)$ with respect to direct sums of $(B_i,f_i)$ reduces its calculation to the following to cases: (1) $f_i=0$ and (2) $f_i\cong {\operatorname{id}}_A$. In case (1) $h$ is isomorphic to ${\operatorname{id}}_{\coprod B_i}$; so it is monomorphic. In case (2) $h$ is monomorphic also since its composition with the projection of $\coprod B_i$ onto $\coprod_{i>0}B_i$ is an isomorphism. Indeed, the inverse to the latter is given by the morphism matrix $$\begin{pmatrix}{\operatorname{id}}_A &{\operatorname{id}}_A &{\operatorname{id}}_A &\dots \\ 0 & {\operatorname{id}}_A &{\operatorname{id}}_A &\dots\\ 0 & 0 &{\operatorname{id}}_A &\dots\\ 0 & 0 &0 &\dots\\ \dots & \dots &\dots &\dots \end{pmatrix}$$ (cf. the proof of [@neebook Lemma 1.6.6]). To prove version (ii) of the assertion note that the composition of $H'(a[1])$ with the obvious monomorphism $\coprod_{i\le j} H'(Y_i[1])\to \coprod_{i\ge 0} H'(Y_i[1])$ is easily seen to be monomorphic for each $j\ge 0$. If ${\underline{A}}$ is an AB5 category then it follows that the morphism $H'(a[1])$ is monomorphic itself. We will also need the following definitions.[^44] \[dses\] 1\. A class ${{\tilde{\mathcal{P}}}}\subset {\operatorname{Obj}}{\underline{C}}$ will be called [*strongly extension-closed*]{} if it contains $0$ and for any $\phi_i:Y_{i}\to Y_{i+1}$ such that $Y_0\in {{\tilde{\mathcal{P}}}}$ and $\operatorname{\operatorname{Cone}}(\phi_i)\in {{\tilde{\mathcal{P}}}}$ for all $i\ge 0$ we have $\operatorname{\varinjlim}_{i\ge 0} Y_i\in {{\tilde{\mathcal{P}}}}$ (i.e. ${{\tilde{\mathcal{P}}}}$ contains all possible cones of the corresponding distinguished triangle; note that these are isomorphic). 2\. The smallest strongly extension-closed Karoubi-closed class of objects of ${\underline{C}}$ that contains a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ and is closed with respect to arbitrary (small) coproducts will be called the [*strong extension-closure*]{} of ${\mathcal{P}}$. 3\. We will write ${\underline{\coprod}{\mathcal{P}}}$ for the closure of ${\mathcal{P}}$ with respect to ${\underline{C}}$-coproducts (and in §\[sperfws\] below we will also use this notation for the full subcategory of ${\underline{C}}$ formed by these objects). Also, we will call the class of the objects of ${\underline{C}}$ that may be presented as homotopy limits of $Y_i$ with $Y_0$ and $\operatorname{\operatorname{Cone}}(\phi_i)\in {\underline{\coprod}{\mathcal{P}}}$ the [*naive big hull*]{} of ${\mathcal{P}}$. We will call the Karoubi-closure of the naive big hull of the class ${\mathcal{P}}$ its [*big hull*]{}. Now we prove a few simple properties of these notions. \[lbes\] Let ${\mathcal{P}}$ be a class of objects of ${\underline{C}}$; denote its strong extension-closure by ${{\tilde{\mathcal{P}}}}$. 1. \[iseses\] ${{\tilde{\mathcal{P}}}}$ is extension-closed in ${\underline{C}}$; it contains the big hull of ${\mathcal{P}}$. 2. \[isesperp\] Let $H$ be a cp functor from ${\underline{C}}$ into a AB4\*-category ${\underline{A}}$, and assume that the restriction of $H$ to ${\mathcal{P}}$ is zero. Then $H$ kills ${{\tilde{\mathcal{P}}}}$ also. In particular, if for some $D\subset {\operatorname{Obj}}{\underline{C}}$ we have $ {\mathcal{P}}\perp D$ then ${{\tilde{\mathcal{P}}}}\perp D$ also. 3. \[isescperp\] Let $H'$ be a cc functor from ${\underline{C}}$ into a AB5-category. Then $H'$ kills ${{\tilde{\mathcal{P}}}}$ whenever it kills ${\mathcal{P}}$. In particular, if $D$ is a class of compact objects in ${\underline{C}}$ and $D\perp {\mathcal{P}}$ then $D\perp {{\tilde{\mathcal{P}}}}$ also. 4. \[izs\] [*Zero classes*]{} of arbitrary families of cp and cc functors (into AB4\* and AB5 categories, respectively) are strongly extension-closed (i.e. for any cp functors $H_i$ and cc functors $H'_i$ of this sort the classes $\{M\in {\operatorname{Obj}}{\underline{C}}:\ H_i(M)=0\ \forall i\}$ and $\{M\in {\operatorname{Obj}}{\underline{C}}:\ H'_i(M)=0\ \forall i\}$ are strongly extension-closed). In particular, if $({\mathcal{LO}},{\mathcal{RO}})$ is a torsion pair in ${\underline{C}}$ then ${\mathcal{LO}}$ is strongly extension-closed. 5. \[ilwd\] Adopt the notation of Definition \[dcoulim\]; let $w$ be a countably smashing weight structure on ${\underline{C}}$. Choose some $w$-decompositions $LZ_i\to Z_i\to RZ_i\to LZ_i[1]$ of $Z_i$ (see Definition \[dcoulim\]) for $i\ge 0$. Then there exists some $w$-decompositions $LY_i\to Y_i\to RY_i\to LY_i[1]$ for $i\ge 0$ and connecting morphisms $l_i:LY_i\to LY_{i+1}$ for $i\ge 0$ such that the corresponding squares commute, $LY_0=LZ_0$, $\operatorname{\operatorname{Cone}}(l_i)\cong LZ_{i+1}$, and there exists a weight decomposition $\operatorname{\varinjlim}LY_i\to Y\to RY\to (\operatorname{\varinjlim}LY_i)[1]$ (for some $RY\in {\underline{C}}_{w\ge 1}$). \[iseses\]. For any distinguished triangle $X\to Y\to Z$ for $X,Z\in {{\tilde{\mathcal{P}}}}$ the object $Y$ is the colimit of $X\stackrel{f}{\to} Y\stackrel{{\operatorname{id}}_Y}{\to} Y \stackrel{{\operatorname{id}}_Y}{\to} Y \stackrel{{\operatorname{id}}_Y}{\to} Y\to \dots$ (see Remark \[rcoulim\](3)). Since a cone of $f$ is $Z$, whereas a cone of ${\operatorname{id}}_Y$ is $0$, ${{\tilde{\mathcal{P}}}}$ is extension-closed indeed. It contains the big hull of ${\mathcal{P}}$ by definition. \[isesperp\]. Since for any $d\in D$ the functor $H_d:{\underline{C}}\to {\underline{\operatorname{Ab}}}$ converts arbitrary coproducts into products, it suffices to verify the first part of the statement. Thus it suffices to verify that $H(Y)=0$ if $Y=\operatorname{\varinjlim}Y_i$ and $H$ kills cones of the connecting morphisms $\phi_i$. Now, $H(Y_j)={\{0\}}$ for any $j\ge 0$ (by obvious induction). Next, the long exact sequence $$\dots \to H(Y_{i+1}[1]) \stackrel{H(\phi_i[1])}{\to} H(Y_i[1]) \to H(\operatorname{\operatorname{Cone}}(\phi_i)) (=0) \to H(Y_{i+1}) \to H(Y_i)\to \dots$$ gives the surjectivity of $H(\phi_i[1])$. Hence $H(Y)\cong \operatorname{\varprojlim}H(Y_i)=0$ according to Lemma \[lcoulim\](1,2). \[isescperp\]. Once again, it suffices to verify the first part of the assertion. Similarly to the previous argument the result easily follows from Lemma \[lcoulim\](3). \[izs\]. The first part of the assertion is immediate from the previous assertions. To deduce the “in particular” part we note that ${\mathcal{LO}}$ is precisely the zero class of the cp functors $\{H_N\}$ for $N$ running through ${\mathcal{RO}}$ and $H_N={\underline{C}}(-,N)$. \[ilwd\]. We can construct $LY_i$ and $l_i$ satisfying the conditions in question expect the last (“colimit”) one inductively using Proposition \[pbw\](\[iwdext\]). Now we consider the commutative square $$\begin{CD} \coprod LY_i@>{La}>>\coprod LY_i\\ @VV{\coprod a_{Y_i}}V@VV{\coprod a_{Y_i}}V \\ D@>{a}>>D \end{CD}$$ where $La=\oplus {\operatorname{id}}_{LY_i}\bigoplus \oplus (-l_i): \coprod LY_i\to \coprod LY_i$ is the morphism corresponding to $\operatorname{\varinjlim}LY_i$, and the remaining notation is from Definition \[dcoulim\]. According to Proposition 1.1.11 of [@bbd], we can complete it to a commutative diagram $$\label{ely}\begin{CD} \coprod LY_i @>{La}>>\coprod LY_i@>{}>> LY@ >{}>>\coprod LY_i[1] \\ @VV{\coprod a_{Y_i}}V@VV{\coprod a_{Y_i}}V @VV{}V@VV{\coprod a_{Y_i}[1]}V\\ D @>{a}>>D@>{}>> Y@>{}>> D[1]\\ @VV{}V @VV{}V @VV{}V@VV{}V \\ \coprod RY_i @>{}>>\coprod RY_i@>{}>> RY @>{}>> \coprod RY_i[1] \\ \end{CD}$$ whose rows and columns are distinguished triangles. Then $LY$ is a homotopy colimit of $LY_i$ (with respect to $l_i$) by definition. Since $LY_0$ and cones of $l_i$ belong to ${\underline{C}}_{w\le 0}$, we also have $LY\in {\underline{C}}_{w\le 0}$ according to assertion \[izs\]. On the other hand, the bottom row of (\[ely\]) gives $RY\in {\underline{C}}_{w\ge 1}$ (since $\coprod RY_i\in {\underline{C}}_{w\ge 1}$). Thus the third column of our diagram is a weight decomposition of $Y$ of the type desired. A classification of compactly generated torsion pairs {#scghop} ----------------------------------------------------- Now we generalize (and extend) Theorem 3.7 of [@postov] to arbitrary triangulated categories that have coproducts.[^45] \[tclass\] Assume that ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ is a set of compact objects (recall that ${\underline{C}}$ has coproducts). Then the following statements are valid. 1. \[iclass1\] The strong extension-closure ${\mathcal{LO}}$ of ${\mathcal{P}}$ and ${\mathcal{RO}}={\mathcal{P}}^\perp$ give a smashing torsion pair $s$ for ${\underline{C}}$ (so, $s$ is the torsion pair generated by ${\mathcal{P}}$). Moreover, ${\mathcal{LO}}$ equals the big hull of ${\mathcal{P}}$, and for any $M\in {\operatorname{Obj}}{\underline{C}}$ there exists a choice of $L_sM$ (see Remark \[rwsts\](3)) belonging to the naive big hull of ${\mathcal{P}}$. 2. \[iclass2\] The class of compact objects in ${\mathcal{LO}}$ equals the ${\underline{C}}$-envelope of ${\mathcal{P}}$ (see §\[snotata\]). 3. \[iclassts\] The correspondence sending a compactly generated torsion pair $s=({\mathcal{LO}},{\mathcal{RO}})$ for ${\underline{C}}$ into ${\mathcal{LO}}\cap{\underline{C}}^{{{\aleph_0}}}$, where ${\underline{C}}^{{{\aleph_0}}}$ is the class of compact objects of ${\underline{C}}$, gives a one-to-one correspondence between the following classes: the class of compactly generated torsion pairs for ${\underline{C}}$ and the class of essentially small Karoubi-closed extension-closed subclasses of ${\underline{C}}^{{{\aleph_0}}}$.[^46] 4. \[iclasst\] If the torsion pair $s$ generated by ${\mathcal{P}}$ is associated to a $t$-structure then the class ${\mathcal{LO}}$ ($={\underline{C}}^{t\le 0}$) equals the naive big hull of ${\mathcal{P}}$. 5. \[iclass5\] Let $H$ be a cp (resp. a cc) functor from ${\underline{C}}$ into an AB4\* (resp. AB5) category ${\underline{A}}$ whose restriction to ${\mathcal{P}}$ is zero. Then $H$ kills all elements of ${\mathcal{LO}}$ also. <!-- --> 1. If $s$ is a torsion pair indeed then it is smashing according to Proposition \[phopft\](III). Since ${\mathcal{P}}\perp {\mathcal{RO}}$, for any $N\in {\mathcal{RO}}$ the cp functor $H_N$ (from ${\underline{C}}$ into ${\underline{\operatorname{Ab}}}$) also kills ${\mathcal{LO}}$ according to Lemma \[lbes\](\[isescperp\]). Hence ${\mathcal{LO}}\perp{\mathcal{RO}}$.[^47] Since ${\mathcal{LO}}$ is Karoubi-closed by definition, Proposition \[phop\](9) (along with Lemma \[lbes\](\[iseses\])) reduces the assertion to the existence for any $M\in {\operatorname{Obj}}{\underline{C}}$ of an $s$-decomposition such that the corresponding $L_sM$ belongs to the naive big hull of ${\mathcal{P}}$. We apply (a certain modification of) the method used in the proof of [@bws Theorem 4.5.2(I)]. We construct a certain sequence of $M_k\in {\operatorname{Obj}}{\underline{C}}$ for $k\ge 0$ by induction in $k$ starting from $M_0=M$. Assume that $M_k$ (for some $k\ge 0$) is constructed; then we take $P_k=\coprod_{(P,f):\,P\in {\mathcal{P}},f\in {\underline{C}}(P,M_k)}P$; $M_{k+1}$ is a cone of the morphism $\coprod_{(P,f):\,P\in {\mathcal{P}},f\in {\underline{C}}(P,M_k)}f:P_k\to M_k$. Now we ’assemble’ $P_k$. The compositions of the morphisms $h_k:M_{k}\to M_{k+1}$ given by this construction yields morphisms $g_i:M\to M_i$ for all $i\ge 0$. Besides, the octahedral axiom of triangulated categories immediately yields $\operatorname{\operatorname{Cone}}(h_k)\cong P_k[1]$. Now we complete $g_k$ to distinguished triangles $$\label{etrproof} L_k\stackrel{b_k}{\to}M \stackrel{g_k}{\to}M_k\stackrel{f_k}{\to} L_k[1].$$ Certainly, $L_0=0$ and the octahedral axiom yields the existence of morphisms $s_i:L_i\to L_{i+1}$ that are compatible with $b_k$ such that $\operatorname{\operatorname{Cone}}(s_i)\cong P_i$ for all $i\ge 0$. We consider $L=\operatorname{\varinjlim}L_k$ and choose a morphism $b: L\to M$ compatible with the morphism system $(b_k)$ (see Remark \[rcoulim\](5)). We complete $b$ to a distinguished triangle $L\stackrel{b}{\to} M\stackrel{a}{\to} R\stackrel{f}{\to} L[1]$. This triangle will be our candidate for an $s$-decomposition of $M$. First we note that $L_0=0$; since $\operatorname{\operatorname{Cone}}(s_i)\cong P_i$ we have $L\in {\mathcal{LO}}$ by the definition of the latter. It remains to prove that $R\in {\mathcal{RO}}$, i.e., that ${\mathcal{P}}\perp R$. For an element $P $ of ${\mathcal{P}}$ we should check that ${\underline{C}}(P,R)={\{0\}}$. The long exact sequence $$\dots \to {\underline{C}}(P,L)\to {\underline{C}}(P,M)\to {\underline{C}}(P,R)\to {\underline{C}}(P, L[1])\to {\underline{C}}(P,M[1])\to\dots$$ translates this into the following assertion: $H^P(b)$ is surjective and $H^P(b[1])$ is injective. Now, by Lemma \[lcoulim\](3(ii)) we have ${\underline{C}}(P,L)\cong\operatorname{\varinjlim}{\underline{C}}(P,L_i)$ and ${\underline{C}}(P,L[1])\cong\operatorname{\varinjlim}{\underline{C}}(P,L_i[1])$. Hence the long exact sequences $$\dots \to H^P(L_k)\to H^P(M)\to H^P(M_k)\to H^P(L_k[1])\to H^P(M[1])\to\dots$$ yield: it suffices to verify that $\operatorname{\varinjlim}{\underline{C}}(P,M_k) ={\{0\}}$ (note here that $h_k$ are compatible with $s_k$). Lastly, ${\underline{C}}(P,P_k)$ surjects onto ${\underline{C}}(P,M_k)$; hence the group ${\underline{C}}(P,M_k)$ dies in ${\underline{C}}(P,M_{k+1})$ for any $k\ge 0$ and we obtain the result. 2. Given the previous assertion, the argument used in the proof of [@postov Theorem 3.7(ii)] goes through without any changes. We will describe another proof of our statement (that does not depend on ibid.) in Remark \[rnz\](1) below. 3. Recall that for any set $D\subset {\operatorname{Obj}}{\underline{C}}$ the smallest strict (full) triangulated subcategory of ${\underline{C}}$ containing $D$ is essentially small by Lemma 3.2.4 of [@neebook]; hence ${\langle}D{\rangle}_{{\underline{C}}}$ (see §\[snotata\] for the notation) is essentially small also (cf. Proposition 3.2.5 of ibid.). Thus for a set ${\mathcal{P}}$ of compact objects of ${\underline{C}}$ its ${\underline{C}}$-envelope ${\mathcal{P}}'$ is essentially small; its elements are compact according to Lemma 4.1.4 of [@neebook]. Since ${\mathcal{P}}'{{}^{\perp}}={\mathcal{P}}{{}^{\perp}}$ (see Proposition \[phop\](1)), the torsion pair $s$ given by assertion \[iclass1\] is also generated by ${\mathcal{P}}'$; hence it suffices to note that ${\mathcal{LO}}\cap {\underline{C}}^{{{\aleph_0}}}={\mathcal{P}}'$ according to assertion \[iclass2\]. 4. Recall from assertion \[iclass1\] that for any $M\in {\operatorname{Obj}}{\underline{C}}$ there exists a choice of $L_sM$ that belongs to the naive big hull of ${\mathcal{P}}$. Now, if $s$ is associated to a $t$-structure and $M\in {\mathcal{LO}}={\underline{C}}_{t\le 0}$ then we certainly have $L_sM=M$ (see Remark \[rtst1\](\[it1\],\[it3\])); this concludes the proof. \[iclass5\]. Immediate from Lemma \[lbes\](\[isesperp\]) (resp. \[isescperp\]). \[rnewt\] 1. As a particular case of part \[iclass1\] of our theorem we obtain that for any set ${\mathcal{P}}$ (of compact objects of ${\underline{C}}$) such that ${\mathcal{P}}[-1]\subset {\mathcal{P}}$ there exists a weight structure on ${\underline{C}}$ such that ${\underline{C}}_{w\ge 1}={\mathcal{P}}^{\perp}$ (cf. Remark \[rwhop\](1); note that this statement was originally proved in [@paucomp]). Thus if ${\underline{C}}$ is compactly or perfectly generated (see Definition \[dwg\](\[idpg\]); in particular, this is certainly the case if ${\underline{C}}={\underline{C}'}$, where the latter is the localizing subcategory of ${\underline{C}}$ generated by ${\mathcal{P}}$) then Theorem \[tadjt\] implies that $({\mathcal{P}}^{\perp}[-1], ({\mathcal{P}}^{\perp})^{\perp})$ is a $t$-structure on ${\underline{C}}$. Moreover, the couple $({\mathcal{P}}^{\perp}[-1]\cap {\operatorname{Obj}}{\underline{C}'}, {\mathcal{P}}^{\perp}\cap {\operatorname{Obj}}{\underline{C}'})^{\perp_{{\underline{C}}'}})$ is a $t$-structure on ${\underline{C}}'$ regardless of any extra restrictions on ${\underline{C}}$ (here one should invoke Propositions \[phopft\](I) and \[pcomp\](II)). So, we obtain a statement on the existence of (cosmashing) $t$-structures that does not mention weight structures! This result appears to be new (unless ${\underline{C}}$ is a compactly generated algebraic triangulated category; see Remark \[rwsym\](1) and Theorem 3.11 of [@postov]). Moreover, the results of the next subsection (see Remark \[revenmorews\]) give an even vaster source of smashing weight structures (and so, of their adjacent $t$-structures as well). 2\. Recall that Theorem 3.7 and Corollary 3.8 of [@postov] give parts \[iclass1\]–\[iclassts\] of our theorem in the case where ${\underline{C}}$ is a “stable derivator” triangulated category ${\underline{C}}$. Moreover, as a consequence of part \[iclassts\] we certainly obtain a bijection between compactly generated $t$-structures (resp. weight structures) and those essentially small Karoubi-closed extension-closed subclasses of ${\underline{C}}^{{{\aleph_0}}}$ that are closed with respect to $[1]$ (resp. $[-1]$); this generalizes Theorem 4.5 of ibid. to arbitrary triangulated categories having coproducts. 3\. Part \[iclasst\] of our theorem generalizes Theorem A.9 of [@kellerw] where “stable derivator” categories were considered (similarly to the aforementioned results of [@postov]). 4\. The question whether all smashing weight structures on a given compactly generated category ${\underline{C}}$ are compactly generated is a certain weight structure version of the (generalized) telescope conjecture (that is also sometimes called the smashing conjecture) for ${\underline{C}}$; this question generalizes its “usual” stable version (see Proposition \[prtst\](\[it4sm\])). As we have noted in Remark \[rnondeg\](3), the main result of [@kellerema] demonstrates that the answer to the shift-stable version of the question is negative for a general ${\underline{C}}$; hence this is only more so for our weight structure version. On the other hand, the answer to our question for ${\underline{C}}=SH$ (the topological stable homotopy category) is not clear. 5\. The description of compact objects in ${\mathcal{LO}}$ provided by part \[iclass2\] of our theorem is important for the continuity arguments in [@bcons]. On perfectly generated weight structures and symmetrically generated $t$-structures {#sperfws} ----------------------------------------------------------------------------------- Now we prove that an arbitrary cosuspended countably perfect set ${\mathcal{P}}$ gives a weight structure; this is an interesting modification of Theorem \[tclass\](\[iclass1\]) (note that any class of compact objects is perfect). \[tpgws\] Let ${\mathcal{P}}$ be a countably perfect (see Definition \[dwg\]) cosuspended (i.e., ${\mathcal{P}}[-1]\subset {\mathcal{P}}$) set of objects of ${\underline{C}}$. Then the strong extension-closure ${\mathcal{LO}}$ of ${\mathcal{P}}$ and ${\mathcal{RO}}={\mathcal{P}}^\perp$ give a weighty torsion pair for ${\underline{C}}$ (i.e., $({\mathcal{LO}},{\mathcal{RO}}[-1])$ is a weight structure). Moreover, ${\mathcal{LO}}$ equals the big hull of ${\mathcal{P}}$. As we have noted in Remark \[rwhop\](1), $s$ is weighty whenever $s$ is a torsion pair and ${\mathcal{LO}}\subset {\mathcal{LO}}[1]$; the latter certainly follows from ${\mathcal{P}}[-1]\subset {\mathcal{P}}$. So, we should prove the remaining assertions. Repeating the beginning of the proof of Theorem \[tclass\](\[iclass1\]), we reduce them to the existence for any $M\in {\operatorname{Obj}}{\underline{C}}$ an $s$-decomposition $L_sM\to M\to R_sM\to L_sM[1]$ with $L_sM$ belonging to the naive big hull of ${\mathcal{P}}$. For this purpose we construct a distinguished triangle $L\stackrel{b}{\to} M\stackrel{a}{\to} R\stackrel{f}{\to} L[1]$ using the method described in the proof of Theorem \[tclass\]; so, $L$ belongs to the naive big hull of ${\mathcal{P}}$ by construction. To finish the proof we should check that $R\in {\mathcal{RO}}$; for this purpose we “mix” the proof of Theorem \[tclass\](\[iclass1\]) with that of [@kraucoh Theorem A]; cf. also Remark \[rigid\](1) below. The idea is to replace the collection of functors $H^P$ for $P\in {\mathcal{P}}$ with a single “more complicated” functor (that would be a wcc one in contrast with the functors $H^P$ in the case of “general” $P$). We will write ${\underline{\coprod}{\mathcal{P}}}$ for the full subcategory of ${\underline{C}}$ formed by the $\coprod$-closure of ${\mathcal{P}}$; following [@kraucoh] (see also [@neebook Definition 5.1.3] and [@auscoh]) we consider the full subcategory ${\operatorname{Coh}_{{\mathcal{P}}}}\subset {\operatorname{PShv}}^{{\mathbb{Z}}}(\underline{\coprod}{\mathcal{P}})=\operatorname{\operatorname{AddFun}}(({\underline{\coprod}{\mathcal{P}}})^{op},{\underline{\operatorname{Ab}}})$ (cf. Remark \[rdetect\]; we will omit the index ${{\mathbb{Z}}}$ in this notation below) of [*coherent functors*]{}. We recall (see [@kraucoh]) that a functor $H\in {\operatorname{Obj}}{\operatorname{PShv}}({\underline{\coprod}{\mathcal{P}}})$ is said to be coherent whenever there exists a ${\operatorname{PShv}}({\underline{\coprod}{\mathcal{P}}})$-short exact sequence ${\underline{\coprod}{\mathcal{P}}}(-,X)\to {\underline{\coprod}{\mathcal{P}}}(-,Y)\to H\to 0$, where $X$ and $Y$ are some objects of ${\underline{\coprod}{\mathcal{P}}}$ (note that this is a projective resolution of $H$ in ${\operatorname{PShv}}({\underline{\coprod}{\mathcal{P}}})$; see [@neebook Lemma 5.1.2]). According to [@kraucoh Lemma 2], the category ${\operatorname{Coh}_{{\mathcal{P}}}}$ is abelian; it has coproducts according to Lemma 1 of ibid. Since any morphisms of (coherent) functors is compatible with some morphism of their (arbitrary) projective resolutions, a ${\operatorname{Coh}_{{\mathcal{P}}}}$-morphism is zero (resp. surjective) if and only if it is surjective in ${\operatorname{PShv}}({\underline{\coprod}{\mathcal{P}}})$. Next, the Yoneda correspondence ${\underline{C}}\to {\operatorname{PShv}}({\underline{\coprod}{\mathcal{P}}})$ (sending $M\in {\operatorname{Obj}}{\underline{C}}$ to the restriction of ${\underline{C}}(-,M)$ to ${\underline{\coprod}{\mathcal{P}}}$) gives a homological functor $H^{{\mathcal{P}}}:{\underline{C}}\to {\operatorname{Coh}_{{\mathcal{P}}}}$ (see Lemma 3 of ibid.). $H^{{\mathcal{P}}}$ is a wcc functor since ${\mathcal{P}}$ is countably perfect (according to that lemma); it also respects arbitrary ${{\underline{\coprod}{\mathcal{P}}}}$-coproducts (very easy; see Lemma 1 of ibid.). Lastly, our discussion of zero and surjective ${\operatorname{Coh}_{{\mathcal{P}}}}$-morphisms certainly yields that $H^{{\mathcal{P}}}(h)$ is zero (resp. surjective) for $h$ being a ${\underline{C}}$-morphism if and only if ${\underline{C}}(N,-)(h)=0$ (resp. surjective) for any $N\in {\operatorname{Obj}}{{\underline{\coprod}{\mathcal{P}}}}$; it is certainly suffices to take $N\in {\mathcal{P}}$ in these “criteria” only. Now we prove that $R\in {\mathcal{RO}}$ using the notation introduced in the proof of Theorem \[tclass\](\[iclass1\]) (see (\[etrproof\])). As we have just proved, $R\in {\mathcal{RO}}$ whenever $H^{{\mathcal{P}}}(R)=0$. Hence the long exact sequence $$\to H^{{\mathcal{P}}}(L)\stackrel{H^{{\mathcal{P}}}(b)} \to H^{{\mathcal{P}}}(M) \to H^{{\mathcal{P}}}(R[1])\to H^{{\mathcal{P}}}(L[1])\stackrel{H^{{\mathcal{P}}}(b[1])} \to H^{{\mathcal{P}}}(M[1])\to$$ reduces the assertion to the surjectivity of $H^{{\mathcal{P}}}(b)$ along with the injectivity of $H^{{\mathcal{P}}}(b[1])$. Next, the vanishing of ${\underline{C}}(P,-)(h_k)$ for all $k\ge 0$ and $P\in {\mathcal{P}}$ implies that the morphisms $H^{{\mathcal{P}}}(h_k)$ are zero also. Thus the corresponding argument used in the proof of Theorem \[tclass\](\[iclass1\]) would carry over to our setting (to yield the assertion) if we knew that $H^{{\mathcal{P}}}(L[1])\cong \operatorname{\varinjlim}H^{{\mathcal{P}}}(L_k[1])$ and $H^{{\mathcal{P}}}(L)$ surjects onto $ \operatorname{\varinjlim}H^{{\mathcal{P}}}(L_k)$.[^48] Now, the surjectivity in question follows immediately from Lemma \[lcoulim\](3). To prove the injectivity statement it suffices to verify that condition (i) of the lemma is fulfilled (for $Y_i=L_i[1]$, $H'=H^{{\mathcal{P}}}$, and some $A,A_i\in {\operatorname{Obj}}{\operatorname{Coh}_{{\mathcal{P}}}}$). Now for all $k\ge 0$ consider the following morphism of ${\operatorname{Coh}_{{\mathcal{P}}}}$-exact sequences: \[1.0\] [$\begin{CD} H^{{\mathcal{P}}}(M[1])@>{H^{{\mathcal{P}}}(g_k[1])}>> H^{{\mathcal{P}}}(M_k[1])@>{H^{{\mathcal{P}}}(f_k[1])}>>H^{{\mathcal{P}}}(L_k[2])@>{H^{{\mathcal{P}}}(b_k[2])}>>H^{{\mathcal{P}}}(M[2])@>{H^{{\mathcal{P}}}(g_k[2])}>>H^{{\mathcal{P}}}(M_k[2]) \\ @VV{=}V @VV{H^{{\mathcal{P}}}(h_k[1])}V@VV{H^{{\mathcal{P}}}(s_k[2])}V@VV{=}V @VV{H^{{\mathcal{P}}}(h_k[2])}V \\ H^{{\mathcal{P}}}(M[1])@>{H^{{\mathcal{P}}}(g_{k+1}[1])}>> H^{{\mathcal{P}}}(M_{k+1}[1])@>{H^{{\mathcal{P}}}(f_{k+1}[1])}>>H^{{\mathcal{P}}}(L_{k+1}[2])@>{H^{{\mathcal{P}}}(b_{k+1}[2])}>>H^{{\mathcal{P}}}(M[2]) @>{H^{{\mathcal{P}}}(g_{k+1}[2])}>>H^{{\mathcal{P}}}(M_{k+1}[2]) \end{CD}$]{} Note that functors $H^{{\mathcal{P}}}\circ [1]$ and $H^{{\mathcal{P}}}\circ [2]$ can be expressed in terms of restricting functors represented by objects of ${\underline{C}}$ to ${\underline{\coprod}{\mathcal{P}}}[-1]$ and ${\underline{\coprod}{\mathcal{P}}}[-2]$, respectively. Since these categories lie in ${\underline{\coprod}{\mathcal{P}}}$, we have $H^{{\mathcal{P}}}(h_k[1])=0=H^{{\mathcal{P}}}(h_k[2])$.[^49] It follows that $H^{{\mathcal{P}}}(g_{k+1}[1])=0=H^{{\mathcal{P}}}(g_{k+1}[2])$ for all $k\ge 0$. Hence for $k\ge 1$ the morphism $H^{{\mathcal{P}}}(f_{k+1}[1])\bigoplus H^{{\mathcal{P}}}(s_{k}[1])$ gives an isomorphism of $\operatorname{\operatorname{Im}}(b_k[2])\bigoplus H^{{\mathcal{P}}}(M_{k+1}[1])$ with $H^{{\mathcal{P}}}(M_{k+1}[1])$, whereas $\operatorname{\operatorname{Im}}(b_k[2])$ is isomorphic to $H^{{\mathcal{P}}}(M[2])$. Thus for $A=H^{{\mathcal{P}}}(M[2])$ and $A_i=H^{{\mathcal{P}}}(M_{i}[1])$ we have $H^{{\mathcal{P}}}(L_i[2])\cong A_i\bigoplus A$ for $i\ge 2$ and these isomorphisms are compatible with $H^{{\mathcal{P}}}(s_i[2])\cong(0:A_i\to A_{i+1}) \bigoplus {\operatorname{id}}_A$. Hence Lemma \[lcoulim\](3)(i) implies that $H^{{\mathcal{P}}}(L[1])\cong \operatorname{\varinjlim}H^{{\mathcal{P}}}(L_k[1])$. \[rigid\] 1. The author was inspired to apply coherent functors in this context by [@salorio]; yet the proof of Theorem 2.2 of ibid. (where coherent functors are applied to the construction of $t$-structures) appears to contain a gap.[^50] The author believes that applying arguments of the sort used in the proof of our theorem in the case of a “general” (countably) perfect set ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ such that ${\mathcal{P}}\subset {\mathcal{P}}[1]$ one can (only) obtain a “semi-$t$-structure” for ${\underline{C}}$, i.e., for any $M\in {\operatorname{Obj}}{\underline{C}}$ there exists a distinguished triangle $L\to M\to R\to L[1]$ such that $L$ belongs to the big hull of ${\mathcal{P}}$ and $R\in {\mathcal{P}}{{}^{\perp}}[1]$.[^51] The author wonders whether this result can be improved, and also whether semi-$t$-structures can be “useful”.[^52] Note also that in Theorem \[tsymt\] below we will prove the existence of a $t$-structure generated by a suspended ${\mathcal{P}}$ whenever ${\mathcal{P}}$ is symmetric to some set ${\mathcal{P}}'\subset {\operatorname{Obj}}{\underline{C}}$; however, the author does not know whether ${\underline{C}}^{t\le 0}$ equals the big hull of ${\mathcal{P}}$ whenever ${\mathcal{P}}$ satisfies these conditions. 2\. The arguments from the proof of our theorem can also be used (and significantly simplified) if instead of requiring ${\mathcal{P}}$ being cosuspended (and countably perfect) we assume that ${{\underline{\coprod}{\mathcal{P}}}}$ is [*rigid*]{}, i.e., ${{\underline{\coprod}{\mathcal{P}}}}\perp {{\underline{\coprod}{\mathcal{P}}}}[1]$. Indeed, then the distinguished triangle $P_0\to M\to M_1\to P_0[1]$ (see the proof of Theorem \[tclass\](\[iclass1\])) is easily seen to be an $s$-decomposition of $M$ (and if we proceed as above then this triangle will actually be equal to $L\to M\to R\to L[1]$). Note also that ${{\underline{\coprod}{\mathcal{P}}}}$ is rigid if and only if ${\mathcal{P}}\perp {{\underline{\coprod}{\mathcal{P}}}}[1]$. Moreover, if ${\mathcal{P}}$ is perfect then these conditions are equivalent to ${\mathcal{P}}\perp {\mathcal{P}}[1]$. The author does not know whether any formulation of this sort is known. 3\. The case ${\mathcal{P}}={\mathcal{P}}[1]$ of our theorem (cf. Remark \[rtst2\]) was essentially treated in the proof of [@kraucoh Theorem A]. 4\. We will say that a weighty torsion pair and the corresponding weight structure are perfectly generated whenever they can be obtained by means of our theorem. Remark \[revenmorews\](1) below will give a “more natural” equivalent of this definition. Note also that instead of assuming that ${\mathcal{P}}$ is a set in the theorem it certainly suffices to assume that ${\mathcal{P}}$ is essentially small. Moreover, Theorem \[twgws\](III.2) states that any smashing weight structure on a well generated triangulated category is perfectly generated. 5. In Theorem \[tsymt\] and Corollary \[csymt\] below we will study examples for Theorem \[tpgws\] that are constructed using “symmetry”; this will yield some new results on $t$-structures. The idea to relate $t$-structures to symmetric sets and Brown-Comenetz duals comes from [@salorio] also; however the author doubts that one can get a “simple description” of a $t$-structure obtained using arguments of this sort (cf. Corollary 2.5 of ibid.). Now we prove a few simple definitions and statements related to countably perfect classes (without claiming much originality in these results); recall that ${\mathcal{P}}$-null morphisms (for ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$) are the ones annihilated by the functors $H^P$ for all $P\in {\mathcal{P}}$ (see Definition \[dhopo\](5)). \[dapprox\] Let ${\mathcal{P}}$ be a subclass of ${\operatorname{Obj}}{\underline{C}}$, $h\in {\underline{C}}(M,N)$ (for some $M,N\in {\operatorname{Obj}}{\underline{C}}$). 1\. We will say that $h$ is a [*${\mathcal{P}}$-epic*]{} whenever for any $P\in {\mathcal{P}}$ the homomorphism $H^P(h)$ is surjective, i.e., if any $g\in {\underline{C}}(P,N)$ for $P\in {\mathcal{P}}$ factors through $h$. We will say that $h$ is [*${\mathcal{P}}$-monic*]{} if all $H^P(h)$ are injective.[^53] 2\. We will say that $h$ is a [*${\mathcal{P}}$-approximation*]{} (of $N$) if it is a ${\mathcal{P}}$-epic and $M$ belongs to ${\operatorname{Obj}}{\underline{\coprod}{\mathcal{P}}}$. 3\. We will say that ${\mathcal{P}}$ is [*contravariantly finite* ]{} (in ${\underline{C}}$) if for any $N\in {\operatorname{Obj}}{\underline{C}}$ there exists its ${\mathcal{P}}$-approximation.[^54] \[lperf\] Let ${\mathcal{P}}$ be a subclass of ${\operatorname{Obj}}{\underline{C}}$; denote by ${{\tilde{\mathcal{P}}}}$ the Karoubi-closure of ${\underline{\coprod}{\mathcal{P}}}$ in ${\underline{C}}$. 1. \[ipercl\] If ${\mathcal{P}}'$ is a subclass of ${{\tilde{\mathcal{P}}}}$ containing ${\mathcal{P}}$ then a ${\underline{C}}$-morphism $h$ is ${\mathcal{P}}'$-null (resp. a ${\mathcal{P}}'$-epic, resp. a ${\mathcal{P}}$-approximation) if and only if it is ${\mathcal{P}}$-null (resp. a ${\mathcal{P}}$-epic, resp. a ${\mathcal{P}}$-approximation). 2. \[iperot\] In a ${\underline{C}}$-distinguished triangle $M\stackrel{h}{\to}N \stackrel{f}{\to}Q\stackrel{g}{\to} M[1]$ the morphism $h$ is a ${\mathcal{P}}$-epic if and only if $f$ is ${\mathcal{P}}$-null; this is also equivalent to $g$ being ${\mathcal{P}}$-monic. 3. \[ipereq\] The class ${\mathcal{P}}$ is (countably) perfect if and only if any (countable) coproduct of ${\mathcal{P}}$-epic morphisms is ${\mathcal{P}}$-epic. Moreover, if this is the case then any (countable) coproduct of ${\mathcal{P}}$-approximations is a ${\mathcal{P}}$-approximation. 4. \[ipertest\] If $h:M\to N$ is a ${\mathcal{P}}$-approximation of $N$ then a ${\underline{C}}$-morphism $g:N\to N'$ is ${\mathcal{P}}$-null if and only if $g\circ h=0$. 5. \[ipercrit\] Assume that for any (countable) collection of $N_i\in {\operatorname{Obj}}{\underline{C}}$ the object $\coprod N_i$ possesses a ${\mathcal{P}}$-approximation being the coproduct of some ${\mathcal{P}}$-approximations of $M_i$. Then ${\mathcal{P}}$ is (countably) perfect. 6. \[iperfcovarf\] If ${\mathcal{P}}$ is a set then it is contravariantly finite. 7. \[iperloc\] Let $F:{\underline{D}}\to {\underline{C}}$ be an exact functor that possesses a right adjoint $G$ respecting (countable) coproducts. Then for any (countably) perfect class ${\mathcal{P}}'$ of objects of ${\underline{D}}$ the class ${\mathcal{P}}=F({\mathcal{P}}')$ is (countably) perfect also. All of the assertions are rather easy. \[ipercl\], \[iperot\]: obvious. Assertion \[ipereq\] follows from assertion \[iperot\] immediately according to Proposition \[pcoprtriang\]. \[ipertest\]. Since $M$ is a coproduct of elements of ${\mathcal{P}}$, the composition of $h$ with any ${\mathcal{P}}$-null morphism is zero. Conversely, since any morphism from ${\mathcal{P}}$ into $N$ factors through $h$, if $g\circ h=0$ then $g$ is ${\mathcal{P}}$-null. \[ipercrit\]. Let $f_i:N_i\to P_i$ be some ${\mathcal{P}}$-null morphisms; choose ${\mathcal{P}}$-approximations $h_i:M_i\to N_i$ such that $\coprod h_i$ is a ${\mathcal{P}}$-approximation of $\coprod N_i$. Then $\coprod f_i \circ \coprod h_i=\coprod (f_i\circ h_i)=0$. Hence $\coprod f_i$ is ${\mathcal{P}}$-null according to assertion \[ipertest\]. \[iperfcovarf\]. Easy and standard: a ${\mathcal{P}}$-approximation of $M\in {\operatorname{Obj}}{\underline{C}}$ is given by $\coprod_{P,h_P}\stackrel{\bigoplus h_P}{\to}M$, where $P$ runs through all elements of ${\mathcal{P}}$ and $h_P$ runs through ${\underline{C}}(P,M)$. \[iperloc\]. The adjunction immediately yields that a ${\underline{C}}$-morphism $h$ is ${\mathcal{P}}$-null if and only if $G(h)$ is ${\mathcal{P}}'$-null. It remains to recall that $G$ respects (countable) coproducts. \[requivdef\] Our definition of perfect classes essentially coincides with the one used in [@modoi]. Moreover, part \[ipereq\] of the lemma gives the equivalence of our definition (\[dcomp\](\[idpc\])) of countably perfect classes to conditions (G1) and (G2) in the definition of perfect generators in [@kraucoh]; hence a category is perfectly generated in the sense of ibid. if and only if it is so in the sense of Definition \[dwg\]. Similarly, our condition \[isym\] in Definition \[dsym\] is equivalent to condition (G3) in [@kraucoh Definition 2]; hence ${\underline{C}}$ is symmetrically generated in the terms of loc.cit. whenever it has products and contains a set ${\mathcal{P}}$ that Hom-generates ${\underline{C}}$ and is symmetric to some set ${\mathcal{P}}'\subset {\operatorname{Obj}}{\underline{C}}$. Furthermore, any class that is $\aleph_1$-perfect in the sense of [@neebook Definition 3.3.1] is countably perfect. We also obtain that our definition of ${\alpha}$-well generated categories is equivalent to the one given in [@krauwg]. Moreover, recall that the latter definition is equivalent to the definition given in [@neebook] according to Theorem A of ibid. We deduce some immediate consequences from the lemma. \[cwftw\] 1. Assume that ${\mathcal{P}}$ is a (countably) perfect cosuspended set of objects of ${\underline{C}}$. Then the weight structure $w$ constructed in Theorem \[tpgws\] is (countably) smashing. 2. Assume that $\{{\mathcal{P}}_i\}$ is a set of (countably) perfect sets of objects of ${\underline{C}}$. Then the couple $w=({\underline{C}}_{w\le 0}, {\underline{C}}_{w\ge 0})$ is a (countably) smashing weight structure on ${\underline{C}}$, where ${\underline{C}}_{w\le 0}$ is the big hull of $\cup_{j\ge 0,i} {\mathcal{P}}_i[-j]$ and ${\underline{C}}_{w\ge 0}=\cap_{j\ge 1,i} ({\mathcal{P}}_i{{}^{\perp}}[-j])$. 3\. Assume that $\{w_i\}$ is a set of perfectly generated weight structures on ${\underline{C}}$, i.e., assume that there exist countably perfect sets ${\mathcal{P}}_i\subset {\operatorname{Obj}}{\underline{C}}$ that generate $w_i$ (see Remark \[rwhop\](1)). Then the couple $w=({\underline{C}}_{w\le 0}, {\underline{C}}_{w\ge 0})$ is a weight structure, where ${\underline{C}}_{w\le 0}$ is the big hull of $\cup_i {\underline{C}}_{w_i\le 0}$ and ${\underline{C}}_{w\ge 0}=\cap_i {\underline{C}}_{w_i\ge 0}$. Moreover, $w$ is perfectly generated in the sense of Remark \[rigid\](4); it is smashing whenever all $w_i$ are. 1\. Recall that we should check whether ${\underline{C}}_{w\ge 0}={\mathcal{P}}{{}^{\perp}}[-1]$ is closed with respect to small (resp. countable) ${\underline{C}}$-coproducts. Hence the statement follows immediately from Proposition \[psym\](\[isymeu\]). 2. $\cup_{j\ge 0;i} {\mathcal{P}}_i[-j]$ is a (countably) perfect set according to Proposition \[psym\](\[isymuni\]); it is certainly cosuspended. Hence $w$ is a weight structure according to Theorem \[tpgws\]. Lastly, the smashing property statements follow immediately from the previous assertion. 3\. According to the previous assertion, the couple $({\underline{C}}_{w'\le 0}, {\underline{C}}_{w'\ge 0})$ is a weight structure on ${\underline{C}}$, where ${\underline{C}}_{w'\le 0}$ is the big hull of $\cup_{j\ge 0,i} {\mathcal{P}}_i[-j]$ and ${\underline{C}}_{w'\ge 0}=\cap_{j\ge 1;i} ({\mathcal{P}}_i{{}^{\perp}}[-j])$. Now we compare $w$ with $w'$. Since ${\mathcal{P}}_i$ generate $w_i$, we certainly have ${\underline{C}}_{w\ge 0}={\underline{C}}_{w'\ge 0}$. Next, ${\underline{C}}_{w\le 0}\perp {\underline{C}}_{w\ge 0}[1]$ according to Lemma \[lbes\](\[isesperp\]). Since ${\underline{C}}_{w\le 0}$ contains ${\underline{C}}_{w'\le 0}$, these classes are equal. Thus $w$ is a perfectly generated weight structure. It is smashing if all $w_i$ are; indeed, ${\underline{C}}_{w\ge 0}$ is coproductive as being the intersection of coproductive classes. \[revenmorews\] 1. In particular, we obtain that $w$ is perfectly generated in the sense of Remark \[rigid\](4) if and only if it is generated by a countably perfect set (i.e., if ${\mathcal{RO}}={\underline{C}}_{w\le -1}$ equals ${\mathcal{P}}^\perp$ for some countably perfect set ${\mathcal{P}}$). Note here that we do not have to assume that ${\mathcal{P}}$ is cosuspended, since $\cup_{i\le 0}{\mathcal{P}}[i]$ generates $w$ whenever ${\mathcal{P}}$ does. Moreover, part 3 of the corollary gives a certain “join” operation on perfectly generated $t$-structures (and so, we obtain a monoid). Note also that the join of any set of smashing weight structures is smashing also. 2\. Thus our corollary gives a vast source of smashing weight structures. Now, the results of §\[sadjt\] allow to construct “new” $t$-structures that are right adjacent to these weight structures (cf. Remark \[rnewt\](1)) and also describe their hearts. Note also that for $t_i$ being right adjacent to (perfectly generated) $w_i$ (for $i\in I$) and the corresponding $w$ and $t$ we obviously have ${\underline{C}}^{t\ge 0}=\cap_{i\in I} {\underline{C}}^{t_i\ge 0}$, whereas ${\underline{C}}^{t\le 0}$ is the big hull of $\cup_{i\in I} {\underline{C}}^{t_i\le 0}$. Now we prove that [**suspended**]{} symmetric sets generate $t$-structures. \[tsymt\] Assume that ${\underline{C}}$ (also) has products; for a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ assume that ${\mathcal{P}}$ is [*suspended*]{} (i.e., ${\mathcal{P}}[1]\subset {\mathcal{P}}$) and ${\mathcal{P}}$ is symmetric (see Definition \[dsym\](\[isym\])) to a [**set**]{} ${\mathcal{P}}'$ (of objects of ${\underline{C}}$). Then the following statements are valid. 1\. There exists a $t$-structure $t$ on ${\underline{C}}$ such that ${\underline{C}}^{t\ge 1}={\mathcal{P}}{{}^{\perp}}$ (i.e., $t$ is generated by ${\mathcal{P}}$) and also a cosmashing weight structure $w$ that is right adjacent to $t$. 2\. ${{\underline{Ht}}}$ has an injective cogenerator and satisfies the AB3\* axiom. Moreover, ${{\underline{Ht}}}$ is naturally anti-equivalent to the subcategory of $\operatorname{\operatorname{AddFun}}({{\underline{Hw}}},{\underline{\operatorname{Ab}}})$ consisting of those functors that respect products; ${{\underline{Hw}}}$ is naturally equivalent to the subcategory of injective objects of ${{\underline{Ht}}}$. 3\. If ${\mathcal{P}}'$ is suspended also then ${\underline{C}}_{w\ge 0}$ equals the big hull of ${\mathcal{P}}'$ in ${\underline{C}}{{^{op}}}$. Firstly we note that ${\mathcal{P}}$ is also symmetric to $\cup_{i\ge 0}{\mathcal{P}}'[i]$ (see Proposition \[psymb\](I.\[iws1\])); hence it suffices to consider the case where ${\mathcal{P}}'$ is suspended. Now we adopt the notation of Proposition \[psymb\](I.\[isym4\]). Assume first that ${\underline{D}}={\{0\}}$ (in the notation of the proposition), and so, ${\underline{C}}'={\underline{C}}$. According to (parts I.\[iwsym1\] and I.\[iws2\] of) the aforementioned proposition, the set ${\mathcal{P}}'$ is perfect in the category ${\underline{C}}{{^{op}}}$. Since ${\mathcal{P}}'$ is certainly cosuspended in ${\underline{C}}{{^{op}}}$, Theorem \[tpgws\] yields a smashing weight structure $w_{{\mathcal{P}}'}^{op}$ that is ${\underline{C}}{{^{op}}}$-generated by ${\mathcal{P}}'$ with ${\underline{C}}{{^{op}}}_{w^{op}\le 0}$ being the ${\underline{C}}{{^{op}}}$-big hull of ${\mathcal{P}}'$. The corresponding weight structure $w_{{\mathcal{P}}'}$ on ${\underline{C}}$ (see Proposition \[pbw\](\[idual\])) will be our candidate for $w$. $w_{{\mathcal{P}}'}$ is certainly cosmashing; hence ${{\underline{Hw}}}_{{\mathcal{P}}'}$ is Karoubian (easy from Remark \[rcoulim\](4)) and closed with respect to ${\underline{C}}$-products. Moreover, ${{\underline{Hw}}}_{{\mathcal{P}}'}$ has a [*cogenerator*]{}, i.e., any its object is a retract of a product of (copies of) some $M\in {\underline{C}}_{{{\underline{Hw}}}_{{\mathcal{P}}'}=0}$; here we apply (the dual to) Corollary \[cvttbrown\](I.2). Next we apply Corollary \[cdualt\] to obtain that there exists a $t$-structure $t_{{\mathcal{P}}'}$ on ${\underline{C}}$ that is left adjacent to $w_{{\mathcal{P}}'}$. Moreover, ${{\underline{Ht}}}_{{\mathcal{P}}'}$ is anti-equivalent to the subcategory of $\operatorname{\operatorname{AddFun}}({{\underline{Hw}}}_{{\mathcal{P}}'},{\underline{\operatorname{Ab}}})$ consisting of those functors that respect products (according to the corollary); ${{\underline{Hw}}}_{{\mathcal{P}}'}$ is equivalent to the subcategory of injective objects of ${{\underline{Ht}}}_{{\mathcal{P}}'}$. Hence ${{\underline{Ht}}}_{{\mathcal{P}}'}$ has an injective cogenerator; it satisfies the AB3\* axiom since ${{\underline{Hw}}}_{{\mathcal{P}}'}$ has products. So to finish the proof in this case it remains to note that $t_{{\mathcal{P}}'}$ is precisely the $t$-structure generated by ${\mathcal{P}}$ according to Proposition \[pwsym\](\[iws4\]). Now we proceed to the general case of our setting (using Proposition \[psymb\](I.\[isym4\]) also). We recall that the corresponding category ${\underline{D}}'$ is closed with respect to ${\underline{C}}$-products and ${\underline{D}}'^{op}$ is perfectly generated by ${\mathcal{P}}'$. Thus (by Theorem \[tpgws\]) there exists a weight structure $w_{{\underline{D}}'}$ on ${\underline{D}}'$ with ${\underline{D}}'_{w_{{\underline{D}}'}\le 0}$ equal to the ${\underline{D}}'^{op}$-big hull of ${\mathcal{P}}'$ and ${\underline{D}}'_{w_{{\underline{D}}'}\ge 0}=({}^{\perp_{{\underline{D}}'}} {\mathcal{P}}')[1]$. Once again, we apply Corollary \[cdualt\] to obtain a $t$-structure $t_{{\underline{D}}'}$ on ${\underline{D}}$ that is left adjacent to $w_{{\underline{D}}'}$. We also obtain that ${{\underline{Ht}}}_{{\underline{D}}'}$ and ${{\underline{Hw}}}_{{\underline{D}}'}$ are related similarly to assertion 2. Now we “extend” $t_{{\underline{D}}'}$ and ${{\underline{Hw}}}_{{\underline{D}}'}$ to ${\underline{C}}$. Recall that the embedding ${\underline{D}}'\to {\underline{C}}$ respects products and possesses a left adjoint; hence (the dual to) Remark \[rwhop\](2) gives the existence of a weight structure $w_{{\mathcal{P}}'}$ as above. Next, we consider the equivalence $j:{\underline{D}}'\to {\underline{C}}'$ induced by the functor $G$ (that is right adjoint to $i:{\underline{C}'}\to{\underline{C}}$) and denote by $t_{{\underline{C}'}}$ the $t$-structure on ${\underline{C}'}$ obtained from $t_{{\underline{D}}'}$ via $j$. Since the embedding $i:{\underline{C}'}\to {\underline{C}}$ possesses an (exact) right adjoint, we obtain (using Proposition \[phopft\](II.1)) that $({\underline{C}'}^{t_{{\underline{C}'}}\le 0},({\underline{C}'}^{t_{{\underline{C}'}}\le 0}){{}^{\perp}}[1])$ is a $t$-structure $t$ on ${\underline{C}}$. Note also that ${{\underline{Ht}}}$ equals $j({{\underline{Ht}}}_{{\underline{D}}'})$, ${{\underline{Hw}}}_{{\mathcal{P}}'}={{\underline{Hw}}}_{{\underline{D}}'}$, and ${\underline{C}}_{w_{{\mathcal{P}}'}\ge 0}={\underline{D}}'_{w_{{\underline{D}}'}\le 0}$; hence assertions 2 and 3 follow from assertion 1, and to prove the latter it suffices to verify that $({\underline{C}'}^{t_{{\underline{C}'}}\le 0}){{}^{\perp}}={\mathcal{P}}{{}^{\perp}}$. Since $i$ possesses a right adjoint, for the latter purpose one should compare ${\mathcal{P}}{{}^{\perp}}\cap {\operatorname{Obj}}{\underline{C}'}$ with $({\underline{C}'}^{t_{{\underline{C}'}}\le 0}){{}^{\perp}}\cap {\operatorname{Obj}}{\underline{C}'}={\underline{C}'}^{t_{{\underline{C}'}}\ge 1}=j({\mathcal{P}}')^{\perp_{{\underline{C}'}}}$ (here we apply Remark \[rwhop\](2)). It remains to recall that ${\mathcal{P}}$ is symmetric to $j({\mathcal{P}}')$ in ${\underline{C}'}$ and apply Proposition \[pwsym\](\[iws4\]) once again. To demonstrate the relevance of our theorem, we apply it to the study of compactly generated $t$-structures. \[csymt\] Let ${\mathcal{Q}}$ be a set of compact objects of ${\underline{C}}$. 1\. Then $t=({\underline{C}}^{t\le 0},{\underline{C}}^{t\ge 0})$ is a smashing $t$-structure on ${\underline{C}}$, where ${\underline{C}}^{t\le 0}$ is the smallest coproductive extension-closed subclass of ${\operatorname{Obj}}{\underline{C}}$ containing $\cup_{i\ge 0}{\mathcal{Q}}[i]$ and ${\underline{C}}^{t\ge 0}=\cap_{i\ge 1} {\mathcal{Q}}{{}^{\perp}}[i]$.[^55] Moreover, ${{\underline{Ht}}}$ has an injective cogenerator and satisfies the AB3\* axiom; it is anti-equivalent to the subcategory of $\operatorname{\operatorname{AddFun}}(({\operatorname{Inj}{{\underline{Ht}}}}){{^{op}}},{\underline{\operatorname{Ab}}})$ consisting of those functors that send ${\operatorname{Inj}{{\underline{Ht}}}}$-products into products of abelian groups.[^56] 2. Assume in addition that ${\underline{C}}$ satisfies the Brown representability condition. Then there exists a weight structure $w$ that is right adjacent to $t$ with ${{\underline{Hw}}}\cong {\operatorname{Inj}{{\underline{Ht}}}}$, and ${\underline{C}}_{w\ge 0}$ equals the ${\underline{C}}{{^{op}}}$-big hull of $\{\hat{Q}[i]_{{\underline{C}}}:\ Q\in {\mathcal{Q}}, i\ge 0\}$, where $\hat{Q}$ is the Brown-Comenetz dual of $Q$ (that represents the functor $M\mapsto {\underline{\operatorname{Ab}}}({\underline{C}}(Q,M),{{\mathbb{Q}}}/{{\mathbb{Z}}})$).[^57] 3\. Assume that the $t$-structure $t$ mentioned in assertion 1 is non-degenerate. Then ${{\underline{Ht}}}$ is Grothendieck abelian. 1\. $t$ is a $t$-structure on ${\underline{C}}$ according to Theorem A.1. of [@talosa]; it is certainly smashing. Next we take ${\underline{C}}'$ being the localizing subcategory of ${\underline{C}}$ generated by ${\mathcal{Q}}$. Then ${\mathcal{Q}}$ is symmetric to the set ${\mathcal{Q}}'_{{\underline{C}'}}$ of ${\underline{C}}'$-Brown-Comenetz duals of elements of ${\mathcal{Q}}$ according to Proposition \[psymb\](II.\[iws23\]). Hence we can apply Theorem \[tsymt\] to the category ${\underline{C}}'$ with the corresponding ${\mathcal{P}}$ and ${\mathcal{P}}'$ being equal to $\cup_{i\ge 0}{\mathcal{Q}}[i]$ and to $\cup_{i\ge 0}{\mathcal{Q}}'_{{\underline{C}'}}[i]$, respectively. This yields the result since for the corresponding $t_{{\underline{C}}'}$ we have ${\underline{C}}^{t\le 0}={\underline{C}}'^{t_{{\underline{C}}'}\le 0}$ and ${\underline{C}}^{t\ge 0}\cap {\operatorname{Obj}}{\underline{C}}'={\underline{C}}'^{t_{{\underline{C}}'}\ge 0}$ (here we note that the embedding ${\underline{C}}'\to {\underline{C}}$ has a right adjoint according to Proposition \[pcomp\](II.2), and apply Remark \[rwhop\](2)). 2\. Once again, it suffices to combine Proposition \[psymb\](II.\[iws23\]) with Theorem \[tsymt\]. 3\. Since $t$ is non-degenerate, the set ${\mathcal{Q}}$ Hom-generates ${\underline{C}}$. Hence the previous assertion gives the existence of $w$ that is right adjacent to $t$. Now to prove the result it suffices to repeat the argument used in the proof of [@humavit Corollary 4.9]. We do so briefly here (without recalling the corresponding definitions). Firstly, the corresponding shifts of the classes ${\underline{C}}^{t\le 0}$, ${\underline{C}}^{t\ge 0}={\underline{C}}_{w\le 0}$, and ${\underline{C}}_{w\ge 0}$ give a [*cosuspended TTF triple*]{}; see Definition 2.3 of ibid.[^58] Next, ${\underline{C}}_{w\le 0}$ is definable in the sense of Definition 4.1 (since it is the zero class of the set $\{{\underline{C}}(Q[i],-),\ Q\in {\mathcal{Q}},\ i>0\}$ of coherent functors; see Lemma \[lbes\](\[izs\]) for the definition of zero classes). Applying Theorem 4.8 of ibid. we obtain that ${\underline{C}}^{t\le 0}={{}^{\perp}}\{I[j],\ j< 0\}$ for some [*pure-injective cosilting object*]{} $I$ of ${\underline{C}}$ (this is where we use the non-degeneracy assumption on $t$). Thus it remains to apply Theorem 3.6 of ibid. \[rsymt\] 1. \[irsymt2\] The author does not know how to deduce the existence of $w$ (in part 2 of our corollary) from the results of §\[sadjw\]. Another interesting fact related to this statement is Theorem 3.11 of [@postov] (where algebraic triangulated categories were considered); cf. Remark \[rwsym\](1) above.[^59] So, we generalize “the $t$-structure case” of loc. cit.; this immediately yields the corresponding generalization of [@humavit Corollary 4.9] (in part 3 of our corollary). Recall also that ${\underline{C}}^{t\le 0}={{}^{\perp}}\{I[j],\ j<0\}$ for some pure-injective cosilting object $I$ under the assumptions of our corollary (see its proof). It is easily seen that any cogenerator of ${{\underline{Hw}}}$ (see the proof of Theorem \[tsymt\]) can be taken for $I$ in this statement. 2. \[irsymtab5\] In Theorem \[tab5\] below we prove under certain restrictions on ${\underline{C}}$ (and without assuming that $t$ is non-degenerate) that ${{\underline{Ht}}}$ is an AB5 category; this argument is completely independent from ibid. (and gives some interesting additional information on ${{\underline{Ht}}}$). On the other hand, the generators of ${{\underline{Ht}}}$ given by part 1 of that theorem are the same as the ones given by the proof of [@humavit Theorem 3.6] (yet they were not specified explicitly in loc. cit.). 3. \[irsymt1\] It appears to be quite difficult to produce perfect and symmetric sets “out of nothing”; note that the existing literature on this subject mostly concentrates on the search of shift-stable sets of perfect generators of triangulated categories. So, the weight structures opposite to those given by Corollary \[csymt\](2) appear to be (essentially) the only known “type” of perfectly generated weight structures that are not compactly generated (yet cf. Remark \[rnondeg\](3)). Let us prove that the corresponding weight structure $w^{op}$ is not compactly generated if the set ${\mathcal{Q}}$ Hom-generates ${\underline{C}}$ (certainly, this condition is fulfilled if and only if ${\mathcal{Q}}$ generates ${\underline{C}}$ as its own localizing subcategory). Indeed, in this case the symmetric set $\hat{Q}$ Hom-generates ${\underline{C}}^{op}$. Then Remark \[rwhop\](8) says that $w^{op}$ is left non-degenerate; thus any class generating $w^{op}$ also Hom-generates ${\underline{C}}^{op}$. On the other hand, ${\underline{C}}^{op}$ is not compactly generated according to Corollary E.1.3 (combined with Remark 6.4.5) of [@neebook]. Certainly, one can also consider the direct sum of an example of this sort with a “compactly generated” one. 4. \[irsymt4\] Probably the most interesting case of Theorem \[tsymt\] and Corollary \[csymt\] is the one where ${\mathcal{P}}$ (resp. ${\mathcal{Q}}$) Hom-generates ${\underline{C}}$; note that in this case ${\underline{C}}$ has products and satisfies the Brown representability condition automatically. Note also that ${\underline{C}}$ is Hom-generated by the corresponding set if and only if $t$ is right non-degenerate (see Definition \[dtstr\](\[ito3\]); this is certainly equivalent to the right non-degeneracy of $w$). 5. \[irsymt5\] It is easily seen that the Brown-Comenetz duals of any family $\{F_i\}$ of cc functors ${\underline{C}}\to {\underline{\operatorname{Ab}}}$ that are also pp ones form a perfect class in ${\underline{C}}{{^{op}}}$. Yet this observation can scarcely give any “new” weight structures since all “known” functors satisfying these conditions appear to be corepresented by compact objects of ${\underline{C}}$ (cf. [@krause Proposition 2.9]). Moreover, when we pass from the weight structure $w$ to its left adjacent $t$ we apply the dual Brown representability condition, whereas the latter says that all pp functors are corepresentable. On well generated weight structures and torsion pairs {#swgws} ----------------------------------------------------- Now we study the relation of (countably) perfect classes to torsion pairs and (especially) to weight structures. In particular, we obtain a complete “description” of smashing weight structures on well generated triangulated categories (see Theorem \[twgws\](III)). We will need some new definitions to deal with well generated categories. Most of them are simple variations of the notions described above; we also recall the notion of ${\beta}$-compact objects. \[dbecomp\] Let ${\beta}$ be a regular infinite cardinal. 1. \[idclass\] We will say that a class ${{\tilde{\mathcal{P}}}}$ of objects of ${\underline{C}}$ is [*${\beta}$-coproductive*]{} if it is closed with respect to ${\underline{C}}$-coproducts of less than ${\beta}$ objects. 2. \[idchop\] We will say that a torsion pair $s=({\mathcal{LO}}',{\mathcal{RO}}')$ for a full triangulated subcategory ${\underline{C}}'$ of ${\underline{C}}$ is [*${\beta}$-coproductive*]{} if both ${\operatorname{Obj}}{\underline{C}}'$ and ${\mathcal{RO}}'$ are ${\beta}$-coproductive. 3. \[idcomp\] We will say that an object $M$ of ${\underline{C}}$ is [*${\beta}$-compact*]{} if it belongs to the maximal perfect class of ${\beta}$-small objects of ${\underline{C}}$ (whose existence is given by Proposition \[psym\](\[isymuni\])). We will write ${\underline{C}}^{{\beta}}$ for the full subcategory of ${\underline{C}}$ formed by ${\beta}$-compact objects. \[rbecomp\] 1. Our definition of ${\beta}$-compact objects is equivalent to the one used in [@krauwg]. Indeed, coproducts of less than ${\beta}$ of ${\beta}$-small objects are obviously ${\beta}$-small; thus ${\operatorname{Obj}}{\underline{C}}^{\beta}$ is ${\beta}$-coproductive. Hence the equivalence of definitions follows from Lemma 4 of ibid. Furthermore, Lemma 6 of ibid. states that (both of) these definitions are equivalent to Definition 4.2.7 of [@neebook] if we assume in addition that ${\underline{C}}^{{\beta}}$ is essentially small. 2\. Now we recall some more basic properties of ${\beta}$-compact objects in an ${\alpha}$-well generated category ${\underline{C}}$ assuming that ${\beta}\ge {\alpha}$. Theorem A of [@krauwg] yields immediately that ${\underline{C}}^{\beta}$ is an essentially small triangulated subcategory of ${\underline{C}}$. Moreover, the union of ${\underline{C}}^{\gamma}$ for $\gamma$ running through all regular cardinals ($\ge {\alpha}$) equals ${\underline{C}}$ (see the Corollary in loc. cit. or Proposition 8.4.2 of [@neebook]). 3. Lastly, we recall a part of [@krauwg Lemma 4]. For any ${\beta}$-coproductive essentially small perfect class ${\mathcal{P}}$ of ${\beta}$-small objects of a triangulated category ${\underline{C}}$ (that has coproducts) it says the following: for any $P\in {\mathcal{P}}$ and any set of $N_i\in {\operatorname{Obj}}{\underline{C}}$ any morphism $P\to \coprod N_i$ factors through the coproduct of some ${\underline{C}}$-morphisms $M_i\to N_i$ with $M_i\in {\mathcal{P}}$. \[twgws\] Let $s=({\mathcal{LO}},{\mathcal{RO}})$ be a torsion pair for ${\underline{C}}$ that is countably smashing, ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$. I. Consider the class $J$ of ${\underline{C}}$-morphisms characterized by the following condition: $h\in {\underline{C}}(M,N)$ (for $M,N\in {\operatorname{Obj}}{\underline{C}}$) belongs to $J$ whenever for any chain of morphisms $ L_sP\stackrel{a_P}{\to} P\stackrel{g}{\to}M \stackrel{h}{\to}N$ its composition is zero if $P\in {\mathcal{P}}$ and $a_P$ is an $s$-decomposition morphism. Then the following statements are valid. 1. \[indep\] The class $J$ will not change if we will fix $a_P$ for any $P\in {\mathcal{P}}$ in this definition. 2. \[icontraf\] Assume that ${\mathcal{P}}$ is contravariantly finite and $s$ is smashing. Then $h$ belongs to $J$ if and only if there exists a ${\mathcal{P}}$-approximation morphism $AM\stackrel{g}{\to} M$ and an $s$-decomposition morphism $a_{AM}: L_sAM\to AM$ such that $h\circ g \circ a_{AM}=0$. Moreover, the latter is equivalent to the vanishing of all compositions of this sort. 3. \[icoprcl\] Assume that ${\mathcal{P}}$ is contravariantly finite and (countably) perfect and $s$ is smashing. Then the class $J$ is closed with respect to (countable) coproducts. 4. \[ilscp\] Assume that for any $P\in {\mathcal{P}}$ there exists a choice of $ L_sP\in {\mathcal{P}}$; denote the class of these choices by ${L_s\mathcal{P}}$. Then $J$ coincides with the class of ${L_s\mathcal{P}}$-null morphisms. 5. \[ilscper\] Assume in addition (to the previous assumption) that ${\mathcal{P}}$ is a (countably) perfect contravariantly finite class and $s$ is smashing. Then ${L_s\mathcal{P}}$ is a (countably) perfect contravariantly finite class also. 6. \[ilscperw\] Assume in addition that $s$ is weighty; assume that the class ${\mathcal{P}}$ is essentially small, contains ${\mathcal{P}}[-1]$, and Hom-generates ${\underline{C}}$. Then the class $L_s{\mathcal{P}}$ generates $s$ and ${\mathcal{LO}}$ is the big hull of $L_s{\mathcal{P}}$; thus $s$ is perfectly generated in the sense of Remark \[rigid\](4). II\. For a regular cardinal ${\beta}$ let $s'=({\mathcal{LO}}',{\mathcal{RO}}')$ be a ${\beta}$-coproductive torsion pair for a full triangulated category ${\underline{C}'}$ of ${\underline{C}}$ such that ${\operatorname{Obj}}{\underline{C}'}$ is a perfect essentially small class of ${\beta}$-small objects. Then ${\mathcal{LO}}'$ is perfect also. Moreover, if $s'$ is weighty in ${\underline{C}'}$ then ${\mathcal{LO}}'$ generates a weighty smashing torsion pair for ${\underline{C}}$. III\. Assume in addition that ${\underline{C}}$ is ${\alpha}$-well generated for some regular cardinal ${\alpha}$, and that $s$ is smashing. 1. Assume that $s$ restricts (see Definition \[dhopo\](4)) to ${\underline{C}}^{{\beta}}$ for a regular cardinal ${\beta}\ge {\alpha}$. Then ${\mathcal{LO}}\cap {\operatorname{Obj}}{\underline{C}}^{\beta}$ is an essentially small perfect class. 2. If $s$ is weighty then it restricts to ${\underline{C}}^{{\beta}}$ for all large enough regular ${\beta}\ge {\alpha}$. Moreover, the class ${\mathcal{LO}}\cap {\operatorname{Obj}}{\underline{C}}^{\beta}$ perfectly generates $s$ for these ${\beta}$. I.\[indep\]. It suffices to note that any $s$-decomposition morphism for $M$ factors through any other one according to Proposition \[phop\](7). \[icontraf\]. We fix $h$ (along with $M$ and $N$). The definition of approximations along with Proposition \[phop\](7) implies that any composition $ L_sP\stackrel{a_P}{\to} P\stackrel{g}{\to}M$ as in the definition of $J$ factors through the composition morphism $L_sAM\to M$. Hence if the composition $L_sAM\to N$ is zero then $h\in J$. Conversely, assume that $h\in J$. Since ${\mathcal{P}}$ is contravariantly finite, we can choose a ${\mathcal{P}}$-approximation morphism $g\in {\underline{C}}(AM, M)$. Present $AM$ as a coproduct of some $P_i\in {\mathcal{P}}$; choose some $s$-decomposition morphisms $L_sP_i\stackrel{a_{P_i}}\to P_i$. Since $s$ is smashing, the morphism $a_{AM}^0=\coprod a_{P_i}$ is an $s$-decomposition one also according to Proposition \[phop\](4). Since $h \circ g\circ a_{P_i}=0$ for all $i$, we also have $h \circ g\circ a_{AM}^0=0$. Lastly, any other choice of $a_{AM}$ factors through $a_{AM}^0$ (by Proposition \[phop\](7); cf. the proof of assertion I.\[indep\]); this gives the “moreover” part of our assertion. \[icoprcl\]. This is an easy consequence of the previous assertion. Indeed, to prove that $\coprod h_i\in J$ for a small (resp. countable) collection of $h_i\in J\cap {\underline{C}}(M,N)$ note that for any choices of ${\mathcal{P}}$-approximations $AM_i\to M_i$ their coproduct is a ${\mathcal{P}}$-approximation of $\coprod M_i$ (by Lemma \[lperf\](\[ipereq\])). The assertion follows easily since the coproduct of any choices of $L_sAM_i\to AM_i$ of $s$-decomposition morphisms is an $s$-decomposition morphism also (according to Proposition \[phop\](4)); thus it remains to apply assertion I.\[icontraf\]. \[ilscp\]. Assertion I.\[indep\] certainly implies that any ${L_s\mathcal{P}}$-null morphism belongs to $J$. The converse implication is immediate from ${L_s\mathcal{P}}\subset {\mathcal{P}}$. \[ilscper\]. This is an obvious combination of the previous two assertions. \[ilscperw\]. Since ${\mathcal{LO}}$ contains ${L_s\mathcal{P}}$, it also contains its big hull (see Lemma \[lbes\](\[iseses\], \[isesperp\])). Thus it suffices to verify the converse inclusion. Now, since ${\mathcal{P}}$ is essentially small, countably perfect, and ${\mathcal{P}}\subset {\mathcal{P}}[1]$, the big hull of ${\mathcal{P}}$ along with ${\mathcal{P}}^\perp$ is a (weighty) torsion pair according to Theorem \[tpgws\]. Since ${\mathcal{P}}^\perp={\{0\}}$, we obtain any object of ${\underline{C}}$ belongs to this big hull. Now let $P$ belong ${\mathcal{LO}}$. As we have just proved, it is a retract of some $P'$ belonging to the naive big hull of ${\mathcal{P}}$. So we present $P'$ as $\operatorname{\varinjlim}Y_i$ so that $Y_0$ and cones of the connecting morphisms $\phi_i$ belong to ${\underline{\coprod}{\mathcal{P}}}$. Thus for $Z_i$ being as in Lemma \[lbes\](\[ilwd\]) we can choose $LZ_i\in {\underline{\coprod} {L_s\mathcal{P}}}$. Applying the lemma we obtain the existence of an $s$-decomposition triangle $L'\to P'\stackrel{n_{P'}}{\to} R'\to L[1]$ with $L'$ belonging to the naive big hull of ${L_s\mathcal{P}}$. Now, the distinguished triangle $P\to P\to 0\to P[1]$ is an $s$-decomposition of $P$. Since ${\operatorname{id}}_P$ can be factored through $P'$, applying Proposition \[phop\](7) to the corresponding morphisms $P\to P'\to P$ we obtain that ${\operatorname{id}}_P$ can be factored through $L'$. II\. Let $f_i\in {\underline{C}}(N_i,Q_i)$ for $i\in J$ be a set of ${\mathcal{LO}}'$-null morphisms; for $N=\coprod N_i$, $f=\coprod f_i$, and $P\in {\mathcal{LO}}'$ we should check that the composition of any $e\in {\underline{C}}(P,N)$ with $f$ vanishes. The ${\beta}$-smallness of $P$ allows us to assume that $J$ contains less than ${\beta}$ elements. Next, Remark \[rbecomp\](3) gives a factorization of $e$ through the coproduct of some $h_i\in {\underline{C}}(M_i,N_i)$ with $M_i\in {\operatorname{Obj}}{\underline{C}'}$. We choose some $s'$-decompositions $L_i\to M_i\to R_i\to L_i[1]$ of $M_i$. Our assumptions easily imply that $\coprod L_i\to \coprod M_i\to \coprod R_i$ is an $s'$-decomposition of $\coprod M_i$ (cf. Proposition \[phop\](4)). Hence part 7 of the proposition implies that $e$ factors through the coproduct $g$ of the corresponding morphisms $L_i\to N_i$. Now, since $f_i$ are ${\mathcal{LO}}'$-null and $L_i\in {\mathcal{LO}}'$ then $f\circ g=0$; hence $f\circ e=0$ also. Lastly, if $s'$ is weighty then ${\mathcal{LO}}'$ is cosuspended. Since it is also essentially small it remains to apply Theorem \[tpgws\] (along with Corollary \[cwftw\]). III\. For ${\beta}\ge {\alpha}$ being a regular cardinal we take ${\mathcal{P}}= {\operatorname{Obj}}{\underline{C}}^{\beta}$. This is certainly a perfect essentially small class that Hom-generates ${\underline{C}}$; we also have ${\mathcal{P}}\subset {\mathcal{P}}[1]$. To prove assertion III.1 it suffices to note that ${\mathcal{LO}}\cap {\operatorname{Obj}}{\underline{C}}^{\beta}$ is a possible choice of $L_s{\mathcal{P}}$ (in the notation of assertion I) and apply assertion I.\[ilscper\]. Next, assertion I.\[ilscperw\] (combined with Remark \[revenmorews\](1)) implies that to prove assertion III.2 it suffices to verify that $s$ restricts to ${\underline{C}}^{{\beta}}$ for all large enough regular ${\beta}\ge {\alpha}$. Now we choose some $L_sM$ for all $M\in {\operatorname{Obj}}{\underline{C}}^{\alpha}$, and choose for ${\alpha}'$ a regular cardinal such that all elements of $L_s{\mathcal{P}}$ belong to ${\underline{C}}^{{\alpha}'}$ (see Remark \[rbecomp\](2)). Then for any regular ${\beta}\ge {{\alpha}'}$ the pair $s$ restricts to ${\underline{C}}^{{\beta}}$, since the corresponding weight decompositions exist according to the “furthermore” part of Proposition \[ppcoprws\](\[icopr7p\]). \[rtkrau\] 1. Our theorem suggests that it makes sense to define (at least) two distinct notions of ${\beta}$-well generatedness for smashing torsion pairs and weight structures in an ${\alpha}$-well generated category ${\underline{C}}$. One may say that $s$ is [*weakly ${\beta}$-well generated*]{} for some regular ${\beta}\ge {\alpha}$ if it is generated by a perfect set of ${\beta}$-compact objects. $s$ is [*strongly ${\beta}$-well generated*]{} if in addition to this condition, $s$ restricts to ${\underline{C}}^{\beta}$. Certainly, compactly generated torsion pairs (see Definition \[dhopo\](3)) are precisely the weakly ${{\aleph_0}}$-well generated ones (since any set of compact objects is perfect; cf. Proposition \[psym\](\[isymcomp\])). Hence our two notions of ${\beta}$-well generatedness are not equivalent (already) in the case ${\alpha}={\beta}={{\aleph_0}}$; this claim follows from [@postov Theorems 4.15, 5.5] (cf. also Corollary 5.6 of ibid.) where (both weakly and strongly ${{\aleph_0}}$-well generated) weight structures on ${\underline{C}}=D({\operatorname{Mod}}-R)$ were considered in detail. Moreover, for $k$ being a field of cardinality $\gamma$ the main subject of [@bgn] gives the following example: the opposite (see Proposition \[pbw\](\[idual\])) to (any version of) the Gersten weight structure over $k$ (on the category ${\underline{C}}$ that is opposite to the corresponding category of [*motivic pro-spectra*]{}; note that ${\underline{C}}$ is compactly generated) is weakly ${{\aleph_0}}$-well generated (by definition) and it does not restrict to the subcategory of ${\beta}$-compact objects for any ${\beta}\le \gamma$. On the other hand, this example is “as bad is possible” for weakly ${{\aleph_0}}$-well generated weight structures in the following sense: combining the arguments used the proof of part IV.2 of our theorem with that for Theorem \[tclass\] one can easily verify that any ${{\aleph_0}}$-well generated weight structure is ${\alpha}$-well generated whenever the set of (all) isomorphism classes of morphisms in the subcategory ${\underline{C}}^{{\aleph_0}}$ compact objects of ${\underline{C}}$ is of cardinality less than ${\alpha}$. 2\. Obviously the join (see Remark \[revenmorews\](1) and Corollary \[cwftw\](3)) of any set of weakly ${\beta}$-well generated weight structures is weakly ${\beta}$-well generated; so, we obtain a filtration (respected by joins) on the “join monoid” of weight structures. The natural analogue of this fact for strongly ${\beta}$-well generated weight structures is probably wrong. Indeed, it is rather difficult to believe that for a general compactly generated category ${\underline{C}}$ the class of weight structures on the subcategory ${\underline{C}}^{{{\aleph_0}}}$ of compact objects (cf. Proposition \[psym\](\[isymcomp\])) would be closed with respect to joins; note that joining compactly generated weight structures $w_i$ on ${\underline{C}}$ corresponds to intersecting the classes ${\underline{C}}_{w_i\ge 0}\cap {\operatorname{Obj}}{\underline{C}}^{{{\aleph_0}}}$. On the other hand, Corollary 4.7 of [@krause] suggests that the filtration of the class of smashing weight structures by the sets of weakly ${\beta}$-well generated ones (for ${\beta}$ running through regular cardinals) may be “quite short”. 3\. According to part III.2 of our theorem, any weight structure on a well generated ${\underline{C}}$ is strongly ${\beta}$-well generated for ${\beta}$ being large enough. Combining this part of the theorem with its part II we also obtain a bijection between strongly ${\beta}$-well generated weight structures on ${\underline{C}}$ and ${\beta}$-coproductive weight structures on ${\underline{C}}^{\beta}$. Note that (even) the restrictions of these results to compactly generated categories appear to be quite interesting. 4. Now assume that a weight structure $w$ on ${\underline{C}}$ is strongly ${{\aleph_0}}$-well generated; this certainly means that ${\underline{C}}$ is compactly generated (see Proposition \[psym\](\[isymcomp\])) and $w$ restricts to its subcategory ${\underline{C}}^{{\aleph_0}}$ of compact objects. Then Proposition \[ppcoprws\](\[icopr7p\]) implies that ${\underline{C}}_{w\ge 0}$ is the smallest coproductive extension-closed subclass of ${\operatorname{Obj}}{\underline{C}}$ that contains ${\underline{C}}_{w\ge 0}\cap {\operatorname{Obj}}{\underline{C}}^{{\aleph_0}}$ (cf. the proof of Theorem \[twgws\](III.2)). Thus the $t$-structure right adjacent to $w$ is generated by the essentially small class ${\underline{C}}_{w\ge 0}\cap {\operatorname{Obj}}{\underline{C}}^{{\aleph_0}}$; so it is compactly generated (and hence smashing). 5\. For ${\underline{C}}$ as above and a weakly ${\beta}$-well generated weight structure $w$ on it one can easily establish a natural weight structure analogue of [@krauwg Theorem B] that will “estimate the size” of an element $M$ of ${\underline{C}}_{w\le 0}$ in terms of the cardinalities of ${\underline{C}}(P,M)$ for $P$ running through ${\beta}$-compact elements of ${\underline{C}}_{w\le 0}$ (modifying the proof of loc. cit. that is closely related to our proof of Theorem \[tpgws\]). Moreover, this result should generalize loc. cit. Note also that there is a “uniform” estimate of this sort that only depends on ${\underline{C}}$ (and does not depend on $w$). This argument should also yield that a weakly ${\beta}$-well generated weight structure is always strongly ${\beta}'$-well generated for a regular cardinal ${\beta}'$ that can be described explicitly. Moreover, similar arguments can possibly yield that any smashing weight structure on a perfectly generated triangulated category ${\underline{C}}$ is perfectly generated (cf. Theorem \[twgws\](III.2)). 6\. Our understanding of “general” well generated torsion pairs is much worse than the one of (well generated) weight structures. In particular, the author does not know which properties of weight structures proved in this section can be carried over to $t$-structures. On torsion pairs orthogonal with respect to dualities {#skan} ===================================================== In this section we study dualities between triangulated categories (generalizing the bifunctor ${\underline{C}}(-,-)$ along with its restrictions to pairs of triangulated subcategories of ${\underline{C}}$). Our main construction tool are Kan extensions of homological functors from triangulated subcategories of ${\underline{C}}$ (to ${\underline{C}}$; we call the resulting functors [*coextensions*]{}); these are interesting for themselves. In §\[scoext\] we study coextensions of homological functors (into an AB5 category ${\underline{A}}$) from a triangulated subcategory ${\underline{C}_0}$ to ${\underline{C}}$ following [@krause]. If ${\underline{C}}$ is compactly generated and ${\underline{C}_0}$ is its subcategory of compact objects then the [*coextended*]{} functors are precisely the cc ones. As an application we demonstrate that for any compactly generated torsion pair $s=({\mathcal{LO}},{\mathcal{RO}})$ there exists an object $N$ of ${\underline{C}}$ such that the functor $H^N$ kills precisely those compact objects of ${\underline{C}}$ that (also) belong to ${\mathcal{LO}}$. In §\[sdual1\] we recall (from [@bger]) the definition of a duality $\Phi:{\underline{C}}{{^{op}}}\times {\underline{C}'}\to {\underline{A}}$ (we are mostly interested in the case ${\underline{C}}={\underline{\operatorname{Ab}}}$). The corresponding notion of [$\Phi$-orthogonal]{} torsion pairs generalizes the notion of adjacent ones. In the case where ${\underline{C}}\subset {\underline{C}'}$ and $\Phi$ is just the restriction of ${\underline{C}'}(-,-)$ to ${\underline{C}}{{^{op}}}\times {\underline{C}'}$ we are able to prove two natural generalizations of Proposition \[psatur\]; so we prove (assuming some additional conditions) that for any weight structure $w$ on ${\underline{C}}$ (resp. on ${\underline{C}}'$) there exists a $t$-structure on ${\underline{C}'}$ (resp. on ${\underline{C}}$) that is right (resp. left) $\Phi$-orthogonal to $w$. The results of [@neesat] and [@roq] demonstrate that these results can be applied for ${\underline{C}}$ being the derived category of perfect complexes and ${\underline{C}}'$ being the bounded derived category of coherent sheaves on a scheme proper over the spectrum of a noetherian ring. In §\[sdual2\] we study in detail the case where ${\underline{C}}$ and ${\underline{C}'}$ contain a common triangulated subcategory ${\underline{C}_0}$ whose objects are cocompact in ${\underline{C}}$ and compact in ${\underline{C}'}$, and $\Phi$ is a certain “bicontinuous biextension” of the bi-functor ${\underline{C}_0}(-,-)$. We obtain that any set ${\mathcal{P}}$ of objects of ${\underline{C}_0}$ gives a couple of $\Phi$-orthogonal torsion pairs (in ${\underline{C}}$ and ${\underline{C}}'$, respectively). For a suspended ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}_0}$ this gives (in §\[sgengroth\]) a $t$-structure $t$ on ${\underline{C}'}$ that is generated by ${\mathcal{P}}$ and a weight structure $w$ on ${\underline{C}}$ that is cogenerated by ${\mathcal{P}}$ and is (left) $\Phi$-orthogonal to $t$. A particular case of this setting is considered in [@bgn] (where our results are applied to the study of various motivic homotopy categories, homotopy $t$-structures, and coniveau spectral sequences). We also apply this result to the study of compactly generated $t$-structures The elements of the heart of $w$ give a faithful family of exact functors ${{\underline{Ht}}}\to {\underline{\operatorname{Ab}}}$ that respect coproducts; hence ${{\underline{Ht}}}$ is an AB5 category. Next an easy argument yields the existence of a generator for ${{\underline{Ht}}}$ (so, it is a Grothendieck abelian category). In §\[sprospectra\] we prove (using the results of [@tmodel]) that taking ${\underline{C}'}$ to be the homotopy category of a proper simplicial stable model category ${\mathcal{M}}$ and ${\underline{C}}$ to be the homotopy category of the category ${\operatorname{Pro}-\mathcal{M}}$ of (filtered) pro-objects of ${\mathcal{M}}$ one obtains an example of the aforementioned setting. Thus we obtain that ${{\underline{Ht}}}$ is an Grothendieck abelian category for any compactly generated $t$-structure on a “topological” triangulated category. Moreover, this pro-object construction is used in [@bgn] for the description of various triangulated categories of [*motivic pro-spectra*]{} (and [*comotives*]{}) and for the study of their relation to the corresponding motivic stable homotopy categories. Another example of the couple $({\underline{C}},{\underline{C}'})$ that may be obtained this way is $(\operatorname{SH}^{op},\operatorname{SH})$; this observation is closely related to the main subject of [@prospect]. In §\[slocoeff\] we recall a few results on “localizing coefficients” in (compactly generated) triangulated categories and the relate this matter to torsion pairs and their orthogonality. We also study (from a similar perspective) decompositions of triangulated categories as direct sums of their subcategories. The section is included here for the purpose of using it in [@bgn]; still some of its results may be interesting for themselves. On Kan extensions of homological functors {#scoext} ------------------------------------------ Now we will study a method of extending of homological functors from a triangulated subcategory of ${\underline{C}}$; this is a version of left Kan extensions that was studied in detail in [@krause §2]. In particular, we obtain a description of all cc functors from ${\underline{C}}$ (into an AB5 abelian category) if it is compactly generated. Since the construction is dual to the one applied in [@bger] and [@bgn] (and it “usually respects coproducts”) we will call the resulting functors [*coextended*]{} ones; we will explain that they are actually Kan extensions below. To formulate some of the properties of the construction we will use (a few times) the following definition. \[drelim\] Let ${\underline{C}_0}$ be a subcategory of ${\underline{C}}$ (or just a class of objects), $M\in {\operatorname{Obj}}{\underline{C}}$; let $L$ be a small (index) category and fix a functor $L\to {\underline{C}}: l\mapsto N_l$. 1\. Let $M$ be a co-cone of this functor (i.e., $M$ is equipped with compatible morphisms from $N_l$ for $l\in {\operatorname{Obj}}L$). Then we will say that $M$ is a [*${\underline{C}_0}$-colimit*]{} of $(N_l)$ if the restriction $H_M$ of the functor ${\underline{C}}(-,M)$ to ${\underline{C}_0}$ equals the colimit of $ H_{N_l}$ (in ${\operatorname{PShv}}({\underline{C}_0})$, i.e., if for any $Y\in {\operatorname{Obj}}{\underline{C}_0}$ the connecting morphisms induce an isomorphism ${\underline{C}}(Y,N)\cong \operatorname{\varinjlim}{\underline{C}}(Y,N_l)$). 2\. If $M$ is a cone of $N$ then we will say that $M$ is a [*${\underline{C}_0}$-limit*]{} of $(N_l)$ if $M$ is a ${\underline{C}_0}^{op}$-colimit of $(N_l)$ in ${\underline{C}}^{op}$. \[rlim\] We will not need much of this definition in the current paper. Moreover, it appears that the most “useful” case of part 1 (resp. 2) of the definition is the filtered one, i.e., the one where $(N_l)$ is an inductive (resp. projective) system. Now let ${\underline{C}_0}$ be an essentially small triangulated subcategory of ${\underline{C}}$; we will also assume that ${\underline{C}}$ has coproducts. We consider the category ${\operatorname{PShv}}({\underline{C}_0})=\operatorname{\operatorname{AddFun}}({\underline{C}}_0^{op},{\underline{\operatorname{Ab}}})$ (cf. the proof of Theorem \[tpgws\]); recall that this is a locally small abelian category. It it easily seen that any $H\in {\operatorname{Obj}}{\operatorname{PShv}({\underline{C}}_0)}$ possesses a (projective) resolution $ \coprod{\underline{C}}_0(-, C_i)\to \coprod {\underline{C}}_0(-,C_j)\to H\to 0 $ where $\{C^i\}$ and $\{C^j\}$ are some families of objects of ${\underline{C}}_0$; cf. Lemma 2.2 of [@krause]. \[pkrause\] Let $H_0:{\underline{C}}_0\to {\underline{A}}$ be a homological functor, where ${\underline{A}}$ is an AB5 abelian category; fix some $N\in {\operatorname{Obj}}{\underline{C}}$. For any $M\in {\operatorname{Obj}}{\underline{C}}$ we fix a resolution (as above) $$\label{ekrause} \coprod_{i\in I}H_{C_M^i}\to \coprod_{j\in J} H_{C_M^j}\to H_M\to 0,$$ where we use the notation $H_M$ for the restriction of the functor ${\underline{C}}(-,M)$ to ${\underline{C}_0}$. Then for the association $H:M\mapsto \operatorname{\operatorname{Coker}}(\coprod H_0(C_M^i)\to \coprod H_0(C_M^j))$ the following statements are valid. 1. \[ikr1\] $H$ is a homological functor ${\underline{C}}\to {\underline{A}}$ that is essentially independent on the choice of resolutions. In particular, the restriction of $H$ to ${\underline{C}_0}$ is canonically isomorphic to $H_0$. We will call $H$ the [*coextension*]{} of $H_0$ to ${\underline{C}}$, and say that it is a [*coextended*]{} (from ${\underline{C}_0}$) functor. 2. \[ikrchar\] For any ${\operatorname{PShv}}({\underline{C}_0})$-resolution $ \coprod H_{C'{}_M^{i}}\to \coprod H_{C'{}_M^{j}} \to H_M\to 0$ of $H_M$ with $C'{}_M^{i}$ and $C'{}_M^{j}$ being some objects of ${\underline{C}_0}$, the object $\operatorname{\operatorname{Coker}}(\coprod H_0(C'{}_M^{i})\to \coprod H_0(C'{}_M^{j}))$ is canonically isomorphic to $H(M)$. Moreover, $H$ is canonically characterized by this condition along with its restriction to ${\underline{C}_0}$. 3. \[ikres\] More generally, we have $H(M)\cong \operatorname{\operatorname{Coker}}(\coprod H(C'{}_{M^i})\to \coprod H(C'{}_{M^j}))$ (also) if the ${\operatorname{PShv}({\underline{C}}_0)}$-sequence $ \coprod H_{C'{}_M^{i}}\to \coprod H_{C'{}_M^{j}} \to H_M\to 0$ is exact for some objects $C'{}_M^{i}$ and $C'{}_M^{j}$ of ${\underline{C}}$. 4. \[ikrtriv\] If $H_0:{\underline{C}_0}\to {\underline{\operatorname{Ab}}}$ is corepresented by some $M_0\in {\operatorname{Obj}}{\underline{C}_0}$ then $H$ is ${\underline{C}}$-corepresented by $M_0$ also. 5. \[ikr7\] For any exact sequence $\coprod H^i_0\to \coprod H^j_0\to H_0\to 0$ the corresponding ${\operatorname{PShv}}({\underline{C}})$-sequence of coextensions $\coprod H^i\to \coprod H^j\to H\to 0$ is exact also. In particular, the coextension of $\coprod H^j_0$ equals $\coprod H^j$. 6. \[ikr2\] $H(N)$ for $N\in {\operatorname{Obj}}{\underline{C}}$ only depends on the restriction of ${\underline{C}}(-,N)$ to ${\underline{C}_0}$. 7. \[ikradj\] Let ${\underline{E}}$ be a full triangulated subcategory of ${\underline{C}}$ that contains ${\underline{C}_0}$ and assume that there exists a right adjoint $G$ to the embedding ${\underline{E}}\to {\underline{C}}$ (so, ${\underline{E}}$ has coproducts). Then we have $H\cong H^{{\underline{E}}}\circ G$,where the functor $H^{{\underline{E}}}:{\underline{E}}\to {\underline{A}}$ is defined on ${\underline{E}}$ using the same construction as the one used for the definition of ${\underline{C}}$. 8. \[ikr3\] If $N$ is a ${\underline{C}_0}$-colimit of some $(N_l)$ (see Definition \[drelim\](1)), then $H(N)\cong \operatorname{\varinjlim}H(N_l)$. Moreover, such a set of $(X_l,f_l)$ exists for any $X\in {\operatorname{Obj}}{\underline{C}}$. 9. \[ikrtransf\] For $H'_0$ being another homological functor ${\underline{C}_0}\to {\underline{A}}$ and the corresponding coextension $H'$ we have the following: the restriction of natural transformations $H\implies H'$ to the subcategory ${\underline{C}_0}$ gives a one-to-one correspondence between them and the transformations $H_0\implies H'_0$. 10. \[ikr6\] Let $H_0\stackrel{f_0}{\to} H'_0 \stackrel{g_0}{\to} H''_0$ be a (three-term) complex of homological functors ${\underline{C}_0}\to {\underline{\operatorname{Ab}}}$ that is exact in the term $H'_0$. Then the complex $H\stackrel{f}{\to} H' \stackrel{g}{\to} H''$ (here $H,H',H'',f,g$ are the corresponding coextensions) is exact in the middle also. 11. \[ikr8\] Assume that all objects of ${\underline{C}_0}$ are compact. Then $H$ is determined (up to a canonical isomorphism) by the following conditions: it respects coproducts, it restriction to ${\underline{C}_0}$ equals $H_0$, and it kills ${\underline{C}_0}^\perp$. \[ikr1\]. Immediate from [@krause Lemma 2.2] (see also Proposition 2.3 of ibid.). \[ikrchar\] –\[ikr3\]. The proofs are straightforward (and very easy); cf. also Remark \[rkrause\](I.1) below. Assertion \[ikrtransf\] is easy also. The injectivity of this restriction correspondence follows easily from the previous assertion and the surjectivity is immediate from the naturality of the coextension construction in $H_0$. \[ikr6\]. We should check that the sequence $H(M)\stackrel{f(M)}{\to} H'(M) \stackrel{g(M)}{\to} H''(M)$ of abelian groups is exact (in the middle term) for any $M\in {\operatorname{Obj}}{\underline{C}}$. Certainly, the functoriality of coextensions gives $g(M)\circ f(M)=0$. Our exactness assertion is obviously valid if $M\in {\operatorname{Obj}}{\underline{C}_0}$. We reduce the general case of this statement to this one. We start from analysing the sequence (\[ekrause\]). The Yoneda lemma immediately implies that the natural transformations in it are given by certain ${\underline{C}}$-morphisms $f_j: C_M^j \to M$ for all $j\in L$ and $g_{ij}:C_M^i\to C_M^j $ for all $i\in I$, $j\in J$; moreover, for any $i\in I$ almost all $g_{ij}$ are $0$ and we have $\sum_{j\in J} f_j\circ g_{ij}=0$. We should check that if for $a\in H'(M)$ we have $g(M)(a)=0$, then $a=f(M)(x)$ for some $x\in H(M)$. Using the additivity of ${\underline{C}_0}$ we can gather finite sets of $C_M^i$ and $C_M^i$ in (\[ekrause\]) into single objects. Hence we can assume that the following assumptions are fulfilled: $a=H'_0(f_{j_0})(a_0)$ for some $j_0\in J$ and $a_0\in H'_0(C_M^{j_0} )$, $g_0(C_M^{j_0})(a_0)=H''_0(g_{i_0j_0}) (b_0)$ for some $i_0\in I$ and $b_0\in H''_0(C_M^{i_0} )$ (recall that $H''(M)$ is defined as the corresponding cokernel!). Moreover, we can assume that $g_{i_0j}=0$ for any $j\neq j_0$; thus $f_{j_0}\circ g_{i_0j_0}=0$. Complete $g_{i_0j_0}$ to a distinguished triangle $C_M^{i_0}$( g\_[i\_0j\_0]{})$ by $Y$ \to C_M^{j_0}\stackrel{{\alpha}}{\to} Y$; then we can assume $Y\in {\operatorname{Obj}}{\underline{C}_0}$, and the equality $f_{j_0}\circ g_{i_0j_0}=0$ implies that $f_{j0}$ can be decomposed as ${\beta}\circ {\alpha}$ for some ${\beta}\in {\underline{C}}(Y,M)$. Since $H''_0$ is homological, $H''({\alpha})(g_0(C_M^{j_0})(a_0))=0$. Applying the exact sequence $H_0\to H_0'\to H_0''$ to $Y$ we obtain that $H'_0({\alpha})(a_0)\in H'(Y)$ can be presented as $f_0(Y)(x_Y)$ for some $x_Y\in H_0(Y)=H(Y)$. Hence $a=f(M)(x)$ for $x=H({\beta})\in H(M)$; see the commutative diagram $$\begin{CD} H_0(C_M^{j_0}) @>{H_0({\alpha})}>> H_0(Y)=H(Y)@>{H({\beta})}>>H(M)\\ @VV{f_0(C_M^{j_0})}V@VV{f_0(Y)=f(Y)}V@VV{f(M)}V \\ H'_0(C_M^{j_0})@>{H'_0({\alpha})}>>H'_0(Y)=H'(Y) @>{H'({\beta})}>>H'(M)\\ @VV{g_0(C_M^{j_0})}V@VV{g_0(Y)=g(Y)}V@VV{g(M)}V \\ H''_0(C_M^{j_0})@>{H''_0({\alpha})}>>H''_0(Y)=H''(Y)@>{H''({\beta})}>>H''(M) \end{CD}$$ \[ikr8\]. In the case where ${\underline{C}_0}$ generates ${\underline{C}}$ as its own localizing category the assertion is given by Proposition 2.3 of [@krause]. Now recall that (in the general case) the embedding of the localizing category generated by ${\underline{C}_0}$ into ${\underline{C}}$ possesses an (exact) right adjoint $G$ that respects coproducts (according to Proposition \[pcomp\](II)). Hence the general case of the assertion reduces to loc. cit. if we apply assertion \[ikradj\]. \[rkrause\] I.1. Now we explain that $H$ is actually the left Kan extension of $H_0$ along the inclusion ${\underline{C}_0}\to {\underline{C}}$. Indeed, we can certainly assume that ${\underline{C}_0}$ is small. Then the standard pointwise construction of the left Kan extension is easily seen to correspond to “the most obvious” resolutions of the functors $H_M$ in ${\operatorname{PShv}({\underline{C}}_0)}$. This observation certainly gives an alternative proof of part \[ikrtransf\] of the proposition. Below we will also mention extensions of cohomological functors ${\underline{C}}\to {\underline{A}}$ obtained via applying the coextension construction to the corresponding (homological) functors ${\underline{C}}{{^{op}}}\to {\underline{A}}$. So they can also be described via right Kan extensions of the opposite homological functors ${\underline{C}}\to {\underline{A}}{{^{op}}}$. 2\. It appears that it is not necessary to assume that ${\underline{C}_0}$ is triangulated to construct $H$ from $H_0$. So, assume that ${\underline{C}_0}$ is an essentially small additive subcategory of ${\underline{C}}$ that satisfies the following condition: for any ${\underline{C}}$-distinguished triangle $$\label{edi} Z[-1]\to X\to Y\to Z$$ the object $X$ belongs to ${\operatorname{Obj}}{\underline{C}_0}$ whenever $Y$ and $Z$ do. Note that any ${\underline{C}_0}$ satisfying this condition is cosuspended, and the extension-closure of any cosuspended subcategory of ${\underline{C}}$ satisfies this condition. Now, our restriction on ${\underline{C}_0}$ certainly implies that it has [*weak kernels*]{} (for all morphisms); hence the subcategory of coherent functors inside ${\operatorname{PShv}}({\underline{C}_0})$ is abelian (see §1.2 of [@krause]). Take an additive functor $H_0:{\underline{C}_0}\to {\underline{A}}$ for an AB5 category ${\underline{A}}$ such that for any triangle (\[edi\]) with $Y,Z\in {\operatorname{Obj}}{\underline{C}_0}$ the sequence $H(X)\to H(Y)\to H(Z)$ is exact (in the middle term). Using [@krause Lemmas 2.1, 2.2] (cf. also the proof of Proposition 2.3 of ibid.) one can easily prove that the (“pointwise Kan”) recipe described in Proposition \[pkrause\] gives a homological functor $H:{\underline{C}}\to {\underline{A}}$ whose restriction to ${\underline{C}_0}$ is isomorphic to $H_0$. Assume in addition that ${\underline{C}_0}$ consists of compact objects of ${\underline{C}}$. Then the functor $H$ (coming from any $H_0$ as above) is certainly a cc functor. Moreover, ${\operatorname{Obj}}{\underline{C}_0}$ generates a weight structure $w$ (see Remark \[rnewt\](1)) and $H$ obviously annihilates ${\mathcal{RO}}={\underline{C}}_{w\ge 1}$. Now we prove that the functors $H$ that can be obtained using coextensions of this sort are precisely the cc functors satisfying this vanishing condition (similarly to Definition \[drange\] one may say that these functors are of weight range $\le 0$). It follows that for $H_0$ being the restriction to ${\underline{C}_0}$ of a cc functor $H':{\underline{C}}\to {\underline{A}}$ we have $H\cong \tau^{\ge 0}H'$ (see Remark \[rwrange\](4)). We start from noting that for any $H'$ as above the definition of $H$ gives a canonical transformation $\Psi: H\to H'$. Now assume that $H'$ annihilates ${\mathcal{RO}}$. First we prove that $\Psi(M_0)$ is injective for any $M_0\in {\operatorname{Obj}}{\underline{C}}$. Considering a $w$-decomposition $(w_{\ge 1}M_0)[1]\to w_{\le 0}M_0\to M\to w_{\ge 1}M_0$ we obtain that $H'(M_0)\cong H'(w_{\le 0}M_0)$ and $H(M_0)\cong H(w_{\le 0}M_0)$. Hence it suffices to verify the injectivity of $\Psi(M)$ for any $M\in {\underline{C}}_{w\le 0}$. We will apply a certain inductive argument with the base being the obvious fact that $\Psi(M)$ is an isomorphism if $M$ is a coproduct of objects of ${\underline{C}_0}$. We use the construction (and adopt the notation) used in the proof of Theorem \[tclass\](\[iclass1\]). Since $M$ belongs to $M\in {\underline{C}}_{w\le 0}$, it is a retract of the corresponding $L$ (according to Proposition \[phop\](7)), whereas the latter is a homotopy colimit of $L_i$ with cones of the connecting morphisms belonging to ${\underline{\coprod}{\mathcal{P}}}$. So, for all $k\ge 0$ we have commutative diagrams $$\begin{CD} H(P_k[-1])@>{}>>H(L_k)@>{}>>H(L_{k+1})@>{}>>H(P_k) \\ @VV{\cong }V@VV{}V@VV{}V @VV{\cong}V\\ H'(P_k[-1])@>{}>>H'(L_k)@>{}>>H'(L_{k+1})@>{}>>H'(P_k) \end{CD}$$ whose rows are exact. It easily follows by induction (starting from $\Psi(L_0)=\Psi(0)=0$) that $\Psi(L_k)$ is injective for any $k\ge 0$. Passing to the colimit (see Lemma \[lcoulim\](3(ii))) we obtain that $\Psi(L)$ is monomorphic; hence $\Psi(M)$ is monomorphic also. Next we prove that $\Psi(M)$ is also epimorphic (for any $M\in {\operatorname{Obj}}{\underline{C}}$). We set $H''(M)= \operatorname{\operatorname{Coker}}(\Psi(M))$. Since $\Psi(M)$ is monomorphic for all $M\in {\operatorname{Obj}}{\underline{C}}$, we obtain that $H''(M)$ is a homological functor (${\underline{C}}\to{\underline{A}}$); certainly it is a cc functor. So we should prove that $H''$ is zero. For any $M$, $L$, $L_k$, and $P_k$ as above we certainly have $H''(P_k)=H''(P_k[-1])=0$ for any $k\ge 0$; hence obvious induction (similar to that used in the proof of the injectivity of $\Psi(M)$) gives the vanishing of $H''(L_k)$ for $k\ge 0$. Passing to the colimit once again we obtain $H''(L)=H''(M)=0$ (for any $M\in {\underline{C}}_{w\le 0}$). Since $H''$ also annihilates ${\mathcal{RO}}$, we obtain that it is zero. Possibly, the author will write these arguments in more detail in a new version of this paper to obtain a set of generators of a (compactly generated) $t$-structure $t$ under the assumptions of Theorem \[tab5\](2) below (this set of generators would be much smaller than the one given by part 1 of this theorem; see Remark \[rab5\](\[ismgen\])). II\. Now assume (once again) that all objects of ${\underline{C}_0}$ are compact in ${\underline{C}}$. 1. Then part \[ikr8\] of our proposition easily implies that $\operatorname{\operatorname{AddFun}}({\underline{C}_0},{\underline{A}})$ is equivalent to the category of those cc functors ${\underline{C}}\to {\underline{A}}$ that kill ${\underline{C}_0}^\perp$. 2. Let $w$ be a smashing weight structure for ${\underline{C}}$ that restricts to ${\underline{C}_0}^\perp$. Since the corresponding virtual $t$-truncations of cc functors are cc ones according to Proposition \[ppcoprws\](\[icopr6\]), we also obtain that virtual $t$-truncations of coextended functors are coextended. It certainly follows that (in this case) virtual t-truncations of coextended functors satisfy the “continuity” property described in part \[ikr3\] of our proposition. Recall also that we have a similar continuity for natural transformations between coextended functors according to part \[ikrtransf\] of the proposition. 3\. Now we describe an interesting application of these observations. We recall that for any (co)homological functor $J:{\underline{C}}\to {\underline{A}}$ and any $N\in {\operatorname{Obj}}{\underline{C}}$ a certain [*weight spectral sequence*]{} $T_w(J,N)$ for $J_*(N)$ is defined. We will not need its definition here. We will only recall (see Theorem 2.3.2 of [@bws]) that this spectral sequence is defined starting from the $E_1$-page; $T_w(J,N)$ becomes independent from any choices (and functorial in $N$) starting from $E_2$ (and we will write $T^{\ge 2}_w(J,N)$ for this “part” of $T_w(J,N)$). Moreover, Theorem 2.4.2(II) of [@bger] immediately implies the following: $T^{\ge 2}_w(J,N)$ can be $N$-functorially described in terms of certain virtual $t$-truncations of $J$ along with canonical transformations between them (and the transformations of the second level essentially come from (\[evtt\]); note that this statement follows from the description of the derived exact couple $(E_2^{**},D_2^{**})$ for the exact couple $(E_1^{**},D_1^{**})$ that gives $T_w(J,N)$). Hence we obtain the following: if $J$ is a coextended functor (and so, ${\underline{A}}$ is an AB5 category) and $N$ is a ${\underline{C}_0}$-colimit of an [**inductive system**]{} $(N_l)$ then $T^{\ge 2}_w(J,N)$ is the direct limit of $T^{\ge 2}_w(J,N_l)$. The author plans to apply (the dual to) this statement in [@bgn] (where weight spectral sequences are described and studied in much more detail). It will be applied to extended (see part I of this remark) cohomological functors from a cocompactly cogenerated category ${\underline{E}}$, where ${\underline{E}}$ is a certain category of [*motivic pro-spectra*]{} (or of [*comotives*]{}). This allows to compute (“generalized”) coniveau spectral sequences for the corresponding cohomology of a projective limits of smooth varieties $X_l$ (over the base field $k$ that is perfect). 4\. Now we give one more proof of the fact that $\operatorname{\varinjlim}T^{\ge 2}_w(J,N_l)\cong T^{\ge 2}_w(J,N) $ (if $N$ is a ${\underline{C}_0}$-colimit of an inductive system $(N_l)$ and $J$ is extended); this argument avoids the consideration of $D_*^{**}$. The functoriality of $T^{\ge 2}_w(J,-)$ yields canonical compatible morphisms between $T^{\ge 2}_w(J,N_l)$ and from them into $T^{\ge 2}_w(J,N)$. Now, since $(N_l)$ form an inductive system, the inductive limit $T'^{\ge 2}_w(J,N)$ of $T^{\ge 2}_w(J,N_l)$ is a spectral sequence (that starts from $E_2$); we also have a canonical morphism $T'^{\ge 2}_w(J,N)\to T^{\ge 2}_w(J,N)$. Thus it remains to verify that this morphism is an isomorphism at $E_2$, i.e., that $\operatorname{\varinjlim}E_2^{**}T_w(J,N_l)\cong E_2^{**}T(J,N_l)$. Now, Theorem 2.3.2 of [@bws] implies immediately that the $E_2$-terms of $T_w(J,-)$ are given by the pure functors $H^{A_{J\circ [i]}}\circ [j]$ for $i,j\in {{\mathbb{Z}}}$ (see Proposition \[ppure\]). It is easily seen that these functors kill ${\underline{C}_0}^\perp$, and they are cc according to Proposition \[ppcoprws\](\[icopr5\]). Hence they are extended according to Proposition \[pkrause\](\[ikr8\])[^60], and it remains to apply Proposition \[pkrause\](\[ikr3\]). Now we combine the results of this section with the ones of [@bsnull]; this gives an alternative proof of the “classification” of compactly generated torsion pairs given by Theorem \[tclass\]. \[rnz\] 1. So, we want to give one more proof of the existence of a one-to-one correspondence between compactly generated torsion pairs for ${\underline{C}}$ and essentially small Karoubi-closed extension-closed subclasses of the class ${\underline{C}}^{{{\aleph_0}}}$ of compact objects of ${\underline{C}}$; the argument should not depend on Theorem \[tclass\](\[iclass2\]). Recall that any essentially small subclass ${\mathcal{P}}$ of ${\underline{C}}^{{{\aleph_0}}}$ generates a compactly generated torsion pair $s=({\mathcal{LO}},{\mathcal{RO}})$ for ${\underline{C}}$ according to Theorem \[tclass\](\[iclass1\]) (this statement is also given by the easier Theorem 4.3 of [@aiya]). Since ${\underline{C}}^{{{\aleph_0}}}$ gives a triangulated subcategory of ${\underline{C}}$ (see Lemma 4.1.4 of [@neebook]), the class ${\underline{C}}^{{{\aleph_0}}}\cap {\mathcal{LO}}$ contains the envelope of ${\mathcal{P}}$. Next (cf. Theorem \[tclass\](\[iclasst\]) or its proof) this envelope is essentially small also. Thus we should prove that ${\underline{C}}^{{{\aleph_0}}}\cap {\mathcal{LO}}$ equals ${\mathcal{P}}$ whenever the generating class ${\mathcal{P}}$ is essentially small, Karoubi-closed, and extension-closed in ${\underline{C}}$. For the latter purpose it suffices to prove the existence of $I\in {\mathcal{RO}}={\mathcal{P}}{{}^{\perp}}$ such that ${{}^{\perp}}I\cap {\underline{C}}^{{{\aleph_0}}}={\mathcal{P}}$. Now, in [@bsnull] zero classes (see Lemma \[lbes\](\[izs\])) of (co)homological functors were studied. Corollary 3.11 of ibid. (applied to the category ${\underline{C}_0}{{^{op}}}$) gives the following remarkable statement: if ${\underline{C}_0}$ is a small triangulated category then a set ${\mathcal{P}}_0$ of its objects is the zero class of some “detecting” homological functor $H_0:{\underline{C}_0}\to {\underline{\operatorname{Ab}}}$ if and only if ${\mathcal{P}}_0$ is extension-closed and Karoubi-closed in ${\underline{C}_0}$. The author believes that this statement will become an important tool of studying (compactly generated) torsion pairs. We take ${\mathcal{P}}_0$ to be a small skeleton of ${\mathcal{P}}$ and take ${\underline{C}_0}$ to be a small skeleton of ${\langle}{\mathcal{P}}{\rangle}_{{\underline{C}}}$ such that ${\operatorname{Obj}}{\underline{C}_0}\cap {\mathcal{P}}={\mathcal{P}}_0$. Since all objects of ${\underline{C}_0}$ are compact in ${\underline{C}}$, the coextension $H$ of the aforementioned functor $H_0$ to ${\underline{C}}$ is a cc functor according to Proposition \[pkrause\](\[ikr8\]). Next we consider the Brown-Comenetz dual functor ${\widehat{H}}$ (see Definition \[dsym\](\[ibcomf\]); recall that ${\widehat{H}}: M\mapsto {\underline{\operatorname{Ab}}}(H(M),{{\mathbb{Q}}}/{{\mathbb{Z}}})$ is a cp functor from ${\underline{C}}$ into ${\underline{\operatorname{Ab}}}$ whose zero class coincides with that of $H$). If we assume in addition that ${\underline{C}}$ equals its own localizing subcategory ${\underline{C}}'$ generated by ${\mathcal{P}}$ then it satisfies the Brown representability condition according to Proposition \[pcomp\](II.1). Thus the functor ${\widehat{H}}$ is ${\underline{C}}$-representable by some $I_0\in {\operatorname{Obj}}{\underline{C}}$ (see Proposition \[psymb\](II.\[iws21\])). Since the zero class of ${\widehat{H}}$ contains ${\mathcal{P}}$, we have $I_0\in {\mathcal{RO}}$. It remains to compute ${{}^{\perp}}I_0\cap {\underline{C}}^{{{\aleph_0}}}$. Now, ${\underline{C}}^{{{\aleph_0}}}$ equals ${\operatorname{Obj}}{\langle}{\mathcal{P}}{\rangle}$ (in this case) according to Lemma 4.4.5 of [@neebook]. Hence ${{}^{\perp}}I_0\cap {\underline{C}}^{{{\aleph_0}}}={\mathcal{P}}$ by construction (since ${\mathcal{P}}$ is the isomorphism-closure of ${\mathcal{P}}_0$ in ${\underline{C}}$). Lastly, to reduce the general case of our assertion (i.e., of ${\underline{C}}^{{{\aleph_0}}}\cap {\mathcal{LO}}={\mathcal{P}}$) to the case ${\underline{C}}={\underline{C}}'$ it suffices to recall that $({\mathcal{LO}},{\mathcal{RO}}\cap {\operatorname{Obj}}{\underline{C}}')$ is a torsion pair for ${\underline{C}}'$ according to Proposition \[phopft\](I.2) . Note also that the existence of a “detector object” $I_0$ is a new result. 2. The aforementioned result of [@bsnull] (along with some of its variations also proved in ibid.) is a sort of Nullenstellensatz for (co)homological functors from (small) triangulated categories (whence the name of the paper). It would certainly be interesting to obtain some analogue of this statement for cc and cp functors; Theorem \[tclass\](\[iclass5\]) is certainly related to this matter. Note however that a priori the intersection of zero classes of all those smashing virtual $t$-truncations of representable functors that contain a given ${\mathcal{P}}\subset {\underline{C}}$ (for ${\underline{C}}$ having coproducts) may be bigger than the strong extension-closure of ${\mathcal{P}}$. Dualities between triangulated categories and orthogonal torsion pairs; applications to categories of coherent sheaves {#sdual1} ---------------------------------------------------------------------------------------------------------------------- Now we study certain pairings between triangulated categories and define the notion of orthogonality of torsion pairs (as well as of weight and $t$-structures) generalizing the one of adjacent structures. We also define the notion of a nice duality; yet we will not use it in the current paper (anywhere except in Proposition \[pnice\]). \[dort\] Let ${\underline{C}}'$ be a triangulated category. 1. \[idu\] We will call a (covariant) bi-functor $\Phi:{\underline{C}}^{op}\times{\underline{C}}'\to {\underline{A}}$ a [*duality*]{} if it is bi-additive, homological with respect to both arguments, and is equipped with a (bi)natural bi-additive transformation $\Phi(A,Y)\cong \Phi (A[1],Y[1])$. 2. \[iorthop\] Suppose that ${\underline{C}}$ is endowed with a torsion pair $s=({\mathcal{LO}},{\mathcal{RO}})$ and ${\underline{C}}'$ is endowed with a torsion pair $s'=({\mathcal{LO}}',{\mathcal{RO}}')$. Then we will say that $s$ is (left) [*orthogonal*]{} to $s'$ with respect to $\Phi$ (or just [*left $\Phi$-orthogonal*]{} to it) if the following [*orthogonality condition*]{} is fulfilled: $\Phi (X,Y)=0$ whenever $X\in {\mathcal{LO}}$ and $Y\in {\mathcal{LO}}'$ or if $X\in {\mathcal{RO}}$ and $Y\in {\mathcal{RO}}'$. 3. \[iorthtw\] Suppose that ${\underline{C}}$ is endowed with a weight structure $w$, ${\underline{C}}'$ is endowed with a $t$-structure $t$. Then we will say that $w$ is (left) [*orthogonal*]{} to $t$ with respect to $\Phi$ if the following condition is fulfilled: $\Phi (X,Y)=0$ if $X\in {\underline{C}}_{w\ge 0}$ and $Y\in {\underline{C}}'^{t \ge 1}$ or if $X\in {\underline{C}}_{w\le 0}$ and $Y\in {\underline{C}}'^{t \le -1}$. 4. \[ini\] We will say that $\Phi$ is [*nice*]{} if for any distinguished triangles $T=A\stackrel{l}{\to} B \stackrel{m}{\to} C\stackrel{n}{\to} A[1]$ in ${\underline{C}}$ and $X\stackrel{f}{\to} Y\stackrel{g}{\to} Z\stackrel{h}{\to}X[1]$ in ${\underline{C}}'$ we have the following: the natural morphism $p$: $$\begin{gathered} \operatorname{\operatorname{Ker}}(\Phi(A,X)\bigoplus \Phi(B,Y) \bigoplus \Phi(C,Z))\\ \xrightarrow{\begin{pmatrix} \Phi(A,-)(f) & -\Phi(-,Y)(l) &0 \\ 0& g(B) &-\Phi(-,Z)(m) \\ - \Phi(-,X)([-1](n)) & 0 &\Phi(C,-)(h) \end{pmatrix}} \\ (\Phi(A,Y) \bigoplus \Phi(B,Z) \bigoplus \Phi(C[-1],X)) \stackrel{p}{\to} \operatorname{\operatorname{Ker}}((\Phi(A,X)\bigoplus \Phi(B,Y))\\ \xrightarrow{\Phi(A,-)(f)\oplus - \Phi(-,Y)(l)} \Phi(A,Y)) \end{gathered}$$ is epimorphic. \[rort\] 1. For ${\underline{C}}'={\underline{C}}$ and $\Phi={\underline{C}}(-,-)$ the definition of orthogonal torsion pairs restricts to the one of adjacent ones. Indeed, in this case the $\Phi$-orthogonality of $s$ and $s'$ means that ${\mathcal{LO}}\perp_{{\underline{C}}} {\mathcal{LO}}'$ and ${\mathcal{RO}}\perp_{{\underline{C}}} {\mathcal{RO}}'$; these inclusions are certainly equivalent to ${\mathcal{LO}}'\subset {\mathcal{RO}}$ and ${\mathcal{RO}}\subset {\mathcal{LO}}'$, respectively. 2\. Similarly to the notions of adjacent (weight and $t$-) structures and torsion pairs, part \[iorthtw\] of Definition \[dort\] is essentially a particular case of part \[iorthop\] if one takes $w$ and $t$ associated with $s$ and $s'$, respectively, and shifts one of them; see Remarks \[rstws\](4), \[rwhop\](1), and \[rtst1\](\[it1\]). Now we give some definitions \[dortt\] 1. We will say that a weight structure $w$ on ${\underline{C}'}$ is [*right $\Phi$-orthogonal*]{} to a $t$-structure $t$ on ${\underline{C}}$ whenever $w$ is left orthogonal to $t$ with respect to the (obviously defined) duality “opposite to $\Phi$”. 2. For an $R$-linear triangulated category ${\underline{C}}$ (where $R$ is an associative commutative unital ring) we will say that an $R$-linear cohomological functor $H$ from ${\underline{C}}$ into $R-{\operatorname{Mod}}$ is [*almost of $R$-finite type*]{} whenever for any $M\in {\operatorname{Obj}}{\underline{C}}$ the $R$-module $H(M)$ is finitely generated and $H(M[i])={\{0\}}$ for $i\ll 0$ (cf. Definition \[dsatur\] and Definition 0.1 of [@neesat]). Now we prove a generalization of Proposition \[psatur\](II) that enables constructing $t$-structures orthogonal to certain weight ones; its formulation is motivated by related results of [@neesat] and [@roq]. \[psaturdu\] Assume that ${\underline{C}}\subset {\underline{C}'}$ are $R$-linear categories (for $R$ being a unital commutative ring; see Definition \[dsatur\] for some of our notation). Denote by $\Phi$ the restriction of the bifunctor ${\underline{C}'}(-,-)$ to ${\underline{C}}{{^{op}}}\times {\underline{C}'}$. 1\. Assume that functors ${\underline{C}'}\to R-{\operatorname{Mod}}$ that are corepresented by objects of ${\underline{C}}$ are precisely those ones that are of $R$-finite type as functors from ${\underline{C}}'{{^{op}}}$. Then for any bounded weight structure $w'$ on ${\underline{C}'}$ the couple $t=(t_1,t_2) =({{}^{\perp}}({\underline{C}'}_{w'\ge 1})\cap {\operatorname{Obj}}{\underline{C}}, {{}^{\perp}}({\underline{C}'}_{w'\le -1})\cap {\operatorname{Obj}}{\underline{C}})$ is a $t$-structure on ${\underline{C}}$. Moreover, $w'$ is right $\Phi$-orthogonal (see Definition \[dortt\]) to this $t$-structure. 2\. Assume that ${\underline{C}}$ is essentially small and there exists a triangulated category ${\underline{D}}$ that satisfies the following conditions: it has coproducts, ${\underline{C}'}\subset {\underline{D}}$, all objects of ${\underline{C}}$ are compact in ${\underline{D}}$, and for $N\in {\operatorname{Obj}}{\underline{D}}$ the restriction of the functor ${\underline{D}}(-,N)$ to ${\underline{C}}$ is of $R$-finite type (resp. almost of $R$-finite type) if and only if $N\in {\operatorname{Obj}}{\underline{C}'}$. Then for any bounded weight structure $w$ on ${\underline{C}}$ the couple $t'=(t_3,t_4) =(({\underline{C}}_{w\ge 1})^{\perp_{{\underline{C}'}}}, ({\underline{C}}_{w\le -1})^{\perp_{{\underline{C}'}}})$ is a $t$-structure on ${\underline{C}'}$. Moreover, $w$ is left $\Phi$-orthogonal to this $t'$. Furthermore, if $R$ is noetherian and the correspondence $N\mapsto H_N$ is a full functor from ${\underline{C}'}$ into $\operatorname{\operatorname{AddFun}_R}({\underline{C}}{{^{op}}},R-{\operatorname{mod}})$ (resp. gives an equivalence of ${\underline{C}'}$ with the subcategory of $\operatorname{\operatorname{AddFun}_R}({\underline{C}}{{^{op}}},R-{\operatorname{mod}})$ consisting of functors of $R$-finite type) then the obvious functor from ${{\underline{Ht}}}'$ into $\operatorname{\operatorname{AddFun}_R}({{\underline{Hw}}}{{^{op}}},R-{\operatorname{mod}})$ is full (resp. gives an equivalence of ${{\underline{Ht}}}'$ with the category $\operatorname{\operatorname{AddFun}_R}({{\underline{Hw}}}, R-{\operatorname{mod}})$). In both assertions the corresponding $\Phi$-orthogonalities are automatic; so we should only check that the corresponding couples are $t$-structures indeed, and study ${{\underline{Ht}}}'$ in assertion 2. 1\. This statement is rather similar to (the dual to) Proposition \[psatur\]. Axioms (i) and (ii) of Definition \[dtstr\] are obvious for $t$. Next, ${}^{\perp_{{\underline{C}}'}}({\underline{C}'}_{w'\ge 1})={\underline{C}}'_{w'\le 0}$; thus $t_1[1]\perp t_2$. It remains to verify the existence of $t$-decompositions. For $M\in {\operatorname{Obj}}{\underline{C}}$ we note that all the virtual $t$-truncations of the functors ${\underline{C}'}(M,-)$ (see Remark \[rwrange\](4)) are of $R$-finite type as considered as cohomological functors from ${\underline{C}}'{{^{op}}}$ according to Proposition \[psatur\](I). Hence they are representable by objects of ${\underline{C}}$ according to our assumptions. Thus arguments similar to that used for the proof of Proposition \[padjt\](\[ile4\]) easily allow us to conclude the proof (note here that similarly to Proposition \[padjt\](\[ile2\]) we can apply the Yoneda lemma here since ${\underline{C}}\subset {\underline{C}}'$). 2\. Since $w$ is bounded, the category ${\underline{C}}$ is densely generated by ${\underline{C}}_{w=0}$ (by Proposition \[pbw\](\[igenlm\]); cf. Remark \[rsatur\](2)) and ${{\underline{Hw}}}$ is negative in it. Moreover, we can assume that ${\underline{D}}$ is generated by ${\operatorname{Obj}}{\underline{C}}$ as its own localizing subcategory (see Remark \[rwhop\](2)); thus ${\operatorname{Obj}}{\underline{C}}^{\perp_{{\underline{D}}}}=(\cup_{i\in {{\mathbb{Z}}}}{\underline{C}}_{w=i})^{\perp_{{\underline{D}}}}={\{0\}}$. Hence we can apply [@bws Theorem 4.5.2(I)] to obtain that $t_{{\underline{D}}}=(({\underline{C}}_{w\ge 1})^{\perp_{{\underline{D}}}}, ({\underline{C}}_{w\le -1})^{\perp_{{\underline{D}}}})$ is a $t$-structure on ${\underline{D}}$.[^61] Next, $w$ is certainly left orthogonal to $t_{{\underline{D}}}$ with respect to the restriction of ${\underline{D}}(-,-)$ to ${\underline{C}}{{^{op}}}\times {\underline{D}}$. Thus Proposition 2.5.4(1) of [@bger] (cf. Proposition \[pwfil\](4)) implies that for $N\in {\operatorname{Obj}}{\underline{D}}$ the restriction of ${\underline{D}}(-,N^{t_{{\underline{D}}}\le 0})$ to ${\underline{C}}$ is isomorphic to $ \tau^{\le 0 }(H_N)$ (where $H_N$ is the restriction of ${\underline{D}}(-,N)$ to ${\underline{C}}$). Now, if $H_N$ is (almost) of $R$-finite type then this virtual $t$-truncation is so as well according to (the corresponding obvious modification of) Proposition \[psatur\](I). Hence $N^{t_{{\underline{D}}}\le 0}$ is an object of $ {\underline{C}'}$ whenever $N$ is. It obviously follows that $t_{{\underline{D}}}$ restricts to ${\underline{C}}'$, i.e., $t'=(t_3,t_4)$ is a $t$-structure on ${\underline{C}}'$. The proof of the “furthermore” part of the assertion is an easy application of Proposition \[phadj\]; see the proof of Proposition \[psatur\](3) where a particular case of this fact is established. \[roq\] 1. Let $X$ be a scheme, and take ${\underline{C}}$ being the triangulated category of perfect complexes on $X$, ${\underline{C}}_1'=D^b(X)$ (the bounded derived category of coherent sheaves on $X$), ${\underline{C}}_2'=D^-(X)$ (the bounded above category), ${\underline{D}}=D(QCoh)$ (the unbounded derived category of quasi-coherent sheaves on $X$). Certainly, ${\underline{C}}\subset {\underline{C}}_1'\subset {\underline{C}}_2'\subset {\underline{D}}$, and objects of ${\underline{C}}$ compactly generate ${\underline{D}}$. Next, assume that $X$ is proper over ${{\operatorname{Spec}\,}}R$, where $R$ is a commutative unital noetherian ring. Then Corollary 0.5 of [@neesat] says that for $N\in {\operatorname{Obj}}{\underline{D}}$ the restriction $H_N$ of the functor ${\underline{D}}(-,N)$ to ${\underline{C}}$ is of $R$-finite type (resp. almost of $R$-finite type) if and only if $N\in {\operatorname{Obj}}{\underline{C}}'_1$ (resp. $N\in {\operatorname{Obj}}{\underline{C}}'_2$). Thus we obtain the existence of the corresponding $t$-structures $t'_i$ both on ${\underline{C}}_1'$ and ${\underline{C}}_2'$; certainly, these $t$-structures are the restrictions of the $t$-structure $t_{{\underline{D}}}$ on ${\underline{D}}$ (see the proof of part 2 of the previous proposition). Moreover, the properties of the correspondence $N\mapsto H_N$ allow us to apply this part 2 to obtain a full functor from ${{\underline{Ht}}}'_2$ into $\operatorname{\operatorname{AddFun}_R}({{\underline{Hw}}}, R-{\operatorname{mod}})$ that restricts to an equivalence of ${{\underline{Ht}}}'_1$ with the latter category. It easily follows that we have ${{\underline{Ht}}}'_1={{\underline{Ht}}}'_2$ in this case. 2\. Assume in addition that $R$ is a field and $X$ is projective over ${{\operatorname{Spec}\,}}R$. Then Lemma 7.49 and Corollary 7.51(ii) of [@roq] (cf. also [@bvdb]\[Theorem A.1\]) also enable us to apply Proposition \[psaturdu\](1) for ${\underline{C}}'={\underline{C}}'_1$. 3\. So, to prove the existence of orthogonal $t$-structures when there is a duality $\Phi:{\underline{C}}{{^{op}}}\times {\underline{C}}'\to {\underline{\operatorname{Ab}}}$ using our methods one needs some version of the Brown representability condition or its dual; i.e., one should have a description of the functors $\Phi(M,-)$ for $M\in {\operatorname{Obj}}{\underline{C}}$ (resp. of $\Phi(-,N)$ for $N\in {\operatorname{Obj}}{\underline{C}}'$) that should be respected by the corresponding virtual $t$-truncations (cf. Remark \[rkrause\](II)). Now we give two simple recipes for constructing nice dualities. \[pnice\] 1. If $F:{\underline{C}}\to {\underline{D}}$ and $F':{\underline{C}}'\to {\underline{D}}$ are exact functors, then $\Phi(X,Y)={\underline{D}}(F(X),F'(Y)):{\underline{C}}{{^{op}}}\times{\underline{C}}\to {\underline{\operatorname{Ab}}}$ is a nice duality. 2\. For triangulated categories ${\underline{C}}$, ${\underline{C}_0}\subset {\underline{C}}'$, ${\underline{C}_0}$ is skeletally small, let $\Phi_0:{\underline{C}}^{op}\times {\underline{C}_0}\to {\underline{A}}$ be a duality. For any $P\in {\operatorname{Obj}}{\underline{C}}'$ denote by $H^P$ the coextension (see Proposition \[pkrause\]) to ${\underline{C}'}$ of the functor $\Phi_0(-,Y)$; denote by $\Phi$ the pairing ${\underline{C}}^{op}\times {\underline{C}'}\to {\underline{\operatorname{Ab}}}:\ \Phi(P,Y)=H^P(Y)$. Then $\Phi$ is a duality (${\underline{C}}^{op}\times {\underline{C}}'\to {\underline{A}}$); it is nice whenever $\Phi_0$ is. 1\. Easy; it suffices to note that the niceness restriction is a generalization of the axiom (TR3) of triangulated categories (any commutative square can be completed to a morphism of distinguished triangles) to the setting of dualities of triangulated categories. 2. This is the categorical dual to Proposition 2.5.6(3) of [@bger]. On “bicontinuous” dualities {#sdual2} --------------------------- Now we describe a interesting type of (nice) dualities of triangulated categories and orthogonal torsion pairs in them. We will apply it (below and in [@bgn]) in the case where ${\underline{C}}$ is a certain category of pro-objects; so it is no wonder that we will consider co-compact objects in it. So we will say that a cohomological functor ${\underline{C}}\to {\underline{A}}$ (${\underline{A}}$ is an abelian category) is pc one if it converts ${\underline{C}}$-products into ${\underline{A}}$-coproducts; recall that an object $M$ of ${\underline{C}}$ is said to be cocompact if the functor ${\underline{C}}(-,M)$ is a pc one (cf. Remark \[rwhop\](5)). \[porthop\] Let ${\underline{C}_0}$ be an essentially small common subcategory of (triangulated categories) ${\underline{C}}$ and ${\underline{C}'}$ whose objects are compact in ${\underline{C}'}$ and cocompact in ${\underline{C}}$ (and so, ${\underline{C}}$ has products and ${\underline{C}'}$ has coproducts); let ${\mathcal{P}}$ be a set of objects of ${\underline{C}_0}$, $M\in {\operatorname{Obj}}{\underline{C}}$, $N\in {\operatorname{Obj}}{\underline{C}'}$. For each $P\in {\operatorname{Obj}}{\underline{C}}$ denote by $H^P$ the coextension (see Proposition \[pkrause\]) to ${\underline{C}'}$ of the restriction to ${\underline{C}_0}$ of the functor ${\underline{C}}(P,-)$; denote by $\Phi$ the pairing ${\underline{C}}^{op}\times {\underline{C}'}\to {\underline{\operatorname{Ab}}}:\ \Phi(P,Y)=H^P(Y)$. Then the following statements are valid. 1. \[icudupa\] $\Phi$ is a nice duality of triangulated categories. 2. \[icuduadj\] Denote by ${\underline{E}}$ the colocalizing triangulated subcategory of ${\underline{C}}$ cogenerated by ${\underline{C}_0}$. Then there exists a left adjoint $L$ to the embedding ${\underline{E}}\to {\underline{C}}$ and $\Phi(-,-)\cong \Phi^{{\underline{E}}}(L(-),-)$, where $\Phi^{{\underline{E}}}$ is the restriction of $\Phi$ to ${\underline{E}}{{^{op}}}\times {\underline{C}'}$. 3. \[icontcu\] Assume that $M\in {\operatorname{Obj}}{\underline{C}}$ is a ${\underline{C}_0}$-limit of some $(M_l)$ (see Definition \[drelim\](2)). Then $\Phi(M,N)\cong \operatorname{\varinjlim}\Phi(M_l,N)$. In particular, the functor $\Phi(-,N)$ is a pc one. 4. \[icontdu\] Assume that $N\in {\operatorname{Obj}}{\underline{C}}$ is a ${\underline{C}_0}$-colimit of some $(N_l)$ (see Definition \[drelim\](1)). Then $\Phi(M,N)\cong \operatorname{\varinjlim}\Phi(M,N_l)$. In particular, the functor $\Phi(M,-):{\underline{C}'}\to {\underline{\operatorname{Ab}}}$ is a cc one. 5. \[ibiext\] The functor $\Phi$ is determined (up to a canonical isomorphism) by the following conditions: it converts ${\underline{C}}$-products and ${\underline{C}'}$-coproducts into direct sums of abelian groups, it restriction to ${\underline{C}_0}{{^{op}}}\times {\underline{C}_0}$ equals ${\underline{C}_0}(-,-)$, and it annihilates both ${}^{\perp_{{\underline{C}}}}{\underline{C}_0}\times {\operatorname{Obj}}{\underline{C}'}$ and ${\operatorname{Obj}}{\underline{C}}\times {\underline{C}_0}^{\perp_{{\underline{C}'}}}$. So, in this case we will say that $\Phi$ is the [*biextension*]{} of ${\underline{C}_0}(-,-)$ to ${\underline{C}}^{op}\times {\underline{C}'}$. 6. \[icupr\] There exists a smashing torsion pair $s'=({\mathcal{LO}}',{\mathcal{RO}}')$ for ${\underline{C}}'$ such that ${\mathcal{LO}}'$ is the ${\underline{C}'}$-strong extension-closure of ${\mathcal{P}}$ and ${\mathcal{RO}}'={\mathcal{P}}^{\perp} $. 7. \[icup\] There exists a cosmashing torsion pair $s=({\mathcal{LO}},{\mathcal{RO}})$ for ${\underline{C}}$ such that ${\mathcal{LO}}={{}^{\perp}}{\mathcal{P}}$ and ${\mathcal{RO}}$ is the strong extension-closure of ${\mathcal{P}}$ in ${\underline{C}}{{^{op}}}$. Moreover, $s$ restricts to a torsion pair $s_{{\underline{E}}}$ for ${\underline{E}}$ (see assertion \[icuduadj\] for the latter notation). 8. \[ihoport\] The torsion pairs $s$ and $s_{{\underline{E}}}$ mentioned in the previous assertion are (respectively) $\Phi$-orthogonal and $\Phi^{{\underline{E}}}$-orthogonal to the torsion pair $s'$ from assertion \[icupr\]. Moreover, ${\mathcal{LO}}={}^{\perp_{\Phi}}({\mathcal{LO}}')$ and ${\mathcal{LO}}_{{\underline{E}}}={}^{\perp_{\Phi^{{\underline{E}}}}}({\mathcal{LO}}')$. 9. \[inters\] Both ${\mathcal{LO}}'\cap {\operatorname{Obj}}{\underline{C}_0}$ and ${\mathcal{RO}}\cap {\operatorname{Obj}}{\underline{C}_0}$ are equal to the ${\underline{C}_0}$-envelope of ${\mathcal{P}}$. <!-- --> 1. Immediate from Proposition \[pnice\]. 2. The existence of $L$ is immediate from Proposition \[pcomp\](II). The rest of the assertion follows from the adjunction property of $L$ immediately. 3. It suffices to note that the coextension construction respects colimits; this is obvious (cf. also Proposition \[pkrause\](\[ikr7\])). 4. Immediate from Proposition \[pkrause\](\[ikres\]). 5. The previous assertions easily imply that $\Phi$ satisfies all the properties desired. Next, Proposition \[pkrause\](\[ikr8\]) implies that $\Phi$ is determined by its restriction to ${\underline{C}_0}{{^{op}}}\times {\underline{C}'}$ along with the conditions that it respects ${\underline{C}'}$-coproducts and annihilates ${\operatorname{Obj}}{\underline{C}}\times {\underline{C}_0}^{\perp_{{\underline{C}'}}}$. Thus one can easily conclude the proof by applying the categorical dual of the aforementioned statement. 6. This is just Theorem \[tclass\](\[iclass1\]). 7. The first part of the assertion is the dual of assertion \[icupr\]. To prove the “moreover” statement we note that the embedding ${\underline{D}}\to {\underline{C}}$ admits an (exact) left adjoint whose kernel is ${\operatorname{Obj}}{\underline{D}}^{\perp_{{\underline{C}}}}={\mathcal{P}}^{\perp_{{\underline{C}}}}$ (by the dual to Proposition \[pcomp\](II.2)). Thus it remains to apply (the dual to) Proposition \[phopft\](II.1). 8. To verify the first part of the assertion it certainly suffices to prove that $s\perp_{\Phi}s'$. Now recall that all the functors of the type $\Phi(M,-):{\underline{C}'}\to {\underline{\operatorname{Ab}}}$ are cc ones and functors of the type $\Phi(-,N):{\underline{C}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ are pc-ones (see assertions \[icontcu\] and \[icontdu\]). Thus Theorem \[tclass\](\[iclass5\]) (along with its dual) reduces the $\Phi$-orthogonality checks to the following ones: $\Phi(X,Y)=0$ if either $X\in {\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ and $Y\in {\mathcal{P}}^{\perp_{{\underline{C}}}}$ or if $X\in {}^{\perp_{{\underline{D}}}}{\mathcal{P}}$ and $Y\in {\mathcal{P}}\subset {\operatorname{Obj}}{\underline{D}}$. Thus it suffices to note that $\Phi(A,B)$ is isomorphic to ${\underline{C}'}(A,B)$ if $A\in {\operatorname{Obj}}{\underline{C}_0}(\subset {\operatorname{Obj}}{\underline{C}})$ and is isomorphic to ${\underline{C}}(A,B)$ if $B\in {\operatorname{Obj}}{\underline{C}_0}(\subset {\operatorname{Obj}}{\underline{C}'})$. The latter statements are immediate from Proposition \[pkrause\] (parts I.\[ikr1\] and I.\[ikrtriv\], respectively). Now, to prove the “moreover” statement it remains to note that ${}^{\perp_{\Phi^{{\underline{E}}}}}({\mathcal{LO}}')\subset {\mathcal{LO}}$ since ${\mathcal{LO}}'$ contains ${\mathcal{P}}$. 9. Immediate from Theorem \[tclass\](\[iclass2\]). \[rcudu\] 1. \[ircocomp\] One may say that all objects of ${\underline{C}}$ are “compact with respect to $\Phi$” and objects of ${\underline{C}'}$ are “cocompact with respect to $\Phi$”; see Proposition \[porthop\](\[icontcu\], \[icontdu\]). Note that both of these properties fail for the duality ${\underline{C}}(-,):{\underline{C}}{{^{op}}}\times {\underline{C}}\to {\underline{\operatorname{Ab}}}$ (and ${\underline{C}'}={\underline{C}}$); so one may say that $\Phi$ is a “regularized Hom bifunctor” that is “bicontinuous”. 2. \[ibcf\] Now assume that $\Phi:{\underline{C}}{{^{op}}}\times {\underline{C}'}\to {\underline{\operatorname{Ab}}}$ is any duality satisfying these bicontinuity conditions and ${\underline{C}'}$ satisfies the Brown representability condition (this is certainly the case whenever ${\underline{C}'}$ is compactly generated). Then one can define a curious functor ${\underline{C}}\to {\underline{C}'}$ as follows. We note that for any $P\in {\operatorname{Obj}}{\underline{C}}$ there exists its “$\Phi$-Brown-Comenetz dual” ${\mathcal{BCD}}(P)\in {\operatorname{Obj}}{\underline{C}'}$ that ${\underline{C}'}$-represents the functor ${\underline{\operatorname{Ab}}}(\Phi(P,-), {{\mathbb{Q}}}/{{\mathbb{Z}}})$ (since this functor is obviously a cp one). Moreover, the correspondence ${\mathcal{BCD}}$ is easily seen to be exact; it also respects products. Next, for the torsion pairs $s$ and $s'$ as above we obviously have ${\mathcal{BCD}}({\mathcal{LO}})\subset {\mathcal{RO}}'{{}^{\perp}}$ and ${\mathcal{BCD}}({\mathcal{RO}})\subset {\mathcal{LO}}'{{}^{\perp}}={\mathcal{RO}}'$ (see Proposition \[psymb\](I.\[isbcd\]). Lastly, if ${\mathcal{P}}$ is suspended then for the corresponding $w$ and $t$ (see Corollary \[cwt\] below) and for the weight structure $w'$ right adjacent to $t$ (whose existence is given by Corollary \[csymt\](2)) we obtain that ${\mathcal{BCD}}$ is weight-exact (with respect to $w$ and $w'$; see Definition \[dwso\](\[idwe\])). Note however that the existence of $\Phi$ is a stronger assumption (in this case) then the existence of a weight-exact functor ${\mathcal{BCD}}$ that respects products since the former condition implies that the image of ${\mathcal{BCD}}$ consists of “Brown-Comenetz duals” only. We will demonstrate the utility of dualities in Theorem \[tab5\] below. 3. \[ir1\] For any ${\underline{C}}$ and ${\underline{C}'}$ as in our proposition and their small subcategories ${\underline{C}}_0$ and ${\underline{C}}_0'$ respectively, one can extend any (nice) duality ${\underline{C}}_0{{^{op}}}\times {\underline{C}}_0'\to {\underline{\operatorname{Ab}}}$ first to a duality ${\underline{C}}{{^{op}}}\times {\underline{C}}_0'\to {\underline{\operatorname{Ab}}}$ using the dual to Proposition \[pkrause\] and next proceed as above to obtain $\Phi: {\underline{C}}{{^{op}}}\times {\underline{C}}'\to {\underline{\operatorname{Ab}}}$. It is easily seen that some of the statements proved above have natural (and easy to prove) analogues for this setting.[^62] Note also that a nice duality ${\underline{C}}_0{{^{op}}}\times {\underline{C}}_0'\to {\underline{\operatorname{Ab}}}$ can be obtained from any exact functors ${\underline{C}}_0\to {\underline{D}}$ and ${\underline{C}}'_0\to {\underline{D}}$ (see Proposition \[pnice\](1)). 4. \[ir2\] Dually, one may start from “coextending” a duality ${\underline{C}}_0{{^{op}}}\times {\underline{C}}_0'\to {\underline{\operatorname{Ab}}}$ to a duality ${\underline{C}_0}{{^{op}}}\times {\underline{C}}'\to {\underline{\operatorname{Ab}}}$ and next extend the result to a duality ${\underline{C}}{{^{op}}}\times {\underline{C}}'\to {\underline{\operatorname{Ab}}}$ (using the dual to Proposition \[pkrause\]). Combining part \[ikrchar\] and \[ikr7\] of this proposition we obtain the duality obtained this way is isomorphic to the one described above (and obtained in the “reverse order”). Note also that this statement easily follows from the corresponding generalization of Proposition \[porthop\](\[ibiext\]) whenever all objects of ${\underline{C}_0}'$ are compact in ${\underline{C}'}$ and objects of ${\underline{C}_0}$ are cocompact in ${\underline{C}}$. 5. \[ir3\] If one applies the “reverse biextension method” of part \[ir2\] of this remark to the duality ${\underline{C}_0}(-,-)$ (that was the “starting one” in our proposition) then the “intermediate” duality ${\underline{C}_0}{{^{op}}}\times {\underline{C}}'\to {\underline{\operatorname{Ab}}}$ would be the restriction of the bi-functor ${\underline{C}'}(-,-)$ to ${\underline{C}_0}{{^{op}}}\times {\underline{C}}'$; see Proposition \[pkrause\](\[ikrtriv\]). 6. \[ikrausl\] An interesting family of examples for our proposition can easily be constructed using Theorem 4.9 of [@krauslender]. An application: hearts of compactly generated $t$-structures are “usually” Grothendieck abelian {#sgengroth} ----------------------------------------------------------------------------------------------- First we list the consequences of Proposition \[porthop\] (essentially) in the case where $s$ is a weight structure and $s'$ is a $t$-structure. \[cwt\] In the setting of the previous proposition (and adopting its notation including the one of its part \[icuduadj\]) assume that ${\mathcal{P}}$ is a suspended subset of ${\operatorname{Obj}}{\underline{C}_0}$ (i.e., ${\mathcal{P}}[1]\subset {\mathcal{P}}$). Then the following statements are valid. 1. \[ic1\] For any $N'\in {\operatorname{Obj}}{\underline{C}'}$ denote by $H^{N'}$ the extension to ${\underline{C}}$ of the functor $H_0^{N'}:{\underline{C}_0}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ (that is the restriction of ${\underline{C}'}(-,N')$ to ${\underline{C}_0}$) obtained via the dual to Proposition \[pkrause\]. Then the bi-functor $\Phi:{\underline{C}}{{^{op}}}\times {\underline{C}'}\to {\underline{\operatorname{Ab}}}: (M,N')\mapsto H^{N'}(M)$ is a nice duality that is naturally isomorphic to the one given by Proposition \[porthop\](\[icudupa\]). 2. \[iccc\] For any $M\in {\operatorname{Obj}}{\underline{C}}$ the functor $\Phi(M,-):{\underline{C}'}\to {\underline{\operatorname{Ab}}}$ is a cc one. 3. \[ic2\] The functor $L:{\underline{C}}\to{\underline{E}}$ (left adjoint to the embedding ${\underline{E}}\to {\underline{C}}$) respects products, ${\underline{C}_0}$-limits, and is identical on ${\operatorname{Obj}}{\underline{E}}$. 4. \[ic15\] The restriction $\Phi^{{\underline{E}}}$ of $\Phi$ to ${\underline{E}}{{^{op}}}\times {\underline{C}'}$ is a nice duality also, and we have $\Phi(-,-)\cong \Phi^{{\underline{E}}}(L(-),-)$. 5. \[ic25\] For any set of $M_i\in {\operatorname{Obj}}{\underline{C}}$ and $N\in {\operatorname{Obj}}{\underline{C}'}$ we have $\Phi(\prod M_i,N)\cong \bigoplus \Phi (M_i,N)$. 6. \[ict\] There exists a smashing $t$-structure $t$ on ${\underline{C}'}$ such that ${\underline{C}'}^{t\le 0}$ is the smallest coproductive extension-closed subclass of ${\operatorname{Obj}}{\underline{C}'}$ containing ${\mathcal{P}}$ and ${\underline{C}'}^{t\ge 0}={\mathcal{P}}{{}^{\perp}}[1]$. Moreover, ${\underline{C}'}^{t\le 0}$ also equals the big hull of ${\mathcal{P}}$ in ${\underline{C}'}$. 7. \[icw\] There exists a cosmashing weight structure $w$ on ${\underline{C}}$ such that ${\underline{C}}_{w\ge 0}$ is the big hull of ${\mathcal{P}}$ in ${\underline{C}}{{^{op}}}$ and ${\underline{C}}_{w\le 0}=({}^{\perp_{{\underline{C}}}}{\mathcal{P}})[1]$. Moreover, ${\underline{C}}_{w\le 0}$ is the extension-closure of $({}^{\perp_{{\underline{E}}}}{\mathcal{P}})[1]\cup {{}^{\perp}}(\cup_{i\in {{\mathbb{Z}}}}{\mathcal{P}})$. 8. \[icwe\] The couple $w_{{\underline{E}}}= (({}^{\perp_{{\underline{E}}}}{\mathcal{P}})[1], {\underline{C}}_{w\ge 0})$ is a cosmashing weight structure on ${\underline{E}}$, $L$ is weight-exact (with respect to $w$ and $w_{{\underline{E}}}$, respectively), and ${\underline{C}}_{w=0}={\underline{E}}_{w_{{\underline{E}}}=0}$. Moreover, if ${\mathcal{P}}$ densely generates ${\underline{C}_0}$ then $w_{{\underline{E}}}$ is right non-degenerate. 9. \[icort\] $w$ and $w_{{\underline{E}}}$ are orthogonal to $t$ with respect to $\Phi$ and $\Phi^{{\underline{E}}}$, respectively. Moreover, ${\underline{C}}_{w\le 0}={}^{\perp_{\Phi}}{\underline{C}'}^{t\le -1}$ and ${\underline{E}}_{w_{{\underline{E}}}\le 0}={}^{\perp_{\Phi^{{\underline{E}}}}}{\underline{C}'}^{t\le -1}$. 10. \[iwheart\] Choose some $w_{{\underline{E}}}$-weight complexes for elements of ${\mathcal{P}}$; denote their terms by $P^k_l$. Then the object $I=\prod P^k_l$ cogenerates ${{\underline{Hw}}}$ (cf. Corollary \[cvttbrown\](I.2)), i.e., any object of ${{\underline{Hw}}}$ is a retract of a product of copies of $I$. 11. \[icint\] The ${\underline{C}_0}$-envelope of ${\mathcal{P}}$ equals both ${\underline{C}'}^{t\le 0}\cap {\operatorname{Obj}}{\underline{C}_0}$ and ${\underline{C}}_{w\ge 0}\cap {\operatorname{Obj}}{\underline{C}_0}$. 12. \[icvirt\]If ${\underline{A}}$ is an AB5 category then for the extension (obtained via the dual to Proposition \[pkrause\]) $H:{\underline{C}}{{^{op}}}\to {\underline{A}}$ of a cohomological functor $H_0$ from ${\underline{C}_0}$ into ${\underline{A}}$ all its virtual $t$-truncations are extended functors (in the same sense as $H$ is). \[ic1\]. Easy from Proposition \[porthop\](\[ibiext\]); see Remark \[rcudu\] (\[ir2\],\[ir3\]). \[iccc\]. Immediate from the previous assertion combined with Proposition \[porthop\](\[icontdu\]). \[ic2\]. Obviously, $L$ respects ${\underline{C}_0}$-limits and is identical on ${\operatorname{Obj}}{\underline{E}}$. Dualizing Proposition \[prtst\](\[it4sm\]) we obtain that $L$ respects products. \[ic15\]. Certainly, (arbitrary) restrictions of nice dualities are nice dualities also. It remains to apply Proposition \[porthop\](\[icuduadj\]). \[ic25\]. Immediate from Proposition \[porthop\](\[icontcu\]). \[ict\]. Certainly, ${\mathcal{P}}$ generates a smashing torsion pair $s'=({\mathcal{LO}}',{\mathcal{RO}}')$ for ${\underline{C}'}$ with ${\mathcal{RO}}'={\underline{C}'}^{t\ge 0}$ and ${\mathcal{LO}}'$ being the big hull of ${\mathcal{P}}[1]$; see Proposition \[porthop\](\[icupr\]). Hence the assertion follows from Corollary \[csymt\](1) easily. \[icw\]. The first part of the assertion is immediate from Proposition \[porthop\](\[icup\]); see Remark \[rwhop\](1). The second part is immediate from Proposition \[phopft\](II.1). \[icwe\]. Proposition \[porthop\](\[icup\]) gives the existence of $w_{{\underline{E}}}$, which certainly implies the weight-exactness of $L$ and ${\underline{C}}_{w=0}={\underline{E}}_{w_{{\underline{E}}}=0}$. To prove the “moreover” part one should note that ${}^{\perp_{{\underline{E}}}}{\operatorname{Obj}}{\langle}{\mathcal{P}}{\rangle}={}^{\perp_{{\underline{E}}}}{\operatorname{Obj}}{\underline{C}_0}={\{0\}}$ according to (the dual to) Proposition \[pcomp\](I.1); it remains to apply (the dual to) Remark \[rwhop\](8). \[icort\]. Immediate from Proposition \[porthop\](\[ihoport\]) (see Remark \[rort\](2)). \[iwheart\]. Immediate from (the dual to) Proposition \[ppcoprws\](\[icopr7\]) (cf. Corollary \[cvttbrown\](I.2)). \[icint\]. This is just a particular case of Proposition \[porthop\](\[inters\]). \[icvirt\]. Immediate from Remark \[rkrause\](II.2). Now we apply this corollary to the study of compactly generated $t$-structures. First we study the AB5 condition. \[tab5\] Let ${\underline{D}}$ be a triangulated category having coproducts and $t$ be a smashing $t$-structure on it. Then the following statements are valid. 1\. Assume moreover that ${{\underline{Ht}}}$ is an AB5 category and ${\underline{D}}$ is compactly generated. Then the (essentially small) class $H_0^t({\underline{D}}^{{{\aleph_0}}})$ generates ${{\underline{Ht}}}$ (and so, the coproduct of any small skeleton of $H_0^t({\underline{D}}^{{{\aleph_0}}})$ generates it also).[^63] 2\. Assume that $t$ is generated by a suspended set ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{D}}$ of compact objects. Denote by ${{\underline{D}}_0}$ the subcategory of ${\underline{D}}$ that is densely generated (see §\[snotata\]) by ${\mathcal{P}}$ and assume that there exists a triangulated category ${\underline{D}}'$ that contains ${\underline{D}}_0^{op}$ as a full subcategory of compact objects. Then ${{\underline{Ht}}}$ is a Grothendieck abelian category, and there exists a faithful exact functor ${\mathcal{S}}:{{\underline{Ht}}}\to {\underline{\operatorname{Ab}}}$ that respects coproducts. 1\. Proposition \[pkrause\](\[ikr8\]) implies that the functor $H_0^t:{\underline{D}}\to {{\underline{Ht}}}$ is coextended (from the subcategory ${\underline{D}}'$ of ${\underline{D}}$ whose object class equals ${\underline{D}}^{{{\aleph_0}}}$). Hence for any $M\in {\operatorname{Obj}}{\underline{D}}$ and a family $C_M^j\in {\underline{D}}^{{{\aleph_0}}}$ as in (\[ekrause\]) we obtain that $H_0^t(M)$ is a ${{\underline{Ht}}}$-quotient of $\coprod H_0^t(C_M^j)$. Hence the class $H_0^t({\underline{D}}^{{{\aleph_0}}})$ generates ${{\underline{Ht}}}$ indeed. 2. The embedding into ${\underline{D}}$ of its localizing subcategory generated by ${{\underline{D}}_0}$ possesses an exact right adjoint (by Proposition \[pcomp\](II.2)); hence Remark \[rwhop\](2) allows us to assume that ${\underline{D}}$ equals this subcategory. Thus ${\underline{D}}$ is compactly generated. Hence assertion 1 says that it suffices to verify whether ${{\underline{Ht}}}$ is an AB5 category. Next, ${{\underline{Ht}}}$ is an AB4 category according to Proposition \[prtst\](\[it2\]). Hence it is cocomplete; thus to check that it is AB5 we should prove that filtered colimits of ${{\underline{Ht}}}$-monomorphisms are ${{\underline{Ht}}}$-monomorphic. Thus it suffices to construct ${\mathcal{S}}$ (since small colimits can be expressed in terms of coproducts). Note also that an exact functor (of abelian categories) is faithful if and only if it is conservative (and this is equivalent to the assumption that ${\mathcal{S}}$ does not kill non-zero objects). Next we note that exact functors ${{\underline{Ht}}}\to {\underline{\operatorname{Ab}}}$ are precisely the restrictions to ${{\underline{Ht}}}$ of those homological functors ${\underline{C}}\to {\underline{\operatorname{Ab}}}$ that annihilate ${\underline{C}}^{t\le -1}\cup {\underline{C}}^{t\ge 1}$. Hence it suffices to find a cc functor ${\mathcal{S}}^{{\underline{D}}}:{\underline{D}}\to {\underline{\operatorname{Ab}}}$ that kills ${\underline{C}}^{t\le -1}\cup {\underline{C}}^{t\ge 1}$ and whose restriction to ${{\underline{Ht}}}$ is conservative. We start with constructing a “big” conservative family of cc functors ${\underline{D}}\to {\underline{\operatorname{Ab}}}$ satisfying the vanishing condition above; we call them [*stalk*]{} ones for the reason that will be explained in Remark \[rsheaves\](1) below.[^64] For this purpose we apply the previous corollary for ${\underline{C}}'$ equal to ${\underline{D}}$ and ${\underline{C}}={\underline{D}}'^{op}$. We obtain the existence of a duality $\Phi:{\underline{C}}{{^{op}}}\times {\underline{D}}\to {\underline{\operatorname{Ab}}}$ and a weight structure $w$ on ${\underline{C}}$ that is (left) $\Phi$-orthogonal to $t$. Our stalk functors are the functors $\Phi(P,-):{\underline{D}}\to {\underline{\operatorname{Ab}}}$ for $P$ running through ${\underline{C}}_{w=0}$. The stalk functors are certainly exact (by the definition of a duality); they annihilate ${\underline{C}}^{t\le -1}\cup {\underline{C}}^{t\ge 1}$ since $w\perp_{\Phi}t$. The stalk functors are cc according to Corollary \[cwt\](\[iccc\]). Let us verify the conservativity of our family. Let $N$ be a non-zero element of ${\underline{D}}^{t=0}$; we should verify that the restriction $A$ of the functor $\Phi_N=\Phi(-, N):{\underline{D}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ to ${{\underline{Hw}}}{{^{op}}}$ is not zero. Now, the $\Phi$-orthogonality of $w$ to $t$ along with Proposition \[pwrange\](\[iwrpure\]) implies that $\Phi_N$ is a pure functor; hence it equals $H_{{\mathcal{A}}}$ in the notation of Definition \[drange\](2). Thus to check that $A\neq 0$ it suffices to prove that $\Phi_N\neq 0$. Now, the class ${\operatorname{Obj}}{{\underline{D}}_0}$ Hom-generates ${\underline{D}}$; hence there exists $M_0\in {\operatorname{Obj}}{{\underline{D}}_0}$ such that ${\underline{D}}(M_0,N)\cong \Phi(M_0, N)\neq{\{0\}}$. Lastly, we recall that ${{\underline{Hw}}}$ has a cogenerator (see Corollary \[cwt\](\[iwheart\])). Thus according to Corollary \[cwt\](\[ic25\]) one can take ${\mathcal{S}}^{{\underline{D}}}$ to be the stalk functor corresponding to this cogenerator. \[rab5\] 1. \[irrel\] Note that in the case where ${{\underline{D}}_0}{{^{op}}}$ embeds into the subcategory of compact objects of ${\underline{D}}$ (in particular, this is the case if the latter category is anti-isomorphic to itself) one can take ${\underline{D}}'={\underline{D}}$. A toy example of this situation is ${\underline{D}}$ being the derived category $D(R-{\operatorname{Mod}})$, where $R$ is a commutative ring; ${{\underline{D}}_0}$ is the category of perfect complexes. More generally, for ${\underline{D}}=R-{\operatorname{Mod}}$, where $R$ is a not (necessarily) commutative ring, one may take ${\underline{D}}'$ being the derived category of right $R$-modules; this example has natural “differential graded” (see §5.2 of [@bger]) and probably “spectral” (see [@schwmod]) generalizations. On the other hand, Corollary \[cgdb\](\[iab5\]) below demonstrates that our theorem can be applied for ${\underline{D}}$ being the homotopy category of an arbitrary proper simplicial stable (Quillen) model category.[^65] Hence our theorem gives a positive answer to Question 3.8 of [@parrasao] for a really wide range of triangulated categories. Recall also that Theorem 3.7 of loc. cit. says that [**countable**]{} colimits in ${{\underline{Ht}}}$ are exact for any compactly generated $t$.[^66] Thus our theorem demonstrates once again that “weight structures can shed some light” on $t$-structures; cf. Corollary \[csymt\]. The author wonders whether one can mimic the argument above (and so, obtain certain stalk functors) using the “naive” category ${\operatorname{Pro}}-{{\underline{D}}_0}$ instead of ${\underline{D}}$ (note that ${\operatorname{Pro}}-{{\underline{D}}_0}$ is a [*pro-triangulated*]{} category that is “very rarely” triangulated). 2. \[ismgen\] The author also wonders whether one can describe a set of generators for ${{\underline{Ht}}}$ that is smaller than that described in part 1 of our theorem. If $t$ is generated by a set ${\mathcal{P}}$ of compact objects then a natural candidate here is the (essentially small) class of $H_0^t(P)$ for $P$ running through elements of ${\underline{D}}^{t\le 0}\cap {\underline{D}}^{{{\aleph_0}}}$. Note that to prove that this class generates ${{\underline{Ht}}}$ is suffices to describe a generating family of stalk functors that could be presented as colimits of functors corepresented by elements of ${\underline{D}}^{t\le 0}\cap {\underline{D}}^{{{\aleph_0}}}$ (see Proposition \[pgen\](\[ipgen1\])). In particular, the proof of [@bondegl Proposition 4.2.10] is essentially an argument of this sort. Moreover, the argument described in Remark \[rkrause\](I.2) above yields that ${{\underline{Ht}}}$ is generated by the (even smaller) class $H_0^t({\mathcal{Q}})$, where $Q$ is the smallest subclass of ${\operatorname{Obj}}{\underline{C}}$ containing ${\mathcal{P}}$ such that for any triangle $Z[-1]\to X\to Y\to Z$ with $Y,Z\in {\mathcal{Q}}$ the object $X$ belongs to ${\mathcal{Q}}$ also. 3. \[irpostov\] Proposition \[porthop\](\[inters\]) (cf. also Remark \[rnewt\](2)) certainly gives a one-to-one correspondence between weight structures (resp. $t$-structures) generated by subsets of ${\operatorname{Obj}}{{\underline{D}}_0}$ in ${\underline{D}}$ and $t$-structures (resp. weight structures) generated by these sets in ${\underline{D}}'$. This observation was very successively applied in [@postov §4]. However, (the proof of) our theorem (cf. also the next part of this remark) demonstrates that introducing a duality between ${\underline{D}}'^{op}$ and ${\underline{D}}$ can give information that can hardly be obtained if ${\underline{D}}$ and ${\underline{D}}'$ are considered “separately” only. 4. \[iremb\] Proposition 6.2.1 of [@bger] suggests the following conjecture: ${{\underline{Ht}}}$ is equivalent to the category of those functors ${{\underline{Hw}}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ that respect coproducts. The conjecture certainly contains more information on ${{\underline{Ht}}}$ than Theorem \[tab5\] (along with its proof); however, this description of ${{\underline{Ht}}}$ may be not that “useful” (since ${{\underline{Hw}}}$ can be rather “complicated”). 5. \[ir4\] An interesting question (that is closely related to the aforementioned conjecture) is which homological functors from ${\underline{D}}$ (resp. from ${\underline{D}}$) are [*$\Phi$-corepresentable*]{}, i.e., have the form $\Phi(-,N)$ (resp. $\Phi(M,-)$) for some $N\in {\operatorname{Obj}}{\underline{D}}$ (resp. $M\in {\operatorname{Obj}}{\underline{D}}'$). Now, the functors of the form $\Phi(-,N):{\underline{D}}'\to {\underline{\operatorname{Ab}}}$ are precisely the “extensions” (obtained using the dual to Proposition \[pkrause\]) of the functors that are ${\underline{D}}$-represented on ${{\underline{D}}_0}$. In the case where ${{\underline{D}}_0}$ is [*countable*]{} (i.e., its object class is essentially countable and all morphism sets are at most countable) all cohomological functors from ${{\underline{D}}_0}$ into ${\underline{\operatorname{Ab}}}$ are represented by objects of ${\underline{D}}$ according to Theorem 5.1 of [@neebrh]; so, we obtain a complete description of $\Phi$-representable functors in this case. Unfortunately, this argument cannot be extended to the case of a general ${{\underline{D}}_0}$. 6. \[istalks\] The stalk functors $\Phi(P,-)$ for $P\in {\underline{D}}'^{op}_{w=0}$ in the proof of Theorem \[tab5\] essentially play the role of functors corepresented by $t$-projective objects of ${\underline{C}}$ (see Definition \[dpt\]). Note however we cannot have “enough” of the latter unless ${{\underline{Ht}}}$ has enough projectives (see Proposition \[pgen\](\[ipgen25\])). If we assume in addition the existence of (a set of) “compact generators” for $P_t$ (that are necessary to obtain enough cc functors) then it would imply that ${{\underline{Ht}}}$ is isomorphic to $\operatorname{\operatorname{AddFun}}({\underline{B}},{\underline{\operatorname{Ab}}})$ for ${\underline{B}}$ being the corresponding small additive category. Certainly, ${{\underline{Ht}}}$ “rarely” can be presented in this form; (cf. Remark \[rsheaves\](1) below). Hence constructing a duality of ${\underline{D}}$ with an “auxiliary” category ${\underline{D}}'$ is necessary for the proof of Theorem \[tab5\]. 7. [irotimes]{} In Proposition \[porthop\] the “starting duality” was ${\underline{C}_0}(-,-)$. It is an interesting question whether Theorem \[tab5\] can be generalized by treating (biextensions of) other dualities (cf. Remark \[rcudu\](\[ir1\],\[ir2\])). Anyway, it appears that one can construct “interesting” dualities using various tensor products. Assume that we are given triangulated categories ${\underline{C}}$ and ${\underline{C}'}$ as above, a triangulated category ${\underline{E}}$ having coproducts, a bi-exact functor $\otimes: {\underline{C}}{{^{op}}}\times {\underline{C}'}\to {\underline{E}}$ that respects coproducts when any of the arguments is fixed, and a cc functor $H:{\underline{E}}\to {\underline{A}}$ (for some abelian ${\underline{A}}$). Then $\Phi(-,-)=H(-\otimes -)$ is a duality ${\underline{C}}{{^{op}}}\times {\underline{C}'}\to {\underline{A}}$ that converts ${\underline{C}}$-products and ${\underline{C}'}$-coproducts into ${\underline{A}}$-coproducts. Certainly, one may take $H$ being the functor corepresented by a compact object of ${\underline{E}}$ (and ${\underline{A}}={\underline{\operatorname{Ab}}}$). Note also that $\Phi$ is canonically characterized by its “coproductivity properties” along with its restriction $\Phi_0$ to ${\underline{C}}'_0{}{{^{op}}}\times {\underline{C}_0}$ whenever ${\underline{C}_0}$ and ${\underline{C}}'_0{}^{op}$ are categories of compact objects in ${\underline{C}}{{^{op}}}$ and ${\underline{C}}'$ (respectively) that Hom-generate these categories (and so, generate them as their own localizing subcategories). The author believes that dualities of this sort may be useful for “computing” dualities of the type treated in Proposition \[porthop\] in the case where ${\underline{C}}{{^{op}}}={\underline{C}'}$, ${\underline{C}_0}$ is self-dual with respect to the tensor product on ${\underline{C}'}$ (cf. part \[irrel\] of this remark) and Hom-generates ${\underline{C}'}$, and the unit object ${{\pmb{1}}}_{{\underline{C}'}}$ is compact in ${\underline{C}'}$. Note that all these condition are fulfilled for ${\underline{C}'}$ being the stable homotopy category of (“topological”) spectra; we will say more on this setting (that was essentially treated in [@prospect]) in Remark \[rsheaves\](4) below. 8. \[krauseideals\] Recall that Corollary 4.7 of [@krause] gives a description of all smashing (see Remark \[rtst2\](\[ismashs\])) subcategories of a compactly generated triangulated category ${\underline{D}}$ in terms of certain ideals of morphisms in its subcategory ${{\underline{D}}_0}$ of compact objects. Along with Theorem 4.9 of ibid. and the example from [@kellerema] this appears to yield an example of a not compactly generated smashing triangulated subcategory $L$ of ${\underline{D}}$ such that the corresponding “shift-stable $t$-structure” (see Remark \[rtst2\](\[ismashs\]) again) possesses a (left) $\Phi$-orthogonal shift-stable weight structure in the corresponding ${\underline{D}}'$. The authors wonders whether one can also obtain similar non-shift-stable examples. Relation to triangulated categories of pro-objects {#sprospectra} -------------------------------------------------- Now we describe a method for constructing a vast family of examples for Proposition \[porthop\] (and so, also of Corollary \[cwt\]) along with Theorem \[tab5\]. Its main ingredient is a construction of a triangulated category of “homotopy pro-objects” for a stable model category that is a straightforward application of the results of [@tmodel]. So let ${\mathcal{M}}$ be a proper simplicial stable (Quillen) model category; denote its homotopy category by ${\underline{C}'}$. We construct another model category ${\operatorname{Pro}-\mathcal{M}}$ whose underlying category is the category of (filtered) pro-objects of ${\mathcal{M}}$ (cf. §5 of [@tmodel]). \[rproobj\] In [@tmodel] pro-objects were defined via filtered diagrams, i.e., via contravariant functors from filtered small categories. However the author prefers (for the reasons of minor notational convenience) to consider inverse limits indexed by filtered sets instead. The latter notion is certainly somewhat more restrictive formally; however, Theorem 1.5 of [@adr] says that any “categorical” filtered colimit can be presented as a certain “set-theoretic” one. It easily follows that it is no difference between the usage of these two notions (for the purposes of the current papers as well as for that of [@bgn]); cf. the discussion in §2 of [@isalim]. We endow ${\operatorname{Pro}-\mathcal{M}}$ with the [*strict*]{} model structure; see §5.1 of ibid. (so, weak equivalences and cofibrations are [*essential levelwise*]{} weak equivalences and cofibrations of pro-objects). An important observation here is that this model structure is a particular case of a [*$t$-model structure*]{} in the sense of §6 of ibid. if one takes the following “totally degenerate” $t$-structure $t_{deg}$ on ${\underline{C}'}$: ${\underline{C}'}^{t_{deg}\ge 0}={\operatorname{Obj}}{\underline{C}'}$, ${\underline{C}'}^{t_{deg}\le 0}={\{0\}}$; see Remark 6.4 of ibid. Indeed, one can take the following functorial factorization of morphisms: for $f\in {\mathcal{M}}(X,Y)$ (for $X,Y\in {\operatorname{Obj}}{\mathcal{M}}$) we can present it as $f\circ {\operatorname{id}}_X$; note that ${\operatorname{id}}_X$ is an [*$n$-equivalence*]{} and $f$ is a [*co-$n$-equivalence*]{} in the sense of Definition 3.2 of ibid. for any $n\in {{\mathbb{Z}}}$ (pay attention to Remark \[rstws\](3)!). Now let us describe some basic properties of ${\operatorname{Pro}-\mathcal{M}}$ and its homotopy category ${\underline{C}}$ (we will apply some of them below, whereas other ones are important for [@bgn]). The pro-object corresponding to a projective system $M_i$ for $i\in I$ where $I$ is an inductive set and $M_i\in {\operatorname{Obj}}{\mathcal{M}}$, will be denoted by $(M_i)$. Note that $(M_i)$ is precisely the (inverse) limit of the system $M_i$ in ${\operatorname{Pro}-\mathcal{M}}$ (by the definition of morphisms in this category). \[pgdb\] Let $X_i,Y_i,Z_i\ i\in I$, be projective systems in ${\mathcal{M}}$. Then the following statements are valid. 1\. ${\operatorname{Pro}-\mathcal{M}}$ is a proper stable simplicial model category; hence ${\underline{C}}$ is triangulated. 2. If some morphisms $X_i\to Y_i$ for all $i\in I$ yield a compatible system of cofibrations (resp. of weak equivalences; resp. some couples of morphisms $X_i\to Y_i\to Z_i$ yield a compatible system of homotopy cofibre sequences) then the corresponding morphism $(X_i)\to (Y_i)$ is a cofibration also (resp. a weak equivalence; resp. the couple of morphisms $(X_i)\to (Y_i)\to (Z_i)$ is a homotopy cofibre sequence). 3. The natural embedding $c:{\mathcal{M}}\to {\operatorname{Pro}-\mathcal{M}}$ is a left Quillen functor; it also respects weak equivalences and fibrations. 4\. For any $N\in {\operatorname{Obj}}{\mathcal{M}}$ we have ${\underline{C}}((X_i),c(N))\cong \operatorname{\varinjlim}{\underline{C}'}(X_i,N)$. In particular, the homotopy functor ${\operatorname{Ho}}(c):{\underline{C}'}\to {\underline{C}}$ is a full embedding, and $(X_i)$ is a ${\underline{C}'}$-limit of $X_i$ in ${\underline{C}}$ (with respect to this embedding; see Definition \[drelim\](2)). 5\. More generally, for any projective system $\{M_i\}$ in $ {\operatorname{Pro}-\mathcal{M}}$ and any $N\in {\operatorname{Obj}}{\underline{C}'}$ the inverse limit of $M_i$ exists in ${\operatorname{Pro}-\mathcal{M}}$ and we have ${\underline{C}}(\operatorname{\varprojlim}M_i,c(N))\cong \operatorname{\varinjlim}_{i\in I} {\underline{C}}(M_i,c(N))$. 6\. ${\underline{C}}$ has products and all objects of ${\operatorname{Ho}}(c)({\underline{C}'})$ are cocompact in ${\underline{C}}$. 7. The class ${\operatorname{Ho}}(c)({\operatorname{Obj}}{\underline{C}'})$ cogenerates ${\underline{C}}$. 1\. Theorems 6.3 and 6.13 of [@tmodel] yield everything except the existence of functorial factorizations for morphisms in ${\operatorname{Pro}-\mathcal{M}}$. The existence of functorial factorizations is given by Theorem 1.3 of [@fufa] (see the text following Remark 1.5 of ibid.). One can also deduce this statement from [@prospect Remark 4.5]. 2\. The first two parts of the assertion are contained in the definition of the strict model structure. The last part follows from the previous ones immediately (recall that pushouts can be computed levelwisely); this fact is also mentioned in the proof of [@tmodel Proposition 9.4]. 3\. The first part of the assertion is given by Lemma 8.1 (an also by §5.1) of [@tmodel]. The second part is immediate from the description of weak equivalences in ${\operatorname{Pro}-\mathcal{M}}$ given in loc. cit. 4\. The first of the statements is immediate from Corollary 8.7 of ibid.; the other ones are its obvious consequences. 5\. The first part of the assertion is provided by Theorem 4.1 of [@isalim]. Since loc. cit. also (roughly) says that filtered limits in ${\operatorname{Pro}-\mathcal{M}}$ can be naturally expressed in terms of limits in ${\mathcal{M}}$, combining this statement with assertion 4 we obtain the second part of assertion 5 immediately. 6\. ${\underline{C}}$ has products since it is the homotopy category of a model category (see Example 1.3.11 of [@hovey]). Hence we should verify that ${\underline{C}}(\prod_{i\in I} Y^i,X)=\bigoplus_{i\in I} {\underline{C}}(Y^i,X)$ for $Y^i$ being fibrant objects of ${\operatorname{Pro}-\mathcal{M}}$, $X\in {\operatorname{Ho}}(c)({\operatorname{Obj}}{\underline{C}'})$. Now (see loc. cit.) the product of $Y^i$ in ${\underline{C}}$ comes from their product in ${\operatorname{Pro}-\mathcal{M}}$. Certainly, if $Y^i=(Y_{j}^i)$ then the product of $Y^i$ in ${\operatorname{Pro}-\mathcal{M}}$ can be presented by the projective system of all $\prod_{i\in J} Y^i_{j_i}$ for finite $J\subset I$. Hence the statement follows from the previous assertion (recall that products are particular cases of inverse limits).[^67] 7\. Theorem 6.1 of [@chor] implies that ${\operatorname{Pro}-\mathcal{M}}$ admits a non-functorial version of the generalized cosmall object argument with respect to $c(f)$. Hence we can apply the dual of the argument used in the proof of Theorem 7.3.1 of [@hovey]. We deduce some consequences from this statement (mostly) using Proposition \[porthop\]. We will consider ${\underline{C}'}$ as a full subcategory of ${\underline{C}}$ (via the embedding $c$ that we will not mention) in this corollary. \[cgdb\] In the setting of the previous proposition assume that ${\underline{C}_0}$ is an essentially small subcategory of ${\underline{C}'}$ consisting of compact objects; let ${\mathcal{P}}$ be a suspended subset of ${\operatorname{Obj}}{\underline{C}_0}$. Denote by ${\underline{E}}$ the colocalizing subcategory of ${\underline{C}}$ cogenerated by ${\operatorname{Obj}}{\underline{C}_0}$; let $X_i$ be a projective system in ${\operatorname{Pro}-\mathcal{M}}$ and $N\in {\operatorname{Obj}}{\underline{C}'}$. Then the following statements are valid. 1. \[iprev\] One can apply Corollary \[cwt\] to this setting. 2. \[iab5\] ${{\underline{Ht}}}$ is a Grothendieck abelian category and there exists a faithful exact functor ${\mathcal{S}}:{{\underline{Ht}}}\to {\underline{\operatorname{Ab}}}$ that respects coproducts. 3. \[ic3\] $\Phi^{{\underline{E}}}(L(\operatorname{\varprojlim}X_i),N)\cong \operatorname{\varinjlim}\Phi^{{\underline{E}}} (L(X_i),N)$. 4. \[icf\] More generally, if ${\underline{A}}$ is an AB5 category and $H$ from ${\underline{C}}$ into ${\underline{A}}$ is an extended functor (i.e., it is obtained via the dual to Proposition \[pkrause\] from a cohomological functor $H_0$ from ${\underline{C}_0}$ into ${\underline{A}}$) then we have $H(\operatorname{\varprojlim}X_i)\cong H(L(\operatorname{\varprojlim}X_i))\cong \operatorname{\varinjlim}H(L(X_i))\cong \operatorname{\varinjlim}H(X_i)$. 5. \[icfig\] $\Phi^{{\underline{E}}}$ is isomorphic to the restriction to ${\underline{E}}{{^{op}}}\times {\underline{C}}'$ of ${\underline{C}}(-,-)$. 6. \[icharw\] We have ${\underline{E}}_{w_{{\underline{E}}}\le 0}=({}^{\perp_{{\underline{C}}}}{\underline{C}'}^{t\le -1})\cap {\operatorname{Obj}}{\underline{E}}$. \[iprev\]. Immediate from the previous proposition. \[iab5\]. According to the previous assertion, we can apply Theorem \[tab5\] to our setting. Assertion \[ic3\] is a particular case of assertion \[icf\] indeed (by the definition of $\Phi$). Next, since $L$ respects ${\underline{C}_0}$-limits, assertion \[icf\] follows from Proposition \[pgdb\](6) according to (the dual to) Proposition \[pkrause\](\[ikres\]). \[icfig\]. According to Proposition \[pkrause\](\[ikr8\]) (cf. also Proposition \[porthop\](\[ibiext\])) it suffices to verify that for any $M\in {\operatorname{Obj}}{\underline{E}}$ the functor ${\underline{C}}(M,-):{\underline{C}}\to {\underline{\operatorname{Ab}}}$ is a cc one. Now, this condition is certainly fulfilled if $M\in {\operatorname{Obj}}{\underline{C}_0}$. Next, the class of objects of ${\underline{E}}$ satisfying this condition is shift-stable; hence it also closed with respect to extensions. Thus it remains to verify for a set of $M_i\in {\operatorname{Obj}}{\underline{E}}$ that the functor ${\underline{C}}(\prod M_i,-):{\underline{C}'}\to {\underline{\operatorname{Ab}}}$ is a cc one if all ${\underline{C}}(M_i,-)$ are. The latter implication follows immediately from Proposition \[pgdb\](6). \[icharw\]. The previous assertion certainly implies that $({}^{\perp_{{\underline{C}}}}{\underline{C}'}^{t\le -1})\cap {\operatorname{Obj}}{\underline{E}}={}^{\perp_{\Phi^{{\underline{E}}}}}{\underline{C}'}^{t\le -1}$. Hence it remains to apply Corollary \[cwt\] (\[icort\]). This proposition is applied in [@bgn] to various motivic homotopy categories. The corresponding $t$ is a (version of) the Voevodsky-Morel homotopy $t$-structure, whereas $w$ is called (a version of) the [*Gersten*]{} weight structure; the corresponding weight filtrations and weight spectral sequences generalize coniveau ones. \[rsheaves\] 1\. Certainly, these methods can be applied for ${\underline{C}'}$ being some (other) triangulated category “constructed from sheaves”; one can use Proposition 8.16 (and other results) of [@jardloc] to present ${\underline{C}'}$ as the homotopy category of a proper stable model category. Note also that a category ${\underline{C}'}$ of this sort is “usually” compactly generated (still cf. [@neeshman] and Remark \[rwg\](2) above); in this case there exist plenty of possible ${\mathcal{P}}$. On the other hand, the heart of $t$ is “quite rarely” of the form $\operatorname{\operatorname{AddFun}}({\underline{B}},{\underline{\operatorname{Ab}}})$ for these examples; this justifies the claim made in Remark \[rab5\](\[istalks\]). Note also that any stalk functor in this case should send a complex of sheaves into a certain “stalk” of the zeroth (co)homology sheaf of this complex; whence the name. Moreover, for the motivic examples considered in [@bgn] the stalk functors are (retracts of coproducts of) certain “twists” of “actual stalks” (cf. also [@bondegl Theorem 3.3.1] for a certain “relative” version of this observation). On the other hand, for other triples $({\underline{D}},t,{\underline{D}}')$ as in Theorem \[tab5\] the stalk functors may give a new (and non-trivial) object of study. In particular, we obtain stalk functors for arbitrary compactly generated $t$-structures on arbitrary motivic homotopy categories (and on their localizing subcategories); these may have quite non-trivial “geometrical meaning” (and do not require any resolution of singularities assumptions in contrast to loc. cit.). 2\. Part \[iab5\] of our corollary generalizes Corollary 4.9 of [@humavit] (where ${\underline{C}}$ was assumed to be [*algebraic*]{} and $t$ was assumed to be non-degenerate; cf. Corollary \[csymt\](3) and Remark \[rsymt\](\[irsymt2\]) above). Note that both of these conditions are rather restrictive if one studies motivic homotopy categories. 3\. Since all objects of ${\underline{C}'}$ are cocompact in ${\underline{C}}$, the class of cocompact objects of ${\underline{C}}$ is not essentially small (in contrast to that for ${\underline{E}}$). 4\. Recall that ${\underline{E}}$ can be presented as the (Verdier) localization of ${\underline{C}}$ by the subcategory ${{}^{\perp}}{\underline{E}}$. In [@prospect] the case ${\underline{C}'}=SH$ (the topological stable homotopy category) was considered, and the corresponding ${\underline{E}}$ was constructed as the homotopy category of a certain right Bousfield localization of ${\underline{C}}$. Moreover, an exact equivalence $F:{\underline{E}}\to {\underline{C}'}{{^{op}}}$ was constructed (in this case). Furthermore, Remark 6.9 of ibid. appears to imply that the duality $\Phi^{{\underline{E}}}$ in this case is isomorphic to the bifunctor $SH(S^0,F(-)\otimes -)$ ($S^0={{\pmb{1}}}_{SH}$ is the sphere spectrum); note that it suffices to construct the restriction of this isomorphism to finite spectra (see Remark \[rab5\](\[irotimes\])). Next, if we take ${\mathcal{P}}=\{S^0[i]: i\ge 0\}$ then the corresponding $t$ is certainly the Postnikov $t$-structure for $SH$, whereas $w_{{\underline{E}}}$ is easily seen to be the opposite (see Proposition \[pbw\](\[idual\])) to the [*spherical*]{} weight structure on ${\underline{E}}{{^{op}}}\cong SH$ (see [@bws §4.6] and [@bkw §2.4]). Moreover, the author hopes that applying Remark \[rab5\](\[irotimes\], \[krauseideals\]) in this context may shed some light on the seminal telescope conjecture. 5\. We conjecture that an isomorphism $F:{\underline{E}}\to {\underline{C}'}{{^{op}}}$ (in the setting of this section) exists for a wide range of stable monoidal model categories such that ${\underline{C}_0}$ is self-dual with respect to $\otimes$ (and it generates ${\underline{C}}'$ as its own localizing subcategory). Note that in the aforementioned particular case ${\underline{C}'}=SH$ the existence of $F$ may be deduced from Theorem 5.3 of [@schwemarg];[^68] cf. also [@bger §6.4] for a certain motivic observation related to our conjecture. Note also that in this case the conjecture stated in Remark \[rab5\](\[iremb\]) is easily seen to be fulfilled also. 6\. Part \[icharw\] of our corollary can easily be “axiomatized” somewhat similarly to Corollary \[cwt\]. However, the existence of a category ${\underline{C}}\supset {\underline{C}}'$ such that all objects of ${\underline{C}}'$ are cocompact in it seems to be rather “exotic”. On localizations of coefficients and “splittings” for triangulated categories {#slocoeff} ----------------------------------------------------------------------------- In this subsection we gather a few results related to localizations of coefficients and decomposing triangulated categories into direct summands for the purpose of applying these statements in [@bgn]. First we recall the “naive” method of localizing coefficients (for a triangulated category). \[rcgws\] 1. Let $S\subset {\mathbb{P}}$ be a set of prime numbers; denote the ring ${{\mathbb{Z}}}[S{^{-1}}]$ by ${\Lambda}$. Then for any triangulated ${\underline{C}}$ one can consider the category ${\underline{C}}\otimes {\Lambda}$ with the same object class and ${\underline{C}}\otimes {\Lambda}(M,N)={\underline{C}}(M,N)\otimes_{{\mathbb{Z}}}{\Lambda}$ for all $M,N\in {\operatorname{Obj}}{\underline{C}}$. The category ${\underline{C}}\otimes {\Lambda}$ has a natural structure of a triangulated category; see Proposition A.2.3 of [@kellyth]. Next, if $s$ is a torsion pair for ${\underline{C}}$ then the Karoubi-closures of the classes $({\mathcal{LO}},{\mathcal{RO}})$ in ${\underline{C}}\otimes {\Lambda}$ give a torsion pair for ${\underline{C}'}$ according to Proposition \[phop\](9). 2\. Certainly, for ${\underline{C}}$ being an $R$-linear category (see Remark \[rsatur\](4)) we can also localize (using any of the methods described in this section) by any multiplicative subset of $R$. However, this method of “localizing coefficients” of a triangulated category does not seem to be “appropriate” if ${\underline{C}}$ is has coproducts. So we describe an alternative construction that “works fine” for compactly generated categories. \[plocoeff\] Assume that ${\underline{C}}$ is compactly generated; denote its subcategory of compact objects by ${\underline{C}}^c$. For a set $S$ as above denote by ${\underline{E}}$ the localizing subcategory of ${\underline{C}}$ generated by cones of $p{\operatorname{id}}_M$ for $M\in {\operatorname{Obj}}{\underline{C}},\ p\in S$; denote by ${\underline{D}}$ the full subcategory of ${\underline{C}}$ whose object class is ${\operatorname{Obj}}{\underline{E}}{{}^{\perp}}$. I. Then the following statements are valid. 1. \[icge\] ${\underline{E}}$ is compactly generated by cones of $p{\operatorname{id}}_N$ for $N$ running through ${\operatorname{Obj}}{\underline{C}}^c$ (and $p\in S$). 2. \[icgd\] ${\operatorname{Obj}}{\underline{D}}$ equals the class of ${\Lambda}$-linear objects of ${\underline{C}}$, i.e., of those $M\in {\operatorname{Obj}}{\underline{C}}$ such that $p{\operatorname{id}}_M$ is an automorphism for any $p\in S$; so, it is closed with respect to ${\underline{C}}$-coproducts. 3. \[icgadj\] The embedding $i:{\underline{D}}\to {\underline{C}}$ possesses an exact left adjoint $l_S$ that gives an equivalence ${\underline{C}}/{\underline{E}}\to {\underline{D}}$. 4. \[icg3\] For any $M\in {\operatorname{Obj}}{\underline{C}}$ and $N\in {\operatorname{Obj}}{\underline{C}}^c$ we have ${\underline{D}}(l(N),l(M))\cong {\underline{C}}(N,M)\otimes_{{\mathbb{Z}}}{\Lambda}$; thus ${\underline{D}}$ contains ${\underline{C}}^c\otimes{\Lambda}$ as a full subcategory. 5. \[icgls\] $l_S$ respects coproducts and converts compact objects into ${\underline{D}}$-compact ones. Moreover, ${\underline{D}}$ is generated by $l_S({\operatorname{Obj}}{\underline{C}}^c)$ as its own localizing subcategory, and the class of compact objects of ${\underline{D}}$ equals ${\operatorname{Obj}}{\langle}l_S ({\operatorname{Obj}}{\underline{C}}^c{\rangle}){\rangle}={\operatorname{Obj}}\operatorname{\operatorname{Kar}}_{{\underline{D}}} ({\underline{C}}^c\otimes {\Lambda})$. 6. \[icgtr\] For any $M\in {\operatorname{Obj}}{\underline{C}}$ there exists a distinguished triangle $$\label{ecdec} N\to M\to i\circ l_S(M)\to N[1]$$ for some $N\in {\operatorname{Obj}}{\underline{E}}$; this triangle is unique up to a canonical isomorphism. 7. \[icgfun\] Let $H:{\underline{C}}\to {\underline{\operatorname{Ab}}}$ be a cc-functor. Then for any $M\in {\operatorname{Obj}}{\underline{C}}$ we have $H( i\circ l_S(M))\cong H(M)\otimes_{{\mathbb{Z}}}{\Lambda}$. 8. \[icgdi\] Assume that ${\underline{C}}'$ is also a compactly generated (triangulated) category; define ${\underline{D}}'$, $i'$ and $l'_S$ as the ${\underline{C}}'$-versions of ${\underline{D}}$, $i$, and $l_S$, respectively. Then any functor $F:{\underline{C}}\to {\underline{C}}'$ that respects coproducts can be canonically completed to a diagram $$\label{ediaf} \begin{CD} {\underline{C}}@>{l_S}>>{\underline{D}}@>{i}>>{\underline{C}}\\ @VV{F}V@VV{G}V@VV{F}V \\ {\underline{C}}'@>{l'_S}>>{\underline{D}}' @>{i'}>>{\underline{C}}' \end{CD}$$ where $G$ is a certain exact functor respecting coproducts. II\. Assume in addition that ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ is a class of compact objects. Denote by $s=({\mathcal{LO}},{\mathcal{RO}})$ the torsion pair generated by ${\mathcal{P}}$ (whose existence is given by Theorem \[tclass\](\[iclass1\]); note that ${\mathcal{P}}$ is essentially small). 1\. Then the couple $s[S{^{-1}}]=(\operatorname{\operatorname{Kar}}_{{\underline{D}}}(l_S({\mathcal{LO}})),\operatorname{\operatorname{Kar}}_{{\underline{D}}}(l_S({\mathcal{RO}})))$ gives the torsion pair generated by $l_S({\mathcal{P}})$ in ${\underline{D}}$. 2\. If $s$ is weighty then $s[S{^{-1}}]$ also is and the functor $l_S$ is weight-exact with respect to the corresponding weight structures $w$ and $w[S{^{-1}}]$, respectively (and so $l_S({\underline{C}}_{w=0})\subset {\underline{D}}_{w[S{^{-1}}]=0}$). 3\. If $s$ is associated to a $t$-structure $t$ then $s[S{^{-1}}]$ also is. Moreover, both $l_S$ and $i$ are $t$-exact (with respect to the corresponding $t$-structures), and so ${{\underline{Ht}}}[S{^{-1}}]\subset {{\underline{Ht}}}$. Furthermore, ${\operatorname{Obj}}{\underline{D}}\cap {\underline{C}}^{t\le 0}={\underline{D}}^{t[S{^{-1}}]\le 0}$ and ${\operatorname{Obj}}{\underline{D}}\cap {\underline{C}}^{t\ge 0}={\underline{D}}^{t[S{^{-1}}]\ge 0}$. 4\. If $N$ is a ${\mathcal{P}}$-colimit of some $(N_j)$ (see Definition \[drelim\]) then $l_S(N)$ is a $l_S({\mathcal{P}})$-colimit of the corresponding $(l_S(N_j))$. III\. Adopt the notation and conventions of Proposition \[porthop\]. Then for ${\underline{D}}'\subset {\underline{C}}'$ (resp. for ${\underline{D}}\subset {\underline{C}}$) being the corresponding subcategories of ${\Lambda}$-linear object the restriction of $\Phi(-,-)$ to ${\underline{D}}{{^{op}}}\times {\underline{D}}'$ is a nice duality $\Phi[S{^{-1}}]:{\underline{D}}{{^{op}}}\times {\underline{D}}'\to {\underline{\operatorname{Ab}}}$. Moreover, one can obtain it by “bi-extending” (see Remark \[rcudu\](\[ir1\])) the duality ${\underline{D}}_0(-,-)$, where ${\underline{D}}_0=l_S({\underline{C}_0})\subset {\underline{D}}'$ is isomorphic to the “naive” $S$-localization of ${\underline{C}_0}$ as described in Remark \[rcgws\](1). Furthermore, the corresponding $w[S{^{-1}}]$ and $t[S{^{-1}}]$ are $\Phi[S{^{-1}}]$-orthogonal. I. This construction was described in detail in Appendix A.2 of [@kellyth]. So ibid. would yield our assertions \[icgd\]–\[icgls\] if we replace ${\underline{E}}$ in it by the subcategory ${\underline{E}}'$ defined by means of assertion \[icge\]. Now, we certainly have ${\underline{E}}'\subset {\underline{E}}$, and the converse inclusion follows from ibid. since $l_S$ is easily seen to kill all $\operatorname{\operatorname{Cone}}(p{\operatorname{id}}_M)$ (for $M\in {\operatorname{Obj}}{\underline{C}},\ p\in S$). Hence ${\underline{E}}={\underline{E}}'$ and we obtain assertions \[icge\]–\[icgadj\]. Assertion \[icgtr\] easily follows from Proposition \[pbouloc\](III.\[ibou1\]). \[icgfun\]. First we note that $H({\operatorname{Obj}}{\underline{D}})$ consists ${\Lambda}$-linear objects of ${\underline{\operatorname{Ab}}}$ (i.e., of ${\Lambda}$-modules) immediately from assertion \[icgd\]. Next, the group $H(\operatorname{\operatorname{Cone}}(p{\operatorname{id}}_M))$ is certainly killed by the multiplication by $p^2$ (for any $M\in {\operatorname{Obj}}{\underline{C}},\ p\in S$). Since the subcategory ${\underline{A}}$ of ${\underline{\operatorname{Ab}}}$ consisting of groups all of whose elements are $S$-torsion (i.e., annihilated by multiplying by some products of elements of $S$) is a Serre subcategory closed with respect to coproducts, and $H$ is a cc functor, we obtain $H({\operatorname{Obj}}{\underline{D}})\subset {\operatorname{Obj}}{\underline{A}}$. Now, for any $M\in {\operatorname{Obj}}{\underline{C}})$ we apply $H$ to the triangle (\[ecdec\]) to obtain an exact sequence $H(N)\to H(M)\to H(i\circ l_S(M))\to H(N[1])$ (for some $N\in {\operatorname{Obj}}{\underline{E}}$). Since ${\Lambda}$ is a flat ${{\mathbb{Z}}}$-module, we can tensor this sequence by ${\Lambda}$ to obtain the exact sequence $H(N)\otimes_{{{\mathbb{Z}}}}{\Lambda}\to H(M)\otimes_{{{\mathbb{Z}}}}{\Lambda}\to H(i\circ l_S(M)) \otimes_{{{\mathbb{Z}}}}{\Lambda}\to H(N[1]) \otimes_{{{\mathbb{Z}}}}{\Lambda}$. Since $H(N)$ and $H(N[1])$ belong to ${\operatorname{Obj}}{\underline{A}}$, they are annihilated by $-\otimes_{{{\mathbb{Z}}}}{\Lambda}$. Lastly, since $H(i\circ l_S(M))$ is ${\Lambda}$-linear, we obtain $H(M)\otimes_{{{\mathbb{Z}}}}{\Lambda}\cong H(i\circ l_S(M)) \otimes_{{{\mathbb{Z}}}}{\Lambda}\cong H(i\circ l_S(M))$. \[icgdi\]. Certainly, $F$ maps ${\Lambda}$-linear objects of ${\underline{C}}$ into ${\Lambda}$-linear ones; hence we can define $G$ as the corresponding restriction of $F$. It remains to verify that the left hand square in (\[ediaf\]) is essentially commutative. Applying assertion I.\[icgtr\] we obtain the following: it suffices to check that for any $M\in {\operatorname{Obj}}{\underline{C}}$ the functor $F$ maps the distinguished triangle (\[ecdec\]) into the corresponding triangle for $F(M)$. Hence it remains to note that $F(N)$ belongs to the corresponding localizing subcategory ${\underline{E}}'$ of ${\underline{C}}$ since $F$ respects coproducts. II.1. For any $M\in {\operatorname{Obj}}{\underline{C}}$ the functor $H_{M,S}: N\mapsto {\underline{D}}(l_S(N),l_S(M))$ is a cp functor from ${\underline{C}}$ into ${\underline{\operatorname{Ab}}}$. Assertion \[icg3\] implies that for any $M\in {\mathcal{RO}}$ the functor $H_{M,S}$ kills ${\mathcal{P}}$; thus it kills ${\mathcal{LO}}$ also (by Theorem \[tclass\](\[iclass5\]). Hence $\operatorname{\operatorname{Kar}}_{{\underline{D}}}(l_S({\mathcal{LO}}))\perp_{{\underline{D}}} \operatorname{\operatorname{Kar}}_{{\underline{D}}}(l_S({\mathcal{RO}}))$. Next, objects of ${\underline{D}}$ certainly possess $s[S{^{-1}}]$-decompositions (since all objects come from ${\underline{C}}$ and one can apply $l_S$ to $s$-decompositions). According to Proposition \[phop\](9), it remains to prove that $l_S({\mathcal{RO}})$ is Karoubi-closed in ${\underline{D}}$.[^69] This certainly reduces to $l_S({\mathcal{RO}})=l_S({\mathcal{P}})^{\perp_{{\underline{D}}}}$ since $l_S$ respects the compactness of objects (see assertion I.\[icgls\]). Now, $l_S({\mathcal{P}})^{\perp_{{\underline{D}}}}\subset {\mathcal{RO}}$ by assertion I.\[icg3\], and it remains to note that $l_S$ maps objects of ${\underline{D}}$ into isomorphic ones. 2. We certainly have ${\mathcal{LO}}[S{^{-1}}]\subset {\mathcal{LO}}[S{^{-1}}][1]$; thus $s[S{^{-1}}]$ is weighty indeed (see Remark \[rwhop\](1)). The remaining parts of the assertion follow immediately. 3\. Similarly to the previous assertion, Remark \[rtst1\](\[it1\]) implies that $s[S{^{-1}}]$ is associated to a $t$-structure. Thus $l$ is $t$-exact. According to assertion I.\[icgd\], to check the $t$-exactness of $i$ we should prove that for any ${\Lambda}$-linear object $M$ of ${\underline{C}}$ its $t$-truncations are ${\Lambda}$-linear also. Now, for any $p\in S$ the functoriality of the $t$-decomposition triangle $L\to M\to R\to L[1]$ implies that ${\underline{C}}(L,L)$ contains a morphism $1/p{\operatorname{id}}_L$ inverse to $p{\operatorname{id}}_M$ and ${\underline{C}}(R,R)$ contains a morphism $1/p{\operatorname{id}}_R$ inverse to $p{\operatorname{id}}_R$. Thus $L$ and $R$ are ${\Lambda}$-linear indeed. It certainly follows that ${{\underline{Ht}}}[S{^{-1}}]\subset {{\underline{Ht}}}$. Lastly, ${\underline{D}}^{t[S{^{-1}}]\le 0}\subset {\operatorname{Obj}}{\underline{D}}\cap {\underline{C}}^{t\le 0}$ and ${\underline{D}}^{t[S{^{-1}}]\ge 0}\subset {\operatorname{Obj}}{\underline{D}}\cap {\underline{C}}^{t\ge 0}$. The converse implication is immediate from the $t$-exactness of $l_S$ (along with the fact that $l_S$ maps objects of ${\underline{E}}$ into isomorphic ones). 4. Immediate from assertion I.\[icg3\]. III\. $\Phi[S{^{-1}}]$ is a nice duality since it is a restriction of a nice duality. Moreover, assertion I.\[icg3\] gives the “description” of ${\underline{D}}_0$ in question. Now, ${\underline{D}}$ has products, ${\underline{D}}'$ has coproducts, and the objects of ${\underline{D}}_0$ are compact in ${\underline{D}}'$ and cocompact in ${\underline{D}}$ according to assertions I.\[icgd\]–\[icgadj\] along with their duals. So we will prove that $\Phi[S{^{-1}}]$ is isomorphic to the corresponding biextension of ${\underline{D}}_0(-,-)$ using Proposition \[porthop\](\[ibiext\]). Since the embeddings ${\underline{D}}'\to {\underline{C}}'$ and ${\underline{D}}{{^{op}}}\to {\underline{C}}{{^{op}}}$ respect coproducts (see assertion I.\[icgd\]), we obtain that $\Phi[S{^{-1}}]$ respects ${\underline{D}}'$-coproducts as well as ${\underline{D}}{{^{op}}}$-ones. Next, assertion I.\[icg3\] (along with assertion I.\[icge\] and the duals of these two) implies that $\Phi[S{^{-1}}]$ annihilates both ${}^{\perp_{{\underline{D}}}}{{\underline{D}}_0}\times {\operatorname{Obj}}{\underline{D}}'$ and ${\operatorname{Obj}}{\underline{D}}\times {{\underline{D}}_0}^{\perp_{{\underline{D}}'}}$. Hence Proposition \[porthop\](\[ibiext\]) implies the “moreover” part of the assertion. It remains to apply part \[ihoport\] of that proposition to obtain the orthogonality result in question. \[rloc\] 1. Combining part II.4 of our proposition (and adopting its notation) with Proposition \[pkrause\](\[ikr3\]) we obtain the following: if ${\underline{C}_0}$ is a full triangulated subcategory of ${\underline{C}}$, ${\operatorname{Obj}}{\underline{C}_0}\subset {\mathcal{P}}$, and $H:{\underline{D}}\to {\underline{A}}$ is an exact functor coextended from $l_S({\underline{C}_0})$ then $H(l_S(N))\cong \operatorname{\varinjlim}H(l_S(N_j))$. Since $i$ is a full embedding, we also obtain that $H'(i(l_S(N)))\cong \operatorname{\varinjlim}H'(i(l_S(N_j)))$ for any functor exact functor $H':{\underline{C}}\to {\underline{A}}$ coextended from ${\underline{C}_0}$. 2\. One can easily generalize part I.\[icgfun\] of our proposition to cc functors into arbitrary AB5 categories. 3\. Certainly, those results of of this subsection that concern compactly generated categories can easily be dualized. Now we study certain “splittings” of triangulated categories. \[psplit\] I. Assume that ${\underline{C}}$ is compactly generated, ${\underline{C}_0}$ is its subcategory of compact objects, and that decomposes into the direct sum of two triangulated subcategories ${\underline{C}_0}^1$ and ${\underline{C}_0}^2$. Denote by ${\underline{C}}^1$ the localizing subcategory of ${\underline{C}}$ generated by ${\underline{C}_0}^1$. Then the following statements are valid. 1. \[idec\] ${\underline{C}}^1$ is a direct summand of the category ${\underline{C}}$; so, there exists an exact functor $l$ projecting ${\underline{C}}$ onto ${\underline{C}}^1$. Moreover, $l$ is essentially the only exact projector functor that respects coproducts and restricts to the projection ${\underline{C}_0}\to {\underline{C}_0}^1$. Furthermore, if $N\in {\operatorname{Obj}}{\underline{C}}$ is a $C$-colimit of some $(N_j)$ (for some $C\in {\operatorname{Obj}}{\underline{C}}$) then $l(N)$ is a $l(C)$-colimit of the corresponding $(l(N_j))$. 2. \[isplcg\] $l$ respects the compactness of objects, ${\underline{C}}^1$ is compactly generated, and ${\underline{C}_0}^1$ is its subcategory of compact objects. 3. \[isplitp\] Assume that $s=({\mathcal{LO}},{\mathcal{RO}})$ is a torsion pair generated by a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$. Then $l({\mathcal{LO}})={\mathcal{LO}}\cap {\operatorname{Obj}}{\underline{C}}^1$, $l({\mathcal{RO}})={\mathcal{RO}}\cap {\operatorname{Obj}}{\underline{C}}^1$, and the couple $l(s)=(l({\mathcal{LO}}),l({\mathcal{RO}}))$ is a torsion pair for ${\underline{C}}^1$. In particular, if $s$ is weighty then the functor $l$ is weight-exact with respect to the corresponding weight structures. Moreover, the pair $l(s)$ is smashing whenever $s$ is. II\. Adopt the notation and conventions of Proposition \[porthop\]; assume in addition that ${\underline{C}_0}$ generates ${\underline{C}}'$ and cogenerates ${\underline{C}}$, and that ${\underline{C}_0}\cong {\underline{C}_0}^1\bigoplus {\underline{C}_0}^2$. Denote by ${\underline{C}}'^1$ (resp. ${\underline{C}}^1$) the subcategory of ${\underline{C}}'$ (resp. of ${\underline{C}}$) that is (co)generated by ${\underline{C}_0}^1$. Then the restriction $\Phi^1$ of $\Phi(-,-)$ to ${\underline{C}}^1{}{{^{op}}}\times{\underline{C}}'^1$ is a nice duality, and one can obtain it by “bi-extending” (see Remark \[rcudu\](\[ir1\])) the duality ${\underline{C}}^1_0(-,-)$. Moreover, the corresponding $w^1$ is $\Phi^1$-orthogonal to $t^1$. I.\[idec\]. Since ${\operatorname{Obj}}{\mathcal{P}}^1\perp {\operatorname{Obj}}{\mathcal{P}}^2$ and vice versa, our compactness assumptions easily imply that the natural functor ${\underline{C}}^1\bigoplus {\underline{C}}^2\to {\underline{C}}$ is an equivalence, where ${\underline{C}}^2$ is the localizing subcategory of ${\underline{C}}$ generated by ${\mathcal{P}}^2$. So, we obtain the existence of $l$. Next, $l$ is essentially unique since it should be identical on ${\underline{C}}^1$ and should kill ${\operatorname{Obj}}{\underline{C}}^1{}^\perp={\operatorname{Obj}}{\underline{C}}^2$. The “furthermore” part of the assertion follows immediately. \[isplcg\]. 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[^1]: The author’s work on sections 1–2 was supported by the RFBR grant No. 15-01-03034-a and by Dmitry Zimin’s Foundation “Dynasty”, whereas the work on sections 3–5 was supported by the Russian Science Foundation grant no. 16-11-10073. [^2]: The relation of pure functors to Deligne’s purity of (singular and étale) cohomology is recalled in Remark \[rwrange\](5). [^3]: In [@bws] $t$ was said to be left adjacent to $w$ in this case; we discuss this distinction in “conventions” in §\[rwsts\] below. [^4]: Thanks to the foundational results of A. Neeman and others, this property is known to hold for several important classes of triangulated categories; in particular, it suffices to assume that either ${\underline{C}}$ or ${\underline{C}}{{^{op}}}$ is compactly generated. [^5]: Here ${{\underline{Hw}}}$ is the heart of $w$; note also that $G: {{\underline{Hw}}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ respects products whenever it converts ${{\underline{Hw}}}$-coproducts into products of groups. [^6]: The author suspects that the content of the paper will not be evenly interesting to the readers. So he suggests the readers not (much) interested in “large” categories to ignore all matters related to infinite coproducts on the first reading (this includes compact objects and smashing torsion pairs). On the other hand, §\[sswc\] (and so, weight complexes and Postnikov towers) are mentioned explicitly in §\[sws\] only. So, the reader is encouraged to look for his personal trajectory through this paper. [^7]: $t$-structures of these type appear to be originally introduced in [@talosa]. They have become a popular object of study recently, with plenty of examples important to various areas of mathematics. [^8]: Recall that $P\in {\operatorname{Obj}}{\underline{C}}$ is said to be compact if the corepresentable functor ${\underline{C}}(P,-)$ respects coproducts. Now, any set of compact objects generates a $t$-structure according to Theorem A.1 of [@talosa]; the corresponding class ${\underline{C}}^{t\le 0}$ is the smallest subclass of ${\operatorname{Obj}}{\underline{C}}$ that is closed with respect to $[1]$, extensions, and coproducts, and contains ${\mathcal{P}}$. [^9]: Recall that Theorem 3.7 of loc. cit. says that countable colimits in the heart ${{\underline{Ht}}}$ are exact for any compactly generated $t$-structure $t$. [^10]: More generally, perfect classes are closely related to smashing torsion pairs; see Proposition \[psym\](\[iperftp\]). [^11]: See Remark \[requivdef\] below for a comparison of this definition with other ones in the literature. [^12]: Recall that this is always the case if the class ${\mathcal{P}}$ [*Hom-generates*]{} ${\underline{C}}$, i.e., $\cap_{i\in {{\mathbb{Z}}}}({\mathcal{P}}[i]{{}^{\perp}})={\{0\}}$. Since all elements of $\cap_i({\mathcal{P}}[i]{{}^{\perp}})$ are [*left degenerate*]{} with respect to $w$, if the Brown representability condition is not fulfilled for ${\underline{C}}$ then $w$ is necessarily “somewhat pathological”. [^13]: Note here that our proof of Theorem \[textw\] (that is somewhat similar to the corresponding proof from [@kraucoh]) does not allow constructing $t$-structures “directly from perfect sets” since it relies crucially on ${\underline{C}}_{w\le 0}[-1]\subset {\underline{C}}_{w\le 0}$ (and does not work for general torsion pairs that will be discussed soon; yet cf. Remark \[rigid\](1)). Still we are able to prove the existence of $t$-structure generated by ${\mathcal{P}}$ whenever ${\mathcal{P}}$ is [*symmetric*]{} to some ${\mathcal{P}}'\subset {\operatorname{Obj}}{\underline{C}}$ (see Theorem \[tsymt\]; note that any such ${\mathcal{P}}$ is certainly perfect). [^14]: Note that in [@postov] torsion pairs were called complete Hom-orthogonal pairs. [^15]: Certainly, if $C$ is triangulated then $X$ is a retract of $Y$ if and only if $X$ is its direct summand. [^16]: These are axioms \[TR5\] and \[TR5\*\] of [@neebook], respectively. [^17]: Another reason of passing to this more general notion is that some of our results are valid for arbitrary torsion pairs. [^18]: In the current paper we are not interested in triangulated categories that only have countable coproducts; yet in [@bsnew] we demonstrate this (weaker) assumption is rather useful also. [^19]: It suffices to assume that ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ and ${\mathcal{P}}^\perp={\mathcal{RO}}$ since then ${\mathcal{P}}$ certainly lies in ${\mathcal{LO}}$. [^20]: Note that the class of ${\mathcal{P}}$-null morphisms is not necessarily shift-stable in contrast to the main examples of the paper [@christ] where this notion was introduced. [^21]: Another way to deal with this discrepancy is to modify the definitions of $t$-structures and weight structures by the corresponding shifts; see Definition 2.1 of [@humavit]. [^22]: So, we don’t have to “shift” ${\mathcal{RO}}$ as we did in Remark \[rtst1\](\[it1\]). [^23]: The class $P_t$ was called the [*coheart*]{} of $t$ in §3 of [@zvon]. [^24]: In [@bws] the axioms of a weight structure also required ${\underline{C}}_{w\le 0}$ and ${\underline{C}}_{w\ge 0}$ to be additive. Yet this is not necessary; see Remark 1.2.3(4) of [@bonspkar]. [^25]: Recall also that D. Pauksztello has introduced weight structures independently (see [@paucomp]); he called them co-t-structures. [^26]: This relation was earlier introduced in [@barrabs]; $m_1$ is [*absolutely homologous*]{} to $m_2$ in the terminology of that paper. Respectively, some of the results below concerning this equivalence relation were proved in ibid. [^27]: The term comes from [@gs]; yet the domain of the weight complex functor in that paper was not triangulated, whereas the target was (“the ordinary”) $K^b({\operatorname{Chow^{eff}}})$. [^28]: Note however that the bounded cases of Proposition \[pdetect\] and Proposition \[pdetectsse\] can also be easily deduced from Theorem 3.3.1(IV) of [@bws]. [^29]: Alternatively, one can apply Proposition \[pwrange\](\[iwrpure\]) below to obtain this duality assertion. [^30]: This result was extended to the case where $k$ is a perfect field of characteristic $p>0$ in [@bzp]; note however that one is forced to invert $p$ in the coefficient ring in this setting. [^31]: Certainly, singular (co)homology (of motives) is only defined if $k$ is a subfield of complex numbers; then it is endowed with Deligne’s weight filtration that can also be computed using $w_{{\operatorname{Chow}}}(k)$ (see Remark 2.4.3 of [@bws]). On the other hand, Deligne’s weight filtration for étale (co)homology can be defined (at least) for $k$ being any finitely generated field; the comparison of the corresponding weight factors with the ones computed in terms of $w_{{\operatorname{Chow}}(k)}$ is carried over in Proposition 4.3.1 of [@bkl]. Note also that in ibid. and in [@brelmot §3.4,3.6] certain “relative perverse” versions of these weight calculations were discussed. [^32]: Certainly, the AB4\* condition for the target category ${\underline{A}}'$ can also be weakened respectively. [^33]: Certainly, in this case ${\underline{C}}$ is also generated by the coproduct of these objects (as its own localizing subcategory). [^34]: Actually, this envelope also equals the extension-closure of $\cup_{i<0} P_t[i]$; see Proposition \[pbw\](\[igenlm\]). [^35]: This is a rather “natural” additional assumption for the corollary (as well as for Theorem \[tadjw\]) since otherwise the corresponding weight structure $w$ would be “rather degenerate” in the following sense: the class $\cap_i{\underline{C}}_{w\ge i}$ would contain the non-zero class $ {}^{\perp_{{\underline{C}}}} {\underline{C}'}$ [^36]: Recall that ${\alpha}$ is said to be regular if it cannot be presented as a sum of less then ${\alpha}$ cardinals that are less than ${\alpha}$. [^37]: In particular, the localization ${\underline{C}}/{\underline{D}}$ is a category, i.e., its Hom-classes are sets. [^38]: Recall that for every combinatorial stable Quillen model category $K$ its homotopy category is well generated; see [@rosibr Proposition 6.10]. [^39]: This is why we use the word “symmetric”. [^40]: Alternatively, Remark \[requivdef\] below allows to deduce this fact from [@kraucoh Theorem B]. [^41]: Recall that any set ${\mathcal{P}}$ of compact objects generates a torsion pair according to Theorem 4.1 of [@aiya]; cf. Theorem \[tclass\](\[iclass1\]) below. [^42]: One can also “unite” symmetric classes (see Proposition \[psymb\](I.\[iws1\])); yet this does not give “really new” essentially small symmetric classes (if ${\underline{C}}$ has coproducts). [^43]: One has to assume in addition that ${\underline{C}}$ satisfies the Brown representability condition; however this is “almost automatic”. [^44]: The terminology we introduce is new; yet big hulls were essentially considered in (Theorem 3.7 of) [@postov]. [^45]: Note however that our reasoning is somewhat more clumsy than that of Pospisil and Šťovíček since we cannot apply their Proposition 2.7 to a general ${\underline{C}}$. Moreover, the proof of Theorem \[tpgws\] below can also be simplified if we assume (following loc. cit.) that ${\underline{C}}$ is a “stable derivator” triangulated category. [^46]: Actually, ${\underline{C}}^{{{\aleph_0}}}$ is essentially small itself in “reasonable” cases; in this case the essential smallness of its classes of objects is automatic. [^47]: This statement was previously proved in [@postov] and our argument is just slightly different from the one of Pospisil and Šťovíček; see Lemma 3.9 of ibid. [^48]: Actually our argument also yields that $H^{{\mathcal{P}}}(L)\cong \operatorname{\varinjlim}H^{{\mathcal{P}}}(L_k)$ easily. [^49]: This is where we need ${\mathcal{P}}$ to be cosuspended! [^50]: An argument even more closely related to our one was used in the proof of [@modoi Lemma 2.2]; yet the assumptions of that lemma appear to require a correction. [^51]: Certainly, the big hull of ${\mathcal{P}}$ is contained in its big extension-closure, whereas the latter (for ${\mathcal{P}}\subset {\mathcal{P}}[1]$) equals the smallest coproductive extension-closed subclass of ${\operatorname{Obj}}{\underline{C}}$ containing ${\mathcal{P}}$; cf. Corollary \[cgdb\](\[ict\]) below. [^52]: Note however that [*weak weight structures*]{} (one replaces the orthogonality axiom in Definition \[dwstr\] by ${\underline{C}}_{w\le 0}\perp{\underline{C}}_{w\ge 2}$) were essentially considered in [@bsosn] (cf. Remark 2.1.2 of ibid.) and in Theorem 3.1.3(2,3) of [@bsnew], whereas in Proposition 3.17 of [@brelmot] it was shown that they are relevant for the study of mixed étale ${{\mathbb{Q}_l}}$-adic sheaves over varieties over finite fields (actually, it was demonstrated that the weight filtration for the category ${D^b_m}(X_0,{{\mathbb{Q}_l}})$ satisfies the somewhat stronger Definition 3.11 of ibid., where $X_0$ is a variety over a finite field of characteristic $\neq l$). [^53]: These definitions (along with the definition of ${\mathcal{P}}$-null morphisms) is taken from [@christ]. [^54]: Actually, the standard convention is to say that ${\underline{\coprod}{\mathcal{P}}}$ is contravariantly finite if this condition is fulfilled; yet our version of this term is somewhat more convenient for the purposes of the current paper. [^55]: Certainly, $t$ is generated by the set $\cup_{i\ge 0}{\mathcal{Q}}[i]$. Hence Theorem \[tclass\](\[iclass1\],\[iclasst\]) gives a “more precise” description of the class ${\underline{C}}^{t\le 0}$. [^56]: This relation of ${{\underline{Ht}}}$ with its subcategory ${\operatorname{Inj}{{\underline{Ht}}}}$ of injective objects does not mention weight structures; yet it appears to follow from the existence of an injective cogenerator along with the AB4 property. [^57]: Recall here that ${\underline{C}}{{^{op}}}$ has coproducts according Proposition \[pcomp\](II.2). [^58]: Note that in Definition 2.1 of ibid. $t$-structures and co-$t$-structures (i.e., weight structures) were defined as the corresponding types of torsion pairs; so our definition differs from loc. cit. by shifts of the corresponding ${\mathcal{RO}}$ (cf. Remarks \[rtst1\](\[it1\]) and \[rwhop\](1)). [^59]: Note also that the reasoning of Pospisil and Šťovíček in the proof of loc. cit. works for arbitrary torsion pairs; this is certainly not the case for our arguments. [^60]: This fact can also be proved by noting that $H^{A_J}$ can be obtained from $J$ “by means” of the corresponding virtual $t$-truncations; see Theorem 2.4.2(II) of [@bger]. [^61]: One can also prove this statement using Remark \[rtkrau\](4); note that is actually not necessary to assume that ${\operatorname{Obj}}{\underline{C}}^{\perp_{{\underline{D}}}}={\{0\}}$ to get $t_{{\underline{D}}}$. Note also that Theorem 1.3 of [@hoshimi] is essentially an important particular case of [@bws Theorem 4.5.2(I.1)]. [^62]: Moreover, it would be interesting to prove some version of part \[ihoport\] of the proposition without assuming that objects of ${\underline{C}_0}$ are compact in ${\underline{C}'}$. [^63]: Recall that ${\underline{D}}^{{{\aleph_0}}}$ is the (essentially small) class of compact objects of ${\underline{D}}$. Following Remark 4.2.11 of [@bondegl], one may call elements of $H_0^t({\underline{D}}^{{{\aleph_0}}})$ [*strongly constructible*]{} objects of ${{\underline{Ht}}}$. [^64]: Note also that our argument has inspired the proof of [@bondegl Corollary 4.2.4]. [^65]: Possibly, a somewhat more general statement of this sort may be obtained by using stable $\infty$-categories; see Denis Nardin’s answer at <http://mathoverflow.net/q/255440>; yet the author does not know much about these matters. Respectively, the author is not sure that the construction of ${\underline{D}}'$ in §\[sprospectra\] is “optimal” for the purposes of Theorem \[tab5\]; yet this argument enables certain “computations” that are important for [@bgn]. [^66]: The author wonders whether some version of our argument can work for arbitrary ${\underline{D}}$. For this purpose it seems necessary to “get rid of ${\underline{D}}'$” in the proof. One may try to construct certain version of the stalk functors $\Phi(P,-)$ “directly”. It appears that a minor modification of the reasoning used in the proof of [@bsnull Proposition 2.1] gives the result whenever ${{\underline{D}}_0}$ is [*countable*]{} (i.e., ${\operatorname{Mor}}{{\underline{D}}_0}$ is a countable set). It is not clear whether the result can be extended to the general case (possibly, using the arguments of §3.1 of ibid.). [^67]: Alternatively, one can combine the argument dual to the one in the proof of Theorem 7.4.3 of [@hovey] with the fact that $c(f)$ for $f$ running through all fibrations in ${\mathcal{M}}$ yield a set of generating fibrations for ${\operatorname{Pro}-\mathcal{M}}$ (see Theorem 6.1 of [@chor]). [^68]: For this purpose one should use the fact that ${\underline{E}}$ has a model; however the Margolis’ uniqueness conjecture (see [@schwemarg §3]) predicts (in particular) that this conditions is fulfilled automatically. [^69]: Note that $l_S({\mathcal{LO}})$ does not necessarily have this property.

--- abstract: 'The resource-constrained nature of the Internet of Things (IoT) devices, poses a challenge in designing a secure, reliable, and particularly high-performance communication for this family of devices. Although side-channel resistant ciphers (either block cipher or stream cipher) are the well-suited solution to establish a guaranteed secure communication, the *energy*-intensive nature of these ciphers makes them undesirable for particularly lightweight IoT solutions. In this paper, we introduce *ExTru*, a novel encrypted communication protocol based on stream ciphers that adds a configurable switching & toggling network (*CSTN*) to not only boost the performance of the communication in lightweight IoT devices, it also consumes far less energy compared with the conventional side-channel resistant ciphers. Although the overall structure of the proposed scheme is leaky against physical attacks, such as side-channel or new scan-based Boolean satisfiability (*SAT*) attack or algebraic attack, we introduce a dynamic encryption mechanism that removes this vulnerability. We demonstrate how each communicated message in the proposed scheme reduces the level of trust. Accordingly, since a specific number of messages, $N$, could break the communication and extract the key, by using the dynamic encryption mechanism, *ExTru* can re-initiate the level of trust periodically after $T$ messages where $T<N$, to protect the communication against side-channel and scan-based attacks (e.g. SAT attack). Furthermore, we demonstrate that by properly configuring the value of $T$, *ExTru* not only increases the strength of security from per “*device*” to per “*message*”, it also significantly improves energy consumption as well as throughput in comparison with an architecture that only uses a conventional side-channel resistant block/stream cipher.' author: - Hadi Mardani Kamali - Kimia Zamiri Azar - | \ Shervin Roshanisefat - Ashkan Vakil - Avesta Sasan bibliography: - 's-bibliography.bib' nocite: - '[@azar2019smt]' - '[@roshanisefat2018srclock]' - '[@kamali2018lut]' title: '*ExTru*: A Lightweight, Fast, and Secure *Ex*pirable *Tru*st for the Internet of Things' --- Introduction ============ The Internet of Things (IoT), which has been foreseen to become the most successful business for the next decade by *International Technology Roadmap for Semiconductors* (ITRS), is an inevitable landmark of smart life providing novel applications and services, ranging from business automation to personal day-to-day life [@gubbi2013internet; @al2015internet; @li2015internet]. The IoT infrastructure is the seamless connection of billions of heterogeneous devices (*“things”*) within a large integrated network (the *“Internet”*). The heterogeneity of IoT constitutes from a wide variety of devices, such as smartwatches, mobile phones, etc, which results in a drastic increase in the number of IoT devices, estimated to be 26 billion connected IoT devices by the end of 2020 [@evans2011internet]. Although IoT devices provide a more efficient, automated, and smart life, from security/privacy perspective, many threats and vulnerabilities have been raised in IoT devices. Many investigations on cyber-based threats demonstrate that there are 176 new cyber-threats every minute, and over 2.5 million within only four months [@frustaci2017evaluating]. Several incidents have highlighted the massive influence of counterfeit/cloned/tampered devices into the supply chain [@guin2014counterfeit; @rostami2014primer]. As an instance, influencing and controlling every connected device within a ZigBee network, which is one of the most prevalent wireless communications in IoT, has been illustrated in [@zillner2015zigbee; @ronen2017iot]. Another recent evaluation by HP demonstrates that 70% of the devices in IoT are vulnerable to different types of threats, including physical attacks [@kumar2016security]. In current IoT applications, almost all proposed IoT devices are working (and communicating) based on a very well known 3-layer hierarchical architecture that is illustrated in Fig. \[IoTarch\]. These three layers, i.e. ***“devices”***, ***“gateways”*** and ***“servers”*** are the main layers in IoT architecture [@atzori2010internet; @da2014internet]. The devices that are responsible for interacting between the physical environment and computer-based systems, called *edge*, can connect with servers through gateways. Accordingly, equipping edge devices with some fundamental components, including sensors, analog to digital (A/D) converters, inter-communication frameworks, memories, and embedded micro-controllers, is required, to provide the capability of collecting, processing, and relaying data in a heterogeneous network. Although several IoT security challenges should be considered meticulously, combating hardware threats that are generally initiated at *edge* (devices layer), requires more attention [@frustaci2017evaluating]. Numerous solutions, including communication standards optimization, more secure configuration, etc, have been introduced to protect IoT devices and their communications against physical threats, which help to prevent the wide variety of conventional attacks [@pinto2017iioteed; @yuan2018reliable]. For instance, the utilization of symmetric-based secret-key ciphers or keyed hash-based authentication code (HMAC) is prevalent in IoT devices to provide integrity and authentication while securely protect the inter-communication of IoT edge devices [@koteshwara2017comparative; @shivraj2015one; @kamali2016fault]. Considering that the power consumption (particularly energy consumption) constraints in resource-constrained edge devices are very strict, the energy overhead of security solutions against hardware threats must be minimized. For instance, tight restrictions in edge devices enforce the designer to employ lightweight ciphers, such as stream ciphers or lightweight block ciphers [@dinu2019triathlon; @beaulieu2015simon]. However, the energy consumption of this breed of encryption architectures is still high for a high portion of IoT edge devices. Also, the performance of these ciphers considerably lower than regular block ciphers. This creates an inevitable security/cost trade-off in lightweight IoT devices, which results in sacrificing one of them, i.e. the security or the cost, which motivates the research community to carry on working/investigating on a low-energy and security-enhanced communication scheme in IoT while the performance is not degraded. In this paper, we introduce a new lightweight, fast, and provably secure *Exp*irable *Tr*ust (*ExTru*) mechanism relied on a configurable switching and toggling network (CSTN) as well as the winner of the Competition for Authenticated Encryption: Security, Applicability, and Robustness (CAESAR) [@caesar2013competition], called ACORN [@wu2016acorn]. *ExTru* provably protects the inter-communication of IoT edge devices while it even obtains higher performance and mitigates the energy consumption compared to the case in which the regular block/stream ciphers have been used. Moreover, we show how *ExTru* engages dynamicity in the circuit to provide guaranteed protection against different types of physical and scan-based attacks, such as side-channel, Boolean satisfiability (SAT) attack, and algebraic attack. We demonstrate that by using this dynamic encryption scheme, the strength of security could be elevated from *per device* to *per message*. The contributions of our paper are as follows: 1. By introducing a near non-blocking configurable switching and toggling network (CSTN), we show how we add dynamicity to the IoT devices intercommunication. 2. We show that this dynamicity along with the fast and efficient ACORN invalidates the possibility of the leakage of each message, which helps to show that this approach is provably resilient against physical attacks such as side-channel, scan-based SAT attack, and algebraic attack. 3. The dynamicity of *ExTru* allows us to relax the responsibility of ACORN, which helps to considerably boost the performance of the communication channel between IoT devices while the possibility of leakage is almost *ZERO*. Also by conveying part of the responsibility to the near non-blocking CSTN, we show that the energy consumption would be mitigated considerably. 4. To depict the efficiency of *ExTru* in terms of security, energy, and performance, we provide a full-detailed post-route evaluation on the proposed scheme compared to conventional IoT inter-communication mechanisms that almost use a conventional side-channel resistant block/stream. The rest of the paper is organized as follows: Section \[sec:related\] presents the previous work. Section \[sec:proposed\] elaborates the overall structure of the proposed dynamically encrypted scheme and how it is able to guarantee the security of IoT communication with significant energy mitigation as well as throughput improvement compared to conventional cipher-based communication schemes. In section \[attacks\], we evaluate the security of ExTru against physical attacks such as side-channel, scan-based SAT, and algebraic attack. In \[sec:results\], the experimental results have been provided and discussed. Finally, Section \[sec:conclusion\] concludes the paper. Related Work {#sec:related} ============ Due to the resource-constrained nature of IoT devices, a big challenge in guaranteeing the security of this group of devices is that the implementation of the security measures must be sufficiently lightweight, which prevents the designers to directly use conventional block ciphers, such as AES-GCM [@dworkin2007sp]. Many studies have been taken by the research community to not only address security issues in IoT networks but also to increase the efficiency by lowering the power (particularly energy) consumption and increasing the throughput. For instance, the fact that the elliptic curves cryptography (ECC) achieves guaranteed security with reduced resource requirements has attracted the research community [@piedra2013extending; @nam2014provably]. The work in [@marin2015optimized] has constructed an optimized ECC for secure communication in heterogeneous IoT devices based on Schnorr signature. Also, a simple key negotiation protocol has been introduced in this work that is based on the Schnorr scheme to demonstrate the usability of the presented ECC optimizations. Based on the desirable features of a physically unclonable function (PUF), such as lightweightedness, unpredictability, unclonability, and uniqueness, many researchers have been motivated to concentrate on the usage of this module to build a secure communication for IoT devices. Among several studies on PUF-based secure communication for IoT devices [@halak2016overview; @chatterjee2017puf; @chatterjee2018building; @liu2019xor], the work in [@chatterjee2017puf] has introduced an authentication, key sharing, and secure communication architecture, in which each IoT device has an integrated PUF. In this work, the identity of each device is created by the challenge-response pair signature of its PUF instance, and by engaging the identity-based encryption scheme proposed in Boneh and Franklin, the security of this approach is proven against attacks like chosen-plaintext/ciphertext attack. Numerous software/hardware implementation of lightweight ciphers suited for IoT devices have been proposed in recent few years, including RECTANGLE [@zhang2015rectangle], PICO [@bansod2016pico], Extended-LILIPUT [@ali2017optimised], SIT [@usman2017sit], SKINNY [@beierle2016skinny], MANTIS [@beierle2016skinny], to name but a few. Some of these ciphers could provide the best performance on software implementation, however, a portion of them have better performance in hardware implementation. For instance, the work in [@usman2017sit] introduces a lightweight 64-bit symmetric block cipher, called SIT, whose implementation is a mixture of Feistel and a uniform substitution-permutation network. The proposed approach uses some logical operations along with some swapping and substitution. Most of the encryption algorithms designed for IoT reduced the number of rounds to make a cost-security trade-off. For instance, SIT uses five rounds of encryption with 5 different keys to improve energy efficiency. The lightweightedness of the stream ciphers, on the other hand, has received fascinated attention from many researchers’ in recent years [@mohd2015survey; @singh2017advanced; @sfar2018roadmap; @manifavas2016survey]. Since IoT being an emerging field requires lightweight cipher designs with robustness, less complexity, and lower energy consumption, stream ciphers are very suited for particularly edge devices. Many studies evaluate the possibility of engaging stream ciphers in IoT devices, such as WG-8 [@fan2013wg], Trivium [@de2005trivium], Quavium [@tian2012quavium], and ACORN [@wu2016acorn]. *ExTru* Infrastructure {#sec:proposed} ====================== *ExTru* consists of four main sub-modules: (1) ACORN as a stream cipher that would be used periodically (The frequency will be discussed further), (2) a configurable switching and toggling network (CSTN) that dynamically permutes/toggles the data based on the configuration generated by TRNG, (3) a random number generator (RNG) that is responsible for generating random data for Threshold Implementation of ACORN as well as for generating the CSTN configuration, and (4) a substitution box placed after CSTN to eliminate the linearity/predictability of the ciphertext. The overall architecture of *ExTru* has been demonstrated in Fig. \[extruarch\] for both transmitter side and receiver side. On the transmitter side, the CSTN is used to permute/toggle the plaintext using the configuration (TRN) generated by the random number generator (RNG). The RNG will periodically change the configuration (TRN) to add dynamicity into the permutation/toggle network (CSTN). Parts of the configuration is fed by the permuted/toggled data (the output of the CSTN) to make the operation stateful (data-dependent). The CSTN is followed only by a substitution-box to eliminate the linearity/predictability of the output. The TRN that is used to configure the CSTN has been also encrypted using the authenticated cipher to be transmitted to the receiver. The key used for authenticated cipher could be pre-stored in the secure memory or produced by a PUF. The output of the transmitter (ciphertext) would be selected from the output of the s-box (permuted/toggled + substituted plaintext) or authenticated cipher output (encrypted TRN). On the receiver side, on the other hand, the reverse CSTN (RCSTN) must be used to recover the permuted/toggled + substituted plaintext. We will show that similar to ACORN that engages only one hardware module for both encryption/decryption, the CSTN hardware is the same for both receiving/sending operations (same hardware for both CSTN and RCSTN). Hence, no duplicated hardware (one for CSTN and one for RCSTN) is required to be added on each side. When TRN is received from the transmitter it must be decrypted using the authenticated cipher to be used as the configuration of the RCSTN. If the received data is not TRN, it first must pass the s-box to accomplish re-substitution, then it must pass the RCSTN to recover the plaintext. Fig. \[expdyn\] depicts the overall structure of dynamic encryption provided by *ExTru*, which has no sign of leaky communication. As shown in Fig. \[expdyn\](b), for each specific number of transmission ($T$), which must be less than $N$, a new CSTN configuration will be sent via side-channel resistant cipher. As it is shown, a secure message ($S$), which contains TRN, will be sent periodically after every $T$ messages ($I$) that are handled by CSTN/RCSTN. Based on different forms of attacks, such as side-channel, scan-based SAT, and algebraic attack, messages ($I$) are leaky. So, periodically changing TRN ($S$) and sending through side-channel resistant ciphers re-intensify the security of the communication. Based on the size of the CSTN/RCSTN (number of I/O), we will show that the maximum feasible update frequency ($N$) would be changed. Consequently, the CSTN configuration (TRN), which is fed by RNG, must be changed dynamically after every $T$ iterations, where $T < N$. Also, the size of CSTN/RCSTN determines the number of configuration bits (size of each $S$) must be generated by the RNG. In the following sub-sections we discuss the details of *ExTru* implementation. Configurable Switching & Toggling Network (CSTN) {#CSTN_section} ------------------------------------------------ The CSTN is a logarithmic routing (permutation) network that could permute the order of the signals at its input pins to its output pins while possibly toggling their logic levels based on its configuration (TRN). Fig. \[CSTN\_arch\](a) captures a simple implementation of an 8$\times$8 CSTN based on *OMEGA* network [@ahmadi1989survey]. The network is constructed using permutation elements, denoted as Re-Routing Blocks (RRB). Each RRB is able to possibly toggle and permute each of the input signals to each of its outputs. The number of RRBs needed to implement this simple CSTN for $N$ inputs ($N$ is a power of 2) is simply $N/2\times logN$. Each CSTN should be paired with an RCSTN. RCSTN must be able to reverse all operations accomplished by CSTN to re-generate the plaintext. Due to the structure of CSTN, RCSTN can be implemented by *vertically flipping* the CSTN without any change in configuration [@goke1973banyan]. In fact, by vertically flipping the CSTN, and then applying the same configuration, we re-generate plaintext. So, implementing RCSTN by vertically flipping the CSTN allows us to use the same configuration for both CSTN and RCSTN. However, to avoid duplicating the hardware (to put one dedicated hardware for CSTN and one dedicated hardware for RCSTN), by flipping the configuration bits (row-pivot reversed TRN), the CSTN would operate as its corresponded reverse CSTN. Hence, only one hardware is enough to operate as both CSTN and RCSTN (using TRN or row-pivot reversed TRN). The *OMEGA* network along with many other networks of such nature (*BUTTERFLY*, etc.) are blocking networks [@ahmadi1989survey], in which we cannot produce all permutations of input at the network’s output pins. This limitation significantly reduces the ability of a CSTN to randomize its input. Also, Evaluation of this permutation networks as a means of obfuscation to defend supply chain shows that the blocking version of this breed of networks could be easily broken by a SAT attack within few iterations [@kamali2019full; @azar2019coma]. Being a blocking or a non-blocking CSTN depends on the number of stages in CSTN. Since no two paths in an RRB are allowed to use the same link to form a connection, for a specific number of RRB columns, only a limited number of permutations is feasible. However, adding extra stages could transform a blocking CSTN into a strictly non-blocking CSTN. Using a strictly non-blocking CSTN not only improves the randomization of propagated messages through the CSTN, but also improves the resiliency of these networks against possible SAT attacks for extraction of a TRN used as the key for a CSTN-RCSTN cipher. A non-blocking logarithmic network could be represented using $LOG_{n, m, p}$, where $n$ is the number of inlets/outlets, $m$ is the number of extra stages, and $p$ indicates the number of copies *vertically cascaded* [@shyy1991log]. According to [@shyy1991log], to have a strictly non-blocking CSTN for an arbitrary $n$, the smallest feasible values of $p$ and $m$ impose very large area/power overhead. For instance, for $n=64$, the smallest feasible values, which make it strictly non-blocking, are $m=3$ and $p=6$, which means there exists more than $5\times$ as much overhead compared to a blocking CSTN with the same $n$, resulting in a significant increase in the area and delay overhead. To avoid such large overhead, we employ a *close to non-blocking CSTN* described in [@shyy1991log] to implement the CSTN-RCSTN pair. This network is able to generate not all, but *almost all* permutations, while it could be implemented using a $LOG_{n, log_2(n) - 2, 1}$ configuration, meaning it needs $log_2(n) - 2$ extra stages and no additional copy. Fig. \[CSTN\_arch\](b), demonstrates an example of such a near non-blocking CSTN with $n = 8$. Based on the structure of CSTN/RCSTN, and the size used for implementation, the size of configuration bits ($S$) would be changed. For instance, for a near non-blocking $LOG_{64, 4, 1}$, the number of selectors is 960 $(2log_2(64) - 2)(32)(3)$ (3 selectors in each $2 \times 2$ switches (Fig. \[CSTN\_arch\])). Based on the size of configuration, and the number of messages that could be sent in each interval ($T$), the overhead (time/energy) would be changed in *ExTru*. However, we show that since ($T$) is large enough, the performance boost, as well as the mitigating of the energy consumption, would be considerably high. Authenticated Encryption with Associated Data {#AEAD} --------------------------------------------- The Authenticated Encryption with Associated Data (AEAD) is used in *ExTru* for the transmission of the CSTN-RCSTN configuration (TRN). Authenticated ciphers incorporate the functionality of confidentiality, integrity, and authentication. The input of an authenticated cipher includes *plaintext* (message), *associated data* (AD), *public message number* (NPUB), and *secret key*. Then, the *ciphertext* is generated as a function of these inputs. A *tag*, which depends on all inputs, is generated after message encryption to assure the integrity and authenticity of the transaction. This tag is then verified after the decryption process. The choice of AEAD could significantly affect the area overhead of the solution, the speed of encrypted communication, and the extra energy/power consumption. To show the performance, power/energy, and area trade-offs, we employ two AEAD solutions: a NIST compliant solution (AES-GCM) [@dworkin2007sp], and a promising lightweight solution (ACORN) [@wu2016acorn]. AES-GCM is the current National Institute of Standards and Technology (NIST) standard for authenticated encryption and decryption as defined in [@dworkin2007sp]. ACORN is one of two finalists of the Competition for Authenticated Encryption: Security, Applicability, and Robustness (CAESAR), in the category of lightweight authenticated ciphers, as defined in [@wu2016acorn]. An 8-bit side-channel protected version of AES-GCM and a 1-bit side-channel protected version of ACORN are implemented as described in [@diehl2018face]. Both implementations comply with lightweight version of the CAESAR HW API [@homsirikamol2015gmu]. Our methodology for side-channel resistant is threshold implementation (TI), which has wide acceptance as a provably secure Differential Power Analysis (DPA) countermeasure [@nikova2006threshold]. In TI, sensitive data is separated into shares and the computations are performed on these shares independently. TI must satisfy three properties: (1) Non-completeness: Each share must lack at least one piece of sensitive data, (2) Correctness: The final recombination of the result must be correct, and (3) Uniformity: An output distribution should match the input distribution. To ensure uniformity, we refresh TI shares after non-linear transformations using randomness. We use a hybrid 2-share/3-share approach, where all linear transformations in each cipher are protected using two shares, which are expanded to three shares only for non-linear transformations. To verify the resistance against DPA, we employ the Test Vector Leakage Assessment methodology in [@gilbert2011testing]. We leverage a “fixed versus random” non-specific t-test, in which we randomly interleave first fixed test vectors and then randomly-generated test vectors, leading to two sequences with the same length but different values. Using means and variances of power consumption for our fixed and random sequences, we compute a figure of merit $t$. If $|t| > 4.5$, we reason that we can distinguish between the two populations and that our design is leaking information. The protected AES-GCM design has a 5-stage pipeline and encrypts one 128-bit input block in 205 cycles. This requires 40 bits of randomness per cycle. In ACORN-1, there are ten 1-bit TI-protected AND-gate modules, which consume a total of 20 random re-share, and 10 random refresh bits per state update. In a two-cycle architecture, 15 random bits are required per clock cycle. **Random Number Generator (RNG)** {#RNG} --------------------------------- A RNG unit is required on both sides to generate random bits for side-channel protection of AEAD units, a random public message number (NPUB) for AEAD, and TRNs for CSTN-RCSTN. We adopted the ERO TRNG core described in [@petura2016survey], which is capable of generating only 1-bit of random data per over 20,000 clock cycles. In our TI implementations, AES-GCM needs 40 and ACORN 15 bits of random data per cycle. So, we employed a hybrid RNG unit combining the ERO TRNG with a Pseudo Random Number Generator (PRNG). TRNG output is used as a 128-bit seed to PRNG. The PRNG generates random numbers needed by other components. The reseeding is performed only once per activation. We adopted two different implementations of PRNG: (1) AES-CTR PRNG, which is based on AES, is compliant with the NIST standard SP 800-90A, and generates 12.8 bits per cycle. (2) Trivium based PRNG, which is based on the Trivium stream cipher described in [@de2005trivium]. The Trivium-based PRNG is significantly smaller in terms of area and much faster than AES-CTR PRNG. It can generate 64 bits of random data per cycle, however, it is not compliant with the NIST standard. Also, the ERO TRNG is equipped with standard-statistical-tests applied post-fabrication, such as Repetition-Count test and the Adaptive-Proportion test, as described in NIST SP 800-90B [@barker2012recommendation], any attempt at weakening the TRNG during regular operation (i.e. fault attack) can be detected by continuously checking the output of a source of entropy for any signs of a significant decrease in entropy, noise source failure, and hardware failure. **Substitution Box (S-Box)** {#sbox} ---------------------------- To eliminate the linearity/predictability in *ExTru*, a non-feistel trial strategy has been used that is based on Khazad block cipher [@barreto2000khazad]. The wide trial strategy is composed of several linear and non-linear transformations that ensures the dependency of output bits on input bits in a complex manner [@daemen1995cipher]. The input and output correlation of this strategy is very large if the linear approximation is done for even one round. Also the transformation is kept uniform which treats every bit in a similar manner and provides opposition to differential attacks. Security Analysis of *ExTru* {#attacks} ============================ Assuming that the attacker can monitor the side-channel information of the chips during normal operations (based on power/current traces), and the possibility of having access to the scan chain to apply any form of scan-based attack, in this section we evaluate the resiliency of *ExTru* against different physical attacks, such as side-channel, the scan-based SAT, and algebraic attack. An Attack objective may be (1) extracting the secret key, or (2) extracting CSTN configuration (TRNs), or (3) eavesdropping on messages exchanged between the devices. \ **Side-Channel Attack (SCA)** ----------------------------- The objective of SCA on *ExTru* is to extract either the secret key used by AEAD (ACORN) or the TRN used by CSTN. Extracting a secret key is sufficient to break the communication. By extracting the secret key, the attacker can decrypt the TRN transmitted between transmitter/receiver, and by knowing the TRN, the plaintext could be recovered. Similarly, extracting the TRN reveals the communicated messages, however, since the TRN would be updated dynamically, extracting the TRN would reveal only part of the messages. It is worth mentioning that assuming that the secret key or TRN is extracted, the functionality of the s-box would be revealed using specific messages. Fig. \[ttest\] captures our assessment of the side-channel resistance of AEAD using a t-test for unprotected and protected implementations of AES-GCM and ACORN [@diehl2018comparison]. As illustrated, both implementations pass the t-test, indicating the guaranteed resistance against SCA. Note that this guaranteed resistance against SCA shows the robustness of communication channel during TRN transmission. In addition, by adding the dynamicity in *ExTru*, any form of attacks, including SCA, the SAT, and algebraic, must be carried out in a limited time while the TRN of the CSTN/RCSTN is unchanged. As soon as the TRN is renewed, the previous side-channel traces or SAT iterations or algebraic calculations are useless. The period of TRN updates introduces a trade-off between energy and security and can be pushed to maximum security by changing the TRN for every new input. **TRN Extraction using the SAT attack** --------------------------------------- Since the attacker might have access to the scan chain to apply any form of scan-based attack, it might be possible to recover and extract the TRN by applying specific inputs to the CSTN and observing the output. This could be done by using the SAT attack that is a very applicable and known attack on logic locking schemes [@zamiri2019threats]. In this scheme, assuming that the TRN is the unknown parameters (such as key in logic locking), based on Table \[omega\_sat\], it is evident that using blocking CSTN, particularly small size CSTN, does not make the design resilient against the SAT attacks. The number of iterations in Table \[omega\_sat\] shows the number ($N$) of specific inputs identified by SAT solver, called Discriminating Inputs (DIPs) [@subramanyan2015evaluating]. Finding $N$ DIPs by SAT solver allows the attacker to find CSTN/RCSTN configuration (TRN), and consequently breaks the scheme. It is evident that increasing the size of CSTN will increase $N$ (e.g. from $N=6$ in size 4 to $N=25$ in size 256). For an *OMEGA*-based CSTN with size 512, SAT is not able to find the TRN after $2\times10^6$ seconds. Even after $2\times10^6$ seconds execution of SAT, it could find only 7 DIPs. However, we expect that for an *OMEGA*-based CSTN with size 512, SAT needs more than 25 DIPs to find TRN. [@ l \*9c @]{} CSTN Size ($n$) & 4 & 8 & 16 & 32 & 64 & 128 & 256 & 512\ SAT Iterations & 6 & 7 & 8 & 12 & 14 & 24 & 25 & TO\ SAT Execution Time $_{(Seconds)}$ & 0.01& 0.03 & 0.2 & 0.8 & 5.9 & 130.5 & 1136.2 & TO\ *TO: Timeout = $2\times10^6$ seconds* Table \[nonblk\_sat\] illustrates that using near non-blocking CSTN considerably enhances the resiliency of this approach against the SAT attack. As shown in Table \[nonblk\_sat\], for a near non-blocking CSTN with a size of 64 ($LOG_{64, 4, 1}$), the SAT is not able to find the TRN after $2\times10^6$ seconds. Even after $2\times10^6$ seconds execution of SAT, it cannot find more than 5 DIPs. However, based on the SAT iterations for $LOG_{32, 3, 1}$, we expect that for a close to non-blocking CSTN with size 64, more than 32 DIPs are required to extract CSTN configuration. [@ l \*9c @]{} CSTN Size ($n$) & 4 & 8 & 16 & 32 & 64\ SAT Iterations & 14 & 18 & 25 & 32 & TO\ SAT Execution Time $_{(Seconds)}$ & 0.01& 0.015 & 2.35 & 79.18 & TO\ *TO: Timeout = $2\times10^6$ seconds* **Algebraic Attacks** --------------------- Algebraic attacks involve (a) expressing the cipher operations as a system of equations, (b) substituting in known data for some variables, and (c) solving for the key. ACORN has been demonstrated to be resistant against all known types of algebraic attacks, including linear cryptanalysis. Therefore, in the absence of any new attacks, the TRN transmission mode is resistant against algebraic attacks. Using CSTN and RCSTN by itself is new and requires more analysis. CSTN can be expressed as an affine function of the data input $x$, of the form $y=A\cdot x + b$, where $A$ is an $n \times n$ matrix and $b$ is an $n \times 1$ vector, with all elements dependent on the input TRN. Although recovering $A$ and $b$ is not equivalent to finding the TRN, it may enable the successful decryption of all blocks encrypted using a given TRN. We protect against this threat in numerous ways: (1) The number of blocks encrypted using a given TRN is set to the value smaller than $n$, which prevents generating and solving a system of linear equations with $A$ and $b$ treated as unknowns, (2) a part of the configuration is data-dependent and is fed from the output of the CSTN (stateful), so the values of $A$ and $b$ are not the same in any two encryptions, without the need of feeding CSTN with two completely different TRN values, (3) the substitution box added after the CSTN will eliminate all linearity/predictability of the CSTN using the algebraic attack. Experimental Setup and Analysis {#sec:results} =============================== For evaluation, all designs have been implemented using Verilog HDL, and have been synthesized for both FPGA and ASIC targets. For ASIC verification, we used Synopsys generic 32nm process. For FPGA verification, we targeted a small FPGA board, Digilent Nexys-4 DDR with Xilinx Artix 7 (XC7A100T-1CSG324). In addition, for SAT evaluation, we employed the Lingling-based SAT attack [@subramanyan2015evaluating] on a Dell PowerEdge R620 equipped with Intel Xeon E5-2670 2.6 GHz and 64GB of RAM. Also, as noted, a run-time limit of $2 \times 10^6$ seconds was set for the SAT solver. For ciphers, we used two side-channel resistant ciphers (AES-GCM128 as a block authenticated cipher, and ACORN as a lightweight stream cipher). We have two modes in *ExTru*: (1) *ExTru* with AES-GCM, compared with its corresponding cipher (AES-GCM), (2) *ExTru* with ACORN, compared with its corresponding cipher (ACORN). All configurations are listed in Table \[extru\_config\]. [@ l \*3c @]{} & Block & Stream\ AEAD & AES-GCM & ACORN\ PRNG & AES-CTR & Trivium\ BUS Width & 8 & 8\ Pins used for Communication & 8 & 8\ CSTN-RCSTN Size & 64 & 64\ Trusted Memory & 4 Kbits & 4 Kbits\ C$_{fix}$: initialization overhead (cycles) & 10,492 & 20,452\ C$_{byte}$: cycles needed for encrypting each byte & 72 & 17\ PRNG$_{perf}$: Throughput of generating TRN & $128bit / 10cycles$ & $64bit / cycle$\ Table \[basic\_ppa\] demonstrates the resource utilization of $LOG_{64, 4, 1}$ compared to both ciphers using Synopsys generic 32nm library, after post-layout (route) verification (PLS). As it can be seen, PLS reports show that the power consumption of $LOG_{64, 4, 1}$ is higher than ACORN. However, based on the area utilization, $LOG_{64, 4, 1}$ is considerably smaller than ACORN and AES-GCM. The main reason is that the switching activity of CSTN is high due to numerous permutation/toggling + substitution which leads to have higher power consumption than ACORN. Additionally, the delay of critical paths in both ciphers is higher than that of CSTN. Based on Fig. \[extruarch\], it is obvious that critical path in *ExTru* is same as that of its corresponding cipher. Consequently, we expect that the delay of critical path in *ExTru* is approximately equal with that of ciphers. Also, Table \[blk\_vs\_nonblk\] depicts area, power, and the delay of CSTNs in both blocking and near non-blocking mode with different sizes in the Synopsys generic 32nm process. As shown, it is evident that using a close to non-blocking CSTN with size 64, $LOG_{64, 4, 1}$, provides the most efficient CSTN structure, which is resilient against SAT attack. It should be noted that due to having extra stages in close to non-blocking CSTNs, the delay of these networks is slightly higher than the blocking CSTNs with the same $n$, which is negligible. [@ l \*9c @]{} & & & Power ($uW$) & & & Area ($nm^2$)& & & Delay ($ns$)\ $LOG_{64, 4, 1}$ & & & 1625.5 & & & 9965.9 & & & 1.74\ AES-GCM & & & 3587.1 & & & 102487.5 & & & 2.48\ ACORN & & & 880.9 & & & 21843.4 & & & 2.3\ [@ l \*9c @]{} & Area ($nm^2$) & Power ($uW$) & Delay ($ns$) & SAT-Resilient\ omega32 & 1013.1 & 44.8 & 1.12 &\ log(32, 3, 1) & 3067.5 & 213.5 & 1.33 &\ omega64 & 2285.5 & 107.1 & 1.22 &\ **log(64, 4, 1)** & **7438.8** & **845.1** & **1.73** & ****\ omega128 & 5081.5 & 250.3 & 1.25 &\ omega256 & 11364.9 & 579.1 & 1.35 &\ **omega512** & **25458.3** & **2308** & **1.42** & ****\ Table \[extru\_ppa\] depicts resource utilization of *ExTru* in each mode of using AES-GCM or ACORN. As we expected, the critical paths of *ExTru* in each mode is same as that of corresponding cipher. In addition, since *ExTru* consists of both CSTN and cipher, it is evident that area and power of *ExTru* in each mode is approximately equal to summation of total area and total power of both sub-modules, i.e. CSTN and corresponding cipher. The active power of each design for different message sizes has been gathered using Synopsys PrimeTime PX. Fig. \[PL\_power\] demonstrates the power breakdown in each design for a 1KB message. As it is shown, the leakage powers are roughly the same. The internal power and switching power of *ExTru* is almost 23% worse. The main reason for increasing switching activity is the structure of CSTN for bit-wise permutation/toggling. Also, internal power has been increased due to merging both CSTN and cipher into one design. [@ l \*9c @]{} & & & Power ($uW$) & & & Area ($nm^2$)& & & Delay ($ns$)\ *ExTru* with AES-GCM & & & 4448.9 & & & 122457.4 & & & 2.48\ *ExTru* with ACORN & & & 1694.6 & & & 33344.7 & & & 2.3\ Energy/Performance Improvement in ExTru --------------------------------------- Although combining CSTN and cipher into *ExTru* imposes area and power overhead by almost 24.5% compared to the corresponding cipher, CSTN can generate {permuted/toggled + substituted} data in only one cycle which provides significant speed-up compared to especially side-channel resistant ciphers that require randomness or complex initialization. Fig. \[time\_exe\] demonstrates the time of preparing data (encryption *or* permutation/toggling + substitution) for different message sizes. Increasing the size of the message, which increases the proportion of $I$ to $S$, significantly (superlinearly) increases the gap between the execution time of *ExTru* compared to its corresponding cipher. As shown, since CSTN prepares each $I$ in one cycle, increasing the size of the message imposes no degradation on *ExTru* performance. The main part of the execution time of *ExTru* is dedicated to encrypting and sending $S$. On the other hand, all data must be encrypted before sending it while only a cipher is used. So, it increases the execution time of ciphers linearly due to encryption time. Note that based on our SAT-based evaluation, the guaranteed number of $I$ messages is 32 (Table \[nonblk\_sat\]). Since we use $LOG_{64, 4, 1}$, each $I$ is 64 bits, so 256KB ($64 \times 32$ = 2Kb = 256KB) is the safe size of sending data through CSTN. The guaranteed speed-up is $3.4\times$ and $1.3\times$ compared to AES-GCM and ACORN, respectively. It is evident that for small messages, *ExTru* works slower than ciphers due to time overhead of sending encrypted TRN. However, *ExTru* can accelerate the execution time up to $25\times$ while the message size is even 2KB. The speed-up gained by *ExTru* depends on the structure of the cipher. For instance, the AES-GCM needs around 300 cycles per each plain data to be first-order side-channel resistant. However, ACORN as a stream cipher needs fewer cycles per data. So, *ExTru* provides better speed-up while the cipher is not streamed/pipelined. \ Table \[energy\_cmp\] depicts energy consumption for different designs, with different message sizes. Since energy is a function of time and power, it is obvious that the energy consumption in *ExTru* is higher for small message sizes due to the time overhead of sending encrypted TRN. However, increasing the size of the network results in significantly less energy consumption in *ExTru* compared to corresponding ciphers. As it can be seen, *ExTru* reduces energy consumption by 94.5% and 67.8% compared to GCM and ACORN, respectively. [@ \*9c @]{} & 32B & 64B & 128B & 256B & 512B & 768B & 1KB & 2KB\ ACORN & 17.01 & 18.69 & 22.06 & 28.79 & 42.26 & 55.73 & 69.20 & 123.1\ *ExTru* with ACORN & 30.66 & 30.80 & 31.09 & 31.67 & 32.82 & 33.98 & 35.13 & 39.75\ AES-GCM & 46.28 & 93.28 & 188.2 & 379.8 & 756.6 & 1143 & 1523 & 3055\ *ExTru* with AES-GCM & 151.1 & 151.4 & 152.1 & 153.3 & 155.8 & 158.4 & 160.9 & 173.1\ *\* Message Size* [@ l \*9c @]{} & & & LUTs & & & Registers & & & Maximum Frequency\ ACORN & & & 1090 & & & 530 & & & 178.5 MHz\ *ExTru* with ACORN & & & 1609 & & & 1573 & & & 172.5 MHz\ AES-GCM & & & 3803 & & & 4418 & & & 158.3 MHz\ *ExTru* with AES-GCM & & & 4376 & & & 5461 & & & 152.4 MHz\ As mentioned previously, *ExTru* has been verified on both ASIC and FPGA. Table \[fpga\_res\] demonstrates the resource utilization of the proposed scheme compared to ciphers on Nexys-4 DDR with Xilinx Artix 7. The results in FPGA are approximately similar to that of ASIC. As expected, ACORN provides higher maximum frequency due to its lightweight structure. However, using more resources in high-performance AES-GCM results in better throughput even with lower frequency. Conclusion {#sec:conclusion} ========== In this paper, we proposed *ExTru* as a dynamic encrypted high speed communication, which is able to provide a level of trust using near non-blocking configurable switching and toggling network (*CSTN*). *ExTru* uses near non-blocking CSTN as a transceiver data. Although the configuration of CSTN will be generated by TRNG, *ExTru* changes the configuration based on a time-interval which is identified by the SAT to guarantee the security of communication. Using this dynamically encrypted mechanism mitigates energy consumption by 94.5% and 67.8% compared to AES-GCM (authenticated) and ACORN (stream) while security is guaranteed. In addition, *ExTru* is able to provide up to $24.4\times$ and $4.3\times$ speed-up for 2KB messages in comparison with AES-GCM and ACORN, respectively.

--- abstract: 'We test several BFKL-like evolution equations for unintegrated gluon distributions against forward-central dijet production at LHC. Our study is based on fitting the evolution scenarios to the LHC data using the high energy factorization approach. Thus, as a by-product, we obtain a set of LHC-motivated unintegrated gluon distributions ready to use. We utilize this application by calculating azimuthal decorrelations for forward-central dijet production and compare with existing data.' author: - | Piotr Kotko$^1$, Wojciech Słomiński$^2$ and Dawid Toton$^3$\ \ $^1$ [*Department of Physics, Penn State University,*]{}\ [*University Park, 16803 PA, USA*]{}\ \ $^2$ [*The M. Smoluchowski Institute of Physics, Jagiellonian University,*]{}\ [*S. Łojasiewicza 11, 30-348 Kraków, Poland*]{}\ \ $^3$ [*The H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences,*]{}\ [*Radzikowskiego 152, 31-342 Kraków, Poland*]{} bibliography: - 'library.bib' title: Unintegrated gluon distributions for forward jets at LHC --- Introduction ============ A typical procedure in applying QCD to hadronic collisions relies on factorization theorems. They consist in two ingredients: a perturbatively calculable hard part and a nonperturbative piece parametrizing hadrons participating in a collision. The most known and tested is the collinear factorization (see e.g. [@Collins:2011zzd] for a review), which applies for a variety of processes, including jet observables in deep inelastic scattering (DIS) and hadron-hadron collisions. Here, the nonperturbative component is parametrized in terms of parton distribution functions (PDFs) which undergo Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations. The key feature of PDFs is the *universality*, i.e. the PDFs that are measured in one process can be used in any other for which the factorization holds. Therefore, for instance one can use PDFs fitted to DIS structure functions and use them to make predictions for jets in hadron-hadron collisions. Although the collinear factorization is powerful and well-tested, it is supposed that for certain observables, e.g. forward jets at high energies, another kind of evolution equations for the PDFs is needed. Namely, the perturbative calculations contain the logarithms of the form $\alpha_{s}\log(1/x)$, where $x$ is the longitudinal fraction of the hadron momentum carried by the parton. At high energies and forward rapidities $x$ is small and these logarithms need to be resumed. This is accomplished by means of various “small $x$” evolution equations, which essentially are various extensions of the pioneering Balitski-Fadin-Kuraev-Lipatov (BFKL) evolution equation (see e.g. [@Lipatov:1996ts]). In the small $x$ domain the transverse momenta of the partons exchanged between the perturbative and nonperturbative parts are not suppressed comparing to the collinear factorization. Therefore, the PDFs have an explicit dependence on the transverse momentum of a parton. Such objects are often referred to, as transverse momentum dependent PDFs (TMDs) or Unintegrated PDFs, although the former are typically used outside the small $x$ physics, and posses unambiguous (though in general process dependent) field theoretic definitions. Actually, at small $x$ one usually deals with initial state gluons only, and thus the object of interest in this paper is an Unintegrated Gluon Distribution (UGD). The UGDs have to be convoluted with a perturbative “hard part” according to so-called $k_{T}$ or High Energy Factorization (HEF). We describe this approach in some more detail in Section \[sec:HEF\]. Here, let us just mention that unlike the collinear factorization, the HEF is not a QCD theorem and actually the universality of UGDs is supposed to be violated for jet production in hadron-hadron collisions. Thus, in principle, the standard procedure of fitting the UGDs to the $F_{2}$ HERA data and using it for jets in hadron-hadron collisions is not correct, but there are no quantitative measures of the factorization violation so far. Actually, HEF is surprisingly quite successful with describing LHC data using UGDs from fits to structure functions, see for instance [@vanHameren:2014ala]. At present, there are several fits to $F_{2}$ data using different small $x$ approaches, see [@Ellis:2008yp; @Ross:2011zzb; @Kutak:2012rf; @Lipatov:2013yra] for more details. In the present work we undertake another path. We make an attempt to fit various BFKL-like UGDs directly to the LHC data for jet forward jet production. It has a twofold purpose. First, we have an opportunity to explore UGDs using relatively exclusive observables. Second, we want to free ourselves from the aforementioned universality problem when transferring UGDs from DIS to the LHC domain. We consider two separate measurements: jet transverse momentum spectra [@Chatrchyan2012] in forward-central jet production and forward-central dijet decorrelations [@CMS:2014oma]. The first measurement consists of two separate sets of data: for the forward jet and for the central jet. Thus, the mutual description of both spectra imposes a strong constraint on the UGDs and we shall use this measurements to make our fits. The second measurement will be used to test the fits. The paper is organized as follows. In Section \[sec:HEF\] we describe the approach of HEF. The small $x$ evolution equations with various components incorporating sub-leading effects are discussed in Section \[sec:evolution\_eqs\]. The fitting procedure and the software used are described in Section \[sec:Procedure\]. We give the results in Section \[sec:Results\]. Having the fits, we test them against recent forward-central dijet decorrelations data in Section \[sec:decorrelations\]. Finally, we discuss our research in Section \[sec:Summary\]. High Energy Factorization {#sec:HEF} ========================= In this introductory section we discuss in more detail issues concerning factorization at small $x$. This task is somewhat complicated, notably because of the various existing approaches and various existing definitions of UGDs. In the following paper the notion of HEF corresponds to a general class of factorization approaches supposed to be valid at small $x$. Below we list some of the existing realizations: 1. the factorization of Gribov, Levin and Ryskin (GLR) [@Gribov1983] for high-$p_{T}$ inclusive gluon production \[HEFGLR\] 2. the factorization of Catani, Ciafaloni and Hautmann (CCH) [@Catani:1990eg; @Catani:1994sq] for heavy quark production in DIS, photo-production and hadron-hadron collisions \[HEFCCH\] 3. the factorization of Collins and Ellis [@Collins1991] for heavy quark production in hadron-hadron collisions \[HEFCE\] 4. the factorization for inclusive gluon production in the saturation regime for proton-nuclei collisions within the Color Glass Condensate (CGC) approach [@Blaizot:2004wu] and color dipole formalism [@Kovchegov:2001sc; @Nikolaev:2004cu] (the equivalence of both approaches was shown in [@Iancu2004]) \[HEFCGC\] In these approaches the nonperturbative part is parametrized in terms of UGDs undergoing BFKL evolution (for GRL, CCH, Collins-Ellis factorizations) or nonlinear Balitsky-Kovchegov evolution [@Balitsky:1995ub; @Kovchegov:1999yj] (for CGC). On the other hand, superficially similar objects to UGDs appear in so-called transverse momentum dependent (TMD) factorization and are called TMD PDFs. One should however realize that the enumerated approaches are valid at leading logarithmic approximation, while the TMD factorizations are valid to all orders in the leading twist approximation. Moreover, unlike most of UGDs in the HEF factorizations, the TMD PDFs have precise operator definitions in terms of matrix elements of nonlocal operators. Those definitions require appropriate Wilson lines to be inserted in order to make the definitions gauge invariant and to resum collinear gluons related to final and initial state interactions. These insertions make the TMD PDFs, in general, process dependent and thus non-universal, breaking the principle of factorization (for more details see e.g. [@Bomhof:2006dp; @Mulders:2011zt]). Only for processes with at most two hadrons the TMD factorization is proved to hold to all orders (for example back-to-back single hadron production in DIS or Drell-Yan scattering). The natural question arises whether the non-universality of TMD PDFs transfers to the small $x$ limit. In ref. [@Xiao:2010sp] an explicit arguments were given that this is the case for dilute-dense collisions (actually the arguments hold for so-called “hybrid” factorization – see also below). Moreover it is known from the CGC approach that at really small $x$, i.e. in the saturation regime, the cross sections cannot be described by just dipoles (averages of two Wilson lines), but also higher correlators are needed [@Dominguez:2012ad], what violates the ordinary logic of factorization. However, for the case of back-to-back dijet production in dilute-dense collisions a generalized factorization has been proposed [@Dominguez:2011wm]; that is, the cross section can be given in terms of hard factors and certain universal pieces. Recently, these results were improved to the case of imbalanced dijets [@Kotko:2015ura]. In particular, when the imbalanced transverse momentum is of the order of transverse momenta of the jets the HEF for dijet production can be derived from the dilute limit of the CGC approach. In the present work we shall constrain ourselves to dijet production in p-p collisions in the linear regime, as the kinematics we are interested in (and where the data exist) do not allow to develop the saturation region. We want to utilize most of the phase space covered by the data, thus we do not constrain ourselves to the back-to-back dijet region analyzed in [@Dominguez:2011wm]. Rather, we shall use the HEF factorization for dijet production. Since this approach is an extension of the CCH formalism, we shall now briefly recall the latter and the required extensions to obtain HEF for dijets. For a direct derivation from CGC approach see [@Kotko:2015ura]. In the CCH high energy factorization, one considers the heavy quark pair produced via the tree-level hard sub-process $g^{*}\left(k_{A}\right)g^{*}\left(k_{B}\right)\rightarrow Q\overline{Q}$ in the axial gauge. The initial state gluons are off-shell and have the momenta of the form $k_{A}=x_A\, p_{A}+k_{TA}$ and $k_{B}=x_B\, p_{B}+k_{TB}$, where $p_{A}$, $p_{B}$ are the momenta of the incoming hadrons and $p_{A}\cdot k_{TA}=p_{B}\cdot k_{TB}=0$. This particular form of the exchanged momenta is a result of the imposed high energy limit. The off-shell gluons have “polarization vectors” that are $p_{A}$ and $p_{B}$ respectively. Thanks to this kinematics the sub-process given by ordinary Feynman diagrams is gauge invariant despite its off-shellness. In CCH approach the factorization formula for heavy quark production reads (see Fig. \[fig:HEF\]A) $$\begin{gathered} d\sigma_{AB\rightarrow Q\overline{Q}}=\int d^{2}k_{T A}\int\frac{dx_{A}}{x_{A}}\,\int d^{2}k_{T B}\int\frac{dx_{B}}{x_{B}}\,\\ \mathcal{F}_{g^{*}/A}\left(x_{A},k_{T A}\right)\,\mathcal{F}_{g^{*}/B}\left(x_{B},k_{T B}\right)\, d\hat{\sigma}_{g^{*}g^{*}\rightarrow Q\overline{Q}}\left(x_{A},x_{B},k_{T A},k_{T B}\right),\label{eq:HEN_fact_1}\end{gathered}$$ where $d\hat{\sigma}_{g^{*}g^{*}\rightarrow Q\overline{Q}}$ is the partonic cross section build up from the gauge invariant $g^{*}g^{*}\rightarrow Q\overline{Q}$ amplitude and $\mathcal{F}_{g^{*}/A}$, $\mathcal{F}_{g^{*}/B}$ are UGDs for hadrons $A$ and $B$. The contributions with off-shell quarks are suppressed. The UGDs are assumed to undergo the BFKL evolution equations. In Ref. [@Catani:1994sq] it was argued that similar factorization holds to all orders for DIS heavy quark structure function, although the argumentation misses the details comparing to collinear factorization proofs [@Collins:2011zzd], especially the definitions of UGDs and complications arising at higher orders in the axial gauge [@Avsar:2012hj].              In the works [@Deak2010],[@Kutak:2012rf],[@VanHameren2013],[@vanHameren2013a; @vanHameren:2014lna; @vanHameren:2014ala] as well as in this paper the CCH factorization was extended to model the cross section for jet production in hadron-hadron collisions. The first difficulty arises because now one has to consider also gluons in the final state, e.g. $g^{*}g^{*}\rightarrow gg$ sub-process for dijet production. The corresponding amplitude is however not gauge invariant when calculated from ordinary Feynman diagrams. A few approaches have been proposed to calculate a gauge invariant extension of such amplitudes [@vanHameren2012; @vanHameren2013a; @vanHameren:2013csa; @vanHameren:2014iua; @Kotko2014a]. These [*gauge invariant off-shell amplitudes*]{} in fact correspond to a vertex that can be calculated from the well-known Lipatov’s effective action [@Lipatov:1995pn; @Antonov:2004hh] (see Fig. \[fig:HEF\]B). The approaches [@vanHameren2012; @vanHameren2013a; @vanHameren:2013csa; @vanHameren:2014iua; @Kotko2014a] were however oriented on practical and efficient computations of multi-particle off-shell amplitudes using helicity method and computer codes. As stated before, in CCH the UGDs undergo BFKL evolution. In our extensions of CCH approach we allow the UGDs to undergo more complicated evolution equations, which are more suitable for jets. More details will be given in Section \[sec:evolution\_eqs\]. Yet another modification of the CCH formula comes from the fact that the present study concerns the system of dijets where one of the jet is forward, while the second is in the central region. From $2\rightarrow2$ kinematics it follows then, that $x_{A}\ll x_{B}$ (or the opposite), except for the small corner of the phase space. Since $x_{B}$ is typically of the order of $0.5$ the usage of small $x$ evolution for $\mathcal{F}_{g^{*}/B}$ is questionable (this is similar to dilute-dense system considered e.g. in [@Dominguez:2011wm]). Therefore we use collinear approach on the $B$ hadron side [@Deak:2009xt]. Technically, one takes the collinear limit in $d\hat{\sigma}_{g^{*}g^{*}\rightarrow2j}$ by sending $k_{TB}\rightarrow0$ to obtain a sub-process with one off-shell gluon $d\hat{\sigma}_{g^{*}g\rightarrow2j}$ (the off-shell amplitudes have well defined on-shell limit). In this limit one has to take into account also sub-processes with initial state on-shell quarks, $d\hat{\sigma}_{g^{*}q\rightarrow2j}$. The remaining integral over $d^{2}k_{TB}$ gives helicity sum for $B$ partons on one hand, and the integrated (collinear) PDF on the other $\int dk_{B}^{2}\,\mathcal{F}_{a^{*}/B}\left(x_{B},k_{TB}\right)=f_{a}\left(x_{B}\right)$. Thus, the final formula for the factorization model reads $$\begin{gathered} d\sigma_{AB\rightarrow2j}=\int d^{2}k_{TA}\int\frac{dx_{A}}{x_{A}}\,\int\frac{dx_{B}}{x_{B}}\,\sum_{b}\\ \mathcal{F}_{g^{*}/A}\left(x_{A},k_{T A},\mu\right)\, f_{b}\left(x_{B},\mu\right)\, d\hat{\sigma}_{g^{*}b\rightarrow2j}\left(x_{A},x_{B},k_{T A},\mu\right),\label{eq:HEN_fact_2}\end{gathered}$$ where we have included the hard scale dependence not only in the collinear PDFs $f_{b}$, but in the UGD as well. Such a dependence turns out to be important for certain exclusive observables involving a hard scale (e.g. large $p_{T}$ of jets; see e.g. [@vanHameren:2014ala]). We note, that when the final states become well separated in rapidity, i.e. when the central jet lies in the opposite hemisphere to the forward jet we start to violate our condition $x_{A}\ll x_{B}$ and different approach should be used. The factorization formula (\[eq:HEN\_fact\_2\]) resembles the linearized approach of [@Dominguez:2011wm] but it extends beyond the correlation limit as here the hard sub-processes have injected a nonzero $k_{T}$. As mentioned before, the formula (\[eq:HEN\_fact\_2\]) has been recently derived from the CGC approach in [@Kotko:2015ura]. Small *x* evolution equations {#sec:evolution_eqs} ============================== Let us now discuss the evolution equations for UGDs which were used in our fits. As described in the preceding section we concentrate on linear evolution equations. Below we list some of them with a short explanation. We consider only gluon UGDs, thus we skip the subscripts in $\mathcal{F}_{g^{*}/A}$. 1. pure BFKL equation\[enu:pure-BFKL-equation\] The equation in the leading logarithmic approximation reads [@Fadin:1975cb; @Kuraev:1977fs] $$\begin{gathered} \mathcal{F}\left(x,k_{T}^{2}\right)=\mathcal{F}_{0}\left(x,k_{T}^{2}\right)\\ +\overline{\alpha}_{s}\int_{x}^{1}\frac{dz}{z}\,\int_{0}^{\infty}dq_{T}^{2}\left[\frac{q_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right)-k_{T}^{2}\mathcal{F}\left(\frac{x}{z},k_{T}^{2}\right)}{\left|q_{T}^{2}-k_{T}^{2}\right|}+\frac{k_{T}^{2}\mathcal{F}\left(\frac{x}{z},k_{T}^{2}\right)}{\sqrt{4q_{T}^{4}+k_{T}^{4}}}\right]\label{eq:BFKL}\end{gathered}$$ where $\overline{\alpha}_{s}=N_{c}\alpha_{s}/\pi$ with $N_{c}$ being the number of colors. The initial condition for the evolution is given by $\mathcal{F}_{0}$. The NLO BFKL equation is also known [@Fadin:1998py; @Ciafaloni:1998gs]. One of the drawbacks of the pure BFKL equation comes from the fact that $q_{T}^{2}$ of the gluons emitted along the ladder is unconstrained. Indeed, since in the BFKL regime the virtuality of the exchanged gluons is dominated by the transverse components, the resulting *kinematic constraint* reads [@Andersson:1995jt; @Kwiecinski:1996a] $$q_{T}^{2}<\frac{1-z}{z}\, k_{T}^{2}\approx\frac{1}{z}\, k_{T}^{2}.\label{eq:kinematic_constr}$$ This constraint is also often referred to as the consistency constraint. 2. BFKL with the kinematic constraint (BFKL+C) To incorporate the consistency constraint one may include the appropriate step function into the real emission part of the BFKL. This operation, actually introduces some higher order corrections into the BFKL equation [@Kwiecinski:1996a]. In addition, one may introduce another class of sub-leading corrections by allowing the strong coupling constant to run with the local scale along the ladder. Finally, one may define the $q_{T}^{2}$ integration region to lie away from the infrared nonperturbative region by separating the $\int_{0}^{k_{T0}^{2}}dq_{T}^{2}$ integration and moving it to the initial condition (the infrared cutoff $k_{T0}^{2}$ is taken to be of the order of $1\,\mathrm{GeV}$). The improved equation reads [@Kwiecinski:1997ee] $$\begin{gathered} \mathcal{F}\left(x,k_{T}^{2}\right)=\mathcal{F}_{0}\left(x,k_{T}^{2}\right)\\ +\overline{\alpha}_{s}\left(k_{T}^{2}\right)\int_{x}^{1}\frac{dz}{z}\,\int_{k_{T0}^{2}}^{\infty}dq_{T}^{2}\left[\frac{q_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right)\Theta\left(k_{T}^{2}-zq_{T}^{2}\right)-k_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right)}{\left|q_{T}^{2}-k_{T}^{2}\right|}+\frac{k_{T}^{2}\mathcal{F}\left(\frac{x}{z},k_{T}^{2}\right)}{\sqrt{4q_{T}^{4}+k_{T}^{4}}}\right].\label{eq:BFKL_constr}\end{gathered}$$ Recently, it has been studied in the context of the Mueller-Navelet jets, that the energy-momentum conservation violation (which above is cured by a “brute force”) becomes less harmful when full NLO corrections are applied [@Ducloue:2014koa]. The effects of the kinematic constraints in the approximate form (\[eq:kinematic\_constr\]) as well as in the full form have been recently analyzed [@Deak:2015dpa] in the context of the CCFM evolution equation [Ciafaloni:1987ur,Catani:1989yc,Catani:1989sg,CCFMd]{}. 3. BFKL with the kinematic constraint in re-summed form (BFKL+CR) The equation (\[eq:BFKL\_constr\]) can be casted in yet another form [@Kutak:2011fu] $$\begin{gathered} \mathcal{F}\left(x,k_{T}^{2}\right)=\tilde{\mathcal{F}}_{0}\left(x,k_{T}^{2}\right)\\ +\overline{\alpha}_{s}\left(k_{T}^{2}\right)\int_{x}^{1}\frac{dz}{z}\,\int_{k_{T0}^{2}}^{\infty}\frac{d^{2}q_{T}}{\pi q_{T}^{2}}\Theta\left(q_{T}^{2}-\mu^{2}\right)\Delta_{R}\left(z,k_{T}^{2},\mu^{2}\right)\mathcal{F}\left(\frac{x}{z},\left|\vec{k}_{T}+\vec{q}_{T}\right|^{2}\right),\label{eq:BFKL+CR}\end{gathered}$$ where $$\Delta_{R}\left(z,k_{T}^{2},\mu^{2}\right)=\exp\left(-\overline{\alpha}_{s}\ln\frac{1}{z}\,\ln\frac{k_{T}^{2}}{\mu^{2}}\right)\label{eq:Regge_ff}$$ is the so-called Regge form factor. This form has been used in Ref. [@Kutak:2011fu] to propose a non-linear extension of the CCFM equation. The scale $\mu$ has been introduced to separate unresolved and resolved emissions in (\[eq:BFKL\_constr\]), i.e. the emissions with $q_{T}^{2}<\mu^{2}$ and $q_{T}^{2}>\mu^{2}$, and further the unresolved part was re-summed to obtain the Regge form factor. Note, that the UGDs undergoing this equation do not explicitly depend on the scale $\mu$ and that the new form of the initial condition has to be used (this is denoted by a tilde sign). 4. BFKL with the kinematic constraint and DGLAP correction (BFKL+CD) In Ref. [@Kwiecinski:1997ee] yet another improvement of (\[eq:BFKL\]) was proposed. One can make an attempt to account for DGLAP-like behaviour by including the non-singular part of the gluon splitting function (the third term below) $$\begin{gathered} \mathcal{F}\left(x,k_{T}^{2}\right)=\mathcal{F}_{0}\left(x,k_{T}^{2}\right)\\ +\overline{\alpha}_{s}\left(k_{T}^{2}\right)\int_{x}^{1}\frac{dz}{z}\,\int_{k_{T0}^{2}}^{\infty}dq_{T}^{2}\left[\frac{q_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right)\Theta\left(k_{T}^{2}-zq_{T}^{2}\right)-k_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right)}{\left|q_{T}^{2}-k_{T}^{2}\right|}+\frac{k_{T}^{2}\mathcal{F}\left(\frac{x}{z},k_{T}^{2}\right)}{\sqrt{4q_{T}^{4}+k_{T}^{4}}}\right]\\ +\overline{\alpha}_{s}\left(k_{T}^{2}\right)\int_{x}^{1}\frac{dz}{z}\,\left(\frac{z}{2N_{c}}P_{gg}\left(z\right)-1\right)\int_{k_{T0}^{2}}^{k_{T}^{2}}dq_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right),\label{eq:BFKL+CD}\end{gathered}$$ where $P_{gg}\left(z\right)$ is the standard gluon splitting function. This correction, similar to the kinematic constraint, accounts for certain sub-leading corrections to the BFKL equation. 5. BFKL with DGLAP correction alone\[enu:lastmodel\] This variant is used to test the significance of the DGLAP term alone. The above UGDs do not involve any hard scale dependence. For observables involving high-$p_{T}$ jets a presence of large scale $\mu^{2}\sim p_{T}^{2}$ in perturbative calculations would involve additional logarithms of the type $\log\left(\mu^{2}/k_{T}^{2}\right)$ which can spoil the procedure. Therefore a re-summation of those logs is desired and it accounts in hard scale dependence for UGDs, c.f. Eq. (\[eq:HEN\_fact\_2\]). The approach which incorporates both $x$, $k_{T}^{2}$ and $\mu^{2}$ dependence in UGDs is provided for example by the CCFM evolution equation (the code available for a practical use is described for example in [@Hautmann:2014uua]). Another approach, so called KMR (Kimber-Martin-Ryskin) procedure [@Kimber:1999xc; @Kimber:2001sc], takes ordinary PDFs and injects $k_{T}$ dependence via the Sudakov form factor taking care of matching to the BFKL evolution at small $x$. A serious advantage of this procedure is that one can use well known PDF sets, fitted to large data sets. Yet another approach was used in [@vanHameren:2014ala] in therms of so-called “Sudakov resummation model”. This procedure reverts, in a sense, the logic used in the KMR and uses the Sudakov form factor to inject the hard scale dependence instead of $k_{T}$. The procedure is parton-shower-like, i.e. it is applied after the MC events are generated and the cross section is known, and is unitary (i.e. the procedure does not change the total cross section). The advantage is that one may use it on the top of UGDs involving nonlinear effects. The basic idea behind the model is that it assigns the Sudakov probability $P$ for events with given $k_T$ and a hard scale $\mu\sim p_T$. Then, the probability of surviving is $1-P$. For events with small $k_T$ and large $\mu$ the emission probability is $P\sim 1$ and the unitarity of the procedure transfers such events to the region $k_T\sim p_T$. There is one more approach proposed in Ref. [@Kutak:2014wga], similar to the one just described, where analogous procedure is applied at the level of UGDs by fixing its integral over $k_{T}$ (it has an advantage of being independent on any software and one may produce grids for a practical usage). In summary, we may consider the following modifications of UGDs \[enu:pure-BFKL-equation\]-\[enu:lastmodel\]: 1. BFKL with the Sudakov (BFKL+S) 2. BFKL with the kinematic constraint and the Sudakov (BFKL+CS) 3. BFKL with DGLAP correction and the Sudakov (BFKL+DS) 4. BFKL with the kinematic constraint in re-summed form and the Sudakov (BFKL+CRS) 5. BFKL with the kinematic constraint, DGLAP correction and the Sudakov (BFKL+CDS)\[enu:BFKLCDS\] Unfortunately, as far as fitting of UGDs is considered, the above Sudakov-based models are not suitable. This is because they require the knowledge of an integral (whether it is a cross section or integrated gluon, c.f. [@vanHameren:2014ala] vs [@Kutak:2014wga]) which is unknown at the stage of fitting. In principle, one could try to use the method of successive approximations with the Sudakov model of Ref. [@vanHameren:2014ala]. We shall report on our attempts in Section \[sec:Results\]. There is one more comment in order here. The Sudakov resummation model is very sensitive to the region $k_{T}\lesssim1\,\mathrm{GeV}$ which is not well described by the practical implementations of the equations \[enu:pure-BFKL-equation\]-\[enu:lastmodel\] as they use certain low-$k_{T}$ cut, $k_{T\,0}$. For $k_{T}<k_{T\,0}$ the UGD is typically modelled or extrapolated by a constant value. Let us now discuss the models for the initial condition $\mathcal{F}_{0}$. In this paper we have tested the following models (in the brackets we give the aliases used below to identify the model): \[eq:param\] 1. exponential model (EXP)\[enu:Models\_ini\] $$\mathcal{F}_{0}\left(x,k_{T}^{2}\right)=N\, e^{-Ak_{T}^{2}}\left(1-x\right)^{a}\left(1-Dx\right)\label{eq:Model_exp}$$ 2. (negative) power-like model with running $\alpha_{s}$ (POW) $$\mathcal{F}_{0}\left(x,k_{T}^{2}\right)=\frac{\overline{\alpha}_{s}\left(k_{T}^{2}\right)}{k_{T}^{2}}\, N \, x^A \left(1-x\right)^{a}\label{eq:Model_ask} (1-Dx)$$ 3. DGLAP-based model (Pgg)\[enu:Models\_fini\] $$\mathcal{F}_{0}\left(x,k_{T}^{2}\right)=\frac{\alpha_{s}\left(k_{T}^{2}\right)}{2\pi k_{T}^{2}}\, \int_{x}^{1}dz \, P_{gg}\left(z\right)\hat{\mathcal{G}}_{0}\left(x\right),\label{eq:Model_Pgg_1}$$ where $$\hat{\mathcal{G}}_{0}\left(x\right)=N \, x^A \left(1-x\right)^{a}\left(1-Dx\right)\label{eq:Model_Pgg_2}$$ is a model for an integrated gluon density. The parameters $N$, $A$, $a$, $D$ are, in general, free parameters and need to be fitted. We see that in principle there are quite a few variants to be fitted. Though not all of the combinations make sense, we are still left with several scenarios to be tested. Fitting procedure {#sec:Procedure} ================= We have used two data samples measured by CMS detector [@Chatrchyan2012] for inclusive forward-central dijet production at CM energy $\sqrt{s}=7\,\mathrm{TeV}$. The central jet is defined to lie within the pseudo-rapidity interval $\left|\eta_{c}\right|<2.8$ while the forward has to lie within $4.9>\left|\eta_{f}\right|>3.2$. Both jets are high-$p_{T}$ jets with $p_{T}>35\,\mathrm{GeV}$. The jets were reconstructed using anti-$k_{T}$ algorithm with radius $R=0.5$. The data samples consist in jet $p_{T}$ spectra for forward and for central jets, $d\sigma_{S}/dp_{T}\Delta\eta_{S}$ with $S=f,c$. There are in total 12 data bins for both forward and central jets. We have applied the following fitting procedure. For each existing experimental data bin $B$ we produce a 2-dimensional normalized histogram $\mathcal{H}^{B}$ with bins in $x$ and $k_{T}$, such that the cross section can be calculated as $$\sigma^{B}=\sum_{i,j}\mathcal{H}_{ij}^{B}\mathcal{F}\left(x\left(i\right),k_{T}\left(j\right)\right),\label{eq:Fit1}$$ where $i,j$ enumerate the bins in $\left(x,k_{T}\right)$. To make the histograms $\mathcal{H}^{B}$ we 1. generate Monte Carlo events for the process under consideration with $\mathcal{F}=\mathcal{F}^{*}$, where $\mathcal{F}^{*}$ is a relatively “broad” trial UGD (evolving according to one of the scenarios \[enu:pure-BFKL-equation\]-\[enu:BFKLCDS\]), 2. make histograms $\mathfrak{h}^{B}$ in $\left(x,k_{T}\right)$ of contributions to each data bin $B$, 3. divide by $\mathcal{F}^{*}\left(x,k_{T}\right)$, i.e. $\mathcal{H}_{ij}^{B}=\mathfrak{h}_{ij}^{B}/\mathcal{F}^{*}\left(x\left(i\right),k_{T}\left(j\right)\right)$. Hence, in principle $\mathcal{H}^{B}$ are independent of $\mathcal{F}^{*}$ used for their generation and are calculated only once. This is advantageous, as the hard cross section calculation is costly in CPU time. The latter is calculated using the Monte Carlo C++ program $\mathtt{LxJet}$ [@Kotko2013a] implementing (\[eq:HEN\_fact\_2\]). The generated events (weighted or unweighted) are stored in a $\mathtt{ROOT}$ [@Brun:1997pa] file for further processing. For the UGD evolution according to scenarios \[enu:pure-BFKL-equation\]-\[enu:lastmodel\] we solve the corresponding integral equations by a straightforward numerical iteration over a grid over $x$ and $k_T$. In order to make the fitting feasible, we need a fast routine to calculate $\mathcal{F}$ used in (\[eq:Fit1\]) for the cross section calculation. However, since our numerical procedure is too slow for that, we prepare grids over which we can interpolate the fitting parameters. Each such grid corresponds to a particular parametrization model and arguments range. Out of four parameters ($N$, $A$, $a$, $D$) of the initial conditions, we fix $D=0$ (see Sec. \[sec:Results\]). Moreover, we note that the solution for $\mathcal{F}$ is linear in $N$. Thus the actual grids are in $A$ and $a$. Results {#sec:Results} ======= We have applied the procedure described in the preceding section to most of the models \[enu:pure-BFKL-equation\]-\[enu:BFKLCDS\] and initial conditions \[enu:Models\_ini\]-\[enu:Models\_fini\]. The best values of [$\chi^{2}\!/\mathrm{NDP}$]{}($\chi^2$ per data point) are listed in Table \[tab:chi2\] for models \[enu:pure-BFKL-equation\]-\[enu:lastmodel\]. Note, that some of the scenarios were unable to describe the data, in particular the pure BFKL and BFKL with the kinematic constraint only. Evidently, the DGLAP correction is essential. The fitted values of the parameters of the initial conditions, $N$, $A$, $a$, for scenarios with ${\ensuremath{\chi^{2}\!/\mathrm{NDP}}\xspace}<2$ are collected in Table \[tab:params\]. The fits are presented in Figs. \[fig:central\]-\[fig:forward\]. For a better comparison we also plot the cross-sections scaled by $p_T^5$. We observe that all the models with the DGLAP correction give excellent description of the central-jet data, while the $p_T$ spectrum of forward jets is reasonably reproduced though less accurately. We also note that the models with lowest $\chi^2$ result in very similar predictions for the $p_T$ spectra. Our attempts to fit the scenarios with the Sudakov resummation can be summarized as follows. First, we observe that the model has a small overall effect on the $p_T$ spectra, although it slightly shifts the theory points away from the data points. We illustrate this in Fig. \[fig:sudeffect\], where we applied the Sudakov model on the top of the events obtained with one of the fits. When we now try to refit the $\mathcal{F}_0$ parameters, we change the total cross section (used already to apply the resummation) and the fit fails. Although we observe that the successive iterations improve the fit, the procedure turns out to be insufficient to make a reliable fit with the Sudakov resummation. $\mathcal{F}_{0}$ BFKL BFKL+C BFKL+D BFKL+CD BFKL+CR ------------------- ------ -------- -------- --------- --------- EXP 2.4 2.2 1.24 1.11 1.52 POW 2.3 1.9 1.02 1.12 Pgg – – 1.13 1.11 : The values of [$\chi^{2}\!/\mathrm{NDP}$]{}for fits of unintegrated gluon density evolving according to various models described in Section \[sec:evolution\_eqs\]. The first column lists the initial condition ansatz, see also Section \[sec:evolution\_eqs\] for details.\[tab:chi2\] model $N$ $A$ $a$ --------------- --------- --------- --------- BFKL+CR (EXP) $0.095$ $0.012$ $0^*$ BFKL+D (EXP) $0.37$ $0.18$ $0.5^*$ BFKL+CD (EXP) $0.68$ $0.14$ $2.5^*$ BFKL+C (POW) $320$ $1.4$ $61.0$ BFKL+D (POW) $12.7$ $0.5^*$ $5.7$ BFKL+CD (POW) $562$ $0.96$ $35.7$ BFKL+D (Pgg) $106$ $1.2$ $2.5$ BFKL+CD (Pgg) $628$ $2.9$ $5.7$ : The values of initial condition \[enu:Models\_ini\]-\[enu:Models\_fini\] parameters obtained from the fits to the CMS data. We list only the scenarios with ${\ensuremath{\chi^{2}\!/\mathrm{NDP}}\xspace}< 2$. The values denoted by a star were fixed — see the main text for details.\[tab:params\] A few comments are in order. The considered jet data are not sufficient to precisely determine all the parameters ($N$, $A$, $a$, $D$) of the initial parametrizations (\[eq:param\]). Thus, first we neglect the $(1-Dx)$ factor, i.e. we take $D=0$. We have checked that we get no improvement when $D$ is a free parameter. Next, in some cases the fits are not sensitive enough to uniquely determine the three remaining free parameters. In these cases we fix $A$ or $a$ at some plausible value (these are marked with a star in Table \[tab:params\]). Actually, besides the initial condition parameters $N$, $A$, $a$, $D$ we have also the boundary values of kinematic parameters $x_{A}$, $k_{T}$ (c.f. (\[eq:HEN\_fact\_2\])), which – to certain extent – are free parameters as well. We set them as follows. First, in order to be in an accordance with the assumptions leading to (\[eq:HEN\_fact\_2\]) we imply the cut $x_{A}<x_{B}$. Next, for all scenarios we set $x_{A\,\mathrm{min}}=0.0001$. For the model with the DGLAP correction we set $x_{A\,\mathrm{max}}=1.0$ while for the others we set $x_{A\,\mathrm{max}}=0.4$. Further we use $k_{T\,\mathrm{min}}=1\,\mathrm{GeV}$ for DGLAP models and $k_{T\,\mathrm{min}}=0.1\,\mathrm{GeV}$ for the others. Finally, we use $k_{T\,\mathrm{max}}=100\,\mathrm{GeV}$ for exponential initial condition and $k_{T\,\mathrm{max}}=400\,\mathrm{GeV}$ for the others. The last comment concerns the hard scale choice: in all fits we have used the average $p_{T}$ of the jets. ![The $p_{T}$ spectra of the central jet calculated using the best fits for individual models versus the CMS data. For the bottom plot the cross sections have been scaled by $p_{T}^{5}$ to better see the differences between the models.\[fig:central\]](Central "fig:"){width="48.00000%"} ![The $p_{T}$ spectra of the central jet calculated using the best fits for individual models versus the CMS data. For the bottom plot the cross sections have been scaled by $p_{T}^{5}$ to better see the differences between the models.\[fig:central\]](Central_scaled "fig:"){width="48.00000%"} ![The $p_{T}$ spectra of the forward jet calculated using the best fits for individual models versus the CMS data. For the bottom plot the cross sections have been scaled by $p_{T}^{5}$ to better see the differences between the models.\[fig:forward\]](Forward "fig:"){width="48.00000%"} ![The $p_{T}$ spectra of the forward jet calculated using the best fits for individual models versus the CMS data. For the bottom plot the cross sections have been scaled by $p_{T}^{5}$ to better see the differences between the models.\[fig:forward\]](Forward_scaled "fig:"){width="48.00000%"} The influence of the Sudakov resummation model is illustrated in Fig. \[fig:sudeffect\]. Here, we have chosen the best fits to illustrate the effect. We see, that the jet spectra are rather weakly affected by the resummation, although the forward jet spectrum becomes steeper than the data. ![An effect of the Sudakov resummation model (BFKL+CDS) when applied to one of our fits for the model BFKL+CD with exponential initial condition. For comparison we plot also the spectra obtained from the unintegrated gluon density with more involved evolution and fitted to HERA data (KS-HERA), see the main text for more details. \[fig:sudeffect\]](bfklcds_exp_pTc "fig:"){width="48.00000%"} ![An effect of the Sudakov resummation model (BFKL+CDS) when applied to one of our fits for the model BFKL+CD with exponential initial condition. For comparison we plot also the spectra obtained from the unintegrated gluon density with more involved evolution and fitted to HERA data (KS-HERA), see the main text for more details. \[fig:sudeffect\]](bfklcds_exp_pTf "fig:"){width="48.00000%"} The obtained UGDs are plotted in one-dimensional plots in Fig. \[fig:gluons\] as a function of $x$ and $k_T$. Note, that in order to better reflect the difference between UGDs we plot $k_T^2\, \mathcal{F}(x,k_T)$. We show results of all the models of Table \[tab:params\], hence also those with rather high $\chi^2$ value (see Table \[tab:chi2\]). All the UGDs with the DGLAP contribution are comparable, which shows that the evolution scenario is more important than a particular shape of the initial parametrization. On the other hand, the differences between UGDs are more pronounced than those in the $p_T$ spectra, which means that the currently available data are not sufficient to discriminate among the models. The two most differing UGDs correspond to the BFKL+C (POW) and BFKL+CR (EXP) models which however have significantly higher ${\ensuremath{\chi^{2}\!/\mathrm{NDP}}\xspace}$ (above 1.5). We compare the new LHC-based UGDs with the one evolving according to a complicated evolution of [@Kwiecinski:1997ee; @Kutak:2004ym] and fitted to HERA data [@Kutak:2012rf] (we abbreviate it as ’KS-HERA’ on the figure). This evolution equation contains the kinematic constraint, full DGLAP correction (including quarks via coupled equations) and a nonlinear term motivated by the Balitsky-Kovchegov equation. The $p_T$ spectra resulting from this gluon density are presented in Fig. \[fig:sudeffect\]. ![Unintegrated gluon distributions evolving due to the models \[enu:pure-BFKL-equation\]-\[enu:lastmodel\] with the initial conditions \[enu:Models\_ini\]-\[enu:Models\_fini\] obtained from the fits to the LHC data as a function of $x$ (top) and $k_T$ (bottom). The UGDs are multiplied by $k_T^2$ to better illustrate the differences between the models. The most differing UGDs are those without the DGLAP correction and with significantly higher ${\ensuremath{\chi^{2}\!/\mathrm{NDP}}\xspace}> 1.5$ (BFKL+C and BFKL+RC). \[fig:gluons\]](gluons){width="90.00000%"} Azimuthal decorrelations {#sec:decorrelations} ======================== In order to apply the fits in practice we have calculated another observable for central-forward dijet production, namely, the differential cross sections in azimuthal angle $\Delta\phi$ between the two jets. At leading order the two jets are produced exactly back-to-back and the distribution is the Dirac delta at $\Delta\phi=\pi$. However, due to QCD emissions of additional partons (either forming additional jets or being soft particles with small $p_{T}$) the two jets are decorrelated. On theory side these decorrelations are well described by QCD-based parton shower algorithms. However, within the HEF there is a natural decorrelation mechanism built-in. Namely, due to the internal transverse momentum $k_{T}$ of a gluon the dijet system with transverse momenta $\vec{p}_{T1}$, $\vec{p}_{T2}$ is unbalanced by the amount $\left|\vec{p}_{T1}+\vec{p}_{T1}\right|=\left|\vec{k}_{T}\right|=k_{T}$. One can think of $k_{T}$ as a cumulative transverse momentum of many gluon emissions. In general, these emissions can be small-$p_{T}$ and large-$p_{T}$ emissions as well. The large-$p_{T}$ emissions may in general contribute a jet, thus we consider an inclusive dijet observables. Using the new fits and the $\mathtt{LxJet}$ program we have calculated the azimuthal decorrelations for the kinematics described in the beginning of Section \[sec:Procedure\]. The results are presented in Fig. \[fig:decorr\]. The bands represent uncertainty that comes from the scale variation by a factor of two. We compare our calculation with the preliminary CMS data [@CMS:2014oma][^1]. ![The results for the azimuthal decorrelations for inclusive forward-central dijet production using our best fits. When the Sudakov resummation model is applied to the generated events we get a better description of the CMS data.\[fig:decorr\]](bfklcds_exp_dphi "fig:"){width="50.00000%"}![The results for the azimuthal decorrelations for inclusive forward-central dijet production using our best fits. When the Sudakov resummation model is applied to the generated events we get a better description of the CMS data.\[fig:decorr\]](bfklcds_pow_dphi "fig:"){width="50.00000%"} ![The results for the azimuthal decorrelations for inclusive forward-central dijet production using our best fits. When the Sudakov resummation model is applied to the generated events we get a better description of the CMS data.\[fig:decorr\]](bfklcd_dphi "fig:"){width="50.00000%"}![The results for the azimuthal decorrelations for inclusive forward-central dijet production using our best fits. When the Sudakov resummation model is applied to the generated events we get a better description of the CMS data.\[fig:decorr\]](bfklcds_dphi "fig:"){width="50.00000%"} Discussion {#sec:Summary} ========== In the present paper we went through a thorough study of various small-$x$ evolution equations analyzing an impact of various effects on jet observables. The effects we mean here, are certain sub-leading corrections to the BFKL equation, such as the kinematic constraint or DGLAP corrections. Our study was based on fitting these evolution scenarios to two samples of LHC data for high-$p_{T}$ spectra for dijet production. These samples consist of separate spectra for the central rapidity and forward rapidity jets. Our findings can be summarized as follows. First observation is that both forward jet and central jet spectra can be simultaneously and reasonably described by the High Energy Factorization approach and BFKL-like evolution. We obtain the best quality fits for BFKL with DGLAP correction and kinematic constraint, with the DGLAP correction being the most important additional ingredient. This matches the fact that the data under consideration can be nicely described by the collinear factorization with a parton shower [@Chatrchyan2012; @CMS:2014oma]. Whereas in the High Energy Factorization the parton shower is – to some extent – simulated by the transverse momentum dependent gluon distribution with the DGLAP correction. For all evolution models we get very good fits to the central jet spectrum, while most of the models have problems with precise reproduction of the shape of the forward jet spectrum. Several models properly describe the dijet data despite some differences in the resulting UGDs. Measurements of some other observables or more differential dijet data could help to discriminate among the models. Using our fits we have calculated azimuthal decorrelations for the same kinematic domain. This observable was also measured by CMS. The comparison of our calculation with the data is reasonably good, especially when using the Sudakov resummation model on the top of the evolution models. Interestingly, the same resummation procedure spoils the forward jet $p_{T}$ spectrum. Our final remark is that although the High Energy Factorization with improved BFKL evolution equation catches the main physical aspects of the jet production at small $x$, one definitely needs higher order corrections. Such calculations exist for certain small $x$ processes like Mueller-Navelet jets [@Ducloue:2013hia; @Ducloue:2014koa] or inclusive hadron production p+A collisions within CGC formalism [@Chirilli:2012jd; @Staato2014], but not for the high-$p_{T}$ dijet observables under consideration. Acknowledgments {#acknowledgments .unnumbered} =============== We thank K. Kutak and A. van Hameren for many fruitful discussions. The work of P.K. and W.S. has been supported by the Polish National Science Center Grant No. . D.T. has been supported by NCBiR Grant No. . P.K. also acknowledges the support of DOE grants No. and . [^1]: We note that the total cross section obtained from [@CMS:2014oma] does not agree with [@Chatrchyan2012]. The ratio of the two is approx. 1.8. If this is a normalization difference only, our predictions should be shifted up by this factor.

--- abstract: 'The classical Julia-Wolff-Carathéodory Theorem characterizes the behaviour of the derivative of an analytic self-map of a unit disc or of a half-plane of the complex plane at certain boundary points. We prove a version of this result that applies to noncommutative self-maps of noncommutative half-planes in von Neumann algebras at points of the distinguished boundary of the domain. Our result, somehow surprisingly, relies almost entirely on simple geometric properties of noncommutative half-planes, which are quite similar to the geometric properties of classical hyperbolic spaces.' address: | CNRS, Institut de Mathématiques de Toulouse\ 118 Route de Narbonne\ F-31062 Toulouse Cedex 09, France. author: - Serban Teodor Belinschi title: 'A noncommutative version of the Julia-Wolff-Carathéodory Theorem' --- Introduction ============ The classical Julia-Wolff-Carathéodory Theorem describes the behaviour of the derivative of an analytic self-map of the unit disc $\mathbb D$ or of the upper half-plane $\mathbb C^+$ of the complex plane $\mathbb C$ at certain boundary points. Numerous generalizations, to self-maps of balls or polydisks in $\mathbb C^n$, analytic functions with values in spaces of linear operators, analytic self-maps on domains in spaces of operators or in more general Banach spaces etc. - see for example [@Rudin; @Fan; @Jafari; @Wlo; @Abate; @Mellon; @MM; @AbateRaissy] (the list is not exhaustive) - are known. This note gives a version of this theorem for noncommutative self-maps of the noncommutative upper half-plane of a von Neumann algebra $\mathcal A$. The result builds on the recent literature in the field - see [@AMcY; @ATY2; @PTD], and falls under the programme aiming to find the noncommutative versions of classical complex analysis results - see for example [@AKV0; @AKV; @AMc1; @AMc3; @AMc2]. In the second section we state our main result, and provide the required background. The third section is dedicated to proving a Schwarz lemma-type result for noncommutative functions. In this same section we give a simple (not necessarily original, though) proof of the classical Julia-Wolff-Carathéodory Theorem in order to make this article self-contained, and some lemmas that make use of it. Finally, in the last section we prove our main result. Noncommutative functions and the Julia-Carathéodory Theorem =========================================================== Noncommutative functions ------------------------ Noncommutative sets and functions originate in [@taylor0; @taylor]. We largely follow [@ncfound] in our presentation below. We refer to this excellent monograph for details on, and proofs of, the statements that follow. First a notation: if $S$ is a nonempty set, we denote by $M_{m\times n} (S)$ the set of all matrices with $m$ rows and $n$ columns having entries from $S$. For simplicity, we let $M_n(S):=M_{n\times n}(S)$. Given C${}^*$-algebra $\mathcal A$, a [*noncommutative set*]{} is a family $\Omega:=(\Omega_n)_{n\in\mathbb N}$ such that 1. for each $n\in\mathbb N$, $\Omega_n\subseteq M_n (\mathcal A);$ 2. for each $m,n\in\mathbb N$, we have $\Omega_m\oplus \Omega_n\subseteq\Omega_{m+n}$. The noncommutative set $\Omega$ is called [*right admissible*]{} if in addition the condition (c) below is satisfied: 1. for each $m,n\in\mathbb N$ and $a\in\Omega_m,c\in \Omega_n,w\in M_{m\times n}(\mathcal A)$, there is an $\epsilon>0$ such that $\left(\begin{array}{cc} a & zw\\ 0 & c\end{array}\right)\in\Omega_{m+n}$ for all $z\in\mathbb C, |z|<\epsilon$. Given C${}^*$-algebras $\mathcal{A,C}$ and a noncommutative set $\Omega\subseteq\coprod_{n=1}^\infty M_n(\mathcal A)$, a [ *noncommutative function*]{} is a family $f:=(f_n)_{n\in\mathbb N}$ such that $f_n\colon\Omega_n\to M_n(\mathcal C)$ and 1. $f_m(a)\oplus f_n(c)=f_{m+n}(a\oplus c)$ for all $m,n\in\mathbb N$, $a\in\Omega_m,c\in\Omega_n$; 2. for all $n\in\mathbb N$, $f_n(T^{-1}aT)=T^{-1}f_n(a)T$ whenever $a\in\Omega_n$ and $T\in GL_n(\mathbb C)$ are such that $T^{-1}aT$ belongs to the domain of definition of $f_n$. These two conditions are equivalent to the single condition 1. For any $m,n\in\mathbb N$, $a\in\Omega_m,c\in\Omega_n$, $S\in M_{m\times n}(\mathbb C)$, one has $$aS=Sc\implies f_m(a)S=Sf_n(c).$$ We shall refer to the indices $n$ of $\Omega_n$ or of $f_n$ as the “levels” of the noncommutative set $\Omega$ or of the noncommutative function $f$. A remarkable result (see [@ncfound Theorem 7.2]) states that, under very mild conditions on $\Omega$, local boundedness for $f$ implies each $f_n$ is analytic as a map between Banach spaces. Indeed, a hint towards the proof of this result is the following essential property of noncommutative functions: if $\Omega$ is admissible, $a\in \Omega_n, c\in\Omega_m, b\in M_{n\times m} (\mathcal A)$, such that $\left(\begin{array}{cc} a & b \\ 0 & c \end{array}\right)\in\Omega_{n+m}$, then there exists a linear map $\Delta f_{n,m}(a,c)\colon M_{n\times m}(\mathcal A)\to M_{n\times m} (\mathcal C)$ such that $$\label{FDQ} f_{n+m}\left(\begin{array}{cc} a & b \\ 0 & c \end{array}\right)=\left(\begin{array}{cc} f_n(a) & \Delta f_{n,m}(a,c)(b) \\ 0 & f_m(c) \end{array}\right).$$ Obviously, this implies in particular that $f_{n+m}$ extends to the set of all elements $\left(\begin{array}{cc} a & b \\ 0 & c \end{array}\right)$ such that $a\in \Omega_n, c\in\Omega_m, b\in M_{n\times m}(\mathcal A)$ (see [@ncfound Section 2.2]). Two properties of this operator that are important for us are $$\label{FDC} \Delta f_{n,n}(a,c)(a-c)=f(a)-f(c)=\Delta f_{n,n}(c,a)(a-c),\quad \Delta f_{n,n}(a,a)(b)=f_n'(a)(b),$$ the classical Frechet derivative of $f_n$ in $a$ aplied to the element $b\in M_n(\mathcal A)$. Moreover, $\Delta f_{n,m}(a,c)$ as functions of $a$ and $c$, respectively, satisfy properties similar to the ones described in items (1), (2) above (see [@ncfound Sections 2.3–2.5] for details). For convenience, from now on we shall suppress the indices denoting the levels for noncommutative functions, as it will almost always be obvious from the context. We provide three examples of noncommutative sets: 1. The noncommutative upper half-plane $H^+(\mathcal A)=(H^+(M_n(\mathcal A)))_{n\in\mathbb N}$, where $H^+(M_n(\mathcal A))=\{b\in M_n(\mathcal A)\colon\Im b>0\}$ (here $\Im b=\frac{b-b^*}{2i},\Re b=\frac{b+b^*}{2}$), 2. The set of nilpotent matrices with entries from $\mathcal A $, and 3. The unit ball $(B(M_n(\mathcal A)))_{n\in\mathbb N}$, where $B(M_n(\mathcal A))=\{b\in M_n(\mathcal A)\colon\|b\|<1\}$. Our paper will focus on the first example. As the domains we consider in this paper are mostly noncommutative subsets of von Neumann algebras given by an order relation, we give below a few of the well-known results we use systematically in the rest of the paper. For them, we refer to [@Bruce; @Paulsen; @SZ]. First, recall that for any C${}^*$-algebra (hence, in particular, von Neumann algebra) $\mathcal A$, if $x\in\mathcal A$, then $\|x\|^2=\|x^*\|^2=\|x^*x\|=\|xx^*\|$. For a selfadjoint element $x=x^*\in\mathcal A$, $\|x\|$ is equal to the spectral radius of $x$. We say that $x\ge0$ in $\mathcal A$ if $x=x^*$ and the spectrum of $x$ is included in $[0,+\infty)$. Equivalently, if $\mathcal H$ is the Hilbert space on which $\mathcal A$ acts as a von Neumann algebra, then a selfadjoint $x\in\mathcal A$ is greater than or equal to zero if and only if $\langle x\xi,\xi\rangle\ge0$ for all $\xi\in\mathcal H$. We say that $x>0$ means that $x\ge0$ and $x$ is invertible (i.e. its spectrum is included in $(0,+\infty)$). We say $x\ge y$ if $x-y\ge0$, and similarly for “$>$.” In particular, it follows that $xx^*\leq\|x\|^2\cdot1_\mathcal A$ and $x^*x\leq\|x\|^2\cdot1_\mathcal A$. Clearly, for and $\varepsilon\in(0,+\infty)$, $xx^*<(\|x\|^2+\varepsilon)\cdot1_\mathcal A$, with strict inequality achieved only when $\varepsilon>0$. As proved in [@Paulsen Lemma 3.1], $$\left(\begin{array}{cc} 1 & a\\ a^* & 1 \end{array}\right)\ge0\text{ in }M_2(\mathcal A)\iff \|a\|\leq1.$$ We claim that $$\left(\begin{array}{cc} 1 & a\\ a^* & 1 \end{array}\right)>0 \text{ in }M_2(\mathcal A)\iff \|a\|<1.$$ Indeed, if $\|a\|<1$, then $1-aa^*$ and $1-a^*a$ are invertible in $\mathcal A$ and $$\left(\begin{array}{cc} 1 & a\\ a^* & 1 \end{array}\right)^{-1}=\left(\begin{array}{cc} (1-aa^*)^{-1} & -(1-aa^*)^{-1}a\\ -a^*(1-aa^*)^{-1} & (1-a^*a)^{-1} \end{array}\right)\ \text{in }M_2(\mathcal A).$$ Conversely, if $\|a\|=1$, then for any $\varepsilon>0$ there exists $\xi_\varepsilon,\eta_\varepsilon\in\mathcal H$ of norm one such that $\langle a\eta_\varepsilon,\xi_\varepsilon\rangle> 1-\varepsilon.$ Then $$\left|\left\langle \left(\begin{array}{cc} 1 & a\\ a^* & 1 \end{array}\right)\left[\begin{array}{c} \xi_\varepsilon\\ \eta_\varepsilon \end{array}\right],\left[\begin{array}{c} \xi_\varepsilon\\ \eta_\varepsilon \end{array}\right] \right\rangle\right|=\|\xi_\varepsilon\|^2_2+\|\eta_\varepsilon\|^2_2 -2\langle a\eta_\varepsilon,\xi_\varepsilon\rangle<2\varepsilon,$$ so that zero belongs to the spectrum of $\left(\begin{array}{cc} 1 & a\\ a^* & 1 \end{array}\right)$. This proves our claim. Observe also that for any selfadjoint $x\in\mathcal A$, we have $x>0$ if and only if for any invertible $a\in\mathcal A$, we have $a^*xa>0$. Indeed, one implication is obvious. Conversely, if $a$ is invertible and $a^*xa>0$, then there is an $\varepsilon_a\in(0,+\infty)$ such that $a^*xa>\varepsilon_a\cdot1_\mathcal A$. For any $\xi\in\mathcal H$, $\langle x\xi,\xi\rangle=\langle x a(a^{-1}\xi),a(a^{-1}\xi)\rangle=\langle a^*x a(a^{-1}\xi),(a^{-1}\xi)\rangle>\varepsilon_a\|(a^{-1}\xi)\|_2^2\ge \varepsilon_a\|a\|^{-2}\|\xi\|_2^2,$ independently of $\xi$, so that $x\ge\varepsilon_a\|a\|^{-2}\cdot1_\mathcal A>0.$ We use these last two results to conclude that $$\left(\begin{array}{cc} u & v\\ v^* & w \end{array}\right)>0\text{ in }M_2(\mathcal A)\iff u,w>0\text{ in }\mathcal A\text{ and } \left\{\begin{array}{c} u>vw^{-1}v^* \\ \text{and/or}\\ w>v^*u^{-1}v \end{array}\right..$$ This follows from the above by writing $$\left(\begin{array}{cc} u & v\\ v^* & w \end{array}\right)= \left(\begin{array}{cc} u^\frac12 & 0\\ 0 & w^\frac12 \end{array}\right) \left(\begin{array}{cc} 1 & u^{-\frac12}vw^{-\frac12}\\ w^{-\frac12}v^*u^{-\frac12} & 1 \end{array}\right) \left(\begin{array}{cc} u^\frac12 & 0\\ 0 & w^\frac12 \end{array}\right)$$ and recalling the chain of equivalences $\|u^{-1/2}vw^{-1/2}(u^{-1/2}vw^{-1/2})^*\|<1\iff \|(u^{-1/2}vw^{-1/2})^*u^{-1/2}vw^{-1/2}\|<1\iff u^{-1/2}vw^{-1/2}(u^{-1/2}vw^{-1/2})^*<1\iff (u^{-1/2}vw^{-1/2})^*u^{-1/2}vw^{-1/2}<1.$ We shall use these facts below without further referencing to them. The Julia-Wolff-Carathéodory Theorem, classical and noncommutative ------------------------------------------------------------------ We state the classical Julia-Wolff-Carathéodory Theorem for analytic self-maps of the upper half-plane $\mathbb C^+$ at a point of the real line $\mathbb R$. In the following we denote by $\displaystyle \lim_{\stackrel{ z\longrightarrow\alpha}{{\sphericalangle}}}$ the nontangential limit at a point $\alpha\in\mathbb R$ (see, for ex. [@garnett]). \[JC\] Let $f\colon\mathbb C^+\to\mathbb C^+$ be a nonconstant analytic function and $\alpha\in \mathbb R$ be fixed. 1. Assume that $$\label{3} c:=\liminf_{z\to\alpha}\frac{\Im f(z)}{\Im z}<\infty.$$ Then $f(\alpha):=\displaystyle\lim_{\stackrel{ z\longrightarrow\alpha}{{ \sphericalangle}}}f(z)$ exists and belongs to $\mathbb R$, and $$\label{4} \lim_{\stackrel{ z\longrightarrow\alpha}{{ \sphericalangle}}}f'(z)= \lim_{\stackrel{ z\longrightarrow\alpha}{{ \sphericalangle}}}\frac{f(z)-f(\alpha)}{z-\alpha}= \lim_{y\downarrow0}\frac{\Im f(\alpha+iy)}{y}=c.$$ 2. Assume that $\displaystyle\lim_{y\downarrow0}f(\alpha+iy)=: f(\alpha)$ exists and belongs to $\mathbb R$. If $$\lim_{\stackrel{ z\longrightarrow\alpha}{{ \sphericalangle}}}\frac{f(z)-f(\alpha)}{z-\alpha}=c\in\mathbb C,$$ then $c\in(0,+\infty)$ and $$c=\liminf_{z\to\alpha}\frac{\Im f(z)}{\Im z}= \lim_{\stackrel{ z\longrightarrow\alpha}{{ \sphericalangle}}}f'(z).$$ 3. Assume that $\displaystyle\lim_{\stackrel{z\longrightarrow \alpha}{{\sphericalangle}}}f'(z)=c\in\mathbb C$ and $\displaystyle\lim_{\stackrel{ z\longrightarrow\alpha}{{ \sphericalangle}}}f(z)=f(\alpha)\in\mathbb R$. Then $$c=\liminf_{z\to\alpha}\frac{\Im f(z)}{\Im z}= \lim_{\stackrel{ z\longrightarrow\alpha}{{ \sphericalangle}}}\frac{f(z)-f(\alpha)}{z-\alpha}\in\mathbb R.$$ The noncommutative version of this theorem becomes quite obvious in light of and of the formulations of the corresponding main result from [@Wlo] as well as the recent work [@PTD]. In the following, when we make a statement about a completely positive map, we usually write the statement for level one, and, unless the contrary is explicitly stated, we mean that the property in question holds for all levels $n$. Thus, for example, the statement $$\lim_{\stackrel{ z\longrightarrow0}{{ \sphericalangle}}}f'(\alpha+zv):=f'(\alpha)$$ exists and is completely positive for $\alpha=\alpha^*\in\mathcal A$ and $v>0$ means that for any $n\in\mathbb N$ and any $v\in M_n(\mathcal A)$, $$\lim_{\stackrel{ z\longrightarrow0}{{ \sphericalangle}}}f'(\alpha\otimes1_n+zv)=f'(\alpha\otimes1_n)= f'(\alpha)\otimes{\rm Id}_n$$ is a positive map on $M_n(\mathcal A)$. \[Main\] Let $\mathcal A$ be a von Neumann algebra and $f\colon H^+(\mathcal A)\to H^+(\mathcal A)$ be a noncommutative analytic map. Fix $\alpha=\alpha^*\in\mathcal A$. 1. Assume that for any $v\in \mathcal A, v>0$ and any state $\varphi$ on $\mathcal A$, $$\label{5} \liminf_{z\to0,z\in\mathbb C^+}\frac{\varphi(\Im f(\alpha +zv))}{\Im z}<\infty.$$ Then there exists $c=c(v)\in\mathcal A$, $c>0$ such that $$\label{6} \lim_{y\downarrow0}\frac{\Im f(\alpha+iyv)}{y}=c$$ in the strong operator [(so)]{} topology. Moreover, $\displaystyle\lim_{\stackrel{z\longrightarrow0}{{\sphericalangle}}} f(\alpha+zv)=f(\alpha)$ exists, does not depend on $v$ and is selfadjoint. The limits $$\label{7} \lim_{\stackrel{ z\longrightarrow0}{{ \sphericalangle}}}\Delta f(\alpha+zv_1,\alpha+zv_2)\quad\text{and}\quad \lim_{\stackrel{z\longrightarrow0}{{\sphericalangle}}}f'(\alpha+zv)$$ exist pointwise [wo]{} for any $v,v_1,v_2>0$, and $\displaystyle\lim_{\stackrel{ z\longrightarrow0}{{ \sphericalangle}}}f'(\alpha+zv)(v)=c(v)$. All statements remain true for any $n\in\mathbb N$, $v,v_1,v_2>0$ in $M_n(\mathcal A)$ and $\alpha$ replaced by $\alpha\otimes 1_n$. 2. Assume in addition to the hypothesis that for any $v,w>0$ in $\mathcal A$ and any state $\varphi$ on $\mathcal A$, the gradient of the two-variable complex function $\{(z,\zeta)\in\mathbb C^2\colon\Im(zv+\zeta w)>0\}\ni(z,\zeta)\mapsto\varphi(f(\alpha+zv+\zeta w)) \in\mathbb C^+$ admits the limit $$\lim_{\stackrel{y_1,y_2\downarrow0}{(y_1,y_2)\in[0,1)^2\setminus\{(0,0)\}}} \left(\varphi(f'(\alpha+iy_1v+iy_2w)(v)),\varphi(f'(\alpha+iy_1v+iy_2w)(w))\right).$$ Then the limits are equal to each other, completely positive and do not depend on $v,v_1,v_2$. 3. Assume that the pointwise [wo]{} limit $\displaystyle\lim_{y\downarrow0}f'(\alpha+iyv):=f'(\alpha)$ exists for any $v>0$, does not depend on $v$ and $f'(\alpha)$ is a completely bounded operator on $\mathcal A$. Then $f'(\alpha)$ is completely positive, $\displaystyle\lim_{\stackrel{z \longrightarrow0}{{\sphericalangle}}}f(\alpha+zv):=f(\alpha)$ exists, does not depend on $v$ and is selfadjoint, and $$f'(\alpha)(v)={\rm so\text{-}} \lim_{\stackrel{ z\longrightarrow0}{{ \sphericalangle}}}\frac{\Im f(\alpha+iyv)}{y}\quad\text{for any } v>0.$$ Unfortunately, unlike in the classical case of Theorem \[JC\], and similar to the case of functions of several complex variables [@Rudin; @Abate], item (1’) above cannot be improved upon. Indeed, it was observed in [@AMcY] that for analytic functions of two complex variables on the bidisk with values in the unit disk, there exist examples that satisfy the commutative equivalent of for the bidisk, and yet the gradient map does not have a nontangential limit. The equivalent of condition implies the existence of all [*directional*]{} derivatives in permissible directions, but these directional derivatives do not necessarily “add up” to a linear map. This commutative example has a natural noncommutative extension, as shown in [@PTD]. It is enough for our purposes to treat a simplified version of this extension. It is shown in [@ATY] that any Loewner map from the $n$-dimensional upper half-plane $(\mathbb C^+)^n$ to $\mathbb C^+$ has a certain operatorial realization: for any such $h\colon(\mathbb C^+)^n\to\mathbb C^+$ there exist Hilbert spaces $\mathcal N,\mathcal M$, a selfadjoint densely defined operator $A$ on $\mathcal M$, a real number $s$ an orthogonal decomposition $P= \{P_1,\dots,P_n\}$ of $\mathcal N\oplus\mathcal M$ (i.e. $P_iP_j=P_jP_i=\delta_{ij}P_j=\delta_{ij}P_j^*$ and $P_1+\cdots+P_n=1_{\mathcal M\oplus\mathcal N}$) and a vector state $\varphi_v(\cdot)=\langle\cdot v,v\rangle$ on the von Neumann algebra of bounded linear operators on $\mathcal N\oplus\mathcal M$ such that $$h(z)=s+\varphi_v(M(z)), \quad z=(z_1,\dots,z_n)\in(\mathbb C^+)^n,1\leq j\leq n,$$ where $$\begin{aligned} M(z) & = & \left(\begin{array}{cc} -i & 0 \\ 0 & 1-iA \end{array}\right)\left(\left(\begin{array}{cc} 1 & 0 \\ 0 & A \end{array}\right)-(z_1P_1+\cdots+z_nP_n) \left(\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}\right)\right)^{-1}\\ & & \mbox{}\times\left((z_1P_1+\cdots+z_nP_n) \left(\begin{array}{cc} 1 & 0 \\ 0 & A \end{array}\right)+ \left(\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}\right)\right) \left(\begin{array}{cc} -i & 0 \\ 0 & 1-iA \end{array}\right)^{-1}.\end{aligned}$$ The $2\times2$ matrix decomposition is realized with respect to the canonical orthogonal decomposition of $\mathcal N\oplus\mathcal M$. We observe that such maps $M\colon(\mathbb C^+)^n\to B(\mathcal N\oplus\mathcal M)$ have a natural noncommutative extension to $H^+(\mathbb C^n):= \coprod_{k\ge1}\{a\in M_k(\mathbb C)\colon \Im a>0\}^n$ given by replacing $(z_1P_1+\cdots+z_nP_n)$ in the above formula of $M(z)$ by $$\sum_{j=1}^n (P_j\otimes1_k) a_j (P_j\otimes 1_k).$$ (While it is not obvious from its formula that $\Im M$ is positive when evaluated on $(\mathbb C^+)^n$, and even less when its amplification is evaluated on $\{a\in M_k(\mathbb C)\colon \Im a>0\}^n$, a careful reading of the proofs of [@ATY Propositions 3.4 and 3.5] allows one to observe that they adapt without modification to show that $\Im M(a_1,\dots, a_n)>0$ for $(a_1,\dots,a_n)\in\{a\in M_k(\mathbb C)\colon \Im a>0\}^n$.) The extension of $h$ becomes $$h_k(a)=s\otimes1_k+(\varphi_v\otimes{\rm Id}_k)(M(a)),$$ for all $ a=(a_1,\dots,a_n)\in \{a\in M_k(\mathbb C)\colon \Im a>0\}^n.$ For $n=2$ [*any*]{} analytic function $h\colon\mathbb C^+\times\mathbb C^+\to\mathbb C^+$ admits such an operatorial realization, and hence it has a noncommutative extension as described above (see [@AMcY; @ATY; @ATY2]). Considering the counterexample $h$ provided in [@AMcY], the map $H\colon H^+(\mathbb C^2)\to H^+(\mathbb C^2)$ defined by $H(a)=(h(a),h(a))$ shows that we cannot dispense of item (1’) in Theorem \[Main\]. However, observe that the noncommutative structure of the function $f$ in Theorem \[Main\] (1) allows for a slightly stronger conclusion than in classical case of [@AMcY; @Abate]: the “directional derivative” becomes a bounded linear operator defined on all of $\mathcal A$. As noted above, a classical analytic function is itself the first level of a noncommutative function, via the classical analytic functional calculus applied to matrices over $\mathbb C$. Relations , , are the obvious consequences of relations and in this context. Thus the statements of Theorem \[Main\] are anything but surprising. Indeed, if $f$ has an analytic extension around $\alpha$, then the proof of Theorem \[Main\] is absolutely trivial. A norm estimate =============== Several slightly different proofs of Julia-Wolff-Carathéodory Theorem can be found in the literature. An essential element in one of them is the Schwarz-Pick Lemma: an analytic self-map of the upper half-plane is a contraction with respect to a “good” metric on $\mathbb C^+$. In the next proposition, we obtain a similar result for noncommutative functions. We think that there is a rather striking resemblance between our result below and [@Mellon Corollary 3.3], but it is not clear to us yet whether the two results can be obtained from each other, or even to what extent they are related. We intend to pursue this question later. \[prop:3.1\] Let $\mathcal A,\mathcal C$ be two von Neumann algebras and $f\colon H^+(\mathcal A) \to H^+(\mathcal C)$ be a noncommutative map. For any $n\in\mathbb N$ and $a,c\in H^+(M_n(\mathcal A))$, the linear map $$M_n(\mathcal A)\ni b\mapsto\left(\Im f(a)\right)^{-\frac12}\Delta f (a,c)\left((\Im a)^{\frac12}b(\Im c)^\frac12\right) \left(\Im f(c)\right)^{-\frac12}\in M_n(\mathcal C)$$ is a complete contraction. In particular, $$\left\|\left(\Im f(a)\right)^{-\frac12}\Delta f (a,c)(b)\left(\Im f(c)\right)^{-\frac12}\right\|_\mathcal C\leq \left\|\left(\Im a\right)^{-\frac12}b\left(\Im c\right)^{-\frac12}\right\|_\mathcal A,$$ so that, by Equation , for $b=a-c$, $$\left\|\left(\Im f(a)\right)^{-\frac12}(f(a)-f(c)) \left(\Im f(c)\right)^{-\frac12}\right\|_\mathcal C\leq\left\|\left( \Im a\right)^{-\frac12}(a-c)\left(\Im c\right)^{-\frac12}\right\|_\mathcal A.$$ The estimate will often be used under the equivalent forms $$\begin{aligned} \lefteqn{\left[(\Im f(a))^{-\frac12}\Delta f(a,c)(b)(\Im f(c))^{-\frac12}\right]^* \left[(\Im f(a))^{-\frac12}\Delta f(a,c)(b)(\Im f(c))^{-\frac12}\right]}\nonumber\\ & \leq & \left\|\left(\Im a\right)^{-\frac12}b\left(\Im c\right)^{-\frac12}\right\|_\mathcal A^2\cdot 1_{M_n(\mathcal C)},\label{est} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \lefteqn{\left[(\Im f(a))^{-\frac12}\Delta f(a,c)(b)(\Im f(c))^{-\frac12}\right] \left[(\Im f(a))^{-\frac12}\Delta f(a,c)(b)(\Im f(c))^{-\frac12}\right]^*}\nonumber\\ & \leq & \left\|\left(\Im a\right)^{-\frac12}b\left(\Im c\right)^{-\frac12}\right\|_\mathcal A^2\cdot 1_{M_n(\mathcal C)},\label{est'} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\Delta f(a,c)(b)^*(\Im f(a))^{-1} \Delta f(a,c)(b)\leq \left\|\left(\Im a\right)^{-\frac12}b\left(\Im c\right)^{-\frac12}\right\|_\mathcal A^2\cdot \Im f(c),\label{est''}$$ $$\Delta f(a,c)(b)(\Im f(c))^{-1} \Delta f(a,c)(b)^*\leq \left\|\left(\Im a\right)^{-\frac12}b\left(\Im c\right)^{-\frac12}\right\|_\mathcal A^2\cdot \Im f(a),\label{est'''}$$ which we give here for convenience. Of course, if $b\in M_n(\mathcal A)$, the notation $\|b\|_\mathcal A$ signifies the C${}^*$-norm of $b$ as an element in $M_n(\mathcal A)$. As $\Im \left(\begin{array}{cc} a & b\\ 0 & c \end{array}\right)= \left(\begin{array}{cc} \Im a & \frac{b}{2i}\\ \left(\frac{b}{2i}\right)^* & \Im c \end{array}\right)$, we have $\left(\begin{array}{cc} a & b\\ 0 & c \end{array}\right)\in H^+(M_{2n}(\mathcal A))$ if and only if $a,c\in H^+(M_{n}(\mathcal A))$ and $b^*(\Im a)^{-1}b<4\Im c$. This last relation is equivalent to $\left[(\Im a)^{-\frac12}b (\Im c)^{-\frac12}\right]^*\left[(\Im a)^{-\frac12}b (\Im c)^{-\frac12}\right]<4$, or $\|(\Im a)^{-\frac12}b (\Im c)^{-\frac12}\|_\mathcal A<2$. Thus, as $f$ maps the noncommutative upper half-plane into itself, and for any $b_0\in M_n(\mathcal A)$ there exists an $\varepsilon_{b_0}=\frac{2}{\|(\Im a)^{-\frac12}b_0 (\Im c)^{-\frac12}\|_\mathcal A}>0$ such that $$\left(\begin{array}{cc} a & \varepsilon {b_0}\\ 0 & c \end{array}\right)\in H^+(M_{2n}(\mathcal A))\quad\text{for all } \varepsilon\in[0,\varepsilon_{b_0}),$$ and so $$\left(\begin{array}{cc} f(a) & \varepsilon\Delta f(a,c)({b_0})\\ 0 & f(c) \end{array}\right)\in H^+(M_{2n}(\mathcal A))\quad\text{for all } \varepsilon\in[0,\varepsilon_{b_0}).$$ In particular $\varepsilon\left\|\left(\Im f(a)\right)^{-\frac12} \Delta f(a,c)({b_0})\left(\Im f(c)\right)^{-\frac12}\right\|_\mathcal C<2$ for $\varepsilon<\frac{2}{\|(\Im a)^{-\frac12}b_0 (\Im c)^{-\frac12}\|_\mathcal A}$. Letting $\varepsilon\to\frac{2}{\|(\Im a)^{-\frac12}b_0 (\Im c)^{-\frac12}\|_\mathcal A}$ from below, we obtain $$\left\|\left(\Im f(a)\right)^{-\frac12} \Delta f(a,c)({b_0})\left(\Im f(c)\right)^{-\frac12}\right\|_\mathcal C\leq \|(\Im a)^{-\frac12}b_0(\Im c)^{-\frac12}\|_\mathcal A.$$ As $b_0\in M_n(\mathcal A)$ has been chosen arbitrarily, we can replace it by $(\Im a)^{\frac12}b(\Im c)^{\frac12}$ to conclude that, as claimed $$\left\|\left(\Im f(a)\right)^{-\frac12}\Delta f (a,c)\left((\Im a)^{\frac12}b(\Im c)^\frac12\right) \left(\Im f(c)\right)^{-\frac12}\right\|_\mathcal C\leq\|b\|_\mathcal A,\quad b\in M_n(\mathcal A).$$ Clearly, the same method can be used to obtain estimates involving $\Delta^jf$ for all $j\in\mathbb N$. We give one such estimate pertaining to a special case of $j=2$. We shall simply apply the above result to appropriately chosen elements in $M_2(\mathcal A)$. Let $$\left(\begin{array}{cccc} a_1 & 0 & 0 & 0 \\ 0 & a_2 & c & 0 \\ 0 & 0 & a_3 & b \\ 0 & 0 & 0 & a_4 \end{array}\right)$$ be such that $\Im a_j>0$ and $ \left(\begin{array}{cc} a_3 & b\\ 0 & a_4 \end{array}\right)\in H^+(M_2(\mathcal A))$. From [@ncfound Section 3] we obtain $$\begin{aligned} \lefteqn{f\left(\begin{array}{cccc} a_1 & 0 & 0 & 0 \\ 0 & a_2 & c & 0 \\ 0 & 0 & a_3 & b \\ 0 & 0 & 0 & a_4 \end{array}\right)}\\ & = & \left(\begin{array}{cccc} f(a_1) & 0 & 0 & 0 \\ 0 & f(a_2) & \Delta f(a_2,a_3)(c) & \Delta^2 f(a_2,a_3,a_4)(c,b) \\ 0 & 0 & f(a_3) & \Delta f(a_3,a_4)(b) \\ 0 & 0 & 0 & f(a_4) \end{array}\right).\quad\quad\end{aligned}$$ Applying Proposition \[prop:3.1\] to $a=\left(\begin{array}{cc} a_1 & 0\\ 0 & a_2 \end{array}\right),c=\left(\begin{array}{cc} a_3 & b\\ 0 & a_4 \end{array}\right)$ and $b=\left(\begin{array}{cc} 0 & 0\\ c & 0 \end{array}\right)$ under the form of provides an estimate for $\Delta^2 f(a_2,a_3,a_4)(c,b)$. As the size of the formula in question becomes quite large, we shall split it. We have $$\begin{aligned} \Im f\left(\begin{array}{cc} a_3 & b\\ 0 & a_4 \end{array}\right)^{-1} & = & \left(\begin{array}{cc} \Im f(a_3) & \frac{\Delta f(a_3,a_4)(b)}{2i}\\ \frac{\Delta f(a_3,a_4)(b)^*}{-2i} & f(a_4) \end{array}\right)^{-1}\\ & = & \left(\begin{array}{cc} e_{11} & e_{12}\\ e_{21} & e_{22} \end{array}\right),\end{aligned}$$ where $$\begin{aligned} e_{11}&=&\left(\Im f(a_3)-\frac{\Delta f(a_3,a_4)(b)(\Im f(a_4))^{-1}\Delta f(a_3,a_4)(b)^*}{4}\right)^{-1}\\ e_{12}&=&\left(\Im f(a_3)-\frac{\Delta f(a_3,a_4)(b)(\Im f(a_4))^{-1}\Delta f(a_3,a_4)(b)^*}{4} \right)^{-1}\\ & & \mbox{}\times\frac{\Delta f(a_3,a_4)(b)}{-2i}(\Im f(a_4))^{-1}\\ e_{21}&=&(\Im f(a_4))^{-1}\frac{\Delta f(a_3,a_4)(b)^*}{2i}\\ & & \mbox{}\times\left(\Im f(a_3)-\frac{\Delta f(a_3,a_4)(b)(\Im f(a_4))^{-1}\Delta f(a_3,a_4)(b)^*}{4} \right)^{-1}=e_{12}^*\\ e_{22}&=&\left(\Im f(a_4)-\frac{\Delta f(a_3,a_4)(b)^*(\Im f(a_3))^{-1}\Delta f(a_3,a_4)(b)}{4}\right)^{-1}.\end{aligned}$$ Thus, in the left-hand side of preserves only one nonzero element, in the position $22$ (lower right corner), namely $$\begin{aligned} \lefteqn{\Delta f(a_2,a_3)(c)e_{11}\Delta f(a_2,a_3)(c)^*+2\Re\left(\Delta f(a_2,a_3)(c)e_{12} \Delta^2 f(a_2,a_3,a_4)(c,b)^*\right)}\\ & & \mbox{}+\Delta^2 f(a_2,a_3,a_4)(c,b)e_{22}\Delta^2 f(a_2,a_3,a_4)(c,b)^* \\ &=&\Delta f(a_2,a_3)(c)e_{11}\Delta f(a_2,a_3)(c)^*\\ & & \mbox{}-\Im\left( \Delta f(a_2,a_3)(c)e_{11}\Delta f(a_3,a_4)(b)(\Im f(a_4))^{-1} \Delta^2 f(a_2,a_3,a_4)(c,b)^*\right)\\ & & \mbox{}+\Delta^2 f(a_2,a_3,a_4)(c,b)e_{22}\Delta^2 f(a_2,a_3,a_4)(c,b)^*.\end{aligned}$$ On the right-hand side of we have the norm $$\left\|\left(\begin{array}{cc} (\Im a_1)^{-\frac12} & 0\\ 0 & (\Im a_2)^{-\frac12} \end{array}\right)\left(\begin{array}{cc} 0 & 0\\ c & 0 \end{array}\right)\left(\begin{array}{cc} \Im a_3 & \frac{b}{2i}\\ \left(\frac{b}{2i}\right)^* & \Im a_4 \end{array}\right)^{-\frac12}\right\|.$$ We use the properties of C${}^*$-norms to conclude that this norm in $M_2(\mathcal A)$ in fact equals the norm $\left\|(\Im a_2)^{-\frac12}c \left(\Im a_3-\frac14b(\Im a_4)^{-1}b^* \right) c^*(\Im a_2)^{-\frac12}\right\|$ in $\mathcal A$. Thus, inequality for elements in $M_2(\mathcal A)$ translates into an inequality of elements in $\mathcal A$ as follows: $$\begin{aligned} \lefteqn{\Delta f(a_2,a_3)(c)e_{11}\Delta f(a_2,a_3)(c)^*}\nonumber\\ & & \mbox{}-\Im\left( \Delta f(a_2,a_3)(c)e_{12}\Delta f(a_3,a_4)(b)(\Im f(a_4))^{-1} \Delta^2 f(a_2,a_3,a_4)(c,b)^*\right)\nonumber\\ & & \mbox{}+\Delta^2 f(a_2,a_3,a_4)(c,b)e_{22}\Delta^2 f(a_2,a_3,a_4)(c,b)^*\nonumber\\ & \leq & \left\|(\Im a_2)^{-\frac12}c \left(\Im a_3-\frac14b(\Im a_4)^{-1}b^* \right)^{-1} c^*(\Im a_2)^{-\frac12}\right\|\Im f(a_2).\label{secondderiv}\end{aligned}$$ However, for now their form seems to be too complicated when $j>2$, and of no significant use for the purposes of this paper. Since the above proposition is applied in this paper only for $\mathcal A=\mathcal C$, from now on we shall eliminate the subscript from the notation of the C${}^*$-norm of $\mathcal A$. \[Hyper\] Fix $n\in\mathbb N$, $r>0$ and $c\in H^+(M_n(\mathcal A))$. Denote $$B^+_n(c,r)=\left\{a\in H^+(M_n(\mathcal A))\colon\left\|(\Im a)^{-1/2}(a-c)(\Im c)^{-1/2}\right\|\leq r \right\}.$$ Then $B_n^+(c,r)$ is a norm-closed norm-bounded convex subset of $H^+(M_n(\mathcal A))$ with nonempty interior, which is bounded away from the topological boundary in the norm topology of $H^+(M_n(\mathcal A))$. Moreover, it is noncommutative. More precisely, $$\label{estimBall} \|a\|\leq\|\Re c\|+\|\Im c\|\left(\frac{r^2+2+r\sqrt{r^2+4}}{2}+r\sqrt{\frac{r^2+2+r\sqrt{r^2+4}}{2}}\right),\ a\in B^+_n(c,r),$$ and $$\label{estimBdry} \Im a\ge\frac{1}{2+r^2}\Im c,\quad a\in B^+_n(c,r).$$ The set $B^+_n(c,r)$ is norm-bounded: $a\in B^+_n(c,r)$ if and only if $(\Im a)^{-\frac12}(a-c) (\Im c)^{-1}(a-c)^*(\Im a)^{-\frac12}\leq r^2\cdot1$, relation which implies $(a-c)(\Im c)^{-1} (a-c)^*\leq r^2\|\Im a\|\cdot1$, which in its own turn implies $\left\|[(a-c)(\Im c)^{-\frac12}] [(a-c)(\Im c)^{-\frac12}]^*\right\|\leq r^2\|\Im a\|$. Recalling that in any C${}^*$-algebra the adjoint (star) operation is isometric and that $\|x^*x\|=\|x\|^2$, this implies that $\left\|[(a-c)(\Im c)^{-\frac12}]^* [(a-c)(\Im c)^{-\frac12}]\right\|\leq r^2\|\Im a\|$, which again implies $(\Im c)^{-\frac12}(a-c)^*(a-c)(\Im c)^{-\frac12}\leq r^2\|\Im a\|\cdot1$. Thus, repeating once again the above computations, we obtain $$\|a-c\|^2\leq r^2\|\Im a\|\|\Im c\|,\quad a\in B^+_n(c,r).$$ Recall that $\Im x=(x-x^*)/2i$, so $\|\Im x\|\leq(\|x\|+\|x^*\|)/2=\|x\|$. Similarly, $\|\Re x\|\leq \|x\|.$ Applying this to $x=a-c$, we obtain $$\left(\|\Im a\|-\|\Im c\|\right)^2\leq\|\Im (a-c)\|^2\leq\|a-c\|^2\leq r^2\|\Im a\|\|\Im c\| ,\quad a\in B^+_n(c,r).$$ Direct computation shows that this relation imposes $$\label{estimIm} \frac{\|\Im c\|}{2}\left(r^2+2-r\sqrt{r^2+4}\right)\leq\|\Im a\|\leq \frac{\|\Im c\|}{2}\left(r^2+2+r\sqrt{r^2+4}\right),$$ for all $a\in B^+_n(c,r).$ Similarly,$\|\Re(a-c)\|^2\leq\|a-c\|^2\leq r^2\|\Im a\|\|\Im c\|$ implies $$\label{estimRe} 0\leq\|\Re a\|\le\|\Re c\|+r\|\Im c\|\sqrt{\frac{r^2+2+r\sqrt{r^2+4}}{2}},\quad a\in B^+_n(c,r).$$ Adding relations and provides the bound $$\|a\|\leq\|\Re c\|+\|\Im c\| \left(\frac{r^2+2+r\sqrt{r^2+4}}{2}+r\sqrt{\frac{r^2+2+r\sqrt{r^2+4}}{2}}\right),$$ as claimed in our remark. Relation is proved by a direct application of one of the equivalent definitions of positivity in a von Neumann algebra and the Cauchy-Buniakovsky-Schwarz inequality in Hilbert spaces. Let $\xi$ be an arbitrary vector in the Hilbert space $\mathcal H^n$ on which $M_n( \mathcal A)$ acts as a von Neumann algebra. As we have seen in the proof of above, $a\in B^+_n(c,r)\iff (a-c)(\Im c)^{-1}(a-c)^*\leq r^2\Im a$. This means that $$\left\langle (a-c)(\Im c)^{-1}(a-c)^*\xi,\xi\right\rangle\leq r^2\langle\Im a\xi,\xi\rangle.$$ Moving $a-c$ to the right with a star and taking real and imaginary parts provides us with $$\begin{aligned} \lefteqn{\left\|(\Im c)^{-\frac12}\Re(a-c)\xi\right\|_2^2+\left\|(\Im c)^{-\frac12}\Im a\xi\right\|_2^2 +\langle\Im c\xi,\xi\rangle}\\ & & \mbox{}+i\left(\left\langle(\Im c)^{-\frac12}\Re(a-c)\xi,(\Im c)^{-\frac12}\Im a\xi\right\rangle -\overline{\left\langle(\Im c)^{-\frac12}\Re(a-c)\xi,(\Im c)^{-\frac12}\Im a\xi\right\rangle}\right)\\ &\leq &(2+r^2)\langle\Im a\xi,\xi\rangle.\end{aligned}$$ Second line above is simply $-2\Im\left\langle(\Im c)^{-\frac12}\Re(a-c)\xi,(\Im c)^{-\frac12}\Im a\xi\right\rangle$, which is clearly greater than $-2\left| \left\langle(\Im c)^{-\frac12}\Re(a-c)\xi,(\Im c)^{-\frac12}\Im a\xi\right\rangle\right|.$ By the Schwarz-Cauchy inequality (applied in the second inequality below) we obtain $$\begin{aligned} \langle\Im c\xi,\xi\rangle & \leq & \langle\Im c\xi,\xi\rangle+\left( \left\|(\Im c)^{-\frac12}\Re(a-c)\xi\right\|_2-\left\|(\Im c)^{-\frac12}\Im a\xi\right\|_2\right)^2\\ & = & \langle\Im c\xi,\xi\rangle+\left\|(\Im c)^{-\frac12}\Re(a-c)\xi\right\|_2^2+\left\|(\Im c)^{-\frac12}\Im a\xi\right\|_2^2\\ & & \mbox{}-2\left\|(\Im c)^{-\frac12}\Re(a-c)\xi\right\|_2\left\|(\Im c)^{-\frac12}\Im a\xi\right\|_2\\ & \leq & \langle\Im c\xi,\xi\rangle+\left\|(\Im c)^{-\frac12}\Re(a-c)\xi\right\|_2^2+\left\|(\Im c)^{-\frac12}\Im a\xi\right\|_2^2\\ & & \mbox{}-2\left| \left\langle(\Im c)^{-\frac12}\Re(a-c)\xi,(\Im c)^{-\frac12}\Im a\xi\right\rangle\right|\\ & \leq & \langle\Im c\xi,\xi\rangle+\left\|(\Im c)^{-\frac12}\Re(a-c)\xi\right\|_2^2+\left\|(\Im c)^{-\frac12}\Im a\xi\right\|_2^2\\ & & \mbox{}-2\Im\left\langle(\Im c)^{-\frac12}\Re(a-c)\xi,(\Im c)^{-\frac12}\Im a\xi\right\rangle\\ &\leq &(2+r^2)\langle\Im a\xi,\xi\rangle.\end{aligned}$$ Since this is true for all vectors $\xi\in\mathcal H^n$, we obtain $\Im c\leq(2+r^2)\Im a$, implying . That $B^+_n(c,r)$ is closed in norm follows even easier: if $a_m\in B^+_n(c,r)$ and $\lim_{m\to\infty} \|a_m-a\|=0$, then $\lim_{m\to\infty}\|a_m^*-a^*\|=0$, and thus $\lim_{m\to\infty} \|\Im a_m-\Im a\|=0$. This also implies that $\Im a\ge\frac{1}{2+r^2}\Im c>0$, so that, by analytic functional calculus, $\lim_{m\to\infty}\left\|(\Im a_m)^{-\frac12}-(\Im a)^{-\frac12}\right\|=0$. A few succesive applications of the product-norm inequalities in C${}^*$-algebras provides $$\begin{aligned} \left\|(\Im a)^{-\frac12}(a-c)(\Im c)^{-\frac12}\right\|&\leq&\left\|(\Im a)^{-\frac12}\right\|\|a_m-a\| +\left\|(\Im a_m)^{-\frac12}-(\Im a)^{-\frac12}\right\|\\ & & \mbox{}\times\left\|(a_m-c)(\Im c)^{-\frac12}\right\|+\left\|(\Im a_m)^{-\frac12}(a_m-c) (\Im c)^{-\frac12}\right\|.\end{aligned}$$ First and second right-hand terms converge to zero as $m\to\infty$, and the last is no more than $r$. Thus, $\left\|(\Im a)^{-\frac12}(a-c)(\Im c)^{-\frac12}\right\|\leq r$, which implies $a\in B^+_n(c,r)$. Midpoint convexity of $B^+_n(c,r)$ follows easily from a direct computation: let $a_1,a_2\in B_n^+(c,r)$. We show that $(a_1+a_2)/2$ is in $B_n^+(c,r)$. As in , this is equivalent to showing that $$\left(\Im\frac{a_1+a_2}{2}\right)^{-\frac12}\left(\frac{a_1+a_2}{2}-c\right)(\Im c)^{-1} \left(\frac{a_1+a_2}{2}-c\right)^*\left(\Im\frac{a_1+a_2}{2}\right)^{-\frac12}\leq r^2\cdot1,$$ which is in its own turn equivalent to $$\label{conv} \left(\frac{a_1-c}{2}+\frac{a_2-c}{2}\right)(\Im c)^{-1}\left(\frac{a_1-c}{2}+\frac{a_2-c}{2}\right)^*\leq \frac{r^2}{2}\Im(a_1+a_2).$$ However, adding the inequalities $(a_1-c)(\Im c)^{-1}(a_1-c)^*\leq r^2\Im a_1$ and $(a_2-c)(\Im c)^{-1}(a_2-c)^*\leq r^2\Im a_2$ (assumed to be true by hypothesis) and dividing by 2, we obtain $$\frac12((a_1-c)(\Im c)^{-1}(a_1-c)^*+\frac12(a_2-c)(\Im c)^{-1}(a_2-c)^*\leq\frac{r^2}{2}\Im(a_1+a_2).$$ Thus, our statement is proved if we show that the left-hand term of is less than or equal to the left-hand term of the inequality above. Expanding the left-hand of and subtracting from the one above yields $$\begin{aligned} & & \frac12(a_1-c)(\Im c)^{-1}(a_1-c)^*+\frac12(a_2-c)(\Im c)^{-1}(a_2-c)^*\\ & & \mbox{}-\frac14(a_1-c)(\Im c)^{-1}(a_1-c)^*-\frac14(a_2-c)(\Im c)^{-1}(a_2-c)^*\\ & & \mbox{}-\frac14(a_1-c)(\Im c)^{-1}(a_2-c)^*-\frac14(a_2-c)(\Im c)^{-1}(a_1-c)^*\\ & = & \frac14\left[(a_1-c)(\Im c)^{-1}(a_1-c-a_2+c)^*+(a_2-c)(\Im c)^{-1}(a_2-c-a_1+c)^*\right]\\ & = & \frac14(a_1-c-a_2+c)(\Im c)^{-1}(a_1-a_2)^*=\frac14(a_1-a_2)(\Im c)^{-1}(a_1-a_2)^*\ge0.\end{aligned}$$ Since $B^+_n(c,r)$ is midpoint convex and closed, it is convex. To conclude, observe that all the above computations hold if $c\in H^+(M_n(\mathcal A))$ is replaced by $c\otimes 1_p\in H^+(M_{np}(\mathcal A))$. Indeed, one only needs to observe that taking imaginary part, inverse and root, as well as multiplication, respect direct sums. Since $\|a\oplus b\|= \max\{\|a\|,\|b\|\}$, we’re done. Estimates and hold on the amplifications of $c$ to any $c\otimes 1_p$, $p\in\mathbb N$, with the same constants. The following lemma will be useful when applying Proposition \[prop:3.1\] to the proof of the main result (compare with the method used in [@BPV1 Remark 2.5]). \[lem:3.2\] Assume that $f$ is a noncommutative self-map of the noncommutative upper half-plane of $\mathcal A$. Let $v_1,v_2>0$ in $\mathcal A$. If $${\rm wo}\text{-}\lim_{y\downarrow0}\frac{\Im f(\alpha+iyv_j)}{y}=c_j\in \mathcal A,\quad j\in\{1,2\},$$ exist, then the set of limit points of $\Delta f(\alpha+zv_1, \alpha+\zeta v_2)(w)$ as $z,\zeta\to 0$ nontangentially is bounded uniformly in norm as $w$ varies in the unit ball of $\mathcal A$. By Proposition \[prop:3.1\], $$\begin{aligned} \lefteqn{\left\|\left(\Im f(\alpha+zv_1)\right)^{-\frac12} \Delta f(\alpha+zv_1,\alpha+\zeta v_2)(w) \left(\Im f(\alpha+\zeta v_2)\right)^{-\frac12}\right\|}\\ &\leq&\left\|(\Im zv_1)^{-\frac12}w(\Im\zeta v_2)^{-\frac12}\right\|. \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ Multiplying by $(\Im z\Im\zeta)^{1/2}$ we obtain $$\left\|\left[\frac{\Im f(\alpha+zv_1)}{\Im z}\right]^{-\frac12} \Delta f(\alpha+zv_1,\alpha+\zeta v_2)(w)\left[\frac{\Im f(\alpha+\zeta v_2)}{\Im\zeta}\right]^{-\frac12} \right\|\leq\left\|v_1^{-\frac12}wv_2^{-\frac12}\right\|.$$ Let $\varepsilon\ge0$ be fixed, and denote $f_\varepsilon(a)= f(a)+\varepsilon a$, i.e. $f_\varepsilon=f+\varepsilon{\rm Id}$. Since $\text{Id}$ is completely positive, $f_\varepsilon$ is still a noncommutative self-map of the noncommutative upper half-plane of $\mathcal A$, so that $$\begin{aligned} \lefteqn{\left \|\left(\frac{\Im f(\alpha+zv_1)}{\Im z}+\varepsilon v_1\right)^{-\frac12} (\Delta f(\alpha+zv_1,\alpha+\zeta v_2)(w)+\varepsilon w)\right.\times}\\ & & \left.\left(\frac{\Im f(\alpha+\zeta v_2)}{\Im\zeta}+\varepsilon v_2 \right)^{-\frac12}\right\|\\ & \leq & \left\|v_1^{-\frac12}wv_2^{-\frac12}\right\|.\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ For simplicity, we denote $A_1(\Im z,\varepsilon)= \frac{\Im f(\alpha+zv_1)}{\Im z}+\varepsilon v_1$, $A_2(\Im\zeta,\varepsilon)= \frac{\Im f(\alpha+\zeta v_2)}{\Im\zeta}+\varepsilon v_2$, $W(z,\zeta,\varepsilon)= \Delta f(\alpha+zv_1,\alpha+\zeta v_2)(w)+\varepsilon w$, and $K=\left\| v_1^{-\frac12}wv_2^{-\frac12}\right\|^2$. As noted in , and following the same procedure as in the proof fo the previous proposition, the above is equivalent to $$A_2(\Im\zeta,\varepsilon)^{-\frac12}W(z,\zeta,\varepsilon)^* A_1(\Im z,\varepsilon)^{-1}W(z,\zeta,\varepsilon)A_2(\Im\zeta,\varepsilon)^{-\frac12} \leq K1.$$ As $A_j(\cdot,\varepsilon)\ge\varepsilon1$, we obtain by the same methods as in the proof of Proposition \[Hyper\] that $$\|W(z,\zeta,\varepsilon)\|^2\leq K\|A_1(\Im z,\varepsilon)\|\|A_2(\Im\zeta,\varepsilon)\|.$$ Let $\mathcal H$ be the Hilbert space on which $\mathcal A$ acts as a von Neumann algebra. By our hypothesis, $\lim_{y\downarrow 0}\frac{\langle\Im f(\alpha+iyv_j)\xi,\xi\rangle}{y}= \lim_{y\downarrow 0}\left\|\left(\frac{\Im f(\alpha+iyv_j)}{y} \right)^\frac12\xi\right\|_2^2$ exist and equal $\langle c_j\xi,\xi\rangle$, finite for any $\xi\in\mathcal H$. Thus, the family $\left\{\left\|\frac{\left(\Im f(\alpha+iyv_j)\right)^{1/2}}{\sqrt{y}} \xi\right\|_2\colon y\in(0,1)\right\}$ is bounded for any $\xi\in\mathcal H$. By the Banach-Steinhaus Theorem and the positivity of the operators $\frac{\Im f(\alpha+iyv_j)}{{y}}$, it follows that $\left\{\left\|\frac{\Im f(\alpha+iyv_j)}{{y}}\right\|\colon y\in(0,1)\right\}$ is a bounded set. Moreover, as it will be seen in the proof of Theorem \[JC\], if $z$ tends to zero nontangentially and $\lim_{y\downarrow 0}\frac{\langle\Im f(\alpha+iyv_j)\xi,\xi\rangle}{y}$ is finite, then $\left\{\frac{\langle\Im f(\alpha+\Im zv_j)\xi,\xi\rangle}{\Im z}\colon|z|<1,z\in\Gamma\right\}$ stays bounded for any closed cone $\Gamma\subset\mathbb C^+\cup\{0\}$. A bound for $c_j$ is $\|c_j\|\leq\limsup_{y\to0}\left\|\frac{\Im f (\alpha+iyv_j)}{{y}}\right\|$. Thus, $\{\|W(z,\zeta,\varepsilon)\|\colon z,\zeta\in\Gamma,|z|,|\zeta|<1\}$ is bounded for any closed cone $\Gamma\subset\mathbb C^+$ with vertex at zero. The lemma follows by letting $\varepsilon\downarrow0$. We note that the bounds depend exclusively on $c_j,v_j (j=1,2),w$. Moreover, the dependence can be bounded (at most) linearly in terms of $\|w\|,\|v_1\|,\|v_2\|,\|v_1^{-1}\|$ and $\|v_2^{-1}\|$. For the sake of completeness, let us use the results of Proposition \[prop:3.1\] to give a short, elementary proof of Theorem \[JC\]. Assume equation holds. By Proposition \[prop:3.1\], $$\left|\frac{f(z)-f(z')}{\sqrt{\Im f(z)\Im f(z')}}\right|\leq \left|\frac{z-z'}{\sqrt{\Im z\Im z'}}\right|,\quad z,z'\in \mathbb C^+.$$ This is equivalent to $$\label{9} \left|\frac{f(z)-f(z')}{z-z'}\right|^2\leq \left|\frac{\Im f(z)\Im f(z')}{\Im z\Im z'}\right|,\quad z,z'\in \mathbb C^+, z\neq z'.$$ Consider a sequence $\{z_n'\}_{n\in\mathbb N}\subset\mathbb C^+$ converging to $\alpha$ such that $\lim_{n\to\infty}\frac{\Im f(z_n')}{ \Im z_n'}=c$. Clearly $\Im f(z_n')\to0$ as $n\to\infty$, and $\{\Re f(z_n')\}_{n\in\mathbb N}$ is a bounded sequence in $\mathbb R$. Moreover, if $\{z_n\}_{n\in\mathbb N}$ and $\{z_n'\}_{n\in\mathbb N}$ are two arbitrary sequences converging to $\alpha$ along which $\Im f(z)/\Im z$ stays bounded, then $\{\Re(f(z_n)-f(z_n'))\}_{ n\in\mathbb N}$ converges to zero. This implies that $\lim_{n\to\infty} f(z_n)$ exists for any sequence $\{z_n\}_{n\in\mathbb N}$ such that $\{\Im f(z_n)/\Im z_n\}_{n\in\mathbb N}$ is bounded and $\lim_{n\to \infty}z_n=\alpha$. We agree to call this limit $f(\alpha)$. Taking limit along $z_n'$ in we obtain $$\left|\frac{f(z)-f(\alpha)}{z-\alpha}\right|^2\leq c\frac{\Im f(z)}{\Im z},\quad z\in\mathbb C^+.$$ Fix an $M\in[0,+\infty)$. Let $D_M=\{z\in\mathbb C^+\colon |\Re z-\alpha|\leq M\Im z\}$. For any $z\in D_M$, this implies $$\begin{aligned} \left(\Re f(z)-f(\alpha)\right)^2 & \leq & c\frac{\Im f(z)}{\Im z}|z- \alpha|^2-\left(\Im f(z)\right)^2\\ & = & \Im f(z)\left(\frac{c|z-\alpha|^2}{\Im z}-\Im f(z)\right)\\ & = & \Im f(z)\left(c\Im z\frac{|\Re z-\alpha|^2}{(\Im z)^2} +c\Im z-\Im f(z)\right)\\ & \leq & \Im f(z)\left(c(M^2+1)\Im z-\Im f(z)\right).\end{aligned}$$ We conclude that $\Im f(z)/\Im z\leq c(M^2+1)$ for all $z\in D_M$ and thus $\displaystyle\lim_{\stackrel{z\longrightarrow0}{{\sphericalangle}}} f(z)=f(\alpha)$. Moreover, for $M=0$ (i.e. $z$ of the form $\alpha+iy$) we have $c\geq\Im f(\alpha+iy)/y$, which together with the definition of $c$ implies $\lim_{y\downarrow0}\frac{\Im f (\alpha+iy)}{y}=c$, so that $$\frac{\left(\Re f(\alpha+iy)-f(\alpha)\right)^2}{y^2}\leq \frac{\Im f(\alpha+iy)}{y} \left(c-\frac{\Im f(\alpha+iy)}{y}\right)\to0\quad\text{as }y\downarrow0.$$ These two facts imply, via direct computation, that $\lim_{y\downarrow0}\frac{f(\alpha+iy)-f(\alpha)}{iy}=c$. Since $$\left|\frac{f(z)-f(\alpha)}{z-\alpha}\right|^2\leq c\frac{\Im f(z)}{\Im z}\leq c^2(M^2+1),\quad z\in D_M, M\ge0,$$ it follows straightforwardly that $$\lim_{\stackrel{z\longrightarrow0}{{\sphericalangle}}} \frac{f(z)-f(\alpha)}{z-\alpha}=c$$ (see for example [@garnett Exercise 5, Chapter I]). Considering the classical definition of the derivative, the above directly implies that $\lim_{y\downarrow0}f'(\alpha+iy)=c.$ Relation implies that $|f'(z)|\leq c(M^2+1)$ for $z\in D_M$, so, by the same [@garnett Exercise 5, Chapter I], $\displaystyle\lim_{\stackrel{z\longrightarrow0}{{\sphericalangle}}} f'(z)=c$. This proves (1). To prove (2), simply observe that $$\left|\frac{f(\alpha+iy)-f(\alpha)}{iy}\right|^2= \left|\frac{(\Re f(\alpha+iy)-f(\alpha))^2+(\Im f(\alpha+iy))^2}{y^2} \right|\ge\frac{(\Im f(\alpha+iy))^2}{y^2},$$ so that $\liminf_{z\to\alpha}\frac{\Im f(z)}{\Im z}<\infty$. Part (2) follows now from part (1). To prove part (3), we apply the classical mean value theorem to bound $\Im f(\alpha+iy)/y$. The result follows then from part (1). We feel it necessary to reiterate that no claim to novelty is made for this proof, and we chose to write it down here for the sake of making the paper more self-contained. Proof of the main result ======================== In this section we prove Theorem \[Main\]. The proof makes use quite often of the results, and sometimes of the proof, of Theorem \[JC\]. For the sake of simplicity, we will isolate some elements of the proof in separate lemmas. For any $n\in\mathbb N$ and any state $\varphi$ on $M_n(\mathcal A)$, $z\mapsto\varphi(f(\alpha+zv))$ is a self-map of $\mathbb C^+$ whenever $\alpha$ is selfadjoint and $v>0$ in $M_n(\mathcal A)$. Thus, Theorem \[JC\] applies to it. In particular, if $\mathcal H$ is the Hilbert space on which the von Neumann algebra $\mathcal A$ acts, the above holds for the vector state corresponding to any $\xi\in \oplus_{j=1}^n\mathcal H$ of $L^2$-norm equal to one. For $n=1$, our hypothesis guarantees that $\liminf_{z\to0}\frac{\langle\Im f(\alpha+zv)\xi,\xi\rangle}{\Im z}$ is finite. Item (1) of Theorem \[JC\] guarantees that $\lim_{y\downarrow 0}\frac{\langle\Im f(\alpha+iyv)\xi,\xi\rangle}{y}= \lim_{y\downarrow 0}\left\|\left(\frac{\Im f(\alpha+iyv)}{y} \right)^\frac12\xi\right\|_2^2$ exists and equals the above $\liminf$, hence it is finite for any $\xi\in \mathcal H$. As in the proof of Lemma \[lem:3.2\], the Banach-Steinhaus Theorem and the positivity of the operators $\frac{\Im f(\alpha+iyv)}{{y}}$ guarantee that $\left\{\left\|\frac{\Im f(\alpha+iyv)}{{y}}\right\|\colon y\in(0,1)\right\}$ is a bounded set. Moreover, the existence of the limits $\lim_{y\downarrow 0}\frac{\langle\Im f(\alpha+iyv)\xi,\xi\rangle}{y}$ for all $\xi\in\mathcal H$ implies, via polarization, the existence of $$\lim_{y\downarrow 0}\frac{\langle\Im f(\alpha+iyv)\xi,\eta\rangle}{y}, \quad \xi,\eta\in\mathcal H.$$ We conclude the existence of a bounded operator $0\le c=c(v)\in\mathcal A$ such that $$\lim_{y\downarrow 0}\frac{\langle\Im f(\alpha+iyv)\xi,\eta\rangle}{y}= \langle c\xi,\eta\rangle,\quad \xi,\eta\in\mathcal H.$$ The bound for $c$ is $\|c\|\leq\limsup_{y\to0}\left\|\frac{\Im f (\alpha+iyv)}{{y}}\right\|$. On the other hand, as seen in the proof of Theorem \[JC\], $\Im\langle f(\alpha+iyv)\xi,\xi\rangle\leq y\langle c\xi,\xi\rangle$ for all $y>0$. Since $f$ takes values in $H^+(\mathcal A)$, applying this relation to $y=1$ guaranteres that $c>0$. Now it follows easily that $\lim_{y\downarrow0}\left\|\left(\frac{\Im f(\alpha+iyv)}{y}-c\right)\xi\right\|=0$ for any $\xi\in\mathcal H$. We show next that the limit $\lim_{y\downarrow 0}f(\alpha+iyv)=f(\alpha)$ exists in $\mathcal A$ (i.e. does not depend on $v$) and is selfadjoint. Indeed, consider again any state $\varphi$ on $\mathcal A $ and define $z\mapsto\varphi(f(\alpha+zv))$. We have seen that this is a self-map of $\mathbb C^+$ to which Theorem \[JC\] applies. Thus, there exists a number $k=k(\varphi,\alpha,v) \in\mathbb R$ such that $\displaystyle\lim_{\stackrel{z\longrightarrow 0}{\sphericalangle}}\varphi(f(\alpha+zv))=k.$ We recall the estimate from Proposition \[prop:3.1\] $$\left| \frac{\varphi(f(\alpha+zv))-\varphi(f(\alpha+z'v))}{z-z'}\right|^2 \leq\frac{\varphi(\Im f(\alpha+zv))\varphi(\Im f(\alpha+z'v))}{\Im z \Im z'}.$$ In this estimate we take $z'=i$ and let $z=iy$ tend to zero. We obtain $$\left|k(\varphi,\alpha,v)-\varphi(f(\alpha+iv))\right|^2 \leq\varphi(c)\varphi(\Im f(\alpha+iv)).$$ Obviously, $|\varphi(f(\alpha+iv))|\leq\|f(\alpha+iv)\|,$ a value independent of $\varphi$. Thus, $$|k(\varphi,\alpha,v)|\leq\|f(\alpha+iv)\|+\sqrt{\|c\| \|\Im f(\alpha+iv)\|},$$ for any state $\varphi$ on $\mathcal A$. By applying as before this result to vector states and using polarization, we find an operator $f_v(\alpha)\in\mathcal A$ such that $$\langle f_v(\alpha)\xi,\eta\rangle=\lim_{y\downarrow 0} \langle f(\alpha+iv)\xi,\eta\rangle,\quad\xi,\eta\in\mathcal H.$$ Since $\|x\|=\sup\{|\varphi(x)|\colon\varphi\text{ state on } \mathcal A\}$, the estimate $$\|f_v(\alpha)\|\leq4\left(\|f(\alpha+iv)\|+\sqrt{\|c\| \|\Im f(\alpha+iv)\|}\right)$$ holds. Since for any state $\varphi$, $k(\varphi,\alpha,v)= \lim_{y\downarrow0}\varphi(f(\alpha+iyv))\in\mathbb R$, it follows that $f_v(\alpha)=f_v(\alpha)^*$. The fact that $f_v(\alpha)$ does not depend on $v$ follows from Proposition \[prop:3.1\] and Lemma \[lem:3.2\]: indeed, $$\begin{aligned} \lefteqn{ \left\|\left(\Im f(\alpha+iy_1v)\right)^{-\frac12} (f(\alpha+iy_1v)-f(\alpha+iy_21))\left(\Im f(\alpha+iy_21)\right)^{-\frac12} \right\|}\\ & \leq &\left\|\left(y_1v\right)^{-\frac12} (iy_1v-iy_21)\left(y_21)\right)^{-\frac12} \right\|\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ is equivalent to $$\begin{aligned} \lefteqn{ \left\|\left(\frac{\Im f(\alpha+iy_1v)}{y_1}\right)^{-\frac12} (f(\alpha+iy_1v)-f(\alpha+iy_21))\left(\frac{\Im f(\alpha+iy_21)}{y_2}\right)^{-\frac12} \right\|}\\ & \leq & \left\|v^{-\frac12}\right\| \left\|y_1v-y_21\right\|.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ We obtain as in the proof of Lemma \[lem:3.2\] $$\begin{aligned} \lefteqn{ \|f(\alpha+iy_1v)-f(\alpha+iy_21)\|}\nonumber\\ & \leq & \left\|v^{-\frac12}\right\| \left\|y_1v-y_21\right\|\sqrt{\left\|\frac{\Im f(\alpha+iy_1v)}{y_1}\right\| \left\|\frac{\Im f(\alpha+iy_21)}{y_2}\right\|}.\label{10}\end{aligned}$$ The two factors under the square root are bounded by hypothesis. Thus, we conclude. This result is similar to results in [@AMcY; @Fan; @Wlo]. We observe that this essentially improves the convergence to norm convergence, without requiring norm convergence in formula . In the classical Julia-Carathéodory Theorem, we noted also that $(\Re f(\alpha+iy)-f(\alpha))/y\to0$ as $y\searrow0$. A similar result holds for general noncommuttive functions. Indeed, using relation with $a=\alpha+iyv,c=\alpha+iy'v$, $b=a-c$ we obtain $$\left(f(\alpha+iyv)-f(\alpha+iy'v)\right)^*\left(\Im f(\alpha+iyv)\right)^{-1} \left(f(\alpha+iyv)-f(\alpha+iy'v)\right)$$ $$\leq\frac{(y-y')^2}{yy'}\Im f(\alpha+iy'v).$$ Letting $y'\searrow0$ we obtain (with the notation from the statement of Theorem \[Main\]) $$\left(f(\alpha+iyv)-f(\alpha)\right)^*\left(\Im f(\alpha+iyv)\right)^{-1} \left(f(\alpha+iyv)-f(\alpha)\right)\leq yc(v).$$ Recalling that $f(\alpha)=f(\alpha)^*$ we conclude that $$\left(\Re f(\alpha+iyv)-f(\alpha)\right)\left(\Im f(\alpha+iyv)\right)^{-1} \left(\Re f(\alpha+iyv)-f(\alpha)\right)\leq yc(v)-\Im f(\alpha+iyv).$$ We divide by $y$ and let $y\searrow0$ to conclude that $$\label{Re0} 0\leq\lim_{y\downarrow0}\frac{\Re f(\alpha+iyv)-f(\alpha)}{y}\left(\frac{\Im f(\alpha+iyv)}{y}\right)^{-1} \frac{\Re f(\alpha+iyv)-f(\alpha)}{y}\leq0.$$ The invertibility of $c(v)$ guarantees that $\lim_{y\downarrow0}\frac{\Re f(\alpha+iyv)-f(\alpha)}{y}=0$ in the so-topology. Thus, ${\displaystyle\lim_{\stackrel{ z\longrightarrow0}{{ \sphericalangle}}}}\frac{\Re f(\alpha+zv)-f(\alpha)}{\Im z}=0$. In order to extend the above result to all levels $n$, we need the following lemma. \[lem:4.2\] Let $f$ be as in Theorem \[Main\]. Fix $\alpha=\alpha^*\in\mathcal A$, $v_1,v_2>0$ in $\mathcal A$, and $b\in\mathcal A$ of norm $\|b\|^2\cdot1<v_2\|v_1^{-1}\|^{-1}$. Then $$\left\{\frac1y\left\|f\left(\begin{array}{cc} \alpha+iyv_1 & iyb \\ 0 & \alpha+iyv_2 \end{array}\right)- f\left(\begin{array}{cc} \alpha+iyv_1 & \frac{iyb}{2} \\ \frac{iyb^*}{2} & \alpha+iyv_2 \end{array}\right) \right\|\colon y\in(0,1)\right\}$$ is bounded Observe that $\|b\|^21<4\|v_1^{-1}\|^{-1}v_2$ implies $\Im\left(\begin{array}{cc} \alpha+iyv_1 & iyb \\ 0 & \alpha+iyv_2 \end{array}\right)>0$ for all $y>0$. We use the same trick as in Lemma \[lem:3.2\]. For simplicity, denote $$\mathfrak D=f\left(\begin{array}{cc} \alpha+iyv_1 & iyb \\ 0 & \alpha+iyv_2 \end{array}\right)- f\left(\begin{array}{cc} \alpha+iyv_1 & \frac{iyb}{2} \\ \frac{iyb^*}{2} & \alpha+iyv_2 \end{array}\right).$$ Proposition \[prop:3.1\] (in the guise of inequality ) applied to $a$ and $c$ equal to the two arguments of the function $f$ in the formula of $\mathfrak D$ above give $$\begin{aligned} \lefteqn{ \Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & {b} \\ 0 & v_2 \end{array}\right)\right)^{-\frac12} \mathfrak D^*\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)\right)^{-1}}\\ & & \mbox{}\times \mathfrak D \Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & {b} \\ 0 & v_2 \end{array}\right)\right)^{-\frac12}\\ & \leq & \left\|\left(\begin{array}{cc} yv_1 & \frac{yb}{2} \\ \frac{yb^*}{2} & yv_2 \end{array}\right)^{-\frac12} \left(\begin{array}{cc} 0 & \frac{-iyb}{2} \\ \frac{iyb^*}{2} & 0 \end{array}\right) \left(\begin{array}{cc} yv_1 & \frac{yb}{2} \\ \frac{yb^*}{2} & yv_2 \end{array}\right)^{-\frac12} \right\|^2\cdot1_{M_2(\mathcal A)}, \quad\quad\quad\quad\end{aligned}$$ for all $y>0$ (we have kept the $y$’s on the right hand side for transparency of the method). As in the proof of Lemma \[lem:3.2\], we “multiply out” the imaginary parts of $f$ on the left to obtain $$\begin{aligned} \lefteqn{\mathfrak D\mathfrak D^*\leq\|\mathfrak D\|^21}\\ & \leq & \left\|y\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)\right)\right\| \left\|\left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12} \left(\begin{array}{cc} 0 & \frac{-ib}{2} \\ \frac{ib^*}{2} & 0 \end{array}\right)\right.\\ & & \mbox{}\times\left. \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12} \right\|^2 \left\|\frac1y\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & {b} \\ 0 & v_2 \end{array}\right)\right) \right\|\cdot1.\end{aligned}$$ The last factor on the right hand side is bounded by the hypothesis, formula , Lemma \[lem:3.2\] and the above arguments. The first factor needs not apriori tend to zero, but it is clearly bounded. However, if this factor is nonzero, consider $\mathcal H$ to be the Hilbert space on which $\mathcal A$ acts as a von Neumann algebra. Then there exists a vector $\xi\in\mathcal H^2$ of norm one such that $\lim_{y\downarrow0}y\varphi_\xi\left( \Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)\right)\right)$ exists and belongs to $(0,+\infty)$, so that necessarily $$\left\|\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)\right)\right\|\to+\infty,\quad y\to0.$$ (Recall that we have denoted by $\varphi_\xi$ the vector state corresponding to $\xi$: $\varphi_\xi(a) =\langle a\xi,\xi\rangle$.) But then $2\|\Im\mathfrak D\|=\|\mathfrak D-\mathfrak D^*\|\le2\|\mathfrak D\|$ is unbounded as $y$ tends to zero, so that $$\begin{aligned} \lefteqn{\|\Im\mathfrak D\|^2\leq\|\mathfrak D\|^2}\nonumber\\ & \leq & \left\|y\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)\right)\right\| \left\|\left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12}\nonumber \left(\begin{array}{cc} 0 & \frac{-ib}{2} \\ \frac{ib^*}{2} & 0 \end{array}\right)\right.\\ & & \mbox{}\times\left. \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12} \right\|^2 \left\|\frac1y\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & {b} \\ 0 & v_2 \end{array}\right)\right) \right\|,\label{12}\end{aligned}$$ making the right hand side unbounded, a contradiction. Thus, $$\lim_{y\to0}\left\|y\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)\right)\right\|=0,$$ so, by a second application of inequality , $$\lim_{y\to0}\left\|\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)\right)\right\|=0.$$ However, more can be concluded from : dividing by $y^2$, one obtains $$\begin{aligned} \lefteqn{\frac{\|\Im \mathfrak D\|^2}{y^2}=\left\|\frac1y\Im f\left(\begin{array}{cc} \alpha+iyv_1 & iyb \\ 0 & \alpha+iyv_2 \end{array}\right)- \frac1y\Im f\left(\begin{array}{cc} \alpha+iyv_1 & \frac{iyb}{2} \\ \frac{iyb^*}{2} & \alpha+iyv_2 \end{array}\right)\right\|^2}\\ & \leq & \left\|\frac1y\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)\right)\right\| \left\|\left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12}\nonumber \left(\begin{array}{cc} 0 & \frac{-ib}{2} \\ \frac{ib^*}{2} & 0 \end{array}\right)\right.\\ & & \mbox{}\times\left. \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12} \right\|^2 \left\|\frac1y\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy \left(\begin{array}{cc} v_1 & {b} \\ 0 & v_2 \end{array}\right)\right) \right\|.\end{aligned}$$ We know from our hypothesis and Lemma \[lem:3.2\] that the set of real positive numbers $\left\{\left\|\frac1y\Im f\left(\left(\begin{array}{cc} \alpha+iyv_1 & iyb \\ 0 & \alpha+iyv_2 \end{array}\right)\right)\right\|\colon y\in(0,1)\right\}$ is bounded. If we assume that the set $\left\{\left\| \frac1y\Im f\left(\left(\begin{array}{cc} \alpha+iyv_1 & \frac{iyb}{2} \\ \frac{iyb^*}{2} & \alpha+iyv_2 \end{array}\right)\right)\right\|\colon y\in(0,1)\right\}$ is unbounded and choose a sequence $\{y_n\}_{n\in\mathbb N}$ converging to zero so that the strictly positive real number $$\ell:=\lim_{n\to\infty}\left\|\frac{1}{y_n}\Im f\left(\left(\begin{array}{cc} \alpha+iy_nv_1 & iy_nb \\ 0 & \alpha+iy_nv_2 \end{array}\right)\right)\right\|\text{ exists, and}$$ $$\lim_{n\to\infty}\left\| \frac{1}{y_n}\Im f\left(\left(\begin{array}{cc} \alpha+iy_nv_1 & \frac{iy_nb}{2} \\ \frac{iy_nb^*}{2} & \alpha+iy_nv_2 \end{array}\right)\right)\right\|=+\infty,$$ then $$\begin{aligned} \lefteqn{\left\| \frac1{y_n}\Im f\left(\left(\begin{array}{cc} \alpha+iy_nv_1 & \frac{iy_nb}{2} \\ \frac{iy_nb^*}{2} & \alpha+iy_nv_2 \end{array}\right)\right)\right\|- \left\|\frac{1}{y_n}\Im f\left(\left(\begin{array}{cc} \alpha+iy_nv_1 & iy_nb \\ 0 & \alpha+iy_nv_2 \end{array}\right)\right)\right\|}\\ & \leq & \left\|\frac{1}{y_n}\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy_n \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)\right)\right\|^\frac12 \left\|\left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12}\nonumber \left(\begin{array}{cc} 0 & \frac{-ib}{2} \\ \frac{ib^*}{2} & 0 \end{array}\right)\right.\\ & & \mbox{}\times\left. \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12} \right\| \left\|\frac{1}{y_n}\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy_n \left(\begin{array}{cc} v_1 & {b} \\ 0 & v_2 \end{array}\right)\right) \right\|^\frac12\end{aligned}$$ becomes $$\begin{aligned} \lefteqn{\left\| \frac{\Im f\left(\left(\begin{array}{cc} \alpha+iy_nv_1 & \frac{iy_nb}{2} \\ \frac{iy_nb^*}{2} & \alpha+iy_nv_2 \end{array}\right)\right)}{y_n}\right\|^\frac12-\frac{ \left\|\frac{1}{y_n}\Im f\left(\left(\begin{array}{cc} \alpha+iy_nv_1 & iy_nb \\ 0 & \alpha+iy_nv_2 \end{array}\right)\right)\right\|}{ \left\| \frac1{y_n}\Im f\left(\left(\begin{array}{cc} \alpha+iy_nv_1 & \frac{iy_nb}{2} \\ \frac{iy_nb^*}{2} & \alpha+iy_nv_2 \end{array}\right)\right)\right\|^\frac12}}\\ & \leq & \left\|\left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12} \left(\begin{array}{cc} 0 & \frac{-ib}{2} \\ \frac{ib^*}{2} & 0 \end{array}\right) \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12} \right\|\\ & & \mbox{}\times \left\|\frac{1}{y_n}\Im f\left(\left(\begin{array}{cc} \alpha & 0 \\ 0 & \alpha \end{array}\right)+iy_n \left(\begin{array}{cc} v_1 & {b} \\ 0 & v_2 \end{array}\right)\right) \right\|^\frac12;\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ by letting $n\to\infty$, we obtain $$\infty-0\leq\ell\left\|\left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12} \left(\begin{array}{cc} 0 & \frac{-ib}{2} \\ \frac{ib^*}{2} & 0 \end{array}\right) \left(\begin{array}{cc} v_1 & \frac{b}{2} \\ \frac{b^*}{2} & v_2 \end{array}\right)^{-\frac12} \right\|,$$ an obvious contradiction. We have thus shown that $\|\Im \mathfrak D\|/y$ stays bounded as $y\searrow0$. By relation , the same holds for $\Re\mathfrak D.$ This proves the lemma. The previous lemma implies more: since $\left\|\frac1y\Im f\left(\left(\begin{array}{cc} \alpha+iyv_1 & iyb \\ 0 & \alpha+iyv_2 \end{array}\right)\right)\right\|$ is bounded as $y\in(0,1),$ it follows immediately from the lemma that $$\liminf_{y\downarrow0}\frac{1}{y}\varphi\left(\Im f\left(\left(\begin{array}{cc} \alpha+iyv_1 & \frac{iyb}{2} \\ \frac{iyb^*}{2} & \alpha+iyv_2 \end{array}\right)\right)\right)<\infty,$$ for all states $\varphi$ on $M_2(\mathcal A)$, and so, as proved above, $$\label{2x2} {\rm so-}\lim_{y\downarrow0}\frac{1}{y}\Im f\left(\left(\begin{array}{cc} \alpha+iyv_1 & \frac{iyb}{2} \\ \frac{iyb^*}{2} & \alpha+iyv_2 \end{array}\right)\right):=C>0 \text{ in }M_2(\mathcal A).$$ In particular, it follows that the finiteness of the liminf in guarantees the boundedness of the sets $\Im f(\alpha\otimes 1_n+iyv)/y, y\in(0,1),$ for all $n\in\mathbb N$, $v>0$ in $M_n(\mathcal A)$, and so the existence of the corresponding so-limits for all $n$, as well as the norm-convergence of $f(\alpha\otimes 1_n+zv)$ to $f(\alpha)\otimes 1_n$ as $z\to0$ nontangentially. We show next the existence of the limit of $\Delta f(\alpha+iyv_1,\alpha+iyv_2)(b)$ as $y\searrow0$. Let $v_1,v_2,b,\alpha$ be as in the above lemma. Fix $\epsilon\in(0,1)$ and denote $V_\epsilon=\left(\begin{array}{cc} 1 & 0 \\ 0 & \sqrt\epsilon \end{array}\right)$. Observe that $$V_\epsilon^{-1}\left(\begin{array}{cc} \alpha+iyv_1 & {iyb} \\ iy\epsilon b^* & \alpha+iyv_2 \end{array}\right)V_\epsilon=\left(\begin{array}{cc} \alpha+iyv_1 & {i\sqrt\epsilon yb} \\ i\sqrt\epsilon yb^* & \alpha+iyv_2 \end{array}\right),$$ so that, by the definition of a noncommutative function, $$f\left(\begin{array}{cc} \alpha+iyv_1 & {iyb} \\ iy\epsilon b^* & \alpha+iyv_2 \end{array}\right)=V_\epsilon f\left(\begin{array}{cc} \alpha+iyv_1 & {i\sqrt\epsilon yb} \\ i\sqrt\epsilon yb^* & \alpha+iyv_2 \end{array}\right) V_\epsilon^{-1},$$ The methods used in the proof of Lemma \[lem:4.2\] allow for an estimate of the form $$\begin{aligned} \lefteqn{\frac{1}{y^2}\left\|f\left(\begin{array}{cc} \alpha+iyv_1 & {i\sqrt\epsilon yb} \\ i\sqrt\epsilon yb^* & \alpha+iyv_2 \end{array}\right)-\left(\begin{array}{cc} f(\alpha) & 0 \\ 0 & f(\alpha) \end{array}\right) \right\|^2}\\ & \leq & \left\|\left(\begin{array}{cc} v_1 & 0 \\ 0 & v_2 \end{array}\right)^{-\frac12}\left(\begin{array}{cc} iv_1 & {i\sqrt\epsilon b} \\ i\sqrt\epsilon b^* & iv_2 \end{array}\right) \left(\begin{array}{cc} v_1 & \frac{\sqrt\epsilon b}{2} \\ \frac{\sqrt\epsilon b^*}{2} & v_2 \end{array}\right)^{-\frac12}\right\|^2\\ & & \mbox{}\times\left\| \left(\begin{array}{cc} c(v_1) & 0 \\ 0 & c(v_2) \end{array}\right) \right\| \left\| \frac1y\Im f\left(\begin{array}{cc} \alpha+iyv_1 & {i\sqrt\epsilon yb} \\ i\sqrt\epsilon yb^* & \alpha+iyv_2 \end{array}\right) \right\|.\end{aligned}$$ If we denote $C_\epsilon:=\lim_{y\downarrow0} \frac1y\Im f\left(\begin{array}{cc} \alpha+iyv_1 & {i\sqrt\epsilon yb} \\ i\sqrt\epsilon yb^* & \alpha+iyv_2 \end{array}\right)$, the above allows us to conclude that $$\|C_\epsilon\|\leq\left\|\left(\begin{array}{cc} v_1 & 0 \\ 0 & v_2 \end{array}\right)^{-\frac12}\left(\begin{array}{cc} v_1 & {\sqrt\epsilon b} \\ \sqrt\epsilon b^* & v_2 \end{array}\right) \left(\begin{array}{cc} v_1 & \frac{\sqrt\epsilon b}{2} \\ \frac{\sqrt\epsilon b^*}{2} & v_2 \end{array}\right)^{-\frac12}\right\|^2\max_{1\le j\le 2}\|c(v_j)\|.$$ Thus, for any $\epsilon\in(0,1)$ we have $\|C_\epsilon\|\leq\textrm{const}(v_1,v_2,b)$. However, a bit more can be obtained: since conjugation by $V_\epsilon$ does not affect diagonal elements, we have $$\begin{aligned} \lefteqn{\frac{1}{y^2}\left\|f\left(\begin{array}{cc} \alpha+iyv_1 & {i yb} \\ i\epsilon yb^* & \alpha+iyv_2 \end{array}\right)-\left(\begin{array}{cc} f(\alpha) & 0 \\ 0 & f(\alpha) \end{array}\right) \right\|^2}\\ & = & \frac{1}{y^2}\left\|V_\epsilon \left(f\left(\begin{array}{cc} \alpha+iyv_1 & {i\sqrt\epsilon yb} \\ i\sqrt\epsilon yb^* & \alpha+iyv_2 \end{array}\right)-\left(\begin{array}{cc} f(\alpha) & 0 \\ 0 & f(\alpha) \end{array}\right)\right)V_\epsilon^{-1} \right\|^2\\ & \leq & \frac1\epsilon \left\|\left(\begin{array}{cc} v_1 & 0 \\ 0 & v_2 \end{array}\right)^{-\frac12}\left(\begin{array}{cc} iv_1 & {i\sqrt\epsilon b} \\ i\sqrt\epsilon b^* & iv_2 \end{array}\right) \left(\begin{array}{cc} v_1 & \frac{\sqrt\epsilon b}{2} \\ \frac{\sqrt\epsilon b^*}{2} & v_2 \end{array}\right)^{-\frac12}\right\|^2\\ & & \mbox{}\times\left\| \left(\begin{array}{cc} c(v_1) & 0 \\ 0 & c(v_2) \end{array}\right) \right\| \left\| \frac1y\Im f\left(\begin{array}{cc} \alpha+iyv_1 & {i\sqrt\epsilon yb} \\ i\sqrt\epsilon yb^* & \alpha+iyv_2 \end{array}\right) \right\|,\end{aligned}$$ as $\|V_\epsilon\|=1,\|V_\epsilon^{-1}\|=\epsilon^{-1/2}$. The existence of the limit $$\ell_\epsilon:=\lim_{y\downarrow0}\frac1y\left[ f\left(\begin{array}{cc} \alpha+iyv_1 & {i\sqrt\epsilon yb} \\ i\sqrt\epsilon yb^* & \alpha+iyv_2 \end{array}\right)-\left(\begin{array}{cc} f(\alpha) & 0 \\ 0 & f(\alpha) \end{array}\right)\right]$$ implies the existence of $$\lim_{y\downarrow0}\frac1y\left[f\left(\begin{array}{cc} \alpha+iyv_1 & {i yb} \\ i\epsilon yb^* & \alpha+iyv_2 \end{array}\right)-\left(\begin{array}{cc} f(\alpha) & 0 \\ 0 & f(\alpha) \end{array}\right)\right]=V_\epsilon\ell_\epsilon V_\epsilon^{-1}.$$ Let us now continue our estimates on the derivative: $$\begin{aligned} \lefteqn{\frac{1}{y^2}\left\|f\left(\begin{array}{cc} \alpha+iyv_1 & {i yb} \\ i\epsilon yb^* & \alpha+iyv_2 \end{array}\right)-f\left(\begin{array}{cc} \alpha+iyv_1 & iyb \\ 0 & \alpha+iyv_2 \end{array}\right) \right\|^2}\\ & \leq & \left\|\left(\begin{array}{cc} yv_1 & \frac{yb}{2} \\ \frac{yb^*}{2} & yv_2 \end{array}\right)^{-\frac12}\left(\begin{array}{cc} 0 & 0 \\ i\epsilon yb^* & 0 \end{array}\right) \left(\begin{array}{cc} yv_1 & \frac{(1+\epsilon)yb}{2} \\ \frac{(1+\epsilon)y b^*}{2} & yv_2 \end{array}\right)^{-\frac12}\right\|^2\\ & & \mbox{}\times\left\| \frac1y\Im f\left(\begin{array}{cc} \alpha+iyv_1 & iyb \\ 0 & \alpha+iyv_2 \end{array}\right) \right\| \left\| \frac1y\Im f\left(\begin{array}{cc} \alpha+iyv_1 & {iyb} \\ i\epsilon yb^* & \alpha+iyv_2 \end{array}\right) \right\|.\end{aligned}$$ The first factor on the right hand side is bounded by $\epsilon^2\textrm{const}(b,v_1,v_2)$, for a constant $\textrm{const}(b,v_1,v_2)\in\mathbb R$, independent of $y,\epsilon\in(0,1)$. The second factor has been shown in Lemma \[lem:3.2\] to be bounded uniformly in $y\in(0,1)$. Finally, the last term is dominated, as seen above, by $\epsilon^{-1}\textrm{const}(b,v_1,v_2)$. Thus, $$\frac{1}{y^2}\left\|f\left(\begin{array}{cc} \alpha+iyv_1 & {i yb} \\ i\epsilon yb^* & \alpha+iyv_2 \end{array}\right)-f\left(\begin{array}{cc} \alpha+iyv_1 & iyb \\ 0 & \alpha+iyv_2 \end{array}\right) \right\|^2\leq\epsilon\textrm{const}(v_1,v_2,b),$$ for any $y,\epsilon\in(0,1)$. By weak compactness of norm-bounded sets, any sequence tending to zero has a subsequence $\{y_n\}$ such that ${\displaystyle\lim_{n\to\infty}}\Delta f(\alpha+iy_nv_1,\alpha+iy_nv_2)(b)$ exists in the weak operator topology. Adding and subtracting $\left(\begin{array}{cc} f(\alpha) & 0 \\ 0 & f(\alpha) \end{array}\right)$ under the norm in the left hand side above and letting $y\searrow0$ along such a sequence provides $$\left\|V_\epsilon\ell_\epsilon V_\epsilon^{-1}-\left(\begin{array}{cc} c(v_1) & {\displaystyle\lim_{n\to\infty}}\Delta f(\alpha+iy_nv_1,\alpha+iy_nv_2)(b) \\ 0 & c(v_2) \end{array}\right) \right\|\leq\sqrt{\epsilon}\textrm{const}(v_1,v_2,b),$$ for any fixed $\epsilon\in(0,1)$. This restricts the diameter of the cluster set of $\Delta f(\alpha+iyv_1,\alpha+iyv_2)(b)$ at zero to a set of norm-diameter of order $\sqrt\epsilon$ for any $\epsilon>0$. Thus, the limit ${\displaystyle\lim_{y\downarrow0}}\Delta f(\alpha+iyv_1,\alpha+iyv_2)(b)$ must exist. We conclude that $\displaystyle\lim_{y\downarrow0}\Delta f(\alpha+iyv_1,\alpha+iyv_2)(b)$ exists and is uniformly bounded as $b\in\mathcal A$ stays in a bounded subset of $\mathcal A$. Clearly the limit depends linearly on $b$, since each of $\Delta f(\alpha+iyv_1,\alpha+iyv_2)(b)$ does. In particular, if $v_1=v_2=v$, $\Delta f(\alpha+iyv,\alpha+iyv)(b)=f'(\alpha+iyv)(b)$ has a limit as $y\to0$, as claimed in part (1) of Theorem \[Main\]. Let now in addition $b=v/4$. For any state $\varphi$ on $\mathcal A$ and $v>0$, $z\mapsto\varphi(f(\alpha+zv))$ is a self-map of $\mathbb C^+$ which satisfies the conditions of Theorem \[JC\] at $z=0$. Thus, $\displaystyle\lim_{y\downarrow0}\varphi(f'(\alpha+iyv)(v))= \lim_{y\downarrow0}\frac{\varphi(\Im f(\alpha+iyv))}{y}$, so that indeed $$\lim_{y\downarrow0}f'(\alpha+iyv)(v)=\lim_{y\downarrow0}\frac{\Im f(\alpha+iyv)}{y}=c(v)>0.$$ Until now we have proved that the finiteness of the liminf in (which is applied to elements in $\mathcal A=M_1(\mathcal A)$) implies not only the existence of $f(\alpha)$ and of limits of $\Delta f(\alpha+iyv_1,\alpha+iyv_2)$ as $y\downarrow0$, but also the existence and finiteness of the liminf in applied to $\alpha$ replaced by $\alpha\otimes 1_{M_2(\mathbb C)}$ and $v$ replaced by a positive in $M_2(\mathcal A)$. Obviously, we now apply the above results to elements in $M_2(\mathcal A)$ to obtain the same conclusion for elements in $M_4(\mathcal A)$ and so on. This, according to [@ncfound Chapters 2 and 3], allows us to conclude the proof of part (1). We prove next part (1’) of Theorem \[Main\]. Let $v,w>0$ be fixed. Recall that we have shown in the proof of part (1) that $\displaystyle\lim_{t\downarrow0}f'(\alpha+ity_1v+ity_2w)$ exists pointwise. Our hypothesis that $$\lim_{y_1,y_2\to0} (\varphi(f'(\alpha+iy_1v+iy_2w)(v)),\varphi(f'(\alpha+iy_1v+iy_2w)(w)))$$ exists and is finite for any state $\varphi$ on $\mathcal A$ implies that $f'(\alpha+iy_1v+iy_2w)(v),f'(\alpha+iy_1v+iy_2w)(w)$ have a weak limit as $(y_1,y_2)\downarrow(0,0)$ in $[0,1)^2\setminus\{(0,0)\}$. Note that the domain $\{(z,\zeta)\in \mathbb C^2\colon\Im(zv+\zeta w)>0\}$ of the function $(z,\zeta)\mapsto\varphi(f(\alpha+zv+\zeta w))$ includes $\overline{\mathbb C^+}\times\mathbb C^+ \cup\mathbb C^+\times\overline{\mathbb C^+}$ (closures taken in $\mathbb C$). In particular, $\{(z,0)\colon z\in\mathbb C^+\}\cup\{(0,\zeta)\colon\zeta\in\mathbb C^+\}\subset \{(z,\zeta)\in\mathbb C^2\colon\Im(zv+\zeta w)>0\}.$ The existence of the above displayed limit thus guarantees that $\lim_{y\downarrow0}\varphi(f'(\alpha+iyw)(v))= \lim_{y\downarrow0}\varphi(f'(\alpha+iyv)(v))$. This means that the limit of $f'(\alpha+iyv)$ as $y\downarrow0$ does not depend on $v$ and is positive. Applying this same result to $M_n(\mathcal A)$ and recalling the properties of noncommutative functions guarantee complete positivity for $f'(\alpha)$. To conclude the proof of (1’), simply observe that $\Delta f(\alpha+iyv_1,\alpha+iyv_2)(b)- f'(\alpha+iyv_1)(b)$ converges to zero as $y\downarrow0$. The proof of (2) is much simpler. Indeed, the existence of the limit $\lim_{y\downarrow0}f'(\alpha+iyv)$ implies the existence of the limit $\lim_{y\downarrow0}\varphi(f'(\alpha+iyv)(v))$ for all states $\varphi$ on $\mathcal A$. An application of Theorem \[JC\] and of parts (1) and (1’) of Theorem \[Main\] allows us to conclude. It might be useful to note that the operator $C$ from equality can be written in terms of the small $c$’s form the statement of Theorem \[Main\], at least when $v_1=v_2$. We use here the condition (A) of the definition of noncommutative functions. Let $v>0$ be fixed and let $b$ be such that $v>b>0$ in $\mathcal A$. Then $$\left(\begin{array}{cc} \alpha+iyv & iyb \\ iyb & \alpha+iyv \end{array}\right)\left(\begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right)= \left(\begin{array}{ccc} \alpha+iy(v+b) & iyb &\alpha+iyv \\ \alpha+iy(v+b) & \alpha+iyv & iyb \end{array}\right),$$ which is in its own turn equal to the product $$\left(\begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right)\left(\begin{array}{ccc} \alpha+iy(v+b) & iyb & 0 \\ 0 & \alpha+iy(v-b) & iyb \\ 0 & 0 & \alpha+iyv \end{array}\right).$$ We recognize in the $2\times2$ matrix the argument of one of the terms involved in the statement of Lemma \[lem:4.2\]. If we denote $$f\left(\begin{array}{cc} \alpha+iyv & iyb \\ iyb & \alpha+iyv \end{array}\right)=\left(\begin{array}{cc} f_{11} & f_{12} \\ f_{21} & f_{22} \end{array}\right),$$ then condition (A) tells us that [$$\begin{aligned} \lefteqn{\left(\begin{array}{ccc} f_{11}+f_{12} & f_{12} & f_{11} \\ f_{21}+f_{22} & f_{22} & f_{21} \end{array}\right)=\left(\begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right)\times}\\ & & \left(\begin{array}{ccc} f(\alpha+iy(v+b)) & \frac{f(\alpha+iy(v+b))-f(\alpha+iy(v-b))}{2} & \Delta^2f \\ 0 & f(\alpha+iy(v-b)) & f(\alpha+iyv)-f(\alpha+iy(v-b)) \\ 0 & 0 & f(\alpha+iyv) \end{array}\right)= \\ & & \left(\begin{array}{ccc} f(\alpha+iy(v+b)) & \frac{f(\alpha+iy(v+b))-f(\alpha+iy(v-b))}{2} & \Delta^2f+f(\alpha+iyv) \\ f(\alpha+iy(v+b)) & \frac{f(\alpha+iy(v+b))+f(\alpha+iy(v-b))}{2} & f(\alpha+iyv)-f(\alpha+iy(v-b))+ \Delta^2f \end{array}\right),\end{aligned}$$]{} where $\Delta^2f$ stands for $\Delta^2f(\alpha+iy(v+b),\alpha+iy(v-b),\alpha+iyv) ( iyb , iyb )$. We obtain immediately the relations $$\begin{aligned} f_{11}=f_{22}&=&\frac{f(\alpha+iy(v+b))+f(\alpha+iy(v-b))}{2}\\ f_{21}=f_{12}&=&\frac{f(\alpha+iy(v+b))-f(\alpha+iy(v-b))}{2}.\end{aligned}$$ It follows that, for $v_1=v_2>0$, $$\begin{aligned} C & = & \lim_{y\downarrow0}\frac1y\Im f\left(\left(\begin{array}{cc} \alpha+iy & iyb \\ iyb & \alpha+iyv \end{array}\right)\right)\\ & = & \frac12\lim_{y\downarrow0} \left(\begin{array}{cc} \frac{\Im f(\alpha+iy(v+b))+\Im f(\alpha+iy(v-b))}{y}&\frac{\Im f(\alpha+iy(v+b))-\Im f(\alpha+iy(v-b))}{y} \\ \frac{\Im f(\alpha+iy(v+b))-\Im f(\alpha+iy(v-b))}{y}&\frac{\Im f(\alpha+iy(v+b))+\Im f(\alpha+iy(v-b))}{y} \end{array}\right).\end{aligned}$$ By considering the functions $z\mapsto\varphi( f(\alpha+z(v\pm b)))$, we obtain on the off-diagonal entries precisely $[f'(\alpha)(v+b)-f'(\alpha)(v-b)]/2$ and on the diagonal entries $[f'(\alpha)(v+b)+f'(\alpha)(v-b)]/2$. 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--- abstract: | We study several aspects of the dynamic programming approach to optimal control of abstract evolution equations, including a class of semilinear partial differential equations. We introduce and prove a verification theorem which provides a sufficient condition for optimality. Moreover we prove sub- and superoptimality principles of dynamic programming and give an explicit construction of $\epsilon$-optimal controls. **Key words**: optimal control of PDE, verification theorem, dynamic programming, $\epsilon$-optimal controls, Hamilton-Jacobi-Bellman equations. **MSC 2000**: 35R15, 49L20, 49L25, 49K20. author: - 'G. Fabbri[^1] F. Gozzi[^2] and A. Świȩch[^3]' title: 'Verification theorem and construction of $\epsilon$-optimal controls for control of abstract evolution equations' --- Introduction ============ In this paper we investigate several aspects of the dynamic programming approach to optimal control of abstract evolution equations. The optimal control problem we have in mind has the following form. The state equation is $$\label{deterministicstateequation} \left\lbrace \begin{array}{l} \dot{x}(t) = Ax(t) + b(t,x(t),u(t)),\\ x(0)=x, \end{array} \right.$$ $A$ is a linear, densely defined maximal dissipative operator in a real separable Hilbert space $\mathcal{H}$, and we want to minimize a cost functional $$\label{deterministiccostfunctional} J(x;u(\cdot))= \int_0^T L(t,x(t),u(t)) {\mathrm{d}}t + h(x(T))$$ over all controls $$u(\cdot)\in\mathcal{U}[0,T]= \{ u\colon [0,T] \to U : \; u \; \hbox{is measurable} \},$$ where $U$ is a metric space. The dynamic programming approach studies the properties of the so called value function for the problem, identifies it as a solution of the associated Hamilton-Jacobi-Bellman (HJB) equation through the dynamic programming principle, and then tries to use this PDE to construct optimal feedback controls, obtain conditions for optimality, do numerical computations, etc.. There exists an extensive literature on the subject for optimal control of ordinary differential equations, i.e. when the HJB equations are finite dimensional (see for instance the books [@BCD; @CLSW; @FR; @FS; @Lo; @Vinter; @YongZhou] and the references therein). The situation is much more complicated for optimal control of partial differential equations (PDE) or abstract evolution equations, i.e. when the HJB equations are infinite dimensional, nevertheless there is by now a large body of results on such HJB equations and the dynamic programming approach ([@B1; @B2; @B3; @B4; @BaBaJe; @BaDaP1; @BaDaP2; @BaDaP3; @BaDaP4; @BaDaPPo; @Ca1; @CaCa; @CaDaP1; @CaDaP2; @CaDiB; @CaGoSo; @CaFr1; @CaFr2; @CaTe1; @CaTe2; @CL4; @CL5; @CL6; @CL7; @DiB; @GSS; @I; @KoSo; @LiYong; @Sh; @Sri; @T1; @T2] and the references therein). Numerous notions of solutions are introduced in these works, the value functions are proved to be solutions of the dynamic programming equations, and various verification theorems and results on existence and explicit forms of optimal feedback controls in particular cases are established. However, despite of these results, so far the use of the dynamic programming approach in the resolution of the general optimal control problems in infinite dimensions has been rather limited. Infinite dimensionality of the state space, unboundedness in the equations, lack of regularity of solutions, and often complicated notions of solutions requiring the use of sophisticated test functions are only some of the difficulties. We will discuss two aspects of the dynamic programming approach for a fairly general control problem: a verification theorem which gives a sufficient condition for optimality, and the problem of construction of $\epsilon$-optimal feedback controls. The verification theorem we prove in this paper is an infinite dimensional version of such a result for finite dimensional problems obtained in [@Zh]. It is based on the notion of viscosity solution (see Definitions \[defdeterministicsubsol\]-\[defdeterministicsol\]). Regarding previous result in this direction we mention [@CaFr1; @CaFr2] and the material in Chapter 6 §5 of [@LiYong], in particular Theorem 5.5 there which is based on [@CaFr1]. We briefly discuss this result in Remark \[remliyo\]. The construction of $\epsilon$-optimal controls we present here is a fairly explicit procedure which relies on the proof of superoptimality inequality of dynamic programming for viscosity supersolutions of the corresponding Hamilton-Jacobi-Bellman equation. It is a delicate generalization of such a method for the finite dimensional case from [@sw]. Similar method has been used in [@CLSS] to construct stabilizing feedbacks for nonlinear systems and later in [@IK] for state constraint problems. The idea here is to approximate the value function by its appropriate inf-convolution which is more regular and satisfies a slightly perturbed HJB inequality pointwise. One can then use this inequality to construct $\epsilon$-optimal piecewise constant controls. This procedure in fact gives the superoptimality inequality of dynamic programming and the suboptimality inequality can be proved similarly. There are other possible approaches to construction of $\epsilon$-optimal controls. For instance under compactness assumption on the operator $B$ (see Section 4) one can approximate the value function by solutions of finite dimensional HJB equations with the operator $A$ replaced by some finite dimensional operators $A_n$ (see [@CL4]) and then use results of [@sw] directly to construct near optimal controls. Other approximation procedures are also possible. The method we present in this paper seems to have some advantages: it uses only one layer of approximations, it is very explicit and the errors in many cases can be made precise, and it does not require any compactness of the operator $B$. It does however require some weak continuity of the Hamiltonian and uniform continuity of the trajectories, uniformly in $u(\cdot)$. Finally we mention that the sub- and superoptimality inequalities of dynamic programming are interesting on their own. The paper is organized as follows. Definitions and the preliminary material is presented in Section 2. Section 3 is devoted to the verification theorem and an example where it applies in a nonsmooth case. In Section 4 we prove sub- and superoptimality principles of dynamic programming and show how to construct $\epsilon$-optimal controls. Notation, definitions and background ==================================== Throughout this paper $\mathcal{H}$ is a real separable Hilbert space equipped with the inner product ${\left\langle}\cdot,\cdot{\right\rangle}$ and the norm $\|\cdot\|$. We recall that $A$ is a linear, densely defined operator such that $-A$ is maximal monotone, i.e. $A$ generates a $C_0$ semigroup of contractions $e^{sA}$, i.e. $$\label{ppp1} \| e^{sA} \| \leq 1 \;\;\; \text{for all $s\geq 0$}$$ We make the following assumptions on $b$ and $L$. \[hpD2onb\] $$b\colon [0,T] \times \mathcal{H} \times U \to \mathcal{H} \; \text{is continuous}$$ and there exist a constant $M>0$ and a local modulus of continuity $\omega(\cdot,\cdot)$ such that $$\begin{array}{ll} \|b(t,x,u) - b(s,y,u)\| \leq M \|x-y\| + \omega(|t-s|,\|x\|\vee \|y\|)\\ \hskip 5cm \text{for all $t,s\in [0,T], \; u\in U \; x,y\in\mathcal{H}$}\\ \|b(t,0,u)\| \leq M \;\; \text{for all $(t,u)\in [0,T] \times U$} \end{array}$$ \[hpD3onLandh\] $$L\colon [0,T] \times \mathcal{H} \times U \to \mathbb{R} \;\;\; and \;\;\; h\colon \mathcal{H} \to \mathbb{R} \;\;\; \text{are continuous}$$ and there exist $M>0$ and a local modulus of continuity $\omega(\cdot,\cdot)$ such that $$\begin{array}{ll} |L(t,x,u) - L(s,y,u)|, \; |h(x)-h(y)| \leq \omega(\|x-y\|+|t-s|,\|x\|\vee \|y\|)\\ \hskip 5cm \text{for all $t,s\in [0,T], \; u\in U \; x,y\in\mathcal{H}$}\\ |L(t,0,u)|, |h(0)| \leq M \;\; \text{for all $(t,u)\in [0,T] \times U$} \end{array}$$ Notice that if we replace $A$ and $b$ by $\tilde A=A-\omega I$ and $b(t,x,u)$ with $\tilde b(t,x,u)= b(t,x,u) + \omega x$ the above assumptions would cover a more general case $$\label{ppp2} \| e^{sA} \| \leq e^{\omega s} \;\;\; \text{for all $s\geq 0$}$$ for some $\omega \geq 0$. However such $\tilde b$ does not satisfy the assumptions of Section 4 and may not satisfy the assumptions needed for comparison for equation (\[deterministicHJB\]). Alternatively, by making a change of variables $\tilde v(t,x)=v(t,e^{\omega t}x)$ in equation (\[deterministicHJB\]) (see [@CL4], page 275) we can always reduce the case (\[ppp2\]) to the case when $A$ satisfies (\[ppp1\]). Following the dynamic programming approach we consider a family of problems for every $t\in[0,T], y\in \mathcal{H}$ $$\label{sydeterministicstate} \left\lbrace \begin{array}{l} \dot{x}_{t,x}(s)=A {x}_{t,x}(s) + b(s,{x}_{t,x}(s),u(s))\\ x_{t,x}(t)=x \end{array} \right.$$ We will write $x(\cdot)$ for $x_{t,x}(\cdot)$ when there is no possibility of confusion. We consider the function $$\label{sydeterministiccost} J(t,x;u(\cdot))= \int_t^T L(s,x(s),u(s)) {\mathrm{d}}t + h(x(T)),$$ where $u(\cdot)$ is in the set of admissible controls $$\mathcal{U}[t,T]= \{ u\colon [t,T] \to U: \; u \hbox{ is measurable} \}.$$ The associated value function $V\colon [0,T]\times\mathcal{H} \to \mathbb{R}$ is defined by $$\label{deterministicvaluefunction} V(t,x)= \inf_{u(\cdot) \in \mathcal{U}[t,T]} J(t,x;u(\cdot)).$$ The Hamilton-Jacobi-Bellman (HJB) equation related to such optimal control problems is $$\label{deterministicHJB} \left\lbrace \begin{array}{l} v_t(t,x) +{\left\langle}Dv(t,x), Ax {\right\rangle}+ H(t,x,Dv(t,x))=0\\ v(T,x)=h(x), \end{array} \right.$$ where $$\left\lbrace \begin{array}{l} H\colon [0,T]\times\mathcal{H}\times\mathcal{H} \to \mathbb{R},\\ H(t,x,p)=\inf_{u\in U} \left ( {\left\langle}p, b(t,x,u) {\right\rangle}+ L(t,x,u) \right ) \end{array} \right.$$ The solution of the above HJB equation is understood in the viscosity sense of Crandall and Lions [@CL4; @CL5] which is slightly modified here. We consider two sets of tests functions: $$\begin{array}{ll} test1=\{ \varphi \in C^1((0,T)\times\mathcal{H}) \; : & \varphi \text{ is weakly sequentially lower}\\ & \text{semicontinuous and } A^*D\varphi\in C((0,T)\times \mathcal{H}) \} \end{array}$$ and $$\begin{array}{ll} test2= \{ g\in C^1((0,T)\times\mathcal{H}) \; :& \exists g_0, \colon [0,+\infty) \to [0,+\infty), \;\\ &and \; \eta\in C^1((0,T)) \text{ positive } \; s.t.\\ &g_0 \in C^1([0,+\infty)), \; g_0'(r) \geq 0 \; \forall r\geq 0, \\ & g_0'(0)=0 \; and \; g(t,x)=\eta(t)g_0(\|x\|) \\ &\forall (t,x)\in (0,T)\times \mathcal{H} \} \end{array}$$ We use test2 functions that are a little different from the ones used in [@CL4]. The extra term $\eta(\cdot)$ in test2 functions is added to deal with unbounded solutions. We recall that $D\varphi$ and $Dg$ stand for the Frechet derivatives of these functions. \[defdeterministicsubsol\] A function $v\in C((0,T]\times\mathcal{H})$ is a (viscosity) *subsolution* of the HJB equation (\[deterministicHJB\]) if $$v(T,x) \leq h(x) \;\;\; for \; all\; x\in\mathcal{H}$$ and whenever $v-\varphi-g$ has a local maximum at $(\bar t, \bar x)\in[0,T)\times\mathcal{H}$ for $\varphi \in test1$ and $g\in test2$, we have $$\label{eqsubsol} \varphi_t(\bar t, \bar x) + g_t(\bar t, \bar x)+{\left\langle}A^* D \varphi(\bar t, \bar x) , \bar x {\right\rangle}+H(\bar t, \bar x, D\varphi(\bar t, \bar x)+ Dg(\bar t, \bar x)) \geq 0.$$ \[defdeterministicsupersol\] A function $v\in C((0,T]\times\mathcal{H})$ is a (viscosity) *supersolution* of the HJB equation (\[deterministicHJB\]) if $$v(T,x) \geq h(x) \;\;\; for \; all\; x\in\mathcal{H}$$ and whenever $v+\varphi+g$ has a local minimum at $(\bar t, \bar x)\in[0,T)\times\mathcal{H}$ for $\varphi \in test1$ and $g\in test2$, we have $$\label{eqsupersol} -\varphi_t(\bar t, \bar x) - g_t(\bar t, \bar x) - {\left\langle}A^* D \varphi(\bar t, \bar x) , \bar x {\right\rangle}+ H(\bar t, \bar x, -D \varphi(\bar t, \bar x)- D g(\bar t, \bar x)) \leq 0.$$ \[defdeterministicsol\] A function $v\in C((0,T]\times\mathcal{H})$ is a (viscosity) *solution* of the HJB equation (\[deterministicHJB\]) if it is at the same time a subsolution and a supersolution. We will be also using viscosity sub- and supersolutions in situations where no terminal values are given in (\[deterministicHJB\]). We will then call a viscosity subsolution (respectively, supersolution) simply a function that satisfies (\[eqsubsol\]) (respectively, (\[eqsupersol\])). \[lemmaphi\] Let Hypotheses \[hpD2onb\] and \[hpD3onLandh\] hold. Let $\phi\in test1$ and $(t,x)\in(0,T)\times\mathcal{H}$. Then the following convergence holds uniformly in $u(\cdot) \in \mathcal{U}[t,T]$: $$\begin{gathered} \lim_{s\downarrow t} \left ( \frac{1}{s-t} \left ( \varphi(s,x_{t,x}(s)) - \varphi(t,x) \right ) - \varphi_t(t,x) - {\left\langle}A^*D\varphi (t,x),x{\right\rangle}\right.\\ \left. - \frac{1}{s-t} \int_t^s {\left\langle}D \varphi (t,x), b(t,x,u(r)) {\right\rangle}{\mathrm{d}}r \right ) =0\end{gathered}$$ Moreover we have for $s-t$ sufficiently small $$\begin{gathered} \label{eq:explicitphi} \varphi(s,x_{t,x}(s))-\varphi(t,x) = \int_t^s \varphi_t(r,x_{t,x}(r)) + {\left\langle}A^*D\varphi (r,x_{t,x}(r)),x_{t,x}(r){\right\rangle}\\ + {\left\langle}D \varphi(r,x_{t,x}(r)), b(r,x_{t,x}(r),u(r)) {\right\rangle}{\mathrm{d}}r\end{gathered}$$ See [@LiYong] Lemma 3.3 page 240 and Proposition 5.5 page 67. \[lemmag\] Let Hypotheses \[hpD2onb\] and \[hpD3onLandh\] hold. Let $g\in test2$ and $(t,x)\in(0,T)\times\mathcal{H}$. Then for $s-t \to 0^+$ $$\begin{gathered} \label{gconv1} \frac{1}{s-t} \left ( g(s,x_{t,x}(s)) - g(t,x) \right ) \leq g_t(t,x) \\ + \frac{1}{s-t} \int_t^s {\left\langle}D g (t,x), b(t,x,u(r)) {\right\rangle}{\mathrm{d}}r + o(1)\end{gathered}$$ where $o(1)$ is uniform in $u(\cdot) \in \mathcal{U}[t,T]$ To prove the statement when $x \ne 0$ we use the fact that, in this case (see [@LiYong] page 241, equation (3.11)), $$\|x_{t,x}(s)\| \leq \|x\| + \int_t^s {\left\langle}\frac{x}{\|x\|}, b(t,x,u(r)) {\right\rangle}{\mathrm{d}}r + o(s-t)$$ So we have $$\begin{gathered} \label{eq:prooflemmag} g(s,x_{t,x}(s))-g(t,x) = \eta(s) g_0(\|x_{t,x}(s)\|) -\eta(t)g_0(\|x\|)\\ \leq \eta(s) g_0\left ( \|x\| + \int_t^s {\left\langle}\frac{x}{\|x\|}, b(t,x,u(r)) {\right\rangle}{\mathrm{d}}r + o(s-t) \right ) - \eta(t)g_0(\|x\|) \\ \leq \eta'(t) g_0(\|x\|) (s-t) + \eta(t) g_0'(\|x\|) \left ( \int_t^s {\left\langle}\frac{x}{\|x\|}, b(t,x,u(r)) {\right\rangle}{\mathrm{d}}r \right ) + o(s-t) \\ = g_t(t,x) (s-t) + \int_t^s {\left\langle}D g(t,x) , b(t,x,u(r)) {\right\rangle}{\mathrm{d}}r + o(s-t)\end{gathered}$$ where $o(s-t)$ is uniform in $u(\cdot)$. When $x=0$, using the fact that $g'_0(0)=0$, we get $$g(s,x_{t,x}(s))-g(t,x)=g_t(t,x) (s-t) + o(s-t+\|x_{t,x}(s)\|)$$ and (\[gconv1\]) follows upon noticing that $\|x_{t,x}(s)\|\le C(s-t)$ for some $C$ independent of $u(\cdot) \in \mathcal{U}[t,T]$. \[thexistence\] Let Hypotheses \[hpD2onb\] and \[hpD3onLandh\] hold. Then the value function $V$ (defined in (\[deterministicvaluefunction\])) is a viscosity solution of the HJB equation (\[deterministicHJB\]). The proof is quite standard and can be obtained with small changes (due to the small differences in the definition of test2 functions) from Theorem 2.2, page 229 of [@LiYong] and the proof of Theorem 3.2, page 240 of [@LiYong] (or from [@CL5]). We will need a comparison result in the proof of the verification theorem. There are various versions of such results for equation (\[deterministicHJB\]) available in the literature, several sufficient sets of hypotheses can be found in [@CL4; @CL5]. Since we are not interested in the comparison result itself we choose to assume a form of comparison theorem as a hypothesis. \[D4deterministiccomparison\] There exists a set $\mathcal{G}\subseteq C([0,T]\times\mathcal{H})$ such that: - the value function $V$ is in $\mathcal{G}$; - if $v_1, v_2 \in \mathcal{G}$, $v_1$ is a subsolution of the HJB equation (\[deterministicHJB\]) and $v_2$ is a supersolution of the HJB equation (\[deterministicHJB\]) then $v_1\leq v_2$. Note that from $(i)$ and $(ii)$ we know that $V$ is the only solution of the HJB equation (\[deterministicHJB\]) in $\mathcal{G}$. We will use the following lemma whose proof can be found in [@YongZhou], page 270. \[lemmaYZ\] Let $g\in C([0,T];\mathbb{R})$. We extend $g$ to a function (still denoted by $g$) on $(-\infty,+\infty)$ by setting $g(t)=g(T)$ for $t>T$ and $g(t)=g(0)$ for $t<0$. Suppose there is a function $\rho \in L^1(0,T;\mathbb{R})$ such that $$\limsup_{h\to 0^+} \frac{g(t+h) - g(t)}{h}\leq \rho(t) \;\;\; a.e. \; t\in[0,T].$$ Then $$g(\beta)-g(\alpha) \leq \int_\alpha^\beta \limsup_{h\to 0^+} \frac{g(t+h) - g(t)}{h} {\mathrm{d}}t\;\;\;\; \forall \; 0\leq\alpha\leq\beta\leq T.$$ We will denote by $B_R$ the open ball of radius $R$ centered at $0$ in $\mathcal{H}$. The verification theorem ======================== We first introduce a set related to a subset of the superdifferential of a function in $C((0,T)\times\mathcal{H})$. Its definition is suggested by the definition of a sub/super solution. We recall that the superdifferential $D^{1,+}v(t,x)$ of $v \in C((0,T)\times\mathcal{H})$ at $(t,x)$ is given by the pairs $(q,p)\in \mathbb{R}\times \mathcal{H}$ such that $v(s,y) - v(t,x) - \left\langle p, y-x \right\rangle - q(s-t) \leq o(\|x-y\| + |t-s|)$, and the subdifferential $D^{1,-}v(t,x)$ at $(t,x)$ is the set of all $(q,p)\in \mathbb{R}\times \mathcal{H}$ such that $v(s,y) - v(t,x) - \left\langle p, y-x \right\rangle - q(s-t) \geq o(\|x-y\| + |t-s|)$. \[defE\] Given $v\in C((0,T)\times\mathcal{H})$ and $(t,x)\in(0,T)\times \mathcal{H}$ we define $E^{1,+} v(t,x)$ as $$\begin{array}{ll} E^{1,+}v(t,x)= \{ (q,p_1,p_2)\in \mathbb{R}\times D(A^*) \times\mathcal{H} : & \exists \varphi\in test1, \; g\in test2\; s.t.\\ & v-\varphi-g \text{ attains a local}\\ & \text{maximum at } (t,x),\\ & \partial_t(\varphi+g)(t,x)=q,\\ & D\varphi(t,x)=p_1, \;\; Dg(t,x)=p_2\\ & and \; v(t,x)= \varphi(t,x)+g(t,x) \} \end{array}$$ If we define $$E^{1,+}_1v(t,x) =\{(q,p)\in \mathbb{R}\times \mathcal{H} \; : \; p=p_1+p_2 \; with \; (q,p_1,p_2) \in E^{1,+}v(t,x) \}$$ then $E^{1,+}_1v(t,x) \subseteq D^{1,+}v(t,x)$ and in the finite dimensional case we have $E^{1,+}_1v(t,x) = D^{1,+}v(t,x)$. Here we have to use $E^{1,+}v(t,x)$ instead of $E^{1,+}_1v(t,x)$ because of the different roles of $g$ and $\varphi$. It is not clear if the sets $E^{1,+}v(t,x)$ and $E^{1,+}_1v(t,x)$ are convex. However if we took finite sums of functions $\eta(t)g_0(\|x\|)$ as $test2$ functions then they would be convex. All the results obtained are unchanged if we use the definition of viscosity solution with this enlarged class of $test2$ functions. A trajectory-strategy pair $\left(x(\cdot), u(\cdot) \right)$ will be called an [admissible couple]{} for $(t,x)$ if $u\in{\cal U}[t,T]$ and $x(\cdot)$ is the corresponding solution of the state equation (\[sydeterministicstate\]). A trajectory-strategy pair $\left(x^*(\cdot),u^*(\cdot) \right)$ will be called an [optimal couple]{} for $(t,x)$ if it is admissible for $(t,x)$ and if we have $$-\infty < J(t,x;u^*(\cdot))\leq J(t,x;u(\cdot))$$ for every admissible control $u(\cdot) \in {\cal U}[t,T]$. We can now state and prove the verification theorem. \[thdeterministicverification\] Let Hypotheses \[hpD2onb\], \[hpD3onLandh\] and \[D4deterministiccomparison\] hold. Let $v\in\mathcal{G}$ be a subsolution of the HJB equation (\[deterministicHJB\]) such that $$\label{terminalconditionvertheorem} v(T,x)=h(x) \;\;\; for\; all\; x\; in \; \mathcal{H}.$$ \(a) We have $v(t,x) \leq V(t,x) \leq J(t,x,u(\cdot))\;\; \forall(t,x) \in (0,T]\times\mathcal{H}, \;u(\cdot)\in\mathcal{U}[t,T]$. \(b) Let $(t,x)\in (0,T)\times H$ and let $(x_{t,x}(\cdot), u(\cdot))$ be an admissible couple at $(t,x)$. Assume that there exist $q\in L^1(t,T;\mathbb{R})$, $p_1\in L^1(t,T;D(A^*))$ and $p_2\in L^1(t,T;\mathcal{H})$ such that $$\label{condE} (q(s),p_1(s),p_2(s)) \in E^{1,+}v(s,x_{t,x}(s)) \; \;\; \text{for almost all } s\in (t,T)$$ and that $$\begin{gathered} \label{condmin} \int_t^T ({\left\langle}p_1(s) + p_2(s), b(s,x_{t,x}(s),u(s)) {\right\rangle}+ q(s) + {\left\langle}A^* p_1(s),x_{t,x}(s){\right\rangle}){\mathrm{d}}t \\ \leq \int_t^T - L(s,x_{t,x}(s),u(s)) {\mathrm{d}}s. $$ Then $(x_{t,x}(\cdot), u(\cdot))$ is an optimal couple at $(t,x)$ and $v(t,x)=V(t,x)$. Moreover we have equality in (\[condmin\]). It is tempting to try to prove, along the lines of Theorem 3.9, p.243 of [@YongZhou], that a condition like (\[condmin\]) can also be necessary if $v$ is a viscosity solution (or maybe simply a supersolution). However this is not an easy task: the main problem is that $E^{1,+}$ and the analogous object $E^{1,-}$ are fundamentally different so a natural generalization of a result like Theorem 3.9, p.243 of [@YongZhou] does not seem possible. Moreover our verification theorem has some drawbacks. Condition (\[condmin\]) implicitly implies that $<p_2(r),Ax_{t,x}(r)>=0$ a.e. if the trajectory is in the domain of $A$. This follows from the fact that we would then have an additional term $<p_2(r),Ax_{t,x}(r)>$ in the integrand of the middle line of (\[acbd\]) so (\[condmin\]) would also have to be an equality with this additional term. Therefore the applicability of the theorem is somehow limited as in practice (\[condmin\]) may be satisfied only if the function is “nice" (i.e. its superdifferential should really only consist of $p_1$). Still it applies in some cases where other results fail (see Remarks \[remliyo\] and \[rm:controesempio\]). Many issues are not fully resolved yet and we plan to work on them in the future. The first statement ($v\le V$) follows from Hypothesis \[D4deterministiccomparison\], it remains to prove second one. The function $$\left\lbrace \begin{array}{l} [t,T]\to \mathcal{H}\times\mathbb{R}\\ s \mapsto (b(s,x_{t,x}(s),u(s)), L(s,x_{t,x}(s),u(s)) \end{array} \right.$$ in view of Hypotheses \[hpD2onb\] and \[hpD3onLandh\] is in $L^1(t,T;\mathcal{H}\times\mathbb{R})$ (in fact it is bounded). So the set of the right-Lebesgue points of this function that in addition satisfy (\[condE\]) is of full measure. We choose $r$ to be a point in this set. We will denote $y= x_{t,x}(r)$. Consider now two functions $\varphi^{r,y}\in test1$ and $g^{r,y}\in test2$ such that (we will avoid the index $^{r,y}$ in the sequel) $v\leq \varphi +g$ in a neighborhood of $(r,y)$, $v(r,y) - \varphi(r,y) - g(r,y) =0$,$(\partial_t)(\varphi+g)(r,y))=q(r)$, $D\phi(r,y)=p_1(r)$ and $D g(r,y)=p_2(r)$. Then for $\tau\in(r,T]$ such that $(\tau-r)$ is small enough we have by Lemmas \[lemmaphi\] and \[lemmag\] $$\frac{v(\tau,x_{t,x}(\tau)) - v(r,y)}{\tau-r} \leq \frac{g(\tau,x_{t,x}(\tau)) - g(r,y) }{\tau-r} + \frac{\varphi(\tau,x_{t,x}(\tau)) - \varphi(r,y)}{\tau-r}$$ $$\begin{gathered} \leq g_t(r,y) + \frac{\int_r^\tau {\left\langle}D g(r,y) , b(r,y,u(s)) {\right\rangle}{\mathrm{d}}s}{\tau-r} \\ + \varphi_t(r,y) + \frac{\int_r^\tau {\left\langle}D\varphi(r,y) , b(r,y,u(s)) {\right\rangle}{\mathrm{d}}s}{\tau-r}+ {\left\langle}A^*D\varphi(r,y),y{\right\rangle}+ o(1).\end{gathered}$$ In view of the choice of $r$ we know that $$\frac{\int_r^\tau {\left\langle}D g(r,y) , b(r,y,u(s)) {\right\rangle}{\mathrm{d}}s}{\tau-r} \xrightarrow{\tau\to r} {\left\langle}D g(r,y) , b(r,y,u(r)) {\right\rangle}$$ and $$\frac{\int_r^\tau {\left\langle}D \varphi(r,y) , b(r,y,u(s)) {\right\rangle}{\mathrm{d}}s}{\tau-r} \xrightarrow{\tau\to r} {\left\langle}D \varphi(r,y) , b(r,y,u(r)) {\right\rangle}.$$ Therefore for almost every $r$ in $[t,T]$ we have $$\begin{gathered} \limsup_{\tau\downarrow r} \frac{v(\tau,x_{t,x}(\tau)) - v(r,x_{t,x}(r)))}{\tau-r}\\ \leq {\left\langle}D g(r,x_{t,x}(r)) + D \varphi(r,x_{t,x}(r)), b(r,x_{t,x}(r),u(r)){\right\rangle}\\ + g_t(r,x_{t,x}(r))+ \varphi_t(r,x_{t,x}(r)) +{\left\langle}A^* D \varphi(r,x_{t,x}(r)),x_{t,x}(r){\right\rangle}\\ = {\left\langle}p_1(r)+p_2(r), b(r,x_{t,x}(r),u(r)){\right\rangle}+ q(r) + {\left\langle}A^* p_1(r),x_{t,x}(r){\right\rangle}.\end{gathered}$$ We can then use Lemma \[lemmaYZ\] and (\[condmin\]) to obtain $$\begin{gathered} \label{acbd} v(T,x_{t,x}(T)) - v(t,x) \\ \leq \int_t^T ({\left\langle}p(r), b(r,x_{t,x}(r),u(r)){\right\rangle}+ q(r) + {\left\langle}A^* p_1(r),x_{t,x}(r){\right\rangle}){\mathrm{d}}r \\ \leq \int_t^T -L(r,x_{t,x}(r),u(r)) {\mathrm{d}}r.\end{gathered}$$ Thus, using (a), we finally arrive at $$\begin{gathered} V(T,x_{t,x}(T)) - V(t,x) = h(x_{t,x}(T)) - V(t,x) \leq h(x_{t,x}(T)) - v(t,x) \\ = v(T,x_{t,x}(T))-v(t,x) \leq \int_t^T -L(r,x_{t,x}(r),u(r)) {\mathrm{d}}r\end{gathered}$$ which implies that $(x_{t,x}(\cdot), u(\cdot))$ is an optimal pair and that $v(t,x)=V(t,x)$. \[remliyo\] In the book [@LiYong] (page 263, Theorem 5.5) the authors present a verification theorem (based on a previous result of [@CaFr2], see also [@CaFr1] for similar results) in which it is required that the trajectory of the system remains in the domain of $A$ a.e. for the admissible control $u(\cdot)$ in question. This is not required here and in fact this is not satisfied in the example of the next section. It is shown in [@LiYong] (under assumptions similar to Hypotheses \[hpD2onb\] and \[hpD3onLandh\]) that the couple $x(\cdot), u(\cdot))$ is optimal if and only if $$\begin{gathered} u(s) \in \bigg\lbrace u\in U \, : \, \lim_{\delta \to 0} \frac{V((s+\delta), x(s)+ \delta(Ax(s) + b(s,x(s),u)) ) - V(s,x(s)) }{\delta} \\ = -L(s,x(s),u) \bigg\rbrace\end{gathered}$$ for almost every $s\in[t,T]$, where $V$ is the value function. An example ---------- We present an example of a control problem for which the value function is a nonsmooth viscosity solution of the corresponding HJB equation, however we can apply our verification theorem. The problem can model a number of phenomena, for example in age-structured population models (see [@Iannelli95; @Anita01; @Iannelli06]), in population economics [@FeichtingerPrskwetzVeliov04], optimal technology adoption in a vintage capital context [@BarucciGozzi98; @BarucciGozzi01]. Consider the state equation $$\label{example1stateequation} \left\lbrace \begin{array}{l} \dot{x}(s) = Ax(s) + Ru(s)\\ x(t)=x \end{array} \right.$$ where$A$ is a linear, densely defined maximal dissipative operator in $\mathcal{H}$, $R$ is a continuous linear operator $R\colon \mathbb{R}\to \mathcal{H}$, so it is of the form $R\colon u\mapsto u \beta$ for some $\beta\in\mathcal{H}$. Let $B$ be an operator as in Section \[subsuper\] satisfying (\[bcond\]). We will be using the notation of Section \[subsuper\]. We will assume that $A^*$ has an eigenvalue $\lambda$ with an eigenvector $\alpha$ belonging to the range of $B$. We consider the functional to be minimized $$\label{example1costfunctional} J(x,u(\cdot))= \int_t^T -\left|\left\langle \alpha,x(s)\right\rangle \right| + \frac{1}{2}u(s)^2 {\mathrm{d}}s.$$ We define $$\bar\alpha(t){\stackrel{def}{=}}\int_t^T e^{(s-t)A^*} \alpha {\mathrm{d}}s$$ and we take $M{\stackrel{def}{=}}\sup_{t\in[0,T]} |{\left\langle}\bar\alpha(t), \beta {\right\rangle}|$. We consider as control set $U$ the compact subset of $\mathbb{R}$ given by $U=[-M-1, M+1]$. So we specify the general problem characterized by (\[deterministicstateequation\]) and (\[deterministiccostfunctional\]) taking $b(t,x,u)=Ru$, $L(t,x,u)= -\left|\left\langle \alpha,x(s)\right\rangle \right| + 1/2 u(t)^2$, $h=0$, $U=[-M-1, M+1]$. The HJB equation (\[deterministicHJB\]) becomes $$\label{example1HJB} \left\lbrace \begin{array}{l} v_t + {\left\langle}Dv, Ax {\right\rangle}-\left|\left\langle \alpha,x\right\rangle \right|+\inf_{u\in U} \left ( {\left\langle}u,R^*Dv{\right\rangle}_{\mathbb{R}} + \frac{1}{2} u^{2} \right)=0\\ v(T,x)=0 \end{array} \right.$$ Note that the operator $R^*\colon \mathcal{H} \to \mathbb{R}$ can be explicitly expressed using $\beta$ which was used to define the operator $R$: $R^*x=\left\langle\beta,x\right\rangle$. Now we observe that for $\left\langle \alpha,x\right\rangle<0$ (respectively $>0$) the HJB equation is the same as the one for the optimal control problem with the objective functional $\int_t^T \left\langle \alpha,x(s)\right\rangle + \frac{1}{2}u(s)^2 {\mathrm{d}}s$ (respectively $\int_t^T -\left\langle \alpha,x(s)\right\rangle + \frac{1}{2}u(s)^2 {\mathrm{d}}s$) and it is known in the literature (see [@FaggianGozzi] Theorem 5.5) that its solution is $$v_1(t,x)= \left\langle \bar\alpha(t),x \right\rangle - \int_t^T \frac{1}{2} \left( R^*\bar\alpha(s)\right)^2 {\mathrm{d}}s$$ (respectively $$v_2(t,x)= -\left\langle \bar\alpha(t),x \right\rangle - \int_t^T \frac{1}{2} \left( R^*\bar\alpha(s)\right)^2 {\mathrm{d}}s).$$ Note that on the separating hyperplane $\left\langle \alpha,x\right\rangle=0$ the two functions assume the same values. Indeed, since $\alpha$ an eigenvector for $A^*$, $$\bar\alpha(t) = G(t) \alpha$$ where $$G(t)= \int_t^T e^{\lambda(s-t)}{\mathrm{d}}s$$ So, if $\left\langle \alpha,x\right\rangle=0$, $${\left\langle}\bar \alpha (t) ,x {\right\rangle}=0\;\;\;\; \text{ for all $t\in [0,T]$}.$$ Therefore we can glue $v_1$ and $v_2$ writing $$W(t,x)=\left\{ \begin{array}{ll} v_1(t,x) & \hbox{if } \left\langle \alpha,x\right\rangle \le 0 \\ v_2(t,x) & \hbox{if } \left\langle \alpha,x\right\rangle>0 \end{array}\right.$$ It is easy to see that $W$ is continuous and concave in $x$. We claim that $W$ is a viscosity solution of (\[example1HJB\]). For $\left\langle \alpha,x\right\rangle<0$ and $\left\langle \alpha,x\right\rangle>0$ it follows from the fact that $v_1$ and $v_2$ are explicit regular solutions of the corresponding HJB equations. For the points $x$ where $\left\langle \alpha,x\right\rangle=0$ it is not difficult to see that $$\left \{ \begin{array}{l} D^{1,+} W(t,x)= \left \{\left(\frac{1}{2}\left( R^*\bar\alpha(t)\right)^2, \gamma G(t) \alpha \right) \; : \; \gamma \in [-1,1] \right \} \subseteq D(A^*)\\ D^{1,-} W(t,x)= \emptyset \end{array} \right .$$ So we have to verify that $W$ is a subsolution on $\left\langle \alpha,x\right\rangle=0$. If $W - \varphi - g$ attains a maximum at $(t,x)$ with $\left\langle \alpha,x\right\rangle=0$ we have that $p{\stackrel{def}{=}}(p_1 + p_2){\stackrel{def}{=}}D(\varphi + g)(t,x)\in \left \{ \gamma G(t) \alpha \; : \; \gamma \in [-1,1] \right \} \subseteq D(A^*)$. From the definition of test1 function $p_1=D\varphi(t,x)\in D(A^*)$ so $\eta(t)g_0'(|x|)\frac{x}{|x|}=p_2=Dg(t,x)\in D(A^*)$. $W(\cdot,x)$ is a $C^1$ function and then, recalling that ${\left\langle}\bar \alpha (t) ,x {\right\rangle}_t={\left\langle}G'(t) \alpha ,x {\right\rangle}=0$, we have $$\label{eq:exampleestimate0} \partial_t (\varphi +g)(t,x)=\partial_t W(t,x)=\frac{1}{2}\left( R^*\bar\alpha(t)\right)^2,$$ and for $p=\gamma \bar\alpha(t)$ we have $$\label{eq:exampleestimate1} \inf_{u\in U} \left ( {\left\langle}Ru,p{\right\rangle}+ \frac{1}{2} u^{2} \right) = - \frac{1}{2} \gamma^2 \left( R^*\bar\alpha(t)\right)^2$$ Moreover, recalling that $g_0'(|x|)\geq 0$ and $-A^*$ is monotone, we have $$\begin{gathered} \label{eq:exampleestimate2} {\left\langle}A^* p_1, x {\right\rangle}= {\left\langle}A^*(p-p_2), x {\right\rangle}= {\left\langle}A^* \gamma G(t) \alpha, x {\right\rangle}- \frac{g_0'(|x|)}{|x|} {\left\langle}A^* x, x {\right\rangle}\geq\\ \geq \gamma G(t) {\left\langle}A^* \alpha, x {\right\rangle}= 0\end{gathered}$$ So, by (\[eq:exampleestimate0\]), (\[eq:exampleestimate1\]) and (\[eq:exampleestimate2\]), $$\begin{gathered} \partial_t (\varphi +g)(t,x) + {\left\langle}A^* p_1 ,x{\right\rangle}- \left| \left\langle \alpha,x\right\rangle\right| + \\ +\inf_{u\in U} \left ( {\left\langle}Ru,D(\varphi +g)(t,x){\right\rangle}+ \frac{1}{2} u^{2} \right) \geq \frac{1}{2} (1-\gamma^2) \left( R^*\bar\alpha(s)\right)^2 \geq 0\end{gathered}$$ and so the claim in proved. It is easy to see that both $W$ and the value function $V$ for the problem are continuous on $[0,T]\times \mathcal{H}$ and moreover $\psi=W$ and $\psi=V$ satisfy $$|\psi(t,x)-\psi(t,y)|\le C\|x-y\|_{-1} \quad\hbox{for all}\,\,t\in[0,T], x,y\in \mathcal{H}$$ for some $C\ge 0$. In particular $W$ and $V$ have at most linear growth as $\|x\|\to\infty$. By Theorem \[thexistence\], the value function $V$ is a a viscosity solution of the HJB equation (\[example1HJB\]) in $(0,T]\times\mathcal{H}$. Moreover, since $\alpha=By$ for some $y\in \mathcal{H}$, comparison holds for equation (\[example1HJB\]) which yields $W=V$ on $[0,T]\times\mathcal{H}$. (Comparison theorem can be easily obtained by a modification of techniques of [@CL5] but we cannot refer to any result there since both $V$ and $W$ are unbounded. However the result follows directly from Theorem 3.1 together with Remark 3.3 of [@Kel]. The reader can also consult the proof of Theorem 4.4 of [@KeSw]. We point out that our assumptions are different from the assumptions of the uniqueness Theorem 4.6 of [@LiYong], page 250). Therefore we have an explicit formula for the value function $V$ given by $V(t,x)=W(t,x)$. We see that $V$ is differentiable at points $(t,x)$ if $\left\langle \alpha,x\right\rangle \ne 0$ and $$DV(t,x)=\left\lbrace \begin{array}{ll} \bar \alpha(t) & if \; \left\langle \alpha,x\right\rangle < 0\\ -\bar \alpha(t) & if \; \left\langle \alpha,x\right\rangle > 0 \end{array} \right.$$ and is not differentiable whenever $\left\langle \alpha,x\right\rangle = 0$. However we can apply Theorem \[thdeterministicverification\] and prove the following result. \[propqui\] The feedback map given by $$u^{op}(t,x) = \left\lbrace \begin{array}{ll} - \left\langle \beta,\bar\alpha(t) \right\rangle & if \; \left\langle \alpha,x\right\rangle \le 0\\ \left\langle \beta,\bar\alpha(t) \right\rangle & if \; \left\langle \alpha,x\right\rangle > 0 \end{array} \right.$$ is optimal. Similarly, also the feedback map $$\bar u^{op}(t,x) = \left\lbrace \begin{array}{ll} - \left\langle \beta,\bar\alpha(t) \right\rangle & if \; \left\langle \alpha,x\right\rangle < 0\\ \left\langle \beta,\bar\alpha(t) \right\rangle & if \; \left\langle \alpha,x\right\rangle \ge 0 \end{array} \right.$$ is optimal. Let $(t,x)\in (0,T]\times \mathcal{H}$ be the initial datum. If $\left\langle \alpha,x\right\rangle \le 0$, taking the control $-\left\langle \beta,\bar\alpha(t) \right\rangle$ the associated state trajectory is $$x^{op}(s)= e^{(s-t)A}x - \int_t^{s}e^{(s-r)A} R(\left\langle \beta,\bar\alpha(r) \right\rangle) {\mathrm{d}}r$$ and it easy to check that it satisfies $\left\langle \alpha,x^{op}(s) \right\rangle \le 0 $ for every $s\ge t$. Indeed, using the form of $R$ and the fact that $\alpha $ is eigenvector of $A^*$ we get $$\left\langle \alpha,x^{op}(s) \right\rangle = e^{\lambda(s-t)} \left\langle \alpha, x \right\rangle - \left\langle \alpha, \beta \right\rangle \int_t^{s}e^{\lambda(s-r)} \left\langle \beta,\bar\alpha(r) \right\rangle {\mathrm{d}}r$$ $$=e^{\lambda(s-t)}\left\langle \alpha, x \right\rangle - \left\langle \alpha, \beta \right\rangle^2 \int_t^{s}e^{\lambda(s-r)} G(r) {\mathrm{d}}r.$$ Similarly if $\left\langle \alpha,x\right\rangle > 0$, taking the control $\left\langle \beta,\bar\alpha(t) \right\rangle$ the associated state trajectory is $$x^{op}(s)= e^{(s-t)A}x + \int_t^{s}e^{(s-r)A} R(\left\langle \beta,\bar\alpha(r) \right\rangle) {\mathrm{d}}r$$ and it easy to check that it satisfies $\left\langle \alpha,x^{op}(s) \right\rangle > 0 $ for every $s\ge t$. We now apply Theorem \[thdeterministicverification\] taking $q(s)=\partial_t V(s, x^{op}(s))$, $$p_1(s)=\left\lbrace \begin{array}{ll} \bar \alpha(s) & if \; \left\langle \alpha,x^{op}(s)\right\rangle \le 0\\ -\bar \alpha(s) & if \; \left\langle \alpha,x^{op}(s)\right\rangle >0 \end{array} \right.$$ and $p_2(s)=0$. It is easy to see that $(q(s), p_1(s), p_2(s)) \in E^{1,+}V(s,x^{op}(s))$. The argument for $\bar u^{op}$ is completely analogous. We continue by giving a specific example of the Hilbert space $\mathcal{H}$, the operator $A$, and the data $\alpha$ and $\beta$. This example is related to the vintage capital problem in economics, see e.g. [@BarucciGozzi01; @BarucciGozzi98]. Let $\mathcal{H}=L^2(0,1)$. Let $\{e^{tA}; \; t \ge 0\}$ be the semigroup that, if we identify the points $0$ and $1$ of the interval $[0,1]$, “rotates” the function: $$e^{tA}f(s) = f(t+s - [t+s])$$ where $[\cdot]$ is the greatest natural number $n$ such that $n\leq t+s$. The domain of $A$ will be $$D(A)= \left\lbrace f\in W^{1,2}(0,1) \; : \; f(0)=f(1) \right\rbrace$$ and for all $f$ in $D(A)$ $A(f)(s) = \frac{{\mathrm{d}}}{{\mathrm{d}}s} f (s)$. We choose $\alpha$ to be the constant function equal to $1$ at every point of the interval $[0,1]$. (We can take for instance $B=(I-\Delta)^{-\frac{1}{2}}$.) Moreover we choose $\beta(s)=\chi_{[0,\frac{1}{2}]}(s) - \chi_{[0,\frac{1}{2}]}(s)$ ($\chi_{\Omega} $ is the characteristic function of a set $\Omega$). Consider an initial datum $(t,x)$ such that $\left\langle \alpha, x\right\rangle =0$. In view of Proposition \[propqui\] an optimal strategy $u^{op}$ is $$u^{op}(s)=-\left\langle \beta,\bar\alpha(s) \right\rangle =0$$ The related optimal trajectory is $$x^{op}(s)= e^{(s-t)A}y.$$ \[rm:controesempio\] We observe that, using such strategy, $\left\langle \alpha,x^{op}(t)\right\rangle = 0$ for all $s\geq t$. So the trajectory remains for a whole interval in a set in which the value function is not differentiable. Anyway, applying Theorem \[thdeterministicverification\], the optimality is proved. Moreover $x$ can be chosen out of the domain of $A$ and so the assumptions of the verification theorem given in [@LiYong] (page 263, Theorem 5.5) are not verified in this case. Sub- and superoptimality principles and construction of $\epsilon$-optimal controls {#subsuper} =================================================================================== Let $B$ be a bounded linear positive self-adjoint operator on $\mathcal{H}$ such that $A^*B$ bounded on $\mathcal{H}$ and let $c_0\leq 0$ be a constant such that $$\label{bcond} {\left\langle}(A^* B + c_0 B)x,x {\right\rangle}\leq 0 \;\;\;\;\;\; for \; all \; x\in\mathcal{H}.$$ Such an operator always exists [@Renardy95] and we refer to [@CL4] for various examples. Using the operator $B$ we define for $\gamma>0$ the space $\mathcal{H}_{-\gamma}$ to be the completion of $\mathcal{H}$ under the norm $$\|x\|_{-\gamma}=\|B^{\frac{\gamma}{2}}x\|.$$ We need to impose another set of assumptions on $b$ and $L$. \[hp:section4\] There exist a constant $K>0$ and a local modulus of continuity $\omega(\cdot,\cdot)$ such that: $$\|b(t,x,u)-b(s,y,u)\| \leq K \|x-y \|_{-1} + \omega(|t-s|, \|x\| \vee \|y\|)$$ and $$|L(t,x,u)-L(s,y,u)| \leq \omega( \|x-y \|_{-1} + |t-s|, \|x\| \vee \|y\|)$$ Let $m\geq 2$. Modifying slightly the functions introduced in [@CL5] we define for a function $w:(0,T)\times \mathcal{H}\to \mathbb{R}$ and $\epsilon,\beta,\lambda>0$ its sup- and inf-convolutions by $$w^{\lambda,\epsilon,\beta}(t,x)=\sup_{(s,y)\in(0,T)\times \mathcal{H}} \left\{w(s,y)-\frac{\|x-y\|_{-1}^2}{2\epsilon} -\frac{(t-s)^2}{2\beta}-\lambda e^{2mK(T-s)}\|y\|^m\right\},$$ $$w_{\lambda,\epsilon,\beta}(t,x)=\inf_{(s,y)\in(0,T)\times \mathcal{H}} \left\{w(s,y)+\frac{\|x-y\|_{-1}^2}{2\epsilon} +\frac{(t-s)^2}{2\beta}+\lambda e^{2mK(T-s)}\|y\|^m\right\}.$$ \[lem2\] Let $w$ be such that $$\label{aaa2} w(t,x)\leq C(1+\|x\|^k)\quad(\hbox{respectively,}\,\,\, w(t,x)\geq -C(1+\|x\|^k))$$ on $(0,T)\times \mathcal{H}$ for some $k\geq 0$. Let $m>k$. Then: - For every $R>0$ there exists $M_{R,\epsilon,\beta}$ such that if $v=w^{\lambda,\epsilon,\beta}$ (respectively, $v=w_{\lambda,\epsilon,\beta}$) then $$\label{aaa6} |v(t,x)-v(s,y)|\leq M_{R,\epsilon,\beta}(|t-s|+\|x-y\|_{-2})\quad \hbox{on}\,\,\,(0,T)\times B_R$$ - The function $$w^{\lambda,\epsilon,\beta}(t,x)+\frac{\|x\|_{-1}^2}{2\epsilon} +\frac{t^2}{2\beta}$$ is convex (respectively, $$w_{\lambda,\epsilon,\beta}(t,x)-\frac{\|x\|_{-1}^2}{2\epsilon} -\frac{t^2}{2\beta}$$ is concave). - If $v=w^{\lambda,\epsilon,\beta}$ (respectively, $v=w_{\lambda,\epsilon,\beta}$) and $v$ is differentiable at $(t,x)\in (0,T)\times B_R$ then $|v_t(t,x)|\leq M_{R,\epsilon,\beta}$, and $Dv(t,x)=Bq$, where $\|q\|\leq M_{R,\epsilon,\beta}$ **(i)** Consider the case $v=w^{\lambda, \epsilon, \beta}$. Observe first that if $\|x\|\le R$ then $$\begin{gathered} \label{eq:suponacompact} w^{\lambda, \epsilon, \beta}(t,x) =\\ = \sup_{(s,y)\in(0,T)\times \mathcal{H} , \; \|y\|\leq N} \left\{w(s,y)-\frac{\|x-y\|_{-1}^2}{2\epsilon} -\frac{(t-s)^2}{2\beta}-\lambda e^{2mK(T-s)}\|y\|^m\right\},\end{gathered}$$ where $N$ depends only on $R$ and $\lambda$. Now suppose $w^{\lambda, \epsilon, \beta}(t,x)\geq w^{\lambda, \epsilon, \beta}(s,y)$. We choose a small $\sigma>0$ and $(\tilde t, \tilde x)$ such that $$w^{\lambda, \epsilon, \beta}(t,x) \leq \sigma + w(\tilde t, \tilde x) - \frac{\| x-\tilde x\|^2_{-1}}{2\epsilon} - \frac{(t-\tilde t)^2}{2\beta} - \lambda e^{2mK(T-\tilde t)} \| \tilde x \|^m.$$ Then $$\begin{gathered} |w^{\lambda, \epsilon, \beta}(t,x)-w^{\lambda, \epsilon, \beta}(s,y)| \leq \sigma - \frac{\| x-\tilde x\|^2_{-1}}{2\epsilon} - \frac{(t-\tilde t)^2}{2\beta} + \frac{\|\tilde x - y \|^2_{-1}}{2\epsilon} + \frac{(\tilde t -s)^2}{2\beta} \\ \leq \sigma - \frac{{\left\langle}B(x-y), x+y{\right\rangle}}{2\epsilon} + \frac{{\left\langle}B(x-y), \tilde x {\right\rangle}}{\epsilon} + \frac{(2\tilde t -t -s)(t-s)}{2\beta} \\ \leq \frac{(2R+N)}{2\epsilon} \|B(x-y)\| + \frac{2T}{2\beta} |t-s| + \sigma\end{gathered}$$ and we conclude because of the arbitrariness of $\sigma$. The case of $w_{\lambda, \epsilon, \beta}$ is similar. **(ii)** It is a standard fact, see for example the Appendix of [@Userguide]. **(iii)** The fact that $|v_t(t,x)|\leq M_{R,\epsilon,\beta}$ is obvious. Moreover if $\alpha>0$ is small and $\|y\|=1$ then $$\alpha M_{R,\epsilon,\beta}\|y\|_{-2}\geq |v(t,x+\alpha y)-v(x)|= \alpha |{\left\langle}Dv(t,x),y{\right\rangle}|+o(\alpha)$$ which upon dividing by $\alpha$ and letting $\alpha\to 0$ gives $$|{\left\langle}Dv(t,x),y{\right\rangle}|\leq M_{R,\epsilon,\beta}\|y\|_{-2}$$ which then holds for every $y\in \mathcal{H}$. This implies that ${\left\langle}Dv(t,x),y{\right\rangle}$ is a bounded linear functional in $\mathcal{H}_{-2}$ and so $Dv(t,x)=Bq$ for some $q\in \mathcal{H}$. Since $|{\left\langle}q,By{\right\rangle}|\leq M_{R,\epsilon,\beta}\|By\|$ we obtain $\|q\|\leq M_{R,\epsilon,\beta}$. \[lem1\] Let Hypotheses \[hpD2onb\], \[hpD3onLandh\] and \[hp:section4\] be satisfied. Let $w$ be a locally bounded viscosity subsolution (respectively, supersolution) of (\[deterministicHJB\]) satisfying (\[aaa2\]). Let $m>k$. Then for every $R,\delta>0$ there exists a non-negative function $\gamma_{R,\delta}(\lambda,\epsilon,\beta)$, where $$\label{aaa3} \lim_{\lambda\to 0}\limsup_{\epsilon\to 0}\limsup_{\beta\to 0} \gamma_{R,\delta}(\lambda,\epsilon,\beta)=0,$$ such that $w^{\lambda,\epsilon,\beta}$ (respectively, $w_{\lambda,\epsilon,\beta}$) is a viscosity subsolution (respectively, supersolution) of $$\label{aaa4} v_t(t,x) +{\left\langle}Dv(t,x), Ax {\right\rangle}+ H(t,x,Dv(t,x))= -\gamma_{R,\delta}(\lambda,\epsilon,\beta)\quad\hbox{in}\,\,\, (\delta,T-\delta)\times B_R$$ (respectively, $$\label{aaa5} v_t(t,x) +{\left\langle}Dv(t,x), Ax {\right\rangle}+ H(t,x,Dv(t,x))= \gamma_{R,\delta}(\lambda,\epsilon,\beta)\quad\hbox{in}\,\,\, (\delta,T-\delta)\times B_R)$$ for $\beta$ sufficiently small (depending on $\delta$). The proof is similar to the proof of Proposition 5.3 of [@CL5]. We notice that $w^{\lambda,\epsilon,\beta}$ is bounded from above. Let $(t_0,x_0)\in (\delta, T-\delta)\times \mathcal{H}$ be a local maximum of $w^{\lambda,\epsilon,\beta}-\phi-g$. We can assume that the maximum is global and strict (see Proposition 2.4 of [@CL5]) and that $w^{\lambda,\epsilon,\beta}-\phi-g\to -\infty$ as $\|x\|\to\infty$ uniformly in $t$. In view of these facts and (\[eq:suponacompact\]) we can choose $S>2\|x_0\|$, depending on $\lambda$ such that, for all $\|x\|+\|y\| >S-1$ and $s,t\in (0,T)$, $$\begin{gathered} \label{eq:lessthen-1} w(s,y) - \frac{1}{2\epsilon} \|(x-y)\|_{-1}^2 - \frac{(t-s)^2}{2\beta} - \lambda e^{2mK(T-s)}\|y\|^m - \phi(t,x) - g(t,x) \\ \leq w(t_0,x_0) - \lambda e^{2mK(T-t_0)}\|x_0\|^m -\phi(t_0,x_0) - g(t_0,x_0) -1.\end{gathered}$$ We can then use a perturbed optimization technique of [@CL5] (see page 424 there) which is a version of the Ekeland-Lebourg Lemma [@EkelandLebourg77] to obtain for every $\alpha>0$ elements $p,q\in\mathcal{H}$ and $a,b\in\mathbb{R}$ with $\|p\|, \|q\|\leq \alpha$ and $|a|,|b|\leq \alpha$ such that the function $$\begin{gathered} \psi(t,x,s,y) {\stackrel{def}{=}}w(s,y) - \frac{1}{2\epsilon} \|(x-y)\|_{-1}^2 - \frac{(t-s)^2}{2\beta} - \lambda e^{2mK(T-s)}\|y\|^m \\ -g(t,x)-\phi(t,x) - {\left\langle}Bp,y {\right\rangle}-{\left\langle}Bq, x {\right\rangle}- at - bs\end{gathered}$$ attains a local maximum $(\bar t, \bar x, \bar s, \bar y)$ over $[\delta/2,T-\delta/2]\times B_S \times [\delta/2,T-\delta/2]\times B_S$. It follows from (\[eq:lessthen-1\]) that if $\alpha$ is sufficiently small then $\|\bar x\|, \|\bar y\| \leq S-1$. By possibly making $S$ bigger we can assume that $(0,T)\times B_S$ contains a maximizing sequence for $$\sup_{(s,y)\in(0,T), \; \|y\|\leq N} \left\{w(s,y)-\frac{\|x_0-y\|_{-1}^2}{2\epsilon} -\frac{(t_0-s)^2}{2\beta}-\lambda e^{2mK(T-s)}\|y\|^m\right\}.$$ Then $$\psi(\bar t, \bar x, \bar s, \bar y) \geq w^{\lambda,\epsilon,\beta} (t_0, x_0) - \phi(t_0, x_0) - g(t_0, x_0) -C\alpha$$ where the constant $C$ does not depend on $\alpha >0$, and $$\psi(\bar t, \bar x, \bar s, \bar y) \leq w^{\lambda,\epsilon,\beta} (\bar t, \bar x) - \phi(\bar t, \bar x) - g(\bar t, \bar x) + C\alpha.$$ Therefore, since $(t_0, x_0)$ is a strict maximum, we have that $(\bar t, \bar x)\xrightarrow{\alpha\downarrow 0} (t_0, x_0)$ and so for small $\alpha$ $\bar t\in(\delta , T-\delta)$. It then easily follows that if $\beta$ is big enough (depending on $\lambda$ and $\delta$) then $\bar s \in (\delta/2,T-\delta/2)$. Moreover, standard arguments (see for instance [@I]) give us $$\label{eq:stimasuepsilon} \lim_{\beta\to 0}\limsup_{\alpha\to 0} \frac{|\bar s -\bar t|^2}{2\beta} =0,$$ $$\label{eq:stimasuepsilon1} \lim_{\epsilon\to 0}\limsup_{\beta\to 0}\limsup_{\alpha\to 0} \frac{|\bar x -\bar y|^2_{-1}}{2\epsilon}=0.$$ We can now use the fact that $w$ is a subsolution to obtain $$\begin{gathered} -\frac{(\bar t-\bar s)}{\beta} - 2\lambda mKe^{2mK(T- \bar s)}\|\bar y\|^m + b - \frac{{\left\langle}A^*B(\bar x - \bar y), \bar y {\right\rangle}}{\epsilon} + {\left\langle}A^*Bp, \bar y {\right\rangle}\\ + H \left (\bar s, \bar y, \frac{1}{\epsilon}B(\bar y - \bar x) + \lambda m e^{2mK(T-\bar s)} \|y\|^{m-1} \frac{y}{\|y\|} + Bp \right ) \geq 0.\end{gathered}$$ We notice that $$-\frac{(\bar t-\bar s)}{\beta}=\phi_t(\bar t , \bar x) + g_t(\bar t, \bar x) +a$$ and $$\frac{1}{\epsilon}B(\bar y - \bar x) = D\phi (\bar t, \bar x) + Dg(\bar t, \bar x) + Bq$$ which in particular implies that $Dg(\bar t, \bar x)\in D(A^*)$, i.e. $\bar x\in D(A^*)$, and so it follows that ${\left\langle}A^*\bar x,Dg(\bar t, \bar x){\right\rangle}\le 0$. Therefore using this, the assumptions on $b$ and $L$, and (\[eq:stimasuepsilon\]) and (\[eq:stimasuepsilon1\]) we have $$\begin{gathered} \phi_t(\bar t , \bar x) + g_t(\bar t, \bar x) + {\left\langle}\bar x, A^*D\phi(\bar t,\bar x){\right\rangle}+ H \left (\bar t, \bar x, D\phi (\bar t, \bar x) + Dg(\bar t, \bar x) \right ) \\ \geq 2\lambda mKe^{2mK(T- \bar s)}\|\bar y\|^m - {\left\langle}A^*Bp, \bar y {\right\rangle}-a -b\\ - {\left\langle}(\bar y - \bar x), A^*\frac{1}{\epsilon}B(\bar y - \bar x){\right\rangle}- {\left\langle}\bar x, A^*Dg(\bar t, \bar x) + A^*Bq){\right\rangle}\\ + H \left (\bar t, \bar x, \frac{1}{\epsilon}B(\bar y - \bar x) -Bq \right ) - H \left (\bar s, \bar y, \frac{1}{\epsilon}B(\bar y - \bar x) + \lambda m e^{2mK(T-\bar s)} \|y\|^{m-1} \frac{y}{\|y\|} \right ) \\ \geq 2\lambda mKe^{2mK(T- \bar s)}\|\bar y\|^m -C_{\lambda,\epsilon}\alpha +\frac{c_0}{\epsilon} \|\bar x - \bar y \|_{-1}^2 \\ -K\|\bar x - \bar y \|_{-1}\frac{\|B(\bar x - \bar y) \|}{\epsilon} -\gamma_{\lambda,\epsilon}(|\bar t-\bar s|) -\lambda m(M+K\|\bar y\|)e^{2mK(T- \bar s)}\|\bar y\|^{m-1}\\ \geq -C_{\lambda,\epsilon}\alpha -\gamma(\lambda,\epsilon,\beta,\alpha)\end{gathered}$$ for some $\gamma(\lambda,\epsilon,\beta,\alpha)$ such that $$\lim_{\lambda\to 0}\limsup_{\epsilon\to 0}\limsup_{\beta\to 0} \limsup_{\alpha\to 0} \gamma(\lambda,\epsilon,\beta,\alpha)=0.$$ We obtain the claim by letting $\alpha\to 0$. The proof for $w_{\lambda,\beta,\epsilon}$ is similar. Similar argument would also work for problems with discounting if $w$ was uniformly continuous in $|\cdot|\times\|\cdot\|_{-1}$ norm uniformly on bounded sets of $(0,T)\times\mathcal{H}$. Moreover in some cases the function $\gamma_{R,\delta}$ could be explicitly computed. For instance if $w$ is bounded and $$\label{aaa12} |w(t,x)-w(s,y)|\leq \sigma(\|x-y\|_{-1})+\sigma_1(|t-s|;\|x\|\vee\|y\|)$$ for $t,s\in(0,T), \|x\|,\|y\|\in \mathcal{H}$, we can replace $\lambda e^{2mK(T- \bar s)}\|\bar y\|^m$ by $\lambda\mu(y)$ for some radial nondecreasing function $\mu$ such that $D\mu$ is bounded and $\mu(y)\to+\infty$ as $\|y\|\to\infty$ (see [@CL5], page 446). If we then replace the order in which we pass to the limits we can get an explicit (but complicated) form for $\gamma_{R,\delta}$ satisfying $$\lim_{\epsilon\to 0}\limsup_{\lambda\to 0}\limsup_{\beta\to 0} \gamma_{R,\delta}(\epsilon,\lambda,\beta)=0.$$ The proofs of Theorem 3.7 and Proposition 5.3 in [@CL5] can give hints how to do this. \[lem3\] Let the assumptions of Lemma \[lem1\] be satisfied. Then: - If $(a,p)\in D^{1,-}w^{\lambda,\epsilon,\beta}(t,x)$ for $(t,x)\in (\delta,T-\delta)\times B_R$ then $$\label{eq:lem3a} a +{\left\langle}A^*p, x {\right\rangle}+ H(t,x,p) \geq -\gamma_{R,\delta}(\lambda,\epsilon,\beta)$$ for $\beta$ sufficiently small. - If in addition $H(s,y,q)$ is weakly lower-semicontinuous with respect to the $q$-variable and $(a,p)\in D^{1,+}w_{\lambda,\epsilon,\beta}(t,x)$ for $(t,x)\in (\delta,T-\delta)\times B_R$ is such that $Dw_{\lambda,\epsilon,\beta}(t_n,x_n)\rightharpoonup p$ for some $(t_n,x_n)\to (t,x)$, $(t_n,x_n)\in(\delta,T-\delta)\times B_R$, then $$a +{\left\langle}A^*p, x {\right\rangle}+ H(t,x,p) \leq \gamma_{R,\delta}(\lambda,\epsilon,\beta)$$ for $\beta$ sufficiently small. The Hamiltonian $H$ is weakly lower-semicontinuous with respect to the $q$-variable for instance if $U$ is compact. To see this we observe that thanks to the compactness of $U$ the infimum in the definition of the Hamiltonian is a minimum. Let now $q_n\rightharpoonup q$ and let $$H(s,y,q_n)= {\left\langle}q_n, b(s, y, u_n){\right\rangle}+ L(s, y, u_n)$$ for some $u_n\in U$. Passing to a subsequence if necessary we can assume that $u_n\longrightarrow \bar u$, and then passing to the limit in the above expression we obtain $$\liminf_{n\to\infty}H(s,y,q_n)= {\left\langle}q, b(s, y, \bar u){\right\rangle}+ L(s, y,\bar u) \geq H(s,y,q).$$ We also remark that since $H$ is concave in $q$ it is weakly upper-semicontinuous in $q$. Therefore in (b) the Hamiltonian $H$ is assumed to be weakly continuous in $q$. [*(of Lemma \[lem3\])*]{} Recall first that for a convex/concave function $v$ its sub/super-differential at a point $(s,z)$ is equal to $$\overline{\hbox{conv}}\{((a,p):v_t(s_n,z_n)\to a, Dv(s_n,z_n)\rightharpoonup p, s_n\to s, z_n\to z\}$$ (see [@Pr], page 319). **(a)** **Step 1**: Denote $v=w^{\lambda,\epsilon,\beta}$. At points of differentiability, it follows from Lemma \[lem2\](iii) and the “semiconvexity" (see Lemma \[lem2\](ii)) of $w^{\lambda,\epsilon,\beta}$ that there exists a test1 function $\varphi$ such that $v-\varphi$ has a local maximum and the result then follows from Lemma \[lem1\]. **Step 2**: Consider first the case $Dv(t_n,x_n)\rightharpoonup p$ with $(t_n,x_n)\to (t,x)$. From Lemma \[lem2\] (iii) $Dv(t_n,x_n)=Bq_n$ with $\|q_n\|\leq M_{R,\epsilon,\beta}$, so, it is always possible to extract a subsequence $q_{n_k}\rightharpoonup q$ for some $q\in \mathcal{H}$. Then $Dv(t_{n_k},x_{n_k})=Bq_{n_k}\rightharpoonup Bq$ and $Bq=p$. Therefore $${\left\langle}A^*B q_{n_k},x_{n_k} {\right\rangle}= {\left\langle}q_{n_k},(A^*B)^*x_{n_k} {\right\rangle}\longrightarrow {\left\langle}q,(A^*B)^*x {\right\rangle}= {\left\langle}A^*B q,x {\right\rangle}= {\left\langle}A^*p,x {\right\rangle}$$ Moreover, since $H$ is concave in $p$ it is weakly upper-semicontinuous so we have $$H(t,x,p)\geq \limsup_{k\to +\infty} H(t_{n_k},x_{n_k},Dv(t_{n_k},x_{n_k}))$$ and we conclude from Step 1. **Step 3**: If $p$ is a generic point of $\overline{\hbox{conv}}\{p:Dv(t_n,x_n)\rightharpoonup p, (t_n,x_n)\to (t,x)\}$, i.e. $p=\lim_{n\to\infty} \sum_{i=1}^n\lambda_i^nBq_i^n$, where $\sum_{i=1}^n\lambda_i^n=1, \|q_i^n\| \leq M_{R,\epsilon,\beta}$, and the $Bq_i^n$ are weak limits of gradients. By passing to a subsequence if necessary we can assume that $\sum_{i=1}^n\lambda_i^nq_i^n\rightharpoonup q$ and $p=Bq$. But then $${\left\langle}A^*\left(\sum_{i=1}^n\lambda_i^nBq_i^n\right), x_n{\right\rangle}={\left\langle}A^*B\left(\sum_{i=1}^n\lambda_i^nq_i^n\right), x_n{\right\rangle}\to {\left\langle}A^*Bq, x {\right\rangle}={\left\langle}A^*p, x {\right\rangle}$$ as $n\to\infty$. The result now follows from Step 2 and the concavity of $$p\mapsto {\left\langle}A^*p, x {\right\rangle}+ H(t,x,p).$$ **(b)** As in (a) at the points of differentiability the claim follows from Lemmas \[lem2\] and \[lem1\]. Denote $v=w_{\lambda, \epsilon, \beta}$. If $Dv(t_n, x_n)\rightharpoonup p$ for some $(t_n,x_n)\to (t,x)$, $(t_n,x_n)\in(\delta,T-\delta)\times B_R$ we have that $$\label{aabba} v_t(t_n, x_n)+{\left\langle}A^*Dv(t_n, x_n), x_n{\right\rangle}+ H(t_n,x_n,Dv(t_n, x_n)) \leq \gamma_{R,\delta}(\lambda,\epsilon,\beta).$$ Observing as in Step 2 of (a) that $${\left\langle}A^*Dv(t_n, x_n), x_n{\right\rangle}\to {\left\langle}A^* p ,x {\right\rangle}$$ we can pass to the limit in (\[aabba\]), using the weak lower semicontinuity of $H$ with respect to the third variable, to get $$a+ {\left\langle}A^* p ,x {\right\rangle}+ H(t, x, p) \leq \gamma_{R,\delta}(\lambda,\epsilon,\beta).$$ \[th1\] Let the assumptions of Lemma \[lem1\] be satisfied and let $w$ be a function such that for every $R>0$ there exists a modulus $\sigma_R$ such that $$\label{aaa1} |w(t,x)-w(s,y)|\leq \sigma_R(|t-s|+\|x-y\|_{-1})\quad\hbox{for}\,\,\, t,s\in(0,T), \|x\|,\|y\|\leq R.$$ Then: - If $w$ is a viscosity subsolution of (\[deterministicHJB\]) satisfying (\[aaa2\]) for subsolutions then for every $0<t<t+h<T$, $x\in \mathcal{H}$ $$\label{aaa7} w(t,x)\leq \inf_{u(\cdot) \in \mathcal{U}[t,T]} \left\{\int_t^{t+h} L(s,x(s),u(s)) {\mathrm{d}}s +w(t+h,x(t+h))\right\}.$$ - Assume in addition that $H(s,y,q)$ is weakly lower-semicontinuous in $q$ and that for every $(t,x)$ there exists a modulus $\omega_{t,x}$ such that $$\label{aaa8} \|{x}_{t,x}(s_2)-{x}_{t,x}(s_1)\|\leq \omega_{t,x}(s_2-s_1)$$ for all $t\leq s_1\leq s_2\leq T$ and all $u(\cdot)\in \mathcal{U}[t,T]$, where ${x}_{t,x}(\cdot)$ is the solution of (\[sydeterministicstate\]). If $w$ is a viscosity supersolution of (\[deterministicHJB\]) satisfying (\[aaa2\]) for supersolutions then for every $0<t<t+h<T, x\in H$, and $\nu>0$ there exists a piecewise constant control $u_{\nu}\in\mathcal{U}[t,T]$ such that $$\label{aaa9} w(t,x)\geq \int_t^{t+h} L(s,x(s),u_{\nu}(s)) {\mathrm{d}}s +w(t+h,x(t+h))-\nu.$$ In particular we obtain the superoptimality principle $$\label{aaa10} w(t,x)\geq \inf_{u(\cdot) \in \mathcal{U}[t,T]} \left\{\int_t^{t+h} L(s,x(s),u(s)) {\mathrm{d}}s +w(t+h,x(t+h))\right\}$$ and if $w$ is the value function $V$ we have existence (together with the explicit construction) of piecewise constant $\nu$-optimal controls. We will only prove $(b)$ as the proof of $(a)$ follows the same strategy after we fix any control $u(\cdot)$ and is in fact much easier. We follow the ideas of [@sw] (that treats the finite dimensional case). **Step 1**. Let $n\geq 1$. We approximate $w$ by $w_{\lambda,\epsilon,\beta}$ with $m>k$. We notice that for any $u(\cdot)$ if $x_{t,x}(\cdot)$ is the solution of (\[sydeterministicstate\]) then $$\sup_{t\le s\le T}\|x_{t,x}(s)\|\le R=R(T,\|x\|).$$ **Step 2**. Take any $(a,p)\in D^{1,+}w_{\lambda,\epsilon,\beta}(t,x)$ as in Lemma \[lem3\]$(b)$ (i.e. $p$ is the weak limit of derivatives nearby). Such elements always exist because $w_{\lambda,\epsilon,\beta}$ is “semiconcave". Then we choose $u_1\in U$ such that $$\label{aaa11} a +{\left\langle}A^*p, x {\right\rangle}+ {\left\langle}p, b(t,x,u_1) {\right\rangle}+ L(t,x,u_1) \leq \gamma_{R,\delta}(\lambda,\epsilon,\beta)+\frac{1}{n^2}.$$ By the “semiconcavity" of $w_{\lambda,\epsilon,\beta}$ $$\label{aaa121} w_{\lambda,\epsilon,\beta}(s,y)\leq w_{\lambda,\epsilon,\beta}(t,x) +a(s-t)+{\left\langle}p, y-x {\right\rangle}+\frac{\|x-y\|_{-1}^2}{2\epsilon} +\frac{(t-s)^2}{2\beta}.$$ But the right hand side of the above inequality is a test1 function so if $s\geq t$ and $x(s)= x_{t,x}(s)$ with constant control $u(s)=u_1$, we can use (\[eq:explicitphi\]) and write $$\begin{gathered} \label{aaa13} \bigg | \frac{a(s-t) + {\left\langle}p, x(s) - x {\right\rangle}+ \frac{\| x(s) - x \|_{-1}^2}{2\epsilon} + \frac{(s-t)^2}{2\beta}}{s-t} \\ - \left ( a+ {\left\langle}p, b(t,x,u_1) {\right\rangle}+ {\left\langle}A^* p , x {\right\rangle}\right ) \bigg | \\ \leq\frac{|t-s|}{2\beta} + \left | \frac{\int_t^s {\left\langle}A^*p, x(r) - x {\right\rangle}{\mathrm{d}}r}{s-t} \right | \\ + \left| \frac{\int_t^s {\left\langle}p, b(r,x(r),u_1) - b(t,x,u_1) {\right\rangle}{\mathrm{d}}r}{s-t} \right | + \left | \frac{\int_t^s {\left\langle}A^*B(x(r) -x), x(r) {\right\rangle}{\mathrm{d}}r}{\epsilon(s-t)} \right | \\ + \left | \frac{\int_t^s {\left\langle}B(x(r) -x), b(r,x(r),u_1) {\right\rangle}{\mathrm{d}}r} {\epsilon(s-t)} \right | \\ \leq \omega_{t,x}' (|s-t| + \sup_{t\le r\le s}\|x(r) - x\|) \leq \tilde\omega_{t,x}(s-t)\end{gathered}$$ for some moduli $\omega'_{t,x}$ and $\tilde\omega_{t,x}$ that depend on $(t,x),\epsilon,\beta$ but not on $u_1$. We can now use (\[aaa11\]), (\[aaa121\]) and (\[aaa13\]) to estimate $$\begin{gathered} \label{wzwza} \frac{w_{\lambda,\epsilon,\beta}(t+\frac{h}{n},x(t+\frac{h}{n}))- w_{\lambda,\epsilon,\beta}(t,x)}{h/n} \\ \leq \tilde\omega_{t,x}\left(\frac{h}{n}\right) + \gamma_{R,\delta}(\lambda,\epsilon,\beta)+\frac{1}{n^2}- L(t,x,u_1)\end{gathered}$$ **Step 3**. Denote $t_i=t+\frac{(t-1)h}{n}$ for $i=1,...,n$. We now repeat the above procedure starting at $x(t_2)$ to abtain $u_2$ satisfying (\[wzwza\]) with $(t_2,x(t_2))$ replaced by $(t_3,x(t_3))$, $(t,x)=(t_1,x(t_1))$ replaced by $(t_2,x(t_2))$, and $u_1$ replaced by $u_2$. After $n$ iterations of this process we obtain a piecewise constant control $u^{(n)},$ where $u^{(n)}(s)=u_i$ if $s\in [t_i,t_{i+1})$. Then if $x(r)$ solves (\[sydeterministicstate\]) with the control $u^{(n)}$ we have $$\begin{gathered} \frac{w_{\lambda,\epsilon,\beta}(t+{h},x(t+{h}))- w_{\lambda,\epsilon,\beta}(t,x)}{h/n}\\ \leq \tilde\omega_{t,x}\left(\frac{h}{n}\right)n + \gamma_{R,\delta}(\lambda,\epsilon,\beta)n+\frac{n}{n^2}- \sum_{i=1}^n L(t_{i-1},x(t_{i-1}),u_i).\nonumber\end{gathered}$$ We remind that (\[aaa8\]) is needed here to guarantee that $\sup_{t_{i-1}\le r\le t_i}\|x(r)-x(t_{i-1})\|$ is independent of $u_i$ and $x(t_{i-1})$ and depends only on $x$ and $t$. We then easily obtain $$\begin{gathered} {w_{\lambda,\epsilon,\beta}(t+{h},x(t+{h}))- w_{\lambda,\epsilon,\beta}(t,x)} \\ \leq \tilde\omega_{t,x}\left(\frac{h}{n}\right)h + \gamma_{R,\delta}(\lambda,\epsilon,\beta)h +\frac{h}{n^2}- \int_t^{t+h} L(r,x(r),u^{(n)}) {\mathrm{d}}r + \tilde\omega'_{t,x} \left(\frac{h}{n}\right)h\end{gathered}$$ for some modulus $\tilde\omega'_{t,x}$, where we have used Hypothesis \[hp:section4\] and (\[aaa8\]) to estimate how the sum converges to the integral. We now finally notice that it follows from (\[aaa1\]) that $$|w_{\lambda,\epsilon,\beta}(s,y)-w(s,y)|\leq \tilde\sigma_R(\lambda + \epsilon + \beta;R) \quad\hbox{for}\,\,\, s\in(\delta,T-\delta),\|y\|\leq R,$$ where the modulus $\tilde\sigma_R$ can be explicitly calculated from $\sigma_R$. Therefore, choosing $\beta,\lambda, \epsilon$ small and then $n$ big enough, and using (\[aaa3\]), we arrive at (\[aaa9\]). We show below one example when condition (\[aaa8\]) is satisfied. Condition (\[aaa8\]) holds for example if $A=A^*$, it generates a differentiable semigroup, and $\|Ae^{tA}\|\le C/t^\delta$ for some $\delta<2$. Indeed under these assumptions, if $u(\cdot)\in \mathcal{U}[t,T]$ and writing $x(s)=x_{t,x}(s)$, we have $$\|(A+I)^{\frac{1}{2}}x(s)\|\leq \|(A+I)^{\frac{1}{2}}e^{(s-t)A}x\| +\int_t^s\|(A+I)^{\frac{1}{2}}e^{(s-\tau)A}b(\tau,x(\tau),u(\tau))\|d\tau$$ However for every $y\in H$ and $0\leq \tau\leq T$ $$\|(A+I)^{\frac{1}{2}}e^{\tau A}y\|^2\leq \|(A+I)e^{\tau A}y\| \; \|y\| \leq \frac{C_1}{\tau^{\delta}}\|y\|^2.$$ This yields $$\|(A+I)^{\frac{1}{2}}e^{\tau A}\|\leq \frac{\sqrt{C_1}} {\tau^{\frac{\delta}{2}}}$$ and therefore $$\|(A+I)^{\frac{1}{2}}x(s)\|\leq C_2\left(\frac{1}{(s-t)^{\frac{\delta}{2}}} +(s-t)^{1-\frac{\delta}{2}}\right)\leq \frac{C_3}{(s-t)^{\frac{\delta}{2}}}.$$ We will first show that for every $\epsilon>0$ there exists a modulus $\sigma_\epsilon$ (also depending on $x$ but independent of $u(\cdot)$) such that $\|e^{(s_2-s_1)A}x(s_1)-x(s_1)\| \leq \sigma_\epsilon(s_2-s_1)$ for all $t+\epsilon\le s_1 < s_2\leq T$. This is now rather obvious since $$e^{(s_2-s_1)A}x(s_1)-x(s_1)=\int_0^{s_2-s_1}Ae^{sA}x(s_1)ds$$ $$= \int_0^{s_2-s_1}(A+I)^{\frac{1}{2}}e^{sA}(A+I)^{\frac{1}{2}}x(s_1)ds -\int_0^{s_2-s_1}e^{sA}x(s_1)ds$$ and thus $$\|e^{(s_2-s_1)A}x(s_1)-x(s_1)\|\leq \|(A+I)^{\frac{1}{2}}x(s_1)\| \int_0^{s_2-s_1}\frac{\sqrt{C_1}}{s^{\frac{\delta}{2}}}ds +(s_2-s_1)\|x(s_1)\|$$ $$\leq \frac{C_4}{\epsilon^{\frac{\delta}{2}}}(s_2-s_1)^{1-\frac{\delta}{2}} +C_5(s_2-s_1).$$ We also notice that there exists a modulus $\sigma$, depending on $x$ and independent of $u(\cdot)$, such that $$\|x(s)-x\|\le \sigma(s-t).$$ Let now $t\le s_1 <s_2\leq T$. Denote $\bar s=\max(s_1,t+\epsilon)$. If $s_2\le t+\epsilon$ then $$\|x(s_2)-x(s_1)\|\le 2\sigma(\epsilon).$$ Otherwise $$\begin{gathered} \|x(s_2)-x(s_1)\|\le 2\sigma(\epsilon)+ \|x(s_2)-x(\bar s)\| \\ \le 2\sigma(\epsilon)+ \|e^{(s_2-\bar s)A}x(s_1)-x(\bar s)\| +\int_{\bar s}^{s_2}\|e^{(s_2-\tau)A}b(\tau,x(\tau),u(\tau))\|d\tau \\ \le 2\sigma(\epsilon)+\sigma_\epsilon(s_2-s_1)+C_4(s_2-s_1)\end{gathered}$$ for some constant $C_4$ independent of $u(\cdot)$. Therefore (\[aaa8\]) is satisfied with the modulus $$\omega_{t,x}(\tau)=\inf_{0<\epsilon<T-t} \left\{2\sigma(\epsilon)+\sigma_\epsilon(\tau)+C(\tau)\right\}.$$ [99]{} S. Ani[ţ]{}a, [*Analysis and control of age-dependent population dynamics*]{}, Kluwer Academic Publishers, Dordrecht, (2001). , [*Optimal feedback controls for a class of nonlinear distributed parameter systems*]{} [SIAM J. Control Optim.]{} 21 (1983), no. 6, 871–894. , [*Hamilton-Jacobi equations and nonlinear control problems*]{}, [J. Math. Anal. 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[^1]: DPTEA, Università *LUISS - Guido Carli* Roma and School of Mathematics and Statistics, UNSW, Sydney e-mail: gfabbri@luiss.it, G.Fabbri was supported by the ARC Discovery project DP0558539. [^2]: Dipartimento di Scienze Economiche ed Aziendali, Università *LUISS - Guido Carli* Roma, e-mail: fgozzi@luiss.it [^3]: School of Mathematics, Georgia Institute of Technology Atlanta, GA 30332, U.S.A., e-mail: swiech@math.gatech.edu. A. Świȩch was supported by NSF grant DMS 0500270.

--- abstract: 'The Cylindrical Algebraic Decomposition (CAD) algorithm is a comprehensive tool to perform quantifier elimination over real closed fields. CAD has doubly exponential running time, making it infeasible for practical purposes. We propose to use the notions of clause normal forms and virtual substitutions to develop a preprocessor for CAD, that will enable an input-level parallelism. We study the performance of CAD in the presence of the preprocessor by extensive experimentation. Since parallelizability of CAD depends on the structure of given prenex formula, we introduce some structural notions to study the performance of CAD with the proposed preprocessor.' author: - | Hari Krishna Malladi    and    Ambedkar Dukkipati\ \ bibliography: - 'harikrishnamalladi.bib' date: - 31 December 2011 - 8 June 2012 title: | **A Preprocessor Based on Clause Normal Forms and Virtual Substitutions to Parallelize\ Cylindrical Algebraic Decomposition** --- Introduction ============ The study of real algebraic geometry deals with the study of the real roots of an equation as, more often than not, the real roots are the most desired solutions to a system of equations. It is important to note that for some problems there are no counterparts in complex algebraic geometry. Given a first-order formula in real algebraic geometry with both quantified and quantifier-free variables, the process of finding an equivalent first-order formula in these quantifier-free variables is called quantifier elimination. When the first-order formula is a boolean combination of polynomial equations and inequalities, we can consider quantifier elimination as a problem in real algebraic geometry. The algorithm specified by Tarski [@tarski] for quantifier elimination is highly resource intensive. So newer and more efficient algorithms have come up and replaced it. The fact that projections upon parameters of a semialgebraic set (semialgebraic set is defined as the solution space of polynomial equations and inequalities) are also semialgebraic (Tarski-Seidenberg principle) has led to Cylindrical Algebraic Decomposition (CAD) [@collins], which eventually became the standard algorithm for quantifier elimination. The time complexity of CAD algorithm is doubly exponential in the number of variables (both quantified and quantifier-free). CAD is a recursive algorithm, which constitutes of a sequence of projections, followed by a sequence of constructions. It has undergone an extensive array of developments, such as Hong’s Partial CAD [@partialcad], Hong’s projection operator [@hongproj], Scott McCallum’s projection operator [@mccallumproj], etc. An application of Gröbner Bases to CAD to improve the time complexity is studied in [@qegrob]. There were also some efforts to parallelize CAD [@issac89]. In spite of all these improvements, the fact that CAD’s time complexity is doubly exponential makes quantifier elimination through CAD impossible for a wide range of real-world applications. Parallelism has appeared to be the trend, while addressing algorithms which have a significant amount of independence in their structure. Speeding up executions by scaling the existing CPU speeds has been found to be inefficient. The maximum possible clock speed has remained at around 3 GHz for the past 12 years, for a desktop microprocessor, in spite of being scaled by about 100 times in the 1990s. Multicores have achieved prominence, as dual-cores and quad-cores have become ubiquitous in the past 7 years. This suggests that the only major way to improve an existing algorithm’s running time is through parallelism. We use this as a motivation to introduce an input-level parallelism in the CAD algorithm. In this paper, we study a possibility to preprocess the input prenex formula, so that several instances of CAD can be run in parallel on it. Input-level parallelism is not directly applicable to CAD because of the constraints imposed by first-order logic itself. Thus, clause normal forms and virtual substitutions are used to ‘separate’ a given input formula so that it’s components can be executed in parallel using existing CAD implementations. We study the preprocessor algorithm through extensive experimentation using the QEPCAD B tool, running several instances on a cluster. The work by Saunders et. al. [@issac89] has brought up the notion of a parallel version of CAD, where they introduce an execution-level parallelism to make the phases of CAD work in parallel. Since our work introduced input-level parallelism, our preprocessor can be used along with Saunders’ approach. The paper is organized as follows. The preliminary notations and the original CAD algorithm are briefly introduced in Section 2. The use of clause normal forms and virtual substitutions and the preprocessor itself are described in Section 3. In Section 4, we introduce some notions to define structure in a prenex formula. An experimental characterization of the preprocessor algorithm, are presented in Section 5. Extensions sought for the proposed algorithm and a few concluding remarks have been made in Section 6. Preliminaries ============= Basic Definitions and Notation ------------------------------ Let $k [x_1,. . .,x_n]$ denote the set of all polynomials in variables, $ x_1,. . .,x_n $ and coefficients from the field $k $. The base field, $k $ is assumed to be the set of real numbers, $\mathbb{R}$ throughout the paper. A formula of the form $$(Q_{1}x_{1}),. . .,(Q_{n}x_{n}) [\psi(y_{1},. . .,y_{m},x_{1},. . .,x_{n})]$$ is called a *prenex* formula, where $Q_{i} \in \{ \exists, \forall \}$ and $\psi$ is a quantifier-free formula in $x_1,,. . .,x_n,y_1,. . .,y_m$. $V \subset \mathbb{R}$ is said to be a semialgebraic set if there exists $ f_1,. . .,f_s \in \mathbb{R} [x_1,. . .,x_n]$ and $g_1,. . .,g_t \in \mathbb{R} [x_1,. . .,x_n]$ such that $V = \{ (a_1,. . .,a_n) \in \mathbb{R} : f_i(a_1,. . .,a_n) = 0, i=1,. . . ,s, g_j(a_1,. . .,a_n) \geq 0, i=1,. . .,t\}$. A connected subset of $\mathbb{R}^{n}$ is called a ‘region’ in $\mathbb{R}^{n}$. Consider a region $R$ and functions $f_i : \mathbb{R}^n \longrightarrow \mathbb{R}, i=1,. . .,l$ satisfying, $f_{1} < f_{2} < ... < f_{l}$. This ordering ensures that these functions do not intersect each other. Graph of $f_{i}$ is called ‘$f_{i}$-section’. In other words, ‘$f_{i}$-section is the set containing all points of the form $(a,b)$, where a $\in \mathbb{R}^{n-1}$ and $b = f(a)$. An ‘(f$_{i}$,f$_{i+1}$)-sector’ is the set of all points $(a,b)$, where a $\in \mathbb{R}^{n-1}$ and $f_{i}(a)$ $< b <$ $f_{i+1}(a)$. A ‘cylinder’ over a region $R$ is the set of all points $(a,b)$, where $a \in R$ and $b \in \mathbb{R}$. A ‘stack’ over a region $R$ is the collection of sections and sectors that occur in the cylinder over $R$. A partition of $\mathbb{R}^{n}$ into semialgebraic partitions is called an ‘algebraic decomposition’ of $\mathbb{R}^{n}$. An algebraic partition of $\mathbb{R}^{n}$ satisfying the following two properties is called a ‘cylindrical algebraic decomposition’: 1. If $n=1$, then the CAD is a set of points and open intervals. 2. If $n>1$, every region of the CAD of $\mathbb{R}^{n-1}$ has a stack over it, which is a disjoint subset of the CAD of $\mathbb{R}^{n}$. A set of polynomials, $F$ is said to be ‘sign invariant’ on the region $R$ iff no polynomial in $F$ changes its sign anywhere on $R$. A CAD $C$ of $\mathbb{R}^n$ is sign invariant with respect to a set of polynomials, $P$ if and only if $P$ is sign invariant on every cell of $C$. The algorithm to generate the CAD of $\mathbb{R}^{n}$(which shall be referred to as ‘CAD Algorithm’) has two phases of execution, namely projection and construction. Projection is specified by a projection operator, whereas construction depends on the projection phase. Cylindrical Algebraic Decomposition Algorithm --------------------------------------------- CAD takes as input, a set of polynomials in $n$ variables and generates a cylindrical algebraic decomposition of the $n$-dimensional real space. This algorithm works recursively to produce a series of projections on lesser dimensions and builds the CAD of each dimension while returning the recurring functions. An outline of CAD algorithm [@20yrs] is presented below. The intricate details have been omitted as the purpose of this presentation is to provide a glimpse into the algorithm, rather than to study the algorithm itself. 1. Project each polynomial onto lesser dimensions. Polynomials in $n$ variables are taken as input and a set of polynomials in $n-1$ variables is obtained as output, upon applying a projection operator. This process is continued till a set of univariate polynomials is obtained. 2. These univariate polynomials can easily be solved (using Sturm’s theorem iteratively or otherwise). The roots of these univariate polynomials and the intervals between them are taken as the regions in the CAD of one dimensional real space. Designate a point in each region as a sample point. 3. For the two dimensional CAD, substitute the sample point found in each one-dimensional region, $R$ in each of the polynomials in the 2-dimensional projection to again get univariate polynomials. Find the roots of these polynomials and create two dimensional regions that form the stack over the region $R$. Designate sample points for these regions as well. Suppose we are given a CAD for $k$-dimensional real space, we use the projected set for $k+1$-dimensions and substitute the sample points of the $k$-dimensional CAD in them to get univariate polynomials. These can now be solved to get $k+1$-dimensional CAD. Now, this information can be used by Algorithm 1 to perform quantifier elimination [@andreas]. Assume that the variables $x_1,...,x_k$ are quantifier-free and $x_{k+1},...,x_n$ are quantified, and let $Q_i$ be the corresponding quantifier for $i\in\{k+1,...,n\}$. 1. For $k \leq i < n$, we have 1. If $Q_{i+1}$ is $\exists$, then a cell,$C \in D_{i}$ is valid if at least one cell in the stack over $C$ is valid. 2. If $Q_{i+1}$ is $\forall$, then a cell,$C \in D_{i}$ is valid if all cells in the stack over $C$ are valid. 2. A region, $C$ in $D_{n}$ is valid if $\psi(t_{C})$ is TRUE, where $t_{C}$ is the sample point of that cell. 3. Obtain the cells of $D_{k}$ which are TRUE. The disjunction of the formulae of these cells is the required quantifier-free formula and is returned. Towards Parallelization ======================= Using Clause Normal Forms ------------------------- In this section we introduce the application of miniscoping and clause normal forms to obtain a preprocessor that can parallelize CAD. Miniscoping is prevalent in the standard literature as a way of localising quantifiers. One can use an algorithm by Nonnengart and Weidenbach [@miniscoping] to compute ‘clause normal forms’. A sentence $\phi$ is said to be in ‘Clause Normal Form’, if $\phi = \forall x_1,. . . ,\forall x_{k}[C_1 \wedge . . . \wedge C_{k}]$ where $C_{i} = L_{i,1} \vee . . . \vee L_{i,l_{i}}$ and each $L_{i,j}$ is a literal. Clause normal forms shall be represented by ‘CNF’ throughout this paper. Their work introduces this algorithm and proves that it terminates in a finite amount of time and that the generated clause normal form is equivalent to the input. The time complexity of this algorithm is polynomial in the number of statements. The algorithm to compute clause normal forms uses the concept of skolemization to eliminate existential quantifiers (by systematic replacement of existentially quantified variables by functions of universally quantified variables). Skolemization may introduce skolem functors of high exponents, but the doubly exponential running time of CAD enables us to cope with them in practice, due to a reduction in the number of variables (brought about by skolemization). CNFs are used to impose a structure on the input prenex formulae, that will make computation easier. But, a well-crafted transformation on a prenex formula may result in a form which performs better than the CNF case. But, such well-crafted transformations usually require a great amount of computational power (and possibly non-determinism) to achieve in practical scenarios. Thus, the CNF has been used to give a measure of speed-up that can be achieved in an average case, as the CNF can be obtained using a deterministic algorithm, which terminates for any input. Using Virtual Substitutions --------------------------- The CAD algorithm benefits from substitutions, which might reduce the number of variables in some clauses, prior to the separation phase. This may decrease the dependence of one clause on another. But substitutions can be non-trivial and in many cases, impossible. Substitutions can be used with linear and quadratic equations and inequalities directly. For cubic and quartic polynomials, substitutions can be used only if it is obtained easily. Newton and Cardano’s formulas for quartics and cubics involve complex roots, and hence cannot be used. For quintic and higher order polynomials, there exists no generic closed form solution, as proved by Abel. We would like to use the algorithm to perform virtual substitutions by Volker Weispfenning [@weispfenning]. While Weispfenning’s work on virtual substitution seeks to find a quantifier-free equivalent by itself, we would like to adapt it to minimize dependencies within the clauses and hence, to increase parallelism. In our notion of substitutability, we classify polynomials into two classes, namely (i) substitutor and (ii) substituend. A ‘substitutor’ polynomial is used to substitute for a variable in other polynomials. A ‘substituend’ polynomial is one, in which a variable is replaced.Assume that $P$ is any polynomial and $T$ is a substitutor. We say that $T$ is ‘substitutable’ in $P$ if, the process of replacing a variable in $P$ by virtual substitution results in a decrease in the number of variables in the polynomial $P$. We use the results proved by Weispfenning and Christopher Brown [@brownsub], which state that virtual substitution leads to an equi-satisfiale formula. The Proposed Preprocessor ------------------------- In this section we present a preprocessor to the CAD algorithm that is motivated by the notion of clause normal forms and virtual substitutions. Virtual substitutions would be performed prior to clause normal form computation, as substitutions might result in an increase in the number of clauses. The clause normal form has been chosen as the preferred format for computational efficiency. The fact that any given prenex formula can be converted to a clause normal form enables us to perform the experimental evaluation on test cases containing solely of clause normal forms. We now list the proposed algorithm. 1. Compute the negation normal form of $f$ and call it $f'$. 2. Perform virtual substitutions after identifying substitutors and substituends and call the resulting formula, $f''$. 3. Compute the clause normal form of $f''$ and call it $f'''$. 4. Call an instance of CAD for each clause in $f'''$, with each instance being an independent thread of execution. 5. Concatenate the outputs of the $k$ instances of CAD using boolean conjunction. Structural Notions of the Prenex Formulae and the Preprocessor ============================================================== The introduction of CNF computation introduces the notion of partitioning the space of all prenex formulae into classes, depending on how many clauses it could be separated into. To interpret the results presented above, we use the following definitions. A prenex formula is said to be separable if it can be split into two or more clauses. Separability is a property of the prenex formula. Separability may result in the loss of structure in a given prenex formula. By this we mean that the interdependencies among the formula’s constituent polynomials might be lost, resulting in wasteful and redundant computations, which would not have been required if the formula was not separated. We need a formal, quantitative definition of ‘structure’ with respect to a given formula to study it’s effects on the running time of the proposed algorithm. The ‘sharing factor’ existing among two given prenex formulae, $f$ and $g$, is the number of variables that are shared between $f$ and $g$. It’s denoted by $T_{f,g}$. The sharing factor essentially captures the distribution of variables among the constituent polynomials of a prenex formula. If formulae $f$ and $g$ have $k$ variables in common, it implies that $f$ imposes conditions on $g$ in $k$ dimensions. The sharing factor gives a measure of structure present in a formula, as the sharing of variables between polynomials causes interdependencies among them. Having defined the sharing factor, we now define the operation, decomposition, where the sharing of variables comes into consideration. Consider a prenex formula, $f$, which has $n$ variables, out of which, there are $k$ quantified variables and $n-k$ quantifier-free variables. We say that the decomposition of $f$, $F$ is the set consisting of all the clauses which occur in the CNF of $f$. This definition of decomposition defines separability as a property of the formula. In the remainder of this section, we assume that all prenex formulae are in variables, $x_1,. . .,x_n$. It is natural to classify prenex formulae into classes based on the number of clauses it can be separated into. Thus, we have the following definition. A prenex formula is defined to be k-separable if its CNF has at least $k$ clauses. The set of all k-separable formulae shall be denoted by $S_k$. According to this definition, we can observe that $S_k \subset S_{k-1}$. The set of 1-separable formulae is the set of all prenex formulae, which shall be denoted by $P$. We assume that we have only a constant number of processors, and we denote this by $K$. This leads to a definition of the class of separable problems. Consider a set $S \subset P$. $S$ is called the ‘separable class’ if the following conditions hold. - $S \subset S_2$ - If $f$ is a prenex formula and $F$ is it’s decomposition, $\forall f \in F$, $\exists x_i \in \{x_1,...,x_n\}$ such that $f$ is independent of $x_i$. The property of separability is not uniform in this separable class. It depends heavily on the sharing factor of the formula. The quantification of the term structure enables us to define the ‘centre of $S_k$’ for $k < K$. The centre of $S_k$ is the set of all prenex formulae, $f$ in $n$ variables ($x_1, . . . ,x_n$) with decomposition $F$, such that the clause $f_i \in F$ contains exactly the variables $x_{\frac{i-1}{k}+1}, . . . ,x_{\frac{i}{k}}$. Unlike the classes $S_1, . . . ,S_k$, the classes $C_1, . . . ,C_k$ do not form a nesting chain structure. In other words, $C_i$ need not contain $C_{i+1}$. This centre of a separable class, $S_i$, is the set of formulae which pose a ‘best case’ scenario for the CAD algorithm to run in parallel for an input prenex formula containing $i$ variables. The lower the maximum sharing factor in a prenex formula, the faster it’s execution would be on the cluster, using the parallel CAD algorithm. This has been demonstrated by considering prenex formulae with zero sharing factor. But, as the sharing factor between two clauses increases, we find that these two clauses become interdependent to a greater extent. Thus, executing them in parallel would lead to greater amount of work being done, as compared to the case where the conditions imposed by one clause influences the computation of CAD on another clause. Experimental Evaluation ======================= The simulation results in this paper utilize the clause normal form (CNF), to ease the process of generating prenex formulae. The test cases are always in the CNF. The simulations are performed on a cluster, comprising of 36 nodes, with 8 Intel Xeon quad-core CPUs in each node. The preprocessor is implemented using C and the OpenMP library. The code used for these experiments has been uploaded at http://algoalgebra.csa.iisc.ernet.in/Preprocessor/. The experimental setup is as follows: (i) The dependence of running time of QEPCAD B on number of terms, highest exponent and total number of terms in a prenex formula, (ii) The distribution of prenex formulae according to the number of clauses in their corresponding CNFs, (iii) The comparison of running times of QEPCAD B, with and without the preprocessor and (iv) The structure imposed be dependence among polynomials is studied in terms of the sharing factor. On the Parameters of CAD ------------------------ Maximum number of terms per polynomial and the highest exponent across all polynomials in a CNF are two of the major parameters which influence the running time of the CAD algorithm. As there is no closed expression for running time of CAD in terms of these parameters, it seems appropriate to plot the variations of these parameters versus the running time. The values presented are averaged over 100 randomly generated prenex formulae. Each prenex formula that is generated, is a CNF, where each clause consists of a set of polynomial equations or inequalities separated by the boolean disjunction. The default random number generator, `rand()` was used.\ \ \ \ \ \ First we fix the maximum number of terms, say T. The number of terms in each polynomial is a random number in between $0$ and $T$. The number of clauses, the highest exponent and number of variables in the prenex formula were kept constant. The increase in running time was observed to be close to exponential in this case, as depicted in Fig. 1.\ This increase is not surprising as an increase in the number of terms per polynomial results in an exponential increase in the total number of terms that are processed, throughout the course of execution. This is due to an increase in number of terms in each set of projected polynomials. The highest exponent, $E$, was kept constant in each stage of the experiment and all other exponents were taken as random numbers between $0$ and $E$. All other parameters such as number of clauses, maximum number of terms and the number of variables were kept constant. The experiment was performed for both, a 2 variable case and a 3 variable case. The increase in the running time was observed to closely mimic Fig. 1, as shown in Fig. 2 and Fig. 3.\ This increase is expected as an increase in the exponent results in an increase in the number of regions, when the construction phase of CAD reaches the variable with the exponent $E$. This is leads to an almost exponential increase in the number of regions, as each previously generated region spawns more regions in the higher dimensions. This would result in a stark increase in running time. One can conclude that number of terms and exponents contribute equally to the running time of CAD. This was conducted for a 2 variable case, with a maximum of $5$ terms per polynomial and a maximum of 2 polynomials per clause. The number of clauses was varied from $2$ to $6$. The results were averaged over 50 randomly generated prenex formulae. An almost exponential increase is observed in this case, as shown in Fig. 4. One could infer from this decrease in gradient that the total number of terms in a collection of projected polynomials depends to a greater extent on the number of terms per polynomial, than on the number of polynomials. On Validity of the Preprocessor ------------------------------- As the analysis is performed on the assumption that the input is in the CNF format, with each prenex formula containing two or more clauses, it is necessary to give an account on the percentage of randomly generated prenex formulae, that can be converted to a CNF containing more than one clause. A CNF is a conjunctive normal form with universally quantified variables. Hence, 100 prenex formulae were randomly generated, which do not conform to the normal form defined previously. They were subjected to a logic converter, which converts them to a minimal conjunctive normal form. We have observed the following from the plot in Fig. 5: only 8% of the formulae were 1-separable. Hence, in 92% of the cases, the algorithm would lead to a parallel execution of CAD.\ \ On the Performance of CAD with the Proposed Preprocessor -------------------------------------------------------- With the above two analyses in place, the next step is to experimentally compare the running times of an implementation of CAD algorithm with and without the preprocessor. The analysis is done for 2,3 and 4 variable cases. In each case, the prenex formula is 100-separable and is in Clause Normal Form. Each clause has at most 5 polynomials and each polynomial has at most 5 terms. The analysis is done for the 2, 3 and the 4 variable case. The implementation uses OpenMP library to divide the processing of the prenex formula into independent threads of execution, each running on a separate Intel Xeon CPU. Each of these threads runs an instance of QEPCAD B. The final results are averaged over 50 different randomly generated prenex formulae.\ \ The comparison between the running times of QEPCAD B with the preprocessor and without are shown in Table 1 and Fig. 6 compares the running times (with and without the preprocessor) of QEPCAD B over 50 different prenex formulae in two variables. Variables No. of terms per polynomial No. of polynomials Time without the preprocessor Time with the preprocessor ----------- ----------------------------- -------------------- ------------------------------- ---------------------------- 2 5 5 3 Sec 0.3 Sec 3 5 5 $\infty$ 92 Sec 4 2 2 77000 Sec 2 Sec \ \ An example of such a randomly generated formula in 2 variables from the separable class $S_{10}$, each with at most 5 polynomials and each polynomial with at most 5 terms is, $(\forall x0)[[[(58) {x_{0}}^{8} {x_{1}}^{2} + (-68) {x_{0}}^{10} {x_{1}}^{4} + (73) + (-4) + (-28) {x_{0}}^{9} = 0] \vee[(44) {x_{0}}^{4} {x_{1}}^{5} + (-88) {x_{0}}^{10} {x_{1}}^{4} + (11) {x_{1}}^{9} + (-70) + (-19) {x_{0}}^{6} {x_{1}}^{5} = 0] \vee[(20) {x_{0}}^{7} + (-61) {x_{0}}^{10} {x_{1}}^{3} + (-71) + (81) {x_{0}}^{3} {x_{1}}^{5} + (14) {x_{0}}^{7} = 0] \vee[(-58) {x_{1}}^{5} + (-35) {x_{0}}^{10} + (5) + (30) + (45) {x_{1}}^{5} = 0] \vee[(-69) {x_{0}}^{5} {x_{1}}^{6} + (71) {x_{0}}^{10} + (40) + (5) + (-90) {x_{1}}^{5} = 0]] \wedge [[(-67) {x_{0}}^{2} {x_{1}}^{6} + (-17) {x_{0}}^{10} + (-18) {x_{1}}^{3} + (14) + (37) {x_{0}}^{6} = 0] \vee[(76) {x_{0}}^{5} {x_{1}}^{1} + (-7) {x_{0}}^{10} + (-44) {x_{0}}^{5} {x_{1}}^{6} + (-96) {x_{0}}^{5} + (-69) {x_{0}}^{1} {x_{1}}^{6} = 0] \vee[(-73) + (-42) {x_{0}}^{10} + (67) {x_{0}}^{7} {x_{1}}^{2} + (-50) {x_{1}}^{9} + (-12) {x_{1}}^{1} = 0] \vee[(63) {x_{0}}^{2} {x_{1}}^{7} + (23) {x_{0}}^{10} + (-74) {x_{1}}^{5} + (31) + (17) {x_{0}}^{7} = 0] \vee[(-47) {x_{0}}^{1} {x_{1}}^{9} + (-31) {x_{0}}^{10} {x_{1}}^{1} + (17) + (17) + (-88) {x_{1}}^{6} = 0]] \wedge [[(78) {x_{0}}^{9} + (-43) {x_{0}}^{10} + (5) {x_{0}}^{6} + (4) {x_{1}}^{7} + (-76) = 0] \vee[(-23) {x_{0}}^{3} + (81) {x_{0}}^{10} + (36) + (35) + (54) {x_{1}}^{1} = 0] \vee[(-29) {x_{0}}^{5} + (54) {x_{0}}^{10} {x_{1}}^{2} + (-20) {x_{0}}^{6} {x_{1}}^{1} + (57) {x_{0}}^{2} + (62) {x_{0}}^{9} {x_{1}}^{4} = 0] \vee[(77) {x_{1}}^{6} + (97) {x_{0}}^{10} {x_{1}}^{6} + (-42) + (-11) {x_{0}}^{1} + (93) {x_{0}}^{3} {x_{1}}^{4} = 0] \vee[(37) {x_{0}}^{7} + (1) {x_{0}}^{10} + (-18) {x_{0}}^{1} {x_{1}}^{9} + (36) + (25) {x_{0}}^{6} = 0]] \wedge [[(-35) {x_{0}}^{3} + (22) {x_{0}}^{10} + (-80) {x_{0}}^{3} + (13) {x_{1}}^{3} + (96) {x_{1}}^{5} = 0] \vee[(-73) + (-25) {x_{0}}^{10} {x_{1}}^{5} + (79) + (-16) + (88) {x_{0}}^{3} {x_{1}}^{7} = 0] \vee[(-94) {x_{0}}^{8} {x_{1}}^{6} + (86) {x_{0}}^{10} + (-52) {x_{0}}^{2} + (15) + (32) {x_{0}}^{2} = 0] \vee[(49) + (-36) {x_{0}}^{10} + (51) {x_{0}}^{5} {x_{1}}^{8} + (59) {x_{0}}^{4} + (-37) = 0] \vee[(99) {x_{0}}^{8} + (71) {x_{0}}^{10} + (73) {x_{1}}^{5} + (68) {x_{0}}^{8} {x_{1}}^{3} + (51) {x_{0}}^{1} {x_{1}}^{1} = 0]] \wedge [[(70) + (-53) {x_{0}}^{10} {x_{1}}^{5} + (73) {x_{0}}^{7} {x_{1}}^{9} + (61) {x_{1}}^{3} + (-59) {x_{0}}^{3} {x_{1}}^{9} = 0] \vee[(-32) {x_{0}}^{3} {x_{1}}^{5} + (-80) {x_{0}}^{10} + (-28) + (-88) {x_{0}}^{4} + (35) {x_{0}}^{1} = 0] \vee[(-65) + (-81) {x_{0}}^{10} + (35) {x_{0}}^{9} {x_{1}}^{6} + (8) {x_{0}}^{7} {x_{1}}^{4} + (-38) {x_{0}}^{3} = 0] \vee[(-24) {x_{1}}^{6} + (26) {x_{0}}^{10} + (15) {x_{0}}^{1} + (80) {x_{1}}^{6} + (93) = 0] \vee[(-42) {x_{0}}^{1} + (84) {x_{0}}^{10} + (-13) {x_{0}}^{5} + (33) + (-17) {x_{0}}^{5} {x_{1}}^{7} = 0]] \wedge [[(75) {x_{0}}^{8} + (26) {x_{0}}^{10} {x_{1}}^{7} + (-47) {x_{0}}^{5} + (8) + (-81) {x_{0}}^{8} {x_{1}}^{9} = 0] \vee[(-96) {x_{0}}^{4} + (24) {x_{0}}^{10} {x_{1}}^{8} + (-78) + (82) {x_{1}}^{6} + (-22) {x_{0}}^{4} = 0] \vee[(69) {x_{1}}^{5} + (12) {x_{0}}^{10} {x_{1}}^{3} + (-9) {x_{1}}^{2} + (63) {x_{0}}^{1} + (-39) {x_{0}}^{2} = 0] \vee[(-27) + (1) {x_{0}}^{10} + (44) {x_{0}}^{9} {x_{1}}^{4} + (-68) {x_{0}}^{8} {x_{1}}^{2} + (-3) = 0] \vee[(94) {x_{1}}^{3} + (-85) {x_{0}}^{10} {x_{1}}^{2} + (-63) + (22) {x_{1}}^{6} + (-74) {x_{1}}^{8} = 0]] \wedge [[(-97) + (-7) {x_{0}}^{10} + (27) {x_{0}}^{8} {x_{1}}^{7} + (71) + (-26) {x_{0}}^{9} {x_{1}}^{1} = 0] \vee[(15) {x_{1}}^{8} + (13) {x_{0}}^{10} + (-15) + (9) + (16) {x_{0}}^{3} {x_{1}}^{2} = 0] \vee[(7) {x_{1}}^{6} + (13) {x_{0}}^{10} {x_{1}}^{1} + (7) {x_{0}}^{9} + (-87) + (13) {x_{1}}^{3} = 0] \vee[(-8) + (-27) {x_{0}}^{10} {x_{1}}^{6} + (-44) {x_{0}}^{3} {x_{1}}^{4} + (-66) {x_{0}}^{2} {x_{1}}^{9} + (42) {x_{1}}^{3} = 0] \vee[(-13) + (-95) {x_{0}}^{10} {x_{1}}^{6} + (48) {x_{0}}^{3} {x_{1}}^{7} + (8) {x_{1}}^{3} + (-95) {x_{0}}^{5} = 0]] \wedge [[(-51) {x_{0}}^{5} + (-40) {x_{0}}^{10} {x_{1}}^{3} + (-18) {x_{0}}^{6} + (22) {x_{1}}^{2} + (-65) = 0] \vee[(-22) {x_{0}}^{7} + (-73) {x_{0}}^{10} {x_{1}}^{5} + (19) {x_{0}}^{9} + (84) {x_{0}}^{7} + (43) {x_{0}}^{8} = 0] \vee[(39) {x_{1}}^{8} + (95) {x_{0}}^{10} {x_{1}}^{3} + (-57) {x_{1}}^{2} + (-1) {x_{1}}^{3} + (21) = 0] \vee[(21) {x_{1}}^{3} + (-80) {x_{0}}^{10} + (-14) + (56) {x_{1}}^{5} + (4) {x_{0}}^{4} {x_{1}}^{9} = 0] \vee[(-56) + (-88) {x_{0}}^{10} {x_{1}}^{1} + (90) {x_{1}}^{9} + (31) + (-63) {x_{0}}^{2} = 0]] \wedge [[(73) {x_{1}}^{3} + (91) {x_{0}}^{10} + (79) {x_{0}}^{9} + (72) {x_{0}}^{8} + (-97) {x_{1}}^{4} = 0] \vee[(-36) {x_{0}}^{8} + (2) {x_{0}}^{10} {x_{1}}^{5} + (-46) {x_{0}}^{7} {x_{1}}^{2} + (11) + (28) {x_{0}}^{7} {x_{1}}^{7} = 0] \vee[(-38) {x_{0}}^{5} + (94) {x_{0}}^{10} {x_{1}}^{9} + (-15) {x_{0}}^{1} + (-91) + (-5) {x_{1}}^{3} = 0] \vee[(93) + (52) {x_{0}}^{10} {x_{1}}^{2} + (-5) {x_{0}}^{3} {x_{1}}^{2} + (-20) + (-5) = 0] \vee[(-47) {x_{0}}^{7} + (80) {x_{0}}^{10} + (76) + (54) {x_{1}}^{3} + (86) {x_{1}}^{4} = 0]] \wedge [[(-21) {x_{0}}^{6} {x_{1}}^{8} + (-83) {x_{0}}^{10} {x_{1}}^{5} + (67) {x_{0}}^{3} + (80) {x_{0}}^{5} + (57) {x_{1}}^{2} = 0] \vee[(-24) {x_{1}}^{6} + (78) {x_{0}}^{10} + (-68) {x_{0}}^{6} + (83) {x_{0}}^{8} + (66) = 0] \vee[(-60) + (97) {x_{0}}^{10} {x_{1}}^{7} + (-6) {x_{0}}^{3} + (53) {x_{1}}^{9} + (-36) {x_{1}}^{4} = 0] \vee[(83) {x_{0}}^{1} {x_{1}}^{5} + (-55) {x_{0}}^{10} {x_{1}}^{6} + (-48) + (69) {x_{0}}^{2} + (-44) = 0] \vee[(15) + (-49) {x_{0}}^{10} + (90) + (-23) {x_{0}}^{6} {x_{1}}^{4} + (85) {x_{1}}^{4} = 0]]].$ An example of the actual input, which consists of 100 clauses could not be provided here due to space constraints (as it would occupy four pages). This also proves the robustness of the algorithm, as CAD in practice relies on the systematic factorization of the polynomials. Using random formulae demonstrates the applicability of the algorithm even if the probability of such factorization is 0. The behaviour of the running time of QEPCAD B with the preprocessor for 50 randomly generated prenex formulae is depicted in Fig. 7. On Effectiveness of the Sharing Factor -------------------------------------- This experiment aims to study the sharing factor and the impact it can have on the amount of computational work done by the CAD algorithm. The space utilized for execution is taken as a metric for computational work. The number of cells utilized is provided by the QEPCAD B tool. A formula from the separable class $S_2$ is considered, with 2 polynomials per clause, 2 terms per polynomial and 6 variables. The sharing factor is varied from 0 to 3. The first clause is kept constant for all the cases and the second clause is varied. The second clause uses $k$ variables, out of those used in the first clause, for a sharing factor of $k$. QEPCAD B is run twice for each sharing factor, the first run being the whole formula and the first clause being truncated in the second run. Any structure imposed by the first clause on the second clause should appear as the difference between the space utilized by both the runs. As the same clause is deleted in all 4 cases, the experiment should not react to factors other than the sharing factor. As shown in Table 2, in the 0 sharing factor case (which is from $C_2$), the space utilized in both cases is almost identical. The difference increases as the sharing factor increases. The case with sharing factor of 1 is taken as an anomaly, where there is a greater amount of dependence on the one variable that is shared. This demonstrates the existence of corner cases. As an example, we provide the formula used for the experiment concerning sharing factor of 3. $(\forall x_0)(\forall x_1)(\forall x_2)(\forall x_3)[[[(85) x_1^1 x_2^2 + (64) x_0^3 x_2^1 = 0] \vee [(41) x_0^1 x_2^1 + (-96) x_0^3 x_1^2 x_2^2 = 0]] \wedge [[(-18) x_4^1 + (-44) x_0^3 x_2^1 x_4^2 x_5^1 = 0] \vee [(-78) x_1^1 x_3^1 x_4^1 x_5^2 + (31) x_0^3 x_1^1 x_2^2 x_4^2 x_5^2 = 0]]]$ The variables $x_0,x_1,x_2$ are shared among the two clauses in this example. The first clause is common to all the four cases studied. **Sharing Factor** **Percentage of space utilized (with the first clause)** **Percentage of Space utilized (without the first clause)** -------------------- ---------------------------------------------------------- ------------------------------------------------------------- 0 shared variables 7.6%(38187 cells) 3.7%(18723 cells) 1 shared variables 44.18%(279058 cells) left after 49 garbage collections 5.14%(25735 cells) 2 shared variables 69.31%(346561 cells) 3.32%(16614 cells) 3 shared variables 65.4%(327124 cells) left after 2 garbage collections 5.04%(25234 cells) : An analysis of the effectiveness of sharing factor to study the structure of a prenex formula Concluding Remarks ================== In this paper, a preprocessor to CAD is proposed based on the notions of clause normal forms and virtual substitutions. This preprocessor uses clause normal forms to impose a structure on an input formula and runs several instances of CAD on these clauses. The effectiveness of this preprocessor is studied experimentally and some theoretic notions of structure inherent in a formula and the consequences of losing it are presented. The paper ends by providing the notion of an ‘idealistic’ scenario and a measure of deviation that a random prenex formula has from it. Though the notion showing factor reflects parallelizability of CAD to some extent it cannot completely describe the best-case scenario for the preprocessor. We seek to improve upon this notion by considering parameters other than just the sharing factor. This will help us to theoretically analyze the time and space complexity of CAD with the proposed preprocessor.

--- author: - 'Arpan Chattopadhyay, Abhishek Sinha, Marceau Coupechoux, and Anurag Kumar \' bibliography: - 'IEEEabrv.bib' - 'arpan-techreport.bib' title: 'Deploy-As-You-Go Wireless Relay Placement: An Optimal Sequential Decision Approach using the Multi-Relay Channel Model[^1][^2][^3] ' --- Introduction {#Introduction} ============ Wireless interconnection of devices such as smart phones, or wireless sensors, to the wireline communication infrastructure is an important requirement. These are battery operated, resource constrained devices. Hence, due to the physical placement of these devices, or due to channel conditions, a direct one-hop link to the infrastructure “base-station” might not be feasible. In such situations, other nodes could serve as *relays* in order to realize a multi-hop path between the source device and the infrastructure. In the wireless sensor network context, the relays could be other wireless sensors or battery operated radio routers deployed specifically as relays. The relays are also resource constrained and a cost might be involved in placing them. Hence, there arises the problem of *optimal relay placement*. Such a problem involves the joint optimization of node placement and operation of the resulting network, where by “operation” we mean activities such as transmission scheduling, power allocation, and channel coding. ![A source and a sink connected by a multi-hop path comprising $N$ relay nodes along a line.[]{data-label="fig:general_line_network"}](general-line-network.pdf){height="1.2cm" width="8cm"} Our work in this paper is motivated by recent interest in problems of [*impromptu (as-you-go)*]{} deployment of wireless relay networks in various situations; for example, “first responders” in emergency situations, or quick deployment (and redeployment) of sensor networks in large terrains, such as forests (see [@souryal-etal07real-time-deployment-range-extension], [@howard-etal02incremental-self-deployment-algorithm-mobile-sensor-network], [@bao-lee07rapid-deployment-ad-hoc-backbone], [@sinha-etal12optimal-sequential-relay-placement-random-lattice-path], [@chattopadhyay-etal13measurement-based-impromptu-placement_wiopt]). In this paper, we are concerned with the situation in which a deployment agent walks [*from the source node to the sink node*]{}, along the line joining these two nodes, and places wireless relays (in an “as-you-go” manner) so as to create a source-to-sink multi-relay channel network with high data rate; see Figure \[fig:general\_line\_network\]. We first consider the scenario where the length $L$ of the line in Figure \[fig:general\_line\_network\] is known; the results of this case are used to formulate the as-you-go deployment in the case where $L$ is a priori unknown, but has exponential distribution with known mean $\overline{L}$. In order to capture the fundamental trade-offs involved in such problems, we consider an information theoretic model. For a placement of the relay nodes and allocation of transmission powers to these relays, we model the “quality” of communication between the source and the sink by the information theoretic achievable rate of the multi-relay channel (see [@cover-gamal79capacity-relay-channel], [@reznik-etal04degraded-gaussian-multirelay-channel] and [@xie-kumar04network-information-theory-scaling-law] for the single and multi-relay channel models). The relays are equipped with full-duplex radios[^4], and carry out decode-and-forward relaying. We consider scalar, memoryless, time-invariant, additive white Gaussian noise (AWGN) channels. We assume synchronous operation across all transmitters and receivers, and consider the exponential path-loss model for radio wave propagation. Related Work {#subsection:related_work} ------------ A formulation of the relay placement problem requires a model of the wireless network at the physical (PHY) and medium access control (MAC) layers. Most researchers have adopted the link scheduling and interference model, i.e., a scheduling algorithm determines radio resource allocation (channel and power) and interference is treated as noise (see [@georgiadis-etal06resource-allocation-cross-layer-control]); treating interference as noise leads to the model that simultaneous transmissions “collide” at receiving nodes, and transmission scheduling aims to avoid collisions. However, node placement for throughput maximization with this model is intractable because the optimal throughput is obtained by first solving for the optimum schedule assuming fixed node locations, followed by an optimization over those locations. Hence, with such a model, there appears to be little work on the problem of jointly optimizing the relay node placement and the transmission schedule. Reference [@firouzabadi-martins08optimal-node-placement] is one such work where the authors considered placing a set of nodes in an existing network such that a certain network utility is optimized subject to a set of linear constraints on link rates, under the link scheduling and interference model. They posed the problem as one of geometric programming assuming exponential path-loss, and proposed a distributed solution. The authors of [@zheng-etal12robust-relay-placement_arxiv] consider relay placement for utility maximization, assuming there are several source nodes, sink nodes and a few candidate locations for placing relays; they ignore interference because of highly directional antennas used in $60$ GHz mmWave networks, which may not always be valid. Relay placement for capacity enhancement has been studied in [@lu-etal11relay-placement-80216], but there interference is mitigated by scheduling transmissions over multiple channels. On the other hand, an information theoretic model for a wireless network often provides a closed-form expression for the channel capacity, or at least an achievable rate region. These results are asymptotic, and make idealized assumptions such as full-duplex radios, perfect interference cancellation, etc., but provide algebraic expressions that can be used to formulate tractable optimization problems which can provide useful insights. In the context of optimal relay placement, some researchers have already exploited this approach. Thakur et al., in [@thakur-etal10optimal-relay-location-power-allocation], report on the problem of placing a single relay node to maximize the capacity of a broadcast relay channel in a wideband regime. Lin et al., in [@lin-etal07relay-placement-80216], numerically solve the problem of a single relay node placement, under power-law path loss and individual power constraints at the source and the relay; however, our work is primarily focused on multi-relay placement, under the exponential path-loss model and a sum power constraint among the nodes. The linear deterministic channel model ([@avestimehr-etal11wireless-network-deterministic]) is used by Appuswamy et al. in [@appuswamy-etal10relay-placement-deterministic-line] to study the problem of placing two or more relay nodes along a line so as to maximize the end-to-end data rate. Our present paper is in a similar spirit; however, we use the achievable rate formulas for the $N$-relay channel (with decode and forward relays) to study the problem of placing relays on a line having length $L$, under a sum power constraint over the nodes. The most important difference of our paper with the literature reported above is that we address the problem of [*sequential placement*]{} of relay nodes along a line of an unknown random length. This paper extends our previous work in [@chattopadhyay-etal12optimal-capacity-relay-placement-line], which presents the analysis for the case of given $L$ and $N$; the study under given $L$ and $N$ is a precursor to the formulation of as-you-go deployment problem, since it motivates an additive cost structure that is essential for the formulation of the sequential deployment problem as a Markov decision process (MDP). The deploy-as-you-go problem has been addressed by previous researchers. For example, Howard et al., in [@howard-etal02incremental-self-deployment-algorithm-mobile-sensor-network], provide heuristic algorithms for incremental deployment of sensors in order to cover a deployment area. Souryal et al., in [@souryal-etal07real-time-deployment-range-extension], propose heuristic deployment algorithms for the problem of impromptu wireless network deployment, with an experimental study of indoor RF link quality variation. The authors of [@bao-lee07rapid-deployment-ad-hoc-backbone] propose a collaborative deployment method for multiple deployment agents, so that the contiguous coverage area of relays is maximized subject to a total number of relays constraint. However, until the work in [@mondal-etal12impromptu-deployment_NCC] and [@sinha-etal12optimal-sequential-relay-placement-random-lattice-path], there appears to have been no effort to rigorously formulate as-you-go deployment problem in order to derive optimal deployment algorithms. The authors of [@mondal-etal12impromptu-deployment_NCC] and [@sinha-etal12optimal-sequential-relay-placement-random-lattice-path] used MDP based formulations to address the problem of placing relay nodes sequentially along a line and along a random lattice path, respectively. The formulations in [@mondal-etal12impromptu-deployment_NCC] and [@sinha-etal12optimal-sequential-relay-placement-random-lattice-path] are based on the so-called “lone packet traffic model" under which, at any time instant, there can be no more than one packet traversing the network, thereby eliminating contention between wireless links. This work was later extended in [@chattopadhyay-etal13measurement-based-impromptu-placement_wiopt] to the scenario where the traffic is still lone packet, but a measurement-based approach is employed to account for the spatial variation of link qualities due to shadowing. In this paper, we consider as-you-go deployment along a line, but move away from the lone-packet traffic assumption by employing information theoretic achievable rate formulas (for full-duplex radios and decode-and-forward relaying). We assume exponential path-loss model (see [@franceschetti-etal04random-walk-model-wave-propagation] and Section \[subsection:motivation-for-exponential-path-loss\]). To the best of our knowledge, there is no prior work that considers as-you-go deployment under this physical layer model. Our Contribution ---------------- - [**Optimal Offline Deployment:**]{} Given the location of $N$ full-duplex relays to connect a source and a sink separated by a given distance $L$, and under the exponential path-loss model and a sum power constraint among the nodes, the optimal power split among the nodes and the achievable rate are expressed (Theorem \[theorem:multirelay\_capacity\]) in terms of the channel gains. We find expression for optimal relay location in the single relay placement problem (Theorem \[theorem:single\_relay\_total\_power\]). For the $N$ relay placement problem, numerical study shows that, the relay nodes are clustered near the source at low attenuation and are placed uniformly at high attenuation. Theorem \[theorem:large\_nodes\_uniform\] shows that, by placing large number of relays uniformly, we can achieve a rate arbitrarily close to the AWGN capacity. Only this part of our current paper was published in the conference version [@chattopadhyay-etal12optimal-capacity-relay-placement-line]. - [**Optimal As-You-Go Deployment:**]{} In Section \[sec:mdp\_total\_power\], we consider the problem of placing relay nodes in a deploy-as-you-go manner, so as to connect a source and a sink separated by an unknown distance, modeled as an exponentially distributed random variable $L$. Specifically, the problem is to start from a source, and walk along a line, placing relay nodes as we go, until the line ends, at which point the sink is placed. With a sum power constraint, the aim is to maximize a capacity limiting term derived from the deployment problem for known $L$, while constraining the expected number of relays. We “relax” the expected number of relays constraint via a Lagrange multiplier, and formulate the problem as a total cost MDP with uncountable state space and non-compact action sets. We prove the existence of an optimal policy and convergence of value iteration (Theorem \[theorem:convergence\_of\_value\_iteration\]); these results for uncountable state space and non-compact action space are not evident from standard literature. We study properties of the value function analytically. This is the first time that the as-you-go deployment problem is formulated to maximize the end-to-end data rate under the full-duplex multi-relay channel model. - [**Numerical Results on As-You-Go Deployment:**]{} In Section \[section:numerical\_work\_information\_theoretic\_model\], we study the policy structure numerically. We also demonstrate numerically that the proposed as-you-go algorithm achieves an end-to-end data rate sufficiently close to the maximum possible achievable data rate for offline placement. This is particularly important since there is no other benchmark in the literature, with which we can make a fair comparison of our policy. - The material in Section \[sec:mdp\_total\_power\] and Section \[section:numerical\_work\_information\_theoretic\_model\] were absent in the conference version [@chattopadhyay-etal12optimal-capacity-relay-placement-line]. Organization of the Paper ------------------------- In Section \[sec:system\_model\_and\_notation\], we describe our system model and notation. In Section \[sec:total\_power\_constraint\], we address the problem of relay placement on a line of known length. Section \[sec:mdp\_total\_power\] deals with the problem of as-you-go deployment along a line of unknown random length. Numerical work on as-you-go deployment has been presented in Section \[section:numerical\_work\_information\_theoretic\_model\]. Some discussions are provided in Section \[section:additional\_discussion\]. Conclusions are drawn in Section \[conclusion\]. System Model and Notation {#sec:system_model_and_notation} ========================= The Multi-Relay Channel ----------------------- The multi-relay channel was studied in [@xie-kumar04network-information-theory-scaling-law] and [@reznik-etal04degraded-gaussian-multirelay-channel] and is an extension of the single relay model presented in [@cover-gamal79capacity-relay-channel]. We consider a network deployed on a line with a source node and a sink node at the end of the line, and $N$ full-duplex relay nodes as shown in Figure \[fig:general\_line\_network\]. The relay nodes are numbered as $1, 2,\cdots,N$. The source and sink are indexed by $0$ and $N+1$, respectively. The distance of the $k$-th node [*from the source*]{} is denoted by $y_{k}:=r_{1}+r_{2}+\cdots+r_{k}$. Thus, $y_{N+1}=L$. As in [@xie-kumar04network-information-theory-scaling-law] and [@reznik-etal04degraded-gaussian-multirelay-channel], we consider the [*scalar, time-invariant, memoryless, AWGN setting.*]{} We use the model that a symbol transmitted by node $i$ is received at node $j$ after multiplication by the (positive, real valued) channel gain $h_{i,j}$ (an assumption often made in the literature, see e.g., [@xie-kumar04network-information-theory-scaling-law] and [@gupta-kumar03capacity-wireless-networks]). The *power gain* from Node $i$ to Node $j$ is denoted by $g_{i,j} = h_{i,j}^2$. We define $g_{i,i}=1$ and $h_{i,i}=1$. The Gaussian additive noise at any receiver is independent and identically distributed from symbol to symbol and has variance $\sigma^2$. An Inner Bound to the Capacity {#subsec:achievable_rate_multirelay_xie_and_kumar} ------------------------------ For the multi-relay channel, we denote the symbol transmitted by the $i$-th node at time $t$ ($t$ is discrete) by $X_{i}(t)$ for $i=0,1,\cdots,N$. $Z_{k}(t) \sim \mathcal{N}(0,\sigma^{2})$ is the additive white Gaussian noise at node $k$ and time $t$, and is assumed to be [*independent and identically distributed across $k$ and $t$.*]{} Thus, at symbol time $t$, node $k, 1 \leq k \leq N+1$ receives: $$Y_{k}(t)= \sum_{j\in \{0,1,\cdots,N\}, j \neq k} h_{j,k}X_{j}(t)+Z_{k}(t) \label{eqn:network_equation}$$ An inner bound to the capacity of this network, under any path-loss model, is given by (see [@xie-kumar04network-information-theory-scaling-law]): $$R=\min_{1 \leq k \leq N+1} C \bigg( \frac{1}{\sigma^{2}×} \sum_{j=1}^{k} ( \sum_{i=0}^{j-1} h_{i,k} \sqrt{P_{i,j}} )^{2} \bigg) \label{eqn:achievable_rate_multirelay}$$ where $C(x):=\frac{1}{2}\log_{2}(1+x)$, and node $i$ transmits to node $j$ at power $P_{i,j}$ (expressed in mW). In Appendix \[section:coding\_scheme\_description\], we provide a descriptive overview of the coding and decoding scheme proposed in [@xie-kumar04network-information-theory-scaling-law]. A sequence of messages are sent from the source to the sink; each message is encoded in a block of symbols and transmitted by using the relay nodes. The scheme involves [*coherent transmission*]{} by the source and relay nodes (this requires [*symbol-level synchronization among the nodes*]{}), and [*successive interference cancellation*]{} at the relay nodes and the sink. A node receives information about a message in two ways (i) by the message being directed to it cooperatively by all the previous nodes, and (ii) by overhearing previous transmissions of the message to the previous nodes. Thus node $k$ receives codes corresponding to a message $k$ times before it attempts to decode the message (a discussion on the practical feasibility of full-duplex decode-and-forward relaying scheme is provided in Section \[subsection:motivation-for-full-duplex-decode-forward\]). Note that, $C \bigg( \frac{1}{\sigma^{2}×} \sum_{j=1}^{k} ( \sum_{i=0}^{j-1} h_{i,k} \sqrt{P_{i,j}} )^{2} \bigg)$ in (\[eqn:achievable\_rate\_multirelay\]), for any $k$, denotes a possible rate that can be achieved by node $k$ from the transmissions from nodes $0,1,\cdots,k-1$. The smallest of these terms becomes the bottleneck, see (\[eqn:achievable\_rate\_multirelay\]). For the single relay channel, $N=1$. Thus, by (\[eqn:achievable\_rate\_multirelay\]), an achievable rate is given by (see also [@cover-gamal79capacity-relay-channel]): $$\begin{aligned} R=\min & \bigg\{ & C \left(\frac{g_{0,1}P_{0,1}}{\sigma^{2}×}\right), \nonumber\\ && C \left( \frac{g_{0,2}P_{0,1}+(h_{0,2}\sqrt{P_{0,2}}+h_{1,2}\sqrt{P_{1,2}})^{2}}{\sigma^{2}×} \right) \bigg\}\label{eqn:single_relay_genaral_capacity_formula}\end{aligned}$$ Here, the first term in the $\min\{\cdot,\cdot\}$ of (\[eqn:single\_relay\_genaral\_capacity\_formula\]) is the achievable rate at node $1$ (i.e., the relay node) due to the transmission from the source. The second term in the $\min\{\cdot,\cdot\}$ corresponds to the possible achievable rate at the sink node due to direct coherent transmission from the source and the relay and due to the overheard transmission from the source to the relay. The higher the channel attenuation, the less will be the contribution of farther nodes, “overheard” transmissions become less relevant, and coherent transmission reduces to a simple transmission from the previous relay. The system is then closer to simple store-and-forward relaying. The authors of [@xie-kumar04network-information-theory-scaling-law] have shown that any rate strictly less than $R$ is achievable through the coding and decoding scheme. This achievable rate formula can also be obtained from the capacity formula of a physically degraded multi-relay channel (see [@reznik-etal04degraded-gaussian-multirelay-channel]), since the capacity of the degraded relay channel is a lower bound to the actual channel capacity. [*In this paper, we will seek to optimize $R$ in (\[eqn:achievable\_rate\_multirelay\]) over power allocations to the nodes and the node locations, keeping in mind that $R$ is a lower bound to the actual capacity. We denote the value of $R$ optimized over power allocation and relay locations by $R^*$.*]{} Path-Loss Model --------------- We model the power gain via the exponential path-loss model: the power gain at a distance $r$ is $e^{-\rho r}$ where $\rho > 0$. This is a simple model used for tractability (see [@firouzabadi-martins08optimal-node-placement], [@appuswamy-etal10relay-placement-deterministic-line]) and [@altman-etal11greec-cellular Section $2.3$] for prior work assuming exponential path-loss). However, for propagation scenarios involving randomly placed scatterers (as would be the case in a dense urban environment, or a forest, for example) analytical and experimental support has been provided for the exponential path-loss model (a discussion has been provided in Section \[subsection:motivation-for-exponential-path-loss\]). We also discuss in Section \[subsection:insights\_for\_power\_law\_from\_exponential\] how the insights obtained from the results for exponential path-loss can be used for power-law path-loss (power gain at a distance $r$ is $r^{-\eta}$, $\eta>0$). Deployment with other path-loss models is left in this paper as a possible future work. Under exponential path-loss, the channel gains and power gains in the line network become multiplicative, e.g., $h_{i,i+2}=h_{i,i+1}h_{i+1,i+2}$ and $g_{i,i+2}=g_{i,i+1}g_{i+1,i+2}$ for $i \in \{0,1,\cdots,N-1\}$. We discuss in Section \[subsection:shadowing-fading\] how shadowing and fading can be taken care of in our model, by providing a fade- margin in the power at each transmitter. Motivation for the Sum Power Constraint {#subsec:sum_power_constraint} --------------------------------------- In this paper we consider the sum power constraint $\sum_{i=0}^{N}P_{i}=P_{T}$ (in mW) over the source and the relays. This constraint has the following motivation. Let the fixed power expended in a relay (for reception and driving the electronic circuits) be denoted by $P_{\mathrm{rcv}}$ (expressed in mW), and the initial battery energy in each node be denoted by $E$ (in mJ unit). The information theoretic approach utilized in this paper requires that the nodes in the network are always on. Hence, the lifetime of node $i, 1 \leq i \leq N$, is $\tau_i=\frac{E}{P_i+P_{rcv}×}$, the lifetime of the source is $\tau_0=\frac{E}{P_0}$, and that of the sink is $\tau_{N+1}=\frac{E}{P_{rcv}}$. The rate of battery replacement at node $i$ is $\frac{1}{\tau_i}$. Hence, the rate at which we have to replace the batteries in the network is $\sum_{i=0}^{N+1}\frac{1}{\tau_i×}=\frac{1}{E×}(\sum_{i=0}^{N}P_i+(N+1)P_{rcv})$. The depletion rate $\frac{P_{rcv}}{E}$ is inevitable at any node, and it does not affect the achievable data rate. Hence, in order to reduce the battery replacement rate, we must reduce the sum transmit power in the entire network. Placement on a Line of Known Length {#sec:total_power_constraint} =================================== As a precursor to addressing the deploy-as-you-go problem over a line of unknown length, in this section we solve the problem of power constrained deployment of [*a given number of relays on a line of known length*]{}. We will often refer to this problem as [*offline deployment problem*]{}. The results of this section provide (i) first insights into the relay placements we obtain using the multi-relay channel model, (ii) a starting point for the formulation of as-you-go deployment problem, and (iii) a benchmark with which we can compare the performance of our as-you-go deployment algorithm. Optimal Power Allocation {#subsec:optimal_power_allocation} ------------------------ In this section, we consider the optimal placement of relay nodes on a line of given length, $L$, so as to to maximize $R$ (see (\[eqn:achievable\_rate\_multirelay\])), subject to a total power constraint on the source and relay nodes given by $\sum_{i=0}^{N}P_{i}=P_{T}$. We will first maximize $R$ in (\[eqn:achievable\_rate\_multirelay\]) over $P_{i,j}, 0 \leq i < j \leq (N+1)$ for any given placement of nodes (i.e., given $y_{1}, y_{2},\cdots,y_{N}$). This will provide an expression of achievable rate in terms of channel gains, which has to be maximized over $y_{1}, y_{2},\cdots,y_{N}$. Let $\gamma_{k}:=\sum_{i=0}^{k-1}P_{i,k}$ for $k \in \{1,2,\cdots,N+1\}$ (expressed in mW). Hence, the sum power constraint becomes $\sum_{k=1}^{N+1}\gamma_{k}=P_{T}$. \[theorem:multirelay\_capacity\] 1. Under the exponential path-loss model, for fixed location of relay nodes, the optimal power allocation that maximizes the achievable rate for the sum power constraint is given by: $$P_{i,j}= \begin{cases} \frac{g_{i,j}}{\sum_{l=0}^{j-1}g_{l,j}×}\gamma_{j}\,\, & \forall 0 \leq i <j \leq (N+1) \\ 0, \,\, &\text{if}\,\, j \leq i \end{cases}\label{eqn:power_gamma_relation}$$ where $$\begin{aligned} \gamma_{1}&=&\frac{P_{T}×}{1+g_{0,1}\sum_{k=2}^{N+1} \frac{(g_{0,k-1}-g_{0,k})}{g_{0,k}g_{0,k-1}\sum_{l=0}^{k-1}\frac{1}{g_{0,l}×}×} ×} \label{eqn:gamma_one}\\ \gamma_{j}&=&\frac{g_{0,1}\frac{(g_{0,j-1}-g_{0,j})}{g_{0,j}g_{0,j-1}\sum_{l=0}^{j-1}\frac{1}{g_{0,l}×}×}×}{1+g_{0,1}\sum_{k=2}^{N+1} \frac{(g_{0,k-1}-g_{0,k})}{g_{0,k}g_{0,k-1}\sum_{l=0}^{k-1}\frac{1}{g_{0,l}×}×} ×} P_{T} \,\,\,\, \forall \, j \geq 2 \label{eqn:gamma_k}\nonumber\end{aligned}$$ 2. The achievable rate optimized over the power allocation for a given placement of nodes is given by: $$R^{opt}_{P_T}(y_1,y_2,\cdots,y_N)=C \bigg( \frac{\frac{P_{T}}{\sigma^{2}×}} {\frac{1}{g_{0,1}×}+\sum_{k=2}^{N+1} \frac{(g_{0,k-1}-g_{0,k})}{g_{0,k}g_{0,k-1}\sum_{l=0}^{k-1}\frac{1}{g_{0,l}×}×} ×} \bigg)\label{eqn:capacity_multirelay}$$ The basic idea is to choose the power levels (i.e., $P_{i,j},\, 0 \leq i<j \leq N+1$) in (\[eqn:achievable\_rate\_multirelay\]) so that all the terms in the $\min\{\cdot\}$ in (\[eqn:achievable\_rate\_multirelay\]) become equal. We provide explicit expressions for $P_{i,j},\,0 \leq i<j \leq N+1$ and the achievable rate (optimized over power allocation) in terms of the power gains. See Appendix \[appendix:proof\_of\_multirelay\_channel\_capacity\_theorem\_after\_power\_allocation\] for the detailed proof. A result on the equality of certain terms under optimal power allocation has also been proved in [@reznik-etal04degraded-gaussian-multirelay-channel] for the coding scheme used in [@reznik-etal04degraded-gaussian-multirelay-channel]. But it was proved in the context of a degraded Gaussian multi-relay channel, and the proof depends on an inductive argument, whereas our proof utilizes LP (linear programming) duality. Recalling the exponential path-loss parameter $\rho$, and the source-sink distance $L$, let us define $\lambda := \rho L$, which can be treated as a measure of attenuation in the line. Let us now comment on the results of Theorem \[theorem:multirelay\_capacity\]: - In order to maximize $R^{opt}_{P_T}(y_1,y_2,\cdots,y_N)$, we need to place the relay nodes such that $\frac{1}{g_{0,1}}+\sum_{k=2}^{N+1}\frac{(g_{0,k-1}-g_{0,k})}{g_{0,k}g_{0,k-1}\sum_{l=0}^{k-1}\frac{1}{g_{0,l}×}×}$ is minimized. This quantity is viewed as the net attenuation the power $P_T$ faces. - When no relay is placed, the attenuation is $e^{\lambda}$. The ratio of attenuation with no relay and attenuation with relays is called the “relaying gain” $G(N,\lambda)$. $$G(N,\lambda):=\frac{e^{\lambda}}{\frac{1}{g_{0,1}×}+\sum_{k=2}^{N+1}\frac{(g_{0,k-1}-g_{0,k})}{g_{0,k}g_{0,k-1}\sum_{l=0}^{k-1}\frac{1}{g_{0,l}×}×}×} \label{eqn:gain_definition}$$ Rate is increasing with the number of relays, and is bounded above by $C(\frac{P_T}{\sigma^2})$. Hence, $G(N,\lambda) \in [1, e^{\lambda}]$. Also, note that $G(N,\lambda)$ does not depend on $P_T$. - By Theorem \[theorem:multirelay\_capacity\], we have $P_{k,j} \geq P_{i,j}$ for $i<k<j$. - Note that we have derived Theorem \[theorem:multirelay\_capacity\] using the fact that $g_{0,k}$ is nonincreasing in $k$. If there exists some $k \geq 1$ such that $g_{0,k}=g_{0,k+1}$, i.e, if $k$-th and $(k+1)$-st nodes are placed at the same position, then $\gamma_{k+1} = 0$, i.e., the nodes $i < k$ do not direct any power specifically to relay $k+1$. However, relay $k+1$ can decode the symbols received at relay $k$, and those transmitted by relay $k$. Then relay $(k+1)$ can transmit coherently with the nodes $l \leq k$ to improve effective received power in the nodes $j > k+1$. ![Single relay placement, total power constraint, exponential path-loss: $\frac{y_{1}^{*}}{L×}$ and optimum $\frac{P_{0,1}}{P_{T}×}$ versus $\lambda$.[]{data-label="fig:single_relay_total_power"}](single-relay-total-power-position-power.pdf){height="3cm" width="8cm"} Optimal Placement of a Single Relay Node {#subsection:optimal_placement_single_relay_sum_power} ---------------------------------------- In the following theorem, we derive the optimal power allocation, node location and data rate when a single relay is placed. \[theorem:single\_relay\_total\_power\] For the single relay node placement problem with sum power constraint and exponential path-loss model, the normalized optimum relay location $\frac{y_{1}^{*}}{L×}$, power allocation and optimized achievable rate are given as follows:[^5] 1. [*For $\lambda \leq \log 3$*]{}, $\frac{y_{1}^{*}}{L×}=0$, $P_{0,1}=\frac{2P_{T}}{e^{\lambda}+1×}$, $P_{0,2}=P_{1,2}=\frac{e^{\lambda}-1}{e^{\lambda}+1×}\frac{P_{T}}{2×}$ and $R^{*}=C \left(\frac{2P_{T}}{(e^{\lambda}+1)\sigma^{2}×}\right)$. 2. [*For $\lambda \geq \log 3$*]{}, $\frac{y_{1}^{*}}{L×}=\frac{1}{\lambda×} \log \left(\sqrt{e^{\lambda}+1}-1\right)$, $P_{0,1}=\frac{P_{T}}{2×}$, $P_{0,2}=\frac{1}{\sqrt{e^{\lambda}+1}×}\frac{P_{T}}{2×}$, $P_{1,2}=\frac{\sqrt{e^{\lambda}+1}-1}{\sqrt{e^{\lambda}+1}×}\frac{P_{T}}{2×}$ and $R^{*}=C \left( \frac{1}{\sqrt{e^{\lambda}+1}-1×}\frac{P_{T}}{2 \sigma^{2}×} \right)$ See Appendix \[appendix:proof\_of\_single\_relay\_sum\_power\_results\]. [*Discussion:*]{} It is easy to check that $R^{*}$ obtained in Theorem \[theorem:single\_relay\_total\_power\] is strictly greater than the AWGN capacity $C \left(\frac{P_{T}}{\sigma^{2}×}e^{-\lambda}\right)$ for all $\lambda>0$. This happens because the source and relay transmit coherently to the sink. $R^{*}$ becomes equal to the AWGN capacity only at $\lambda=0$. At $\lambda=0$, we do not use the relay since the sink can decode any message that the relay is able to decode. The variation of $\frac{y_{1}^{*}}{L×}$ and $\frac{P_{0,1}}{P_{T}}$ with $\lambda$ has been shown in Figure \[fig:single\_relay\_total\_power\]. We observe that (from Figure \[fig:single\_relay\_total\_power\] and Theorem \[theorem:single\_relay\_total\_power\]) $\lim_{\lambda \rightarrow \infty} \frac{y_{1}^{*}}{L}=\frac{1}{2×}$, $\lim_{\lambda \rightarrow \infty}P_{0,2}=0$ and $\lim_{\lambda \rightarrow 0}P_{0,1}=P_{T}$. For large values of $\lambda$, source and relay cooperation provides negligible benefit since source to sink attenuation is very high. So it is optimal to place the relay at a distance $\frac{L}{2×}$. The relay works as a repeater which forwards data received from the source to the sink. For small $\lambda$, the gain obtained from coherent transmission is dominant, and, in order to receive sufficient information (required for coherent transmission) from the source, the relay is placed near the source. Optimal Placement for a Multi-Relay Channel {#subsection:properties_of_gain} ------------------------------------------- As we discussed earlier, we need to place $N$ relay nodes such that $\frac{1}{g_{0,1}×}+\sum_{k=2}^{N+1} \frac{(g_{0,k-1}-g_{0,k})}{g_{0,k}g_{0,k-1}\sum_{l=0}^{k-1}\frac{1}{g_{0,l}×}×}$ is minimized. Here $g_{0,k}=e^{-\rho y_{k}}$. We have the constraint $0 \leq y_{1} \leq y_{2} \leq \cdots \leq y_{N} \leq y_{N+1}=L$. Now, writing $z_{k}=e^{\rho y_{k}}$, and defining $z_{0}:=1$, we arrive at the following problem: $$\begin{aligned} & & \min \bigg\{ z_{1}+\sum_{k=2}^{N+1} \frac{z_{k}-z_{k-1}}{\sum_{l=0}^{k-1} z_{l}}\bigg\}\nonumber\\ & s.t & \,\, 1 \leq z_{1} \leq \cdots \leq z_{N} \leq z_{N+1} = e^{\lambda} \label{eqn:multirelay_optimization}\end{aligned}$$ The objective function is convex in each of the variables $z_{1}, z_{2},\cdots, z_{N}$. The objective function is sum of linear fractionals, and the constraints are linear. [*Remark:*]{} From optimization problem (\[eqn:multirelay\_optimization\]) we observe that optimum $z_{1},z_{2},\cdots,z_{N}$ depend only on $\lambda:=\rho L$. Since $z_{k}=e^{\lambda \frac{y_{k}}{L}}$, the normalized optimal distance of relays from the source depend only on $\lambda$ and $N$. \[theorem:capacity\_increasing\_with\_N\] For fixed $\rho$, $L$ and $\sigma^{2}$, the optimized achievable rate $R^{*}$ for a sum power constraint *strictly* increases with the number of relay nodes. See Appendix \[appendix:proof\_of\_capacity\_increases\_in\_N\]. \[theorem:G\_increasing\_in\_lambda\] For any fixed number of relays $N \geq 1$, $G(N,\lambda)$ is increasing in $\lambda$. See Appendix \[appendix:proof\_of\_G\_increasing\_in\_lambda\]. [**A numerical study of multi-relay placement:**]{} We discretize the interval $[0,L]$ and run a search program to find normalized optimal relay locations for different values of $\lambda$ and $N$. The results are summarized in Figure \[fig:multirelay-optimal-location\]. We observe that at low attenuation (small $\lambda$), relay nodes are clustered near the source node and are often at the source node, whereas at high attenuation (large $\lambda$) they are almost uniformly placed along the line. For large $\lambda$, the effect of long distance between any two adjacent nodes dominates the gain obtained by coherent relaying. Hence, it is beneficial to minimize the maximum distance between any two adjacent nodes and thus multihopping is a better strategy in this case. For small $\lambda$, the gain obtained by coherent transmission is dominant. In order to allow this, relays should be able to receive sufficient information from their previous nodes. Thus, they tend to be clustered near the source. In Figure \[fig:G\_vs\_N\] we plot the relaying gain $G(N,\lambda)$ in dB vs. the number of relays $N$, for various values of $\lambda$. As proved in Theorem \[theorem:capacity\_increasing\_with\_N\], we see that $G(N,\lambda)$ increases with $N$ for fixed $\lambda$. On the other hand, $G(N,\lambda)$ increases with $\lambda$ for fixed $N$, as proved in Theorem \[theorem:G\_increasing\_in\_lambda\]. ![$G (N,\lambda)$ vs $N$ for total power constraint.[]{data-label="fig:G_vs_N"}](optimal-placement-fixed-line.pdf){height="3cm" width="8cm"} Uniformly Placed Relays, Large $N$ ---------------------------------- When the relays are uniformly placed, the behaviour of $R^{opt}_{P_T}(y_1,\cdots,y_N)$ (called $R_{N}$ in the next theorem) for large number of relays is captured by the following: \[theorem:large\_nodes\_uniform\] For exponential path-loss and sum power constraint, if $N$ relay nodes are placed uniformly between the source and the sink, resulting in $R_{N}$ achievable rate, then $ \lim_{N \rightarrow \infty} R_N= C \left(\frac{P_{T}}{\sigma^{2}×}\right)$. See Appendix \[appendix:proof\_of\_large\_nodes\_uniform\]. [*Remark:*]{} From Theorem \[theorem:large\_nodes\_uniform\], it is clear that we can achieve a rate arbitrarily close to $C(\frac{P_{T}}{ \sigma ^{2}})$ (i.e., the effect of path-loss can completely be mitigated) by placing a large enough number of relay nodes. In this context, we would like to mention that the variation of broadcast capacity as a function of the number of nodes $N$ (located randomly inside a unit square) was studied in [@zheng06information-dissemination]; but the broadcast capacity in their paper increases with $N$ since they assume per-node power constraint. As-You-Go Deployment of Relays on a Line of Unknown Length {#sec:mdp_total_power} ========================================================== Having developed the problem of placing a given number of relays over a line of fixed, given length, we now turn to the deploy-as-you-go problem. An agent walks along a line, starting from the source and heading towards the sink which is at an unknown distance from the source location, deploying relays as he goes, so as to achieve a multi-relay network when he encounters the sink location (and places the sink there). We model the distance from the source to sink as an exponentially distributed random variable with mean $\overline{L}=\frac{1}{\beta}$.[^6] The deployment objective is to achieve a high data rate from the source to the sink, subject to a total power constraint and a constraint on the expected number of relays placed (note that, the number of relay nodes, $N$, is a random variable here, due to the randomness in $L$). Using the rate expression from Theorem \[theorem:multirelay\_capacity\], we formulate the problem as a total cost MDP. Such a deployment problem could be motivated by a situation where it is required to place a sensor (say, a video camera) to monitor an event or an object from a safe distance (e.g., the battlefront in urban combat, or a suspicious object that needs to be detonated, or a group of animals in a forest). In such a situation, the deployment agent, after placing the sensor, walk away from the scene of the event, along a forest trail, or a road, or a building corridor, placing relays as he walks, until a suitable safe sink location is found, in such a way that the number of relays is kept small while the end-to-end data rate is maximized. Formulation as an MDP {#subsection:mdp_formulation} --------------------- We now formulate the as-you-go deployment problem as an MDP. ### Deployment Policies In the as-you-go placement problem, the person carries a number of nodes and places them as he walks, under the control of a placement policy. A deployment policy $\pi$ is a sequence of mappings $\{\mu_1,\mu_2, \mu_3, \cdots \}$ from the state space to the action space; at the $k$-th decision instant (i.e., after placing the $(k-1)$-st relay), $\mu_k$ provides the distance at which the next relay should be placed (provided that the line does not end before that point), given the system state which is a function of the locations of previously placed nodes. Thus, the decisions are made based on the locations of the relays placed earlier. The first decision instant is the start of the line, and the subsequent decision instants are the placement points of the relays. Let $\Pi$ denote the set (possibly uncountable) of all deployment policies. Let $\mathbb{E}_{\pi}$ denote the expectation under policy $\pi$. ### The Unconstrained Problem We recall from (\[eqn:capacity\_multirelay\]) that for a fixed length $L$ of the line and a fixed $N$, $e^{\rho y_1}+\sum_{k=2}^{N+1}\frac{e^{\rho y_k}-e^{\rho y_{k-1}}}{1+e^{\rho y_1}+\cdots+e^{\rho y_{k-1}}×}$ has to be minimized in order to maximize $R^{opt}_{P_T}(y_1,y_2,\cdots,y_N)$. $e^{\rho y_1}+\sum_{k=2}^{N+1}\frac{e^{\rho y_k}-e^{\rho y_{k-1}}}{1+e^{\rho y_1}+\cdots+e^{\rho y_{k-1}}×}$ is basically a scaling factor which captures the effect of attenuation and relaying on the maximum possible SNR $\frac{P_T}{\sigma^2}$. Let $\xi>0$ be the cost of placing a relay. We are interested in solving the following problem: $$\begin{aligned} \inf_{\pi\in \Pi } \mathbb{E}_{\pi} \left( \left(e^{\rho y_1}+ \sum_{k=2}^{N+1}\frac{e^{\rho y_k}-e^{\rho y_{k-1}}}{1+e^{\rho y_1}+ \cdots+e^{\rho y_{k-1}}×} \right)+\xi N \right)\label{eqn:unconstrained_mdp}\end{aligned}$$ The “cost” function inside the outer parentheses in (\[eqn:unconstrained\_mdp\]) has two terms, one is the denominator of $G(N,\lambda)$ in (\[eqn:gain\_definition\]), and the other is a linear multiple of the number of relays. Thus, the cost function captures the tradeoff between the cost of placing relays (quantified as $\xi$ per relay), and the need to achieve high end-to-end data rate by making the denominator of $G(N,\lambda)$ small. Note that, due to the randomness in the length of the line, the $y_k, k \geq 1,$ and $N$ are all random variables.[^7] We will see in Theorem \[theorem:convergence\_of\_value\_iteration\] that an optimal policy always exists for this problem. ### The Constrained Problem Solving the problem in (\[eqn:unconstrained\_mdp\]) also helps in solving the following constrained problem: $$\begin{aligned} & & \inf_{\pi\in \Pi } \mathbb{E}_{\pi} \left(e^{\rho y_1}+ \sum_{k=2}^{N+1}\frac{e^{\rho y_k}-e^{\rho y_{k-1}}}{1+e^{\rho y_1}+\cdots+e^{\rho y_{k-1}}×}\right)\nonumber\\ &\textit{s.t., }&\mathbb{E}_{\pi}(N) \leq M \label{eqn:constrained_mdp}\end{aligned}$$ where $M>0$ is a constraint on the expected number of relays. [^8] The following standard result (see [@beutler-ross85optimal-policies-controlled-markov-chains-constraint Theorem $4.3$]) gives the optimal $\xi^*$: \[lemma:choice-of-xi\] If there exists $\xi^*>0$ and a policy $\pi_{\xi^*}^* \in \Pi$ such that $\pi_{\xi^*}^*$ is an optimal policy for the unconstrained problem (\[eqn:unconstrained\_mdp\]) under $\xi^*$ and $\mathbb{E}_{\pi_{\xi^*}^*}N=M$, then $\pi_{\xi^*}^*$ is also optimal for the constrained problem (\[eqn:constrained\_mdp\]). The motivation behind formulation (\[eqn:constrained\_mdp\]) is as follows. Suppose that one seeks to solve the following problem: $$\begin{aligned} && sup_{\pi } \mathbb{E}_{\pi} \log_2 \bigg(1+\frac{\frac{P_T}{\sigma^2×}}{e^{\rho y_1}+ \sum_{k=2}^{N+1}\frac{e^{\rho y_k}-e^{\rho y_{k-1}}}{1+e^{\rho y_1}+\cdots+e^{\rho y_{k-1}}×}}\bigg)\nonumber\\ &&\textit{s.t., } \mathbb{E}_{\pi}(N) \leq M \label{eqn:constrained_mdp_actual}\end{aligned}$$ Since $\log_2 (1+\frac{1}{x})$ is convex in $x$, we can argue by Jensen’s inequality that by solving (\[eqn:constrained\_mdp\]) we actually find a relay placement policy that maximizes a lower bound to the expected achievable data rate obtained from (\[eqn:constrained\_mdp\_actual\]). But formulation (\[eqn:constrained\_mdp\]) (and hence formulation (\[eqn:unconstrained\_mdp\]), by Lemma \[lemma:choice-of-xi\]) allows us to write the objective function as a summation of hop-costs; this motivates us to formulate the as-you-go deployment problem as an MDP, resulting in a substantial reduction in policy computation. However, in Section \[section:numerical\_work\_information\_theoretic\_model\], we will show numerically that solving (\[eqn:constrained\_mdp\]) is a reasonable approach to deal with the computational complexity of (\[eqn:constrained\_mdp\_actual\]); we will see that formulation (\[eqn:constrained\_mdp\]) allows us to achieve a reasonable performance. We now formulate the above “as-you-go" relay placement problem (\[eqn:unconstrained\_mdp\]) as a total cost Markov decision process. ### State Space, Action Space and Cost Structure Let us define $s_0:=1$, $s_k:=\frac{e^{\rho y_k}}{1+e^{\rho y_1}+\cdots+e^{\rho y_k}×}\, \forall \, k \geq 1$. Also, recall that $r_{k+1}=y_{k+1}-y_k$. Thus, we can rewrite (\[eqn:unconstrained\_mdp\]) as: $$\begin{aligned} \inf_{\pi\in \Pi } \mathbb{E}_{\pi} \left(1+\sum_{k=0}^{N}s_k(e^{\rho r_{k+1}}-1)+\xi N \right) \label{eqn:unconstrained_mdp_with_state}\end{aligned}$$ When the person starts walking from the source along the line, the state of the system is set to $s_0:=1$. At this instant the placement policy provides the location at which the first relay should be placed. The person walks towards the prescribed placement point. If the sink placement location is encountered before reaching this point, the sink is placed; if not, then the first relay is placed at the placement point. In general, the state after placing the $k$-th relay is denoted by $s_k$ (a function of the location of the nodes up to the $k$-th instant), for $k=1,2,\cdots$. At state $s_k$, the action is the distance $r_{k+1}$ where the next relay has to be placed (action $\infty$ means that no further relay will be placed). If the line ends before this distance, the sink node has to be placed at the end. [*The randomness is coming from the random residual length of the line.*]{} Let $l_k$ denote the residual length at the $k$-th instant. With this notation, the state of the system evolves as: $$\begin{aligned} s_{k+1}= \begin{cases} \frac{s_k e^{\rho r_{k+1}}}{1+s_k e^{\rho r_{k+1}}×}, \text{ if $l_k > r_{k+1}$} , \\ \mathbf{EOL}, \text{ \hspace{7mm} else}. \label{eqn:state-evolution} \end{cases}\end{aligned}$$ Here $\mathbf{EOL}$ denotes the end of the line, i.e., the termination state. The single stage cost (for problem (\[eqn:unconstrained\_mdp\_with\_state\])) for state $s$, action $a$ and residual length $l$, is: $$\begin{aligned} c(s,a,l)= \begin{cases} \xi + s(e^{\rho a}-1), \text{ if $l>a$},\\ s(e^{\rho l}-1), \text{\hspace{6.5mm} else}. \label{eqn:single-stage-cost} \end{cases}\end{aligned}$$ Also, $c(\mathbf{EOL},a,\cdot)=0$ for all $a$. From (\[eqn:state-evolution\]), it is clear that the next state $s_{k+1}$ depends on the current state $s_k$, the current action $r_{k+1}$ and the residual length of the line. Since the length of the line is exponentially distributed, from any placement point, the residual line length is exponentially distributed, and independent of the history of the process. The cost incurred at the $k$-th decision instant is given by (\[eqn:single-stage-cost\]), which depends on $s_k$, $r_{k+1}$ and $l_k$. Hence, our formulation in (\[eqn:unconstrained\_mdp\_with\_state\]) is an MDP with state space $\mathcal{S}:=(0,1]\cup \{\mathbf{EOL}\}$ and action space $\mathcal{A}\cup \{\infty \}$ where $\mathcal{A}:=[0,\infty)$. [*Remark:*]{} An optimal policy (if it exists) for the problem (\[eqn:unconstrained\_mdp\]) will be used to place relay nodes along a line whose length is a sample from an exponential distribution with mean $\frac{1}{\beta×}$. After the deployment is over, the power $P_T$ will be shared optimally among the source and the deployed relay nodes (according to Theorem \[theorem:multirelay\_capacity\]). Optimal Value Function {#subsection:analysis_of_mdp} ---------------------- Suppose $s_k=s$ for some $k \geq 0$. Then, the optimal value function (cost-to-go) at state $s$ is defined by: $$J_{\xi}(s)=\inf_{\pi \in \Pi} \mathbb{E} \left(\sum_{n=k}^{\infty} c(s_n, a_n, l_n)|s_k=s \right)$$ If we decide to place the next relay at a distance $a<\infty$ and follow the optimal policy thereafter, the expected cost-to-go at a state $s \in (0,1]$ becomes: $$\int_{0}^{a}\beta e^{-\beta z}s(e^{\rho z}-1)dz + e^{-\beta a}\bigg(s(e^{\rho a}-1)+\xi+J_{\xi}\left(\frac{se^{\rho a}}{1+se^{\rho a}×}\right)\bigg)\label{eqn:cost-to-go}$$ The first term in (\[eqn:cost-to-go\]) corresponds to the case in which the line ends at a distance less than $a$ and we are forced to place the sink node. The second term corresponds to the case where the residual length of the line is greater than $a$ and a relay is placed at a distance $a$. Note that our MDP has an uncountable state space $\mathcal{S}=(0,1] \cup \{\mathbf{EOL}\}$ and a non-compact action space $\mathcal{A}=[0,\infty) \cup \{\infty \}$. Several technical issues arise in this kind of problems, such as the existence of optimal or $\epsilon$-optimal policies, measurability of the policies, etc. We, therefore, invoke the results provided by Schäl [@schal75conditions-optimality], which deal with such issues. Our problem is one of minimizing total, undiscounted, non-negative costs over an infinite horizon. Equivalently, in the context of [@schal75conditions-optimality], we have a problem of total reward maximization where the rewards are the negative of the costs. Thus, our problem specifically fits into the negative dynamic programming setting of [@schal75conditions-optimality] (i.e., the $\mathsf{N}$ case where single-stage rewards are non-positive). Now, the state $\mathbf{EOL}$ is absorbing. Also, no action is taken at this state and the cost at this state is $0$. Hence, we can think of this state as state $0$ in order to make our state space a Borel subset of the real line. \[thm:schal\_bellman\_eqn\] \[[@schal75conditions-optimality], Equation (3.6)\] The optimal value function $J_{\xi}(\cdot)$ satisfies the Bellman equation. [$\Box$]{} Thus, $J_\xi(\cdot)$ satisfies the following Bellman equation for each $s \in (0,1]$: $$\begin{aligned} J_{\xi}(s) &=& \min \bigg \{\inf_{a \geq 0} \bigg[\int_{0}^{a}\beta e^{-\beta z}s(e^{\rho z}-1)dz \nonumber\\ && + e^{-\beta a}\bigg(s(e^{\rho a}-1)+\xi+J_{\xi}\left(\frac{se^{\rho a}}{1+se^{\rho a}×}\right)\bigg)\bigg], \nonumber\\ && \int_{0}^{\infty}\beta e^{-\beta z}s(e^{\rho z}-1)dz \bigg \} \label{eqn:bellman_unbroken} \end{aligned}$$ where the second term inside $\min\{\cdot, \cdot \}$ is the cost of not placing any relay (i.e., $a=\infty$). We analyze the MDP for $\beta>\rho$ and $\beta \leq \rho$. ### Case I ($\beta>\rho$) We observe that the cost of not placing any relay (i.e., $a=\infty$) at state $s \in (0,1]$ is given by: $$\begin{aligned} \int_{0}^{\infty} \beta e^{-\beta z}s(e^{\rho z}-1)dz=\theta s\end{aligned}$$ where $\theta:=\frac{\rho}{\beta-\rho×}$ (using the fact that $\beta>\rho$). Since not placing a relay (i.e., $a = \infty$) is a possible action for every $s$, it follows that $J_{\xi}(s)\leq \theta s $. The cost in (\[eqn:cost-to-go\]), upon simplification, can be written as: $$\begin{aligned} \theta s + e^{-\beta a}\bigg(-\theta s e^{\rho a}+\xi+ J_{\xi}\left(\frac{se^{\rho a}}{1+se^{\rho a}×}\right)\bigg)\label{eqn:cost_to_go_beta_greater_than_rho}\end{aligned}$$ Since $J_{\xi}(s) \leq \theta$ for all $s \in (0,1]$, the expression in (\[eqn:cost\_to\_go\_beta\_greater\_than\_rho\]) is strictly less that $\theta s$ for large enough $a<\infty$. Hence, according to (\[eqn:bellman\_unbroken\]), it is not optimal to not place any relay and the Bellman equation (\[eqn:bellman\_unbroken\]) can be rewritten as: $$\begin{aligned} J_{\xi}(s)= \theta s + \inf_{a \geq 0} e^{-\beta a}\bigg(-\theta s e^{\rho a}+ \xi+J_{\xi}\left(\frac{se^{\rho a}}{1+se^{\rho a}×}\right)\bigg)\label{eqn:bellman_equation_simplified_in_a}\end{aligned}$$ ### Case II ($\beta \leq \rho$) Here the cost in (\[eqn:cost-to-go\]) is $\infty$ if we do not place a relay (i.e., if $a=\infty$). Let us consider a policy $\pi_1$ where we place the next relay at a fixed distance $0 <a <\infty$ from the current relay, irrespective of the current state. If the residual length of the line is $z$ at any state $s$, we will place less than $\frac{z}{a×}$ additional relays, and for each relay a cost less than $(\xi+(e^{\rho a}-1))$ is incurred (since $s \leq 1$). At the last step when we place the sink, a cost less than $(e^{\rho a}-1)$ is incurred. Thus, the value function of this policy is upper bounded by: $$\begin{aligned} && \int_{0}^{\infty} \beta e^{-\beta z} \frac{z}{a×}(\xi+(e^{\rho a}-1)) dz+(e^{\rho a}-1) \nonumber\\ &=& \frac{1}{\beta a ×}\left(\xi+(e^{\rho a}-1) \right)+(e^{\rho a}-1)\label{eqn:upper_bound_on_cost_beta_leq_rho}\end{aligned}$$ Hence, $J_{\xi}(s) \leq \frac{1}{\beta a ×}\left(\xi+(e^{\rho a}-1) \right)+(e^{\rho a}-1) < \infty$. Thus, by the same argument as in the case $\beta > \rho$, the minimizer in the Bellman equation lies in $[0,\infty)$, i.e., the optimal placement distance lies in $[0,\infty)$. Hence, (\[eqn:bellman\_unbroken\]) can be rewritten as: $$\begin{aligned} J_{\xi}(s) &=& \inf_{a \geq 0} \bigg\{\int_{0}^{a}\beta e^{-\beta z}s(e^{\rho z}-1)dz + \nonumber\\ && e^{-\beta a}\bigg(s(e^{\rho a}-1)+\xi+J_{\xi}\left(\frac{se^{\rho a}}{1+se^{\rho a}×}\right)\bigg)\bigg\} \label{eqn:bellman_beta_leq_rho} \end{aligned}$$ Upper Bound on the Optimal Value Function ----------------------------------------- \[prop:upper\_bound\_on\_cost\_beta\_geq\_rho\] If $\beta>\rho$, then $J_{\xi}(s) < \theta s$ for all $s \in (0,1]$. We know that $J_{\xi}(s) \leq \theta s \leq \theta$. Now, let us consider the Bellman equation (\[eqn:bellman\_equation\_simplified\_in\_a\]). It is easy to see that $(-\theta s e^{\rho a}+\xi+J_{\xi}(\frac{se^{\rho a}}{1+se^{\rho a}×}))$ is strictly negative for sufficiently large $a$. Hence, the R.H.S of (\[eqn:bellman\_equation\_simplified\_in\_a\]) is strictly less than $\theta s$. For $\beta>\rho$, $\lim_{s \rightarrow 0} J_{\xi}(s) \rightarrow 0$ for any $\xi>0$. Follows from Proposition \[prop:upper\_bound\_on\_cost\_beta\_geq\_rho\]. \[prop:upper\_bound\_on\_cost\] If $\beta>0$ and $\rho>0$ and $0<a<\infty$, then $J_{\xi}(s) <\frac{1}{\beta a ×}\left(\xi+(e^{\rho a}-1) \right)+(e^{\rho a}-1)$ for all $s \in (0,1]$. Follows from (\[eqn:upper\_bound\_on\_cost\_beta\_leq\_rho\]), since the analysis is valid even for $\beta>\rho$. Convergence of the Value Iteration ---------------------------------- The value iteration for our MDP is given by: $$\begin{aligned} J_{\xi}^{(k+1)}(s) &=& \inf_{a \geq 0} \bigg\{\int_{0}^{a}\beta e^{-\beta z}s(e^{\rho z}-1)dz + e^{-\beta a}\bigg(s(e^{\rho a}-1) \nonumber\\ &&+\xi+J_{\xi}^{(k)}\left(\frac{se^{\rho a}}{1+se^{\rho a}×}\right)\bigg)\bigg\}\label{eqn:value_iteration}\end{aligned}$$ Here $J_{\xi}^{(k)}(s)$ is the $k$-th iterate of the value iteration. Let us start with $J_{\xi}^{(0)}(s):=0$ for all $s \in (0,1]$. We set $J_{\xi}^{(k)}(\mathbf{EOL})=0$ for all $k \geq 0$. $J_{\xi}^{(k)}(s)$ is the optimal value function for a problem with the same single-stage cost and the same transition structure, but with the horizon length being $k$ (instead of infinite horizon as in our original problem) and $0$ terminal cost. Here, by horizon length $k$, we mean that there are $k$ number of relays available for deployment. Let $\Gamma_k(s)$ be the set of minimizers of (\[eqn:value\_iteration\]) at the $k$-th iteration at state $s$, if the infimum is achieved at some $a<\infty$. Let $\Gamma_{\infty}(s):=\{a \in \mathcal{A}:a$ be an accumulation point of some sequence $\{a_k\}$ where each $a_k \in \Gamma_{k}(s)\}$. Let $\Gamma^*(s)$ be the set of minimizers in (\[eqn:bellman\_beta\_leq\_rho\]). In Appendix \[appendix:sequential\_placement\_total\_power\], we show that $\Gamma_k(s)$ for each $k \geq 1$, $\Gamma_{\infty}(s)$ and $\Gamma^*(s)$ are nonempty. \[theorem:convergence\_of\_value\_iteration\] The value iteration given by (\[eqn:value\_iteration\]) has the following properties: 1. $J_{\xi}^{(k)}(s)\rightarrow J_{\xi}(s)$ for all $s \in (0,1]$, i.e., the value iteration converges to the optimal value function. 2. $\Gamma_{\infty}(s) \subset \Gamma^*(s)$. 3. There is a stationary optimal policy $f^{\infty}=\{f,f,f,\cdots\}$ where $f:(0,1] \rightarrow \mathcal{A}$ and $f(s) \in \Gamma_{\infty}(s)$ for all $s \in (0,1]$. The proof is given in Appendix \[appendix:proof\_of\_value\_iteration\_convergence\]. It uses some results from [@schal75conditions-optimality], which have been discussed first. Next, we provide a general theorem (Theorem \[thm:value\_iteration\_general\]) on the convergence of value iteration, which has been used to prove Theorem \[theorem:convergence\_of\_value\_iteration\]. [*Remark:*]{} Since the action space is noncompact, it is not obvious from standard results whether the optimal policy exists. However, we are able to show that in our problem, for each state $s \in (0,1]$, the optimal action will lie in a compact set of the from $[0,a(s)]$, where $a(s)$ is continuous in $s$, and $a(s)$ could possibly go to $\infty$ as $s \rightarrow 0$. The results of [@schal75conditions-optimality] allow us to work with the scenario where for each state $s$, it is sufficient to focus only on a compact action space $[0,a(s)]$. Properties of the Value Function $J_{\xi}(s)$ --------------------------------------------- \[prop:increasing\_concave\_in\_s\] $J_{\xi}(s)$ is increasing and concave over $s \in (0,1]$. \[prop:increasing\_concave\_in\_lambda\] $J_{\xi}(s)$ is increasing and concave in $\xi$ for all $s \in (0,1]$. \[prop:continuity\_of\_cost\] $J_{\xi}(s)$ is continuous in $s$ over $(0,1]$ and continuous in $\xi$ over $(0,\infty)$. See Appendix \[appendix:proof\_of\_propositions\] for the proofs of these propositions. A Useful Normalization {#subsection:a_useful_normalization} ---------------------- Note that, $\beta L$ is exponentially distributed with mean $1$. Defining $\Lambda:=\frac{\rho}{\beta×}$ and $\tilde{z}_k:=\beta y_k$, $k=1,2,\cdots,(N+1)$, we can rewrite (\[eqn:unconstrained\_mdp\]) as follows: $$\begin{aligned} \inf_{\pi\in \Pi } \mathbb{E}_{\pi} \left(e^{\Lambda \tilde{z}_1}+ \sum_{k=2}^{N+1}\frac{e^{\Lambda \tilde{z}_k}-e^{\Lambda \tilde{z}_{k-1}}}{1+e^{\Lambda \tilde{z}_1}+ \cdots+e^{\Lambda \tilde{z}_{k-1}}×}+\xi N \right)\label{eqn:unconstrained_mdp_beta_one}\end{aligned}$$ Note that, $\Lambda$ plays the same role as $\lambda$ played in the known $L$ case (see Section \[subsection:optimal\_placement\_single\_relay\_sum\_power\]). Since $\frac{1}{\beta×}$ is the mean length of the line, $\Lambda$ can be considered as a measure of attenuation in the network. We can think of the new problem (\[eqn:unconstrained\_mdp\_beta\_one\]) in the same way as (\[eqn:unconstrained\_mdp\]), but with the length of the line being exponentially distributed with mean $1$ ($\beta'=1$) and the path-loss exponent being changed to $\rho'=\Lambda=\frac{\rho}{\beta×}$. The relay locations are also normalized ($\tilde{z}_k=\beta y_k$). One can solve the new problem (\[eqn:unconstrained\_mdp\_beta\_one\]) and obtain the optimal policy. Then the solution to (\[eqn:unconstrained\_mdp\]) can be obtained by multiplying each control distance (from the optimal policy of (\[eqn:unconstrained\_mdp\_beta\_one\])) with the constant $\frac{1}{\beta×}$. Hence, it suffices to work with $\beta=1$. A Numerical Study of As-You-Go Deployment {#section:numerical_work_information_theoretic_model} ========================================= ![$\beta=1$, $\Lambda:=\frac{\rho}{\beta}=2$; $a^*$ vs. $s$.[]{data-label="fig:control_vs_state"}](control_vs_state.pdf){height="3cm" width="8cm"} ![$\beta=1$, $\Lambda:=\frac{\rho}{\beta×}=2$; $a^*$ vs. $\xi$.[]{data-label="fig:control_vs_xi"}](control_vs_xi.pdf){height="3cm" width="8cm"} ![$\beta=1$, $\xi=0.01$; $a^*$ vs. $\Lambda$.[]{data-label="fig:control_vs_rho"}](control_vs_rho.pdf){height="3cm" width="8cm"} Let us recall that the state of the system after placing the $k$-th relay is given by $s_k=\frac{e^{\Lambda \tilde{z}_k}}{\sum_{i=0}^{k}e^{\Lambda \tilde{z}_i}}$. The action is the normalized distance of the next relay to be placed from the current location. The single stage cost function for our total cost minimization problem is given by (\[eqn:single-stage-cost\]). In our numerical work, we discretized the state space $(0,1]$ into $100$ steps as $\{0.01,0.02,\cdots,0.99,1\}$, and discretized the action space into steps of size $0.001$, i.e., the action space becomes $\{0,0.001,0.002,\cdots\}$. Structure of the Optimal Policy {#subsection:policy_structure_numerical} ------------------------------- We performed numerical experiments to study the structure of the optimal policy obtained through value iteration for $\beta=1$ and some values of $\Lambda:=\frac{\rho}{\beta}$. The value iteration in these experiments converged and we obtained a stationary optimal policy, though Theorem \[theorem:convergence\_of\_value\_iteration\] does not guarantee the uniqueness of the stationary optimal policy. Figure \[fig:control\_vs\_state\] shows that the [*normalized*]{} optimal placement distance $a^*$ is decreasing with the state $s \in (0,1]$. This can be understood as follows. The state $s$ (at a placement point) is small only if a sufficiently large number of relays have already been placed.[^9] Hence, if several relays have already been placed and $\sum_{i=0}^{k}e^{\Lambda \tilde{z}_i}$ is sufficiently large compared to $e^{\Lambda \tilde{z}_k}$ (i.e., $s_k$ is small), the $(k+1)$-st relay will be able to receive sufficient amount of power from the previous nodes, and hence does not need to be placed close to the $k$-th relay. A value of $s_k$ close to $1$ indicates that there is a large gap between relay $k$ and relay $k-1$, the power received at the next relay from the previous relays is small and hence the next relay must be placed closer to the previous one. On the other hand, $a^*$ is increasing with $\xi$ (see Figure \[fig:control\_vs\_xi\]). Recall that $\xi$ is the price of placing a relay. This figure confirms the intuition that if the relay price is high, then the relays should be placed less frequently. -------------------- ----------------------------------------------------------- -------- $\mathbf{\Lambda}$ Normalised Optimal distances of the nodes No. of from the source relays $0.01$ 0, 0, 8.4180, 10.0000 3 $0.1$ 0, 0, 0, 0, 0, 0, 0, 0.2950, 0.5950, 0.9810, 1.3670, 33 1.7530, 2.1390, $\cdots,$ 9.0870, 9.4730, 9.8590, 10.0000 $5$ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.0020, 0.0080, 0.0140, 1677 0.0200, $\cdots$, 9.9860, 9.9920, 9.9980, 10.0000 -------------------- ----------------------------------------------------------- -------- : Sequential placement on a line of length $10$ for various $\Lambda$, using the corresponding optimal policies for $\xi=0.001$.[]{data-label="table:effect_of_rho_on_placement_1"} $\mathbf{\Lambda}$ Evolution of state in the process of sequential placement -------------------- ------------------------------------------------------------------------ $0.01$ 1, 0.5, 0.34, 0.27 $0.1$ 1, 0.5, 0.34, 0.26, 0.21, 0.18, 0.16, 0.14, 0.13, 0.12, 0.12, $\cdots$ $5$ 1, 0.5, 0.34, 0.26, 0.21, 0.18, 0.16, 0.14, 0.13, 0.12, 0.11, 0.1, 0.1, 0.1, $\cdots$ : Evolution of state in the process of sequential placement on a line of length $10$ for various values of $\Lambda$, using the corresponding optimal policies for $\xi=0.001$.[]{data-label="table:state-evolution_1"} -------------------- ----------------------------------------------------------- -------- $\mathbf{\Lambda}$ Normalised Optimal distances of the nodes No. of from the source relays $0.01$ 10 0 $0.1$ 5.3060, 10.0000 1 $5$ 0, 0.0050, 0.0510, 0.1220, 0.1930, 143 0.2640,$\cdots$, 9.9910, 10.0000 $8$ 0, 0.003, 0.019, 0.06, 0.101, $\cdots$, 9.982, 10 246 $20$ 0, 0.001, 0.003, 0.016, 0.031, 0.046, $\cdots$, 9.991, 10 669 -------------------- ----------------------------------------------------------- -------- : Sequential placement on a line of length $10$ for various $\Lambda$, using the corresponding optimal policies for $\xi=0.1$.[]{data-label="table:effect_of_rho_on_placement_3"} $\mathbf{\Lambda}$ Evolution of state in the process of sequential placement -------------------- ----------------------------------------------------------- $0.01$ 1 $0.1$ 1, 0.63 $5$ 1, 0.5, 0.34, 0.3, 0.3, $\cdots$ $8$ 1, 0.5, 0.34, 0.28, 0.28, $\cdots$ $20$ 1, 0.5, 0.34, 0.27, 0.26, 0.26, $\cdots$ : Evolution of state in the process of sequential placement on a line of length $10$ for various values of $\Lambda$, using the corresponding optimal policies for $\xi=0.1$.[]{data-label="table:state-evolution_3"} Figure \[fig:control\_vs\_rho\] shows that $a^*$ is decreasing with $\Lambda$, for fixed values of $\xi$ and $s$. This happens because increased attenuation will require frequent placement of the relays. Relay Placement Patterns {#subsection:numerical_relay_plecement_pattern_as_you_go} ------------------------ The policy that we use corresponds to a line having exponentially distributed length with mean $1$, but it is applied to the scenario where the actual realization of the (normalised) length (see Section \[subsection:a\_useful\_normalization\]) of the line is $10$. Tables \[table:effect\_of\_rho\_on\_placement\_1\], \[table:effect\_of\_rho\_on\_placement\_3\], and \[table:effect\_of\_rho\_on\_placement\_4\] illustrate some examples of as-you-go placement of relay nodes along a line of normalised length $10$, using various values of $\Lambda$ and $\xi$. Tables \[table:state-evolution\_1\], \[table:state-evolution\_3\], and \[table:state-evolution\_4\] illustrate the corresponding evolution of state as the relays are placed in the examples in Tables \[table:effect\_of\_rho\_on\_placement\_1\], \[table:effect\_of\_rho\_on\_placement\_3\] and \[table:effect\_of\_rho\_on\_placement\_4\]. If the line actually ends at some point before (normalised) distance $10$, the process would end there with the corresponding placement of relays (as can be obtained from Tables \[table:effect\_of\_rho\_on\_placement\_1\], \[table:effect\_of\_rho\_on\_placement\_3\], and \[table:effect\_of\_rho\_on\_placement\_4\]) before the sink being placed at the end-point. Thus, for example, reading from Table \[table:effect\_of\_rho\_on\_placement\_3\] for $\xi=0.1$ and $\Lambda=5$, if the actual normalised length of the line is $0.99$, then one relay will be placed at $0$ (the source itself), followed by $15$ relays at normalised distances $0.005, 0.051, 0.122, 0.193, 0.264, \cdots, 0.974$ from the source, and finally the sink is placed at a normalised distance $0.99$, the end of the line. We observe that as $\Lambda$ increases, more relays need to be placed since the optimal control decreases with $\Lambda$ for each $s$ (see Figure \[fig:control\_vs\_rho\]). On the other hand, the number of relays decreases with increasing $\xi$ (the relay cost); this is in confirmation of the observations from Figure \[fig:control\_vs\_xi\]. Note that, initially one or more relays are placed at or near the source if $a^*(s=1)$ is $0$ or small. But, after some relays have been placed, the relays are placed equally spaced apart. We see that this happens because, after a few relays have been placed, the state, $s$, does not change, hence, resulting in the relays being subsequently placed equally spaced apart. This phenomenon is evident in Table \[table:state-evolution\_1\], Table \[table:state-evolution\_3\], Table \[table:state-evolution\_4\], and Figure \[fig:state\_evolution\]. The state $s$ will remain unchanged after a relay placement if $s=\lceil{\frac{se^{\Lambda a^*(s)}}{0.01(1+se^{\Lambda a^*(s)})×}}\rceil \times 0.01$, since we have discretized the state space. After some relays are placed, the state becomes equal to a fixed point $s'$ of the function $\lceil{\frac{se^{\Lambda a^*(s)}}{0.01(1+se^{\Lambda a^*(s)})×}}\rceil \times 0.01$. Note that the deployment starts from $s_{0}:=1$, but for any value of $s_{0}$ (even with $s_{0}$ smaller than $s'$), we numerically observe the same phenomenon. Hence, $s'$ is an absorbing state. ------- ------------------------------------------------- -------- $\xi$ Normalised Optimal distances of the nodes No. of from the source relays $0.2$ 0 , 0.008, 0.03, 0.052, $\cdots$, 9.996, 10 456 $1$ 0.022, 0.069, 0.116, $\cdots$, 9.986, 10 213 $2$ 0.042, 0.103, 0.163, 0.223, $\cdots$, 9.943, 10 166 $10$ 0.099, 0.205, 0.311, $\cdots$, 9.957, 10 94 ------- ------------------------------------------------- -------- : Sequential placement on a line of length $10$ for of $\xi$, using the corresponding optimal policies for $\Lambda=20$.[]{data-label="table:effect_of_rho_on_placement_4"} $\xi$ Evolution of state in the process of sequential placement ------- ----------------------------------------------------------- $0.2$ 1, 0.5, 0.37, 0.37, $\cdots$ $1$ 1, 0.61, 0.61, $\cdots$ $2$ 1, 0.7, 0.71, 0.71, $\cdots$ $10$ 1, 0.88, 0.88, $\cdots$ : Evolution of state in the process of sequential placement on a line of length $10$ for various values of $\xi$, using the corresponding optimal policies for $\Lambda=20$.[]{data-label="table:state-evolution_4"} Numerical Examples for Practical Deployment {#subsection:numerical-example-practical-deployment} ------------------------------------------- In order to provide a more concrete illustration we adopt a path loss parameter from [@franceschetti-etal04random-walk-model-wave-propagation]. Figure $4$ of [@franceschetti-etal04random-walk-model-wave-propagation] shows that the attenuation in the received signal power in a dense urban environment is roughly $50$ dB when we move from $50$ m distance to $300$ m distance away from the transmitter. This yields a value of $\rho$ to be $0.04$ per meter for the exponential path-loss (see the discussion in Section \[subsection:motivation-for-exponential-path-loss\] on the motivation for choosing the exponential path-loss model in the light of the results from [@franceschetti-etal04random-walk-model-wave-propagation]). Then, $\frac{1}{\beta}=200$ m corresponds to $\Lambda=8$, and $\frac{1}{\beta}=500$ m corresponds to $\Lambda=20$. For $\Lambda=20$, normalised relay locations and state evolution $\{s_k\}_{k \geq 1}$ are available in Tables \[table:effect\_of\_rho\_on\_placement\_3\]-\[table:state-evolution\_4\], and, for $\Lambda=8$, normalised relay locations and state evolution $\{s_k\}_{k \geq 1}$ are available in Tables \[table:effect\_of\_rho\_on\_placement\_3\]-\[table:state-evolution\_3\]. Note that, under $\rho=0.04$ per meter and $\Lambda=20$, one unit normalised distance in the tables correspond to $500$ m distance in the dense urban environment (due to the normalization as in Section \[subsection:a\_useful\_normalization\]). For the sake of illustration, let us consider the sample deployment for $\xi=10$, $\Lambda=20$ (Table \[table:effect\_of\_rho\_on\_placement\_4\]). In this case, the first relay will be placed at a distance $0.099 \times 500=49.5$ m from the source, the second relay will be placed at a distance $0.205 \times 500=102.5$ m from the source, etc. Also, if we choose $\xi$ such that few relays will be placed on a typical line whose length is several hundreds of meters, then the relays will be placed almost uniformly on the line. But, for small $\xi$, more relays will be placed and some of them will be clustered near the source (see the deployment for $\Lambda=8$ and $\xi=0.1$ in Table \[table:effect\_of\_rho\_on\_placement\_3\]). ![Evolution of the state $s_k$ with $k$: initial state $s_0=1$.[]{data-label="fig:state_evolution"}](state_evolution.pdf){height="3cm" width="8cm"} ------- ----------- ------------ ------------- ------------- ------------ Average Mean Number of Maximum $\xi$ $\Lambda$ percentage number of cases where percentage difference relays used no relay difference was used 0.001 0.01 0.0068 2.0002 0 0.7698 0.001 0.1 0.3996 9.4849 0 6.8947 0.01 0.01 0 0 10000 0 0.01 0.1 0.3517 2.2723 0 4.6618 0.01 0.5 1.5661 7.7572 0 4.7789 0.1 0.01 0 0 10000 0 0.1 0.1 0.1259 0.0056 9944 25.9098 0.1 0.5 2.9869 1.8252 0 12.5907 0.1 2 4.7023 7.1530 0 9.0211 0.1 20 3.5472 27.9217 0 6.6223 0.1 8 4.0097 21.0671 0 7.8264 1 8 8.0286 7.8886 495 27.7362 1 20 5.2158 11.2342 402 26.0026 5 20 10.3341 7.1950 597 61.7460 ------- ----------- ------------ ------------- ------------- ------------ : Comparison of the performance (in terms of $H$; see text) of optimal sequential placement over a line of random length, with the optimal placement if the length was known. Results from 10000 samples of exponentially distributed line lengths.[]{data-label="table:comparison_optimal_mdp"} Comparison with Optimal Offline Deployment {#subsubsection:numerical_performance_as_you_go} ------------------------------------------ Since there is no prior work in the literature with which we can make a fair comparison of our as-you-go deployment policy for the full-duplex wireless multi-relay network, we compare the performance of our policy with optimal offline deployment. Thus, the numerical experiments reported in Table \[table:comparison\_optimal\_mdp\] are a result of asking the following question: how does the cost of as-you-go deployment over a line of exponentially distributed length compare with the cost of placing the same number of relays optimally over the line, once the length of the line has been revealed? For several combinations of $\Lambda$ and $\xi$, we generated $10000$ random numbers independently from an exponential distribution with parameter $\beta=1$. Each of these numbers was considered as a possible realization of the length of the line. Then we computed the placement of the relay nodes for each realization by optimal sequential placement policy, which gave us $H=\frac{1}{g_{0,1}×}+\sum_{k=2}^{N+1}\frac{(g_{0,k-1}-g_{0,k})}{g_{0,k}g_{0,k-1}\sum_{l=0}^{k-1}\frac{1}{g_{0,l}×}×}$, a quantity that we use to evaluate the quality of the relay placement. The significance of $H$ can be recalled from (\[eqn:capacity\_multirelay\]) where we found that the rate $C(1+\frac{P_T/\sigma^2}{H})$ can be achieved if total power $P_T$ is available to distribute among the source and the relays; i.e., $H$ can be interpreted as the net effective attenuation after power has been allocated optimally over the nodes. Also, for each realization, we computed $H$ for optimal relay placement, assuming that the length of the line is known before deployment and that the number of relays available is the [*same*]{} as the number of relays used by the corresponding sequential placement policy. For a given combination of $\Lambda$ and $\xi$, for the $k$-th realization of the length of the line, let us denote the two $H$ values by $H_{\mathsf{sequential}}^{(k)}$ and $H_{\mathsf{optimal}}^{(k)}$. Then the percentage [*difference*]{} for the $k$-th realization is: $$e_k:= \frac{|H_{\mathsf{optimal}}^{(k)}-H_{\mathsf{sequential}}^{(k)}|}{H_{\mathsf{optimal}}^{(k)}} \times 100 \label{eqn:error_or_difference_expression}$$ The average percentage difference in Table \[table:comparison\_optimal\_mdp\] is the quantity $\frac{\sum_{k=1}^{10000}e_k}{10000×}$. The maximum percentage difference is the quantity $\max_{k \in \{1,2,\cdots,10000\}}e_k$. **Discussion of Table \[table:comparison\_optimal\_mdp\]:** 1. For small enough $\xi$, some relays will be placed at the source itself. For example, for $\Lambda=0.01$ and $\xi=0.001$, we will place two relays at the source (Table \[table:effect\_of\_rho\_on\_placement\_1\]). After placing the first relay, the next state will become $s=0.5$, and $a^*(s=0.5)=0$. The state after placing the second relay becomes $s=0.34$, for which $a^*(s=0.34)=8.41$ (see the placement in Table \[table:effect\_of\_rho\_on\_placement\_1\]). Now, the line having an exponentially distributed length with mean $1$ will end before $a^*(s=0.34)=8.41$ distance with high probability, and the probability of placing the third relay will be very small. As a result, the mean number of relays will be $2.0002$. In case only $2$ relays are placed by the sequential deployment policy and we seek to place $2$ relays optimally for the same length of the line (with the length known), the optimal locations for both relays are close to the source location if the length of the line is small (i.e., if the attenuation $\lambda$ is small, recall the definition of $\lambda$ from Section \[subsection:optimal\_placement\_single\_relay\_sum\_power\]). If the line is long (which has a very small probability), the optimal placement will be significantly different from the sequential placement. Altogether, the difference (from (\[eqn:error\_or\_difference\_expression\])) will be small. 2. For $\Lambda=0.01$ and $\xi=0.1$, $a^*(1)$ is so large that with high probability the line will end in a distance less than $a^*(1)$ and no relay will be placed. 3. From (\[eqn:capacity\_multirelay\]) we know that for a given placement of relays on a line of given length $L$, the optimal power allocation yields an achievable rate $\log_2(1+\frac{P_T/\sigma^2}{H})$. At the end of as-you-go deployment the power is allocated optimally among the nodes deployed, and a rate $\log_2(1+\frac{P_T/\sigma^2}{H_{sequential}})$ can be achieved. If the same number of relays are optimally placed over the same line, with the same total power, then the inner bound is given by $\log_2(1+\frac{P_T/\sigma^2}{H_{optimal}})$. We seek to compare these two rates numerically. The maximum fractional difference in Table \[table:comparison\_optimal\_mdp\] is less than $\frac{2}{3}$, and substantially smaller than $\frac{2}{3}$ in most cases. Since, in (\[eqn:error\_or\_difference\_expression\]), $H_{sequential}^{(k)}$ is always greater than $H_{optimal}^{(k)}$, we have $H_{sequential}^{(k)} \leq \frac{5}{3} H_{optimal}^{(k)}$ for all $k \geq 1$ (i.e., for all realizations of $L$ in the simulation). Now, by the monotonicity of $\log_2(\cdot)$: $$\begin{aligned} && \frac{1}{2} \log_2 \bigg(1+\frac{P_T/\sigma^2}{H_{optimal}^{(k)}} \bigg)- \frac{1}{2} \log_2 \bigg(1+\frac{P_T/\sigma^2}{H_{sequential}^{(k)}} \bigg) \nonumber\\ & \leq & \frac{1}{2} \log_2 \bigg(1+\frac{P_T/\sigma^2}{H_{optimal}^{(k)}} \bigg)- \frac{1}{2} \log_2 \bigg(1+\frac{P_T/\sigma^2}{\frac{5}{3} H_{optimal}^{(k)}} \bigg) \nonumber\\\end{aligned}$$ Since $\log_2(\cdot)$ is a concave function, for any $x>y>0$, we have $\log_2 (1+x)-\log_2(1+y) \leq \log_2 (x) -\log_2 (y)$. Using this inequality, we can upper bound the difference in achievable rate from the previous equation by: $$\begin{aligned} \frac{1}{2} \log_2 \bigg(\frac{P_T/\sigma^2}{H_{optimal}^{(k)}} \bigg)- \frac{1}{2} \log_2 \bigg(\frac{P_T/\sigma^2}{\frac{5}{3} H_{optimal}^{(k)}} \bigg) = 0.3685\end{aligned}$$ This calculation implies that, for the large number of cases reported in Table \[table:comparison\_optimal\_mdp\], by using the approximation in (\[eqn:constrained\_mdp\]) and by using the corresponding optimal policy for as-you-go deployment, we lose at most $0.3685$ bits per channel use, compared to the case when the realization of the exponentially distributed source to sink distance is known apriori and when we use the same number of relays as used in the as-you-go deployment case. Note that, the statement of this claim holds [*with high probability*]{} since the maximum difference is taken over $10000$ sample deployments. Hence, it is reasonable to solve (\[eqn:constrained\_mdp\]) instead of (\[eqn:constrained\_mdp\_actual\]) which is intractable. Discussion {#section:additional_discussion} ========== Exponential Path-Loss Model {#subsection:motivation-for-exponential-path-loss} --------------------------- Exponential path-loss model has been used before in the context of relay placement (see [@firouzabadi-martins08optimal-node-placement], [@appuswamy-etal10relay-placement-deterministic-line]) and in the context of cellular networks (see [@altman-etal11greec-cellular Section $2.3$]). Analytical and experimental support for the exponential path-loss model have been provided by Franceschetti et al. ([@franceschetti-etal04random-walk-model-wave-propagation]). Franceschetti et al. used a random scattering model (applicable to an urban environment, or a forest environment) to show that the path-loss in such an environment is the product of an exponential function and a power function of the distance (see [@franceschetti-etal04random-walk-model-wave-propagation Equation $(14)$]). Figure $4$ of their paper, which is obtained from measurements made in an urban environment, shows that path-loss (in dB) varies linearly with distance beyond a distance of $40-50$ meters, which implies exponential path-loss for longer distance. These distances are practical for urban scenarios where the network is deployed over several hundreds of meters or several kilometers. Exponential path-loss was also proposed by Marano and Franceschetti for urban environment, and validated by theory and experiment (see [@marano-franceschetti05ray-propagation-random-lattice Figure $10$]).[^10] Incorporating Shadowing and Fading {#subsection:shadowing-fading} ---------------------------------- Shadowing (which is typically viewed as being static once a link is deployed) and time varying fading, can be incorporated in our setting by providing a fade-margin in the power at each transmitter. Thus, when expressed in dBm, the actual transmit power for any transmitter-receiver pair is the fade-margin plus the power used in the information theoretic capacity formulas; this fade-margin does not depend on the distance between the transmitter-receiver pair. Note that, this approach, though very conservative in nature, can remove the complexity in analysis arising out of fading in the network. Also note that, if the actual power gain between two nodes $r$ distance apart is $c_0e^{-\rho r}$ with $c_0 > 0$, then $c_0$ can be absorbed in the fade margin. Full-Duplex Decode-and-Forward Relaying {#subsection:motivation-for-full-duplex-decode-forward} --------------------------------------- Full-duplex radios might become a reality soon; see [@khandani13two-way-full-duplex-wireless], [@khandani10spatial-multiplexing-two-way-channel], [@choi-etal10single-channel-full-duplex], [@jain-etal11real-time-full-duplex] for recent efforts to realize them. Decode-and-forward relaying requires symbol-level synchronous operation across all nodes in the network. The requirement of globally coherent transmission and reception seems to be restrictive at the moment, but this problem will be solved with the advent of better clocks (with less drift) and efficient clock synchronization algorithms. Any research on impromptu deployment assuming imperfect synchronization, or half-duplex communication, or no interference cancellation, can use this paper as a benchmark for performance analysis. Insights on Power-Law Path-Loss {#subsection:insights_for_power_law_from_exponential} ------------------------------- In [@chattopadhyay-etal12optimal-capacity-relay-placement-line], we studied the problem of single-relay placement under a per-node power constraint at the source and the relay, for both exponential and power-law path-loss models. The variation of optimal relay location, as the amount of attenuation in the network varies, follow slightly different (but mostly similar) trends (see Figures $2$ and $3$ of [@chattopadhyay-etal12optimal-capacity-relay-placement-line]) because of the fact that power-law model allows unbounded power gain (unlike the exponential model) when the distance $r$ tends to $0$ ($\lim_{r \rightarrow 0} r^{-\eta}=\infty$). The findings are even more similar when we bound the power gain from above by some constant value in case of the power-law model (power gain is $\min\{r^{-\eta}, b^{-\eta} \}$ for some $b > 0$); see the similarity between Figures $2$ and $4$ in [@chattopadhyay-etal12optimal-capacity-relay-placement-line]. The results on the fixed node power case provide the insight that when the power gain is $r^{-\eta}$ or $\min\{r^{-\eta}, b^{-\eta} \}$; under the sum power constraint, the variation of the relay locations as a function of attenuation will follow a pattern similar to that in case of exponential path-loss. Conclusion ========== Motivated by the problem of as-you-go deployment of wireless relay networks, we first studied the problem of placing relay nodes along a line, in order to connect a sink at the end of the line to a source at the start of the line, so as to maximize the end-to-end achievable data rate. For the multi-relay channel with exponential path-loss and sum power constraint, we derived an expression for the achievable rate in terms of the power gains among all possible node pairs, and formulated an optimization problem in order to maximize the end-to-end data rate. Numerical work for the fixed source-sink distance suggests that at low attenuation the relays are mostly clustered close to the source in order to be able to cooperate among themselves, whereas at high attenuation they are uniformly placed and work as repeaters. Next, the deploy-as-you-go sequential placement problem was addressed; a sequential relay placement problem along a line having unknown random length was formulated as an MDP, the value function was characterized analytically, and the policy structure was investigated numerically. We found numerically that at the initial stage of the deployment process the inter-relay distances are smaller, and, as deployment progresses, the inter-relay distances increase gradually, and finally the relays start being placed at regular intervals. Our results are based on information theoretic achievable rate results. In order to utilize currently commercially available wireless devices, we have also been exploring non-information theoretic, packet forwarding models for optimal relay placement, with the aim of obtaining placement algorithms that can be easily reduced to practice (see [@chattopadhyay-etal13measurement-based-impromptu-placement_wiopt] for reference). The study of as-you-go deployment under the information theoretic model and under the packet forwarding model provides two complementary approaches for two different conditions in the physical layer and the MAC layer, and provides a more comprehensive development of the problem. [Arpan Chattopadhyay]{} obtained his B.E. in Electronics and Telecommunication Engineering from Jadavpur University, Kolkata, India in the year 2008, and M.E. and Ph.D in Telecommunication Engineering from Indian Institute of Science, Bangalore, India in the year 2010 and 2015, respectively. He is currently working in INRIA, Paris as a postdoctoral researcher. His research interests include networks and machine learning. [Abhishek Sinha]{} is currently a graduate student in the Laboratory for Information and Decision Systems (LIDS), at Massachusetts Institute of Technology, Cambridge, MA. Prior to joining MIT, he completed his Master’s studies in Telecommunication Engineering at the Indian Institute of Science, Bangalore, in the year 2012. His areas of interests include stochastic processes, information theory and network control. [Marceau Coupechoux]{} is an Associate Professor at Telecom ParisTech since 2005. He obtained his master from Telecom ParisTech in 1999 and from University of Stuttgart, Germany in 2000, and his Ph.D. from Institut Eurecom, Sophia-Antipolis, France, in 2004. From 2000 to 2005, he was with Alcatel-Lucent (Bell Labs former Research & Innovation and then in the Network Design department). In the Computer and Network Science department of Telecom ParisTech, he is working on cellular networks, wireless networks, ad hoc networks, cognitive networks, focusing mainly on layer 2 protocols, scheduling and resource management. From August 2011 to August 2012 he was a visiting scientist at IISc Bangalore. [Anurag Kumar]{} obtained his B.Tech. degree from the Indian Institute of Technology at Kanpur, and the PhD degree from Cornell University, both in Electrical Engineering. He was then with Bell Laboratories, Holmdel, N.J., for over 6 years. Since 1988 he has been on the faculty of the Indian Institute of Science (IISc), Bangalore, in the Department of Electrical Communication Engineering. He is currently also the Director of the Institute. From 1988 to 2003 he was the Coordinator at IISc of the Education and Research Network Project (ERNET), India’s first wide-area packet switching network. His area of research is communication networking, specifically, modeling, analysis, control and optimisation problems arising in communication networks and distributed systems. Recently his research has focused primarily on wireless networking. He is a Fellow of the IEEE, of the Indian National Science Academy (INSA), of the Indian Academy of Science (IASc), of the Indian National Academy of Engineering (INAE), and of The World Academy of Sciences (TWAS). He is a recepient of the Indian Institute of Science Alumni Award for Engineering Research for 2008. A Brief Description of the Coding Scheme of [@xie-kumar04network-information-theory-scaling-law] {#section:coding_scheme_description} ================================================================================================ Transmissions take place via block codes of $T$ symbols each. The transmission blocks at the source and the $N$ relays are synchronized. The coding and decoding scheme is such that a message generated at the source at the beginning of block $b, b \geq 1,$ is decoded by the sink at the end of block $b + N$, i.e., $N+1$ block durations after the message was generated (with probability tending to 1, as $T \to \infty)$. Thus, at the end of $B$ blocks, $B \geq N+1$, the sink is able to decode $B-N$ messages. It follows, by taking $B \to \infty$, that, if the code rate is $R$ bits per symbol, then an information rate of $R$ bits per symbol can be achieved from the source to the sink. As mentioned earlier, we index the source by $0$, the relays by $k, 1 \leq k \leq N$, and the sink by $N+1$. There are $(N+1)^2$ independent Gaussian random codebooks, each containing $2^{TR}$ codes, each code being of length $T$; these codebooks are available to all nodes. At the beginning of block $b$, the source generates a new message $w_b$, and, at this stage, we assume that each node $k, 1 \leq k \leq N+1,$ has a reliable estimate of all the messages $w_{b-j}, j \geq k$. In block $b$, the source uses a new codebook to encode $w_b$. In addition, relay $k, 1 \leq k \leq N,$ and [*all*]{} of its previous transmitters (indexed $0 \leq j \leq k-1$), use [*another*]{} codebook to encode $w_{b-k}$ (or their estimate of it). Thus, if the relays $1,2,\cdots,k$ have a perfect estimate of $w_{b-k}$ at the beginning of block $b$, they will transmit the same codeword for $w_{b-k}$. Therefore, in block $b$, the source and relays $1, 2, \cdots, k$ *coherently transmit* the codeword for $w_{b-k}$. In this manner, in block $b$, transmitter $k, 0 \leq k \leq N,$ generates $N+1 - k$ codewords, corresponding to $w_{b-k}, w_{b-k-1}, \cdots, w_{b-N}$, which are transmitted with powers $P_{k,k+1}, P_{k,k+2}, \cdots, P_{k,N+1}$. In block $b$, node $k, 1 \leq k \leq N+1,$ receives a superposition of transmissions from all other nodes. Assuming that node $k$ knows all the powers, and all the channel gains, and recalling that it has a reliable estimate of all the messages $w_{b-j}, j \geq k$, it can subtract the interference from transmitters $k+1, k+2, \cdots, N$. At the end of block $b$, after subtracting the signals it knows, node $k$ is left with the $k$ received signals from nodes $0, 1, \cdots, (k-1)$ (received in blocks $b, b-1, \cdots, b-k+1$), which all carry an encoding of the message $w_{b-k+1}$. These $k$ signals are then jointly used to decode $w_{b-k+1},$ using joint typicality decoding. The codebooks are cycled through in a manner so that in any block all nodes encoding a message (or their estimate of it) use the same codebook, but different (thus, independent) codebooks are used for different messages. Under this encoding and decoding scheme, any rate strictly less than $R$ displayed in (\[eqn:achievable\_rate\_multirelay\]) is achievable. Proof of Theorem \[theorem:multirelay\_capacity\] {#appendix:proof_of_multirelay_channel_capacity_theorem_after_power_allocation} ================================================= We want to maximize $R$ given in (\[eqn:achievable\_rate\_multirelay\]) subject to the total power constraint, assuming fixed relay locations. Let us consider $C (\frac{1}{\sigma^{2}×} \sum_{j=1}^{k} ( \sum_{i=0}^{j-1} h_{i,k} \sqrt{P_{i,j}} )^{2})$, i.e., the $k$-th term in the argument of $\min \{\cdots\}$ in (\[eqn:achievable\_rate\_multirelay\]). By the monotonicity of $C(\cdot)$, it is sufficient to consider $\sum_{j=1}^{k} ( \sum_{i=0}^{j-1} h_{i,k} \sqrt{P_{i,j}} )^{2}$. Now since the channel gains are multiplicative, we have: $$\sum_{j=1}^{k} ( \sum_{i=0}^{j-1} h_{i,k} \sqrt{P_{i,j}} )^{2}=g_{0,k}\sum_{j=1}^{k} \bigg( \sum_{i=0}^{j-1}\frac{\sqrt{P_{i,j}}}{h_{0,i}×}\bigg)^{2}\nonumber$$ Thus our optimization problem becomes: $$\begin{aligned} & & \max \, \min_{k \in \{1,2,\cdots,N+1\}} g_{0,k} \sum_{j=1}^{k} \bigg( \sum_{i=0}^{j-1}\frac{\sqrt{P_{i,j}}}{h_{0,i}×}\bigg)^{2}\nonumber\\ & \textit{s.t} & \, \sum_{j=1}^{N+1}\gamma_{j} \leq P_{T} \,\, \textit{and} \,\sum_{i=0}^{j-1}P_{i,j}=\gamma_{j} \, \forall \, 1 \leq j \leq (N+1) \label{eqn:optimization_problem}\end{aligned}$$ Let us fix $\gamma_{1}, \gamma_{2},\cdots,\gamma_{N+1}$ such that their sum is equal to $P_{T}$. We observe that $P_{i,N+1}$ for $i \in \{0,1,\cdots,N\}$ appear in the objective function only once: for $k=N+1$ through the term $( \sum_{i=0}^{N}\frac{\sqrt{P_{i,N+1}}}{h_{0,i}×})^{2}$. Since we have fixed $\gamma_{N+1}$, we need to maximize this term over $P_{i,N+1},\, i \in \{0,1,\cdots,N\}$. So we have the following optimization problem: $$\begin{aligned} \max \sum_{i=0}^{N} \frac{\sqrt{P_{i,N+1}}}{h_{0,i}×} \,\,\,\,\, \textit{s.t} \,\,\,\,\, \sum_{i=0}^{N} P_{i,N+1}=\gamma_{N+1}\label{eqn:problem}\end{aligned}$$ By Cauchy-Schwartz inequality, the objective function in this optimization problem is upper bounded by (using the fact that $g_{0,i}=h_{0,i}^2$ $\forall i \in \{0,1,\cdots,N\}$): $$\sqrt{(\sum_{i=0}^{N} P_{i,N+1})(\sum_{i=0}^{N}\frac{1}{g_{0,i}×})}=\sqrt{\gamma_{N+1}\sum_{i=0}^{N}\frac{1}{g_{0,i}}}$$ The upper bound is achieved if there exists some $c>0$ such that $\frac{\sqrt{P_{i,N+1}}}{\frac{1}{h_{0,i}×}×}=c$ $\forall i \in \{0,1,\cdots,N\}$. So we have: $$P_{i,N+1}=\frac{c^2}{g_{0,i}×} \,\, \forall i \in \{0,1,\cdots,N\}\nonumber\\$$ Since $\sum_{i=0}^{N}P_{i,N+1}=\gamma_{N+1}$, we obtain $c^2=\frac{\gamma_{N+1}}{\sum_{l=0}^{N}\frac{1}{g_{0,l}×}×}$. Thus, $ P_{i,N+1}=\frac{\frac{1}{g_{0,i}×}}{\sum_{l=0}^{N}\frac{1}{g_{0,l}×}×}\gamma_{N+1}$. Here we have used the fact that $h_{0,0}=1$. Now $\{P_{i,N}: i=0,1,\cdots,(N-1)\}$ appear only through the sum $\sum_{i=0}^{N-1}\frac{\sqrt{P_{i,N}}}{h_{0,i}×}$, and it appears twice: for $k=N$ and $k=N+1$. We need to maximize this sum subject to the constraint $\sum_{i=0}^{N-1}P_{i,N}=\gamma_{N}$. This optimization can be solved in a similar way as before. Thus by repeatedly using this argument and solving optimization problems similar in nature to (\[eqn:problem\]), we obtain: $$P_{i,j}=\frac{\frac{1}{g_{0,i}×}}{\sum_{l=0}^{j-1}\frac{1}{g_{0,l}×}×}\gamma_{j} \,\, \forall 0 \leq i < j \leq (N+1)$$ Substituting for $P_{i,j},0 \leq i < j \leq (N+1)$ in (\[eqn:optimization\_problem\]), we obtain the following optimization problem: $$\begin{aligned} & & \max \min_{k \in \{1,2,\cdots,N+1\}} g_{0,k} \sum_{j=1}^{k}\bigg(\gamma_{j} \sum_{i=0}^{j-1}\frac{1}{g_{0,i}×} \bigg)\nonumber\\ & \textit{s.t.} & \,\, \sum_{j=1}^{N+1}\gamma_{j} \leq P_{T}\end{aligned}$$ Let us define $b_{k}:=g_{0,k}$ and $a_{j}:=\sum_{i=0}^{j-1}\frac{1}{g_{0,i}×}$. Observe that $b_{k}$ is decreasing and $a_{k}$ is increasing with $k$. Let us define: $$\tilde{s}_{k}(\gamma_{1},\gamma_{2},\cdots,\gamma_{N+1}) := b_{k} \sum_{j=1}^{k} a_{j} \gamma_{j}$$ With this notation, our optimization problem becomes: $$\begin{aligned} \max \min_{1 \leq k \leq N+1} \tilde{s}_{k}(\gamma_{1},\gamma_{2},\cdots,\gamma_{N+1}) \,\,\,\, \textit{s.t.} \,\, \sum_{j=1}^{N+1}\gamma_{j} \leq P_{T} \label{eqn:modified_optimization_problem}\end{aligned}$$ Under optimal allocation of $\gamma_{1}, \gamma_{2},\cdots,\gamma_{N+1}$ for the optimization problem (\[eqn:modified\_optimization\_problem\]), $\tilde{s}_{1}=\tilde{s}_{2}=\cdots=\tilde{s}_{N+1}$. [$\Box$]{} (\[eqn:modified\_optimization\_problem\]) can be rewritten as: $$\begin{aligned} && \max \zeta \nonumber\\ \textit{s.t} && \zeta \leq b_{k}\sum_{j=1}^{k}a_{j}\gamma_{j} \, \forall \, k \in \{1,2,\cdots,N+1\}, \nonumber\\ && \sum_{j=1}^{N+1}\gamma_{j} \leq P_{T}, \,\,\, \gamma_{j} \geq 0 \,\, \forall \, 1 \leq j \leq N+1\label{eqn:equality_problem_primal}\end{aligned}$$ The dual of this linear program is given by: $$\begin{aligned} &&\min P_{T}\theta \nonumber\\ \textit{s.t} && \sum_{k=1}^{N+1}\mu_{k}=1,\,\,\, \theta \geq 0,\nonumber\\ && a_{l}\sum_{k=l}^{N+1}b_{k}\mu_{k}+\nu_{l}=\theta \, \forall \, l \in \{1,2,\cdots,N+1\},\nonumber\\ && \mu_{l} \geq 0, \nu_{l} \geq 0 \, \forall \, l \in \{1,2,\cdots,N+1\} \label{eqn:equality_problem_dual}\end{aligned}$$ Now, let us consider a primal feasible solution $(\{\gamma_j^*\}_{1 \leq j \leq N+1}, \zeta^*)$ which satisfies: $$\begin{aligned} && b_{k}\sum_{j=1}^{k}a_{j}\gamma_{j}^*=\zeta^* \, \forall \, k \in \{1,2,\cdots,N+1\}, \nonumber\\ && \sum_{j=1}^{N+1}\gamma_{j}^* = P_{T}\end{aligned}$$ Thus we have, $b_1 a_1 \gamma_1^*=\zeta^*$, i.e., $\gamma_1^*=\frac{\zeta^*}{b_1 a_1×}$. Again, $b_2(a_1 \gamma_1^*+a_2 \gamma_2^*)=\zeta^*$, which implies $ \frac{b_2}{b_1×}\zeta^*+b_2 a_2 \gamma_2^*=\zeta^*$. Thus we obtain $\gamma_2^*=\frac{\zeta^*}{a_2×}(\frac{1}{b_2×}-\frac{1}{b_1×})$. In general, we can write: $$\gamma_k^*=\frac{\zeta^*}{a_k×}\left(\frac{1}{b_k×}-\frac{1}{b_{k-1}×}\right) \, \forall k \in \{1,2,\cdots,N+1\}\nonumber\\$$ with $\frac{1}{b_{0}×}:=0$. Now, since $\sum_{k=1}^{N+1}\gamma_k^*=P_T$, we obtain: $$\begin{aligned} \zeta^*&=&\frac{P_{T}}{\sum_{k=1}^{N+1}\frac{1}{a_{k}×}(\frac{1}{b_k×}-\frac{1}{b_{k-1}×})×}\nonumber\\ \gamma_j^*&=&\frac{ \frac{1}{a_j×} \left(\frac{1}{b_j×}-\frac{1}{b_{j-1}×}\right) } {\sum_{k=1}^{N+1}\frac{1}{a_{k}×}(\frac{1}{b_k×}-\frac{1}{b_{k-1}×})×}P_{T}, \, j \in \{1,2,\cdots,N+1\} \label{eqn:primal_optimal}\end{aligned}$$ It should be noted that since $b_{k}$ is nonincreasing in $k$, the primal variables above are nonnegative and satisfies feasibility conditions. Again, let us consider a dual feasible solution $(\{\mu_j^*,\nu_j^*\}_{1 \leq j \leq N+1}, \theta^*)$ which satisfies: $$\begin{aligned} && \sum_{k=1}^{N+1}\mu_{k}^*=1, \,\,\, \nu_l^*=0 \, \forall \, l \in \{1,2,\cdots,N+1\}\nonumber\\ && a_{l}\sum_{k=l}^{N+1}b_{k}\mu_{k}^*+\nu_{l}^*=\theta^* \, \forall \, l \in \{1,2,\cdots,N+1\}\end{aligned}$$ Solving these equations, we obtain: $$\begin{aligned} \theta^*&=&\frac{1}{\sum_{k=1}^{N+1}\frac{1}{b_k×}\left(\frac{1}{a_k×}-\frac{1}{a_{k+1}×}\right)×}\nonumber\\ \mu_j^*&=&\frac{\frac{1}{b_j×}(\frac{1}{a_j×}-\frac{1}{a_{j+1}×})} {\sum_{k=1}^{N+1}\frac{1}{b_k×}\left(\frac{1}{a_k×}-\frac{1}{a_{k+1}×}\right)×},\, j \in \{1,2,\cdots,N+1\} \label{eqn:dual_optimal}\end{aligned}$$ where $\frac{1}{a_{N+2}×}:=0$. Since $a_k$ is increasing in $k$, all dual variables are feasible. It is easy to check that $\zeta^*=P_T \theta^*$, which means that there is no duality gap. Since the primal is a linear program, the solution $(\gamma_1^*, \gamma_2^*,\cdots,\gamma_{N+1}^*, \zeta^*)$ is primal optimal. Thus we have established the claim, since the primal optimal solution satisfies it. So let us obtain $\gamma_{1},\gamma_{2},\cdots,\gamma_{N+1}$ for which $\tilde{s}_{1}=\tilde{s}_{2}=\cdots=\tilde{s}_{N+1}$. Putting $\tilde{s}_{k}=\tilde{s}_{k-1}$, we obtain $b_{k} \sum_{j=1}^{k} a_{j} \gamma_{j}=b_{k-1} \sum_{j=1}^{k-1} a_{j} \gamma_{j}$. Thus, we obtain, $ \gamma_{k}=\frac{(b_{k-1}-b_{k})}{b_{k}×} \frac{1}{a_{k}×} \sum_{j=1}^{k-1}a_{j}\gamma_{j}$ Let $d_{k}:=\frac{(b_{k-1}-b_{k})}{b_{k}×} \frac{1}{a_{k}×}$. Hence, $ \gamma_{k}=d_{k} \sum_{j=1}^{k-1}a_{j}\gamma_{j}$. From this recursive equation, we have $\gamma_{2}=d_{2}a_{1}\gamma_{1}$, $\gamma_{3}=d_{3}(a_{1}\gamma_{1}+a_{2}\gamma_{2})=d_{3}a_{1}(1+a_{2}d_{2})\gamma_{1}$, and, in general for $k \geq 3$, $$\gamma_{k}=d_{k}a_{1}\Pi_{j=2}^{k-1}(1+a_{j}d_{j})\gamma_{1}\label{eqn:gamma_k_gamma_1}$$ Using the fact that $\gamma_{1}+\gamma_{2}+\cdots+\gamma_{N+1}=P_{T}$, we obtain: $$\gamma_{1}=\frac{P_{T}}{1+d_{2}a_{1}+ \sum_{k=3}^{N+1}d_{k}a_{1} \Pi_{j=2}^{k-1}(1+a_{j}d_{j}) ×} \label{eqn:gamma_1}$$ Thus if $\tilde{s}_{1}=\tilde{s}_{2}=\cdots=\tilde{s}_{N+1}$, there is a unique allocation $\gamma_{1},\gamma_{2},\cdots,\gamma_{N+1}$. So this must be the one maximizing $R$. Hence, optimum $\gamma_{1}$ is obtained by (\[eqn:gamma\_1\]). Then, substituting the values of $\{a_{k}:k=0,1,\cdots,N\}$ and $d_{k}:k=1,2,\cdots,N+1$ in (\[eqn:gamma\_k\_gamma\_1\]) and (\[eqn:gamma\_1\]), we obtain the values of $\gamma_{1},\gamma_{2},\cdots,\gamma_{N+1}$ as shown in Theorem \[theorem:multirelay\_capacity\]. Now under these optimal values of $\gamma_{1},\gamma_{2},\cdots,\gamma_{N+1}$, all terms in the argument of $\min \{\cdots\}$ in (\[eqn:achievable\_rate\_multirelay\]) are equal. So we can consider the first term alone. Thus we obtain the expression for $R$ optimized over power allocation among all the nodes for fixed relay locations as : $R_{P_T}^{opt}(y_1,y_2,\cdots,y_N)=C \left(\frac{g_{0,1}P_{0,1}}{\sigma^{2}×}\right)=C \left(\frac{g_{0,1}\gamma_{1}}{\sigma^{2}×}\right)$. Substituting the expression for $\gamma_{1}$ from (\[eqn:gamma\_one\]), we obtain the achievable rate formula (\[eqn:capacity\_multirelay\]).[$\Box$]{} $$\begin{aligned} \label{eqn:capacity_increasing_with_N} & & z_{1}^{*}+\frac{z_{2}^{*}-z_{1}^{*}}{1+z_{1}^{*}×}+\cdots+\frac{e^{\rho y}-z_{i}^{*}}{1+z_{1}^{*}+\cdots+z_{i}^{*}×}+\frac{z_{i+1}^{*}-e^{\rho y}}{1+z_{1}^{*}+\cdots+z_{i}^{*}+e^{\rho y}×}+\cdots.+\frac{e^{\rho L}-z_{N}^{*}}{1+z_{1}^{*}+\cdots+z_{i}^{*}+e^{\rho y}+z_{i+1}^{*}+\cdots+z_{N}^{*}×}\nonumber\\ & & < z_{1}^{*}+\frac{z_{2}^{*}-z_{1}^{*}}{1+z_{1}^{*}×}+\cdots+\frac{e^{\rho L}-z_{N}^{*}}{1+z_{1}^{*}+\cdots+z_{i}^{*}+z_{i+1}^{*}+\cdots+z_{N}^{*}×} \label{eqn:intermediate_eqn:capacity_increasing_in_N}\end{aligned}$$ ------------------------------------------------------------------------ Proof of Theorem \[theorem:single\_relay\_total\_power\] {#appendix:proof_of_single_relay_sum_power_results} ======================================================== Here we want to place the relay node at a distance $r_{1}$ from the source to minimize $\bigg\{\frac{1}{g_{0,1}×}+\frac{g_{0,1}-g_{0,2}}{g_{0,2}(1+g_{0,1})×}\bigg\}$ (see Equation (\[eqn:capacity\_multirelay\])). Hence, our optimization problem becomes : $$\min_{r_{1} \in [0,L]} \bigg\{e^{\rho r_{1}}+\frac{e^{-\rho r_{1}}-e^{-\rho L}}{e^{-\rho L}(1+e^{-\rho r_{1}})×}\bigg\}\nonumber\\$$ Writing $z_{1}=e^{\rho r_{1}}$, the problem becomes : $$\min_{z_{1} \in [1,e^{\rho L}]} \bigg\{z_{1}-1+\frac{e^{\rho L}+1}{z_{1}+1×}\bigg\}\nonumber\\$$ This is a convex optimization problem. Equating the derivative of the objective function to zero, we obtain $1-\frac{e^{\rho L}+1}{(z_{1}+1)^{2}×}=0$. Thus the derivative becomes zero at $z_{1}'=\sqrt{1+e^{\rho L}}-1>0$. Hence, the objective function is decreasing in $z_{1}$ for $z_{1} \leq z_{1}'$ and increasing in $z_{1} \geq z_{1}'$. So the minimizer is $z_{1}^{*}=\max \{z_{1}',1 \}$. So the optimum distance of the relay node from the source is $y_{1}^{*}=r_{1}^{*}=\max \{0,r_{1}' \}$, where $r_{1}'=\frac{1}{\rho×} \log (\sqrt{1+e^{\rho L}}-1)$. Hence, $\frac{y_{1}^{*}}{L×}=\max \{\frac{1}{\lambda×} \log \left(\sqrt{e^{\lambda}+1}-1 \right),0\}$. Now $r_{1}' \geq 0$ if and only if $\lambda \geq \log 3$. Hence, $\frac{y_{1}^{*}}{L×}=0$ for $\lambda \leq \log 3$ and $\frac{y_{1}^{*}}{L×}=\frac{1}{\lambda×} \log \left(\sqrt{e^{\lambda}+1}-1 \right)$ for $\lambda \geq \log 3$. [*For $\lambda \leq \log 3$*]{}, the relay is placed at the source. Then $g_{0,1}=1$ and $g_{0,2}=g_{1,2}=e^{-\lambda}$. Then $P_{0,1}=\gamma_{1}=\frac{2P_{T}}{e^{\lambda}+1×}$ (by Theorem $1$) and $R^{*}=C \left(\frac{2P_{T}}{(e^{\lambda}+1)\sigma^{2}×}\right)$. Also $\gamma_{2}=\frac{e^{\lambda}-1}{e^{\lambda}+1×}P_{T}$. Hence, $P_{0,2}=P_{1,2}=\frac{e^{\lambda}-1}{e^{\lambda}+1×}\frac{P_{T}}{2×}$. [*for $\lambda \geq \log 3$*]{}, the relay is placed at $r_{1}'$. Substituting the value of $r_{1}'$ into Equation ($\ref{eqn:power_gamma_relation})$, we obtain $P_{0,1}=\gamma_{1}=\frac{P_{T}}{2×}$, $P_{0,2}=\frac{1}{\sqrt{e^{\lambda}+1}×}\frac{P_{T}}{2×}$, $P_{1,2}=\frac{\sqrt{e^{\lambda}+1}-1}{\sqrt{e^{\lambda}+1}×}\frac{P_{T}}{2×}$. So in this case $R^{*}=C \left(\frac{g_{0,1}P_{0,1}}{\sigma^{2}×} \right)$. Since $P_{0,1}=\frac{P_{T}}{2×}$, we have $R^{*}=C \left( \frac{1}{\sqrt{e^{\lambda}+1}-1×}\frac{P_{T}}{2 \sigma^{2}×} \right)$. [$\Box$]{} Proof of Theorem \[theorem:capacity\_increasing\_with\_N\] {#appendix:proof_of_capacity_increases_in_N} ========================================================== For the $N$-relay problem, let the minimizer in (\[eqn:multirelay\_optimization\]) be $z_{1}^{*}, z_{2}^{*},\cdots,z_{N}^{*}$ and let $y_{k}^{*}=\frac{1}{\rho×} \log z_{k}^{*}$. Clearly, there exists $i \in \{0,1,\cdots,N\}$ such that $y_{i+1}^{*}>y_{i}^{*}$. Let us insert a new relay at a distance $y$ from the source such that $y_{i}^{*}<y<y_{i+1}^{*}$. Now we find that we can easily reach (\[eqn:capacity\_increasing\_with\_N\]) (see next page) just by simple comparison. For example, $$\begin{aligned} & & \frac{e^{\rho y}-z_{i}^{*}}{1+z_{1}^{*}+\cdots+z_{i}^{*}×}+\frac{z_{i+1}^{*}-e^{\rho y}}{1+z_{1}^{*}+\cdots+z_{i}^{*}+e^{\rho y}×} \\ & < & \frac{e^{\rho y}-z_{i}^{*}}{1+z_{1}^{*}+\cdots+z_{i}^{*}×}+\frac{z_{i+1}^{*}-e^{\rho y}}{1+z_{1}^{*}+\cdots+z_{i}^{*}×}\\ & =& \frac{z_{i+1}^{*}-z_{i}^{*}}{1+z_{1}^{*}+\cdots+z_{i}^{*}×}\end{aligned}$$ First $i$ terms in the summations of L.H.S (left hand side) and R.H.S (right hand side) of (\[eqn:capacity\_increasing\_with\_N\]) are identical. Also sum of the remaining terms in L.H.S is smaller than that of the R.H.S since there is an additional $e^{\rho y}$ in the denominator of each fraction for the L.H.S. Hence, we can justify (\[eqn:capacity\_increasing\_with\_N\]). Now R.H.S is precisely the optimum objective function for the $N$-relay placement problem (see (\[eqn:multirelay\_optimization\])). On the other hand, L.H.S is a particular value of the objective in (\[eqn:multirelay\_optimization\]), for $(N+1)$-relay placement problem. This clearly implies that by adding one additional relay we can strictly improve from $R^{*}$ of the $N$ relay channel. Hence, $R^{*}(N+1)>R^{*}(N)$. [$\Box$]{} Proof of Theorem \[theorem:G\_increasing\_in\_lambda\] {#appendix:proof_of_G_increasing_in_lambda} ====================================================== Consider the optimization problem as shown in (\[eqn:multirelay\_optimization\]). Let us consider $\lambda_1$, $\lambda_2$, with $\lambda_1<\lambda_2$, the respective minimizers being $(z_{1}^{*},\cdots,z_{N}^{*})$ and $(z_{1}',\cdots,z_{N}')$. Clearly, $$\begin{aligned} G(N,\lambda_1)=\frac{e^{\lambda_1}}{ z_{1}^{*}+\sum_{k=2}^{N+1} \frac{z_{k}^{*}-z_{k-1}^{*}}{\sum_{l=0}^{k-1} z_{l}^{*}}×}\end{aligned}$$ with $z_{N+1}^{*}=e^{\lambda_1}$ and $z_{0}^{*}=1$. With $N \geq 1 $, note that $z_{1}^{*}-\frac{z_{N}^{*}}{1+z_{1}^{*}+\cdots+z_{N}^{*}×} \geq 0$, since $z_{1}^{*} \geq 1$ and $\frac{z_{N}^{*}}{1+z_{1}^{*}+\cdots+z_{N}^{*}×} \leq 1$. Hence, it is easy to see that $\frac{e^{\lambda}}{ z_{1}^{*}-\frac{z_{N}^{*}}{1+z_{1}^{*}+\cdots+z_{N}^{*}×}+\sum_{k=2}^{N} \frac{z_{k}^{*}-z_{k-1}^{*}}{\sum_{l=0}^{k-1} z_{l}^{*}}+\frac{e^{\lambda}}{1+z_{1}^{*}+\cdots+z_{N}^{*}×}×}$ is increasing in $\lambda$ where $(z_{1}^{*},\cdots,z_{N}^{*})$ is the optimal solution of (\[eqn:multirelay\_optimization\]) with $\lambda=\lambda_1$. Hence, $$\begin{aligned} G(N,\lambda_1)&=&\frac{e^{\lambda_1}}{ z_{1}^{*}+\sum_{k=2}^{N} \frac{z_{k}^{*}-z_{k-1}^{*}}{\sum_{l=0}^{k-1} z_{l}^{*}}+\frac{e^{\lambda_1}-z_{N}^{*}}{\sum_{l=0}^{N} z_{l}^{*}}×}\nonumber\\ &\leq & \frac{e^{\lambda_2}}{ z_{1}^{*}+\sum_{k=2}^{N} \frac{z_{k}^{*}-z_{k-1}^{*}}{\sum_{l=0}^{k-1} z_{l}^{*}}+\frac{e^{\lambda_2}-z_{N}^{*}}{1+z_{1}^{*}+\cdots+z_{N}^{*}×}×}\nonumber\\ &\leq & \frac{e^{\lambda_2}}{ z_{1}^{'}+\sum_{k=2}^{N} \frac{z_{k}^{'}-z_{k-1}^{'}}{\sum_{l=0}^{k-1} z_{l}^{'}}+\frac{e^{\lambda_2}-z_{N}^{'}}{1+z_{1}^{'}+\cdots+z_{N}^{'}×}×}\nonumber\\ &=& G(N,\lambda_2)\end{aligned}$$ The second inequality follows from the fact that $(z_{1}',\cdots,z_{N}')$ minimizes $z_{1}+\sum_{k=2}^{N+1} \frac{z_{k}-z_{k-1}}{\sum_{l=0}^{k-1} z_{l}}$ subject to the constraint $1 \leq z_1 \leq z_2 \leq \cdots \leq z_N \leq z_{N+1} =e^{\lambda_2}$. Hence, $G(N,\lambda)$ is increasing in $\lambda$ for fixed $N$. [$\Box$]{} Proof of Theorem \[theorem:large\_nodes\_uniform\] {#appendix:proof_of_large_nodes_uniform} ================================================== When $N$ relay nodes are uniformly placed along a line, we will have $y_{k}=\frac{kL}{N+1×}$. Then our formula for achievable rate $R_{P_T}^{opt}(y_1,y_2,\cdots,y_N)$ for sum power constraint becomes: $R_{N}=C(\frac{P_{T}}{\sigma^{2}×}\frac{1}{f(N)×})$ where $f(N)=a_N+\sum_{k=2}^{N+1}\frac{a_N^{k}-a_N^{k-1}}{1+a_N+\cdots+a_N^{k-1}×}$ with $a_N=e^{ \frac{\rho L}{N+1×}}=e^{\frac{\lambda}{N+1×}}$. Since $a_N>1$ for all $N<\infty$ and $\rho>0$, we have $f(N) > a_N$ for all $N \geq 1$ and hence, $\liminf_N f(N) \geq \lim_{N \rightarrow \infty} a_N =1$. Now, $$\begin{aligned} f(N)&=& a_N+\sum_{k=1}^{N}\frac{a_N^{k+1}-a_N^{k}}{1+a_N+\cdots+a_N^{k}×} \nonumber\\ &=& a_N+ (a_N-1)^{2}\sum_{k=1}^{N} \frac{a_N^{k}}{a_N^{k+1}-1×}\nonumber\\ &\leq& a_N+ (a_N-1)^{2}\sum_{k=1}^{N} \frac{a_N^{k}}{a_N^{k}-1×}\nonumber\\ &=& e^{\frac{\lambda}{N+1×}}+ (e^{\frac{\lambda}{N+1×}}-1)^{2}\sum_{k=1}^{N} \frac{e^{\frac{k \lambda}{N+1×}}}{e^{\frac{k \lambda}{N+1×}}-1×}\nonumber\\ &\leq& e^{\frac{\lambda}{N+1×}}+ (e^{\frac{\lambda}{N+1×}}-1)^{2}\sum_{k=1}^{N} \frac{e^{\frac{k \lambda}{N+1×}}}{\frac{k \lambda}{N+1×}}\nonumber\\ &=& e^{\frac{\lambda}{N+1×}}+ (e^{\frac{\lambda}{N+1×}}-1)^{2} \frac{(N+1)}{\lambda×} \sum_{k=1}^{N} \frac{e^{\frac{k \lambda}{N+1×}}}{k} \label{eqn:inequality_of_fN}\end{aligned}$$ where the first inequality follows from the fact that $a_N>1$ and the second inequality follows from the fact that $e^{\frac{k \lambda}{N+1×}} \geq 1+ \frac{k \lambda}{N+1×}$. Now, by Cauchy-Schwartz inequality, $$\begin{aligned} \sum_{k=1}^{N} \frac{e^{\frac{k \lambda}{N+1×}}}{k} \leq \sqrt{(\sum_{k=1}^{N}e^{\frac{2 k \lambda}{N+1×}}) (\sum_{k=1}^{N} \frac{1}{k^2×})}\end{aligned}$$ Since $\sum_{k=1}^{\infty}\frac{1}{k^2×}=\frac{\pi^2}{6×}$, we can write: $$\begin{aligned} \sum_{k=1}^{N} \frac{e^{\frac{k \lambda}{N+1×}}}{k} \leq \sqrt{(\sum_{k=1}^{N}e^{\frac{2 k \lambda}{N+1×}}) \frac{\pi^2}{6×}} \label{eqn:inequality_using_the_series}\end{aligned}$$ Hence, by (\[eqn:inequality\_using\_the\_series\]) and (\[eqn:inequality\_of\_fN\]), $$\begin{aligned} f(N) &\leq& e^{\frac{\lambda}{N+1×}}+ (e^{\frac{\lambda}{N+1×}}-1)^{2} \frac{(N+1)\pi}{\sqrt{6}\lambda×} \sqrt{\sum_{k=1}^{N}e^{\frac{2 k \lambda}{N+1×}}} \nonumber\\ &=& e^{\frac{\lambda}{N+1×}}+ (e^{\frac{\lambda}{N+1×}}-1)^{2} \frac{(N+1)\pi}{\sqrt{6}\lambda×} \sqrt{e^{\frac{2 \lambda}{N+1×}} \frac{(e^{\frac{2 N \lambda}{N+1×}}-1)}{(e^{\frac{2 \lambda}{N+1×}}-1)×}}\nonumber\\\end{aligned}$$ Now, since $e^{\frac{2 N \lambda}{N+1×}}-1 \leq e^{\frac{2 N \lambda}{N+1×}} \leq e^{2\lambda}$, we obtain: $$\begin{aligned} f(N) &\leq& e^{\frac{\lambda}{N+1×}}+ (e^{\frac{\lambda}{N+1×}}-1)^{2} \frac{(N+1)\pi}{\sqrt{6}\lambda×} e^{\lambda} e^{\frac{\lambda}{N+1×}} \sqrt{\frac{1}{(e^{\frac{2 \lambda}{N+1×}}-1)×}} \nonumber\\ &=& e^{\frac{\lambda}{N+1×}}+ (e^{\frac{\lambda}{N+1×}}-1)^{\frac{3}{2×}} \frac{(N+1)\pi}{\sqrt{6}\lambda×} e^{\lambda} e^{\frac{\lambda}{N+1×}} \sqrt{\frac{1}{(e^{\frac{ \lambda}{N+1×}}+1)×}} \nonumber\\\end{aligned}$$ Hence, $$\begin{aligned} && \limsup_N f(N) \nonumber\\ & \leq & 1+ \frac{\pi e^{\lambda}}{\sqrt{12}×} \lim_{N \rightarrow \infty} \frac{(N+1)}{\lambda} (e^{\frac{\lambda}{N+1×}}-1)^{\frac{3}{2×}} \nonumber\\\end{aligned}$$ Putting $q=\frac{\lambda}{N+1×}$, $$\begin{aligned} \limsup_N f(N) &\leq& 1+ \frac{\pi e^{\lambda}}{\sqrt{12}×} \lim_{q \rightarrow 0} \sqrt{q} \lim_{q \rightarrow 0} (\frac{e^q-1}{q×})^{\frac{3}{2×}} \nonumber\\ &=& 1 \nonumber\\\end{aligned}$$ Now, we have proved that $\limsup_N f(N) \leq 1 \leq \liminf_N f(N)$ and hence $\lim_{N \rightarrow \infty} f(N)=1$. Hence, $\lim_{N \rightarrow \infty} R_N = C(\frac{P_T}{\sigma^{2}})$ and the theorem is proved. [$\Box$]{} Proof of Theorem \[theorem:convergence\_of\_value\_iteration\] {#appendix:sequential_placement_total_power} ============================================================== \[appendix:proof\_of\_value\_iteration\_convergence\] As we have seen in Section \[sec:mdp\_total\_power\], our problem is a negative dynamic programming problem (i.e., the $\mathsf{N}$ case of [@schal75conditions-optimality], where single-stage rewards are non-positive). It is to be noted that Schäl [@schal75conditions-optimality] discusses two other kind of problems as well: the $\mathsf{P}$ case (single-stage rewards are positive) and the $\mathsf{D}$ case (the reward at stage $k$ is discounted by a factor $\alpha^k$, where $0<\alpha<1$). In this appendix, we first state a general-purpose theorem for the value iteration (Theorem \[thm:value\_iteration\_general\]), prove it by some results of [@schal75conditions-optimality], and then we use this theorem to prove Theorem \[theorem:convergence\_of\_value\_iteration\]. A General Result (Derived from [@schal75conditions-optimality]) {#appendix_subsection_schal-discussion} --------------------------------------------------------------- Consider an infinite horizon total cost MDP whose state space $\mathcal{S}$ is an interval in $\mathbb{R}$ and the action space $\mathcal{A}$ is $[0,\infty)$. Let the set of possible actions at state $s$ be denoted by $\mathcal{A}(s)$. Let the single-stage cost be $c(s,a,w) \geq 0$ where $s$, $a$ and $w$ are the state, the action and the disturbance, respectively. Let us denote the optimal expected cost-to-go at state $s$ by $V^*(s)$. Let the state of the system evolve as $s_{k+1}=h(s_k,a_k,w_k)$, where $s_k$, $a_k$ and $w_k$ are the state, the action and the disturbance at the $k$-th instant, respectively. Let $s^{*} \in \mathcal{S}$ be an absorbing state with $c(s^{*},a, w)=0$ for all $a$, $w$. Let us consider the value iteration for all $s \in \mathcal{S}$, with $V^{(0)}(\cdot)= 0 $: $$\begin{aligned} V^{(k+1)}(s)&=&\inf_{a \in [0,\infty)} \mathbb{E}_{w} \bigg( c(s,a,w)+V^{(k)}(h(s,a,w)) \bigg), s \neq s^{*}\nonumber\\ V^{(k+1)}(s^{*})&=& 0\label{eqn:value_iteration_general}\end{aligned}$$ We provide some results and concepts from [@schal75conditions-optimality], which will be used later to prove Theorem \[theorem:convergence\_of\_value\_iteration\]. \[thm:schal\_convergence\_value\_iteration\] \[[*Theorem 4.2 ([@schal75conditions-optimality])*]{}\] $V^{(k)}(s)\rightarrow V^{(\infty)}(s)$ for all $s \in \mathcal{S}$, i.e., the value iteration (\[eqn:value\_iteration\_general\]) converges. [$\Box$]{} Let us recall that $\Gamma_k(s)$ is the set of minimizers of (\[eqn:value\_iteration\_general\]) at the $k$-th iteration at state $s$, if the infimum is achieved at some $a<\infty$. $\Gamma_{\infty}(s):=\{a \in \mathcal{A}:a$ is an accumulation point of some sequence $\{a_k\}$ where each $a_k \in \Gamma_{k}(s)\}$. $\Gamma^*(s)$ is the set of minimizers in the Bellman Equation. Let $\mathcal{C}(\mathcal{A})$ be the set of nonempty compact subsets of $\mathcal{A}$. The Hausdorff metric $d$ on $\mathcal{C}(\mathcal{A})$ is defined as follows: $$d(C_1,C_2)=\max \{ \sup_{c \in C_1}\rho(c,C_2), \, \sup_{c \in C_2}\rho(c,C_1) \}$$ where $\rho(c,C)$ is the minimum distance between the point $c$ and the compact set $C$. \[prop:separable\_hausdorff\] \[Proposition 9.1([@schal75conditions-optimality])\] $(\mathcal{C}(\mathcal{A}),d)$ is a separable metric space. A mapping $\phi:\mathcal{S}\rightarrow \mathcal{C}(\mathcal{A})$ is called measurable if it is measurable with respect to the Borel $\sigma$-algebra of $(\mathcal{C}(\mathcal{A}),d)$. $\hat{\mathcal{F}}(\mathcal{S} \times \mathcal{A})$ is the set of all measurable functions $v:\mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}$ which are bounded below and where every such $v(\cdot)$ is the limit of a non-decreasing sequence of measurable, bounded functions $v_k:\mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}$. We will next present a condition followed by a theorem. The condition, if satisfied, implies the convergence of value iteration (\[eqn:value\_iteration\_general\]) to the optimal value function (according to the theorem). \[condition\_A\] \[[*Derived from Condition A in [@schal75conditions-optimality]*]{}\] 1. $\mathcal{A}(s)\in \mathcal{C}(\mathcal{A})$ for all $s \in \mathcal{S}$ and $\mathcal{A}:\mathcal{S}\rightarrow \mathcal{C}(\mathcal{A})$ is measurable. 2. $\mathbb{E}_{w}(c(s,a,w)+V^{(k)}(h(s,a,w))) $ is in $\hat{\mathcal{F}}(\mathcal{S}\times \mathcal{A})$ for all $k \geq 0$.[$\Box$]{} \[thm:schal\_main\_theorem\] \[[*Theorem $13.3$, [@schal75conditions-optimality]*]{}\] If $c(s,a,w) \geq 0$ for all $s,a,w$ and Condition \[condition\_A\] holds: 1. $V^{(\infty)}(s)=V^*(s)$, $s \in \mathcal{S}$. 2. $\Gamma_{\infty}(s) \subset \Gamma^*(s)$. 3. There is a stationary optimal policy $f^{\infty}$ where $f:\mathcal{S} \rightarrow \mathcal{A}$ and $f(s) \in \Gamma_{\infty}(s)$ for all $s \in \mathcal{S}$.[$\Box$]{} The next condition and theorem deal with the situation where the action space is noncompact. \[condition\_B\] \[[*Condition B ([@schal75conditions-optimality])*]{}\] There is a measurable mapping $\underline{\mathcal{A}}:\mathcal{S} \rightarrow \mathcal{C}(\mathcal{A})$ such that: 1. $\underline{{\mathcal{A}}}(s)\subset {\mathcal{A}}(s)$ for all $s \in \mathcal{S}$. 2. for all $k \geq 0$. [$\Box$]{} This condition requires that for each state $s$, there is a compact set $\underline{\mathcal{A}}(s)$ of actions such that no optimizer of the value iteration lies outside the set $\underline{\mathcal{A}}(s)$ at any stage $k \geq 0$. \[thm:schal\_compact\_action\] \[[*Theorem $17.1$, [@schal75conditions-optimality]*]{}\] If Condition \[condition\_B\] is satisfied and if the three statements in Theorem \[thm:schal\_main\_theorem\] are valid for the modified problem having admissible set of actions $\underline{\mathcal{A}}(s)$ for each state $s \in \mathcal{S}$, then those statements are valid for the original problem as well.[$\Box$]{} Now we will provide an important theorem which will be used to prove Theorem \[theorem:convergence\_of\_value\_iteration\]. \[thm:value\_iteration\_general\] If the value iteration (\[eqn:value\_iteration\_general\]) satisfies the following conditions: 1. For each $k$, $\mathbb{E}_{w}\bigg(c(s,a,w)+V^{(k)}(h(s,a,w))\bigg)$ is jointly continuous in $a$ and $s$ for $s \neq s^{*}$. 2. The infimum in (\[eqn:value\_iteration\_general\]) is achieved in $[0,\infty)$ for all $s \neq s^{*}$. 3. For each $s \in \mathcal{S}$, there exists $a(s)<\infty$ such that $a(s)$ is continuous in $s$ for $s \neq s^{*}$, and no minimizer of (\[eqn:value\_iteration\_general\]) lies in $(a(s), \infty)$ for each $k \geq 0$. Then the following hold: 1. The value iteration converges, i.e., $V^{(k)}(s) \rightarrow V^{(\infty)}(s)$ for all $s \neq s^{*}$. 2. $V^{(\infty)}(s)=V^*(s)$ for all $s \neq s^{*}$. 3. $\Gamma_{\infty}(s) \subset \Gamma^*(s)$ for all $s \neq s^{*}$. 4. There is a stationary optimal policy $f^{\infty}$ where $f:\mathcal{S} \setminus \{s^{*}\} \rightarrow \mathcal{A}$ and $f(s) \in \Gamma_{\infty}(s) \,\,\forall \,\, s \neq s^{*}$.[$\Box$]{} ***Proof of Theorem \[thm:value\_iteration\_general\]:*** By Theorem \[thm:schal\_convergence\_value\_iteration\], the value iteration converges, i.e., $V^{(k)}(s)\rightarrow V^{(\infty)}(s)$. Moreover, $V^{(k)}(s)$ is the optimal cost for a $k$-stage problem with zero terminal cost, and the cost at each stage is positive. Hence, $V^{(k)}(s)$ increases in $k$ for every $s \in \mathcal{S}$. Thus, for all $s \in \mathcal{S}$, $V^{(k)}(s) \uparrow V^{(\infty)}(s)$. Now, Condition \[condition\_B\] and Theorem \[thm:schal\_compact\_action\] say that if no optimizer of the value iteration in each stage $k$ lies outside a compact subset $\underline{\mathcal{A}}(s)$ of $\mathcal{A}(s) \subset \mathcal{A}$, then we can deal with the modified problem having a new action space $\underline{\mathcal{A}}(s)$. If the value iteration converges to the optimal value in this modified problem, then it will converge to the optimal value in the original problem as well, provided that the mapping $\underline{\mathcal{A}}:\mathcal{S}\rightarrow \mathcal{C}(\mathcal{A})$ is measurable. Let us choose $\underline{\mathcal{A}}(s):=[0,a(s)]$, where $a(s)$ satisfies hypothesis (c) of Theorem \[thm:value\_iteration\_general\]. Since $a(s)$ is continuous at $s \neq s^{*}$, for any $\epsilon>0$ we can find a $\delta_{s,\epsilon}>0$ such that $|a(s)-a(s')|<\epsilon$ whenever $|s-s'|<\delta_{s,\epsilon}$, $s \neq s^{*}$, $s' \neq s^{*}$. Now, when $|a(s)-a(s')|<\epsilon$, we have $d([0,a(s)],[0,a(s')])<\epsilon$. Hence, the mapping $\underline{\mathcal{A}}:\mathcal{S}\rightarrow \mathcal{C}(\mathcal{A})$ is continuous at all $s \neq s^{*}$, and thereby measurable in this case. Hence, the value iteration (\[eqn:value\_iteration\_general\]) satisfies Condition \[condition\_B\]. Thus, the value iteration for $s \neq s^{*}$ can be modified as: $$V^{(k+1)}(s)=\inf_{a \in [0,a(s)]} \mathbb{E}_{w} ( c(s,a,w)+V^{(k)}(h(s,a,w)) ) \label{eqn:modified_value_iteration_general}$$ Now, $\mathbb{E}_{w} ( c(s,a,w)+V^{(k)}(h(s,a,w)) )$ is continuous (can be discontinuous at $s=s^{*}$, since this quantity is $0$ at $s=s^*$) on $\mathcal{S} \times \mathcal{A}$ (by our hypothesis). Hence, $\mathbb{E}_{w} ( c(s,a,w)+V^{(k)}(h(s,a,w)) )$ is measurable on $\mathcal{S} \times \mathcal{A}$. Also, it is bounded below by $0$. Hence, it can be approximated by an increasing sequence of bounded measurable functions $\{v_{n,k}\}_{n \geq 1}$ given by $v_{n,k}(s,a)=\min \{\mathbb{E}_{w} ( c(s,a,w)+V^{(k)}(h(s,a,w)) ),\,n \}$. Hence, $\mathbb{E}_{w} ( c(s,a,w)+V^{(k)}(h(s,a,w)) )$ is in $\hat{\mathcal{F}}(\mathcal{S} \times \mathcal{A})$. Thus, Condition \[condition\_A\] is satisfied for the modified problem and therefore, by Theorem \[thm:schal\_main\_theorem\], the modified value iteration in (\[eqn:modified\_value\_iteration\_general\]) converges to the optimal value function. Now, by Theorem \[thm:schal\_compact\_action\], we can argue that the value iteration (\[eqn:value\_iteration\_general\]) converges to the optimal value function in the original problem and hence $V^{(\infty)}(s)=V^*(s)$ for all $s \in \mathcal{S} \setminus s^*$. Also, $\Gamma_{\infty}(s) \subset \Gamma^*(s)$ for all $s \in \mathcal{S} \setminus s^*$ and there exists a stationary optimal policy $f^{\infty}$ where $f(s) \in \Gamma_{\infty}(s)$ for all $s \in \mathcal{S} \setminus s^*$ (by Theorem \[thm:schal\_main\_theorem\]). Proof of Theorem \[theorem:convergence\_of\_value\_iteration\] {#appendix_subsection-convergence-value-iteration-proof} -------------------------------------------------------------- This proof uses the results of Theorem \[thm:value\_iteration\_general\] provided in this appendix. Remember that the state $\mathbf{EOL}$ is absorbing and $c(\mathbf{EOL}, a, w)=0$ for all $a$, $w$. We can think of it as state $0$ so that our state space becomes $[0,1]$ which is a Borel set. We will see that the state $0$ plays the role of the state $s^*$ as mentioned in Theorem \[thm:value\_iteration\_general\]. We need to check whether the conditions (a), (b), and (c) in Theorem \[thm:value\_iteration\_general\] are satisfied for the value iteration (\[eqn:value\_iteration\]). Of course, $J_{\xi}^{(0)}(s)=0$ is concave, increasing in $s \in (0,1]$. Suppose that $J_{\xi}^{(k)}(s)$ is concave, increasing in $s$ for some $k \geq 0$. Also, for any fixed $a \geq 0$, $\frac{se^{\rho a}}{1+se^{\rho a}×}$ is concave and increasing in $s$. Thus, by the composition rule for the composition of a concave increasing function $J_{\xi}^{(k)}(\cdot)$ and a concave increasing function $\frac{se^{\rho a}}{1+se^{\rho a}×}$, for any $a \geq 0$ the term $J_{\xi}^{(k)}\left(\frac{se^{\rho a}}{1+se^{\rho a}×}\right)$ is concave, increasing over $s \in (0,1]$. Hence, $\int_{0}^{a}\beta e^{-\beta z}s(e^{\rho z}-1)dz + e^{-\beta a}\bigg(s(e^{\rho a}-1)+\xi+J_{\xi}^{(k)}\left(\frac{se^{\rho a}}{1+se^{\rho a}×}\right)\bigg)$ (in (\[eqn:value\_iteration\])) is concave increasing over $s \in (0,1]$. Since the infimization over $a$ preserves concavity, we conclude that $J_{\xi}^{(k+1)}(s)$ is concave, increasing over $s \in (0,1]$. Hence, for each $k$, $J_{\xi}^{(k)}(s)$ is continuous in $s$ over $(0,1)$, since otherwise concavity w.r.t. $s$ will be violated. Now, we must have $J_{\xi}^{(k)}(1) \leq \lim_{s \uparrow 1} J_{\xi}^{(k)}(s)$, since otherwise the concavity of $J_{\xi}^{(k)}(s)$ will be violated. But since $J_{\xi}^{(k)}(s)$ is increasing in $s$, $J_{\xi}^{(k)}(1) \geq \lim_{s \uparrow 1} J_{\xi}^{(k)}(s)$. Hence, $J_{\xi}^{(k)}(1) = \lim_{s \uparrow 1} J_{\xi}^{(k)}(s)$. Thus, $J_{\xi}^{(k)}(s)$ is continuous in $s$ over $(0,1]$ for each $k$. Hence, $\int_{0}^{a}\beta e^{-\beta z}s(e^{\rho z}-1)dz + e^{-\beta a}(s(e^{\rho a}-1)+\xi+J_{\xi}^{(k)}(\frac{se^{\rho a}}{1+se^{\rho a}×}) )$ is continuous in $s,a$ for $s \neq 0$. Hence, condition (a) in Theorem \[thm:value\_iteration\_general\] is satisfied. Now, we will check condition (c) in Theorem \[thm:value\_iteration\_general\]. By Theorem \[thm:schal\_convergence\_value\_iteration\], the value iteration converges, i.e., $J_{\xi}^{(k)}(s)\rightarrow J_{\xi}^{(\infty)}(s)$. Also, $J_{\xi}^{(\infty)}(s)$ is concave, increasing in $s \in (0,1]$ and hence continuous. Moreover, $J_{\xi}^{(k)}(s)$ is the optimal cost for a $k$-stage problem with zero terminal cost, and the cost at each stage is positive. Hence, $J_{\xi}^{(k)}(s)$ increases in $k$ for every $s \in (0,1]$. Thus, for all $s \in (0,1]$, $J_{\xi}^{(k)}(s) \uparrow J_{\xi}^{(\infty)}(s)$. Again, $J_{\xi}^{(k)}(s)$ is the optimal cost for a $k$-stage problem with zero terminal cost. Hence, it is less than or equal to the optimal cost for the infinite horizon problem with the same transition law and cost structure. Hence, $J_{\xi}^{(k)}(s)\leq J_{\xi}(s)$ for all $k \geq 1$. Since $J_{\xi}^{(k)}(s)\uparrow J_{\xi}^{(\infty)}(s)$, we have $J_{\xi}^{(\infty)}(s) \leq J_{\xi}(s)$. Now, consider the following two cases: ### $\beta>\rho$ Let us define a function $\psi:(0,1]\rightarrow \mathbb{R}$ by $\psi(s)=\frac{J_{\xi}^{(\infty)}(s)+\theta s}{2×}$. By Proposition \[prop:upper\_bound\_on\_cost\_beta\_geq\_rho\], $J_{\xi}(s)<\theta s$ for all $s \in (0,1]$. Hence, $J_{\xi}^{(\infty)}(s)<\psi(s)<\theta s$ and $\psi(s)$ is continuous over $s \in (0,1]$. Since $\beta>\rho$ and $J_{\xi}^{(k)}(s) \in [0, \theta]$ for any $s$ in $(0,1]$, the expression $\theta s + e^{-\beta a}\bigg(-\theta s e^{\rho a}+\xi+J_{\xi}^{(k)}\left(\frac{se^{\rho a}}{1+se^{\rho a}×}\right)\bigg)$ obtained from the R.H.S of (\[eqn:value\_iteration\]) converges to $\theta s$ as $a \rightarrow \infty$. A lower bound to this expression is $\theta s + e^{-\beta a}(-\theta s e^{\rho a})$. With $\beta > \rho$, for each $s$, there exists $a(s)<\infty$ such that $\theta s + e^{-\beta a}(-\theta s e^{\rho a})> \psi(s)$ for all $a > a(s)$. But $\theta s + \inf_{a \geq 0} e^{-\beta a}\bigg(-\theta s e^{\rho a}+\xi+J_{\xi}^{(k)}\left(\frac{se^{\rho a}}{1+se^{\rho a}×}\right)\bigg)$ is equal to $J_{\xi}^{(k+1)}(s)<\psi(s)$. Hence, for any $s \in (0,1]$, the minimizers for (\[eqn:bellman\_equation\_simplified\_in\_a\]) always lie in the compact interval $[0,a(s)]$ for all $k \geq 1$. Since $\psi(s)$ is continuous in $s$, we can choose $a(s)$ as a continuous function of $s$ on $(0,1]$. ### $\beta \leq \rho$ Fix $A$, $0 <A <\infty$. Let $K:=\frac{1}{\beta A ×}\left(\xi+(e^{\rho A}-1) \right)+(e^{\rho A}-1)$. Then, by Proposition \[prop:upper\_bound\_on\_cost\], $J_{\xi}(s) \leq K$ for all $s \in (0,1]$. Now, we observe that the objective function (for minimization over $a$) in the R.H.S of (\[eqn:value\_iteration\]) is lower bounded by $\int_{0}^{a}\beta e^{-\beta z}s(e^{\rho z}-1)dz$, which is continuous in $s,a$ and goes to $\infty$ as $a \rightarrow \infty$ for each $s \in (0,1]$. Hence, for each $s \in (0,1]$, there exists $0<a(s)<\infty$ such that $\int_{0}^{a}\beta s e^{-\beta z}(e^{\rho z}-1)dz >2K$ for all $a>a(s)$ and $a(s)$ is continuous over $s \in (0,1]$. But $J_{\xi}^{(k+1)}(s) \leq J_{\xi}(s) \leq K$ for all $k$. Hence, the minimizers in (\[eqn:value\_iteration\]) always lie in $[0,a(s)]$ where $a(s)$ is independent of $k$ and continuous over $s \in (0,1]$. Let us set $a(0)=a(1)$.[^11] Then, the chosen function $a(s)$ is continuous over $s \in (0,1]$ and can be discontinuous only at $s=0$. Thus, condition (c) of Theorem \[thm:value\_iteration\_general\] has been verified for the value iteration (\[eqn:value\_iteration\]). Condition (b) of Theorem \[thm:value\_iteration\_general\] is obviously satisfied since a continuous function over a compact set always has a minimizer. [$\Box$]{} [*Remark:*]{} Observe that in our value iteration (\[eqn:value\_iteration\]) it is always sufficient to deal with compact action spaces, and the objective functions to be minimized at each stage of the value iteration are continuous in $s$, $a$. Hence, $\Gamma_{k}(s)$ is nonempty for each $s \in (0,1]$, $k \geq 0$. Also, since there exists $K>0$ such that $J_{\xi}^{(k)}(s) \leq K$ for all $k \geq 0$, $s \in (0,1]$, it is sufficient to restrict the action space in (\[eqn:value\_iteration\]) to a set $[0, a(s)]$ for any $s \in (0,1]$, $k \geq 0$. Hence, $\Gamma_k (s) \subset [0,a(s)]$ for all $s \in (0,1]$, $k \geq 0$. Now, for a fixed $s \in (0,1]$, any sequence $\{a_k\}_{k \geq 0}$ with $a_k \in \Gamma_k (s)$, in bounded. Hence, the sequence must have a limit point. Hence, $\Gamma_{\infty}(s)$ is nonempty for each $s \in (0,1]$. Since $\Gamma_{\infty}(s) \subset \Gamma^*(s)$, $\Gamma^*(s)$ is nonempty for each $s \in (0,1]$. Proofs of Propositions \[prop:increasing\_concave\_in\_s\], \[prop:increasing\_concave\_in\_lambda\] and \[prop:continuity\_of\_cost\] {#appendix_subsection_policy-structure} ====================================================================================================================================== \[appendix:proof\_of\_propositions\] Proof of Proposition \[prop:increasing\_concave\_in\_s\] -------------------------------------------------------- Fix $\xi$. Consider the value iteration (\[eqn:value\_iteration\]). Let us start with $J_{\xi}^{(0)}(s):=0$ for all $s \in (0,1]$. Clearly, $J_{\xi}^{(1)}(s)$ is concave and increasing in $s$, since pointwise infimum of linear functions is concave. Now let us assume that $J_{\xi}^{(k)}(s)$ is concave and increasing in $s$. Then, by the composition rule, it is easy to show that $J_{\xi}^{(k)}(\frac{se^{\rho a}}{1+se^{\rho a}×})$ is concave and increasing in $s$ for any fixed $a\geq 0$. Hence, $J_{\xi}^{(k+1)}(s)$ is concave and increasing, since pointwise infimum of a set of concave and increasing functions is concave and increasing. By Theorem \[theorem:convergence\_of\_value\_iteration\], $J_{\xi}^{(k)}(s)\rightarrow J_{\xi}(s)$. Hence, $J_{\xi}(s)$ is concave and increasing in $s$.[$\Box$]{} Proof of Proposition \[prop:increasing\_concave\_in\_lambda\] ------------------------------------------------------------- Consider the value iteration (\[eqn:value\_iteration\]). Since $J_{\xi}^{(0)}(s):=0$ for all $s \in (0,1]$, $J_{\xi}^{(1)}(s)$ is obtained by taking infimum (over $a$) of a linear, increasing function of $\xi$. Hence, $J_{\xi}^{(1)}(s)$ is concave, increasing over $\xi \in (0, \infty)$. If we assume that $J_{\xi}^{(k)}(s)$ is concave and increasing in $\xi$, then $J_{\xi}^{(k)}(\frac{se^{\rho a}}{1+se^{\rho a}×})$ is also concave and increasing in $\xi$ for fixed $s$ and $a$. Thus, $J_{\xi}^{(k+1)}(s)$ is also concave and increasing in $\xi$. Now, $J_{\xi}^{(k)}(s) \rightarrow J_{\xi}(s)$ for all $s \in \mathcal{S}$, and $J_{\xi}^{(k)}(s)$ is concave, increasing in $\xi$ for all $k \geq 0$, $s \in \mathcal{S}$. Hence, $J_{\xi}(s)$ is concave and increasing in $\xi$. [$\Box$]{} Proof of Proposition \[prop:continuity\_of\_cost\] -------------------------------------------------- Clearly, $J_{\xi}(s)$ is continuous in $s$ over $(0,1)$, since otherwise concavity w.r.t. $s$ will be violated. Now, since $J_{\xi}(s)$ is concave in $s$ over $(0,1]$, we must have $J_{\xi}(1) \leq \lim_{s \uparrow 1}J_{\xi}(s)$. But since $J_{\xi}(s)$ is increasing in $s$, $J_{\xi}(1) \geq \lim_{s \uparrow 1}J_{\xi}(s)$. Hence, $J_{\xi}(1) = \lim_{s \uparrow 1}J_{\xi}(s)$. Thus, $J_{\xi}(s)$ is continuous in $s$ over $(0,1]$. Again, for a fixed $s \in (0,1]$, $J_{\xi}(s)$ is concave and increasing in $\xi$. Hence, $J_{\xi}(s)$ is continuous in $\xi$ over $\xi \in (0,c),\, \forall \, c>0$. Hence, $J_{\xi}(s)$ is continuous in $\xi$ over $(0,\infty)$. [$\Box$]{} [^1]: This work was supported by the Department of Science and Technology (DST), India, through the J.C. Bose Fellowship, by an Indo-Brazil cooperative project on “WIreless Networks and techniques with applications to SOcial Needs (WINSON)," and by a project funded by the Department of Electronics and Information Technology, India, and NSF, USA, titled “Wireless Sensor Networks for Protecting Wildlife and Humans in Forests.” [^2]: This paper is an extension of [@chattopadhyay-etal12optimal-capacity-relay-placement-line], and is also available in [@chattopadhyay-etal12optimal-capacity-relay-placement-line-arxiv-17April2013]. [^3]: Arpan Chattopadhyay and Anurag Kumar are with the Electrical Communication Engineering (ECE) Department, Indian Institute of Science (IISc), Bangalore-560012, India (e-mail: arpanc.ju@gmail.com, anurag@ece.iisc.ernet.in). Abhishek Sinha is with the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA 02139 (e-mail: sinhaa@mit.edu). Marceau Coupechoux is with Telecom ParisTech and CNRS LTCI, Dept. Informatique et Réseaux, 23, avenue d’Italie, 75013 Paris, France (e-mail: marceau.coupechoux@telecom-paristech.fr). This work was done during the period when he was a Visiting Scientist in the ECE Deparment, IISc. [^4]: Full-duplex radios are becoming practical; see [@khandani13two-way-full-duplex-wireless], [@khandani10spatial-multiplexing-two-way-channel], [@choi-etal10single-channel-full-duplex], [@jain-etal11real-time-full-duplex]. [^5]: $\log (\cdot)$ in this paper will mean the natural logarithm unless the base is specified. [^6]: A motivation for the use of the exponential distribution, given the prior knowledge of the mean length $\overline{L}$, is that it is the maximum entropy continuous probability density function with the given mean. By using the exponential distribution, we are leaving the length of the line as uncertain as we can, given the prior knowledge of its mean. [^7]: Recall Section \[subsec:sum\_power\_constraint\]. The battery depletion rate $\frac{P_{rcv}}{E}$ of a node due to the receive power alone can be absorbed into the relay cost $\xi$. [^8]: The constraint on the mean number of relays can be justified if we consider the relay deployment problem for multiple source-sink pairs over several lines of mean length $\overline{L}$, given a large pool of relays, and we are only interested in keeping small the total number of relays over all these deployments. [^9]: $\frac{e^{\Lambda \tilde{z}_k}}{\sum_{i=0}^{k}e^{\Lambda \tilde{z}_i}} \geq \frac{e^{\Lambda \tilde{z}_k}}{(k+1)e^{\Lambda \tilde{z}_k}}=\frac{1}{k+1}$; hence, if $s_k$ is small, $k$ must be large enough. [^10]: Marano and Franceschetti ([@marano-franceschetti05ray-propagation-random-lattice]) modeled a city as a random lattice, and the distance from the transmitter to the receiver is measured along the edges of the lattice instead of the Euclidean distance. Hence, this result renders the analysis in our paper valid even for deployment along the streets of a city with turns; deployment algorithm in that case will only consider the distances along the streets and not on the actual Euclidean distances. [^11]: Remember that at state $0$ (i.e., state $\mathbf{EOL}$), the single stage cost is $0$ irrespective of the action, and that this state is absorbing. Hence, any action at state $0$ can be optimal.

--- abstract: 'The radio access network (RAN) is regarded as one of the potential proposals for massive Internet of Things (mIoT), where the random access channel (RACH) procedure should be exploited for IoT devices to access to the RAN. However, modelling of the dynamic process of RACH of mIoT devices is challenging. To address this challenge, we first revisit the frame and minislot structure of the RAN. Then, we correlate the RACH request of an IoT device with its queue status and analyze the queue evolution process. Based on the analysis result, we derive the closed-form expression of the RA success probability of the device. Besides, considering the agreement on converging different services onto a shared infrastructure, we further investigate the RAN slicing for mIoT and bursty ultra-reliable and low latency communications (URLLC) service multiplexing. Specifically, we formulate the RAN slicing problem as an optimization one aiming at optimally orchestrating RAN resources for mIoT slices and bursty URLLC slices to maximize the RA success probability and energy-efficiently satisfy bursty URLLC slices’ quality-of-service (QoS) requirements. A slice resource optimization (SRO) algorithm exploiting relaxation and approximation with provable tightness and error bound is then proposed to mitigate the optimization problem.' author: - 'Peng Yang, Xing Xi, Tony Q. S. Quek, , Jingxuan Chen, Xianbin Cao, , Dapeng Wu, [^1]' bibliography: - 'Network\_slicing.bib' title: 'RAN Slicing for Massive IoT and Bursty URLLC Service Multiplexing: Analysis and Optimization' --- Massive IoT, random access channel, bursty URLLC, RAN slicing Introduction ============ the explosive growth of the Internet of Things (IoT), massive IoT (mIoT) devices, the number of which is predicted to reach 20.8 billion by 2020, will access to the wireless networks for implementing advanced applications, such as e-health, public safety, smart traffic, virtual navigation/management, remote maintenance and control, and environment monitoring. To address the IoT market, the third-generation partnership project (3GPP) has identified mIoT as one of the three main use cases of 5G and has already initiated several task groups to standardize several solutions including extended coverage GSM (EC-GSM), LTE for machine-type communication (LTE-M), and narrowband IoT (NB-IoT) [@Cellular-Ericsson; @LTE-Nokia]. For establishing massive connections among the wireless networks and mIoT devices, the investigation of reliable and efficient access mechanisms should be prioritized. In accomplishing the massive connections, when an active IoT device wants to transmit signal in the uplink, it randomly chooses a random access (RA) preamble from an RA preamble pool and transmits it through an RA channel (RACH). If more than one device tries to access to a base station (BS) simultaneously, then interference occurs at the RRH. During the past few years, a rich body of works on RA mechanisms has been developed [@xing2019novel; @jang2016non; @jiang2019reinforcement; @zhang2019dnn; @jiang2018analyzing; @grau2019preamble; @jang2019recursive; @jin2017recursive; @jang2019versatile] to mitigate interference and improve the RA success probability or reduce the access delay of an IoT device. Most of the studies [@xing2019novel; @jang2016non; @jiang2019reinforcement; @zhang2019dnn; @jiang2018analyzing; @grau2019preamble; @jang2019recursive; @jin2017recursive; @jang2019versatile], however, assumed that the whole network resources were reserved for the IoT service and did not investigate the case of the coexistence of IoT service and many other services such as enhanced mobile broadband (eMBB) and ultra-reliable and low latency communications (URLLC). The research of the coexistence of IoT service and other services is essential as future networks are convinced to converge variety of services with different latency, reliability, and throughput requirements onto a shared physical infrastructure rather than deploying individual network solution for each service [@alliance20155g]. What is more, owing to the shared characteristic of network resources, some conclusions obtained in the case of providing sole IoT service may become inapplicable if multiple types of services are required to be supported by the networks. Network slicing is considered as a promising technology in future networks for providing scalability and flexibility in allocating network resources to various services. Recently, many network slicing frameworks have been developed to provide performance guarantees to IoT or massive machine-type communications (mMTC) service, eMBB service, and URLLC service [@popovski20185g; @budhiraja2019tactile; @ksentini2017toward; @rost2017network; @wen2018robustness; @buyakar2018poster; @guo2019enabling]. Different from previous works, this paper investigates the mIoT and bursty URLLC service multiplexing via slicing the radio access network (RAN). This study is highly challenging because i) performance requirements of a massive number of IoT devices should be satisfied. Yet, the typical 5G cellular IoT, NB-IoT can admit only 50,000 devices per cell [@3GPP15Cellular]; ii) RAN slicing operation (e.g., creating, activating, and releasing slices) has to be conducted in a timescale of minutes to hours to keep pace with the upper layer slicing. However, the wireless channel generally changes in a timescale of millisecond to seconds. Results of the RAN slicing operation are desired to be achieved based on the time-varying channel. Thus, the RAN slicing should tackle a two timescale issue [@tang2019service]. These challenges motivate us to investigate the RAN slicing for mIoT and bursty URLLC service provision to maximize the utility of mIoT slices and that of bursty URLLC slices. The main contributions of this paper can be summarized as the following: - We revisit the frame and minislot structure for mIoT transmission to accommodate more RA requests from a massive number of IoT devices. - We adopt a queueing model to track the IoT packet arrival, accumulate and departure processes and analyze the queue evolution process by employing probability and stochastic geometry theories. Based on the analysis result, we derive the closed-form expression of the RA success probability of a randomly chosen IoT device. - We define mIoT slice utility and bursty URLLC slice utility and formulate the RAN slicing for mIoT and bursty URLLC service multiplexing as a resource optimization problem. The objective of the optimization problem is to maximize the total mIoT and URLLC slice utilities, subject to limited physical resource constraints. The mitigation of this problem is difficult due to the existence of indeterministic objective function and thorny non-convex constraints and the requirement of tackling a two timescale issue as well. - To mitigate this thorny optimization problem, we propose a slice resource optimization (SRO) algorithm. In this algorithm, we first exploit a sample average approximate (SAA) technique and an alternating direction method of multipliers (ADMM) to tackle the indeterministic objective function and the two timescale issue. Then, a semidefinite relaxation (SDR) scheme joint with a Taylor expansion scheme are leveraged to approximate the non-convex problem as a convex one. The tightness of the SDR scheme and the error bound of the Taylor expansion are also analyzed. The remaining of this paper is organized as follows. We review the prior arts in Section II. In Section III, we describe our system model and formulate the service multiplexing problem in Section IV. The problem-mitigating algorithm is presented in Sections V and VI. In Section VII, we give the simulation results and conclude this paper in Section VIII. Prior arts ========== Recently, many researches have been conducted to increase RA success probabilities and/or reduce the access delay of mIoT devices. They can be generally classified into two groups: traffic detection and estimation based algorithms and algorithms without traffic detection and estimation. The fundamental idea of the traffic detection and estimation based algorithms is to design an RA algorithm based on the detected and/or estimated users’ activity and traffic congestion situation and so on. For example, to reduce the access delay, a grant-free non-orthogonal RA system relying on the accurate user activity detection and channel estimation was proposed in [@zhang2019dnn]. A traffic-aware spatiotemporal model for the contention-based RA analysis is conducted for mIoT networks in [@jiang2018analyzing]. With the spatiotemporal model, a hybrid power ramping and back-off RA scheme was then developed to improve the RA success probability. Besides, an extended pseudo-Bayesian backlog estimation scheme was exploited in [@jang2019versatile] to estimate the number of backlogged nodes to attempt access. A versatile access control mechanism was then designed to reduce the access delay based on the estimation results. For algorithms without detection and estimation, they design RA schemes without detecting users’ activity or estimating the statistical characteristic of traffic. For instance, the work in [@xing2019novel] proposed to improve the RA success probability of an IoT device by exploiting a distributed queue mechanism and then proposed an access resource grouping mechanism to reduce the access delay caused by the queuing process of the distributed queue mechanism. To increase RA success probability, the work in [@jang2016non] proposed to increase the number of preambles at the first step of the RA procedure by utilizing a spatial group mechanism and improve resource utilization through non-orthogonally allocating uplink channel resources at the second step of the RA procedure. Additionally, without knowing the statistical characteristic of traffic, a reinforcement learning-based algorithm was proposed in [@jiang2019reinforcement] to determine the uplink resource configuration for RA such that the average number of served IoT devices was maximized while ensuring a high RA success probability. Except for the IoT service, future networks are envisioned to simultaneously support different services and applications with significantly different requirements on reliability, latency and bandwidth. As a result, researchers are now paying more attention to the service multiplexing of IoT/mMTC and many other services such as eMBB and URLLC. For example, instead of slicing the RAN via orthogonal resource allocation among different services, the work in [@popovski20185g; @budhiraja2019tactile] studied the potential advantages of allowing for non-orthogonal RAN resources sharing in uplink communications from a collection of mMTC, eMBB, and URLLC devices to the same BS. The work in [@ksentini2017toward] developed a two-level scheduling process to allocate dynamically dedicated bandwidth to each network slice according to workload demand and slices’ quality of service (QoS) requirement such that flexible resource allocation could be implemented. The work in [@rost2017network] proposed to maintain slice-specific radio resource control elements with which the RAN protocol stacks and different slices were configured. Besides, the work in [@wen2018robustness] aimed to optimize the virtual network functions and infrastructure resources such as the system bandwidth to implement slice recovery and reconfiguration for mMTC and eMBB service provision. The work in [@buyakar2018poster] proposed to maintain slice isolation between mMTC and eMBB slices and meet the performance requirements of these slices through limiting and dynamically updating the amount of resources allocated to each slice and monitoring the resource usage of each slice. After representing the slice performance requirements as the required amount of resources per deadline interval, an idea of earliest-deadline and first-scheduling was exploited in [@guo2019enabling] to allocate radio resources to mMTC, eMBB, and URLLC slices effectively. System model ============ We consider a coordinated-multipoint-enabled RAN slicing system for mIoT and bursty URLLC multiplexing service provision. From the viewpoint of the infrastructure composition, the system mainly includes one baseband unit (BBU) pool and multiple remote radio heads (RRHs) that connect to the BBU via fronthaul links. From the perspective of network slicing, two types of inter-slices, i.e., mIoT slices and URLLC slices, are exploited in this system with $\mathcal{S}^I$ and $\mathcal{S}^u$ representing the sets of mIoT slices and URLLC slices, respectively. We focus on the modelling of uplink IoT data transmission in mIoT slices and the modelling of downlink URLLC data transmission in URLLC slices. The IoT devices are spatially distributed in ${\mathbb R}^2$ according to an independent homogeneous Poisson point process (PPP) $\Phi_s = \{u_{i,s}; s \in \mathcal{S}^I, i = 1, 2, \ldots\}$ with intensity $\lambda_s^I$, where $u_{i,s}$ denotes the location of the $i$-th IoT device in the $s$-th mIoT slice. There are also $N^u$ URLLC devices that are randomly and evenly distributed in ${\mathbb R}^2$. The RRHs are spatially distributed in ${\mathbb R}^2$ according to an independent PPP $\Phi_R = \{v_j; j = 1, 2, \ldots\}$ with intensity $\lambda_R$, where $v_j$ represents the location of the $j$-th RRH. The number and locations of IoT devices and RRHs will be fixed once deployed. Besides, each RRH is equipped with $K$ antennas, and each device is equipped with a single antenna. In IoT network slices, each IoT device is assumed to connect to its geographically closest RRH [@jiang2018analyzing]; thus, the cell area of each RRH constitutes a Voronoi tessellation. In URLLC network slices, RRHs cooperate to transmit signals to a URLLC device to improve its signal-to-noise ratio (SNR). A flexible frequency division multiple access (FDMA) technique is utilized to achieve the inter-slice and intra-slice interference isolation [@tang2019service]. The system time is discretized and partitioned into time slots and minislots with a time slot consisting of $T$ minislots. On the one hand, at the beginning of each time slot, a RAN slicing coordinator [@How2019yang] will decide whether to accept or reject received network slice requests which will be defined in the following subsections. Once a slice request is accepted, a network slice management will be responsible for activating or creating a virtual slice that is well resource-configured to satisfied the QoS requirements of devices in the slice [@How2019yang]. The slice configuration process is time costly and will generally be conducted in a timescale of minutes to hours [@tang2019service]. On the other hand, at the beginning of each minislot, each active IoT device may try to connect to its associated RRH, and RRHs will generate cooperated beamformers based on sensed channel coefficients. mIoT slice model ---------------- By referring to the concept of a network slice [@rost2017network], especially from the viewpoint of the QoS requirement of a slice, we can define a mIoT slice request as follows. \[def:IoT\_slice\_definition\] A mIoT slice request is defined as a tuple $\{\lambda_s^I, \gamma_s^{th}, N_{a,s}\}$ for any slice $s \in {\mathcal S}^I$, where $\gamma_s^{th}$ is the requirement of data transfer rate from an IoT device in $s$ to its associated RRH, $N_{a,s}$ denotes the number of accumulated packets in a queue of an IoT device in $s$. In this paper, all mIoT slice requests are always accepted by the RAN slicing coordinator. IoT devices with the same data transfer rate are assigned to the similar slice. For an IoT device in $s$, if it has the opportunity to send its endogenous arrival packets to the corresponding RRH, then it will randomly select a preamble (e.g., orthogonal Zadoff¨CChu sequences) from a BBU-maintained preamble pool and transmit the preamble to the RRH at the data rate $\gamma_s^{th}$. Just like the literature [@soorki2017stochastic; @gharbieh2017spatiotemporal], if the RRH can successfully decode the preamble, then a connection between the IoT device and the RRH are considered to be set up although the whole connection establishment process usually follows an RA four-step procedure [@grau2019preamble]. In other word, the RA success probability is regarded as the probability of successfully transmitting a preamble in this paper. Next, we will analyze an IoT device queue evolution model, with the analysis of which the RA success probability of the IoT device will be derived. ### Queue evolution model The queue evolution process consists of the packet arrival process, packet accumulate process, and packet departure process. During minislot $t$, a Poisson distribution with intensity (or arrival rate) $\epsilon_{w,s}(t)$ is exploited to model the random, mutually independent endogenous packet arrivals in an IoT device in slice $s$. Then during minislot $t$ with a duration $\tau$, the arrival intensity of new packets can be expressed as $\mu_{w,s}(t) = \epsilon_{w,s}(t)\tau$. Once arrived, new packets will not be sent out immediately in general and will enter a queue, which is modelled as an $M/M/k$ queue with unlimited capacity, to wait for their scheduling. In the $M/M/k$ queue, packets will be scheduled according to the first-come, first-served (FCFS) basis. The unlimited queue capacity indicates that the age of information [@kaul2011minimizing; @kaul2012real] of new arrivals will not be considered, and packets will not be dropped before sending out. Besides, owing to the RA behavior of a slotted-ALOHA protocol, new arrivals during $t$ will only be counted at minislot $t + 1$. Thus, the accumulated number of packets $N_{a,s}(t)$ of a randomly selected IoT device in slice $s$ at $t$ is determined by the accumulated number of packets and the number of new arrivals at $t - 1$ and whether the preamble of the device can be successfully decoded by its associated RRH. Table \[table:queue\_evolution\] shows the evolution of accumulated packets in an IoT device. In this table, $x_s = \gamma_s^{th}/L$ packets at the head of the queue will be popped out if the corresponding RA succeeds, where $L$ denotes the IoT packet length; otherwise, they will be kept in the queue and wait for the opportunity of re-transmission at the next minislot. The operation $[x]^+ = \max(x,0)$. Value Success Failure ---------------- ------------------------------------------------------ ----------------------------------------- $N_{a,s}(1)$ $0$ $0$ $N_{a,s}(2)$ $[N_{w,s}{(1)}$ - $x_s]^+$ $N_{w,s}{(1)}$ $N_{a,s}{(3)}$ $[N_{a,s}{(2)}$ + $N_{w,s}{(2)}$ - $x_s]^+$ $N_{a,s}{(2)}$ + $N_{w,s}{(2)}$ $\ldots$ $\ldots$ $\ldots$ $N_{a,s}{(t)}$ $ [N_{a,s}(t$ - $1)$ + $N_{w,s}(t$ - $1)$ - $x_s]^+$ $N_{a,s}(t$ - $1)$ + $N_{w,s}(t$ - $1)$ : Accumulated packets evolution in an IoT device[]{data-label="table:queue_evolution"} With the evolution of accumulated packets, we can define the non-empty probability of the queue of an IoT device in $s$ as the following. \[def:non\_empty\_prob\] At minislot $t$, for a randomly selected IoT device in slice $s \in \mathcal{S}^I$, the probability that its queue is not empty can be defined as $$\label{eq:non_empty_prob} P_{ne,s}{(t)} = {\mathbb P}\{N_{a,s}(t) > 0\}, \forall s \in {\mathcal{S}^I}$$ (\[eq:non\_empty\_prob\]) implicitly reflects that new arrival packets at $t$ will not be sent out immediately. According to the evolution of $N_{a,s}(t)$, it can be observed that $P_{ne,s}(t)$ is determined by the probability distribution of $N_{a,s}(t-1)$ and the RA success probability. Since these probabilities and their correlations are unknown, the derivation of the explicit expression of $P_{ne,s}(t)$ is difficult. Next, we describe the packet departure process combined with a frame and minislot structure for mIoT packets transmission. As mentioned above, partly because of the limitation on the frame and minislot structure, NB-IoT and LTE-M can only admit 50,000 devices. For NB-IoT, only one physical resource block (PRB) with a bandwidth of $180$ KHz in the frequency domain is allocated for IoT transmission, and each physical channel occupies the whole PRB. For LTE-M, although the physical channels are time and frequency multiplexed, it only reserves six in-band PRBs with a total bandwidth of $1.08$ MHz in the frequency domain for IoT data transmission. Therefore, the frame and minislot structure for mIoT transmission should be revisited if more RA requests from IoT devices want to be accepted. Fig. \[fig:fig\_frame\_minislot\_structure\] depicts a frame and minislot structure for mIoT transmission in each mIoT slice[^2]. In this structure, both the frequency division multiplexing scheme and code division multiplexing scheme are leveraged to admit more IoT devices in the way of alleviating the mutual device interference. Particularly, the frequency division multiplexing scheme alleviates signal interference through orthogonal frequency allocation, and the code division multiplexing scheme mitigates the co-channel signal interference via reducing the cross-correlation of simultaneous transmissions. The combination of the two schemes may significantly mitigate interference experienced at an RRH. In this way, the QoS requirements of more IoT devices may be satisfied, and the RAN slicing system may support more IoT devices. For a mIoT slice $s \in \mathcal{S}^I$, each subframe includes $F_s$ orthogonal uplink physical RA channels (PRACHs). A single tone mode with a tone spacing of size of $a$ MHz is adopted for each uplink PRACH, which indicates that each PRACH occupies a PRB. At the beginning of each minislot, an active IoT device, i.e., the device’s queue is non-empty, will randomly choose a preamble from a set of non-dedicated RA preambles of size $\xi$ and transmit the preamble through a randomly selected PRACH. For each preamble, it has an equal probability ${\frac{1}{\xi}}$ to be chosen by each IoT device. Similarly, each PRACH has an equal probability $\frac{1}{F_s}$ to be selected. Thus, the average number of IoT devices in mIoT slice $s \in \mathcal{S}^I$ choosing the same PRACH and the same preamble is $\frac{\lambda_s^I}{\xi F_s}$. Notably, a greater $\xi F_s$ may significantly reduce signal interference experienced at each RRH. Then, the following question should be tackled: *how many PRBs should be reserved for mIoT transmission?* To improve the resource utilization, the resource allocated to mIoT should be determined according to the requirements of mIoT and other coexistence services. It motivates us to optimize the resources orchestrated for the mIoT service except for analyzing the RACH procedure of IoT devices. The optimization procedure will be discussed in detail in the next section. ![The frame and minislot structure. ’R’ and ’D’ denote the resource block reserved for preamble and IoT data transmission. PBCH, PSS and SSS represent the PRBs for physical broadcast channel, primary synchronization signal and secondary synchronization signal transmission, respectively.[]{data-label="fig:fig_frame_minislot_structure"}](frame_slot_structure.eps){width="2.9in"} ### Access control scheme In a mIoT network slice, as the slotted-ALOHA protocol allows all active IoT devices to request for RA at the beginning of each minislot without checking the status of channels, IoT devices may simultaneously transmit preambles. It may incur severe slice congestion that may lower the RA success probabilities of IoT devices and degrade the system performance. Access control has been considered as an efficient proposal of alleviating congestion, and many access control schemes have been proposed [@jiang2018analyzing; @Study163GPP]. In this paper, we aim at illustrating the performance difference between a network slicing system without access control and with access control. Therefore, we adopt the following two schemes [@jiang2018analyzing]: - **Unrestricted scheme:** each active IoT device requests the RACH at the beginning of minislot $t$ without access restriction. If mIoT slices are not crowded or in a light-crowded condition, then this scheme may quickly flush queues of IoT devices. However, if a heavy-crowded condition is encountered, then this scheme may result in a high packet queueing delay. - **Access class barring (ACB) scheme:** at the beginning of $t$, each active IoT device throws a random number $q \in [0, 1]$ and can request the RACH only if $q < P_{ACB}$, where $P_{ACB}$ is an ACB factor determined by RRHs based on the slice congestion condition. The ACB scheme can relieve slice congestion to some extent by reducing RACH requests of active IoT devices. With the introduced access control schemes, we can define the non-restriction probability of a randomly selected IoT device in $s$ as follows. \[mydef:unrestricted\_prob\] At minislot $t$, for a randomly selected IoT device in slice $s \in \mathcal{S}^I$, the probability that its RACH request is not restricted is defined as $$\label{eq:non_restriction_prob} P_{nr,s}{(t)} = {\mathbb P}\{{\rm Unrestricted} \text{ } {\rm RACH} \text{ } {\rm requests}\}, \forall s \in {\mathcal{S}^I}$$ For all $s \in \mathcal{S}^I$ at any minislot $t$, we have $P_{nr,s}(t) = 1$ for the unrestricted scheme and $P_{nr,s}(t) = P_{ACB}$ for the ACB scheme. ### Analysis of RA success probability For an RRH, two significant reasons may lead to an error preamble decoding i) the achieved preamble transfer rate at the RRH is less than a preset threshold; ii) the RRH simultaneously decodes at least two similar co-channel preambles, and thus preamble collision occurs. The research of the mitigation of preamble collision has been well conducted in [@jang2019versatile; @jiang2018collision]. Just like [@jiang2018random], we focus on the exploration of enabling successful single preamble transmission that is discussed in detail as follows. We utilize a power-law path-loss model to calculate the path-loss between an IoT device and its RRH in mIoT slices and utilize a truncated channel inversion power control scheme to eliminate the ’near-far’ effect. In the power-law path-loss model, the IoT device transmit power decays at the rate of $r^{-{\varphi}}$ with $r$ representing the propagation distance and $\varphi$ denoting the path-loss exponent. In the power control scheme, IoT devices associated with the same RRH compensate for the path-loss to maintain the average received signal power at the RRH equal to a threshold $\rho_o$. Without loss of generality, the cutoff threshold $\rho_o$ is set to be the same for all RRHs. Owing to the channel deep fading, severe co-channel interference, and insufficient transmit power, an IoT device may experience uplink preamble transmission outage. The following definition describes the definition of the probability that a randomly selected IoT device can successfully transmit a chosen preamble to its corresponding RRH. \[mydef:preamble\_trans\_suc\_prob\] At minislot $t$, for a randomly selected active IoT device in slice $s \in \mathcal{S}^I$, its RA success probability is defined as $$\label{eq:preamble_trans_suc_prob} P_{s}{(t)} = {\mathbb P}\{r_s(t) \ge \gamma_s^{th}\}, \forall s \in {\mathcal{S}^I}$$ where $r_s(t) = a{\log _2}(1 + SIN{R_s}(t))$ denotes the achieved preamble transfer rate at the IoT device’s associated RRH and $SIN{R_s}(t)$ is the signal-to-interference-plus-noise ratio (SINR). Then, for any active IoT device in $s$, its QoS requirement is given by $$\label{eq:mMTC_QoS} P_s(t) \ge {\pi _s}, \forall s \in {\mathcal{S}^I}$$ where $\pi_s$ denotes a threshold of the required RA success probability. This definition shows that the QoS requirement of each active IoT device in $s$ should be satisfied if the slice request of $s$ is accepted. The definition also states that $P_{s}{(t)}$ is correlated with the non-empty probability $P_{ne,s}{(t)}$. Recall that the RA success probability of an IoT device impacts its non-empty probability, we can know that the RA success probability and the non-empty probability are intertwined. Additionally, $SIN{R_s}(t)$ is a function of complicated co-channel interference. Thus, it is hard to obtain the closed-form expression of $P_{s}{(t)}$. Without any loss in generality, we perform the analysis of RA success probability on an RRH located at the origin. According to Slivnyak’s theorem [@haenggi2012stochastic], the analysis holds for a generic RRH located at a generic location. For a randomly selected IoT device with non-empty queue in $s \in \mathcal{S}^I$, the theoretical preamble transfer rate experienced at the RRH located at the origin can take the form $$\label{eq:preamble_trans_suc_prob} r_s(t) = a \log_2 \left (1 + \frac{{\rho_o {h_o}}}{{{\sigma ^2} + {{\cal I}_{s}}(t)}} \right ), \forall s \in {\mathcal{S}^I}$$ where $\sigma^2$ represents the noise power, ${{\cal I}_{s}}(t)$ denotes signal interference received at the RRH, the useful signal power equals to $\rho_o h_o$ due to the truncated channel inversion power control[^3] [@elsawy2014stochastic] with $h_o$ denoting the channel power gain between the IoT device and the RRH. It is noteworthy that the channel power gain experienced at a generic RRH is related to the spatial locations of both the RRH and its associated IoT devices. Nevertheless, we drop the spatial indices for notation lightening. Besides, just like [@elsawy2014stochastic], all channel gains are assumed to be known and be independent of each other, independent of the spatial locations, symmetric and are identically distributed (i.i.d.). Considering both the particular IoT device deployment environment and the convenience of theoretical analysis, the Rayleigh fading is assumed, and the channel power gain $h_o$ is assumed to be exponentially distributed with unit mean. Based on the following five facts, we next present the analytical expression of signal interference - **Fact 1**: the average signal received from any single IoT device belonging to inter-cells is strictly less than $\rho_o$. - **Fact 2**: the average interference signal received from any single interfering IoT device associated with the origin RRH strictly equals to $\rho_o$. - **Fact 3**: IoT devices choosing the same co-channel preamble as the randomly selected IoT device may become an interfering IoT device. - **Fact 4**: at each minislot, IoT devices with non-empty queue may become interfering IoT devices. - **Fact 5**: IoT devices in difference slices may not mutually interfere. Note that Fact 1 and Fact 2 are direct consequences of the device-RRH association policy and power control scheme. Fact 5 holds due to the exploration of intra-slice isolation. Therefore, the aggregate interference received at the origin RRH can take the following form $$\label{eq:I_intra} \begin{array}{l} {\cal I}_{s}(t) = \sum\limits_{{u_{m,s}} \in \Phi_s \backslash \{ o\} } {{\mathbbm 1}({p_m}||{d_m}|{|^{ - \varphi }} = {\rho _o}){\mathbbm 1}(N_{a,s}{(t)} > 0)} \times \\ \qquad \quad \text{ } {{\mathbbm 1}({f_m} = {f_o}){\rho _o}{h_m}}, \forall s \in {\mathcal{S}^I} \end{array}$$ where $o$ represents the randomly selected IoT device associated with the RRH at the origin, $p_m$ represents the transmit power of the $m$-th IoT device, $||d_m||$ is the distance between the $m$-th IoT device and the origin RRH, $f_o$ denotes the preamble and channel chosen by the randomly selected IoT device, $f_o = f_m$ indicates that the randomly selected IoT device and the $m$-th IoT device select the same preamble and channel. ${\mathbbm 1}(\cdot)$ is the indicator function that equals to one if the statement ${\mathbbm 1}(\cdot)$ is ture; otherwise, it equals to zero. Just like [@zhang2015resource], in (\[eq:I\_intra\]), co-channel inter-cell interference is assumed as a part of thermal noise mainly because of the severe wall penetration loss. Then, for the randomly selected IoT device in $s \in {\mathcal{S}^I}$, we can rewrite (\[eq:preamble\_trans\_suc\_prob\]) as the following form with (\[eq:I\_intra\]) $$\label{eq:QoS_analysis} \begin{array}{l} P_s(t) = {\mathbb P}\{ SINR_s(t) \ge \theta _s^{th}\} \\ \qquad \text{ } = {\mathbb P}\{ {h_o} \ge \frac{{\theta _s^{th}}}{{{\rho _o}}}({\sigma ^2} + {{\cal I}_{s}(t)})\} \\ \qquad \text{ } \mathop = \limits^{(a)} {{\mathbb E}}\left[ {\exp \left\{ { - \frac{{\theta _s^{th}}}{{{\rho _o}}}({\sigma ^2} + {{\cal I}_{s}(t)})} \right\}} \right]\\ \qquad \text{ } = \exp \left\{ { - \frac{{\theta _s^{th}}}{{{\rho _o}}}{\sigma ^2}} \right\}{\cal L}_{{{\mathcal I}_{s}(t)}}\left(\frac{\theta_s^{th}}{\rho_o}\right), \forall s \in {\mathcal{S}^I} \end{array}$$ where $\theta_s^{th} = {2^{\gamma _s^{th}/a}} - 1$. (a) follows from the full probability law over ${\cal I}_{s}(t)$, and ${\cal L}_{{{\cal I}_{s}(t)}} (\cdot)$ denotes the Laplace transform (LT) of the probability density function (PDF) of the random variable ${\cal I}_{s}(t)$. Note that the notation ${\cal L}_{{{\cal I}_{s}(t)}} (\cdot)$ is a terminology that is a slight abuse of subscript ${{\cal I}_{s}(t)}$. The following lemma characterizes the LT of aggregate interference $\mathcal{I}_s(t)$. \[lem:LT\_interference\_expression\] For the origin RRH, the LT of its received aggregate interference from active IoT devices associated with it is given by $$\label{eq:LT_interference_expression} {\cal L}_{{{\cal I}_{s}(t)}} \left(\varpi_s\right) = {\frac{{1 + \varpi_s{\rho _o}}}{{{{\left( {1 + \alpha_s \varpi_s{\rho _o}/\left( {1 + \varpi_s{\rho _o}} \right)} \right)}^{3.5}}}} - \frac{{1 + \varpi_s{\rho _o}}}{{{{\left( {1 + \alpha_s } \right)}^{3.5}}}}}$$ where $\varpi_s = \frac{\theta_s^{th}}{\rho_o}$, $\alpha_s = \frac{{{P_{nr,s}}(t){P_{ne,s}}(t){\lambda _{{s}}^I}}}{{3.5{\lambda _R}{\xi F_s}}}$, for all $s \in {\mathcal{S}^I}$. Please refer to Appendix A. With the conclusion in Lemma \[lem:LT\_interference\_expression\], we can then obtain the mathematical expression of the RA success probability of a randomly selected IoT device at $t$ in the following corollary. \[lem:QoS\_prob\_analysis\] For a randomly selected IoT device in a mIoT slice $s \in {\cal S}^I$, its RA success probability at minislot $t$ is given by $$\label{eq:QoS_prob_analysis} {P_s}(t) = {\frac{{(1 + \varpi_s{\rho _o})}{{e^{-\varpi_s{\sigma ^2}}}}}{{{{\left( {1 + \alpha_s \varpi_s{\rho _o}/\left( {1 + \varpi_s{\rho _o}} \right)} \right)}^{3.5}}}} - \frac{{(1 + \varpi_s{\rho _o})}{{e^{-\varpi_s{\sigma ^2}}}}}{{{{\left( {1 + \alpha_s } \right)}^{3.5}}}}}$$ By substituting (\[eq:LT\_interference\_expression\]) into (\[eq:QoS\_analysis\]), we can obtain (\[eq:QoS\_prob\_analysis\]). Although Corollary \[lem:QoS\_prob\_analysis\] presents a mathematical expression of $P_s(t)$, the expression is not in the closed-form as it is a function of $P_{ne,s}(t)$ the closed-form expression of which is not obtained. Next, we derive the closed-form expression of $P_{ne,s}(t)$. ### Analysis of non-empty probability According to the definition of non-empty probability, $P_{ne,s}(t)$ is correlated with the number of accumulated packets $N_{a,s}(t)$ of the randomly selected IoT device in mIoT slice $s$. Thus, we theoretically analyze the non-empty probability of the randomly selected IoT device as the following. As the number of the accumulated packets in the queue of a randomly selected IoT device in slice $s \in \mathcal{S}^I$ at the $1^{\rm st}$ minislot is empty, its non-empty probability $P_{ne,s}^{1}$ at the $1^{\rm st}$ minislot can take the form $$\label{eq:non_empty_prob_1} P_{ne,s}^{1} = {\mathbb P}\{N_{a,s}^1 > 0\} = 0, \forall s \in {\mathcal{S}^I}$$ where we write $x^t$ instead of $x(t)$ to lighten the notation. The following lemma presents the closed-form expression of the non-empty probability of a randomly selected IoT device served by the origin RRH when minislot $t > 1$. \[lem:non\_empty\_prob\_lemma\] The number of accumulated packets of a randomly selected IoT device served by the origin RRH at minislot $t > 1$ may be approximately Poisson distributed. Therefore, for any mIoT slice $s \in \mathcal{S}^I$, we approximate the number of accumulated packets $N_{a,s}^{t}$ at minislot $t$ as a Poisson distribution with intensity $\mu_{a,s}^{t}$, which is given by $$\label{eq:mu_accumulated_packets} \mu _{a,s}^t = \left [\mu _{w,s}^{t - 1} + \mu _{a,s}^{t - 1} - P_{s}^{t - 1}\left( {1 - {e^{ - \mu _{w,s}^{t - 1} - \mu _{a,s}^{t - 1}}}} \right) \right ]^+, \forall s \in {\mathcal{S}^I}$$ Then, the non-empty probability of the device at minislot $t$ can be written as $$\label{eq:non_empty_prob_m} P_{ne,s}^t = 1 - {e^{ - \mu _{a,s}^t}}, \forall s \in {\mathcal{S}^I}$$ Please refer to Appendix B. Combine with (\[eq:QoS\_prob\_analysis\]) and (\[eq:mu\_accumulated\_packets\]), the closed-form expression of $P_s(t)$ ($s \in \mathcal{S}^I$) can be obtained. Bursty URLLC slice model ------------------------ Similar to the definition of a mIoT slice request, a bursty URLLC slice request can be defined as the following. \[mydef:URLLC\_slice\_request\] A bursty URLLC slice request is defined as four tuples $\{I_s^u, D_s, \alpha, \beta\}$ for slice $s \in {\mathcal S}^u$, where $I_s^u$ denotes the number of URLLC devices in $s$, $D_s$ denotes the transmission latency requirement of each URLLC device in $s$, $\alpha$ and $\beta$ represent the packet blocking probability and the codeword error decoding probability of each URLLC device, respectively. In this definition, URLLC devices are grouped into $|{\mathcal S}^u|$ clusters according to the transmission latency requirement of each device. Owing to the low latency requirement URLLC packets should be immediately scheduled upon arrival; thus, all URLLC slice requests will always be accepted by the RAN coordinator. Except for the low packet error decoding probability that has been emphasized for URLLC transmission in a plenty of works [@tang2019service], this paper attempts to orchestrate slice resources to reduce the packet blocking probability for bursty URLLC transmission. This is because the bursty characteristic of URLLC traffic [@How2019yang] may lead to the packet blocking in URLLC slices, which may significantly reduce the reliability of URLLC transmission. Therefore, the indicators $\alpha$ and $\beta$ are involved to reflect the ultra-reliable requirement of URLLC transmission jointly. Then, we address the following question: *how to orchestrate slice resources for reducing packet blocking probability and codeword error decoding probability?* ### Reduction of packet blocking probability As mentioned above, the bursty feature of URLLC traffic is the crucial factor that leads to the URLLC packet blocking for URLLC transmission. Therefore, we next model the bursty URLLC traffic based on which we discuss how to orchestrate slice resources to alleviate the impact of bursty URLLC traffic. During minislot $t$, an independent homogeneous Poisson distribution with intensity ${\bm \lambda} = \{\lambda_s; s \in {\mathcal{S}^u}\}$ is utilized to model the number of bursty URLLC packets aggregated at RRHs, where $\lambda_s$ denotes the intensity of new arrivals destined to devices belonging to URLLC slice $s$. Once arrived, new URLLC arrivals will enter a queue maintained by an RRH to be immediately scheduled. An $M/M/W^u$ queueing system with limited bandwidth $W^u$ is exploited to model the queue. Without loss of any generality, we assume that each RRH maintains the same queue due to the exploration of cooperated transmission. In the queue, a packet destined to URLLC device $i \in {\mathcal{I}_s^u}$, $s \in {\mathcal{S}^u}$ will be allocated with a block of system bandwidth $\omega_{i,s}^u(t)$ for a period of time $d_s \le D_s$ at $t$. Owing to stochastic variations in the bursty packet arrival process, the limited bandwidth may not be enough to serve new arrivals occasionally. As such, URLLC packet blocking may happen. To reduce the probability of URLLC packet blocking, the redesign of URLLC frame and minislot structure may be required. At minislot $t$, let $P_b(\bm \omega^u(t), \bm \lambda, \bm d, W^u(t))$ denote the packet blocking probability experienced at an RRH, where ${{\bm \omega}^u(t) = \{\omega_{1,1}^u(t), \ldots, \omega_{i,s}^u(t), \ldots, \omega_{I_{|{\mathcal S}^u|}^u,|{\mathcal S}^u|}^u(t)\}}$ and ${\bm d} = \{d_1, \ldots, d_s \ldots, d_{|{\mathcal S}^u|}\}$. The Theorem 1 in [@anand2018resource] provides us with a clue of redesigning the URLLC frame and minislot structure in the time-frequency plane for bursty URLLC traffic transmission. This theorem indicates that if we narrow the PRB of the URLLC frame in the frequency domain while widening it in the time domain, then the number of concurrent transmissions will be increased. As a result, the packet blocking probability is reduced. Therefore, for a URLLC packet destined to device $i \in {\mathcal I}_s^u$, $s \in \mathcal{S}^u$, we should scale up $d_s$ and choose $d_s$ and $\omega_{i,s}^u(t)$ at $t$ using the following equation $$\label{eq:omega_s} d_{{s}} = D_s \ {\rm and} \ \omega_{{i,s}}^{u}(t) = \frac{b_{i,s}^u(t){r_{{i,s}}^{u}(t)}}{{\kappa D_s}}, \forall i \in {\mathcal I}_s^u, s \in {\mathcal S}^u$$ where $r_{i,s}^u(t)$ denotes channel uses for transmitting a URLLC packet, $\kappa$ is a constant reflecting the number of channel uses per unit time per unit bandwidth of FDMA frame structure and numerology, $b_{i,s}^u(t)$ is an indicator variable reflecting whether the QoS requirement of device $i$ in slice $s$ can be satisfied at $t$. As network resources are limited and shared by all network slices, not all URLLC devices can be guaranteed to be served at every minislot. Certainly, we can adjust the slice priority weight that will be introduced in the following section to guide the resource orchestration for enforcing the entire URLLC devices coverage. Based on the result in (\[eq:omega\_s\]) and the conclusion of the Lemma 3.2 in our previous work [@How2019yang], we can derive the minimum upper bound of bandwidth orchestrated for URLLC slices in the following lemma. \[lem:upper\_bound\_urllc\_bandwidth\] At minislot $t$, for a given $M/M/W^u$ queue with packet arrival intensity $\bm \lambda$ and a family of packet transmit rates $\{{\kappa/r_{i,s}^u(t)}\}$, let $W^u({\bm r}(t))$ denote the minimum upper bound of bandwidth orchestrated for URLLC slices such that $P_Q^{M/M/W^u}\le \varsigma$ and $P_b({\bm \omega}^u(t), {\bm \lambda}, {\bm D}$, $W^u({\bm r}(t)))$ is of the order of $\alpha$, where $P_Q^{M/M/W^u}$ represents the queueing probability, and ${\bm D} = \{D_1, \ldots, D_{|{\mathcal S}^u|}\}$. If $\varsigma > \alpha$, then we have $$\label{eq:URLLC_bandwidth} \begin{array}{l} W^u(\bm r(t)) \approx \sum\limits_{s \in {\mathcal S}^u } {\sum\limits_{i \in {\mathcal I}_s^u} {{\lambda _{s}}b_{i,s}^u(t)\frac{{{{r_{i,s}^u(t)}}}}{\kappa}} } + \\ \frac{{{\alpha} - \varsigma {\alpha}}}{{\varsigma - {\alpha}}}\sqrt {\frac{\left({\sum\limits_{s \in {{\mathcal S}^u}} {\sum\limits_{i \in {\mathcal I}_s^u}{b_{i,s}^u(t)\lambda _s^2D_s^2}} }\right)\left( {\sum\limits_{s \in {\mathcal S}^u} {\sum\limits_{i \in {\mathcal I}_s^u} {{\lambda _{s}}b_{i,s}^u(t)\frac{{r_{i,s}^u(t)}^2}{{{\kappa ^2}{D_s}}}} }}\right)}{{\mathop {\min }\limits_{s \in {{\mathcal S}^u}} \{ {\lambda _s}{D_s}\} }}} \end{array}$$ We omit the proof here as the similar proof can be found in the proof section of Lemma 3.2 in [@How2019yang]. ### Reduction of codeword error decoding probability The crucial factor that impacts the codeword error decoding probability is the network capacity. Next, we discuss the relationship between the network capacity and codeword error decoding probability. For any URLLC slice $s \in {\cal S}^u$, during minislot $t$, let $x_{i,s}^u(t)$ be the original data symbol destined to a URLLC device $i \in {\mathcal{I}_s^u}$ with ${\mathbb E}[|x_{i,s}^u(t)|^2] = 1$, $\bm g_{ij,s}(t) \in {\mathbb C}^K$ be the transmit beamformer pointing at the device $i$ from the $j$-th RRH and $\bm h_{ij,s}(t) \in \mathbb{C}^K$ be the channel coefficient between the $i$-th URLLC device and the $j$-th RRH. The channel coefficient may change over minislots. However, it is assumed to be i.i.d. over each minislot and remain unchanged during each minislot. Then, the received signal at device $i$ in $s$ during minislot $t$ is given by $$\label{eq:URLLC_receiving_signal} {\hat x_{i,s}^u(t) = }{\sum\limits_{j \in {\cal J}} {{\bm h}_{ij,s}^{\rm{H}}} (t){{\bm g}_{ij,s}}(t)x_{i,s}^u(t) + {\sigma _{i,s}}(t)}, \forall i \in {\mathcal I}_s^u, s \in {\mathcal S}^u$$ where the first term is the useful signal for $i$ and $\sigma_{i,s}(t) \sim {\cal CN}(0, \sigma_{i,s}^2) $ is the additive white Gaussian noise (AWGN) experienced at $i$. Similar to [@tang2019service], interference signal is not involved in (\[eq:URLLC\_receiving\_signal\]) due to the utilization of a flexible FDMA mechanism. Then the SNR received at device $i$ in $s$ at minislot $t$ can be written as $$\label{eq:URLLC_SINR} SINR_{i,s}^u(t) = \frac{{|\sum\nolimits_{j \in {\cal J}} {{{\bm h}_{ij,s}^{\rm H}}{{(t)}}{{\bm g}_{ij,s}}(t)} {|^2}}}{{\phi \sigma _{i,s}^2}}, \forall i \in {\mathcal I}_s^u, s \in {\mathcal S}^u$$ where $\phi > 1$ is an SNR loss coefficient. The perception of perfect channel status information (CSI) or accurate channel coefficients requires the information exchange between an RRH and its associated device before data transmission, the process of which is generally time consuming. URLLC packets, however, have a stringent latency requirement. As a result, perfect CSI or accurate channel parameters may be unavailable for URLLC transmission, which may incur the SNR loss. The coefficient $\phi$ is then utilized to characterize the SNR loss [@hou2018burstiness]. Shannon capacity formula is created under a crucial assumption of transmitting a block with long enough blocklength. However, URLLC packets are typically very short to satisfy the ultra-low latency requirement. Thus, the famous Shannon capacity formula cannot be utilized to model the URLLC transmission data rate and capture the corresponding codeword error decoding probability. For URLLC transmission, we resort to the capacity analysis for a finite blocklength channel coding regime derived in [@yang2014quasi]. For any device $i \in {\mathcal{I}_s^u}$, $s \in \mathcal{S}^u$, the number of transmitted information bits $L_{i,s}^u(t)$ at minislot $t$ using $r_{i,s}^u(t)$ channel uses in AWGN channel can be accurately correlated with the codeword error decoding probability $\beta$ according to the following equation $$\label{eq:URLLC_bit_length} \begin{array}{l} L_{i,s}^u(t) \approx r_{i,s}^u(t)C(SNR_{i,s}^u(t)) - \\ \quad {Q^{ - 1}}(\beta)\sqrt {r_{i,s}^u(t)V(SNR_{i,s}^u(t))}, \forall i \in {\mathcal I}_s^u, s \in {\mathcal S}^u \end{array}$$ where $C(SNR_{i,s}^u(t)) = \log_2(1 + SNR_{i,s}^u(t) )$ is the AWGN channel capacity under infinite blocklength assumption, $V(SNR_{i,s}^u(t)) = \ln^2 2\left( {1 - \frac{1}{{{{(1 + SNR_{i,s}^u(t))}^2}}}} \right)$ is the channel dispersion, $Q(\cdot)$ is the $Q$-function. It is noteworthy that a URLLC packet will usually be coded before transmission and the generated codeword will be transmitted in the air interface such that the transmission reliability can be improved. The complicated expression of $V(SNR_{i,s}^u(t))$ in (\[eq:URLLC\_bit\_length\]) significantly hinders the theoretical analysis of network resources orchestrated for URLLC slices. Fortunately, as $V(SNR_{i,s}^u(t))$ is maximized by $\ln^2 2$, the closed-form expression of the minimum upper bound of $r_{i,s}^u(t)$ ($i \in {\mathcal I}_s^u, s \in {\mathcal S}^u $) with a codeword error decoding probability $\beta$ can be given by [@How2019yang] $$\label{eq:URLLC_channel_use} \begin{array}{l} r_{i,s}^u(t) = \frac{{L_{i,s}^u(t)}}{{C(SINR_{i,s}^u(t))}} + \frac{{{{({Q^{ - 1}}(\beta))}^2}}}{{2{{(C(SINR_{i,s}^u(t)))}^2}}}\\ \quad + \frac{{{{({Q^{ - 1}}(\beta))}^2}}}{{2{{(C(SINR_{i,s}^u(t)))}^2}}}\sqrt {1 + \frac{{4L_{i,s}^u(t)C(SINR_{i,s}^u(t))}}{{{{({Q^{ - 1}}(\beta))}^2}}}} \end{array}$$ Problem formulation =================== This section aims to formulate the problem of RAN slicing for mIoT and bursty URLLC service multiplexing based on the above models. In mIoT slices, each RRH may transmit feedback signal to its connected IoT devices for the connection establishment according to an RA four-step procedure [@grau2019preamble]. Meanwhile, in URLLC slices, each RRH may transmit URLLC packets to URLLC devices. As the transmit power $E_j$ of each RRH is limited, we have the following transmit power constraint $$\label{eq:RRH_energy} \begin{array}{l} \sum\limits_{s \in {\cal S}} {{(1 + \alpha_g)\frac{\lambda_s^I}{\lambda_R}{{\hat E}_{j}^I}}} + \\ \qquad \sum\limits_{s \in {\cal S}^u} {\sum\limits_{i \in {\cal I}_s^u} {b_{i,s}^u(t){{\bm g}_{ij,s}^{\rm H}}{{(t)}}{{\bm g}_{ij,s}}(t)} } \le {E_j}, \forall j \in \mathcal{J} \end{array}$$ where ${\hat E}_{j}^I$ is assumed to be a constant and denotes the transmit power of the $j$-th RRH for connecting to its associated IoT devices over downlink, $\alpha_g$ represents a safety margin coefficient. As a PPP with intensity $\lambda_s^I$ is utilized to model the distribution of IoT devices, the actual number of IoT devices may be greater than $\lambda_s^I$ once deployed. As a result, the coefficient $\alpha_g$ is introduced to reserve transmit power for exceeded IoT devices. In the RAN slicing system, as the total limited system bandwidth $W$ will be shared by mIoT slices and URLLC slices, we have the following bandwidth constraint $$\label{eq:total_bandwidth} \sum\limits_{s \in {\cal S}^I} {(1+\alpha_g)\omega_s(\bar t)} + W^u(\bm r(t)) \le W$$ where $\omega_s(\bar t)$ denotes the bandwidth allocated to mIoT slice $s \in \mathcal{S}^I$ that is correlated with $F_s$ by means of $F_s = \left\lfloor {{{\omega _s}(\bar t)}/{{a}}} \right\rfloor$, and $\alpha_g \omega_s(\bar t)$ denotes a block of reserved bandwidth. In (\[eq:total\_bandwidth\]), $F_s$ is an integer, and some integer variable recovery schemes [@tang2019systematic] can be leveraged to obtain the suboptimal $F_s$. However, considering the high computational complexity of optimizing an integer variable and the utilization of the scheme of spectrum safety margin, we directly relax the integer variable into a continuous one, i.e., let $F_s = {{{\omega _s}(\bar t)}/{{a}}}$. Without loss of any generality, we regard $\omega_s(\bar t)$ as an independent variable below. Besides, as at least one PRB should be allocated to mIoT slices, we have $$\label{eq:mMTC_bandwidth} \omega_s(\bar t) \ge a, \forall s \in {\cal S}^I$$ Owing to the exploration of mIoT and bursty URLLC service multiplexing, we should orchestrate network resources for all mIoT slices and URLLC slices to simultaneously maximize the utilities of mIoT slices and URLLC slices. For a mIoT slice $s \in \mathcal{S}^I$, its primary goal is to offload as many data packets as possible from IoT devices. In this way, the number of accumulated packets in each IoT device should be kept at a low level. Considering that a great RA success probability of an IoT device will lead to a low number of accumulated packets in its queue, we define the utility of a mIoT slice as the following. \[mydef:IoT\_slice\_utility\] Over a time slot of duration $T$, the mIoT slice utility is defined as the time-average of RA success probabilities of IoT devices in all mIoT slices, which is given by $$\label{eq:IoT_utility} {{\bar U}^I} = \frac{1}{T}\sum\limits_{t = 1}^T {{U^I}(t)} = \frac{1}{T}\sum\limits_{t = 1}^T {\tilde P(t) }$$ where $\tilde P(t) = \sum\limits_{s \in {{\mathcal S}^I}} {\frac{{{\lambda _s^I}{P_s}(t)}}{{\sum\nolimits_{s \in {{\mathcal S}^I}} {{\lambda _s^I}} }}}$ with the numerator $\lambda_s^I P_s (t)$ representing the expected sum of RA success probabilities of IoT devices in slice $s \in \mathcal{S}^I$ and the denominator $\sum\nolimits_{s\in\mathcal{S}^I}{\lambda_s^I}$ denoting a normalization coefficient. In (\[eq:IoT\_utility\]), $\frac{\lambda_s^I}{\sum\nolimits_{s \in \mathcal{S}^I} {\lambda_s^I}}$ can be regarded as an intra-slice priority coefficient. A mIoT slice serving more IoT devices will be orchestrated with more network resources. For a URLLC slice $s \in \mathcal{S}^u$, its primary objective is to maximize the slice gain that is reflected by the parameters in the slice request at a low cost. Therefore, we define an energy-efficient utility for URLLC slices, as presented below. \[mydef:URLLC\_slice\_utility\] Over a time slot of duration $T$, the bursty URLLC slice utility is defined as the time-average energy efficiency for serving all URLLC devices, which is given by $$\label{URLLC_long_term_utility} \begin{array}{l} {{\bar U}^u} = \frac{1}{T}\sum\limits_{t = 1}^T {{U^u}(t)} = \frac{1}{T}\sum\limits_{t = 1}^T {\sum\limits_{s \in {{\mathcal S}^u}} {U_s^u({D_s},{\bm g_{ij,s}}(t))} } \\ \qquad = \frac{1}{T}\sum\limits_{t = 1}^T {\sum\limits_{s \in {{\mathcal S}^u}} {\sum\limits_{i \in {\mathcal I}_s^u} \frac{b_{i,s}^u(t)}{{1 - {e^{ - {D_s}}}}} } } - \\ \qquad \quad \frac{\eta }{T}\sum\limits_{t = 1}^T {\sum\limits_{s \in {{\mathcal S}^u}} {\sum\limits_{j \in {\mathcal J}} {\sum\limits_{i \in {\mathcal I}_s^u} {b_{i,s}^u(t)\bm g_{ij,s}^{\rm{H}}(t){\bm g_{ij,s}}(t)} } } } \end{array}$$ where $\eta$ is an energy efficiency coefficient reflecting a tradeoff between the URLLC slice gain and the RRH energy consumption. Then, over a time slot of duration $T$, the RAN slicing problem for mIoT and URLLC service multiplexing can be formulated as follows \[eq:original\_problem\] $$\begin{aligned} {2} & \mathop {\rm maximize}\limits_{{b_{i,s}^u(t)},{{\omega_s}}(\bar t),{{\bm g}_{ij,s}}(t)} {{\bar U}^I} + {\tilde \rho} {{\bar U}^u} \\ & {\rm subject \text{ } to:} \nonumber \\ & b_{i,s}^u(t) \in \{ 0,1\}, \forall s \in {\cal S}^u, i \in {\cal I}_s^u \\ & \rm {constraints \text{ } (\ref{eq:mMTC_QoS}), (\ref{eq:RRH_energy})-(\ref{eq:mMTC_bandwidth}) \text{ } are \text{ } satisfied.}\end{aligned}$$ where $\tilde \rho$ is an inter-slice priority coefficient reflecting the priority of orchestrating network resources for mIoT slices and URLLC slices. The mitigation of (\[eq:original\_problem\]) is quite challenging mainly because - **indeterministic objective function**: (\[eq:original\_problem\]) should be optimized at the beginning of the $1^{\rm st}$ minislot. The time-averaged objective function of (\[eq:original\_problem\]) can only be exactly computed according to the future channel information. Therefore, the value of the objective function is indeterministic at the beginning of the $1^{\rm st}$ minislot. - **two timescale issue**: the creation of a network slice is performed at a timescale of time slot. Thus, the variable $\omega_s(\bar t)$ should be determined at the beginning of the time slot $\bar t$ and kept unchanged over the whole time slot. The channel, however, is time-varying. As a result, the beamformer $\bm g_{ij,s}(t)$ should be optimized at each minislot. In summary, the variables in (\[eq:original\_problem\]) should be optimized at two different timescales. - **thorny optimization problem**: at each minislot $t$, the constraint (\[eq:mMTC\_QoS\]) is non-convex over $\omega_s(\bar t)$, and the constraints (\[eq:RRH\_energy\]), (\[eq:total\_bandwidth\]) are non-convex over $\bm g_{ij,s}(t)$, which together lead to a non-convex problem. Problem solution with system generated channel ============================================== This section aims to tackle these challenges by exploiting of an SAA technique [@kim2015guide], an ADMM method [@boyd2011distributed], a semidefinite relaxation scheme and a Taylor expansion scheme. Sample average approximation and alternating direction method of multipliers ---------------------------------------------------------------------------- As mIoT slices and URLLC slices share the network resources, both $\bar U^I$ and $\bar U^u$ may be determined by channel coefficients experienced by URLLC slices. At each minislot $t$, due to the i.i.d. assumption on the channel coefficients of URLLC slices, we have $$\label{objfun_approx} \frac{1}{T}\sum\limits_{t = 1}^T {{U^I}(t)} + \frac{1}{T}\sum\limits_{t = 1}^T {\tilde \rho {U^u}(t)} \approx {E_{\hat {\bm h}}}\left[ {{{\hat U}^I} + \tilde \rho {{\hat U}^u}} \right]$$ where $\hat {\bm h}$ represents the channel samples of URLLC slices collected at the beginning of the time slot $\bar t$. Given a collection of channel samples $\{\bm h_m\}$ with $\bm h_m = [\bm h_{11,1m};\ldots;\bm h_{1J,sm};\ldots;\bm h_{N^uJ,|\mathcal{S}^u|m}]$ and $m \in \mathcal{M}=\{1,\ldots,M\}$, Just like [@How2019yang], as constraints (\[eq:original\_problem\]b) and (\[eq:original\_problem\]c) construct a nonempty compact set, the conclusion of Proposition 5.1 in [@How2019yang] is applicable to this paper by exploiting the SAA technique. The conclusion indicates that if the number of channel samples $M$ is reasonably large, then $\frac{1}{M}\sum\limits_{m = 1}^M {U_m^I } + \frac{\tilde \rho}{M}\sum\limits_{m = 1}^M {U_m^u }$ converges to ${E_{\bm {\hat h}}}\left[ {{{\hat U}^I} + {\tilde \rho {\hat U}^u}} \right]$ uniformly on the nonempty compact set almost surely. In other words, the SAA technique enables us to use the channel samples collected at the beginning of a time slot to approximate the unknown channel coefficients over the time slot. For notation lightening, we write $x_m$ instead of $x(m)$ that represents a variable corresponding to the channel sample $\bm h_m$. Recall that the variable $\omega_s(\bar t)$ will be kept unchanged over the time slot $\bar t$ and the beamformer $\bm g_{ij,s}(t)$ should be calculated at each minislot $t$, we can further consider (\[eq:original\_problem\]) as a global consensus problem, which can be effectively mitigated by an ADMM method. In this problem, $\omega_s(\bar t)$ is a global consensus variable that should be maintained in consensus for all $\bm h_m$, and $\bm g_{ij,sm}$ that is calculated based on $\bm h_m$ is a local variable. The fundamental principle of an ADMM method is to impose augmented penalty terms characterizing global consensus constraints on the objective function of an optimization problem. In this way, the local variables can be driven into the global consensus while still attempting to maximize the objective function. Let ${\bm G}_{i,sm} = {\bm g}_{i,sm} {\bm g}_{i,sm}^{\rm H} \in {\mathbb R}^{JK \times JK}$, ${\bm H}_{i,sm} = {\bm h}_{i,sm} {\bm h}_{i,sm}^{\rm H} \in {\mathbb R}^{JK \times JK}$, where ${\bm g}_{i,sm} = [{\bm g}_{i1,sm};\ldots;{\bm g}_{iJ,sm}] \in {\mathbb C}^{JK \times 1}$ and ${\bm h}_{i,sm} = [{\bm h}_{i_1,sm};\ldots;{\bm h}_{i_J,sm}] \in {\mathbb C}^{JK \times 1}$. By applying the matrix property ${{{\bm G}_{i,sm}} = {{\bm g}_{i,sm}}{\bm g}_{i,sm}^{\rm H} \Leftrightarrow {{\bm G}_{i,sm}} \succeq 0,} \text{ } {{\rm rank}({{\bm G}_{i,sm}}) \le 1}$ and utilizing the conclusions of SAA and ADMM, we can approximate (\[eq:original\_problem\]) as the following problem at the beginning of the time slot $\bar t$ \[eq:SAA\_admm\_problem\] $$\begin{aligned} {2} & \mathop {{\rm{minimize}}}\limits_{\{ \omega _{sm},\omega_s(\bar t),{b_{i,sm}^u},{\bm G_{i,sm}}\}\hfill} \sum\limits_{m = 1}^M {\left[ { - \frac{{U_m^I}}{M} - \frac{{\tilde \rho U_m^u}}{M}} \right]} + \nonumber \\ & \underbrace {\sum\limits_{m = 1}^M {\sum\limits_{s \in {{\mathcal S}^I}} {\left[ {{\psi _{sm}}\left( {{\omega _{sm}} - {\omega _s}(\bar t)} \right) + \frac{\mu }{2}\left\| {{\omega _{sm}} - \omega{_s}(\bar t)} \right\|_2^2} \right]} } }_{{\rm augmented} \text{ } {\rm penalty} \text{ } {\rm terms}} \\ & {\rm subject \text{ } to:} \nonumber \\ & P_{sm} \ge \pi_s, \forall s \in {\cal S}^I, m \in {\cal M} \\ & \sum\limits_{s \in {\cal S}^I} {{(1 + \alpha_g)\frac{\lambda_s^I}{\lambda_R}{\hat E}_{j}^I}} + \sum\limits_{s \in {{\cal S}^u}} {\sum\limits_{i \in {\cal I}_s^u} {b_{i,sm}^u {\rm tr}({{\bm Z}_j}{{\bm G}_{i,sm}})} } \le {E_j} \nonumber \\ & \qquad \qquad \forall j\in {\cal J},m\in {\cal M} \\ & \sum\limits_{s \in {{\cal S}^I}} {(1+\alpha_g) \omega _{s}(\bar t)} + W^u(\bm r_m) \le W, m \in {\cal M} \\ &{{\bm G}_{i,sm}} \succeq 0, \forall s \in {\cal S}^u, i \in {\cal I}_s^u, m \in {\cal M} \\ &{{\rm rank}({{\bm G}_{i,sm}}) \le 1}, \forall s \in {\cal S}^u, i \in {\cal I}_s^u, m \in {\cal M} \\ & b_{i,sm}^u \in \{0,1\}, \forall s \in {\cal S}^u, i \in {\cal I}_s^u, m \in {\cal M}\end{aligned}$$ where $\psi_{sm}$ is the Lagrangian multiplier, $\mu$ is a penalty coefficient, ${\bm Z}_j$ is a square matrix with $J \times J$ blocks, and each block in ${\bm Z}_j$ is a $K \times K$ matrix. In ${\bm Z}_j$, the block in the $j$-th row and $j$-th column is a $K \times K$ identity matrix, and all other blocks are zero matrices. (\[eq:original\_problem\]) is now reduced to a deterministic single timescale problem (\[eq:SAA\_admm\_problem\]). What is more, (\[eq:SAA\_admm\_problem\]) can be split into $M$ separate problems that can be optimized in parallel as its objective function is separable. Thus, the following ADMM-based framework from (\[eq:arg\_lagarangian\]) to (\[eq:psi\_update\]) can be exploited to mitigate (\[eq:SAA\_admm\_problem\]) \[eq:arg\_lagarangian\] $$\begin{aligned} {2} & \left\{ {\begin{array}{*{20}{l}} {\omega _{sm}^{(k + 1)},b_{i,sm}^{u(k+1)}}\\ {\bm G_{i,sm}^{(k + 1)}} \end{array}} \right\} = \mathop {{\rm{argmin}}}\limits_{\left\{ {\scriptstyle\omega _{sm},b_{i,sm}^{u},\hfill\atop \scriptstyle{\bm G_{i,sm}}\hfill} \right\}} \overline {\cal L} (\omega _{sm},{{\bm G}_{i,sm}}) \\ & {\rm subject \text{ } to:} \nonumber \\ & {\rm for} \text{ } {\rm the} \text{ } m-{\rm th} \text{ } {\rm sample}, (\ref{eq:SAA_admm_problem}b)-(\ref{eq:SAA_admm_problem}g) \text{ } \rm{are} \text{ } satisfied.\end{aligned}$$ $$\label{eq:omega_update} \omega _s^{(k + 1)}(\bar t) = \frac{1}{M}\sum\limits_{m = 1}^M {\left( {\omega _{sm}^{(k + 1)} + \frac{1}{\mu }\psi _{sm}^{(k)}} \right)}, \text{ } \forall s\in {\cal S}^I$$ $$\label{eq:psi_update} \psi _{sm}^{(k + 1)} = \psi _{sm}^{(k)} + \mu \left( {\omega _{sm}^{(k + 1)} - \omega _s^{(k + 1)}(\bar t)} \right), \text{ } \forall s\in {\cal S}^I$$ where the augmented partial Lagrangian function $$\label{eq:average_Lagrangian_func} \begin{array}{l} \bar{\mathcal{L}}(\omega _{sm},\bm {\bm G}_{i,sm}) = {- \frac{{U_m^{I(k)}}}{M} - \frac{{\tilde \rho U_m^{u(k)}}}{M} +} \\ { \sum\limits_{s \in {{\cal S}^I}} {\left[ {\psi _{sm}^{(k)}\left( {\omega _{sm} - \omega _s^{(k)}(\bar t)} \right) + \frac{\mu }{2}{{\left\| {\omega _{sm} - \omega _s^{(k)}(\bar t)} \right\|}_2^2}} \right]} } \end{array}$$ This ADMM-based framework can be executed on multiple processors. Each processor is responsible for optimizing (\[eq:arg\_lagarangian\]) and calculating (\[eq:psi\_update\]) with a global value as an input. (\[eq:omega\_update\]) is centrally updated in such a way that local variables converge to the global value, which is the solution of (\[eq:SAA\_admm\_problem\]). Unfortunately, (\[eq:arg\_lagarangian\]) is a mixed-integer non-convex optimization problem as there are zero-one variables, continuous variables and non-convex constraints in (\[eq:arg\_lagarangian\]). As a result, the optimization of (\[eq:arg\_lagarangian\]) is quite difficult. We next discuss how to handle this hard problem. Alternative optimization ------------------------ In this subsection, we exploit a widely applied scheme, i.e., an alternative optimization scheme, to handle the mixed-integer non-convex optimization problem. Specifically, we first assume that continuous variables are known and attempt to mitigate a zero-one optimization problem. Given the zero-one variables, we then try to optimize a non-convex optimization problem. The process is alternatively conducted until convergence. ### URLLC device associations Given continuous variables $\{{\bm G}_{i,sm}^{(k)}, \omega_{sm}^{(k)}\}$ at the $k$-th iteration, the association problem of URLLC devices in URLLC slices can take the following form \[eq:arg\_lagarangian\_b\_isu\] $$\begin{aligned} {2} & \{ b_{i,sm}^{u(k+1)} \} = \mathop {{\rm{argmin}}}\limits_{\{b_{i,sm}^u\}} - \frac{{\tilde \rho U_m^{u(k)}}}{M} \\ & {\rm subject \text{ } to:} \nonumber \\ & {\rm for} \text{ } m, \text{ } (\ref{eq:SAA_admm_problem}c), (\ref{eq:SAA_admm_problem}d),(\ref{eq:SAA_admm_problem}g) \text{ } \rm{are} \text{ } satisfied.\end{aligned}$$ This problem is non-linear and hard to be handled. In theory, an exhaustive algorithm can obtain the optimal solution of (\[eq:arg\_lagarangian\_b\_isu\]). The computation complexity of this algorithm is $O(2^{N^u})$ that may be impractical in implementation. Therefore, a greedy scheme of the computational complexity $O(N^u)$, which is summarized as the following, is proposed to obtain $\{b_{i,sm}^{u(k+1)}\}$ a) initialize two device sets, i.e., candidate device set ${\mathcal{I}}^{u-} = {\mathcal I}^u$, association device set ${\mathcal I}^{u+} = \emptyset$. b) select the device that maximizes $\frac{{\tilde \rho U_m^{u(k)}}}{M}$ from ${\mathcal{I}}^{u-}$, remove it from ${\mathcal{I}}^{u-}$, and add it to ${\mathcal{I}}^{u+}$. Given ${\mathcal{I}}^{u+}$, check the feasibility of (\[eq:arg\_lagarangian\_b\_isu\]). If (\[eq:arg\_lagarangian\_b\_isu\]) is feasible, then accept the device; otherwise, remove the device from ${\mathcal{I}}^{u+}$. Continue till ${\mathcal{I}}^{u-} = \emptyset$. ### joint bandwidth and beamforming optimization Given the obtained $b_{i,sm}^{u(k+1)}$, (\[eq:SAA\_admm\_problem\]) will be reduced to the following joint bandwidth and beamforming problem. \[eq:bandwidth\_beamforming\_problem\] $$\begin{aligned} {2} & \left\{ {\omega _{sm}^{(k + 1)}}, {\bm G_{i,sm}^{(k + 1)}} \right\} = \mathop {{\rm{argmin}}}\limits_{\{\omega _{sm},{\bm G_{i,sm}}\}} \overline {\cal L} (\omega _{sm},{{\bm G}_{i,sm}}) \\ & {\rm subject \text{ } to:} \nonumber \\ & {\rm for} \text{ } m, (\ref{eq:SAA_admm_problem}b)-(\ref{eq:SAA_admm_problem}f) \text{ } \rm{are} \text{ } satisfied.\end{aligned}$$ In (\[eq:bandwidth\_beamforming\_problem\]), the low-rank constraint (\[eq:SAA\_admm\_problem\]f) is non-convex, and its objective function is not convex and even not quasi-convex w.r.t. ${\omega_{sm}}$, the tackling of which is quite tricky. To tackle the non-convex low-rank constraint (\[eq:SAA\_admm\_problem\]e), we resort to the semidefinite relaxation technique. The primary procedures of SDR are i) directly drop the low-rank constraint; ii) solve the optimization problem without the low-rank constraint to obtain the solution; iii) owing to the relaxation, the obtained solution cannot satisfy the low-rank constraint in general. If it is, its principal component is the optimal solution to the problem. If not, then some manipulations such as randomization/scale [@ma2010semidefinite] are needed to perform on the solution to impose the low-rank constraint. For the tricky objective function, we are reminded of the art of dealing with a non-convex function, i.e., study the structure of the function if it is non-convex. A crucial observation is that $P_{sm}$ is quasi-concave w.r.t. $\omega_{sm}$ although the objective function is not quasi-convex w.r.t. $\omega_{sm}$. Therefore, we resort to the Taylor expansion to approximate the tricky objection function. The following analysis is built under the following two facts - the value of the objective function of (\[eq:bandwidth\_beamforming\_problem\]) is mainly determined by that of ${\tilde P_{m}^{(k)}}$ (or $U_m^{I(k)}$); - for all $s \in \mathcal{S}^u$, $m \in {\mathcal{M}}$, the solution $\omega_{sm}$ maximizing ${\tilde P_{m}^{(k)}}$ must locate in the range of $[\omega_{sm}^{lb}, S_{sm}^{\star}]$ shown in Fig. \[fig:fig\_quasi\_convex\_structure\], where $\omega_{sm}^{lb}$ denotes the lower bound of $\omega_{sm}$ satisfying the constraint (\[eq:SAA\_admm\_problem\]b), $S_{sm}^{\star}$ is the $\omega_{sm}$ maximizing $P_{sm}$, and the notation $P_{sm}|_{\omega_{sm}}$ is utilized to explicitly indicate that $P_{sm}$ is a function of $\omega_{sm}$. Fact 1 holds because the linear terms w.r.t. $\omega_{sm}$ will donate little to the objective function as the consensus constraint is active. Besides, the quadratic terms pull local values towards the consensus; thus, they will also donate little to the objective function. Fact 2 holds because the total bandwidth is limited and shared. For example, given a value $\omega_{sm, 2} \in [S_{sm}^{\star}+\delta_{\omega}, W]$ with $\delta_{\omega}$ being a small positive constant, there must exist a value $\omega_{sm,1} \in [\omega_{sm}^{lb}, S_{sm}^{\star}]$ such that $P_{sm}|_{\omega_{sm,1}} = P_{sm}|_{\omega_{sm,2}}$. Thus, a small $\omega_{sm}$ will be preferred as it indicates that more bandwidth can be allocated to URLLC slices to further improve the objective function. For all $s \in \mathcal{S}^I$, it can be proved that $P_{sm}$ is concave in the interval $(a_1, a_2]$ by evaluating the second-order derivative of $P_{sm}$. Therefore, we can leverage the $2^{\rm nd}$ degree Taylor expansion to approximate $P_{sm}$ in this interval. Considering that $P_{sm}$ is convex in the interval $[\omega_{sm}^{lb}, a_1]$, the $1^{\rm st}$ degree Taylor expansion is always leveraged to obtain the lower bound of $P_{sm}$. However, this interval is usually rather narrow, and the value of $P_{sm}$ in this interval is much lower than the value of that in the interval $(a_1, a_2]$. What is more, the error bound of the $1^{\rm st}$ degree Taylor expansion is greater than that of the $2^{\rm nd}$ expansion. Therefore, we explore the $2^{\rm nd}$ degree Taylor expansion to approximate $P_{sm}$ in the interval $[\omega_{sm}^{lb}, a_2]$. Fig. \[fig\_Taylor\_expansion\] shows an example of the $2^{\rm nd}$ degree Taylor expansion of $P_{sm}$. Given a local point $\bm \omega_{m}^{(k,q)}$ at the $q$-th iteration, the Taylor expansion of $-\tilde P_{m}^{(k)}$ at the local point can be given by $$\label{eq:P_sm_Taylor_expansion} \begin{array}{*{20}{l}} { - {{\tilde P}_{m}^{(k)}} \approx - \tilde P_{m}^{(k,q)} - \nabla \tilde P_m^{(k,q)}{{({{\bm \omega} _m} - {\bm \omega} _m^{(k,q)})}^{\rm T}} - }\\ {\frac{1}{2}({{\bm \omega} _m} - {\bm \omega} _m^{(k,q)})H({\bm \omega} _{m}^{(k,q)}){{({\bm \omega _m} - {\bm \omega} _m^{(k,q)})}^{\rm T}} - R_2(\bm \omega_{m})} \end{array}$$ where ${\bm \omega}_m = [\omega_{1m}, \ldots, \omega_{|{\mathcal S}^I|m}]$, $\nabla \tilde P_m^{(k,q)}$ is the gradient of $\tilde P_m^{(k)}$ over ${\bm \omega_m}$ at the local point ${\bm \omega_m^{(k,q)}}$ with $$\label{eq:first_order_Taylor_expansion} \begin{array}{l} \frac{{\partial P_{sm}^{(k)}}}{{\partial \omega _{sm}^{(k,q)}}} = \frac{{{\lambda _s^I}(1 + {\varpi _s}{\rho _o}){e^{ - {\varpi _s}{\sigma ^2}}}}}{{\sum\nolimits_{s \in {{\cal S}^I}} {{\lambda _s^I}} }} \times \\ \qquad \qquad \left[ {\frac{{3.5{y_{sm}}{z_s}\omega _{sm}^{2.5(k,q)}}}{{{{({y_{sm}}{z_s} + \omega _{sm}^{(k,q)})}^{4.5}}}} - \frac{{3.5{y_{sm}}\omega _{sm}^{2.5(k,q)}}}{{{{({y_{sm}} + \omega _{sm}^{(k,q)})}^{4.5}}}}} \right] \end{array}$$ and $H({\bm \omega}_m^{(k,q)})$ is a Hessian matrix with $$\label{eq:second_order_Taylor_expansion} \begin{array}{l} \frac{{{\partial ^2}P_{sm}^{(k)}}}{{\partial \omega _{sm}^{2(k,q)}}} = \frac{{\lambda _s^I(1 + {\varpi _s}{\rho _o}){e^{ - {\varpi _s}{\sigma ^2}}}}}{{\sum\nolimits_{s \in {{\mathcal S}^I}} {\lambda _s^I} }}\left[ {\frac{{15.75y_{sm}^2z_s^2\omega _{sm}^{1.5(k,q)}}}{{{{({y_{sm}}{z_s} + \omega _{sm}^{(k,q)})}^{5.5}}}} - } \right.\\ \frac{{7{y_{sm}}{z_s}\omega _{sm}^{1.5(k,q)}}}{{{{({y_{sm}}{z_s} + \omega _{sm}^{(k,q)})}^{4.5}}}} + \frac{{7{y_{sm}}\omega _{sm}^{1.5(k,q)}}}{{{{({y_{sm}} + \omega _{sm}^{(k,q)})}^{4.5}}}} - \left. {\frac{{15.75y_{sm}^2\omega _{sm}^{1.5(k,q)}}}{{{{({y_{sm}} + \omega _{sm}^{(k,q)})}^{5.5}}}}} \right] \end{array}$$ $$\label{eq:second_order_intersection_term} \frac{{{\partial ^2}P_{sm}^{(k)}}}{{\partial \omega _{sm}^{(k,q)}\partial \omega _{s'm}^{(k,q)}}} = 0, \forall s \ne s'$$ $y_{sm} = \frac{{a{P_{nr,sm}}{P_{ne,sm}}{\lambda_s^I}}}{{3.5{\lambda _R}}}$, ${z_s} = \frac{{\theta_s^{th}}}{{ 1 + \theta_s^{th} }}$. Besides, we write $\omega_{sm}^{2.5(k,q)}$ rather than ${\left( {\omega _{sm}^{(k,q)}} \right)^{2.5}}$ for lightening the notation. \[lemma:error\_bound\] Let the function $\tilde P_{m}^{(k)}: \mathbb{R}^{|{\mathcal S}^I|} \to \mathbb{R}$ be three times differentiable in a given interval $[\omega_{sm}^{lb}, S_{sm}^{\star}]$ for all $s \in {\mathcal{S}^I}$, then the error bound of the $2^{\rm nd}$ degree Taylor expansion of $\tilde P_{m}^{(k)}$ at the local point $\bm \omega_{sm}^{(k,q)}$ with $\omega_{sm}^{(k,q)} \in [\omega_{sm}^{lb}, S_{sm}^{\star}]$ is given by $$\label{eq:error_bound_lemma} {R_2}({\bm \omega _m}) = \frac{1}{{3!}}{\left[ {\sum\limits_{s \in {{\cal S}^I}} {\left( {{\omega _{sm}} - \omega _{sm}^{(k,q)}} \right)\frac{\partial }{{\partial \omega _{sm}^{(k,q)}}}} } \right]^3} \tilde P_m^{(k)}{{|_{\bm \omega _m^{lb}}^{{\bm S_m^ \star }}}}$$ where $\tilde P_m^{(k)}{{|_{\bm \omega _m^{lb}}^{{\bm S_m^ \star }}}} = \max \left\{ {\tilde P_m^{(k)}{|_{\bm \omega _m^{lb}}},\tilde P_m^{(k)}{|_{\bm S_m^ \star }}} \right\}$, $\bm \omega_{m}^{lb} = [\omega_{1m}^{lb}, \ldots, \omega_{|\mathcal{S}^I|m}^{lb}]$ and $\bm S_{m}^{\star} = [S_{1m}^{\star}, \ldots, S_{|\mathcal{S}^I|m}^{\star}]$. Please refer to Appendix C. After conducting the $2^{\rm nd}$ degree Taylor approximation, the objective function becomes a convex function. Although the constraint (\[eq:SAA\_admm\_problem\]b) is $P_{sm}$ related, we need not to conduct the Taylor approximation on (\[eq:SAA\_admm\_problem\]b) as $P_{sm}$ is quasi-concave and unimodal. In fact, the probability constraint (\[eq:SAA\_admm\_problem\]b) is equivalent to the following inequality $$\label{eq:variable_range} \omega_{sm}^{lb} \le \omega_{sm} \le \omega_{sm}^{ub}, \forall s \in \mathcal{S}^I$$ where $\omega_{sm}^{ub} \le W$ represents the upper bound of $\omega_{sm}$ satisfying (\[eq:SAA\_admm\_problem\]b). Next, a low-complexity bisection-search-based scheme, the main procedures of which are described below, is developed to obtain $\omega_{sm}^{lb}$, $S_{sm}^{\star}$, and $\omega_{sm}^{ub}$ a) let the function $Q_{sm} = P_{sm} - \pi_s$. Perform the bisection search method [@yang2018three] on $Q_{sm} = 0$ to obtain $\omega_{sm}^{lb}$ and $\omega_{sm}^{ub}$ that are the two zero points of $Q_{sm}$. b) with the obtained $\omega_{sm}^{lb}$ and $\omega_{sm}^{ub}$, find the maximum value $S_{sm}^{\star}$ of $P_{sm}$ using the bisection search method again. According to the above analysis, at the $q$-th iteration, we can rewrite (\[eq:bandwidth\_beamforming\_problem\]) as \[eq:SCA\_bandwidth\_beamforming\_problem\] $$\begin{aligned} {2} & \left\{ {\omega _{sm}^{(k + 1,q+1)}},{\bm G_{i,sm}^{(k + 1,q+1)}} \right\} = \mathop {{\rm{argmin}}}\limits_{\{\omega _{sm},{\bm G_{i,sm}}\}} \bar {\cal L}^{(q)} (\omega _{sm},{\bm G_{i,sm}}) \\ & {\rm subject \text{ } to:} \nonumber \\ & {\rm for} \text{ } m, (\ref{eq:SAA_admm_problem}c)-(\ref{eq:SAA_admm_problem}e),(\ref{eq:variable_range}) \text{ } \rm{are} \text{ } satisfied.\end{aligned}$$ where $$\label{eq:average_Lagrangian_func} \begin{array}{l} \bar {\mathcal{L}}^{(q)}(\omega _{sm},\bm G_{i,sm}) = -{ \frac{1}{M}{\tilde P_{m}^{(k)}} - \frac{{\tilde \rho U_m^{u(k)}}}{M} +} \\ { \sum\limits_{s \in {{\cal S}^I}} {\left[ {\psi _{sm}^{(k,q)}\left( {\omega _{sm} - \omega _s^{(k,q)}(\bar t)} \right) + \frac{\mu }{2}{{\left\| {\omega _{sm} - \omega _s^{(k,q)}(\bar t)} \right\|}_2^2}} \right]} } \nonumber \end{array}$$ In (\[eq:SCA\_bandwidth\_beamforming\_problem\]), the objective function is convex, (\[eq:SAA\_admm\_problem\]c) is affine, and the constraint (\[eq:SAA\_admm\_problem\]d) can be proved to be convex w.r.t. both $\omega_{sm}$ and $\bm G_{i,sm}$ [@How2019yang]. Therefore, (\[eq:SCA\_bandwidth\_beamforming\_problem\]) is a convex problem that can be effectively mitigated by some standard convex optimization tools such as CVX [@CVX] and MOSEK [@MOSEK]. Then we can summarize the main steps of mitigating the problem (\[eq:SAA\_admm\_problem\]) in Algorithm \[alg1\]. **Initialization:** Randomly initialize $\bm G_{i,s}^{(0,0)}$, $\{\omega_{s}^{(0,0)}\}$, let $k_{\rm max}=250$, $q_{\rm max}=250$, $q = 0$, $k = 0$, and generate channel samples {$\bm H_{i,sm}$}. Given $\bm G_{i,sm}^{(k,q)}$, $\omega_{sm}^{(k,q)}$, call the greedy scheme to obtain $b_{i,sm}^{u(k,q+1)}$. Optimize (\[eq:SCA\_bandwidth\_beamforming\_problem\]) with obtained $b_{i,sm}^{u(k,q+1)}$ to achieve $\bm G_{i,sm}^{(k,q+1)}$ and $\omega_{sm}^{(k,q+1)}$. Update $q = q + 1$. Let $\omega_{sm}^{(k+1,q+1)} = \omega_{sm}^{(k,q+1)}$, update $\psi _{sm}^{(k + 1)}$ using (\[eq:psi\_update\]). Call (\[eq:omega\_update\]) to update $\omega_{s}^{(k+1)}(\bar t)$. Update $k = k + 1$. \[lemma\_5\] For all $i\in\mathcal{I}_s^u$, $s \in {\mathcal{S}^u}$, and $m \in {\mathcal{M}}$, the obtained power matrix $\bm G_{i,sm}^{(k,q)}$ by Algorithm 1 at the $(k,q)$-th iteration satisfies the low-rank constraint, i.e., the SDR for the power matrix utilized in Algorithm 1 is tight. Please refer to Appendix D. Optimization of beamforming with system sensed channels ======================================================= In Section V, we obtained a family of global consensus variables $\{\omega_{s}(\bar t)\}$ with the system generated channel samples. The time-varying actual channels may require the re-optimization of beamformers and device associations at each minislot. According to system sensed channels at each minislot, we next discuss how to calculate beamforms and device associations. At each minislot $t$, given the global consensus variables $\{\omega_s(\bar t)\}$, the original problem (\[eq:original\_problem\]) will be reduced to the following problem \[eq:mini\_time\_scale\_transformed\_problem\] $$\begin{aligned} {2} & \mathop {{\rm{maximize}}}\limits_{\{b_{i,s}^u(t), {\bm G}_{i,s}(t)\}} \text{ } {\tilde \rho} {{U}^u(t)} \\ & {\rm subject \text{ } to:} \nonumber \\ & \rm {constraints \text{ } (\ref{eq:RRH_energy}), (\ref{eq:total_bandwidth}),(\ref{eq:original_problem}b) \text{ } are \text{ } satisfied.}\end{aligned}$$ In (\[eq:mini\_time\_scale\_transformed\_problem\]), the channels are system sensed ones at $t$. According to the convexity analysis in Section V, (\[eq:mini\_time\_scale\_transformed\_problem\]) is a mixed-integer non-convex programming problem with positive semidefinite matrices, which is hard to be mitigated. Therefore, the alternative optimization scheme presented in subsection V-B can be leveraged to achieve the solutions $b_{i,s}^u(t)$ and $\bm G_{i,s}(t)$ of (\[eq:mini\_time\_scale\_transformed\_problem\]). Lemma \[lemma\_5\] indicates that the achieved ${\rm rank}(\bm G_{i,s}(t)) \le 1$. Thus, we can obtain the beamformers $\bm g_{i,s}(t)$ by performing the eigendecomposition on $\bm G_{i,s}(t)$. To sum up, over a time slot $\bar t$, the slice resource optimization algorithm designed for the RAN slicing system can be summarized as follows. **Initialization:** $\{{\bm H}_{i,s}(t)\}$, $\forall i \in {{\mathcal I}^u}$, $s \in {{\mathcal S}^u}$, and let $P_{s}^{1} \in [0,1]$, $\mu_{a,s}^{1}=0$, $\forall s \in {{\mathcal S}^I}$. Call Algorithm \[alg1\] to obtain $\{\omega_s(\bar t)\}$, for all $s\in {\mathcal S}^I$. Given $\{\omega_s(\bar t)\}$, mitigate (\[eq:mini\_time\_scale\_transformed\_problem\]) by exploiting the alternative optimization scheme to obtain beamformers $\{{\bm g}_{i,s}(t)\}$ and URLLC device associations $b_{i,s}^u(t)$ for all $i \in {\mathcal I}_s^u$, $s \in {\mathcal S}^u$. Simulation results ================== In this section, we aim at evaluating the proposed algorithm via extensive simulations. Comparison algorithms and parameter setting ------------------------------------------- We compare the following three algorithms to verify the effectiveness of the proposed algorithm and to explain the impact of access control schemes on the RAN system performance intuitively i) SRO algorithm that adopts the unrestricted access control scheme; ii) SRO-ACB$_{\rm I}$ algorithm that utilizes the ACB access control scheme with $P_{ACB} = 0.9$; iii) SRO-ACB$_{\rm II}$ algorithm that adopts the ACB access control scheme with $P_{ACB} = 0.5$. The parameter setting is as follows: RRHs and IoT devices are deployed following independent PPPs in a one km$^2$ area. URLLC devices are randomly and uniformly distributed in this area. There are three mIoT slices and two URLLC slices in the RAN slicing system. For the mIoT slices, set the new endogenous packet arrivals rate $\epsilon_{w,s}(t) = [1.5, 1.0, 0.5]$ packets/minislot, $\pi_s = 0.5$, $\forall s,t$. Let the path-loss component $\varphi = 4$, $L = 2000$ bits, $\tau = 1$ unit, $\sigma^2 = -90$ dBm, $\rho_o = -90$ dBm, ${\hat E}_{j}^I = 0.03$ mW, $\lambda_R = 3$ RRHs/km$^2$, $\{\lambda_{s}^I\} = [18000, 18000, 18000]$ IoT devices/km$^2$, $\{\gamma_s^{th}\} = \{5.8, 4.35, 2.9\}$ Kbits/minislot. For the URLLC slices, the transmit antenna gain at each RRH is set to be $5$ dB, and a log-normal shadowing path-loss model is leveraged to simulate the path-loss between an RRH and a URLLC device with the log-normal shadowing parameter being $10$ dB. A path-loss is computed by $h({\rm dB}) = 128.1 + 37.6\log_{10}d$, where $d$ (in km) represents the distance between a device and an RRH. Let $L_{i,s}^u = 160$ bits, $\sigma_{i,s}^2 = -100$ dBm, $\lambda_{s} = \lambda = 0.1$ packets/minislot, $\forall i,s$, $\{I_s^u\} = \{3, 5\}$ devices, and $\{D_s\} = \{1, 2\}$ milliseconds, $E_j = 3$ W, $\forall j$ [@tang2019service]. Other simulation parameters are shown in Table \[table\_simulation\_parameters\]. Para. Value Para. Value Para. Value ---------- ----------- ------------ -------------------- --------------- ----------------------- $J$ $3$ $K$ $2$ $\tilde \rho$ $1$ $\eta$ $100$ $T$ $60$ $W$ $60$ MHz $M$ $100$ $\phi$ $1.5$ $a$ $0.18$ MHz $\xi$ $54$ $\alpha_g$ $0.05$ $\kappa$ $5.12 \times 10^{-4}$ $\alpha$ $10^{-5}$ $\beta$ $2 \times 10^{-8}$ $\varsigma$ $2 \times 10^{-5}$ : Simulation parameters[]{data-label="table_simulation_parameters"} Performance evaluation ---------------------- To evaluate the comparison algorithms, the following performance indicators are utilized i) RA success probability $P_s(t)$ that is computed using (\[eq:QoS\_prob\_analysis\]); ii) expected queue length per IoT device at minislot $t$, $E[Q_s(t)] = \mu_{a,s}(t)$; iii) total slice utility $\bar U$ that is the objective function of (\[eq:original\_problem\]). We first evaluate the convergence of the proposed SRO algorithm. Fig. \[fig:fig\_convergence\] illustrates the convergence of SRO with $\Delta_{\omega} = {\rm{ }}\sum\nolimits_{s \in {{\mathcal S}^I}} {\left| {\omega _s^{(k + 1)}(\bar t) - \omega _s^{(k)}(\bar t)} \right|} $. It shows that SRO can converge after several iterations. ![The convergence curve of the proposed SRO algorithm.[]{data-label="fig:fig_convergence"}](Convergence_curve.eps){width="2.9in"} We next plot the tendency of the RA success probability $P_s(t)$ and the corresponding expected queue length $E[Q_s(t)]$ during a time slot in Fig. \[fig:fig\_queueLen\_Pst\]. Fig. \[fig:fig\_queueLen\_Pst\](a) and \[fig:fig\_queueLen\_Pst\](c) show the tendency of $P_s(t)$ and $E[Q_s(t)]$ in the case of $\{\gamma_s^{th}\} = \{1.8, 1.35, 0.9\}$. Fig. \[fig:fig\_queueLen\_Pst\](b) and \[fig:fig\_queueLen\_Pst\](d) depict the tendency of $P_s(t)$ and $E[Q_s(t)]$ in the case $\{\gamma_s^{th}\} = \{5.8, 4.35, 2.9\}$. From Fig. \[fig:fig\_queueLen\_Pst\], we obtain the following interesting conclusions: the queue of each IoT device is not stable when the queue serving rate $\gamma_s^{th}$ is small. In this case, the average queue length monotonously increases over minislot $t$. On the contrary, the queue of each IoT device is periodically flushed when a great queue serving rate is configured. ![Trends of $P_s(t)$ and $E[Q_s(t)]$. (a) and (c) are results of the parameter setting $\{\gamma_s^{th}\} = \{1.8, 1.35, 0.9\}$ Kbits/minislot; (b) and (d) correspond to the parameter setting $\{\gamma_s^{th}\} = \{5.8, 4.35, 2.9\}$ Kbits/minislot.[]{data-label="fig:fig_queueLen_Pst"}](Ps_EQs_vs_t.eps){width="3.1in"} Let the IoT device intensity $\bm \lambda^I = [900 n, 900 n, 900 n]$ with $n \in \{6, 8, \ldots, 26\}$. Under the existence of both mIoT and URLLC slices, we plot trends of the total slice utility $\bar U$ and bursty URLLC slice utility $\bar U^u$ w.r.t. $n$ in Fig. \[fig:fig\_utility\_vs\_lamdba\_IoT\] to understand the impact of the mIoT slices on the performance of all comparison algorithms. In this figure, $B = [b_{11}^u, \ldots, b_{31}^u, b_{12}^u, \ldots, b_{52}^u]$, $\omega^I=[\omega_{SRO}^I, \omega_{ACB_{\rm I}}^I, \omega_{ACB_{\rm II}}^I]$ MHz with $\omega_{SRO}^I$, $\omega_{ACB_{\rm I}}^I$ and $\omega_{ACB_{\rm II}}^I$ representing the bandwidth allocated to mIoT slices by executing SRO, SRO-ACB$_{\rm I}$, and SRO-ACB$_{\rm II}$ algorithms, respectively, and $\bar U^I = [\bar U_{SRO}^I, \bar U_{ACB_{\rm I}}^I, \bar U_{ACB_{\rm II}}^I]$ with $\bar U_{SRO}^I$ denoting the achieved mIoT slice utility of SRO. \ The following observations can be obtained from Fig. \[fig:fig\_utility\_vs\_lamdba\_IoT\]: i) when $n < 16$, all algorithms almost obtain the same total slice utility, and the obtained utilities are robust to the average number of IoT devices; ii) when $16 \le n \le 26$, the conclusion changes. For the SRO algorithm, its achieved $\bar U$ decreases with an increasing $n$ due to increasing interference. A great $n$, however, does not cause a significant decrease in the total slice utilities obtained by SRO-ACB$_{\rm I}$ and SRO-ACB$_{\rm II}$. Thanks to the exploration of an access control scheme, both SRO-ACB$_{\rm I}$ and SRO-ACB$_{\rm II}$ can achieve greater $\bar U$ than SRO. For example, compared with SRO, SRO-ACB$_{\rm II}$ improves $\bar U$ by $6.65\%$ when $n = 24$; iii) when $n = 26$, which means that the total average number of IoT devices reaches $70,200$ devices, the RAN slicing system fails to create and manage mIoT slices as the QoS requirements of mIoT slices serving such a massive average number of devices cannot be simultaneously satisfied. In this case, all system resources are allocated to URLLC slices, and the maximum bursty URLLC slice utility is obtained; iv) as mIoT slices and URLLC slices share the system resources, an increasing $n$ results in a decreasing bursty URLLC slice utility $\bar U^u$; Besides, it is interesting to find that the two access-control-based algorithms may not outperform SRO in terms of obtaining $\bar U^u$. It indicates that URLLC slices do not benefit from access control schemes of mIoT slices when changing $n$; v) the RAN slicing system can always accommodate the QoS requirements of all URLLC devices. Next, to understand the impact of URLLC slices on the performance of all comparison algorithms, we plot the trends of the total slice utilities and the mIoT slice utilities obtained by all comparison algorithms w.r.t. URLLC packet arrival rate $\lambda$ with $\lambda = \{0.1,0.5,1.0,\ldots,4.5,5.0\}$ packets per unit time in Fig. \[fig:fig\_utility\_vs\_lamdba\_URLLC\]. Similarly, the following notations are involved in this figure: $\omega^u = [\omega_{SRO}^u, \omega_{ACB_{\rm I}}^u, \omega_{ACB_{\rm II}}^u]$, $\bar U^u = [\bar U_{SRO}^u, \bar U_{ACB_{\rm I}}^u, \bar U_{ACB_{\rm II}}^u]$ with $\omega_{SRO}^u$ and $\bar U_{SRO}^u$ denoting the bandwidth allocated to URLLC slices and the URLLC slice utility obtained by running the SRO algorithm, respectively. \ From Fig. \[fig:fig\_utility\_vs\_lamdba\_URLLC\], we can observe that: i) the obtained $\bar U$ of all algorithms decrease with $\lambda$ mainly due to the decrease of the bursty URLLC slice utility. Two algorithms adopting the access control scheme always achieve greater utilities $\bar U$ than SRO. For example, when $\lambda = 5$, compared with the SRO algorithm, the obtained $\bar U$ of SRO-ACB$_{\rm II}$ is increased by $29.41\%$; ii) for all algorithms, the computed bandwidth for URLLC slices increases with an increasing $\lambda$. However, their obtained URLLC slice utilities $\bar U^u$ are reduced owing to the increase of energy consumption; iii) SRO-ACB$_{\rm II}$ may achieve greater $\bar U$ than SRO-ACB$_{\rm I}$ as a greater $\bar U^I$ is obtained by reducing the number of interfering IoT devices; iv) the obtained mIoT slice utilities $\bar U^I$ of SRO-ACB$_{\rm I}$ and SRO-ACB$_{\rm II}$ are robust to the URLLC packet arrival rate. The obtained $\bar U^I$ of SRO decreases with an increasing $\lambda$; v) an important observation is that the $\bar U^I$ of the access-control-based SRO-ACB$_{\rm I}$ algorithm is $1.65$ times that of the SRO algorithm when $\lambda = 5$. It explicitly reflects that mIoT slices can still benefit from access control schemes even though $\lambda$ is changed. Figs. \[fig:fig\_utility\_vs\_lamdba\_IoT\] and \[fig:fig\_utility\_vs\_lamdba\_URLLC\] illustrate the situation of a given total system bandwidth. We next change the total bandwidth $W$ and plot its impact on the obtained total slice utilities of all algorithms in Fig. \[fig:fig\_total\_utility\_vs\_bandwidth\]. ![Trend of achieved total slice utility vs. system bandwidth.[]{data-label="fig:fig_total_utility_vs_bandwidth"}](total_slice_utility_vs_bandwidth.eps){width="3.1in"} The following conclusions can be obtained from this figure i) when $W=45$ MHz, the QoS requirements of all IoT devices cannot be simultaneously satisfied. As a result, the total bandwidth is allocated to URLLC slices; ii) when $W$ locates in the range of $(45, 55]$ MHz, the achieved total slice utilities $\bar U$ of SRO and SRO-ACB$_{\rm I}$ increase with $W$. Owing to the utilization of the access control scheme, SRO-ACB$_{\rm I}$ and SRO-ACB$_{\rm II}$ obtain higher $\bar U$ than SRO. For example, compared with the SRO algorithm, the SRO-ACB$_{\rm II}$ algorithm improves the achieved total slice utility by $6.66\%$ when $W = 50$ MHz. iii) when $W > 55$ MHz, all algorithms cannot remarkably improve $\bar U$. At last, we discuss other crucial parameters’ impact on the performance of the comparison algorithms. We reconfigure $\{\gamma_s^{th}\}$ of mIoT slices as $\gamma_1^{th} = 3.6 m$, $\gamma_2^{th} = 2.7 m$ and $\gamma_3^{th} = 1.8 m$ Kbits/minislot with $m \in \{1.5, 1.6, \ldots, 2.1\}$ and $\{D_s\}$ of URLLC slices as $D_1 = 0.00025 d$ second and $D_2 = 0.0005 d$ second with $d \in \{2, 3, \ldots, 10\}$. The impact of QoS requirements of network slices on the total slice utility is plotted in Figs. \[fig:fig\_total\_utility\_vs\_gammath\] and \[fig:fig\_total\_utility\_vs\_Ds\]. The impact of energy efficiency coefficient $\eta$ is plotted in Fig. \[fig:fig\_total\_utility\_vs\_eta\]. In this figure, we denote the energy consumption of RRHs of all algorithms by $E^u = [E_{SRO}^u, E_{ACB_{\rm I}}^u, E_{ACB_{\rm II}}^u]$ with $E_{SRO}^u = \sum\nolimits_{t = 1}^T {\sum\limits_{s \in {{\mathcal S}^u}} {\sum\limits_{i \in {\mathcal I}_s^u} {b_{i,s}^u{\rm tr}({\bm G_{i,s}})} } } $. ![Trend of achieved total slice utility vs. $m$.[]{data-label="fig:fig_total_utility_vs_gammath"}](total_slice_utility_vs_Csth.eps){width="3.1in"} ![Trend of achieved total slice utility vs. $d$.[]{data-label="fig:fig_total_utility_vs_Ds"}](total_slice_utility_vs_Ds.eps){width="3.1in"} ![Trend of achieved total slice utility vs. $\eta$.[]{data-label="fig:fig_total_utility_vs_eta"}](total_slice_utility_vs_eta.eps){width="3.1in"} From these figures, the following observations can be achieved: i) the obtained utilities $\bar U$ of all algorithms decrease with an increasing $m$. This is because a great $m$ indicates that the accumulated IoT packets in the queue of each IoT device can be quickly emptied, and then a small $P_s(t)$ is obtained; ii) a great $D_s$ will reduce RRHs’ energy consumption. However, it also reduces the URLLC slice gain. Then, it may be hard to conclude the trend of $\bar U^u$ w.r.t. $D_s$ as the energy efficiency coefficient $\eta$ significantly affects the value of $\bar U^u$; iii) it is also uneasy to conclude the trend of $\bar U^u$ w.r.t. $\eta$. An increasing $\eta$ causes a decrease of RRHs’ energy consumption. Yet, the value of $\bar U^u$ is determined by the multiplier of $\eta$ and $E^u$; iv) the SRO-ACB$_{\rm II}$ algorithm may perform better than the SRO algorithm. However, the performance of the other access-control-based algorithm, SRO-ACB$_{\rm I}$, is slightly worse than the SRO algorithm. Besides, it cannot ensure that the $\bar U^I$ obtained by the access-control-based algorithms are always higher than that of SRO. At sometimes, access control schemes may drag down the utility of the mIoT service. To sum up, in the case of service multiplexing, RA control schemes for alleviating signal interference and enhancing mIoT slice utility may be preferred for mIoT slices. However, considering both the CAPEX and the improvement of slice utility, RA control schemes should be carefully designed and employed because some RA control schemes may worsen the mIoT and even the total slice utilities. Conclusion ========== In this paper, we revisited the frame and minislot structure of a RAN slicing system to admit more IoT devices and proposed a queue evolution model to analyze the RACH of a randomly chosen IoT device. Based on the analysis result, we derived the closed-form expression of the RA success probabilities of the device with unrestricted access control scheme and ACB access control scheme. Next, we formulated the RAN slicing for mIoT and bursty URLLC service multiplexing as an optimization problem to optimally orchestrating RAN resources for mIoT slices and URLLC slices, and efficient mechanisms such as SAA and ADMM were exploited to mitigate the optimization problem. Simulation results showed that RA control schemes should be carefully designed and employed in the case of service multiplexing. Proof of Lemma 1 ---------------- For the origin RRH, the LT of its aggregate interference from interfering IoT devices in $s \in {\mathcal{S}^I}$ can be derived as $$\label{eq:LT_intra_interference} \begin{array}{*{20}{l}} {{{\mathcal L}_{{{\mathcal I}_s}(t)}}({\varpi _s}) = {E_{{{\mathcal I}_s}(t)}}\left[ {{e^{ - {\varpi _s}{{\mathcal I}_s}(t)}}} \right]}\\ = {E_{{{\mathcal I}_s}(t)}}\left[ {\exp ( { - {\varpi _s}\sum\limits_{{u_{m,s}} \in {\Phi _s}\backslash \{ o\} } {{\mathbbm 1}({p_m}||{d_m}|{|^{ - \varphi }} = {\rho _o}) \times } } } \right.\\ \text{ } \left. {\left. {{\mathbbm 1}({N_{a,s}}(t) > 0){\mathbbm 1}({f_m} = {f_o}){\rho _o}{h_m}} \right)} \right] \\ \mathop = \limits^{(a)} {E_{{\Phi _s}}}\left[ {{\prod _{{u_{m,s}} \in {\Phi _s}\backslash \{ o\} }}{E_{{h_m}}}\left[ {\exp \left( { - {\varpi _s} \times } \right.} \right.} \right.\\ \text{ } \left. {\left. {{\mathbbm 1}({p_m}||{u_{m,s}}|{|^{ - \varphi }} = {\rho _o}){\mathbbm 1}({N_{a,s}}(t) > 0){\mathbbm 1}({f_m} = {f_o}){\rho _o}{h_m}} \right)} \right] \\ {\mathop = \limits^{(b)} \sum\limits_{n = 0}^\infty {P\{ |{Z_s}| = n\} \prod\limits_{{u_{m,s}} \in {Z_s}} {{E_{{h_m}}}\left[ {{e^{ - {\varpi _s}{\rho _o}{h_m}}}} \right]} } }\\ {\mathop = \limits^{(c)} P\{ |{Z_s}| = 0\} + \sum\limits_{n = 1}^\infty {P\{ |{Z_s}| = n\} {{\left( {\frac{1}{{1 + {\varpi _s}{\rho _o}}}} \right)}^n}} }\\ {\mathop = \limits^{(d)} {{\tilde P}_{{X_s}}}\{ {X_s} = 1\} + \left\{ {\sum\limits_{n' = 0}^\infty {{{\tilde P}_{{X_s}}}\{ {X_s} = n'\} {{\left( {\frac{1}{{1 + {\varpi _s}{\rho _o}}}} \right)}^{n'}}} - } \right.}\\ \text{ } {\left. {\sum\limits_{n' = 0}^1 {{{\tilde P}_{{X_s}}}\{ {X_s} = n'\} {{\left( {\frac{1}{{1 + {\varpi _s}{\rho _o}}}} \right)}^{n'}}} } \right\}(1 + {\varpi _s}{\rho _o})} \end{array}$$ where $\varpi_s = \frac{\theta_s^{th}}{\rho_o}$, $Z_s$ denotes the set of interfering IoT devices in mIoT slice $s$, $X_s$ represents the number of active IoT devices associated with the origin RRH in $s$. According to the conclusion of Lemma 1 in [@yu2013downlink], the probability mass function (PMF) ${\tilde P}_{X_s} \{ X_s = n'\}$ can be written as $$\label{eq:intra_interference_proba} {\tilde P}_{X_s}\{ X_s = n'\} = \frac{{{{3.5}^{3.5}}\Gamma (n' + 3.5){{(\frac{{{P_{nr,s}}(t)P_{ne,s}(t){\lambda _{s}^I}}}{{{\lambda _R}\xi F_s}})}^{n' }}}}{{\Gamma (3.5)(n')!{{(\frac{{P_{nr,s}}(t){P_{ne,s}(t){\lambda _{s}^I}}}{{{\lambda _R} \xi F_s}} + 3.5)}^{n' + 3.5}}}}$$ with $\Gamma(\cdot)$ being the gamma function. Besides, in (\[eq:LT\_intra\_interference\]), (a) follows from the i.i.d. distribution of $h_m$ and its further independence from the Poisson point process $\Phi_s$; (b) follows from the expectation of a discrete random variable; (c) follows from the LT over $h_m$; (d) follows from the fact that the number of active IoT device in a specific Voronoi cell is one more than the number of active interfering IoT devices in this cell. From (\[eq:intra\_interference\_proba\]), we can deduce that $X_s$ ($s\in \mathcal{S}^I$) is a gamma¨CPoisson random variable with $X_s \sim {\rm gamma-Poisson}(\alpha_s, 3.5)$ and $\alpha_s = \frac{{{P_{nr,s}}(t){P_{ne,s}}(t){\lambda _{s}^I}}}{{3.5{\lambda _R}\xi F_s}}$. For a gamma¨CPoisson random variable $X_s \sim {\rm gamma-Poisson}(\alpha, \beta)$, the following expression holds: $E[e^{X_s}] = (1+\alpha-\alpha e)^{-\beta}$. Thus, we can rewrite (\[eq:LT\_intra\_interference\]) as (\[eq:LT\_interference\_expression\]). This completes the proof. Proof of Lemma 2 ---------------- As new endogenous packet arrivals in any IoT device at each minislot $t$ is modelled as a Poisson distribution, the departure process of packets can be regarded as an approximated thinning process of new arrivals, where the thinning factor is related to the RA success probability. The number of accumulated packets in the queue of any IoT device can then be approximated as a Poisson distribution with intensity $\mu_{a,s}^{t}$ ($s\in \mathcal S^I$) after the thinning process in a specific minislot $t$ ($t > 1$) [@jiang2018random]. Thus, we can derive the expression of $\mu_{a,s}^{t}$ ($t > 1$) via combining with the following facts - [**Fact 1:**]{} the accumulated packets during the $t-1$-th minislot will contribute to the accumulated packets at the $m$-th minislot. - [**Fact 2:**]{} the arrival packets during the $t-1$-th minislot will also contribute to the accumulated packets in the queue of an IoT device at the $m$-th minislot. - [**Fact 3:**]{} an IoT device can send packets only if its preamble is successfully transmitted. - [**Fact 4:**]{} at the same minislot, the new packet arrival process and the packet accumulated process are independent. Similar as the Theorem 2 in [@jiang2018random], we can infer that at the $2^{\rm nd}$ minislot, for all $s \in \mathcal{S}^I$, $\mu_{a,s}^{2}$ depends on the intensity of new packet arrivals $\mu_{w,s}^{1}$ and the probability $P_{s}^{1}$ of a randomly selected IoT device at the $1^{\rm st}$ minislot, which is given by $$\label{eq:mu_acc_2_1} \begin{array}{l} \mu _{a,s}^2 = \mu _{w,s}^1 - x_s P_s^1\left( {1 - {e^{ - \mu _{w,s}^1}}} \right) \end{array}$$ The detailed proof of (\[eq:mu\_acc\_2\_1\]) is omitted for brevity, and a similar proof can be found in the proof section of Theorem 2 in [@jiang2018random]. Considering that $\mu_{a,s}(t)$ is non-negative at each minislot $t$, we have $$\label{eq:mu_acc_2} \begin{array}{*{20}{l}} {\mu _{a,s}^2 = } \end{array}{\left[ {\mu _{w,s}^1 - {x_s}P_s^1\left( {1 - {e^{ - \mu _{w,s}^1}}} \right)} \right]^ + }$$ Then, according to the definition of non-empty probability and the Poisson approximation, the non-empty probability of a randomly selected IoT device in mIoT slice $s \in \mathcal{S}^I$ at the $2^{\rm nd}$ minislot can be approximated as $$\label{eq:non_empty_prob_2_expression} P_{ne,s}^2 = 1 - {e^{ - \mu _{a,s}^2}}$$ At the $3^{\rm rd}$ minislot, the intensity of accumulated data packets in the queue of a randomly selected IoT device can be derived as the following $$\label{eq:mu_acc_3} \begin{array}{l} \mu _{a,s}^3 = P_s^2\left( {\sum\nolimits_{n = 1}^\infty {( {{{[n - x_s]}^ + } \sum\nolimits_{z = 0}^n {{P_{N_{w,s}^2}}(z){P_{N_{a,s}^2}}(n - z)} } )} } \right)\\ \quad + (1 - P_s^2)\left( {\sum\nolimits_{n = 1}^\infty {n \sum\nolimits_{z = 0}^n {{P_{N_{w,s}^2}}(z){P_{N_{a,s}^2}}(n - z)} } } \right)\\ \mathop = \limits^{(a)} P_s^2\left [ {\sum\nolimits_{n = 1}^\infty {\sum\nolimits_{z = 0}^n {\frac{{{{\left( {\mu _{w,s}^2} \right)}^z}{e^{ - \mu _{w,s}^2}}}}{{z!}}\frac{{{{\left( {\mu _{a,s}^2} \right)}^{n - z}}{e^{ - \mu _{a,s}^2}}}}{{(n - z)!}}} \times n} - } \right.\\ \quad \left. {x_s\sum\nolimits_{n = 1}^\infty {\sum\nolimits_{z = 0}^n {\frac{{{{\left( {\mu _{w,s}^2} \right)}^z}{e^{ - \mu _{w,s}^2}}}}{{z!}}\frac{{{{\left( {\mu _{a,s}^2} \right)}^{n - z}}{e^{ - \mu _{a,s}^2}}}}{{(n - z)!}}} } } \right ]^+ + \\ \quad (1 - P_s^2)\sum\nolimits_{n = 1}^\infty {\sum\nolimits_{z = 0}^n {\frac{{{{\left( {\mu _{w,s}^2} \right)}^z}{e^{ - \mu _{w,s}^2}}}}{{z!}}\frac{{{{\left( {\mu _{a,s}^2} \right)}^{n - z}}{e^{ - \mu _{a,s}^2}}}}{{(n - z)!}}} \times n} \\ \mathop = \limits^{(b)} \left [ \mu _{w,s}^2 + \mu _{a,s}^2 - x_sP_s^2\left( {1 - {e^{ - \mu _{w,s}^2 - \mu _{a,s}^2}}} \right) \right ]^+ \end{array}$$ where $P_{N_{w,s}^2}$ and $P_{N_{a,s}^2}$ represent the PMFs of new arrival packets and accumulated packets at the $2^{\rm nd}$ minislot, respectively. Besides, (a) follows from the fact: for any two independent Poisson distribution $\Phi_{X_1}$ and $\Phi_{X_2}$, $P_{X_1,X_2}({X_1} + {X_2} = x) = \sum\nolimits_{y = 0}^x {P_{X_1}({X_1} = y)P_{X_2}(X_2 = x - y)} $; (b) holds as $\Phi_{X_1,X_2}$ is a two dimensional Poisson distribution with an intensity $\lambda_{X_1} + \lambda_{X_2}$, and $\sum\nolimits_{x = 1}^\infty {P_{X_1, X_2}({X_1} + {X_2} = x)} = 1 - P_{X_1,X_2}({X_1} + {X_2} = 0)$. Similarly, we have $$\label{eq:non_empty_prob_3} P_{ne,s}^3 = 1 - {e^{ - \mu _{a,s}^3}}$$ When $t > 3$, since the accumulated packets evolution model of the queue of any IoT device is the similar as that at $t = 3$, we can directly extend the conclusion obtained at $t = 3$ to that at $t > 3$. Therefore, we can obtain the closed-form expression of $\mu_{a,s}^t$ for all $s \in \mathcal{S}^I$ at $t > 1$ with $$\label{eq:mu_acc_m_expression_proof} \mu _{a,s}^t = \left [ \mu _{w,s}^{t - 1} + \mu _{a,s}^{t - 1} - x_sP_s^{t - 1}\left( {1 - {e^{ - \mu _{w,s}^{t - 1} - \mu _{a,s}^{t - 1}}}} \right) \right ]^+$$ and $$\label{eq:non_empty_prob_m_expression_proof} P_{ne,s}^t = 1 - {e^{ - \mu _{a,s}^t}}$$ This completes the proof. Proof of Lemma 4 ---------------- The $2^{\rm nd}$ degree Taylor expansion of $\tilde P_{m}^{(k)}$ at the local point $\bm \omega_{m}^{(k,q)}$ is $$\tilde P_{2,m}^{(k)} = \sum\limits_{j = 0}^2 {\frac{1}{{j!}}{{\left[ {\sum\limits_{s \in {\mathcal S}^I} {\left( {{\omega _{sm}} - \omega _{sm}^{(k,q)}} \right)\frac{\partial }{{\partial \omega _{sm}^{(k,q)}}}} } \right]}^j}{\tilde P_m^{(k)}}|_{\bm \omega_m^{(k,q)}}}$$ The $3^{\rm rd}$ degree Taylor expansion of $\tilde P_{m}^{(k)}$ at $\bm \omega_{m}^{(k,q)}$ must be more accurate than $\tilde P_{2,m}^{(k)}$ with $$\tilde P_{3,m}^{(k)} = \tilde P_{2,m}^{(k)} + {\frac{1}{{3!}}{{\left[ {\sum\limits_{s \in {\mathcal S}^I} {\left( {{\omega _{sm}} - \omega _{sm}^{(k,q)}} \right)\frac{\partial }{{\partial \omega _{sm}^{(k,q)}}}} } \right]}^3}{\tilde P_m^{(k)}}|_{\bm \omega_m^{(k,q)}}}$$ Since the error of $\tilde P_{2,m}^{(k)}$ is no greater than the maximum difference between $\tilde P_{3,m}^{(k)}$ and $\tilde P_{2,m}^{(k)}$, we have $$\label{eq:error_R2} R_{2}(\bm \omega_m) = \max \{ {\frac{1}{{3!}}{{[ {\sum\limits_{s \in {\mathcal S}^I} {( {{\omega _{sm}} - \omega _{sm}^{(k,q)}} )\frac{\partial }{{\partial \omega _{sm}^{(k,q)}}}} } ]}^3}{\tilde P_m^{(k)}}|_{\bm \omega_m^{(k,q)}}} \}$$ In (\[eq:error\_R2\]), $\bm \omega_{m}^{(k,q)}$ is a constant vector, the $\max$ operation will not affect the constant vector and the vector $\bm \omega_{m}$. For any $s\in \mathcal{S}^I$, the maximum value obtainable by $\frac{{{\partial ^3}{\tilde P_m^{(k)}}|_{\bm \omega_m^{(k,q)}}}}{{\partial \omega _{sm}^{3(k,q)}}}$ will not exceed the greatest value of that derivative in the interval $[\omega_{sm}^{lb}, S_{sm}^{\star}]$. Additionally, the maximum value of $\frac{{{\partial ^3}{P_m}|_{\bm \omega_m^{(k,q)}}}}{{\partial \omega _{sm}^{3(k,q)}}}$ will generally occur at one of the endpoints of the interval $[\omega_{sm}^{lb}, S_{sm}^{\star}]$. Therefore, we obtain (\[eq:error\_bound\_lemma\]). This completes the proof. Proof of Lemma 5 ---------------- For all $i \in {\mathcal{I}^u}$, $s \in {\mathcal{S}^u}$, $m \in {\mathcal{M}}$, a feasible way of proving that ${\rm rank}(\bm G_{i,sm}) \le 1$ is to utilize the Lagrange method. However, owing to the complicated expression of $W^{u}(\bm r_m)$ w.r.t. $\bm G_{i,sm}$ it will be uneasy to do that. Fortunately, we find that the proof can be conducted if a family of auxiliary variables is introduced. For the constraint (\[eq:SAA\_admm\_problem\]d), if we introduce the auxiliary variables $\{\nu_{i,sm}\}$ and let $$\label{eq:linear_snr_tau_ism} \frac{{\rm{tr}}({\bm H_{i,sm}}{{\bm G}_{i,sm}})}{\phi \sigma _{i,s}^2} \ge \nu _{i,sm},\forall i \in {\mathcal I}_s^u,s \in {{\mathcal S}^u},m \in {\mathcal{M}}$$ then (\[eq:SAA\_admm\_problem\]d) is equivalent to $$\label{eq:} \sum\limits_{s \in {{\cal S}^I}} {(1+\alpha_g)\omega _{sm}(\bar t)} + W^u(\bm f_m) \le W, \text{ } {\rm and} \text{ } {\rm (\ref{eq:linear_snr_tau_ism})}$$ where $\bm f_m = \{f_{i,sm};i \in {\mathcal{I}_s^u}, s \in {\mathcal{S}^u}\}$ and $$\begin{array}{*{20}{l}} {{f_{i,sm}} = \frac{{L_{i,s}^u}}{{{{\log }_2}(1 + {\nu _{i,sm}})}} + \frac{{{{({Q^{ - 1}}(\beta))}^2}}}{{2\log _2^2(1 + {\nu _{i,sm}})}}}\\ {\qquad \quad + \frac{{{{({Q^{ - 1}}(\beta))}^2}}}{{2\log _2^2(1 + {\nu _{i,sm}})}}\sqrt {1 + \frac{{4L_{i,s}^u{{\log }_2}(1 + {\nu _{i,sm}})}}{{{{({Q^{ - 1}}(\beta))}^2}}}} } \end{array}$$ We omit the proof of the equivalence as a similar proof can be found in the proof section of constraints’ equivalence in [@How2019yang]. The partial Lagrangian function of (\[eq:SCA\_bandwidth\_beamforming\_problem\]) can be written as $$\label{eq:lagrangian_func} \begin{array}{l} L( \ldots ) = \sum\limits_{s \in {{\mathcal S}^u}} {\sum\limits_{i \in {\mathcal I}_s^u} {\left[ {\frac{{\tilde \rho \eta }}{M}{\rm{tr}}({\bm G_{i,sm}}) - {{\bar \mu }_{i,sm}}\frac{{{\rm{tr}}({\bm H_{i,sm}}{\bm G_{i,sm}})}}{{\phi \sigma _{i,s}^2}} + } \right.} } \\ \qquad \qquad \left. {\sum\limits_{j \in {\mathcal J}} {{{\bar \lambda }_{jm}}{\rm{tr}}(b_{i,sm}^{u(k,q)}{\bm Z_j}{\bm G_{i,sm}})} - {{\bm {\bar X}}_{i,sm}}{\bm G_{i,sm}}} \right] \end{array}$$ where $\bar \lambda_{jm}$, $\bar \mu_{i,sm}$, and $\bar {\bm X}_{i,sm}$ are Lagrangian multipliers corresponding to constraints (\[eq:SAA\_admm\_problem\]c), (\[eq:linear\_snr\_tau\_ism\]) and (\[eq:SAA\_admm\_problem\]e). Besides, only terms related to ${\bm G}_{i,sm}$ are included in this function for brevity. According to the Karush-Kuhn-Tucker (KKT) conditions, the necessary condition for obtaining the optimal matrix power at the $(k,q)$-th iteration ${\bm G}_{i,sm}^{(k,q)\star}$ is given by $$\label{eq:KKT_condition_2} \begin{array}{l} \frac{{\partial L( \ldots )}}{{\partial {\bm G}_{i,sm}^{(k,q)\star}}} = \frac{{\tilde \rho \eta }}{M}{\bm I_{i,sm}} + \frac{\bar \mu_{i,sm}{{\bm H_{i,sm}}}}{{\phi \sigma _{i,s}^2}} - \\ \qquad \qquad \text{ } \sum\limits_{j \in {\mathcal J}} {{{\bar \lambda }_{jm}}{b_{i,sm}^{u(k,q)}\bm Z_j}} - {{\bm X}_{i,sm}} = 0 \end{array}$$ where ${\bm I_{i,sm}} \in \mathbb{R}^{JK \times JK}$ is an identity matrix. Then, we can conclude that ${\rm rank}({\bm X}_{i,sm}) \ge JK - 1$. The reasons are i) ${{\bar \lambda }_{jm}}$, $b_{i,sm}^{u(k,q)}$, and $\bar \mu _{i,sm}$ are nonnegative and the matrix $\bm I_{i,sm}$ is full rank; ii) ${\rm rank}(\bm H_{i,sm}) \le 1$. Next, according to the complementary slackness condition, we have $$\label{eq:complementary_slack_condition_2} {{\bm X}_{i,sm}}{\bm G}_{i,sm}^{(k,q)\star} = 0$$ Based on (\[eq:complementary\_slack\_condition\_2\]) and the rank result of $\bm X_{i,sm}$, we can conclude that ${\rm rank}(\bm G_{i,sm}^{(k,q)\star}) \le 1$. This completes the proof. [^1]: P. Yang and T. Q. S. Quek are with the Information Systems Technology and Design, Singapore University of Technology and Design, 487372 Singapore. X. Xi, J. Chen, and X. Cao are with the School of Electronic and Information Engineering, Beihang University, Beijing 100083, China, and also with the Key Laboratory of Advanced Technology, Near Space Information System (Beihang University), Ministry of Industry and Information Technology of China, Beijing 100083, China. D. Wu is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville FL 32611 USA. [^2]: We do not show all channels in this figure as the detailed research of the physical layer supporting the mIoT service is out of the scope of this paper. [^3]: Owing to the truncated channel inversion power control, not all of the IoT devices in mIoT slices can communicate in the uplink when the cutoff threshold is relatively high [@elsawy2014stochastic]. However, this paper assumes that the transmit power of each IoT device is large enough such that the IoT device will not experience preamble outage resulting from the insufficient power.

--- abstract: 'We exploit the geometric approach to the virtual fundamental class, due to Fukaya-Ono and Li-Tian, to compare Gromov-Witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order. Various special cases of the comparison theorem in this paper have long been used in the algebraic category; some of them have also appeared in the symplectic setting. Combined with the inherent flexibility of the symplectic category, the main theorem leads to a confirmation of Pandharipande’s Gopakumar-Vafa prediction for GW-invariants of Fano classes in 6-dimensional symplectic manifolds. The proof of the main theorem uses deformations of the Cauchy-Riemann equation that respect the submanifold and Carleman Similarity Principle for solutions of perturbed Cauchy-Riemann equations. In a forthcoming paper, we apply a similar approach to relative Gromov-Witten invariants and the absolute/relative correspondence in genus $0$.' author: - 'Aleksey Zinger[^1]' title: | A Comparison Theorem for Gromov-Witten Invariants\ in the Symplectic Category --- Introduction {#intro_sec} ============ Gromov-Witten invariants are certain counts of pseudo-holomorphic curves in symplectic manifolds that play prominent roles in symplectic topology, algebraic geometry, and string theory. These are usually rational numbers, and their precise relations with some sort of integer enumerative counts of curves are rarely clear. However, it is well-known that genus $0$ GW-invariants of Fano manifolds are precisely counts of rational curves; this observation is key to enumerating rational curves in projective space in [@KoM Section 5] and [@RT Section 10]. String theory predicts an amazing integral structure for GW-invariants of Calabi-Yau threefolds. These predictions originate in [@AsMo], [@GV1], and [@GV2] and are extended to all threefolds in [@P1].\ GW-invariants of a symplectic manifold $X$ are obtained by evaluating natural coholomogy classes on the virtual fundamental class () of the space of stable $J$-holomorphic maps to $X$. The main statement of this paper, Theorem \[main\_thm\], compares GW-invariants counting stable maps meeting specified constraints in the ambient manifold with analogous counts of such maps to a submanifold containing the images of all such constrained maps to first order. It leads immediately to Corollary \[LeP\_crl\], which in a way is a succinct re-formulation of the main conclusion of [@LeP], and with a bit more work to Theorem \[FanoGV\_thm\], which confirms the “Fano case" of the Gopakumar-Vafa prediction of [@P1 Section 0.2]. Theorem \[main\_thm\] is obtained by deforming the Cauchy-Riemann equation in two stages so that the first stage respects the submanifold. Carleman Similarity Principle is used to take advantage of properties of solutions of Cauchy-Riemann equations that are preserved by a large class of perturbations of the equations. In a forthcoming paper [@divisorGWs], we will apply similar geometric principles to study relative GW-invariants and the absolute/relative correspondence in genus $0$ with applications to birational geometry in the spirit of Hu-Li-Ruan ([@HuLtR], [@HuR], [@LtR]) and McDuff ([@Mc]).\ The author would like to thank R. Pandharipande for bringing the “Fano case" of the Gopakumar-Vafa prediction to the author’s attention, D. McDuff for detailed comments and suggestions on an earlier version of this paper, and T. Graber, T.-J. Li, D. Maulik, and Y. Ruan for related discussions. A comparison theorem for GW-invariants {#AbsGW_subs} -------------------------------------- We will denote by $\bar\Z^+$ the set of non-negative integers. Let $(X,\om)$ be a compact symplectic manifold. If $g\!\in\!\bar\Z^+$, $S$ is a finite set, $\be\!\in\!H_2(X;\Z)$, and $J$ is an $\om$-tame[^2] almost complex structure on $X$, denote by $\ov\M_{g,S}(X,\be;J)$ the moduli space of equivalence classes of stable $S$-marked genus $g$ degree $\be$ $J$-holomorphic maps to $X$. For each $j\!\in\!S$, there is a well-defined evaluation map \_j: \_[g,S]{}(X,;J)X.As standard in GW-theory, we will denote by $$\psi_j\in H^2\big(\ov\M_{g,S}(X,\be;J)\big)$$ the first chern class of the universal cotangent line bundle for the $j$-th marked point. The space $\ov\M_{g,S}(X,\be;J)$ carries a natural VFC, which is independent of $J$ and will be denoted by $[\ov\M_{g,S}(X,\be)]^{vir}$. If the (real) dimension of $X$ is $2n$, then \^[vir]{} =\_[g,S]{}(X,) 2(+(n-3)(1-g)+|S|).If $J$ is regular[^3], then $\ov\M_{0,S}(X,\be;J)$ is a topological manifold with a preferred choice of orientation and $$\big[\ov\M_{0,S}(X,\be)\big]^{vir}=\big[\ov\M_{0,S}(X,\be;J)\big].$$\ If $a_j\!\in\!\bar\Z^+$ and $\ka_j\!\in\!H_*(X;\Z)$ for each $j\!\in\!S$, let ((\_[a\_j]{}\_j)\_[jS]{})\_[g,]{}\^X \_[jS]{} (\_j\^[a\_j]{}\_j\^\*(\_X\_j)),\^[vir]{},where $\PD_X\ka_j\!\in\!H^*(X;\Z)$ is the Poincare dual of $\ka_j$ in $X$.[^4] In order to avoid any sign ambiguities, we define the number in \_ref[GWdfn\_e]{} to be $0$ if the dimension of $\ka_j$ is odd for some $j$. By \_ref[virdim\_e]{}, this number is zero unless \_[jS]{}(2a\_j+2n-\_j)=\_[g,S]{}(X,).The number \_ref[GWdfn\_e]{} can be expressed as an integral on a “smaller" moduli space as follows. Choose cobordism representatives $f_j\!:M_j\!\lra\!X$ for $\ka_j$, with $j\!\in\!S$.[^5] Let \_[g,]{}(X,;J)={(\[u\],(w\_j)\_[jS]{}) \_[g,S]{}(X,;J)\_[jS]{}M\_j: \_j(\[u\])=f\_j(w\_j) jS}.The space $\ov\M_{g,\f}(X,\be;J)$ of constrained stable maps also carries a virtual fundamental class and $$\big((\tau_{a_j}\ka_j)_{j\in S}\big)_{g,\be}^X = \bigg\lan\prod_{j\in S}\psi_j^{a_j},\big[\ov\M_{g,\f}(X,\be;J)\big]^{vir}\bigg\ran.$$ The subject of this section is a reduction of this GW-invariant of $X$ to a combination of GW-invariants for its submanifolds. \[propinter\_dfn\] Let $Y$ be a submanifold of $X$. A smooth map $f\!:M\!\lra\!X$ intersects $Y$ if $f^{-1}(Y)\!\subset\!M$ is a smooth orientable even-dimensional submanifold of $M$ and $$d_wf\big(T_w\big(f^{-1}(Y)\big)\big)= d_w(TM)\cap T_{f(w)}Y$$ for every $w\!\in\!f^{-1}(Y)$. If $f\!:M\!\lra\!X$ intersects $Y\!\subset\!X$ transversally and $M$, $X$, and $Y$ are orientable of even total dimension, then $f$ intersects $Y$ properly. However, a proper intersection need not be transverse. For example, any two real lines in $\R^n$ intersect properly, but not transversally if $n\!\ge\!3$. Two curves that are tangent to each other do not intersect properly.\ If $f\!:M\!\lra\!X$ intersects $Y\!\subset\!X$ properly and $NY\!\lra\!Y$ is the normal bundle of $Y$ in $X$, the homomorphisms $$d_w^{NY}f\!: T_wM\lra N_{f(w)}Y, \quad v\lra d_wf(v)+T_{f(w)}Y, \qquad w\!\in\!f^{-1}(Y),$$ have constant rank; the kernel of $d_w^{NY}f$ is $T_w(f^{-1}(Y))$. If $M$, $X$, and $Y$ are oriented, an orientation on $f^{-1}(Y)$ then induces an orientation on the vector bundle $$N^fY\equiv f^*NY\big/(\Im\, d^{NY}f)\lra f^{-1}(Y).$$ Note that N\^fY=(X-M)-(Y-f\^[-1]{}(Y)).\ Let $Y$ be a compact symplectic submanifold of $X$ and $$\io_{Y*}\!: H_*(Y;\Z)\lra H_*(X;\Z)$$ the homomorphism induced by the inclusion $\io_Y\!:Y\!\lra\!X$. If $\be_Y\!\in\!H_2(Y;\Z)$ and $J$ is an $\om$-tame almost complex structure on $X$ which preserves $TY\!\subset\!TX|_Y$, then $\io_Y$ induces an embedding $$\ov\M_{g,S}(Y,\be_Y;J)\hookrightarrow \ov\M_{g,S}(X,\io_{Y*}\be_Y;J).$$ If $f_j\!:M_j\!\lra\!X$, $j\!\in\!S$, are smooth maps as above, let $$\ov\M_{g,\f}(Y,\be_Y;J)=\big\{\big([u],(w_j)_{j\in S}\big) \in \ov\M_{g,\f}(X,\io_{Y*}\be_Y;J)\!:\, [u]\!\in\!\ov\M_{g,S}(Y,\be_Y;J)\big\}.$$ If in addition $u\!:\Si_u\!\lra\!Y$ is a $J$-holomorphic map from a nodal Riemann surface (see Section \[NRS\_subs\]), let $\H_u$ denote the space of deformations of the complex structure on $\Si_u$. The linearization of the $\dbar_J$-operator for maps to $X$, $$D_{J;u}^X\!:\H_u\oplus L^p_1(\Si_u;u^*TX)\lra L^p(\Si_u;T^*\Si_u^{0,1}\!\otimes\!_{\C}u^*TX), \quad p\!>\!2,$$ induces a generalized Cauchy-Riemann operator $$D_{J;u}^{NY}\!: L^p_1(\Si_u;u^*NY)\lra L^p(\Si_u;T^*\Si_u^{0,1}\!\otimes\!_{\C}u^*NY).$$ For each $j\!\in\!S$, define $$\ti\ev_j\!:\ker D_{J;u}^{NY}\lra N_{z_j(u)}Y \qquad\hbox{by}\quad \xi\lra \xi(z_j(u))+T_{z_j(u)}Y,$$ where $z_j(u)\!\in\!\Si_u$ is the $j$-th marked point; this homomorphism is the composition of the differential of the evaluation map \_ref[evmap\_e]{} with the projection to the normal bundle. \[main\_thm\] Suppose $(X,\om)$ is a compact symplectic $2n$-manifold, $g\!\in\!\bar\Z^+$, $S$ is a finite set, $\be\!\in\!H_2(X;\Z)$, $a_j\!\in\!\bar\Z^+$ for each $j\!\in\!S$, and $f_j\!:M_j\!\lra\!X$ is a cobordism representative for $\ka_j\!\in\!H_*(X;\Z)$ for each $j\!\in\!S$. If $J$ is an $\om$-tame almost complex structure on $X$, $Y$ is a compact almost complex submanifold of $(X,J)$, and $\be_Y\!\in\!H_2(Y;\Z)$ are such that 1. $\io_{Y*}(\be_Y)\!=\!\be$ and $f_j$ intersects $Y$ properly for each $j\!\in\!S$; 2. for every $([u],(w_j)_{j\in S})\!\in\!\ov\M_{g,\f}(Y,\be_Y;J)$, the homomorphism (D\_[J;u]{}\^[NY]{})\_[jS]{}N\_[f\_j(w\_j)]{}\^[f\_j]{}Y, (\_j()+(d\_[w\_j]{}f\_j))\_[jS]{},is an isomorphism, then 1. the space $\ov\M_{g,\f}(Y,\be_Y;J)$ carries a natural VFC (dependent on the orientations of $f_j^{-1}(Y)$) with \^[vir]{} &=\_[g,S]{}(X,)-\_[jS]{}(2n-\_j)\ &+\_[jS]{}N\^[f\_j]{}Y-2(+\_NY(1-g)); 2. the vector spaces $\cok(D_{J;u}^{NY})$ form a natural oriented vector orbi-bundle $$\cok\big(D_J^{NY}\big) \lra \ov\M_{g,\f}(Y,\be_Y;J)$$ with \_ (D\_[J;u]{}\^[NY]{})= \_[jS]{}N\^[f\_j]{}Y-2(+\_NY(1-g)); 3. $\ov\M_{g,\f}(Y,\be_Y;J)$ is a union of connected components of $\ov\M_{g,\f}(X,\be;J)$ and its contribution to the number \_ref[GWdfn\_e]{} is given by \_[g,]{}(Y,\_Y)= e((D\_J\^[NY]{}))\_[jS]{}\_j\^[a\_j]{}, \^[vir]{}.\ \[CY\_eg\] Suppose $(X,J)$ is a Calabi-Yau $3$-fold and $Y\!\subset\!X$ is a smooth isolated rational curve with $NY\!\approx\!\O(-1)\!\oplus\!\O(-1)$. We can then apply Theorem \[main\_thm\] with $S\!=\!\eset$, $g\!=\!0$, and $\be\!=\!d\io_{Y*}([Y])$ for any $d\!\in\!\Z^+$. The assumption on the normal bundle implies that $\ker(D_{J;u}^{NY})$ is trivial and thus Condition (b) is satisfied. The right-hand side of \_ref[mainthm\_e2]{} is then the famous multiple-cover contribution of $1/d^3$ ([@AsMo], [@MirSym Section 27.5], [@Voisin]). \[inY\_eg\] If the image of each map $f_j$ in Theorem 1.2 lies in $Y$, the second part of Condition (a) is automatically satisfied. Condition (b) is equivalent to the homomorphisms $$\bigoplus_{j\in S}\ti\ev_j\!: \ker(D_{J;u}^{NY})\lra \bigoplus_{j\in S}N_{z_j(u)}Y, \qquad ([u],w)\in \ov\M_{g,\f}(X,\be;J),$$ being isomorphisms. For example, this is the case if $X\!=\!\P^n$, $Y\!=\!\P^1\!\subset\!X$, $S\!=\!\{1,2\}$, $g\!=\!0$, $\be\!=\!\io_{Y*}([Y])$ is the homology class of a line, $a_1,a_2\!=\!0$, and $f_1,f_2\!:pt\!\lra\!Y$ are maps to two distinct points. In this particular case, $$\ov\M_{0,\f}(X,\be;J)=\ov\M_{0,\f}(Y,\be_Y;J),$$ where $\be_Y\!=\![Y]$, and $\cok(D_J^{NY})$ is the zero vector bundle. Thus, $$\big(pt,pt\big)_{0,\be}^{\P^n}= \big((\tau_{a_j}\ka_j)_{j\in S}\big)_{0,\be}^X =\bC_{0,\f}(Y,\be_Y) =\,^{\pm}\big|\ov\M_{g,\f}(Y,\be_Y;J)\big| =\big(pt,pt\big)_{0,\be_Y}^{\P^1} =1,$$ as expected.[^6] \[transversetoY\_eg\] If each map $f_j$ in Theorem 1.2 is transverse to $Y$, the second part of Condition (a) is again automatically satisfied. Condition (b) is equivalent to the injectivity of the operators $D_{J;u}^{NY}$ whenever $([u],w)\!\in\!\ov\M_{g,\f}(X,\be;J)$. For example, this is the case if $X$ is the blowup of $\P^n$, with $n\!\ge\!2$, at a point, $Y\!\approx\!\P^{n-1}$ is the exceptional divisor, $S\!=\!\{1,2\}$, $g\!=\!0$, $\be_Y\!\in\!H_2(Y;\Z)$ is the homology class of a line in the exceptional divisor, $\be\!=\!\io_{Y*}(\be_Y)$, $a_1,a_2\!=\!0$, and $f_1,f_2\!:\P^1\!\lra\!X$ are parametrizations of proper transforms of two distinct lines in $\P^n$ passing through the center of the blowup. In this particular case, $$\ov\M_{0,\f}(X,\be;J)=\ov\M_{0,\f}(Y,\be_Y;J)$$ and $\cok(D_J^{NY})$ is the zero vector bundle. Thus, if $\bar\ell$ denotes the homology class of $f_1$ and $f_2$, $$\big(\bar\ell,\bar\ell\big)_{0,\be}^X= \big((\tau_{a_j}\ka_j)_{j\in S}\big)_{0,\be}^X =\bC_{0,\f}(Y,\be_Y) =\,^{\pm}\big|\ov\M_{g,\f}(Y,\be_Y;J)\big| =\big(\bar\ell\cap Y,\bar\ell\cap Y\big)_{0,\be_Y}^Y =1;$$ see Footnote \[inY\_ft\]. Various special cases of Theorem \[main\_thm\], such as those in Examples \[CY\_eg\]-\[transversetoY\_eg\], are standard in the algebraic setting and are used in [@BP], [@KlP], and [@P1], for example. Some special cases of Theorem \[main\_thm\] have appeared in the symplectic setting as well, including in [@LiZ], [@McTo], and [@Taubes]. Examples \[inY\_eg\] and \[transversetoY\_eg\] generalize Example \[CY\_eg\] in two opposite directions. Corollary \[LeP\_crl\] below, which applies this theorem in the setting of [@LeP], is yet another special case of Example \[transversetoY\_eg\]. The full statement of Theorem \[main\_thm\] mixes the two extreme cases of Examples \[inY\_eg\] and \[transversetoY\_eg\].\ The striking conclusion of [@LeP] is that all GW-invariants of a Kahler surface $X$ of general type localize to a canonical divisor. The situation is particularly beautiful if $X$ admits a smooth canonical divisor $\K_X$. If $X$ is minimal, the GW-invariants of $X$ in degrees other than multiples of $\K_X$ vanish. The GW-invariants of $X$ in degrees $\K_X$ and $2\K_X$ are computed in [@KLi] via an algebraic reformulation of [@LeP] and shown to satisfy a conjecture of [@MaP]. In the next paragraph we review the relevant statements from [@LeP].\ Let $(X,J_0)$ be a minimal Kahler surface of general type and $\al$ the real part of a non-zero holomorphic $(2,0)$-form such that $Y\!\equiv\!\al^{-1}(0)$ is smooth (and reduced). Since $X$ is minimal, $Y$ is connected. With $\lr{\cdot,\cdot}$ denoting the Riemannian metric on $X$, define $$\begin{gathered} K_{\al}\in \Ga\big(X;\Hom_{\R}(TX,TX)\big), \quad R_{\al}\in \Ga\big(Y;\Hom_{\R}(TY\!\otimes\!_{\C}NY,NY)\big), \qquad\hbox{by}\notag\\ \label{Rdfn_e}\begin{split} \lr{v_1,K_{\al}v_2}&=\al(v_1,v_2)~~\forall\,v_1,v_2\!\in\!T_xX,\,x\!\in\!X;\\ R_{\al}(v_1,v_2)&= J_0\big\{\na_{v_2}K_{\al}\big\}(v_1)+T_xY ~~\forall\,v_1\!\in\!T_xY,\,v_2\!\in\!T_xX,\,x\!\in\!X. \end{split}\end{gathered}$$ By [@LeP Lemmas 2.1,8.2], $R_{\al}$ is well-defined. The almost complex structure $J_{\al}$ on $X$ described in [@LeP Section 2] agrees with $J_0$ along the smooth complex curve $Y$. By [@LeP Lemma 2.3], every non-constant $J_{\al}$-holomorphic map $u\!:\Si_u\!\lra\!X$ is in fact a $J_0$-holomorphic map to $Y$ and so lies in the homology class $dY$ for some $d\!\in\!\Z^+$. By [@LeP Section 8], the operator on the normal bundle $NY$ of $Y$ induced by the linearization of the $\dbar_{J_{\al}}$-operator for maps to $X$ at such a map $u$ is given by D\_[J\_;u]{}\^[NY]{}=\_[u\^\*NY]{}+R\_(du,): L\^p\_1(\_u;u\^\*NY) L\^p(\_u;T\^\*\_u\^[0,1]{}\_u\^\*NY),where $\dbar_{u^*NY}$ is the $\dbar$-operator in the holomorphic bundle $u^*(NY,J_0)\!\lra\!\Si_u$. By [@LeP Proposition 8.6], $D_{J_{\al};u}^{NY}$ is injective. In light of Theorem \[main\_thm\], Corollary \[LeP\_crl\] below is thus simply a re-formulation of the main conclusion of [@LeP]. \[LeP\_crl\] Suppose $(X,J_0)$ is a minimal Kahler surface of general type, $\al$ is the real part of a non-zero holomorphic $(2,0)$-form such that $Y\!\equiv\!\al^{-1}(0)$ is smooth, $g\!\in\!\bar\Z^+$, $d\!\in\!\Z^+$, $S$ is a finite set, $S_2\!\subset\!S$, $a_j\!\in\!\bar\Z^+$ for each $j\!\in\!S$, and $\ka_j\!\in\!H_2(X;\Z)$ for each $j\!\in\!S_2$. If $R_{\al}$ is defined by \_ref[Rdfn\_e]{}, then the cokernels of the operators \_ref[DSurf\_e]{} form a natural oriented vector orbi-bundle $$\cok\big(D_{\al}^{NY}\big) \lra \ov\M_{g,S}(Y,dY)$$ and $$\begin{split} &\big((\tau_{a_j}\ka_j)_{j\in S_2},(\tau_{a_j}1)_{j\in S-S_2}\big)_{g,d\K_X}^X\\ &\qquad =\bigg(\prod_{j\in S_2}\lr{c_1(T^*X),\ka_j}\bigg) \bigg\lan e\big(\cok(D_{\al}^{NY})\big)\!\prod_{j\in S_2}\!\!\!\big(\ev_j^*PD_Y(pt)\big) \prod_{j\in S}\!\psi_j^{a_j}, \big[\ov\M_{g,S}(Y,dY)\big]^{vir}\bigg\ran. \end{split}$$ The Fano case of the Gopakumar-Vafa prediction {#FanoGV_subs} ---------------------------------------------- GW-invariants are generally not integers. On the other hand, at least in the case of projective $3$-folds (symplectic $6$-manifolds), certain combinations of them are believed to be integers. Ideally these combinations would be precisely counts of curves of fixed genus and degree and passing through appropriate constraints. A projective $3$-fold $X$ is never ideal in this sense, but one might hope that $X$ becomes ideal if its Kahler complex structure is replaced with a generic almost complex one. We show that this is indeed the case in the “Fano" case.\ If $(X,\om)$ is a compact symplectic manifold, $g\!\in\!\bar\Z^+$, $S$ is a finite set, $\be\!\in\!H_2(X;\Z)$, and $J$ is an $\om$-tame almost complex structure on $X$, let $$\M_{g,S}^*(X,\be;J)\subset\ov\M_{g,S}(X,\be;J)$$ denote the subspace consisting of maps, i.e. $J$-holomorphic maps $u\!:\Si_u\!\lra\!X$ such that $\Si_u$ is a smooth (connected) Riemann surface and $u^{-1}(u(z))\!=\!\{z\}$ and $d_zu\!\neq\!0$ for some $z\!\in\!\Si_u$. These conditions imply that $u$ does not factor through a $d$-fold cover $\Si_u\!\lra\!\Si$, with $d\!>\!1$; see [@McS Section 2.5]. If $f_j\!: M_j\!\lra\!X$, $j\!\in\!S$, are smooth maps from compact oriented manifolds of even dimensions, let $$\M_{g,\f}^*(X,\be;J)= \ov\M_{g,\f}(X,\be;J)\cap \bigg(\M_{g,S}^*(X,\be;J)\times \prod_{j\in S}M_j\bigg),$$ with $\ov\M_{g,\f}(X,\be;J)$ defined by \_ref[Mfdfn\_e]{}. If $\M_{g,\f}^*(X,\be;J)$ is a finite set consisting of regular pairs $([u],(w_j)_{j\in S})$, we will denote its signed cardinality by $E_{g,\be}^X(J,\f)$.\ If the (real) dimension of $X$ is $6$, the expected dimension of the moduli space $\ov\M_{g,S}(X,\be;J)$ is independent of the genus $g$; see \_ref[virdim\_e]{}. Thus, one can mix curve counts of different genera passing through the same constraints. Furthermore, if $\be\!\in\!H_2(X;\Z)$ and $\lr{c_1(TX),\be}\!<\!0$, all degree $\be$ GW-invariants are zero, since the moduli space of unmarked maps has negative expected dimension. This leaves the “Calabi-Yau" case, $\lr{c_1(TX),\be}\!=\!0$, and the “Fano" case, $\lr{c_1(TX),\be}\!>\!0$. If $g,h\!\in\!\bar\Z^+$, define $C_{h,\be}^X(g)\!\in\!\Q$ by \_[g=0]{}\^[i]{}C\_[h,]{}\^X(g)t\^[2g]{} =()\^[2h-2+]{}. \[FanoGV\_thm\] Suppose $(X,\om)$ is a compact symplectic $6$-fold, $\be\!\in\!H_2(X;\Z)$, $g\!\in\!\bar\Z^+$, $S$ is a finite set, and $\ka_j\!\in\!H_*(X;\Z)$ for $j\!\in\!S$ are such that \_ref[dimcond\_e]{} is satisfied with $a_j\!=\!0$. If $\lr{c_1(TX),\be}\!>\!0$, 1. there exists a dense open subset $\cJ_{\reg}(g,\be)$ of the space of smooth $\om$-tame almost complex structures on $X$ such that for all $h\!\le\!g$: - the moduli space $\M_{h,S}^*(X,\be;J)$ consists of regular maps; - for a generic choice of pseudocycle representatives $f_j\!:M_j\!\lra\!X$ for $\ka_j$, $\M_{h,\f}^*(X,\be;J)$ is a finite set of regular pairs $([u],(w_j)_{j\in S})$ such that $u$ is an embedding; 2. the numbers $E_{h,\be}^X(\f,J)$, with $h\!\le\!g$, are independent of the choice of $J\!\in\!\cJ_{\reg}(g,\be)$ and $f_j$ and can thus be denoted $E_{h,\be}^X((\ka_j)_{j\in S})$; 3. if $C_{g,\be}^X(h)$ is defined by \_ref[Cdfn\_e]{}, ((\_j)\_[jS]{})\_[g,]{}\^X= \_[h=0]{}\^[h=g]{} C\_[h,]{}\^X(g-h) E\_[h,]{}\^X((\_j)\_[jS]{}).\ For $g\!=\!0,1$, \_ref[FanoGV\_e]{} gives ((\_j)\_[jS]{})\_[0,]{}\^X&=E\_[0,]{}\^X((\_j)\_[jS]{}),\ ((\_j)\_[jS]{})\_[1,]{}\^X&=E\_[1,]{}\^X((\_j)\_[jS]{}) +E\_[0,]{}\^X((\_j)\_[jS]{}). The first identity expresses the well-known fact that the genus $0$ GW-invariants of a Fano manifold are enumerative. The second identity in \_ref[lowgenusGV\_e]{} is the $n\!=\!3$ case of the relation between the standard genus $1$ GW-invariants and the reduced genus $1$ GW-invariants constructed in [@g1comp2] for all symplectic manifolds.\ By the proof of [@McS Theorem 3.1.5], for a generic almost complex structure $J$ on $X$ all moduli spaces $\M_{h,\eset}^*(X,\be';J)$ are smooth and of the expected dimension, $2\lr{c_1(TX),\be'}$. In particular, &lt;0 \_[h,S]{}\^\*(X,’;J),\_[h,S]{}(X,’;J)=.By a similar argument, for a generic $J$ on $X$ the evaluation maps $$\ev_1,\ev_2\!: \M_{g,\{1,2\}}^*(X,\be;J)\lra X$$ are transverse, while the bundle section $$\M_{g,\{1\}}^*(X,\be;J)\lra L_1^*\!\otimes\!\ev_1^*TX, \qquad [u]\lra d_{z_1(u)}u\,,$$ where $L_1\!\lra\!\M_{g,\{1\}}^*(X,\be;J)$ is the universal tangent line bundle at the marked point and $z_1(u)\!\in\!\Si_u$ is the marked point of $u$, is transverse to the zero set. Thus, $$\M_{g,S}^{sing}(X,\be;J)\equiv \big\{[u]\!\in\!\M_{g,S}^*(X,\be;J)\!:~ u~\textnormal{is not an embedding}\big\}$$ is the image of a smooth map from a smooth manifold of (real) dimension two less than the dimension of $\M_{g,S}^*(X,\be;J)$. It follows that for a generic choice of pseudocycle representatives $f_j\!:M_j\!\lra\!X$ for $\ka_j$, $\M_{g,\f}^*(X,\be;J)$ is a $0$-dimensional oriented sub-manifold of $$\big(\M_{g,S}^*(X,\be;J)-\M_{g,S}^{sing}(X,\be;J)\big) \times\prod_{j\in S}\!M_j.$$\ We next show that $\M_{g,\f}^*(X,\be;J)$ is a finite set. If not, there is a sequence $([u_r],(w_{r,j})_{j\in S})$ in $\M_{g,\f}^*(X,\be;J)$ converging to some $$\big([u],(w_j)_{j\in S}\big)\in \ov\M_{g,\f}(X,\be;J)- \M_{g,\f}^*(X,\be;J).$$ The image of $u$ is a connected $J$-holomorphic curve in $X$ of genus $h\!\le\!g$ with $k\!\ge\!1$ irreducible components of degrees $\be_1,\ldots,\be_k\!\in\!H_2(X;\Z)$ such that $$d_1\be_1+\ldots+d_k\be_k=\be \qquad\hbox{for some}\quad d_1,\ldots,d_k\in\Z^+.$$ By \_ref[emptyspace\_e]{}, $\lr{c_1(TX),\be_i}\!\ge\!0$ for all $i\!=\!1,\ldots,k$. Thus, $$\sum_{i=1}^{i=k}\lr{c_1(TX),\be_i}\le \lr{c_1(TX),\be}.$$ The dimension-counting argument of [@McS Section 6.6] then shows that $k\!=\!1$ and $d_1\!=\!1$. It then follows that the image of $u$ is an irreducible $J$-holomorphic curve of degree $\be$ and genus $h\!<\!g$ that meets each of the maps $f_j$ with $j\!\in\!S$.\ While degree $\be$ genus $h\!<\!g$ $J$-holomorphic curves meeting the maps $f_j$ can certainly exist for a generic $J$, they cannot be limits of other degree $\be$ curves meeting the maps $f_j$ by the $\nu_r\!=\!0$ case of Proposition \[horreg\_prp\] for the following reason. If $$\big([u],(w_j)_{j\in S}\big)\in\ov\M_{g,\f}(X,\be;J) -\M_{g,\f}^*(X,\be;J),$$ the domain of $u$ consists of two or more irreducible components. Furthermore, by the previous paragraph, the restriction of $u$ to all components, except for one, is constant; let $u_{\eff}$ denote the of $u$, i.e. the non-constant restriction. The domain $\Si_{u_{\eff}}$ of $u_{\eff}$ is a smooth curve of genus $h\!<\!g$ with distinct points $(z_j(u_{\eff}))_{j\in S}$ that are mapped to $(\ev_j(u))_{j\in S}$ by $u_{\eff}$. Thus, $$\big([u_{\eff}],(w_j)_{j\in S}\big)\in\M_{h,\f}^*(X,\be;J);$$ by the previous paragraph, $u_{\eff}$ is an embedding onto a smooth $J$-holomorphic curve $Y$ of genus $h$ degree $\be$ meeting the maps $f_j$. This implies that removing a node from $\Si_{u_{\eff}}$ disconnects $\Si_u$.[^7] Since the total evaluation map $$\Bev\!\equiv\!\prod_{j\in S}\ev_j: \M_{h,S}^*(X,\be;J)\lra X^S$$ is transverse to $\f$, (D\_[J;u\_]{}\^[NY]{})\_[jS]{}N\_[f\_j(w\_j)]{}\^[f\_j]{}Y, ((z\_j(u\_))+T\_[f\_j(w\_j)]{}Y+(d\_[w\_j]{}f\_j))\_[jS]{},is surjective; see Section \[AbsGW\_subs\] for the notation. Since $u_{\eff}$ is a regular map, $$\begin{split} \dim \ker(D_{J;u_{\eff}}^{NY}) =\ind\big(D_{J;u_{\eff}}^{NY}\big) &=2\big(\lr{c_1(NY),Y}+2(1\!-\!h)\big)=2\lr{c_1(TX),\be}\\ &=\sum_{j\in S}(4\!-\!\dim\,M_j) \le\sum_{j\in S}\dim\,N^{f_j}_{f_j(w)}Y;\end{split}$$ the second-to-last equality holds by \_ref[dimcond\_e]{}. Thus, the homomorphism in \_ref[Ycount\_e]{} is an isomorphism. On the other hand, $D_{J;u}^{NY}$ is the restriction of the operator $\bigoplus_i D_{J;u_i}^{NY}$ to $$L^p_1(\Si_u;u^*NY)\subset\bigoplus_i L^p_1\big(\Si_{u;i};u_i^*NY\big),$$ where $\{\Si_{u;i}\}$ are the irreducible components of $\Si$ and $u_i\!=\!u|_{\Si_{u;i}}$. If $u_i$ is a constant map, then $D_{J;u_i}^{NY}$ is the usual $\dbar$-operator on the space of functions on $\Si_{u_i}$ with values in $N_{u_i(\Si_{u;i})}Y\!\approx\!\C^2$. Since $\Si_u$ is a connected nodal Riemann containing $\Si_{u_{\eff}}$ as a component, $u|_{\Si_{\eff}}\!=\!u_{\eff}$, and $u$ is constant on each of the irreducible components of $\Si_u\!-\!\Si_{u_{\eff}}$, it follows that the projection homomorphism D\_[J;u]{}\^[NY]{}D\_[J;u\_]{}\^[NY]{}, |\_[\_[u\_]{}]{},is an isomorphism. Thus, the homomorphism $$\ker(D_{J;u}^{NY})\lra \bigoplus_{j\in S}N_{f_j(w_j)}^{f_j}Y, \qquad \xi\lra \big(\xi(z_j(u))+T_{f_j(w_j)}Y+(\Im\,d_{w_j}f_j)\big)_{j\in S}\,,$$ is an isomorphism, since the homomorphism \_ref[Ycount\_e]{} is. Therefore, by Proposition \[horreg\_prp\] there is no sequence in $$\ov\M_{g,\f}(X,\be;J)-\ov\M_{g,\f}(Y,[Y];J)\supset \M_{g,\f}^*(X,\be;J)$$ converging to $([u],(w_j)_{j\in S})$.\ We have thus shown that $\M_{g,\f}^*(X,\be;J)$ is a compact oriented $0$-dimensional manifold and its signed cardinality $E_{g,\be}^X(\f,J)$ is well-defined. The independence of $E_{g,\be}^X(\f,J)$ of the choices of $J$ and $f_j$ follows from \_ref[FanoGV\_e]{}, with $E_{h,\be}^X((\ka_j)_{j\in S})$ replaced by $E_{h,\be}^X(\f,J)$. In turn, this identity follows from Theorem \[main\_thm\] and the proof of [@P1 Theorem 3]. Let $Y$ be a degree $\be$ $J$-holomorphic curve of genus $h\!\le\!g$ meeting each $f_j$. By the above, the assumptions of Theorem \[main\_thm\] are satisfied. By definition (see Section \[CR\_subs2\]), the orbi-bundle $\cok(D_J^{NY})$ is dual to the bundle $\ker((D_J^{NY})^*)$ of kernels of the dual operators $(D_J^{NY})^*$. For each $$\big([u],(w_j)_{j\in S}\big)\in\ov\M_{g,\f}(Y,[Y];J) \subset\ov\M_{g,\f}(X,\be;J),$$ the operator $(D_{J;u}^{NY})^*$ is the natural extension of the operator $\bigoplus_i (D_{J;u_i}^{NY})^*$ to $(1,0)$-forms on $\Si_u$ with poles at the nodes such that the residues at each node sum up to $0$. Since $(D_{J;u_{\eff}}^{NY})^*$ is injective by the regularity of $u_{\eff}$, the projection $$\eta\lra \bigoplus_{\Si_{u;i}\neq\Si_{u_{\eff}}}\eta|_{\Si_{u;i}}$$ to the contracted components is injective. Since $(D_{J;u_i}^{NY})^*\!=\!\dbar^*$ if $u_i$ is constant, the image of this homomorphism is determined by $\Si_u$ and is independent of $D_{J;u_{\eff}}^{NY}$ (as long as $D_{J;u_{\eff}}^{NY}$ is surjective). Thus, $\cok(D_J^{NY})$ is isomorphic to the restriction to $\ov\M_{g,\f}(Y,[Y];J)$ of the obstruction bundle in [@P1 Section 3], i.e. the bundle of cokernels of the operators $D_{J;u}^{NY}$ as above, but for a holomorphic vector bundle $NY$. Thus, \_[g,]{}(Y,\_Y) &=e((D\_J\^[NY]{})), \^[vir]{}\ &=C\_[h,]{}\^X(g-h)(\[u\_\],(w\_j)\_[jS]{}) by \_ref[mainthm\_e2]{} and [@P1 Theorem 3]. Since $$\ov\M_{g,\f}(X,\be;J) =\bigsqcup_{h=0}^{h=g}\bigsqcup_{([u],(w_j)_{j\in S})\in\M_{h,\f}^*(X,\be;J)} \!\!\!\!\!\! \ov\M_{g,\f}(\Im\,u,[\Im\, u];J),$$ the identity \_ref[FanoGV\_e]{} follows from \_ref[P2\_e]{}.\ Theorem \[FanoGV\_thm\] confirms (a stronger version of) the Fano case of [@P2 Conjecture 2(i)], i.e. that the numbers $E_{h,\be}^X((\ka_j)_{j\in S})$ [*defined from GW-invariants by \_ref[FanoGV\_e]{}*]{} are integers. The Calabi-Yau case is fundamentally more difficult as it involves multiple covers of curves.[^8] On the other hand, it might be possible to approach [@P2 Conjecture 2(ii)], i.e. that $E_{h,\be}^X((\ka_j)_{j\in S})\!=\!0$ for a fixed $\be$ and all sufficiently large $g$ if $X$ is projective, by studying possible limits of $J_t$-holomorphic curves with $J_t\!\in\!\cJ_{\reg}(g,\be)$ as $J_t$ approaches the standard complex structure on $X\!\subset\!\P^n$ and using the Castelnuovo bound [@ACGH p116].\ An algebro-geometric approach to Theorem \[FanoGV\_thm\] has recently been proposed in [@KKO], at least in the usual, more narrow, meaning of [*Fano*]{} in algebraic geometry. The stable-map style invariants of smooth projective varieties defined in [@KKO] are a priori integers in the case of Fano varieties, just like the numbers $E_{h,\be}^X((\ka_j)_{j\in S})$. In addition, in this Fano case, they are non-negative integers and satisfy the vanishing prediction of [@P2 Conjecture 2(ii)]. However, it remains to be shown that they are related to the GW-invariants in the required way, i.e. as in \_ref[FanoGV\_e]{}. Analytic Preliminaries {#analysis_sec} ====================== In this section, we collect a number of background statements concerning solutions of perturbed Cauchy-Riemann equations. For the rest of the paper, fix a real number $p\!>\!2$. If $\Si$ is a $2$-dimensional manifold, this condition implies that any $L^p_1$-map $\Si\!\lra\!\R$ is continuous and in particular has a well-defined value at each point. Nodal Riemann surfaces {#NRS_subs} ---------------------- Let $(E,\fI)\!\lra\!\Si$ be an $L^p_1$-complex vector bundle over a smooth Riemann surface, i.e. a one-dimensional complex manifold. If $z\!\in\!\Si$ and $$A_z\in\Hom_{\R}(E_z,T_z^*\Si^{0,1}\!\otimes_{\C}\!E_z),$$ we define $$\begin{gathered} A_z^*\in\Hom_{\R}(T_z^*\Si^{1,0}\!\otimes_{\C}\!E_z^*,T_z^*\Si^{1,1}\!\otimes_{\C}\!E_z^*) \qquad\hbox{by}\\ \Re\big(v\wedge (A_z^*w)\big)= \Re\big((A_zv)\wedge w\big) \in \La_{\R}^2(T_z^*\Si) \qquad\forall\, v\!\in\!E_z,\,w\!\in\!T_z^*\Si^{1,0}\!\otimes_{\C}\!E_z^*\,.\end{gathered}$$ Since $\La_{\R}^2(T_z^*\Si)$ is one-dimensional, $A_z^*$ is well-defined. If $$A\in L^p\big(\Si;\Hom_{\R}(E,T^*\Si^{0,1}\!\otimes_{\C}\!E)\big),$$ this construction gives rise to an element $$\begin{gathered} A^*\in L^p\big(\Si;\Hom_{\R}(T^*\Si^{1,0}\!\otimes_{\C}\!E^*, T^*\Si^{1,1}\!\otimes_{\C}\!E^*)\big) \qquad\hbox{s.t.} \notag\\ \label{Aadj_e} \bllrr{\xi,A^*\eta}\equiv \Re\Big(\int_{\Si}\xi\wedge (A^*\eta)\Big) =\Re\Big(\int_{\Si}(A\xi)\wedge\eta\Big) \equiv \bllrr{A\xi,\eta}\end{gathered}$$ for all $\xi\!\in\!L^p_1(\Si;E)$ and $\eta\!\in\!L^p_1(\Si;T^*\Si^{1,0}\!\otimes\!E^*)$.\ Let $E\!\lra\!\Si$ be as above. If $S$ is a finite subset of $\Si$, denote by $$L_k^p\big(\Si;E(S)\big)\subset L^p_{k,loc}(\Si\!-\!S;E)$$ the subspace of sections $\eta$ of $E$ such that for every $z_0\!\in\!S$ there exist a neighborhood $U$ of $z_0$ in $\Si$ and a coordinate $w\!:U\!\lra\!\C$ such that $$w(z_0)=0 \qquad\hbox{and}\qquad w\cdot\eta|_U\in L_k^p(U;E).$$ If $k\!\ge\!1$, an element $\eta$ of $L_k^p(\Si;T^*\Si^{1,0}\!\otimes_{\C}\!E(S))$ has a well-defined residue at $z_0\!\in\!S$ given by $$\Res_{z=z_0}\eta=\xi(z_0)\in E_{z_0} \qquad\hbox{if}\qquad \eta(z)=\frac{dw}{w(z)}\otimes\xi(z)~~\forall~z\!\in\!U,~ \xi\in L_1^p(U;E).\footnotemark$$ If $\vr$ is a function assigning to each element $z_0\!\in\!S$ a real subspace $E_{z_0}'\!\subset\!E_{z_0}$, let $$L_1^p\big(\Si;T^*\Si^{1,0}\!\otimes_{\C}\!E(\vr)\big)=\big\{ \eta\!\in\!L_1^p\big(\Si;E(S)\big)\!: \, \Res_{z=z_0}\eta\!\in\!E_{z_0}'~\forall\,z_0\!\in\!S\big\}.$$\ By a $\Si$ we will mean a compact complex one-dimensional manifold with pairs of distinct points identified. In other words, =/\~, x\_i\^[(1)]{}\~x\_i\^[(2)]{}  i=1,…,m,for some smooth compact Riemann surface $\ti\Si$ and distinct points $x_i^{(1)},x_i^{(2)}\!\in\!\ti\Si$. The quotient map $$\si\!: \ti\Si\lra\Si$$ is determined by $\Si$ up to an isomorphism. We will denote by $$\Si_{\sing}\equiv\big\{\si(x_i^{(1)})\!:\,i\!=\!1,\ldots,m\big\} \subset\Si \qquad\hbox{and}\qquad \ti\Si_{\sing}\equiv\big\{x_i^{(1)},x_i^{(2)}\!:\,i\!=\!1,\ldots,m\big\} \subset\ti\Si$$ the subset of of $\Si$ and its preimage under $\si$, respectively. Let $\Si^*\!\subset\!\Si$ be the subspace of , i.e. the complement of $\Si_{\sing}$.\ If $Y$ is a smooth manifold and $\Si$ is a Riemann surface as above, an is an $L^p_1$-map $$\ti{u}\!:\ti\Si\lra Y \qquad\hbox{s.t.}\quad \ti{u}\big(x_i^{(1)}\big)=\ti{u}\big(x_i^{(2)}\big)~~\forall\,i=1,\ldots,m.$$ By a $E\!\lra\!\Si$, we will mean a topological complex vector bundle such that $\si^*E\!\lra\!\ti\Si$ is an $L^p_1$-complex vector bundle. Let $$\begin{split} L_1^p(\Si;E) &=\big\{\xi\!\in\!L_1^p(\ti\Si;\si^*E\big)\!:\, \xi(x_i^{(1)})\!=\!\xi(x_i^{(2)})~\forall\,i\!=\!1,\ldots,m\big\};\\ L^p\big(\Si;T^*\Si^{0,1}\!\otimes\!_{\C}E\big) &=L^p\big(\ti\Si;T^*\ti\Si^{0,1}\!\otimes\!_{\C}\si^*E\big). \end{split}$$ If $S$ is a finite subset of $\Si^*$, let $\ti{S}\!=\!\si^{-1}(S)$ and define $$\label{dualsheafdfn_e}\begin{split} L_1^p\big(\Si;\K_{\Si}\!\otimes_{\C}\!E(S)\big) &=\Big\{\eta\!\in\! L_1^p\big(\ti\Si;T^*\ti\Si^{1,0}\!\otimes_{\C}\! \si^*E(\ti{S}\!\cup\!\ti\Si_{\sing})\big)\!: \\ &\qquad\qquad\qquad \sum_{\ti{z}_0\in\si^{-1}(z_0)}\!\!\!\!\!\!\Res_{z=\ti{z}_0}\eta(\ti{z}_0)\!=\!0~~\forall\, z_0\!\in\!\Si_{\sing}\Big\},\\ L^p\big(\Si;T^*\Si^{0,1}\!\otimes_{\C}\!\K_{\Si}\!\otimes\!_{\C}E(S)\big) &=L^p\big(\ti\Si;T^*\ti\Si^{0,1}\!\otimes_{\C}\!T^*\ti\Si^{1,0}\!\otimes_{\C}\! \si^*E(\ti{S}\!\cup\!\ti\Si_{\sing})\big). \end{split}$$ If $\vr$ is a function assigning to each element $z_0\!\in\!S$ a real subspace $E_{z_0}'\!\subset\!E_{z_0}$, let L\_1\^p(;\_\_E())={ L\_1\^p(;\_\_E(S)): \_[z=\^[-1]{}(z\_0)]{}E\_[z\_0]{}’ z\_0S}.Similarly, we define $$\begin{split} L_1^p\big(\Si;E(-S)\big) &=\big\{\xi\!\in\!L_1^p\big(\Si;E)\!:\,\xi(z_0)\!=\!0~\forall\,z_0\!\in\!S\big\},\\ L_1^p\big(\Si;E^*(-\vr)\big) &=\big\{\xi\!\in\!L_1^p\big(\Si;E^*)\!:\,\xi(z_0)\!\in\!\Ann(E_{z_0}') ~\forall\,z_0\!\in\!S\big\}, \end{split}$$ where $\Ann(E_{z_0}')\!\subset\!\Hom_{\R}(E_{z_0},\R)$ is the annihilator of $E_{z_0}'\!\subset\!E_{z_0}$. The real pairings in \_ref[Aadj\_e]{} extend to pairings $$\begin{split} L_1^p\big(\Si;E\big)\!\otimes\! L^p\big(\Si;T^*\Si^{0,1}\!\otimes_{\C}\!\K_{\Si}\!\otimes_{\C}\!E^*(S)\big) &\lra\R,\\ L^p\big(\Si;T^*\Si^{0,1}\!\otimes_{\C}\!E\big)\!\otimes\! L_1^p\big(\Si;\K_{\Si}\!\otimes_{\C}\!E^*(S)\big) &\lra\R. \end{split}$$ Furthermore, the equality in \_ref[Aadj\_e]{} holds for all $\eta\in L_1^p\big(\Si;\K_{\Si}\!\otimes_{\C}\!E^*(S)\big)$. Generalized Cauchy-Riemann operators {#CR_subs} ------------------------------------ \[CR\_dfn\] Let $(Y,J)$ be an almost complex manifold and $(N,\fI)\!\lra\!(Y,J)$ a smooth vector bundle. 1. A is a $\C$-linear map $$\dbar\!: \Ga(Y;N)\lra \Ga^{0,1}(Y;N)\equiv \Ga\big(Y;T^*Y^{0,1}\!\otimes\!_{\C}N\big)$$ such that $$\dbar\big(f\xi)=(\dbar{f})\!\otimes\!\xi+f(\dbar\xi) \qquad\forall~f\!\in\!C^{\i}(Y),~\xi\!\in\!\Ga(Y;N).$$ 2. A (or ) on $(N,\fI)$ is a differential operator of the form D=+A: (Y;N)\^[0,1]{}(Y;N),where $\dbar$ is a $\dbar$-operator on $(N,\fI)$ and $$A\in \Ga\big(Y;\Hom_{\R}(N,T^*Y^{0,1}\!\otimes_{\C}\!N)\big).$$\ If $\na$ is an affine connection in $(N,\fI)$, the operator (Y;N)\^[0,1]{}(Y;N), (+J),is a $\dbar$-operator on $(N,\fI)$. Furthermore, any $\C$-linear CR-operator on $(N,\fI)$ is a $\dbar$-operator, and any $\dbar$-operator on $(N,\fI)$ is of the form \_ref[ClinCR\_e0]{} for some (not unique) connection $\na$ in $(N,\fI)$. In particular, $A$ in the decomposition \_ref[CRsplit\_e0]{} can be assumed to be $\C$-anti-linear.\ Let $\na^J$ be the $J$-linear connection in $TY$ obtained from a Levi-Civita connection $\na$ on $Y$ and $A_Y(\cdot,\cdot)$ the of $J$: $$\begin{split} \na^J_{\xi_1}\xi_2&=\frac{1}{2}\Big(\na_{\xi_1}\xi_2-J\na_{\xi_1}(J\xi_2)\Big)\\ A_Y(\xi_1,\xi_2)&=\frac{1}{4}\Big([\xi_1,\xi_2]+J[\xi_1,J\xi_2]+J[J\xi_1,\xi_2] -[J\xi_1,J\xi_2]\Big) \end{split} \qquad\forall\,\xi_1,\xi_2\!\in\!\Ga(Y;TY).$$ We identify $A_Y$ with the element $$A_Y\in \Ga\big(Y;\Hom_{\R}(TY,T^*Y^{0,1}\!\otimes_{\C}\!TY)\big), \qquad v\lra A_Y(\cdot,v).$$ Then, $$\dbar_Y\!\equiv \frac{1}{2}\Big(\na^J\xi+J\na^J\circ J\Big), \, D_Y\!\equiv\! \dbar_Y\!+\!A_Y\!: \Ga(Y;TY)\lra \Ga^{0,1}(Y;TY)$$ are a $\dbar$-operator on $TY$ and a smooth CR-operator on $TY$, respectively. \[CR\_dfn2\] Let $(E,\fI)$ be an $L^p_1$ complex vector bundle over a Riemann surface $(\Si,\fJ)$. 1. A is a $\C$-linear map $$\dbar\!: L^p_1(\Si;E)\lra L^p\big(\Si;T^*\Si^{0,1}\!\otimes\!_{\C}E\big)$$ such that $$\dbar\big(f\xi)=(\dbar{f})\!\otimes\!\xi+f(\dbar\xi) \qquad\forall~f\!\in\!C^{\i}(\Si),~\xi\!\in\!\Ga(\Si;E).$$ 2. A (or ) on $(E,\fI)$ is a differential operator of the form D=+A: L\^p\_1(;E)L\^p(;T\^\*\^[0,1]{}\_E), where $\dbar$ is a $\dbar$-operator on $(E,\fI)$ and AL\^p(;\_(E,T\^\*\^[0,1]{}\_E)).\ If $\na$ is an affine connection in $(E,\fI)$, the operator L\^p\_1(;E)L\^p(;T\^\*\^[0,1]{}\_E), (+),is the usual $\dbar$-operator for a unique holomorphic structure in $(E,\fI)$. Furthermore, any $\C$-linear CR-operator is of the form \_ref[ClinCR\_e]{}.\ If $\Si$ and $N\!\lra\!Y$ are as above, an $L^p_1$-map $u\!:\Si\!\lra\!Y$ pulls back a smooth CR-operator $D$ on $N$ to a CR-operator $D_u$ on $u^*N\!\lra\!\Si$ as follows. Suppose $D$ is presented as in \_ref[CRsplit\_e0]{} with $\C$-anti-linear $A$ and $\na$ is a connection in $(N,\fI)$ inducing the corresponding $\dbar$-operator. Let $\ti{u}\!:\ti\Si\!\lra\!Y$ be the map corresponding to $u$ as in Section \[NRS\_subs\] and $$\ti\na\!: L^p_1(\ti\Si;\ti{u}^*N)\lra L^p(\ti\Si;T^*\ti\Si\!\otimes\!_{\R}\ti{u}^*N)$$ the connection induced by $\na$. Then, $$D_{\ti{u}}=\frac{1}{2}\Big(\ti\na+\fI\ti\na\circ\fJ\Big) +A\circ\partial_J \ti{u}, \qquad\hbox{where}\quad \partial_J \ti{u}=\frac{1}{2}\Big(du-J d\ti{u}\circ \fJ\Big),$$ is a generalized CR-operator on $\ti{u}^*(N,\fI)$; $D_{\ti{u}}$ is independent of the choice of $\na$ if $u$ is $(J,\fJ)$-holomorphic.\ Suppose $(Y,J)$ is an almost complex manifold and $D_Y$ is as above. If $(\Si,\fJ)$ is a Riemann surface and $u\!:\Si\!\lra\!Y$ is a $(J,\fJ)$-holomorphic $L^p_1$-map, then $D_{J;u}\!\equiv\!u^*D_Y$ is the linearization of the $\dbar_J$-operator on the space of $L^p_1$-maps from $\Si$, with complex structure fixed, to $Y$; see [@McS Section 3.1]. If in addition, $(Y,J)$ is an almost complex submanifold of an almost complex manifold $(X,J)$, then $$D_{J;u}\!\equiv\!D_{J;u}^Y\!\equiv\!u^*D_Y\!: L^p_1(\Si;u^*TY)\lra L^p\big(\Si;T^*\Si^{0,1}\!\otimes\!_{\C}u^*TY\big)$$ is the restriction of $$D_{J;u}^X\!\equiv\!u^*D_X\!: L^p_1(\Si;u^*TX)\lra L^p\big(\Si;T^*\Si^{0,1}\!\otimes\!_{\C}u^*TX\big).$$ Thus, $D_{J;u}^X$ induces a CR-operator $$D_{J;u}^{NY}\!: L^p_1(\Si;u^*NY)\lra L^p\big(\Si;T^*\Si^{0,1}\!\otimes\!_{\C}u^*NY\big),$$ where $NY\!\equiv\!TX|_Y/TY$ is the complex normal bundle of $Y$ in $X$.\ The next lemma extends Serre duality from $\dbar$-operators to CR-operators. If $D$ is as in \_ref[CRdfn\_e]{}, let $$D^*=\dbar-A^*\!: L^p_1(\Si;\K_{\Si}\!\otimes\!_{\C}E^*)\lra L^p(\Si;T^*\Si^{0,1}\!\otimes_{\C}\K_{\Si}\!\otimes\!_{\C}E^*);$$ see \_ref[Aadj\_e]{} and \_ref[dualsheafdfn\_e]{} for notation. If $S\!\subset\!\Si$ is a finite subset of smooth points of $\Si$ and $\vr$ is a function assigning to $z_0\!\in\!S$ a complex subspace of $E_{z_0}^*$, $D^*$ extends to an operator $$D_{\vr}^*\!: L^p_1\big(\Si;\K_{\Si}\!\otimes\!_{\C}E^*(\vr)\big)\lra L^p\big(\Si;T^*\Si^{0,1}\!\otimes_{\C}\K_{\Si}\!\otimes\!_{\C}E^*(S)\big);$$ see \_ref[dualsheafdfn\_e2]{}. Let $D_{\vr}$ be the restriction of $D$ to the closed subspace $L^p_1(\Si;E(-\vr))$ of $L^p_1(\Si;E)$. \[serre\_lmm\] Let $D$ be a CR-operator on a complex vector bundle $(E,\fI)$ over a Riemann surface $(\Si,\fJ)$. If $S$ is a finite subset of smooth points of $\Si$ and $\vr$ is a function assigning to $z_0\!\in\!S$ a real subspace of $E_{z_0}^*$, the homomorphism D\_\_(D\_\^\*,), ,is an isomorphism. [*Proof:*]{} If $\Si$ is smooth and $S\!=\!\eset$, this is [@IvSh Lemma 2.3.2]. Furthermore, by the twisting construction of [@Sh Lemma 2.4.1][^9], the elements $z_0$ of $S$ for which $\vr(z_0)\!=\!E_{z_0}^*$ can be omitted from $S$. In the general case, the proof of [@IvSh Lemma 2.3.2] shows that the homomorphisms D\_\^\*\_( D\_,), D\_\_( D\_\^\*,),induced by the pairings \_ref[Aadj\_e]{} are well-defined and injective. It follows that $$\ind\,D_{\vr}+ \ind\,D_{\vr}^*\le0$$ and equality holds if and only if the homomorphisms \_ref[serre\_e1]{} are isomorphisms. On the other hand, if $\ti{D}_{\vr}$ and $\ti{D}_{\vr}^*$ are the operators corresponding to $D_{\vr}$ and $D_{\vr}^*$ over the normalization $\si\!:\ti\Si\!\lra\!\Si$, dropping any matching conditions at the nodes and the other restricting conditions at the points of $S$, then $$\begin{split} \ind\, D_{\vr}&=\ind\,\ti{D}_{\vr}-2k m-\|\vr\|, \\ \ind\, D_{\vr}^*&=\ind\,\ti{D}_{\vr}^*-2k m-2k|S|+\|\vr\|, \end{split}$$ where $k$ is the complex rank of $E$, $m$ is the number of nodes in $\Si$, and $$\|\vr\|=\sum_{z_0\in S}\dim_{\R}\vr(z_0).$$ Since the kernel and cokernel of $\ti{D}_{\vr}^*$ are isomorphic to the kernel and cokernel of a CR-operator on $T^*\ti\Si\!\otimes\!\si^*E^*$ twisted by the preimages of the nodes and the elements of $S$, $$\ind\,\ti{D}_{\vr}^*=-\ind\,\ti{D}_{\vr}+4km+2k|S|.$$ It follows that $\ind\,D_{\vr}^*=-\ind\,D_{\vr}$ and thus the injective homomorphisms in \_ref[serre\_e1]{} are in fact isomorphisms. Families of nodal Riemann surfaces {#RSF_subs} ---------------------------------- By a (), we will mean a topological space $\ov\M$ together with a partition $$\ov\M=\bigsqcup_{l=0}^{l=k}\M^{(l)}$$ such that $\M^{(l)}$ is a smooth manifold of (real) dimension $k\!-\!l$ and $$\ov\M^{(l)}-\M^{(l)}\subset\bigsqcup_{l'=l+1}^{l=k}\!\!\!\M^{(l')}\,.$$ If $U$ is an open subspace of a stratified space $\ov\M$ as above, then $$U=\bigsqcup_{l=0}^{l=k}(\M^{(l)}\!\cap\!U)$$ is also a stratified space. If $\ov\M_1$ and $\ov\M_2$ are stratified spaces, $\ov\M_1\!\times\!\ov\M_2$ is a stratified space with the strata given by unions of the products of the strata of $\ov\M_1$ and $\ov\M_2$. A continuous map $\pi\!:\ov\M_1\!\lra\!\ov\M_2$ between stratified spaces will be called a if the restriction of $\pi$ to each stratum of $\ov\M_1$ is a smooth map to a stratum of $\ov\M_2$. A stratified map $\pi_V\!:V\!\lra\!\ov\M$ will be called a if $\pi_V$ is a topological vector bundle with fiber $\C^k$ and the transition maps from open subsets of $\ov\M$ to $\GL_k\C$ are stratified.\ For the purposes of Definition \[flatfam\_dfn\] below, we set $$\pi_{\std}\!\equiv\!\pi_1: \fU_{\std}\equiv\big\{(t,u,v)\!\in\!\C^3\!:~ uv\!=\!t\big\}\lra\C$$ to be the projection to the first component. This is a stratified map with respect to the stratifications $$\C=\C^*\sqcup\{0\}, \qquad \fU_{\std}=\pi_{\std}^{-1}(\C^*)\sqcup \big(\pi_{\std}^{-1}(0)\!-\!0\big) \sqcup\{0\}.$$ For each $t\!\in\!\C^*$, define $$\rho_t\!:\Si_t\!\equiv\!\pi_{\std}^{-1}(t)\lra\R^+ \qquad\hbox{by}\quad \rho_t(t,u,v)=u^2+v^2\,.$$ If in addition $\ep\!\in\!\R^+$, let $$\Si_{t,\ep}=\big\{(t,u,v)\!\in\!\Si_t\!:\, |u|^2\!+\!|v|^2<\ep\big\}.$$ If $E\!\lra\!\Si_t$ is a normed vector bundle and $\eta\!\in\!L^p(\Si_t;E)$, let $$\|\eta\|_{t,\ep}=\bigg(\int_{\Si_{t,\ep}}|\eta|^p\bigg)^{1/p} +\bigg(\int_{\Si_{t,\ep}}\rho_t^{-\frac{p-2}{p}}|\eta|^2\bigg)^{1/2}\,.$$ \[flatfam\_dfn\] A stratified map $\pi\!:\fU\!\lra\!\ov\M$ is a  if - each fiber $\Si_u\!\equiv\!\pi^{-1}(u)$ is a (possibly nodal) Riemann surface; - if $z_0\!\in\!\Si_{u_0}$ is a smooth point, there are neighborhoods $U_{z_0}$ of $u_0$ in $\ov\M$ and $\ti{U}_{z_0}$ of $z_0$ in $\fU$ and a stratified isomorphism of fiber bundles $$\ti\phi_{z_0}\!:\ti{U}_{z_0}\lra U_{z_0}\!\times\!(\Si_{u_0}\!\cap\!\ti{U}_{z_0})$$ over $U_{z_0}$ such that the restriction of $\ti\phi_{z_0}$ to each fiber of $\pi$ is holomorphic and the restriction of $\ti\phi_{z_0}$ to $\Si_{u_0}\!\cap\!\ti{U}_{z_0}$ is the identity; - if $z_0\!\in\!\Si_{u_0}$ is a node, there are neighborhoods $U_{z_0}$ of $u_0$ in $\ov\M$ and $\ti{U}_{z_0}$ of $z_0$ in $\fU$, a stratified space $U_{z_0}'$, and stratified embeddings $$\phi_{z_0}\!: U_{z_0}\lra U_{z_0}'\times\C \qquad\hbox{and}\qquad \ti\phi_{z_0}\!:\ti{U}_{z_0}\lra U_{z_0}'\times\!\fU_{\std}$$ such that the diagram $$\xymatrix{ \ti{U}_{z_0} \ar[d]^{\pi} \ar[r]^-{\ti\phi_{z_0}}& U_{z_0}'\!\times\!\fU_{\std} \ar[d]^{\id\times\pi_{\std}}\\ U_{z_0} \ar[r]^-{\phi_{z_0}}& U_{z_0}'\!\times\!\C }$$ commutes and the restriction of $\ti\phi_{z_0}$ to each fiber of $\pi$ is holomorphic. \[flatfam\_df2\] If $S$ is a finite set, a stratified map $\pi\!:\fU\!\lra\!\ov\M$ with stratified sections $z_j\!:\ov\M\!\lra\!\fU$, $j\!\in\!S$, is a  if - $\pi\!:\fU\!\lra\!\ov\M$ is a flat stratified family of Riemann surfaces; - $z_j(u)\!\in\!\Si_u$ is a smooth point for every $u\!\in\!\ov\M$ and $j\!\in\!S$; - $z_{j_1}(u)\!\neq\!z_{j_2}(z)$ for every $u\!\in\!\ov\M$, $j_1,j_2\!\in\!S$ with $j_1\!\neq\!j_2$. \[flatfam\_df3\] If $\pi\!:\fU\!\lra\!\ov\M$ is a flat stratified family of $S$-marked Riemann surfaces and $Y$ is a smooth manifold, a continuous map $F\!:\fU\!\lra\!Y$ is a if - for every $u\!\in\!\ov\M$, the restriction of $F$ to $\Si_u\!\equiv\!\pi^{-1}(u)$ is an $L^p_1$-map; - if $z_0\!\in\!\Si_{u_0}$ is a smooth point and $U_{z_0}$, $\ti{U}_{z_0}$, and $\ti\phi_{z_0}$ are as in Definition \[flatfam\_dfn\], there exists a compact neighborhood $K_{z_0}(F)$ of $z_0$ in $\Si_{u_0}\!\cap\!\ti{U}_{z_0}$ such that $F\circ\ti\phi_{z_0}^{-1}|_{u\times K_{z_0}(F)}$ converges to $F|_{K_{z_0}(F)}$ in the $L^p_1$-norm as $u\!\in\!U_{z_0}$ approaches $u_0$; - if $z_0\!\in\!\Si_{u_0}$ is a node and $U_{z_0}$, $\ti{U}_{z_0}$, $\phi_{z_0}$, and $\ti\phi_{z_0}$ are as in Definition \[flatfam\_dfn\], $$\lim_{\ep\lra0}\lim_{\underset{(u',t)\in\phi_{z_0}(U_{z_0})}{(u',t)\lra\phi_{z_0}(u)}} \big\|d(F\circ\ti\phi_{z_0}^{-1}|_{u'\times\Si_t})\big\|_{t,\ep}=0\,.$$ In the case of interest to us, $\ov\M$ will be a family of $S$-marked stable maps to a smooth manifold $Y$. The fiber of $\fU\!\lra\!\ov\M$ over a point $u\!:\Si_u\!\lra\!Y$ will be the Riemann surface $\Si_u$. Families of generalized CR-operators {#CR_subs2} ------------------------------------ Let $D$ be a smooth CR-operator on a vector bundle $(N,\fI)$ over an almost complex manifold $(Y,J)$. Suppose $\fU\!\lra\!\ov\M$ is a flat stratified family of $S$-marked Riemann surfaces, $F\!: \fU\!\lra\!Y$ is a flat family of maps, $S_0\!\subset\!S$, and $\vr$ is a function assigning to each $z_0\!\in\!S_0$ a real subbundle of $\ev_j^*N^*$. For each $u\!\in\!\ov\M$ and $z_0\!\in\!S$, let $\vr_u(z_0)$ be the fiber of $\vr(z_0)$ over $u$. Denote by $\ker_{\vr;u}^F(D)$ and $\ker_{\vr;u}^F(D^*)$ the kernels of the operators $$\begin{split} \big\{\big(F|_{\Si_u}\big)^*D\big\}_{\vr_u}\!:\, & L^p_1\big(\Si_u;\{F|_{\Si_u}^*N\}(-\vr_u)\big) \lra L^p\big(\Si_u;T^*\Si^{0,1}\!\otimes\!_{\C}F|_{\Si_u}^*N\big),\\ \big\{\big(F|_{\Si_u}\big)^*D\big\}_{\vr_u}^*\!:\, & L^p_1\big(\Si_u;\K_{\Si_u}\!\otimes\!_{\C}\{F|_{\Si_u}^*N\}(\vr_u)\big)\\ &\qquad\qquad\qquad \lra L^p\big(\Si_u;T^*\Si_u^{0,1}\!\otimes_{\C}\K_{\Si_u}\!\otimes\!_{\C} \{F|_{\Si_u}^*N\}(\{z_j(u)\}_{j\in S_0})\big), \end{split}$$ respectively.\ We topologize the sets $$\ker_{\vr}^F(D)\equiv\bigsqcup_{u\in\ov\M}\ker_{\vr;u}^F(D) \qquad\hbox{and}\qquad \ker_{\vr}^F(D^*)\equiv\bigsqcup_{u\in\ov\M}\ker_{\vr;u}^F(D^*)$$ by point-wise convergence on compact subsets of the complement of the special (nodal and marked) points of the fiber. In other words, suppose $u_r\!\in\!\ov\M$, $r\!\in\!\Z^+$, is a sequence converging to $u_0\!\in\!\ov\M$ and $\xi_r\!\in\!\ker_{\vr;u_r}^F(D')$ for $r\!\in\!\bar\Z^+$, where $D'\!=\!D,D^*$ and $\bar\Z^+\!=\!\{0\}\!\sqcup\!\Z$. The sequence $\{u_r\}$ if for every smooth point $z_0\!\in\!\Si_{u_0}$, with $z_0\!\neq\!z_j(u)$ for $j\!\in\!S$, there exists a compact neighborhood $K_{z_0}(F)$ as in Definition \[flatfam\_df3\] such that $\xi_r\circ\ti\phi_{z_0}^{-1}|_{u_r \times K_{z_0}(F)}$ converges pointwise to $\xi_0|_{K_{z_0}(F)}$.\ By Carleman Similarity Principle [@FHoS Theorem 2.2], if the restriction of an element $\xi$ of $\ker_{\vr;u}^F(D')$ to an open subset of a component $\Si_{u;i}$ of $\Si_u$ vanishes, then the restriction of $\xi$ to $\Si_{u;i}$ is zero as well. This implies that the above convergence topology on $\ker_{\vr}^F(D)$ is the topology inherited from the convergence topology on the bundle over $\ov\M$ with fibers $L^p_1(\Si_u;u^*N)$ described in [@LiT Section 3].[^10] Furthermore, if the dimension of $\ker_{\vr;u}^F(D)$ is independent of $u$, then $\ker_{\vr}^F(D)\!\lra\!\ov\M$ is a vector bundle. By [@RT Section 6], the analogous statement holds for $\ker_{\vr;u}^F(D^*)$.[^11] Lemma \[serre\_lmm\] then implies that $\ker_{\vr}^F(D^*)\!\lra\!\ov\M$ is a vector bundle if the dimension of $\ker_{\vr;u}^F(D)$ is independent of $u\!\in\!\ov\M$. If in addition, the vector bundles $\ker_{\vr}^F(D)\!\lra\!\ov\M$ and $\vr(z_0)$, $z_0\!\in\!S$, are oriented (and $S$ is ordered if any of the bundles $\vr(z_0)$ is of odd rank), then the vector bundle \_\^F(D\^\*)has a canonical induced orientation, since $\ker_{\vr;u}^F(D)$ and $(\ker_{\vr;u}^F(D^*))^*$ are the kernel and cokernel of an operator obtained by a zeroth-order deformation from a first-order complex-linear Fredholm operator; the determinant line of such an operator has a canonical orientation defined via a homotopy of Fredholm operators (see the proof of [@McS Theorem 3.1.5]). Proof of Theorem \[main\_thm\] {#GW_sec} ============================== The first claim of Theorem \[main\_thm\] is immediate from the assumption that $f_j^{-1}(Y)$ is a smooth oriented manifold. Thus, $$\big[\ov\M_{g,\f}(Y,\be_Y;J)\big]^{vir} =\bigg( \prod_{j\in S}\big\{\ev_j\!\times\!(f_j\!\circ\!\pi_j)\!\big\}^*\big(\PD_{Y^2}(\De_Y)\big) \bigg) \cap \bigg[\ov\M_{g,S}(Y,\be_Y;J)\times\prod_{j\in S}f_j^{-1}(Y)\bigg]^{vir}\,,$$ where $\De_Y\!\subset\!Y^2$ is the diagonal and $\pi_j\!:\prod_{j\in S}f_j^{-1}(Y)\lra f_j^{-1}(Y)$ is the projection onto the $j$-th component; the identity \_ref[Ydim\_e]{} now follows from \_ref[rkNf\_e]{}. Sections \[CR\_subs\] and \[CR\_subs2\] imply the second claim of Theorem \[main\_thm\]. Since the vector spaces ((D\_[J;u]{}\^[NY]{})\^\*)(D\_[J;u]{}\^[NY]{})\^\* have constant rank and are oriented via the isomorphism \_ref[mainthm\_e]{}, they form natural oriented bundles over the uniformizing charts for $\ov\M_{g,\f}(Y,\be_Y;J)$ described in [@LiT Section 3]. These bundles glue together to form an oriented vector orbi-bundle over $\ov\M_{g,\f}(Y,\be_Y;J)$.[^12] In the notation of Sections \[CR\_subs\] and \[CR\_subs2\], this is also the bundle of the cokernels of the injective operators $D_{J,\vr;\u}^{NY}\!\equiv\!(D_{J;u}^{NY})_{\vr}$, where \[u\](\[u\],(w\_j)\_[jS]{})\_[g,]{}(Y,\_Y;J)and $\vr$ is the function assigning to each element $j\!\in\!S$ the subbundle $\Ann(\ev_j^*(\Im\, d^{NY}f_j),\R)$ of $\ev_j^*NY^*$. The identity \_ref[cokrk\_e]{} is immediate from \_ref[mainthm\_e]{} and the Index Theorem. The first part of the third claim follows immediately from Proposition \[horreg\_prp\] below in light of assumption (b) in Theorem \[main\_thm\].\ We note that the second part of the third claim of Theorem \[main\_thm\] is consistent with the divisor relation for GW-invariants [@RT2 (3.4)] in the following sense. Let $$\pi_0\!: \ov\M_{g,\{0\}\sqcup S}(Y,\be_Y;J)\lra \ov\M_{g,S}(Y,\be_Y;J)$$ be the forgetful map dropping the 0-th marked point and $f_0\!:M_0\!\lra\!X$ a cobordism representative for some $\ka_0\!\in\!H_{2n-2}(X;\Z)$ so that $f_0$ is transverse to $Y$; the last assumption implies that $N_{f_0(w_0)}^{f_0}Y\!=\!\{0\}$ for all $w_0\!\in\!f_0^{-1}(Y)$. With $$\begin{split} \ov\M_{g,f_0\sqcup\f}(Y,\be_Y;J) \equiv\big\{\big([u],w_0,(w_j)_{j\in S}\big)\in \ov\M_{g,\{0\}\sqcup S}(Y,\be_Y;J)\!\times\!M_0\!\times\!\prod_{j\in S}\!M_j\!: \qquad\qquad\qquad&\\ \ev_j([u])\!=\!f_j(w_j)~\forall\,j\!\in\!\{0\}\!\sqcup\!S\big\}\,,& \end{split}$$ let $$\ti\pi_0\!:\ov\M_{g,f_0\sqcup\f}(Y,\be_Y;J)\lra \ov\M_{g,\f}(Y,\be_Y;J)$$ be the map induced by $\pi_0$. If $[u]\!\in\!\ov\M_{g,\{0\}\sqcup S}(Y,\be_Y;J)$ and $\pi_0$ contracts component $\Si_{u;i_0}$ of $\Si_u$, then $\Si_{u;i_0}$ is $\P^1$, contains precisely two nodes, say $0$ and $\i$, along with the $0$-th marked point and no other marked points, and $u|_{\Si_{u;i_0}}$ is constant. Therefore, if $[u']\!=\!\pi_0(u)$ and $\chi_u$ is the set of components of $\Si_u$, then the homomorphisms $$\begin{aligned} {2} \label{kerisom_e} &\ker(D_{J;u}^{NY})\lra \ker(D_{J;u'}^{NY}), &\qquad &(\xi_i)_{i\in\chi_u}\lra (\xi_i)_{i\in\chi_u-i_0}\,,\\ \label{cokisom_e} &\ker\big((D_{J;u}^{NY})^*\big)\lra \ker\big((D_{J;u'}^{NY})^*\big), &\qquad &(\eta_i)_{i\in\chi_u}\lra (\eta_i)_{i\in\chi_u-i_0}\,,\end{aligned}$$ are well-defined and are in fact isomorphisms. Since \_ref[kerisom\_e]{} is an isomorphism, $f_0\!\sqcup\!\f$ satisfies the assumptions of Theorem \[main\_thm\] if and only if $\f$ does. Since the total spaces of the cokernel bundles are topologized using convergence of elements of $\ker(D_{J;u}^{NY})^*$ on compact subsets of smooth points, \_ref[cokisom\_e]{} induces an isomorphism of orbi-bundles (D\_J\^[NY]{})\_0\^\*(D\_J\^[NY]{}) over $\ov\M_{g,f_0\sqcup\f}(Y,\be_Y;J)$; it extends over a neighborhood of $\ov\M_{g,f_0\sqcup\f}(Y,\be_Y;J)$ in the space of $L^p_1$-maps via the construction described at the end of Section \[config\_subs\]. Thus, by the standard divisor relation, $$\begin{split} &\bigg\lan e\big(\cok(D_J^{NY})\big)\prod_{j\in S}\psi_j^{a_j}, \big[\ov\M_{g,f_0\sqcup\f}(Y,\be_Y;J)\big]^{vir}\bigg\ran\\ &\hspace{1in} =\blr{\PD_Y\ka_0,\be}\cdot \bigg\lan e\big(\cok(D_J^{NY})\big)\prod_{j\in S}\psi_j^{a_j}, \big[\ov\M_{g,\f}(Y,\be_Y;J)\big]^{vir}\bigg\ran. \end{split}$$ In particular, it is sufficient to verify \_ref[mainthm\_e2]{} under the assumption that $2g\!+\!|S|\!\ge\!3$; this slightly simplifies the presentation.\ For the remainder of the paper, we assume that $2g\!+\!|S|\!\ge\!3$. Section \[config\_subs\] sets up notation for the configuration spaces that play a central role in [@FuO] and [@LiT]. The main geometric observation used in the proof of Theorem \[main\_thm\] is Proposition \[horreg\_prp\], stated and proved in Section \[subman\_subs\]. Our approach to \_ref[mainthm\_e2]{} is illustrated in Section \[semipos\_subs\], where \_ref[mainthm\_e2]{} is verified in some cases, including the case of Theorem \[FanoGV\_thm\]. The general case is the subject of Section \[mainpf\_subs\]. Configuration spaces {#config_subs} -------------------- Let $X$ be a compact manifold, $\be\!\in\!H_2(X;\Z)$, $g$ a non-negative integer, and $S$ a finite set. We denote by $\X_{g,S}(X,\be)$ the space of equivalence classes of stable $L^p_1$-maps $u\!:\Si_u\!\lra\!X$ from genus $g$ Riemann surfaces with $S$-marked points, which may have simple nodes, to $X$ of degree $\be$, i.e. $$u_*[\Si_u]=\be\in H_2(X;\Z).$$ Let $\X_{g,S}^0(X,\be)$ be the subset of $\X_{g,S}(X,\be)$ consisting of the stable maps with smooth domains. The space $\X_{g,S}(X,\be)$ is topologized in [@LiT Section 3] using $L^p_1$-convergence on compact subsets of smooth points of the domain and certain convergence requirements near the nodes. The space $\X_{g,S}(X,\be)$ is stratified by subspaces $\X_{\T}(X)$ of stable maps from domains of the same geometric type and with the same degree distribution between the components of the domain. Each stratum is the quotient of a smooth Banach manifold $\ti\X_{\T}(X)$ by a finite-dimensional Lie group $G_{\T}$; the restriction of the $G_{\T}$-action to any finite-dimensional submanifold of $\ti\X_{\T}(X)$ consisting of smooth maps and preserved by $G_{\T}$ is smooth. The closure of the main stratum, $\X_{g,S}^0(X,\be)$, is $\X_{g,S}(X,\be)$. If $f_j\!:M_j\!\lra\!X$ for $j\!\in\!S$ are smooth maps, let $$\X_{g,\f}(X,\be)=\big\{\big([u],(w_j)_{j\in S}\big) \in \X_{g,S}(X,\be)\times\prod_{j\in S}M_j\!:~ u(z_j(u))\!=\!f_j(w_j)~\forall\,j\!\in\!S\big\}.$$\ If $J$ is an almost complex structure on $X$, let $$\Ga_{g,S}^{0,1}(X,\be;J)\!\lra\!\X_{g,S}(X,\be)$$ be the family of $(TX,J)$-valued $(0,1)$ $L^p$-forms. In other words, the fiber of $\Ga_{g,S}^{0,1}(X,\be;J)$ over a point $[u]$ in $\X_{g,S}(X,\be)$ is the space $$\Ga_{g,S}^{0,1}(X,\be;J)\big|_{[u]}=\Ga^{0,1}(X,u;J)\big/\hbox{Aut}(u), \quad\hbox{where}\quad \Ga^{0,1}(X,u;J)=L^p\big(\Si_u;T^*\Si_u^{0,1}\!\otimes\!_{\C}u^*TX\big).$$ The total space of this family is topologized in [@LiT Section 3] using $L^p$-convergence on compact subsets of smooth points of the domain and certain convergence requirements near the nodes. The restriction of $\Ga_{g,S}^{0,1}(X,\be;J)$ to each stratum $\X_{\T}(X)$ is the quotient of a smooth Banach vector bundle $\ti\Ga_{\T}^{0,1}(X;J)$ over $\ti\X_{\T}(X)$ by $G_{\T}$. The smooth sections of the bundles $\ti\Ga_{\T}^{0,1}(X;J)\lra \ti\X_{\T}(X)$ given by $$\dbar_J\big([\Si_u,\fJ_u;u]\big) = \dbar_{J,\fJ_u}u = \frac{1}{2}\big(du+J\!\circ\!du\!\circ\!\fJ_u\big)$$ induce sections of $\Ga_{g,S}^{0,1}(X,\be;J)$ over $\X_{\T}(X)$, which define a continuous section $\bar\partial_J$ of the family $$\Ga_{g,S}^{0,1}(X,\be;J) \lra \X_{g,S}(X,\be).$$ The zero set of this section is the moduli space $\ov\M_{g,S}(X,\be;J)$ of equivalence classes of stable $J$-holomorphic degree $\be$ maps from genus-$g$ curves with $S$-marked points into $X$. The section $\dbar_J$ over $\ti\X_{\T}(X)$ is Fredholm, i.e. its linearization has finite-dimensional kernel and cokernel at every point of the zero set. The index of the linearization $D_{J;u}$ of $\dbar_J$ at $u\!\in\!\ti\X_{\T}(X)$ such that $$[u]\in\M_{g,S}(X,\be;J)\equiv \ov\M_{g,S}(X,\be;J)\cap \X_{g,S}^0(X,\be)$$ is the expected dimension $\dim_{g,S}(X,\be)$ of the moduli space $\ov\M_{g,S}(X,\be;J)$.\ If $f_j\!:M_j\!\lra\!X$ for $j\!\in\!S$ are smooth maps, $Y\!\subset\!X$ is a submanifold, $\be_Y\!\in\!H_2(X;\Z)$ is such that $\io_{Y*}\be_Y\!=\!\be$, and $\T$ is any combinatorial type of maps to $X$ or $Y$ of degree $\be$ or $\be_Y$, respectively, let $$\begin{split} \X_{g,\f}(X,\be)&=\big\{\big([u],(w_j)_{j\in S}\big)\in \X_{g,S}(X,\be)\!\times\!\prod_{j\in S}\!M_j\!:\, \ev_j([u])\!=\!f_j(w_j)\,\forall\,j\!\in\!S\big\},\\ \X_{g,\f}(Y,\be_Y)&=\X_{g,\f}(X,\be) \cap \bigg(\X_{g,S}(Y,\be_Y)\!\times\!\prod_{j\in S}\!M_j\bigg),\\ \X_{\T,\f}(X)&=\X_{g,\f}(X,\be) \cap \bigg(\X_{\T}(X)\!\times\!\prod_{j\in S}\!M_j\bigg),\\ \X_{\T,\f}(Y)&=\X_{g,\f}(Y,\be_Y)\cap \bigg(\X_{\T}(Y)\!\times\!\prod_{j\in S}\!M_j\bigg). \end{split}$$ With $\pi\!:\X_{g,\f}(X,\be)\!\lra\!\X_{g,S}(X,\be)$ denoting the projection map, let $$\begin{split} \Ga^{0,1}_{g,\f}(X,\be;J)&=\pi^*\Ga^{0,1}_{g,S}(X,\be;J)\lra \X_{g,\f}(X,\be);\\ \Ga^{0,1}_{g,\f}(Y,\be_Y;J)&=\pi^*\Ga^{0,1}_{g,S}(Y,\be_Y;J)\lra \X_{g,\f}(Y,\be_Y). \end{split}$$ With $a_j$, $j\!\in\!S$, as in Theorem \[main\_thm\], let $$\bL_{\a,\f}\equiv\bigoplus_{j\in S}a_j\pi^*L_j^*\lra\X_{g,\f}(X,\be),$$ where $L_j\!\lra\!\X_{g,S}(X,\be)$ is the tautological line bundle for the $j$-th marked point.\ If $J$ is an almost complex structure on $X$ preserving $Y$, let $g_J$ be a $J$-invariant metric on $X$, $\na^J$ the $J$-linear connection of $g_J$ induced by the Levi-Civita connection of $g_J$, $TY^{\v}\!\subset\!TX|_Y$ the $g_J$-orthogonal complement of $TY$, and $\pi^{\h}\!:TX|_Y\!\lra\!TY$ the orthogonal projection map. Define $$\begin{gathered} \ti\na^J\!: \Ga(Y;TX)\lra \Ga(Y;T^*Y\!\otimes_{\R}\!TX) \qquad\hbox{by}\\ \ti\na^J_v(\xi^{\h}+\xi^{\v})= \pi^{\h}\big(\na_v^J\xi^{\h}\big)+\na_v^J\xi^{\v} \qquad\forall~v\!\in\!TY,~\xi^{\h}\!\in\!\Ga(Y;TY),~\xi^{\v}\!\in\!\Ga(Y;TY^{\v}).\end{gathered}$$ This connection in $TX|_Y$ gives rise to a $\C$-linear connection $\na^{\perp}$ on $NY$ and thus to a $\dbar$-operator $\dbar^{\perp}$ on $NY$. Define $$D^{NY}\!:\Ga(Y;NY)\lra \Ga(Y;T^*Y^{0,1}\!\otimes_{\C}\!NY) \qquad\hbox{by}\qquad D^{NY}\xi=\dbar^{\perp}\xi+A_X^{\perp}(\cdot,\xi),$$ where $A_X^{\perp}$ is the composition of the Nijenhuis tensor of $J$ on $X$ with the projection to $NY$. If $[u]\!\in\!\X_{g,S}(Y,\be_Y)$, let $$D^{NY}_{J;u}\!: L^p_1(\Si_u;u^*NY)\lra L^p\big(\Si_u;T^*\Si_u^{0,1}\!\otimes_{\C}\!u^*NY\big)$$ be the pull-back of $D^{NY}$ by $u$ with respect to the connection $\na^{\perp}$ as in Section \[CR\_subs\]. If $[u]$ is an element of $\ov\M_{g,S}(Y,\be_Y;J)$, this definition agrees with the one in Section \[AbsGW\_subs\]. Thus, under the assumptions of Theorem \[main\_thm\], the dimension of $\cok(D^{NY}_{J;u})$ is fixed on a neighborhood of $\ov\M_{g,\f}(Y,\be_Y;J)$ in $\X_{g,\f}(Y,\be_Y)$. By Section \[CR\_subs2\], the vector spaces $\cok(D^{NY}_{J;u})$ form a vector orbi-bundle over such a neighborhood. Symplectic submanifolds and pseudo-holomorphic maps {#subman_subs} --------------------------------------------------- \[Jtub\_dfn\] If $(X,J)$ is an almost complex manifold and $Y\!\subset\!X$ is an almost complex submanifold, a tuple $(\pi_Y\!:U_Y\!\lra\!Y,TU_Y^h)$ is a  if - $U_Y$ is a tubular neighborhood of $Y$ in $X$; - $\pi_Y\!:U_Y\!\lra\!Y$ is a vector bundle such that $\pi_Y|_Y\!=\!\id_Y$ and $\ker d_y\pi_Y$ is a complex subspace of $(T_yX,J)$ for every $y\!\in\!Y$; - $TU_Y^h\!\lra\!U_Y$ is a complex subbundle of $(TU_Y,J)$ such that $d_x\pi_Y\!:TU_Y^h\!\lra\!T_{\pi_Y(x)}Y$ is an isomorphism of real vector spaces for every $x\!\in\!U_Y$ and is the identity for every $x\!\in\!Y$.\ Every embedded almost complex submanifold $Y$ of an almost complex manifold $(X,J)$ admits a $J$-regularized tubular neighborhood. Let $g$ be a $J$-invariant Riemannian metric on $X$ and $\exp^g\!:TX\!\lra\!X$ the exponential map with respect to the Levi-Civita connection of the metric $g$. Identifying $NY$ with the $g$-orthogonal complement of $TY$ in $TX|_Y$, we obtain a smooth map $$\exp^Y\!: NY\lra X$$ by restricting $\exp^g$. Since $Y$ is an embedded submanifold of $X$, there exist tubular neighborhoods $U_Y'$ and $U_Y$ of $Y$ in $NY$ and in $Y$, respectively, such that the map $$\exp\!\equiv\!\exp^Y\big|_{U_Y'}\!: U_Y'\lra U_Y$$ is a diffeomorphism. Furthermore, $\exp|_Y\!=\!\id_Y$ and $d_y\exp\!:T_yNY\lra T_yX$ is $\C$-linear for every $y\!\in\!Y$. Thus, $$\pi_Y=\pi_{NY}\!\circ\!\exp|_{U_Y'}^{\,-1}\!: U_Y\lra Y,$$ where $\pi_{NY}\!: NY\!\lra\!Y$ is the bundle projection map, satisfies the middle condition in Definition \[Jtub\_dfn\]. Furthermore, if $(\ker d\pi_Y)^{\perp}$ is the $g$-orthogonal complement of $\ker d\pi_Y$ in $TU_Y$, $$d_x\pi_Y\!:(\ker d_x\pi_Y)^{\perp}\lra T_{\pi_Y(x)}Y$$ is an isomorphism and induces a complex structure $J_Y$ in the vector bundle $(\ker d\pi_Y)^{\perp}\!\lra\!U_Y$ (which may differ from $J$). Let $$T_xU_Y^h=\big\{v\!-\!JJ_Yv\!: v\!\in\!(\ker d_x\pi_Y)^{\perp}\big\}.$$ Note that $T_xU_Y^h$ is a complex linear subspace of $(T_xU_Y,J_x)$ for each $x\!\in\!U_Y$. Since $(\ker d_y\pi_Y)^{\perp}\!=\!T_yY$ and $J_Y|_y\!=\!J|_{T_yY}$ for every $y\!\in\!Y$, $$d_y\pi_Y=\id\!: T_yU_Y^h\lra T_{\pi_Y(y)}Y$$ for every $y\!\in\!Y$. Thus, $$d_x\pi_Y\!: T_xU_Y^h\lra T_{\pi_Y(x)}Y$$ is an isomorphism for every $x\!\in\!U_Y$ if $U_Y$ is sufficiently small. We conclude that $TU_Y^h$ satisfies the final condition in Definition \[Jtub\_dfn\]. \[horreg\_prp\] Suppose $(X,\om)$ is a compact symplectic manifold, $g\!\in\!\bar\Z^+$, $S$ is a finite set, $\be\!\in\!H_2(X;\Z)$, and $f_j\!:M_j\!\lra\!X$ is a smooth map for each $j\!\in\!S$. Let $J$ be an $\om$-tame almost complex structure on $X$, $Y$ a compact almost complex submanifold of $(X,J)$, and $(\pi_Y\!:U_Y\!\lra\!Y,TU_Y^h)$ a $J$-regularized tubular neighborhood of $Y$ in $X$. If $([u_r],(w_{r,j})_{j\in S})\in\X_{g,\f}(X,\be)$ is a sequence such that $$\begin{gathered} \label{dbarhor_e} u_r(\Si_{u_r})\not\subset Y, \qquad \dbar_Ju_r\big|_{u_r^{-1}(U_Y)}\in L^p\big(u_r^{-1}(U_Y);T^*(u_r^{-1}(U_Y))^{0,1}\!\otimes_{\C}\!u_r^*TU_Y^h\big),\\ \lim_{r\lra\i}\big([u_r],(w_{r,j})_{j\in S})=\big([u],(w_j)_{j\in S}) \in\ov\M_{g,\f}(Y,\be_Y;J)\subset\X_{g,\f}(X,\be) \notag\end{gathered}$$ for some $\be_Y\!\in\!H_2(Y;\Z)$, then $$\exists~~ \xi\!\in\!\ker\,D_{J;u}^{NY}, ~~ v_j\!\in\!T_{w_j}M_j~ \forall\,j\!\in\!S \qquad\hbox{s.t.}\qquad \xi\neq0, \qquad \xi\big(z_j(u)\big)=d_{w_j}f_j(v_j) \quad\forall\,j\!\in\!S.$$\ The rest of this subsection is dedicated to the proof of this proposition by adopting a now-standard rescaling argument. It is sufficient to consider the case $X\!=\!NY$ as smooth manifolds and $\pi_Y\!:NY\!\lra\!Y$ is the bundle projection map. After passing to a subsequence, it can be assumed that the topological types of the domains $\Si_{u_r}$ of $u_r$ are the same (but not necessarily the same as the topological type of $\Si_u$). The desired vector field $\xi$ and tangent vectors $v_j$ will be constructed by re-scaling $u_r$ in the normal direction to $Y$ and then taking the limit.\ For each $j\!\in\!S$, let $N_jY\!\subset\!T_{w_j}M$ be a complement of $T_{w_j}(f_j^{-1}(Y))$ and $$\exp_j\!: T_{w_j}M_j\lra M_j$$ a diffeomorphism onto a neighborhood of $w_j$ in $M_j$ such that $$\exp_j(0)=w_j, \qquad d_0\exp_j=\Id, \qquad \exp_j(v)\in f_j^{-1}(Y)~~\forall\,v\!\in\!T_{w_j}(f_j^{-1}(Y)).$$ For each $r\!\in\!\Z^+$, define $$v_{r,j}^h\oplus v_{r,j}^{\perp}\in T_{w_j}(f_j^{-1}(Y))\oplus N_jY=T_{w_j}M_j \qquad\hbox{by}\quad \exp_j\big(v_{r,j}^h\!+\!v_{r,j}^{\perp}\big)=w_{r,j}\,.$$ Choose metrics on $NY$ and $N_jY$, $j\!\in\!S$. By our assumptions, $$\ep_r\equiv\sup_{z\in\Si_{u_r}}\!\!\!\big|u_r(z)\big|\in\R^+, \quad \lim_{r\lra\i}\!\!\ep_r=0, \quad \lim_{r\lra\i}\!v_{r,j}^h=0~~\forall\,j\!\in\!S, \quad \big|v_{r,j}^{\perp}\big|\le C\ep_r~~\forall\,r\!\in\!\Z^+,\,j\!\in\!S,$$ for some $C\!\in\!\R^+$ independent of $r$ and $j$. By the last condition, for each $j\!\in\!S$ (a subsequence of) the sequence $$\ti{v}_{r,j}^{\perp}=\ep_r^{-1}v_{r,j}^{\perp}, \quad r\!\in\!\Z^+,$$ converges to some $v_j\!\in\!N_jY\!\subset\!T_{w_j}M_j$.\ For each $r\!\in\!\Z^+$, we define $$\begin{aligned} {2} &m_r\!: NY\lra NY &\qquad &\hbox{by}\quad m_r(x)=\ep_r\cdot x;\\ &J_r\in\Ga\big(NY;\Hom(T(NY),T(NY))\big)&\qquad &\hbox{by}\quad J_r|_x= \big\{d_x m_r\big\}^{-1}\circ J_{\ep_rx}\circ d_x m_r;\\ &\ti{u}_r\!: \Si_{u_r}\lra NY&\qquad &\hbox{by}\quad \ti{u}_r(z)=\ep_r^{-1}\cdot u_r(z);\\ &\eta_r\in L^p(\Si_{u_r};T^*\Si_{u_r}^{0,1}\!\otimes\!_{\C}\ti{u}_r^*T(NY))&\qquad &\hbox{by}\quad \eta_r= \big\{d_{u_r(\cdot)}m_r\big\}^{-1}\circ\dbar_Ju_r.\end{aligned}$$ If in addition $j\!\in\!S$, define $\ti{f}_{r,j}\!:T_{w_j}M_j\!\lra\!NY$ by $$\ti{f}_{r,j}\big(v^h+v^{\perp}\big)= \ep_r^{-1}\cdot f_j\big(\exp_j(v^h+\ep_rv^{\perp})\big) \quad\forall\,v^h\!\in\!T_{w_j}(f_j^{-1}(Y)),\,v^{\perp}\!\in\!N_jY.$$ Then, for all $r\!\in\!\Z^+$, \_[J\_r]{}\_r=\_r, \_[z\_[\_r]{}]{}|\_r(z)|=1, \_r(z\_j(u\_r))=\_[r,j]{}(v\_[r,j]{}\^h+\_[r,j]{}\^)   jS.By the following paragraph, the sequence of almost complex structures $J_r$ $C^{\infty}$-converges on compact subsets of $NY$ to an almost complex structure $\ti{J}$ such that $\ti{J}|_{TY}\!=\!J|_{TY}$ and $$\dbar_{\ti{J}}\xi=0 ~~\Llra~~ D_{J;u}^{NY}\xi=0 \qquad\forall~\xi\in\Ga\big(\Si_u;u^*NY).$$ Furthermore, the sequence $\eta_r$ converges to $0$. Thus, by \_ref[rescalecond\_e]{}, $\ti{u}_r$ converges to some $$\begin{gathered} [\ti{u}]\in \ov\M_{g,S}(NY,\be;\ti{J})\subset \X_{g,S}(NY,\be) \qquad\hbox{s.t.}\\ \ti{u}(\Si_{\ti{u}})\not\subset Y, ~~~ \ti{u}(x_j(\ti{u}))=d_{w_j}f_j(v_j)\in N_{f_j(w_j)}Y~~\forall\,j\!\in\!S.\end{gathered}$$ Since we must have $\pi_Y\!\circ\!\ti{u}=\!u$, $\ti{u}$ corresponds to a section $\xi$ of $u^*NY\!\lra\!\Si_u$ as needed.\ It remains to prove the two local claims made above. It is sufficient to assume that $$\pi_Y\!=\!\pi_1\!: NY=Y\!\times\!\C^k\lra Y$$ as vector bundles over $Y$, and there exists $$\begin{gathered} \al\in\Ga(Y\!\times\!\C^k;\Hom_{\R}(\pi_1^*TY,\pi_2^*T\C^k)\big) \qquad\st \notag\\ \al|_{Y\times0}=0, \qquad \label{Jregulr_e2} T_{(y,w)}U_Y^h=\big\{\big(y',\al_{(y,w)}(y')\big)\!: y'\!\in\!T_yY\big\} \qquad \forall~(y,w)\in Y\!\times\!\C^k.\end{gathered}$$ Thus, by assumption on $u_r$, $$\dbar_Ju_r=(\nu^{\h},\al_u\nu^h) \qquad\hbox{for some}\quad \nu^h\in L^p(\Si_{u_r};T^*\Si_{u_r}\!\otimes_{\R}\!u_r^{\h*}TY\big),$$ where $u_r^{\h}=\pi_1\circ u_r$. Let $$J=\left(\begin{array}{cc} J^{\h\h}& J^{\h\v}\\ J^{\v\h}& J^{\v\v}\end{array}\right)\!: TU_Y\!=\!\pi_1^*TY\!\oplus\!\pi_2^*T\C^k\lra \pi_1^*TY\!\oplus\!\pi_2^*T\C^k$$ be the almost complex structure. By Definition \[Jtub\_dfn\], $J^{\h\v}|_{Y\times0}\!=\!0$ and $J^{\v\h}|_{Y\times0}\!=\!0$; we can also assume that $J^{\v\v}|_{Y\times0}\!=\!\fI$ is the standard complex structure on $\C^k$. If $\vec\na$ is the gradient with respect to the standard coordinates on $\C^k$, it follows that $$\begin{gathered} \al_{(y,w)}=\ti\al_{(y,w)}w, \qquad J^{\v\h}_{(y,w)}=\ti{J}^{\v\h}_{(y,w)}w, \qquad J^{\v\v}_{(y,w)}=\fI+\ti{J}^{\v\v}_{(y,w)}w, \qquad\hbox{where}\\ \ti\al_{(y,w)}=\int_0^1\vec\na \al_{(y,tw)}\,dt, \qquad \ti{J}^{\v\h}_{(y,w)}=\int_0^1\vec\na J^{\v\h}_{(y,tw)}\,dt, \qquad \ti{J}^{\v\v}_{(y,w)}=\int_0^1\vec\na J^{\v\v}_{(y,tw)}\,dt.\notag\end{gathered}$$ This gives $$\begin{split} \eta_r&=\left(\begin{array}{c}\nu^{\h}\\ \ep_r^{-1}\{\ti\al_{u_r}u_r\}\nu_r^{\h} \end{array}\right)\lra0\,,\\ J_r|_{(y,w)}&= \left(\begin{array}{cc} J_{(y,\ep_rw)}^{\h\h}& \ep_r J_{(y,\ep_rw)}^{\h\v}\\ \ep_r^{-1} J_{(y,\ep_rw)}^{\v\h}& J_{(y,\ep_rw)}^{\v\v}\end{array}\right) \lra \left(\begin{array}{cc} J_{T_yY}& 0\\ \ti{J}_{(y,0)}^{\v\h}w& \fI\end{array}\right)\equiv\ti{J}_{(y,w)}\,,\\ D_{J;u}\left(\begin{array}{c}\xi^{\h}\\ \xi^{\v}\end{array}\right) &=\left(\begin{array}{c}\dbar\xi^{\h}\\ \dbar\xi^{\v}+\frac{1}{2}\{\ti{J}_{(y,0)}^{\v\h}\xi^{\v}\}du\circ\fJ\end{array}\right); \end{split}$$ the last identity is a special case of [@McS (3.1.4)]. This concludes the proof of Proposition \[horreg\_prp\]. Geometric motivation for \_ref[mainthm\_e2]{} {#semipos_subs} --------------------------------------------- In this section we give a rough argument for \_ref[mainthm\_e2]{} before translating it into the virtual setting of [@FuO] and [@LiT] in Section \[mainpf\_subs\]. As explained at the end of this section, this argument suffices in some cases. We continue with the notation of Theorem \[main\_thm\] and Section \[config\_subs\]. For the remainder of the paper, we assume that \_ref[dimcond\_e]{} holds; otherwise, the left-hand side of \_ref[mainthm\_e]{} vanishes by definition, while the right-hand side vanishes by \_ref[Ydim\_e]{} and \_ref[cokrk\_e]{}. Our assumption implies that \_[g,]{}(Y,\_Y)\^[vir]{} =2\_ja\_j+\_(D\_J\^[NY]{}).We also assume that $a_j\!\ge\!0$ for every $j\!\in\!S$.\ If $\nu$ is a sufficiently small multi-section of $\Ga^{0,1}_{g,\f}(X,\be;J)$ over $\X_{g,\f}(X,\be)$, the space $$\ov\M_{g,\f}(X,\be;J,\nu) =\{\dbar_J\!+\!\nu\}^{-1}(0)\subset \X_{g,\f}(X,\be)$$ is compact, because $\ov\M_{g,\f}(X,\be;J)$ is. If in addition $\nu$ is smooth and generic in the appropriate sense, $\ov\M_{g,\f}(X,\be;J,\nu)$ is stratified by smooth branched orbifolds of even dimensions. If $\vph$ is a multi-section of the orbi-bundle $\bL_{\a,\f}\lra\X_{g,\f}(X,\be)$, let \_[g,]{}\^(X,;J,)=\_[g,]{}(X,;J,)\^[-1]{}(0).If $\nu$ is sufficiently small and generic and $\vph$ is generic, the left-hand side of \_ref[mainthm\_e2]{} is the number of elements of $\ov\M_{g,\f}^{\vph}(X,\be;J,\nu)$ counted with appropriate multiplicities that lie in a small neighborhood of $$\ov\M_{g,\f}^{\vph}(Y,\be_Y;J)\equiv \ov\M_{g,\f}(Y,\be_Y;J)\cap\vph^{-1}(0)$$ in $\X_{g,\f}(X,\be)$.\ In order to verify \_ref[mainthm\_e2]{}, fix a $J$-regularized tubular neighborhood $(\pi_Y\!:U_Y\!\lra\!Y,TU_Y^h)$. We will take $\nu\!=\!\nu_Y\!+\!\nu_X$ so that - for every $\u\!=\!([u],(w_j)_{j\in S})\!\in\!\X_{g,\f}(X,\be)$ with $[u]\!\in\!\X_{g,S}(U_Y,\be_Y)$, $$\nu_Y(\u)\in L^p(\Si_u;T^*\Si_u^{0,1}\!\otimes_{\C}\!TU_Y^h);$$ - $\nu_Y|_{\X_{g,\f}(Y,\be_Y)}$ is generic, so that $\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)$ is stratified by smooth branched manifolds of the expected dimensions and the dimension of the main stratum $$\M_{g,\f}(Y,\be_Y;J,\nu_Y) \equiv \ov\M_{g,\f}(Y,\be_Y;J,\nu_Y) \cap \bigg(\X_{g,S}^0(Y,\be_Y)\times\prod_{j\in S}\!M_j\bigg)$$ is $\dim_{g,\f}(Y,\be_Y)$; - $\nu_X$ is generic and small relative to $\nu_Y$. Using $\pi_Y$, $d\pi_Y|_{TU_Y^h}^{\,-1}$, and a bump function around $Y$ with support in $U_Y$, any section of $$\pi^*\Ga^{0,1}_{g,S}(Y,\be_Y;J)\lra \X_{g,S}(Y,\be_Y)\times\prod_{j\in S}\!M_j$$ can be extended to a section of $\Ga^{0,1}_{g,\f}(X,\be;J)$ over $\X_{g,\f}(X,\be)$ satisfying the middle condition above. In light of Proposition \[horreg\_prp\], the first condition implies that there exists an open neighborhood $\U(\nu_Y)$ of $\ov\M_{g,\f}(Y,\be_Y;J)$ in $\X_{g,\f}(X,\be)$ such that $$\ov\M_{g,\f}(X,\be;J,\nu_Y)\cap \U(\nu_Y) = \ov\M_{g,\f}(Y,\be_Y;J,\nu_Y).$$ In addition, choose a multi-section $\vph$ of the bundle $\bL_{\f,\a}\lra\X_{g,\f}(X,\be)$ so that $\vph$ is transverse to the zero set on every stratum of $\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)$ and every stratum of $\ov\M_{g,\f}(X,\be;J,\nu)$. This implies that the dimension of every stratum of $\ov\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)$ is at most the rank \_ref[cokrk\_e]{} of the bundle $\cok(D_J^{NY})$ over $\ov\M_{g,\f}(Y,\be_Y;J)$ and the equality holds only for the main stratum.\ By the middle assumption on $\nu_Y$ above, for every element $[u]$ of $\ov\M_{g,S}(Y,\be_Y;J,\nu_Y)$ the linearization $$D_{J,\nu_Y;u}^X\!: \H_u\oplus L^p_1(\Si_u;u^*TX)\lra L^p(\Si_u;T^*\Si_u^{0,1}\!\otimes\!_{\C}u^*TX)$$ of the section $\dbar_J\!+\!\nu_Y$ for maps to $X$ restricts to the linearization $$D_{J,\nu_Y;u}^Y\!: \H_u\oplus L^p_1(\Si_u;u^*TY)\lra L^p(\Si_u;T^*\Si_u^{0,1}\!\otimes\!_{\C}u^*TY)$$ of the section $\dbar_J\!+\!\nu_Y$ for maps to $Y$. Thus, $D_{J,\nu_Y;u}^X$ descends to a Fredholm operator $$D_{J,\nu_Y;u}^{NY}\!: L^p_1(\Si_u;u^*NY)\lra L^p(\Si_u;T^*\Si_u^{0,1}\!\otimes\!_{\C}u^*NY).$$ If $\nu_Y$ is sufficiently small, by the last assumption in Theorem \[main\_thm\] the operator $$\begin{split} D_{J,\nu_Y,\vr;\u}^{NY}\!\equiv\!\big(D_{J,\nu_Y;u}^{NY}\big)_{\vr}\!: \big\{\xi\!\in\!L^p_1(\Si_u;u^*NY)\!:\,\xi(z_j(u))\!\in\!\Im\,d_{w_j}^{NY}f_j ~\forall\,j\!\in\!S\big\} \qquad\qquad&\\ \lra L^p(\Si_u;T^*\Si_u^{0,1}\!\otimes\!_{\C}u^*NY)& \end{split}$$ is injective for every $[\u]\!\in\!\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)$ as in \_ref[udfn\_e]{}. Thus, the cokernels of these operators still form an oriented vector orbi-bundle over $\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)$ of rank \_ref[cokrk\_e]{}, which will be denoted by $\cok(D_{J,\nu_Y,\vr}^{NY})$. Furthermore, $\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)$ is compact (because $\ov\M_{g,\f}(Y,\be_Y;J)$ is) and is a union of connected components of $\ov\M_{g,\f}(X,\be;J,\nu_Y)$ by Proposition \[horreg\_prp\].\ The left-hand side of \_ref[mainthm\_e2]{} is the number of elements of $$\ov\M_{g,\f}^{\vph}(X,\be;J,\nu_Y\!+\!\nu_X) \subset \X_{g,S}(X,\be)\times\prod_{j\in S}M_j$$ that lie in a small neighborhood of $\ov\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)$ for any sufficiently small and generic $\nu_X$. The map component of any such element must be of the form $\exp_{u_{\ups}}\!\xi$, where - $([u],(w_j)_{j\in S})\!\in\!\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)$ is an element of a fixed stratum, i.e. the topological structure of $\Si_u$ is fixed; - $\ups$ is a small gluing parameter for $\Si_u$ consisting of the smoothings of the nodes of $\Si_u$; - $u_{\ups}\!:\Si_{u_{\ups}}\!\lra\!Y$ is the approximately $(J,\nu_Y)$-map corresponding to $\ups$ as in [@gluing Section 3]; - $\xi\!\in\!L^p_1(\Si_{\ups};u_{\ups}^*TX)$ is small with respect to the $\|\cdot\|_{\ups,p,1}$-norm of [@LiT Section 3] and satisfies & {\_J+\_Y}u\_ + D\_[J,\_Y;u\_]{}+\_X(u\_)+N\_()=0,\ &(z\_j(u\_))(d\_[w\_j]{}\^[NY]{}f\_j)+T\_[f\_j(w\_j)]{}Y jS, where $N_{\ups}$ is a combination of a term quadratic in $\xi$ and a term which is linear in $\xi$ and $\nu_X$.\ Projecting \_ref[XvsY\_e0]{} to $NY$, we obtain & D\_[J,\_Y;u\_]{}\^[NY]{}+\_X\^(u\_)+N\_\^()=0,\ & L\^p\_1(\_[u\_]{};u\_\^\*NY),(z\_j(u\_))(d\_[w\_j]{}\^[NY]{}f\_j)    jS. This equation has no small solutions in $\vph^{-1}(0)$ away from the subset of elements $$\u\!\equiv\!([u],(w_j)_{j\in S})\in\ov\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)$$ for which $\nu_X^{\perp}(\u)$ lies in the image of $D_{J,\nu_Y,\vr;\u}^{NY}$, i.e. the projection $\bar\nu_X(\u)$ of $\nu_X(\u)$ to $\cok(D_{J,\nu_Y,\vr;\u}^{NY})$ is zero. For dimensional reasons, all zeros of $\bar\nu_X$ lie in the main stratum $$\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)\equiv \ov\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)\cap \M_{g,\f}(Y,\be_Y;J,\nu_Y).$$ Thus, only $\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)$ contributes to the left-hand side in \_ref[mainthm\_e2]{}. In this case equation \_ref[XvsY\_e1]{} no longer involves $\ups$ and thus $u_{\ups}\!=\!u$. Since $\vph$ vanishes transversally on $\M_{g,\f}(Y,\be_Y;J,\nu_Y)$ and $\bar\nu_X$ on $\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)$, the contribution of the main stratum to the left-hand side is the signed cardinality of the oriented zero-dimensional orbifold $$\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)\cap\bar\nu_X^{-1}(0).$$ As $\bar\nu_X$ extends to a continuous multi-section of the orbi-bundle (D\_[J,\_Y,]{}\^[NY]{})\_[g,]{}\^(Y,\_Y;J,\_Y),which is transverse to the zero set over every stratum, the left-hand side of \_ref[mainthm\_e2]{} is the euler class of the bundle \_ref[cokeuler\_e0]{} evaluated on $\ov\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)$. While the operators $D_{J,\nu_Y;\u}^{NY}$ and $D_{J;u}^{NY}$ are not the same, they are homotopic through operators keeping the dimension of the cokernels fixed and thus define orbi-bundles with the same euler class, as needed.\ The above argument requires some notion of smoothness for the strata of $\X_{\T,\f}(X)$ or at least $\X_{\T,\f}(Y)$. If the domain curve $\Si_u$ of $[u]$ with its marked points is stable for every element $([u],(w_j)_{j\in S})$ of $\ov\M_{g,\f}(Y,\be_Y;J)$, then every stratum $\X_{\T,\f}(X)$ meeting $\ov\M_{g,\f}(Y,\be_Y;J)$ is a smooth Banach orbifold. The topological aspects of the resulting setting are sorted out in [@Mc0], and the above argument suffices in such cases. These include the cases of Theorem \[FanoGV\_thm\] (with $2g\!+\!|S|\!\ge\!3$, which can be assumed) and Corollary \[LeP\_crl\] (since the genus of $Y_{\al}$ is positive), but not of Example \[CY\_eg\] or the specific cases of Examples \[inY\_eg\] or \[transversetoY\_eg\].\ In general, $\X_{\T}(X)$ is a subspace of a product of main strata $\X_{g_i,S_i}^0(X,\be_i)$ for some $g_i$, $S_i$, and $\be_i$ and the restriction of $\Ga^{0,1}_{g,S}(X,\be;J)$ is the direct sum of the pull-backs of the corresponding bundles over the components of the product. If for every $([u],(w_j)_{j\in S})\!\in\!\ov\M_{g,\f}(Y,\be_Y;J)$ and every unstable component $\Si_{u;i}$ of $\Si_u$ the restriction of $u$ to $\Si_{u;i}$ is regular in the appropriate sense, then $\nu$ can be taken to be a smooth section of the components of $\Ga^{0,1}_{g,S}(X,\be;J)$ coming from the “stable parts" of $\T$; as in the previous paragraph there is a well-defined notion of smoothness over these components. This is done explicitly in [@RT2 Section 2]. The resulting extension of the previous paragraph then covers the specific cases of Examples \[inY\_eg\] and \[transversetoY\_eg\].\ Finally, for an arbitrary symplectic manifold $(X,\om)$, the “notion" of smoothness is described by introducing smooth finite-dimensional approximations to $\ov\M_{g,S}(X,\be;J)$. This is done in the next section. Virtual setting {#mainpf_subs} --------------- Continuing with the notation of Section \[config\_subs\], we now recall the virtual fundamental class setup of [@FuO] and [@LiT] and then reformulate the argument of Section \[semipos\_subs\] for \_ref[mainthm\_e2]{} in the general case.\ An is a collection $\{(\U_{\al},E_{\al})\}_{\al\in\A}$, where - $\{\U_{\al}\}_{\al\in\A}$ is an open cover of $\ov\M_{g,S}(X,\be;J)$ in $\X_{g,S}(X,\be)$ and $E_{\al}\!\subset\!\Ga_{g,S}^{0,1}(X,\be;J)|_{\U_{\al}}$ is a topological (finite-rank) vector orbi-bundle over $\U_{\al}$; - $\dbar_J^{-1}(E_{\al})$ is a smooth orbifold and $\dbar_J^{-1}(E_{\al})\cap\X_{T}(X)$ is a smooth sub-orbifold of $\dbar_J^{-1}(E_{\al})$ of the codimension corresponding to $\T$ (twice the number of nodes) for every stratum $\X_{T}(X)$; - the restriction of $E_{\al}$ to $\dbar_J^{-1}(E_{\al})$ is a smooth vector orbi-bundle and the restriction of $\dbar_J$ to $\dbar_J^{-1}(E_{\al})$ is a smooth section of $E_{\al}|_{\dbar_J^{-1}(E_{\al})}$; - for every $[u]\!\in\!\ov\M_{g,S}(X,\be;J)\!\cap\!\dbar_J^{-1}(E_{\al})\!\cap\! \dbar_J^{-1}(E_{\al'})$, there exists $\ga\!\in\!\A$ such that $$[u]\in\U_{\ga}\subset\U_{\al}\cap\U_{\al'}\,, \qquad E_{\al},E_{\al'}\big|_{U_{\ga}}\subset E_{\ga}\,,$$ the restrictions of $E_{\al}$ and $E_{\al'}$ to $\dbar_J^{-1}(E_{\ga})\cap\X_{T}(X)$ are smooth orbifold subbundles of the restriction of $E_{\ga}$, and the restriction of $\dbar_J$ to $\dbar_J^{-1}(E_{\ga})\cap\X_{T}(X)$ is transverse to $E_{\al}$ and $E_{\al'}$; - for every $[u]\!\in\!\ov\M_{g,S}(X,\be;J)$, \^[0,1]{}(X,u;J)= {D\_[J;u]{}:L\^p\_1(\_u;u\^\*TX)}+\_|\_u, where $\ti{E}_{\al}|_u\!\subset\!\ti\Ga^{0,1}_{\T}(X;J)|_u$ is the preimage of $E_{\al}|_u$ under the quotient map $$\ti\Ga^{0,1}_{\T}(X;J)|_u\!\lra\!\Ga^{0,1}_{g,S}(X,\be;J)|_{[u]}\,.$$ Such collections $\{(\U_{\al},E_{\al})\}_{\al\in\A}$ are described in [@FuO Section 12] and [@LiT Section 3]. An atlas for $\ov\M_{g,\f}(X,\be;J)$ is defined similarly, with the domain of $D_{J;u}$ in \_ref[chartcond\_e]{} replaced by $$\big\{\xi\!\in\!L^p_1(\Si_u;u^*TX)\!:\, \xi(z_j(u))\in\Im\,d_{w_j}f_j~\forall\,j\!\in\!S\big\}$$ for an element $([u],(w_j)_{j\in S})$ of $\ov\M_{g,\f}(X,\be;J)$. Such an atlas induces a compatible atlas for the total space of the restriction of the bundle $\bL_{\a,\f}$ to $\ov\M_{g,\f}(X,\be;J)$.\ A is a continuous multi-section such that the restriction of $\nu$ to $\dbar_J^{-1}(E_{\al})$ is a smooth section of $E_{\al}$. Similarly, a [multi-section $\vph$ of $\bL_{\a,\f}$ for $\{(\U_{\al},E_{\al})\}_{\al\in\A}$]{} is a continuous multi-section such that the restriction of $\vph$ to $\dbar_J^{-1}(E_{\al})$ is smooth. A multi-section $\nu$ as above is if the restriction of $\nu$ to $\dbar_J^{-1}(E_{\al})\!\cap\!\X_{\T,\f}(X)$ is transverse to the zero set in $E_{\al}$ for every $\al$ and $\T$. If $(\{(\U_{\al},E_{\al})\}_{\al\in A},\nu)$ is regular, $\ov\M_{g,\f}(X,\be;J,\nu)$ is stratified by smooth branched orbifolds of even dimensions. The existence of regular multi-sections for a refinement of a subatlas is the subject of [@FuO Chapter 1] and [@Mc1 Section 4].[^13] If $\nu$ is sufficiently small and regular and $\vph$ is generic, the left-hand side of \_ref[mainthm\_e2]{} is again the weighted number of elements of $$\ov\M_{g,\f}^{\vph}(X,\be;J,\nu)\equiv \ov\M_{g,\f}(X,\be;J,\nu)\cap\vph^{-1}(0)$$ that lie in a small neighborhood of $$\ov\M_{g,\f}^{\vph}(Y,\be_Y;J)\equiv \ov\M_{g,\f}(Y,\be_Y;J)\cap\vph^{-1}(0)$$ in $\X_{g,\f}(X,\be)$.\ By [@FuO Chapter 3] and [@LiT Section 3], pairs $(\U_{Y;\al},E_{Y;\al})$ for an atlas for $$\ov\M_{g,S}(Y,\be_Y;J)\times\prod_{j\in S}\!M_j$$ that restrict to an atlas for $\ov\M_{g,\f}(Y,\be_Y;J)$ can be obtained in the following way. Given $\u\!=\!([u],(w_j)_{j\in S})$, choose - a neighborhood $V_{Y;u}$ of $u(\Si_u)$ in $Y$; - a representative $u\!:\Si_u\!\lra\!Y$ for $[u]$; - universal family of deformations $\W_u\!\lra\!\De_u$ of $\Si_u$ with its marked points (thus $\Si_u\!\subset\!\W_u$); - a finite-dimensional subspace $$\cE_{Y;\u}\subset \Ga_c\big(\W_u^*\!\times\!V_{Y;u}; \pi_1^*(T^*\W_u^v)^{0,1}\!\otimes_{\C}\!\pi_2^*TY\big),$$ where $\W_u^*\!\subset\!\W_u$ is the subspace of smooth points of the fibers, $T\W_u^v\!\subset\!T\W_u$ is the vertical tangent space, and $\Ga_c$ denotes the space of smooth compactly supported bundle sections, such that $$\Ga(\Si_u;T^*\Si_u^{0,1}\!\otimes_{\C}\!u^*TY)= \big\{D_u\xi\!: \xi\!\in\!\Ga(\Si_u;u^*TY),\,\xi(z_i(u))\!\in\!\Im\,d_{w_j}f_j\, \forall\,j\!\in\!S\big\} +\{\id\!\times\!u\}^*\cE_{Y;\u}\,$$ if $\u\!\in\!\ov\M_{g,\f}(Y,\be_Y;J)$; if $\u\!\not\in\!\ov\M_{g,\f}(Y,\be_Y;J)$, the point-wise condition on $\xi$ is omitted. If $\u'\!=\!([u'],(w_j')_{j\in S})$ with $[u']\!\in\!\X_{g,S}(V_{Y;u},\be_Y)$ and $\Si_{u'}\!\in\!\De_u$, let $$\ti{E}_{Y;\u}|_{\u'}=\{\id\!\times\!u'\}^*\cE_{Y;\u}\,.$$ By [@FuO Chapter 3] and [@LiT Section 3], $\U_{Y;\al}$ can be taken to be the image of a sufficiently small neighborhood $\ti\U_{Y;\al}$ of $\u$ in the space of $L^p_1$-maps from the fibers of $\W_u\!\lra\!\De_u$ to $X$ under the equivalence relation and $E_{Y;\al}$ the image of the bundle formed by the spaces $\ti{E}_{Y;\u}|_{\u'}$ over $\ti\U_{Y;\al}$. With these choices, $\dbar_J^{-1}(E_{Y;\al})$ consists of equivalence classes of smooth maps to $Y$.\ Fix a $J$-regularized tubular neighborhood $(\pi_Y\!:U_Y\!\lra\!Y,TU_Y^h)$ of $Y$ in $X$. Using $\pi_Y$ and $d\pi_Y|_{TU_Y^h}^{-1}$, each $\cE_{Y;\u}$ can be extended to a finite-dimensional subspace $$\cE_{X|Y;\u}\subset \Ga_c\big(\W_u^*\!\times\!V_{X;u}; \pi_1^*(T^*\W_u^v)^{0,1}\!\otimes_{\C}\!\pi_2^*TU_Y^h\big) \subset \Ga_c\big(\W_u^*\!\times\!V_{X;u}; \pi_1^*(T^*\W_u^v)^{0,1}\!\otimes_{\C}\!\pi_2^*TX\big)$$ for a neighborhood $V_{X;u}$ of $V_{Y;u}$ in $U_Y\!\subset\!Y$. A larger subspace $$\cE_{X;\u} \subset \Ga_c\big(\W_u^*\!\times\!V_{X;u}; \pi_1^*(T^*\W_u^v)^{0,1}\!\otimes_{\C}\!\pi_2^*TX\big)$$ can then be chosen so that $$\Ga(\Si_u;T^*\Si_u^{0,1}\!\otimes_{\C}\!u^*TX)= \big\{D_u\xi\!: \xi\!\in\!\Ga(\Si_u;u^*TX),\,\xi(z_i(u))\!\in\!\Im\,d_{w_j}f_j\, \forall\,j\!\in\!S\big\} +\{\id\!\times\!u\}^*\cE_{X;\u}\,,$$ whenever $[u]\!\in\!\ov\M_{g,\f}(Y,\be_Y;J)$. This gives rise to a pair $(\U_{X;\al},E_{X;\al})$ for an atlas for $\ov\M_{g,\f}(X,\be;J)$; the union of such pairs covers $\ov\M_{g,\f}(Y,\be_Y;J)$. Since $\ov\M_{g,\f}(Y,\be_Y;J)$ is a union of components of $\ov\M_{g,\f}(X,\be;J)$, this sub-collection of an atlas is sufficient for determining the left-hand side of \_ref[mainthm\_e2]{}. Similarly, using $\pi_Y$, $d\pi_Y|_{TU_Y^h}^{-1}$, and a bump function around $Y$ with support in $U_Y$, any multi-section of $$\pi_1^*\Ga_{g,S}^{0,1}(Y,\be_Y;J)\lra \X_{g,S}(Y,\be_Y) \times\prod_{j\in S}\!M_j$$ for the atlas $(\{(\U_{Y;\al},E_{Y;\al})\}_{\al\in\A})$ gives rise to a multi-section $\nu$ of $$\Ga_{g,\f}^{0,1}(X,\be;J)\lra \X_{g,\f}(X,\be)$$ for the atlas $(\{(\U_{X;\al},E_{X;\al})\}_{\al\in\A})$ such that for every element $[\u]\!\in\!\X_{g,\f}(X,\be)$ $$\nu([\u])\in L^p\big(\Si_u;T^*\Si_u^{0,1}\!\otimes\!u^*TU_Y^h\big)$$ for every $[\u]\!=\!([u],(w_j)_{j\in S})\!\in\!\X_{g,\f}(U_Y,\be_Y)$.\ Let $\nu\!=\!\nu_Y\!+\!\nu_X$ be a regular multi-section of $\Ga^{0,1}_{g,\f}(X,\be)$ for atlas for $\ov\M_{g,\f}(X,\be;J)$ as above so that - for every $\u\!=\!([u],(w_j)_{j\in S})\!\in\!\X_{g,\f}(X,\be)$ with $[u]\!\in\!\X_{g,S}(U_Y,\be_Y)$, $$\nu_Y(\u)\in L^p(\Si_u;T^*\Si_u^{0,1}\!\otimes_{\C}\!u^*TU_Y^h);$$ - $\nu_Y|_{\X_{g,\f}(Y,\be_Y)}$ is a regular multi-section of $\Ga^{0,1}_{g,\f}(Y,\be_Y)$ so that $\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)$ is stratified by smooth branched orbifolds of the expected dimensions and the dimension of the main stratum $$\M_{g,\f}(Y,\be_Y;J,\nu_Y) \equiv \ov\M_{g,\f}(Y,\be_Y;J,\nu_Y) \cap \bigg(\X_{g,S}^0(Y,\be_Y)\times\prod_{j\in S}\!M_j\bigg)$$ is $\dim_{g,\f}(Y,\be_Y)$; - $\nu_X$ is small relative to $\nu_Y$. The previous paragraph implies that such multi-sections $\nu_Y$ exist. By Proposition \[horreg\_prp\], the first condition implies that there exists an open neighborhood $\U(\nu_Y)$ of $\ov\M_{g,\f}(Y,\be_Y;J)$ in $\X_{g,\f}(X,\be)$ such that $$\ov\M_{g,\f}(X,\be;J,\nu_Y)\cap \U(\nu_Y) = \ov\M_{g,\f}(Y,\be_Y;J,\nu_Y).$$ In addition, choose a multi-section $\vph$ of the bundle $\bL_{\f,\a}\lra\X_{g,\f}(X,\be)$ for the above atlas so that $\vph$ is transverse to the zero set on every stratum of $\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)$ and every stratum of $\ov\M_{g,\f}(X,\be;J,\nu)$.\ For each $\al\!\in\!\A$ and $\u\!\in\!\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)\!\cap\!\U_{Y;\al}$, let $$\cD_{\nu_Y,\al;\u}\!: T_{\u}\dbar_J^{-1}(E_{X;\al}) \lra E_{X;\al}$$ be the linearization of the section $\dbar_J\!+\!\nu_Y$ over $\dbar_J^{-1}(E_{X;\al})$ along the zero set. The kernel of $\cD_{\nu_Y,\al;\u}$ is the tangent space of $\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)$ at $\u$. If $\al$ and $\ga$ are as in the overlap condition in the definition of an atlas above, then $$\begin{gathered} E_{X;\al}\cap \Im\,\cD_{\nu_Y,\ga;\u}=\Im\,\cD_{\nu_Y,\al;\u} \quad\forall\,\u\in\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)\cap\U_{Y;\ga}\,,\\ \dim\dbar_J^{-1}(E_{X;\ga})-\dim\dbar_J^{-1}(E_{X;\al}) =\rk\, E_{X;\ga}-\rk\, E_{X;\al}.\end{gathered}$$ Thus, the inclusion $T\dbar_J^{-1}(E_{X;\al})\lra T\dbar_J^{-1}(E_{X;\ga})$ over $\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)\!\cap\!\U_{Y;\ga}$ induces isomorphisms $$\cok(\cD_{\nu_Y,\al;\u})\lra \cok(\cD_{\nu_Y,\ga;\u}).$$ It follows that these vector spaces form an orbi-bundle $\cok(\cD_{\nu_Y})$ over $\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)$. By the last requirement in the definition of an atlas and condition (b) in Theorem \[main\_thm\], the homomorphism $$\cok(\cD_{\nu_Y,\al;\u})\lra\cok(D_{J;\u}^{NY})$$ induced by the inclusion $E_{X;\al}\lra\Ga^{0,1}_{g,\f}(X,\be;J)$ followed by the projections to $NY$ and the cokernel is surjective for all $\u\!\in\!\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)\!\cap\!\U_{Y;\al}$, if $\nu_Y$ is sufficiently small. A dimension count then shows that this homomorphism is an isomorphism (the injectivity also follows from Proposition \[horreg\_prp\]). Thus, the orbi-bundles $$\cok(\cD_{\nu_Y}),\cok(D_J^{NY})\lra\ov\M_{g,\f}(Y,\be_Y;J,\nu_Y)$$ are isomorphic.\ The left-hand side of \_ref[mainthm\_e2]{} is the number of elements of $$\ov\M_{g,\f}^{\vph}(X,\be;J,\nu_Y\!+\!\nu_X) \subset \X_{g,S}(X,\be)\times\prod_{j\in S}\!M_j$$ that lie in a small neighborhood of $\ov\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)$ for a small generic multi-section $\nu_X$. The number of such elements near $\ov\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)\!\cap\!\U_{Y;\al}$ is the number of solutions of $$\cD_{\nu_Y,\al;\u}\xi+\nu_X(\u)+N_{\al}(\xi)=0, \qquad \xi\in T_{\u}\dbar_J^{-1}(E_{X;\al}),$$ with small $\xi$, where $N_{\al}$ is a combination of a term quadratic in $\xi$ and a term which is linear in $\xi$ and $\nu_X$. This equation has no solutions in $\vph^{-1}(0)$ away from the subset of elements $$\u\in\ov\M_{g,\f}^{\vph}(Y,\be_Y;J,\nu_Y)$$ for which $\nu_X(\u)$ lies in the image of $\cD_{\nu_Y,\al;\u}$, i.e. the projection $\bar\nu_X(\u)$ to $\cok(\cD_{\nu_Y,\al;\u})$ is zero. 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[^1]: Partially supported by a Sloan fellowship and DMS Grant 0604874 [^2]: an almost complex structure on $(X,\om)$ is if $\om(v,Jv)>0$ for all $v\!\in\!TX$ with $v\!\neq\!0$ [^3]: an almost complex structure $J$ is if for every $J$-holomorphic map $u\!:\Si\!\lra\!X$, where $\Si$ is a tree of Riemann spheres, the linearization $D_{J;u}$ of the $\dbar_J$-operator at $u$ is surjective [^4]: In the descriptions of Sections \[semipos\_subs\] and \[mainpf\_subs\], $\big[\ov\M_{g,S}(X,\be)\big]^{vir}$ is a homology class in an arbitrarily small neighborhood of $\ov\M_{g,S}(X,\be;J)$ in the space of equivalence classes of $L^p_1$-maps to $X$; there are well-defined evaluation maps $\ev_j$ and cohomology classes $\psi_j$ on this space as well. [^5]: We can assume that this is possible, since each $\ka_j$ can be replaced by a multiple for our purposes. [^6]: \[inY\_ft\]This is the number of lines through $2$ points in $\P^n$. In this particular case, each operator $D_{J;u}^{NY}$ is $\C$-linear and its zero-dimensional kernel is positively oriented. In general, this need not be the case; see [@LeP Sections 9,10] for explicit sign computations. [^7]: This observation implies that the homomorphism \_ref[kerrestr\_e]{} is surjective. [^8]: Theorem \[FanoGV\_thm\] and its proof also apply to the cases when $\lr{c_1(TX),\be}\!=\!0$, but $\be$ is not a non-trivial integer multiple of another element of $H_2(X;\Z)$. [^9]: This construction extends the usual procedure of twisting a holomorphic vector bundle by a divisor to generalized CR-operators; it can be seen as a manifestation of Carleman Similarity Principle [@FHoS Theorem 2.2]. [^10]: While [@LiT Section 3] concerns only the case $N\!=\!TY$, it applies to any vector bundle $N\!\lra\!Y$. [^11]: While [@RT Section 6] concerns only the case $N\!=\!TY$ and $S_0\!=\!\eset$, the argument applies to any vector bundle $N\!\lra\!Y$. Furthermore, the twisting construction of [@Sh Lemma 2.4.1] reduces the situation to the case $S_0\!=\!\eset$. By [@Si Chapter 4], which builds on [@SeSi], there are Fredholm operators defining these vector spaces that form a continuous family over $\ov\M$ and thus define a K-theory class; however, this statement is stronger than needed here. [^12]: Neither the topologies of the bundles over the uniformizing charts nor the isomorphisms \_ref[kercok\_e]{} depend on the Riemannian metrics over the uniformizing charts of [@LiT Section 3]. [^13]: It is also shown in [@FuO] and [@Mc1] that a regular multi-section $\nu$ determines a rational homology class; however, this notion of virtual fundamental class is not necessary for defining GW-invariants or comparing the two sides of \_ref[mainthm\_e2]{}.

--- author: - 'M. K. Volkov' - 'A. A. Pivovarov' - 'A. A Osipov' date: '30 January 2018 / Revised version: date' title: '$\tau \to f_1 (1285)\pi^{-} \nu_{\tau}$ decay in the extended Nambu – Jona-Lasinio model' --- [leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore Introduction {#intro} ============ The recent measurements of the branching fractions of three-prong $\tau$ decay modes made by BABAR Collaboration [@Lees12] contain important new results on the decay $\tau \to f_1(1285) \pi^{-} \nu_{\tau}$ providing confidence that the precise information on the corresponding invariant mass distributions will also soon become available. Together with the large data sets obtained on this mode in the past by BABAR [@Aubert08; @Aubert05] and CLEO [@Bergfeld97] Collaborations this calls for improved understanding of the theoretical description of the process. The main purpose of our paper is to make a step in this direction. The $\tau \to f_1\,\pi^{-} \nu_{\tau}$ decay is driven by the hadronization of the QCD axial-vector currents involved. The details of this mechanism are not yet clearly understood due to poor knowledge of the QCD dynamics at low-energies. Indeed, the invariant mass of the $(f_1, \pi )$-system belongs to the interval $m_{f_1}+m_\pi\leq \sqrt{s}\leq m_\tau$, so it is too low to apply the QCD perturbation theory, but it is too large that the original chiral perturbation theory (ChPT) of QCD would be applicable [@Weinberg79; @Gasser84; @Gasser85a; @Gasser85b]. In addition, an order of magnitude of energies involved seems to indicate that not only the ground axial-vector $a_1(1260)$ state contributes to the pertinent hadronic axial-vector current. It is not excluded that the first radial excitation, $a_1(1640)$, of the ground resonance state may be also important. We have found that both resonances affect the form of the spectral function, giving an interesting interference picture. Presently there is no clear understanding of the nature of the $a_1(1260)$ and $f_1(1285)$ mesons. In our work we hold the view that these resonances are the standard quark-antiquark bound states. Our reasoning is based on the large-$N_c$ expansion of QCD [@Hooft74; @Witten79], indicating that at $N_c=\infty$ mesons are pure $q\bar q$ states. Phenomenologically these states can be described by the local effective meson Lagrangians [@Schwinger67; @Wess67; @Weinberg68; @Gasiorowicz69]. These Lagrangians are not known from first principles, however, they basically can be constructed on the chiral symmetry grounds. Comprehensive reviews of such attempts can be found in [@Meissner88; @Bando88]. The general problem of including these states in ChPT has been addressed in [@Ecker89; @Pich89]. Notice, that there is a different interpretation of $a_1(1260)$ and $f_1(1285)$. Assuming that these resonances not belong to the large-$N_c$ ground state of QCD (i.e. the $qq\bar q\bar q$ states), one can generate them in ChPT by implementing unitarity in coupled-channels (see, for instance, [@Lutz04; @Roca05; @Zhou14; @Xie15] and references therein). This approach can provide an alternative platform for studying of the $\tau\to f_1\pi^-\nu_\tau$ decay. Over last years, not much has been done with respect to the theoretical study of this particular mode of the $\tau$-decay. One can indicate just a few attempts. This is an approach based on a meson dominance model considered in [@Calderon13], which leads to a strong disagreement with the data of BABAR Collaboration, namely the calculated branching ratio Br$(\tau \to f_1\, \pi^{-} \nu_{\tau})=1.3\times 10^{-4}$ is three times less of the experimental value Br$(\tau \to f_1\, \pi^{-} \nu_{\tau})=(3.9\pm 0.5)\times 10^{-4}$ reported by the Particle Data Group [@Patrignani16]. More encouraging but relatively old result, Br$(\tau \to f_1\, \pi^{-} \nu_{\tau})=2.91\times 10^{-4}$, gives the model [@Li97] based on the hypothesis of $a_1(1260)$-meson dominance. Both ideas are naturally realized under the framework of the Nambu – Jona-Lasinio (NJL) model [@Eguchi76; @Volkov82; @Ebert83; @Volkov84; @Volkov86; @Ebert86; @Vogl91; @Klevansky92; @Volkov93; @Bijnens93; @Hatsuda94; @Volkov94; @Osipov96]. The first attempt to apply this model to describe the hadron part of the $\tau \to f_1 (1285) \pi^{-} \nu_{\tau}$ decay is presented in [@Vishneva14]. Though an analysis made there allows to reproduce the experimental value of branching ratio, the picture presented cannot be considered as a fully satisfactory description. Here we improve it on the following aspects. First, as opposed to [@Vishneva14] we demonstrate that the pseudoscalar channel does not give contribution to the decay $\tau \to f_1 \pi^{-} \nu_{\tau}$. The latter is a direct consequence of the anomaly structure of the corresponding quark triangle diagram. Indeed, for the pseudoscalar-exchange channel the hadron axial-vector current, $J_\mu^A$, is proportional to a gradient of the pion field: $J_\mu^A\propto\partial_\mu\pi$. If the pion turns into the $f_1\pi$ couple, the corresponding $f_1\pi\pi$ vertex vanishes because it has an anomaly structure $e_{\mu\nu\alpha\beta}\epsilon^\mu_{f_1} p_1^\nu p_2^\alpha p_3^\beta$, where the antisymmetric Levi-Civita symbol is contracted with a polarization vector of the axial-vector field, $\epsilon^\mu_{f_1}$, and with three momenta $p_1, p_2, p_3$ of particles involved. However, due to the conservation of the total momenta, there are only two independent vectors. The linear dependence between $p_1, p_2$ and $p_3$ makes the product to be zero. On the other hand, if pion turns into $a_1$ (the $\pi-a_1$ transition is described by the Lagrangian density $\propto \partial_\nu \vec\pi\vec a_1^\nu$ [@Osipov17je; @Osipov17ap; @Morais17]), the amplitude of the $a_1\to f_1\pi$ transition, which is not zero by itself $e_{\mu\nu\alpha\beta}\epsilon^\mu_{f_1} \epsilon^\nu_{a_1} p_1^\alpha p_2^\beta\neq 0$, will be changed to $\epsilon^\nu_{a_1}\to p^\nu_{\pi}$ and this vanishes the product for the reasons just mentioned above. Second, we show that it is necessary to take into account the first radially-excited state of the $a_1(1260)$-meson, that is the $a_1(1640)$ resonance. This state has not been considered in [@Vishneva14]. The evidence of this hadron resonance has been recently approved by the new data of COMPASS collaboration [@Wallner17]. To take the $a_1(1640)$ into account, we carry out the calculations in the framework of the extended NJL model [@VolkovW97; @Volkov97; @VolkovE97; @VolkovYu00; @Volkov06; @Volkov16; @Volkov17]. The model allows one to describe both the ground and the first radially-excited meson states in accord with the chiral symmetry requirements. And finally, we study the influence of the $\pi -a_1$ transitions on the amplitude and demonstrate their significance. Notice the essential difference between present calculations of $\pi -a_1$ effects, and the previous ones, presented in [@Vishneva14]. In our work we take into account the $\pi -a_1$ transition on the external pion line, that has not been done in [@Vishneva14]. On the contrary, we show that the $\pi -a_1$ transition on the virtual axial-vector line, considered in that paper, does not contribute. Thus, we properly account for $\pi -a_1$ mixing effects in the $\tau \to f_1 \pi^{-} \nu_{\tau}$ decay amplitude. As a result we obtain a reasonable theoretical description of the branching ratio of the $\tau \to f_1\, \pi^{-} \nu_{\tau}$ decay. We believe that as soon as new experimental data on the spectral functions of this process will be available a more detailed test of the extended NJL model will be possible. The material of this paper is distributed as follows. In Section 2 we establish some convenient notations and review the basic properties of the extended NJL model including its Lagrangian, which is the basis of all our calculations. In Section 3 we give a derivation of the $\tau \to f_1 \pi^{-} \nu_{\tau}$ decay amplitude which does not take into account the $\pi -a_1$ mixing effects. To trace the numerical effect coming out of $\pi-a_1$ transitions, we intentionally delayed this material up to the Section 4, where we also calculate the spectral distribution of $f_1\pi$ pair and find the two particle decay width of $a_1(1640)\to f_1(1285)\pi$. In Section 5 we summarize our results and make conclusions. In the Appendix we present the integral form of the amplitude describing the $\tau \to f_1 \pi^{-} \nu_{\tau}$ decay and collect some general expressions for the quark-loop-integrals considered. The quark-meson Lagrangian of the extended NJL model {#sec:1} ==================================================== Let us review the main ingredients of the model which we apply to study the $\tau \to f_1\, \pi^{-} \nu_{\tau}$ decay. Its dynamics is described by the quark Lagrangian density ${\cal L}(x)$ with the effective $U(2)_L\times U(2)_R$ chiral symmetric four-quark interactions $$\begin{aligned} \label{qint} {\cal L}(x)&=&\bar q(x) \left(i\hat\partial-m^0 \right)q(x)+{\cal L}_{int}(x), \nonumber \\ {\cal L}_{int}(x)&=&\frac{G_S}{2}\sum_{i=1}^{2}\sum_{a=0}^{3}\left\{[j^S_{a(i)}(x)]^2+[j^P_{a(i)}(x)]^2\right\} \nonumber \\ &-&\frac{G_V}{2}\sum_{i=1}^{2}\sum_{a=0}^{3}\left\{[j^V_{a(i)}(x)]^2+[j^A_{a(i)}(x)]^2\right\},\end{aligned}$$ where $q=(u,d)$ are the up and down current quark fields of mass $m^0=(m^0_u, m^0_d)$; $G_S$ is a coupling determining the strength of the four-quark interactions of scalar and pseudoscalar types, $G_V$ is a coupling of vector and axial-vector interactions. The general form of quark currents is $$j^{S,P,V,A}_{a(i)}(x)=\!\!\int\! d^4x_1 d^4 x_2 \bar q(x_1) F^{S,P,V,A}_{a(i)}(x; x_1,x_2)q(x_2).$$ At each value of index $i$ the sum over $a$ in (\[qint\]) is invariant with respect to chiral transformations. Thus, there are four independent $U(2)_L\times U(2)_R$ invariant interactions. The local interactions with $i=1$ represent the conventional NJL type model (see, for instance, [@Volkov86; @Bijnens93]) which describes the physics of ground-state mesons with quantum numbers $J^{PC}=0^{++}, 0^{-+}, 1^{--}$ and $1^{++}$. The covariant form factors of local interactions are given by $$\begin{aligned} &&F^{S,P,V,A}_{a(1)}(x; x_1,x_2)=F^{S,P,V,A}_{a} \delta(x-x_1)\delta(x-x_2), \nonumber \\ &&F^{S,P,V,A}_{a} =\tau_a\, (1,i\gamma_5,\gamma_\mu, \gamma_\mu\gamma_5 ),\end{aligned}$$ where the flavour matrices $\tau_a=(1, \vec\tau)$, with the standard notation of the isospin Pauli matrices $\vec\tau$. The non-local part of the Lagrangian density $i=2$ represents the first radial excitations of ground-states. The corresponding covariant non-local form factors have been constructed in [@VolkovW97] with the use of the following transformation $$\begin{aligned} F^{S,\ldots}_{a(2)}(x; x_1,x_2)&=&\!\int\!\frac{d^4k}{(2\pi )^4}\frac{d^4p}{(2\pi )^4} \exp i\left[ p\left(x-\frac{x_1+x_2}{2}\right) \right.\nonumber \\ &-& \left. k\left(x_1-x_2\right)\right] F^{S,\ldots}_{a(2)}(k_\perp ),\end{aligned}$$ where $k_\mu$ is a relative momentum of the quark-antiquark pair, and $p_\mu$ is a 4-momentum of their center of mass reference frame, i.e. a meson momentum. The total momentum $p_\mu$ of the composite hadron provides a naturally preferred direction which forms the basis for a covariant three dimensional support to the interaction kernel $F^{S,\ldots}_{a(2)}(k_\perp )$ which is implemented here in accord with the concept of bilocal fields. The dependence of the interaction kernel on the transverse part of the quark momentum $k$, i.e. $k_\perp =k-(kp)p/p^2$ is a consequence of a subsidiary condition [@Markov40; @Yukawa50], which as it was shown in [@Lukierski77] is equivalent to a ’gauge principle’ and expresses the redundance of the longitudinal component of the relative momentum $k$ for the physical interaction between the quark-antiquark constituents. The covariant kernels of non-local interactions are $$\label{ff2} F^{S,P,V,A}_{a(2)}(k_\perp^2)=F^{S,P,V,A}_{a} c^{S,P,V,A} f (k_\perp^2 )\theta (\Lambda_3-|k_\perp |).$$ The coefficients $c^{S,P,V,A}$ renormalize the couplings of four-quark interactions $G_S$ and $G_V$ increasing a strength of these forces for the non-local interactions. They are fixed from the empirical values of meson masses. The step function $\theta (\Lambda_3-|k_\perp |)$, where $\Lambda_3$ is a covariant cutoff, restricts the integration over relative momentum of a bound quark-antiquark pair to the size of the bag. The function $$\label{fk} f (k_\perp^2) =1+d |k_\perp |^2, \quad |k_\perp |=\sqrt{-k_\perp^2},$$ being a Lorentz scalar, can be calculated in any convenient reference frame. In the following we use the instantaneous rest frame of the meson. In this case $k_\perp =(0, \vec k)$, and $f (k_\perp^2)= 1+d \vec k^2\equiv f(\vec k^2)$. The slope parameter $d$ is fixed by the requirement that the numerical values of the quark condensate and constituent quark masses are not changed due to an inclusion of the radially excited states. Equivalently, one can require that the single excited quark-antiquark states averaged over vacuum are vanishing (an absence of the vacuum tadpoles). It gives $d = -1.784 \,\textrm{GeV}^{-2}$. There is a simple argument in favour of this requirement: the non-local bound states do not survive in the large $N_c$ limit, therefore they cannot affect the main characteristics of the QCD ground state. The form factor $f (k_\perp^2)$ has for $d\leq \Lambda_3^{-2}$ the form of an excited-state wave function, with a node in the interval $0\leq |k_\perp |\leq \Lambda_3$. In (\[fk\]) we consider only the first two terms in a series of polynomials in $|k_\perp |^2$; inclusion of higher excited states would require polynomials of higher degree. The Lagrangian density (\[qint\]) has to be bosonized. The boson variables can be introduced in the two stages. On the first stage, the four quark interactions can be equivalently rewritten as a Yukawa type quark-antiquark-meson interactions. In this form the model Lagrangian has a structure of a linear sigma model. On the second stage, one should integrate out the quark fields completely. Exactly this way the Lagrangian of the extended NJL model has been worked out in [@Volkov97; @VolkovYu00]. Its numerous applications were reviewed in [@Volkov17]. Let us stress the main features of such calculations: (a) The model reveals the mechanism of spontaneous chiral symmetry breaking. Starting from some critical value of coupling $G_S\geq G_{crit}$ the Wigner-Weyl ground state of the system is changed to the Nambu-Goldstone phase. The transition is described by the gap equation. In particular, the current quark mass $m^0$ is replaced on the constituent quark mass $m$. It is assumed that the non-local sector of the model does not contribute to $m$, and does not affect the value of the quark condensate. This assumption help us to fix a slope parameter $d$; (b) Several mixing effects take place at the level of free Lagrangian. First, as a consequence of the phase transition, the mixing between $J^{P}=0^-$ and $J^P=1^+$ states, the so-called $\pi -a_1$ transitions, occurs. Second, there are mixings between ground and excited states with the same quantum numbers; (c) The effective meson vertices and corresponding coupling constants follow from the one-quark-loop calculations. It means that one should separate divergences in a regularized form, and renormalize the meson fields. All these effects are taken into account in our calculations. The description of collective bound states can be facilitated if we introduce, as it was discussed above, the set of bosonic variables and pass to the semi-bosonized effective meson action which is responsible for the $\tau \to f_1\, \pi^{-} \nu_{\tau}$ decay, namely, the part which describes both the interactions of the ground pseudoscalar $\pi^{\pm}$, axial-vector $a_1^{\pm}(1260)$, $f_1(1285)$ mesons and their first radially excited states with light constituent quarks. In momentum representation the action takes the form $$\begin{aligned} \label{Lagrangiane} S &=&\!\int\!\frac{d^4p}{(2\pi )^4}\bar q(p) \left(\hat p-m \right)q(p)+ \Delta S_{mass}+ \Delta S_{int}, \nonumber \\ \Delta S_{int} &=&\! \int\!\frac{d^4p}{(2\pi )^4}\!\int\!\frac{d^4k}{(2\pi )^4}\, \bar{q}\left(k+\frac{p}{2}\right) \nonumber \\ &\times& \left[ \frac{1}{2} \gamma^{\mu}\gamma^{5}\left(A_{f_{1}}f_{1\mu}(p) + B_{f_{1}}f^{'}_{1\mu}(p)\right) \right.\nonumber \\ &+& \frac{1}{2} \gamma^{\mu}\gamma^{5} \vec\tau \left(A_{a_{1}}\vec a_{1\mu}(p) + B_{a_{1}}\vec a'_{1\mu}(p)\right) \nonumber \\ &+& \left. \gamma^{5} \vec\tau \left(A_{\pi}\vec \pi (p) + B_{\pi}\vec \pi'\right)\right]q\left(k-\frac{p}{2}\right) .\end{aligned}$$ We shall not discuss here the mass part of the action $\Delta S_{mass}$ (see, for instance, [@Volkov17] or references therein). Our notations are as follows: $f_{1\mu}$, $\vec a_{1\mu}$ and $\vec \pi $ are the fields corresponding to the axial-vector and pseudoscalar mesons, the related excited states are marked with a prime. The constituent quark fields, $q$, have equal masses $m_{u} = m_{d} = 280$ MeV [@VolkovYu00; @Volkov17]. The summation over colour index is assumed. The couplings of the ground state meson $M=(\pi, a_1, f_1)$ with quarks, $A_M$, and the corresponding couplings of its radially excited state $M'=(\pi', a_1', f_1')$, $B_M$, can be written in a form $$\begin{aligned} \label{coefficients} &&\!\!\!\! A_M = A_M^0\left[g_M \sin\theta_{M}^+ +g'_M f(k_\perp^2)\sin\theta_M^ -\right]\! , \nonumber \\ &&\!\!\!\! B_M =-A_M^0\left[g_M \cos\theta_{M}^+ +g'_{M} f (k_\perp^2)\cos\theta_M^ -\right].\end{aligned}$$ These expressions are obtained as a result of several procedures. The renormalization factors $g_M$ and $g_M'$ eliminate the divergent parts of the amplitudes describing the one-quark-loop self-energy transitions $M\to M$ and $M'\to M'$ correspondingly. In the NJL model they are fully determined by the requirement that such transitions generate a free Lagrangian of meson fields. Their specific values are expressed through the divergent quark-loop integrals $$I_{2, n} = -i\frac{N_{c}}{(2\pi)^{4}}\int\frac{f^{n}(k_\perp^2)}{(m^{2} - k^2)^{2}}\theta(\Lambda_{3}^2 - k_\perp^2) \mathrm{d}^{4}k,$$ which are regularized by imposing the three-dimensional cutoff on $|k_\perp|\leq \Lambda_3 = 1.03$ GeV (in accord with eq.(\[ff2\])), after carrying out the $k_0$ integration in the self-energy diagrams [@Volkov97; @Volkov17]. In particular, we have $$\begin{aligned} &&g_{f_1} = g_{a_{1}}=\sqrt{\frac{3}{2I_{2,0}}}, \quad g_{\pi}=\sqrt{\frac{Z_\pi}{4I_{2,0}}}, \nonumber\\ &&g'_{f_1} = g'_{a_1} = \sqrt{\frac{3}{2I_{2,2}}}, \quad g'_{\pi}=\sqrt{\frac{1}{4I_{2,2}}}.\end{aligned}$$ Note that $Z_{\pi}$ is the factor induced by a diagonalization of the free Lagrangian for ground state mesons (the $\pi -a_{1}$ transitions). To avoid this mixing the unphysical axial-vector fields $\vec a_{1\mu}(x)$ should be redefined $$\label{pa-trans} \vec a_{1\mu}(x)=\vec a_{1\mu}^{phys}(x)+\sqrt{\frac{2Z_\pi}{3}}\kappa m\partial_\mu\vec\pi (x),$$ where $$\begin{aligned} Z_{\pi} = \left(1 - \frac{6m^{2}}{M^{2}_{a_{1}}}\right)^{-1} \approx 1.45. \end{aligned}$$ A dimensional parameter $\kappa$ is fixed by requiring that the free meson Lagrangian does not contain the unphysical $\vec\pi-\vec a_{1\mu}$ transitions; it gives $\kappa =3/M_{a_1}^2$, where $M_{a_1}$ is a mass of the $a_1(1260)$ meson, $M_{a_1}=1230\pm 40\,\mbox{MeV}$. The replacement (\[pa-trans\]) does not affect the Green function of the axial-vector field, but it affects the kinetic term of a pion free Lagrangian. Consequently, the pion field wave-function is additionally renormalized by the factor $\sqrt{Z_\pi}$ in (\[pa-trans\]). There are other mixings between $J^P=1^+$ and $J^P=0^-$ states. The $\pi' -a_1'$ mixing does not contribute to the $\tau\to f_1\pi\nu_\tau$ decay. In the following we neglect the $a_1'\to\pi$ mixing due to a heavy mass of the $a_1'$ state which is associated with $a_1(1640)$ meson. Eq. (\[coefficients\]) includes the angles, $\theta_{M}^{0}$ and $\theta_{M}$ [@Volkov97; @Volkov17]. These parameters appear due to $M\to M'$ mixing ($\theta_{M}^{0}$), and as a result of diagonalization, which aimed to avoid such a mixing ($\theta_{M}$). We arranged them in the following combinations: $A_M^0=1/\sin(2\theta_{M}^{0})$, $\theta_M^\pm = \theta_{M} \pm \theta_{M}^{0}$. The numerical values of mixing angles follow from the meson mass formulae. They are $$\begin{aligned} \label{angles} && \theta_{f_{1}} = \theta_{a_{1}} = 81.8^{\circ}, \quad \theta_{\pi} = 59.48^{\circ}, \nonumber\\ && \theta_{f_{1}}^{0} = \theta_{a_{1}}^{0} = 61.5^{\circ}, \quad \theta_{\pi}^{0} = 59.12^{\circ}. \end{aligned}$$ Notice, that all parameters of the model are determined from the empirical data, which are not related with the characteristics of the processes considered in this work. Therefore, our estimations can be used to test the predictive power of the extended NJL model. The decay amplitude without $\pi -a_{1}$ transitions {#sec:2} ==================================================== The decay amplitude of the process $\tau \to f_1(1285)\, \pi^- \nu_\tau$ is described by two types of diagrams which are shown in Fig.\[Contact\], and Fig.\[Intermediate\] (the $\pi -a_1$ transitions are neglected there). The first diagram describes the so-called direct contribution to the amplitude. This means that $W$-boson decays directly to the final products of the reaction, $f_1\pi$ - pair, i.e. without a resonance exchange. The latter is taken into account by the second diagram. Notice, that hadrons are alway interact through the one-quark-loop in accord with our action (\[Lagrangiane\]). In addition, we would like to point out that these loop integrals are expanded in momenta of external fields, and only the divergent parts are kept (in the case of anomalies, which lead to the finite result we take the first term of such expansion). This approximation is qualitatively justified by the $1/N_c$ expansion which states that meson physics in the large $N_c$ limit is described by the local vertices [@Hooft74; @Witten79]. It is also well known from the sigma model that divergent parts of radiative corrections have a strictly chiral-symmetric structure [@Eguchi78]. The amplitude corresponding to the graphs shown in Fig.\[Contact\] and Fig.\[Intermediate\] is presented in an Appendix. Here we show the result obtained after derivative expansion of quark vertices, as it was discussed above. Note also that from (\[angles\]) it follows that the angle $\theta_\pi^-$ is small, and $\theta_\pi\simeq \theta^0_\pi$. Neglecting $\theta_\pi^-$, one concludes from (\[coefficients\]) that $A_{\pi} \simeq g_{\pi}$. As a result, an amplitude takes the form $$\begin{aligned} \label{M-without} \mathcal{M} & = & 4mG_{F}V_{ud}\,g_\pi l^\mu e_{\mu\nu\alpha\beta}\epsilon^{\nu}(p_{f_1})p_{f_1}^\alpha p_{\pi}^\beta\left\{ I_{3}^{A_{f_{1}}} \right. \nonumber \\ &+& \frac{C_{a_1}}{g_{a_{1}}} I_{3}^{A_{f_{1}}A_{a_{1}}} \frac{s - 6m^2}{M_{a_{1}}^2 - s - i \sqrt{s}\Gamma_{a_{1}}} \nonumber\\ & + & \left. \frac{C_{a'_1}}{g_{a_{1}}} I_{3}^{A_{f_{1}}B_{a_{1}}} \frac{s - 6m^2}{M_{a'_{1}}^2 - s - i \sqrt{s}\Gamma_{a'_{1}}} \right\}\!.\end{aligned}$$ Here $G_F = 1.1663787(6) \times 10^{-11}$ MeV$^{-2}$ is the Fermi coupling constant; $V_{ud} = 0.97417\pm 0.00021$ is the Cabbibo-Kobayashi-Maskawa quark-mixing matrix element; $l_{\mu}=\bar\nu_\tau(Q')\gamma_\mu(1-\gamma_5)\tau (Q)$ is a lepton current, where $Q$ and $Q'$ are momenta of the tau-lepton and neutrino; $\epsilon_{\nu}(p_{f_{1}})$ is a polarization vector of the $f_1$ meson with the momentum $p_{f_1}$; $s = (p_{f_1} + p_{\pi})^2$ is a square of the invariant mass of the $f_1\pi$-pair, the masses and widths of resonances are $M_{a_{1}} = 1230$ MeV, $M_{a'_{1}} = 1640$ MeV, $\Gamma_{a_{1}} \approx 400$ MeV, $\Gamma_{a'_{1}} = 254$ MeV [@Patrignani16]. The factors $C_{a_1}$ and $C_{a'_1}$ are the remaining of functions $A_{a_1}$ and $B_{a_1}$ (see eq.(\[coefficients\])) after integration over an internal momentum in the $W-a_1$ and $W-a_1'$ quark-loops (see Fig.\[Intermediate\]) correspondingly. $$\begin{aligned} C_{a_1} & = & A^0_{a_1} \left( \sin\theta_{a_1}^+ +R_V\sin\theta_{a_1}^- \right), \nonumber \\ C_{a'_1} & = & -A^0_{a_1} \left(\cos\theta_{a_1}^+ +R_V\cos\theta_{a_1}^-\right), \nonumber \\ R_{V} & = & \frac{I_{2,1}}{\sqrt{I_{2,0}I_{2,2}}}\, .\end{aligned}$$ The integrals $I_3^{A_{f_1}\dots}$ come out from the evaluation of the one-loop-quark triangle diagrams of Fig.\[Contact\] and Fig.\[Intermediate\]. They correspond to the case $n=3$ of the general expression (in the next section the case $n=4$ will be also required) $$\label{In} I_{n}^{A_{M}B_{M}\dots} = \frac{-iN_c}{(2\pi)^4}\!\int\! \frac{A_M (k_\perp^2) B_M (k_\perp^2)\dots}{(m^2 - k^2)^n}\theta(\Lambda_3^2 - k_\perp^2) \mathrm{d}^{4}k,$$ where $A_{M}, B_{M}$ are given in (\[coefficients\]). The amplitude (\[M-without\]) has a conventional form $\mathcal{M}=G_{F}V_{ud}l^\mu H_\mu$, where we have found the hadron current $H_\mu$ with the help of the extended NJL model. The square of this amplitude has a simple form $$|\mathcal{M}|^2=4G_{F}^2V_{ud}^2\left[2(QH)(Q'H)-H^2 (QQ')\right].$$ Thus one can easily find the decay width of the process $$\Gamma = \frac{1}{32M_\tau^3(2\pi )^3}\!\! \int\limits_{(M_{f_1}+M_\pi )^2}^{M_\tau^2}\!\!\!\!\! ds \int\limits_{t_-}^{t_+} dt \,| {\cal M} |^2$$ by performing the numerical integration over kinematical variables $t=(Q-p_{f_1})^2$ and $s=(p_{f_1}+p_\pi )^2$. Here a boundary of the physical region at fixed value of $s$ belongs to the interval $t_-\leq t\leq t_+$, where $$\begin{aligned} &&t_\pm = \frac{1}{2} \left( t_0\pm\sqrt{D}\right), \nonumber \\ && t_0= M_\tau^2+M_{f_1}^2+M_\pi^2 -s -\frac{M_\tau^2}{s}\left(M_{f_1}^2-M_\pi^2\right), \nonumber \\ &&\sqrt{D}=\frac{1}{s}\left(M_\tau^2-s\right)\sqrt{\lambda (s,M_{f_1}^2,M_\pi^2 )}, \nonumber \\ &&\lambda (x, y, z )=[ x - (\surd y-\surd z )^2] [ x - (\surd y+ \surd z )^2 ]. \end{aligned}$$ Finally we arrive at the following result for the branching ratio Br$(\tau\to f_1\pi^-\nu_\tau )=6.04 \times 10^{-4}$. It can be schematically presented in terms of individual contributions as follows $$\begin{aligned} &&10^{4}\times \mbox{Br}(\tau\to f_1\pi^-\nu_\tau)= 6.04 \nonumber\\ &&= |c|^2+|a_1|^2+|a'_1|^2+2\mbox{Re}(a_1 c^*+a'_1 c^*+a'_1a_1^*) \nonumber\\ &&= 2.25+8.43+0.78-7.34+0.53+1.39.\end{aligned}$$ where $c, a_1, a'_1$ represent the contributions from the contact (direct) term, and from $a_1$ and $a'_1$ exchanges correspondingly. One can see that if one neglects the $\pi -a_1$ transitions (as we did here) the result overestimates the experimental value Br$(\tau \to f_1\, \pi^{-} \nu_{\tau})=(3.9\pm 0.5) \times 10^{-4}$. The effect of $\pi -a_{1}$ transitions ====================================== The replacement (\[pa-trans\]) in the quark-meson Lagrangian originates new vertices of the axial-vector type with the gradient of the pion field. This can be equivalently considered as a creation of the final pion via old $a_1$ meson field ($\pi -a_1$ transitions). To take them into account one has to append the diagrams shown in Fig. \[pi-a1-contact\] and Fig. \[pi-a1-intermediate\]. The corresponding additional contribution to the amplitude is given in the Appendix (see eq.(\[add-pi-a1\])). It can be simplified after the following observation. Let us consider the trace of the quark triangles corresponding to these diagrams $$\mbox{tr}[\gamma^{\nu}\gamma^{5}(\hat{k} + \hat{p}_{f_{1}} + m)\gamma^{\mu}\gamma^{5}(\hat{k} - \hat{p}_{\pi} + m) \gamma^{\lambda}\gamma^{5}(\hat{k} + m)].$$ It can be written as a sum $$\begin{aligned} \label{trace} && \mbox{tr}\,[\gamma^{\nu}(\hat{k} + \hat{p}_{f_{1}} + m)\gamma^{\mu}\gamma^{5}(\hat{k} - \hat{p}_{\pi} + m)\gamma^{\lambda}(\hat{k} + m) \nonumber\\ && - 2m \gamma^{\nu}(\hat{k} + \hat{p}_{f_{1}} + m)\gamma^{\mu}\gamma^{5}(\hat{k} - \hat{p}_{\pi} + m)\gamma^{\lambda}].\end{aligned}$$ It is not difficult to see that the first term coincides with a trace from the triangle of the process $f_{1}(1285) \to \rho \gamma$ considered in our previous work [@Osipov17]. After integration over loop-momentum $k$ in the corresponding quark triangle expression (see, for instance, eq.(\[afa\])) one finds that it contributes to the amplitude as $$\begin{aligned} &&e^{\mu\nu\lambda}_{\, . \, . \ . \, \alpha} \, p^\alpha_{\pi} \left(2 p_{f_{1}}^{2} + p_\pi p_{f_{1}}\right) - e^{\mu\nu\lambda}_{\, . \, . \ . \, \alpha}\, p^\alpha_{f_{1}} \! \left( p_\pi p_{f_{1}} \right) \nonumber \\ && + e^{\mu\nu}_{\, . \, .\, \alpha\beta}\, p^\alpha_{\pi} p^\beta_{f_{1}} p_{f_{1}}^{\lambda} - e^{\lambda\mu}_{\, . \, .\, \alpha\beta}\, p^\alpha_{f_1} p^\beta_{\pi} p_{\pi}^{\nu}.\end{aligned}$$ Due to $\pi -a_1$ transitions this result is multiplied by the 4-momentum $p_{\pi}^{\lambda}$. That vanishes it. The second term in eq. (\[trace\]) is easily calculated, giving $-8im^2e^{\mu\nu\lambda\alpha} (2k+p_{f_1}-p_\pi )_\alpha$. The result of its integration over $k$ depends on the structure of the integrand. In the case considered, we have three similar structures which differ only by the functions $A_M$ and $B_M$ at the vertices of anomalous triangle diagrams. These alternatives are absorbed by the corresponding coefficient $I_{4}^{A_{f_1}A_{\alpha_1}\dots}$ in the following common structure $$\begin{aligned} 8 m^{4} I_{4}^{A_{f_1}A_{\alpha_1}\dots} e^{\mu\nu\lambda}_{\, . \, . \ . \, \alpha}\, p^\alpha_{f_{1}},\end{aligned}$$ where the integrals $I_{4}^{A_{f_1}A_{\alpha_1}\dots}$ are given by (\[In\]). Taking into account these remarks, one obtains the corrections induced by the $\pi -a_1$ transitions to the total amplitude of the $\tau \to f_1 \pi^{-} \nu_{\tau}$ decay. As a result the total amplitude is $$\begin{aligned} \label{Mtot} && \mathcal{M}_{tot} = G_{F}V_{ud}l^\mu 4mg_\pi \left\{ \left[I_{3}^{A_{f_{1}}} - \frac{C_{a_1}6m^4}{g_{a_1}M_{a_1}^2} I_4^{A_{f_{1}}A_{a_{1}}} \right] \right. \nonumber\\ &&+ \frac{C_{a_1}}{g_{a_1}} \left[I_{3}^{A_{f_{1}}A_{a_{1}}} - \frac{C_{a_1}6m^4}{g_{a_1}M_{a_1}^2} I_{4}^{A_{f_{1}}A_{a_{1}}A_{a_{1}}}\right] \nonumber \\ &&\times \frac{s - 6m^2}{M_{a_{1}}^2 - s - i \sqrt{s}\Gamma_{a_{1}}} \nonumber\\ &&+ \frac{C_{a'_1}}{g_{a_{1}}} \left[I_{3}^{A_{f_{1}}B_{a_{1}}} - \frac{C_{a_1}6m^4}{g_{a_1}M_{a_1}^2} I_{4}^{A_{f_{1}}B_{a_{1}}A_{a_{1}}}\right] \nonumber\\ &&\times\left.\frac{s - 6m^2}{M_{a'_{1}}^2 - s - i \sqrt{s}\Gamma_{a'_{1}}} \right\} e_{\mu\nu\alpha\beta} \epsilon^{\nu}(p_{f_{1}}) p^\alpha_{f_{1} }p^\beta_{\pi}.\end{aligned}$$ Here, in the first square brackets, the contact contribution is presented. The second and third square brackets contain the contributions of $a_1(1260)$ and $a_1(1640)$ resonances correspondingly. The numerical integration over the $ f_1\pi\nu_\tau$ three-body phase space with the amplitude (\[Mtot\]) gives the branching ratio Br$(\tau\to f_1\pi^-\nu_\tau )=3.98 \times 10^{-4}$. The result can be schematically presented in terms of the individual contributions as follows $$\begin{aligned} &&10^{4}\times \mbox{Br}(\tau\to f_1\pi^-\nu_\tau)= 3.98 \nonumber\\ &&= |c|^2+|a_1|^2+|a'_1|^2+2\mbox{Re}(a_1 c^*+a'_1 c^*+a'_1a_1^*) \nonumber\\ &&=1.62+5.92+0.45-5.22+0.33 +0.88,\end{aligned}$$ where $c, a_1, a'_1$ represent the contributions from the contact term, and from $a_1$ and $a'_1$ exchanges correspondingly. One can see that if one takes into account the $\pi -a_1$ transitions the branching ratio is in a good agreement with the experimental value Br$(\tau \to f_1\, \pi^{-} \nu_{\tau})=(3.9\pm 0.5) \times 10^{-4}$. The differential decay distribution, $d\Gamma /d \sqrt s$, is shown in Fig. \[distribution\]. It reaches its maximum value near the $a_1(1640)$ resonance mass shell. Therefore, it appears worth while to estimate the decay width of the state, that dominates the spectral function. The amplitude of the $a_1(1640)\to f_1(1285)\pi$ decay is given by $$\begin{aligned} \mathcal{M} & = & 4img_\pi e_{\mu\nu\alpha\beta }\, \epsilon^{\mu}(p_{a_{1}})\epsilon^{\nu}(p_{f_{1}}) p^\alpha_{f_{1}}p^\beta_{\pi} \nonumber \\ &\times& \left\{I_{3}^{A_{f_{1}}B_{a_{1}}} - \frac{C_{a_1}6m^4}{g_{a_{1} M_{a_1}^2}} I_{4}^{A_{f_{1}}B_{a_{1}}A_{a_{1}}} \right\}. \end{aligned}$$ It follows then that $$\Gamma \left[a_1(1640)\to f_1(1285)\pi \right]=14.1\,\mbox{MeV}.$$ The future measurements will show how good is the extended NJL model in its predictions here. Probably, the tau decay mode studied in this work can serve us with a detailed information on the nature of $a_1(1640)$ state. Conclusions =========== The main purpose of our calculations was to apply the extended NJL model to the decay $\tau\to f_1(1285)\,\pi^- \nu_\tau$. Presently available phenomenological data on this mode give a rare opportunity to test the model. As a result we have found that hadronic part of the amplitude is sensitive to the four types of different contributions which are equally important. These are the contact interaction, the exchange by the $a_1(1260)$ meson, the exchange by the first radially excited state, $a_1(1640)$, and the pion creation by the intermediate $a_1(1260)$ meson (the $\pi a_1$-transitions). Indeed, the contact term alone gives the value Br$\ =2.25\times 10^{-4}$. The $a_1(1260)$ ground state exchange alone gives a larger number Br$\ =8.43\times 10^{-4}$. The sum of these two contributions is Br$\ =3.34\times 10^{-4}$. The radially excited state increase this value up to Br$\ =6.04\times 10^{-4}$. And finally taking into account the $\pi -a_1$ transitions we come to the final number Br$\ =3.98\times 10^{-4}$ which agrees with presently known empirical values. Our result shows that both $a_1(1260)$ and $a_1(1640)$ exchanges are very important for the description of data, and it indicates that many ideas about description of the first radial exited meson states in the NJL model seems to be correct. Let us stress that we did not introduce any new parameters to get the consistent result. All model parameters were fixed with the use of other input data, which are not related with the process $\tau\to f_1(1285)\,\pi^- \nu_\tau$. Note, that this is not the only case where the model predictions correspond to the empirical values. We refer to the recent review [@Volkov17], which contains many other examples. Those include the meson production processes in $e^+e^-$ collisions and tau decay modes. Of some relevance may be the fact that our result does not leave the place for the contribution of a new axial-vector resonance $a_1(1420)$ observed recently by the COMPASS collaboration [@Adolph15]. Our study shows that even if there is a contribution due to $a_1(1420)$ exchange to the process discussed here, this contribution is negligible. It indicates that most probably $a_1(1420)$ is not the $q\bar q$ state. Some reasonings in favour of multi-quark structure of this state are recently given in [@Ivanov17]. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank A. B. Arbuzov for his interest to this work and useful discussions. Appendix {#appendix .unnumbered} ======== Here we present the general expressions for the amplitude of $\tau\to f_1(1285)\pi^-\nu_\tau$ decay shown in Figs.\[Contact\]-\[pi-a1-intermediate\]. In the text we make derivative expansions of these expressions to obtain effective meson vertices in the long wavelength approximation. We consider the process $$\tau (Q)\to \nu_\tau (Q') f_1(p_{f_1}) \pi (p_\pi )$$ with the quantities in the parentheses denoting the four moments of the particles. The amplitude, corresponding to diagrams of Fig.\[Contact\]-\[Intermediate\], has the following form $$\mathcal{M} = G_{F} V_{ud} l_{\mu} \mathcal{H}^\mu ,$$ where a hadron current $\mathcal{H}^\mu$ is $$\begin{aligned} \mathcal{H}^\mu & = & \frac{g_{\pi}}{2} \epsilon_{\nu}(p_{f_{1}}) \left\{ I_{Wf_{1}\pi 1}^{\mu\nu} + I_{Wf_{1}\pi 2}^{\mu\nu} \right. \nonumber\\ & + & \frac{I_{Wa_{1}}^{\mu\lambda}\left(I_{a_{1}f_{1}\pi1}^{\lambda\nu} + I_{a_{1}f_{1}\pi2}^{\lambda\nu}\right)}{M_{a_{1}}^2 - s - i \sqrt{s}\Gamma_{a_{1}}} \nonumber \\ & + & \left. \frac{I_{Wa'_{1}}^{\mu\lambda}\left(I_{a'_{1}f_{1}\pi1}^{\lambda\nu} + I_{a'_{1}f_{1}\pi2}^{\lambda\nu}\right) }{M_{a'_{1}}^2 - s - i \sqrt{s}\Gamma_{a'_{1}}} \right\}.\end{aligned}$$ The first two integrals $I_{W f_1\pi 1}^{\mu\nu}$ and $I_{Wf_{1}\pi 2}^{\mu\nu}$ describe the direct contribution from the transition $W^\mu\to f_1^\nu\pi$ generated by the triangle quark loop (see Fig.\[Contact\]). In accord with two different directions for the loop momenta we specify them by indices 1 (clockwise) and 2 (counter-clockwise). Their expressions are $$\begin{aligned} && I_{Wf_{1}\pi1}^{\mu\nu} = N_c \int \frac{\mathrm{d}^4k}{(2\pi)^4} A_{f_1}(k_\perp^2) \\ &&\times\frac{ \mbox{tr}[\gamma^\nu\gamma^5(\hat{k} + \hat{p}_{f_{1}} + m)\gamma^{\mu}\gamma^{5}(\hat{k} - \hat{p}_{\pi} + m)\gamma^{5}(\hat{k} + m)]}{[(k + p_{f_{1}})^2 - m^2][(k - p_{\pi})^2 - m^2](k^2 - m^2)}, \nonumber \end{aligned}$$ $$\begin{aligned} &&I_{Wf_{1}\pi2}^{\mu\nu} = N_c \int \frac{\mathrm{d}^4k}{(2\pi)^4} A_{f_{1}} (k_\perp^2) \\ &&\times\frac{\mbox{tr}[\gamma^{5}(\hat{k} + \hat{p}_{\pi} + m)\gamma^{\mu}\gamma^{5}(\hat{k} - \hat{p}_{f_{1}} + m)\gamma^{\nu}\gamma^{5}(\hat{k} + m)]}{[(k - p_{f_{1}})^2 - m^2][(k + p_{\pi})^2 - m^2](k^2 - m^2)} . \nonumber\end{aligned}$$ The other two integrals $I_{Wa_1}^{\mu\lambda}$ and $I_{Wa'_1}^{\mu\lambda}$ describe the $W^\mu\to a_1^\lambda$ and $W^\mu\to {a'_1}^\lambda$ transitions correspondingly (see Fig.\[Intermediate\]). The first of them is equal to $$\begin{aligned} \label{IWa} &&I_{Wa_{1}}^{\mu\lambda} = N_c \int \frac{\mathrm{d}^4k}{(2\pi)^4}A_{a_{1}} (k_\perp^2) \nonumber \\ &&\times\frac{ \mbox{tr} [\gamma^{\lambda}\gamma^{5}(\hat{k} + \frac{\hat{q}}{2} + m)\gamma^{\mu}\gamma^5(\hat{k} - \frac{\hat{q}}{2} + m)]}{[(k + \frac{q}{2})^2 - m^2][(k - \frac{q}{2})^2 - m^2]}, \end{aligned}$$ with $q=Q-Q'=p_{f_1}+p_\pi$, and $q^2=s$. The second one can be obtained from (\[IWa\]) by the replacement $A_{a_{1}} (k_\perp^2) \to B_{a_{1}} (k_\perp^2)$. The anomalous quark triangle integrals $I_{a_{1}f_{1}\pi1}^{\lambda\nu}$ and $I_{a_{1}f_{1}\pi2}^{\lambda\nu}$ of Fig.\[Intermediate\] differ from other similar pair $I_{a'_{1}f_{1}\pi1}^{\lambda\nu}$ and $I_{a'_{1}f_{1}\pi2}^{\lambda\nu}$ by the replacement $A_{a_{1}}(k_\perp^2)\to B_{a_{1}}(k_\perp^2)$. Therefore, we give here only the first two expressions $$\begin{aligned} && I_{a_{1}f_{1}\pi1}^{\lambda\nu} = N_c \int \frac{\mathrm{d}^4k}{(2\pi)^4}A_{a_{1}} (k_\perp^2) A_{f_{1}}(k_\perp^2) \\ &&\times\frac{\mbox{tr} [\gamma^{\nu}\gamma^{5}(\hat{k} + \hat{p}_{f_{1}} + m)\gamma^{\lambda}\gamma^{5}(\hat{k} - \hat{p}_{\pi} + m)\gamma^{5}(\hat{k} + m)]}{[(k + p_{f_{1}})^2 - m^2][(k - p_{\pi})^2 - m^2](k^2 - m^2)},\nonumber\end{aligned}$$ $$\begin{aligned} &&I_{a_{1}f_{1}\pi2}^{\lambda\nu} = N_c \int \frac{\mathrm{d}^4k}{(2\pi)^4} A_{a_{1}} (k_\perp^2) A_{f_{1}}(k_\perp^2) \\ &&\times\frac{ \mbox{tr}[\gamma^{5}(\hat{k} + \hat{p}_{\pi} + m)\gamma^{\lambda}\gamma^{5}(\hat{k} - \hat{p}_{f_{1}} + m)\gamma^{\nu}\gamma^{5}(\hat{k} + m)]}{[(k - p_{f_{1}})^2 - m^2][(k + p_{\pi})^2 - m^2](k^2 - m^2)}. \nonumber \end{aligned}$$ Consider now the diagrams shown in Fig.\[pi-a1-contact\]-\[pi-a1-intermediate\]. They give an additional contribution to the axial current $\mathcal{H}^\mu$ by taking into account the $\pi -a_1$ transitions $$\begin{aligned} \label{add-pi-a1} \Delta\mathcal{H}^\mu &=& \frac{g_{\pi}}{4} \epsilon_{\nu}(p_{f_{1}})\frac{I_{a_{1}\pi}^{\lambda}}{M_{a_1}^2}\left\{ I_{Wf_{1}a_{1}1}^{\mu\nu\lambda} + I_{Wf_{1}a_{1}2}^{\mu\nu\lambda} \right. \nonumber\\ &+& \frac{I_{Wa_{1}}^{\mu\delta}\left(I_{a_{1}f_{1}a_{1}1}^{\delta\nu\lambda} + I_{a_{1}f_{1}a_{1}2}^{\delta\nu\lambda}\right)}{M_{a_{1}}^2 - s - i \sqrt{s}\Gamma_{a_{1}}} \nonumber\\ &+& \left. \frac{I_{Wa'_{1}}^{\mu\delta} \left(I_{a'_{1}f_{1}a_{1}1}^{\delta\nu\lambda} + I_{a'_{1}f_{1}a_{1}2}^{\delta\nu\lambda}\right)}{M_{a'_{1}}^2 - s - i \sqrt{s}\Gamma_{a'_{1}}} \right\}\end{aligned}$$ Here, the integrals with two Lorentz indices have been already defined in (\[IWa\]). The one-index integral describes the quark loop corresponding to the $\pi -a_1$ transition, i.e. $$\begin{aligned} && I_{a_{1}\pi}^{\lambda} = N_c \int \frac{\mathrm{d}^4k}{(2\pi)^4} A_{a_{1}} (k_\perp^2) \nonumber \\ &&\times \frac{ \mbox{tr} [\gamma^{5}(\hat{k} + \frac{\hat{p}_{\pi}}{2} + m)\gamma^{\lambda}\gamma^{5}(\hat{k} - \frac{\hat{p}_{\pi}}{2} + m)]}{[(k + \frac{p_{\pi}}{2})^2 - m^2][(k - \frac{p_{\pi}}{2})^2 - m^2]} . \end{aligned}$$ The integral $I_{Wf_{1}a_{1}2}^{\mu\nu\lambda}$ can be obtained from the integral $$\begin{aligned} && I_{Wf_{1}a_{1}1}^{\mu\nu\lambda} = N_c \int \frac{\mathrm{d}^4k}{(2\pi)^4} A_{f_{1}}(k_\perp^2)A_{a_{1}} (k_\perp^2) \\ && \frac{ \mbox{tr}[\gamma^{\nu}\gamma^{5}(\hat{k} + \hat{p}_{f_{1}} + m)\gamma^{\mu}\gamma^{5}(\hat{k} - \hat{p}_{\pi} + m)\gamma^{\lambda}\gamma^{5}(\hat{k} + m)]}{(k^2 - m^2)[(k + p_{f_{1}})^2 - m^2][(k - p_{\pi})^2 - m^2]} \nonumber\end{aligned}$$ by replacements $\nu\leftrightarrow \lambda$ and $p_{f_1} \leftrightarrow p_\pi$. Both describe the direct transition $W^\mu\to f_1^\nu a_1^\lambda$. The last four integrals can be obtained, for instance, from the integral $$\begin{aligned} \label{afa} && I_{a_{1}f_{1}a_{1}1}^{\delta\nu\lambda} = N_c \int \frac{\mathrm{d}^4k}{(2\pi)^4} A_{a_{1}} (k_\perp^2) A_{a_{1}}(k_\perp^2) A_{f_{1}}(k_\perp^2) \\ && \frac{\mbox{tr}[\gamma^{\nu}\gamma^{5}(\hat{k} + \hat{p}_{f_{1}} + m)\gamma^{\delta}\gamma^{5}(\hat{k} - \hat{p}_{\pi} + m)\gamma^{\lambda}\gamma^{5}(\hat{k} + m)]}{(k^2 - m^2)[(k + p_{f_{1}})^2 - m^2][(k - p_{\pi})^2 - m^2]}. \nonumber \end{aligned}$$ In this case $I_{a_{1}f_{1}a_{1}2}^{\delta\nu\lambda}$ is given by the replacement $\nu\leftrightarrow \lambda$ and $p_{f_1} \leftrightarrow p_\pi$; the integral $I_{a'_{1}f_{1}a_{1}1}^{\delta\nu\lambda}$ follows from (\[afa\]) by the replacement of one of the two functions $A_{a_{1}} (k_\perp^2) $ by $B_{a_{1}} (k_\perp^2) $; and finally, the integral $I_{a'_{1}f_{1}a_{1}2}^{\delta\nu\lambda}$ can be obtained from (\[afa\]) by the replacement of one of the two functions $A_{a_{1}} (k_\perp^2) $ by $B_{a_{1}} (k_\perp^2) $ together with $\nu\leftrightarrow \lambda$ and $p_{f_1} \leftrightarrow p_\pi$. 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--- abstract: | Let $M$ be a $2\times2$ real matrix with both eigenvalues less than 1 in modulus. Consider two self-affine contraction maps from $\mathbb R^2 \to \mathbb R^2$, $$T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M v + u,$$ where $u\neq0$. We are interested in the properties of the attractor of the iterated function system (IFS) generated by $T_m$ and $T_p$, i.e., the unique non-empty compact set $A$ such that $A = T_m(A) \cup T_p(A)$. Our two main results are as follows: - If both eigenvalues of $M$ are between $2^{-1/4}\approx 0.8409$ and $1$ in absolute value, and the IFS is non-degenerate, then $A$ has non-empty interior. - For almost all non-degenerate IFS, the set of points which have a unique address is of positive Hausdorff dimension – with the exceptional cases fully described as well. This paper continues our work begun in [@HS]. address: - | Department of Pure Mathematics\ University of Waterloo\ Waterloo, Ontario\ Canada N2L 3G1 - | School of Mathematics\ The University of Manchester\ Oxford Road, Manchester M13 9PL\ United Kingdom. author: - 'Kevin G. Hare' - Nikita Sidorov title: 'Two-dimensional self-affine sets with interior points, and the set of uniqueness' --- [^1] [^2] Introduction {#sec:intro} ============ Consider two self-affine linear contraction maps $T_m, T_p: {\mathbb{R}}^2 \to {\mathbb{R}}^2$: $$\label{eq:main} T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M v + u,$$ where $M$ is a $2\times2$ real matrix with both eigenvalues less than 1 in modulus and $u\neq0$. Here “$m$” is for “minus” and “$p$” is for “plus”. We are interested in the iterated function system (IFS) generated by $T_m$ and $T_p$. Then, as is well known, there exists a unique non-empty compact set $A$ such that $A = T_m(A) \cup T_p(A)$. The properties that we are interested in (non-empty interior of $A$ and the set of uniqueness) do not change if we consider a conjugate of $T_m$ and $T_p$. That is, if we consider $g \circ T_i \circ g^{-1}$ instead of the $T_i$ where $g$ is any invertible linear map from ${\mathbb{R}}^2 \to {\mathbb{R}}^2$. As such, we can assume that $M$ is a $2 \times 2$ matrix in one of three forms: $$\begin{pmatrix} {\lambda}& 0 \\ 0 & \mu \end{pmatrix}, \begin{pmatrix} \nu & 1 \\ 0 & \nu \end{pmatrix}, \begin{pmatrix} a & b \\ -b & a \end{pmatrix}.$$ We will call the first of these [*the real case*]{}, the second [*the Jordan block case*]{}, and the last [*the complex case*]{}. We will say that the IFS is [*degenerate*]{} if it is restricted to a one-dimensional subspace of ${\mathbb{R}}^2$. This will occur if any of the eigenvalues are $0$. It will also occur if ${\lambda}= \mu$ in the real case, or (equivalently) if $b = 0$ in the complex case. If our IFS is non-degenerate, then $u$ can be chosen to be a cyclic vector for $M$, i.e., such that the span of $\{M^n u \mid n\ge0\}$ is all of $\mathbb R^2$ (which we will assume henceforth). In [@HS] the authors studied the real case with ${\lambda}, \mu > 0$. Properties of the complex case have been studied extensively since the seminal paper [@BH] - see, e.g., [@Cal] and references therein. Note that most authors concentrate on the connectedness locus, i.e., pairs $(a,b)$ such that the attractor $A$ is connected. In the present paper we study all three of the above cases, allowing us to make general claims. Our main result is \[thm:interior\] If all eigenvalues of $M$ are between $2^{-1/4} \approx 0.8409$ and $1$ in absolute value, and the IFS is non-degenerate, then the attractor of the IFS has non-empty interior. More precisely, - If $0.832 < {\lambda}< \mu < 1$ then $A_{{\lambda}, \mu}$, the attractor for the (positive) real case, has non-empty interior [@HS Corollary 1.3]. - If $2^{-1/2} \approx 0.707 < {\lambda}< \mu < 1$ then $A_{-{\lambda}, \mu}$, the attractor for the (mixed) real case, has non-empty interior. - If $0.832 < \nu < 1$ then $A_{\nu}$, the attractor for the Jordan block case, has non-empty interior. - If $2^{-1/4} \approx 0.841 < |\kappa| < 1$ with $\kappa = a+ b i \not\in {\mathbb{R}}$ then $A_{{\kappa}}$, the attractor for the complex case, has non-empty interior. The remaining three cases are shown in Section \[sec:interior\]. The last case relies upon an argument of V. Kleptsyn [@MO]. Some non-explicit results are known in the complex case. Let $\kappa = a + b i$ and consider the attractor $A'_{\kappa}$ satisfying $A'_{\kappa}= {\kappa}A'_{\kappa}\cup ({\kappa}A'_{\kappa}+ 1)$ which is clearly similar to $A_{\kappa}$. Let ${\kappa}\not\in {\mathbb{R}}$ be sufficiently close to $1$ in absolute value. Then $A'_{{\kappa}}\supset \{z : |z| \le 1\}$. Since $A'_{\kappa}$ tends to a segment in $\mathbb R$ as the imaginary part of ${\kappa}$ tends to 0 in the Hausdorff metric (with any fixed real part), it is clear that there cannot be an absolute bound in a result like this. In fact, a detailed analysis of the proof indicates that the actual condition the authors use is $|{\kappa}|>1-C|\arg({\kappa})|$ with some absolute constant $C>0$. That is, “sufficiently close to 1” means “for any $\theta\in(0,\pi)\cup(\pi,2\pi)$ there exists $\delta$ such that $A_{\kappa}$ contains the closed unit disc for all ${\kappa}$ with $\arg({\kappa})=\theta$ and $|{\kappa}|>1-\delta$” [^3]. Theorem \[thm:interior\] overcomes this obstacle. Given the two maps $T_m$ and $T_p$, there is a natural projection map from the set of all $\{m,p\}$ sequences to points on $A$. We define $\pi:\{m,p\}^{\mathbb{N}}\to A$ by $\pi(a_0 a_1 a_2 \dots) = \lim_{n\to\infty}T_{a_0} \circ T_{a_1} \circ\dots\circ T_{a_n}(0,0)$. Note that because both $T_m$ and $T_p$ are contraction maps, this yields a well defined point in $A$. We call $a_0a_1\dots$ an [*address*]{} for $(x,y)\in A$ if $\pi(a_0a_1\dots)=(x,y)$. We say that a point $(x,y) \in A$ is a [*point of uniqueness*]{} if it has a unique address. The question on when this IFS has a large number of points of uniqueness depends somewhat on the nature of the eigenvalues. If $M$ has two complex eigenvalues, $\kappa$ and $\overline\kappa$ where $\mathrm{arg}(\kappa)/\pi \in {\mathbb{Q}}$ then it is possible for the IFS to have a small number of points of uniqueness (see Theorem \[thm:uniq-rational\]). With the exception of this case, all other IFS will have a continuum of points of uniqueness. Our second result is \[thm:uniq\] For all non-degenerate IFS not explicitly mentioned in Theorem \[thm:uniq-rational\], the set of points of uniqueness is uncountable, and with positive Hausdorff dimension. Again, the real case where ${\lambda}>0, \mu > 0$ has been shown in [@HS]. We prove the remaining cases in Section \[sec:unique\]. \[ex:rauzy\] ![Points of uniqueness for the Rauzy fractal[]{data-label="fig:Dragon"}](rauzy-uniq.eps){width="350pt"} As an example, consider the famous Rauzy fractal introduced in [@Rauzy]. Let ${\kappa}$ be one of the complex roots of $z^3-z^2-z-1$, i.e., ${\kappa}\approx -0.419 + 0.606i$. Consider the attractor $A_{\kappa}$ satisfying $A_{\kappa}= ({\kappa}A_{\kappa}-1) \cup ({\kappa}A_{\kappa}+ 1)$. It follows from the results of [@Mess] that the unique addresses in this case are precisely those which do not contain three consecutive identical symbols. It is easy to show by induction that the number of $m$-$p$ words of length $n$ with such a property which start with $m$ is the $n$th Fibonacci number. Consequently, the set of unique addresses has topological entropy equal to $\log\tau$, where $\tau=\frac{1+\sqrt5}2$. Hence the Hausdorff dimension of the set of uniqueness is $-\frac{\log{\tau}}{\log{|\kappa|}} \approx 1.579354467$. See Figure \[fig:Dragon\] for the attractor (grey) and points of uniqueness (black). It is interesting to note that since the Hausdorff dimension of the boundary here is approximately $1.093$ (see [@IK]), “most” points of uniqueness of the Rauzy fractal are interior points, whereas our general construction only uses boundary points - see Section \[sec:unique\]. \[ex:twindragon\] ![The twin dragon curve $A_{\frac{1+i}2}$[]{data-label="fig:twindragon"}](twindragon.eps){width="300pt"} Another famous complex fractal is the twin dragon curve which in our notation is $A_{\kappa}$ with ${\kappa}=\frac{1+i}2$ – see Figure \[fig:twindragon\]. The grey half corresponds to all points in $A_{\kappa}$ whose address begins with $m$ and the black half – with $p$. Their intersection is a part of the boundary of either half, which has the same Hausdorff dimension as the boundary of $A_{\kappa}$, approximately $1.524$ (see, e.g., [@dragon]). Clearly, if a point in $A_{\kappa}$ has a non-unique address $a_0a_1\dots$, then $\pi(a_na_{n+1}\dots)$ must lie in the aforementioned intersection for some $n$. This means that the complement of the set of uniqueness in this case has dimension $\approx1.524$; on the other hand, it is well known that $A_{\kappa}$ has non-empty interior (see, e.g. [@Gilbert] and references therein). Consequently, a.e. point of $A_{\kappa}$ has a unique address. Notation {#sec:notation} ======== For the real case we will consider two subcases. Let $0 < {\lambda}\leq \mu < 1$ and consider $$M = \begin{pmatrix} {\lambda}& 0 \\ 0 & \mu \end{pmatrix}.$$ This we will call the [*positive real case*]{}. This was the case considered in [@HS]. The second subcase is $$M = \begin{pmatrix}-{\lambda}& 0 \\ 0 & \mu \end{pmatrix},$$ which we will call the [*mixed real case*]{}. In both cases we take $u = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$, which is clearly cyclic. In the real positive case, $\pi: \{m,p\}^{\mathbb{N}}\to A_{{\lambda}, \mu}$, we have $\pi(a_0 a_1 a_2 \dots) = \left(\sum_{i=0}^\infty a_i {\lambda}^i, \sum_{i=0}^\infty a_i \mu^i\right) \in {\mathbb{R}}^2$, whereas in the real mixed case, $\pi: \{0,1\}^{\mathbb{N}}\to A_{-{\lambda}, \mu}$, we have $$\pi(a_0 a_1 a_2 \dots) = \left( \sum_{i=0}^\infty a_i (-{\lambda})^i, \sum_{i=0}^\infty a_i \mu^i\right).$$ It is easy to see that all other real cases can be reduced to one of these two. For example, there is a symmetry from $(-{\lambda}, \mu)$ to $({\lambda}, -\mu)$. To see this, write $(x,y) = \left(\sum a_i (-{\lambda})^i, \sum a_i \mu^i\right) \in A_{-{\lambda}, \mu}$. Taking $a'_i = (-1)^i a_i \in \{\pm1\}$ we see that $(x,y) = \left(\sum a'_i {\lambda}^i, \sum a'_i (-\mu)\right) \in A_{{\lambda}, -\mu}$. For the Jordan block case we will assume that $0 < \nu < 1$. In this case we take $u = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, which is, again, clearly a cyclic vector. We have $$\pi(a_0 a_1 a_2 \dots) = \left(\sum_{i=0}^\infty i a_i \nu^{i-1},\sum_{i=0}^\infty a_i\nu^i\right)$$ (see Lemma \[lem:jordan-exp\] below). There is a symmetry to the $\nu < 0$ case such that $A_{\nu}$ and $A_{-\nu}$ share all of the desired properties. To see this, write $(x,y) = (\sum i a_i \nu^{i-1} \sum a_i \nu^i) \in A_{-\nu}$. Taking $a'_i = (-1)^i a_i \in \{m, p\}$, we see that $(-x,y) = (\sum a'_i i (-\nu)^i, \sum a'_i (-\nu)) \in A_{-\nu}$. Hence $A_\nu$ and $A_{-\nu}$ are reflections of each other across the $y$-axis. For the complex case, we let ${\kappa}= a+bi$ and consider $v = \begin{pmatrix} x\\ y \end{pmatrix}$ as $z = x+ yi$. We see that the maps in with $u = \begin{pmatrix}1 \\ 0 \end{pmatrix}$, are equivalent to the maps in ${\mathbb{C}}$, namely, $$T_m (z) = {\kappa}z - 1 \ \ \mathrm{or}\ \ T_p(z) = {\kappa}z + 1.$$ In the complex case we have $\pi(a_0 a_1 a_2 \dots) = \sum_{j=0}^\infty a_j \kappa^j \in {\mathbb{C}}$, i.e., the attractor $A_{\kappa}$ is the set of expansions in complex base ${\kappa}$ with “digits” 0 and 1. Note that if ${\kappa}\in {\mathbb{R}}$ then the resulting IFS is real (and degenerate). Throughout we will refer to $[i_1\dots i_k]$ as the [*cylinder*]{} of all $(a_i)_0^\infty\in\{m,p\}^{\mathbb N}$ such that $a_j = i_j$ for $j = 1, \dots, k$. We note that this is a compact subset of $\{m,p\}^{\mathbb{N}}$ under the usual product topology. Attractors with interior {#sec:interior} ======================== The first question that we are interested in is, when does $A$ have interior. For the real and Jordan block case we look at a related, albeit somewhat easier, question: when is $(0,0)$ contained in the interior of $A$? We will say that $(-{\lambda}, \mu)$ for the mixed real case is in ${{\mathcal{Z}}}_{\mathbb{R}}$ if $(0,0) \in \mathrm{int}(A_{-{\lambda}, \mu})$. An equivalent definition is given for ${{\mathcal{Z}}}_J$ for the Jordan block case. In fact, the real case (both mixed and positive) and the Jordan block case are both special cases of a more general result – see Theorem \[thm:tool2\] below. Consider a contraction matrix $M$ with all real eigenvalues such that any duplicate eigenvalue is within the same Jordan block. That is, let $J_{{\lambda}, k}$ be the $k \times k$ Jordan block $$J_{{\lambda}, k} = \begin{pmatrix} {\lambda}& 1 & & & 0 \\ & {\lambda}& \ddots & & \\ & & \ddots & \ddots& \\ & & & {\lambda}& 1 \\ 0 & & & & {\lambda}\end{pmatrix}$$ and write $M$ as $$M = \begin{pmatrix} J_{{\lambda}_1, k_1} & & & 0 \\ & J_{{\lambda}_2, k_2} & & \\ & & \ddots & \\ 0 & & & J_{{\lambda}_r, k_r} \end{pmatrix}, \label{eq:M}$$ where all $\lambda_i$ are distinct and $0 < |\lambda_i| < 1$ for all $i$. Then $M$ will have dimensions $N \times N$ where $N = k_1 + k_2 + \dots + k_r$. We consider the two affine maps $$T_m(v) = M v - \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \\ \vdots \\ \vdots \\ 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix} \ \ \mathrm{and} \ \ T_p(v) = M v + \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \\ \vdots \\ \vdots \\ 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}.$$ Here there are $k_1-1$ copies of $0$s follows by one $1$, then $k_2-1$ copies of $0$s follows by one $1$, and so on. Consider the case with $M$ as a single $k \times k$ Jordan block $J_{{\lambda},k}$. \[lem:jordan-exp\] We have $$\pi(a_0 a_1 a_2 \dots) = \begin{pmatrix} \frac{1}{(k-1)!}\frac{d^{k-1}}{d {\lambda}^{k-1}} \sum_{i=0}^\infty a_i {\lambda}^i \\ \frac{1}{(k-2)!}\frac{d^{k-2}}{d {\lambda}^{k-2}} \sum_{i=0}^\infty a_i {\lambda}^i \\ \vdots \\ \frac{d}{d {\lambda}} \sum_{i=0}^\infty a_i {\lambda}^i \\ \sum_{i=0}^\infty a_i {\lambda}^i \end{pmatrix}.$$ It suffices to show that $$\label{eq:jordan} T_{a_0}\dots T_{a_n}\begin{pmatrix} 0\\ \vdots \\ 0\end{pmatrix}= \begin{pmatrix} \sum_{i=0}^n \binom{i}{k-1} a_i{\lambda}^{i-k+1} \\ \sum_{i=0}^n \binom{i}{k-2} a_i{\lambda}^{i-k+2}\\ \vdots \\ \sum_{i=0}^n ia_i{\lambda}^{i-1}\\ \sum_{i=0}^n a_i {\lambda}^i \end{pmatrix},$$ with the usual convention that $\binom ij=0$ if $i<j$. We prove this by induction: for $n=0$ we have $$T_{a_0} \begin{pmatrix} 0\\ \vdots \\ 0 \\ 0\end{pmatrix} = \begin{pmatrix} 0\\ \vdots \\ 0 \\ a_0 \end{pmatrix},$$ which is what we need. Assume (\[eq:jordan\]) holds for $n-1$; then, given that $T_{a_0}(v)=Mv+a_0(0,0,\dots,0,1)^T$, $$\begin{aligned} T_{a_0}\begin{pmatrix} \sum_{i=0}^{n-1} \binom{i}{k-1} a_{i+1}{\lambda}^{i-k+1} \\ \sum_{i=0}^{n-1} \binom{i}{k-2} a_{i+1}{\lambda}^{i-k+2}\\ \vdots \\ \sum_{i=0}^{n-1} ia_{i+1}{\lambda}^{i-1}\\ \sum_{i=0}^{n-1} a_{i+1} {\lambda}^i \end{pmatrix} &= \begin{pmatrix} \sum_{i=0}^{n-1} \left(\binom{i}{k-1}+\binom{i}{k-2}\right) a_{i+1}{\lambda}^{i-k+2} \\ \sum_{i=0}^{n-1} \left(\binom{i}{k-2}+\binom{i}{k-3}\right) a_{i+1}{\lambda}^{i-k+3}\\ \vdots \\ \sum_{i=0}^{n-1} (i+1)a_{i+1}{\lambda}^i\\ \sum_{i=0}^{n-1} a_{i+1} {\lambda}^{i+1}+a_0 \end{pmatrix} \\ &=\begin{pmatrix} \sum_{i=0}^n \binom{i}{k-1} a_i{\lambda}^{i-k+1} \\ \sum_{i=0}^n \binom{i}{k-2} a_i{\lambda}^{i-k+2}\\ \vdots \\ \sum_{i=0}^n ia_i{\lambda}^{i-1}\\ \sum_{i=0}^n a_i {\lambda}^i \end{pmatrix},\end{aligned}$$ as required. It is easy to see how this would generalize to multiple Jordan blocks. Return to the general case of $M$ given by (\[eq:M\]). The following theorem is along the lines of [@HS Theorem 3.1] and is based on the ideas from [@Gunturk] (originally) and [@DJK]. \[thm:tool2\] Let $P(x) = x^n + b_{n-1} x_{n-1} + \dots + b_0$ with $n\ge N$. Assume that 1. $P(1/{\lambda}_i) = P'(1/{\lambda}_i) = \dots = P^{(k_i-1)}(1/{\lambda}_i) = 0$ for $i = 1, \dots, r$. 2. $\sum_{j=0}^{n-1} |b_j| \leq 2$. 3. There exists a non-singular $N \times N$ submatrix of the matrix $B$ (defined by (\[eq:B\]) below). Then there exists a neighbourhood of $(\underbrace{0,0,\dots, 0}_{N})$ contained in $A$. Let $$B_t(y) = \sum_{k=0}^t b_k y^{t-k}$$ for $t = 0, 1, \dots, n-1$. Define the matrix $B$ as follows: $$\label{eq:B} B := \begin{pmatrix} B_{0}^{(k_1-1)}({\lambda}_1) & B_{1}^{(k_1-1)}({\lambda}_1) & \dots & B_{{n-1}}^{(k_1-1)}({\lambda}_1) \\ \vdots & \vdots & & \vdots \\ B_0^{(1)}({\lambda}_1) & B_1^{(1)}({\lambda}_1) & \dots & B_{n-1}^{(1)}({\lambda}_1) \\ B_0({\lambda}_1) & B_1({\lambda}_1) & \dots & B_{n-1}({\lambda}_1) \\ \vdots & \vdots & & \vdots \\ \vdots & \vdots & & \vdots \\ B_{0}^{(k_r-1)}({\lambda}_r) & B_{1}^{(k_r-1)}({\lambda}_r) & \dots & B_{{n-1}}^{(k_r-1)}({\lambda}_r) \\ \vdots & \vdots & & \vdots \\ B_0^{(1)}({\lambda}_r) & B_1^{(1)}({\lambda}_r) & \dots & B_{n-1}^{(1)}({\lambda}_r) \\ B_0({\lambda}_r) & B_1({\lambda}_r) & \dots & B_{n-1}({\lambda}_r) \\ \end{pmatrix}.$$ Here $B^{(s)}_t(y) = \frac{1}{s!} \frac{d^s}{d y^s} B_t(y)$. Notice that $B$ is an $N \times n$ matrix. Let $P$ have the required properties and let $u_{-n}, \dots, u_{-1}$ satisfy $$\label{eq:un} \begin{pmatrix}x_1 \\ x_2 \\ \vdots \\ x_N \end{pmatrix} = B \begin{pmatrix}u_{-n} \\ u_{-n+1} \\ \vdots \\ u_{-1} \end{pmatrix}.$$ So long as some $N\times N$ sub-matrix of $B$ has non-zero determinant, we have that for all $x_i$ sufficiently close to $0$, there is a solution of (\[eq:un\]) with small $u_j$. Specifically, we can choose $\delta$ such that if $|x_i| < \delta$, then there is a solution with $|u_j| \leq 1$. Fix a vector $(x_1, x_2, \dots, x_N)$ in the neighbourhood of $(0,\dots,0)$, where each $|x_i| < \delta$. We will construct a sequence $(a_j)$ with $a_j \in \{-1, 1\}$ such that $$\pi (a_1 a_2 a_3 \dots) = (x_1, x_2, \dots, x_n),$$ which will prove the result. To do this, we first solve equation  for $u_{-n}, \dots, u_{-1}$ with $|u_i| \leq 1$. We will then choose the $u_j$ and $a_j$ for $j = 0, 1, 2, 3, \dots$ by induction, such that $$u_j := a_j - \sum_{k=0}^{n-1} b_{k} u_{j+k-n}$$ and such that $u_j \in [-1,1]$ and $a_j \in \{-1, +1\}$. We see that this is possible, as, by induction, all $u_j$ with $j < 0$ are such that $|u_j| \leq 1$. Furthermore, $$\begin{aligned} \left|\sum_{k=0}^{n-1} b_j u_{j+k-n}\right| & \leq & \sum_{k=0}^{n-1} |b_k u_{j+k-n}| \\ & \leq & \sum_{k=0}^{n-1} |b_k| \\ & \leq & 2,\end{aligned}$$ by our assumption on the $b_k$. Hence there is a choice of $a_j$, either $+1$ or $-1$, such that $u_j = a_j - \sum_{k=0}^{n-1} b_k u_{j+k-n} \in [-1, 1]$. We claim that this sequence of $a_j$ has the desired properties. To see this, we first consider the base case (we put $b_n = 1$ for ease of notation). Observe that: $$\begin{aligned} \sum_{j=0}^\infty a_j y^j & = & \sum_{j=0}^\infty \left( \sum_{k=0}^n b_k u_{j+k-n} \right) y^j \\ & = & \sum_{k=0}^n b_k y^{-k} \sum_{j=0}^\infty u_{j+k-n} y^{j+k} \\ & = & \sum_{k=0}^n b_k y^{-k} \left(\sum_{t=k}^{n-1} u_{t-n} y^{t} + \sum_{t=n}^\infty u_{t-n} y^{t} \right) \\ & = & \sum_{k=0}^n \sum_{t=k}^{n-1} b_k y^{t-k} u_{t-n} + P(y^{-1}) \sum_{t=n}^\infty u_{t-n} y^{t}.\end{aligned}$$ Evaluating at $y = {\lambda}_i$ and observing that $P({\lambda}_i^{-1}) = 0$, this simplifies to $$\sum_{t=0}^{n-1} u_{t-n} B_t({\lambda}_i). \label{eq:base}$$ We further see that $$\frac{1}{s!} \frac{d^s}{d y^s} \left( \sum_{j=0}^\infty a_j y^j \right) = \frac{1}{s!} \frac{d^s}{d y^s} \left( \sum_{k=0}^n \sum_{t=k}^{n-1} b_k y^{t-k} u_{t-n} + P(y^{-1}) \sum_{t=n}^\infty u_{t-n} y^{t} \right).$$ Taking derivatives and evaluating at ${\lambda}_i$, this simplifies to $$\sum_{t=0}^{n-1} u_{t-n} B_t^{(s)}({\lambda}_i), \label{eq:induct}$$ Combining equations , with Lemma \[lem:jordan-exp\] gives $$\begin{aligned} \pi(a_0 a_1 a_2 \dots) & = \begin{pmatrix} \frac{1}{(k_1-1)!}\frac{d^{k_1-1}}{d {\lambda}_1^{k_1-1}} \sum_{i=0}^\infty a_i {\lambda}_1^i \\ \frac{1}{(k_1-2)!}\frac{d^{k_1-2}}{d {\lambda}_1^{k_1-2}} \sum_{i=0}^\infty a_i {\lambda}_1^i \\ \vdots \\ \frac{d}{d {\lambda}_1} \sum_{i=0}^\infty a_i {\lambda}_1^i \\ \sum_{i=0}^\infty a_i {\lambda}_1^i \\ \vdots \\ \frac{1}{(k_r-1)!}\frac{d^{k_r-1}}{d {\lambda}_r^{k_r-1}} \sum_{i=0}^\infty a_i {\lambda}_r^i \\ \frac{1}{(k_r-2)!}\frac{d^{k_r-2}}{d {\lambda}_r^{k_r-2}} \sum_{i=0}^\infty a_i {\lambda}_r^i \\ \vdots \\ \frac{d}{d {\lambda}_r} \sum_{i=0}^\infty a_i {\lambda}_r^i \\ \sum_{i=0}^\infty a_i {\lambda}_r^i \end{pmatrix} & = \begin{pmatrix} \sum_{t=0}^{n-1} u_{t-n} B_t^{(k_1-1)}({\lambda}_1) \\ \sum_{t=0}^{n-1} u_{t-n} B_t^{(k_1-2)}({\lambda}_1) \\ \vdots \\ \sum_{t=0}^{n-1} u_{t-n} B_t^{(1)}({\lambda}_1) \\ \sum_{t=0}^{n-1} u_{t-n} B_t({\lambda}_1) \\ \vdots \\ \sum_{t=0}^{n-1} u_{t-n} B_t^{(k_1-1)}({\lambda}_r) \\ \sum_{t=0}^{n-1} u_{t-n} B_t^{(k_1-2)}({\lambda}_r) \\ \vdots \\ \sum_{t=0}^{n-1} u_{t-n} B_t^{(1)}({\lambda}_r) \\ \sum_{t=0}^{n-1} u_{t-n} B_t({\lambda}_r) \end{pmatrix} \\ & = B \begin{pmatrix} u_{-n} \\ u_{-n+1} \\ \vdots \\ u_{-1} \end{pmatrix} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_N \end{pmatrix},\end{aligned}$$ which proves the desired result. \[rmk:nonempty\] It is worth observing that if $M$ is an $N \times N$ matrix with distinct eigenvalues sufficiently close to (but less than) $1$ in absolute value, then the $N$-dimensional attractor $A$ will have non-empty interior. Here “sufficiently close” depends only on $N$. This follows from essentially the same proof as in [@HS Theorem 3.4] using the polynomial $$P(x) = x^{m n + 2} - x^{nm} + b_{m-1} x^{(m-1) n} + b_{m-2} x^{(m-2) n} + \dots + b_{0}$$ and $n$ even. We can choose the $b_i$ of this polynomial such that $\sum |b_i| < 2$ and $(x^2-1)^m | P(x)$. Letting $P(x) = Q(x) (x^2-1)^m$, we have for $\lambda_i$ sufficiently close to $1$ that $$\begin{aligned} P^*(x) &=& Q(x) (x^2-1/\lambda_1^2) (x^2-1/\lambda_2^2) \cdots (x^2-1/\lambda_m^2) \\ &=& x^{m n + 2} + b^*_{nm+1} x^{nm+1} + \cdots + b_0^*\end{aligned}$$ will also have $\sum |b_i^*| < 2$. It seems highly likely that the same would be true for the case where $M$ contains non-trivial Jordan blocks, although the analysis becomes much messier. The mixed real case ------------------- Here we apply Theorem \[thm:tool2\] with roots $-\lambda$ and $\mu$ and $k_1 = k_2 = 1$. The polynomial we use is ($N=n=2$): $$P(x) = x^2 + \left(\frac{1}{{\lambda}}-\frac{1}{\mu}\right) x - \frac{1}{\mu{\lambda}}.$$ Observe that $P(-1/{\lambda}) = P(1/\mu) = 0$. The matrix $B$ in this case is $$B = \begin{pmatrix} B_0(-{\lambda}) & B_1(-{\lambda}) \\ B_0(\mu) & B_1(\mu) \end{pmatrix} = \begin{pmatrix} - \frac{1}{{\lambda}\mu} & \frac{1}{{\lambda}} \\ - \frac{1}{{\lambda}\mu} & -\frac{1}{\mu} \end{pmatrix}.$$ We see that this has determinant $\frac{{\lambda}+\mu}{{\lambda}^2 \mu^2} \neq 0$, as we are assuming both ${\lambda}, \mu > 0$. Since $$\left|\frac1{{\lambda}}-\frac1{\mu}\right|+\frac1{|\mu{\lambda}|}\le2, \quad \frac1{\sqrt2}\le {\lambda},\mu\le1,$$ we infer For all $\frac1{\sqrt2} \leq {\lambda}, \mu \leq 1$ we have that $(0,0)$ lies in the interior of $A_{-{\lambda}, \mu}$. The above gives us a sufficient condition for checking whether a point $(-{\lambda}, \mu) \in {{\mathcal{Z}}}_{{\mathbb{R}}}$. To show a point $(-{\lambda}, \mu) \not\in {{\mathcal{Z}}}_{{\mathbb{R}}}$ it suffices to show that $(0,0) \not\in A$. This can be done utilizing information about the convex hull of $A$ and using the techniques described in [@HS]. In particular, let $K = K_0$ be the convex hull of $A$ and let $K_n = T_p(K_{n-1}) \cup T_m(K_{n-1})$. It is easy to see that $A \subset K_n$ for all $n$. Hence if there exists an $n$ such that $(0,0) \not\in K_n$ then $(0,0) \not\in A$. A precise description of $K$ is given in Section \[sec:unique\]. See Figure \[fig:Z2\] for illustration. ![Points in ${{\mathcal{Z}}}_{\mathbb R}$ (red) and points not in ${{\mathcal{Z}}}_{\mathbb R}$ (black)[]{data-label="fig:Z2"}](Z20.eps){width="275pt" height="275pt"} The Jordan block case --------------------- Consider the polynomial ($n=8, N=2$) $$P(x) = x^8 - \frac{8}{7 \nu} x^7 + \frac{1}{7 \nu^8}.$$ A quick check shows that $P(1/\nu) = P'(1/\nu) = 0$. Furthermore, for all $\nu \geq 0.831458513$ then we have $$\left| \frac{8}{7 \nu} \right| + \left|\frac{1}{7 \nu^8}\right| \leq 2.$$ In this case, the matrix from the Proof is $$B=\begin{pmatrix} 0 & \frac{1}{7 \nu^8} & \frac{2}{7 \nu^7} & \frac{3}{7 \nu^6} & \frac{4}{7 \nu^5} & \frac{5}{7 \nu^4} & \frac{6}{7 \nu^3} & \frac{7}{7 \nu^2} \\ \frac{1}{7 \nu^8} & \frac{1}{7 \nu^7} & \frac{1}{7 \nu^6} & \frac{1}{7 \nu^5} & \frac{1}{7 \nu^4} & \frac{1}{7 \nu^3} & \frac{1}{7 \nu^2} & -\frac{1}{\nu} \end{pmatrix}.$$ Clearly, the first $2\times 2$ minor of $B$ in this case is non-zero. It is shown in [@ShSo Theorem 2.6] that if $\nu < 0.6684$ that $A_\nu$ is disconnected, and hence totally disconnected, whence $\nu \not \in {{\mathcal{Z}}}_J$. Here we have that if $\nu > 0.8315$ then $\nu \in {{\mathcal{Z}}}_J$. Where exactly this dividing line is between these two conditions is still unclear. For that matter, it is not even clear if ${{\mathcal{Z}}}_J$ is a connected set, so the term “dividing line” might not be an accurate description of the boundary. The complex case ---------------- Theorem \[thm:tool2\] does not seem to be applicable here, so we use a different method. Notice that this method works for the other three cases as well (and even higher-dimensional ones – see [@HS-multi]) but gives worse bounds. If $A_{{\kappa}^2}$ is connected, then $A_{\kappa}$ has non-empty interior. In particular, this is the case if $|\kappa| \ge 2^{-1/4}$. Note first that if $|\kappa^2| \ge 1/\sqrt{2}$ then $A_{\kappa^2}$ is connected – see [@BH Proposition 1]. Moreover, by the Hahn-Mazurkiewicz theorem, $A_{{\kappa}^2}$ is path connected. Let $a, b \in A_{{\kappa}^2}$ and $\gamma$ the path connecting them. Consider ${\kappa}a, {\kappa}b \in {\kappa}A_{{\kappa}^2}$ and let $\gamma'$ be the path between them. As ${\kappa}\not \in {\mathbb{R}}$, we see that $\gamma$ and $\gamma'$ cannot be parallel lines. By observing that $\sum a_i \kappa^i = \sum a_{2i} \kappa^{2i} + {\kappa}\sum a_{2i +1} \kappa^{2i}$, we have $A_{\kappa}= A_{{\kappa}^2} + {\kappa}A_{{\kappa}^2}$ (the Minkowski sum). In particular, $A_{\kappa}$ will contain $\gamma + \gamma'$. By Theorem \[thm:Victor\] below, $\gamma+ \gamma'$ contains points in its interior, whence so does $A_{\kappa}$. Hence if $|{\kappa}|\ge 2^{-1/4}$, then $A_{\kappa}$ has non-empty interior. A great deal of information is known about the set $\mathcal{M}$ of all ${\kappa}$ for which $A_{\kappa}$ is connected – see [@Cal] and references therein. The following proof is by V. Kleptsyn (via Mathoverflow [@MO]). \[thm:Victor\] If $\gamma$ and $\gamma'$ are two paths in ${\mathbb{R}}^2$, not both parallel lines, then $\gamma + \gamma'$ has non-empty interior. See Appendix. Unique addresses and convex hulls {#sec:unique} ================================= Recall that a point $(x,y) \in A$ has a unique address (notation: $(x,y)\in\mathcal U$) if there is a unique sequence $(a_i)_0^\infty \in \{p, m\}^{\mathbb N}$ such that $(x,y) = \pi (a_1 a_2 a_3 \dots)$. These have been studied in [@HS] for the positive real case and in [@GS] for the one-dimensional real case. We say the set of all such points in $A$ is the [*set of uniqueness*]{} and denote it by ${{\mathcal{U}}}_{-{\lambda}, \mu}$, ${{\mathcal{U}}}_{\nu}$ and ${{\mathcal{U}}}_{\kappa}$ for the mixed real case the Jordan block case, and the complex case respectively. The purpose of this section is to provide a proof of Theorem \[thm:uniq\] by considering all three cases. The main outline of all three of these proofs are the same: - find the vertices for the convex hull of $A$; - show that these vertices have unique addresses; - using these vertices, in combination with Lemma \[thm:unique tool\] below, construct a set of points with unique addresses that have positive Hausdorff dimension. \[thm:unique tool\] Denote $\overline m=p, \overline p=m$ and assume that $u = a_1 a_2 \dots a_\ell$, $v = b_1 b_2 \dots b_k$ and $w = c_1 c_2 \dots c_n$ satisfy - $\pi [a_i a_{i+1} \dots a_\ell b_1 b_2 \dots b_k a_1 a_2 \dots a_\ell] \cap \pi [\overline{a_i}] = {\varnothing}$; - $\pi [b_j b_{j+1} \dots b_k a_1 a_2 \dots a_\ell] \cap \pi [\overline{b_j}] = {\varnothing}$; - $\pi [a_i a_{i+1} \dots a_\ell c_1 c_2 \dots c_n a_1 a_2 \dots a_\ell] \cap \pi [\overline{a_i}] = {\varnothing}$; - $\pi [c_j c_{j+1} \dots c_n a_1 a_2 \dots a_\ell] \cap \pi [\overline{c_j}] = {\varnothing}$. Then the images of $\{uv, uw\}^*$ under $\pi$ all have unique addresses. That is, the images of all infinite words of the form $t_1 t_2 t_3 \dots$ with $t_i \in \{uv, uw\}$ under $\pi$ all have unique addresses. We see that any shift of a word from $\{uv, uw\}^*$ is such that it’s prefix will be of one of the four forms listed above. Further, by assumption, the first term is uniquely determined. By applying $T_m^{-1}$ or $T_p^{-1}$ as appropriate, we get that all terms are uniquely determined, which proves the result. If the conditions of Lemma \[thm:unique tool\] are satisfied and $\{uv, uw\}^*$ is unambiguous, then $\dim_H\mathcal U>0$. We recall that $\{uv, uw\}$ is ambiguous if there exists two sequences $(t_1, t_2, t_3, \dots) \neq (s_1, s_2, s_3, \dots)$ with $t_i, s_i \in \{uv, uw\}$ where $t_1 t_2 t_3 \dots = s_1 s_2 s_3 \dots$. If no such sequence exists, then this language is unambiguous. For example, $\{mpmp, mp\}^*$ would be ambiguous, whereas $\{m, pp\}^*$ would be unambiguous. This is completely analogous to [@HS Corollary 4.3]. We say that a language $\mathcal{L}$ has positive topological entropy if the size of the set of prefixes of length $n$ of $\mathcal{L}$ grows exponentially in $n$. In brief, if we consider closure of all the shifts of sequences from $\{uv, uw\}^*$, then this set will clearly have positive topological entropy, and the injective projection $\pi$ of this set will have positive Hausdorff dimension. The mixed real case ------------------- We first assume that ${\lambda}\neq\mu$. The case when they are equal is considered in subsection \[sub:equal\] below. Let $0 < {\lambda}< \mu < 1$. The vertices of the convex hull of $A_{-{\lambda}, \mu}$ are given by $\pi((pm)^k p^\infty), \pi((mp)^k p^\infty), \pi((pm)^k m^\infty)$, and $\pi((mp)^k m^\infty)$, where $k\ge0$. It suffices to show that the lines from $\pi((pm)^k p^\infty)$ to $\pi((pm)^{k+1} p^\infty)$, and similarly from $\pi((mp)^k p^\infty)$ to $\pi((mp)^{k+1} p^\infty)$, from $\pi((pm)^k m^\infty)$ to $\pi((pm)^{k+1} m^\infty)$, and from $\pi((mp)^k m^\infty)$ to $\pi((mp)^{k+1} m^\infty)$ are support lines for $A_{-{\lambda}, \mu}$ and that their union is homeomorphic to a circle. We will do the first case only. The other cases are similar. We will proceed by induction. Consider first the line from $\pi(p^\infty)$ to $\pi(pmp^\infty)$. This will be in the direction $\pi(p^\infty) - \pi(pmp^\infty) = (2 {\lambda}, -2\mu)$, with slope $-\mu/{\lambda}$. Consider now the line from $\pi(p^\infty)$ to any other point $(x,y)=\pi(a_0a_1\dots)\in A_{-{\lambda},\mu}$. This will have a direction of the form $$\begin{aligned} \pi(p^\infty) - \pi(a_0 a_1 \dots) &= \left(\sum_{i=0}^\infty (1-a_i) (-{\lambda})^i, \sum_{i=0}^\infty (1-a_i) \mu^i\right)\\ &= \left(\sum_{i\text{ even}} (1-a_i) {\lambda}^i, \sum_{i\text{ even}} (1-a_i) \mu^i\right) \\ & + \left(-\sum_{i\text{ odd}} (1-a_i) {\lambda}^i, \sum_{i\text{ odd}} (1-a_i) \mu^i\right).\end{aligned}$$ Clearly, no point in $A_{-{\lambda},\mu}$ can have larger $y$-coordinate that $p^\infty$, whence the second coordinate is always non-negative. If the first coordinate is positive as well, then we are done; so, let us assume that it is negative. We notice that the slope has the form: $$\frac{ \sum_{i\text{ even}} (1-a_i) \mu^i + \sum_{i\text{ odd}} (1-a_i) \mu^i} { \sum_{i\text{ even}} (1-a_i) {\lambda}^i -\sum_{i\text{ odd}} (1-a_i) {\lambda}^i } \le -\frac{ \sum_{i\text{ odd}} (1-a_i) \mu^i} { \sum_{i\text{ odd}} (1-a_i) {\lambda}^i }$$ (since ${\lambda}<\mu$). We want $$\frac{ \sum_{i\text{ odd}} (1-a_i) \mu^i} { \sum_{i\text{ odd}} (1-a_i) {\lambda}^i } \le -\frac{\mu}{{\lambda}}.$$ Cross multiplying, this will occur if $${\lambda}\sum_{i\text{ odd}} (1-a_i) \mu^i \geq \mu \sum_{i\text{ odd}} (1-a_i) {\lambda}^i$$ or, equivalently, $$\sum_{i\text{ odd}} (1-a_i) \mu^{i-1} \geq \sum_{i\text{ odd}} (1-a_i) {\lambda}^{i-1}.$$ This is clearly true, as ${\lambda}< \mu$. This proves the base case $k=0$. ![Convex hull for $A_{-0.55, 0.8}$[]{data-label="fig:hull-mixed"}](conv-minus055-08.eps){width="275pt" height="275pt"} Assume the line from $\pi((pm)^j p^\infty)$ to $\pi((pm)^{j+1} p^\infty)$ is a support hyperplane for $A_{-{\lambda},\mu}$ for all $j<k$. Consider the line from $\pi((pm)^k p^\infty)$ to $\pi((pm)^{k+1} p^\infty)$. This will have slope $-\frac{\mu^k}{{\lambda}^k}$ Consider any $(x,y)=\pi(a_0a_1\dots) \in A_{-{\lambda}, \mu}$. Note that without loss of generality we can assume that $a_0a_1\dots a_{2k}=(pm)^k$, in view of the fact that the sequence of slopes, $-\frac{\mu^k}{{\lambda}^k}$, is is a decreasing negative sequence, so if $a_0\dots a_{2k}\neq (pm)^k$, then we can apply the inductive hypothesis for some $j<k$. As before, we see that the slope of this point is $$\frac{ \sum_{i\text{ even}} (1-a_i) \mu^i + \sum_{i\text{ odd}} (\varepsilon_i-a_i) \mu^i} { \sum_{i\text{ even}} (1-a_i) {\lambda}^i -\sum_{i\text{ odd}} (\varepsilon_i-a_i) {\lambda}^i } < -\frac{ \sum_{i\text{ odd}} (\varepsilon_i-a_i) \mu^i} { \sum_{i\text{ odd}}(\varepsilon_i-a_i) {\lambda}^i },$$ where $\varepsilon_i = -1$ if $i < 2k$ and $1$ otherwise. We want $$\frac{ \sum_{i\text{ odd}} (\varepsilon_i-a_i) \mu^i} { \sum_{i\text{ odd}} (\varepsilon_i-a_i) {\lambda}^i } < -\frac{\mu^{2k}}{{\lambda}^{2k}}.$$ Cross multiplying, this will occur if $${\lambda}^{2k} \sum_{i\text{ odd}} (\varepsilon_i-a_i) \mu^i \geq \mu^{2k} \sum_{i\text{ odd}} (\varepsilon_i-a_i) {\lambda}^i$$ or, equivalently, $$\sum_{i\text{ odd}} (\varepsilon_i-a_i) \mu^{i-2k} \geq \sum_{i\text{ odd}} (\varepsilon_i-a_i) {\lambda}^{i-2k}.$$ We see that ${\lambda}< \mu$ and hence $1/\mu < 1/{\lambda}$, from which it follows that $$\sum_{i\text{ odd}, i<2k} (\varepsilon_i-a_i) \mu^{i-2k} \geq \sum_{i\text{ odd}, i<2k} (\varepsilon_i-a_i) {\lambda}^{i-2k}$$ and $$\sum_{i\text{ odd}, i>2k} (\varepsilon_i-a_i) \mu^{i-2k} \geq \sum_{i\text{ odd}, i>2k} (\varepsilon_i-a_i) {\lambda}^{i-2k}.$$ Thus, we have shown that the line from $\pi((pm)^k p^\infty)$ to $\pi((pm)^{k+1} p^\infty)$ is a support line for $A_{-{\lambda},\mu}$ and that $A_{-{\lambda},\mu}$ lies below it. The remaining three cases (see the beginning of the proof) are similar, and once it is established whether $A_{-{\lambda},\mu}$ lies below or above these, the claim about their union being a topological circle becomes trivial. We leave the details to the reader. See Figure \[fig:hull-mixed\] for illustration. There exists an $L$ such that for all $k_1, k_2 > 0$ we have $u = m p^L$, $v = p^{k_1}$ and $w = p^{k_2}$ satisfy the conditions of Lemma \[thm:unique tool\]. We claim that there exists an $L$ such that for all $k \geq 0$ and $1 \leq i \leq L + k$ we have $$\label{eq:emp1} \pi[m p^{L+k} m p^L] \cap \pi[p]={\varnothing}.$$ and $$\label{eq:emp2} \pi[p^i m p^L] \cap \pi[m]={\varnothing}.$$ Consequently, using $u = m p^L$, $v = p^{k_1}$ and $w = p^{k_2}$ with $k_1, k_2 \geq 0$ in Lemma \[thm:unique tool\] proves the result. To prove (\[eq:emp1\]), we observe that $\pi(m p^\infty)$ is a point of uniqueness. Therefore, there exists an $L_1$ such $\pi[m p^{L_1}]$ will be disjoint from $\pi[p]$. To establish (\[eq:emp2\]), we observe that the point in $\pi[m]$ with maximal second coordinate is $\pi(mp^\infty)$. Denote this maximal second coordinate by $e$. We also observe that $\pi(p m p^\infty)$ has second coordinate strictly greater than $e$. Hence there exists an $L_2$ such that the minimal second coordinate of $\pi[p m p^{L_1}]$ is greater than $e$. By observing that the minimal second coordinate of $\pi[p^{i+1} m p^{L_2}]$ is always greater than that of $\pi[p^{i} m p^{L_1}]$, we see that $\pi[p^i m p^{L_2}]$ is disjoint from $\pi[m]$ for all $i$. Taking $L = \max(L_1, L_2)$ proves the claim. The set ${{\mathcal{U}}}_{-\mu, {\lambda}}$ has positive Hausdorff dimension. The Jordan block case --------------------- The vertices of the convex hull of $A_{\nu}$ are given by $\pi(m^k p^\infty)$, and $\pi(p^k m^\infty)$, where $k\ge0$. Recall that $\pi(a_0 a_1 a_2 \dots) = (\sum i a_i \nu^{i-1}, \sum a_i \nu^i)$. Consider the map taking an address $a_0 a_1\dots$ to $(x+y, y)$, because it will simplify our argument. Thus, we have $${\widetilde\pi}(a_0a_1\dots)=\left(\sum_{i=1}^\infty i(a_i+1)\nu^{i-1},\sum_{i=0}^\infty a_i\nu^i\right).$$ Note first that $${\widetilde\pi}(p m^\infty) - {\widetilde\pi}(m^\infty) = (0,2)$$ and for $w=a_0a_1\dots$, $${\widetilde\pi}(w) - {\widetilde\pi}(m^\infty) = \left(\sum_{i=1}^\infty i (a_i+1)\nu^{i-1}, \sum_{i=0}^\infty (a_i+1)\nu^i\right).$$ We notice that the first coordinate of ${\widetilde\pi}(w)-{\widetilde\pi}(m^\infty)$ is clearly nonnegative, which is enough to prove that $w$ is to the right of the vertical line from ${\widetilde\pi}(m^\infty)$ to ${\widetilde\pi}(p m^\infty)$. Proceed by induction and assume that for all $j<k$, the straight line passing through ${\widetilde\pi}(p^jm^\infty)$ and ${\widetilde\pi}(p^{j+1}m^\infty)$ is a support hyperplane for $A_\nu$ which lies to the left of the attractor – see Figure \[fig:jordan-hull\]. ![Convex hull for $A_{0.7}$[]{data-label="fig:jordan-hull"}](conv-jordan-07.eps){width="300pt"} Consider now the case $j=k$; we have $${\widetilde\pi}(p^{k+1} m^\infty) - {\widetilde\pi}(p^k m^\infty) = (2 k\nu^{k-1},2 \nu^k).$$ This sequence has the slopes $\nu/k$, which is clearly decreasing. Thus, we can assume that $a_i\equiv p,\ 0\le i\le k-1$, otherwise we appeal to a case $j<k$. We see that the desired result is true if the slope of ${\widetilde\pi}(w) - {\widetilde\pi}(p^k m^\infty)$ is less than or equal to $\nu/k$. After simplifying, this is equivalent to $$\frac{\sum_{i=k+1}^\infty (a_i+1)\nu^{i-k}}{\sum_{i=k+1}^\infty i(a_i+1)\nu^{i-k-1}} \le \frac{\nu}k,$$ which is clearly true, since $i>k$. The following claim is trivial. \[lem:uniq-general\] There exists an $L$ such that for all $k_1, k_2 \geq L$ we have $u = m^{k_1}$, $v = p^{k_1}$ and $w = p^{k_2}$ satisfy the conditions of Theorem \[thm:unique tool\]. The set of uniqueness $A_\nu$ has positive Hausdorff dimension. The complex case ---------------- For each $\phi \in [0, 2 \pi)$ define $p_\phi:{\mathbb{C}}\to {\mathbb{R}}$ by $p_\phi(z) = \Re(z {\mathrm{e}}^{-i \phi})$. This measures the distance of $z$ in the ${\mathrm{e}}^{i \phi}$ direction. We define the set $Z_\phi$ as those $z \in A$ such that $p_\phi(z)$ is maximized. We note that this set is well defined as $A$ is a compact set. If points $z \in Z_\phi$ then $z = \pi(a_1 a_2 a_3 \dots) = \sum_{j=0}^\infty a_{j}^{(\phi)} {\kappa}^j$, where $$a_j^{(\phi)} = \left\{ \begin{array}{ll} -1 & \mathrm{if}\ \Im({\kappa}^j {\mathrm{e}}^{i \phi}) < 0 \\ +1 & \mathrm{if}\ \Im({\kappa}^j {\mathrm{e}}^{i \phi}) > 0 \\ -1\ \mathrm{or}\ +1 & \mathrm{if}\ \Im({\kappa}^j {\mathrm{e}}^{i \phi}) = 0 \end{array} \right. \ \ \mathrm{for\ all}\ j.$$ These have been studied in [@Mess Sections 5–7] in the cases of the Rauzy fractal and the twin dragon curve (see Examples \[ex:twindragon\] and \[ex:rauzy\] above). We will distinguish two cases, depending on whether $\arg({\kappa})/\pi$ is irrational or rational. ### Case 1 – irrational Let $\mathcal {E}_\phi = \{(a_1 a_2 a_3 \dots) \mid \pi(a_1 a_2 a_3 \dots) \in Z_\phi\}$. We see that $|\mathcal{E}_\phi| = 1$ or $2$, as there is at most one $j$ where $\Im({\kappa}^j {\mathrm{e}}^{i \phi}) = 0$. All points $z \in {{\mathcal{Z}}}_\phi$ are points of uniqueness. Let $\bar{\mathcal{E}}_\phi$ denote the closure of the orbit of $\mathcal{E}_\phi$ under the shift transformation. Notice that any $z\in\pi(\bar{\mathcal{E}}_\phi)$ has a unique address, since for any $w\in A_{\kappa}$ we have $\Im (w e^{i\phi})\le \Im (z e^{i\phi})$, with the equality if and only if $w\in \bar{\mathcal{E}}_\phi$ (whose elements are all distinct). \[prop:Ephi\] Put $\mathcal E=\bigcup_{\phi} \bar{\mathcal{E}}_\phi$. We have - $\mathcal{E}$ is closed under the standard product topology. - $\mathcal{E}$ is uncountable. - $\mathcal{E}$ is a shift-invariant. - For each $(a_i)$ in $\mathcal{E}$ we have that $(a_i)$ is recurrent. - The image $\pi(\mathcal E)$ is a closed compact subset of $A_\kappa$. Notice that $(a_j^{(\phi)})_0^\infty$ is closely related to the irrational rotation of the circle $\mathbb R/\mathbb Z$ by $\arg({\kappa})/2\pi$, namely, $$a_j^{(\phi)}= \begin{cases} +1, & j\arg({\kappa})/2\pi\in\left(-\frac{\phi}{2\pi}-\frac14,-\frac{\phi}{2\pi}+\frac14\right)\bmod1,\\ -1, & j\arg({\kappa})/2\pi\in\left(-\frac{\phi}{2\pi}+\frac14,-\frac{\phi}{2\pi}-\frac14\right)\bmod1,\\ +1\text{ or } -1, & \text{otherwise} \end{cases}$$ (the third case can only occur for one $j$). In other words, each $(a_j^{(\phi)})$ is a hitting sequence for some semi-circle. Since our rotation is irrational, it is uniquely ergodic, whence $\bar{\mathcal{E}}_\phi$ is recurrent. The remaining properties are obvious. The sequences $(a_j^{(\phi)})$ are known to have subword complexity $2n$ (for $n$ large enough). Such sequences are studied in detail in [@Rote]. In particular, $\pi(\mathcal E_\phi)$ has zero Hausdorff dimension for all $\phi$. By the last property in Proposition \[prop:Ephi\], there exists $d > 0$ such that $$\mathrm{dist}(\pi(\mathcal{E} \cap [m]), \pi[p]) > d$$ and $$\mathrm{dist}(\pi(\mathcal{E} \cap [p]), \pi[m]) > d.$$ By taking $K$ such that $\frac{|\kappa|^{K+1}}{1-|\kappa|} < d$, we observe that for all $(a_i) \in \mathcal{E}$, $$\pi [a_0 a_1 \dots a_K] \cap \pi [\overline{a_0}] = {\varnothing}.$$ As the sequence is recurrent, for any $(a_i) \in \mathcal{E}$ there will exist two subwords of length $L > K$ of the form $b_1 b_2 \dots b_L b_{L+1}$ and $b_1 b_2 \dots b_L \overline{b_{L+1}}$. (If two such words did not exist, then the sequence would necessarily be periodic.) Since this sequence is recurrent, there exist $c_1 \dots c_m$ and $d_1 \dots d_n$ such that $$b_1 b_2 \dots b_L b_{L+1} c_1 \dots c_m b_1 b_2 \dots b_L$$ and $$b_1 b_2 \dots b_L \overline{b_{L+1}} d_1 \dots d_n b_1 \dots b_L$$ are both subwords of $(a_i)$. It is easy to see that $u = b_1 \dots b_L$, $v = b_{L+1} c_1 \dots c_m$ and $w = \overline{b_{L+1}} d_1 \dots d_n$ satisfy the conditions of Lemma \[thm:unique tool\], from which it follows that the images of $\{uv, uw\}^*$ will all have unique address. As this set has positive topological entropy, we have that the set of uniqueness has positive Hausdorff dimension. Thus, in this case the points $z_\phi$ are all points of uniqueness. Furthermore, they are the vertices of the convex hull of $A_{\kappa}$. The proof is essentially the same as that of [@Mess Théorème 7], so we omit it. ### Case 2 – rational {#sec:complex rational unique} Let now ${\kappa}= \rho {\mathrm{e}}^{2\pi i p/q}$ with $(p,q)=1$. Put $$q'=\begin{cases} q,& q\ \text{odd},\\ q/2,& q\ \text{even} \end{cases}$$ and $$\label{eq:beta} {\beta}=\rho^{-q'}>1.$$ If ${\beta}\le2$, then $A_{\kappa}$ is a convex polygon. Put $$J=\left\{\sum_{k=0}^\infty b_k{\beta}^{-k}\mid b_k\in\{\pm1\}\right\}.$$ Since ${\beta}\le2$, we have $J=\bigl[-\frac{{\beta}}{{\beta}-1},\frac{{\beta}}{{\beta}-1}\bigr]$. Now the claim follows from the fact that $A_{\kappa}$ can be expressed as the following Minkowski sum: $$A_{\kappa}=J+{\kappa}J+\dots +{\kappa}^{q'-1} J.$$ Let $U_{\beta}$ denote the set of all unique addresses for $x=\sum_{k=0}^\infty b_k{\beta}^{-k}$ with $b_k\in\{\pm1\}$. \[lem:uniq2uniq\]We have: 1. if $(a_k)_{k=0}^\infty$ is a unique address in $A_{\kappa}$, then $(a_{q'j+\ell})_{j=0}^\infty\in U_{\beta}$ for all $\ell\in\{0,1,\dots,q'-1\}$; \[i\] 2. if $(a_{q'j})_{j=0}^\infty$ belongs to $U_{\beta}$, then there exists $(b_k)_{k=0}^\infty$ such that $b_{q'j}=a_{q'j}$ for all $j\ge0$, and $(b_k)_{k=0}^\infty$ is a unique address in $A_{\kappa}$. \[ii\] \(i) If $(a_{q'j+\ell})$ were not unique, there would exist $(b_{q'j+\ell})$ such that $\sum_{j=0}^\infty a_{q'j+\ell}{\beta}^{-j}=\sum_{j=0}^\infty b_{q'j+\ell}{\beta}^{-j}$, i.e., $\sum_{j=0}^\infty a_{q'j+\ell}{\kappa}^{q'j}=\sum_{j=0}^\infty b_{q'j+\ell}{\kappa}^{q'j}$, whence $(a_k)$ could not be a unique address. \(ii) Let $q$ be odd; the even case is similar. Put for $k\not\equiv0\bmod q$, $$b_k=\begin{cases} +1, & \Im({\kappa}^k)>0,\\ -1, & \Im({\kappa}^k)<0. \end{cases}$$ Clearly, this sequence is well defined, since $\Im({\kappa}^k)\neq0$ if $k\not\equiv0\bmod q$. Now put $b_{qj}=a_{qj}$ for all $j\ge0$. We claim that the resulting sequence $(b_k)_{k=0}^\infty$ is a unique address. Indeed, by our construction, $\Im(\sum_{k=0}^\infty b_k'{\kappa}^k)\le \Im(\sum_{k=0}^\infty b_k{\kappa}^k)$ for any $(b_k')$, with the equality if and only if $b_k'\equiv b_k$ for all $k\not\equiv0\bmod q$. If such an equality takes place, then $\sum_{k=0}^\infty (b_k'-b_k){\kappa}^k$ is real. Moreover, $\sum_{k=0}^\infty (b_k'-b_k){\kappa}^k=\sum_{j=0}^\infty (b'_{qj}-a_{qj}){\beta}^{-j}\neq0$, since $(a_{qj})_{j=0}^\infty\in U_{\beta}$. This yields the following result. \[lem:finite-finite\] The set of uniqueness $\mathcal U_{\kappa}$ is finite if and only if $U_{\beta}$ is. If these sets are infinite, then their cardinalities are equal. Furthermore, $\dim_H \mathcal U_{\kappa}>0$ if and only if the topological entropy of $U_{\beta}$ is positive. By Lemma \[lem:uniq2uniq\] part \[i\], we see that the cardinality of $\mathcal{U}_\kappa$ is bounded above by the cardinality of $\underbrace{U_\beta \times \dots \times U_\beta}_{q'}$, and hence by the cardinality of $U_\beta$. By part \[ii\] we see that the cardinality of $\mathcal{U}_\kappa$ is bounded below by the cardinality of $U_\beta$. This proves the first two statements. If $\dim \mathcal{U}_\kappa > 0$ then $\mathcal{U}_\kappa$ has positive topological entropy, and hence so does $\underbrace{U_\beta \times \dots \times U_\beta}_{q'}$. The other direction is similar. Let ${\beta}_*=1.787231650\dots$ denote the *Komornik-Loreti constant* introduced by V. Komornik and P. Loreti in [@KL], which is defined as the unique solution of the equation $\sum_{n=1}^{\infty}\mathfrak{m}_{n}x^{-n+1}=1$, where $\mathfrak{m}=(\mathfrak{m}_n)_1^\infty$ is the Thue-Morse sequence $$\mathfrak{m}=0110\,\,1001\,\,1001\,\,0110\,\,1001\,\,0110\dots,$$ i.e., the fixed point of the substitution $0\to01,\ 1\to10$. Put $G=\frac{1+\sqrt5}2$. The following result gives a complete description of the set $U_{\beta}$. \[thm:GS\] The set $U_{\beta}$ is: 1. $\bigl\{-\frac{{\beta}}{{\beta}-1},\frac{{\beta}}{{\beta}-1}\bigr\}$ if ${\beta}\in (1,G]$; 2. infinite countable for ${\beta}\in(G,{\beta}_*)$; 3. an uncountable set of zero Hausdorff dimension if ${\beta}={\beta}_*$; and 4. a set of positive Hausdorff dimension for ${\beta}\in ({\beta}_*,\infty)$. Lemma \[lem:finite-finite\] and Theorem \[thm:GS\] yield \[thm:uniq-rational\] Let ${\beta}$ be given by (\[eq:beta\]). Then the set of uniqueness $\mathcal U_{\kappa}$ for the rational case is: 1. finite non-empty if ${\beta}\in (1,G]$; 2. infinite countable for ${\beta}\in(G,{\beta}_*)$; 3. an uncountable set of zero Hausdorff dimension if ${\beta}={\beta}_*$; and 4. a set of positive Hausdorff dimension for ${\beta}\in ({\beta}_*,\infty)$. ![Convex hulls for $A_{0.7i}$ (a square), $A_{0.7 {\mathrm{e}}^{2 \pi i/5}}$ (a decagon), $A_{0.7 {\mathrm{e}}^{\pi i/3}}$ (a hexagon) and $A_{0.4 + 0.5i}$ (an “infinite polygon”)[]{data-label="fig:conv-complex"}](Picab07_4.eps "fig:"){width="175pt" height="175pt"} ![Convex hulls for $A_{0.7i}$ (a square), $A_{0.7 {\mathrm{e}}^{2 \pi i/5}}$ (a decagon), $A_{0.7 {\mathrm{e}}^{\pi i/3}}$ (a hexagon) and $A_{0.4 + 0.5i}$ (an “infinite polygon”)[]{data-label="fig:conv-complex"}](Picab07_5.eps "fig:"){width="175pt" height="175pt"}\ ![Convex hulls for $A_{0.7i}$ (a square), $A_{0.7 {\mathrm{e}}^{2 \pi i/5}}$ (a decagon), $A_{0.7 {\mathrm{e}}^{\pi i/3}}$ (a hexagon) and $A_{0.4 + 0.5i}$ (an “infinite polygon”)[]{data-label="fig:conv-complex"}](Picab07_6.eps "fig:"){width="175pt" height="175pt"} ![Convex hulls for $A_{0.7i}$ (a square), $A_{0.7 {\mathrm{e}}^{2 \pi i/5}}$ (a decagon), $A_{0.7 {\mathrm{e}}^{\pi i/3}}$ (a hexagon) and $A_{0.4 + 0.5i}$ (an “infinite polygon”)[]{data-label="fig:conv-complex"}](Picab0405.eps "fig:"){width="175pt" height="175pt"} Note that if $\arg({\kappa})/\pi \in {\mathbb{Q}}$ and ${\beta}>2$, then the convex hull of $A_{\kappa}$ is still a $2q'$-gon. This follows directly from [@SW Theorem 4.1]. See also [@SW2] for further discussion on the convex hull of $A$. If ${\beta}\le G$, then we have a bound $\#\mathcal U_{\kappa}\le 2^{q'}$. In fact, one can show that $\#\mathcal U_{\kappa}=2q'$ - more precisely, only the extreme points of $A_{\kappa}$ are points of uniqueness. This is completely analogous to [@S07 Theorem 2.7] which deals with the self-similar IFS without rotations. We leave a proof to the interested reader. See Figure \[fig:conv-complex\] for illustration. The remaining mixed real case {#sub:equal} ----------------------------- Finally let $$M = \begin{pmatrix} -{\lambda}& 0 \\ 0 & {\lambda}\end{pmatrix}$$ for $0 < {\lambda}< 1$. \[lem:-la la\] 1. If $\lambda < 1/\sqrt{2}$ then $A_{-{\lambda}, {\lambda}}$ is totally disconnected. 2. If $\lambda \geq 1/\sqrt{2}$ then $A_{-{\lambda}, {\lambda}}$ is a parallelogram. We have that $x = \sum_{k=0}^\infty a_k (-{\lambda})^k$ and $y = \sum_{k=0}^\infty a_k {\lambda}^k$. Make a change of coordinates $(x,y) \to \left(\frac{x+y}{2}, \frac{x-y}{2}\right)$. Then $$\begin{aligned} x& = \sum_{k=0}^\infty a_{2k}\lambda^{2k} \\ y& = \sum_{k=0}^\infty a_{2k+1}\lambda^{2k+1},\end{aligned}$$ where $a_j\in\{0,1\},\ j\ge0$. If ${\lambda}< \frac{1}{\sqrt{2}}$ then the set of $x$’s and $y$’s are both Cantor sets, and hence $A_{-{\lambda}, {\lambda}}$ is disconnected. If ${\lambda}> \frac{1}{\sqrt{2}}$ then $x \in \left[ \frac{-1}{1-{\lambda}^2}, \frac{1}{1-{\lambda}^2}\right]$ and $y \in \left[ \frac{-{\lambda}}{1-{\lambda}^2}, \frac{{\lambda}}{1-{\lambda}^2}\right]$, with $x$ and $y$ independent, and taking all values in these intervals. Thus, under this change of variables, the attractor is a rectangle. Inverting the change of variables proves the result. Lemma \[lem:-la la\] implies that the bound $1/\sqrt{2}$ in Theorem \[thm:interior\] is sharp for the mixed real case. Let $\beta = {\lambda}^{-2}$. The set $U_{-{\lambda}, {\lambda}}$ is: 1. finite non-empty if ${\beta}\in (1,G]$; 2. infinite countable for ${\beta}\in(G,{\beta}_*)$; 3. an uncountable set of zero Hausdorff dimension if ${\beta}={\beta}_*$; and 4. a set of positive Hausdorff dimension for ${\beta}\in ({\beta}_*,\infty)$. Notice that if $(a_{2k})_0^\infty \in U_{\beta}$, then $(a_k)_0^\infty\in U_{-{\lambda},{\lambda}}$ with $a_{2k+1}\equiv -1, k\ge0$. The rest of the proof goes exactly like in the previous subsection, so we omit it. Appendix: proof of Theorem \[thm:Victor\] ========================================= \[lem:Victor\] Let $\gamma$ and $\gamma'$ be two paths in ${\mathbb{C}}$. Let $\delta$ be the diameter of $\gamma([s_1,s_2])$, and assume that there is no point with nonzero index with respect to the loop $\sigma=\{\gamma(s) + \gamma'(t) :s,t\in\partial([s_1, s_2] \times [0,1])$. Then the sets $\gamma(s_1) + \gamma'([0,1])$ and $\gamma(s_2) + \gamma'(([0,1])$ coincide outside $\delta$-neighbourhoods of $\gamma([s_1, s_2])+\gamma'(0)$ and $\gamma([s_1, s_2])+\gamma'(1)$. Assume the contrary and let $z$ be a point of the curve $\widetilde\gamma:=\gamma([s_1,s_2])+\gamma(t_1)$ that lies outside the above neighbourhoods and that does not belong to the $\gamma([s_1,s_2])+\gamma(t_2)$. By continuity, there is $\varepsilon$-neighbourhood of $z$ that the latter curve does not intersect. Now, by the Jordan curve Theorem, in this neighbourhood one can find two points “on different sides” with respect to $\widetilde\gamma$. These two points have thus different indices with respect to the loop $\sigma$. Hence, for at least one of them this index is non-zero. Let $\gamma$ and $\gamma'$ be two paths in ${\mathbb{C}}$ with $\gamma(0) = a$, $\gamma(1) = b$, $\gamma'(0) = c$ and $\gamma'(1) = d$. Consider the loop $$\omega:=\{\gamma(s) + \gamma'(t) : (s,t)\in\partial([0,1] \times [0,1])\}.$$ Any point not on $\omega$ that has non-zero index with respect to this loop is contained in $\gamma([0,1]) + \gamma'([0,1])$. This yields a point in the interior of $\gamma([0,1]) + \gamma'([0,1])$. Hence it suffices to show that there exists a point of non-zero index. Let $\delta=\delta(s_1,s_2)$ be the diameter of $\gamma([s_1, s_2])$ for $s_1, s_2 \in [0,1]$. Clearly, $\delta \to 0$ as $s_1 \to s_2$. Pick $s_1$ and $s_2$ sufficiently close so the diameter of $\gamma'([0,1])$ is greater than $2\delta$. Hence there exists a point on the curve $\gamma(s_1) + \gamma'([0,1])$ that is neither in the $\delta$-neighbourhood of $\gamma([s_1, s_2]) + \gamma'(0)$ nor in the $\delta$-neighbourhood of $\gamma([s_1, s_2]) + \gamma'(1)$. By Lemma \[lem:Victor\], either there exists a point not on this curve of non-zero index, or $\gamma(s_1) + \gamma'([0,1])$ and $\gamma(s_2) + \gamma'([0,1])$ coincide outside the $\delta$-neighbourhoods of $\gamma([s_1, s_2]) + \gamma'(0)$ and $\gamma([s_1, s_2]) + \gamma'(1)$. Taking $s_1 \to s_2$ and assuming that there is never a point of non-zero index gives that $\gamma'([0,1])$ admits an arbitrarily small translation symmetry outside its endpoints, and hence is a straight line. Reversing the roles of $\gamma$ and $\gamma'$ gives that either there is a point of non-zero index, or $\gamma([0,1])$ is also a straight line. If $\gamma$ and $\gamma'$ are both straight lines, then $\gamma+\gamma'$ is a parallelogram, and will only have empty interior if $\gamma$ and $\gamma'$ are parallel. By assumption, $\gamma$ and $\gamma'$ are not parallel lines, and hence $\gamma + \gamma'$ contains a point in its interior. Open questions {#sec:conc} ============== [**1.**]{} Let $d\ge3$ and let $M$ be a $d\times d$ real matrix whose eigenvalues are all less than 1 in modulus. Denote by $A_M$ the attractor for the contracting self-affine iterated function system (IFS) $\{Mv-u, Mv+u\}$, where $u$ is a cyclic vector. The following result is proved in our most recent paper on the subject to date [@HS-multi]. If $$|\det M|\ge 2^{-1/d},$$ then the attractor $A_M$ has non-empty interior. In particular, this is the case when each eigenvalue of $M$ is greater than $2^{-1/d^2}$ in modulus. Clearly, this is generalisation of Theorem \[thm:interior\] to higher dimensions (albeit with different constants). Is it true that $A_M$ contains no holes if all the eigenvalues are close enough to 1? [**2.**]{} Is there a closed description of $\mathcal B:=\partial A$? In particular, does $\mathcal B$ always have Hausdorff dimension greater than 1? The known examples for the complex case involve ${\kappa}$ which are Galois conjugates of certain Pisot numbers (algebraic integers greater than 1 whose other conjugates are less than 1 in modulus) – e.g. the Rauzy fractal for the tribonacci number or the fractal associated with the smallest Pisot number [@AkSad] – in which case one can generate the boundary via a self-similar IFS. [**3.**]{} Denote $\mathcal B_m=\partial T_m(A), \mathcal B_p=\partial T_p(A)$ and $\mathcal B_0=\mathcal B \cap \mathcal B_m\cap \mathcal B_p$. If $z\in \mathcal B_0$, then clearly, $z\notin \mathcal U$. ![The attractor $A_{0.5+0.58i}$. It appears that $z$ and $z'$ are the only points in $\mathcal B_0$. If this is indeed the case, then all except a countable set of points of the boundary have a unique address.[]{data-label="fig:intersection"}](complex-05-058.eps){width="350pt"} The set $$\mathcal B_0':=\bigcup_{\substack{n\ge0\\ (i_1,\dots,i_n)\in\{m,p\}^n}} T_{i_1}\dots T_{i_n}(\mathcal B_0)$$ lies in $\mathcal B$. Moreover, $\mathcal B\setminus \mathcal B_0'\subset \mathcal U$. Note first that $T_i(\mathcal B)\subset \mathcal B$ for $i\in\{m,p\}$, whence follows the first claim. Now, suppose $z\in\mathcal B\setminus \mathcal B_p$, say. Then the first symbol of any address of $z$ has to be $m$. Let us shift this address, which corresponds to applying $T_m^{-1}$ to $z$ in the plane. If the resulting point is in $\mathcal B\setminus \mathcal B_p$ or $\mathcal B\setminus \mathcal B_m$, then the first symbol of its address is also unique, etc. Hence follows the second claim. 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Rote, *Sequences with subword complexity $2n$*, J. Number Theory [**46**]{} (1972), 196-–213. P. Shmerkin and B. Solomyak, *Zeros of $\{-1,0,1\}$ power series and connectedness loci for self-affine sets*, Exp. Math. [**15**]{} (2006), 499–511. N. Sidorov, *Combinatorics of linear iterated function systems with overlaps*, Nonlinearity [**20**]{} (2007), 1299–1312. R. S. Strichartz and Y. Wang, *Geometry of self-affine tiles I*, Indiana Univ. Math. J. [**48**]{} (1999), 1–-23. R. Kenyon, J. Li, R. S. Strichartz and Y. Wang, *Geometry of self-affine tiles II*, Indiana Univ. Math. J. [**48**]{} (1999), 25–42. [^1]: Research of K. G. Hare was supported by NSERC Grant RGPIN-2014-03154 [^2]: Computational support provided in part by the Canadian Foundation for Innovation, and the Ontario Research Fund. [^3]: In [@KL2007] V. Komornik and P. Loreti obtained a similar result for the condition $A'_{{\kappa}}\supset \{z : |z| \le R\}$ for an arbitrary $R>1$.

--- abstract: 'Time Reversal Violation (TRV) interactions between quarks which appear in Standard Model (SM) and beyond-SM theories can induce TRV components in the nucleon-nucleon potential. The effects of these components can be studied by measuring the electric dipole moment (EDM) of light nuclei. In this work we present a complete derivation of the TRV nucleon-nucleon and three-nucleon potential up to next-to-next-to leading order (N2LO) in a chiral effective field theory ($\chi$EFT) framework. The TRV interaction is then used to evaluate the EDM of $\det$, $\tri$ and $\hel$ focusing in particular on the effects of the TRV three-body force and on the calculation of the theoretical errors. In case of a measurement of the EDM of these nuclei, the result of present work would permit to determine the values of the low energy constants and to identify the source of TRV.' author: - 'A. Gnech$^{\,{\rm a,b}}$, M. Viviani$^{\,{\rm b}}$' title: Time Reversal Violation in Light Nuclei --- Introduction ============ The violation of parity (P) with the conservation of charge conjugation (C) generates a CP violation that, using the CPT theorem, results in a time reversal violation (TRV). TRV is a key ingredient in the explanation of the observed matter-antimatter asymmetry in the Universe  [@AS67]. The Standard Model (SM) has a natural source of CP-violation in the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix, however this mechanism is not sufficient for explaining the observed asymmetry [@AC93]. This discrepancy opens a window in possible TRV effect in the SM, such as the $\theta$-term in the Quantum Chromodynamics (QCD) sector [@GH76], or in other sources of beyond-SM (BSM) theories [@MP05]. The measurement of Electric Dipole Moments (EDMs) of particles is the most promising observable for studying TRV beyond CKM mixing matrix effects. Indeed, the EDM induced by the complex phase of the CKM matrix are suppressed since the EDM does not involve flavour changing [@MP05; @AC97; @MU12; @MU13]. Therefore, any non-vanishing EDM of a nuclear or an atomic system would highlight TRV effects beyond the CKM mixing matrix. The present experimental upper bounds on the EDMs of neutron and proton are $|d_n|<2.9\cdot 10^{-13}\ e$ fm [@CA06] and $|d_p|<7.9\cdot 10^{-12}\ e$ fm, where the proton EDM has been inferred from a measurement of the diamagnetic ${}^{199}{\rm Hg}$ atom [@WC09] using a calculation of the nuclear Schiff moment [@VF03]. For the electron, the most recent upperbound is $|d_e|<8.7\cdot10^{-16}\ e$ fm [@JB14], derived from the EDM of the ThO molecule. In this context, there are proposals for the direct measurement of EDMs of electrons, single nucleons and light nuclei in dedicated storage rings  [@YF06; @YK11; @AL13; @JP13; @FR13]. This new approach plans to reach an accuracy of $\sim 10^{-16}\ e$ fm, improving the sensitivity in particular in the hadronic sector. Any measurement of a non-vanishing EDM of this magnitude would be the evidence of TRV beyond CKM effects. However, a single measurement will not be sufficient to identify the source of TRV. For this reason, the measurement of EDM of various light nuclei such as $\det$, $\tri$ and $\hel$ can impose constrains on the TRV sources. On the other hand, also spin rotation of polarized neutrons along the $y$-axis can be used as probe for TRV [@PK82; @LS82; @CP04; @CP06; @JDB14]. This explorative study is motivated by the new sensitivity reached by various cold neutron facilities such as the Los Alamos Neutron Science Center, the National Institute of Standards and Technology (NIST) Center for Neutron Research, and the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory. Even if the sensitivity of the spin-rotation on TRV is far away from the present sensitivity of the EDM, this observable addresses the same physics of the EDM, posing, for the future, as a complementary and independent test for TRV in nuclei. Also other TRV observables in proton- and neutron-nucleus scattering have been proposed as probes of TRV effects [@YH11; @JDB14; @YNU16; @VG18; @PF19]. The use of light nuclei to study TRV results to be advantageous because the nuclear physics of the systems is theoretically under control and so the TRV effects can be easily highlighted. In particular, the chiral effective field theory ($\chi$EFT) has provided a practical and successful scheme to study two and many-nucleon interactions [@DR15; @EE15] and the interaction of electroweak probes with nuclei  [@Park96; @Park03; @Koelling09; @Pastore09; @Koelling11; @Pastore11; @Piarulli13]. The $\chi$EFT approach is based on the observation that the chiral symmetry exhibited by QCD has a noticeable impact in the low-energy regime. Therefore, the form of the strong interactions of pions among themselves and other particles is severely constrained by the transformation properties of the fundamental Lagrangian under Lorentz, parity, time-reversal, and chiral symmetry [@W66; @CCWZ69]. The Lagrangian terms can be organized as an expansion in powers of $Q/\Lx$, where $\Lx\simeq1$ GeV specifies the chiral symmetry breaking scale and $Q$ is the exchanged pion momentum. Each term is associated to a low-energy constants (LECs) which are then determined by fits to experimental data. The $\chi$EFT method permits to construct also an effective TRV Lagrangian treating all possible sources of TRV. The TRV Lagrangian induced by the $\theta$-term was derived in Refs. [@EM10; @JB150]. Also BSM terms such as supersymmetry, multi-Higgs scenarios, left-right symmetric model [*etc.*]{} induce TRV operators at the quark-gluon level which appear at level of dimension six (see for example Ref. [@BG10]). The $\chi$EFT Lagrangians for these sources were derived in Refs. [@JB150; @JV13; @WD13; @JV11]. This approach permits not only to determine the TRV interactions but also to estimate the chiral order of the LECs and their values as function of the fundamental parameters, providing a direct connection between the fundamental theories and the nuclear observables  [@EM10; @JB150; @JV13]. To be noticed that the chiral order of the Lagrangian terms, which is determined by the products of the chiral order of the dynamical part and that of the LECs, really depends on the particular source of TRV. Therefore, when the LECs will be determined experimentally it will be possible to identify the dynamical properties of the TRV source [@JV11; @WD14]. Starting from the TRV Lagrangian, de Vries [*et al.*]{} [@CM11] and also Bsaisou [*et al.*]{} [@JB13] derived the chiral potential up to next-to-next leading order (N2LO) including only nucleon-pion interaction and contact interactions. In both works also the electromagnetic currents which play a role at N2LO for the EDM were derived but only in Ref. [@JB13] they were used to evaluate the EDM of the deuteron. In Ref. [@JV11b] the calculation of the EDM of $\det$, $\tri$ and $\hel$ was performed using only the one-pion-exchange part of the TRV potential coupled with phenomenological potential for the parity conserving (PC) part of the interaction. Subsequent works showed the presence in the TRV Lagrangian of a three-pion term [@JB150], which was included in the calculation for the first time by Bsaisou [*et al.*]{} [@JB151]. This term generates at next-to-leading order (NLO) also a TRV three body force, which contribution to the $\tri$ and $\hel$ EDM was found to be smaller than expected by the chiral counting. The calculation reported in Ref. [@JB151] was also the first to use a complete chiral approach including the TRV potential up to NLO and the PC potential up to N2LO. The aim of this work is twofold: First, the construction of a TRV potential up to N2LO considering all possible TRV interaction terms in the $\chi$EFT without making any assumption for the chiral order of the LECs. In this way all possible sources of TRV can be studied just setting the LECs to their estimated values and turning on and off the various terms in the Lagrangian. The second is the study of the EDM of $\det$, $\tri$ and $\hel$, providing a suitable framework for the future determination of the LECs. In our calculation the contribution of the TRV three-body force is found to be sizably larger than reported in Ref. [@JB151]. Finally, it is worthy to mention that exists a different approach to the derivation of the TRV nuclear forces based on meson-exchange model [@CP04]. This model includes pion and vector-meson exchange with 10 unknown meson constants. Such a theory, which has a wider energy range of validity but it is less systematic and with no direct connection with the fundamental Lagrangian, has been used to study the EDM of light nuclei [@CP04; @IS08; @NY15; @YH13] and the neutron spin rotation $\vec{n}-\vec{p}$ [@CP06] and $\vec{n}-\vec{d}$  [@CP04; @YH11] scattering. The present paper is organized as follows. In Sec. \[sec:trvlag\] we will present the TRV chiral Lagrangian up to order $Q^2$ relevant for the calculation of the TRV potential, while in Sec. \[sec:trvpot\] we derive the TRV potential at N2LO. In Sec. \[sec:res\], we report the results obtained for the EDM of $\det$, $\tri$ and $\hel$ using the N2LO potential. Finally, in Sec. \[sec:conc\] we present our conclusions and perspectives. The technical details relative to the contributions of the various diagrams, and of the derivation of the potential in configuration space are given in Appendices \[app:vertex\],  \[app:pot\] and \[app:rpot\]. Moreover, in Appendix \[app:NNNconv\] we will give some details of the calculation of the trinucleon wave function negative-parity component and about the convergence of the TRV three body force contribution. The TRV Lagrangian {#sec:trvlag} ================== The various possible sources of TRV in SM induce a TRV $NN$ and $NNN$ potential. This potential can be constructed starting from a pion-nucleon effective Lagrangian which includes, in principle, an infinite set of terms which violates the chiral symmetry as the fundamental (quark-level) Lagrangian. The effective Lagrangian can be ordered by a power counting scheme which permits to select the most important interactions. In literature the chiral order of the TRV Lagrangian is determined by considering the estimated order of the LECs [@EM10; @JV13; @JB150] which, however, is source dependent. In this section we present only the TRV Lagrangian terms which can give some contribution to TRV $NN$ and $NNN$ potential up to N2LO in terms of pion field. In order to remain source independent we consider isoscalar, isovector and isotensor terms and we determine the chiral order of the TRV Lagrangian considering only the dynamical part. Namely, in the following we will consider all the TRV LECs to be equally important. To deal with a specific source of TRV, it will be sufficient to set some of the LECs to be zero, etc. At order $Q^0$ the TRV pion-nucleon Lagrangian includes three terms  [@EM10; @JB150] $$\begin{aligned} {\cal{L}}_{\rm TRV}^{\pi N\,(0)}=g_0\overline{\psi}\vec{\pi}\cdot\vec{\tau} \psi+g_1\overline{\psi}\pi_3\psi+g_2\overline{\psi}\pi_3\tau_3\psi\,,\end{aligned}$$ where $\vec{\pi}$ is the pion field and $\psi$ is the nucleon field. To be noticed that the isotensor term is usually considered of higher order. At the same order a purely pionic interaction appears [@JB150], which reads $$\begin{aligned} {\cal{L}}_{\rm TRV}^{3\pi\,(0)}=M\Delta_3 \pi_3\pi^2\,,\end{aligned}$$ where $M=938.88$ MeV is the average nucleon mass. At order $Q$ we have only one term that can give contribution to the potential up to N2LO and it reads, given explicitly as, $$\begin{aligned} {\cal{L}}_{\rm TRV}^{\pi N\,(1)}=\frac{g_V^{(1)}}{2M\fp}\big[\overline{\psi} \partial_\mu(\vec{\pi}\times\vec{\tau})_3\partial^\mu\psi+\ h.c.\ \big]\,, \label{eq:ltrv1}\end{aligned}$$ while no Lagrangian terms for the pure pionic interaction are allowed. At order $Q^2$ we get six new contributions, =1.0pt $$\begin{aligned} {\cal{L}}_{\rm TRV}^{\pi N\,(2)}&=& {g_{S1}^{(2)}\over\fp^2}\overline{\psi}\partial_\mu\partial^\mu (\pi\cdot\tau)\psi\nonumber\\ &+&{g_{S2}^{(2)}\over 2M^2\fp^2}\big[\overline{\psi}\partial_\mu\partial_\nu (\pi\cdot\tau)\partial^\mu\partial^\nu\psi+{\rm h.c.} \big]\nonumber\\ &+&{g_{V1}^{(2)}\over\fp^2}\overline{\psi}\partial_\mu\partial^\mu \pi_3\psi\nonumber\\ &+&{g_{V2}^{(2)}\over 2M^2\fp^2}\big[\overline{\psi}\partial_\mu\partial_\nu \pi_3\partial^\mu\partial^\nu\psi+{\rm h.c.} \big]\nonumber\\ &+&{g_{T1}^{(2)}\over\fp^2}\overline{\psi}\partial_\mu\partial^\mu \pi_3\tau_3\psi\nonumber\\ &+&{g_{T2}^{(2)}\over 2M^2\fp^2}\big[\overline{\psi}\partial_\mu\partial_\nu \pi_3\tau_3\partial^\mu\partial^\nu\psi+{\rm h.c.} \big]\ , \label{eq:ltrv2}\end{aligned}$$ where $S$, $V$, $T$ stand for isoscalar, isovector and isotensor respectively and $\fp\simeq92$ MeV is the pion decay constant. The three-pion interaction that appears at this level gives contributions of order $Q^2$ to the TRV nuclear potential which is beyond our purpose, therefore we do not consider it. The Lagrangian contains also four-nucleon contact terms ${\cal L}^{\rm CT}_{\rm TRV}$ representing interactions originating from excitation of resonances and exchange of heavy mesons. At lowest order ${\cal L}^{\rm CT}_{\rm TRV}$ contains only five independent four-nucleon contact terms with a single gradient. In the following we also need the PC Lagrangian up to N2LO: =1.0pt $$\begin{aligned} {\cal L}^{PC}&=& {\cal L}_{\pi\pi}^{(2)}+\ldots\nonumber\\ &+& {\cal L}_{N\pi}^{(1)}+{\cal L}_{N\pi}^{(2)}+{\cal L}_{N\pi}^{(3)}+\ldots+ {\cal L}^{PC}_{CT}\ ,\label{eq:Lpc} \\ {\cal L}_{\pi\pi}^{(2)}&=& {f_\pi^2\over 4} \langle\nabla_\mu U^\dag \nabla^\mu U + \chi^\dag U+\chi U^\dag\rangle\ , \label{eq:Lpc_pipi2}\\ {\cal L}_{N\pi}^{(1)} &=& \overline{\psi}\Bigl(i\gamma^\mu D_\mu -M +{g_A\over 2} \gamma^\mu\gamma^5 u_\mu \Bigr)\psi\ , \label{eq:Lpc_pin1}\\ {\cal L}_{N\pi}^{(2)} &=& c_1 \overline{\psi}\langle \chi_+\rangle \psi - \frac{c_2}{8M^2}\big[\overline{\psi}\langle u_\mu u_\nu \rangle D^\mu D^\nu \psi +\ {\rm h.c.}\big]\nonumber\\ &+&c_3\overline{\psi}\frac{1}{2}\langle u_\mu u^\mu \rangle \psi+ c_4 \overline{\psi}\frac{i}{4}[u_\mu,u_\nu]\sigma^{\mu\nu}\psi+ \ldots \ ,\label{eq:Lpc_pin2}\\ {\cal L}_{N\pi}^{(3)} &=& d_{16} \overline{\psi}{1\over 2} \gamma^\mu\gamma^5 u_\mu \langle \chi_+\rangle \psi\nonumber \\ &+& d_{18} \overline{\psi}{i\over 2} \gamma^\mu\gamma^5 [D_\mu, \chi_-] \psi+\ldots\ ,\label{eq:Lpc_pin3}\end{aligned}$$ where the building blocks $U$, $u_\mu$, $\chi_+$, $\chi_-$ and the covariant derivative $D_\mu$ are defined in Ref. [@viviani14]. We have omitted all the terms not relevant in the present work (for the complete expression for Lagrangian $ {\cal L}_{N\pi}^{(2)}$ and ${\cal L}_{N\pi}^{(3)}$ in Ref. [@NF00]). Four-nucleon contact terms (see, for example, Refs.[@EE09; @RM11]) are lumped into ${\cal L}^{PC}_{CT}$. The parameters $c_1$, $c_2$, $c_3$, $c_4$, $d_{16}$, and $d_{18}$ are LECs entering the PC Lagrangian. All the constants entering the terms discussed in this section have to be considered as “bare” parameters (i.e. unrenormalized). The TRV potential up to order Q {#sec:trvpot} =============================== In this section, we discuss the derivation of the TRV $NN$ and $NNN$ potential at N2LO. We provide, order by order in power counting the formal expressions for it in terms of the time-ordered perturbation theory (TOPT) amplitudes, following the scheme presented in Ref. [@viviani14]. Then, the various diagrams associated with these amplitudes are discussed (additional details are given in Appendix \[app:pot\]). We will give also some hints in the renormalization of the coupling constants. From amplitudes to potentials ----------------------------- We start considering the conventional $NN$ scattering amplitude ${\langle}N'N' |T|NN{\rangle}$, where $|NN{\rangle}$ and $|N'N'{\rangle}$ represent the initial and final two nucleon state and $T$ can be written as, $$\begin{aligned} T = H_I \sum_{n=1}^\infty \left( \frac{1}{E_i -H_0 +i\, \eta } H_I \right)^{n-1} \ , \label{eq:pt}\end{aligned}$$ where $E_i$ is the initial energy of the two nucleons, $H_0$ is the Hamiltonian describing free pions and nucleons, and $H_I$ is the Hamiltonian describing interactions among these particles. To be noticed that in Eq. (\[eq:pt\]) the interaction Hamiltonian $H_I$ is in the Schrödinger picture and that, at the order of interest here, it follows simply from $H_I=-\int {\rm d} \bmx\; {\cal L}_I (t=0,\bmx)$, where ${\cal L}_I$ is the interaction Lagrangian in interaction picture. Vertices from $H_I$ are listed in Appendix \[app:vertex\]. The $NN$ scattering amplitude can be organized as an expansion in powers of $Q/{\Lambda_\chi}\ll 1$, where ${\Lambda_\chi}\simeq1$ GeV is the typical hadronic mass scale, $${\langle}N'N' |T|NN{\rangle}=\sum_n T^{(n)}\,, \label{eq:teft}$$ where $T^{(n)}\sim Q^n$. The $T$ matrix in Eq. (\[eq:teft\]) is generated, order by order in the power counting, by the $NN$ potential $V$ from iterations of it in the Lippmann-Schwinger (LS) equation, $$V+V\, G_0\, V+V\, G_0 \, V\, G_0 \, V+\dots \ , \label{eq:lse}$$ where $G_0$ denotes the free two-nucleon propagator. If we assume that, $$\langle N^\prime N^\prime |V|NN\rangle=\sum_n V^{(n)}\ ,$$ with $V^{(n)}$ of order $Q^n$, it is possible to assign to any $T^{(n)}$ the terms in the LS equation that are of the same order. Generally in term like $[V^{(m)}G_0V^{(n)}]$ is of order $Q^{m+n+1}$ because $G_0$ is of order $Q^{-2}$ and the implicit loop integration brings a factor $Q^3$ (for a more detailed discussion, see Ref. [@Pastore11]). In our case the two nucleons interact via a PC potential plus a very small TRV component. The $\chi$EFT Lagrangian implies the following expansion in powers of $Q$ for $T= T_{PC} + T_{TRV}$: $$\begin{aligned} T_{PC}&=&T^{(0)}_{PC}+T^{(1)}_{PC}+T^{(2)}_{PC}+\ldots ,\label{tpc}\\ T_{TRV}&=&T^{(-1)}_{TRV}+T^{(0)}_{TRV}+T^{(1)}_{TRV}+\ldots .\label{ttrv}\end{aligned}$$ Assuming that the potential $V=V_{PC}+V_{TRV}$ have a similar expansion, $$\begin{aligned} V_{PC} & = & V_{PC}^{(0)}+V_{PC}^{(1)}+V_{PC}^{(2)}+\dots \\ V_{TRV} & = & V_{TRV}^{(-1)}+V_{TRV}^{(0)}+V_{TRV}^{(1)}+\dots ,\end{aligned}$$ we can match order by order the $T$ and the terms in the LS equation obtaining the form definition of the TRV $NN$ potential from the scattering amplitude, $$\begin{aligned} V_{TRV}^{(-1)} & = & T_{TRV}^{(-1)}\ ,\label{eq:vtrvml}\\ V_{TRV}^{(0)} & = & T_{TRV}^{(0)}-\left[V_{TRV}^{(-1)}G_0 V_{PC}^{(0)}\right] -\left[V_{PC}^{(0)}G_0 V_{TRV}^{(-1)}\right]\ ,\label{eq:vtrv0}\\ V_{TRV}^{(1)} & = & T_{PV}^{(1)}-\left[V_{TRV}^{(0)}G_0 V_{PC}^{(0)}\right] -\left[V_{PC}^{(0)}G_0 V_{TRV}^{(0)}\right]\nonumber\\ & - & \left[V_{TRV}^{(-1)}G_0 V_{PC}^{(1)}\right]- \left[V_{PC}^{(1)}G_0 V_{TRV}^{(-1)}\right]\nonumber\\ & - & \left[V_{TRV}^{(-1)}G_0 V_{PC}^{(0)}G_0 V_{PC}^{(0)}\right] - \left[V_{PC}^{(0)}G_0 V_{TRV}^{(-1)}G_0 V_{PC}^{(0)}\right] \nonumber\\ & - & \left[V_{PC}^{(0)}G_0 V_{PC}^{(0)}G_0 V_{TRV}^{(-1)}\right] \ .\label{eq:vtrv1}\end{aligned}$$ The generalization for the $NNN$ TRV potential is straightforward. The $NN$ TRV potential ---------------------- We define the following momenta, $$\begin{aligned} &&\bmK_j=(\bmp_j'+\bmp_j)/2\ , \quad \bmk_j=\bmp_j'-\bmp_j\ ,\label{eq:notjb1}\end{aligned}$$ where $\bmp_j$ and $\bmp'_j$ are the initial and final momenta of nucleon $j$. From the overall momentum conservation $\bmp_1+\bmp_2=\bmp_1'+\bmp_2'$, we can define $\bmk=\bmk_1=-\bmk_2$. We also define $\bmK=(\bmK_1-\bmK_2)/2$, $\bmP=\bmp_1+\bmp_2=\bmK_1+\bmK_2$, in this way is possible to write the TRV $NN$ potential as, $$V_{TRV}(\bmk,\bmK_1,\bmK_2)= V_{TRV}^{(c.m.)}(\bmk,\bmK)+ V^{(\bmP)}_{TRV}(\bmk,\bmK)\ ,\label{eq:widetildev2}$$ where the term $ V^{(\bmP)}_{TRV}(\bmk,\bmK)$ represents boost corrections to $ V^{(c.m.)}_{TRV}(\bmk,\bmK)$ [@girlanda10], the potential in the center-of-mass (c.m.) frame. Below we ignore these boost corrections and provide expressions for $ V^{(c.m.)}_{TRV}(\bmk,\bmK)$ only. ![ \[fig:diagNN\] Time-ordered diagrams contributing to the TRV potential (only a single time ordering is shown). Nucleons and pions are denoted by solid and dashed lines, respectively. The open (solid) circle represents a PC (TRV) vertex.](diagNN.eps "fig:"). The diagrams that give contribution to the $NN$ TRV potential are shown in Fig. \[fig:diagNN\]. We do not consider diagrams which give contribution only to the renormalization of the LECs. In this section we write the final expression of the $NN$ TRV potential $ V^{(c.m.)}_{TRV}$ as, $$\begin{aligned} V^{(c.m.)}_{TRV}&=& V^{ ({\rm OPE})}_{TRV} + V^{ ({\rm TPE})}_{TRV} + V^{ ({\rm 3\pi,0})}_{TRV} + V^{ ({\rm 3\pi,1})}_{TRV} \nonumber\\ &&+ V^{ ({\rm RC})}_{TRV} + V^{ ({\rm 3\pi,RC})}_{TRV} + V^{ ({\rm CT})}_{TRV} \ ,\label{eq:dec}\end{aligned}$$ namely as a sum of terms due to one-pion exchange (OPE), two-pion exchange (TPE), three-pion exchange at NLO (3$\pi$,0) and at N2LO (3$\pi$,1), relativistic corrections derived from the OPE (RC) and from the $3\pi$-exchange ($3\pi$,RC), and contact contributions (CT). From now on we define also $g_0^*=g_0+g_2/3$ (see Appendix \[app:pot\] for details). The OPE term is the contribution of diagram (a) of order $Q^{-1}$ (LO) and it reads, =0.7pt $$\begin{aligned} {V_{TRV}}^{(\rm OPE)} &=&\frac{\overline{g}_A\overline{g}_0^*}{2\overline{f}_\pi}({\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2})\frac{i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})}{\omk^2}\nonumber\\ &+& \frac{\overline{g}_A\overline{g}_1}{4\overline{f}_\pi}\Big[(\tau_{1z}+\tau_{2z}) \frac{i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})}{\omega_k^2}\nonumber\\ &&\qquad +(\tau_{1z}-\tau_{2z})\frac{i \bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})}{\omega_k^2}\Big] \nonumber \\ &+&\frac{\overline{g}_A\overline{g}_2}{6\overline{f}_{\pi}} (3\tau_{1z}\tau_{2z}-{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2})\frac{i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})}{\omk^2}\ ,\nonumber\\ \label{eq:isotope}\end{aligned}$$ where there are an isoscalar, an isovector and an isotensor components and $\overline{g}_0^*=\overline{g}_0+\overline{g}_2/3$. The coupling constants $\overline{g}_A/\overline{f}_{\pi}$, $\overline{g}_0^*$, $\overline{g}_1$, $\overline{g}_2$ are renormalized coupling constants, having reabsorbed the various infinites generated by loops and given as combinations of the “bare” LECs entering the Lagrangian. The expression for $\overline{g}_A/\overline{f}_{\pi}$ is the same as reported in Ref. [@viviani14]. The LECs $g_{S1}^{(2)}$, $g_{S2}^{(2)}$, etc. enter in Eq. (\[eq:isotope\]) through the renormalized constants $\overline{g}_0$, etc. (see Appendix \[app:pot\] for more details). Note that, as mentioned before, the correct expressions of the renormalized constants in term of the bare ones should contain also the contributions of additional diagrams not considered here (see Ref. [@viviani14] for the procedure to be followed for the PC and PV potentials, respectively). The TPE term comes from the not singular contribution of panels (e)-(h) of Fig.  \[fig:diagNN\], taking also into account the subtracting terms given in Eq. (\[eq:vtrv1\]). As discussed in Appendix \[app:pot\] this term has no isovector component, in agreement with the result reported in [@CM11]. Therefore, as reported in Eqs. (\[eq:pote\]) and (\[eq:box\]), we have, =1.0pt $$\begin{aligned} {V_{TRV}}^{(\rm TPE)}&=\frac{\ga{g}_0^*}{\fp\Lx^2} {\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\ i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\ L(k)\nonumber\\ &+\frac{\ga^3 {g}_0^*}{\fp\Lx^2}\ {\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\ i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}) \ (H(k)-3L(k))\nonumber\\ &-\frac{\ga {g}_2}{3\fp\Lx^2} (3\tau_{1z}\tau_{2z}-{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}) \ i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\ L(k)\nonumber\\ &-\frac{\ga^3 {g}_2}{3\fp\Lx^2}(3\tau_{1z}\tau_{2z}-{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2})\nonumber\\ &\qquad\qquad\times i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\ (H(k)-3L(k))\ ,\end{aligned}$$ where the loop functions $L(k)$ and $H(k)$ are defined in Eqs. (\[eq:sL\]) and  (\[eq:H\]). In this and the following NLO and N2LO terms the bare coupling constants $\ga$, $\fp$, $g_0^*$, $\ldots$ can be safely replaced by the corresponding physical (renormalized) values. The $3\pi$-exchange term gives a NLO contribution through the diagram (i) of Fig. \[fig:diagNN\], =1.0pt $$\begin{aligned} {V_{TRV}}^{(3\pi,0)}=&-\frac{5\ga^3\deltatre M}{4\fp\Lx^2}\pi \Big[(\tau_{1z}+\tau_{2z}) \frac{i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})}{\omega_k^2}\nonumber\\ &+ (\tau_{1z}-\tau_{2z})\frac{i \bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})}{\omega_k^2}\Big] \nonumber\\ &\qquad\times\Big(\big(1-\frac{2\mp^2}{s^2}\big)s^2A(k)+\mp\Big)\label{eq:3pnlo}\ ,\end{aligned}$$ where $A(k)$ is defined in Eq. (\[eq:A\]). The expression obtained in Eq. (\[eq:3pnlo\]) is in agreement with the expression derived in Refs. [@JV13; @JB150]. The diagram in panel (l) of Fig. \[fig:diagNN\] contributes to ${V_{TRV}}^{(3\pi)}$ at N2LO. The expression for this diagram is derived in Appendix \[app:pot\], here we report the final result only, =1.0pt $$\begin{aligned} &{V_{TRV}}^{(3\pi,1)}=\frac{5\ga\deltatre Mc_1}{2\fp\Lx^2} \Big[(\tau_{1z}+\tau_{2z})i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &\qquad+(\tau_{1z}-\tau_{2z})i \bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})\Big] \ 4\frac{\mp^2}{\omega_k^2}L(k)\nonumber\\ &\qquad-\frac{5\ga\deltatre Mc_2}{6\fp\Lx^2} \Big[(\tau_{1z}+\tau_{2z})i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &\qquad+(\tau_{1z}-\tau_{2z}) i \bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})\Big] \Big(2L(k)+6\frac{\mp^2}{\omega_k^2}L(k)\Big)\nonumber\\ &\qquad-\frac{5\ga\deltatre Mc_3}{4\fp\Lx^2} \Big[(\tau_{1z}+\tau_{2z})i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &\qquad+(\tau_{1z}-\tau_{2z})i \bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})\Big] \Big(3L(k)+5\frac{\mp^2}{\omega_k^2}L(k)\Big)\label{eq:3pc3}\ .\end{aligned}$$ Note in Eq. (\[eq:3pc3\]) the presence of the $c_1$, $c_2$ and $c_3$ LECs, which belong to the PC sector. In Eqs. (\[eq:3pnlo\]) and (\[eq:3pc3\]) the $\deltatre$ is the renormalized LEC to the relative order of the expressions. The ${V_{TRV}}^{ ({\rm RC})}$ term of the potential takes into account contributions from the RC of the vertices in the OPE of panel (a), $$\begin{aligned} {V_{TRV}}^{(\rm RC)}&=&\frac{\ga {g}_0^*}{8\fp M^2} {\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\ \frac{1}{\omk^2}\nonumber\\ &&\times\Big[-\frac{i}{2} \left(8K^2+k^2\right)\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &&+\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}-\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big]\nonumber\\ &+&\frac{\ga {g}_1}{16\fp M^2}\ \frac{1}{\omk^2}\nonumber\\ &&\times\Big\{(\tau_{1z}-\tau_{2z}) \Big[-\frac{i}{2} \left(8K^2+k^2\right)\bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})\nonumber\\ &&+\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}+\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big]\nonumber\\ &&+(\tau_{1z}+\tau_{2z}) \Big[-\frac{i}{2} \left(8K^2+k^2\right)\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &&+\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}-\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big]\Big\}\nonumber\\ &+&\frac{\ga {g}_2}{24\fp M^2}(3\tau_{1z}\tau_{2z}-{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}) \ \frac{1}{\omk^2}\nonumber\\ &&\times\Big[-\frac{i}{2} \left(8K^2+k^2\right)\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &&+\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}-\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big]\label{eq:rco2}\ ,\nonumber\\\end{aligned}$$ where, as for the OPE, we have an isoscalar, an isovector and an isotensor term. Also the diagram of Fig. \[fig:diagNN\] (i) gives a contribution to the at N2LO, both from the RC of the vertices, and from NLO in the pion propagators (see Appendix \[app:pot\]). The final expression we obtain is $$\begin{aligned} {V_{TRV}}&^{(\rm 3\pi,RC)}=-\frac{5\ga^3\deltatre}{16\fp\Lx^2} \Big[(\tau_{1z}+\tau_{2z})i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &+(\tau_{1z}-\tau_{2z})i \bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})\Big]\nonumber\\ &\times\Big(\frac{25}{6}L(k)-\frac{7}{2}\frac{\mp^2}{\omega_k^2}L(k) +2\frac{\mp^2}{\omega_k^2}H(k)\Big)\nonumber\\ &-\frac{25\ga^3\deltatre}{12\fp\Lx^2}\frac{1} {\omega_k^2}\ \Big\{(\tau_{1z}+\tau_{2z}) \Big[\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}\nonumber\\ &+\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big] +(\tau_{1z}+\tau_{2z})\nonumber\\ &\times\Big[\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}-\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big]\Big\}\label{eq:rc3p}\ , \nonumber\\\end{aligned}$$ where in Eqs. (\[eq:rco2\]) and (\[eq:rc3p\]) the TRV coupling constants are renormalized to order $Q^2$. Last, the potential $V^{ ({\rm CT})}_{TRV}$, derived from the $NN$ contact diagrams (b) of Fig. \[fig:diagNN\], reads $$\begin{aligned} V^{\text{(CT)}}_{TRV}&=&\frac{1}{\Lambda_\chi^2f_\pi}\big\{\overline{C}_1\ i\bmk\cdot \left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right)\nonumber\\ &+&\overline{C}_2\ i\bmk\cdot \left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right){\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\nonumber\\ &+&{\overline{C}_3\over 2}\ \big[i\bmk\cdot\left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right)\left(\tau_{z1}+\tau_{z2}\right) \nonumber\\ &&-i\bmk\cdot\left({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2}\right) \left(\tau_{z1}-\tau_{z2}\right)\big]\nonumber\\ &+&{\overline{C}_4\over 2}\ \big[i\bmk\cdot\left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right)\left(\tau_{z1}+\tau_{z2}\right) \nonumber\\ &&+i\bmk\cdot\left({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2}\right) \left(\tau_{z1}-\tau_{z2}\right)\big]\nonumber\\ &+&\overline{C}_5\ i\bmk\cdot \left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right)\left(3\tau_{z1}\tau_{z2}-{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\right) \big\}\label{eq:ct}\ .\end{aligned}$$ Above $\overline{C}_1$, $\overline{C}_2$, $\overline{C}_3$, $\overline{C}_4$ and $\overline{C}_5$ are renormalized LECs since they have reabsorbed various singular terms coming from the TPE diagrams, 3$\pi$ diagrams and the relativistic corrections. Note that it is possible to write ten operators which can enter $V_{TRV}^{(\text{CT})}$ at order $Q$ but only five of them are independent. In this work we have chosen to write the operators in term of $\bmk$, so that the $r$-space version of $V_{TRV}^{(\text{CT})}$ will assume a simple local form with no gradients. We want also to remark that the $C_2$, $C_4$ and $C_5$ LECs are needed in order to reabsorb the divergences coming from the TPE and $3\pi$ exchange diagrams. In the calculation of the EDM in Sec. \[sec:res\], the configuration space version of the potential is needed. This formally follows from =1.0pt $$\begin{aligned} \!\!\!\!\!\!\!\!\!\!\!\! \langle \bmr_1'\bmr_2'|V|\bmr_1\bmr_2\rangle&=& \delta (\bmR-\bmR') \int {d^3k\over (2\pi)^3} {d^3K\over (2\pi)^3} \nonumber\\ &\times& e^{i(\bmK+\bmk/2)\cdot\bmr'} V(\bmk,\bmK) e^{-i(\bmK-\bmk/2)\cdot\bmr}\ , \label{eq:vrsp}\end{aligned}$$ where $\bmr=\bmr_1-\bmr_2$ and $\bmR=(\bmr_1+\bmr_2)/2$, and similarly for the primed variables. In order to carry out the Fourier transforms above, the integrand is regularized by including a cutoff of the form $$C_\Lambda(k)={\rm e}^{-(k/\Lambda)^4}\ ,\label{eq:cutoff}$$ where the cutoff parameter $\Lambda$ is taken in the range 450–550 MeV. With such a choice the $V^{ ({\rm OPE})}_{TRV}$, $V^{ ({\rm TPE})}_{TRV}$, $V^{ ({\rm 3\pi,0})}_{TRV}$, $V^{ ({\rm 3\pi,1})}_{TRV}$, and $V^{ ({\rm CT})}_{TRV}$ components of the resulting potential are local, i.e., $\langle \bmr_1'\bmr_2'|V|\bmr_1\bmr_2\rangle= \delta (\bmR-\bmR')\, \delta (\bmr-\bmr') V(\bmr)$, while the RC component contains mild non-localities associated with linear and quadratic terms in the relative momentum operator $-i {{\bm \nabla}}$. Explicit expressions for all these components are listed in Appendix \[app:rpotNN\]. The $NNN$ TRV potential {#sec:NNNtrv} ----------------------- The $3\pi$ TRV vertex gives rise to a three body contribution through the diagram (m) in Fig. \[fig:diagNN\]. The lowest contribution appears at NLO while at N2LO the various time ordering cancel out (see Appendix \[app:pot\]). The final expression for the NLO of the $NNN$ TRV potential is, $$\begin{aligned} {V_{TRV}}^{\rm{NNN}}&=&\frac{\deltatre g_A^3 M}{4\fp^3}({\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\ \tau_{3z}+ {\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_3}\ \tau_{2z}+{\boldsymbol{\tau}_2}\cdot{\boldsymbol{\tau}_3}\ \tau_{1z})\nonumber\\ &&\times\frac{(i\bmk_1\cdot{\boldsymbol{\sigma}_1})\ (i\bmk_2\cdot{\boldsymbol{\sigma}_2})\ (i\bmk_3\cdot{\boldsymbol{\sigma}_3})} {\omega^2_{k_1}\omega^2_{k_2}\omega^2_{k_3}}\ , \label{eq:NNNpot}\end{aligned}$$ which is in agreement with the expression of Ref. [@JV13; @JB150]. Also in this case we need the Fourier transform of the potential. Using the overall momentum conservation $\bmp_1+\bmp_2+\bmp_3=\bmp_1'+\bmp_2'+\bmp_3'$, which give us that $\bmk_3=-\bmk_1-\bmk_2$, and defining $\bmQ=\bmk_1+\bmk_2$ and $\bmq=\bmk_1-\bmk_2$ the Fourier transform becomes, $$\begin{aligned} \langle \bmr_1'\bmr_2'\bmr_3'|& V|\bmr_1\bmr_2\bmr_3\rangle= \delta (\bmr_1-\bmr_1')\delta (\bmr_2-\bmr_2')\delta (\bmr_3-\bmr_3') \nonumber\\ &\times\int {d^3q\over (2\pi)^3} {d^3Q\over (2\pi)^3} V(\bmq,\bmQ) e^{-i(\bmq/2)\cdot\bmx_2}\,e^{-i\bmQ\cdot\bmx_1}\ , \label{eq:vrnnn}\nonumber\\\end{aligned}$$ where $\bmx_1=\bmr_2-\bmr_1$ and $\bmx_2=\bmr_3-(\bmr_1+\bmr_2)/2$. Note that with this choice of $\bmQ$ and $\bmq$, the form of the Jacobi vectors appear automatically. We use the regularization function reported in Eq. (\[eq:cutoff\]), where we replace $k$ with $Q$, namely, $$C_\Lambda(Q)={\rm e}^{-(Q/\Lambda)^4}\ ,\label{eq:cutoffnnn}$$ which gives a local form of the $NNN$ TRV potential. A complete description of how to carry out the integration in the Fourier transform is given in Appendix \[sec:NNNr\]. Results {#sec:res} ======= In this section, we report results for the EDM of $\det$, $\tri$ and $\hel$. Hereafter, we do not use the barred notation for the renormalized LECs but all the coupling constant must be considered as renormalized. The calculations are based on the TRV $NN$ potential derived in the previous section (and summarized in Appendix \[app:pot\]) and on the (strong interaction) PC $NN$ potential obtained by Entem and Machleidt at next-to-next-to-next-to-next-to-leading order (N4LO) [@DR15]. These potentials are regularized with a cutoff function depending on a parameter $\Lambda$; its functional form, however, is different from the adopted here for $V_{TRV}$. Below we consider the versions with $\Lambda=450$ MeV, $500$ MeV, and $550$ MeV. The calculations of $\tri$ and $\hel$ EDMs also include the $NNN$ TRV nuclear potential derived in Sec. \[sec:NNNtrv\] and the PC $NNN$ potential derived in $\chi$EFT at N2LO. As for the $NN$ PC potential, it depends on a cutoff parameter $\Lambda$ which is chosen to be consistent with those in the PC and TRV $NN$ potentials. The three-nucleon PC potential depends, in addition, on two unknown LECs, denoted as $c_D$ and $c_E$ and also on the LECs $c_1$, $c_3$ and $c_4$. In this work, we use the values reported in Table III of Ref. [@LE18]. The ${V_{TRV}}^{(3\pi,1)}$ term of the TRV potential depends on the LECs $c_1$, $c_2$ and $c_3$ which are taken from Table II of Ref. [@DR15] and summarized here in Table \[tab:cecd\]. PC interactions $c_1$ $c_2$ $c_3$ ----------------- ------- ------- ------- N2LO -0.74 - -3.61 N3LO -1.07 3.20 -5.32 N4LO -1.10 3.57 -5.54 : \[tab:cecd\] Values of the coefficients and $c_1$, $c_2$, $c_3$ in unit of GeV$^{-1}$ taken from Ref. [@DR15]. In the following the values $g_A=1.267$ and $\fp=92.4$ MeV are adopted. This section is organized as follows. In Sec. \[sec:dedm\], we present the general expression for the EDM operator and the results for the deuteron EDM, while in Sec. \[sec:3Hedm\] we present the calculation of the $\tri$ and $\hel$ EDMs. Deuteron EDM {#sec:dedm} ------------ The EDM operator ${\hat{\boldsymbol{D}}}$ is composed by two parts, $${\hat{\boldsymbol{D}}}={\hat{\boldsymbol{D}}}_{\rm PC}+{\hat{\boldsymbol{D}}}_{\rm TRV}.$$ The ${\hat{\boldsymbol{D}}}_{\rm PC}$ receives contribution at LO from the nuclear EDM polarization operator $${\hat{\boldsymbol{D}}}_{\rm PC}=e\sum_i\frac{1+\tau_z(i)}{2}\boldsymbol{r}_i, \label{eq:dpc}$$ where $e>0$ is the electric unit charge, $\tau_z(i)$ and $\bmr_i$ are the $z$ component of the isospin and the position of the i-th particle. The ${\hat{\boldsymbol{D}}}_{\rm TRV}$ LO contribution comes from the intrinsic nucleon EDM, $${\hat{\boldsymbol{D}}}_{\rm TRV}=\frac{1}{2}\sum_i\left[(d_p+d_n)+(d_p-d_n)\tau_z(i) \right]\boldsymbol{\sigma}(i)\ , \label{eq:dtrv}$$ where $d_p$ and $d_n$ are the EDM of proton and neutron respectively and $\boldsymbol{\sigma}(i)$ is the spin operator which act on the i-th particle. As discussed in Refs. [@JV11b; @JB13] both the ${\hat{\boldsymbol{D}}}_{\rm PC}$ and ${\hat{\boldsymbol{D}}}_{\rm TRV}$ receive contributions from transition currents at N2LO. Of course, a complete treatment of the EDM up to N2LO needs to take care of them but hereafter we neglect their contribution, showing only the effects of N2LO TRV potential. In future work we plan to include the N2LO current contributions in the calculations. The nucleon EDM of an $A$ nucleus can be expressed as $$\begin{aligned} d^A & = & {\langle}\psi^A_{+} | \hat{D}_{\rm TRV} | \psi^A_{+} {\rangle}+2\, {\langle}\psi^A_{+} | \hat{D}_{\rm PC} | \psi^A_{-} {\rangle}\nonumber\\ & \equiv & d_{\rm TRV}^A+d_{\rm PC}^A\ ,\end{aligned}$$ where $|\psi^A_{+}{\rangle}$ $(|\psi^A_{-}{\rangle})$ is defined to be the even-parity (odd-parity) component of the wave function. In general, due to the smallness of the LECs, the EDM can be expressed as linear on the TRV LECs, $$\begin{aligned} d^A_{\rm TRV}&=&d_pa_p+d_na_n\\ d^A_{\rm PC}&=&g_0a_0+g_1a_1+g_2a_2+\Delta_3 a_\Delta\nonumber\\ &+&C_1A_1+C_2A_2+C_3A_3+C_4A_4+C_5A_5\ , \label{eq:da}\end{aligned}$$ where the $a_i$ for $i=0,1,2$, $a_\Delta$, $A_i$ for $i=1,\dots,5$ and $a_p$, $a_n$ are numerical coefficients independent on the LECs values (however, they do depend on the cutoff $\Lambda$ in the PC and TRV chiral potentials). We evaluate also the theoretical errors associated with the chiral expansion of the nuclear potential. We express the error on the numerical coefficients for the deuteron as, $$\left(\delta a_i\right)^2=\left(\delta a_i^{\text{PC}}\right)^2 +\left(\delta a_i^{\text{TRV}}\right)^2\ , \label{eq:errd}$$ where $\delta a_i^{\text{PC}}$ is the error associated to the chiral expansion of the PC potential and $\delta a_i^{\text{TRV}}$ the error associated with the chiral expansion of the TRV potential. Both the contributions were evaluated following the prescriptions of Epelbaum [*et al.*]{} [@EE15] where as reference momentum in the calculation of the errors we used the mass of the pion. It is straightforward to understand that the errors are dominated by the TRV part because we are using the N2LO potential for it and the N4LO potential for the PC part. The coefficients for the deuteron, evaluated with the N4LO PC potential and the N2LO TRV potential and the associated errors for the three different choices of the cutoff parameters, are given in Table \[tab:2hdpc\]. The coefficients $a_p$ and $a_n$ multiplying the intrinsic neutron and proton EDM, as already pointed out first in Ref. [@NY15] and then in Ref. [@JB151], are given by, $$a_n=a_p=\left(1-\frac{3}{2}P_D\right)\, ,$$ where $P_D$ is the percentage of D-wave present in the deuteron wave function. The values of $a_n$ and $a_p$ obtained using the Entem and Machleidt at N4LO for three different choices of the cut off are reported in Table \[tab:2hdpc\]. The operator ${\hat{\boldsymbol{D}}}_{\rm PC}$ for the deuteron reduces to ${\hat{\boldsymbol{D}}}_{\rm PC}=(\tau_z(1)-\tau_z(2))\bmr$ therefore $d^2_{\rm PC}$ in Eq. (\[eq:da\]) receives contribution only from the component $^3P_1$ which has $T=1$. This component is generated by the TRV potential components proportional to $({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})$, namely those proportional to $g_1$, $\Delta_3$, $C_3$ and $C_4$. $\Lambda$(MeV) $450$ $500$ $550$ ---------------- ------------------- ------------------- ------------------- $a_n(a_p)$ $\m0.934\pm0.001$ $\m0.939\pm0.001$ $\m0.938\pm0.001$ $a_1$ $\m0.192\pm0.006$ $\m0.197\pm0.004$ $\m0.194\pm0.003$ $a_\Delta$ $ -0.306\pm0.174$ $ -0.341\pm0.153$ $ -0.349\pm0.137$ $A_3$ $\m0.013\pm0.004$ $\m0.013\pm0.004$ $\m0.013\pm0.004$ $A_4$ $ -0.013\pm0.004$ $ -0.013\pm0.004$ $ -0.013\pm0.004$ : \[tab:2hdpc\] Values of the deuteron coefficients $a_n$ and $a_p$ in units of $d_p$ and $d_n$ and of $a_1$, $a_\Delta$, $A_3$, $A_4$ in units of $e$ fm for the three different choices of cutoff parameters $\Lambda$. The coefficient $a_\Delta$ can be written as, $$a_\Delta=a_\Delta(0)+\big(c_1a_\Delta(1)+c_2a_\Delta(2)+c_3a_\Delta(3)\big)+ a_\Delta(RC)\ ,\label{eq:adelta}$$ where the first term comes from $V_{TRV}^{(3\pi,0)}$, the term in parenthesis from $V_{TRV}^{(3\pi,1)}$, where the LECs $c_1$, $c_2$ and $c_3$ appear, and the last term from $V_{TRV}^{(3\pi,{\rm RC})}$. In Table \[tab:2had\] we report the values of the coefficients $a_\Delta(i)$ evaluated using the N4LO PC potential for the various cut-off. $\Lambda$(MeV) $450$ $500$ $550$ ---------------- ----------- ----------- ----------- $a_\Delta(0) $ $ -0.872$ $ -0.894$ $ -0.894$ $a_\Delta(1) $ $\m0.117$ $\m0.120$ $\m0.120$ $a_\Delta(2) $ $ -0.119$ $ -0.119$ $ -0.117$ $a_\Delta(3) $ $ -0.209$ $ -0.207$ $ -0.203$ $a_\Delta(RC)$ $ -0.042$ $ -0.037$ $ -0.032$ : \[tab:2had\] Values of the various components of the deuteron coefficients $a_\Delta$ as given in Eq. (\[eq:adelta\]) in units of $e$ fm evaluated using the N4LO PC potential for the three different choices of cutoff parameters $\Lambda$. We observe that, taking individually the coefficients which multiplies the LECs $c_i$, their correction to the NLO is between $13-23\%$ which is in line with what we expect from the chiral expansion. However, using the values of $c_i$ given in Table \[tab:cecd\] the total correction of $V^{(3\pi,1)}_{TRV}$ is about $66\% $, which is very large compared to what we expect adding a new order in the potential. Therefore, the value of $a_\Delta$ is very sensitive to the $V_{TRV}^{(3\pi,1)}$ component of the potential and on the choice of the values of the constant $c_i$ and in particular of $c_3$. This is reflected also in the large error associated with this coefficient reported in Table \[tab:2hdpc\]. On the other hand, the $V_{TRV}^{(3\pi,{\rm RC})}$ contribution is about $\sim4\%$ as expected for a relativistic corrections. We notice also that the values of the coefficients for the three different cut-off are compatible within the error bars. We now compare our results with the values reported in Ref. [@JB151] where the Authors used the same TRV potential at NLO with a N2LO PC potential with three-body forces [@EE09; @EE05]. In Table \[tab:2hdcomp\] we compare our results obtained with our TRV potential up to NLO and N2LO with a cutoff $\Lambda=500$ MeV with the ones reported in Ref. [@JB151]. TRV pot. $a_n(=a_p)$ $a_1$ $a_\Delta$ $A_3$ $A_4$ -------------------- ------------- --------- ------------ ----------- ---------- Ref. [@JB151](NLO) $0.939$ $0.183$ $-0.748$ $\m0.006$ $-0.006$ This work (NLO) $0.939$ $0.200$ $-0.893$ $-$ $-$ This work (N2LO) $0.939$ $0.197$ $-0.341$ $0.013$ $-0.013$ : \[tab:2hdcomp\] Comparison of the coefficients $a_1$, $a_\Delta$, $A_3$ and $A_4$ for the deuteron with the result of Refs. [@JB151]. To be notice that in this work we use $e>0$. For this work we report the calculation up to NLO and N2LO. In order to compare the values of the $A_i$ coefficient we divide the reported values for $\frac{2(\hbar c)^3}{\Lx^2\fp}$ which permits to connect the two potentials. As can be seen from Table \[tab:2hdcomp\], our values at NLO seem to be systematically larger compared to the values reported in Ref. [@JB151]. Even if we use a different PC potential, the reason should be found in the different regularization function. However, from a qualitative point of view there is a substantial agreement with Ref. [@JB151]. Similar agreement has been founded for $a_1$ with the result reported in Refs. [@CP04; @NY15; @YH13] while are smaller compare the results reported in Refs. [@JV11; @IS08]. $\tri$ and $\hel$ EDMs {#sec:3Hedm} ---------------------- In this section we report the results obtained for the EDM of the $\tri$ and $\hel$. The wave functions of $\tri$ and $\hel$ have been obtained with the hyperspherical harmonics (HH) [@AK08; @LE09] from the Hamiltonians N4LO/N2LO-500, N4LO/N2LO-450 and N4LO/N2LO-550 discussed in Sec. \[sec:res\]. Moreover, we evaluated also the errors on the numerical coefficients due to the chiral expansion as, $$\left(\delta a_i\right)^2=\left(\delta a_i^{\text{PC}}\right)^2 +\left(\delta a_i^{\text{TRV}}\right)^2+\left(\delta a_i^{\psi}\right)^2\ , \label{eq:errh}$$ where $\delta a_i^{\text{PC}}$ and $\delta a_i^{\text{TRV}}$ are the same as in Eq. (\[eq:errd\]), while $\delta a_i^{\psi}$ is the error associated to the numerical accuracy of the 3-body wave function which we estimated to be of the order of $\sim 1\%$. The calculated values of the numerical coefficients for the three choices of the cutoff with their associated errors are reported in Table \[tab:3htot\]. The $a_\Delta$ can be written as, $$\begin{aligned} a_\Delta&=&a_\Delta(0)+\big(c_1a_\Delta(1)+c_2a_\Delta(2)+c_3a_\Delta(3)\big)+ \nonumber\\ &&a_\Delta(RC)+a_\Delta(3N) ,\label{eq:adelta3}\end{aligned}$$ where $a_\Delta(0)$, $a_\Delta(1)$, $a_\Delta(2)$, $a_\Delta(3)$ and $a_\Delta(RC)$ are defined as in Eq. (\[eq:adelta\]) while $a_\Delta(3N)$ represents the TRV 3-body potential contribution. In Table \[tab:3had\] we report the values of the coefficients $a_\Delta(i)$ evaluated using the N4LO/N2LO-$\Lambda$ PC potential for the various cut-off. ---------------- ----------- ----------- ----------- ----------- ----------- ----------- $\Lambda$(MeV) $450$ $500$ $550$ $450$ $500$ $550$ $a_\Delta(0) $ $ -0.716$ $ -0.751$ $ -0.758$ $ -0.716$ $ -0.749$ $ -0.755$ $a_\Delta(1) $ $\m0.093$ $\m0.098$ $\m0.099$ $\m0.093$ $\m0.098$ $\m0.099$ $a_\Delta(2) $ $ -0.107$ $ -0.110$ $ -0.110$ $ -0.106$ $ -0.109$ $ -0.109$ $a_\Delta(3) $ $ -0.194$ $ -0.198$ $ -0.198$ $ -0.192$ $ -0.196$ $ -0.196$ $a_\Delta(RC)$ $ -0.048$ $ -0.046$ $ -0.042$ $ -0.048$ $ -0.044$ $ -0.041$ $a_\Delta(3N)$ $ -0.202$ $ -0.190$ $ -0.205$ $ -0.193$ $ -0.180$ $ -0.196$ ---------------- ----------- ----------- ----------- ----------- ----------- ----------- : \[tab:3had\] Values of the various components of $a_\Delta$ as given in Eq. (\[eq:adelta3\]) for $\tri$ and $\hel$ in units of $e$ fm evaluated using the N4LO/N2LO PC potential for the three different choices of cutoff parameters $\Lambda$. The $a_\Delta(3N)$ give a correction to $a_\Delta(0)$ of the order of the $\sim25\%$, which is in line with the chiral perturbation theory prediction because both the contributions appear at the same order. For completeness we report in Table \[tab:3Nconv\] of Appendix \[app:NNNconv\] the convergence pattern of this contribution as function of the HH basis expansion. For both $\tri$ and $\hel$, using the values of the $c_i$ reported in Table \[tab:cecd\], we have found a correction to $a_\Delta(0)$ due to ${V_{TRV}}^{(3\pi,1)}$ of $\sim79\%$, and of order $\sim6\%$ due to ${V_{TRV}}^{(3\pi,{\rm RC})}$. While the RC are in line with what we expect, the ${V_{TRV}}^{(3\pi,1)}$ corrections have much more impact on the values of $a_\Delta$ than predicted by the chiral perturbation theory. The large correction due to ${V_{TRV}}^{(3\pi,1)}$ that appears at N2LO is also reflected in the large uncertainties associated with this coefficient. Again, the estimated uncertainties depends critically on the adopted values of $c_1$, $c_2$, and $c_3$ (see Table \[tab:cecd\]). The contribution of the TPE diagrams to $a_0$ and $a_2$ are of the order of $\sim45\%$ and $\sim40\%$ respectively which is larger than expected from the chiral convergence and they are due mainly to the box diagram in Fig. \[fig:diagNN\]. On the other hand, the RC to $a_0$, $a_1$ and $a_2$ are of the order of $\sim1-3\%$ perfectly consistent with the prediction of the chiral perturbation theory. The effects of the PC $NNN$ potential on the values of all the coefficients is around $\sim2\%$. From Table \[tab:3htot\] it is possible also to observe that the values of the numerical coefficients are mostly equal in modulus between $\tri$ and $\hel$ except $a_p$ and $a_n$. Moreover, it is possible to observe that the isovector terms have the same sign for $\tri$ and $\hel$. The isovector part of the TRV potential depends on the third component of the isospin, therefore the $|\psi^A_{-}{\rangle}$ wave function component generated has a sign $-$ for $\tri$ and a $+$ for $\hel$. However, in the ${\hat{\boldsymbol{D}}}_{\rm PC}$ there is the $\tau_z(i)$ operator which bring again a sign $-$ for $\tri$ and $+$ for $\hel$ giving in total a $+$. In Table \[tab:3hcomp\] we compare our calculations at NLO for a value of the cutoff $\Lambda=500$ MeV with the values reported in Ref. [@JB151]. As can be seen inspecting the values of the coefficients evaluated at NLO, there is a nice agreement with the results of Ref. [@JB151]. The numerical differences (which however are within the error bars reported in Ref. [@JB151]) must be searched in the different PC potential and in the different regularization function in the TRV potential employed here. On the other hand for $a_\Delta^{(3)}$, the pure three body part of $a_\Delta$, the difference is of one order of magnitude which can not be explained by the different regularization function used. We found a good agreement for the values of $a_0$ and $a_1$ with the results obtained using a phenomenological potential and the meson exchange TRV potential in Refs. [@NY15; @YH13] while we obtain smaller values compare to Refs. [@JV11; @IS08]. In the case of $\tri$ and $\hel$ we studied also the dependence of the values of the coefficients on the chiral order of the PC potential. As example, in Fig. \[fig:a0coeff\] and \[fig:Dcoeff\] we show the values of the coefficient $a_0$ and $a_\Delta^{(3)}$ for $\tri$ evaluated using the N2LO TRV potential with different chiral order of the PC potential and for the three different choices of the cutoff. ![\[fig:a0coeff\] Values of the coefficient $a_0$ for the $\tri$ nucleus and for the three choice of the cutoff when varying the chiral order of the PC potential.](figure2.eps){width="8cm"} ![\[fig:Dcoeff\] The same as Fig. \[fig:a0coeff\] but for the coefficient $a_\Delta^{(3)}$.](figure3.eps){width="8cm"} In all the case we studied, the values evaluated with the N2LO, N3LO and N4LO PC potential differ by less than $5\%$ which confirm the robustness of the calculation. ----------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- $\Lambda$ (MeV) $450$ $500$ $550$ $450$ $500$ $550$ $a_n$ $ -0.032\pm0.001$ $ -0.033\pm0.001$ $ -0.035\pm0.001$ $\m0.900\pm0.009$ $\m0.908\pm0.009$ $\m0.906\pm0.009$ $a_p$ $\m0.901\pm0.009$ $\m0.909\pm0.009$ $\m0.907\pm0.009$ $ -0.032\pm0.001$ $ -0.033\pm0.001$ $ -0.034\pm0.001$ $a_0$ $ -0.052\pm0.012$ $ -0.055\pm0.013$ $ -0.052\pm0.012$ $\m0.053\pm0.012$ $\m0.056\pm0.013$ $\m0.052\pm0.012$ $a_1$ $\m0.147\pm0.005$ $\m0.154\pm0.004$ $\m0.155\pm0.003$ $\m0.148\pm0.005$ $\m0.155\pm0.004$ $\m0.155\pm0.003$ $a_2$ $ -0.114\pm0.010$ $ -0.120\pm0.009$ $ -0.121\pm0.008$ $\m0.112\pm0.009$ $\m0.118\pm0.009$ $\m0.119\pm0.008$ $a_\Delta$ $ -0.378\pm0.105$ $ -0.388\pm0.101$ $ -0.407\pm0.088$ $ -0.373\pm0.106$ $ -0.383\pm0.102$ $ -0.402\pm0.089$ $A_1$ $\m0.005\pm0.001$ $\m0.006\pm0.001$ $\m0.006\pm0.001$ $ -0.005\pm0.002$ $ -0.006\pm0.002$ $ -0.006\pm0.001$ $A_2$ $ -0.009\pm0.003$ $ -0.010\pm0.003$ $ -0.010\pm0.003$ $\m0.009\pm0.003$ $\m0.010\pm0.003$ $\m0.010\pm0.002$ $A_3$ $ -0.008\pm0.002$ $ -0.008\pm0.002$ $ -0.008\pm0.002$ $ -0.008\pm0.003$ $ -0.008\pm0.002$ $ -0.008\pm0.002$ $A_4$ $\m0.012\pm0.004$ $\m0.013\pm0.004$ $\m0.013\pm0.004$ $\m0.012\pm0.004$ $\m0.013\pm0.004$ $\m0.013\pm0.003$ $A_5$ $ -0.021\pm0.006$ $ -0.022\pm0.006$ $ -0.022\pm0.006$ $\m0.020\pm0.006$ $\m0.022\pm0.006$ $\m0.022\pm0.005$ ----------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- : \[tab:3htot\] Values of the numerical coefficients for $\tri$ and $\hel$ in units of $e$ fm ($a_n$ $(a_p)$ in units of $d_n$ ($d_p$)) for the three different choices of the cutoff. ------------------ ----------- ----------- --------------- ----------- ----------- --------------- This work This work Ref. [@JB151] This work This work Ref. [@JB151] (NLO) (N2LO) (NLO) (NLO) (N2LO) (NLO) $a_n$ $ -0.033$ $ -0.033$ $ -0.030$ $\m0.908$ $\m0.908$ $\m0.904$ $a_p$ $\m0.909$ $\m0.909$ $\m0.918$ $ -0.033$ $ -0.033$ $ -0.029$ $a_0$ $ -0.101$ $ -0.055$ $ -0.108$ $\m0.101$ $\m0.056$ $\m0.111$ $a_1$ $\m0.158$ $\m0.154$ $\m0.139$ $\m0.158$ $\m0.155$ $\m0.139$ $a_2$ $\m0.087$ $\m0.120$ n.a. $\m0.086$ $\m0.118$ n.a. $a_\Delta^{(2)}$ $ -0.751$ $ -0.198$ $-0.598$ $ -0.749$ $ -0.202$ $-0.608$ $a_\Delta^{(3)}$ $ -0.190$ $ - $ $-0.017$ $ -0.180$ $ - $ $-0.017$ $A_1$ $-$ $\m0.006$ $\m0.005$ $-$ $ -0.006$ $ -0.005$ $A_2$ $-$ $ -0.010$ $ -0.011$ $-$ $\m0.010$ $\m0.011$ $A_3$ $-$ $ -0.008$ $ -0.005$ $-$ $ -0.008$ $ -0.005$ $A_4$ $-$ $\m0.013$ $\m0.009$ $-$ $\m0.013$ $\m0.009$ $A_5$ $-$ $ -0.022$ n.a. $-$ $\m0.022$ n.a. ------------------ ----------- ----------- --------------- ----------- ----------- --------------- : \[tab:3hcomp\] Comparison of the values of the coefficients for $\tri$ and $\hel$ obtained for $\Lambda=500$ MeV with the results of Ref. [@JB151]. $a_\Delta^{(2)}$ and $a_\Delta^{(3)}$ correspond respectively to the 2-body and 3-body contribution to $a_\Delta$. To be noticed that in this work we use $e>0$. Conclusions {#sec:conc} =========== In this work we derived the TRV $NN$ and $NNN$ potential at N2LO. In order to derive the potential we have considered the most generic Lagrangian without any specific hypothesis for the TRV source. With the derived potential, we have calculated the EDM of $\det$, $\tri$ and $\hel$ investigating the effect of the N2LO components. We have found that the sensitivity of the light nuclei EDM to the LEC $\Delta_3$ found at NLO, is well reduced by the N2LO contribution which is a quite unexpected behavior inside the chiral perturbation framework. We also checked the robustness of our calculation, evaluating the EDM of the nuclei using different chiral orders in the PC potential. The discrepancy between the use of the N2LO and the N4LO PC potential is approximately $5\%$. We have compared our results with the existing other values reported in literature and in particular with the calculation of Ref. [@JB151]. We have found a substantial agreement with the results reported in Ref. [@JB151] where the small numerical differences can be originated by the different function used to regularize the potential. We have found a qualitative agreement with those of Refs. [@CP04; @NY15; @YH13] while we have obtained smaller values compared to Refs. [@JV11; @IS08]. Our results depend on eleven coupling constants that should be determined by comparing with experimental data. As many authors already pointed out  [@EM10; @JB150; @JV13], the size of the coupling constant depends on the CP violating model. Using our study it will be possible, in case of more than one measurements, to determine the LECs and then individuate the TRV source which generates the EDM by comparing the values of the calculated LECs and their predicted sizes. In future, we plan to use $\chi$EFT to derive the TRV currents which give contribution at N2LO [@JV11b; @JB13]. This would allow to have a fully consistent calculation of the EDM of light nuclei up to N2LO. We also plan to study the $\vec{n}-\vec{p}$ and $\vec{n}-\vec{d}$ spin rotation for an independent and complementary study of TRV effects respect to EDM. acknowledgments {#acknowledgments .unnumbered} =============== Computational resources provided by the INFN-Pisa Computer Center are gratefully acknowledged. Interaction vertices {#app:vertex} ==================== It is convenient to decompose the interaction Hamiltonian $H_I$ as follows =0.5pt $$\begin{aligned} H_I\!=\!H^{00}\!+\!H^{01}\!+\!H^{10}\!+\!H^{02}\!+\!H^{11}\!+\!H^{20}\!+\!\cdots\ ,\end{aligned}$$ where $H^{nm}$ has $n$ creation and $m$ annihilation operators for the pion. Explicitly, $$\begin{aligned} H^{00} & = & {1\over \Omega}\sum_{\a_1' \a_1\a_2' \a_2} b^{\dag}_{\a_1'}b_{\a_1} b^{\dag}_{\a_2'} b_{\a_2} M^{00}_{\a_1'\a_1\a_2'\a_2} \delta_{\bmp_1'+\bmp_2' ,\bmp_1+\bmp_2}\ , \label{eq:m00}\nonumber\\ \\ H^{01}& = & \frac{1}{\sqrt{\Omega}} \sum_{\a' \a}\sum_{\bmq \, a} b^{\dag}_{\a'}b_{\a}a_{\bmq\, a} M^{01}_{\a'\a,\bmq\, a} \delta_{\bmq+\bmp,\bmp'}\ ,\label{eq:m01}\\ H^{10}& = & \frac{1}{\sqrt{\Omega}} \sum_{\a' \a}\sum_{\bmq\, a} b^{\dag}_{\a'}b_{\a}a^\dag_{\bmq\, a} M^{10}_{\a'\a,\bmq\, a} \delta_{\bmq+\bmp',\bmp}\ ,\label{eq:m10}\\ H^{02} &= & \frac{1}{\Omega}\sum_{\a' \a}\sum_{\bmq' a'\,\bmq\, a} b_{\a'}^{\dag}b_{\a}a_{\bmq' a'}a_{\bmq\, a} M^{02}_{\a'\a,\bmq' a'\, \bmq \, a} \delta_{\bmq+\bmq'+\bmp,\bmp'}\ ,\label{eq:m02}\nonumber\\ \\ H^{11} &= & \frac{1}{\Omega}\sum_{\a' \a}\sum_{\bmq' a'\, \bmq\, a} b_{\a'}^{\dag}b_{\a}a^\dag_{\bmq' a'}a_{\bmq\,a } M^{11}_{\a'\a,\bmq'a'\bmq \, a} \delta_{\bmq+\bmp,\bmq'+\bmp'}\ ,\label{eq:m11}\nonumber\\ \\ H^{20} &= & \frac{1}{\Omega}\sum_{\a' \a}\sum_{\bmq'a' \,\bmq\,a} b_{\a'}^{\dag}b_{\a}a^\dag_{\bmq' a'}a^\dag_{\bmq \, a} M^{20}_{\a'\a,\bmq' a'\, \bmq \,a} \delta_{\bmp,\bmq+\bmq'+\bmp'}\ ,\label{eq:m20}\nonumber\\\end{aligned}$$ etc. Here $\alpha_j\equiv \bmp_j,s_j,t_j$ denotes the momentum, spin projection, isospin projection of nucleon $j$ with energy $E_j=\sqrt{p_j^2+M^2}$, $\bmq$ and $a$ denote the momentum and isospin projection of a pion with energy $\omega_q=\sqrt{\bmq^2+m_\pi^2}$, and $M^{nm}$ are the vertex functions listed below. The various momenta are discretized by assuming periodic boundary conditions in a box of volume $\Omega$. We note that in the expansion of the nucleon field $\psi$ we have only retained the nucleon degrees of freedom, since anti-nucleon contributions do not enter the TRV $NN$ and $NNN$ potential at the order of interest here. We note also that in general the creation and annihilation operators are not normal-ordered. After normal-ordering them, tadpole-type contributions result, which are relevant only for renormalization, therefore we discard them hereafter. The vertex functions $M^{nm}$ involve products of Dirac 4-spinors, which are expanded non-relativistically in powers of momenta, and terms up to order $Q^3$ are retained. Useful formulas are reported in Appendix F of Ref. [@viviani14]. ![ \[fig:vertex\] Vertices entering the TRV potential at N2LO. The solid (dash) lines represent nucleons (pions). The open (solid) symbols denote PC (TRV) vertices.](vertex.eps) The interaction vertices needed for the construction of the TRV potential, without considering renormalization contributions, are summarized in Fig. \[fig:vertex\]. Note that in the power counting of these vertices below, we do not include the $1/\sqrt{\omega_k}$ normalization factors in the pion fields. We obtain: 1. $\pi NN$ vertices. The PC interaction term is derived in Appendix F of Ref. [@viviani14]. For completeness, here we report the final formula for the PC vertex up to order $Q^3$ that reads, $$\begin{aligned} {}^{PC}M^{\pi NN,01}_{\alpha' \alpha, \bmq\,a} &=& {g_A \over 2 f_\pi} {\tau_a\over \sqrt{2\omega_q}}\Bigl[ i\, \bmq\cdot{{\bm \sigma}}-{i\over M}\omega_q\;\bmK\cdot{{\bm \sigma}}\nonumber \\ &+& {i\over 4M^2}\Bigl(2\bmK\cdot\bmq\;\bmK\cdot{{\bm \sigma}}- 2K^2\;\bmq\cdot{{\bm \sigma}}\nonumber\\ && \qquad -{1\over 2}\bmk\cdot{{\bm \sigma}}\; \bmq\cdot\bmk\Bigr)\Bigr] \nonumber\\ &+& {m_\pi^2 \over f_\pi}(2d_{16}-d_{18}) {\tau_a\over \sqrt{2\omega_q}}\; i\bmq\cdot{{\bm \sigma}}\ ,\label{eq:MpiNN01b}\\ {}^{PC}M^{\pi NN, 10}_{\alpha'\alpha, \bmq\,a} &= & -{}^{PC}M^{\pi NN,01}_{\alpha' \alpha,\bmq\,a} . \label{eq:MpiNN10b} \end{aligned}$$ In diagrams, these PC vertex functions are represented as open circles. The TRV $\pi NN$ vertices are due to interaction terms proportional to LECs $g_0$, $g_1$ and $g_2$ which corresponds to isoscalar, isovector and isotensor interaction, plus terms which derive from ${\cal L}_{\text{TRV}}^{\pi N\, (1)}$ and ${\cal L}_{\text{TRV}}^{\pi N\, (2)}$ given in Eqs. (\[eq:ltrv1\]) and  (\[eq:ltrv2\]). They read (up to order $Q^2$), $$\begin{aligned} {}^{TRV}&M^{\pi NN,01}_{\alpha' \alpha, \bmq\,a}=-\frac{(g_0 \tau_a+g_1\delta_{a,3}+g_2\tau_3\delta_{a,3})} {\sqrt{2\omega_q}}\nonumber\\ &\qquad\times\Bigl[ 1-{1\over 4M^2}\Bigl( 2K^2+i\ \bmk\times\bmK\Bigr)\Bigr]\nonumber\\ &\qquad-ig^{(1)}_V\frac{\epsilon_{ab3}\tau_b}{\fp\sqrt{2\omega_q}}\big[\omega_q -{1\over 2M}\bmK\cdot\bmq\big]\nonumber\\ &\qquad+\frac{\mp^2}{\fp^2\sqrt{2\omega_q}}\big(g^{(2)}_{S1}\tau_a +g^{(2)}_{V1}\delta_{a,3}+g^{(2)}_{T1}\tau_3\delta_{a,3}\big)\nonumber\\ &\qquad-\frac{\omega_q^2}{\fp^2\sqrt{2\omega_q}}\big(g^{(2)}_{S2}\tau_a +g^{(2)}_{V2}\delta_{a,3}+g^{(2)}_{T2}\tau_3\delta_{a,3}\big) \ ,\nonumber\\\label{eq:MpiNN01a} \end{aligned}$$ $$\begin{aligned} {}^{TRV}&M^{\pi NN, 10}_{\alpha'\alpha, \bmq\,a}=-\frac{(g_0 \tau_a+g_1\delta_{a,3}+g_2\tau_3\delta_{a,3})} {\sqrt{2\omega_q}}\nonumber\\ &\qquad\times\Bigl[ 1-{1\over 4M^2}\Bigl( 2K^2+i\ \bmk\times\bmK\Bigr)\Bigr]\nonumber\\ &\qquad+ig^{(1)}_V\frac{\epsilon_{ab3}\tau_b}{\fp\sqrt{2\omega_q}}\big[\omega_q -{1\over 2M}\bmK\cdot\bmq\big]\nonumber\\ &\qquad+\frac{\mp^2}{\fp^2\sqrt{2\omega_q}}\big(g^{(2)}_{S1}\tau_a +g^{(2)}_{V1}\delta_{a,3}+g^{(2)}_{T1}\tau_3\delta_{a,3}\big)\nonumber\\ &\qquad-\frac{\omega_q^2}{\fp^2\sqrt{2\omega_q}}\big(g^{(2)}_{S2}\tau_a +g^{(2)}_{V2}\delta_{a,3}+g^{(2)}_{T2}\tau_3\delta_{a,3}\big)\ ,\nonumber\\ \label{eq:MpiNN10a} \end{aligned}$$ 2. $\pi\pi NN$ vertices. The PC interaction is needed up to NLO in the NR expansion at the order we are interested. The corresponding vertex functions will receive contribution from the Weinberg-Tomozawa term and from ${\cal L}^{(2)}_{N\pi}$. The vertex functions read, $$\begin{aligned} {}^{PC}M&^{\pi\pi NN,02}_{\alpha' \alpha,\bmq' a'\,\bmq\, a}= {i\over 8f_\pi^2} {\epsilon_{aa'b}\tau_b \over\sqrt{2\omega_q}\sqrt{2\omega_{q'}}}\Big[(\omega_q-\omega_{q'}) \nonumber\\ &-\frac{2\bmK\cdot(\bmq-\bmq')-i(\bmk\times{{\bm \sigma}})\cdot(\bmq-\bmq')} {2M}\Big]\nonumber\\ &+\frac{\delta_{ij}}{\fp^2}\frac{c_1(2m_\pi^2)+c_2\omega_q\omega_{q'}+c_3 (\omega_q\omega_{q'}-\bmq\cdot\bmq')}{\sqrt{2\omega_q}\sqrt{2\omega_{q'}}} \nonumber\\ &-\frac{c_4}{2\fp^2}{\epsilon_{aa'b}\tau_b \over\sqrt{2\omega_q}\sqrt{2\omega_{q'}}}(\bmq\times\bmq')\cdot{{\bm \sigma}}\ , \\ {}^{PC}M&^{\pi\pi NN, 20}_{\alpha' \alpha,\bmq' a'\,\bmq\, a}= {i\over 8f_\pi^2} {\epsilon_{aa'b}\tau_b \over\sqrt{2\omega_q}\sqrt{2\omega_{q'}}}\Big[(\omega_{q'}-\omega_{q}) \nonumber\\ &-\frac{2\bmK\cdot(\bmq'-\bmq)-i(\bmk\times{{\bm \sigma}})\cdot(\bmq'-\bmq)} {2M}\Big]\nonumber\\ &+\frac{\delta_{ij}}{\fp^2}\frac{c_1(2m_\pi^2)+c_2\omega_q\omega_{q'}+c_3 (\omega_q\omega_{q'}-\bmq\cdot\bmq')}{\sqrt{2\omega_q}\sqrt{2\omega_{q'}}} \nonumber\\ &-\frac{c_4}{2\fp^2}{\epsilon_{aa'b}\tau_b \over\sqrt{2\omega_q}\sqrt{2\omega_{q'}}}(\bmq\times\bmq')\cdot{{\bm \sigma}}\ . \end{aligned}$$ The ${}^{PC}M^{\pi\pi NN,11}_{\alpha' \alpha,\bmq' a'\,\bmq\, a}$ vertex is not needed for the evaluation of the time-ordered diagrams. 3. $3\pi$ vertices. The TRV Lagrangian has a three-pion interaction term shown in diagram (4) of Fig. \[fig:vertex\]. The vertices, neglecting the tadpole terms, are given by $H_I^{3\pi}=H_I^{3\pi,03}+H_I^{3\pi,12} +H_I^{3\pi,21}+H_I^{3\pi,30}$, where $$\begin{aligned} H_I^{3\pi,03} &={1\over\Omega^{3/2}} \sum_{\substack{\bmq a\,\bmq'a'\\ \bmp b}} a_{\bmq\,a}a_{\bmq'\,a'}a_{\bmp\,b}\; M^{3\pi,03}_{\bmq a\,\bmq'a'\,\bmp b}\,\delta_{\bmq+\bmq'+\bmp,0}\ ,\nonumber\\ \\ H_I^{3\pi,12} &= {1\over\Omega^{3/2}} \sum_{\substack{\bmq a\,\bmq'a'\\ \bmp b}} a_{\bmq\,a}a_{\bmq'\,a'}a_{\bmp\,b}^\dagger\; M^{3\pi,12}_{\bmq a\,\bmq'a'\, \bmp b}\,\delta_{\bmq+\bmq',\bmp}\ ,\nonumber\\ \\ H_I^{3\pi,21} &= {1\over\Omega^{3/2}} \sum_{\substack{\bmq a\,\bmq'a'\\\bmp b}} a_{\bmq\,a}a_{\bmq'\,a'}^\dagger a_{\bmp\,b}^\dagger\; M^{3\pi,21}_{\bmq a\,\bmq'a'\,\bmp b}\,\delta_{\bmq,\bmq'+\bmp}\ ,\nonumber\\ \\ H_I^{3\pi,30} & ={1\over\Omega^{3/2}} \sum_{\substack{\bmq a\,\bmq'a'\\ \bmp b}} a_{\bmq\,a}^\dagger a_{\bmq'\,a'}^\dagger a_{\bmp\,b}^\dagger\; M^{3\pi,30}_{\bmq a\,\bmq'a'\,\bmp b}\,\delta_{0,\bmq+\bmq'+\bmp}\ ,\nonumber\\ \end{aligned}$$ with, $$\begin{aligned} {}^{TRV}M^{3\pi,03}_{\bmq a\,\bmq'a'\,\bmp b}&=-\frac{\Delta M} {3\sqrt{2\omega_q\,2\omega_{q'}\,2\omega_p}}\nonumber\\ &\times\bigl(\delta_{a,a'}\delta_{b,3}+ \delta_{a,b}\delta_{a',3}+\delta_{a',b}\delta_{a,3}\bigr)\ ,\nonumber\\ \\ {}^{TRV}M^{3\pi,12}_{\bmq a\,\bmq'a'\,\bmp b}&= 3\,{}^{TRV}M^{3\pi,03}_{\bmq a\,\bmq'a'\,\bmp b}\ ,\\ {}^{TRV}M^{3\pi,21}_{\bmq a\,\bmq'a'\,\bmp b}&= 3\,{}^{TRV}M^{3\pi,03}_{\bmq a\,\bmq'a'\,\bmp b}\ ,\\ {}^{TRV}M^{3\pi,30}_{\bmq a\,\bmq'a'\,\bmp b}&= {}^{TRV}M^{3\pi,03}_{\bmq a\,\bmq'a'\,\bmp b}\ . \end{aligned}$$ 4. $4N$ contact interaction. The EFT Hamiltonian includes also the term given in Eq. (\[eq:m00\]) derived from a contact Lagrangian. We only need its TRV part of order $Q$, which includes five independent interaction terms. With a suitable choice of the LECs, the vertex function ${}^{TRV}M^{00}$ can be written as $$\begin{aligned} {}^{TRV}&M^{00}_{\a_1'\a_1\a_2'\a_2}= {1 \over 2\Lambda_\chi^2 f_\pi} \Bigl[ C_1 i \bmk_1\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}) \nonumber\\ &\qquad+ C_2\,{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\, i\bmk_1\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &\qquad+ {C_3\over 2}\, \bigl(\left(\tau_{1z}+\tau_{2z}\right) \, i \bmk_1\cdot\left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right)\nonumber\\ &\qquad\qquad-\left(\tau_{1z}-\tau_{2z}\right)\, i\bmk_1\cdot\left({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2}\right)\bigr)\nonumber\\ &\qquad+ {C_4\over 2}\, \bigl(\left(\tau_{1z}+\tau_{2z}\right) \, i \bmk_1\cdot\left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right)\nonumber\\ &\qquad\qquad+\left(\tau_{1z}-\tau_{2z}\right)\, i\bmk_1\cdot\left({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2}\right)\bigr)\nonumber\\ &\qquad+C_5\, \left(3\tau_{1z}\tau_{2z}-{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\right) i\bmk_1\cdot \left({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}\right) \Bigr]\ , \label{eq:m00trv}\nonumber\\ \end{aligned}$$ where $\bmk_1=\bmp_1'-\bmp_1=-\bmp_2'+\bmp_2$. The TRV $NN$ potential {#app:pot} ====================== In this section we discuss the derivation of the TRV $NN$ and $NNN$ potential, providing explicit expressions of the diagrams given in Fig. \[fig:diagNN\]. The power counting is as follows: (i) a PC (TRV) $\pi NN$ vertex is of order $Q$ ($Q^{0}$); (ii) a PC $\pi\pi NN$ vertex is of order $Q^1$; (iii) a TRV $3\pi$ vertex is of order $Q^{-3}$; (iv) a PC (TRV) $NN$ contact vertex is of order $Q^0$ ($Q$); (v) an energy denominator without (with one or more) pions is of order $Q^{-2}$ ($Q^{-1}$); (vi) factors $Q^{-1}$ and $Q^{3}$ are associated with, respectively, each pion line and each loop integration. The momenta are defined as given in Eq. (\[eq:notjb1\]), and in what follows use is made of the fact that $\bmk\cdot\bmK$ vanishes in the c.m. frame. It is useful to define the isospin operator as, $$\begin{aligned} \TO &=&{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\ ,\\ \ttp&=&(\tau_{1z}+\tau_{2z})\ ,\\ \ttm&=&(\tau_{1z}-\tau_{2z})\ ,\\ \tten&=&(3\tau_{1z}\tau_{2z}-{\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2})\ .\end{aligned}$$ The TRV $NN$ potential is derived from the amplitudes in Fig. \[fig:diagNN\] via Eqs. (\[eq:vtrvml\])–(\[eq:vtrv1\]). Up to order $Q$ included, we obtain for the OPE component in panel (a) of Fig. \[fig:diagNN\] : $$V({\rm a}) = V^{(-1)}({\rm NR})+ V^{(1)}({\rm RC}) + V^{(1)}({\rm LEC})\ , \label{eq:ope2}$$ where, $$\begin{aligned} V^{(-1)}&({\rm NR})=\frac{\ga g_0^*}{2\fp}\TO\frac{i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})}{\omk^2} \nonumber\\ &+\frac{g_Ag_1}{4f_\pi}\Big[\ttp \frac{i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})}{\omega_k^2}+ \ttm\frac{i \bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})}{\omega_k^2}\Big]\nonumber\\ &+\frac{\ga g_2}{6\fp}\tten\frac{i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})}{\omk^2} \!\!\ ,\label{eq:opeloapp} \end{aligned}$$ $$\begin{aligned} {V_{TRV}}^{(1)}&{(\rm RC)}=\frac{\ga g_0^*}{8\fp M^2} \TO\ \frac{1}{\omk^2}\nonumber\\ &\times\Big[-\frac{i}{2} \left(8K^2+k^2\right)\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &+\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}-\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big]\nonumber\\ &+\frac{\ga g_1}{16\fp M^2}\ \frac{1}{\omk^2}\nonumber\\ &\times\Big\{\ttm \Big[-\frac{i}{2} \left(8K^2+k^2\right)\bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})\nonumber\\ &+\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}+\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big]\nonumber\\ &+\ttp \Big[-\frac{i}{2} \left(8K^2+k^2\right)\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &+\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}-\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big]\Big\}\nonumber\\ &+\frac{\ga g_2}{24\fp M^2}\tten \ \frac{1}{\omk^2} \Big[-\frac{i}{2} \left(8K^2+k^2\right)\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &+\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}-\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\Big]\label{eq:rcopeapp}\ ,\\ {V_{TRV}}^{(1)}&({\rm LEC})={V_{TRV}}^{(-1)}({\rm NR}) \frac{2m^2_\pi}{\ga}(2d_{16}-d_{18})\nonumber\\ &+\frac{\ga}{2\fp^3}\TO i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}) \Big[g^{(2)}_{S2}-g^{(2)}_{S1}\frac{\mp^2}{\omk^2}\Big]\nonumber\\ &+\frac{g_A}{4f_\pi^3}\Big[\ttp i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})+ \ttm i \bmk\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})\Big]\nonumber\\ &\qquad \times\Big[g^{(2)}_{V2}-g^{(2)}_{V1}\frac{\mp^2}{\omk^2}\Big]\nonumber\\ &+\frac{\ga}{6\fp^3}\tten i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}) \Big[g^{(2)}_{T2}-g^{(2)}_{T1}\frac{\mp^2}{\omk^2}\Big]\nonumber\\ &+\frac{g^{(1)}_Vg_A}{2\fp^2M}({\boldsymbol{\tau}_1}\times{\boldsymbol{\tau}_2})_z\,\bmK\cdot({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})\, \label{eq:lec},\end{aligned}$$ where $g_0^*=g_0+g_2/3$. The contribution given in the first line of Eq. (\[eq:lec\]) renormalizes the coupling constants $g_0^*$, $g_1$ and $g_2$ in the OPE term. The terms of Eq. (\[eq:lec\]) which are multiplied by the factor $\mp^2/\omk^2$ are also reabsorbed in the constant $g_0^*$, $g_1$ and $g_2$ while all the other terms are reabsorbed in the LECs $C_2$, $C_4$ and $C_5$ in Eq. (\[eq:ct\]). Regarding the last term in Eq. (\[eq:lec\]), it is possible to use a Fierz transformation obtaining a combination of the operators which multiplies the LECs $C_3$ and $C_4$. Therefore all ${V_{TRV}}^{(1)}(\text{LEC})$ can be reabsorbed in the OPE and contact potentials. The factor $k^2=\omega_k^2-m_\pi^2$ in the isoscalar, isovector and isotensor component of $V^{(1)}({\rm RC})$ leads to a piece that can be reabsorbed in the contact term proportional to $C_2$, $C_4$ and $C_5$ in Eq. (\[eq:ct\]) and a piece proportional to $m_\pi^2$ that simply renormalizes the LECs $g_0^*$, $g_1$ and $g_2$. The component of the TRV potential coming from the contact terms in panel (b) of Fig. \[fig:diagNN\] derives directly from the vertex function ${}^{TRV}M^{00}$ given in Eq. (\[eq:m00trv\]). The final expression has already been given in Eq. (\[eq:ct\]). The diagrams reported in panels (c) and (d) contain a combination of a contact interaction with the exchange of a pion. However, it can be shown that their contribution is at least of order $Q^3$ . Next we consider the TPE components in panels (e)-(h). The contribution from diagrams (e) reads, $$\begin{aligned} V^{(1)}(\rm e)&=&-\frac{\ga g_0^*}{4\fp^3}\TO \, i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\int_\bmq\frac{1}{\omp\omm(\omp+\omm)} \nonumber\\ &&+\frac{\ga g_2}{12\fp^3}\tten \, i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\int_\bmq\frac{1}{\omp\omm(\omp+\omm)}\ , \nonumber\\ \end{aligned}$$ where $\omega_\pm=\sqrt{(\bmk\pm \bmq)^2 + 4 \, m_\pi^2}$ and $\int_\bmq=\int \frac{d\bmq}{(2\pi)^3}$. The isovector component of the OPE vertex vanishes since the integrand is proportional to $\bmq$, therefore there is no isovector component from panel (e). Dimensional regularization allows one to obtain the finite part as, $$\begin{aligned} \overline{V}^{(1)}(\rm e)&=&\frac{\ga g_0^*}{\fp\Lx^2}\TO i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}) L(k)\nonumber\\ &&-\frac{\ga g_2}{3\fp\Lx^2}\tten i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}) L(k)\ ,\label{eq:pote} \end{aligned}$$ where $\Lambda_\chi=4\pi f_\pi$ and the loop function $L(k)$ is defined as $$L(k)= {1\over 2} {s\over k} \ln\left({s+k\over s-k}\right)\ ,\quad s=\sqrt{k^2+4\, m^2_\pi}\ .\label{eq:sL}$$ The singular part is given by, $$\begin{aligned} V^{(1)}_\infty({\rm e}) &=&\frac{\ga g_0^*}{2\fp\Lx^2}\TO i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})(\deps-2)\nonumber\\ &&-\frac{\ga g_2}{6\fp\Lx^2}\tten i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})(\deps-2)\ , \end{aligned}$$ where $$\deps = -{2\over \epsilon}+\gamma-\ln \pi +\ln\left({m^2_\pi\over\mu^2}\right) \ ,$$ $\epsilon=3-d$, $d$ being the number of dimensions ($d\rightarrow 3$), and $\mu$ is a renormalization scale. This singular contribution is absorbed in the ${V_{TRV}}^{({\rm CT})}$ term proportional to $C_2$ for the isoscalar part and to $C_5$ for the isotensor part. The contributions from panels (f)-(h) in Fig. \[fig:diagNN\] are collectively denoted as “box” below, and the non-iterative pieces in reducible diagrams of type (h) are identified via Eq. (\[eq:vtrv1\]). From the panel (f) we obtain, $$\begin{aligned} V^{(1)}&({\rm f})=\Big\{-\frac{\ga^3 g_0^*}{16\fp^3}(3+2\ \TO)i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &+\frac{\ga^3 g_2}{48\fp^3}\tten \ i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &-\frac{\ga^3 g_1}{32\fp^3} \Big[\ttp \ i\bmk\cdot({{\bm \sigma}}_1-{{\bm \sigma}}_2)+ \ttm \ i\bmk\cdot({{\bm \sigma}}_1+{{\bm \sigma}}_2)\Big]\Big\}\nonumber\\ &\times\int_\bmq\frac{\omp^2+\omp\omm+\omm^2}{\omp^3\omm^3(\omp+\omm)} (k^2-q^2)\ ,\label{eq:panelf}\end{aligned}$$ while the contribution of panel (g) results, $$\begin{aligned} V^{(1)}&({\rm g})=\Big\{\frac{\ga^3 g_0^*}{16\fp^3}(3-2\ \TO)i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &+\frac{\ga^3 g_2}{48\fp^3}\tten \ i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &+\frac{\ga^3 g_1}{32\fp^3} \Big[\ttp \ i\bmk\cdot({{\bm \sigma}}_1-{{\bm \sigma}}_2)+ \ttm \ i\bmk\cdot({{\bm \sigma}}_1+{{\bm \sigma}}_2)\Big]\Big\}\nonumber\\ &\times\int_\bmq\frac{\omp^2+\omp\omm+\omm^2}{\omp^3\omm^3(\omp+\omm)} (k^2-q^2)\ .\label{eq:panelg}\end{aligned}$$ The complete “box” contribution is given by the sum of $V({\rm f})$ and $V({\rm g})$ and it reads, $$\begin{aligned} V^{(1)}({\rm box})&=\Big\{-\frac{\ga^3 g_0^*}{4\fp^3}\ \TO \ i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &+\frac{\ga^3 g_2}{12\fp^3}\ \tten \ i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\Big\}\nonumber\\ &\times\int_\bmq\frac{\omp^2+\omp\omm+\omm^2}{\omp^3\omm^3(\omp+\omm)} (k^2-q^2)\ ,\label{eq:panelbox}\end{aligned}$$ where all the isovector terms cancel out. After dimensional regularization, the finite part reads, $$\begin{aligned} \overline{V}^{(1)}({\rm box})&=\frac{\ga^3 g_0^*}{\fp\Lx^2}\ \TO \ i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}) [H(k)-3\, L(k)]\nonumber\\ &-\frac{\ga^3 g_2}{3\fp\Lx^2}\ \tten \ i\bmk\cdot ({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2}) [H(k)-3\, L(k)]\ ,\label{eq:box}\nonumber\\\end{aligned}$$ where $$H(k)= {4\, m^2_\pi\over s^2} L(k)\ ,\label{eq:H}$$ while the singular part is given by, $$\begin{aligned} V^{(1)}_{\infty}&({\rm box})=-\frac{\ga^3 g_0^*}{\fp\Lx^2}\TO\ i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\Big(\frac{3}{2}\deps-1\Big)\nonumber\\ &+\frac{\ga^3 g_2}{3\fp\Lx^2}\tten\ i\bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\Big(\frac{3}{2}\deps-1\Big)\ .\end{aligned}$$ The latter is absorbed in the ${V_{TRV}}^{({\rm CT})}$ term proportional to $C_2$ for the isoscalar part and to $C_5$ for the isotensor part. Now we consider the contributions that come from the panels (i) and (l) of Fig.  \[fig:diagNN\]. At NLO the contributions of the panel (l) cancel out due to the isospin structure of the vertices. The contribution of diagrams (i) result, $$\begin{aligned} V^{(0)}({\rm i})&=-\frac{5\ga^3\Delta_3M}{32\fp^3}\frac{1}{\omk^2} \Big[\ttp\ i\bmk\cdot({{\bm \sigma}}_1-{{\bm \sigma}}_2)\nonumber\\ &+\ttm \ i \bmk\cdot({{\bm \sigma}}_1+{{\bm \sigma}}_2)\Big] \int_\bmq \frac{k^2-q^2}{\omp^2\omm^2}\ .\label{eq:3piapp}\end{aligned}$$ Using dimensional regularization we obtain, $$\begin{aligned} \overline{V}^{(0)}&({\rm i})=-\frac{5\ga^3\Delta_3 M}{4\fp\Lx^2} \frac{\pi}{\omega_k^2} \big(\ttm\ i \bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &+\ttp\ i \bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\big) \Big[\Big(1-\frac{2\mp^2}{s^2}\Big)s^2A(k)+\mp\Big]\ ,\nonumber\\\end{aligned}$$ where, $$\begin{aligned} A(k)=\frac{1}{2k}\arctan\Big({\frac{k}{2\mp}}\Big)\ .\label{eq:A}\end{aligned}$$ To be noticed that the use of dimensional regularization does not give the divergent part of the integral in Eq. (\[eq:3piapp\]). This is due to the fact that the dimensional regularization cannot deal with linear divergences. To explicit the linear divergence we use a simple regularization of Eq. (\[eq:3piapp\]), namely we integrate over $q$ up to a (large) value $\Lambda_R$. The result for the non divergent part is equal to the one reported in Eq. (\[eq:3piapp\]) while the divergent part reads, $$\begin{aligned} V^{(0)}_{\infty}&({\rm i})=-\frac{5\ga^3\Delta_3 M}{4\fp\Lx^2} \frac{1}{\omega_k^2} \big(\ttm\ i \bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\nonumber\\ &+\ttp\ i \bmk\cdot({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\big) \Big[\Lambda_R+{\cal O}\big(\frac{k^2}{\Lambda_R}\big)\Big]\ ,\end{aligned}$$ where spurious contributions of order $Q^2/\Lambda_R$ or more appear but they can be neglected for $\Lambda_R\rightarrow\infty$. The divergent part can be reabsorbed in the ${V_{TRV}}^{({\rm CT})}$ term proportional to $C_4$. At N2LO the contribution of panel (i) comes both from the second order in the pion propagator (PP) and in the pion-nucleon vertex (PNV). For the former we obtain, $$\begin{aligned} &V^{(1)}({\rm i-PP})=-\frac{5\ga^3\Delta_3}{128\fp^3}\frac{1}{\omk^2} \Big[\ttp\ i\bmk\cdot({{\bm \sigma}}_1-{{\bm \sigma}}_2)\nonumber\\ &\quad+\ttm \ i \bmk\cdot({{\bm \sigma}}_1+{{\bm \sigma}}_2)\Big] \int_\bmq \frac{\omp^2+\omp\omm+\omm^2}{\omp^3\omm^3(\omp+\omm)} \big(k^2-q^2\big)^2\nonumber\\ &\quad+\frac{5\ga^3\Delta_3}{8\fp^3}\frac{1}{\omk^2} \int_\bmq \frac{\omp^2+\omp\omm+\omm^2}{\omp^3\omm^3(\omp+\omm)}\nonumber\\ &\qquad\qquad\times \big[\bmk\cdot{\boldsymbol{\sigma}_1}(\bmq\times\bmk)\cdot{\boldsymbol{\sigma}_2}(\bmq\cdot\bmK)\tau_{1z} \nonumber\\ &\qquad\qquad -\bmk\cdot{\boldsymbol{\sigma}_2}(\bmq\times\bmk)\cdot{\boldsymbol{\sigma}_1}(\bmq\cdot\bmK)\tau_{2z}\big]\end{aligned}$$ while for the latter, $$\begin{aligned} &V^{(1)}({\rm i-PNV})=\frac{5\ga^3\Delta_3}{64\fp^3}\frac{1}{\omk^2} \Big[\ttp\ i\bmk\cdot({{\bm \sigma}}_1-{{\bm \sigma}}_2)\nonumber \\ &\qquad+\ttm \ i \bmk\cdot({{\bm \sigma}}_1+{{\bm \sigma}}_2)\Big] \int_\bmq \frac{k^2+q^2}{\omp\omm(\omp+\omm)}\nonumber\\ &\qquad+\frac{5\ga^3\Delta_3}{16\fp^3}\frac{1}{\omk^2}\nonumber\\ &\qquad\times\Big[\big(\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}+\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\big)\ttm\nonumber\\ &\qquad+\big(\bmk\cdot{\boldsymbol{\sigma}_1}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}-\bmk\cdot{\boldsymbol{\sigma}_2}(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\big)\ttp\Big]\nonumber\\ &\qquad\times\int_\bmq \frac{1}{\omp\omm(\omp+\omm)}\ .\end{aligned}$$ Also in this case we use the dimensional regularization that permits us to write the finite contribution as, $$\begin{aligned} \overline{V}&^{(1)}({\rm i})=-\frac{5\ga^3\Delta_3}{16\fp\Lx^2} (\ttm\ i\bmk\cdot({{\bm \sigma}}_1+{{\bm \sigma}}_2) \nonumber\\ &+\ttp\ i\bmk\cdot({{\bm \sigma}}_1-{{\bm \sigma}}_2))\nonumber\\ &\times\Big(\frac{25}{6}L(k)-\frac{7}{2}\frac{\mp^2L(k)}{\omega_k^2} +2\frac{\mp^2H(k)}{\omega_k^2}\Big) -\frac{25\ga^3\Delta_3}{12\fp\Lx^2}\frac{1} {\omega_k^2}\nonumber\\ &\times\Big[\ttm \ \big((\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}\bmk\cdot{\boldsymbol{\sigma}_1}+(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\bmk\cdot{\boldsymbol{\sigma}_2}\big)\nonumber\\ &+\ttp \ \big((\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_2}\bmk\cdot{\boldsymbol{\sigma}_1}-(\bmk\times\bmK)\cdot{\boldsymbol{\sigma}_1}\bmk\cdot{\boldsymbol{\sigma}_2}\big)\Big]\ ,\end{aligned}$$ while the divergent part reads, $$\begin{aligned} V^{(1)}_\infty&({\rm i})= \frac{5\ga^3\Delta_3}{16\fp\Lx^2} (\ttm\ i\bmk\cdot({{\bm \sigma}}_1+{{\bm \sigma}}_2)\nonumber\\ &+\ttp\ i\bmk\cdot({{\bm \sigma}}_1-{{\bm \sigma}}_2))\nonumber\\ &\times\Big[\frac{\mp^2} {\omega_k^2}\Big(\frac{151}{12}\deps-\frac{305}{18}\Big) -\frac{25}{12}\deps-\frac{2}{9}\Big]\ . \label{eq:divpanli}\end{aligned}$$ The divergences present in Eq. (\[eq:divpanli\]) are reabsorbed in the ${V_{TRV}}^{({\rm OPE})}$ term proportional to $g_1$ for the part which multiply $\mp^2/\omega_k^2$ and in the ${V_{TRV}}^{({\rm CT})}$ term proportional to $C_4$ for the rest. All the divergences related to the term where $\bmK$ is present cancel out. The N2LO contribution of panel (l) in Fig. \[fig:diagNN\] is proportional to the LECs $c_1$, $c_2$ and $c_3$ of the PC sector and it reads, $$\begin{aligned} V^{(1)}&({\rm l})=-\frac{5\ga\Delta_3M}{2\fp^3}\frac{1}{\omk^2} \Big[\ttp \ i\bmk\cdot({{\bm \sigma}}_1-{{\bm \sigma}}_2)\nonumber\\ &+\ttm \ i \bmk\cdot({{\bm \sigma}}_1+{{\bm \sigma}}_2)\Big] \Big(c_1 m_\pi^2\int_\bmq\frac{1}{\omp\omm(\omp+\omm)}\nonumber\\ &+\frac{c_2+c_3}{8}\int_\bmq\frac{1}{\omp+\omm} -\frac{c_3}{8}\int_\bmq\frac{k^2-q^2}{\omp\omm(\omp+\omm)}\Big)\ .\end{aligned}$$ Using dimensional regularization we obtain for the finite part, $$\begin{aligned} \overline{V}^{(1)}&({\rm l})=\frac{5\ga\Delta_3M}{2\fp\Lx^2} (\ttm\ i\bmk\cdot({{\bm \sigma}}_1+{{\bm \sigma}}_2)\nonumber\\ &+\ttp\ i\bmk\cdot({{\bm \sigma}}_1+{{\bm \sigma}}_2))\nonumber\\ &\times\Big[4c_1\ \frac{\mp^2L(k)}{\omega_k^2} -\frac{c_2}{3}\ \Big(2L(k)+6\frac{\mp^2}{\omega_k^2}L(k)\Big)\nonumber\\ &-\frac{c_3}{2}\ \Big(3L(k)+5\frac{\mp^2}{\omega_k^2}L(k)\Big)\Big]\ ,\end{aligned}$$ while the divergent part is given by, $$\begin{aligned} V^{(1)}_\infty&(l)=\frac{5\ga\Delta_3M}{2\fp\Lx^2} (\ttm\ i\bmk\cdot({{\bm \sigma}}_1+{{\bm \sigma}}_2)\nonumber\\ &+\ttp\ i\bmk\cdot({{\bm \sigma}}_1-{{\bm \sigma}}_2)) \Big[c_1\frac{1}{\omega_k^2}(-2\deps+4)\nonumber\\ &-\frac{c_2}{3}\Big(\frac{\mp^2}{\omega_k^2} \big(5\deps-\frac{19}{3}\big) +\big(\deps-\frac{5}{3}\big)\Big)\nonumber\\ &-\frac{c_3}{2}\Big(\frac{\mp^2} {\omega_k^2}\big(\frac{11}{2}\deps-\frac{25}{3}\big) +\big(\frac{3}{2}\deps-\frac{2}{3}\big)\Big)\Big]\ .\end{aligned}$$ As for the panel (i) the divergences which multiply $\mp^2/\omega_k^2$ are reabsorbed in the $V^{({\rm OPE})}$ term proportional to $g_1$ while all the others in the $V^{({\rm CT})}$ term proportional to $C_4$. At N2LO panel (l) can receive contribution also from the second order in the $\pi NN$ and $\pi\pi NN$ vertices but due to the isospin structure these contributions vanish. As regarding the $NNN$ TRV potential all the contributions come from diagrams (m) of Fig. \[fig:diagNN\]. The expression we obtain at NLO is given in Eq. (\[eq:NNNpot\]). The N2LO component would come from NLO PC $\pi NN$ vertex or in the pion propagators. In both cases the different time-order diagrams cancel out each-other completely. The potential in configuration space {#app:rpot} ==================================== The $NN$ potential {#app:rpotNN} ------------------ In this subsection we present the $NN$ potential part in the configuration space which follows directly from Eq. (\[eq:vrsp\]) and it reads, $$\begin{aligned} V_{TRV}(\bmr,\bmp)&=& V^{({\rm OPE})}(\bmr)+V^{({\rm TPE})}(\bmr)+V^{(3\pi,0)}(\bmr) \nonumber\\ &+&V^{(3\pi,1)}(\bmr)+ V^{({\rm CT})}(\bmr)+V^{({\rm RC})}(\bmr,\bmp) \nonumber\\ &+&V^{(3\pi,{\rm RC})}(\bmr,\bmp)\ , \label{eq:trvnnr}\end{aligned}$$ where $\bmp=-i{{\bm \nabla}}$ is the relative momentum operator. It is convenient to define the operators, $$\begin{aligned} \ssp&=&({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})\cdot\bmvr\ ,\\ \ssm&=&({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\cdot\bmvr\ ,\\ \slp&=&-i({\boldsymbol{\sigma}_1}+{\boldsymbol{\sigma}_2})\cdot(\bmvr\times\hat{\bm L})\ ,\\ \slm&=&-i({\boldsymbol{\sigma}_1}-{\boldsymbol{\sigma}_2})\cdot(\bmvr\times\hat{\bm L})\ ,\\ \srx&=&({\boldsymbol{\sigma}_1}\times{\boldsymbol{\sigma}_2})\cdot\bmvr\ ,\\ \srlp&=&{\boldsymbol{\sigma}_1}\cdot\bmvr\, {\boldsymbol{\sigma}_2}\cdot\hat{\bm L}+ {\boldsymbol{\sigma}_2}\cdot\bmvr\, {\boldsymbol{\sigma}_1}\cdot\vec{\bm L}\ ,\\ \srlm&=&{\boldsymbol{\sigma}_1}\cdot\bmvr\, {\boldsymbol{\sigma}_2}\cdot\hat{\bm L}- {\boldsymbol{\sigma}_2}\cdot\bmvr\, {\boldsymbol{\sigma}_1}\cdot\vec{\bm L}\ ,\end{aligned}$$ where $\hat\bmL=\hat\bmr\times\bmp$ is the “reduced” orbital angular momentum operator. In terms of these, $V_{TRV}(\bmr,\bmp)$ can be written as =1.0pt $$\begin{aligned} V^{({\rm OPE})}(\bmr) &=\frac{\ga g_0^*\mp}{2\fp}\TO\ \ssm \ g'(r)\nonumber\\ &+ \frac{g_Ag_1\mp}{4f_\pi}\left(\ttp\ \ssm+\ttm\ \ssp\right)\ g'(r)\nonumber\\ &+\frac{\ga g_2\mp}{6\fp}\tten\ \ssm \ g'(r)\ ,\\ V^{({\rm TPE})}(\bmr) &= \frac{\ga g_0^*\mp^3}{\fp\Lx^2}\TO\ \ssm\ L'(r)\nonumber\\ &+\frac{\ga^3 g_0^*\mp^3}{\fp\Lx^2}\TO\ \ssm\ (H'(r)-3L'(r))\nonumber\\ &-\frac{\ga g_2\mp^3}{3\fp\Lx^2}\tten\ \ssm\ L'(r)\nonumber\\ &-\frac{\ga^3 g_2\mp^3}{3\fp\Lx^2}\tten\ \ssm\ (H'(r)-3L'(r))\ ,\\ V^{(3\pi,0)}(\bmr)&= -\frac{5\ga^3\Delta_3 M \mp^2}{4\fp\Lx^2}\pi\nonumber\\ &\times(\ttp\ \ssm+\ttm\ \ssp)(A'(r)+g'(r))\\ V^{(3\pi,1)}(\bmr)&=\frac{5\ga\Delta_3M\mp^3}{2\fp\Lx^2}(\ttp\ \ssm+\ttm\ \ssp) \nonumber\\ &\Big[4c_1\ L'_\omega(r)-\frac{c_2}{3}(2L'(r)+6L'_\omega(r))\nonumber\\ &\qquad-\frac{c_3}{2}(3L'(r)+5L'_\omega(r))\Big]\ ,\\ V^{({\rm CT})}(\bmr) &= {m_\pi^2 \over \Lambda_\chi^2 f_\pi} \Bigl[ C_1\ \ssm\ Z'(r)+C_2\ \TO\ \ssm\ Z'(r)\nonumber\\ &+{C_3 \over 2}\ (\ttp\ \ssm-\ttm\ \ssp)\ Z'(r)\nonumber\\ &+{C_4 \over 2}\ (\ttp\ \ssm+\ttm\ \ssp)\ Z'(r)\nonumber\\ &+C_5\ \tten\ \ssm\ Z'(r) \Bigr]\ ,\end{aligned}$$ $$\begin{aligned} V^{({\rm RC})}&(\bmr,\bmp)=\nonumber\\ &+\frac{\ga g_0^*\mp^3}{2\fp M^2 }\ \TO\ \Big[\frac{\ssm}{4}\Big(g'''(r) +2\frac{g''(r)}{r}-2\frac{g'(r)}{r^2}\Big)\nonumber\\ &+\slm\frac{g'(r)}{r^2}-\frac{g'(r)}{r^2}\ssm L^2 -\frac{1}{4}\Big(\frac{g''(r)}{r} +\frac{g'(r)}{r^2}\Big)\srlm\nonumber\\ &+\Big(\ssm\Big(g''(r)+2\frac{g'(r)}{r}\Big) -i\frac{\srx}{2}\frac{g'(r)}{r}\Big) \frac{d}{dr}\nonumber\\ &+g'(r)\ssm\frac{d^2}{dr^2}\Big]\frac{1}{\mp^2}\nonumber\\ &+\frac{\ga g_1\mp^3}{4\fp M^2}\ \ttp\ \Big[\frac{\ssm}{4}\Big(g'''(r) +2\frac{g''(r)}{r}-2\frac{g'(r)}{r^2}\Big)\nonumber\\ &+\slm\frac{g'(r)}{r^2}-\frac{g'(r)}{r^2}\ssm L^2 -\frac{1}{4}\Big(\frac{g''(r)}{r} +\frac{g'(r)}{r^2}\Big)\srlm\nonumber\\ &+\Big(\ssm\Big(g''(r)+2\frac{g'(r)}{r}\Big) -i\frac{\srx}{2}\frac{g'(r)}{r}\Big) \frac{d}{dr}\nonumber\\ &+g'(r)\ssm\frac{d^2}{dr^2}\Big]\frac{1}{\mp^2}\nonumber\\ &+\frac{\ga g_1\mp^3}{4\fp M^2}\ \ttm\ \Big[\frac{\ssp}{4}\Big(g'''(r) +2\frac{g''(r)}{r}-2\frac{g'(r)}{r^2}\Big)+\nonumber\\ &+\slp\frac{g'(r)}{r^2}-\frac{g'(r)}{r^2}\ssp L^2 -\frac{1}{4}\Big(\frac{g''(r)}{r} -\frac{g'(r)}{r^2}\Big)\srlp\nonumber\\ &+\ssp\Big(g''(r)+2\frac{g'(r)}{r}\Big) \frac{d}{dr}+g'(r)\ssp\frac{d^2}{dr^2}\Big]\frac{1}{\mp^2}\nonumber\\ &+\frac{\ga g_2\mp^3}{6\fp M^2}\ \tten\ \Big[\frac{\ssm}{4}\Big(g'''(r) +2\frac{g''(r)}{r}-2\frac{g'(r)}{r^2}\Big)\nonumber\\ &+\slm\frac{g'(r)}{r^2}-\frac{g'(r)}{r^2}\ssm L^2 -\frac{1}{4}\Big(\frac{g''(r)}{r} +\frac{g'(r)}{r^2}\Big)\srlm\nonumber\\ &+\Big(\ssm\Big(g''(r)+2\frac{g'(r)}{r}\Big) -i\frac{\srx}{2}\frac{g'(r)}{r}\Big) \frac{d}{dr}\nonumber\\ &+g'(r)\ssm\frac{d^2}{dr^2}\Big]\frac{1}{\mp^2}\ ,\label{eq:rcoper}\end{aligned}$$ $$\begin{aligned} &V^{(3\pi-{\rm RC})}(\bmr,\bmp)= -\frac{5\ga^3\Delta_3\mp^3}{16\fp\Lx^2}(\ttp\ \ssm+\ttm\ \ssp)\nonumber\\ &\qquad \Big(\frac{25}{6}L'(r)-\frac{7}{2}L'_\omega(r)+2H'_\omega(r)\Big) \nonumber\\ &\qquad+\frac{25\ga^3\Delta_3\mp^3}{12\fp\Lx^2}\Big[\ttp\ \Big(\Big(\frac{g''(r)}{r} +\frac{g'(r)}{r^2}\Big)\srlm\nonumber\\ &\qquad+2i\srx\frac{g'(r)}{r}\frac{d}{dr}\Big) +\ttm\ \Big(\frac{g''(r)}{r} -\frac{g'(r)}{r^2}\Big)\srlp\Big]\frac{1}{\mp^2}\ ,\label{eq:rc3pr}\end{aligned}$$ with $$\begin{aligned} g(r) &=& \int {{\rm d}^3k\over (2\pi)^3}\; {C_\Lambda(k)\over \mp} \frac{1}{\omega_k^2}\, e^{i\bmk\cdot\bmr} \ ,\label{eq:g}\\ L(r) &=& \int {{\rm d}^3k\over (2\pi)^3}\; {C_\Lambda(k)\over m_\pi^3} L(k)\, e^{i\bmk\cdot\bmr} \ ,\label{eq:lr}\\ H(r) &=& \int {{\rm d}^3k\over (2\pi)^3}\; {C_\Lambda(k)\over m_\pi^3} H(k)\, e^{i\bmk\cdot\bmr} \ ,\label{eq:hr}\\ A(r) &=& \int {{\rm d}^3k\over (2\pi)^3}\; {C_\Lambda(k)\over m_\pi^2} {s^2\ A(k)\over \omega_k^2}\, e^{i\bmk\cdot\bmr} \ ,\label{eq:ar}\\ L_\omega(r) &=& \int {{\rm d}^3k\over (2\pi)^3}\; {C_\Lambda(k)\over m_\pi} {L(k) \over \omega_k^2}\, e^{i\bmk\cdot\bmr} \ ,\label{eq:lrw}\\ H_\omega(r) &=& \int {{\rm d}^3k\over (2\pi)^3}\; {C_\Lambda(k)\over m_\pi} {H(k) \over \omega_k^2}\, e^{i\bmk\cdot\bmr} \ ,\label{eq:hrw}\\ Z(r) &=& \int {{\rm d}^3k\over (2\pi)^3}\; {C_\Lambda(k)\over m_\pi^2}\, e^{i\bmk\cdot\bmr} \ .\label{eq:z}\end{aligned}$$ The functions $g(r)$, $L(r)$, $H(r)$, $A(r)$, $L_\omega(r)$, $H_\omega(r)$ and $Z(r)$ are calculated numerically by standard quadrature techniques. The $NNN$ potential {#sec:NNNr} ------------------- In this section we present the explicit derivation of the $NNN$ potential in configuration space. Writing explicitly the integral in Eq. (\[eq:vrnnn\]), neglecting the deltas for simplicity, we get, $$\begin{aligned} V&(\bmx_1,\bmx_2)=\nonumber\\ &\quad-\frac{\Delta_3 g_A^3 M}{4\fp^3}\ T_3\ \frac{1}{2}\int {d^3q\over (2\pi)^3} {d^3Q\over (2\pi)^3} e^{-i(\bmq/2)\cdot\bmx_2}\,e^{-i\bmQ\cdot\bmx_1}\,\nonumber\\ &\quad\times\frac{i(\bmQ+\bmq)\cdot{\boldsymbol{\sigma}_1}\, i(\bmQ-\bmq)\cdot{\boldsymbol{\sigma}_2}\, i\bmQ\cdot{\boldsymbol{\sigma}_3}} {\omega_+^2\omega_-^2\omega_Q^2}\ ,\label{eq:qQxx}\end{aligned}$$ where $T_3={\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_2}\ \tau_{3z}+ {\boldsymbol{\tau}_1}\cdot{\boldsymbol{\tau}_3}\ \tau_{2z}+{\boldsymbol{\tau}_2}\cdot{\boldsymbol{\tau}_3}\ \tau_{1z}$. The momenta $\bmq$ and $\bmQ$ in Eq. (\[eq:qQxx\]) can be rewritten applying the gradient $i{{\bm \nabla}}$ to the exponential functions and it reads, $$\begin{aligned} V&(\bmx_1,\bmx_2)=\nonumber\\ &\frac{\Delta_3 g_A^3 M}{4\fp^3}\ T_3\ \frac{1}{2}\big[ ({{\bm \nabla}}_{\bmx_1}+2{{\bm \nabla}}_{\bmx_2})\cdot{\boldsymbol{\sigma}_1}\ ({{\bm \nabla}}_{\bmx_1}-2{{\bm \nabla}}_{\bmx_2})\cdot{\boldsymbol{\sigma}_2}\nonumber\\ &\times{{\bm \nabla}}_{\bmx_1}\cdot{\boldsymbol{\sigma}_3}\big] \int{d^3Q\over (2\pi)^3}\ \frac{e^{-i\bmQ\cdot\bmx_1}}{\omega_Q^2} \ \int {d^3q\over (2\pi)^3}\ \frac{e^{-i(\bmq/2)\cdot\bmx_2}} {\omega_+^2\omega_-^2}\ .\label{eq:qQxxd} \end{aligned}$$ The integral in $\bmq$ in Eq. (\[eq:qQxxd\]) can be solved using the Feynman tricks. The final result is $$\begin{aligned} V(\bmx_1,&\bmx_2)= \frac{\Delta_3 g_A^3 M}{4\fp^3}\ T_3\ \frac{1}{32\pi}\big[ ({{\bm \nabla}}_{\bmx_1}+2{{\bm \nabla}}_{\bmx_2})\cdot{\boldsymbol{\sigma}_1}\nonumber\\ &\times({{\bm \nabla}}_{\bmx_1}-2{{\bm \nabla}}_{\bmx_2})\cdot{\boldsymbol{\sigma}_2}\ {{\bm \nabla}}_{\bmx_1}\cdot{\boldsymbol{\sigma}_3}\big] \int{d^3Q\over (2\pi)^3}\ \frac{e^{-i\bmQ\cdot\bmx_1}}{\omega_Q^2}\nonumber\\ &\times \int_{-1}^1 dx\ e^{i(x/2)\bmQ\bmx_2} \frac{e^{-(L/2)x_2}} {L}\ ,\label{eq:qxxxd} \end{aligned}$$ where $L=\sqrt{Q^2(1-x^2)+4\mp^2}$. The derivative are then evaluated obtaining five integral operators $I_i$, $$\begin{aligned} V(\bmx_1,\bmx_2)&=&\frac{\Delta_3 g_A^3 M}{4\fp^3}\ T_3\ \sum_{i=1,5} I_i \end{aligned}$$ where, $$\begin{aligned} I_1&=-\frac{i}{16\pi}\int{d^3Q\over (2\pi)^3}\ \frac{e^{-i\bmQ\cdot\bmx_1}}{\omega_Q^2} \int_{-1}^1 dx\, \nonumber\\ &\quad\times e^{i(x/2)\bmQ\cdot\bmx_2}\frac{e^{-(L/2)x_2}}{x_2} \,({\boldsymbol{\sigma}_1}\cdot{\boldsymbol{\sigma}_2})\bmQ\cdot{\boldsymbol{\sigma}_3}\ ,\\ I_2&=+\frac{i}{32\pi}\int{d^3Q\over (2\pi)^3}\ \frac{e^{-i\bmQ\cdot\bmx_1}}{\omega_Q^2} \int_{-1}^1 dx\, (1-x^2) \nonumber\\ &\quad\times e^{i(x/2)\bmQ\cdot\bmx_2}\frac{e^{-(L/2)x_2}}{L} \,\bmQ\cdot{\boldsymbol{\sigma}_1}\,\bmQ\cdot{\boldsymbol{\sigma}_2}\,\bmQ\cdot{\boldsymbol{\sigma}_3}\ ,\\ I_3&=-\frac{1}{32\pi}\int{d^3Q\over (2\pi)^3}\ \frac{e^{-i\bmQ\cdot\bmx_1}}{\omega_Q^2} \int_{-1}^1 dx\, (1-x)\nonumber\\ &\quad\times e^{i(x/2)\bmQ\cdot\bmx_2}e^{-(L/2)x_2} \bmQ\cdot{\boldsymbol{\sigma}_1}\,\bmx_2\cdot{\boldsymbol{\sigma}_2}\,\bmQ\cdot{\boldsymbol{\sigma}_3}\ ,\\ I_4&=+\frac{1}{32\pi}\int{d^3Q\over (2\pi)^3}\ \frac{e^{-i\bmQ\cdot\bmx_1}}{\omega_Q^2} \int_{-1}^1 dx\, (1+x)\nonumber\\ &\quad\times e^{i(x/2)\bmQ\cdot\bmx_2}e^{-(L/2)x_2} \,\bmx_2\cdot{\boldsymbol{\sigma}_1}\,\bmQ\cdot{\boldsymbol{\sigma}_2}\,\bmQ\cdot{\boldsymbol{\sigma}_3}\ ,\\ I_5&=+\frac{i}{32\pi}\int{d^3Q\over (2\pi)^3}\ \frac{e^{-i\bmQ\cdot\bmx_1}}{\omega_Q^2} \int_{-1}^1 dx\, \Big(L+\frac{2}{x_2}\Big)\nonumber\\ &\quad\times e^{i(x/2)\bmQ\cdot\bmx_2} \frac{e^{-(L/2)x_2}}{L} \,\bmx_2\cdot{\boldsymbol{\sigma}_1}\,\bmx_2\cdot{\boldsymbol{\sigma}_2}\,\bmQ\cdot{\boldsymbol{\sigma}_3}\ , \end{aligned}$$ To be noticed that integral $I_4=I_3$ exchanging particle 1 and 2. In order to divide the angular integration in $\hat{\bmQ}$ from the radial integration in $Q$, the exponential functions are expanded in plane waves. Therefore the integrals $I_i$ result, $$I_i=\sum_{l,l'}O_i^{ll'}J_i^{ll'}\ ,$$ where $J_i$ is the integral over $Q$ while in $O_i$ the integral is over $\hat{Q}$. The expressions of $J_i$ are explicitly given below: $$\begin{aligned} J_1^{ll'}&=\frac{1}{8\pi}\int_{-1}^1 dx\ \int_0^\infty dQ \frac{Q^2}{\omega_Q^2} \ C_\Lambda(Q)e^{-(L/2)x_2}\nonumber\\ &\qquad\qquad\times j_l\big({xQx_2\over 2}\big)j_{l'}(Qx_1)\frac{Q}{x_2} \ ,\\ J_2^{ll'}&=\frac{1}{16\pi}\int_{-1}^1 dx\ \int_0^\infty dQ \frac{Q^2}{\omega_Q^2} \ C_\Lambda(Q)e^{-(L/2)x_2}\nonumber\\ &\qquad\qquad\times j_l\big({xQx_2\over 2}\big)j_{l'}(Qx_1) \frac{Q^3}{L}(1-x^2)\ ,\\ J_3^{ll'}&=\frac{1}{16\pi}\int_{-1}^1 dx\ \int_0^\infty dQ \frac{Q^2}{\omega_Q^2} \ C_\Lambda(Q)e^{-(L/2)x_2}\nonumber\\ &\qquad\qquad\times j_l\big({xQx_2\over 2}\big)j_{l'}(Qx_1) Q^2(1-x)\ ,\\ J_4^{ll'}&=\frac{1}{16\pi}\int_{-1}^1 dx\ \int_0^\infty dQ \frac{Q^2}{\omega_Q^2} \ C_\Lambda(Q)e^{-(L/2)x_2}\nonumber\\ &\qquad\qquad\times j_l\big({xQx_2\over 2}\big)j_{l'}(Qx_1) Q^2(1+x)\ ,\\ J_5^{ll'}&=\frac{1}{16\pi}\int_{-1}^1 dx\ \int_0^\infty dQ \frac{Q^2}{\omega_Q^2} \ C_\Lambda(Q)e^{-(L/2)x_2}\nonumber\\ &\qquad\qquad\times j_l\big({xQx_2\over 2}\big)j_{l'}(Qx_1) Q\Big(L-\frac{2}{x_2}\Big)\ , \end{aligned}$$ where $j_l(x)$ are spherical Bessel functions and we include a regularization function $C_\Lambda(Q)$ which is defined in Eq. (\[eq:cutoffnnn\]). The functions $J_i$ depend only on the modules of $\bmx_1$ and $\bmx_2$ and they are evaluated numerically by standard quadrature techniques. For the $O_i$ integrals we get, $$\begin{aligned} O_1^{ll'}&=(-i)i^{l'}(-i)^{l}\hat{l'}\hat{l}\int d\hat{Q} \Big[Y_{l'}(\hat{Q})Y_{l'}(\hat{x}_1)\Big]_0\nonumber\\ &\times \Big[Y_{l }(\hat{Q})Y_{l}(\hat{x}_2)\Big]_0 {\boldsymbol{\sigma}_1}\cdot{\boldsymbol{\sigma}_2}\, \hat{Q}\cdot{\boldsymbol{\sigma}_3}\ ,\\ O_2^{ll'}&=(+i)i^{l'}(-i)^{l}\hat{l'}\hat{l}\int d\hat{Q} \Big[Y_{l'}(\hat{Q})Y_{l'}(\hat{x}_1)\Big]_0\nonumber\\ &\times \Big[Y_{l }(\hat{Q})Y_{l}(\hat{x}_2)\Big]_0 \hat{Q}\cdot{\boldsymbol{\sigma}_1}\,\hat{Q}\cdot{\boldsymbol{\sigma}_2}\,\hat{Q}\cdot{\boldsymbol{\sigma}_3}\ ,\\ O_3^{ll'}&=-i^{l'}(-i)^{l}\hat{l'}\hat{l}\int d\hat{Q} \Big[Y_{l'}(\hat{Q})Y_{l'}(\hat{x}_1)\Big]_0\nonumber\\ &\times \Big[Y_{l }(\hat{Q})Y_{l}(\hat{x}_2)\Big]_0 \hat{Q}\cdot{\boldsymbol{\sigma}_1}\,\hat{x}_2\cdot{\boldsymbol{\sigma}_2}\,\hat{Q}\cdot{\boldsymbol{\sigma}_3}\ ,\\ O_4^{ll'}&=+i^{l'}(-i)^{l}\hat{l'}\hat{l}\int d\hat{Q} \Big[Y_{l'}(\hat{Q})Y_{l'}(\hat{x}_1)\Big]_0\nonumber\\ &\times \Big[Y_{l }(\hat{Q})Y_{l}(\hat{x}_2)\Big]_0 \hat{x}_2\cdot{\boldsymbol{\sigma}_1}\,\hat{Q}\cdot{\boldsymbol{\sigma}_2}\,\hat{Q}\cdot{\boldsymbol{\sigma}_3}\ ,\\ O_5^{ll'}&=(+i)i^{l'}(-i)^{l}\hat{l'}\hat{l}\int d\hat{Q} \Big[Y_{l'}(\hat{Q})Y_{l'}(\hat{x}_1)\Big]_0\nonumber\\ &\times \Big[Y_{l }(\hat{Q})Y_{l}(\hat{x}_2)\Big]_0 \hat{x}_2\cdot{\boldsymbol{\sigma}_1}\,\hat{x}_2\cdot{\boldsymbol{\sigma}_2}\,\bmQ\cdot{\boldsymbol{\sigma}_3}\ , \end{aligned}$$ which depend only to the angular part of the spatial coordinates $\bmx_1$ and $\bmx_2$. The matrix element of the $O_i$ operators between HH functions can easily expressed in terms of products of 9-j, 6-j and 3-j Wigner symbols. Details of the calculation and convergence of the $a_\Delta(3N)$ coefficient {#app:NNNconv} ============================================================================ In order to compute the even-parity $|\psi^A_{+}{\rangle}$ and odd-parity $|\psi^A_{-}{\rangle}$ component of the wave function we need to solve the following eigenvalue problem, $$\begin{aligned} \label{eq:egp} \begin{pmatrix} V_{\rm PC}^{++} & V_{\rm TRV}^{+-}\\ V_{\rm TRV}^{-+} & V_{\rm PC}^{--} \end{pmatrix} \begin{pmatrix} |\psi^A_{+}{\rangle}\\ |\psi^A_{-}{\rangle}\end{pmatrix} =E \begin{pmatrix} |\psi^A_{+}{\rangle}\\ |\psi^A_{-}{\rangle}\end{pmatrix}\,. \end{aligned}$$ Using the fact that $||V_{\rm TRV}||<<||V_{\rm PC}||$ we can rewrite the eigenvalue problem of Eq. (\[eq:egp\]) as, $$\begin{aligned} \label{eq:egp2} \begin{cases} V_{\rm PC}^{++}|\psi^A_{+}{\rangle}=E|\psi^A_{+}{\rangle}\\ |\psi^A_{-}{\rangle}=-(V_{\rm PC}^{--}-E)^{-1} V_{\rm TRV}^{-+} |\psi^A_{+}{\rangle}\end{cases}\,, \end{aligned}$$ where the first equation is the standard eigenvalue problem, while with the second we can compute the odd component of the wave function. This approach from the numerical point of view results more stable than solving directly Eq. (\[eq:egp\]) and permits to study the odd component of the wave function without solving every time the eigenvalue problem. Let us study now the convergence pattern of the $a_\Delta(3N)$ coefficient, defined in Eq. (\[eq:adelta3\]), in term of the grandangular momentum $K$ of the HH basis (for more details, see Ref. [@AK08; @LE09]). Increasing $K$ is equivalent of enlarging the expansion basis. To be definite, in this appendix, we have considered the $\tri$ case and used the N4LO/N2LO-500 PC interaction. Let us denote with $K^+$ ($K^-$) the maximum value of the grandangular momentum of the HH functions used to describe the even (odd) part of the wave function $|\psi^A_{+}{\rangle}$ ($|\psi^A_{-}{\rangle}$). We have computed the even part of the wave function up to complete convergence ($K^+=50$) obtaining a binding energy $B(\tri)=8.476$ MeV using the first formula in Eq. (\[eq:egp2\]). For solving the second equation of Eq. (\[eq:egp2\]) we have performed different calculations varying $K^-$ and we have reported the corresponding results for $a_\Delta(3N)$ in Table \[tab:3Nconv\]. As can be seen by inspecting the table, the pattern of convergence is very smooth. A safe convergence at the third digit is reached for $K^-\sim21$, which was the value also selected for performing the final calculations. On the other hand, the value of $a_\Delta(3N)$ is not much sensitive to $K^+$, in particular when $K^+>20$. Therefore, the value of the coefficient $a_\Delta(3N)$ appears to be well under control in our calculation. $K^-$ $a_\Delta(3N)$ ------- ---------------- 5 -0.1728 9 -0.1842 13 -0.1879 17 -0.1892 21 -0.1897 : \[tab:3Nconv\] Convergence pattern of the $\tri$ $a_\Delta(3N)$ coefficient as function of $K^-$, the maximum grandangular momentum of the HH functions used for constructing the odd-parity component of the $\tri$ wave function. 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--- abstract: 'In this review, we discuss the role of the various experimental programs taking part in the broader effort to identify the particle nature of dark matter. In particular, we focus on electroweak scale dark matter particles and discuss a wide range of search strategies being carried out and developed to detect them. These efforts include direct detection experiments, which attempt to observe the elastic scattering of dark matter particles with nuclei, indirect detection experiments, which search for photons, antimatter and neutrinos produced as a result of dark matter annihilations, and collider searches for new TeV-scale physics. Each of these techniques could potentially provide a different and complementary set of information related to the mass, interactions and distribution of dark matter. Ultimately, it is hoped that these many different tools will be used together to conclusively identify the particle or particles that constitute the dark matter of our universe.' author: - 'Dan Hooper Edward A. Baltz' title: | Strategies for Determining the Nature of Dark Matter\ ------------------------------------------------------------------------ --- psfig.sty dark matter INTRODUCTION ============ There exists a wide array of evidence in support of the conclusion that most of the matter in our universe is non-luminous. This includes observations of the rotational speeds of galaxies [@rotationcurves], the orbital velocities of galaxies within clusters [@clusters], gravitational lensing [@lensing], the cosmic microwave background [@wmap], the light element abundances [@bbn] and large scale structure [@lss]. Despite these many observational indications of dark matter, it is clear that it does not consist of baryonic material or other known forms of matter. For the time being, we remain ignorant of the particle identity of this substance. In this review, we summarize some of the most promising strategies and techniques being pursued to elucidate the nature of dark matter. These efforts include direct detection experiments designed to observe the elastic scattering of dark matter particles with nuclei, indirect detection experiments which hope to detect the annihilation products of dark matter such gamma rays, neutrinos, positrons, antiprotons, antideuterons, synchrotron radiation and X-rays, and collider searches for dark matter and associated particles. The material presented here is not intended to be an exhaustive summary of the field of dark matter physics. Such reviews can be found elsewhere [@Bertone:2004pz]. In this article, we limit our discussion to the case of candidate dark matter particles with electroweak scale masses and couplings. Furthermore, for the sake of length, we do not discuss every experimental approach being pursued, but instead focus on several of the most promising direct, indirect and collider efforts, and on the interplay and complementarity between these various programs. We would like to emphasize that the detection of dark matter particles in any one of the experimental channels discussed here will not alone be sufficient to conclusively identify the nature of dark matter. The direct or indirect detection of the dark matter particles making up our galaxy’s halo is highly unlikely to provide enough information to reveal the underlying physics (supersymmetry, etc.) behind these particles. In contrast, collider experiments may identify a long-lived, weakly interacting particle, but will not be able to test its cosmological stability or abundance. Only by combining the information provided by many different experimental approaches is the mystery of dark matter’s particle nature likely to be solved. Although the detection of dark matter in any one search channel would constitute a discovery of the utmost importance, it would almost certainly leave many important questions unanswered. THE WIMP HYPOTHESIS =================== In this review, we limit our discussion to dark matter candidates which are heavy, electrically neutral and weakly-interacting. This class of particles, collectively known as WIMPs, are particularly well motivated, especially when their mass and couplings are tied to the physics of the electroweak scale. Before we discuss the experimental techniques for detecting dark matter particles, we will briefly discuss some of the most compelling motivations for electroweak-scale dark matter. The challenge of stabilizing the mass of the Higgs boson ([*ie.*]{} the hierarchy problem) leads us to expect new forms of matter to appear at or near the electroweak scale. The nature of any physics beyond the Standard Model which might appear at the TeV scale, however, is tightly constrained by the precision electroweak measurements made at LEP. In particular, new discrete symmetries are required of most phenomenologically viable models of TeV scale physics [@hier]. Such symmetries naturally lead to a stable particle or particles, which may potentially constitute the dark matter of our universe. A number of extensions of the Standard Model have been proposed which introduce new particle content at or near the electroweak scale, and which include a discrete symmetry of the form required to stabilize a potential dark matter candidate. The most well studied example is the lightest neutralino in supersymmetric models. Others examples include Kaluza-Klein hypercharge gauge bosons in models with universal extra dimensions [@kkdm], and the lightest T-parity odd particle in little Higgs theories [@lh]. Each of these candidates have similar masses and couplings, and thus will undergo similar thermal histories in the early universe. At high temperatures, WIMPs are abundant, being freely created and annihilated in pairs. As the universe expands and the temperature drops below the WIMPs’ production threshold, however, the number density of these particles becomes rapidly suppressed. Ultimately, the WIMPs will “freeze out” and remain as a thermal relic of the universe’s hot youth. The resulting density of WIMPs is given by: \_h\^2 = [s\_0\_c/h\^2]{} ( [45g\_\*]{})\^[1/2]{} [ x\_f m\_]{}[1]{}, where $s_0$ is the current entropy density of the universe, $\rho_c$ is the critical density, $h$ is the (scaled) Hubble constant, $g_*$ is the effective number of relativistic degrees of freedom at the time that the dark matter particle goes out of thermal equilibrium, $m_\Pl$ is the Planck mass, $x_f = m/T_f \approx 25$ is the inverse freeze-out temperature in units of the WIMP mass, and $\VEV{\sigma v}$ is the thermal average of the dark matter pair annihilation cross section times the relative velocity. In order for this process to yield a thermal abundance of dark matter within the range measured by WMAP ($0.095 < \Omega h^2 < 0.129$) [@wmap], the thermally averaged annihilation cross section is required to be $\sigv \approx 3 \times 10^{-26}$ cm$^3$/s (or alternatively, $\sigv \approx 0.9$ pb). Remarkably, this is quite similar to the value obtained for a generic electroweak mass particle annihilating through the exchange of the electroweak gauge or Higgs bosons. In particular, we notice that $\VEV{\sigma v} = \pi \alpha^2/8m^2$ leads us to a WIMP mass on the order of $m \sim 100$ GeV. We conclude that if a stable, weakly interacting, electroweak-scale particle exists, then it is likely to be present in the universe today with an abundance similar to the measured dark matter density. With this in mind, we focus our dark matter search strategy on this particularly well motivated scenario in which the dark matter particle has electroweak interactions and a mass near the electroweak scale. DIRECT DETECTION {#sec:section5} ================ Experiments such as XENON [@xenon], CDMS [@cdmssi; @cdmssd], ZEPLIN [@zeplin], Edelweiss [@edelweiss], CRESST [@cresst], WARP [@warp] and COUPP [@coupp] are designed to detect dark matter particles through their elastic scattering with nuclei. This class of techniques is collectively known as direct detection, in contrast to indirect detection efforts which attempt to observe the annihilation products of dark matter particles. The role played by direct detection is important for a number of reasons. Firstly, although collider experiments may be capable of detecting dark matter particles, they will not be able to distinguish a cosmologically stable WIMP from a long-lived but unstable particle. More generally speaking, colliders will not inform us as to the cosmological abundance of a WIMP they might observe. Furthermore, while the mass of the dark matter particle could potentially be measured by a collider experiment such as the LHC, its couplings are much more difficult to access in this way. Direct detection experiments, in contrast, provide a valuable probe of the dark matter’s couplings to the Standard Model. Finally, the uncertainties involved in direct detection are likely to be significantly smaller than in most indirect detection channels. Whereas indirect detection rates rely critically on the distribution of dark matter, especially in high density regions, and on other astrophysical properties such as the galactic magnetic and radiation fields, direct detection experiments rely only on the local dark matter density and velocity distribution. The density of dark matter in the local neighborhood is inferred by fitting observations to models of the galactic halo. These observations including the rotational speed of stars at the solar circle and other locations, the total projected mass density (estimated by considering the motion of stars perpendicular to the galactic disk), peak-to-trough variations in the rotation curve ([*ie.*]{} the ‘flatness constraint’), and microlensing. Taken together, these constraints can be used to estimate the local halo density to lie between 4 $\times$ 10$^{-25}$ g/cm$^{-3}$ and 13 $\times$ 10$^{-25}$ g/cm$^3$ ($0.22-0.73\,$GeV/cm$^3$) [@GGTurner]. Limits on the density of MACHO microlensing objects imply that at least 80% of this is cold dark matter. The velocity of the WIMPs is expected be close to the galactic rotation velocity, $230 \pm 20$ km/sec [@Drukier]. These observations, however, only constrain the dark matter density as averaged over scales larger than a kiloparsec or so. In contrast, the solar system moves a distance of $\sim$$10^{-3}$ parsecs relative to the dark matter halo each year. If dark matter is distributed in an inhomogeneous way over milliparsec scales ([*ie.*]{} as a collection of dense clumps and voids), then the density along the path of the Earth, as seen by direct detection experiments, could be much larger or smaller than is inferred by the rotational dynamics of our galaxy. Throughout most of our galaxy’s halo, however, inhomogeneities in the small scale dark matter distribution are not anticipated to be large. The vast majority of the dark matter in the inner regions of our galaxy has been in place for $\sim$$10^{10}$ years; ample time for the destruction of clumps through tidal interactions. Using high-resolution simulations, Helmi, White and Springel find that the dark matter in the solar neighborhood is likely to consist of a superposition of hundreds of thousands of dark matter streams, collectively representing a very smooth and homogeneous distribution [@white]. That being said, if we happen to find our Solar System residing in a overdense clump or stream of dark matter, high direct detection rates could lead us to mistakenly infer an artificially large WIMP-nucleon elastic scattering cross section. The nuclear physics involved in WIMP-nuclei elastic scattering also introduces uncertainties which may ultimately limit the accuracy to which the dark matter’s couplings to the Standard Model can be measured. In many models, including many supersymmetric models, the WIMP-nucleon scattering cross section is dominated by the $t$-channel exchange of a Higgs boson. The coupling of the Higgs boson to the proton receives its dominant contributions from two sources, the coupling of the Higgs to gluons through a heavy quark loop and the direct coupling of the Higgs to strange quarks [@jungman]. That means that this coupling depends on the parameter $$f_{Ts} = {\bra{p} m_s \bar s s \ket{p} \over \bra{p} H_{QCD} \ket{p}} \ , \label{fTsdefin}$$ that is, the fraction of the mass of the proton that arises from the mass of the non-valence strange quarks in the proton wavefunction. It has been known for some time that there is significant uncertainty in this quantity [@KaplanNelson], and several recent papers have pointed out the uncertainty this introduces to calculations of the WIMP-nucleon elastic scattering cross section [@Bottino; @EllisUpdate]. In particular, in the case of WIMPs which couple dominantly to the strange content of the nucleon, this can lead to an uncertainty in the direct detection cross section of a factor of 4 or even larger [@Bottino]. It is possible that this uncertainty could be reduced in the future through the use of lattice gauge theory [@UKQCD; @Weise]. The processes of WIMP-nuclei elastic scattering can be naturally divided into spin-dependent and spin-independent contributions. The spin-independent, or coherent scattering, term is enhanced in WIMP-nucleus cross sections by factors of $A^2$, making it advantageous to use targets consisting of heavy nuclei. This enhancement is due to the fact that the WIMP wavelength is of order the size of the nucleus, thus the scattering amplitudes on individual nucleons add coherently. The spin-dependent contribution, in contrast, couples to the spin of the target nuclei and scales with $J (J+1)$. Naively, this could be considered a coherent subtraction of amplitudes of opposite signs of pairs of nucleons. As the current spin-dependent scattering constraints are not strong enough to test many dark matter models, we devote our attention primarily to the process of spin-independent scattering. The spin-independent WIMP-nucleus elastic scattering cross section is given by: $$\label{sig} \sigma \approx \frac{4 m^2_{X} m^2_{T}}{\pi (m_{X}+m_T)^2} [Z f_p + (A-Z) f_n]^2,$$ where $m_T$ is the mass of the target nucleus, $m_X$ is the WIMP’s mass and $Z$ and $A$ are the atomic number and atomic mass of the nucleus. $f_p$ and $f_n$ are the WIMP’s couplings to protons and neutrons, given by: $$f_{p,n}=\sum_{q=u,d,s} f^{(p,n)}_{T_q} a_q \frac{m_{p,n}}{m_q} + \frac{2}{27} f^{(p,n)}_{TG} \sum_{q=c,b,t} a_q \frac{m_{p,n}}{m_q}, \label{feqn}$$ where $a_q$ are the WIMP-quark couplings and $f^{(p,n)}_{T_q}$ denote the quark content of the nucleon. The first term in Eq. \[feqn\] corresponds to interactions with the quarks in the target nuclei. In the case of neutralino dark matter, this can occur through either $t$-channel CP-even Higgs exchange, or $s$-channel squark exchange: (222,254)(3,4.3) (5.,20.)(20.,10.)(0.,)[/Straight]{}[0]{} (0.,18.0)\[b\][$\chi^0$]{} (20.,-5.0)(20.,10.0)(0.,)[/ScalarDash]{}[0]{} ( 5.,-15.0)(20.,-5.0)(0.,)[/Straight]{}[+1]{} (0.,-18.0)\[b\][$q$]{} (25.,0.0)\[b\][$H, h$]{} (20.,10.)(35.,20.)(0.,)[/Straight]{}[0]{} (20.,-5.)(35.,-15.0)(0.,)[/Straight]{}[+1]{} (40.,18.0)\[b\][$\chi^0$]{} (40.,-18.)\[b\][$q$]{} (70.,14.)(85.,3.)(0.,)[/Straight]{}[0]{} (65.,12.0)\[b\][$\chi^0$]{} (70.,-7.0)(85.,3.0)(0.,)[/Straight]{}[+1]{} (65.,-9.0)\[b\][$q$]{} (85.,3.)(105.,3.0)(0.,)[/ScalarDash]{}[0]{} (95.,9.)\[t\][$\tilde{q}$]{} (120.,14.)(105.,3.0)(0.,)[/Straight]{}[0]{} (125.,12.)\[b\][$\chi^0$]{} (120.,-7.)(105.,3.0)(0.,)[/Straight]{}[-1]{} (125.,-9.)\[b\][$q$]{} The second term corresponds to interactions with the gluons in the target through a loop diagram (a quark/squark loop in the case of supersymmetry). $f^{(p)}_{TG}$ is given by $1 -f^{(p)}_{T_u}-f^{(p)}_{T_d}-f^{(p)}_{T_s} \approx 0.84$, and analogously, $f^{(n)}_{TG} \approx 0.83$. Besides its mass, the only thing we need to know about the WIMP itself to calculate this cross section are its couplings to quarks, $a_q$. In the case of neutralino dark matter, the value of this coupling depends on many features of the supersymmetric spectrum. The contribution resulting from Higgs exchange depends on the neutralino composition, as well as the Higgs masses and couplings. In the case of heavy squarks, small wino component and little mixing between the CP-even Higgs bosons ($\cos \alpha \approx 1$), neutralino-nuclei elastic scattering is dominated by $H$ exchange with strange and bottom quarks, leading to a neutralino-nucleon cross section approximately given by: $$\sigma_{\chi N} \sim \frac{g^2_1 g^2_2 f_{\tilde{B}} f_{\tilde{H}} \,m^4_N}{4\pi m^2_W \cos^2 \beta \, m^4_H} \bigg(f_{T_s}+\frac{2}{27}f_{TG}\bigg)^2, \,\,\,\, (m_{\tilde{q}}\, \rm{large}, \cos \alpha \approx 1), \label{case1}$$ where $f_{\tilde{B}}$ and $f_{\tilde{H}}$ denote the bino and higgsino fractions of the lightest neutralino and $\tan \beta$ is the ratio of the vevs of the two Higgs doublets in the MSSM. Note that the coupling involves the product of $f_{\tilde{B}}$ and $f_{\tilde{H}}$. Neutralinos that are purely gaugino-like or purely higgsino-like have zero cross section with nuclei. The fundamental reason for this is that the relevant vertex is gaugino-higgsino-Higgs. If the heavier of the two CP-even Higgs bosons is very heavy and/or $\tan \beta$ is small, scattering with up-type quarks through light Higgs exchange can dominate: $$\sigma_{\chi N} \sim \frac{g^2_1 g^2_2 f_{\tilde{B}} f_{\tilde{H}} \, m^4_N}{4\pi m^2_W \, m^4_h} \bigg(f_{T_u}+\frac{4}{27}f_{TG}\bigg)^2, \,\,\,\, (m_{\tilde{q}}, m_H\, \rm{large}, \cos \alpha \approx 1). \label{case2}$$ If $\tan \beta$ and $m_H$ are large and the squarks somewhat light, elastic scattering can instead be dominated by squark exchange: $$\sigma_{\chi N} \sim \frac{g^2_1 g^2_2 f_{\tilde{B}} f_{\tilde{H}} \, m^4_N}{4\pi m^2_W \cos^2 \beta \, m^4_{\tilde{q}}} \bigg(f_{T_s}+\frac{2}{27}f_{TG}\bigg)^2, \,\,\,\, (\tilde{q}\,\, \rm{dominated}, \tan \beta \gg 1). \label{case3}$$ From these expressions [@scatteraq], it is clear that the direct detection of dark matter alone will not be very capable of revealing much about supersymmetry or the other underlying physics. There are a large number of degeneracies which can lead to a given value of the WIMP-nucleon cross section. Only by combining this information with collider and/or indirect detection data can one hope to infer the nature of the dark matter particle. Currently, the strongest limits on the spin-independent WIMP-nucleon cross section come from the XENON [@xenon] and CDMS [@cdmssi] experiments, which have obtained an upper bound on the cross section of a $\sim$100 GeV WIMP at the $\sim$$10^{-7}$ pb level. These constraints, along with those of other experiments, are shown in Fig. \[directcurrent\]. There is currently a great deal of progress being made in the experimental field of direct detection. Within the next several months (early 2008), the CDMS collaboration is expected to release a new limit which will likely be the most stringent (assuming no detection is made). In the meantime, the XENON collaboration is preparing for a run with a larger detector, with results expected within a year or so of this time. Beyond the next year or two, it is difficult to foresee which experiment(s) will be leading this search. It is still not clear whether detectors using liquid noble elements or cryogenic technologies will advance most rapidly. For the time being, there are clear advantages to proceeding with multiple technologies. Despite our inability to predict how this field will develop, it is reasonable to expect that by 2010 or so direct detection experiments will reach the $\sim 10^{-9}$ pb level of sensitivity. Roughly speaking, such cross sections are sufficient to test many, if not most, supersymmetric models, as well as many WIMPs candidates in other particle physics frameworks. Given the rate at which direct dark matter experiments are developing, it is interesting to recognize that such experiments are likely to see their first evidence for WIMPs within the same time frame that the Large Hadron Collider is expected to reveal the presence of the associated physics. In such a scenario, it will be essential to compare the mass of the WIMP observed in each experimental program. Direct detection experiments can determine the mass of the WIMP by measuring the distribution of the recoil energy, $E_R$ [@Green:2007rb]. This varies with the mass of the WIMP, with a resonance where the WIMP mass equals the target mass. Roughly, one expects , where $m_T$ is the target mass and $v$ is the WIMP velocity, with corrections depending on the precise target material and the properties of the detector [@LewinSmith]. Assuming the standard velocity distribution in smooth halo models, with approximately 10% uncertainty, an experiment with a Xenon or Germanium target that detects 100 signal events for a WIMP of mass $m_X = 100$ GeV can expect to measure the mass of this particle at the 20% level, thus potentially confirming the cosmological stability (and abundance) of a WIMP detected at the LHC. If the WIMP mass inferred in a direct detection experiment was not consistent with that measured at the LHC, this could imply that different particle species are being observed, or could be the result of a nonstandard dark matter velocity distribution. In the future, directional dark matter detectors may help to clear up such a scenario. INDIRECT DETECTION ================== In parallel to direct detection experiments, a wide range of indirect detection programs have been developed to search for the annihilation products of dark matter particles. In particular, searches are underway to detect neutrinos from dark matter annihilations in the core of the Sun, antimatter particles from dark matter annihilations in the galactic halo, and photons from dark matter annihilations in the halo of the Milky Way, galactic substructure and the dark matter distribution integrated over cosmological volumes. In this section, we briefly describe the role of these experimental programs in the overall strategy to reveal dark matter’s identity. Gamma Rays ---------- Searches for prompt photons generated in dark matter annihilations have a key advantage over other indirect detection channels in that they travel essentially unimpeded from their production site. In particular, gamma rays are not deflected by magnetic fields, and thus can potentially provide valuable angular information. For example, point sources of dark matter annihilation radiation might appear from high density regions such as the Galactic Center or dwarf spheroidal galaxies. Furthermore, over galactic distance scales, gamma rays are not attenuated, and thus retain their spectral information. In other words, the spectrum observed at Earth is the same spectrum that was generated in the dark matter annihilations. The spectrum of photons produced in dark matter annihilations depends on the details of the WIMP being considered. Supersymmetric neutralinos, for example, typically annihilate to final states consisting of heavy fermions and gauge or Higgs bosons [@jungman]. Generally speaking, each of these annihilation modes typically result in a very similar spectrum of gamma rays (see, however, Ref. [@bringmann]). The gamma ray spectrum from a WIMP which annihilates to light leptons can be quite different, however. This can be particularly important in the case of Kaluza-Klein dark matter in models with one universal extra dimension, for example, in which dark matter particles annihilate significantly to $e^+ e^-$ and $\mu^+ \mu^-$ [@kkdm]. The Galactic Center has long been considered to be one of the most promising regions of the sky in which to search for gamma rays from dark matter annihilations [@gchist]. The prospects for this depend, however, on a number of factors including the nature of the WIMP, the distribution of dark matter in the region around the Galactic Center, and our understanding of the astrophysical backgrounds. The gamma ray spectrum produced through dark matter annihilations is given by $$\Phi_{\gamma}(E_{\gamma},\psi) = \frac{1}{2}\sigv \frac{dN_{\gamma}}{dE_{\gamma}} \frac{1}{4\pi m^2_X} \int_{\rm{los}} \rho^2(r) dl(\psi) d\psi. \label{flux1}$$ Here, $\sigv$ is the thermally averaged WIMP annihilation cross section, $m_X$ is the mass of the WIMP, $\psi$ is the angle observed relative to the direction of the Galactic Center, $\rho(r)$ is the dark matter density as a function of distance to the Galactic Center, and the integral is performed over the line-of-sight. $dN_{\gamma}/dE_{\gamma}$ is the gamma ray spectrum generated per WIMP annihilation. Averaging over a solid angle centered around a direction, $\psi$, we arrive at $$\Phi_{\gamma}(E_{\gamma}) \approx 2.8 \times 10^{-12} \, \rm{cm}^{-2} \, \rm{s}^{-1} \, \frac{dN_{\gamma}}{dE_{\gamma}} \bigg(\frac{\sigv}{3 \times 10^{-26} \,\rm{cm}^3/\rm{s}}\bigg) \bigg(\frac{1 \, \rm{TeV}}{m_{\rm{X}}}\bigg)^2 J(\Delta \Omega, \psi) \Delta \Omega, \label{flux2}$$ where $\Delta \Omega$ is the solid angle observed. The quantity $J(\Delta \Omega, \psi)$ depends only on the dark matter distribution, and is the average over the solid angle of the quantity: $$J(\psi) = \frac{1}{8.5 \, \rm{kpc}} \bigg(\frac{1}{0.3 \, \rm{GeV}/\rm{cm}^3}\bigg)^2 \, \int_{\rm{los}} \rho^2(r(l,\psi)) dl.$$ $J(\psi)$ is normalized such that a completely flat halo profile, with a density equal to the value at the solar circle, integrated along the line-of-sight to the Galactic Center would yield a value of one. In dark matter distributions favored by N-body simulations, however, this value can be much larger. The Narvarro-Frenk-White profile [@nfw], which is a commonly used benchmark halo model, leads to values of $J(\Delta \Omega=10^{-5} \, {\rm sr}, \psi=0) \sim 10^5$. The effects adiabatic contraction due to the cooling of baryons is further expected to increase this quantity [@ac]. The recent discovery of a bright, very high-energy gamma ray source in the galactic center region by the Atmospheric Cerenkov Telescopes HESS [@hess], MAGIC [@magic], WHIPPLE [@whipple] and CANGAROO-II [@cangaroo] has made efforts to identify gamma rays from dark matter annihilations more difficult. This source appears to be coincident with the dynamical center of the Milky Way (Sgr A$^*$) and has no detectable angular extension (less than 1.2 arcminutes). Its spectrum is well described by a power-law, $dN_{\gamma}/dE_{\gamma} \propto E_{\gamma}^{-\alpha}$, where $\alpha=2.25 \pm 0.04 (\rm{stat}) \pm 0.10 (\rm{syst})$ over the range of 160 GeV to 20 TeV. Although speculations were initially made that this source could be the product of annihilations of very heavy ($\gsim 10$ TeV) dark matter particles [@actdark], the spectral shape appears inconsistent with a dark matter interpretation. The source of these gamma rays is more likely an astrophysical accelerator associated with our Galaxy’s central supermassive black hole [@hessastro]. Although this gamma ray source represents a formidable background for GLAST and other experiments searching for dark matter annihilation radiation [@gabi], it may be possible to reduce the impact of this and other backgrounds by studying the angular distribution of gamma rays from this region of the sky [@Dodelson:2007gd]. The prospects for identifying dark matter annihilation radiation from the Galactic Center depends critically on the unknown dark matter density within the inner parsecs of the Milky Way and on the properties of the astrophysical backgrounds present. If these characteristics are favorable, then the Galactic Center is very likely to be the most promising region of the sky to study. If not, other regions with high dark matter densities may be more advantageous. Dwarf spheroidal galaxies within and near the Milky Way provide an opportunity to search for dark matter annihilation radiation with considerably less contamination from astrophysical backgrounds. The flux of gamma rays from dark matter annihilations in such objects, however, is also expected to be lower than from a cusp in the center of the Milky Way [@Evans:2003sc; @Bergstrom:2005qk; @Strigari:2007at]. As a result, planned experiments are likely to observe dark matter annihilation radiation from dwarf galaxies only in the most favorable range of particle physics models. The integrated gamma ray signal from dark matter annihilations throughout the cosmological distribution of dark matter may also provide an opportunity to identify the products of dark matter annihilations. The ability of future gamma ray telescopes to identify a dark matter component of the diffuse flux depends strongly on the fraction of the extragalactic gamma ray background observed by the EGRET experiment which will be resolved as individual sources, such as blazars. If a large fraction of this background is resolved, the remaining extragalactic signal could potentially contain identifiable signatures of dark matter annihilations [@egdiffuse]. The telescopes potentially capable of detecting gamma rays from dark matter annihilations in the near future include the satellite-based experiment GLAST [@glast], and a number ground based Atmospheric Cerenkov Telescopes, including HESS, MAGIC and VERITAS. The roles played by each of these two classes of experiments in the search for dark matter are quite different. GLAST will continuously observe a large fraction of the sky, but with an effective area far smaller than possessed by ground based telescopes. Ground based telescopes, in contrast, study the emission from a small angular field, but with far greater exposure. Furthermore, while ground based telescopes can only study gamma rays with energy greater than $\sim$100 GeV, GLAST will be able to directly study gamma rays with energies over the range of 100 MeV to 300 GeV. As a result of the different energy ranges accessible by these experiments, searches for dark matter particles lighter than a few hundred GeV are most promising with GLAST, while ground based telescopes are better suited for heavier WIMPs. The large field-of-view of GLAST also makes it well suited for measurements of the diffuse gamma ray background. GLAST is also expected to detect a number of unidentified sources, some of which could potentially be signals of dark matter substructures. Follow up observations with ground based gamma ray telescopes would be very useful for clarifying the nature of such sources. Antimatter ---------- WIMP annihilations in the galactic halo generate charged anti-matter particles: positrons, anti-protons and anti-deuterons. Unlike gamma rays, which travel along straight lines, charged particles move under the influence of the Galactic Magnetic Field, diffusing and losing energy, resulting in a diffuse spectrum at Earth. By studying the cosmic anti-matter spectra, satellite-based experiments such as PAMELA [@pamela] and AMS-02 [@ams02] may be able to identify signatures of dark matter. PAMELA began its three-year satellite mission in June of 2006. AMS-02 is planned for later deployment onboard the International Space Station. As compared to antiprotons and antideuterons, cosmic positrons are attractive probes of dark matter for several reasons. In particular, positrons lose the majority of their energy over typical length scales of a few kiloparsecs or less [@baltzpos]. The cosmic positron spectrum, therefore, samples only the local dark matter distribution and is thus subject to considerably less uncertainty than the other anti-matter species. Additionally, data from the HEAT [@heat] and AMS-01 experiments [@ams01] contain features which could plausibly be the consequence of dark matter annihilations in the local halo. The spectral shape of the cosmic positron spectrum generated in dark matter annihilation depends on the leading annihilation modes of the WIMP in the low velocity limit. Bino-like neutralinos, for example, typically annihilate to heavy fermion pairs: $b\bar{b}$ with a small $\tau^+ \tau^-$ admixture, along with a fraction to $t\bar{t}$ if $m_{\chi} \gsim m_t$. Wino or higgsino-like neutralinos annihilate most efficiently to combinations of Higgs and gauge bosons. In other particle dark matter candidates, such as Kaluza-Klein dark matter in models with universal extra dimensions, annihilation to light charged leptons can lead to a much harder positron spectrum than is expected from neutralinos [@kkpos]. Once positrons are injected into the local halo through dark matter annihilations, they propagate under the influence of galactic magnetic fields, gradually losing energy through synchrotron emission and inverse Compton scattering with radiation fields, such as starlight and the cosmic microwave background. The spectrum observed at Earth is found by solving the diffusion-loss equation [@diffusion]: $$\begin{aligned} \frac{\partial}{\partial t}\frac{dn_{e^{+}}}{dE_{e^{+}}} = \vec{\bigtriangledown} \cdot \bigg[K(E_{e^{+}},\vec{x}) \vec{\bigtriangledown} \frac{dn_{e^{+}}}{dE_{e^{+}}} \bigg] + \frac{\partial}{\partial E_{e^{+}}} \bigg[b(E_{e^{+}},\vec{x})\frac{dn_{e^{+}}}{dE_{e^{+}}} \bigg] + Q(E_{e^{+}},\vec{x}), \label{dif}\end{aligned}$$ where $dn_{e^{+}}/dE_{e^{+}}$ is the number density of positrons per unit energy, $K(E_{e^{+}},\vec{x})$ is the diffusion constant, $b(E_{e^{+}},\vec{x})$ is the rate of energy loss and $Q(E_{e^{+}},\vec{x})$ is the source term, which contains all of the information about the dark matter annihilation modes, cross section, and distribution. To solve the diffusion-loss equation, a set of boundary conditions must be adopted. In this application, the boundary condition is described as the distance from the galactic plane at which the positrons can freely escape, $L$. These diffusion parameters can be constrained by studying the spectra of various species of cosmic ray nuclei, most importantly the boron-to-carbon ratio [@btoc]. In Fig. \[heat\], the ratio of positrons to positrons plus electrons in the cosmic ray spectrum is shown as a function of energy, including a possible contribution from dark matter annihilations. Also shown are the measurements from the HEAT experiment [@heat], which may possibly contain an excess in comparison to standard astrophysical expectations at energies above 7 GeV or so. While positrons from dark matter annihilations are indeed able to generate this possible excess, it requires a somewhat larger annihilation rate than is typically expected. In particular, if a smooth dark matter halo and an annihilation cross section of $\sigma v \approx 3\times 10^{-26}$ cm$^{3}$/s (as required to thermally produce the observed dark matter abundance via S-wave processes) are assumed, the annihilation rate will be a factor of 50 or more too low to generate the spectrum measured by HEAT. Fluctuations in the local dark matter density, however, could lead to enhancements in the local annihilation rate, known as the “boost factor”. It is typically expected that this quantity could be as large as 5 to 10. Although boost factors of 50 or more are not impossible, such large values would be somewhat surprising. If the positron flux observed by HEAT is in fact the result of annihilating dark matter, then the corresponding spectrum will be precisely measured by PAMELA [@pamela] and AMS-02 [@ams02]. If not, then the detection of positrons from dark matter annihilations will be more difficult, but perhaps still possible [@silkpos]. Unlike gamma ray measurements of the Galactic Center or dwarf galaxies, observations of the cosmic positron spectrum (as well as the antiproton and antideuteron spectra) could potentially provide a measurement of the dark matter annihilation rate over large volumes of space. Such a measurement, therefore, could be used to determine the product of the WIMP’s annihilation cross section and its density squared, averaged over the sampled volume (roughly a few cubic kiloparsec region, corresponding to the distance a typical positron travels from its point of origin before losing the majority of its energy). As a result of this limited range, only the dark matter distribution in the local halo is relevant to the observed cosmic positron flux. Assuming there are no very large and unknown clumps of dark matter in the surrounding kiloparsecs (which, although not impossible, is very unlikely [@hoopertaylorsilk]), a measurement of the cosmic positron spectrum could be used to infer the dark matter particle’s annihilation cross section (in the low velocity limit) with a comparatively modest degree of uncertainty coming from the unknown distribution of dark matter. Neutrino Telescopes ------------------- Although dark matter annihilations in the galactic halo produce too few neutrinos to be detected [@neutrinoshalo], annihilations which occur in the center of the Sun could potentially generate an observable flux of high energy neutrinos [@neutrinosun]. Dark matter particles scatter elastically with and become captured in the Sun at a rate given by [@capture] $$C^{\odot} \approx 3.35 \times 10^{19} \, \mathrm{s}^{-1} \left( \frac{\sigma_{\mathrm{H, SD}} +\, \sigma_{\mathrm{H, SI}} + 0.07 \, \sigma_{\mathrm{He, SI}} } {10^{-7}\, \mathrm{pb}} \right) \left( \frac{100 \, \mathrm{GeV}}{m_{X}} \right)^2 , \label{capture}$$ where $m_{X}$ is the dark matter particle’s mass. $\sigma_{\mathrm{H, SD}}$, $\sigma_{\mathrm{H, SI}}$ and $\sigma_{\mathrm{He, SI}}$ are the spin dependent (SD) and spin independent (SI) elastic scattering cross sections of the WIMP with hydrogen and helium nuclei, respectively. The factor of $0.07$ reflects the solar abundance of helium relative to hydrogen and well as dynamical factors and form factor suppression. Notice that the capture rate is suppressed by two factors of the WIMP mass. One of these is simply the result of the depleted number density of WIMPs in the local halo ($n \propto 1/m$) while the second factor is the result of kinematic suppression for the capture of a WIMP much heavier than the target nuclei, in this case hydrogen or helium. If the WIMP’s mass were comparable to the masses of hydrogen or helium nuclei, these expressions would no longer be valid. For WIMPs heavy enough to generate neutrinos detectable in the high-energy neutrino telescopes, Eq. \[capture\] should be applicable. If the capture rate and annihilation cross sections are sufficiently large, equilibrium will be reached between these processes. For a number of WIMPs in the Sun, $N$, the rate of change of this quantity is given by $$\dot{N} = C^{\odot} - A^{\odot} N^2 ,$$ where $C^{\odot}$ is the capture rate and $A^{\odot}$ is the annihilation cross section times the relative WIMP velocity per volume. $A^{\odot}$ can be approximated by $$A^{\odot} \approx \frac{\sigv}{V_{\mathrm{eff}}},$$ where $V_{\mathrm{eff}}$ is the effective volume of the core of the Sun determined roughly by matching the core temperature with the gravitational potential energy of a single WIMP at the core radius. This was found in Refs. [@equ1; @equ2] to be $$V_{\rm eff} \approx 5.7 \times 10^{27} \, \mathrm{cm}^3 \left( \frac{100 \, \mathrm{GeV}}{m_{X}} \right)^{3/2} \;.$$ The present WIMP annihilation rate in the Sun is given by $$\Gamma = \frac{1}{2} A^{\odot} N^2 = \frac{1}{2} \, C^{\odot} \, \tanh^2 \left( \sqrt{C^{\odot} A^{\odot}} \, t_{\odot} \right) \;,$$ where $t_{\odot} \approx 4.5$ billion years is the age of the solar system. The annihilation rate is maximized when it reaches equilibrium with the capture rate. This occurs when $$\sqrt{C^{\odot} A^{\odot}} t_{\odot} \gg 1 \; .$$ If this condition is met, the final annihilation rate (and corresponding neutrino flux and event rate) has no further dependence on the dark matter particle’s annihilation cross section. WIMPs can generate neutrinos through a wide range of annihilation channels. Annihilations to heavy quarks, tau leptons, gauge bosons and Higgs bosons can all generate neutrinos in the subsequent decay. In some models, WIMPs can also annihilate directly to neutrino pairs. Once produced, neutrinos can travel to the Earth where they can be detected. The muon neutrino spectrum at the Earth from WIMP annihilations in the Sun is given by: $$\frac{dN_{\nu_{\mu}}}{dE_{\nu_{\mu}}} = \frac{ C_{\odot} F_{\rm{Eq}}}{4 \pi D_{\rm{ES}}^2} \bigg(\frac{dN_{\nu}}{dE_{\nu}}\bigg)^{\rm{Inj}}, \label{wimpflux}$$ where $C_{\odot}$ is the WIMP capture rate in the Sun, $F_{\rm{Eq}}$ is the non-equilibrium suppression factor ($\approx 1$ for capture-annihilation equilibrium), $D_{\rm{ES}}$ is the Earth-Sun distance and $(\frac{dN_{\nu}}{dE_{\nu}})^{\rm{Inj}}$ is the neutrino spectrum from the Sun per WIMP annihilating. Due to $\nu_{\mu}-\nu_{\tau}$ vacuum oscillations, the muon neutrino flux from WIMP annihilations in the Sun observed at Earth is the average of the $\nu_{\mu}$ and $\nu_{\tau}$ components. Muon neutrinos produce muons in charged current interactions with ice or water nuclei inside or near the detector volume of a high energy neutrino telescope. The rate of neutrino-induced muons observed in a high-energy neutrino telescope is estimated by: $$N_{\rm{events}} \simeq \int \int \frac{dN_{\nu_{\mu}}}{dE_{\nu_{\mu}}}\, \frac{d\sigma_{\nu}}{dy}(E_{\nu_{\mu}},y) \,R_{\mu}((1-y)\,E_{\nu})\, A_{\rm{eff}} \, dE_{\nu_{\mu}} \, dy,$$ where $\sigma_{\nu}(E_{\nu_{\mu}})$ is the neutrino-nucleon charged current interaction cross section, $(1-y)$ is the fraction of neutrino energy which goes into the muon, $A_{\rm{eff}}$ is the effective area of the detector, $R_{\mu}((1-y)\,E_{\nu})$ is the distance a muon of energy, $(1-y)\,E_{\nu}$, travels before falling below the muon energy threshold of the experiment (ranging from $\sim$1 to 100 GeV), called the muon range. The spectrum and flux of neutrinos generated in WIMP annihilations depends on the annihilation modes which dominate, and thus is model dependent. For most annihilation modes, however, the variation from model-to-model is not dramatic. In Fig. \[ratecompare\], the event rate in a kilometer-scale neutrino telescope (with a 50 GeV muon energy threshold) is shown as a function of the WIMP’s effective elastic scattering cross section for a variety of annihilation modes [@Halzen:2005ar]. The effective elastic scattering cross section is defined as $\sigma_{\rm{eff}} = \sigma_{\mathrm{H, SD}} +\, \sigma_{\mathrm{H, SI}} + 0.07 \, \sigma_{\mathrm{He, SI}}$, following Eq. \[capture\]. These rates are indicative of that expected for experiments such as IceCube at the South Pole [@icecube], or a future kilometer-scale neutrino telescope built in the Mediterranean Sea [@km3]. To detect neutrinos from WIMP annihilations in the Sun over the background of atmospheric neutrinos, a rate in the range of 10-100 events per square-kilometer, per year is required. ![The event rate in a kilometer-scale neutrino telescope such as IceCube as a function of the WIMP’s effective elastic scattering cross section in the Sun for a variety of annihilation modes. The effective elastic scattering cross section is defined as $\sigma_{\rm{eff}} = \sigma_{\mathrm{H, SD}} +\, \sigma_{\mathrm{H, SI}} + 0.07 \, \sigma_{\mathrm{He, SI}}$, following Eq. \[capture\]. The dashes, solid and dotted lines correspond to WIMPs of mass 100, 300 and 1000 GeV, respectively.[]{data-label="ratecompare"}](indirectratebb.ps "fig:"){width="2.2in"} ![The event rate in a kilometer-scale neutrino telescope such as IceCube as a function of the WIMP’s effective elastic scattering cross section in the Sun for a variety of annihilation modes. The effective elastic scattering cross section is defined as $\sigma_{\rm{eff}} = \sigma_{\mathrm{H, SD}} +\, \sigma_{\mathrm{H, SI}} + 0.07 \, \sigma_{\mathrm{He, SI}}$, following Eq. \[capture\]. The dashes, solid and dotted lines correspond to WIMPs of mass 100, 300 and 1000 GeV, respectively.[]{data-label="ratecompare"}](indirectrateWW.ps "fig:"){width="2.2in"}\ ![The event rate in a kilometer-scale neutrino telescope such as IceCube as a function of the WIMP’s effective elastic scattering cross section in the Sun for a variety of annihilation modes. The effective elastic scattering cross section is defined as $\sigma_{\rm{eff}} = \sigma_{\mathrm{H, SD}} +\, \sigma_{\mathrm{H, SI}} + 0.07 \, \sigma_{\mathrm{He, SI}}$, following Eq. \[capture\]. The dashes, solid and dotted lines correspond to WIMPs of mass 100, 300 and 1000 GeV, respectively.[]{data-label="ratecompare"}](indirectratett.ps "fig:"){width="2.2in"} ![The event rate in a kilometer-scale neutrino telescope such as IceCube as a function of the WIMP’s effective elastic scattering cross section in the Sun for a variety of annihilation modes. The effective elastic scattering cross section is defined as $\sigma_{\rm{eff}} = \sigma_{\mathrm{H, SD}} +\, \sigma_{\mathrm{H, SI}} + 0.07 \, \sigma_{\mathrm{He, SI}}$, following Eq. \[capture\]. The dashes, solid and dotted lines correspond to WIMPs of mass 100, 300 and 1000 GeV, respectively.[]{data-label="ratecompare"}](indirectratetautau.ps "fig:"){width="2.2in"} Currently, the Super-Kamiokande experiment has placed the strongest bounds on high-energy neutrinos from the direction of the Sun [@superk]. In this application, Super-K has two primary advantages over other neutrino detectors. Firstly, they have analyzed data over a longer period than most of their competitors, a total of nearly 1700 live days. Secondly, Super-K was designed to be sensitive to low energy ($\sim$GeV) neutrinos, which gives them an advantage in searching for lighter WIMPs. Super-K’s upper limit on neutrino-induced muons above 1 GeV from WIMP annihilations in the Sun is approximately 1000 to 2000 per square kilometer per year for WIMPs heavier than 100 GeV, and approximately 2000 to 5000 per square kilometer per year for WIMPs in the 20 to 100 GeV range. The precise value of these limits depends on the WIMP annihilation mode considered. The Amanda-II [@amanda], Baksan [@baksan] and Macro [@macro] experiments have each placed limits on the flux of neutrino-induced muons from the Sun that are only slightly weaker than Super-Kamiokande’s. The limit placed by the Amanda experiment resulted from only 144 live days of data. Having operated the detector for seven years, Amanda is expected to produce significantly improved bounds in the future. In addition to these experiments, the next generation neutrino telescopes IceCube and Antares are currently under construction at the South Pole and in the Mediterranean, respectively. IceCube, with a full cubic kilometer of instrumented volume, will be considerably more sensitive to WIMP annihilations in the Sun than other planned or existing experiments [@icecube]. Antares, with less than one tenth of the effective area of IceCube, will have the advantage of a lower energy threshold, and may thus be more sensitive to low mass WIMPs [@antares]. Beyond Antares, there are also plans to build a kilometer-scale detector in the Mediterranean Sea [@km3]. From Fig. \[ratecompare\], we see that a WIMP-proton elastic scattering cross section on the order of $10^{-6}$ pb or greater is needed if kilometer-scale neutrino telescopes are to detect a signal from dark matter annihilations. Elastic scattering cross sections of this size are constrained by the absence of a positive signal in direct detection experiments, however. Currently, the strongest constraints on the WIMP-nucleon, spin-independent elastic scattering cross section have been made by the XENON [@xenon] and CDMS experiments [@cdmssi], who each place limits below $10^{-6}$ pb. Therefore, if current or planned neutrino telescopes are to detect neutrinos from dark matter annihilations in the Sun, they must scatter elastically with nuclei in the Sun via spin-dependent interactions, which are far less strongly constrained by direct detection experiments. The strongest bounds on the WIMP-proton spin-dependent cross section have been made by the NAIAD experiment [@naiad]. This result limits the spin-dependent cross section with protons to be less than approximately 0.3 pb for a WIMP in the mass range of 50-100 GeV and less than 0.8 pb ($m_{X}$/500 GeV) for a heavier WIMP. The PICASSO [@picasso] and CDMS [@cdmssd] experiments have placed limits on the spin-dependent WIMP-proton cross section roughly one order of magnitude weaker than the NAIAD result. A WIMP with a largely spin-dependent scattering cross section with protons may thus be capable of generating large event rates in high energy neutrino telescopes. Considering, for example, a 300 GeV WIMP with an elastic scattering cross section near the experimental limit, Fig. \[ratecompare\] suggests that rates as high as $\sim 10^6$ per year could be generated if purely spin-dependent scattering contributes to the capture rate of WIMPs in the Sun. The relative size of the spin-independent and spin-dependent elastic scattering cross sections depend on the nature of the WIMP in question. For a neutralino, these cross sections depend on its composition and on the mass spectrum of the exchanged Higgs bosons and squarks. Spin dependent, axial-vector, scattering of neutralinos with quarks within a nucleon is made possible through the t-channel exchange of a $Z$, or the s-channel exchange of a squark. Spin independent scattering occurs at the tree level through s-channel squark exchange and t-channel Higgs exchange, and at the one-loop level through diagrams involving a loop of quarks and/or squarks. For higgsino-like or mixed higgsino-bino neutralinos, the spin dependent cross section can be somewhat larger than the spin independent, which is potentially well suited for the prospects for indirect detection. In particular, spin-dependent cross sections as large as $\sim 10^{-3}$pb are possible even in models with very small spin-independent scattering rates. Such neutralinos would go easily undetected in all planned direct detection experiments, while still generating on the order of $\sim 1000$ events per year at IceCube. Synchrotron Emission -------------------- As described in Sec. \[antimatter\], electrons and positrons produced in dark matter annihilations travel under the influence of the Galactic Magnetic Field, losing energy through Compton scattering off of starlight, cosmic microwave background photons and far infrared emission from dust, and through synchrotron emission from interactions with the Galactic magnetic field. The relative importance of these processes depends on the energy densities of radiation and magnetic fields. The processes of synchrotron emission and inverse Compton scattering each lead to potentially observable byproducts [@syndm]. For dark matter particles with electroweak scale masses, the resulting synchrotron photons typically fall in the microwave frequency band, and thus are well suited for study with cosmic microwave background experiments [@haze]. The inverse Compton scattering of highly relativistic electrons and positrons with starlight photons, on the other hand, can generate photons with MeV-GeV energies. THE ROLE OF COLLIDERS ===================== Among other new states, particles with TeV scale masses and QCD color are generic features of models of electroweak symmetry breaking. These particles appear as counterparts to the quarks to provide new physics associated with the generation of the large top quark mass. In many scenarios, including supersymmetry, electroweak symmetry breaking arises as a result of radiative corrections due to these particles, enhanced by the large coupling of the Higgs boson to the top quark. Any particle with these properties will be pair-produced at the Large Hadron Collider (LHC) with a cross section of tens of picobarns [@sqgl]. That particle (or particles) will then subsequently decay to particles including quark or gluon jets and the lightest particle in the new sector ([*ie.*]{} the dark matter candidate) which proceeds to exit the detector unseen. For any such model, the LHC experiments are, therefore, expected to observe large numbers of events with many hadronic jets and an imbalance of measured momentum. These ‘missing energy’ events are signatures of a wide range of models that contains an electroweak scale candidate for dark matter. If TeV-scale supersymmetry exists in nature, it will very likely be within the discovery reach of the Large Hadron Collider (LHC). The rate of missing energy events depends strongly on the mass of the colored particles that are produced and only weakly on other properties of the model. In Fig. \[fig:ATLAS\], the estimates of the ATLAS collaboration are shown for the discovery of missing energy events [@Tovey]. If squarks or gluinos have masses below 1 TeV, the missing energy events can be discovered with an integrated luminosity of 100 pb$^{-1}$, about 1% of the LHC first-year design luminosity. Thus, we will know very early in the LHC program that a WIMP candidate is being produced. By studying the decays of squarks and/or gluinos it will also be possible to discover other superpartners at the LHC. For example, in many models, decays of the variety, $\tilde{q} \rightarrow \chi^0_2 q \rightarrow \tilde{l}^{\pm} l^{\mp} q \rightarrow \chi^0_1 l^+ l^- q$, provide a clean signal of supersymmetry in the form of $l^+l^- +\, \rm{jets}\, + \,\,\rm{missing}\,\, E_T$. By studying the kinematics of these decays, the quantities $m_{\tilde{q}}$, $m_{\chi^0_2}$, $m_{\tilde{l}}$ and $m_{\chi^0_1}$ could each be potentially reconstructed [@recon; @drees; @fitter]. More generally speaking, the LHC is, in most models, likely to measure the mass of the lightest neutralino to roughly 10% accuracy, and may also be able to determine the masses of one or more of the other neutralinos, and any light sleptons [@lhc]. Charginos are more difficult to study at the LHC. The heavy, neutral Higgs bosons of the MSSM ($A$, $H$), can also be potentially produced and studied at the LHC. In particular, in models with large $\tan \beta$, heavy Higgs bosons have enhanced couplings to down-type fermions, thus leading to potentially observable di-tau final states. If enough of these events are observed, the masses of the heavy Higgs bosons could be potentially reconstructed, and $\tan \beta$ measured [@ditau; @higgsmeasure]. Prospects for the discovery of supersymmetry at the Tevatron, although not nearly as strong as at the LHC, are also exciting. The most likely discovery channel at the Tevatron is probably through clean tri-lepton plus missing energy events originating from the production of a chargino and a heavy neutralino, followed by a decay of the form, $\chi^{\pm} \chi^0_2 \rightarrow \tilde{\nu} l^{\pm} l^{+} \tilde{l^{-}} \rightarrow \nu \chi^0_1 l^{\pm} l^+ l^- \chi^0_1$ [@fermilab]. Only models with rather light gauginos (neutralinos and charginos) and sleptons can be discovered in this way, however. For some of the recent results from supersymmetry searches at the Tevatron, see Ref. [@recenttevatron]. Measurements of particle masses and other properties at the LHC can provide an essential cross check for direct and indirect detection channels. In particular, neither direct nor indirect detection experiments provide information capable of identifying the overall cosmological abundance of a WIMP, but instead infer only combinations of density and interaction cross section, leaving open the possibility that an observation may be generated by a sub-dominant component of the cosmological dark matter with a somewhat larger elastic scattering or annihilation cross section. Collider measurements can help to clarify this situation. In the left frame of Fig. \[LCC2\], we show the ability of the LHC to infer the thermal neutralino relic abundance from measurements of sparticle masses and other properties. The results shown are for a specific benchmark supersymmetric model (see Ref. [@Baltz:2006fm]), but are not atypical. In this case, the LHC can infer $\Omega_{\chi} h^2$ to lie roughly within 0.05 to 0.2 (assuming properties such as R-parity conservation), which, along with a detection in either a direct or indirect channel, would provide a strong confirmation that the observed neutralino does in fact constitute the bulk of the cosmological dark matter. In the right frame of Fig. \[LCC2\], the same supersymmetric model is considered, but instead showing the LHC’s ability to determine the neutralino-nucleon elastic scattering cross section. This information would be very useful in combination with a direct detection signal. In particular, it would enable uncertainties in the local dark matter density to be reduced with confidence. Also shown in each frame of Fig. \[LCC2\] are the results which could be obtained from a future 500 GeV or 1 TeV (center-of-mass) $e^+ e^-$ linear collider, such as the International Linear Collider (ILC) [@ilc]. Such an experiment would have considerable advantages over hadron colliders such as the Tevatron or the LHC. Although hadron colliders can reach very high center-of-mass energies, and thus play an essential role as discovery machines, lepton colliders are best suited for lower energy, precision measurements. In particular, at an electron-positron collider, the process $\ee \to X\bar X$ can provide an exquisite diagnostic of the quantum numbers of the massive particle, $X$. As long as only the diagrams with annihilation through $\gamma$ and $Z$ are relevant, the angular distribution and threshold shape of the reaction are characteristic for each spin, and the normalization of the cross section directly determines the $SU(2)\times U(1)$ quantum numbers. These tests can be applied to any particles with electric or weak charge whose pair-production thresholds lie in the range of the collider. Such a measurement could be used to pin down the spin and quantum numbers of a given particle and bring us a long way toward the qualitative identification of the underlying model. We cannot emphasize enough the importance of the complementary roles played by each of the LHC and ILC programs. Whereas the LHC can more easily reach high energies and offers very large cross sections for specific states of a model of new physics, the ILC will likely reach fewer states in the new particle spectrum, but will provide extremely incisive measurements of the properties of the particles that are available to it. Furthermore, the particles within the ILC reach are typically the ones on which the dark matter density depends most strongly. Although both the LHC and ILC can make precision measurements, the measurements at the ILC typically have a more direct interpretation in terms of particle masses and couplings. SUMMARY ======= In this review, we have attempted to summarize the diverse and complementary roles played by the various direct, indirect and collider searches for particle dark matter. As of 2008, there has not yet been a clear or conclusive detection of dark matter’s non-gravitational interactions. There is reason to be optitmistic, however, that such a detection will be made within the next few years, moving the field beyond the discovery phase and into the measurement phase of the quest to reveal dark matter’s nature and particle identity. As next generation direct detection experiments such as Super-CDMS, XENON-plus, LUX and others come online, most TeV-scale models containing a viable WIMP candidate will become within reach of these programs. Indirect detection experiments, including GLAST, VERITAS, HESS, MAGIC, PAMELA, AMS-02, IceCube and others are also rapidly advancing, and may see the first signals of dark matter annihilations. As the Large Hadron Collider begins its operation later this year, a new window into high-energy phenonoma will be opened. If dark matter is associated with physics of the electroweak scale, it is very likely to be within the discovery reach of this experiment. The various experimental programs described in this review are each potentially capable of bringing very different measurments to the table. Although any one of these programs may be the first to discover particle dark matter, no single experiment or observation will answer all of our questions concerning this substance. Only by combining several of these detection methods together will it be possible to conclusively identify the dark matter of our universe. Acknowledgments {#acknowledgments .unnumbered} =============== DH is supported by the United States Department of Energy and NASA grant NAG5-10842. 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--- abstract: 'Let $\omega$ be a differential $q$-form defining a foliation of codimension $q$ in a projective variety. In this article we study the singular locus of $\omega$ in various settings. We relate a certain type of singularities, which we name *persistent*, with the unfoldings of $\omega$, generalizing previous work done on foliations of codimension $1$ in projective space. We also relate the absence of persistent singularities with the existence of a connection in the sheaf of $1$-forms defining the foliation. In the latter parts of the article we extend some of these results to toric varieties by making computations on the Cox ring and modules over this ring.' author: - 'César Massri[^1]' - 'Ariel Molinuevo[^2]' - 'Federico Quallbrunn$^*$' title: Foliations with persistent singularities --- Introduction {#introduction .unnumbered} ============ Overview of the subject and existing work {#overview-of-the-subject-and-existing-work .unnumbered} ----------------------------------------- Foliations of arbitrary codimension over algebraic varieties have been considered at least since the seminal works of Malgrange [@malgrange; @malgrange2] in the local case, and Jouanolou [@jou] in a more global approach. Aside from the main result of [@malgrange2] and general definitions, most of the early theorems about foliations on projective algebraic varieties have been formulated for codimension $1$ foliations on the projective space ${\mathbb{P}}^n$. In those articles, codimension $q$ foliations were defined locally by $1$-forms $\omega_1,\dots,\omega_q$ satisfying Frobenius integrability equations: $d\omega_i\wedge\omega_1\wedge\dots\wedge\omega_q=0$ for $i=1,\dots,q$. Later, de Medeiros observed that this definition is not general enough for singular foliations of codimension $q$, as singular foliations by curves in dimension $n\geq 3$ cannot be given by $n-1$ forms even locally, see [@deMed1] and Example \[exampleCodimension2inA3\]. The correct definition is given by a $q$-form verifying the Plücker relations and Frobenius integrability (see below for definitions). As for why many results were stated with ${\mathbb{P}}^n$ as ambient variety, notice that working in ${\mathbb{P}}^n$ allows the use of homogeneous coordinates and so one can define a codimension $1$ foliation with an integrable polynomial $1$-form $\omega=\sum_i f_i(x) dx_i$, that is a $1$-form verifying $\omega\wedge d\omega=0$ and $\sum_i x_i f_i(x)=0$. Such a setting can give concrete examples of foliations which may be hard to produce and study in more general contexts, see for instance the book [@omegarlibro]. Going beyond isolated examples there is the problem of establishing irreducible components for the space $\mathcal{F}ol(X)$ parameterizing all integrable forms on a variety $X$. The existence of this space for a projective variety $X$ follows from the existence of the $\mathrm{Quot}_X(\Omega^1_X)$ scheme and have been settled in [@quallbrunn]. In the case of codimension $1$ and $X={\mathbb{P}}^n$ there are several known examples of such components, the first known examples were established in [@jou] and up to the present is a very active research subject, in [@omegarlibro; @fj] one can find (non-exhaustive) lists of components. In the case of codimension $1$ and a general variety $X$ much less is known. In [@omegar] Calvo-Andrade proves that for a variety $X$ with $H^1(X,{\mathbb{C}})=0$ generic logarithmic $1$-forms give rise to integrable $1$-forms that are stable under small perturbations, *i.e.*: that there is an irreducible component of the space of integrable forms whose generic member defines a logarithmic foliation. Besides stability, another important problem is the local and global characterization of the singularities of a foliation. Local results include the main theorems of [@malgrange] in codimension $1$ and of [@saito] and [@malgrange2] in higher codimension. Global studies have been made in the case of logarithmic foliations in [@fmi] and in the case of foliations defined by polynomial representations of affine lie algebras in [@ocgln] among others. An important type of singularity of a holomorphic foliation was discovered by Ivan Kupka in [@kupka]. A Kupka singularity for an integrable $1$-form $\omega$ is a point $p$ such that $\omega(p)=0$ and $d\omega(p)\neq 0$. Kupka showed that this type of singularity of codimension $1$ foliation is stable, meaning that if $\omega_t$ is a family of integrable $1$-forms parameterized by $t$ and $\omega_0$ has a Kupka singularity then $\omega_t$ also has a Kupka singularity for small enough $t$. Also if a foliation have a Kupka singularity then there is a codimension $2$ subvariety whose points are singular points of the foliation. Kupka singularities were generalized to arbitrary codimension by de Medeiros in [@deMedTesis], where stability for this singularities is proved in general. In codimension $q$ Kupka singularities come in subvarieties of codimension less or equal than $q+1$. In codimension $1$ there are many results relating the geometry of the variety of Kupka points with the global properties of the foliation, see *e.g.*: [@omegar-ivan; @omegar-kupka]. In higher codimension there is the work of Calvo-Andrade [@omegar2009]. A third subject we look upon in this work is the study of the unfoldings of a foliation. Unfoldings in the context of foliations were introduced independently by Suwa and Mattei in different contexts, see [@suwa-review] for a survey on the subject. Unfoldings of foliations where computed mostly in some codimension $1$ cases, locally by Suwa (see *loc. cit.*) and on ${\mathbb{P}}^n$ by Molinuevo in [@moli]. Recently we have related the study of unfoldings and singularities of a codimension $1$ foliation on ${\mathbb{P}}^n$. Indeed, in [@mmq] we define a homogeneous ideal $I(\omega)$ defining a subscheme of the singular scheme of $\omega$ (see below for precise definitions), the elements of degree equal to the degree of $\omega$ in $I(\omega)$ are in natural correspondence with the infinitesimal unfoldings of $\omega$. Under generic conditions we can prove that if $K(\omega)$ is the ideal defining the closure of the variety of Kupka points then $\sqrt{I(\omega)}=\sqrt{K(\omega)}$, using this result we were able to compute the unfoldings of foliations of codimension $1$ on ${\mathbb{P}}^n$ with split tangent sheaf and also prove the existence of Kupka points for every foliation in ${\mathbb{P}}^n$ with reduced singular scheme. Main results {#main-results .unnumbered} ------------ Our aim in this article is to generalize previous results on the relation of unfoldings and singular points of a foliation to arbitrary codimension and to foliations on a non-singular projective variety. We also begin a study of this relation in toric varieties by making use of the Cox ring of the variety and modules over this ring. In codimension $1$ there is a direct relation between unfoldings and a certain type of singularities which we call persistent singularities. In this respect we prove Proposition \[JJcIIcKK\] relating Kupka and persistent singularities: Let ${\mathscr{J}}$ be the ideal sheaf of the singular locus of $\omega$, ${\mathcal{K}}$ the ideal of the Kupka singularities of $\omega$ and ${\mathscr{I}}$ the ideal of persistent singularities. Then the following inclusions hold, $${\mathscr{J}}\subseteq {\mathscr{I}}\subseteq {\mathcal{K}}.$$ and Theorem \[teoKnotempty\] stating the existence of Kupka points under certain hypotheses: Let $X$ be a projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ a foliation of codimension $1$ such that ${\mathscr{J}}(\omega)$ is a sheaf of radical ideals and such that $c_1({\mathcal{L}})\neq 0$ and $H^1(X,{\mathcal{L}})=0$. Then $\omega$ has Kupka singularities. In higher codimension the relation of persistent and Kupka singularities is not so clear, specially in the case where the foliation is not given locally by a complete intersection of $1$-forms, as in Example \[exampleCodimension2inA3\]. However, under suitable cohomological conditions the absence of persistent singularities impose very strong consequences on the foliation. If ${\mathcal{E}}$ is the sheaf of $1$-forms defining the foliation, and if ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}}, \mathrm{Sym}^2{\mathcal{E}})=0$ then the absence of persistent singularities implies the existence of a connection on ${\mathcal{E}}$, see Theorem \[teoConnectionE\]: Let $X$ be a projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega_X^q$ be an integrable $q$-form and ${\mathcal{E}}$ be the associated subsheaf of $1$-forms ${\mathcal{E}}\subseteq \Omega^1_X$. Let $\mathrm{Sym}^2({\mathcal{E}})$ denote the symmetric power of ${\mathcal{E}}$ and suppose ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=0$. If ${\mathscr{I}}(\omega)={\mathcal{O}}_X$ then ${\mathcal{E}}$ admits a holomorphic connection, in particular is locally free and every Chern class of ${\mathcal{E}}$ vanishes. In the case of toric varieties we can extend some of the results known for ${\mathbb{P}}^n$ by making use of the Cox ring, in this regard we obtain Theorem \[prop1notinI\]: Let $X_\Sigma$ be a projective simplicial toric variety, $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$ a codimension $q$ foliation and let $\mathcal{E}=\{\eta\in\widehat\Omega^1\,\colon\,\omega\wedge\eta=0\}$. Assume that $\mathcal{E}$ is locally free (*e.g.* $q=1$), $H^1(X_\Sigma,\mathcal{E})=0$ and $\beta$ not a torsion element. Then $\omega$ has persistent singularities. Furthermore, if instead ${\mathcal{E}}$ locally free, we require $\widehat\Omega^1/{\mathcal{E}}$ reflexive, the same conclusion holds. Kupka scheme in the Projective space for codimension 1 foliations ================================================================= Along this section we will revisit some definitions that we used in [@mmq], among them we will define the Kupka variety as a projective scheme ${\mathpzc{Kup}(\omega)}$ over ${\mathbb{P}}^n$ and $I=I(\omega)$ the *ideal of persistent singularities* (*a.k.a.* *unfoldings ideal*) of $\omega$. Then we will recall some results that we proved in *loc. cit.* that we will generalize later. The scheme ${\mathpzc{Kup}(\omega)}$ and the ideal $I$ were of central importance in those results. We refer the reader to [@mmq] for a full overview of this subjects. With the exception of Theorem \[teodivision\] through this section we will restrict to the projective space ${\mathbb{P}}^n$. So let us denote $S={\mathbb{C}}[x_0,\dots,x_n]$ to the homogeneous coordinate ring of ${\mathbb{P}}^n$ and $\Omega^1_{{\mathbb{P}}^n}(e)$ the sheaf of twisted differential 1-forms in ${\mathbb{P}}^n$ of degree $e$. With ${\mathrm{Sing}(\omega)}_{set}$ we will denote the (set theoretic) singular set of $\omega\in H^0({\mathbb{P}}^n, \Omega^1_{{\mathbb{P}}^n}(e))$ in ${\mathbb{P}}^n$, $${\mathrm{Sing}(\omega)}_{set} = \{p\in {\mathbb{P}}^n: \omega(p) = 0 \}\ .$$ \[foliation\] Let ${\mathcal{L}}\simeq {\mathcal{O}}_{{\mathbb{P}}^n}(-e)$, $e\geq 2$, be a line bundle and $\omega:{\mathcal{L}}\to \Omega^1_{{\mathbb{P}}^n}$ be a morphism of sheaves, we will say that $\omega$ defines an *algebraic foliation of codimension 1* on ${\mathbb{P}}^n$, if $\Omega^1_{{\mathbb{P}}^n}/{\mathcal{L}}$ is torsion free and the morfism is generated by a non zero global section $\omega\in H^0({\mathbb{P}}^n,\Omega^1_{{\mathbb{P}}^n}(e))$ such that $\omega\wedge d\omega = 0$. We recall that such foliations have *geometric degree* $e-2$. The condition of $\Omega^1_{{\mathbb{P}}^n}/{\mathcal{L}}$ to be torsion free in the definition of a foliation is equivalent to ask the singular set to have codimension greater than 2. Indeed, this is the same to ask that $\omega$ is not of the form $f.\omega'$, for some global section $f\in H^0({\mathbb{P}}^n,{\mathcal{O}}_{{\mathbb{P}}^n}(d))$ and a 1-form $\omega'\in H^0({\mathbb{P}}^n,\Omega^1_{{\mathbb{P}}^n}(e-d))$. Also, integrable differential 1-forms define the same foliation up to scalar multiplication. Then, we will denote the set of codimension 1 foliations of geometric degree $e-2$ as $$\label{torsion} {\mathcal{F}}^1({\mathbb{P}}^n,e) := \left\{\omega\in{\mathbb{P}}\left(H^0({\mathbb{P}}^n,\Omega^1_{{\mathbb{P}}^n}(e))\right):\ \omega\wedge d\omega=0,\ {\mathrm{codim}}({\mathrm{Sing}(\omega)}_{set})\geq 2 \right\}.$$ \[IJ\] We define the graded ideals of $S$ associated to $\omega$ as $$\begin{aligned} I(\omega) &:= \left\{ h\in S:\ h\ d\omega = \omega\wedge\eta\text{ for some } \eta\in\Omega^1_S\right\}\\ J(\omega) &:= \left\{ i_X(\omega)\in S:\ X\in T_S \right\}.\end{aligned}$$ We will name $I(\omega)$ the *ideal of persistent singularities* of $\omega$. We will also denote them $I=I(\omega)$ and $J=J(\omega)$ if no confusion arises. \[1notinI\] Notice that $1\not\in I$, since the class of $d\omega$ in the Koszul complex of $\omega$, $H^2(\omega)$ is not zero, see Definition \[koszulcomplex\]. Also $J(\omega)$ equals the ideal defining the singular locus of $\omega$. This last thing, can be seen by contracting with the vector fields ${\partial}/{\partial x_i}$. The definition given for $J(\omega)$ is better suited for our schematic approach that we will develop next. For $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$, we define the *Kupka set* as the subset of the singular set $$\mathpzc{K}_{set} = \overline{\{p\in {\mathrm{Sing}(\omega)}_{set} : d\omega(p)\neq 0 \}}\ .$$ Notice that the definition above it is not the standard definition of the Kupka set. Usually it is defined just as the set of points in ${\mathrm{Sing}(\omega)}_{set}$ such that $d\omega(p)\neq 0$. Instead, we consider the closure of that set. \[KK\] For $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$, we define the *Kupka scheme* ${\mathpzc{Kup}(\omega)}$ as the scheme theoretic support of $d\omega$ at $\Omega^2_{S}\otimes_S (S\big/J)$. Then, ${\mathpzc{Kup}(\omega)}={\mathrm{Proj}}(S/K(\omega))$ where $K(\omega)$ is the homogeneous ideal defined as $$K(\omega)={\mathrm{ann}}(\overline{d\omega})+J(\omega)\subseteq S,\quad \overline{d\omega}\in \Omega^2_{S}\otimes_S \left(S\big/J(\omega)\right).$$ We will denote $K=K(\omega)$ if no confusion arises. We recall the notion of *ideal quotient* of two $S$-modules $M$ and $N$ as $$(N:M) := \left\{a\in S: a.M\subseteq N\right\},$$ then, one could also define $K(\omega)$ as $K(\omega)=(J\cdot \Omega^2_S: d\omega)$. Also, given that $\Omega^2_S$ is free, we can also write $$\label{Kbis} K(\omega)=(J(\omega):J(d\omega)),$$ where $J(d\omega)$ denotes the ideal generated by the polynomial coefficients of $d\omega$. From the properties of ideal quotient, it follows that if $J$ is radical, then $K$ is radical as well. With the Example 4.5 in [@mmq]\[p. 1034\] we showed that the algebraic geometric approach is indeed necessary, since the reduced structure associated to the Kupka scheme ${\mathcal{K}}$ differs from the reduced variety associated to $\mathpzc{K}_{set}$. With the following lemma we show that the Kupka scheme and the Kupka set coincide when the singular locus it is radical. ([@mmq]\[Lemma 4.6, p.1034\])\[K=Kset\] Let $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$ such that $J=\sqrt{J}$. Then $${\mathpzc{Kup}(\omega)}= \mathpzc{K}_{set}.$$ We have the following chain of inclusions, see Proposition \[JJcIIcKK\] and Proposition \[JJcIIcKK2\] for a generalization, in the codimension one and codimension $q$ case, respectively: ([@mmq]\[Proposition 4.7, p. 1035\])\[incl2\] Let $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$. Then, we have the following relations $$J\subseteq I\subseteq K\ .$$ Let $\mathfrak{p}$ be a point in ${\mathbb{P}}^n$, *e.g.*, an homogeneous prime ideal in $S$ different from the *irrelevant ideal* $(x_0.\ldots,x_n)$, and let $\omega$ be an integrable differential 1-form. We will denote with a subscript $\mathfrak{p}$ the localization at the point $\mathfrak{p}$ and with $\widehat{S}_\mathfrak{p}$ the completion of the local ring $S_\mathfrak{p}$ with respect to the maximal ideal defined by $\mathfrak{p}$. We say that $\mathfrak{p}\in{\mathbb{P}}^n$ is a *division point of $\omega$* if $1\in I(\omega)_\mathfrak{p}$. We now define a subset of the moduli space of foliations on which we are going to state our next result. \[generic\] We define the set ${\mathcal{U}}\subseteq{\mathcal{F}}^1({\mathbb{P}}^n,e)$ as $${\mathcal{U}}= \left\{\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e): \ \forall \mathfrak{p}\not\in{\mathpzc{Kup}(\omega)},\,\mathfrak{p}\text{ is a division point of }\omega\right\}.$$ See Theorem \[propP=K\] for a generalization of the following: ([@mmq]\[Theorem 4.12, p. 1036\])\[teo1\] Let $\omega\in\mathcal{U}\subseteq{\mathcal{F}}^1({\mathbb{P}}^n,e)$. Then, $$\sqrt{I}=\sqrt{K}.$$ Furthermore, if $\sqrt{I}=\sqrt{K}$ then $\omega\in{\mathcal{U}}$. See Theorem \[teoKnotempty\] for a generalization of the following: ([@mmq]\[Theorem 4.24, p. 1041\])\[teo3\] Let $\omega\in{\mathcal{F}}^1({\mathbb{P}}^n,e)$ such that $J=\sqrt{J}$. Then $${\mathpzc{Kup}(\omega)}=\mathpzc{K}_{set}\neq \emptyset.$$ The following statement is valid in a non-singular variety $X$ and we will use it later. We will consider a $1$-form $\omega$ on $X$ with singular set of codimension equal or greater than 2. And we will denote with $\mathcal{J}$ the ideal sheaf of ${\mathrm{Sing}(\omega)}$. ([@mmq]\[Theorem 2.7, p. 1030\])\[teodivision\] Let $\omega$ be an integrable $1$-form in a non-singular variety $X$ and let ${\mathfrak{p}}\in{\mathrm{Sing}(\omega)}$ be such that ${\mathscr{J}}_{\mathfrak{p}}$ is radical and such that $d\omega_{\mathfrak{p}}\in {\mathscr{J}}_{\mathfrak{p}}\cdot \Omega^2_{X,{\mathfrak{p}}}$. Then there is a formal $1$-form $\eta$ such that $d\omega=\omega\wedge\eta$. Unfoldings over schemes {#unf-schemes} ======================= Along this section we will give the definition of *codimension $q$ foliation* on a smooth variety $X$. Then we will redefine the *singular locus* with a scheme theoretic approach. Finally we define an *unfolding* of a codimension $q$ foliation. For the rest of the article, until Section \[section\_toric\], let us consider $X$ as a non-singular projective variety unless stated otherwise. If $\Xi \in \Gamma(U, \bigwedge^p TX)$ is a multivector and $\varpi\in \Gamma(U,\Omega^q_X)$ a $q$-form we will denote by $i_\Xi \varpi \in \Gamma(U,\Omega^{q-p}_X)$ the contraction. Recall that the *Plücker relations* for $\varpi$ are given by $$i_\Xi \varpi\wedge \varpi=0$$ for any $\Xi\in\bigwedge^{q-1} TX$. When $\varpi({\mathfrak{p}})\neq 0$ for some closed point ${\mathfrak{p}}\in X$ then $\varpi$ is *locally decomposable* as a product $\varpi=\varpi_1\wedge\dots\wedge\varpi_q$ of $q$ $1$-forms. Let ${\mathcal{L}}$ be a line bundle and $\omega:{\mathcal{L}}\to \Omega^q_X$ be a morphism of sheaves, we will say that the morphism is *integrable* if - $\Omega^q_X/{\mathcal{L}}$ is torsion free. - The map $$i_\Xi \omega\wedge \omega:{\mathcal{L}}\to\Omega^{q+1}_X\otimes{\mathcal{L}}^{-1}$$ is zero for every local section $\Xi$ of $\bigwedge^{q-1} TX$. - For every local section $s$ of ${\mathcal{L}}$ and $\Xi$ of $\bigwedge^{q-1} TX$, $\omega(s)$ verifies $$\label{frobenius2} d (i_\Xi \omega(s))\wedge \omega(s)=0.$$ We also say that $\omega$ determines a *codimension $q$ foliation*. By using Equation (\[frobenius2\]) with $q=1$ we recover the definition of codimension one foliation as in Definition \[foliation\]. The fact that $\omega(s)$ is locally decomposable as a product of $q$ $1$-forms $\varpi_1,\dots,\varpi_q$ implies that there exist a rank $q$ vector bundle $\mathcal{E}\hookrightarrow \Omega^1_X$, locally generated by $\omega_1(s),\dots,\omega_q(s)$ and such that ${\mathcal{L}}\simeq \bigwedge^q\mathcal{E}$. Reciprocally, given a locally free sheaf of rank $q$, $\mathcal{E}$ and a map $\mathcal{E}\hookrightarrow \Omega^1_X$, we have that $\bigwedge^q\mathcal{E}$ is a line bundle ${\mathcal{L}}$ and a map ${\mathcal{L}}\to\Omega^q_X$. The condition that $\Omega^q_X/{\mathcal{L}}$ is torsion free is equivalent to $\Omega^1_X/\mathcal{E}$ being torsion free. Example \[exampleCodimension2inA3\] shows that the condition *locally free* is necessary for this equivalence. Let $\omega:{\mathcal{L}}\to\Omega^q$ be a integrable $q$-form. Then, we can consider two maps, $$\xymatrix{ \bigwedge^{q-1} TX\otimes {\mathcal{L}}\ar[r]^<<<<<{i_{(\cdot)}\omega} &\Omega^1_X\ar[r]^<<<<<{(\cdot)\wedge\omega}&\Omega_X^{q+1}\otimes{\mathcal{L}}^{-1} }$$ The integrability condition on $\omega$ implies that this diagram is a complex and it is easy to check that its homology is supported over the points where $\omega$ is not decomposable. We define the sheaf associated to $\omega$, denoted ${\mathcal{E}}={\mathcal{E}}(\omega)$, as the kernel of $(\cdot)\wedge\omega$. By definition, ${\mathcal{E}}$ is a reflexive sheaf. \[exampleCodimension2inA3\] Let $X=\mathbb{A}^3$ or, in the holomorphic case, a polydisk of dimension $3$. We take $v\in \Gamma(X,TX)$ a vector field, generic in the sense that in a coordinate system $(x_1,x_2,x_3)$ we can write $v=f_1\frac{\partial}{\partial x_1}-f_2\frac{\partial}{\partial x_2}+f_3\frac{\partial}{\partial x_3}$ with $f_1, f_2, f_3 \in k[x_1,x_2,x_3]$ and such that the ideal $(f_1, f_2,f_3)\subseteq k[x_1,x_2,x_3]$ is a complete intersection, that is, there are no nontrivial relations among the $f_i$’s. The vector field $v$ generates a codimension $2$ foliation in $X$, this foliation is determined by a $2$-form $\omega$ such that $i_v\omega=0$. One such $\omega$ is given by $$\omega= f_3 dx_1\wedge dx_2 + f_2 dx_1\wedge dx_3 + f_1 dx_2\wedge dx_3.$$ It can be verified that this $\omega$ satisfies Plücker relations, is integrable, $i_v\omega=0$ and that $\Omega^2_X/(\omega)$ is torsion free. Therefore $\omega$ determines the same foliation of codimension $2$ as $v$. If we now look at the $1$-forms annihilated by $v$ we get the subsheaf generated by the forms $$\omega_1= f_3 dx_2+f_2dx_3,\quad \omega_2=f_3 dx_1- f_1 dx_3\ \text{ and } \omega_3=f_2 dx_1 + f_1 dx_2.$$ These generators satisfy the relation $f_1\omega_1 + f_2 \omega_2= f_3\omega_3$. The subsheaf $\mathcal{E}=(\omega_1,\omega_2,\omega_3)$ is generically of rank $2$ outside the zeros of the ideal $(f_1,f_2,f_3)$ but $\mathcal{E}\otimes k({\mathfrak{p}})$ is of rank $3$ when ${\mathfrak{p}}$ is in the zeros of this ideal. Therefore $\mathcal{E}$ is *not* locally free. Moreover when we compute the determinant of $\mathcal{E}$ we get $\wedge^2\mathcal{E}= (f_1,f_2,f_3)\cdot(\omega)\subseteq \Omega^2_X$, $$\omega_1\wedge\omega_2=f_3\omega,\quad \omega_3\wedge\omega_1=f_2\omega,\quad \omega_2\wedge\omega_3=f_1\omega.$$ In particular $\omega$ is not in $\wedge^2\mathcal{E}$. But by [@GH Lemma, p. 210], if $\omega$ is locally decomposable, then $\omega\in \wedge^2\mathcal{E}$. Then $\omega$ is *not* locally decomposable around the zeros of the ideal $(f_1,f_2,f_3)$. Composing a morphism $\omega:{\mathcal{L}}\to \Omega^q$ with the contraction of forms with vector fields give us a morphism $$\bigwedge^q TX\otimes {\mathcal{L}}\to{\mathcal{O}}_X.$$ The ideal sheaf ${\mathscr{J}}(\omega)$ is defined to be the sheaf-theoretic image of the morphism $\bigwedge^q TX\otimes {\mathcal{L}}\to{\mathcal{O}}_X$. The subscheme it defines is called *the singular set* of $\omega$ and denoted ${\mathrm{Sing}(\omega)}\subseteq X$. We will denote it just as ${\mathscr{J}}$ if no confusion arises. This definition agrees with Remark \[1notinI\], where we said that the ideal $J(\omega)$ gives the ideal defining the singular locus of $\omega$. From [@suwa-review]\[(4.6) Definition, p. 192\] we get the following definition for a codimension $q$ foliation: Let $S$ be a scheme, $p\in S$ a closed point, and ${\mathcal{L}}\xrightarrow{\omega} \Omega^q_X$ a codimension $q$ foliation on $X$. An *unfolding* of $\omega$ is a codimension $q$ foliation $\widetilde{{\mathcal{L}}}\xrightarrow{\widetilde{\omega}}\Omega^q_{X\times S}$ on $X\times S$ such that $\widetilde{\omega}|_{X\times\{p\}}\cong \omega$. In the case $S=\mathrm{Spec}(k[x]/(x^2))$ we will call $\widetilde{\omega}$ a *first order infinitesimal unfolding*. Kupka scheme in general for codimension 1 foliations ==================================================== Over this section we will restate the definition of *persistent singularities* and of the *Kupka scheme*, through its ideal sheaf, in a more general setting, see Definition \[II2\] and Definition \[KK2\], respectively. In [@mmq] we showed that persistent singularities are related to unfoldings in codimension one. We want to extend this relation to higher codimension. First we prove Proposition \[JJcIIcKK\], generalizing Proposition \[incl2\] in the codimension one case. Then we define the Kupka scheme and we prove Theorem \[propP=K\] and Theorem \[teoKnotempty\], generalizing Theorems \[teo1\] and \[teo3\]. Given a line bundle ${\mathcal{L}}$ and a global section $\omega\in H^0\left(X,\Omega^1_X\otimes {\mathcal{L}}^{-1}\right)$ we will consider the Koszul complex associated with $\omega$, $$\label{koszulcomplex} \xymatrix{K(\omega): & {\mathcal{O}}_X\ar[r]^-{\wedge\omega}& \Omega^1_X\otimes{\mathcal{L}}^{-1} \ar[r]^-{\wedge\omega} & \dots\ar[r] & \Omega^i\otimes{\mathcal{L}}^{-i}\ar[r]& \dots}$$ where we are following [@GKZ Chapter 2, B, p. 51] and using the identification $\bigwedge^k\left(\Omega^1_X\otimes {\mathcal{L}}^{-1}\right)\simeq \left(\bigwedge^k \Omega^1_X \right)\otimes \left({\mathcal{L}}^{-k}\right)$. We will denote the cohomology sheaves of this complex by $H^\bullet(\omega)$, the *Koszul cohomology sheaves* of $\omega$. We can use $K(\omega)$ to compute the codimension of ${\mathrm{Sing}(\omega)}$ by the well known result, see [@eisenbud Theorem 17.4, p. 424]: \[koszul\] Let $\omega\in H^0\left(X,\Omega^1_X\otimes {\mathcal{L}}^{-1}\right)$. The following statements are equivalent: - $codim({\mathrm{Sing}(\omega)})\geq k$ - $H^\ell(\omega)=0$ for all $\ell <k$ \[notation\] Suppose now the morphism $\omega:{\mathcal{L}}\to \Omega^1_X$ defines a foliation on $X$. Given a trivializing open set $U$ and a choice of a trivialization ${\mathcal{O}}_X|_U\cong {\mathcal{L}}|_U$, we take a local generator $\varpi$ of ${\mathcal{L}}(U)$ (we think about it as a $1$-form through the morphism ${\mathcal{L}}\to \Omega^1_X$) and take the differential $d \varpi$. This defines a ${\mathbb{C}}$-linear morphism ${\mathcal{L}}(U)\to \Omega^2_X(U)$, which in turn we can compose with the projection $\Omega^2_X\to H^2(\omega)\otimes {\mathcal{L}}^{\otimes 2}$. Note that the submodule ${\mathcal{O}}_X(U)\cdot (d\varpi)$ of $\left(H^2(\omega)\otimes{\mathcal{L}}^{\otimes 2}\right)(U)$ is independent of the choice of the trivialization. In this way one gets a morphism of coherent sheaves, $${\mathcal{L}}\to H^2(\omega)\otimes{\mathcal{L}}^{\otimes 2}.$$ Or, equivalently, a (non trivial) global section of $H^2(\omega)\otimes{\mathcal{L}}$. We will denote the global section or the morphism indistinctly by $[d\omega]$. By Theorem \[koszul\] above, we conclude that $codim({\mathrm{Sing}(\omega)})\geq 2$. \[II2\] The subscheme of *persistent singularities* of $\omega$ is the one defined by the ideal sheaf ${\mathscr{I}}(\omega):={\mathrm{ann}}([d\omega])$, for $[d\omega]\in H^0\left(X,H^2(\omega)\otimes{\mathcal{L}}\right)$. We will denote it just as ${\mathscr{I}}$ if no confusion arises. \[remarkI\] Let $\varpi\in \Omega^1_X(U)$ be a local generator of the image of ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$, then the local sections of ${\mathscr{I}}(\omega)$ in $U$ are given by $${\mathscr{I}}(U)=\{h\in {\mathcal{O}}_X(U): \text{there is a section } \eta\in\Gamma(U,\Omega^1_X/{\mathcal{L}})\text{ s.t. } hd \varpi=\varpi\wedge\eta\}.$$ \[formalremark\] For a regular local ring $(R,\mathfrak{m})$, an $R$-module $M$ and an element $m\in M$, let us denote $\widehat{R}$ the $\mathfrak{m}$-adic completion of $R$ and $\widehat{M}=M\otimes\widehat{R}$. The element $m\otimes 1\in\widehat{M}$ has as annihilator the ideal ${\mathrm{ann}}(m)\otimes \widehat{R}$. Setting $R={\mathcal{O}}_{X,{\mathfrak{p}}}$, $M= \left(H^2(\omega)\otimes{\mathcal{L}}\right)_{\mathfrak{p}}$ and $m=[d\omega]_{\mathfrak{p}}$, and following the notation of Remark \[notation\], we have that $${\mathrm{ann}}([d\omega]_{\mathfrak{p}}\otimes 1)=\{h\in \widehat{{\mathcal{O}}_{X,{\mathfrak{p}}}}: \text{there is a formal $1$-form } \eta \ \text{ s.t. } hd \varpi=\varpi\wedge\eta\}.$$ Let ${\mathfrak{p}}\in X$ be a point in ${\mathrm{Sing}(\omega)}$, ${\mathcal{O}}_{X,{\mathfrak{p}}}$ the local ring around ${\mathfrak{p}}$, and $X_{\mathfrak{p}}=\mathrm{Spec}({\mathcal{O}}_{X,{\mathfrak{p}}})$. Then ${\mathfrak{p}}$ is in the subscheme of persistent singularities if and only if for any infinitesimal first order unfolding $\widetilde{\omega}$ of $\omega$ in $X_{\mathfrak{p}}$, the point $({\mathfrak{p}}, 0)\in X_{\mathfrak{p}}\times \mathrm{Spec}(k[\varepsilon]/(\varepsilon^2))$ is a singular point of $\widetilde{\omega}$. Let $S=\mathrm{Spec}(k[\varepsilon]/(\varepsilon^2))$, $0\in S$ be its closed point, $p:X\times S\to S$ be the projection and $\iota:X\cong X\times\{0\}\hookrightarrow X\times S$ be the inclusion. Then the sheaf $\Omega^1_{X\times S}$ can be decomposed as direct sum of $\iota_*({\mathcal{O}}_X)$-modules as $$\Omega^1_{X\times S}\cong \iota_*(\Omega^1_X)\oplus \epsilon\cdot \iota_*(\Omega^1_X)\oplus \iota_*({\mathcal{O}}_X)d\epsilon.$$ A point ${\mathfrak{p}}\in{\mathrm{Sing}(\omega)}$ is *not* a persistent singularity if and only if $1\in {\mathscr{I}}_{\mathfrak{p}}\subseteq {\mathcal{O}}_{X,{\mathfrak{p}}}$ which, by Remark \[remarkI\], means that there is an open neighborhood $U\subseteq X$ of ${\mathfrak{p}}$, a local generator $\varpi$ of the image of ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$, and a section $\eta\in\Gamma(U,\Omega^1_X/{\mathcal{L}})$ such that $d\omega=\omega\wedge \eta$. By shrinking $U$ if necessary we can take a lifting of $\eta$ in $\Omega^1_X$ which by abuse of notation we also call $\eta$ and define $$\widetilde{\omega}= \omega + \varepsilon \eta + d\varepsilon.$$ Thus $\widetilde{\omega}$ is a form in $\Omega^1_{X\times S}$ and $\widetilde{\omega}({\mathfrak{p}},0)= d\varepsilon\neq 0$, so ${\mathfrak{p}}\times \{0\}$ is *not* a singular point of $\widetilde{\omega}$. Reciprocally, if there is an unfolding $\widetilde{\omega}$ of $\omega|_U$, then $$\widetilde{\omega}({\mathfrak{p}}, 0)=\omega({\mathfrak{p}}) + h(0) d\varepsilon.$$ As ${\mathfrak{p}}$ is a singular point of $\omega$, we have $\omega({\mathfrak{p}})$, so if $({\mathfrak{p}},0)$ is not a singular point of $\widetilde{\omega}$, then $h(0)\neq 0$, so again shrinking $U$ if necessary we have that $h$ is a unit, hence $1\in {\mathscr{I}}_{\mathfrak{p}}$. Most of the known families of foliations on algebraic varieties present persistent singularities, see [@gmln; @omegar; @celn; @pullback; @fji; @fmi; @fj; @mmq]. As it happens the absence of persistent singularities impose some restrictions on the line bundle ${\mathcal{L}}$. To explain this we have to make explicit use of a result that is implied in the proof of Lefschetz Theorem on $(1,1)$ classes as is proved in [@GH Chapter 1.1 p.:  141]. Let ${\mathcal{L}}$ be a line bundle. Choose a trivialization $(U_i, \phi_i)_{i\in I}$ of ${\mathcal{L}}$ with gluing data $g_{ij}\in {\mathcal{O}}^*_X(U_{ij})$. The Čech cocycle $\frac{1}{2\pi i }[d\log g_{ij}]\in Z^1(\Omega^1_X)$ represents the Chern class $c_1({\mathcal{L}})$ of ${\mathcal{L}}$ in $H^1(X,\Omega^1_X)$. The claim follows from a careful reading of the proof of the Proposition in page 141 of [@GH Chern classes of line bundles, Chapter 1.1, p. 141], as we will show next. There is shown that de Rham’s Theorem for $\mathcal{C}^\infty$-forms with complex coefficients give us exact sequence of sheaves $$0\to {\mathbb{C}}\to \mathcal{A}^0\to \mathcal{Z}^1_d\to 0,\qquad 0\to\mathcal{Z}^1_d\to\mathcal{A}^1\to \mathcal{Z}^2_d\to 0,$$ where $\mathcal{A}^i$ are $\mathcal{C}^\infty$ $i$-forms with complex coefficients and $\mathcal{Z}^i_d$ are the cycles of the de Rham complex. These exact sequences give us boundary isomorphisms $$\frac{H^0(\mathcal{Z}^2)}{dH^0(\mathcal{A}^1)}\xrightarrow{\delta_1}H^1(\mathcal{Z}^1_d),\qquad H^1(\mathcal{Z}^1_d) \xrightarrow{\delta_2} H^2({\mathbb{C}}).$$ As explained in *loc. cit.* the multiple of the Chern class $-(2\pi i)c_1({\mathcal{L}})$ can be calculated as $\delta_2\delta_1(\Theta)$, where $\Theta$ is the curvature form of a connection on ${\mathcal{L}}$. It also follows from *loc. cit.* that $\delta_1(\Theta)$ is represented by the cocycle $-d\log g_{ij}$. \[propc1L=0\] Let $X$ be a smooth projective variety over ${\mathbb{C}}$. If ${\mathcal{L}}$ is a line bundle such that $H^1(X,{\mathcal{L}})=0$ and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ is a foliation without persistent singularities then $c_1({\mathcal{L}})=0$, where $c_1({\mathcal{L}})$ is the Chern class of the line bundle viewed in $H^2(X,{\mathbb{C}})$. Let $(U_i,\phi_i)$ be a trivialization of ${\mathcal{L}}$ with gluing data $g_{ij}\in {\mathcal{O}}^*_X(U_{ij})$. On each $U_i$ we have a local generator of ${\mathcal{L}}(U_i)$, namely $\phi_i^{-1}(1)$, we denote by $\omega_i$ the image under $\omega$ of this generator. The fact that the foliation defined by $\omega$ has no persistent singularities means that on each $U_i$ there is a local section $\eta_i$ of $\Omega^1_X/{\mathcal{L}}(U_i)$ such that $d\omega_i=\omega_i\wedge\eta_i$. On $U_{ij}$ the restriction of the local $1$-form $\omega_i$ satisfies $$\omega_i=g_{ij}\omega_j.$$ So computing the de Rham differential of this forms on $U_{ij}$ gives us, $$\begin{aligned} \omega_i\wedge\eta_i=d\omega_i=d(g_{ij}\omega_j)=\\ =g_{ij}d\omega_j+dg_{ij}\wedge \omega_j=\\ =g_{ij} \omega_j\wedge\eta_j + dg_{ij}\wedge \omega_j=\\ =g_{ij}\omega_j\wedge\left(\eta_j-\frac{dg_{ij}}{g_{ij}}\right).\end{aligned}$$ Subtracting both sides of the equality we get that, on $U_{ij}$, $$\eta_i-\eta_j=\frac{dg_{ij}}{g_{ij}},$$ as sections of $\Gamma(U_{ij},\Omega^1_X /{\mathcal{L}})$. Therefore we get a Čech cochain $(\eta_i)_{i\in I}$ of $C^0(\Omega^1_X/{\mathcal{L}})$ whose border is $$\partial (\eta)_{ij}=d\log g_{ij}\in B^1(\Omega^1/{\mathcal{L}}).$$ As the cocycle $(d\log g_{ij})\in Z^1(\Omega^1_X)$ represents $(2\pi i )c_1({\mathcal{L}})$, the existence of the cochain $(\eta_i)$ implies $c_1({\mathcal{L}})$ is in the kernel of the map $H^1(\Omega^1_X)\to H^1(\Omega^1_X/{\mathcal{L}})$ induced by the short exact sequence of sheaves $$0\to{\mathcal{L}}\xrightarrow{\omega}\Omega^1_X\to \Omega^1_X/{\mathcal{L}}\to 0.$$ The hypothesis $H^1({\mathcal{L}})=0$ then implies $c_1({\mathcal{L}})=0$. Let $X$ be a smooth projective variety over ${\mathbb{C}}$ such that every line bundle ${\mathcal{L}}$ verifies $H^1(X,{\mathcal{L}})=0$ and such that $Pic(X)$ is torsion-free (*e.g.*: $X$ smooth complete intersection). Then every foliation on $X$ have persistent singularities. From the exponential sequence and the hypothesis $H^1(X,{\mathcal{O}}_X)=0$ it follows that $c_1:Pic(X)\to H^2(X,{\mathbb{Z}})$ is injective. Assume that $\omega:{\mathcal{L}}\to\Omega^1_X$ is a foliation without persistent singularities. Then the above Proposition imply that $c_1({\mathcal{L}})$ is a torsion element in $H^2(X,{\mathbb{Z}})$. But given that $Pic(X)$ is torsion-free, we get ${\mathcal{L}}\cong{\mathcal{O}}_X$. In particular, $\omega$ is a global differential $1$-form which contradicts the fact that $H^0(X,\Omega^1_X)=H^1(X,{\mathcal{O}}_X)=0$. \[[domega]{}\] Given a trivializing open set $U$, a choice of a trivialization ${\mathcal{O}}_X|_U\cong {\mathcal{L}}|_U$ and a local generator $\varpi$ of ${\mathcal{L}}(U)$, the mapping $\varpi\mapsto d\varpi$ defines a ${\mathcal{O}}_X$-linear morphism ${\mathcal{L}}\to \Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}$. We will denote by $\{d\omega\}$ this morphism or equivalently the global section of $\Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}\otimes{\mathcal{L}}^{-1}$ it defines. \[KK2\]The subscheme of *Kupka singularities* of $\omega$ is the one defined by the ideal sheaf ${\mathcal{K}}(\omega):={\mathrm{ann}}(\{d\omega\})\in\Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}\otimes{\mathcal{L}}^{-1}$. We will denote it just as ${\mathcal{K}}$ if no confusion arises. \[JJcIIcKK\] Let ${\mathscr{J}}$ be the ideal sheaf of the singular set of $\omega$, ${\mathcal{K}}$ the ideal of the Kupka singularities of $\omega$ and ${\mathscr{I}}$ the ideal of persistent singularities. Then the following inclusions hold, $${\mathscr{J}}\subseteq {\mathscr{I}}\subseteq {\mathcal{K}}.$$ Let $U\subseteq X$ be an open subscheme such that ${\mathcal{L}}|_U\simeq{\mathcal{O}}_X$, and $\varpi$ a local generator of ${\mathcal{L}}(U)$. Suppose $h\in{\mathscr{J}}(U)\subseteq{\mathcal{O}}_X(U)$ is a local section. By shrinking $U$ if necessary we may assume that there is a vector field $v\in T_X(U)$ such that $h= i_v(\omega)$. Then we have $$0=i_v(\varpi\wedge d\varpi)=i_v(\varpi)d\varpi - \varpi \wedge i_v(d\varpi).$$ So, calling $\eta=i_v(d\varpi)$, we get $hd\varpi=\varpi\wedge \eta$. Hence $h$ is in ${\mathscr{I}}(U)$, which proves the first inclusion. Now assume $h\in {\mathscr{I}}(U)$, then again by shrinking $U$ if necessary, we may assume that there is a $\eta\in \Omega^1_X/{\mathcal{L}}(U)$ such that $hd\varpi= \varpi\wedge\eta$. By definition we have $\varpi \in {\mathscr{J}}(U)\cdot \Omega^1_X(U)$, then $hd\varpi\in {\mathscr{J}}(U)\cdot \Omega^2_X(U)$ so $h$ is in the annihilator of $\{d\omega\}$ in $\Omega^2_X\otimes {\mathcal{O}}_{{\mathrm{Sing}(\omega)}}$. Then $h\in {\mathcal{K}}(U)$, which proves the second inclusion. With the following results we can generalize Theorem \[teo1\] and Theorem \[teo3\] giving conditions for the existence of Kupka singularities: Let $X$ be a smooth projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ a foliation of codimension $1$, we are going to call ${\mathpzc{Per}(\omega)}\subseteq X$ the subschemes of persistent singularities. \[propP=K\] Let $X$ be a smooth projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ a foliation of codimension $1$ such that ${\mathscr{J}}(\omega)$ is a sheaf of radical ideals. Let ${\mathpzc{Per}(\omega)}\subseteq X$ and ${\mathpzc{Kup}(\omega)}\subseteq X$ be the subschemes of persistent and Kupka singularities respectively. Then ${\mathpzc{Per}(\omega)}_{\mbox{red}}={\mathpzc{Kup}(\omega)}_{\mbox{red}}$. We are going to prove that $X\setminus{\mathpzc{Per}(\omega)}=X\setminus{\mathpzc{Kup}(\omega)}$. By Proposition \[JJcIIcKK\] we have ${\mathpzc{Kup}(\omega)}\subseteq{\mathpzc{Per}(\omega)}$, so $X\setminus {\mathpzc{Per}(\omega)}\subseteq X\setminus {\mathpzc{Kup}(\omega)}$. Now suppose ${\mathfrak{p}}$ is a point *not* in ${\mathpzc{Kup}(\omega)}$, by abuse of notation we will call $\omega$ a local generator of ${\mathcal{L}}_{\mathfrak{p}}$ viewed as a $1$-form. As ${\mathfrak{p}}$ is not in ${\mathpzc{Kup}(\omega)}$ then $d\omega \in {\mathscr{J}}_{\mathfrak{p}}\cdot\Omega^2_{X,{\mathfrak{p}}}$. By hypothesis ${\mathscr{J}}_{\mathfrak{p}}$ is radical and so by Theorem \[teodivision\] we have that $d\omega$ decomposes as $\omega\wedge\eta$ for some formal $1$-form $\eta$, this implies $1\in {\mathscr{I}}_{\mathfrak{p}}$, so ${\mathfrak{p}}$ is not in ${\mathpzc{Per}(\omega)}$ (see Remark \[formalremark\]). \[teoKnotempty\] Let $X$ be a smooth projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega^1_X$ a foliation of codimension $1$ such that ${\mathscr{J}}(\omega)$ is a sheaf of radical ideals and such that $c_1({\mathcal{L}})\neq 0$ and $H^1(X,{\mathcal{L}})=0$. Then $\omega$ has Kupka singularities. This follows from Proposition \[propc1L=0\] and Theorem \[propP=K\], as a foliation with $c_1({\mathcal{L}})\neq 0$ and $H^1(X,{\mathcal{L}})=0$ has persistent singularities on one hand, and having radical singular ideal implies the reduced scheme defined by persistent singularities is equal to the reduced scheme of Kupka singularities, in particular this last scheme is not empty. Infinitesimal unfoldings in codimension $q$ =========================================== Along this section we review the definition of unfolding of a codimension $q$ foliation on a variety $X$. We will also generalize the definitions of persistent singularities and of Kupka singularities for codimension $q$ foliations, see Definition \[defII\] and Definition \[defKK\], respectively. We classify which singular points of $\omega$ are such that they extend to singular points of every unfolding $\widetilde{\omega}$ (Proposition \[prop\]) and then, we generalize Proposition \[incl2\] and Proposition \[JJcIIcKK\] to the codimension $q$ case (Proposition \[JJcIIcKK2\]). Finally, with Theorem \[teoConnectionE\] we establish that the absence of persistent singularities implies the existence of a connection on ${\mathcal{E}}$, the sheaf of 1-forms defining the foliation. Let $S=\mathrm{Spec}(k[\varepsilon]/(\varepsilon^2))$, $0\in S$ be its closed point, $p:X\times S\to S$ be the projection and $\iota:X\cong X\times\{0\}\hookrightarrow X\times S$ be the inclusion. Then the sheaf $\Omega^q_{X\times S}$ can be decomposed as direct sum of $\iota_*({\mathcal{O}}_X)$-modules as $$\Omega^q_{X\times S}\cong \iota_*\Omega^q_X\oplus \varepsilon\cdot (\iota_*\Omega^q_X)\oplus \iota_*\Omega^{q-1}_X\wedge d\varepsilon.$$ Given a codimension $q$ foliation determined by a morphism ${\mathcal{L}}\xrightarrow{\omega}\Omega^q_X$, and a first order infinitesimal unfolding $\widetilde{\omega}: \widetilde{{\mathcal{L}}}\to \Omega^q_{X\times S}$ of $\omega$, we take local generators $\varpi$ of ${\mathcal{L}}(U)$ and $\widetilde{\varpi}$ of $\widetilde{{\mathcal{L}}}(U\times S)$. Suppose $\omega$ and $\widetilde{\omega}$ are locally decomposable, then we may take $U$ small enough such that $\varpi$ and $\widetilde{\varpi}$ decompose as products $$\varpi=\varpi_1\wedge\dots\wedge\varpi_q, \qquad \widetilde{\varpi}=\widetilde{\varpi}_1\wedge\dots\wedge \widetilde{\varpi}_q.$$ Then we can write $\widetilde{\varpi}_i=\varpi_i + \varepsilon \eta_i + h_i d\varepsilon$ and the equations $d\widetilde{\varpi}_i\wedge \widetilde{\varpi}=0$ for $i=1,\dots, q$ are equivalent to the equations $$\left\{ \begin{aligned} &d\eta_i\wedge \varpi+ d\varpi_i\wedge \left(\sum_{j=1}^q (-1)^j\eta_j \varpi_{\widehat{j}}\right)=0,\qquad (i=1,\dots,q), \\ &(dh_i-\eta_i)\wedge \varpi+ d\varpi_i\wedge \left(\sum_{j=1}^q (-1)^jh_j \varpi_{\widehat{j}}\right)=0,\qquad (i=1,\dots,q), \end{aligned} \right.$$ where $ \varpi_{\widehat{j}}=\varpi_1\wedge\dots\wedge\varpi_{j-1}\wedge\varpi_{j+1}\wedge\dots\wedge\varpi_q\in \Omega^{q-1}_X(U)$. As is shown in [@suwa-review proof of (6.1) Theorem, p. 199] the second equation implies the first. So we finally get that the equations $d\widetilde{\varpi}_i\wedge \widetilde{\varpi}=0$ for $i=1,\dots, q$ are equivalent to $$\label{equnfcodq} \begin{aligned} \left\{ (dh_i-\eta_i)\wedge \varpi+ d\varpi_i\wedge \left(\sum_{j=1}^q (-1)^jh_j \varpi_{\widehat{j}}\right)=0,\qquad (i=1,\dots,q)\ . \right. \end{aligned}$$ \[prop\] Suppose ${\mathfrak{p}}$ is a singular point of $\omega$. Then there exist an infinitesimal unfolding $\widetilde{\omega}$ of $\omega$ in $X_{\mathfrak{p}}$ such that $({\mathfrak{p}},0)$ is *not* a singular point of $\widetilde{\omega}$ if and only if $\omega$ is decomposable locally around ${\mathfrak{p}}$, not all $\varpi_{\widehat{j}}({\mathfrak{p}})$ vanish and there are $1$-forms $\alpha_{ij}$ such that $$d\varpi_i=\sum_{j=1}^q \alpha_{ij}\wedge \varpi_j,\qquad (i=1,\dots,q).$$ Given local forms $\alpha_{ij}\in \Omega^1_{X,{\mathfrak{p}}}$ such that $d\varpi_i=\sum_{j=1}^q \alpha_{ij}\wedge \varpi_j, (i=1,\dots,q)$ we may take local sections $h_i\in{\mathcal{O}}_{X,{\mathfrak{p}}}$ such that $\sum_{i=1}^q(-1)^ih_i({\mathfrak{p}})\varpi_{\widehat{i}}({\mathfrak{p}})\neq 0$. With that choice of $h_i$’s we take $\eta_i:=dh_i+\sum_{j=1}^q (-1)^j h_j\alpha_{ij}$. We will see that the $\eta_i$’s and $h_i$’s determine an unfolding of $\omega$ locally around ${\mathfrak{p}}$. For that we need to verify the Equation (\[equnfcodq\]) above. Indeed we have $$\begin{aligned} (dh_i-\eta_i)&\wedge\varpi+d\varpi_i\wedge\left(\sum_{j=1}^q (-1)^jh_j\varpi_{\widehat{j}}\right)=\\ &= (dh_i-\eta_i)\wedge\varpi+\left( \sum_{k=1}^q\alpha_{ik}\wedge \varpi_k \right)\wedge\sum_{j=1}^q (-1)^j\varpi_{\widehat{j}}=\\ &=(dh_i-\eta_i)\wedge\varpi + \left(\sum_{j,k=1}^1(-1)^j h_j\alpha_{ik}\wedge\varpi_k\wedge\varpi_{\widehat{j}}\right)=\\ &=(dh_i-\eta_i)\wedge \varpi + \left(\sum_{j=1}^q (-1)^j\alpha_{ij}\wedge \varpi\right) =\\ &=\left((dh_i-\eta_i)+\sum_{j=1}^q (-1)^j h_j\alpha_{ij} \right)\wedge \varpi . \end{aligned}$$ And from the definition of the $\eta_i$ we have that $$\begin{aligned} &\left((dh_i-\eta_i)+\sum_{j=1}^q (-1)^j h_j\alpha_{ij} \right)\wedge \varpi =\\ &= \left(-\sum_{j=1}^q (-1)^j h_j\alpha_{ij} +\sum_{j=1}^q (-1)^j h_j\alpha_{ij} \right)\wedge \varpi= 0\\ \end{aligned}$$ Then we have an unfolding $\widetilde{\omega}$ given locally around ${\mathfrak{p}}$ by $$\bigwedge_{i=1}^q (\varpi_i+\varepsilon \eta_i + h_i d\varepsilon)= \varpi+\varepsilon \left(\sum_{i=1}^q \eta_i\wedge \varpi_{\widehat{j}}\right)+ \left(\sum_{j=1}^q (-1)^jh_j \varpi_{\widehat{j}}\right)\wedge d\varepsilon.$$ As $\sum_{j=1}^q (-1)^jh_j \varpi_{\widehat{j}}\neq 0$ then $\widetilde{\omega}$ does not vanishes on $({\mathfrak{p}},0)$. Reciprocally, let us suppose there is an unfolding $\widetilde{\omega}$ such that $\widetilde{\omega}({\mathfrak{p}},0)\neq 0$. As $\widetilde{\omega}$ satisfies Plücker relations and does not vanish in ${\mathfrak{p}}$, then it decomposes as a product of $1$-forms $\varpi_i+\varepsilon\eta_i+h_id\varepsilon$, $i=1,\dots,q$. As $\widetilde{\omega}|_{X\times\{0\}}=\omega$ then any local generator $\varpi$ of the image of $\omega$ is locally decomposable as $\varpi_1\wedge\dots\wedge\varpi_q$. We want to prove that the class $[d\varpi_i]$ of $d\varpi_i$ in $\Omega^2_{X,{\mathfrak{p}}}/((\varpi_1,\dots,\varpi_q)\wedge\Omega^1_{X,{\mathfrak{p}}})$ is zero for $i=1,\dots, q$. Let $\mathfrak{q}$ be a point in the support of $[d\varpi]$, then $\omega$ is singular in $\mathfrak{q}$, for otherwise $[d\varpi]=0$ because of the Frobenius condition $d\varpi_i\wedge \varpi = 0$, for $i=1,\ldots,q$. By Equation (\[equnfcodq\]), we have $\sum_{j=1}^q (-1)^jh_j \varpi_{\widehat{j}}({\mathfrak{p}})\neq 0$, in particular not all of the $\varpi_{\widehat{j}}({\mathfrak{p}})$ vanishes. Without any loss of generality, we may assume $\varpi_{\widehat{1}}({\mathfrak{p}})$ does not vanish. Then also $\varpi_{\widehat{1}}(\mathfrak{q})\neq 0$. But $\varpi(\mathfrak{q})=0$, therefore $\varpi_2(\mathfrak{q}),\dots,\varpi_q(\mathfrak{q})$ are linearly independent and $\varpi_1(\mathfrak{q})$ is a linear combination of them. Hence $\varpi_{\widehat{j}}(\mathfrak{q})=f_j \varpi_{\widehat{1}}(\mathfrak{q})$. Then evaluating Equation (\[equnfcodq\]) in $\mathfrak{q}$, and adding the term $h_{1}(\mathfrak{q}) d\varpi_i(\mathfrak{q}) \wedge \varpi_{\widehat{1}}(\mathfrak{q})$, gives $$h_{1}(\mathfrak{q}) d\varpi_i(\mathfrak{q}) \wedge \varpi_{\widehat{1}}(\mathfrak{q})=(dh_i-\eta_i)\wedge\varpi(\mathfrak{q}) +\left(\sum_{j=2}^q h_j(\mathfrak{q}) f_j(\mathfrak{q})\right)d\varpi_i(\mathfrak{q})\wedge\varpi_{\widehat{1}}(\mathfrak{q}) \ .$$ So, after clearing $h_1(\mathfrak{q})\neq 0$, there is a $1$-form $\alpha_{i1}$ such that $$d\varpi_i\wedge \varpi_{\widehat{1}}(\mathfrak{q})= \alpha_{i1}\wedge \varpi_1\wedge \varpi_{\widehat{1}}(\mathfrak{q}) ,$$ then we have $(d\varpi_i-\alpha_{i1}\wedge\varpi_1)\wedge \varpi_{\widehat{1}}(\mathfrak{q})=0$, but as $\varpi_{\widehat{1}}(\mathfrak{q})\neq 0$ , this implies that there are forms $\alpha_{ij}$ such that $$(d\varpi_i-\alpha_{i1}\wedge\varpi_1)(\mathfrak{q})=\sum_{j\neq 1}\alpha_{ij}\wedge \varpi_j(\mathfrak{q}).$$ Hence $[d\varpi_i]=0$ in any point of its support, a contradiction, so $[d\varpi_i]=0$ in $\Omega^2_{X,{\mathfrak{p}}}/((\varpi_1,\dots,\varpi_q)\wedge\Omega^1_{X,{\mathfrak{p}}})$. Let ${\mathcal{L}}\xrightarrow{\omega}\Omega^q_X$ be an integrable morphism determining a subsheaf $\mathcal{E}\to\Omega^1_X$. Composing $\omega$ with wedge product gives a morphism ${\mathcal{L}}\otimes\Omega^2_X\xrightarrow{\omega\wedge -} \Omega^{q+2}_X$. As $\omega$ is integrable the sheaf $\mathcal{E}\otimes\Omega^1_X$ is in the kernel of $\omega\wedge -$. We define the sheaf $H^2(\omega)$ as $$H^2(\omega):= \ker(\omega\wedge -) /\mathcal{E}\otimes \Omega^1_X.$$ The restriction of the de Rham differential to $\mathcal{E}$ gives a sheaf map $\mathcal{E}\to \Omega^2_X$ which is not ${\mathcal{O}}_X$-linear but whose image is in $ \ker(\omega\wedge -) $ as $\omega$ is integrable. The projection of this map to $H^2(\omega)$ is however ${\mathcal{O}}_X$-linear as $d g\varpi\cong g d\varpi \mod \mathcal{E}\otimes\Omega^1_X$ for every local section $\varpi$ of $\mathcal{E}$. Let us fix ${\mathcal{L}}\xrightarrow{\omega}\Omega^q_X$ be an integrable morphism determining a subsheaf $\mathcal{E}\to\Omega^1_X$. Then we have the following definitions: \[defII\] The subscheme of *persistent singularities* of $\omega$ is the one defined by the ideal sheaf ${\mathscr{I}}(\omega)$ to be the annihilator of $d(\mathcal{E})$ in $H^2(\omega)$. In other words the local sections of ${\mathscr{I}}(\omega)$ in an open set $U\subseteq X$ are given by $${\mathscr{I}}(\omega)(U)=\{ h\in {\mathcal{O}}_X(U): \forall \varpi\in\mathcal{E}(U),\ hd\varpi=\sum_j \alpha_j\wedge\omega_j \},$$ for some local $1$-forms $\alpha_j$ and forms $\omega_j$ in $\mathcal{E}(U)$. We will denote it just as ${\mathscr{I}}$ if no confusion arises. With the following example we are showing that the ideal ${\mathscr{I}}(\omega)$ can have codimension greater than $2$. Let us consider the $2$-form in ${\mathbb{P}}^3$: $$\begin{aligned} \omega&=(-288 x_2+2880 x_3) dx_0 dx_1+(288 x_1-96 x_3) dx_0 dx_2+\\ &\hspace{.3cm}+(-288 x_0-1152 x_3) dx_1 dx_2+(-2880 x_1+96 x_2) dx_0 dx_3+\\ &\hspace{.3cm}+(2880 x_0+1152 x_2) dx_1 dx_3+(-96 x_0-1152 x_1) dx_2 dx_3 \end{aligned}$$ Suche a differential form it is locally decomposable and locally integrable and has singular locus of codimension 3. The ideal of persistent singularities has also codimension 3. We did the computations using the software [DiffAlg]{}, see [@diffalg]. We can consider an extension of Remark \[[domega]{}\] for $\omega\in\Omega^q_X$. Then: \[defKK\] The subscheme of *Kupka singularities* of $\omega$ is the one defined by the ideal sheaf ${\mathcal{K}}(\omega):= {\mathrm{ann}}(\{d\omega\})\in \Omega^{q+1}_X\otimes{\mathcal{O}}_{{\mathrm{Sing}(\omega)}}\otimes {\mathcal{L}}^{-1}$. We will denote it just as ${\mathcal{K}}$ if no confusion arises. We would like to notice that both definitions above coincide to the ones given in the codimension 1 case, as the reader can see by comparing them to Definition \[II2\] and Remark \[remarkI\] and to Defintion \[KK2\], respectively. \[pluckerFiltration\] Given a short exact sequence of modules $$0\to M\to P\to N \to 0,$$ there is a filtration in $\bigwedge^q P$. $$\bigwedge^q P=F^0 \supseteq F^1\supseteq \dots \supseteq F^{q+1}=(0),$$ such that $$F^i/F^{i+1}\cong \bigwedge^{q-i}N\otimes \bigwedge^{i}M.$$ The result follows from defining $F^i\subseteq \bigwedge^q P$ to be the submodule generated by the elements of the form $(m_1\wedge\dots\wedge m_i\wedge a_{i+1}\wedge\dots\wedge a_q)$ where $m_j\in M$. \[JJcIIcKK2\] Given an integrable morphism ${\mathcal{L}}\xrightarrow{\omega}\Omega^q_X$ we have the inclusions ${\mathscr{J}}(\omega)\subseteq{\mathscr{I}}(\omega)$ and ${\mathscr{J}}(\omega) \subseteq {\mathcal{K}}(\omega)$. If moreover $\omega$ is locally decomposable (*i.e.*: if ${\mathcal{E}}$ is locally free) then we have ${\mathscr{J}}(\omega)\subseteq{\mathscr{I}}(\omega)\subseteq {\mathcal{K}}(\omega)$. To ease the notation let us set ${\mathscr{J}}={\mathscr{J}}(\omega)$, and likewise with ${\mathscr{I}}$ and ${\mathcal{K}}$. Let $h$ be a local section of ${\mathscr{J}}$, and by abuse of notation we will call $\omega$ a local generator of the image of the morphism $\omega:{\mathcal{L}}\to\Omega^q_X$, then by definition of ${\mathscr{J}}$ there is a local $q-1$-vector $v\in \bigwedge^{q}T_X$ such that $h=i_v\omega$. Then taking the filtration $\Omega^2_X=F^0\supseteq F^1\supseteq F^2\supseteq F^3=0$ of lemma \[pluckerFiltration\], we can say that $h$ is in ${\mathscr{I}}$ if and only if for every local section $\varpi \in {\mathcal{E}}$ we have $d\varpi\in F^1$. To establish this we recall that for every local section $\varpi$ of ${\mathcal{E}}$ the equation $d\varpi\wedge\omega=0$ holds. Then contracting with $v$ we get $$\begin{aligned} 0= i_v(d\varpi\wedge\omega)=\\ =d\varpi\wedge i_v\omega + \sum_{\substack{a_j\in T_X,\ b_j\in \bigwedge^{q-1}T_X\\ a_1\wedge b_1+\dots +a_r\wedge b_r=v}} i_a d\varpi\wedge i_b \omega+ \sum_{\substack{c_j\in \bigwedge^2 T_X,\ d_j\in \bigwedge^{q-2}T_X\\ c_1\wedge d_1+\dots +c_r\wedge d_s=v}} i_c d\varpi \wedge i_d\omega. \end{aligned}$$ By definition of ${\mathcal{E}}$ we have that $i_b \omega$ is a local section of ${\mathcal{E}}$, so every summand of the form $i_a d\varpi \wedge i_b \omega$ is in $\Omega^1_X\otimes{\mathcal{E}}$. Hence, to see that $hd\omega\in F^1$ it suffices to show that $i_d\omega$ is in $F^1$ for every $d\in \bigwedge^{q-2}T_X$. To see this we can calculate the class of $i_d\omega$ in $\Omega^2_X/F^1=F^0/F^1\cong \bigwedge^2(\Omega^1_X/{\mathcal{E}})$. The dual sheaf $(\Omega^1_X/{\mathcal{E}})^\vee \subseteq T_X$ is the distribution defined by $\omega$, that is, is the sheaf of vector fields $w$ such that $i_w\omega=0$. Then, when we evaluate $i_d \omega$ in a section $w_1\wedge w_2\in \bigwedge^2(\Omega^1_X/{\mathcal{E}})^\vee$ we get $0$. As $\bigwedge^2(\Omega^1_X/{\mathcal{E}})$ is torsion-free then the class of $i_d \omega$ in $\Omega^2_X/F^1$ is zero, then $i_d\omega\in F^1$, which means $h d\omega$ is in $F^1$ as we wanted to show. The second assertion is clear by definition, as ${\mathcal{K}}$ is the annihilator of a section whose support is contained in ${\mathrm{Sing}(\omega)}$. Now suppose ${\mathcal{E}}$ is locally free. So we can take local generators $\varpi_1,\dots,\varpi_q$ of ${\mathcal{E}}$, this sections verify that $\omega=\varpi_1\wedge\dots\wedge\varpi_q$. Then for every section $h$ of ${\mathscr{I}}$ there are local $1$-forms $\alpha_{ij}$ such that $$hd\varpi_i=\sum_{j} \alpha_{ij}\wedge\varpi_j.$$ Therefore we have $$\begin{aligned} hd\omega&=d(\varpi_1\wedge\dots\wedge\varpi_q)=\sum_i (-1)^i \varpi_1\wedge\dots\wedge d\varpi_i \wedge \varpi_{i+1}\wedge\dots\wedge\varpi_q=\\ &=\sum_{i} (-1)^i \varpi_1\wedge\dots\wedge (\sum_j \alpha_{ij})\wedge\varpi_j \wedge \varpi_{i+1}\wedge\dots\wedge\varpi_q=\\ &=\sum_i \alpha_{ii}\wedge \varpi_1\wedge\dots\wedge\varpi_q=(\sum_i \alpha_{ii})\wedge\omega. \end{aligned}$$ In particular $hd\omega$ vanishes in ${\mathrm{Sing}(\omega)}$ so $h$ is in ${\mathcal{K}}$. Let $\omega\in \Omega^2_{\mathbb{A}^3}$ be like in Example \[exampleCodimension2inA3\] so we write $$\omega= f_3 dx_1\wedge dx_2+f_2 dx_1\wedge dx_3+f_1 dx_2\wedge dx_3.$$ So we have $$d\omega = \left(\frac{\partial f_3 }{\partial x_3}- \frac{\partial f_2 }{\partial x_2}+ \frac{\partial f_1 }{\partial x_1}\right) dx_1\wedge dx_2\wedge dx_3.$$ For a general choice of the $f_i$’s the restriction $d\omega|_{{\mathrm{Sing}(\omega)}}$ does not vanish, so ${\mathscr{J}}={\mathcal{K}}$. However, by setting for instance $f_3=f_3(x_1,x_2)$, $f_2=f_2(x_1,x_3)$ and $f_1=f_1(x_2,x_3)$, we get a form $\omega$ such that $d\omega=0$. With this choice of $\omega$ we have ${\mathcal{K}}={\mathcal{O}}_X$. When computing the ideal ${\mathscr{I}}$ for this case we need to check that $h d\omega_i= \alpha_{i1}\wedge\omega_1+\alpha_{i2}\wedge\omega_2+\alpha_{i3}\wedge\omega_3$ for $i=1,2,3$, where the $\omega_i$’s are the generators of ${\mathcal{E}}$ of Example \[exampleCodimension2inA3\] and $h\in {\mathcal{O}}_X$. Further specializing our choice of $\omega$ we can take $f_3=x_2$ and $f_2=-x_1$, in order to get $d\omega_1=dx_1\wedge dx_2 + dx_1\wedge dx_3$, so clearly $1\notin {\mathscr{I}}(\omega)$. So we see that there are cases where ${\mathcal{K}}={\mathcal{O}}_X$ and $1\notin {\mathscr{I}}$. This is in stark contrast to the situation in codimension $1$ where, from Theorem \[teodivision\], follows that the condition ${\mathscr{J}}=\sqrt{{\mathscr{J}}}$ implies $\sqrt{{\mathscr{I}}}=\sqrt{{\mathcal{K}}}$. \[teoConnectionE\] Let $X$ be a projective variety and ${\mathcal{L}}\xrightarrow{\omega}\Omega_X^q$ be an integrable $q$-form and ${\mathcal{E}}$ be the associated subsheaf of $1$-forms ${\mathcal{E}}\subseteq \Omega^1_X$. Let $\mathrm{Sym}^2({\mathcal{E}})$ denote the symmetric power of ${\mathcal{E}}$ and suppose ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=0$. If ${\mathscr{I}}(\omega)={\mathcal{O}}_X$ then ${\mathcal{E}}$ admits a holomorphic connection, in particular is locally free (in other words the foliation is locally decomposable) and every Chern class of ${\mathcal{E}}$ vanishes. In order to prove the vanishing of the Chern classes of ${\mathcal{E}}$ we are going to use Atiyah’s classical result [@atiyah Theorem 4, p. 192] which states that if a holomorphic vector bundle on a compact Kähler manifold admits a holomorphic connection, then its Chern classes are all zero. We will then produce a holomorphic connection for ${\mathcal{E}}$ in this case. The condition ${\mathscr{I}}(\omega)={\mathcal{O}}_X$ implies that for every local section $\varpi$ of ${\mathcal{E}}$ we have $d\varpi=\sum_i\alpha_i\wedge\omega_i$ for some local $1$-forms $\alpha_i$ and $\omega_i \in {\mathcal{E}}$. In other words, de Rham differential applied to sections of ${\mathcal{E}}$ give us a map $d:{\mathcal{E}}\to F^1\subseteq \Omega^2_X$ such that $d(f\varpi)=df\wedge\varpi+fd\varpi$, that is a differential operator of order $1$ between ${\mathcal{E}}$ and $F^1$. We will call $\mathrm{Diff}^{\leq 1}(A, B)$ the set of differential operators of order $\leq 1$ between two sheaves $A$ and $B$. Let us denote with $\mathcal{PE}$ the sheaf of principal parts of ${\mathcal{E}}$ of order $1$, this sheaf is defined by the universal property ${\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},M)=\mathrm{Diff}^{\leq 1}({\mathcal{E}}, M)$ for every coherent sheaf $M$. So the de Rham differential defines a coherent sheaves morphism $[\nabla]:\mathcal{PE}\to F^1$. To see if we can lift $[\nabla]$ to a morphism $\nabla:\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$ defining a connection, we observe that the short exact sequence $0\to\mathrm{Sym}^2({\mathcal{E}})\to\Omega^1_X\otimes {\mathcal{E}}\to F^1\to 0$ gives an exact sequence of modules $$\begin{aligned} 0 &\to {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))\to {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},\Omega^1_X\otimes{\mathcal{E}})\to {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1 )\xrightarrow{\delta} \\ &\xrightarrow{\delta} {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))\to {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\Omega^1_X\otimes{\mathcal{E}})\to\cdots \end{aligned}$$ So $[\nabla]$ lifts to a morphism $\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$ if and only if is in the kernel of $ {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1)\xrightarrow{\delta} {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))$. In order to compute ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))$ recall the short exact sequence of sheaves $$0\to \Omega^1_X\otimes{\mathcal{E}}\to \mathcal{PE} \to {\mathcal{E}}\to 0,$$ which give rise to an exact sequence $$\begin{aligned} \cdots\to {\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))&\to {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))\to \\ &\to{\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\Omega^1_X\otimes{\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))\to \cdots\end{aligned}$$ Recall that the group ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))$ can be regarded as the group of isomorphism classes of extensions of $\mathcal{PE}$ by $\mathrm{Sym}^2({\mathcal{E}})$. Viewed like this, the morphism $\delta: {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1 )\to {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))$ evaluated at an element $a\in {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1)$ returns the isomorphism class of the extension $0 \to \mathrm{Sym}^2({\mathcal{E}})\to A \to \mathcal{PE}\to 0$ where $A$ is the pull-back of the diagram $$\xymatrix{ A \ar[d] \ar[r] & \mathcal{PE}\ar[d]^a \\ \Omega^1_X\otimes{\mathcal{E}}\ar[r] & F^1 }$$ In particular the composition $${\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1 )\xrightarrow{\delta} {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))\to {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\Omega^1_X\otimes{\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))$$ evaluated at the element $[\nabla] \in {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},F^1 )$ returns the isomorphism class of the extension $0\to \mathrm{Sym}^2({\mathcal{E}}) \to B \to \Omega^1_X\otimes{\mathcal{E}}$ where $B$ is the pull-back of the diagram $$\xymatrix{ B \ar[d] \ar[r] & \Omega^1_X\otimes{\mathcal{E}}\ar[d]^{[\nabla]\circ i} \\ \Omega^1_X\otimes{\mathcal{E}}\ar[r] & F^1 }$$ where $i:\Omega^1_X\otimes{\mathcal{E}}\to\mathcal{PE}$ is the canonical immersion. Now to compute $[\nabla]\circ i:\Omega^1_X\otimes{\mathcal{E}}\to F^1$ recall that $[\nabla]$ is defined by being the unique ${\mathcal{O}}_X$-linear morphism making the following diagram commute, $$\xymatrix{ {\mathcal{E}}\ar[d]^{d^{(1)}} \ar[r]^d & F^1 \\ \mathcal{PE} \ar[ru]_{[\nabla]} & }$$ where $d^{(1)}:{\mathcal{E}}\to \mathcal{PE}$ is the universal differential operator of order $1$. Then, as follows from the formulas of [@atiyah p. 193] explicitly describing the ${\mathcal{O}}_X$-module structure of $\mathcal{PE}$, given local sections $f$ of ${\mathcal{O}}_X$ and $\varpi$ of ${\mathcal{E}}$ we have $$[\nabla](df\otimes \varpi) = d(f\varpi)-fd(\varpi).$$ So, $[\nabla]\circ i$ is just the exterior product of forms, hence the sequence $0\to \mathrm{Sym}^2({\mathcal{E}}) \to B \to \Omega^1_X\otimes{\mathcal{E}}\to 0$ splits, then the class of $\delta([\nabla])$ in ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\Omega^1_X\otimes{\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))$ is zero. Therefore $\delta([\nabla])$ is in the image of ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))\to {\mathrm{Ext}}^1_{{\mathcal{O}}_X}(\mathcal{PE},\mathrm{Sym}^2({\mathcal{E}}))$. Hence if ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=(0)$ then $\delta([\nabla])=0$. So, if ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=(0)$, then there is a morphism $\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$ lifting $[\nabla]$. What we need to prove to conclude is that among the morphisms $\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$, there is one $\nabla:\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$ such that $\nabla|_{\Omega^1_X\otimes {\mathcal{E}}}$ is the identity. To do this we consider the short exact sequence $0\to \Omega^1_X\otimes{\mathcal{E}}\to \mathcal{PE}\to {\mathcal{E}}\to 0$ and the exact sequence of ${\mathrm{Hom}}$ groups $$\begin{aligned} {\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{E}},\Omega_X\otimes{\mathcal{E}})\to{\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{PE},\Omega_X\otimes{\mathcal{E}})&\to{\mathrm{Hom}}_{{\mathcal{O}}_X}(\Omega_X\otimes{\mathcal{E}},\Omega_X\otimes{\mathcal{E}})\xrightarrow{\delta} \\ &\xrightarrow{\delta}{\mathrm{Ext}}^1({\mathcal{E}},\Omega_X\otimes{\mathcal{E}})\to\cdots \end{aligned}$$ The identity is an element $\mathrm{id}\in {\mathrm{Hom}}_{{\mathcal{O}}_X}(\Omega_X\otimes{\mathcal{E}},\Omega_X\otimes{\mathcal{E}})$ and we want to show that it is the restriction of some morphism $\nabla:\mathcal{PE}\to \Omega_X\otimes{\mathcal{E}}$, which is equivalent to the condition $\delta(\mathrm{id})=0$. We already know that there is a morphism $\tilde{\nabla}:\mathcal{PE}\to \Omega^1_X\otimes{\mathcal{E}}$ lifting $[\nabla]$, so the restriction of $\tilde{\nabla}$ to $\Omega^1_X\otimes{\mathcal{E}}$, which we also denote $\tilde{\nabla}$, makes the following diagram commute. $$\xymatrix{ 0 \ar[r] & \mathrm{Sym}^2({\mathcal{E}}) \ar[d] \ar[r] & \Omega^1_X\otimes{\mathcal{E}}\ar[d]^{\tilde{\nabla}} \ar[r] & F^1 \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathrm{Sym}^2({\mathcal{E}}) \ar[r] & \Omega^1_X\otimes{\mathcal{E}}\ar[r] & F^1 \ar[r] & 0 }$$ Being a restriction we have $\delta(\tilde{\nabla})=0$. If we can prove that $\delta(\tilde{\nabla}-\mathrm{id})=0$ then $\delta(\mathrm{id})=0$ and we are set. The image of $\tilde{\nabla}-\mathrm{id}$ is in $\mathrm{Sym}^2({\mathcal{E}})$ so the element $\delta(\tilde{\nabla}-\mathrm{id})\in {\mathrm{Ext}}^1({\mathcal{E}},\Omega_X\otimes{\mathcal{E}})$ is the class of the extension in the last row of the diagram: $$\xymatrix{ 0 \ar[r] & \Omega^1_X\otimes{\mathcal{E}}\ar[d]^{\tilde{\nabla}-\mathrm{id}} \ar[r] & \mathcal{PE} \ar[d] \ar[r] & {\mathcal{E}}\ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathrm{Sym}^2({\mathcal{E}}) \ar[d] \ar[r] & I \ar[d] \ar[r] & {\mathcal{E}}\ar[d] \ar[r] & 0 \\ 0 \ar[r] & \Omega^1_X\otimes{\mathcal{E}}\ar[r] & II \ar[r] & {\mathcal{E}}\ar[r] & 0 }$$ Where $I$ and $II$ are push-forwards. If ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=(0)$ then $[I]=0\in {\mathrm{Ext}}^1({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))$ and so $\delta(\tilde{\nabla}-\mathrm{id})=[II]=0\in {\mathrm{Ext}}^1({\mathcal{E}},\Omega^1_X\otimes{\mathcal{E}})$. So the condition ${\mathrm{Ext}}^1_{{\mathcal{O}}_X}({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=(0)$ implies that there is a connection on ${\mathcal{E}}$. Let $X$ be a smooth projective variety over ${\mathbb{C}}$ such that every line bundle ${\mathcal{L}}$ verifies $H^1(X,{\mathcal{L}})=0$ and such that $Pic(X)$ is torsion-free (*e.g.*: $X$ smooth complete intersection). And let ${\mathcal{L}}\xrightarrow{\omega}\Omega^q_X$ be a foliation such that ${\mathcal{E}}\cong\bigoplus_i {\mathcal{L}}_i$ for some line bundles ${\mathcal{L}}_i$. Then ${\mathcal{L}}$ has persistent singularities. As ${\mathcal{E}}$ is a direct sum of line bundles the group ${\mathrm{Ext}}^1_X({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))$ decomposes as $${\mathrm{Ext}}^1_X({\mathcal{E}},\mathrm{Sym}^2({\mathcal{E}}))=\bigoplus_{i}\bigoplus_{j\le k} H^1(X,{\mathcal{L}}_i^{-1}\otimes {\mathcal{L}}_j\otimes{\mathcal{L}}_k)=0.$$ So, as $\omega$ is decomposable, if it does not posses persistent singularities then ${\mathscr{I}}={\mathcal{O}}_X$ so by Theorem \[teoConnectionE\] there is a connection on ${\mathcal{E}}$. This implies that the Chern classes of the line bundle ${\mathcal{L}}_1,\dots,{\mathcal{L}}_q$ are all zero. Then ${\mathcal{L}}_i\cong{\mathcal{O}}_X$ for $i=1,\dots,q$, giving global sections of $\Omega^1_X$, contradicting the fact that $H^0(X,\Omega^1_X)=H^1(X,{\mathcal{O}}_X)=0$. Applications to toric varieties {#section_toric} =============================== In this section we try to generalize some of our results to foliations on varieties that may have singularities. We focus our attention in normal toric varieties and make use of the Cox ring of such a variety to generalize what is known in the case of projective space. Let $X_{\Sigma}$ be a toric variety with no torus factors and let $S$ be its Cox ring graded by $\text{Cl}(X_{\Sigma})$, $$S={\mathbb{C}}[x_\rho\,\colon\,\rho\in\Sigma(1)].$$ The graded pieces $S_\beta$ for $\beta\in \text{Cl}(X_{\Sigma})$ consist of toric-homogeneous polynomials. Given $\phi\in{\mathrm{Hom}}_{{\mathbb{Z}}}(\text{Cl}(X_{\Sigma}),{\mathbb{Z}})$, let us define the vector field $R_{\phi}$, $$R_\phi:=\sum_{\rho\in\Sigma(1)}\phi(\rho)x_\rho\frac{\partial}{\partial x_\rho}.$$ Denote $Lie_\phi$ to the Lie derivative with respect to $R_\phi$, $$Lie_\phi=di_{R_{\phi}}+i_{R_{\phi}}d.$$ Let $X_{\Sigma}$ be a toric variety with no torus factors and let $S$ be its Cox ring. A $q$-form $\omega=\sum a_I dx_I$ with coefficients in $S$ has (homogeneous) *degree* $\beta\in\text{Cl}(X_{\Sigma})$ if $a_Ix_I\in S_{\beta}$ for all multi-index $|I|=q$. Clearly if $\omega'$ is a $q'$-form of degree $\beta'$, then $\omega\wedge\omega'$ has degree $\beta+\beta'$. Furthermore, the degree of $d\omega$ and of $i_{R_{\phi}}(\omega)$ is also $\beta$. Let $X_{\Sigma}$ be a toric variety with no torus factors and free class group $\text{Cl}(X_{\Sigma})$. The following are equivalent for a $q$-form $\omega$, - $\omega$ has degree $\beta$. - $Lie_\phi(\omega)=\phi(\beta)\omega$ for all $\phi\in{\mathrm{Hom}}_{{\mathbb{Z}}}(\text{Cl}(X_{\Sigma}),{\mathbb{Z}})$ Given that $Lie_\phi$ is a derivation and commutes with $d$, the result follows from the $0$-form case, see [@COX Exercise 8.1.8, p. 357], $$Lie_{\phi}(f)=\phi(\beta)f,\quad \forall f\in S_\beta.$$ Let $X_{\Sigma}$ be a toric variety with no torus factors and let $S$ be its Cox ring. A $q$-form $\omega$ with coefficients in $S$ is said to *descend to $X_{\Sigma}$* if $i_{R_\phi}\omega=0$ for all $\phi\in{\mathrm{Hom}}_{{\mathbb{Z}}}(\text{Cl}(X_{\Sigma}),{\mathbb{Z}})$. Let $X_{\Sigma}$ be a simplicial toric variety with no torus factors and let $\omega$ be a $q$-form of degree $\beta$. Let $\widehat{\Omega}^q=(\Omega^q)^{\vee\vee}$ be the sheaf of Zariski $q$-forms, [@COX Equation 8.0.5, p. 347]. The following are equivalent, - $\omega$ descends to $X_{\Sigma}$. - $\omega\in H^0(X_{\Sigma},\widehat{\Omega}^q(\beta))$. If $\omega$ is a $1$-form, the result follows from [@COX Corollary 8.1.5, p. 354] and [@COX Theorem 8.1.6, p. 355]. The general case follows from [@COX Corollary 8.2.17, p. 368] and [@COX Theorem 8.2.16, p. 367]. \[prop1notinI\] Let $X_\Sigma$ be a projective simplicial toric variety (hence with no torus factors) and let $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$. If $d\omega=\omega\wedge\eta$ for some $1$-form $\eta$, then $\beta$ is a torsion element and $d\omega=0$. Let us apply the contraction with respect to the radial field $R_\phi$, where $\phi:\text{Cl}(X_\Sigma)\to{\mathbb{Z}}$ is some linear function, $$i_{R_\phi}(d\omega)=i_{R_\phi}(\omega\wedge\eta)\Longrightarrow \phi(\beta)\omega+\omega i_{R_\phi}(\eta)=0\Longrightarrow \phi(\beta)+i_{R_\phi}(\eta)=0.$$ Given that the degree of $d\omega$ is equal to the degree of $\omega$, it follows that the degree of $\eta$ is $0\in\text{Cl}(X_\Sigma)$, $$\eta=\sum_{\rho\in\Sigma(1)} a_{\rho}dx_{\rho}\Longrightarrow -\phi(\beta)=\sum_{\rho\in\Sigma(1)} \phi(\rho)a_{\rho}x_{\rho}\Longrightarrow \phi(\beta)\in\langle x_\rho\,\colon\,\rho\in\Sigma(1)\rangle.$$ Since $X_{\Sigma}$ has no torus factors, it follows $\phi(\beta)=0$ for all $\phi$ and this implies $\beta$ a torsion element and $\eta\in H^0(X_\Sigma,\widehat{\Omega}^1)$, but from [@materov Theorem 2.14], $H^0(X_\Sigma,\widehat{\Omega}^1)=0$. Hence, $\eta=0$ and then $d\omega=0$. It is a priori unclear what a foliation is when the ambient space is singular. At the very least, we should ask a foliation on a singular space $X$ to restrict to a foliation on its maximal non-singular subscheme $U$. When $X$ is a normal variety, which is the case we will consider here, $U$ is an open subscheme whose complement $X\setminus U$ is a closed subset of codimension at least $2$. So whatever a codimension $q$ foliation on such an $X$ is, it should determine a morphism $L\xrightarrow{\omega}\Omega^q_{U}$ for some line bundle on $U$ verifying the Plücker relations and integrability. As $X$ is normal and the complement of $U$ is of codimension at least $2$ then restriction of line bundles defines an isomorphism $\mathrm{Pic}(X)\simeq\mathrm{Pic}(U)$, so there is a line bundle ${\mathcal{L}}$ on $X$ such that ${\mathcal{L}}|_U\simeq L$. So the foliation on $U$ defines a section (which we also call $\omega$ by abuse of notation) $\omega \in \Gamma(U, \Omega^q_X\otimes {\mathcal{L}}^{-1}|_U)$. Would $\Omega^q_X$ be a reflexive sheaf, a section on $U$ would extend in a unique way to a section on the whole $X$, this needs not to be the case if $X$ has singularities. To remedy this we can take the reflexive hull of $\Omega^q_X$, this sheaf is known as the *sheaf of Zariski $q$-forms on $X$*, we denote it by $\widehat{\Omega}^q_X$ and is defined as the double dual $\widehat{\Omega}^q_X:=({\Omega^q_X})^{\vee\vee}$. As $U$ is by definition a non-singular open subcheme we have identifications $\Omega^q_U=\Omega^q_X|_U=\widehat{\Omega}^q_X|_U$. Therefore a foliation on $X$ defines a section $\omega\in\Gamma(U, \widehat{\Omega}^q_X\otimes{\mathcal{L}}^{-1})$, as $X$ is normal, $\widehat{\Omega}^q_X\otimes{\mathcal{L}}^{-1}$ is a reflexive sheaf on $X$ and $\omega$ is a section defined outside a codimension $2$ subset, there is a unique way in which $\omega$ extends to a global section of $\widehat{\Omega}^q_X\otimes{\mathcal{L}}^{-1}$. As $\omega|_U$ verify the Plücker relations and the integrability condition, so does its global extension. We see in this way that considering twisted Zariski forms which are integrable is general enough and, as we will see bellow, is also manageable enough. Let $X_\Sigma$ be a toric variety, let $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$. We say that $\omega$ defines a *codimension $q$ foliation* if ${\mathrm{codim}}({\mathrm{Sing}(\omega)})\ge 2$ and for all $\eta\in E$, $$\begin{array}{rcl} i_\Xi(\omega)\wedge\omega=0,&\forall\,\Xi\in\bigwedge^{q-1}T_S&\text{(Pl\"ucker relations),}\\ i_\Xi(\omega)\wedge d\omega=,0&\forall\,\Xi\in\bigwedge^{q-1}T_S&\text{(locally integrable).} \end{array}$$ Notice that if the first condition is true, then the second condition is equivalent to $d(i_\Xi(\omega))\wedge\omega=0$. The ideal of *singularities* of $\omega$ is the ideal in the Cox ring $S$ generated by the coefficients of $\omega$, $$J(\omega)=\{i_\Xi(\omega)\,\colon\, \Xi\in\bigwedge^{q}T_S\}\subseteq S.$$ Analogously, define $J(d\omega)$ as the ideal of coefficients of $d\omega\in\widehat{\Omega}^{q+1}_S$. Notice that if $\beta$ is not a torsion element, there exists $\phi:\text{Cl}(X_\Sigma)\to{\mathbb{Z}}$ such that $\phi(\beta)\neq 0$ and then contracting with $R_{\phi}$, the equation $i_{R_{\phi}}d\omega=\phi(\beta)\omega$ implies $J(\omega)\subseteq J(d\omega)$. Let $$E:=\{\eta\,\colon\,\eta\wedge\omega=0\} \subseteq\widehat\Omega^1_S.$$ The ideal of *persistent singularities* is $$I(\omega):= (E\wedge\widehat{\Omega}^1_S\,:dE)= \bigcap_{\eta\in E}(E\wedge\widehat{\Omega}^1_S\,:d\eta)$$ and the Kupka ideal is defined as the colon ideal, $$K(\omega):=(J\cdot \widehat{\Omega}^{q+1}_S:d\omega). $$ The *subscheme of persistent singularities* ${\mathpzc{Per}(\omega)}$ is defined by the homogeneous ideal $I(\omega)\subseteq S$, the *subscheme of singularities* ${\mathrm{Sing}(\omega)}$ by $J(\omega)$ and the *Kupka subscheme* ${\mathpzc{Kup}(\omega)}$ by $K(\omega)$. In this simple example we show that the ideals $J(\omega)$ and $I(\omega)$ might be nontrivial ideals of the Cox ring although they may define empty subschemes. Let $\omega$ be the following $1$-form $$\omega=-x_3dx_2+x_2dx_3$$ and let $R_1$ and $R_2$ be the vector fields, $$R_1= x_0\partial_{x_0}+x_1\partial_{x_1} ,\quad R_2= x_2\partial_{x_2}+x_3\partial_{x_3}.$$ Notice that, $$d\omega= 2dx_2\wedge dx_3 ,\quad i_{R_1}d\omega= 0 =0\omega ,\quad i_{R_2}d\omega= -2x_3dx_2+2x_2dx_3 =2\omega.$$ So clearly $\omega\in H^0({\mathbb{P}}^1\times{\mathbb{P}}^1,\Omega^1(0,2))$. Indeed, $\omega$ is the pull-back $\pi_2^*\eta$, where $\pi_2:{\mathbb{P}}^1\times{\mathbb{P}}^1\to{\mathbb{P}}^1$ is the projection on the second factor and $\eta\in H^0({\mathbb{P}}^1,\Omega^1_{{\mathbb{P}}^1}(2))$ is the unique (up to multiplication by a constant) global $1$-form of degree $2$ in ${\mathbb{P}}^1$. Then the foliation defined by $\omega$ is non-singular, its leaves being the subvarieties ${\mathbb{P}}^1\times\{p\}$ with $p\in{\mathbb{P}}^1$. Also note that in this case $I(\omega)=J(\omega)=(x_3,x_2)$ is a non-trivial ideal of the Cox ring although it contains the irrelevant ideal of ${\mathbb{P}}^1\times{\mathbb{P}}^1$ (so they define the empty subscheme in $X_{\Sigma}$) in accordance to the foliation defined by $\omega$ being non-singular. Let $X_\Sigma$ be a toric variety and let $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$. The sheaves associated to $J(\omega)$, $K(\omega)$ and $I(\omega)$ are $\mathscr{J}(\omega)$, $\mathscr{K}(\omega)$ and $\mathscr{I}(\omega)$ respectively. First of all, if $\omega=\sum a_Idx_I$, then $J(\omega)$ is the ideal generated by $\{a_I\}$. Hence, $\Gamma_{*}(\mathscr{J}(\omega))=J(\omega)$. Now, let ${\mathfrak{p}}\in X_\Sigma$ and consider the stalk $K(\omega)_{{\mathfrak{p}}}$. Then, $K(\omega)_{{\mathfrak{p}}}=(J_{\mathfrak{p}}\cdot \widehat{\Omega}^{q+1}_{\mathfrak{p}}:d\omega)= \{f\in S_{\mathfrak{p}}\,\colon\,fd\omega\in J_{\mathfrak{p}}\cdot \widehat{\Omega}^{q+1}_{\mathfrak{p}}\}$ and this is equal to the annihilator of $d\omega$ in $\widehat{\Omega}^{q+1}_{\mathfrak{p}}/(J_{\mathfrak{p}}\cdot \widehat{\Omega}^{q+1}_{\mathfrak{p}})\cong \widehat{\Omega}^{q+1}_{\mathfrak{p}}\otimes (S/J)_{\mathfrak{p}}$. Hence, $\Gamma_{*}(\mathscr{K}(\omega))=K(\omega)$. Finally, $I(\omega)_{\mathfrak{p}}=\{f\in S_p\,\colon\,fd\varpi=\sum \alpha_j\wedge\varpi_j \text{ where } \varpi,\varpi_j\in E_{\mathfrak{p}}\}=\mathscr{I}(\omega)_{\mathfrak{p}}$. Then, $\Gamma_{*}(\mathscr{I}(\omega))=I(\omega)$. Let $X_\Sigma$ be a projective simplicial toric variety and let $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$ be a codimension $q$ foliation such that $\omega$ is locally decomposable on $X_\Sigma$ (*e.g.* $q=1$). If $H^1(X_\Sigma,\mathcal{E})=0$ and $\beta$ not a torsion element, then ${\mathpzc{Per}(\omega)}\neq\emptyset$, where $\mathcal{E}=\{\eta\in\widehat\Omega^1\,\colon\,\omega\wedge\eta=0\}$. Let $\{U_r\}$ be an open cover trivializing ${\mathcal{O}}_X(\beta)$ and let $(g_{rs},U_{rs})$ be a 1-cocycle in $\check{H}^1(X_\Sigma,{\mathcal{O}}_{X_\Sigma}^*)$ such that $\omega_{r}=g_{rs}\omega_s$ in $\Gamma(U_{rs},\widehat{\Omega}^q)$, where $U_{rs}:=U_r\cap U_s$. Assume ${\mathpzc{Per}(\omega)}=\emptyset$. On $U_r$, the $q$-form $\omega_r$ is decomposable and there exists $\eta_r\in\Gamma(U_r,\widehat\Omega^1)$ such that $d\omega_r=\eta_r\wedge\omega_r$. Indeed, if $\omega_r=\varpi_1\wedge\ldots\wedge\varpi_q$, then $d\varpi_i=\sum\theta_{ij}\wedge\varpi_j$ because ${\mathpzc{Per}(\omega)}=\emptyset$. Then $d\omega_r=(\sum\theta_{ii})\wedge\omega_r$. Now, on $U_{rs}$ we have, $$\eta_r\wedge\omega_r=d\omega_r=d(g_{rs}\omega_s)=dg_{rs}\wedge\omega_s+g_{rs}d\omega_s= dg_{rs}\wedge\omega_s+g_{rs}\eta_s\wedge\omega_s\Longrightarrow$$ $$(\eta_r-dg_{rs}/g_{rs}-\eta_s)\wedge\omega_s=0\Longrightarrow \overline\eta_r-\overline\eta_s=\overline{dg_{rs}/g_{rs}}\in \Gamma(U_{rs},\widehat\Omega^1/\mathcal{E}).$$ Hence, the 1-cocycle $(dg_{rs}/g_{rs},U_{rs})$ is in the kernel of the map $$\check{H}^1(X_{\Sigma},\widehat\Omega^1)\to \check{H}^1(X_{\Sigma},\widehat\Omega^1/\mathcal{E})$$ which is $0=\check{H}^1(X_\Sigma,\mathcal{E})$. Then, $0=[(dg_{\alpha\beta}/g_{\alpha\beta},U_{\alpha\beta})] \in \check{H}^1(X_{\Sigma},\widehat\Omega^1)$. Now, let us recall several results from complete simplicial toric varieties. By [@COX proof of Theorem 12.3.11, p. 588] $H^2(X_\Sigma,\mathbb{C})\cong H^1(X_{\Sigma},\widehat\Omega^1)$, by [@COX Theorem 12.3.2, p. 577] $H^2(X_\Sigma,\mathbb{C})\cong \text{Pic}(X_\Sigma)_{\mathbb{C}}$ and by [@COX Proposition 4.2.7, p. 180] $\text{Pic}(X_\Sigma)_{\mathbb{C}}\cong\text{Cl}(X_\Sigma)_{\mathbb{C}}$. Then, $$H^1(X_{\Sigma},\widehat\Omega^1)\cong\text{Cl}(X_\Sigma)_{\mathbb{C}}.$$ Hence, $[\beta]=0$ in $\text{Cl}(X_\Sigma)_{\mathbb{C}}$ and this implies $\beta$ is a torsion element. Let $X_\Sigma$ be a projective simplicial toric variety and let $\omega\in H^0(X_\Sigma,\widehat{\Omega}^q(\beta))$ be a codimension $q$ foliation such that $\widehat{\Omega}^1/{\mathcal{E}}$ is reflexive. If $H^1(X_\Sigma,\mathcal{E})=0$ and $\beta$ not a torsion element, then ${\mathpzc{Per}(\omega)}\neq\emptyset$. Let $\mathrm{Sing}(X_\Sigma)$ be the singular locus of $X_\Sigma$ and let $Z:=\text{Sing}(X_\Sigma)\cup{\mathrm{Sing}(\omega)}$ which has codimension $\ge 2$. Let $\{U_r\}$ be an open cover trivializing ${\mathcal{O}}_X(\beta)$ and let $(g_{rs},U_{rs})$ be a 1-cocycle in $\check{H}^1(X_\Sigma,{\mathcal{O}}_{X_\Sigma}^*)$ such that $\omega_{r}=g_{rs}\omega_s$ in $\Gamma(U_{rs},\widehat{\Omega}^q)$, where $U_{rs}:=U_r\cap U_s$. Assume ${\mathpzc{Per}(\omega)}=\emptyset$. Take an open cover $\{V_a\}$ of $U_r\setminus Z$ small enough such that the $q$-form $\omega_r$ is decomposable. Hence, as before, there exists $\zeta_a\in\Gamma(V_a,\widehat\Omega^1/{\mathcal{E}})$ such that $d\omega_r=\zeta_a\wedge\omega_r$ on $V_a$. Notice that $\zeta_a-\zeta_b=0$ in $\Gamma(V_a\cap V_b,\widehat\Omega^1/{\mathcal{E}})$. Then, $\zeta_r:=\{(\zeta_a,V_a)\}$ defines a section in $\check H^0(U_r\setminus Z,\widehat\Omega^1/{\mathcal{E}})$. Being $\widehat\Omega^1/{\mathcal{E}}$ reflexive, there exists a unique extension $\eta_r\in\check H^0(U_r,\widehat\Omega^1/{\mathcal{E}})$ and it satisfies $d\omega_r-\eta_r\wedge\omega_r\in \Gamma_Z(U_r,\widehat\Omega^{q+1})$. From the reflexivity of $\widehat\Omega^{q+1}$, we get $\Gamma_Z(U_r,\widehat\Omega^{q+1})=0$. Then, $d\omega_r=\eta_r\wedge\omega_r$ in $\Gamma(U_r,\widehat\Omega^{q+1})$. The result follows by repeating the arguments of the previous proof. Let $\omega$ be the following $1$-form $$\omega= x_2^2x_0dx_1-x_2^2x_1dx_0+x_0^2x_2dx_3-x_0^2x_3dx_2$$ and let $R_1$ and $R_2$ be the vector fields, $$R_1= x_0\partial_{x_0}+x_1\partial_{x_1} ,\quad R_2= x_2\partial_{x_2}+x_3\partial_{x_3}.$$ Clearly $\omega\in H^0({\mathbb{P}}^1\times{\mathbb{P}}^1,\Omega^1(2,2))$ and, as $H^1({\mathbb{P}}^1\times{\mathbb{P}}^1,{\mathcal{O}}(-2,-2))=0$ it follows from the previous corollary that $\omega$ must have persistent singularities. Indeed $$\begin{aligned} d\omega&= 2x_2^2dx_0\wedge dx_1+(2x_1x_2-2x_0x_3)dx_0\wedge dx_2\\ &-2x_0x_2dx_1\wedge dx_2+ 2x_0x_2dx_0\wedge dx_3+2x_0^2dx_2\wedge dx_3,\\ i_{R_1}d\omega&= -2x_1x_2^2dx_0+2x_0x_2^2dx_1-2x_0^2x_3dx_2+2x_0^2x_2dx_3 =2\omega,\\ i_{R_2}d\omega&= -2x_1x_2^2dx_0+2x_0x_2^2dx_1-2x_0^2x_3dx_2+2x_0^2x_2dx_3 =2\omega.\end{aligned}$$ And from this one can compute the ideal $I(\omega)$, it turns out to be $$I(\omega)= (x_0^2, x_3 x_0 + x_4 x_1, x_4 x_0, x_4^2),\quad \sqrt{I(\omega)}=(x_0,x_4).$$ This computations were done with [DiffAlg]{}, see [@diffalg]. The singular ideal in this case is $J(\omega)=(x_0^2, x_4^2)$, so ${\mathpzc{Per}(\omega)}$ and ${\mathrm{Sing}(\omega)}$ have equal reduced structure although their scheme structure is not the same. Here we provide an example were the subscheme ${\mathpzc{Per}(\omega)}$ is supported on a proper closed subset of ${\mathrm{Sing}(\omega)}$. This example was done with [DiffAlg]{}, see [@diffalg]. Let $\omega$ be the following $1$-form $$\omega = x_1x_2x_3dx_0+x_0x_2x_3dx_1+2x_0x_1x_3dx_2-2x_0x_1x_2dx_3$$ and let $R_1$ and $R_2$ be the vector fields, $$R_1= x_0\partial_{x_0}+x_1\partial_{x_1}+x_3\partial_{x_3} ,\quad R_2= x_2\partial_{x_2}+x_3\partial_{x_3}.$$ Then, it is easy to see that $i_{R_1}\omega=i_{R_2}\omega=0$, hence $\omega$ defines a $1$-form over the Hirzebruch surface $\mathcal{H}_1\cong \text{Bl}_{p}({\mathbb{P}}^2)$ of bi-degree $(2,3)$. From $$\begin{aligned} d\omega&= x_1x_3dx_0\wedge dx_2+ x_0x_3dx_1\wedge dx_2 -3x_1x_2dx_0\wedge dx_3\\ &-3x_0x_2dx_1\wedge dx_3-4x_0x_1dx_2\wedge dx_3,\\ i_{R_1}d\omega&= 3x_1x_2x_3dx_0+3x_0x_2x_3dx_1+6x_0x_1x_3dx_2-6x_0x_1x_2dx_3 =3\omega,\\ i_{R_2}d\omega&= 2x_1x_2x_3dx_0+2x_0x_2x_3dx_1+4x_0x_1x_3dx_2-4x_0x_1x_2dx_3 =2\omega,\end{aligned}$$ follows that $$I(\omega)=(x_2,x_1)\cap (x_1,x_0)\cap (x_2,x_3),\quad J(\omega)=(x_2,x_1)\cap (x_1,x_0)\cap (x_2,x_3)\cap (x_0, x_3).$$ [CACGLN04]{} M. F. Atiyah. 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Publ., River Edge, NJ, 1995. ----------------------------- -- César Massri$^*$ Ariel Molinuevo$^\dag$ Federico Quallbrunn$^\ddag$ ----------------------------- -- ---------------------------------------- -- $^*$[Departamento de Matemática]{} [Pabellón I]{} [Ciudad Universitaria]{} [CP C1428EGA]{} [Buenos Aires]{} [Argentina]{} $^\dag$Instituto de Matemática Universidade Federal do Rio de Janeiro Caixa Postal 68530 CEP. 21945-970 Rio de Janeiro - RJ BRASIL ---------------------------------------- -- ------------------------------------- -- $^\ddag$ Departamento de Matemática Universidad CAECE Av. de Mayo 866 CP C1084AAQ [Buenos Aires]{} [Argentina]{} ------------------------------------- -- [^1]: The author was fully supported by CONICET, Argentina. [^2]: The author was fully supported by Universidade Federal do Rio de Janeiro, Brasil.

--- abstract: 'We consider the evolution of a flat, isotropic and homogeneous Friedmann-Robertson-Walker Universe, filled with a causal bulk viscous cosmological fluid, that can be characterized by an ultra-relativistic equation of state and bulk viscosity coefficient obtained from recent lattice QCD calculations. The basic equation for the Hubble parameter is derived under the assumption that the total energy in the Universe is conserved. By assuming a power law dependence of bulk viscosity coefficient, temperature and relaxation time on energy density, an approximate solution of the field equations has been obtained, in which we utilized equations of state from recent lattice QCD simulations QCD and heavy-ion collisions to derive an evolution equation. In this treatment for the viscous cosmology, we found no evidence for singularity. For example, both Hubble parameter and scale factor are finite at $t=0$, $t$ is the comoving time. Furthermore, their time evolution essentially differs from the one associated with non-viscous and ideal gas. Also thermodynamic quantities, like temperature, energy density and bulk pressure remain finite as well. In order to prove that the free parameter in our model does influence the final results, qualitatively, we checked out other particular solutions.' author: - | A. Tawfik$^{1}$, M. Wahba$^1$, H. Mansour$^2$ and T. Harko$^3$\ [$^1$Egyptian Center for Theoretical Physics (ECTP), MTI University, Cairo-Egypt]{}\ [$^2$Department of Physics, Faculty of Science, Cairo University, Giza-Egypt]{}\ [$^3$Department of Physics and Center for Theoretical and Computational Physics,]{}\ [University of Hong Kong, Hong Kong]{}\ title: | \ Viscous Quark-Gluon Plasma in the Early Universe --- Introduction {#sec:intro} ============ The dissipative effects, including both bulk and shear viscosity, are supposed to play a very important role in the early evolution of the Universe. The first attempts at creating a theory of relativistic fluids were those of Eckart [@Ec40] and Landau and Lifshitz [@LaLi87]. These theories are now known to be pathological in several respects. Regardless of the choice of equation of state, all equilibrium states in these theories are unstable and in addition signals may be propagated through the fluid at velocities exceeding the speed of light. These problems arise due to the first order nature of the theory, that is, it considers only first-order deviations from the equilibrium leading to parabolic differential equations, hence to infinite speeds of propagation for heat flow and viscosity, in contradiction with the principle of causality. Conventional theory is thus applicable only to phenomena which are quasi-stationary, i.e. slowly varying on space and time scales characterized by mean free path and mean collision time. A relativistic second-order theory was found by Israel [@Is76] and developed by Israel and Stewart [@IsSt76], Hiscock and Lindblom [HiLi89]{} and Hiscock and Salmonson [@HiSa91] into what is called “transient” or “extended” irreversible thermodynamics. In this model deviations from equilibrium (bulk stress, heat flow and shear stress) are treated as independent dynamical variables, leading to a total of 14 dynamical fluid variables to be determined. For general reviews on causal thermodynamics and its role in relativity see [@Ma95]. Causal bulk viscous thermodynamics has been extensively used for describing the dynamics and evolution of the early Universe or in an astrophysical context. But due to the complicated character of the evolution equations, very few exact cosmological solutions of the gravitational field equations are known in the framework of the full causal theory. For a homogeneous Universe filled with a full causal viscous fluid source obeying the relation $\xi \sim \rho ^{1/2}$, with $\rho $ the energy density of the cosmological fluid, exact general solutions of the field equations have been obtained in [@ChJa97; @MaHa99a; @MaHa99b; @MaHa00a; @MaTr97]. It has also been proposed that causal bulk viscous thermodynamics can model on a phenomenological level matter creation in the early Universe [@ChJa97]. Exact causal viscous cosmologies with $\xi \sim \rho ^{s},s\neq 1/2$ have been considered in Ref. [@MaHa99a]. Because of technical reasons, most investigations of dissipative causal cosmologies have assumed Friedmann-Robertson-Walker (FRW) symmetry (i.e. homogeneity and isotropy) or small perturbations around it [@MaTr97]. The Einstein field equations for homogeneous models with dissipative fluids can be decoupled and therefore are reduced to an autonomous system of first order ordinary differential equations, which can be analyzed qualitatively [@CoHo95]. The role of a transient bulk viscosity in a FRW space-time with decaying vacuum has been discussed in [@AbVi97]. Models with causal bulk viscous cosmological fluid have been considered recently [@ArBe00]. They obtained both power-law and inflationary solutions, with the gravitational constant an increasing function of time. The dynamics of a viscous cosmological fluids in the generalized Randall-Sundrum model for an isotropic brane were considered in [@Chen01]. The renormalization group method was applied to the study of homogeneous and flat FRW Universes, filled with a causal bulk viscous cosmological fluid, in [@Be03]. A generalization of the Chaplygin gas model, by assuming the presence of a bulk viscous type dissipative term in the effective thermodynamic pressure of the gas, was investigated recently in [@Pun08]. Recent RHIC results give a strong indication that in the heavy-ion collisions experiments, a hot dense matter can be formed [@reff1]. Such an experimental evidence might agree with the “new state of matter” as predicted in the Lattice QCD simulations [@reff5]. However, the experimentally observed elliptic flow in peripheral heavy-ion collisions seems to indicate that a thermalized collective QCD matter has been produced. In a addition to that, the success of ideal fluid dynamics in explaining several experimental data e.g. transverse momentum spectra of identified particles, elliptic flow [@reff6], together with the string theory motivated that the shear viscosity $\eta$ to the entropy $s$ would have the lower limit $\approx 1/4\pi$ [@reff7] leading to a paradigm that in heavy- ion collisions, that a [*nearly*]{} perfect fluid likely be created and the quarks and gluons likely go through relatively rapid equilibrium characterized with a thermalization time less than $1$ fm/c [@mueller1]. According to recent lattice QCD simulations [@mueller2], the bulk viscosity $ \xi $ is not negligible near the QCD critical temperature $T_c$. It has been shown that the bulk and shear viscosity at high temperature $T$ and weak coupling $\alpha_s $, $\xi\sim \alpha_s^2 T^3/\ln \alpha_s^{-1}$ and $\eta\sim T^3/(\alpha_s^2 \ln \alpha_s^{-1})$ [@mueller3]. Such a behavior obviously reflects the fact that near $T_c$ QCD is far from being conformal. But at high $T$, QCD approaches conformal invariance, which can be indicated by low trace anomaly $(\epsilon-3p)/T^4$  [@karsh09], where $\epsilon$ and $p$ are energy and pressure density, respectively. In the quenched lattice QCD, the ratio $\zeta/s$ seems to diverge near $T_c$ [@meyer08]. To avoid the mathematical difficulties accompanied with the Abel second type non-homogeneous and non-linear differential equations [@TawCosmos], one used to model the cosmological fluid as an ideal (non-viscous) fluid. No doubt that the viscous treatment of the cosmological background should have many essential consequences [@taw08]. The thermodynamical ones, for instance, can profoundly modify the dynamics and configurations of the whole cosmological background [@conseq1]. The reason is obvious. The bulk viscosity is to be expressed as a function of the Universe energy density $\rho$ [@conseq2]. Much progress has been achieved in relativistic thermodynamics of dissipative fluids. The pioneering theories of Eckart [@Ec40] and Landau and Lifshitz [@LaLi87] suffer from lake of causality constrains. The currently used theory is the Israel and Stewart theory [@Is76; @IsSt76], in which the causality is conserved and theory itself seems to be stable [@HiLi89; @Ma95]. In this article, we aim to investigate the effects that bulk viscosity has on the Early Universe. We consider a background corresponding to a FRW model filled with ultra-relativistic viscous matter, whose bulk viscosity and equation of state have been deduced from recent heavy-ion collisions experiments and lattice QCD simulations. The present paper is organized as follows. The basic equations of the model are written down in Section \[field\]. In Section \[approx\] we present an approximate solution of the evolution equation. Section \[part1\] is devoted to one particular solution, in which we assume that $H=const.$ The results and conclusions are given in Sections \[final\] and  \[final2\], respectively. Evolution equations {#field} =================== We assume that geometry of the early Universe is filled with a bulk viscous cosmological fluid, which can be described by a spatially flat FRW type metric given by $$\label{1} ds^{2}=dt^{2}-a^{2}\left( t\right) \left[ dr^{2}+r^{2}\left( d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}\right) \right] .$$ The Einstein gravitational field equations are: $$R_{ik}-\frac{1}{2}g_{ik}R=8\pi GT_{ik}. \label{ein}$$ In rest of this article, we take into consideration natural units, i.e., $c=1$, for instance. The energy-momentum tensor of the bulk viscous cosmological fluid filling the very early Universe is given by $$T_{i}^{k}=\left( \rho +p+\Pi\right) u_{i}u^{k}-\left( p+\Pi\right) \delta_{i}^{k},\label{1_a}$$ where $i,k$ takes $0,1,2,3$, $\rho$ is the mass density, $p$ the thermodynamic pressure, $\Pi $ the bulk viscous pressure and $u_{i}$ the four velocity satisfying the condition $u_{i}u^{i}=1$. The particle and entropy fluxes are defined according to $N^{i}=nu^{i}$ and $S^{i}=sN^{i}-\left( \tau\Pi^{2}/2\xi T\right) u^{i}$, where $n$ is the number density, $s$ the specific entropy, $T\geq0$ the temperature, $\xi$ the bulk viscosity coefficient, and $\tau\geq0$ the relaxation coefficient for transient bulk viscous effect (i.e. the relaxation time), respectively. The evolution of the cosmological fluid is subject to the dynamical laws of particle number conservation $N_{\text{ };i}^{i}=0$ and Gibbs’ equation $Td\rho=d\left( \rho /n\right) +pd\left( 1/n\right) $. In the following we shall also suppose that the energy-momentum tensor of the cosmological fluid is conserved, that is $T_{i;k}^{k}=0$. The bulk viscous effects can be generally described by means of an effective pressure $\Pi $, formally included in the effective thermodynamic pressure $p_{eff}=p+\Pi $ [@Ma95]. Then in the comoving frame the energy momentum tensor has the components $T_{0}^{0}=\rho ,T_{1}^{1}=T_{2}^{2}=T_{3}^{3}=-p_{eff}$. For the line element given by Eq. (\[1\]), the Einstein field equations read $$\begin{aligned} \label{2} \left( \frac{\dot{a}}{a}\right)^{2} &=& \frac{8\pi}{3}G \;\rho, \\ \frac{\ddot{a}}{a} &=& -\frac{4\pi}{3}G \; \left( 3p_{eff}+\rho \right), \label{3}\end{aligned}$$ where one dot denotes derivative with respect to the time $t$, $G$ is the gravitational constant and $a$ is the scale factor. Assuming that the total matter content of the Universe is conserved, $T_{i;j}^j=0$, the energy density of the cosmic matter fulfills the conservation law: $$\label{5} \dot{\rho}+3H\left( p_{eff}+\rho \right) =0,$$ where we introduced the Hubble parameter $H=\dot{a}/a$. In presence of bulk viscous stress $\Pi $, the effective thermodynamic pressure term becomes $p_{eff}=p+\Pi $. Then Eq. (\[5\]) can be written as $$\label{6} \dot{\rho}+3H\left( p+\rho \right) =-3\Pi H.$$ For the evolution of the bulk viscous pressure we adopt the causal evolution equation [@Ma95], obtained in the simplest way (linear in $\Pi)$ to satisfy the $H$-theorem (i.e., for the entropy production to be non-negative, $S_{;i}^{i}=\Pi^{2}/\xi T\geq0$ [@Is76; @IsSt76]). According to the causal relativistic Israel-Stewart theory, the evolution equation of the bulk viscous pressure reads [@Ma95] $$\label{8} \tau \dot{\Pi}+\Pi =-3\xi H-\frac{1}{2}\tau \Pi \left( 3H+\frac{\dot{\tau}}{\tau }-\frac{\dot{\xi}}{\xi }-\frac{\dot{T}}{T}\right).$$ In order to have a closed system from equations (\[2\]), (\[6\]) and (\[8\]) we have to add the equations of state for $p$ and $T$. As shown in Appendix A, the equation of state, the temperature and the bulk viscosity of the quark-gluon plasma (QGP), can be determined approximately at high temperatures [@karsch07] from recent lattice QCD calculations [@Cheng:2007jq], as $$\label{13} P = \omega \rho,\hspace*{1cm}T = \beta \rho^r,\hspace*{1cm}\xi = \alpha \rho + \frac{9}{\omega_0} T_c^4,$$ with $\omega = (\gamma-1)$, $\gamma \simeq 1.183$, $r\simeq 0.213$, $\beta\simeq 0.718$, $$\alpha = \frac{1}{9\omega_0} \frac{9\gamma^2-24\gamma+16}{\gamma-1},$$ and $\omega_0 \simeq 0.5-1.5$ GeV. In the following we assume that $\alpha \rho >> 9/\omega_0 T_c^4$, and therefore we take $\xi \simeq \alpha \rho$. In order to close the system of the cosmological equations, we have also to give the expression of the relaxation time $\tau $, for which we adopt the expression [@Ma95], $$\label{tau} \tau=\xi\rho^{-1}\simeq\alpha .$$ Eqs. (\[13\]) are standard in the study of the viscous cosmological models, whereas the equation for $\tau$ is a simple procedure to ensure that the speed of viscous pulses does not exceed the speed of light. Eq. (\[tau\]) implies that the relaxation time in our treatment is constant but strongly depends on EoS. These equations are without sufficient thermodynamical motivation, but in the absence of better alternatives, we shall follow the practice of adopting them in the hope that they will at least provide some indication of the range of bulk viscous effects. The temperature law is the simplest law guaranteeing positive heat capacity. With the use of Eqs. (\[8\]), (\[13\]) and (\[tau\]), respectively, we obtain the following equation describing the cosmological evolution of the Hubble function $H$ $$\begin{aligned} \label{init} \ddot H + \frac{3}{2} [1+(1-r) \gamma] H\dot H + \frac{1}{\alpha}\dot H - (1+r) H^{-1} \dot H^2 + \frac{9}{4}(\gamma -2) H^3 + \frac{3}{2}\frac{\gamma}{\alpha} H^2 &=& 0.\end{aligned}$$ An approximate solution {#approx} ======================= We introduce the transformation $u=\dot{H}$, so that Eq. (\[init\]) is transformed into a first order ordinary differential equation, $$\label{init2} u\frac{du}{dH}-(1+r)H^{-1}u^{2}+\left(\frac{3}{2}[1+(1-r)\gamma ]H+\alpha ^{-1}\right) u+\frac{9}{4}\frac{1}{(\gamma)}H^{3}+\frac{3}{2}\frac{\gamma}{\alpha} H^{2}=0.$$ We can rewrite Eq. (\[init2\]) in the form $$\label{OmegH1} \Omega \frac{d\Omega }{dH} = F_1(H)\Omega + F_0(H),$$ where $$\begin{aligned} \Omega &=& u \; E \;\; = u\; \exp\left(-\int \frac{1+r}{H} dH\right), \nonumber \\ F_1(H) &=& -\left( \frac{3}{2} [1+(1-r)\gamma]H + \frac{1}{\alpha}\right)E, \nonumber \\ F_0(H) &=& -\left(\frac{9}{4}(\gamma-2) H^3 + \frac{3}{2} \frac{\gamma}{\alpha} H^2\right)E^2. \nonumber\end{aligned}$$ By introducing a new independent variable $z=\int F_1(H)\,dH$, we obtain $$\Omega \frac{d\Omega}{dz} - \Omega = g(z),$$ with $g(z)$ is defined parametrically as, $$\label{gofzz1} g(z) = \frac{F_0}{F_1}.$$ As shown in Appendix B, $g(z)$ can be approximated as a simple function of $z$ $$g(z)\approx {\cal C}\; z,$$ where ${\cal C}$ is a constant. We proceed with this approximation to get solvable differential equations. Keeping the parametric solution of $g(z)$, Eq. (\[fullgofz\]), results in much more complicated differential equations. This would be the subject of a future work. From the definitions of $\Omega$ and $z$ we have $$\begin{aligned} \Omega &=& H^{1+r}\dot H, \label{Eq1} \\ z &=& H^{2+r} \left(\frac{-3[1+(1-r)\gamma]H}{2(1-r)} +\frac {1}{\alpha r}\right), \label{Eq2}\end{aligned}$$ Analogous to the solution of reduced Abel type canonical equation, $$\label{abel1} y \frac{dy}{dx} - y = a x$$ (see Appendix C) we obtain the relation $\Omega = z/{\cal P}$. Therefore, from Eqs. (\[Eq1\]) and (\[Eq2\]) we obtain the following first order differential equation for Hubble parameter $H$, $$\label{init-polyn} {\cal P} \dot H = \frac{-3[1+(1-r)\gamma]}{2(1-r)} H^2 +\frac{1}{\alpha r}H$$ with the solution $$H(t) = \frac{B}{\exp(-Bt/{\cal P})-A} \label{eq:mysolut1}$$ where $$\label{paramsab} A=\frac{-3[1+(1-r)\gamma]}{2(1-r)}, \hspace*{1cm} B=\frac{1}{\alpha r},$$ and ${\cal P}$ is taken as a free parameter. We can assign any real value to ${\cal P}$. For the results presented in this work, we used a negative value. This negative sign is necessarily to overcome the sign from the integral limits. The geometric and thermodynamic quantities of the Universe read $$\begin{aligned} a(t) &=& a_0\left(\frac{\exp(-B t/{\cal P})}{\exp(-B t/{\cal P})-A}\right)^{{\cal P}/A} \label{approx-a}, \\ \rho(t) &=& 3\, H^2= 3 \left(\frac{B}{\exp(-Bt/{\cal P})-A}\right)^2 \label{approx-rho},\\ T(t) &=& \beta \rho^{r}=\beta \left(3 \frac{B^{2}}{[\exp(-Bt/{\cal P})-A]^{2}}\right)^r \label{approx-T},\\ \Pi(t) &=& -2\dot{H}-3\gamma H^{2}=-\frac{B^2}{{\cal P}} \left(\frac{2\exp(-Bt/{\cal P})+3\gamma{\cal P}}{[\exp(-Bt/{\cal P})-A]^2}\right) \label{approx-Pi},\\ q(t)&=&\frac{d}{dt}H^{-1}-1=-\frac{1}{{\cal P}}\exp(-Bt/{\cal P})-1. \label{approx-q}\end{aligned}$$ $a_0$ is an arbitrary constant of the integration. The sign of $q$ indicates whether the Universe decelerates (positive) or accelerates (negative). $q$ can also be given as a function of the thermodynamic, gravitational and cosmological quantities $q(t)=[\rho(t) +3p(t)+3\Pi(t)]/2\rho(t)$ [@kolbBook]. de Sitter Universe {#part1} =================== Besides the approximation in $g(z)$, previous solution apparently depends on the free parameter ${\cal P}$. In this section, we suggest a particular solution to overcome ${\cal P}$. Eq. (\[init\]) can easily be obtained by assuming that $H$ doesn’t depend one $t$, i.e, de Sitter Universe. With a simple calculation, we get an estimation for $H$ $$\label{partcH} H=\frac{4}{9}\frac{\alpha ^{-1}\gamma}{2-\gamma }.$$ The geometric and thermodynamic parameters of the Universe are given by $$\begin{aligned} a(t) &=& a_{0}\exp \left[ \frac{4\alpha ^{-1}\gamma }{9(2-\gamma )}t\right], \label{partca} \\ \rho(t) &=& 3\left[ \frac{4\alpha ^{-1}\gamma }{9(2-\gamma )}\right] ^{2}, \label{partcrho}\\ T(t) &=& 3^{r}\beta \left[ \frac{4\alpha ^{-1}\gamma }{9(2-\gamma )}\right] ^{2r}, \label{partcT}\\ \Pi(t) &=& -3\gamma \left[ \frac{4\alpha ^{-1}\gamma }{9(2-\gamma )}\right] ^{2}, \label{partcPi}\\ q(t) &=& -1. \label{partcq}\end{aligned}$$ Although we have assumed here that the cosmic background is filled with viscous matter, the assumption that $H=const$ results in an exponential scale parameter, Eq. (\[partca\]). This behavior characterizes the de Sitter space, when $\Lambda=k=0$. $\rho$ and $T$ are finite at small $t$ as given in Fig. \[Figg2\]. Particular Solution {#part2} =================== Another particular solution for Eq. (\[init\]) can be obtained, when assuming that the dependence of $u$ on $H$ can be given by the polynomial in Eq. (\[init-polyn\]) $$\label{init-partc2} u=b_{1}H^{2}+b_{2}H,$$ where $b_{1}$ and $b_{2}$ are constants. Some simple calculations show that this form is a solution of the initial equation, Eq. (\[init2\]), if $$\begin{aligned} b_1 &=&-\frac{3}{2}\frac{1+\gamma }{1-r}, \\ b_2 &=& \frac{1}{r\alpha }.\end{aligned}$$ $b_2$ is identical to $B$ in Eq. (\[paramsab\]). $r$ and $\gamma$ have to satisfy the compatibility relation $$r=\frac{2-\gamma}{2+\gamma^2}.$$ Integrating Eq. (\[init-partc2\]) results in $$\begin{aligned} \label{Eq:Ht} H(t) &=& \frac{b_2\exp(-b_2 t)}{1-b_1\exp(-b_2 t)},\end{aligned}$$ where minus sign in the exponential function refers to flipping the integral limits. This was not necessary while deriving the expressions given in Section \[approx\]. The free parameter [P]{} compensates it. The geometric and thermodynamic quantities of the Universe read $$\begin{aligned} a(t)&=&a_0\left(\frac{\exp(b_2t)-b_1}{\exp(b_2t)}\right)^{1/b_1}, \label{partc2a} \\ \rho(t)&=& 3 \left(\frac{b_2\exp(-b_2 t)}{1-b_1\exp(-b_2 t)}\right)^2, \label{partc2rho}\\ T(t)&=& 3^r\;\beta \left(\frac{b_2\exp(-b_2 t)}{1-b_1\exp(-b_2 t)}\right)^{2r}, \label{partc2T}\\ \Pi(t)&=& \frac{b_2^2 \left[2\exp(b_2t)-3\gamma\right]}{\left[\exp(b_2t)-b_1\right]^2}, \label{partc2Pi}\\ q(t) &=& \exp(b_2t)-1. \label{partc2q}\end{aligned}$$ Obviously , we notice that the scale parameter in Eq. (\[partc2a\]) looks like Eq. (\[approx-a\]), which strongly depends on the free parameter ${\cal P}$. The other geometric and thermodynamic quantities find similarities in Eq. (\[approx-rho\]) - (\[approx-q\]), respectively. Deceleration parameter $q$ seems to be positive everywhere. Results {#final} ======= In present work, we have considered the evolution of a full causal bulk viscous flat, isotropic and homogeneous Universe with bulk viscosity parameters and equation of state taken from recent lattice QCD data and heavy-ion collisions. Three classes of solutions of the evolution equation have been obtained. In Fig. \[Figg1\], $H(t)$ and $a(t)$ are depicted in dependence on the comoving time $t$. We compare $H(t)$, given by Eq. (\[eq:mysolut1\]), and $a(t)$, given by Eq. (\[approx-a\]), with the counterpart parameters obtained in the case when the background matter is assumed to be an ideal and non-viscous fluid, described by the equations of state of the non-interacting ideal gas, $$\begin{aligned} H(t) &=& \frac{1}{2t} \label{htideal1}, \\ a(t) &=& \sqrt{t}. \label{atideal1}\end{aligned}$$ In the left panel of Fig. \[Figg1\], $H(t)=\dot a/a$ has an exponential decay, whereas in the non-viscous case, $H(t)$ is decreasing according to Eq. (\[htideal1\]). The latter is much slower than the former, reflecting the nature of the exponential and linear dependencies. The other difference between the two cases is obvious at small $t$. We notice a divergence, or singularity, associated with the ideal non-viscous fluid, Eq. (\[htideal1\]). The viscous fluid results in finite $H$ even at vanishing $t$, as can be seen from Eq. (\[eq:mysolut1\]). The scale factor $a(t)$ also shows differences in both cases. $a(t)$ in a Universe with an ideal and non-viscous background matter depends on $t$ according to Eq. (\[atideal1\]), which simply implies that $a(t)$ is directly proportional to $t$, and $a(t)$ vanishes at $t=0$, which shows the existence of a singularity of $H$. Assuming that the background matter is described by a viscous fluid results in different $a(t)$-behaviors with increasing $t$. At $t=0$, $a(t)$ remains finite. Correspondingly, $H(t)$ remains also finite. In general, the dependence on $t$ is much more complicated than in Eq. (\[atideal1\]). Here we have an $A/{\cal P}$ root of an exponential function. If $\exp(-Bt/{\cal P})>>A$, $a$ remains constant. Fig. \[Figg2\] illustrates the dependence of the two thermodynamical quantities, $\rho $ and $T$, on the comoving time. The non-viscous Universe shows a singular behavior in $\rho$ at vanishing $t$, as shown in the left panel of Fig. \[Figg2\]. This is not obvious in the case where we have taken into consideration a finite viscosity coefficient, i.e., $\rho$ is finite at $t=0$. In both cases, $\rho$ is decreasing with increasing $t$, reflecting that the Early Universe was likely expanding. Also the life time of the thermal viscous Universe seems to be shorter than for the non-viscous Universe. Almost the same behavior is observed in the right panel of Fig. \[Figg2\]. The temperature $T$ seems to be finite at vanishing $t$ in the viscous Universe. The $T$-singularity is only present, if we assume that the background matter is non-viscous ideal gas. In left panel of Fig. \[Figg3\], we show the dependence of the bulk viscous pressure $\Pi$ on $t$. $\Pi$ takes negative values at very small $t$. Then it switches to positive values at some values of $t$. After reaching the maximum value, $\Pi$ decays exponentially with increasing $t$. At larger $t$, $\Pi$ entirely vanishes. The deceleration parameter $q$, given by Eq. (\[approx-q\]), is depicted in the right panel of Fig. \[Figg3\], and it is compared with $q$ for a non-viscous fluid, $q=-3$. The approximate solution, given by Eq. (\[approx-q\]), results in negative $q$ at small $t$, referring to expansion era. $q$ from the particular solution, Eq. (\[partcq\]) is negative everywhere.\ For the particular solution, only the scale factor depends on $t$, Eq. (\[partca\]). The results are given in the right panel of Fig. \[Figg1\]. All cosmological and thermodynamical quantities given by Eq. (\[partcH\]) and Eqs. (\[partcrho\])-(\[partcq\]) are constant in time. Conclusions {#final2} =========== It is obvious that the bulk viscosity plays an important role in the evolution of the Early Universe. Despite of the simplicity of our model, it shows that a better understanding of the dynamics of our Universe is only accessible, if we use reliable equation of state to characterize the matter filling out the cosmic background. We conclude that the causal bulk viscous Universe described by the approximate solution starts its evolution from an initial non-singular state with a non-zero initial value of Hubble parameter $H(t)$ and scale factor $a(t)$, where $t$ is the comoving time. In this treatment, $t$ is given in GeV$^{-1}$. Also the thermodynamical quantities, energy density $\rho$ for instance, are finite at vanishing $t$. Even the temperature $T$ itself shows no singularity at $t=0$. The Hubble parameter $H$ decreases monotonically with $T$ similar to $\rho$. The bulk viscous pressure $\Pi$ likely satisfies the condition that $\Pi<0$ at very small $t$ indicating to inflationary era. Then $\Pi $ switches to positive value. It reaches a maximum value and then decays and vanishes, exponentially, at large $t$. The deceleration parameter $q$ shows an expanding behavior in the case of non-viscous ideal gas and first particular solution. For second particular solution, $q$ starts from zero and increases, exponentially. According to this solution, the Universe was decelerating. The approximate solution shows an interesting behavior in $q(t)$, Eq. (\[approx-q\]). At small $t$, the values of $q$ are negative, i.e. the Universe was accelerating (expansion). At larger $t$, a non-inflationary behavior sets on, $q>0$, i.e., the Universe switched to a decelerating evolution. In this treatment, we assumed that the Universe is flat, $k=0$, and the background geometry is filled out with QCD matter (QGP) with a finite viscosity coefficient. The resulting Universe is obviously characterized by a shortly increasing and afterward constant scale factor and a fast vanishing Hubble parameter. At $t=0$, both $a(t)$ and $H(t)$ remain finite, i.e., there is no singularity. The validity of our treatment depends on the validity of the equations of states, Eq. \[13\], which we have deduced from the lattice QCD simulations at temperatures larger than $T_c\approx 0.19~$GeV. Below $T_c$, as the Universe cooled down, not only the degrees of freedom suddenly increase [@Tawfik03] but also the equations of state turn to be the ones characterizing the hadronic matter. Such a phase transition - from QGP to hadronic matter - would characterize one end of the validity of our treatment. 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Lett. B [**623**]{}, 48 (2005). F. Karsch, K. Redlich and A. Tawfik, Eur. Phys. J. [**C29**]{} 549 (2003), e-Print: hep-ph/0303108; Phys. Lett. [**B571**]{} 67 (2003), e-Print: hep-ph/0306208; K. Redlich, F. Karsch and A. Tawfik, J. Phys. [**G30**]{} S1271 (2004), e-Print: nucl-th/0404009; A. Tawfik, Phys. Rev. [**D71**]{} 054502, (2005), e-Print: hep-ph/0412336. Appendix A: Viscosity coefficient $\xi (T)$ from LQCD {#App:C .unnumbered} ===================================================== Following the discussion presented in [[@Cheng:2007jq]]{}, the bulk viscosity of QGP can be calculated from the lattice QCD by Eq. (13) in that paper. We assume that the decay factors for pions and kaons are vanishing above the critical temperature of the phase transition QGP-hadrons. The quark-antiquark condensates can be neglected at temperatures higher than the critical one [@TawDom]. Therefore, Eq. (22) of Ref. [[@Cheng:2007jq]]{} would be reduced to $$\label{ze} 9\,\omega_0\,\xi = T\, s\, \left(\frac{1}{c_s^2}-3\right)-4(\rho-3p) +16|\epsilon_v|$$ where $\rho$ is the energy density and $c_s^2=dp/d\rho $ is the square of the speed of sound. The parameter $\omega_o$ is a scale depending on the temperature $T$, and defines the validity of the underlying perturbation theory. In this relation, the viscosity is assumed to have a thermal part which can be determined through lattice calculations, and a vacuum contributing part, which can be fixed using quark and gluon condensates. The vacuum part would take the value $$16 |\epsilon_v| (1 + \frac{3}{8} \cdot 1.6) \simeq (560\ {\rm MeV})^4 \simeq (3 \,T_c)^4 \,$$ Our algorithm is the following. Using lattice QCD results on trace anomaly, $(\epsilon-3p)/T^4$, and other thermodynamical quantities, we can determine the bulk viscosity. To make use of the lattice QCD results, it is useful to make a suitable fit to the data at high temperatures. Then we obtain the following equations of state $$\begin{aligned} \label{EoS} p &=&\omega \rho, \hspace*{2cm} T =\beta \rho^r, \hspace*{2cm} c_s^2 = \omega \nonumber\end{aligned}$$ where $\omega=0.319$, $\beta=0.718\pm 0.054$ and $r=0.23\pm 0.196$. Using the equations of state, Eq. (\[EoS\]) in Eq. (\[ze\]), we obtain $$\label{zeta} \xi(\epsilon)=\frac{1}{9\omega_o}\frac{9\gamma^2-24\gamma+16}{\gamma-1}\rho+\frac{9}{\omega_o}T_c^4.$$ Appendix B: Estimations of $g(z)$ {#App:A .unnumbered} ================================= For analytical purposes, the function $g(z)$, which is defined in $z$ parameter as $g(z)=F_0/F_1$ in Eq. (\[gofzz1\]), can be numerically estimated depending on the parameter $z$ by using the following procedure. First, we plot it parametrically depending on the parameter $H$, Fig. (\[Figg4\]). Then we fit the resulting curve to various functions. Based on least-square fit, best choice would be a mixture of polynomial and exponential functions, $$\label{fullgofz} g(z)= a + b\, z + c \frac{\exp(d\, z)+e}{\left[\exp(d\, z)+f\right]^2},$$ where the coefficients read $a=-2.078\pm0.117$, $b=0.091\pm0.007$ and $c=17.332\pm1.553$, $d=0.189\pm0.003$, $e=-0.814\pm0.162$ and $f=2.849\pm0.02$. At small values of $z$, it is clear that the dependence is linear, $$\label{lineargofz2} g(z) = c + {\cal C} z.$$ Obviously, the intersect $c$ is much smaller than the slope ${\cal C}$. The sign of $g(z)$ can be flipped regarding to the sign of its independent variable $z$. Accordingly, we get $$\label{lineargofz} g(z)\approx {\cal C} z.$$ To prove this dependence, algebraically, we try to estimate $g(z)$ directly from the division of $F_0$ by $F_1$, which can be approximated by including their first terms only, i.e. $$\label{eq.A1} g(H)\approx \frac{3(\gamma-2)}{2[1+(1-r)\gamma]}\; H^{1-r},$$ Then, we approximate $z(H)$ to the form, $$\label{eq.A2} z(H)\approx-\frac{3[1+(1-r)\gamma]}{2(1-r)}\; H^{1-r}.$$ Finally, we now able to derive an approximate estimation for $g(z)$. According to Eq. (\[eq.A1\]) and (\[eq.A2\]), we get $$g(z)\approx \frac{(1-r)(\gamma-2)}{[1+(1-r)\gamma]^2}\; z$$ Amazingly, this expression looks the same as the one we obtained from the numerical approximation with $${\cal C} = \frac{(1-r)(\gamma-2)}{\left[1+(1-r)\gamma\right]^2}.$$ Appendix C: Solution of Abel equation $y\dot y -y = ax$ {#App:B .unnumbered} ======================================================= To solve Eq. (\[abel1\]) we divide the whole equation by $y^3$ and introduce a new variable $v=1/y$. Then Eq. (\[abel1\]) reads $$\frac{dv}{dx}+v^{2}+axv^{3}=0.$$ We then introduce the function $v=w/x$. $$x\frac{dw}{dx}=w-w^{2}-aw^{3}, \label{abel2}$$ Previous differential equation can be solved by separation of variables $$\int \frac{dw}{w-w^{2}-aw^{3}}=\ln C^{-1}x,$$ where $C$ is an arbitrary constant of integration. To calculate the integral, we write the function to be integrated as $$\frac{1}{w-w^{2}-aw^{3}}=\frac{1}{w}-\frac{aw}{aw^{2}+w-1}-\frac{1}{aw^{2}+w-1}.$$ Let us assume that $\Delta =1+4a>0$ (this implies that $a>0$). $$\int \frac{dw}{w-w^{2}-aw^{3}}=-\frac{1}{2\sqrt{\Delta }}\ln \frac{2aw-\sqrt{\Delta }+1}{2aw+\sqrt{\Delta }+1}-\frac{1}{2}\ln \left( aw^{2}+w-1\right) +\ln w.$$ Therefore the general solution of Eq. (\[abel2\]) can be written as $$x=C\frac{w}{\sqrt{aw^{2}+w-1}}\left( \frac{2aw+\sqrt{\Delta }+1}{2aw-\sqrt{\Delta }+1}\right) ^{1/2\sqrt{\Delta }},$$ leading to $$y=\frac{1}{v}=\frac{x}{w}=C\frac{1}{\sqrt{aw^{2}+w-1}}\left( \frac{2aw+\sqrt{\Delta }+1}{2aw-\sqrt{\Delta }+1}\right) ^{1/2\sqrt{\Delta }}.$$

--- abstract: 'We consider distributed iterative algorithms for the averaging problem over time-varying topologies. Our focus is on the convergence time of such algorithms when complete (unquantized) information is available, and on the degradation of performance when only quantized information is available. We study a large and natural class of averaging algorithms, which includes the vast majority of algorithms proposed to date, and provide tight polynomial bounds on their convergence time. We then propose and analyze distributed averaging algorithms under the additional constraint that agents can only store and communicate quantized information, converge to the average of the initial values of the agents within some error. We establish bounds on the error and tight bounds on the convergence time, as a function of the number of quantization levels.' author: - 'Angelia Nedić[^1], Alex Olshevsky, Asuman Ozdaglar, and John N. Tsitsiklis[^2]' bibliography: - 'distributed\_alex.bib' title: ' **On Distributed Averaging Algorithms and Quantization Effects[^3]**' --- Introduction ============ There has been much recent interest in distributed control and coordination of networks consisting of multiple, potentially mobile, agents. This is motivated mainly by the emergence of large scale networks, characterized by the lack of centralized access to information and time-varying connectivity. Control and optimization algorithms deployed in such networks should be completely distributed, relying only on local observations and information, and robust against unexpected changes in topology such as link or node failures. A canonical problem in distributed control is the [*consensus problem*]{}. The objective in the consensus problem is to develop distributed algorithms that can be used by a group of agents in order to reach agreement (consensus) on a common decision (represented by a scalar or a vector value). The agents start with some different initial decisions and communicate them locally under some constraints on connectivity and inter-agent information exchange. The consensus problem arises in a number of applications including coordination of UAVs (e.g., aligning the agents’ directions of motion), information processing in sensor networks, and distributed optimization (e.g., agreeing on the estimates of some unknown parameters). The [*averaging problem*]{} is a special case in which the goal is to compute the exact average of the initial values of the agents. A natural and widely studied consensus algorithm, proposed and analyzed by Tsitsiklis [@T84] and Tsitsiklis [*et al.*]{} [@TBA86], involves, at each time step, every agent taking a weighted average of its own value with values received from some of the other agents. Similar algorithms have been studied in the load-balancing literature (see for example [@C89]). Motivated by observed group behavior in biological and dynamical systems, the recent literature in cooperative control has studied similar algorithms and proved convergence results under various assumptions on agent connectivity and information exchange (see [@VCBJCS95], [@OSM04], [@RB05], [@M05], [@LR06]). In this paper, our goal is to provide tight bounds on the convergence time (defined as the number of iterations required to reduce a suitable Lyapunov function by a constant factor) of a general class of consensus algorithms, as a function of the number $n$ of agents. We focus on algorithms that are designed to solve the averaging problem. We consider both problems where agents have access to exact values and problems where agents only have access to quantized values of the other agents. Our contributions can be summarized as follows. In the first part of the paper, we consider the case where agents can exchange and store continuous values, which is a widely adopted assumption in the previous literature. We consider a large class of averaging algorithms defined by the condition that the weight matrix is a possibly nonsymmetric, doubly stochastic matrix. For this class of algorithms, we prove that the convergence time is $O(n^2/\eta)$, where $n$ is the number of agents and $\eta$ is a lower bound on the nonzero weights used in the algorithm. To the best of our knowledge, [*this is the best polynomial-time bound on the convergence time of such algorithms*]{}. We also show that this bound is tight. Since this result implies an $O(n^3)$ bound on convergence time. In Section \[matrixpicking\], we present a distributed algorithm that selects the weights dynamically, using three-hop neighborhood information. Under the assumption that the underlying connectivity graph at each iteration is undirected, we establish an improved $O(n^2)$ upper bound on convergence time. This matches the best available convergence time guarantee for the much simpler case of static connectivity graphs . In the second part of the paper, we impose the additional constraint that agents can only store and transmit quantized values. This model provides a good approximation for communication networks that are subject to communication bandwidth constraints. We focus on a particular quantization rule, which rounds down the values to the nearest quantization level. We propose a distributed algorithm that uses quantized values and, using a slightly different Lyapunov function, we show that the algorithm guarantees the convergence of the values of the agents to a common value. In particular, we prove that all agents have the same value after $O((n^2/\eta)\log (nQ))$ time steps, where $Q$ is the number of quantization levels per unit value. Due to the rounding-down feature of the quantizer, this algorithm does not preserve the average of the values at each iteration. However, we provide bounds on the error between the final consensus value and the initial average, as a function of the number $Q$ of available quantization levels. In particular, we show that the error goes to 0 at a rate of $(\log Q)/Q$, as the number $Q$ of quantization levels increases to infinity. Other than the papers cited above, our work is also related to [@KBS06] and [@CFSZ05; @CFFTZ07], which studied the effects of quantization on the performance of averaging algorithms. In [@KBS06], Kashyap [*et al.*]{} proposed randomized [*gossip-type*]{} quantized averaging algorithms under the assumption that each agent value is an integer. They showed that these algorithms preserve the average of the values at each iteration and converge to approximate consensus. They also provided bounds on the convergence time of these algorithms for specific static topologies (fully connected and linear networks). In the recent work [@CFFTZ07], Carli [*et al.*]{} proposed a distributed algorithm that uses quantized values and preserves the average at each iteration. They showed favorable convergence properties using simulations on some static topologies, and provided performance bounds for the limit points of the generated iterates. Our results on quantized averaging algorithms differ from these works in that [*we study a more general case of time-varying topologies, and provide tight polynomial bounds on both the convergence time and the discrepancy from the initial average, in terms of the number of quantization levels*]{}. The paper is organized as follows. In Section \[agreementalg\], we introduce a general class of averaging algorithms, and present our assumptions on the algorithm parameters and on the information exchange among the agents. In Section \[convtimesection\], we present our main result on the convergence time of the averaging algorithms under consideration. In Section \[matrixpicking\], we present a distributed averaging algorithm for the case of undirected graphs, which picks the weights dynamically, resulting in an improved bound on the convergence time. In Section \[qanalysis\], we propose and analyze a quantized version of the averaging algorithm. In particular, we establish bounds on the convergence time of the iterates, and on the error between the final value and the average of the initial values of the agents. Finally, we give our concluding remarks in Section \[conclusions\]. A Class of Averaging Algorithms {#agreementalg} =============================== We consider a set $N=\{1,2,\ldots,n \}$ of agents, which will henceforth be referred to as “nodes.” Each node $i$ starts with a scalar value $x_i(0)$. At each nonnegative integer time $k$, node $i$ receives from some of the other nodes $j$ a message with the value of $x_j(k)$, and updates its value according to: $$x_i(k+1) = \sum_{j=1}^n a_{ij}(k) x_j(k), \label{noquant}$$ where the $a_{ij}(k)$ are nonnegative weights with the property that $a_{ij}(k)>0$ only if node $i$ receives information from node $j$ at time $k$. We use the notation $A(k)$ to denote the [*weight matrix*]{} $[a_{ij}(k)]_{i,j=1,\ldots,n}$, Given a matrix $A$, we use $\E(A)$ to denote the set of directed edges $(j,i)$, including self-edges $(i,i)$, such that $a_{ij}>0$. At each time $k$, the nodes’ connectivity can be represented by the directed graph $G(k)=(N,\E(A(k)))$. Our goal is to study the convergence of the iterates $x_i(k)$ to the average of the initial values, $(1/n)\sum_{i=1}^n x_i(0)$, as $k$ approaches infinity. In order to establish such convergence, we impose some assumptions on the weights $a_{ij}(k)$ and the graph sequence $G(k)$. For each $k$, the weight matrix $A(k)$ is a doubly stochastic matrix[^4] with positive diagonal . Additionally, there exists a constant $\eta> 0$ such that if $a_{ij}(k)>0$, then $a_{ij}(k) \geq \eta$.\[weights\] The double stochasticity assumption on the weight matrix guarantees that the average of the node values remains the same at each iteration (cf. the proof of Lemma \[vl\] below). The second part of this assumption states that each node gives significant weight to its values and to the values of its neighbors at each time $k$. Our next assumption ensures that the graph sequence $G(k)$ is sufficiently connected for the nodes to repeatedly influence each other’s values. There exists an integer $B \geq 1$ such that the directed graph $$\Big(N, \E(A(kB)) \bigcup \E(A(kB+1))\bigcup \cdots \bigcup \E(A((k+1)B-1))\Big)$$ is strongly connected for all nonnegative integers $k$. \[connectivity\] Any algorithm of the form given in Eq. (\[noquant\]) with the sequence of weights $a_{ij}(k)$ satisfying Assumptions \[weights\] and \[connectivity\] solves the averaging problem. This is formalized in the following . \[uqc\] Let Assumptions \[weights\] and \[connectivity\] hold. Let $\{x(k)\}$ be generated by the algorithm (\[noquant\]). Then, for all $i$, we have $$\lim_{k \rightarrow \infty} x_i(k) = \frac{1}{n} \sum_{j=1}^n x_j(0).$$ This is a minor modification of known results in [@T84; @TBA86; @JLM03; @BHOT05], where the convergence of each $x_i(k)$ to the same value is established under weaker versions of Assumptions \[weights\] and \[connectivity\]. The fact that the limit is the average of the entries of the vector $x(0)$ follows from the fact that multiplication of a vector by a doubly stochastic matrix preserves the average of the vector’s components. Recent research has focused on methods of choosing weights $a_{ij}(k)$ that satisfy Assumptions \[weights\] and \[connectivity\], and minimize the convergence time of the resulting averaging algorithm (see [@XB04] for the case of static graphs, see [@OSM04] and [@BFT05] for the case of symmetric weights, i.e., weights satisfying $a_{ij}(k)=a_{ji}(k)$, and also see [@C06; @boyd]). For static graphs, some recent results on optimal time-invariant algorithms may be found in [@OT06]. Convergence time \[convtimesection\] ==================================== In this section, we give an analysis of the convergence time of averaging algorithms of the form (\[noquant\]). Our goal is to obtain tight estimates of the convergence time, under Assumptions \[weights\] and \[connectivity\]. As a convergence measure, we use the “sample variance” of a vector $x \in \R^n$, defined as $$V(x) = \sum_{i=1}^n (x_i - \bar{x} )^2,$$ where $\bar{x}$ is the average of the entries of $x$: $$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i.$$ Let $x(k)$ denote the vector of node values at time $k$ \[i.e., the vector of iterates generated by algorithm (\[noquant\]) at time $k$\]. We are interested in providing an upper bound on the number of iterations it takes for the “sample variance” $V(x(k))$ to decrease to a small fraction of its initial value $V(x(0))$. We first establish some technical preliminaries that will be key in the subsequent analysis. In particular, in the next subsection, we explore several implications of the double stochasticity assumption on the weight matrix $A(k)$. Preliminaries on Doubly Stochastic Matrices ------------------------------------------- We begin by analyzing how the sample variance $V(x)$ changes when the vector $x$ is multiplied by a doubly stochastic matrix $A$. The next lemma shows that $V(Ax) \leq V(x)$. Thus, under Assumption \[weights\], the sample variance $V(x(k))$ is nonincreasing in $k$, and $V(x(k))$ can be used as a Lyapunov function. \[vl\] Let $A$ be a doubly stochastic matrix. Then,[^5] for all $x \in \R^n$, $$V(Ax) = V(x) - \sum_{i<j} w_{ij} (x_i - x_j)^2,$$ where $w_{ij}$ is the $(i,j)$-th entry of the matrix $A^T A$. Let ${\bf 1}$ denote the vector in $\R^n$ with all entries equal to $1$. The double stochasticity of $A$ implies $$A {\bf 1}= {\bf 1}, \qquad {\bf 1}^T A = {\bf 1}^T.$$ Note that multiplication by a doubly stochastic matrix $A$ preserves the average of the entries of a vector, i.e., for any $x\in\rn$, there holds $$\overline{Ax} = \frac{1}{n}\, {\bf 1}^T Ax =\frac{1}{n}\, {\bf 1}^T x = \bar{x}.$$ We now write the quadratic form $V(x)-V(Ax)$ explicitly, as follows: $$\begin{aligned} V(x) - V(Ax) & = & (x - \bar{x} {\bf 1})^T (x-\bar{x}{\bf 1}) - (Ax-\overline{Ax}{\bf 1})^T (Ax - \overline{Ax}{\bf 1}) \nonumber \\ & = & (x - \bar{x} {\bf 1})^T (x-\bar{x}{\bf 1}) - (Ax-\bar{x}A{\bf 1})^T (Ax - \bar{x} A {\bf 1}) \nonumber \\ & = & (x - \bar{x} {\bf 1})^T (I-A^T A) (x-\bar{x}{\bf 1}). \label{vdec} \end{aligned}$$ Let $w_{ij}$ be the $(i,j)$-th entry of $A^T A$. Note that $A^T A$ is symmetric and stochastic, so that $w_{ij}=w_{ji}$ and $w_{ii} = 1 - \sum_{j \neq i} w_{ij}$. Then, it can be verified that $$A^T A = I - \sum_{i<j} w_{ij} (e_i - e_j) (e_i - e_j)^T, \label{asquareform}$$ where $e_i$ is a unit vector with the $i$-th entry equal to 1, and all other entries equal to 0 . By combining Eqs. (\[vdec\]) and (\[asquareform\]), we obtain $$\begin{aligned} V(x) - V(Ax) & =& (x - \bar{x} {\bf 1})^T \Big(\sum_{i<j} w_{ij} (e_i - e_j) (e_i - e_j)^T\Big) (x-\bar{x}{\bf 1}) \nonumber \\ & = & \sum_{i<j} w_{ij} (x_i - x_j)^2. $$ Note that the entries $w_{ij}(k)$ of $A(k)^T A(k)$ are nonnegative, because the weight matrix $A(k)$ has nonnegative entries. In view of this, Lemma \[vl\] implies that $$V(x(k+1))\le V(x(k))\qquad \hbox{for all }k.$$ Moreover, the amount of variance decrease is given by $$V(x(k)) - V(x(k+1)) = \sum_{i<j} w_{ij}(k) (x_i(k) - x_j(k))^2.$$ We will use this result to provide a lower bound on the amount of decrease of the sample variance $V(x(k))$ in between iterations. Since every positive entry of $A(k)$ is at least $\eta$, it follows that every positive entry of $A(k)^T A(k)$ is at least $\eta^2$. Therefore, it is immediate that $$\hbox{if }\quad w_{ij}(k) > 0, \quad\hbox{then }\quad w_{ij}{(k)} \geq \eta^2.$$ In our next lemma, we establish a stronger lower bound. In particular, we find it useful to focus not on an individual $w_{ij}$, but rather on all $w_{ij}$ associated with edges $(i,j)$ [that]{} cross [a particular]{} cut in the graph $(N,\E(A^TA))$. For such groups of $w_{ij}$, we prove a lower bound which is linear in $\eta$, as seen in the following. \[db\] Let $A$ be a stochastic matrix with positive diagonal , and assume that is at least $\eta$. Also, let $(S^{-},S^{+})$ be a partition of the set $N=\{1,\ldots,n\}$ into two disjoint sets. If $$\sum_{i \in S^{-}, ~j \in S^{+}} w_{ij} > 0,$$ then $$\sum_{i \in S^{-}, ~j \in S^{+}} w_{ij} \geq \frac{\eta}{2}.$$ Let $\sum_{i \in S^{-}, ~j \in S^{+}} w_{ij} >0$. From the definition of the weights $w_{ij}$, we have $w_{ij}=\sum_k a_{ki}a_{kj}$, which shows that there exist $i\in S^-$, $j\in S^+$, and some $k$ such that $a_{ki}>0$ and $a_{kj}>0$. For either case where $k$ belongs to $S^-$ or $S^+$, we see that there exists an edge in the set $\E(A)$ that crosses the cut $(S^{-},S^{+})$. Let $(i^*,j^*)$ be such an edge. Without loss of generality, we assume that $i^* \in S^{-}$ and $j^* \in S^{+}$. We define $$\begin{aligned} C_{j^*}^{+} & = & \sum_{i \in S^{+}} a_{j^*i}, \\ C_{j^*}^{-} & =& \sum_{i \in S^{-}} a_{j^*i}. \end{aligned}$$ See Figure \[cdef\](a) for an illustration. Since $A$ is a -stochastic matrix, we have $$C_{j^*}^{-} + C_{j^*}^{+} = 1, $$implying that at least one of the following is true: $$\begin{aligned} \mbox{ Case (a): } & & C_{j^*}^{-} \geq \frac{1}{2}, \\ \mbox{ Case (b): } & & C_{j^*}^{+} \geq \frac{1}{2}.\end{aligned}$$ We consider these [two cases]{} separately. In both cases, we focus on a subset of the edges and we use the fact that the elements $w_{ij}$ correspond to paths of length $2$, with one step in $\E(A)$ and another in $\E(A^T)$. [ Intuitively, $C_{j^*}^{+}$ measures how much weight $j^*$ assigns to nodes in $S^{+}$ (including itself), and $C_{j^*}^{-}$ measures how much weight $j^*$ assigns to nodes in $S^{-}$. Note that the [edge]{} $(j^*,j^*)$ is also present, but not shown. (b) For the case [where]{} $C_{j^*}^{-} \geq 1/2$, we only focus on two-hop paths between $j^*$ and elements $i \in S^{-}$ obtained by taking $(i,j^*)$ as the first step and the self-edge $(j^*,j^*)$ as the second step. (c) For the case where $C_{j^*}^{+} \geq 1/2$, we only focus on two-hop paths between $i^*$ and elements $j\in S^{+}$ obtained by taking $(i^*,j^*)$ as the first step in $\E(A)$ and $(j^*,j)$ as the second step in $\E(A^T)$.](csnew.eps "fig:"){width="4cm"}  Intuitively, $C_{j^*}^{+}$ measures how much weight $j^*$ assigns to nodes in $S^{+}$ (including itself), and $C_{j^*}^{-}$ measures how much weight $j^*$ assigns to nodes in $S^{-}$. Note that the [edge]{} $(j^*,j^*)$ is also present, but not shown. (b) For the case [where]{} $C_{j^*}^{-} \geq 1/2$, we only focus on two-hop paths between $j^*$ and elements $i \in S^{-}$ obtained by taking $(i,j^*)$ as the first step and the self-edge $(j^*,j^*)$ as the second step. (c) For the case where $C_{j^*}^{+} \geq 1/2$, we only focus on two-hop paths between $i^*$ and elements $j\in S^{+}$ obtained by taking $(i^*,j^*)$ as the first step in $\E(A)$ and $(j^*,j)$ as the second step in $\E(A^T)$.](cplusbignew.eps "fig:"){width="4cm"}  Intuitively, $C_{j^*}^{+}$ measures how much weight $j^*$ assigns to nodes in $S^{+}$ (including itself), and $C_{j^*}^{-}$ measures how much weight $j^*$ assigns to nodes in $S^{-}$. Note that the [edge]{} $(j^*,j^*)$ is also present, but not shown. (b) For the case [where]{} $C_{j^*}^{-} \geq 1/2$, we only focus on two-hop paths between $j^*$ and elements $i \in S^{-}$ obtained by taking $(i,j^*)$ as the first step and the self-edge $(j^*,j^*)$ as the second step. (c) For the case where $C_{j^*}^{+} \geq 1/2$, we only focus on two-hop paths between $i^*$ and elements $j\in S^{+}$ obtained by taking $(i^*,j^*)$ as the first step in $\E(A)$ and $(j^*,j)$ as the second step in $\E(A^T)$.](c2bignew.eps "fig:"){width="4cm"} ]{} [*Case (a):*]{} $C_{j^*}^{-} \geq 1/2.$\ We focus on those $w_{ij}$ with $i\in S^-$ and $j=j^*$. Indeed, since all $w_{ij}$ are nonnegative, we have $$\label{wsubset} \sum_{i \in S^{-}, ~j \in S^{+}} w_{ij} \geq \sum_{i \in S^{-}} w_{ij^*}.$$ For each element in the sum on the right-hand [side]{}, we have $$w_{ij^*} = \sum_{k=1}^n a_{ki}\ a_{kj^*} \geq a_{j^*i}\ a_{j^* j^*} \geq a_{j^*i}\ \eta,$$ where the inequalities follow from the facts that $A$ has nonnegative entries, its diagonal entries are positive, and its positive entries are at least $\eta$. Consequently, $$\sum_{i \in S^{-}} w_{ij^*} \geq \eta\ \sum_{i \in S^{-}} a_{j^*i} = \eta\, C_{j^*}^{-}. \label{wlowerbound}$$ Combining Eqs. (\[wsubset\]) and (\[wlowerbound\]), and recalling [the assumption]{} $C_{j^*}^{-} \geq 1/2$, the [result]{} follows. [An]{} illustration of [this argument]{} can be found in Figure \[cdef\][(b)]{}. [*Case (b):*]{} $C_{j^*}^{+} \geq 1/2.$\ We focus on those $w_{ij}$ with $i=i^*$ and $j \in S^{+}$. We have $$\label{wsubset2} \sum_{i \in S_{-}, ~j \in S^{+}} w_{ij} \geq \sum_{j \in S^{+}} w_{i^* j},$$ since all $w_{ij}$ are nonnegative. For each element in the sum on the right-hand side, we have $$w_{i^* j} = \sum_{k=1}^n a_{ki^*}\ a_{k j} \geq a_{j^* i^*}\ a_{j^* j} \geq \eta\, a_{j^* j},$$ where the inequalities follow [because all entries of $A$ are nonnegative, and because the choice $(i^*, j^*) \in \E(A)$ implies that $a_{j^* i^*} \geq{\eta}$.]{} Consequently, $$\sum_{j \in S^{+}} w_{i^* j} \geq \eta \sum_{j \in S^{+}} a_{j^*j} = \eta\, C_{j^*}^{+}. \label{wlowerbound2}$$ [Combining Eqs. (\[wsubset2\]) and]{} (\[wlowerbound2\]), and recalling [the assumption]{} $C_{j^*}^+ \geq 1/2$, the [result]{} follows. An illustration of this argument can be found in Figure \[cdef\][(c)]{}. A Bound on Convergence Time {#convergtime} --------------------------- With the preliminaries on doubly stochastic matrices in place, we [can now proceed to derive]{} bounds on the decrease of $V(x(k))$ in between iterations. We will first somewhat relax our connectivity assumptions. In particular, we consider the following relaxation of Assumption \[connectivity\]. \[weakconnect\] Given an integer $k\ge 0$, suppose that the components of $x(kB)$ have been reordered so that they are in nonincreasing order. We assume that for every $d\in\{1,\ldots,n-1\}$, we either have $x_d(kB)=x_{d+1}(kB)$, or there exist some time $t\in\{kB,\ldots,(k+1)B-1\}$ and some $i\in\{1,\ldots,d\}$, $j\in\{d+1,\ldots,n\}$ such that $(i,j)$ or $(j,i)$ belongs to $\E(A(t))$. \[as2implas3\] Assumption \[connectivity\] implies Assumption \[weakconnect\], [with the same value of $B$.]{} If Assumption \[weakconnect\] does not hold, then $$\Big(N,\E({A}(kB)) \bigcup \E(A(kB+1)){\bigcup} \cdots \bigcup \E(A((k+1)B-1))\Big)$$ is disconnected, which violates Assumption 2. For our convergence time results, we will use the weaker Assumption \[weakconnect\], rather than the stronger Assumption 2. Later on, in Section \[matrixpicking\], we will exploit the sufficiency of Assumption \[weakconnect\] to [design]{} a decentralized algorithm for selecting the weights $a_{ij}(k)$, which satisfies Assumption \[weakconnect\], but not Assumption 2. We now proceed to bound the decrease of our Lyapunov function $V(x(k))$ during the interval $[kB, (k+1) B-1]$. In what follows, we denote by $V(k)$ the sample variance $V(x(k))$ at time $k$. \[vardiff\] Let Assumptions \[weights\] and \[weakconnect\] hold. Let $\{x(k)\}$ be generated by the update rule (\[noquant\]). Suppose that the components $x_i(kB)$ of the vector $x(kB)$ have been ordered from largest to smallest, with ties broken arbitrarily. Then, we have $$V(kB) - V((k+1)B) \geq \frac{\eta}{2} \sum_{i=1}^{n-1} (x_{i}(kB) - x_{i+1}(kB))^2.$$ By Lemma \[vl\], we have for all $t$, $$V(t)-V(t+1) = \sum_{i<j} w_{ij}(t) (x_i(t)-x_j(t))^2, \label{firstvdec0}$$ where $w_{ij}(t)$ is the $(i,j)$-th entry of $A(t)^T A(t)$. Summing up the variance differences $V(t)-V(t+1)$ over different values of $t$, we obtain $$V(kB) - V((k+1)B) = \sum_{t=kB}^{(k+1)B-1} \sum_{i<j} w_{ij}(t) (x_i(t)-x_j(t))^2. \label{firstvdec}$$ - For all $d\in\{1,\ldots,n-1\}$, let $t_d$ be the first time larger than or equal to $kB$ (if it exists) at which there is a communication between two nodes belonging to the two sets $\{1,\ldots,d\}$ and $\{d+1,\ldots,n\}$, to be referred to as a communication across the cut $d$. - For all $t\in \{kB,\ldots,(k+1)B-1\}$, let $D(t)=\{d\mid t_d=t\}$, i.e., $D(t)$ consists of “cuts" $d\in \{1,\ldots,n-1\}$ such that time $t$ is the first communication time larger than or equal to $kB$ between nodes in the sets $\{1,\ldots,d\}$ and $\{d+1,\ldots,n\}$. Because of Assumption \[weakconnect\], the union of the sets $D(t)$ includes all indices $1,\ldots,n-1$, except possibly for indices for which $x_d(kB)=x_{d+1}(kB)$. - For all $d\in \{1,\ldots,n-1\}$, let $C_d=\{(i,j) \alexo{,~(j,i)} \mid i\leq d, \ d+1\leq j\}$. - For all $t\in \{kB,\ldots,(k+1)B-1\}$, let $F_{ij}(t) =\{d\in D(t)\ |\ (i,j) \alexo{\mbox{ or } (j,i)} \in C_d\}$, i.e., $F_{ij}(t)$ consists of all cuts $d$ such that the edge $(i,j)$ at time $t$ is the first communication across the cut at a time larger than or equal to $kB$. - To simplify notation, let $y_i=x_i(kB)$. By assumption, we have $y_1\geq\cdots\geq y_n$. We make two observations, as follows: - Suppose that $d\in D(t)$. Then, for some $(i,j)\in C_d$, we have either $a_{ij}(t)>0$ or $a_{ji}(t)>0$. by Lemma \[db\], we obtain $$\sum_{(i,j)\in C_d} w_{ij}(t)\geq \frac{\eta}{2}. \label{eq:w}$$ - Fix some $(i,j)$, with $i<j$, and time $t\in \{kB,\ldots,(k+1)B-1\}$, and suppose that $F_{ij}(t)$ is nonempty. Let $F_{ij}\ao{(t)}=\{d_1,\ldots,d_k\}$, where the $d_j$ are arranged in increasing order. Since $d_1\in F_{ij}\ao{(t)}$, we have $d_1\in D(t)$ and therefore $t_{d_1}=t$. By the definition of $t_{d_1}$, this implies that there has been no communication between a node in $\{1,\ldots,d_1\}$ and a node in $\{d_1+1,\ldots,n\}$ during the time interval $[kB,t-1]$. It follows that $x_i(t) \geq y_{d_1}$. By a symmetrical argument, we also have $$x_j(t)\leq y_{d_k+1}.\label{cutbound}$$ These relations imply that $$x_i(t)-x_j(t) \geq y_{d_1}-y_{d_k+1} \ao{\geq } \sum_{d\in F_{ij}\ao{(t)}} (y_d-y_{d+1}),$$ Since the components of $y$ are sorted in nonincreasing order, we have $y_d-y_{d+1}\geq 0$, for every $d\in F_{ij}\ao{(t)}$. For any nonnegative numbers $z_i$, we have $$(z_1+\cdots+z_k)^2\geq z_1^2+\cdots+z_k^2,$$ which implies that $$(x_i(t)-x_j(t))^2 \geq \sum_{d\in F_{ij}\ao{(t)}} (y_d-y_{d+1})^2. \label{eq:dx}$$ We now use these two observations to provide a lower bound on the expression on the right-hand side of Eq. (\[firstvdec0\]) at time $t$. We use Eq. (\[eq:dx\]) and then Eq. (\[eq:w\]), to obtain $$\begin{aligned} \sum_{i<j} w_{ij}(t)(x_i(t)-x_j(t))^2& \geq& \sum_{i<j} w_{ij}(t) \sum_{d\in F_{ij}\ao{(t)}} (y_d-y_{d+1})^2\\ &=&\sum_{d\in D(t)} \sum_{(i,j)\in C_d} w_{ij}(t) (y_d-y_{d+1})^2\\ &\geq& \frac{\eta}{2} \sum_{d\in D(t)}(y_d-y_{d+1})^2.\end{aligned}$$ We now sum both sides of the above inequality for different values of $t$, and use Eq. (\[firstvdec\]), to obtain $$\begin{aligned} V(kB)-V((k+1)B)&=& \sum_{t=kB}^{(k+1)B-1} \sum_{i<j} w_{ij}(t)(x_i(t)-x_j(t))^2\\ &\geq& \frac{\eta}{2} \sum_{t=kB}^{(k+1)B-1}\sum_{d\in D(t)}(y_d-y_{d+1})^2\\ &=&\frac{\eta}{2}\sum_{d=1}^{n-1} (y_d-y_{d+1})^2,\end{aligned}$$ where the last inequality follows from the fact that the union of the sets $D(t)$ is only missing those $d$ for which $y_d=y_{d+1}$. We next establish a bound on [the]{} variance decrease that plays [a]{} key role in our convergence analysis. \[lboundvar\] Let Assumptions \[weights\] and \[weakconnect\] hold, and suppose that $V(kB)>0$. Then, $$\frac{V(kB)-V((k+1)B)}{V(kB)} \geq \frac{\eta}{2 n^2}\qquad \hbox{for all }k.$$ [Without loss of generality, we assume that the components of $x(kB)$ have been sorted in nonincreasing order.]{} By Lemma \[vardiff\], we have $$V(kB) - V((k+1)B) \geq \frac{\eta}{2}\, \sum_{i=1}^{n-1} (x_{i}(kB) - x_{i+1}(kB))^2.$$ This implies that $$\frac{V(kB)-V((k+1)B)}{V(kB)}\ge\frac{\eta}{2} \, \frac{\sum_{i=1}^{n-1} (x_{i}(kB) - x_{i+1}(kB))^2}{\sum_{i=1}^n (x_i(kB)-\bar{x}(kB))^2}.$$ Observe that the right-hand side does not change when we add a constant to every $x_i(kB)$. We can therefore assume, without loss of generality, that $\bar{x}(kB)=0$, [so that]{} $$\frac{V(kB)-V((k+1)B)}{V(kB)} \geq \frac{\eta}{2}\, \min_{{x_1\ge x_2\ge \cdots\ge x_n \atop \sum_i x_i=0}} \frac{\sum_{i=1}^{n-1} (x_{i} - x_{i+1})^2}{\sum_{i=1}^n x_i^2}.$$ Note that the right-hand side is unchanged if we multiply each $x_i$ by the same constant. Therefore, we can assume, without loss of generality, that $\sum_{i=1}^n x_i^2=1$, [so that]{} $$\label{eq:vv} \frac{V(kB)-V((k+1)B)}{V(kB)} \geq \frac{\eta}{2}\, \min_{{x_1\ge x_2\ge \cdots\ge x_n \atop \sum_i x_i=0, \ \sum_i x_i^2 = 1}} ~~\sum_{i=1}^{n-1} (x_{i} - x_{i+1})^2.$$ The requirement $\sum_i x_i^2 = 1$ implies that the average value of $x_i^2$ is $1/n$, which implies that there exists some $j$ such that $|x_j| \ge 1/\sqrt{n}$. Without loss of generality, let us suppose that this $x_j$ is positive.[^6] The rest of the proof relies on a technique from [@LO81] to provide a lower bound on the right-hand side of Eq. (\[eq:vv\]). Let $$z_i = x_{i} - x_{i+1} \ \ \hbox{for } i<n,\quad \hbox{and}\quad z_n=0.$$ Note that $z_i \geq 0$ for all $i$ Since $x_j\ge 1/\sqrt{n}$ for some $j$, Combining with [Eq. (\[eq:vv\])]{}, we obtain $$\frac{V(kB)-V((k+1)B)}{V(kB)} \geq \frac{\eta}{2} \min_{z_i \geq 0,\ \sum_i z_i \geq 1/\sqrt{n}} \sum_{i=1}^{n} z_i^2.$$ [The minimization problem on the right-hand side is a symmetric convex optimization problem, and therefore has a symmetric optimal solution, namely]{} $z_i = 1/n^{1.5}$ for all $i$. This results in an optimal value of $1/n^2$. Therefore, $$\frac{V(kB)-V((k+1)B)}{V(kB)} \geq \frac{\eta}{2 n^2},$$ which [is]{} the desired result. We are now ready for our main result, which establishes that the convergence time of the sequence of vectors $x(k)$ generated by Eq.(\[noquant\]) is of order $O(n^2B/\eta)$. \[uqbound\] Let Assumptions \[weights\] and \[weakconnect\] hold. Then, there exists an absolute constant[^7] $c$ such that we have $$V({k}) \leq \epsilon V(0)\qquad \hbox{for all } k\ge {c} (n^2/\eta)B \log (1/\epsilon).$$ The result follows immediately from Lemma \[lboundvar\]. Recall that, according to Lemma \[as2implas3\], Assumption \[connectivity\] implies Assumption \[weakconnect\]. In view of this, the convergence time bound of Theorem \[uqbound\] holds for any $n$ and any sequence of weights satisfying Assumptions \[weights\] and \[connectivity\]. In the next subsection, we show that this bound is tight when the stronger Assumption \[connectivity\] holds. Tightness --------- The next shows that the convergence time bound of Theorem \[uqbound\] is tight under Assumption \[connectivity\]. \[unquanttight\] [There exist constants $c$ and $n_0$ with the following property. For any $n\geq n_0$,]{} nonnegative integer $B$, $\eta < 1/2$, and $\epsilon < 1$, there exist a sequence of weight matrices $A(k)$ satisfying Assumptions \[weights\] and \[connectivity\], and an initial value $x(0)$ such that [if]{} $V(k)/V(0) \leq \epsilon$, [then]{} $$k \geq {c}\, \frac{n^2}{\eta}\, B \log \frac{1}{\epsilon}.$$ Let $P$ be the circulant shift operator defined by $P e_i = e_{i+1}$, $P e_n = e_1$, where $e_i$ is a unit vector with the $i$-th entry equal to 1, and all other entries equal to 0. Consider the symmetric circulant matrix defined by $$A = (1 - 2 \eta) I + \eta P + \eta P^{-1}.$$ Let $A({k})=A$, when ${k}$ is a multiple of $B$, and $A({k})=I$ otherwise. Note that this sequence satisfies Assumptions 1 and 2. The second largest eigenvalue of $A$ is $$\lambda_2(A) = 1 - 2 \eta + 2 \eta \cos \frac{2 \pi}{n},$$ (see Eq. (3.7) of [@G06]). Therefore, using the inequality $\cos x \geq 1- x^2/2$, $$\lambda_2(A) \geq 1 - \frac{4 \eta \pi^2}{n^2}.$$ [For $n$ large enough,]{} the quantity on the right-hand side is nonnegative. Let the initial [vector]{} $x(0)$ be the eigenvector corresponding to [$\lambda_2(A)$.]{} Then, $$\frac{V(kB)}{V(0)} = \lambda_2(A)^{2k} \geq \Big(1 - \frac{8 \eta \pi^2}{n^2}\Big)^k.$$ [For the right-hand side to become]{} less than $\epsilon$, we need $k = \Omega( (n^2/\eta) \log (1/\epsilon))$. [This implies that for $V(k)/V(0)$ to become less than $\epsilon$, we need]{} $k=\Omega( (n^2/\eta) B \log (1/\epsilon))$. Saving a [factor of]{} $n$: faster averaging on undirected graphs {#matrixpicking} ================================================================= In the previous section, we have shown that a large class of averaging algorithms have $O(B (n^2/\eta) ~\alexo{ \log 1/\epsilon}) $ In this section, we consider decentralized ways of synthesizing the weights $a_{ij}(k)$ while satisfying Assumptions \[weights\] and \[weakconnect\]. We assume that the communications of the nodes are governed by an exogenous sequence of graphs $\Gb(k)=(N,\Eb(k))$ that provides strong connectivity over time periods of length $B$. in particular, we require that $a_{ij}(k)=0$ if $(j,i)\notin\Eb(k)$. Naturally, we assume that $(i,i)\in\Eb(k)$ for every $i$. Several such decentralized protocols exist. For example, each node may assign $$\begin{aligned} a_{ij}{(k)} & = & \epsilon, \qquad\qquad \mbox{ \ \ if } (j,i) \in \Eb(k) {\mbox{ and } i\neq j,} \\ a_{ii}{(k)} & = & 1 - \epsilon\cdot \mbox{deg}(i),\end{aligned}$$ where [deg]{}($i$) is the degree of $i$ in $\Gb(k)$. If $\epsilon$ is small enough and the graph $\Gb(k)$ is undirected [\[i.e., $(i,j)\in\Eb(k)$ if and only if $(j,i)\in\Eb(k)$\],]{} this results in a nonnegative, doubly stochastic matrix (see [@OSM04]). However, [if a node has $\ao{\Theta}(n)$ neighbors, $\eta$ will be of order $\ao{\Theta}(1/n)$, resulting in $\ao{\Theta}(n^3)$ convergence time.]{} Moreover, this argument applies to all protocols in which nodes assign equal weights to all their neighbors; see [@XB04] and [@BFT05] for more examples. In this section, we examine whether it is possible to synthesize the weights $a_{ij}(k)$ in a decentralized manner, so that $a_{ij}(k)\geq \eta$ whenever $a_{ij}(k)\neq 0$, where $\eta$ is a positive constant independent of $n$ and $B$. We show that this is indeed possible, under the additional assumption that the graphs $\Gb(k)$ are undirected. Our algorithm is data-dependent, in that $a_{ij}(k)$ depends not only on the graph $\Gb(k)$, but also on the data vector $x(k)$. Furthermore, it is a decentralized 3-hop algorithm, in that $a_{ij}{(k)}$ depends only on the data at nodes within a distance of at most $3$ from $i$. Our algorithm is such that the resulting sequences of vectors $x(k)$ and graphs $G(k)=(N,\E(k))$, with $\E(k)=\{(j,i)\mid a_{ij}(k)> 0\}$, satisfy Assumptions \[weights\] and \[weakconnect\]. Thus, a convergence time result can be obtained from Theorem \[uqbound\]. The algorithm ------------- The algorithm we present here is a variation of an old [*load balancing*]{} algorithm (see [@C89] and Chapter 7.3 of [@BT89]).[^8] At each step of the algorithm, each node offers some of its value to its neighbors, and accepts or rejects such offers from its neighbors. Once an offer from $i$ to $j$, of size $\delta>0$, has been accepted, the updates $x_i \leftarrow x_i - \delta$ and $x_j \leftarrow x_j + \delta$ are executed. We next describe the formal steps the nodes execute at each time ${k}$. For clarity, we refer to the node executing the steps below as node $\ao{C}$. Moreover, the instructions below sometimes refer to the neighbors of node $\ao{C}$; this always means current neighbors at time $k$, when the step is being executed, [as determined by the current graph $\Gb(k)$.]{} We assume that at each time $k$, all nodes execute these steps in the order described below, while the graph remains unchanged. [**Balancing Algorithm:**]{} 1. Node $\ao{C}$ broadcasts its current value $x_{\ao{C}}$ to all its neighbors. 2. Going through the values it just received from its neighbors, Node $\ao{C}$ finds the smallest value that is less than [its own]{}. Let $\ao{D}$ be a neighbor with this value. Node $\ao{C}$ makes an offer of $(x_{\ao{C}} - x_{\ao{D}})/3$ to node $D$. If no [neighbor of $\ao{C}$]{} has a value smaller than $x_{\ao{C}}$, node $C$ does nothing at this stage. 3. Node $\ao{C}$ goes through the incoming offers. It sends an acceptance to the sender of [a]{} largest offer, and a rejection to all the other senders. It updates the value of $x_{\ao{C}}$ by adding the value of the accepted offer. If node $C$ did not receive any offers, it does nothing at this stage. 4. If an acceptance arrives [for]{} the offer made by node $\ao{C}$, node $\ao{C}$ updates $x_{\ao{C}}$ by subtracting the value of the offer. Note that the new value of each node is a linear combination of the values of its neighbors. Furthermore, the weights $a_{ij}(k)$ are completely determined by the data and the graph at most $3$ hops from node $i$ in $\Gb(k)$. Performance analysis -------------------- . \[savingn\] Consider the balancing algorithm, and suppose that $\Gb(k)=(N,\Eb(k))$ is a sequence of undirected graphs such that $(N,\Eb(kB)\cup\Eb(kB+1)\cup \cdots \cup \Eb((k+1)B-1))$ is connected, for all integers $k$. There exists an absolute constant $c$ such that we have $$V({k}) \leq \epsilon V(0)\qquad \hbox{for all }k\ge {c} n^2B \log (1/\epsilon).$$ Note that with this algorithm, the new value at some node $i$ is a convex combination of the previous values of itself and its neighbors. Furthermore, the algorithm keeps the sum of the nodes’ values constant, because every accepted offer involves an increase at the receiving node equal to the decrease at the offering node. These two properties imply that the algorithm can be written in the form $$x(k+1) = A(k) x(k),$$ where $A(k)$ is a doubly stochastic matrix, determined by $\Gb(k)$ and $x(k)$. It can be seen that the diagonal entries of $A(k)$ are positive and, furthermore, all nonzero entries of $A(k)$ are larger than or equal to 1/3; thus, $\eta=1/3$. We claim that the algorithm \[in particular, the sequence $\E(A(k))$\] satisfies Assumption \[weakconnect\]. Indeed, suppose that at time $kB$, the nodes are reordered so that the values $x_i(kB)$ are nonincreasing in $i$. Fix some $d\in\{1,\ldots,n-1\}$, and suppose that $x_d(kB)\neq x_{d+1}(kB)$. Let $S^+=\{1,\ldots,d\}$ and $S^-=\{d+1,\ldots,n\}$. Because of our assumptions on the graphs $\Gb(k)$, there will be a first time $t$ in the interval $\{kB,\ldots,(k+1)B-1\}$, at which there is an edge in $\Eb(t)$ between some $i^*\in S^+$ and $j^*\in S^-$. Note that between times $kB$ and $t$, the two sets of nodes, $S^+$ and $S^-$, do not interact, which implies that $x_i(t)\geq x_d(kB)$, for $i\in S^+$, and $x_j(t) < x_d(kB)$, for $j\in S^-$. At time $t$, node $i^*$ sends an offer to a neighbor with the smallest value; let us denote that neighbor by $k^*$. Since $(i^*,j^*)\in \Eb(t)$, we have $x_{k^*}(t)\leq x_{j^*}(t)<x_d(kB)$, which implies that $k^*\in S^-$. Node $k^*$ will accept the largest offer it receives, which must come from a node with a value no smaller than $x_{i^*}(t)$, ; hence the latter node belongs to $S^+$. It follows that $\E(A(t))$ contains an edge between $k^*$ and some node in $S^{+}$, showing that Assumption \[weakconnect\] is satisfied. The claimed result follows from Theorem \[uqbound\], because we have shown that all of the assumptions in that theorem are satisfied . Quantization Effects \[qanalysis\] ================================== In this section, we consider a quantized version of the update rule (\[noquant\]). This model is a good approximation for a network of nodes communicating through [finite bandwidth channels, so that]{} at each time instant, only a finite number of bits can be transmitted. We incorporate this constraint in our algorithm by assuming that each node, upon receiving the values of its neighbors, computes the convex combination $\sum_{j=1}^{{n}} a_{ij}(k) x_j(k)$ and quantizes it. This update rule also captures [a]{} constraint that each node can only store quantized values. Unfortunately, [under Assumptions \[weights\] and \[connectivity\], if the output of Eq. (\[noquant\]) is rounded to the nearest integer, the sequence $x(k)$]{} is not guaranteed to converge to consensus; see [@KBS06]. We therefore choose a quantization rule that rounds the values down, according to $$\label{quantupdate} x_i(k+1) = \left\lfloor \sum_{j=1}^{{n}} a_{ij}(k) x_j(k) \right\rfloor,$$ where $\lfloor \cdot \rfloor$ represents rounding [*down*]{} to the nearest multiple of $1/Q$, and where $Q$ is some positive integer. We adopt the natural assumption that the initial values are [already]{} quantized. For all $i$, $x_i(0)$ is a multiple of $1/Q$.\[quantizedinitials\] For convenience we define $$U = \max_i x_i(0), \qquad L = \min_i x_i(0).$$ We use $K$ to denote the total number of [relevant]{} quantization levels, i.e., $$K = (U-L)Q,$$ which is an integer by Assumption \[quantizedinitials\]. A quantization level dependent bound ------------------------------------ We first present a [convergence time bound that depends on the quantization level $Q$.]{} \[simpleq\] Let Assumptions \[weights\], \[connectivity\], and \[quantizedinitials\] hold. Let $\{{x(k)}\}$ be generated by the update rule (\[quantupdate\]). If $k \geq nBK$, [then]{} all [components]{} of $x(k)$ are equal. Consider the nodes [whose initial value is $U$.]{} There are at most $n$ of them. [As long as]{} not all entries of ${x(k)}$ are equal, then every $B$ iterations, at least one [node]{} must use a value strictly less than $U$ [in]{} an update; node will have its value decreased to $U-1/Q$ or less. It follows that after $nB$ iterations, the largest node [value will be]{} at most $U-1/Q$. Repeating this argument, we [see]{} that at most $nBK$ iterations are possible before [all the nodes have]{} the same value. Although the above bound gives informative results for small $K$, it becomes weaker as [$Q$ (and, therefore, $K$)]{} increases. On the other hand, as $Q$ approaches infinity, the quantized system approaches the unquantized system; the availability of convergence time bounds for the unquantized system suggests that similar bounds should be possible for the quantized one. Indeed, in the next subsection, obtain a bound on the convergence time [which is]{} independent of the total number of quantization levels. A quantization level independent bound -------------------------------------- We adopt a slightly different measure of convergence for the analysis of the quantized consensus algorithm. For any $x\in\R^n$, we define $m(x)=\min_i x_i$ and $$\underline{V}(x)=\sum_{i=1}^n (x_i - m(x))^2.$$ We will also use the simpler notation $m(k)$ and $\underline{V}(k)$ to denote $m(x(k))$ and $\underline{V}(x(k))$, respectively, where it is more convenient to do so. The function $\underline{V}$ will be our Lyapunov function for the analysis of the quantized consensus algorithm. The reason for not using our earlier Lyapunov function, $V$, is that for the quantized algorithm, $V$ is not guaranteed to be monotonically nonincreasing in time. On the other hand, we have that $V(x)\leq \underline{V}(x) \leq \alexo{4} \ao{n}V(x)$ for any[^9] $x \in \R^n$. As a consequence, any convergence time bounds expressed in terms of $\underline{V}$ translate to essentially the same bounds expressed in terms of $V$, . Before proceeding, we [record an elementary fact which will allow us to]{} relate the variance decrease $V(x)-V(y)$ to the decrease, $\underline{V}(x)- \underline{V}(y)$, of our new Lyapunov function. The proof involves [simple algebra,]{} and is therefore omitted. \[sumlemma\] Let $u_1,\ldots,u_{n}$ and $w_1,\ldots,w_n$ be real numbers satisfying $$\sum_{i=1}^n u_i = \sum_{i=1}^n w_i.$$ [Then, the expression]{} $$f(z) = \sum_{i=1}^n (u_i - z)^2 - \sum_{i=1}^n (w_i - z)^2$$ is [a constant,]{} independent of the scalar $z$. Our next lemma places a bound on [the decrease of]{} the Lyapunov function $\underline{V}(t)$ between times $kB$ and $(k+1)B-1$. \[quantdiff\] Let Assumptions \[weights\], \[weakconnect\], and \[quantizedinitials\] hold. Let $\{x{(k)}\}$ be generated by the update rule (\[quantupdate\]). Suppose that the components $x_i(kB)$ of the vector $x(kB)$ have been ordered from largest to smallest, with ties broken arbitrarily. Then, we have $$\underline{V}(kB) - \underline{V}((k+1)B) \geq \frac{\eta}{2}\, \sum_{i=1}^{n-1} (x_{i}(kB) - x_{i+1}(kB))^2.$$ For all $k$, we view Eq. (\[quantupdate\]) as the composition of two operators: $$y(k) = A(k) x(k),$$ where $A(k)$ is [a]{} doubly stochastic matrix, and $$x(k+1) = \lfloor y(k) \rfloor,$$ where the quantization $\lfloor \cdot \rfloor$ is [carried out]{} componentwise. We apply Lemma \[sumlemma\] with the identification $u_i=x_{{i}}(k)$, $w_i=y_{{i}}(k)$. Since multiplication by a doubly stochastic matrix preserves the mean, the condition $\sum_i u_i = \sum_i w_i$ is satisfied. By considering two different choices for the scalar $z$, namely, $z_1=\bar{x}(k)=\bar{y}(k)$ and $z_2=m(k)$, we obtain $$\label{firststep} V(x(k)) - V(y(k)) = \underline{V}(\ao{x(k)}) - \sum_{i=1}^n (y_i(k)-m(k))^2.$$ Note that $x_i(k+1)-m(k)\leq y_i(k)-m(k)$. Therefore, $$\label{secondstep} \underline{V}(\ao{x(k)}) - \sum_{i=1}^n (y_i(k)-m(k))^2 \leq \underline{V}(\ao{x(k)}) - \sum_{i=1}^n (x_i(k+1)-m(k))^2.$$ Furthermore, note that $x_i(k+1)-m(k+1)\leq x_i(k+1)-m(k)$. Therefore, $$\label{thirdstep} \underline{V}(\ao{x(k)}) - \sum_{i=1}^n (x_i(k+1)-m(k))^2 \leq \underline{V}(\ao{x(k)}) - \underline{V}(\ao{x(k+1)}).$$ By combining Eqs. (\[firststep\]), (\[secondstep\]), and (\[thirdstep\]), we obtain $$V(x(t)) - V(y(t)) \leq \underline{V}(\ao{x(t)}) - \underline{V}(\ao{x(t+1)})\qquad \hbox{for all }t.$$ Summing the preceding relations over $t=kB,\ldots, (k+1)B-1$, we further obtain $$\sum_{t=kB}^{(k+1)B-1} \Big(V(x(t))-V(y(t))\Big) \leq \underline{V}(\ao{x(kB)}) - \underline{V}(\ao{x((k+1)B))}.$$ To complete the proof, we provide a lower bound on the expression $$\sum_{t=kB}^{(k+1)B-1} \Big(V(x(t)) - V(y(t))\Big).$$ Since $y(t)=A(t)x(t)$ for all $t$, it follows from Lemma \[vl\] that for any $t$, $$V(x(t))-V(y(t)) = \sum_{i<j} w_{ij}(t) (x_i(t)-x_j(t))^2,$$ where $w_{ij}(t)$ is the $(i,j)$-th entry of $A(t)^T A(t)$. Using this relation and following the same line of analysis used in the proof of Lemma \[vardiff\] \[where the relation $\alexo{x_i(t) \geq y_{d_1}}$ holds in view of the assumption that $x_i(kB)$ is a multiple of $1/Q$ for all $k\ge 0$, cf. Assumption \[quantizedinitials\]\] , we obtain the desired result. The next theorem contains our main result on the convergence time of the quantized algorithm. \[qbound\] Let Assumptions \[weights\], \[weakconnect\], and \[quantizedinitials\] hold. Let $\{x{(k)}\}$ be generated by the update rule (\[quantupdate\]). Then, there exists an absolute constant $c$ such that we have $$\underline{V}(k) \leq \epsilon \underline{V}(0)\qquad \hbox{for all } k\ge c\, (n^2/\eta) B \log(1/\epsilon).$$ Let us assume that $\underline{V}(kB)>0$. From Lemma \[quantdiff\], we have $$\underline{V}(kB)- \underline{V}((k+1)B) \geq \frac{\eta}{2} \sum_{i=1}^{n-1} (x_{i}(kB) - x_{i+1}(kB))^2,$$ where [the components]{} $x_i(kB)$ are ordered from largest to smallest. Since $\underline V(kB) = \sum_{i=1}^n (x_i(kB) - x_n(kB))^2$, we have $$\frac{\underline{V}(kB)-\underline{V}((k+1)B)}{\underline{V}(kB)} \geq \frac{\eta}{2} \frac{\sum_{i=1}^{n-1} (x_{i}(kB) - x_{i+1}(kB))^2}{\sum_{i=1}^n (x_i(kB)-x_n(kB))^2}.$$ [Let]{} $y_i = x_i(kB)-x_n(kB)$. Clearly, $y_i \geq 0$ for all $i$, and $y_n=0$. Moreover, the monotonicity of $x_i(kB)$ implies the monotonicity of $y_i$: $$y_1 \geq y_2 \geq \cdots \geq y_n=0.$$ Thus, $$\frac{\underline{V}(kB)-\underline{V}((k+1)B)}{\underline{V}(kB)} \geq \frac{\eta}{2} \min_{{y_1 \geq y_2\ge \cdots\ge y_n\atop y_n=0}}\frac{\sum_{i=1}^{n-1} (y_i - y_{i+1})^2}{\sum_{i=1}^n y_i^2}.$$ $$\frac{\eta}{2} \min_{{y_1 \geq y_2\geq \cdots \geq y_n\atop \sum_i y_i^2=1}} \sum_{i=1}^{n-1} (y_i - y_{i+1})^2 \geq \frac{\eta}{2} \min_{z_i \geq 0, \sum_i z_i \geq 1/\sqrt{n}} \sum_{i=1}^n z_i^2.$$ The minimization problem on the right-hand side has an optimal value of at least $1/n^2$, and the desired result follows. Extensions and modifications ---------------------------- In this subsection, we comment briefly on some corollaries of Theorem \[qbound\]. First, we note that the results of Section \[matrixpicking\] immediately carry over to the quantized case. Indeed, in Section \[matrixpicking\], we showed how to pick the weights $a_{ij}(k)$ in a decentralized manner, based only on local information, so that Assumptions 1 and \[weakconnect\] are satisfied, with $\eta \geq 1/3$. When using a quantized version of the balancing algorithm, we once again \[savingnq\] For the quantized version of the balancing algorithm, and under the same assumptions as in Theorem \[savingn\], if $k\geq c\, n^2B \log (1/\epsilon))$, then $\underline{V}(k ) \leq \epsilon \underline{V}(0)$, where $c$ is an absolute constant. Second, we note that Theorem \[qbound\] [can be used]{} to obtain a bound on the time until [the values of all nodes are equal. Indeed, we observe that in the presence of quantization, once the condition $\underline{V}(k) <1/Q^2$ is satisfied, all components of $x(k)$ must be equal.]{} \[qboundequal\] Consider the quantized algorithm (\[quantupdate\]), and assume that Assumptions \[weights\], \[weakconnect\], and \[quantizedinitials\] hold. If $k\geq c(n^2/\eta) B \big[\log Q + \log \underline{V}(0)\big]$, then all components of $x(k)$ are equal, where $c$ is an absolute constant. Tightness --------- [We now]{} show that the bound in Theorem \[qbound\] is tight, Assumption \[weakconnect\] is replaced with Assumption \[connectivity\]. \[quanttight\] There absolute constant $c$ with the following property. For [any]{} nonnegative integer $B$, $\eta < 1/2$, $\epsilon < 1$, and , there exist a sequence of weight matrices $A(k)$ satisfying Assumptions \[weights\] and \[connectivity\], and an initial value $x(0)$ satisfying Assumption \[quantizedinitials\], such that under the dynamics of Eq. (\[quantupdate\]), [if]{} $\underline{V}(k)/\underline{V}(0) \leq \epsilon$, [then]{} $$k\geq c\, \frac{n^2}{\eta}\, B \log \frac{1}{\epsilon}.$$ We have demonstrated in \[unquanttight\] a similar result for the unquantized algorithm. Namely, we have shown that for [$n$ and for any]{} $B$, $\eta<1/2$, and $\epsilon<1$, there exists a weight sequence $a_{ij}(k)$ and an initial [vector]{} $x(0)$ such that the first time when $V(t) \leq \epsilon V(0)$ occurs after $\Omega((n^2/\eta) B \log (1/\epsilon))$ steps. Let $T^*$ be this first time. [Consider the quantized algorithm under]{} the exact same sequence $a_{ij}(k)$, [initialized at $\lfloor x(0) \rfloor$.]{} Let $\hat{x}_i(t)$ refer to the value of node $i$ at time $t$ in the quantized algorithm [under]{} this scenario, [as opposed to]{} $x_i(t)$ which [denotes the]{} value in the unquantized algorithm. Since quantization can [only]{} decrease a nodes value by at most $1/Q$ at each iteration, it is easy to show, by induction, that $$x_i(t) \geq \hat{x}_i(t) \geq x_i(t) - t/Q$$ We can pick $Q$ large enough so that, [for $t < T^*$,]{} the vector $\hat{x}(t)$ is [as close as desired to]{} $x(t)$. Therefore, for $t < T^*$ and for large enough $Q$, $\underline{V}(\hat{x}(t))/\underline{V}(\hat{x}(0))$ will be arbitrarily close to $\underline{V}(x(t))/\underline{V}(x(0))$. [From the proof of \[unquanttight\], we see that]{} $x(t)$ is always a [scalar]{} multiple of $x(0)$. Since $\underline{V}(x)/V(x)$ is invariant under multiplication by a constant, it follows that $\underline{V}(x(t))/\underline{V}(x(0)) = V(x(t))/V(x(0))$. Since this last quantity is above $\epsilon$ for $t<T^*$, it follows that provided $Q$ is large enough, $\underline{V}(\hat{x}(t))/\underline{V}(\hat{x}(0))$ is also above $\epsilon$ for $t<T^*$. This proves the proposition. Quantization error ------------------ Despite favorable convergence properties of our quantized [averaging]{} algorithm (\[quantupdate\]), the update rule does not preserve the average of the values at each iteration. Therefore, the [common limit of the sequences $x_i(k)$, denoted by $x_f$,]{} need not be equal to the exact average of the initial values. We next provide an upper bound on the error between [$x_f$]{} and the initial average, as a function of the number of quantization levels. \[qerror\] There is an absolute constant $c$ such that for the common limit $x_f$ of the values $x_i(k)$ generated by the quantized algorithm (\[quantupdate\]), we have $$\left|x_f - \frac{1}{n} \sum_{i=1}^n x_i(0)\right| \le {c\over Q} \ {n^2\over\eta}\, B\log (Qn(U-L)).$$ By \[qboundequal\], after $O\Big((n^2/\eta) B \log (Q \underline{V}(x(0)))\Big)$ iterations, all nodes will have the same value. Since $\underline{V}(x(0)))\le n(U-L)^2$ and the average decreases by at most $1/Q$ at each iteration, the result follows. Let us assume that the parameters $B$, $\eta$, and $U-L$ are fixed. \[qerror\] implies that as $n$ increases, the number of bits [used for each communication,]{} which is proportional to $\log Q$, needs to grow only as $O(\log n)$ to make the error negligible. Furthermore, this is true even if the [parameters]{} $B$, ${1}/{\eta}$, and $U-L$ grow polynomially in $n$. [For a converse, it can]{} be seen that ${\Omega}(\log n)$ bits are needed. Indeed, consider $n$ nodes, with $n/2$ [nodes initialized at]{} $0$, and $n/2$ [nodes initialized at]{} $1$. [Suppose]{} that ${Q} < n/2$; we connect the nodes by [forming]{} a complete subgraph over all the nodes with value $0$ and exactly [one]{} node with value $1$; see Figure \[ic\] for an example with $n=6$. Then, each node [forms the]{} average [of]{} its neighbors. This brings one of the nodes with [an initial value of]{} $1$ down to $0$, without raising the value of any [other]{} nodes. We can repeat this [process,]{} to bring all of the nodes with [an initial value of]{} $1$ down to $0$. Since the true average is $1/2$, the final result is $1/2$ away from the true average. [Note now that $Q$ can]{} grow linearly with $n$, and still satisfy the inequality ${Q}<n/2$. [Thus,]{} the number of bits can grow as $\Omega(\log n)$, and yet, independent of $n$, the error remains $1/2$. ![\[ic\] Initial configuration. Each node takes the average value of its neighbors. ](newlb.eps){width="12cm"} Conclusions =========== We studied distributed algorithms for the [averaging]{} problem over networks with time-varying topology, [with a focus on]{} tight bounds on the convergence time of a general class of [averaging]{} algorithms. We first considered [algorithms for the case where]{} agents can exchange and store continuous values, [and established tight convergence time bounds.]{} We next studied averaging algorithms under the additional constraint that agents can [only]{} store and send quantized values. We showed that these algorithms guarantee convergence of the agents values to consensus within some error from the average of the initial values. We provided a bound on the error that highlights the dependence on the number of quantization levels. Our paper is a contribution to the growing literature on distributed control of multi-agent systems. Quantization effects are an integral part of such systems but, with the exception of a few recent studies, have not attracted much attention in the vast literature on this subject. In this paper, we studied a quantization scheme that guarantees consensus at the expense of some error from the initial average value. We used this scheme to study the effects of the number of quantization levels on the convergence time of the algorithm and the distance from the true average. The framework provided in this paper motivates a number of further research directions: - The algorithms studied in this paper assume that there is no delay in receiving the values of the other agents, which is a restrictive assumption in network settings. Understanding the convergence of averaging algorithms and implications of quantization in the presence of delays is an important topic for future research. - We studied a quantization scheme with favorable convergence properties, that is, rounding down to the nearest quantization level. Investigation of other quantization schemes and their impact on convergence time and error is left for future work. - The quantization algorithm we adopted implicitly assumes that the agents can carry out computations with continuous values, but can store and transmit only quantized values. Another interesting area for future work is to incorporate the additional constraint of finite precision computations into the quantization scheme. - [^1]: A. Nedić is with the Industrial and Enterprise Systems Engineering Department, University of Illinois at Urbana-Champaign, Urbana IL 61801 (e-mail:angelia@illinois.edu) [^2]: A. Olshevsky, A. Ozdaglar, and J. N. Tsitsiklis are with the Laboratory for Information and Decision Systems, Electrical Engineering and Computer Science Department, Massachusetts Institute of Technology, Cambridge MA, 02139 (e-mails: alex\_o@mit.edu, asuman@mit.edu, jnt@mit.edu) [^3]: This research was partially supported by the National Science Foundation under grants ECCS-0701623, CMMI 07-42538, and DMI-0545910, and by DARPA ITMANET program [^4]: [^5]: In the sequel, the notation $\sum_{i<j}$ will be used to denote the double sum $\sum_{j=1}^n\sum_{i=1}^{j-1}$. [^6]: Otherwise, we can replace $x$ with $-x$ and subsequently reorder to maintain the property that the components of $x$ are in descending order. It can be seen that these operations do not affect the objective value. [^7]: [^8]: This algorithm was also considered in [@OT06], but in the absence of a result such as Theorem \[uqbound\], a weaker convergence time bound was derived. [^9]:

--- abstract: 'We investigate the relationship between the quenching of star formation and the structural transformation of massive galaxies, using a large sample of photometrically-selected post-starburst galaxies in the UKIDSS UDS field. We find that post-starburst galaxies at high-redshift ($z>1$) show high Sérsic indices, significantly higher than those of active star-forming galaxies, but with a distribution that is indistinguishable from the old quiescent population. We conclude that the morphological transformation occurs before (or during) the quenching of star formation. Recently quenched galaxies are also the most compact; we find evidence that massive post-starburst galaxies (M$_{\ast}> 10^{10.5} ~$M$_{\sun}$) at high redshift ($z>1$) are on average smaller than comparable quiescent galaxies at the same epoch. Our findings are consistent with a scenario in which massive passive galaxies are formed from three distinct phases: (1) gas-rich dissipative collapse to very high densities, forming the proto-spheroid; (2) rapid quenching of star formation, to create the “red nugget” with post-starburst features; (3) a gradual growth in size as the population ages, perhaps as a result of minor mergers.' author: - | Omar Almaini$^{1}$[^1], Vivienne Wild$^{2}$, David T. Maltby$^{1}$, William G. Hartley$^{3}$, Chris Simpson$^{4}$, Nina A. Hatch$^{1}$, Ross J. McLure$^{5}$, James S. Dunlop$^{5}$, Kate Rowlands$^{2}$\ $^{1}$School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, U.K.\ $^2$School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, KY16 9SS, U.K.\ $^3$ETH Zürich, Institut für Astronomie, HIT J 11.3, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland\ $^4$Gemini Observatory, Northern Operations Center, 670 N. A’ohuku Place, Hilo, HI96720, USA\ $^5$Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, U.K.\ date: 'Accepted 2017 July 28. Received 2017 July 24; in original form 2016 February 3.' title: 'Massive post-starburst galaxies at $z>1$ are compact proto-spheroids' --- \[firstpage\] galaxies: evolution – galaxies: formation – galaxies: fundamental parameters – galaxies: structure – galaxies: high-redshift Introduction {#sec:intro} ============ Galaxies in the local Universe display a striking bimodality in their morphological and spectral characteristics; massive galaxies (M$_{\ast}> 10^{10.5} ~$M$_{\sun}$) are typically spheroidal with old stellar populations, while lower mass galaxies are typically disc-dominated with blue, younger stellar populations (e.g. Strateva et al. 2001; Hogg et al. 2002). Deep surveys have revealed that the most massive galaxies were formed at high redshift ($z>1$; e.g. Fontana et al. 2004; Kodama et al. 2004; Cirasuolo et al. 2010), but we still do not understand why their star formation was abruptly terminated. Feedback from AGN (e.g. Silk & Rees 1998; Hopkins et al. 2005) or starburst-driven superwinds (e.g. Diamond-Stanic et al. 2012) are leading contenders for rapidly quenching distant galaxies, while jet-mode AGN feedback may be required to maintain the “red and dead” phase (e.g. Best et al. 2006). In addition to the quenching of star formation, massive galaxies must also undergo a dramatic structural transformation, to produce the spheroid-dominated population we see today. The transition appears to occur at $z>1$ for most galaxies with M$_{\ast}> 10^{10.5} ~$M$_{\sun}$ (Mortlock et al. 2013; Bruce et al. 2014), but it is unclear if quenching and structural transformation occurred during the same event. Over the last 10 years it has also emerged that quiescent galaxies were significantly more compact in the early Universe compared to the present day (e.g. Daddi et al. 2005; Trujillo et al. 2006; van Dokkum et al. 2008; Belli et al. 2014). Plausible explanations for the dramatic size growth include minor mergers (e.g. Bezanson et al. 2009; Naab, Johansson & Ostriker 2009) or expansion due to mass loss (Fan et al. 2008). For the population of quiescent galaxies as a whole, however, there may also be an element of progenitor bias; galaxies quenched at lower redshift tend to be larger than their counterparts at early times, which may drive much of the observed size evolution (e.g. Poggianti et al. 2013; Carollo et al. 2013). From a theoretical perspective, the formation of ultra-compact massive spheroids requires the concentration of vast reservoirs of cool gas via dissipation, which can radiate and collapse to very high densities. A variety of models have arisen to explain spheroid formation in detail, with some invoking gas-rich mergers (e.g. Hopkins et al. 2009; Wellons et al. 2015) while others use the inflow of gas through cold streams, feeding an extended disc that eventually becomes unstable and contracts (e.g. Dekel et al. 2009; Zolotov et al. 2015). Outflows driven by AGN or star formation may then terminate the star formation by expelling the remainder of the gas (e.g. Hopkins et al. 2005). In this work we explore the relationship between the quenching of distant galaxies and their structural transformation. We focus in particular on the rare class of “post-starburst” (PSB) galaxies, which are observed a few hundred Myr after a major episode of star formation was rapidly quenched. In the local Universe PSBs are identified from characteristic strong Balmer absorption lines (Dressler & Gunn 1983; Wild et al. 2009), due to the strong contribution from A stars, but until recently very few were spectroscopically identified at $z>1$ (e.g. Vergani et al. 2010). Two photometric methods have therefore been developed to identify this population. Whitaker et al. (2012) used medium-band photometry from the NEWFIRM Medium-Band Survey to identify “young red-sequence” galaxies using rest-frame UVJ colour-colour diagrams. In Wild et al. (2014) an alternative approach was used, based on a Principal Component Analysis (PCA) applied to the deep multi-wavelength photometry in the UDS field. Three spectral shape parameters (“supercolours”) were found to provide a compact representation for a wide range of photometric SEDs. In addition to cleanly separating quiescent and star-forming galaxies, the PCA method identifies “post-starburst” galaxy candidates in a distinct region of supercolour space, corresponding to galaxies in which a significant amount of mass was formed within the last Gyr but then rapidly quenched. The method was recently verified with deep 8m spectroscopy from VLT, which established that between 60% and 80% of photometric candidates are spectroscopically confirmed post-starburst galaxies, depending on the specific criteria adopted (Maltby et al. 2016)[^2]. In terms of completeness, the photometric method was found to identify approximately 60% of galaxies that would be spectroscopically classified as PSBs. Overall, these figures confirm that photometric PCA techniques can be used to identify large and relatively clean samples of recently quenched galaxies. So far the spectroscopic confirmation is restricted to $z<1.4$, given the requirement to detect H$\delta$ with optical spectroscopy. Future near-infrared spectroscopy (e.g. with the [*James Webb Space Telescope*]{}) will allow a detailed investigation of the population at higher redshift. The identification of large PSB samples has allowed the first study of the PSB galaxy mass function and its evolution to $z=2$ (Wild et al. 2016). Strong evolution was observed, with the implication that a large fraction of massive galaxies are rapidly quenched and pass through a PSB phase. In this paper we use the unique PCA sample described in Wild et al. (2016) to explore the structural properties of post-starburst galaxies. As newly quenched systems, our primary aim is to investigate if this population is structurally similar to star-forming galaxies at the same epoch, or if they already show evidence for the compact spheroid-dominated morphology of well-established quiescent galaxies. We assume a cosmology with $\Omega_M=0.3$, $\Omega_\Lambda=0.7$ and $h=0.7$. All magnitudes are given in the AB system. Data and sample selection ========================= The UDS K-band galaxy sample ---------------------------- Our study is based on deep $K$-band imaging from the UKIRT Infrared Deep Sky Survey (UKIDSS; Lawrence et al. 2007) Ultra-Deep Survey (UDS; Almaini et al., in preparation). The UDS is the deepest of the UKIDSS surveys, covering $0.77$ square degrees in the $J$, $H$ and $K$ bands. We use the 8th UDS data release (Hartley et al. 2013), reaching depths of $J=24.9$, $H=24.2$ and $K=24.6$ (AB, 5$\sigma$, 2-arcsec apertures). The final UDS data release (June 2016) achieved estimated depths $J=25.4, H=24.8, K=25.3$, and will be used to extend our PSB studies in future work. To complement the near-infrared imaging from UKIDSS, the UDS has deep optical coverage from [*Subaru*]{} Suprime-CAM, to depths of $B= 27.6, ~ V= 27.2, ~ R = 27.0, ~ i^{\prime} = 27.0$ and $ z^{\prime} = 26.0$ (AB, 5$\sigma$, $2$ arcsec), as described in Furusawa et al. (2008). Additional $u^{\prime}$-band imaging is provided by the Canada-France-Hawaii Telescope (CFHT) MegaCam instrument, reaching $ u^{\prime} = 26.75$ (AB, 5$\sigma$, $2$ arcsec). Deep imaging at longer near-infrared wavelengths is provided by the [*Spitzer*]{} UDS Legacy Program (SpUDS; PI: Dunlop), achieving depths of 24.2 and 24.0 (AB) using the IRAC camera at 3.6$\mu$m and 4.5$\mu$m respectively. The resulting area with full multiwavelength coverage, following the masking of bright stars and artefacts, is 0.62 square degrees. Further details on the construction of the multiwavelength DR8 catalogue can be found in Hartley et al. (2013). We determined photometric redshifts using the techniques outlined in Simpson et al. (2013). The 11-band photometric data were fit using a grid of galaxy templates, assembled using simple stellar populations from Bruzual & Charlot (2003; hereafter BC03), with a logarithmic spacing of ages between 30 Myr and 10 Gyr, and the addition of younger templates with dust-reddened spectral energy distributions (SEDs). The additional templates consist of a mildly reddened ($A_V=0.25$ mag) version of the two youngest templates, plus a version of the 30 Myr template with heavier reddening ($A_V=1.0$ mag). The resulting photometric redshifts show a normalized median absolute deviation $\sigma_{{\rm NMAD}} = 0.027$. The stellar masses used in this work differ slightly from those presented in Simpson et al. (2013), and instead are based on supercolour templates (see Section \[sec:masses\]). As noted in Section \[sec:uncertainties\], we investigated a range of alternative stellar masses, including those from Simpson et al. (2013), and found no significant impact on our conclusions. Classification using PCA supercolours ------------------------------------- We classify galaxies using a Principal Component Analysis (PCA) method applied to the broad-band photometric data, using the techniques outlined in Wild et al. (2014; hereafter W14). The aim of the PCA analysis is to describe the variation in galaxy SEDs using the linear combination of only a small number of components. The components are derived using a library of spectral synthesis models from BC03, using a wide range of stochastic star formation histories, metallicities, and dust-reddening. We found that only three principal components (effectively low-resolution “eigenspectra”) are required, in linear combination, to account for $>$99.9% of the variance in photometric SEDs. The amplitude of each component defines a “supercolour”, analogous to a traditional colour but defined using all available photometric bands. Supercolours allow the comparison of SEDs without the need for model fitting, and galaxies with extreme properties are free to have colours that differ substantially from any of the input model components. Using the supercolour technique, we separated the UDS galaxy population into three categories; passive galaxies (with low specific star-formation rates), star-forming galaxies, and post-starburst galaxies. Post-starburst galaxies are identified as a well-defined stream of galaxies in supercolour space, consistent with quiescent stellar populations in which a large fraction ($>$10%) of the stellar mass was formed within the last $\sim$1 Gyr, with the star formation then rapidly quenched (W14; see also Wild et al. 2016). The precise boundary between the passive and post-starburst population is set by the ability to observe strong Balmer absorption lines in optical spectroscopy (W14; Maltby et al. 2016). As outlined in Wild et al. (2016), evolutionary tracks suggest that not all photometric PSBs may have necessarily undergone a short-lived ‘burst’ of star formation. The population may also include galaxies that have undergone more extended ($\la$ 3 Gyr) periods of star formation, but the key characteristic is the rapid quenching of star formation within the last $\sim$1 Gyr. In W14, four categories of star-forming galaxies were identified (SF1, SF2, SF3 and dusty), which we combine for the purposes of this work. As outlined in W14 (see also Section \[sec:uvj\]) the classification by supercolours is in very good agreement with the separation of galaxies using the more traditional rest-frame UVJ technique for separating star-forming and passive galaxies (Labbé et al. 2005; Wuyts et al. 2007). There is also good agreement with the UVJ method of Whitaker et al. (2012), who identified recently-quenched candidates at the blue end of the passive UVJ sequence. ![The distribution of stellar mass as a function of photometric redshift for the three primary galaxy populations. The curves show the corresponding 95% completeness limits, determined using the method of Pozzetti et al. (2010). Details of the galaxy classification and stellar mass determination can be found in Wild et al. (2016). In this paper we focus on the structural properties of galaxies in the redshift range $1<z<2$.[]{data-label="fig:zmass"}](zmass.pdf){width="0.9\linewidth"} ![A comparison of size measurements for galaxies measured with ground-based $K$-band imaging and HST $H$-band imaging, using the subsample of $\sim$9% of galaxies within the UDS CANDELS mosaic. The upper panel compares half-light radii for individual galaxies. We find that ground-based imaging can provide robust measurements of galaxy sizes, although the relationship is not precisely 1:1. A linear fit to all points suggests that ground-based sizes are $\sim$13% smaller on average (dashed line). A characteristic uncertainty on individual measurements is shown, representing the median error on the PSB sizes. The lower (binned) distribution compares mean sizes for the three key populations, with unfilled symbols representing mean values determined from fewer than 5 galaxies. We find no evidence for a significant systematic bias in ground-based size determinations with spectral type. Galaxies shown have photometric redshifts in the range $1<z<2$ and stellar masses with log (M$_{\ast}$/M$_{\sun}$)$>10$, to match the samples used in this work.[]{data-label="fig:size-test"}](rtest.pdf "fig:"){width="0.9\linewidth"} ![A comparison of size measurements for galaxies measured with ground-based $K$-band imaging and HST $H$-band imaging, using the subsample of $\sim$9% of galaxies within the UDS CANDELS mosaic. The upper panel compares half-light radii for individual galaxies. We find that ground-based imaging can provide robust measurements of galaxy sizes, although the relationship is not precisely 1:1. A linear fit to all points suggests that ground-based sizes are $\sim$13% smaller on average (dashed line). A characteristic uncertainty on individual measurements is shown, representing the median error on the PSB sizes. The lower (binned) distribution compares mean sizes for the three key populations, with unfilled symbols representing mean values determined from fewer than 5 galaxies. We find no evidence for a significant systematic bias in ground-based size determinations with spectral type. Galaxies shown have photometric redshifts in the range $1<z<2$ and stellar masses with log (M$_{\ast}$/M$_{\sun}$)$>10$, to match the samples used in this work.[]{data-label="fig:size-test"}](rtest_bin_mean.pdf "fig:"){width="0.9\linewidth"} Stellar masses {#sec:masses} -------------- We determined stellar masses for each galaxy using a Bayesian analysis, to account for the degeneracy between physical parameters. Further details may be found in Wild et al. (2016). A library of 10’s of thousands of population synthesis models was created using BC03 and fit to the supercolours to obtain a probability density function for each physical property. A wide range of star formation histories, dust properties, and metallicities were explored, including exponentially declining star formation rates with superimposed stochastic starbursts. Stellar masses were calculated assuming a Chabrier initial mass function (IMF), defined as the stellar mass at the time of observation, i.e. allowing for the fraction of mass in stars returned to the interstellar medium due to mass loss and supernovae. The resulting stellar mass uncertainties from the Bayesian fits are typically $\pm ~0.1$ dex for all populations, assuming BC03 stellar population synthesis models and allowing for uncertainties in the photometric redshifts. The potential impact of stellar mass errors (random and systematic) is discussed further in Section \[sec:uncertainties\]. Figure \[fig:zmass\] shows the resulting distribution of stellar mass as a function of redshift, separated into the three primary populations. Mass completeness limits (95%) were determined as a function of redshift using the method of Pozzetti et al. (2010). We note that the 95% completeness limit for star-forming galaxies appears to be surprisingly high in this diagram (formally slightly higher than the PSB population). This is caused by the wide range in mass-to-light ratios within the star-forming population. As outlined in Wild et al. (2016), the PSB mass function evolves strongly with redshift, so that the comoving space density of massive PSBs (M$_{\ast}> 10^{10} ~$M$_{\sun}$) is several times higher at $z\sim 2$ than at $z\sim 0.5$. This trend is apparent in Figure \[fig:zmass\], which shows a sharp decline in the number of massive PSBs at $z<1$. The majority of PSBs at $z<1$ are close to the 95% completeness limit (see Figure \[fig:zmass\]) and typically very faint; the median $K$-band magnitude for PSBs at $z<1$ is $K=23.0$, compared to $K=21.8$ at $z>1$. In this work we therefore concentrate on the structural properties of PSBs at $z>1$, and defer an examination of the low-redshift ground-based sample to future work using deeper $K$-band imaging. In the redshift range $1<z<2$, our initial sample consists of 24,880 star-forming galaxies, 2043 passive galaxies and 502 PSBs, of which 9183, 2001 and 385, respectively, have stellar masses M$_{\ast}> 10^{10} ~$M$_{\sun}$. ![ A comparison of Sérsic index measurements for galaxies measured with ground-based $K$-band imaging and HST $H$-band imaging, using the subsample of $\sim$9% of galaxies within the UDS CANDELS mosaic. The upper panel compares individual galaxies, with a characteristic error bar denoting the median uncertainty in $\log_{10}(n)$ for the PSB population. We find that ground-based determinations of Sérsic indices are sufficient to broadly distinguish populations with high average values from those with low average values. The lower (binned) distribution compares the mean Sérsic indices, binned as a function of $n_{\rm CANDELS}$. Open symbols denote mean values determined from fewer than 5 galaxies. There are deviations from a 1:1 relation, but no evidence for a systematic bias in ground-based Sérsic determinations with spectral type. Galaxies shown have photometric redshifts in the range $1<z<2$ and stellar masses with log (M$_{\ast}$/M$_{\sun}$)$>10$, to match the samples used in this work. []{data-label="fig:sersic-test"}](ntest.pdf "fig:"){width="0.9\linewidth"} ![ A comparison of Sérsic index measurements for galaxies measured with ground-based $K$-band imaging and HST $H$-band imaging, using the subsample of $\sim$9% of galaxies within the UDS CANDELS mosaic. The upper panel compares individual galaxies, with a characteristic error bar denoting the median uncertainty in $\log_{10}(n)$ for the PSB population. We find that ground-based determinations of Sérsic indices are sufficient to broadly distinguish populations with high average values from those with low average values. The lower (binned) distribution compares the mean Sérsic indices, binned as a function of $n_{\rm CANDELS}$. Open symbols denote mean values determined from fewer than 5 galaxies. There are deviations from a 1:1 relation, but no evidence for a systematic bias in ground-based Sérsic determinations with spectral type. Galaxies shown have photometric redshifts in the range $1<z<2$ and stellar masses with log (M$_{\ast}$/M$_{\sun}$)$>10$, to match the samples used in this work. []{data-label="fig:sersic-test"}](ntest_bin_mean.pdf "fig:"){width="0.9\linewidth"} CANDELS-UDS ----------- Throughout this work we compare our ground-based determinations of galaxy size and Sérsic index with measurements from the Hubble Space Telescope (HST) CANDELS survey (Grogin et al. 2011; Koekemoer et al. 2011), using measurements provided in van der Wel et al. (2012). The UDS is one of the three targets for the CANDELS Wide survey, with imaging in the $J$ and $H$ bands taken with the Wide Field Camera 3 (WFC3). The CANDELS imaging covers only $\sim$7% of the UDS field ($\sim$9% of the area used in our analysis), but this is sufficient to provide an independent test and calibration for our ground-based structural parameters. Full details of the <span style="font-variant:small-caps;">GALFIT</span> measurement of structural parameters within CANDELS are given in van der Wel et al. (2012). {width="0.45\linewidth"} {width="0.45\linewidth"} {width="0.45\linewidth"} {width="0.45\linewidth"} ![We compare the distribution of $\Sigma_{1.5}$ values for the three galaxy populations in the redshift range $1<z<2$, with stellar masses M$_{\ast}> 10^{10.5} ~$M$_{\sun}$. This parameter ($\Sigma_{1.5} \equiv$M$_{\ast}/R_e^{1.5}$), defined by Barro et al. (2013), effectively removes the slope of the galaxy mass/size relation; high values of $\Sigma_{1.5}$ correspond to galaxies that are compact for their stellar mass. A KS test rejects the null hypothesis that passive galaxies and PSBs are drawn from the same underlying distribution in $\Sigma_{1.5}$, with a significance of $>99.99\%$.[]{data-label="fig:sigma15"}](sigma15.pdf){width="0.9\linewidth"} {width="0.45\linewidth"} {width="0.45\linewidth"} Ground-based measurements of size and Sérsic index {#sec:sizesersic} ================================================== We determined structural parameters for the $K$-band galaxy sample using the <span style="font-variant:small-caps;">GALAPAGOS</span> software (Barden et al. 2012). The package allows the automated use of <span style="font-variant:small-caps;">GALFIT</span> (Peng et al. 2002) to fit Sérsic light profiles (Sérsic 1968) to all galaxies in the UDS, parameterized with a Sérsic index, $n$, and effective radius, $R_e$, measured along the semi-major axis. We acknowledge that many high-redshift galaxies are described by more complex morphologies (see Bruce et al. 2014), but single Sérsic fits provide a simple parameterization to allow us to compare the bulk properties of the galaxy populations. An accurate determination of the point-spread function (PSF) is critical for this process, as galaxies at $z>1$ typically have half-light radii below 0.5 arcsec. Following the work of Lani et al. (2013), we investigated PSF variations across the UDS field and found that most variation occurred between WFCAM detector boundaries within the UDS mosaic (Casali et al. 2007). Testing revealed that we could obtain consistent results by splitting the UDS field into 16 overlapping sub-regions, corresponding to the $4\times 4$ WFCAM tiling pattern. Within each region the light profiles of approximately 100 stars were stacked to provide the local PSF measurement, with variations across the field in the range $0.75-0.81$ arcsec (FWHM). Considerable care was taken to mask sources in the vicinity of the stars used for PSF measurement. From the resulting measurements of size and Sérsic index we rejected $\sim$9% of galaxies where GALFIT failed to converge on a Sérsic solution. The rejection rate was similar for the star-forming, passive and post-starburst populations. A further $\sim$1% of galaxies were rejected (a-priori) if the fits were formally very poor ($\chi^2_\nu>100$), which typically corresponded to highly blended objects on the $K$-band image. Matching the output from <span style="font-variant:small-caps;">GALAPAGOS</span> with our Supercolour catalogue, we obtain a final sample of 8098 star-forming galaxies, 1829 passive galaxies, and 348 PSBs in the redshift range $1<z<2$ with M$_{\ast}> 10^{10} ~$M$_{\sun}$. In Figures \[fig:size-test\] and \[fig:sersic-test\] we display the resulting size and Sérsic measurements for the subset of UDS galaxies within the HST CANDELS survey. Ground-based measurements are compared with those obtained using H-band CANDELS measurements, as published in van der Wel et al. (2012). For the size measurements, we find a tight relationship between the ground-based $K$-band and CANDELS $H$-band sizes, as previously found by Lani et al. (2013). On average, the sizes obtained from CANDELS are systematically $\sim$13% larger, which is consistent with previous comparisons of size measurements as a function of waveband (e.g. Kelvin et al. 2012). The systematic offset is consistent among the three galaxy populations studied here so we apply no corrections for this effect. The characteristic scatter in $\delta R/R$, given by the normalized median absolute deviation ($\sigma_{{\rm NMAD}}$) is 17%, 16% and 24% for the star-forming, passive and PSB populations respectively. Ground-based measurements of Sérsic index are far more uncertain for a given galaxy (Figure \[fig:sersic-test\]), and we find a significant degree of scatter when comparing ground-based and CANDELS measurements. Formally, the characteristic scatter in $\delta n/n$, given by the normalized median absolute deviation ($\sigma_{{\rm NMAD}}$) is 45%, 39%, and 35% for the star-forming, passive and PSB populations respectively. Nevertheless, there is a clear correlation, and we find that ground-based determinations are sufficient to distinguish populations with “high” Sérsic indices (e.g. $n>2$) from those with “low” Sérsic indices ($n<2$). Comparing the three primary galaxy types, we find consistent results; the passive and post-starburst populations show consistently high Sérsic indices (from ground or spaced-based measurements), while star-forming galaxies are concentrated at lower values. The binned distribution demonstrates that the correlation between ground and space-based Sérsic indices is not perfectly 1:1, but we see no systematic differences in this relation between the three populations. We conclude that ground-based measurements can be used to broadly compare the Sérsic indices for our galaxy populations. In addition, we will use the subset of galaxies with HST measurements ($\sim$9%) to verify any conclusions drawn from the larger ground-based sample. The Sérsic distributions will be compared further (and as a function of stellar mass) in Section \[sec:sersic\]. The sizes of post-starburst galaxies ==================================== The size–mass relation {#sec:sizemass} ---------------------- In Figure \[fig:size-mass\] we compare the size versus stellar-mass relation for galaxies in the redshift range $1<z<2$. Individual galaxies are shown, along with mean values as a function of stellar mass (in bins of 0.25 dex). In the upper panels we show the results from the ground-based $K$-band imaging, while the lower panels are based on independent sizes from CANDELS $H$-band imaging (covering $\sim$9% of the sample). The 95% mass completeness limits are shown (see Section \[sec:masses\]), determined at $z=2$ to provide conservative limits. In determining mean values for the ground-based sample we applied a 5$\sigma$ clip (with one iteration), to remove extreme outliers, but removing this constraint has no significant influence. Representative error bounds for individual galaxies are shown on the left panels, based on the median errors on the PSB sample. For the CANDELS sizes, data are taken from van der Wel et al. (2012), which include estimates for random and systematic errors from GALFIT. For the ground-based individual errors, we add in quadrature the scatter in $\delta R/R$ determined by the comparison with CANDELS (Section \[sec:sizesersic\]). The representative uncertainty on stellar masses is based on our Bayesian mass-fitting analysis described in Wild et al. (2016), as briefly outlined in Section \[sec:masses\]. The size–mass relations show the expected trends for star-forming and passive galaxies, consistent with previous studies (e.g. Daddi et al. 2005; Trujillo et al. 2006; van Dokkum et al. 2008; van der Wel et al. 2014; McLure et al. 2013). On average, passive galaxies appear significantly more compact than star-forming galaxies of equivalent stellar mass, but show a steeper size–mass relation, leading to convergence at the highest masses (M$_{\ast} \sim 10^{11.5} ~$M$_{\sun}$). Intriguingly, we find that post-starburst galaxies at this epoch are also extremely compact; they are comparable in size to the established passive galaxies, with evidence that they are smaller on average at high mass (M$_{\ast}> 10^{10.5} ~$M$_{\sun}$). These trends are apparent with the large ground-based sample and with the smaller CANDELS sample. We performed a bootstrap analysis as a simple test of significance, randomly sampling the (ground-based) populations within each mass bin, with replacement. For the 4 bins above $10^{10.5} ~$M$_{\sun}$, the fraction of the resampled populations in which the passive galaxies show mean sizes equal to (or smaller than) the mean of the PSB population are  $0.07$, $5\times10^{-5}$, $1.2\times 10^{-4}$, $4\times10^{-5}$ (low to high mass, respectively). We note that the final bin contains 86 passive galaxies, but only 3 PSBs, so the bootstrap comparison for this bin may be unreliable. Overall, assuming no systematic errors, we find evidence that massive post-starburst galaxies at $z>1$ are significantly more compact, on average, than passive galaxies of comparable mass. Repeating the analysis using the median sizes produced very similar trends. The median analysis and further tests of robustness are presented in Appendix A. Our results are consistent with the findings of Yano et al. (2016), who also found evidence that high-redshift post-starburst galaxies are very compact. As an additional comparison, in Figure \[fig:sigma15\] we display the distribution of $\Sigma_{1.5}\equiv M_{\ast}/R_e^{1.5}$ for the three galaxy populations, measured for the redshift range $1<z<2$ and stellar masses M$_{\ast}> 10^{10.5} ~$M$_{\sun}$. Following Barro et al. (2013), we use this parameter to effectively remove the slope in the galaxy size/mass relation. Fitting a function of the form M$_{\ast} = \Sigma ~ R_e^{\alpha}$ to the passive population, we find a best fit with $\alpha=1.55$, in very good agreement with the value $\alpha=1.5$ assumed by Barro et al. (2013). A simple Kolmogorov-Smirnov (KS) test rejects the null hypothesis that passive galaxies and PSBs are drawn from the same underlying distribution in $\Sigma_{1.5}$ with a significance $>99.99$%, with the same significance obtained with either value of $\alpha$. Given the strong evolution of the PSB mass function (Wild et al. 2016), a concern is that massive PSBs are more common at higher redshifts, which may bias the size–mass comparison when measured over a wide redshift range. In Figure \[fig:size-mass-z12\] we therefore display the size–mass relation in two narrower redshift bins, $1.0<z<1.5$ and $1.5<z<2.0$. With two independent samples, the results confirm that post-starburst galaxies, on average, show smaller half-light radii than the passive population at high mass (M$_{\ast}> 10^{10.5} ~$M$_{\sun}$). An additional test was performed using a weighted mean, with a redshift-dependent weight determined for each PSB using the ratio of the $n(z)$ distributions for passive galaxies and PSBs. The resulting size–mass relations were barely changed, with only a slight reduction in the significance of the differences reported above. A natural interpretation of our findings is that quiescent galaxies are most compact when they are newly-quenched, but then grow with cosmic time. Given the short-lived nature of the PSB phase, we expect the majority of passive galaxies to have gone through a similar stage in their past (Wild et al. 2016). Our results therefore provide evidence for the genuine growth of individual galaxies, suggesting that the growth of the population as a whole is not purely caused by a progenitor bias. We discuss the implications further in Section \[sec:discussion\]. A simple calculation allows us to compare the size differences with the observed cosmological growth. Based on population synthesis models, we estimate that the established passive population quenched approximately 0.5–1 Gyr before the PSB population at these redshifts (Wild et al. 2016), and the characteristic difference in size (e.g. based on the shift in $\Sigma_{1.5}$) is approximately 25% on average at M$_{\ast}> 10^{10.5} ~$M$_{\sun}$. The implied growth rate is similar to the observed [*cosmological*]{} growth, parameterized by van Dokkum et al. (2010) in the form $r_e \propto (1+z)^{-1.3}$ (i.e. $\sim$25% per Gyr at $z=1.5-2$). An improvement on these tentative conclusions will require more accurate age-dating of the stellar populations, which will soon be possible with growing spectroscopic samples. {width="0.45\linewidth"} {width="0.45\linewidth"} Stellar mass uncertainties {#sec:uncertainties} -------------------------- In this section we explore the potential impact of both random and systematic errors on our stellar masses. The typical uncertainty from our Bayesian stellar mass fitting is $\sigma \simeq$ 0.1 dex for all galaxy types, allowing for the degeneracy between fitted parameters and the uncertainties on photometric redshifts. To investigate the impact on our conclusions, we performed Monte Carlo realisations, allowing the stellar masses to shift randomly within a Gaussian probability distribution in log M$_{\ast}$. We found no impact on any of the results presented in Section \[sec:sizemass\]. Comparing the resulting distributions in $\Sigma_{1.5}$, the significance of the difference between the PSB and passive galaxy populations was unchanged, suggesting that the sample overall is large enough to minimise the influence of random errors. To investigate the influence of fitting methods, we re-evaluated the size–mass relations using two independent sets of stellar masses derived by Simpson et al. (2013) and Hartley et al. (2013), the latter also using an independent set of photometric redshifts. No significant differences were found. As a note of caution, however, we acknowledge that the observed differences between PSBs and passive galaxies could arise if our stellar masses are [*systematically*]{} overestimated for younger stellar populations. Our stellar masses are based on population synthesis models from BC03, which may underestimate the influence of thermally-pulsing asymptotic giant branch (TP-AGB) stars (Maraston et al. 2006). Such stars may have a major contribution to the rest-frame near-infrared light for stellar populations in the age range 0.2 to 2 Gyr, potentially leading to overestimated stellar masses for passive galaxy populations. The influence of TP-AGB stars is discussed further in Wild et al. (2016), where it was found that the influence of TP-AGB stars is strongest for galaxies with BC03-determined ages $>1$ Gyr, for which the ages and mass-to-light ratios are reduced using the models of Maraston et al. (2006). The net effect would be to move our passive galaxies to lower masses relative to the younger PSB population, which would [*enhance*]{} the differences in the size–mass relation. In summary, our findings appear robust to known sources of random and systematic error, but we acknowledge the possibility that unknown systematic uncertainties in the stellar mass determination may contribute to the difference in size–mass relations presented in our work. Future deep infrared spectroscopy may allow a more detailed investigation of the inherent uncertainties in determining stellar masses from photometric data. A comparison with UVJ selection {#sec:uvj} ------------------------------- To allow a comparison of our supercolour technique with previous work, in Figure \[fig:uvjfig\] (left) we present a rest-frame UVJ colour-colour diagram for UDS galaxies in the redshift range $1<z<2$. As previously shown in W14, the classification of galaxies using supercolours agrees very well with the more traditional UVJ colour selection (Labbé et al. 2005; Wuyts et al. 2007). Post-starburst galaxies are generally found at the blue end of the passive UVJ region, in good agreement with the findings of Whitaker et al. (2012). The distribution in $U-V$ alone confirms that PSBs predominantly lie in the classic “green valley”, intermediate between passive and star-forming galaxies, but the addition of the $V-J$ colour isolates this population in the distinct region corresponding to the youngest red-sequence galaxies. In the right-hand panel of Figure \[fig:uvjfig\] we illustrate the average effective radii in colour-colour bins. We select the subset of galaxies over the mass range $10.5< \log ~($M$_{*}/$M$_{\sun})<11.5$ to minimise the effects of the size–mass relation. The trends confirm that “young quiescent” galaxies selected by the UVJ technique show smaller average sizes, in good agreement with our analysis based on supercolours. Our results are consistent with a similar recent analysis by Yano et al. (2016). The Sérsic indices of post-starburst galaxies {#sec:sersic} ============================================= {width="0.32\linewidth"} {width="0.32\linewidth"} {width="0.32\linewidth"} In Figure \[fig:sersicfig\] we compare the Sérsic indices for star-forming, passive and post-starburst galaxies at $z>1$. The distributions obtained from the ground-based $K$-band data are consistent with those obtained for the smaller CANDELS sample. In both cases, we find that star-forming galaxies show a distribution peaking sharply at $n\simeq 1$, while passive and post-starburst galaxies show very different distributions, peaking at significantly higher values. A Kolmogorov-Smirnov test confirms these findings, rejecting the null hypothesis that either passive or post-starburst galaxies are drawn from the same distribution as star-forming galaxies to a high level of significance ($>>99.99$%, using the ground-based sample). In contrast, the Sérsic distributions for passive and post-starburst galaxies do not appear significantly different. The right-hand panel in Figure \[fig:sersicfig\] presents the median Sérsic indices as a function of stellar mass. We find evidence for a slight increase in the median Sérsic index with stellar mass for all populations, but in all mass bins the post-starburst galaxies show significantly higher Sérsic indices than star-forming galaxies, and values consistent with the passive population. Using the mean produced very similar trends, but Sérsic indices were slightly higher in all cases (by $\delta n\simeq 0.5$). We caution that single Sérsic fits provide only a crude parameterisation of the data, as it is now well-established that most galaxies at this epoch have more complex morphologies (e.g. Bruce et al. 2014). A high Sérsic index does not necessarily imply a purely spheroidal system, and low Sérsic indices do not necessarily imply the presence of an established disc (Mortlock et al. 2013). Nevertheless, it is clear from our data that the post-starburst galaxies are structurally very different to the actively star-forming population, and more comparable to ultra-compact equivalents of the passive population. We explore multiple-component fitting in future work (Maltby et al., in preparation). Discussion {#sec:discussion} ========== We have presented evidence that massive (M$_{\ast}> 10^{10.5} ~$M$_{\sun}$) recently-quenched (post-starburst) galaxies at high redshift ($z>1$) are exceptionally compact. Furthermore, they show high Sérsic indices, indistinguishable from the established passive population at the same epoch. We conclude that the structural transformation must have occurred before (or during) the event that quenched their star formation. Given that the majority of massive passive galaxies at $z>1$ are thought to have passed through a post-starburst phase (Wild et al. 2016), our findings suggest a strong link between quenching and the formation of a compact spheroid. Our results confirm the findings of Whitaker et al. (2012), who found evidence that younger passive galaxies are more compact at $z>1$, and the more recent HST CANDELS study by Yano et al. (2016). At intermediate redshifts ($z\sim 1$) previous studies have found conflicting results on the relationship between stellar age and the compactness of passive galaxies, with indications that progenitor bias may be playing a role at this epoch (Keating et al. 2015; Williams et al. 2017). Our findings may be explained if high-redshift post-starburst galaxies are formed from the dramatic collapse of gas at high redshift, formed from either a gas-rich merger (e.g. Hopkins et al. 2009; Wellons et al. 2015), or from gas inflow feeding a massive disc, which becomes unstable and collapses by “compaction” (e.g. Dekel et al. 2009; Zolotov et al. 2015; Tacchella et al. 2016). Star formation must then be rapidly quenched, either by a central AGN or feedback from highly nucleated star formation (e.g. Hopkins et al. 2005; Diamond-Stanic et al. 2012), leaving an ultra-compact post-starburst remnant. It may be possible to test these evolutionary scenarios by comparing the properties of post-starburst galaxies with their likely active progenitors, i.e. star-forming galaxies caught during the merging or “compaction” phase. Current candidates include submillimetre galaxies, many of which appear to be highly compact at $\sim$250 $\mu$m in the rest-frame (e.g. Simpson et al. 2015), and the high-redshift “blue nuggets” (e.g. Barro et al. 2013; Mei et al. 2015; Barro et al. 2017). Whatever the true progenitors for our PSBs, the most likely explanation is that structural transformation occurred immediately prior to quenching. The fact that the PSBs and passive galaxies have indistinguishable Sérsic indices (Figure \[fig:sersicfig\]) would suggest that most of the structural change is already established when the star formation is quenched, unless the structural transformation occurs on a much shorter timescale than the $\sim$500 Myr lifetime for the PSB phase. Following the formation of the proto-spheroid, there are currently two leading explanations for the observed growth in passive galaxies with cosmic time. Minor gas-free mergers provide a plausible physical mechanism (e.g. Bezanson et al. 2009; Naab, Johansson & Ostriker 2009), and indeed there is evidence that high-redshift passive galaxies are larger in dense environments, where such interactions are more likely (e.g. Lani et al. 2013). Alternatively, progenitor bias may mimic the observed growth, since passive galaxies formed at later times are typically larger (e.g. Poggianti et al. 2013; Carollo et al. 2013). Our finding that post-starburst galaxies are more compact than older passive galaxies, on average, would suggest that we are observing an earlier phase in the lifetime of steadily-growing spheroids. Thus progenitor bias is unlikely to be the primary cause for the observed growth at early times; our observed trends show precisely the opposite behaviour (with younger galaxies being more compact). On the other hand, if massive post-starburst galaxies represent newly-formed “red nuggets”, it is notable that their abundance is a strong function of redshift; massive (M$_{\ast}> 10^{10.5} ~$M$_{\sun}$) post-starburst galaxies are several times more abundant at $z\sim 2$ compared to $z\sim 0.5$ (Wild et al. 2016). Thus, while the majority of high-mass passive galaxies at $z\sim 2$ are likely to have been through the ultra-compact post-starburst phase (Wild et al. 2016), this may become an increasingly less dominant channel towards lower redshifts, when progenitor bias may play a more significant role in explaining size evolution (e.g. see Fagioli et al. 2016, Williams et al. 2017). Even at low redshift, however, there is evidence that the most compact quiescent galaxies have evolved from post-starburst progenitors (Zahid et al. 2016). Finally, we note that the mass function for PSBs shows a very distinctive evolution in shape (Wild et al. 2016). At high redshift ($z\sim 2$) the mass function resembles that of quiescent galaxies, dominated by high-mass systems with a sharp decline in space density above a mass of M$_{\ast}\sim 10^{10.5} ~$M$_{\sun}$. At low redshift ($z<1$) the population is dominated by lower-mass systems, with a shape resembling the mass function for star-forming galaxies. These features are interpreted as evidence for two distinct formation channels for post-starburst galaxies; high-mass systems formed from gas-rich dissipative collapse, and low-mass systems formed from environmental quenching or the merging of normal disc galaxies (Wild et al. 2016). Our structural findings are in good agreement with this scenario, with the PSBs above the same characteristic mass displaying distinctive, ultra-compact morphologies, consistent with a highly-dissipative, gas-rich origin. We will present a detailed study of the structural parameters for low-mass PSBs in future work (Maltby et al., in preparation). Conclusions =========== We present a study of the structural parameters for a large sample of photometrically-selected post-starburst galaxies in the redshift range $1<z<2$, recently identified in the UKIDSS UDS field. These rare transition objects provide the ideal sample for understanding the links between the quenching of star formation and the structural transformation of massive galaxies. We demonstrate that deep ground-based near-infrared imaging can be used to obtain robust sizes and Sérsic indices for large samples of high-redshift galaxies. From the resulting size–mass relation, we find that massive (M$_{\ast}> 10^{10.5} ~$M$_{\sun}$) post-starburst galaxies are exceptionally compact at $z>1$, with evidence that they are more compact on average than established passive galaxies at the same epoch. Since most high-mass passive galaxies at $z>1$ are likely to have been through a post-starburst phase (Wild et al. 2016), the implication is that quiescent galaxies are most compact when they are newly quenched, thereafter growing with cosmic time. An important caveat, however, is to acknowledge the considerable uncertainty in stellar mass estimation, as discussed in Section \[sec:uncertainties\]. As an avenue for future research, it will be important to determine whether stellar masses are systematically overestimated for recently quenched stellar populations. We also find that post-starburst galaxies show high Sérsic indices, significantly higher than star-forming galaxies on average, but statistically indistinguishable from the Sérsic indices of established passive galaxies at the same epoch. We conclude that massive post-starburst galaxies represent newly-formed compact proto-spheroids. Furthermore, the structural transformation of these galaxies must have occurred before (or during) the event that quenched their star formation. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Louis Abramson, Steven Bamford, Dale Kocevski, Mike Merrifield, and Ian Smail for useful discussions. We extend our gratitude to the staff at UKIRT for their tireless efforts in ensuring the success of the UDS project. We also wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has within the indigenous Hawaiian community. We were most fortunate to have the opportunity to conduct observations from this mountain. V.W. and K.R. and acknowledge support from the European Research Council Starting Grant (PI Wild). RJM acknowledges the support of the European Research Council via the award of a consolidator grant (PI McLure). 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Throughout this work we have measured average sizes when comparing galaxy populations, yielding evidence that post-starburst galaxies are typically smaller than passive galaxies of comparable stellar mass. The use of a mean may produce misleading results, however, if either population is skewed by a significant number of outliers (e.g. misclassified interlopers from other galaxy categories). In Figure \[fig:appenfig1\] (left panel) we therefore reproduce the size–mass relation from Figure \[fig:size-mass\], but this time using median size values. In most bins the median sizes are slightly smaller than the mean, as the size distributions show a slight tail to high values. The significant differences between the passive and post-starburst galaxies remain, however. At high-mass (M$_{\ast}> 10^{10.5} ~$M$_{\sun}$) the overall significance of the trends is essentially unchanged compared to the results outlined in Section \[sec:sizemass\]. We conclude that the use of a mean has not biased our primary conclusions. As a further source of error, we considered the possibility that the passive and post-starburst samples are contaminated by star-forming galaxies, e.g. due to uncertainties in classification. If the contaminating fraction is higher for the passive population (for reasons unknown), this could skew the size measurements upwards. We tested for this effect by applying a cut in Sérsic index. Noting the very different distributions in Sérsic index between star-forming and passive samples (see Figure \[fig:sersicfig\]), we re-evaluated the size–mass relations using only passive and post-starburst galaxies with $n>2$ (see Figure \[fig:appenfig1\], right panel), again using the median size to further reduce the impact of interlopers. The striking difference in sizes remains, with a negligible reduction in significance. Using a mean estimator yields the same result, with a size–mass relation that is almost identical to the upper-right panel of Figure \[fig:size-mass\]. We conclude that “contamination” from galaxies with low Sérsic indices (whether passive or star-forming) is not affecting our conclusions. As an additional test, we investigated the influence of using a cleaner (though less complete) sample of post-starburst galaxies. As outlined in Section \[sec:intro\] and Maltby et al. (2016), the primary source of contamination is between post-starburst galaxies and “normal” passive galaxies. Depending on the precise selection criteria, between 20–40% galaxies in the PSB category would be classified as passive (rather than PSB) using spectroscopy, while 6–10% of the passive category would be spectroscopically classified as PSBs (Maltby; private communication). Based on Figure 3 in Maltby et al. (2016), we therefore identify a “cleaner” PSB sample by selecting galaxies further from the passive/PSB boundary, with supercolours $SC2$&gt;6. In this regime, formally 100% of PSB candidates (15/15) were confirmed with $W_{\rm H\delta}>5$Å. Using this new sub-sample, the resulting size–mass relation (evaluated using the standard mean estimator) is shown in Figure \[fig:appenfig2\] (left panel). We find that the difference in size compared to passive galaxies remains, and in fact is slightly enhanced; the mean PSB size is formally smaller in five out of six bins. We conclude that contamination of the PSB category by older passive galaxies can only act to dilute the differences we observe. As another test for contamination, we combined the PCA classification with the classic UVJ criteria (e.g. see Figure \[fig:uvjfig\]), to exclude PSBs and passive galaxies that are classified as “star-forming” using rest-frame UVJ colours. The aim is to remove red star-forming galaxies that may have been misclassified by the PCA technique. Using these joint criteria removes 26% of passive galaxies and 34% of PSBs from our primary sample ($z>1$, M$_{\ast}>10^{10} ~$M$_{\sun}$). The resulting size–mass relations are shown in Figure \[fig:appenfig2\] (right panel). We find that the difference between the PSBs and passive galaxies at high mass remains, and in fact is slightly enhanced. Finally, we performed a variety of tests using more stringent cuts on the structural parameters derived for our ground-based galaxy sample, using the CANDELS dataset as a calibrating sample. No significant differences were found. We noted, however, that ground-based size measurements become increasingly unreliable when GALFIT assigns a very low axis ratio, $q<0.1$. Approximately 6% of the galaxies in our sample are affected, mostly among the star-forming galaxies, but also affecting 3–4% of the passive and post-starburst sample (mostly at low mass; M$_{\ast}< 10^{10.5} ~$M$_{\sun}$). Removing these galaxies had no major influence on the size-mass relations; in fact, the difference in size between passive and post-starburst galaxies became marginally more significant. {width="0.45\linewidth"} {width="0.45\linewidth"} \[lastpage\] [^1]: E-mail: omar.almaini@nottingham.ac.uk [^2]: From a sample of 24 PSB candidates, 19 galaxies ($\sim$80%) showed strong Balmer absorption lines ($W_{\rm H\delta}>5$Å), dropping to 14 confirmations ($\sim$60%) if stricter criteria are used to exclude galaxies with significant \[O<span style="font-variant:small-caps;">ii</span>\] emission. The fraction of spectroscopic PSBs among the passive and star-forming PCA classes was estimated to be $<$10% and $<$1% respectively (Maltby; private communication).

--- abstract: | We present results on the compact steep-spectrum quasar 3C 48 from observations with the Very Long Baseline Array (VLBA), the Multi-Element Radio Linked Interferometer Network (MERLIN) and the European VLBI Network (EVN) at multiple radio frequencies. In the 1.5-GHz VLBI images, the radio jet is characterized by a series of bright knots. The active nucleus is embedded in the southernmost VLBI component A, which is further resolved into two sub-components A1 and A2 at 4.8 and 8.3 GHz. A1 shows a flat spectrum and A2 shows a steep spectrum. The most strongly polarized VLBI components are located at component C $\sim$0.25 arcsec north of the core, where the jet starts to bend to the northeast. The polarization angles at C show gradual changes across the jet width at all observed frequencies, indicative of a gradient in the emission-weighted intrinsic polarization angle across the jet and possibly a systematic gradient in the rotation measure; moreover, the percentage of polarization increases near the curvature at C, likely consistent with the presence of a local jet-ISM interaction and/or changing magnetic-field directions. The hot spot B shows a higher rotation measure, and has no detected proper motion. These facts provide some evidence for a stationary shock in the vicinity of B. Comparison of the present VLBI observations with those made 8.43 years ago suggests a significant northward motion for A2 with an apparent transverse velocity $\beta_{app}=3.7\pm0.4\, c$. The apparent superluminal motion suggests that the relativistic jet plasma moves at a velocity of $\gtrsim0.96\, c$ if the jet is viewed at an inclination angle less than $20\degr$. A simple precessing jet model and a hydrodynamical isothermal jet model with helical-mode Kelvin-Helmholtz instabilities are used to fit the oscillatory jet trajectory of 3C 48 defined by the bright knots. author: - | T. An$^{1}$[^1], X.Y. Hong$^{1}$, M.J. Hardcastle$^{2,3}$, D.M. Worrall$^{3}$, T. Venturi$^{4}$, T.J. Pearson$^{5}$, Z.-Q. Shen$^{1}$, W. Zhao$^{1}$ and W.X. Feng$^{6}$\ $^{1}$ Shanghai Astronomical Observatory, Chinese Academy of Sciences, 200030, Shanghai, China\ $^{2}$ School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB\ $^{3}$ Department of Physics, University of Bristol, Tyndall Avenue, Bristol BS8 1TL\ $^{4}$ INAF – Istituto di Radioastronomia, I-40129, Bologna, Italy\ $^{5}$ California Institute of Technology, Mail Stop 105-24, Pasadena, CA 91125, USA\ $^{6}$ Liaocheng University, 252059, Liaocheng, China title: 'Kinematics of the parsec-scale radio jet in 3C 48' --- \[firstpage\] galaxies: active, galaxies: kinematics, galaxies: jets, quasars: individual: 3C 48 INTRODUCTION {#section:intro} ============ Compact Steep Spectrum (CSS) sources are a population of powerful radio sources with projected linear size less than 20 kpc and steep high radio frequency spectrum $\alpha<-0.5$ [^2] (Peacock & Wall 1982, Fanti et al. 1990, and review by O’Dea 1998 [**and Fanti 2009**]{}). Kinematical studies of the hot spots and analysis of the high-frequency turnover in the radio spectrum due to radiative cooling imply ages for CSS sources in the range 10$^2$–10$^5$ yr (e.g., Owsianik, Conway & Polatidis 1998; Murgia et al. 1999). The sub-galactic size of CSS sources has been used to argue that CSS sources are probably young radio sources, (the ‘youth’ model: Fanti et al. 1995; Readhead et al. 1996). However, another interpretation attributes the apparent compactness of the CSS sources to being strongly confined by the dense ISM in the host galaxy (the ‘frustration’ model: van Breugel, Miley & Heckman 1984). Spectroscopic observations of CSS sources provide evidence for abundant gas reservoirs in the host galaxies and strong interaction between the radio sources and the emission-line clouds [@ODe02]. Some CSS sources have been observed to have high-velocity clouds (as high as $\sim500$ kms$^{-1}$) in the Narrow-Line Region (NLR), presumably driven by radio jets or outflows; an example is 3C 48 [@Cha99; @Sto07]. In addition, many CSS sources show distorted radio structures, suggestive of violent interaction between the jet and the ambient interstellar medium [@Wil84; @Fan85; @Nan91; @Spe91; @Nan92; @Aku91]. The ample supply of cold gas in their host galaxies and their strong radio activity, which results in a detection rate as high as $\sim30$ per cent in flux-density limited radio source surveys [@Pea82; @Fan90], make CSS sources good laboratories for the study of AGN triggering and feedback. 3C 48 ($z=0.367$) is associated with the first quasar to be discovered [@Mat61; @Gre63] in the optical band. Its host galaxy is brighter than that of most other low redshift quasars. The radio source 3C 48 is classified as a CSS source due to its small size and steep radio spectrum [@Pea82]. Optical and NIR spectroscopic observations suggest that the active nucleus is located in a gas-rich environment and that the line-emitting gas clouds are interacting with the jet material [@Can90; @Sto91; @Cha99; @Zut04; @Kri05; @Sto07]. VLBI images [@Wil90; @Wil91; @Nan91; @Wor04] have revealed a disrupted jet in 3C 48, indicative of strong interactions between the jet flow and the dense clouds in the host galaxy. Although some authors [@Wil91; @Gup05; @Sto07] have suggested that the vigorous radio jet is powerful enough to drive massive clouds in the NLR at speeds up to 1000 km s$^{-1}$, the dynamics of the 3C 48 radio jet have yet to be well constrained. Due to the complex structure of the source, kinematical analysis of 3C 48 through tracing proper motions of compact jet components can only be done with VLBI observations at 4.8 GHz and higher frequencies, but until now the required multi-epoch high-frequency VLBI observations had not been carried out. In order to study the kinematics of the radio jet for comparison with the physical properties of the host galaxy, we observed 3C 48 in full polarization mode with the VLBA at 1.5, 4.8 and 8.3 GHz in 2004, and with the EVN and MERLIN at 1.65 GHz in 2005. Combined with earlier VLBA and EVN observations, these data allow us to constrain the dynamics of the jet on various scales. Our new observations and our interpretation of the data are presented in this paper. The remainder of the paper is laid out as follows. Section 2 describes the observations and data reduction; Section 3 presents the total intensity images of 3C 48; and Section 4 discusses the spectral properties and the linear polarization of the components of the radio jet. In Section 5, we discuss the implications of our observations for the kinematics and dynamics of the radio jet. Section 6 summarizes our results. Throughout this paper we adopt a cosmological model with Hubble constant $H_0$=70 km s$^{-1}$ Mpc$^{-1}$, $\Omega_m=0.3$, and $\Omega_{\Lambda}=0.7$. Under this cosmological model, a 1-arcsec angular separation corresponds to a projected linear size of 5.1 kpc in the source frame at the distance of 3C 48 ($z=0.367$). OBSERVATIONS AND DATA REDUCTION =============================== The VLBA observations (which included a single VLA antenna) of 3C 48 were carried out at 1.5, 4.8, and 8.3 GHz on 2004 June 25. The EVN and MERLIN observations at 1.65 GHz were simultaneously made on 2005 June 7. Table \[tab:obs\] lists the parameters of the VLBA, EVN and MERLIN observations. In addition to our new observations, we made use of the VLBA observations described by Worrall et al. (2004) taken in 1996 at 1.5, 5.0. 8.4 and 15.4 GHz. VLBA observations and data reduction ------------------------------------ The total 12 hours of VLBA observing time were evenly allocated among the three frequencies. At each frequency the effective observing time on 3C 48 is about 2.6 hours. The data were recorded at four observing frequencies (IFs) at 1.5 GHz and at two frequencies at the other two bands, initially split into 16 channels each, in full polarization mode. The total bandwidth in each case was 32 MHz. The detailed data reduction procedure was as described by Worrall et al. (2004) and was carried out in [aips]{}. We used models derived from our 1996 observations to facilitate fringe fitting of the 3C 48 data. Because the source structure of 3C 48 is heavily resolved at 4.8 and 8.3 GHz, and missing short baselines adds noise to the image, the initial data were not perfectly calibrated. We carried out self-calibration to further correct the antenna-based phase and amplitude errors. This progress improves the dynamic range in the final images. Polarization calibration was also carried out in the standard manner. Observations of our bandpass calibrator, 3C 345, were used to determine the R-L phase and delay offsets. The bright calibrator source DA 193 was observed at a range of parallactic angles and we used a model image of this, made from the Stokes $I$ data, to solve for instrumental polarization. Our observing run included a snapshot observation of the strongly polarized source 3C 138. Assuming that the polarization position angle (or the E-Vector position angle in polarization images, ‘EVPA’) of 3C 138 on VLBI scales at 1.5 GHz is the same as the value measured by the VLA, and we used the measured polarization position angle of this source to make a rotation of $94\degr$ of the position angles in our 3C 48 data. We will show later that the corrected EVPAs of 3C 48 at 1.5 GHz are well consistent with those derived from the 1.65-GHz EVN data that are calibrated independently. At 4.8 GHz, 3C 138 shows multiple polarized components; we estimated the polarization angle for the brightest polarized component in 3C 138 from Figure 1 in Cotton et al. 2003, and determined a correction of $-55\degr$ for the 3C 48 data. After the rotation of the EVPAs, the polarized structures at 4.8 GHz are basically in agreement with those at 1.5 GHz. At 8.3 GHz the polarized emission of 3C 138 is too weak to be used to correct the absolute EVPA; we therefore did not calibrate the absolute EVPAs at 8.3 GHz. EVN observations and data reduction ----------------------------------- The effective observing time on 3C 48 was about 8 hours. Apart from occasional RFI (radio frequency interference), the whole observation ran successfully. The data were recorded in four IFs. Each IF was split into 16 channels, each of 0.5-MHz channel width. In addition to 3C 48 we observed the quasars DA 193 and 3C 138 for phase calibration. 3C 138 was used as a fringe finder due to its high flux density of $\sim$9 Jy at 1.65 GHz. The amplitude of the visibility data was calibrated using the system temperatures, monitored during the observations, and gain curves of each antenna that were measured within 2 weeks of the observations. The parallactic angles were determined on each telescope and the data were corrected appropriately before phase and polarization calibration. We corrected the ionospheric Faraday rotation using archival ionospheric model data from the CDDIS. DA 193 and OQ 208 were used to calibrate the complex bandpass response of each antenna. We first ran fringe fitting on DA 193 over a 10-minute time span to align the multi-band delays. Then a full fringe fitting using all calibrators over the whole observing time was carried out to solve for the residual delays and phase rates. The derived gain solutions were interpolated to calibrate the 3C 48 visibility data. The single-source data were split for hybrid imaging. We first ran phase-only self-calibration of the 3C 48 data to remove the antenna-based, residual phase errors. Next we ran three iterations of both amplitude and phase self-calibration to improve the dynamic range of the image. DA 193 is weakly polarized at centimetre wavelengths (its fractional polarization is no more than 1 per cent at 5 GHz, Xiang et al. 2006), and was observed over a wide range of parallactic angles to calibrate the feed response to polarized signals. The instrumental polarization parameters of the antenna feeds (the so-called ‘D-terms’) were calculated from the DA 193 data and then used to correct the phase of the 3C 48 data. The absolute EVPA was then calibrated from observations of 3C 138 [@Cot97b; @Tay00]. A comparison between the apparent polarization angle of 3C 138 and the value from the VLA calibrator monitoring program (i.e., $-15\degr$ at 20 cm wavelength) leads to a differential angle $-22\degr$, which was applied to correct the apparent orientation of the E-vector for the 3C 48 data. After correction of instrumental polarization and absolute polarization angle, the cross-correlated 3C 48 data were used to produce Stokes $Q$ and $U$ images, from which maps of linear polarization intensity and position angle were produced. MERLIN observations and data reduction -------------------------------------- The MERLIN observations of 3C 48 were performed in the fake-continuum mode: the total bandwidth of 15 MHz was split into 15 contiguous channels, 1 MHz for each channel. A number of strong, compact extragalactic sources were interspersed into the observations of 3C 48 to calibrate the complex antenna gains. The MERLIN data were reduced in [aips]{} following the standard procedure described in the MERLIN cookbook. The flux-density scale was determined using 3C 286 which has a flux density of 13.7 Jy at 1.65 GHz. The phases of the data were corrected for the varying parallactic angles on each antenna. Magnetized plasma in the ionosphere results in an additional phase difference between the right- and left-handed signals, owing to Faraday rotation. This time-variable Faraday rotation tends to defocus the polarized image and to give rise to erroneous estimates of the instrumental polarization parameters. We estimated the ionospheric Faraday rotation on each antenna based on the model suggested in the [aips]{} Cookbook, and corrected the phases of the visibilities accordingly. DA 193, OQ 208, PKS 2134+004 and 3C 138 were used to calibrate the time- and elevation-dependent complex gains. These gain solutions from the calibrators were interpolated to the 3C 48 data. The calibrated data were averaged in 30-second bins for further imaging analysis. Self-calibration in both amplitude and phase was performed to remove residual errors. The observations of OQ 208 were used to calculate the instrumental polarization parameters of each antenna assuming a point-source model. The derived parameters were then applied to the multi-source data. We compared the right- and left-hand phase difference of the 3C 286 visibility data with the phase difference value derived from the VLA monitoring program (i.e., $66\degr$ at 20 cm, Cotton et al. 1997b; Taylor & Myers 2000), and obtained a differential angle of $141\degr$. This angle was used to rotate the EVPA of the polarized data for 3C 48. Combination of EVN and MERLIN data ---------------------------------- After self-calibration, the EVN and MERLIN data of 3C 48 were combined to make an image with intermediate resolution and high sensitivity. The pointing centre of the MERLIN observation was offset by 0.034 arcsec to the West and 0.378 arcsec to the North with respect to the EVN pointing centre (Table \[tab:obs\]). Before combination, we first shifted the pointing centre of the MERLIN data to align with that of the EVN data. The Lovell and Cambridge telescopes took part in both the EVN and MERLIN observations. We compared the amplitude of 3C 48 on the common Lovell–Cambridge baseline in the EVN and MERLIN data, and re-scaled the EVN visibilities by multiplying them by a factor of 1.4 to match the MERLIN flux. After combination of EVN and MERLIN visibility data, we performed a few iterations of amplitude and phase self-calibration to eliminate the residual errors resulting from minor offsets in registering the two coordinate frames and flux scales. RESULTS – total intensity images ================================ Figures \[fig:MERcont\] and \[fig:vlbimap\] exhibit the total intensity images derived from the MERLIN, VLBA and EVN data. The final images were created using the [aips]{} and [miriad]{} software packages as well as the [mapplot]{} program in the Caltech VLBI software package. MERLIN images ------------- Figure \[fig:MERcont\] shows the total intensity image of 3C 48 from the MERLIN observations. We used the multi-frequency synthesis technique to minimize the effects of bandwidth smearing, and assumed an optically thin synchrotron spectral index ($\alpha=-0.7$) to scale the amplitude of the visibilities with respect to the central frequency when averaging the data across multiple channels. The final image was produced using a hybrid of the Clark (BGC CLEAN) and Steer (SDI CLEAN) deconvolution algorithms. The image shows that the source structure is characterized by two major features: a compact component contributing about half of the total flux density (hereafter referred to as the ‘compact jet’), and an extended component surrounding the compact jet like a cocoon (hereafter called the ‘extended envelope’). The compact jet is elongated in roughly the north-south direction, in alignment with the VLBI jet. The galactic nucleus corresponding to the central engine of 3C 48 is associated with VLBI component A [@Sim90; @Wil91]. It is embedded in the southern end of the compact jet. The emission peaks at a location close to the VLBI jet component D; the second brightest component in the compact jet is located in the vicinity of the VLBI jet component B2 (Figure \[fig:vlbimap\]: see Section \[section:vlbimap\]). The extended envelope extends out to $\sim$1 arcsec north from the nucleus. At $\sim$0.25 arcsec north of the nucleus, the extended component bends and diffuses toward the northeast. The absence of short baselines ($uv<30 k\lambda$) results in some negative features (the so-called ‘negative bowl’ in synthesis images) just outside the outer boundary of the envelope. The integrated flux density over the whole source is 14.36$\pm$1.02 Jy (very close to the single-dish measurement), suggesting that there is not much missing flux on short spacings. The uncertainty we assign includes both the systematic errors and the [*r.m.s.*]{} fluctuations in the image. Since the calibrator of the flux density scale, 3C 286, is resolved on baselines longer than 600 k$\lambda$ [@An04; @Cot97a], a model with a set of CLEAN components was used in flux density calibration instead of a point-source model. We further compared the derived flux density of the phase calibrator DA 193 from our observations with published results [@Sta98; @Con98]. The comparison suggests that the flux density of DA 193 from our MERLIN observation was consistent with that from the VLBI measurements to within 7 per cent. We note that this systematic error includes both the amplitude calibration error of 3C 286 and the error induced by the intrinsic long-term variability of DA 193; the latter is likely to be dominant. The optical and NIR observations [@Sto91; @Cha99; @Zut04] detect a secondary continuum peak, denoted 3C 48A, at $\sim$1 arcsec northeast of the optical peak of 3C 48. Although MERLIN would be sensitive to any compact structure with this offset from the pointing centre, we did not find any significant radio emission associated with 3C 48A. There is no strong feature at the position of 3C 48A even in high-dynamic-range VLA images [@Bri95; @Feng05]. It is possibly that the radio emission from 3C 48A is intrinsically weak if 3C 48A is a disrupted nucleus of the companion galaxy without an active AGN [@Sto91] or 3C 48A is an active star forming region [@Cha99]. In either case, the emission power of 3C 48A would be dominated by thermal sources and any radio radiation would be highly obscured by the surrounding interstellar medium. VLBA and EVN images {#section:vlbimap} ------------------- Figure \[fig:vlbimap\] shows the compact radio jet of 3C 48 on various scales derived from the VLBA and EVN observations. Table \[tab:figpar\] gives the parameters of the images. The VLBI data have been averaged on all frequency channels in individual IFs to export a single-channel dataset. The visibility amplitudes on each IF have been corrected on the assumption of a spectral index of $-0.7$. The total-intensity images derived from the 1.5-GHz VLBA and 1.65-GHz EVN data are shown in Figures \[fig:vlbimap\]-a to \[fig:vlbimap\]-c. The jet morphology we see is consistent with other published high-resolution images [@Wil90; @Wil91; @Nan91; @Wor04; @Feng05]. The jet extends $\sim$0.5 arcsec in the north-south direction, and consists of a diffuse plume in which a number of bright compact knots are embedded. We label these knots in the image using nomenclature consistent with the previous VLBI observations [@Wil91; @Wor04] (we introduce the labels B3 and D2 for faint features in the B and D regions revealed by our new observations). The active nucleus is thought to be located at the southern end of the jet, i.e., close to the position of component A [@Sim90; @Wil91]. The bright knots, other than the nuclear component A, are thought to be associated with shocks that are created when the jet flow passes through the dense interstellar medium in the host galaxy [@Wil91; @Wor04; @Feng05]. Figure \[fig:vlbimap\]-b enlarges the inner jet region of the 3C 48, showing the structure between A and B2. At $\sim$0.05 arcsec north away from the core A, the jet brightens at the hot spot B. B is in fact the brightest jet knot in the VLBI images. Earlier 1.5-GHz images (Figure 1 : Wilkinson et al. 1991; Figure 5 : Worrall et al. 2004) show only weak flux ($\sim4\sigma$) between A and B, but in our high-dynamic-range image in Figure \[fig:vlbimap\]-b, a continuous jet is distinctly seen to connect A and B. From component B, the jet curves to the northwest. At $\sim$0.1 arcsec north of the nucleus, there is a bright component B2. After B2, the jet position angle seems to have a significant increase, and the jet bends into a second curve with a larger radius. At 0.25 arcsec north of the nucleus, the jet runs into a bright knot C which is elongated in the East-West direction. Here a plume of emission turns toward the northeast. The outer boundary of the plume feature is ill-defined in this image since its surface brightness is dependent on the [*r.m.s.*]{} noise in the image. The compact jet still keeps its northward motion from component C, but bends into an even larger curvature. Beyond component D2, the compact VLBI jet is too weak to be detected. At 4.8 and 8.3 GHz, most of the extended emission is resolved out (Figures \[fig:vlbimap\]-d to \[fig:vlbimap\]-g) and only a few compact knots remain visible. Figure \[fig:vlbimap\]-e at 4.8 GHz highlights the core-jet structure within 150 pc ($\sim$30 mas); the ridge line appears to oscillate from side to side. At the resolution of this image the core A is resolved into two sub-components, which we denote A1 and A2. Figure \[fig:vlbimap\]-g at 8.3 GHz focuses on the nuclear region within 50 pc ($\sim$10 mas) and clearly shows two well-separated components. Beyond this distance the brightness of the inner jet is below the detection threshold. This is consistent with what was seen in the 8.4- and 15.4-GHz images from the 1996 VLBA observations [@Wor04]. Figure \[fig:core\] focuses on the core A and inner jet out to the hot spot B. Figure \[fig:core\]-a shows the 1.5-GHz image from 2004. Unlike the image already shown in Figure \[fig:vlbimap\]-b, this image was produced with a super-uniform weighting of the $uv$ plane (see the caption of Figure \[fig:core\] for details). The high-resolution 1.5-GHz image reveals a quasi-oscillatory jet extending to a distance of $\sim$40 mas ($\sim$200 pc) to the north of the core A. Interestingly, Figure \[fig:core\]-b shows similar oscillatory jet structure at 4.8-GHz on both epoch 2004 (contours) and epoch 1996 (grey-scale, Worrall et al. 2004). The consistency of the jet morphology seen in both 1.5- and 4.8-GHz images and in both epochs may suggest that the oscillatory pattern of the jet seen on kpc scales (Figure \[fig:vlbimap\]) may be traced back to the innermost jet on parsec scales. Figure \[fig:core\]-c shows the 8.3-GHz images in 2004 (contours) and 1996 (grey scale, Worrall et al. 2004). In 1996 (the image denoted ‘1996X’) the core is only slightly resolved into the two components A1 and A2, while these are well separated by 3.5 mas (2 times the synthesized beam size) in the 2004 observations (‘2004X’). Direct comparison of 1996X and 2004X images thus provides evidence for a northward position shift of A2 between 1996 and 2004. Figure \[fig:core\]-d overlays the 2004X contour map on the 1996U (15.4 GHz, Worrall et al. 2004) grey-scale map. Neglecting the minor positional offset of A1 between 1996U and 2004X, possibly due to opacity effects, this comparison of 1996U and 2004X maps is also consistent with the idea that A2 has moved north between 1996 and 2004. We will discuss the jet kinematics in detail in Section \[section:pm\]. image analysis ============== Spectral index distribution along the radio jet ----------------------------------------------- In order to measure the spectral properties of the 3C 48 jet, we re-mapped the 4.99-GHz MERLIN data acquired on 1992 June 15 [@Feng05] and compared it with the 1.65-GHz EVN+MERLIN data described in the present paper. The individual data sets were first mapped with the same [*uv*]{} range, and convolved with the same 40$\times$40 (mas) restoring beam. Then we compared the intensities of the two images pixel by pixel to calculate the spectral index $\alpha^{4.99}_{1.65}$. The results are shown in Figure \[fig:spix\]. Component A shows a rather flat spectrum with a spectral index $\alpha^{4.99}_{1.65}=-0.24\pm0.09$. All other bright knots show steep spectral indices, ranging from $-0.66$ to $-0.92$. The extended envelope in general has an even steeper spectrum with $\alpha\lesssim-1.10$. Spectral steepening in radio sources is a signature of a less efficient acceleration mechanism and/or the depletion of high-energy electrons through synchrotron/Compton radiation losses and adiabatic losses as a result of the expansion of the plasma as it flows away from active acceleration region. The different spectral index distribution seen in the compact jet and extended envelope may indicate that there are different electron populations in these two components, with the extended component arising from an aged electron population. Linear polarization images -------------------------- ### MERLIN images {#section:rm-reference} Figure \[fig:MERpol\] displays the polarization image made from the MERLIN data. The majority of the polarized emission is detected in the inner region of the source, in alignment with the compact jet. The polarized intensity peaks in two locations. The brightest one is near the VLBI jet component C, with an integrated polarized intensity of 0.31 Jy and a mean percentage of polarization (defined as $\frac{\Sigma \sqrt{Q_i^2+U_i^2}}{\Sigma I_i}$, where $i$ represents the $i$th polarized sub-components) of $m=5.8$ per cent. The secondary one is located between VLBI jet components B and B2, with an integrated polarized intensity of 0.23 Jy and a mean degree of polarization $m=9.5$ per cent. Both of the two peaks show clear deviations from the total intensity peaks in Figure \[fig:MERcont\]. These measurements of polarization structure and fractional polarization are in good agreement with those observed with the VLA at 2-cm wavelength with a similar angular resolution [@Bre84]. The integrated polarized flux density in the whole source is 0.64$\pm$0.05 Jy and the integrated fractional polarization is (4.9$\pm$0.4) per cent. Since the integrated polarized intensity is in fact a vector sum of different polarized sub-components, the percentage polarization calculated in this way represents a lower limit. We can see from the image (Figure \[fig:MERpol\]) that the percentage of the polarization at individual pixels is higher than 5 per cent, and increases toward the south of the nucleus. A maximum value of $m\gtrsim30$ per cent is detected at $\sim$0.045 arcsec south of the nucleus. The fractional polarization ($m>4.9$ per cent) measured from our MERLIN observation at 18 cm is at least an order of magnitude higher than the VLA measurement at 20 cm, although it is consistent with the values measured by the VLA at 6 cm and shorter wavelengths. This difference in the fractional polarization at these very similar wavelengths is most likely to be an observational effect due to beam depolarization, rather than being due to intrinsic variations in the Faraday depth (R. Perley, private communication). The averaged polarization angle (or EVPA) is $-18\degr\pm5\degr$ in the polarization structure. On the basis of the new measurements of the Rotation Measure (RM) towards 3C 48 by Mantovani et al. (2009), i.e., RM=$-64$ rad m$^{-2}$ and intrinsic position angle $\phi_0=116\degr$ [@Sim81; @Man09], we get a polarization angle of $-4\degr$ at 1.65 GHz. This result suggests that the absolute EVPA calibration of 3C 48 agrees with the RM-corrected EVPA within 3$\sigma$. We show in Figure \[fig:MERpol\] the RM-corrected EVPAs. The EVPAs are well aligned in the North-South direction, indicating an ordered magnetic field in the Faraday screen. ### EVN and VLBA images At the resolution of the EVN, most of the polarized emission from extended structures is resolved out. In order to map the polarized emission with modest sensitivity and resolution, we created Stokes $Q$ and $U$ maps using only the European baselines. Figure \[fig:VLBIpol\]-a shows the linear polarization of 3C 48 from the 1.65 GHz EVN data. The polarized emission peaks at two components to the East (hereafter, ‘C-East’) and West (hereafter, ‘C-West’) of component C. The integrated polarized flux density is 24.8 mJy in ‘C-West’ and of 22.9 mJy in ‘C-East’, and the mean percentage polarization in the two regions is 6.3 per cent and 10.7 per cent respectively. The real fractional polarization at individual pixels is much higher, for the reasons discussed above (Section 4.2.1). There is clear evidence for the existence of sub-components in ‘C-West’ and ‘C-East’; these polarized sub-components show a variety of EVPAs, and have much higher fractional polarization than the ‘mean’ value. The polarization is as high as 40 per cent at the inner edge of the knot C, which would be consistent with the existence of a shear layer produced by the jet-ISM interaction and/or a helical magnetic field (3C 43: Cotton et al. 2003; 3C 120: Gómez et al. 2008). Component B, the brightest VLBI component, however, is weakly polarized with an intensity $<$4.0 mJy beam$^{-1}$ (percentage polarization less than 1 per cent). The nucleus A shows no obvious polarization. The 20-cm VLBA observations were carried out in four 8-MHz bands, centred at 1404.5, 1412.5, 1604.5 and 1612.5 MHz. In order to compare with the 1.65-GHz EVN polarization image, we made a VLBA polarization image (Figure \[fig:VLBIpol\]-b) using data in the latter two bands. This image displays a polarization structure in excellent agreement with that detected at 1.65 GHz with the EVN, although the angular resolution is 3 times higher than the latter: the polarized emission mostly comes from the vicinity of component C and the fractional polarization increases where the jet bends; the hot spot B and the core A are weakly polarized or not detected in polarization. The 1.65- and 1.61-GHz images show detailed polarized structure in the component-C region on a spatial scale of tens of parsecs: the polarization angle (EVPA) shows a gradual increase across component C, with a total range of $160\degr$, and the percentage of polarization gradually increases from 5 per cent to $\gtrsim$30 per cent from the Western edge to the Eastern edge at both ‘C-West’ and ‘C-East’. Figure \[fig:VLBIpol\]-c and \[fig:VLBIpol\]-d show the 4.8- and 8.3-GHz polarization images made with the VLBA data. Both images were made by tapering the visibility data using a Gaussian function in order to increase the signal-to-noise of the low-surface brightness emission. Similar to what is seen in the 1.65 and 1.61-GHz images, component ‘C-West’ shows a polarization angle that increases by $80\degr$ across the component, but these images show the opposite sense of change of fractional polarization – fractional polarization decreases from 60 per cent down to 20 per cent from the northwest to the southeast. Another distinct difference is that hot spot B shows increasing fractional polarization toward the higher frequencies, $m \sim2.0$ per cent at 4.8 GHz and $m\sim12$ per cent at 8.3 GHz in contrast with $m\lesssim1$ per cent at 1.6 GHz. The difference in the fractional polarizations of B at 1.6/4.8 GHz and 8.3 GHz imply that a component of the Faraday screen is unresolved at 1.6 and 4.8 GHz and/or that some internal depolarization is at work. The non-detection of polarization from the core A at all four frequencies may suggest a tangled magnetic field at the base of the jet. EVPA gradient at component C and RM distribution ------------------------------------------------ We found at all four frequencies that the polarization angles undergo a rotation by $\gtrsim 80\degr$ across the jet ridge line at both the ‘C-East’ and ‘C-West’ components. There are four possible factors that may affect the observed polarization angle: (1) the calibration of the absolute EVPAs; (2) Faraday rotation caused by Galactic ionized gas; (3) Faraday rotation due to gas within the 3C 48 system and (4) intrinsic polarization structure changes. The correction of absolute EVPAs applies to all polarization structure, so it can not explain the position-dependent polarization angle changes at component C; in any case, the fact that we see similar patterns at four different frequencies, calibrated following independent procedures, rules out the possibility of calibration error. Galactic Faraday rotation is non-negligible (Section \[section:rm-reference\]; $-64$ rad m$^{-2}$ implies rotations from the true position angle of $168\degr$ at 1.4 GHz, $129\degr$ at 1.6 GHz, $14.3\degr$ at 4.8 GHz and $4.8\degr$ at 8.3 GHz), and means that we expect significant differences between the EVPA measured at our different frequencies; however, the Galactic Faraday screen should vary on much larger angular scales than we observe. Only factors (3) and (4), which reflect the situation internal to the 3C 48 system itself, will give rise to a position-dependent rotation of the EVPAs. The EVPA gradient is related to the gradient of the RM and the intrinsic polarization angle by: $\frac{{\rm d}\phi}{{\rm d}x}=\lambda^2\frac{{\rm d}(RM)}{{\rm d}x}+\frac{{\rm d}\phi_0}{{\rm d}x}$, where the first term represents the RM gradient and the latter term represents the intrinsic polarization angle gradient. If the systematic gradient of EVPAs, $\frac{{\rm d}\phi}{{\rm d}x}$, were solely attributed to an RM gradient, then $\frac{d\phi}{dx}$ would show a strong frequency dependence; on the other hand, if $\frac{{\rm d}\phi}{{\rm d}x}$ is associated with the change of the intrinsic polarization angle, there is no frequency-dependence. We compared the $\frac{{\rm d}\phi}{{\rm d}x}$ at 1.6 and 4.8 GHz and found a ratio $\frac{{\rm d}\phi/{\rm d}x (1.6GHz)}{{\rm d}\phi/{\rm d}x(4.8GHz)}=1.8$. This number falls between 1.0 (the value expected if there were no RM gradient) and 8.8 (the ratio of $\lambda^2$), suggesting that a combination of RM and intrinsic polarization angle gradients are responsible for the systematic gradient of EVPAs at C. Accordingly, it is worthwhile to attempt to measure the RM in the VLBI components of 3C 48. The first two bands of the 20-cm VLBA data (centre frequency 1.408 GHz) are separated from the last two bands (centre frequency 1.608 GHz) by 200 MHz, indicating a differential polarization angle of $\sim 40\degr$ across the passband. The low integrated rotation measure means that the effects of Faraday rotation are not significant ($<10\degr$) between 4.8 and 8.3 GHz, while the absolute EVPA calibration at 8.3 GHz is uncertain; moreover, the [*uv*]{} sampling at 8.3 GHz is too sparse to allow us to image identical source structure at 1.5 and 4.8 GHz. Therefore we used the 1.408, 1.608 and 4.78 GHz data to map the RM distribution in 3C 48. We first re-imaged the Stokes $Q$ and $U$ data at the three frequencies with a common [*uv*]{} cutoff at $>$400 k$\lambda$ and restored with the same convolving beam. We tapered the $uv$ plane weights when imaging the 4.78-GHz data in order to achieve a similar intrinsic resolution to that of the images at the two lower frequencies. We then made polarization angle images from the Stokes $Q$ and $U$ maps. The three polarization angle images were assembled to calculate the RM (using [aips]{} task RM). The resulting RM image is shown in Figure \[fig:RM\]. The image shows a smooth distribution of RM in the component-C region except for a region northeast of ‘C-West’. The superposed plots present the fits to the RM and intrinsic polarization angle ($\phi_0$, the orientation of polarization extrapolated at $\lambda=0$) at four selected locations. The polarization position angles at individual frequencies have multiples of $\pi$ added or subtracted to remove the $n\pi$ amibiguity. The errors in the calculated RMs and $\phi_0$ are derived from the linear fits. We note that the systematic error due to the absolute EVPA calibration feeds into the error on the observed polarization angle. All four fits show a good match with a $\lambda^2$ law. The fitted parameters at ‘P4’ in the ‘C-East’ region are consistent with those derived from the single-dish measurements for the overall source [@Man09]. The western component (‘C-West’) shows a gradient of RM from $-95$ rad m$^{-2}$ at ‘P1’ to $-85$ rad m$^{-2}$ at ‘P3’, and the intrinsic polarization angle varies from $123\degr$ (or $-57\degr$) at ‘P1’, through $146\degr$ (or $-34\degr$) at ‘P2’ to $5\degr$ at ‘P3’. This result is in good agreement with the qualitative analysis of the EVPA gradients above. A straightforward interpretation of the gradients of the RMs and the intrinsic polarization angles is that the magnetic field orientation gradually varies across the jet ridge line; for example, a helical magnetic field surrounding the jet might have this effect. An alternative interpretation for the enhancement of the rotation measurement at the edge of the jet is that it is associated with thermal electrons in a milliarcsec-scale Faraday screens surrounding or inside the jet due to jet-ISM interactions [@Cot03; @Gom08]. More observations are needed to investigate the origins of the varying RM and $\phi_0$. The hot spot B shows a much larger difference of EVPAs between 4.8 and 8.3 GHz than is seen in component C. This might be a signature of different RMs at B and C. A rough calculation suggests a RM of $-330\pm60$ rad m$^{-2}$ at B. The high rotation measure and high fractional polarization (Section 4.2) is indicative of a strong, ordered magnetic field in the vicinity of B. This might be expected in a region containing a shock in which the line-of-sight component of the magnetic field and/or the density of thermal electrons are enhanced; in fact, the proper motion of B (Section 4.5) does provide some evidence for a stationary shock in this region. Physical properties of compact components in VLBI images -------------------------------------------------------- In order to make a quantitative study of the radiation properties of the compact VLBI components in 3C 48, we fitted the images of compact components in the VLBI images from our new observations and from the VLBA data taken in 1996 [@Wor04] with Gaussian models. Measurements from the 1996 data used mapping parameters consistent with those for the 2004 images. Table \[tab:model\] lists the fitted parameters of bright VLBI components in ascending frequency order. The discrete compact components in the 4.8- and 8.3-GHz VLBA images are well fitted with Gaussian models along with a zero-level base and slope accounting for the extended background structure. The fit to extended emission structure is sensitive to the [*uv*]{} sampling and the sensitivity of the image. We have re-imaged the 1.5-GHz VLBA image using the same parameters as for the 1.65-GHz EVN image, i.e., the same [*uv*]{} range and restoring beam. At 1.5 and 1.65 GHz, Gaussian models are good approximations to the emission structure of compact sources with high signal-to-noise ratio, such as components A and B. For extended sources (i.e., components B2 to D2) whose emission structures are either not well modelled by Gaussian distribution, or blended with many sub-components, model fitting with a single Gaussian model gives a larger uncertainty for the fitted parameters. In particular, the determination of the integrated flux density is very sensitive to the apparent source size. The uncertainties for the fitted parameters in Table \[tab:model\] are derived from the output of the [aips]{} task [JMFIT]{}. These fitting errors are sensitive to the intensity fluctuations in the images and source shapes. In most cases, the fitting errors for the peak intensities of Gaussian components are roughly equal to the [ *r.m.s.*]{} noise. We note that the uncertainty on the integrated flux density should also contain systematic calibration errors propagated from the amplitude calibration of the visibility data, in addition to the fitting errors. The calibration error normally dominates over the fitting error. The amplitude calibration for the VLBI antennas was made from the measurements of system temperature ($T_{sys}$) at two-minute intervals during the observations combined with the antenna gain curves measured at each VLBI station. For the VLBA data, this calibration has an accuracy $\lesssim$5 per cent of the amplitude scale[^3]. Because of the diversity of the antenna performance of the EVN elements, we adopted an averaged amplitude calibration uncertainty of 5 per cent for the EVN data. The positions of the VLBI core A1 at 4.8, 8.3 and 15.4 GHz show good alignment within 0.4 mas at different frequencies and epochs. The positions of the unresolved core A at 1.5 and 1.65 GHz show a systematic northward offset by 2–4 mas relative to the position of A1 at higher frequencies. Due to the low resolution and high opacity at 1.5 GHz, the position of A at this frequency reflects the centroid of the blended emission structure of the active galactic nucleus and inner 40-pc jet. The parameters that we have derived for the compact components A, B and B2 in epoch 1996 are in good agreement with those determined by Worrall et al. (2004) at the same frequency band. The results for fitting to extended knots at 1.5 and 1.65 GHz are in less good agreement. This is probably because of the different [*uv*]{} sampling on short spacings, meaning that the VLBA and EVN data sample different extended structures in the emission. The integrated flux densities of the VLBI components A1 and A2 in 1996X (8.3 GHz) are higher than those in 2004X (8.3 GHz) by $\sim$100 per cent (A1) and $\sim$60 per cent (A2), respectively. The large discrepancy in the flux densities of A1 and A2 between epochs 1996X and 2004X can not easily be interpreted as an amplitude calibration error of larger than 60 per cent since we do not see a variation at a comparable level in the flux densities of components B, B2 and D. Although the [*total*]{} flux densities of CSS sources in general exhibit no violent variability at radio wavelengths, the possibility of small-amplitude ($\lesssim$100 per cent) variability in the VLBI core and inner jet components is not ruled out. Component A1 has a flat spectrum with $\alpha^{8.3}_{4.8}=-0.34\pm0.04$ between 4.8 and 8.3 GHz in epoch 2004; component A2 has a rather steeper spectrum with $\alpha^{8.3}_{4.8}=-1.29\pm0.16$ (epoch 2004). The spectral properties of these two components support the idea that A1 is associated with the active nucleus and suffers from synchrotron self-absorption at centimetre radio wavelengths; in this picture, A2 is the innermost jet. The spectral indices of components B and B2 in epoch 2004 are $\alpha^{8.3}_{4.8}=-0.82\pm0.10$ (B) and $\alpha^{8.3}_{4.8}=-0.79\pm0.10$ (B2), respectively. This is consistent with the measurements from the 1.65 and 4.99 GHz images (Figure \[fig:spix\]). Component D shows a relatively flatter spectrum in epoch 2004 with $\alpha^{8.3}_{4.8}=-0.46\pm0.06$, in contrast to the other jet knots. While this spectral index is consistent with those of the shock-accelerated hot spots in radio galaxies, the flattening of the spectrum in D might also arise from a local compression of particles and magnetic field. Table \[tab:tb\] lists the brightness temperatures ($T_b$) of the compact VLBI components A1, A2 and B. All these VLBI components have a brightness temperatures ($T_b$) higher than $10^8$K, confirming their non-thermal origin. These brightness temperatures are well below the $10^{11-12}$ K upper limit constrained by the inverse Compton catastrophic [@KP69], suggesting that the relativistic jet plasma is only mildly beamed toward the line of sight. The $T_b$ of A1 is about 3 times higher than that of A2 at 4.8 and 8.3 GHz, and the $T_b$ of A1 decreases toward higher frequencies. Together with the flat spectrum and variability of A1, the observed results are consistent with A1 being the self-absorbed core harbouring the AGN. $T_b$ is much higher in 1996X than 2004X for both A1 and A2, a consequence of the measured flux density variation between the two epochs. Proper motions of VLBI components {#section:pm} --------------------------------- The Gaussian fitting results presented in Table \[tab:model\] may be used to calculate the proper motions of VLBI components. In order to search for proper motions in 3C 48, maps at different epochs should be aligned at a compact component such as the core [@Wor04]. However, thanks to our new VLBI observations we know that aligning the cores at 1.5-GHz is not likely to be practical, since the core structure appears to be changing on the relevant timescales. Even at 4.8 GHz, the core still blends with the inner jet A2 in epoch 1996C (Figure \[fig:core\]). In contrast to these two lower frequencies, the 8.3-GHz images have higher resolution, better separation of A1 and A2, and less contamination from extended emission. These make 8.3-GHz images the best choice for the proper motion analysis. In the following discussion of proper motion measurements we rely on the 8.3 GHz images. We have already commented on the shift of the peak of A2 to the north from epochs 1996X to 2004X in Figure \[fig:core\]. A quantitative calculation based on the model fitting results gives a positional variation of 1.38 mas to the North and 0.15 mas to the West during a time span 8.43 yr, assuming that the core A1 is stationary. That corresponds to a proper motion of $\mu_\alpha = -0.018\pm0.007$ mas yr$^{-1}$ (’minus’ mean moving to the West) and $\mu_\delta=0.164\pm0.015$ mas yr$^{-1}$, corresponding to an apparent transverse velocity of $v_\alpha = -0.40\pm0.16 \,c$ and $v_\delta=3.74\pm0.35 \,c$. The error quoted here includes both the positional uncertainty derived from Gaussian fitting and the relative offset of the reference point ([*i.e.*]{}, A1). That means that we detect a significant ($>10\sigma$) proper motion for A2 moving to the north. The apparent transverse velocity for A2 is similar to velocities derived from other CSS and GPS sources in which apparent superluminal motions in the pc-scale jet have been detected, e.g., 3.3–9.7$c$ in 3C 138 [@Cot97b; @She01]. We also searched for evidence for proper motions of the other jet knots. The proper motion measurement is limited by the accuracy of the reference point alignment, our ability to make a high-precision position determination at each epoch, and the contamination from extended structure. We found only a $3\sigma$ proper motion from B2, which shows a position change of $\Delta\alpha=0.22\pm0.07$ mas and $\Delta\delta=0.48\pm0.13$ mas in 8.43 yr, corresponding to an apparent velocity of $\beta_{app}=1.43\pm0.33\,c$ to the northeast. The measurements of the position variation of the hot spot B between 1996X and 2004X show no evidence for proper motion with $\mu_\alpha = 0.012\pm0.007$ mas yr$^{-1}$ and $\mu_\delta=0.005\pm0.015$ mas yr$^{-1}$. Worrall et al. (2004) earlier reported a $3\sigma$ proper motion for B by comparing the the 1.5-GHz VLBA image taken in 1996 with Wilkinson et al’s 1.6-GHz image from 11.8 years previously. However, as mentioned above, the 1.5-GHz measurements are subject to the problems of lower angular resolution, poor reference point alignment and contamination from structural variation. In particular, if we extrapolate the observed angular motion of A2 back, the creation of jet component A2 took place in 1984, therefore in 1996 A2 would still have been blended with A1 in the 1.5-GHz image within $\frac{1}{4}$ beam. The fitting of a Gaussian to the combination of A1 and A2 at 1.5 GHz on epoch 1996 would then have suffered from the effects of the structural changes in the core due to the expansion of A2. For these reasons we conclude that the hot spot B is stationary to the limit of our ability to measure motions. For the other jet components, the complex source structure does not permit any determination of proper motions. Kinematics of the radio jet {#section:kinematics} =========================== Geometry of the radio jet ------------------------- Most CSS sources show double or triple structures on kpc scales, analogous to classical FR I or FR II galaxies. However, some CSS sources show strongly asymmetric structures. At small viewing angles, the advancing jet looks much brighter than the receding one, due to Doppler boosting. The sidedness of radio jets can be characterized by the jet-to-counterjet intensity ratio $R$. In VLA images [@Bri95; @Feng05], 3C 48 shows two-sided structure in the north-south direction. The southern (presumably receding) component is much weaker than the north (advancing) one. In VLBI images (Wilkinson et al. 1991; Worrall et al. 2004; the present paper) 3C 48 shows a one-sided jet to the north of the nucleus. If the non-detection of the counterjet is solely attributed to Doppler deboosting, the sideness parameter $R$ can be estimated from the intensity ratio of jet knots to the detection limit (derived from the $3\sigma$ off-source noise). Assuming the source is intrinsically symmetric out to a projected separation of 600 pc (the distance of B2 away from A1), the sideness parameter would be $>200$ for B2 and B in the 1.5-GHz image (Figure \[fig:vlbimap\]-a). In the highest-sensitivity image on epoch 2004C (Figure \[fig:vlbimap\]-d), the off-source noise in the image is 40 $\mu$Jy beam$^{-1}$, so that the derived $R$ at component B could be as high as $\gtrsim$900. For a smooth jet which consists of a number of unresolved components, the jet-to-counterjet brightness ratio $R$ is related to the jet velocity ($\beta$) and viewing angle ($\Theta$) by $$R=\left(\frac{1+\beta\cos\Theta}{1-\beta\cos\Theta}\right)^{2-\alpha}$$ Assuming an optically thin spectral index $\alpha=-1.0$ for the 3C 48 jet (Figure \[fig:spix\]), the sideness parameter $R\gtrsim900$ estimated above gives a limit of $\beta\cos\Theta>0.81\,c$ for the projected jet velocity in the line of sight. Using only the combination of parameters $\beta\cos\Theta$ it is not possible to determine the kinematics (jet speed $\beta$) and the geometry (viewing angle $\Theta$) of the jet flow. Additional constraints may come from the apparent transverse velocity, which is related to the jet velocity by $\beta_{app} = \frac{\beta\sin\Theta}{1-\beta\cos\Theta}$. In Section \[section:pm\] we determined the apparent velocities for components B and B2, $\beta_{app}(B)=3.74c\pm0.35c$, $\beta_{app}(B2)=1.43c\pm0.33c$, and so we can combine $\beta\cos\Theta$ and $\beta_{app}$ to place a constraint on the kinematics and orientation of the outer jet. The constraints to the jet velocity and source orientation are shown in Figure \[fig:viewangle\]. The results imply that the 3C 48 jet moves at $v>0.85c$ along a viewing angle less than $35\degr$. Helical radio jet structure {#section:helic} --------------------------- As discussed in Section \[section:vlbimap\] the bright jet knots define a sinusoidal ridge line. This is the expected appearance of a helically twisted jet projected on to the plane of the sky. Helical radio jets, or jet structure with multiple bends, can be triggered by periodic variations in the direction of ejection (e.g., precession of the jet nozzle), and/or random perturbations at the start of the jet (e.g., jet-cloud collisions). For example, the wiggles in the ballistic jets in SS 433 are interpreted in terms of periodic variation in the direction of ejection [@Hje81]. Alternatively, small perturbations at the start of a coherent, smooth jet stream might be amplified by the Kelvin-Helmholtz (K-H) instability and grow downstream in the jet. In this case, the triggering of the helical mode and its actual evolution in the jet are dependent on the fluctuation properties of the initial perturbations, the dynamics of the jet flow, and the physical properties of the surrounding interstellar medium [@Har87; @Har03]. In the following subsections we consider these two models in more detail. ### Model 1 – precessing jet We use a simple precession model [@Hje81], taking into account only kinematics, to model the apparently oscillatory structure of the 3C 48 radio jet. Figure \[fig:sketch\] shows a sketch map of a 3-D jet projected on the plane of the sky. The X- and Y-axis are defined so that they point to the Right Ascension and Declination directions, respectively. In the right-handed coordinate system, the Z-axis is perpendicular to the XOY plane and the minus-Z direction points to the observer. The jet axis is tilting toward the observer by an inclination angle of ($90-\theta$). The observed jet axis lies at a position angle $\alpha$. In the jet rest frame, the kinematic equation of a precessing jet can be parameterized by jet velocity ($V_j$), half-opening angle of the helix cone ($\varphi$) and angular velocity (or, equivalently, precession period $P$). To simplify the calculations, we assume a constant jet flow velocity $V_j$, a constant opening angle $\varphi$ of the helix, and a constant angular velocity. We ignore the width of the jet itself, so we are actually fitting to the ridge line of the jet. The jet thickness does not significantly affect the fitting unless it is far wider than the opening angle of the helix cone. (We note that, although we have measured lower proper motion velocities in B and B2 than the velocity in the inner jet A2, this does not necessarily imply deceleration in the outer jet flow, since the brightening at B, and to some extent at B2, may arise mostly from stationary shocks; the proper motions of B and B2 thus represent a lower limit on the actual bulk motions of the jet.) We further assume the origin of the precession arises from the central black hole and accretion disk system, so that ($X_0$,$Y_0$,$Z_0$) can be taken as zero. In the observer’s frame the jet trajectory shown in the CLEAN image can be acquired by projecting the 3-D jet on the plane of the sky and then performing a rotation by an angle $\alpha$ in the plane of the sky so that the Y-axis aligns to the North (Declination) and X-axis points to the East (Right Ascension). In addition to the above parameters, we need to define a rotation sign parameter $s_{rot}$ ($s_{rot}=+1$ means counterclockwise rotation) and jet side parameter ($s_{jet}=+1$ means the jet moves toward the observer). Since we are dealing with the advancing jet, the jet side parameter is set to 1. Based on our calculations, we found that a clockwise rotation pattern ($s_{rot}=-1$) fits the 3C 48 jet. To estimate the kinematical properties of the precessing jet flow, we use the proper motion measurements of component A2 as an estimate of the jet velocity and orientation (Figure \[fig:viewangle\]). We have chosen a set of parameters consistent with the curve for $V_{\rm app,j}=3.7c$ and a viewing angle of $17\degr$. Other combinations of angles to the line of sight and velocities give qualitatively similar curves. For example, if we use a lower flow speed instead, a similar model structure can be produced by adjusting other parameters accordingly, e.g. by increasing the precessing period by the same factor. The high-resolution VLBI images (Figure \[fig:core\]) show that the innermost jet aligns to the North. So an initial position angle $\alpha=0$ should be a reasonable estimate. The VLBI images (Figure \[fig:vlbimap\]) suggest that the position angle of the jet ridge line shows an increasing trend starting from the hot spot B. Moreover, we found that a model with a constant position angle does not fit simultaneously to both the inner and outer jet. To simplify the calculation, we introduced a parameter $\frac{{\rm d}\alpha}{{\rm d}t}$ to account for the increasing position angle in the outer jet. The fitted jet ridge line is shown (thick green line) in the upper panel of Figure \[fig:helicalfit\] overlaid on the total intensity image. The assumed and fitted parameters are listed in Table \[tab:helicalfit\]. The modelled helix fits the general wiggling jet structure with at least two complete periods of oscillation. The fitted opening angle of $2.0\degr$ suggests that the line of sight falls outside the helix cone. The initial phase angle $\phi_0$ is loosely constrained; it is related to the reference time of the ejection of the jet knot, $\phi_0 = 2\pi t_{ref}/P$. The fits suggest that the reference time is $t_{ref}=-480$ yr. In the presence of the gradual tilting of the jet axis as well as the helical coiling around the jet axis, the fits most likely represents a superposition of the precession of the jet knots and the nutation of the jet axis, analogous to SS 433 (e.g. Katz et al. 1982; Begelman, King & Pringle 2006). The fitted period of 3500 yr is then a nutation period, about 0.4 times the dynamical time scale of the jet, assuming a flow speed of $0.965c$, while the precession period is much longer. From the rate of the jet axis tilting, we estimate a precession period of $\sim2\times10^5$ yr. The ratio of the estimated precession period to the nutation period is 57:1, 2.2 times the ratio in SS 433 (which has a 162-day periodic precession and 6.3-day nodding motion: see Begelman, King & Pringle 2006 and references therein). The precessing jet model predicts a smooth structure on small scales, and a constant evolution of the wavelength so long as the jet kinetic energy is conserved and the helix cone is not disrupted (the opening angle of the helix cone is constant). However, the real 3C 48 jet probably does not conserve kinetic energy, as it is characterized by a disrupted jet and violent jet-ISM interactions. In particular, the inner-kpc jet is seen to be physically interacting with a massive gas system, and the observed blue-shifted NIR clouds could be driven by the radio jet to move at velocities up to 1000 km s$^{-1}$ [@Cha99; @Gup05; @Sto07]. The 3C 48 radio jet thus might lose a fraction of its kinetic energy, resulting in a slowing down of the jet flow and the shrinking of the wavelength in the outer jet, assuming that the precessing periodicity is not destroyed. ### Model 2 – Kelvin-Helmholtz instabilities We next investigate the interpretation of a hydrodynamic or magnetized jet instability for a helical structure [@Har87; @Cam86]. We used the simple analytic model described in Steffen et al. (1995) to fit to the helical jet trajectory in 3C 48. The kinematic equations of this toy model are solved on the basis of the conservation of kinetic energy $E_{kin}$ and the specific momentum in the jet motion direction (Case 2 : Steffen et al. 1995). It is in fact identical to the isothermal hydrodynamic model [@Har87] under the condition of a small helix opening angle. Model fitting with an adiabatically expanding jet can basically obtain similar helical twisting jet as well, but the initial amplitude growth is much faster [@Har87] than that of the isothermal jet. In this analysis we confine our discussion to the isothermal case. To make the calculations simple but not to lose generality, we used similar assumptions to those of Model 1 on the jet kinematics and geometry. (We should note that although we used an apparent velocity $V_{\rm app,j}$ with same value in Model 1, the jet speed $V_j$ in the K-H model is the pattern speed, and therefore the real flow speed and the viewing angle in the K-H model are more uncertain than for the ballistic case.) In addition, we assume that the initial perturbations originate from a region very close to the central engine. The calculations thus start from an initial distance of zero along the jet axis and a small displacement $r_0$ in the rotation plane away from the jet axis. Moreover, we assumed an initial position angle $\alpha_0=0\degr$ , and again introduced a rate ${\rm d}\alpha/{\rm d}t$ to explain the eastward tilting of the jet axis. The half opening angle, which is a parameter to be fitted, is assumed constant. This assumption is plausible since the jet width seems not to change much within 0.5 arcsec, indicating that the trajectory of the jet is not disrupted even given the occurrence of a number of jet-ISM interactions. In addition to the above morphological assumptions, the model also assumes the conservation of specific momentum and kinetic energy $E_{\rm kin}$ along the jet axis. The conservation of specific momentum is equivalent to a constant velocity along the jet axis if mass loss or entrainment are negligible. The combination of the conservation of specific momentum and kinetic energy along the jet axis results in a constant pitch angle along the helical jet. Furthermore, the constant jet opening angle and pitch angle lead to a helical geometry in which the oscillatory wavelength linearly increases with time. The parameter $r_0$ controls how fast the wavelength varies (Equation 12 : Steffen et al. 1995). The model describes a self-similar helical trajectory with a number of revolutions as long as the helical amplitude is not dampened too rapidly. The modelled curve is exhibited in the lower panel of Figure \[fig:helicalfit\]. The assumed and fitted parameters are listed in Table \[tab:helicalfit\]. As mentioned above, this K-H instability model predicts that, when the helical amplitude is not dampened and the opening angle $\varphi$ is small ($\varphi\ll \arctan{\frac{r_0}{\lambda_0}}$), the oscillating wavelength (or period) along the jet axis increases linearly with time. The fits give an initial wavelength of 60 mas and initial period of 370 yr. The period increases to $1.3\times10^4$ yr at the end of the plot window of 9000 yr. The fitted curve displays more oscillations in the inner part of the jet and smoother structure in the outer part, due to the decreasing angular velocity downstream. The initial transverse distance $r_0$ represents the location where the K-H instability starts to grow in the surface of the jet. It is associated with the varying rate of the wavelength. A value of $r_0=1.8$ mas corresponds to a projected linear distance of 9.2 pc off the jet axis. As discussed above, the major discrepancy between the helical model and the real 3C 48 jet could be the assumption of the conservation of kinetic energy $E_{kin}$. We have tried to fit the helical model without the conservation of kinetic energy but with conserved angular momentum, which is in principle similar to Case 4 in Steffen et al. (1995). However, in this case, the modelled helix rapidly evolves into a straight line, and thus fails to reproduce the observed 3C 48 jet on kpc scales. ### Comparison of the two models Both two models give fits to the overall jet structure of 3C 48 within 0.45 arcsec with 2–3 complete revolutions, but they have some differences in detail. The helical shape of the precessing jet is a superposition of ballistic jet knots modulated by a nodding motion (nutation). In this case, the whole jet envelope wiggles out and shows a restricted periodicity. The observed jet structure displays a smooth shape on rather smaller scales. If, alternatively, the coherent, smooth jet stream is initially disturbed at the jet base, and is amplified by the Kelvin-Helmholtz instability downstream in the jet, the jet stream itself is bent. The resulting helical jet flow rotates faster at the start and gradually slows down as it moves further away. If the twisted inner jet morphology detected at 1.5 and 4.8 GHz (Figure \[fig:core\]) is real, this would support the K-H instability model. Further high-dynamic-range VLBI maps of the inner jet region could test this scenario. In addition to the morphological discrepancy, the two models require different physical origins. In the precessing-jet model, ballistic knots are ejected in different directions which are associated with an ordered rotation in the jet flow direction in the vicinity of the central engine. If the precession results from a rotating injector at the jet base (see discussion in Worrall et al. 2007), the precession period of 0.2 million yr requires a radius of $17 \times (\frac{M_{\bullet} }{10^9 M_{\odot}})^{1/3}$ pc, assuming the injector is in a Keplerian motion around the black hole. This size scale is much larger than the accretion disk, and so we may simply rule out the possibility of an injection from the rotating accretion disk. Instead the long-term precession can plausibly take place in a binary SMBH system or a tilting accretion disk (e.g. [@Beg80; @Lu05]). For example, the precessing period caused by a tilting disk is $\sim 2\times10^5$ yr, assuming a $3\times10^9 M_{\sun}$ SMBH for 3C 48, a dimensionless viscosity parameter $\alpha=0.1$ and the dimentionless specific angular momentum of the black hole $a=0.5$ [@Lu05]. In this scenario, the short-term nodding motion can then be triggered by the tidally-induced torque on the outer brim of the wobbling accretion disk, analogous to SS 433 [@Kat82; @Bat00]. On the other hand, the helical K-H instabilities modes can be triggered by ordered or random perturbations to the jet flow. The fits with Model 2 give an initial perturbation period $\sim370$ yr, which leads to a radius of $\sim 0.25 \times (\frac{M_{\bullet} }{10^9 M_{\odot}})^{1/3}$ pc where perturbations take place. This radius is still larger than the size of the accretion disk, but at this size scale it is still plausible for the perturbations to be due to interactions between the jet flow and the broad-line-region clouds (e.g. 3C120: [@Gom00]). However, the high Faraday depth and/or the possible internal depolarization structure in the radio core A makes it difficult to investigate this scenario through VLBI polarimetric measurements. In addition, K-H instabilities would not only produce simple helical modes, but also many other instability modes mixed together; the K-H interpretation of the oscillatory 3C 48 jet on both pc and kpc scales requires a selection of modes or a simple mix of low-order modes. However, it is difficult to see how these required modes are excited while others with higher growth rates are suppressed (see the discussion of the wiggling filament in NGC 315 by Worrall et al. 2007). Moreover, the K-H model does not have a ready explanation for the observed large-scale gradual bend of the jet axis. Simple kinematical models, such as a reflection by an oblique shock or a pressure gradient in the Narrow-Line-Region ISM, may not be adequate to explain the bends of the robust ($\gtrsim0.9c$) jet flow. Summary ======= We have observed 3C 48 at multiple frequencies with the VLBA, EVN and MERLIN with spatial resolutions between tens and hundreds of parsec. Our principal results may be summarized as follows: \(1) The total-intensity MERLIN image of 3C 48 is characterized by two components with comparable integrated flux density. A compact component aligns with the VLBI jet, while an extended envelope surrounds it. The extended emission structure becomes diffuse and extends toward the northeast at $\sim$0.25 arcsec from the nucleus. The extended component shows a steeper spectrum than the compact jet. \(2) In the VLBA and EVN images, the compact jet seen in the MERLIN image is resolved into a series of bright knots. Knot A is further resolved into two smaller features A1 and A2 in 4.8- and 8.3-GHz VLBA images. A1 shows a flat spectrum with spectral index $\alpha^{4.8}_{8.3}=-0.34\pm0.04$. A2 shows a steep spectrum with $\alpha^{4.8}_{8.3}=-1.29\pm0.16$, and may be identified with the inner jet. The brightness temperature of A1 is $>10^9$ K and much higher than the $T_b$ of A2. The flux densities of A1 and A2 in epoch 2004 show a 100 and 60 per cent decrease compared with those in 1996. The high brightness temperature, flat spectrum and variability imply that A1 is the synchrotron self-absorbed core found close to the active nucleus. \(3) Comparison of the present VLBA data with those of 1996 January 20 strongly suggests that A2 is moving, with an apparent velocity $3.7c\pm0.4c$ to the North. Combining the apparent superluminal motion and the jet-to-counterjet intensity ratio yields a constraint on the jet kinematics and geometry: the jet is relativistic ($>0.85c$) and closely aligned to the line of sight ($<35\degr$). \(4) We present for the first time VLBI polarization images of 3C 48, which reveal polarized structures with multiple sub-components in component C. The fractional polarization peaks at the interface between the compact jet and the surrounding medium, perhaps consistent with a local jet-induced shock. The systematic gradient of the EVPAs across the jet width at C can be attributed to the combination of a gradient in the emission-weighted intrinsic polarization angle across the jet and possibly a systematic gradient in the RM. Changing magnetic field directions are a possible interpretation of the RM gradient, but other alternatives can not be ruled out. The fractional polarization of the hot spot B increases towards higher frequencies, from $\sim1$ per cent (1.6 GHz), $\sim2.0$ per cent (4.8 GHz) to $12$ per cent (8.3 GHz). The relatively low degree of polarization at lower frequencies probably results from a unresolved Faraday screen associated with the NLR clouds and/or the internal depolarization in the jet itself. Hot spot B has a higher RM than C, which can perhaps be attributed to a stationary shock in the vicinity of B. The core A at all frequencies is unpolarized, which may be the result of a tangled magnetic field in the inner part of the jet. \(5) The combined EVN+MERLIN 1.65-GHz image and 1.5-GHz VLBA images show that the bright knots trace out a wave-like shape within the jet. We fitted the jet structure with a simple precession model and a K-H instability model. Both models in general reproduce the observed oscillatory jet trajectory, but neither of them is able to explain all the observations. More observations are required to investigate the physical origin of the helical pattern. Further monitoring of the proper motion of the inner jet A2 should be able to constrain the ballistic motion in the framework of the precessing jet. High-resolution VLBI images of the inner jet region will be required to check whether or not the jet flow is oscillating on scales of tens of mas, which might give a morphological means of discriminating between the two models. Sophisticated simulations of the jet would be needed to take into account the deceleration of the jet flow due to kinetic energy loss via jet-cloud interaction and radiation loss, but these are beyond the scope of the present paper. Acknowledgments {#acknowledgments .unnumbered} =============== TA and XYH are grateful for partial support for this work from the National Natural Science Foundation of PR China (NSFC 10503008, 10473018) and Shanghai Natural Science Foundation (09ZR1437400). MJH thanks the Royal Society (UK) for support. We thank Mark Birkinshaw for helpful discussions on the jet kinematics. The VLBA is an instrument of the National Radio Astronomy Observatory, a facility of the US National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The European VLBI Network (EVN) is a joint facility of European, Chinese, South African and other radio astronomy institutes funded by their national research councils. MERLIN is a National Facility operated by the University of Manchester at Jodrell Bank Observatory on behalf of the UK Science and Technology Facilities Council (STFC). 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The [*r.m.s.*]{} noise in the image measured in an off-source region is $\sim$1.3 mJy b$^{-1}$, corresponding to a dynamic range of $\sim$2800:1 in the image. The contours are 6 mJy b$^{-1}\times$(-2, 1, 2, 4, ..., 512). The cross denotes the location of the hidden AGN. The square marks the region in which compact jet dominates the emission structure. []{data-label="fig:MERcont"}](Fig1.ps) ![Total intensity (Stokes $I$) image of 3C 48 from VLBA and EVN observations. The phase centres of all images have been shifted to RA=01$^h$37$^d$41$^s$.29949, Dec=$+$330935.1338. Table \[tab:figpar\] presents the image parameters. The horizon bars in the subpanels illuminates the length scale in projection. A number of bright components are labeled in the images. []{data-label="fig:vlbimap"}](Fig2.eps) ![Total intensity (Stokes $I$) image of 3C 48 from VLBA observations in 2004 and 1996. [**top left:**]{} 1.5 GHz image on epoch 2004. The image was made with a super-uniform weighting (ROBUST=$-4$ and UVBOX=3 in [aips]{} task IMAGR) and restored with a beam of 6.2$\times$3.9 (mas), PA=$-30\degr$ Contours: 0.4 mJy b$^{-1}\times$ (1,2,...,128); [**top right:**]{} 4.8 GHz contours (epoch 1996) superposited on the grey-scale image (epoch 2004); [**bottom left:**]{} 8.3 GHz contours (epoch 2004) and grey-scale (epoch 1996), [**bottom right:**]{} 8.3 GHz contours (epoch 2004) and 15.4 GHz grey-scale (epoch 1996). []{data-label="fig:core"}](Fig3.eps){width="\textwidth"} ![Spectral index map (gray scale) of 3C 48 between 1.65 and 4.99 GHz. The 4.99 GHz total intensity map (contours) is derived from the MERLIN observations on 1992 June 15 (Feng et al. 2005). The restoring beam of the 4.99 GHz image is 40$\times$40 (mas). The lowest contour is 0.7 mJy b$^{-1}$, increasing in a step of 4. The 1.65 GHz data are obtained from the combined EVN and MERLIN data observed on 2005 June 7 (the present paper). The two images are re-produced using visibility data on the common [*uv*]{} range, and restored with the same 40$\times$40 (mas) beam. Compact VLBI components are labeled in the image. []{data-label="fig:spix"}](Fig4.ps){width="\textwidth"} ![Polarization structure of 3C 48 from the MERLIN observations at 1.65 GHz. The contours map is Stokes $I$ image (Figure \[fig:MERcont\]). The polarization image is derived from Stokes $Q$ and $U$ images above a 4$\sigma$ cutoff (1$\sigma$=6 mJy b$^{-1}$). The wedge at the top indicates the percentage of the polarization. The length of the bars represents the strength of polarized emission, 1 arcsec represents 0.5 Jy b$^{-1}$. The orientation of the bar indicates the RM-corrected EVPA. []{data-label="fig:MERpol"}](Fig5.ps){width="\textwidth"} ![ Polarization structure of 3C 48 derived from the EVN and VLBA data. The contours show the total intensity (Stoke I) emission, and the grey scale indicates the fractional polarization. The length of the bars indicates the strength of the linear polarization intensity, and the orientation of the bars indicates the polarization angle, which has been corrected by the RM on the basis of the measurements by Mantovani et al. (2009). We should note that the VLBI images show more complex polarization structure than that shown in MERLIN image (Figure \[fig:MERpol\]): quantitative calculations (Figure \[fig:RM\]) show that the RMs in the component C region is about 1.4 times the value measured from the overall source; moreover, the intrinsic polarization angles rotate by $\sim 60\degr$ from the northwest edge of component C to the southeast edge. Therefore the correction based on the overall-source RM might not be sufficiently accurate to all sub-components, while this uncertainty tends to small toward the higher frequencies. [**(a)**]{}: the EVN image at 1.65 GHz. Contours are 4 mJy b$^{-1}\times$ (1,4,16,64,256). The Stokes $Q$ and $U$ maps were convolved with a 20-mas circular beam, and we used intensities above 4$\sigma$ to calculate the polarized intensity and polarization angle; [**(b)**]{}: contours : 1 mJy b$^{-1}\times$ (1,4,16,64,256); [**(c)**]{}: contours : 1 mJy b$^{-1}\times$ (1,4,16,64,256); [**(d)**]{}: contours : 1.2 mJy b$^{-1}\times$ (4,16,64,256); the contours in the inset panel are 1.2 mJy b$^{-1}\times$ (1,4,8,16,32,64,128,256,512). []{data-label="fig:VLBIpol"}](Fig6a.ps "fig:"){width="45.00000%"} ![ Polarization structure of 3C 48 derived from the EVN and VLBA data. The contours show the total intensity (Stoke I) emission, and the grey scale indicates the fractional polarization. The length of the bars indicates the strength of the linear polarization intensity, and the orientation of the bars indicates the polarization angle, which has been corrected by the RM on the basis of the measurements by Mantovani et al. (2009). We should note that the VLBI images show more complex polarization structure than that shown in MERLIN image (Figure \[fig:MERpol\]): quantitative calculations (Figure \[fig:RM\]) show that the RMs in the component C region is about 1.4 times the value measured from the overall source; moreover, the intrinsic polarization angles rotate by $\sim 60\degr$ from the northwest edge of component C to the southeast edge. Therefore the correction based on the overall-source RM might not be sufficiently accurate to all sub-components, while this uncertainty tends to small toward the higher frequencies. [**(a)**]{}: the EVN image at 1.65 GHz. Contours are 4 mJy b$^{-1}\times$ (1,4,16,64,256). The Stokes $Q$ and $U$ maps were convolved with a 20-mas circular beam, and we used intensities above 4$\sigma$ to calculate the polarized intensity and polarization angle; [**(b)**]{}: contours : 1 mJy b$^{-1}\times$ (1,4,16,64,256); [**(c)**]{}: contours : 1 mJy b$^{-1}\times$ (1,4,16,64,256); [**(d)**]{}: contours : 1.2 mJy b$^{-1}\times$ (4,16,64,256); the contours in the inset panel are 1.2 mJy b$^{-1}\times$ (1,4,8,16,32,64,128,256,512). []{data-label="fig:VLBIpol"}](Fig6b.ps "fig:"){width="45.00000%"}\ ![ Polarization structure of 3C 48 derived from the EVN and VLBA data. The contours show the total intensity (Stoke I) emission, and the grey scale indicates the fractional polarization. The length of the bars indicates the strength of the linear polarization intensity, and the orientation of the bars indicates the polarization angle, which has been corrected by the RM on the basis of the measurements by Mantovani et al. (2009). We should note that the VLBI images show more complex polarization structure than that shown in MERLIN image (Figure \[fig:MERpol\]): quantitative calculations (Figure \[fig:RM\]) show that the RMs in the component C region is about 1.4 times the value measured from the overall source; moreover, the intrinsic polarization angles rotate by $\sim 60\degr$ from the northwest edge of component C to the southeast edge. Therefore the correction based on the overall-source RM might not be sufficiently accurate to all sub-components, while this uncertainty tends to small toward the higher frequencies. [**(a)**]{}: the EVN image at 1.65 GHz. Contours are 4 mJy b$^{-1}\times$ (1,4,16,64,256). The Stokes $Q$ and $U$ maps were convolved with a 20-mas circular beam, and we used intensities above 4$\sigma$ to calculate the polarized intensity and polarization angle; [**(b)**]{}: contours : 1 mJy b$^{-1}\times$ (1,4,16,64,256); [**(c)**]{}: contours : 1 mJy b$^{-1}\times$ (1,4,16,64,256); [**(d)**]{}: contours : 1.2 mJy b$^{-1}\times$ (4,16,64,256); the contours in the inset panel are 1.2 mJy b$^{-1}\times$ (1,4,8,16,32,64,128,256,512). []{data-label="fig:VLBIpol"}](Fig6c.eps "fig:"){width="45.00000%"} ![ Polarization structure of 3C 48 derived from the EVN and VLBA data. The contours show the total intensity (Stoke I) emission, and the grey scale indicates the fractional polarization. The length of the bars indicates the strength of the linear polarization intensity, and the orientation of the bars indicates the polarization angle, which has been corrected by the RM on the basis of the measurements by Mantovani et al. (2009). We should note that the VLBI images show more complex polarization structure than that shown in MERLIN image (Figure \[fig:MERpol\]): quantitative calculations (Figure \[fig:RM\]) show that the RMs in the component C region is about 1.4 times the value measured from the overall source; moreover, the intrinsic polarization angles rotate by $\sim 60\degr$ from the northwest edge of component C to the southeast edge. Therefore the correction based on the overall-source RM might not be sufficiently accurate to all sub-components, while this uncertainty tends to small toward the higher frequencies. [**(a)**]{}: the EVN image at 1.65 GHz. Contours are 4 mJy b$^{-1}\times$ (1,4,16,64,256). The Stokes $Q$ and $U$ maps were convolved with a 20-mas circular beam, and we used intensities above 4$\sigma$ to calculate the polarized intensity and polarization angle; [**(b)**]{}: contours : 1 mJy b$^{-1}\times$ (1,4,16,64,256); [**(c)**]{}: contours : 1 mJy b$^{-1}\times$ (1,4,16,64,256); [**(d)**]{}: contours : 1.2 mJy b$^{-1}\times$ (4,16,64,256); the contours in the inset panel are 1.2 mJy b$^{-1}\times$ (1,4,8,16,32,64,128,256,512). []{data-label="fig:VLBIpol"}](Fig6d.eps "fig:"){width="45.00000%"} ![RM distribution in the component-C region. The patchy morphology is because at some pixels polarization was not detected at all three frequencies simultaneously. The contours represent the total intensity: 1.0 mJy b$^{-1}\times$(1,4,8,16). The wedge at the top indicates the RM in the observer’s frame, in unit of rad m$^{-2}$. The insets show the measured values of the observed polarization angle for four selective locations as a function of $\lambda^2$ along with a linear fitting of rotation measure.[]{data-label="fig:RM"}](Fig7.eps){width="\textwidth"} ![ Constraints on source orientation and jet velocity from the VLBI observations. The shaded region indicates the parameter space constrained by the proper motion measurements and the jet-to-counterjet intensity ratios.[]{data-label="fig:viewangle"}](Fig8.ps) ![Sketch plot of the helical jet in 3C 48. The jet knots move on the surface of a helical cone. The XOY plane in the plot represents the projected sky plane, and the X-axis points to the RA direction and Y-axis to the DEC direction. The Z-axis is perpendicular to the plane of the sky and points away from the observer. The half of the opening angle of the helix is $\varphi$. The jet axis ’OP’ is inclining by an angle of ($90-\theta$) with respect to the line of sight. The angle $\alpha$ between the OP’ and Y-axis is defined as the position angle in the 2-Dimension CLEAN image. []{data-label="fig:sketch"}](Fig9.eps) ![Helical model fits overlaid on the total intensity images. [**Upper panel**]{} : the ridge line of the fitted precessing jet (thick green lines) overlaid on the 1.5-GHz VLBA (left) and 1.65-GHz EVN+MERLIN images. The CLEAN image parameters are referred to Table \[tab:figpar\]. [**Lower panel**]{} : the fitted jet trajectory (thick green line) from the K-H instability model. []{data-label="fig:helicalfit"}](Fig10.eps){width="\textwidth"} Array$^a$ VLBA EVN MERLIN ------------------ -------------------- --------------------- ---------------------------- R.A.$^b$ 01 37 41.29943 01 37 41.29949 01 37 41.29675 Dec.$^b$ 33 09 35.1330 33 09 35.1338 33 09 35.5117 Date 2004 June 25 2005 June 7 2005 June 7 Time(UT) 08:00–20:00 02:00–14:00 02:00–14:00 $\tau$(hour)$^c$ 2.6/2.6/2.6 8.0 8.0 Freq(GHz)$^d$ 1.5/4.8/8.3 1.65 1.65 BL(km)$^e$ 20–8600 130–8800 0.3–220 Calibrators DA193, 3C138,3C345 DA193, 3C138, 3C286 DA193, 3C138, 3C286, OQ208 BW(MHz) 32 32 15 Correlator Socorro (VLBA) JIVE (MK IV) Jodrell Bank : Observational parameters of 3C 48 \ $^a$ : Participating EVN telescopes were Jodrell Bank (Lovell 76-m), Westerbork (phased array), Effelsberg, Onsala (25-m), Medicina, Noto, Torun, Shanghai, Urumqi, Hartebeesthoek and Cambridge; the MERLIN array consistsed of Defford, Cambridge, Knockin, Darnhall, MK2, Lovell and Tabley; all ten telescopes of the VLBA and a single VLA telescope participated in the VLBA observations;\ $^b$ : pointing centre of the observations;\ $^c$ : total integrating time on 3C 48;\ $^d$ : the central frequency of the observing band. The VLBA observations were carried out at three frequency bands of 1.5, 5 and 8 GHz;\ $^e$ : the projected baseline range of the array in thousands of wavelengths.\ \[tab:obs\] --------------- ----------- ---------- ----------- ---------- ---------------- -------------------------------- Label Frequency rms noise Contours (GHz) Maj(mas) Min(mas) PA(deg) (mJy b$^{-1}$) (mJy b$^{-1}$) Figure 1 1.65 138 115 65.1 1.3 6.0$\times$(-2,1,2,4,8,...512) Figure 2-a 1.51 8.3 5.3 1.0 0.25 1.0$\times$(1,2,4,8,16,64,256) Figure 2-b 1.65 5.0 5.0 0.0 0.30 1.0$\times$(1,4,16,64,256) Figure 2-c 4.78 2.7 1.7 0.0 0.040 0.16$\times$(1,4,16,64,256) Figure 2-d 8.31 1.8 1.1 9.7 0.060 0.24$\times$(1,4,16,64,256) Figure 10$^*$ 1.65 16 10 $-$4.4 1.6 6.0$\times$(1,4,16,32,64,128) --------------- ----------- ---------- ----------- ---------- ---------------- -------------------------------- : Parameters of total intensity maps of Figure \[fig:vlbimap\] \ \[tab:figpar\] Note: all images are registered to the phase centre of the 2005 EVN image. $^*$: the parameters are for the 1.65-GHz EVN+MERLIN image in the right panel. [llrrrrrrr]{} Freq.&Comp. &RA(J2000)$^a$&Dec(J2000)$^a$& S$_p$ & S$_i$ & $\theta_{maj}$ & $\theta_{min}$ & P.A.\ (GHz)& &($^h$,$^m$,$^s$)&($\degr$,$\arcmin$,$\arcsec$) &(mJy b$^{-1}$) & (mJy) & (mas) & (mas) & (degree)\ (1) & (2) & (3) & (4) & (5) & (6) & (7) &(8) &(9)\ \ 1.53 &A &01 37 41.2994260 &33 09 35.021073 & 59.94$\pm$0.84& 93.29$\pm$1.97& 4.77$\pm$0.14 & 2.58$\pm$0.18 &173.0$\pm$2.7\ &B &$-$2.69 & 53.61 &153.56$\pm$0.84&224.45$\pm$1.89& 4.02$\pm$0.06 & 2.72$\pm$0.07 &157.2$\pm$1.8\ &B2 & 0.58 &113.73 & 93.20$\pm$0.78&529.51$\pm$5.17&13.46$\pm$0.13 & 8.53$\pm$0.10 & 22.8$\pm$0.8\ &B3 &$-$20.97 & 84.80 & 37.04$\pm$0.78&309.78$\pm$7.23&20.38$\pm$0.45 & 8.62$\pm$0.24 &131.8$\pm$1.0\ &C & 37.56 &247.68 &24.42$\pm$0.62&385.70$\pm$10.30&27.94$\pm$0.73 &12.98$\pm$0.38 & 70.1$\pm$1.3\ &D & 70.64 &331.18 & 86.88$\pm$0.78&518.96$\pm$5.40&14.39$\pm$0.15 & 8.43$\pm$0.10 &123.7$\pm$0.8\ &D2 & 50.59 &378.93 & 21.07$\pm$0.76&219.28$\pm$8.62&19.56$\pm$0.75 &11.87$\pm$0.51 & 31.4$\pm$3.1\ \ 1.51 &A &01 37 41.2993864 &33 09 35.018436 & 72.33$\pm$0.71& 99.91$\pm$1.53& 4.87$\pm$0.14& 0.00$\pm$0.00&173.2$\pm$1.1\ &B &$-$2.57 &52.94 &264.06$\pm$0.72&308.76$\pm$1.38& 2.56$\pm$0.06& 1.43$\pm$0.05& 0.7$\pm$1.4\ &B2 & 0.73 &112.98 &130.17$\pm$0.66&657.32$\pm$3.92&13.17$\pm$0.13& 7.43$\pm$0.06& 20.6$\pm$0.4\ &B3 &$-$21.79 &84.88 & 47.78$\pm$0.65&371.90$\pm$5.67&18.56$\pm$0.45& 8.80$\pm$0.16&127.6$\pm$0.8\ &C & 38.82 &246.04 & 28.84$\pm$0.57&376.94$\pm$7.94&28.05$\pm$0.73& 10.31$\pm$0.25& 65.4$\pm$0.8\ &D & 70.82 &329.60 &119.13$\pm$0.66&616.05$\pm$3.99&13.49$\pm$0.15& 7.46$\pm$0.06&125.4$\pm$0.5\ &D2 & 50.52 &377.20 & 26.99$\pm$0.65&235.54$\pm$6.27&18.01$\pm$0.75& 10.54$\pm$0.31& 62.5$\pm$2.0\ \ 1.65 &A &01 37 41.2993690 &33 09 35.018826 & 75.23$\pm$0.29&108.27$\pm$0.64&4.94$\pm$0.04 &1.09$\pm$0.10 &178.0$\pm$0.5\ &B &$-$2.56 & 52.92 &279.84$\pm$0.30&318.86$\pm$0.56&2.40$\pm$0.01 &1.18$\pm$0.02 & 23.7$\pm$0.6\ &B2 & 2.24 &111.56 &155.31$\pm$0.27&771.75$\pm$1.59&13.44$\pm$0.03 &7.08$\pm$0.02 & 12.1$\pm$0.1\ &B3 &$-$20.48 & 85.23 &50.78$\pm$0.27&297.29$\pm$1.82&17.98$\pm$0.10 &6.04$\pm$0.05 &126.9$\pm$0.2\ &C & 40.60 &249.55 &53.14$\pm$0.27&388.12$\pm$2.20&26.65$\pm$0.14 &4.51$\pm$0.05 & 62.6$\pm$0.1\ &D & 70.31 &329.77 &190.60$\pm$0.28&597.33$\pm$1.10&9.79$\pm$0.02 &5.08$\pm$0.02 &109.9$\pm$0.1\ &D2 & 53.48 &377.67 &46.10$\pm$0.27&186.20$\pm$1.34&11.92$\pm$0.08 &6.00$\pm$0.06 & 40.6$\pm$0.4\ \ 4.99 &A &01 37 41.2993853 &33 09 35.016904 &46.80$\pm$0.04 & 60.77$\pm$0.08 &1.80$\pm$0.01 &0.43$\pm$0.01 &174.4$\pm$ 0.1\ &B &$-$2.71 & 54.85 &88.73$\pm$0.04 &135.70$\pm$0.08 &2.11$\pm$0.01 &1.37$\pm$0.01 & 36.3$\pm$ 0.2\ &B2 & 0.19 &115.75 &17.99$\pm$0.03 &162.30$\pm$0.34 &9.37$\pm$0.02 &4.99$\pm$0.01 & 44.8$\pm$ 0.1\ &B3 &$-$22.32 & 90.35 & 4.04$\pm$0.03 & 46.77$\pm$0.39&10.21$\pm$0.08 &6.21$\pm$0.06 &149.3$\pm$ 0.7\ &D & 71.33 &331.53 &11.75$\pm$0.02 &300.35$\pm$0.56&15.82$\pm$0.03 &9.28$\pm$0.02 &121.1$\pm$ 0.1\ \ 4.78 &A1 &01 37 41.2993814 &33 09 35.016523 & 26.75$\pm$0.05 & 28.80$\pm$0.10& 1.00$\pm$0.02 &0.56$\pm$0.02 &176.3$\pm$ 2.2\ &A2 &$-$0.28 & 3.39 & 16.69$\pm$0.05 & 22.86$\pm$0.11& 2.97$\pm$0.02 &0.57$\pm$0.03 &175.7$\pm$ 0.3\ &B &$-$2.63 & 55.19 &105.06$\pm$0.05 &140.80$\pm$0.11& 2.00$\pm$0.01 &0.96$\pm$0.01 & 39.8$\pm$ 0.2\ &B2 &$-$0.02 &116.06 & 23.11$\pm$0.05 &161.84$\pm$0.37& 9.12$\pm$0.02 &4.61$\pm$0.02 & 42.9$\pm$ 0.2\ &B3 &$-$22.39 & 90.01 & 4.07$\pm$0.04 & 52.92$\pm$0.57&12.42$\pm$0.13 &6.92$\pm$0.09 &112.9$\pm$ 0.8\ &D & 71.39 &331.59 & 14.88$\pm$0.03 &322.44$\pm$0.72&16.36$\pm$0.04 &9.22$\pm$0.03 &121.1$\pm$ 0.2\ \ 8.41 &A1 &01 37 41.2993871&33 09 35.016521 & 30.55$\pm$0.06 & 47.91$\pm$0.15& 1.39$\pm$0.01 &0.22$\pm$0.01&176.0$\pm$0.2\ &A2 &$-$0.14 & 2.03 & 11.30$\pm$0.06 & 18.27$\pm$0.15& 1.39$\pm$0.03 &0.34$\pm$0.02&174.0$\pm$0.5\ &B &$-$2.73 & 55.20 & 46.00$\pm$0.06 & 89.00$\pm$0.17& 1.38$\pm$0.01 &0.71$\pm$0.01& 46.8$\pm$0.3\ &B2 &$-$0.34 &115.65 & 5.04$\pm$0.05 & 96.36$\pm$0.86& 6.08$\pm$0.06 &3.54$\pm$0.03& 46.2$\pm$0.5\ &D & 71.46 &331.98 & 4.41$\pm$0.02 & 271.57$\pm$1.46&13.10$\pm$0.07 &8.09$\pm$0.04&121.5$\pm$0.4\ \ 8.31 &A1 &01 37 41.2993822 &33 09 35.016514& 18.09$\pm$0.06 & 23.85$\pm$0.12& 1.18$\pm$0.01 &0.38$\pm$0.02&178.7$\pm$0.6\ &A2 &$-$0.29 & 3.41 & 6.89$\pm$0.05 & 11.19$\pm$0.14& 1.80$\pm$0.03 &0.41$\pm$0.04&175.1$\pm$0.8\ &B &$-$2.63 & 55.24 & 55.84$\pm$0.06 & 89.38$\pm$0.14& 1.34$\pm$0.01 &0.72$\pm$0.01& 37.8$\pm$0.3\ &B2 &$-$0.12 &116.13 & 6.61$\pm$0.04 & 104.47$\pm$0.74& 7.14$\pm$0.05 &3.60$\pm$0.03& 43.4$\pm$0.4\ &D & 70.99 &331.82 & 4.16$\pm$0.02 & 250.24$\pm$1.39&13.11$\pm$0.07 &7.89$\pm$0.04&120.8$\pm$0.4\ \ 15.36&A1 &01 37 41.2993868& 33 09 35.016511& 9.22$\pm$0.20 & 14.59$\pm$0.48& 1.05$\pm$0.04 & 0.44$\pm$0.06 & 11.3$\pm$3.0\ &A2 &$-$0.10 & 1.76 & 5.48$\pm$0.18 & 7.50$\pm$0.44& 0.93$\pm$0.08 & 0.20$\pm$0.04 & 19.4$\pm$5.0\ &B &$-$2.72 & 55.20 & 16.27$\pm$0.17 & 27.83$\pm$0.51& 1.18$\pm$0.03 & 0.47$\pm$0.03 & 41.8$\pm$1.6\ &B2 &$-$0.07 &116.01 & 1.63$\pm$0.14 & 29.50$\pm$2.65& 6.59$\pm$0.57 & 2.52$\pm$0.25 & 39.8$\pm$3.3\ \[tab:model\] $^a$ : for individual data sets, the Right Ascension and Declination positions of the nuclear component A (A1) in J2000.0 coordinate frame are presented; the relative positions of jet components are given with respect to the nuclear component A (A1). $T_b$ A1 A2 B ------- ------ ----- ------ 2004C 37.4 9.8 53.4 1996X 74.2 8.9 17.2 2004X 12.8 3.7 22.4 1996U 2.2 2.7 23.2 : Brightness temperature ($T_b$) of compact VLBI components \ Note : T$_b$ are given in units of $10^8$ K. \[tab:tb\] ------------- ---------- ------------- ------------ ---------------- ----------- ---------- ------------ ---------- $V_j$ $90-\theta$ $\alpha_0$ $d\alpha/dt$ $\varphi$ $\psi_0$ P $r_0$ (mas/yr) (deg) (deg) (deg/$10^3$yr) (deg) (deg) ($10^3$yr) (mas) Model 1$^a$ 0.164 17 0.0 1.8 2.0 50.0 3.5 $\cdots$ Model 2$^b$ 0.164 17 0.0 1.6 1.5 50.0 0.366 1.8 ------------- ---------- ------------- ------------ ---------------- ----------- ---------- ------------ ---------- : Parameters of helical jet models \ $^a$ : a precessing jet model;\ $^b$ : a helical-mode K-H instability model;\ $V_j$: in the precessing model, $V_j$ represents the flow speed in the observer’s frame, taking into account relativistic aberration effects; in the K-H model, $V_j$ denotes the pattern velocity. The velocity is expressed in terms of proper motion in order to agree with the coordinates used in CLEAN images;\ $\theta$: the angle between the jet axis and the sky plane;\ $\alpha_0$: the initial position angle of the jet axis, measured from north to east ;\ $d\alpha/dt$: the rate of change of position angle with time. In the precessing model, it gives an estimate of the angular velocity of the precession;\ $\varphi$: half of the opening angle of the helix cone ;\ $\psi_0$: initial phase angle of the helical jet flow;\ $P$: in the ’Model 1’, the fitted $P$ is actually the nutation period, see discussion in Section 5.2.1; in the K-H model, $P$ represents an initial period for the triggered perturbations;\ $r_0$: the initial radius where the K-H instabilities starts to grow;\ \[tab:helicalfit\] [^1]: E-mail: antao@shao.ac.cn [^2]: In the present paper, the spectral index is defined as $S_\nu\propto\nu^\alpha$. [^3]: See the online VLBA status summary at http://www.vlba.nrao.edu/astro/obstatus/current/obssum.html .

--- abstract: 'In this contribution we review the large body of work carried out over the past two decades to probe the dark matter in the local universe using redshift survey and peculiar velocity data. While redshift surveys have evolved rapidly over the years, gathering suitable peculiar velocity data and understanding the short-comings of different analyses have proven to be a difficult task. These difficulties have led to conflicting results which have casted some doubts on the usefulness of cosmic flows to constrain cosmological models. Recently, however, a consistent picture seems to be finally emerging with various methods of analyses applied to different data sets yielding concordant results. These favor a low-density universe, with constraints which are in good agreement with those obtained from LSS, high-redshift supernovae and CMB studies.' author: - 'L. da Costa' title: Matter in the Local Universe --- \#1 Introduction ============ LSS studies of the nearby universe are arguably ideal to address the question posed by the title of this conference. Indeed, if all the mass in the universe were locked into galaxies, complete redshift surveys of galaxies would provide the data required to fully characterize the matter distribution. However, we have learned that the luminous matter associated to galaxies represents a small fraction of the mass density of the universe, and that galaxies may be biased relative to the underlying distribution of matter. Still, if structures grow as a result of gravity alone, observation of the peculiar velocity of galaxies provides the means to probe the distribution of the total matter. In the standard picture for the formation of cosmic structures via gravitational instability the peculiar velocity of a galaxy is generated by fluctuations in the mass distribution. For galaxies outside virialized systems, linear perturbation theory predicts $$\vvec (\rvec) \approx {\Omega^{0.6}H_o \over 4\pi} \int{ d^3r^\prime \delta_m {(\rvec^\prime - \rvec) \over \vert \rvec^\prime - \rvec \vert ^3}} \; . \label{lingrav}$$ This can also be expressed in the following differential form $$\nabla \cdot {\bf v} = -\Omega^{0.6} \delta_m, \label{divv}$$ where $\Omega$ is the mass density parameter, $H_o$ is the Hubble constant and $\delta_m$ is the mass density fluctuation field. If galaxies are fair tracers of the underlying mass distribution and galaxy biasing is linear then $\delta_g = b \delta_m$, where $\delta_g$ is the galaxy density contrast and $b$ is the bias parameter for a given population of mass tracers. The above equations show that by mapping the peculiar velocity field one can determine the distribution of mass and measure the parameter $\beta=\Omega^{0.6}/b$ by comparing the reconstructed density field with that observed for galaxies or by comparing the measured velocity field with the predicted gravity field generated by fluctuations of the galaxy density field. These simple ideas have been the underlying motivation for the major wide-angle redshift surveys of optical and infrared galaxies and the Tully-Fisher (TF) and $D_n-\sigma$ redshift-distance surveys conducted over the past two decades. In this contribution we review all of these efforts, giving special emphasis to the results obtained from recently completed redshift-distance surveys. In section \[z\], we briefly mention the redshift surveys that have contributed to our understanding of the local galaxy distribution and those which have played a major role in the analysis of peculiar velocity data. In Section \[zv\], we review the redshift-distance surveys and the peculiar velocity catalogs that have been used to map the peculiar velocity field in the nearby universe. In section \[results\], the most recent surveys are used to reconstruct the velocity and density fields and to measure $\beta$. These results are also compared with those obtained in earlier works. Finally, in Section \[summary\] we briefly summarize the current status of the field. Galaxy Distribution {#z} =================== Over the past two decades the number of redshift surveys and redshift data has greatly increased and a complete review is beyond the scope of the present contribution and can be found elsewhere [@strauss-willick][@dacostampaeso]. Here we point out two classes of surveys that have had a strong bearing on some of the issues discussed here. The first class consists of wide-angle, dense sampling surveys such as the CfA2 [@gellerhuchra] and SSRS2 [@dacostassrs] which revealed for the first time the full complexity of the galaxy distribution. The discovery of extended, coherent wall-like structures and of large regions devoid of luminous matter with scales comparable to the survey depth represented a serious challenge to the prevailing theories of structure formation and evolution. Furthermore, these surveys probed relatively large volumes which allowed for reasonable estimates of the power-spectrum of the galaxy density fluctuations to be made for the first time [@gott][@vogeley][@dacostaps]. Even though unable to reach very large scales, when COBE normalized, comparison with N-body simulations demonstrated that the results were consistent with a low-$\Omega$ cosmological model and an unbiased galaxy distribution. The PS was well described by a shape parameter $\Gamma=0.2$, consistent with other determinations [@peakcockdodds]. Attempts were also made to study the small-scale velocity field by analyzing the redshift-space distortions. However, the small number of independent structures within the sampled volume made the results extremely sensitive to shot-noise [@marzke]. These early surveys were followed by the considerably deeper LCRS [@lcrs] which demonstrated unambiguously that the largest scales of inhomogeneities had finally been reached. Quantitative analyses of the LCRS, by and large confirmed earlier statistical results, albeit with considerably smaller errors. More recently, the first results of the 2dFGRS project have become available. The survey consists of over 100,000 galaxies to a depth comparable to the LCRS, allowing for precise measurements of redshift-space distortions and large-scale power spectrum. Analysis of the redshift distortions caused by large-scale infall velocities yields a value of $\beta_o=0.43$ [@peakcock2df], where $\beta_o$ refers to optical galaxies. Assuming a relative bias $b_o/b_I\sim 1.3$ between optical and galaxies this implies $\beta_I\sim0.56$. The derived galaxy power-spectrum [@percival] was found to be well-represented by a shape parameter $\Gamma=0.2$, in good agreement with previous determinations. These results provide important constraints on the mass power-spectrum which can later be compared with those obtained using cosmic flows to test for consistency. The second class of redshift surveys worth mentioning in the present context is that involving galaxy samples extracted from the [*IRAS*]{} survey such as the 1.9 Jy[@strauss1.9], the 1.2 Jy[@fisher1.2] and the  [@saunderspscz] redshift surveys. While sparsely sampling the galaxy distribution, these surveys provide a sky coverage unmatched by optical surveys. This all-sky coverage allows a more reliable determination of the gravity field induced by fluctuations of the galaxy density field which can be compared to the measured peculiar velocity field to estimate the parameter $\beta$. From the above discussion, it is clear that by themselves redshift surveys are more useful for studying the properties of galaxies than as cosmological probes. However, combining them with redshift independent distances to map out the peculiar velocity field of galaxies and to predict the peculiar velocity field from galaxy density fluctuations provide powerful tools to probe the nature of the matter distribution and its relation with the galaxy distribution. Mapping the Peculiar Velocity Field {#zv} =================================== The radial component of the peculiar velocity is given by $$U = cz - d$$ where $d$ is an estimate of the galaxy distance derived from a secondary distance indicator. Most of the available samples rely on the TF and Fundamental Plane relations for spirals and early-type galaxies, respectively, with typical errors in distance of $\sim 20\%$. However, samples based on distance indicators with significant smaller errors, such as those based on surface brightness fluctuations and nearby Type Ia supernovae, are slowly growing and have already been successfully used to measure $\beta$[@tonry][@riess]. In contrast to the rapid growth of samples with complete redshift information, redshift-distance samples have been difficult to gather. There are various practical reasons for that. First, to ensure the uniformity of the data and of the sky coverage requires coordinated observations in both hemispheres. Second, TF distances require the measurement of the rotational velocity of the galaxy either from the HI line width, which can only be efficiently measured in the northern hemisphere, or from measurements of optical rotation curves, a challenging observation. Third, for early-type galaxies, high signal-to-noise spectra are required for accurate measurements of the velocity dispersion. Finally, both distance indicators require high-quality photometric data. Table \[rd\] summarizes the redshift-distance surveys conducted to date. The table includes only wide-angle redshift-distance surveys and the number of objects is just indicative of the sample size. Not included are the various surveys conducted to measure distances and peculiar velocities of clusters of galaxies which have been used to constrain the amplitude of the bulk flow on very large scales. [lrll]{}\ Survey & $ N_{obj}$ & Type & Coverage\ \ Aaronson & 300 & spirals & all-sky\ Tonry & Davis & 300 & early & north\ 7 Samurai & 400 & early & all-sky\ Willick & 320 & spirals & Perseus-Pisces\ Courteau & 380 & Sb-Sc & north\ Mathewson & 2000+ & spirals & south\ SFI & 1300 & Sbc-Sc & all-sky\ ENEAR & 1600 & early & all-sky\ Shellflow & 300 & Sb-Sc & all-sky\ \ \ Early studies [@tonry-davis][@aaronson] focused on the properties of the flow field near Virgo. However, it was soon realized that the assumption of a spherical infall was too restrictive and that Virgo alone could not explain the motion of the Local Group relative to the CMB. A major contribution to the field was the work of the 7 Samurai[@lynden-bell], the first to probe well beyond the local supercluster, albeit sparsely. Analyses of this sample led to startling results such as the measurement of a large amplitude bulk flow, and the discovery of the Great Attractor, a large mass concentration associated with the Hydra-Centaurus complex. Among the main conclusions of this work was that the large peculiar velocities measured implied large values of $\Omega$, a result that placed cosmic flows at odds with several other analyses. By the end of the 80’s the first attempt to produce a homogeneous catalog by merging different peculiar velocity data set was made (Mark II) and used to obtain the first map of the dark matter in the nearby universe [@potent]. Even though providing an important first glimpse of the dark matter distribution, this early map showed that important regions of the sky were severely undersampled. In subsequent years a major effort was made to expand the available sample to confirm the conclusions of the 7S and to improve the mass maps. These efforts included small surveys of specific areas of the sky [@courteausurvey][@willickdata] and major TF surveys of spirals, such as those carried out by Matthewson and collaborators [@mat1][@mat2] and the SFI survey [@haynes1][@haynes2], and FP surveys of early-type galaxies, such as the recently completed ENEAR survey [@enear] Trying to capitalize as much as possible on all of the available data, Willick and collaborators[@markiii] assembled the data from these different surveys into a catalog (Mark III) consisting of about 3000 galaxies, predominantly spirals, with measured peculiar velocities. The Mark III catalog, which does not include the more recent all-sky SFI and ENEAR surveys, has been extensively used in the analyses of peculiar velocity data. While considerable effort was made to ensure uniformity, it is a compilation of heterogeneous data sets. As illustrated in figure 11 of Kollat  [@kollat], it lacks uniformity in sky coverage due to the uneven coverage of the main data sets included in the compilation. While the availability of this catalog prompted the development of several techniques to analyze peculiar velocity data and efforts to understand possible bias, its use has led to conflicting results. The reasons for the discrepancies are not understood and could indicate limitations of the data or of the methods used. Efforts to re-calibrate this catalog using new observations [@shellflow] are still underway. In this context the completion of the SFI I-band TF survey of late spirals and the ENEAR $D_n-\sigma$ survey of early-type galaxies are important additions, providing homogeneous samples of comparable sizes. Figure \[fig:skydist\] shows the projected distribution of galaxies in these two surveys. In contrast to the Mark III compilation, the sky coverage of both surveys is remarkably uniform and nearly all of the data consist of new measurements reduced in a uniform way. Also note that the surveys nicely complement each other: ENEAR galaxies probe high density regions and delineate large-scale structures more sharply; SFI galaxies probe lower density regimes and are more uniformly distributed across the sky. Another important point is that the peculiar velocities in these catalogs are measured using distinct distance estimators based on different observed quantities. Therefore, to take full advantage of these characteristics these samples have been analyzed separately, to test the reproducibility of the results, and combined into the SEcat catalog to produce a fair sample probing a wide range of density regimes. The results of analyses based on these new catalogs of peculiar velocity data are reviewed below and compared to those obtained using Mark III. Results ======= Reconstructed density and velocity fields ----------------------------------------- An underlying assumption of all methods used in estimating $\beta$ is that galaxies, even though biased, are fair tracers of the mass distribution. This hypothesis can be, in principle, directly tested by comparing the galaxy density field as derived from redshift surveys and the mass density field reconstructed from peculiar velocity data. POTENT, Wiener Filter[@wf] and more recently the Unbiased Minimal Variance estimator (UMV)[@umv] are examples of methods developed to reconstruct the three-dimensional velocity and density fields from the observed radial component of the peculiar velocity. All methods assume that on the scales of interest the perturbations are small and non-linear effects can be neglected. The various methods have also been extensively tested using mock catalogs drawn from simulations that mimic the nearby universe. Recently, the UMV method has been applied to the SEcat catalog of peculiar velocities and to the redshift survey data. Figure \[fig:mass\] shows the map of the PSCz galaxy density field (left panel) and the mass density field (right panel) along the Supergalactic plane, the latter obtained from the SEcat data using a Gaussian smoothing radius of 1200 . The main features of our local universe are easily identified in these maps, including the Great Attractor (GA) on the left and the Perseus-Pisces supercluster (PP) in the lower right. There is also a hint of the Coma cluster, which lies just outside the volume probed by SEcat, in the upper part on the map. The similarity between the mass and galaxy density fields is striking, especially considering the limitations to the peculiar velocity data imposed by the Zone of Avoidance. Furthermore, even though different in details, the gross features of the mass density field are similar to those obtained by applying either the same or the POTENT formalism to the Mark III catalog[@wfmark][@potentm3] and SFI catalogs[@dacostasc]. This is an outstanding result considering the different ways these catalogs were constructed and the peculiar velocities measured. Current results are also a remarkable improvement over those obtained from earlier catalogs. In particular, it is worth mentioning the prominence of the Perseus-Pisces region, completely absent in the earlier maps, and the well-defined voids, well-known features in redshift surveys which are now clearly seen in the reconstructed mass distribution. The reconstructed three-dimensional velocity field can also be used to measure other quantities of interest. For instance, the amplitude of the bulk flow is found to vary from $V_{B} =300 \pm$ 70  for a sphere of $R=20\hmpc$ to $160 \pm$ 60  for $R=60\hmpc$. This value is in good agreement with that obtained from a direct fit to the radial peculiar velocities for the SFI[@sdipole] and the ENEAR[@edipole] samples. This result disagrees with the bulk flow determined for the Mark III survey, which has an amplitude of roughly twice this value[@wfmark]. The small amplitude of the bulk flow recently measured is in marked contrast to earlier claims of large amplitude coherent motions over scales of the order of 100$h^{-1}$ Mpc[@courteausurvey], which at face value would imply excess power on very large scales. This result is in line with the results of recent redshift surveys which have not detected inhomogeneities on very large scales. Greater insight on the characteristic of the flowfield can be obtained by decomposing the 3-D velocity field into two components, one which is induced by the local mass distribution and a tidal component due to mass fluctuations external to the volume considered[@hoffman][@wfmark][@wfenear]. Figure \[fig:tidal\] shows the results of this decomposition applied to the ENEAR survey[@wfenear], where the local volume is a sphere of $ 80\hmpc$ centered on the Local Group. The plots show the full velocity field (upper left panel), the divergent (upper right panel) and the tidal (lower left panel) components. To further understand the nature of the tidal field, its bulk velocity component has been subtracted and the residual is shown in the lower right panel. This residual is clearly dominated by a quadrupole component. In principle, the analysis of this residual field can shed light on the exterior mass distribution. For the ENEAR catalog we find that the local dynamics is hardly affected by structure on scales larger than its depth. For this sample not only the bulk velocity at large radii is small but so is the $rms$ value of the tidal field, estimated to be of the order of 60 . This is in marked contrast to the results obtained from the analysis of the Mark III survey which yields a much stronger tidal field, pointing (in the sense of its quadrupole moment) towards the Shapley concentration. For Mark III, the tidal field contributes $\sim$ 200 to the total bulk velocity. Estimates of $\beta$ -------------------- Equations (\[lingrav\]) and (\[divv\]) show that there are two alternative ways for estimating $\beta$ - velocity-velocity or density-density comparisons. In the first case, the observed galaxy distribution is used to infer a mass density field from which peculiar velocities can be predicted and compared to the observed ones. In the second case, the three-dimensional velocity field is obtained from the observed radial velocities and used to infer a self-consistent mass density field and thus a galaxy distribution, via linear biasing. The latter is then compared to the one obtained from large all-sky redshift surveys. A particularly useful method for performing a velocity-velocity comparison is the modal expansion method[@nd]. This method expands the velocity fields by means of smooth functions (Bessel and spherical harmonics) defined in redshift space, thus alleviating the Malmquist biases inherent in real space analysis. Furthermore, the modal expansion smooths the observed and predicted velocities in the same way, so that the smoothed fields can be compared directly. Because the number of modes is substantially smaller than the number of data points, the method also provides the means of estimating $\beta$ from a likelihood analysis carried out on a mode-by-mode basis, instead of galaxy-by-galaxy. The similar smoothing and the mode-by-mode comparison substantially simplify the error analysis. The modal expansion method has been used in comparisons between the 1.2 Jy predicted velocities and observed velocities inferred from TF measurements[@dnw] in the Mark III catalog, yielding $\beta_I\sim$ 0.4. However, examination of the residual field showed a strong dipole signature suggesting a significant mismatch between the Mark III and the fields. The reasons for the mismatch are still not well-understood. More recently, the same method has been employed in the comparison of the 1.2 Jy and SFI[@dacostanusser] and of the and ENEAR velocity fields[@nusserdacosta]. Figure \[fig:12sfi\] shows the smoothed velocity field predicted from the 1.2 Jy survey (left), adopting the best-fit value of $\beta_I=0.6$, and the measured SFI field (right). The infall to Virgo ($l =284^\circ, b = 74^\circ$) dominates the nearby SFI flow. In the middle panel, the field exhibits a dipole pattern corresponding to the reflex motion of the Local Group with infalling galaxies in the Hydra-Centaurus direction and an outward flow in the Perseus-Pisces direction, as seen in the LG restframe. Comparing the two fields one immediately sees that the general pattern of the velocity fields is remarkably similar with excellent agreement in the location of outflows and inflows and with only a few nearby galaxies having large residuals. This result gives confidence in the determination of $\beta_I$. Most encouraging is the absence of large regions of coherent residuals such as the dipole signature seen in the Mark III analysis at intermediate and distant redshift shells. Similar analysis has been performed using the survey and the ENEAR catalog of peculiar velocities. Figure \[fig:psczenear\] shows the corresponding smoothed velocity fields, for an adopted value of $\beta_I=0.5$. Comparison of the right-side of Figures \[fig:12sfi\] and \[fig:psczenear\] shows that the general flow pattern of the SFI and ENEAR velocity fields is remarkably similar. In the ENEAR case, very few prominent structures are probed by bright ellipticals in the innermost shell. However, in the next two shells a strong dipole pattern can be easily recognized, having an amplitude comparable to that observed in SFI. The agreement between the and ENEAR velocity fields is also very good with only a few more distant galaxies having large residuals. The above results demonstrate that the velocity fields of both SFI and ENEAR are similar and well described by the gravity fields of the 1.2 Jy and surveys, yielding comparable values of $\beta_I$. Consistent values of $\beta_I$ have also been obtained from similar analysis of the SBF survey of galaxy distances ($\beta_I=0.42$)[@sbfbeta] and from the peculiar velocities measured for a sample of nearby Type Ia supernovae ($\beta_I=0.4$) [@riess]. Another method to carry out a velocity-velocity comparison considered is VELMOD, a maximum likelihood method which takes as input the distance indicator observables and galaxy redshifts and determines the parameters describing the distance relation and the velocity model adopted. The method does not require smoothing and it is constructed for high-resolution analysis. The method has been used to analyze sub-samples of spiral galaxies extracted from the Mark III [@velmod][@velmod1] and the SFI data[@branchini], yielding $\beta_I=0.49$ and $\beta_I=0.42$, respectively. These results show that the value of $\beta_I$ obtained from velocity-velocity comparisons is independent not only of the data set considered but also of the method used, with all estimates being in the range $0.4\lsim \beta_I \lsim 0.6$. Unfortunately, until recently there has been a disparity between the results obtained from velocity-velocity comparisons and other methods such as density-density comparisons and maximum-likelihood estimates of the power-spectrum (PS) of mass fluctuations derived from peculiar velocity data [@zaroubips]. For instance, density-density comparisons using different data sets have invariably led to high values of $\beta$[@sigad], consistent with unity. In particular, comparison of the 1.2 Jy and POTENT reconstructed density field, based on the Mark III catalog, yields $\beta_I=0.89$. Similarly, estimates based on the PS derived from peculiar velocity data using Mark III[@zaroubips], SFI[@freudling] and ENEAR[@wfenear] have yielded values of $\beta$ in the range 0.82-1.1. The nature of this discrepancy is unknown. Both density-density and velocity-velocity methods have been carefully tested using mock catalogs extracted from N-body simulations and have been shown to provide unbiased estimates of $\beta$. Possible reasons for the discrepancy are non-linear effects, scale dependence of the biasing, poorly understood errors and/or problems with the data. However, attempts to evaluate their impact have so far failed to explain the discrepancy. In general, velocity-velocity comparisons are considered more robust as they depend more on redshift data, while density-density comparisons uses less reliable peculiar velocity data. Recently, a new attempt to carry out a density-density comparison has been made using the SEcat catalog mentioned earlier and the UMV method to reconstruct the 3-D velocity and density fields. These reconstructed fields were then used to determine the value of $\beta$ from direct velocity-velocity and density-density comparisons with the corresponding fields predicted from the redshift survey[@secat]. Figure \[fig:umv\] shows the results of the density-density (left) and velocity-velocity (right) comparisons, which give $\beta_I=0.56 \pm 0.1$ and $\beta_I=0.51 \pm 0.05$, respectively. This result is remarkable since it is the first time a good agreement is found for $\beta$ values derived from these two methods. This encouraging new result, which apparently resolves a long-standing dispute, may be due either to the new method used in the reconstruction of the fields or to the more homogeneous peculiar velocity data used or a combination of both. High values of $\beta$ have also been derived from applying a maximum-likelihood technique to the peculiar velocity data to derive the power-spectrum of mass density fluctuations. These results are summarized in Figure \[fig:ps\]. In the left panel the PS obtained from the ENEAR sample[@wfenear] with those measured for Mark III[@zaroubips] and SFI[@freudling]. The right panel shows the contour map of the likelihood (in the $\Gamma-\eta_8$ plane) for a $\Gamma$ model fit to the ENEAR data, where $\eta_8=\sigma_8\Omega^{0.6}$ and $\sigma_8$ is the $rms$ fluctuation amplitude within a sphere of $8~h^{-1}$ Mpc radius. It is clear from the figure that all data sets lead to similar high-amplitude PS, equivalent to high values of $\beta$. From the figure one also can see that while the likelihood analysis poses a strong constraint on $\eta_8$, the value of $\Gamma$ is poorly determined. Note, however, that low values of $\Gamma$, such as those required from the analyses of redshift survey data ($\Gamma \sim0.2$), are excluded at about the $3\sigma$ level. It is important to recall that the likelihood method used in estimating the PS involves the use of model power-spectra to compute the velocity correlation tensor which is then compared to that computed from the peculiar velocity data to determine the fit parameters. An equivalent way of exploring the same information is to use the scalar velocity correlation function, computed under the assumption of a homogeneous and isotropic flow. The results of this analysis can then be compared directly to model predictions using linear theory and an ensemble average of cosmic flow realizations for different cosmological models. The statistics of the model velocity field is parameterized by the amplitude, $\eta_8$, and by the shape parameter, $\Gamma$, of a CDM–like power spectrum. Applying the velocity correlation statistics to the SFI[@borgani1] and ENEAR[@borgani2] data sets one finds $\eta_8=0.34$ (SFI) and $\eta_8=0.51$ (ENEAR) for $\Gamma=0.25$. These values translate to $\beta_I=0.45-0.67$, assuming $b_I/b_o\sim 1.3$, results which agree within the uncertainties with the lower values of $\beta$ obtained by other methods presented above. More importantly, in contrast to the PS analyses, the region of acceptable solutions comfortably overlaps with other constraints on $\eta_8$ derived from the $rms$ of cluster peculiar velocities and cluster abundances, and on $\Gamma$ as determined for the galaxy power spectrum. One possible explanation for the discrepancy between the results of the PS analysis and the velocity correlation statistics is the different way the errors in the distance measurements are taken into account. An important clue is the weak constraint imposed on the shape parameter by the PS analysis. This suggests that the available samples may not be sufficiently deep for this type of analysis, making the method insensitive to the effect that large scale power may have in inducing velocities on small scales. 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--- abstract: 'Coherent transport promises to be the basis for an emerging new technology. Notwithstanding, a mechanistic understanding of the fundamental principles behind optimal scattering media is still missing. Here, complex network analysis is applied for the characterization of geometries that result in optimal coherent transport. The approach is tailored towards the elucidation of the subtle relationship between transport and geometry. Investigating systems with a different number of elementary units allows us to identify classes of structures which are common to all system sizes and which possess distinct robustness features. In particular, we find that small groups of two or three sites closely packed together that do not carry excitation at any time are fundamental to realize efficient and robust excitation transport. Features identified in small systems recur also in larger systems, what suggests that such strategy can efficiently be used to construct close-to-optimal transport properties irrespective of the system size.' author: - Stefano Mostarda - Federico Levi - 'Diego Prada-Gracia' - Florian Mintert - Francesco Rao bibliography: - 'mostarda2012bisse.bib' title: 'Optimal, robust geometries for coherent excitation transport' --- Introduction ============ Energy and charge transport are of fundamental importance for technological innovation as well as biological processes such as photosynthesis [@Scholes2011; @Renger2001]. If the dynamics is coherent, transport can be enhanced due to constructive interference. This, however, relies on well defined phase relations which get modified easily if the scattering medium is subject to external or internal sources of noise, even for small perturbations. Consequently, interference is destructive in most “real world" cases so that the efficiency is reduced to the point where transport might be completely suppressed [@Lattices1956; @Kramer1993; @Chin2010; @Rebentrost2009c]. The relationship between the detailed spatial configuration of the medium and its functional dynamical properties is subtle [@Baumann1986; @Renger2009]: two structures with similar geometries can possess strongly different transport properties and, vice versa, two structures with comparable transport properties may not share any evident common geometrical feature [@Scholak2011]. Clearly, a mechanistic understanding of the relationship between structure and transport efficiency would be necessary to use quantum coherence as a physical mechanism to develop new technological applications as well as understand photosynthesis at a fundamental level [@Li2012; @Leegwater1996; @Chachisvilis1997]. A recent application of complex network analysis on a set of randomly arranged excitable sites provided a systematic framework to characterize the structural properties of efficient transport [@Mostarda2013]. With much of a surprise, results provided strong evidence for the positive role of a structural motif formed by pair sites that are tightly packed together; although never significantly excited, they assure high transport efficiency and robustness against random displacements of the sites. This partition into excitation carriers and inactive pairs defines a dynamical separation that is reflected in the Hamiltonian, which is approximately composed of two weakly coupled blocks. While not necessarily emerging from the same geometrical features, such a dynamical arrangement has been located in some natural light harvesting complexes such as FMO [@Brixner2005; @Adolphs2006a]. It is then interesting to understand whether such a active/inactive modular arrangement is a truly general principle or if it is rather a peculiarity of systems of [*e.g.*]{} specific size. In this contribution, we therefore consider a paradigmatic system with variable number of randomly disposed excitable sites. Structures with outstanding transport properties are scrutinized, their common geometrical features determined through complex network analyses and their dynamical characteristics studied via inverse participation ratio and eigenvalue distributions. Comparison of the results obtained for different system sizes confirms the presence of specific structural classes for efficient transport that can be differentiated by their robustness properties. This outcome reinforces the idea that tightly packed sites which are not actively involved in the excitation transfer play a fundamental role in the transport, as they make the whole system efficient and robust under perturbations. Methods ======= Tight binding model ------------------- We analyze the transport properties of discrete systems, comprised by a set of $N$ excitable sites that are modeled as two-level systems. The interactions are described by the tight-binding Hamiltonian $$H=\sum_{i\neq j}^N \frac{Jr^{3}_{0}}{|\vec r_i-\vec r_j|^3}\sigma_i^{-}\sigma_j^{+}\ , \label{eq:ham}$$ where $J$ is the dimensionless coupling constant and $\sigma_i^{-/+}$ describe the annihilation/creation of an excitation at site $i$. The interaction rate decays cubically with the inter-site distance in accordance with dipole-dipole interaction. Within this model, a *structure* is defined by the positions of the $N$ sites. The initially excited site (input) and the site where the excitation is sought to arrive (output) are located at the diagonally opposite corners of a cube of side $r_0$, while the remaining $N-2$ sites are placed randomly within this cube. The system is initialized with an excitation on the input site; transport efficiency is defined as the maximal probability to find the excitation at the output site within a short time interval after initialization $$\epsilon=\mathrm{max}_{t\in[0,\mathcal{T}]}|\langle in|\mathrm{e}^{iHt}|out\rangle|^2\ . \label{eq:emaxtau}$$ The states $| in \rangle$/ $| out \rangle$ denote the situation where the input/ output site is excited and all other sites are in their ground state. In order to target exclusively fast transport that necessarily results from constructive interference, we choose $\mathcal{T} = \frac{1}{10} \frac{2\pi \hbar}{J} \frac{r_{in-out}^3}{r_{0}^3}$, [*i.e.*]{} a time-scale ten times shorter than the interval associated with direct interaction between input and output sites [@Scholak2011; @Mostarda2013]. For longer times the excitation would oscillate back and forth between input and output because the dynamics is purely coherent. With a sufficiently short time window, however, only a single oscillation is taken into consideration. Inverse Participation Ratio (IPR) --------------------------------- Under a coherent dynamics induced by a Hamiltonian of the form given in equation (\[eq:ham\]) the excitation will get delocalized over the sites of the system. This delocalization can be quantified in terms of the inverse participation ratio (IPR) defined as $$\mathrm{IPR}(t)=\frac{1}{\sum_{i=1}^{N} q_{i}^2(t)}\ , \label{eq:ipr}$$ where $q_i$ is the probability for site $i$ to be excited. A value for the IPR which is larger than $K-1$ implies that the excitation is delocalised over at least $K$ sites. The maximum value of the IPR is $N$, which is obtained in the case of even delocalization over the whole $N$ constituents. On the other hand, if the excitation is completely localized (e.g. at $t=0$ in our case), the IPR adopts its minimal value of 1. Efficiency Network ------------------ To unravel the structure-dynamics relationship, we apply a set of tools based on complex networks. Originally, these tools had been developed for the characterization of molecular systems [@Rao2004; @Gfeller2007]. However, since these methods are designed to analyze large ensembles of configurations, they prove very useful for our present purposes as they allow a systematic classification of structures which lead to exceptional transport. We generate a complex network where structures with $\epsilon>0.9$ represent the nodes and a link is placed between them if two structures are geometrically similar independently on the specific dynamics of the excitation. The parameter used to estimate structural similarity depends on the relative distances of the excitable sites of two structures under comparison. The sites are indistinguishable, thus all different permutations of the site labels need to be performed [^1]. In addition, a rotational symmetry around the in-out axis and an additional mirror symmetry has to be taken into consideration. The measure $S$ of similarity between configurations A and B is thus defined as $$S^2=\min\sum^n_{i=1} \frac{d_i ^2}{n}\ ,$$ where $d_i$ is the difference of the coordinates of the $i-$th site in the two configurations, and the minimization is performed over all permutations, rotations around the in-out axis and the mirror symmetry. A link is placed in the network only if $S$ lies below a certain threshold value $S^{*}$ which is going to be discussed in detail in the next sections. ![The structural superposition algorithm (here on the first cluster with $N=6$, 10731 structures) makes geometrical features emerge from the noise.[]{data-label="fig:superimposition"}](superimposition_1.pdf){width="47.50000%"} Network Clusterization ---------------------- Densely connected regions of the network indicate the presence of groups of structures with common geometrical motifs [@Mostarda2013]. We identify these regions using a network clusterization algorithm based on a self-consistency criterion in terms of network random walks, the Markov clusterization algorithm (MCL). The network is in this way split into different clusters comprised of structures with similar sites arrangements [@Gfeller2007; @Van2008]. The method consists of four steps: (a) start with the transition matrix $A$ of the network, where each column is normalized to 1; (b) compute $A^{2}$; (c) take the [*p*]{}-th power ($p>1$) of every element of $A^{2}$, normalize each column; (d) go back to step (b). After some iterations of the MCL, $A$ converges to $A_{MCL}$, where only one entry for each column is non-zero. Clusters are defined by the connected regions of the percolation network. In the limit of $p=1$, only one cluster is detected. On the other hand, the parameter [*p*]{} is related to the granularity of the clustering process. Large values of [*p*]{} generate several small clusters. Structural superposition ------------------------ A structural representation of the clusters is obtained in the following way: for each cluster, the most connected structure is taken as reference and all the others are superimposed. For each structure, we represent the one obtained with the combination of labeling, rotation and mirror state which minimizes the similarity parameter $S$ (see [*Network Creation*]{} section). In order to reduce noise, the coordinates of the sites are averaged with the ones from two other structures of the cluster taken at random. Structural rendering is done with VMD [@humphrey1996vmd]. An example of the effects of such algorithm is shown in Fig. \[fig:superimposition\]. We follow this procedure to depict all the clusters in Supp. Fig. \[fig:supp:VMDrepr1\],\[fig:supp:VMDrepr2\],\[fig:supp:VMDrepr3\]. ![Distribution of the similarity parameter $S$ for systems with different number of sites. (a) The average similarity between two given structures decreases ($S$ increases) with the number of sites. (b) The chosen cutoffs (in grey) show a dependence on N that is similar to that of the average similarity $S$ (in black).[]{data-label="fig:Sdistrib"}](Sdistribuwithcutoffs_2.png){width="47.50000%"} Consistency parameter $C$ -------------------------- In order to monitor whether the clusterization procedure is consistent while varying the granularity parameter $p$, we introduce here a “consistency parameter” $C$. If the clusterization is accurate, an increase in the granularity breaks big clusters into smaller clusters, without mixing them. In fact, every cluster obtained for a given value of $p$ should be fully (or almost fully) included in only one single cluster generated with a smaller value $p -\Delta p$. If this is the case, the clusterization is consistent and the value for $C$ will be maximal. On the other hand, the worst case is a completely random clusterization: the structures of each cluster for a given $p$ are equally distributed between the $n$ clusters generated with $p-\Delta p$. This would correspond to the minimal value of $C$. For the computation of $C$, we first perform the clusterization analysis from $p=1.1$ to $p=2.2$ in steps of $\Delta p = 0.1$ (from very low to very high values of $p$). For every step $p$, we calculate for each cluster $\mathcal{I}$ the largest portion $C_{\mathcal{I}}$ of its population $\mathcal{P}_{\mathcal{I}}$ included in a single cluster obtained at $p - \Delta p$. $C$ is then calculated as the average of $C_{\mathcal{I}}$ weighted over the relative populations $\mathcal{P}_{\mathcal{I}} / \sum_{\mathcal{I}} \mathcal{P}_{\mathcal{I}}$. In this way, the maximum value of $C$ is always 1, which corresponds to a perfectly consistent clusterization. The minimum of $C$ at a given $p$ is $1/n$, where $n$ is the number of clusters generated at $p -\Delta p$. To make the value of $C$ independent of $n$, we normalize it such that $1/n$ corresponds to 0 and rescale the $[1,1/n]$ segment linearly to $[1,0]$. Practical advices for the parameters choice =========================================== Network creation and clusterization depend on two parameters: the similarity cutoff $S^{*}$ which sets the accepted degree of similarity between different structures and the granularity parameter $p$ which determines the degree of coarse-graining in the clusterization. It is important to note that finding the correct value of these parameters for structural comparison is an open and unsolved problem in the broader field of complex systems. Apparently, there is no single [*right*]{} choice, as those parameters probe the system at different resolutions. Best practice suggests a scanning in parameter space in order to asses the robustness of the observations on a particular data set. In this section, we discuss cut-off choices in some detail. In Fig. \[fig:Sdistrib\]-a the distributions of $S$ are shown for the most efficient structures ($\epsilon>0.9$) obtained for $N=4-8$. Interestingly, two behaviors are present. The case $N=4$ is compatible with an almost homogeneous ensemble, where any two structures are very similar to each other ($S<0.15$ for the 94% of the links). On the other hand, in the systems with $N=6,7,8$ the number of pairs of compatible structures is instead very small, i.e. the ensemble is deeply heterogeneous ($S>0.15$ for the 92%, 98% and 99% of the links for $N=6,7,8$, respectively). The case $N=5$ shows an intermediate behavior. ![Parameters choice in the clusterization procedure for $N=6$. (a) Relative cluster populations for $p=1.2,1.4,1.6$ are shown in black, dark and light grey, respectively. Significant clusters separate from the noise which results in an exponential tail (fitted dashed lines). (b) Consistency parameter $C$ as a function of $p$.[]{data-label="fig:loyaltyparam"}](consistencyparam_3.pdf){width="47.50000%"} If only one system is considered, the cut-off needs just to be self consistent, i.e. the results should not vary too much with $S^{*}$. The problem arises when one wishes to compare different networks, [*i.e.*]{} different distance distributions. A fixed value for $S^{*}$ for the different cases would create networks with very different connectivities, which makes the comparison very hard. In order to set the thresholds in a compatible way, $S^{*}$ is taken as the minimal value of $S$ for which the networks are fully connected (99.9% of nodes have been considered). The resulting values are $0.02$, $0.06$, $0.11$, $0.16$ and $0.18 \ r_0$ for $N=4-8$, respectively. These values increase in a similar manner as the average value of the distance $S$ (see Fig. \[fig:Sdistrib\]-b). $S^{*}$ lies just above the tails of the pairwise distance distributions. Consequently, only the most similar structures are linked together. Lower values of the cut-off would generate a disconnected network, while values too close to the maximum of the distributions would put links between structures that are not very similar. For the clusterization process the goal is to separate the bulk of the signal from the statistical noise. To this aim, one can look at the population of the clusters obtained, ranked by decreasing size. Typically, the signal is formed by a small number of populated clusters, while the noise is composed of a large number of small clusters which follow an exponential tail. Fig. \[fig:loyaltyparam\]-a depicts as an example the results for the $N=6$ case obtained with different $p$. At $p=1.2$ (black curve) the algorithm detects two big clusters with 74.6% and 25.1% of the population plus two satellites due to noise with 0.3% of cumulative population. With $p=1.4$ (dark grey curve) the second cluster at $p=1.2$ splits into eight clusters with smaller relative populations ranging from 1.0% to 7.8%, while the noise is composed by the remaining 20 clusters (nicely fitted by an exponential function in Fig. \[fig:loyaltyparam\]-a). At $p=1.6$ (light grey curve) only four significant clusters are detected. Their populations are 41.5%, 14.9%, 8.5% and 7.1% of the total population (cumulatively the 72.0%), while the remaining 80 clusters have a cumulative relative population of 28.0% and constitute noise. With even higher values of $p$ the network breaks more and more into small noisy clusters. These three scenarios show how changing the granularity parameter $p$ leads to different signal to noise ratio. This behaviour is not necessarily monotonic: incrementing $p$ at first increases the number of significative clusters up to a maximum after which the noise grows and becomes dominant. However, similarly to the choice of $S^{*}$, our priority is to compare different networks. Therefore, the choice of $p$ which maximizes the signal to noise ratio for each network might not be the best for this purpose. We thus employ the *consistency parameter* $C$, which we calculate for a wide range of $p$ (see Methods for details). This quantity monitors whether the clusterization procedure is accurate and provides a way to consistently compare different networks. {width="99.00000%"} Let us illustrate the behaviour of $C$ for the $N=6$ case in detail as an example (see Fig. \[fig:loyaltyparam\]-b): at $p=1$ only one cluster is present, so $C=1$ by definition. At $p=1.1$ we have 2 clusters, but they are obviously both fully contained in the cluster of the former step, thus $C=1$. The first non trivial value of $C$ is at $p=1.2$ where the 4 clusters are quite well identifiable with the 2 clusters at $p=1.1$. There is a slight drop of $C$, but the value $C=0.99$ is sufficiently close to unity to warrant consistency. This regime is valid up to $p=1.4$, while at $p=1.5$ the value of $C$ drops to $0.95$ (the values of $C$ are normalized); this implies that the clusterization loses some consistency. The biggest drop of $C$ occurs at $p=1.7$, where $C=0.87$ and the consistency of the clusterization process is lost. For each choice of $N$, $C$ has a different behavior (this can be seen in Fig. \[figsupp:fidelityparam\] in Supp.Mat.). This means that we cannot choose a unique value $p$ to use in all the clusterization processes, but we need to investigate case by case the dependence of $C$ on $p$. We then select the highest value of $p$ for which $C=1$ for a given system size $N$: this systematic choice of $p$ allows a first qualitative understanding of the geometrical characterization of the system. The values correspond to $p=1.3,1.3,1.1,1.1$ for $N=4-7$, respectively. For the case $N=8$, the choice of $p=1.4$ obtained following the mentioned criterion leads to a single cluster. This is probably due to the high value of the $S^{*}$ chosen for this system, which creates a more densely connected network, hard to break into clusters ([*i.e.*]{} a higher value of $p$ would be needed). We therefore increase in this case the value slightly to $p=1.5$. Results ======= Structural characterization of the clusters {#sect:STRCHAR} ------------------------------------------- We analyze quantum transport for a large sample ($10^8$) of randomly generated structures with different number of sites ($N=4-8$, see Methods for details). The case $N=3$ has not been studied, since it never leads to efficiencies higher than 37% [@scholakphd]. Our analysis focuses on structures with $\epsilon>0.9$. Within this reduced set, the number of efficient structures is 3530, 7368, 14280, 5896, 6688 for $N=4$ to $N=8$, respectively (in Fig. \[figsupp:effdistrib\] in Supp. Mat. the probability of generating efficient structures is shown for $N=4-8$). Most of the sets of efficient structures are highly *heterogeneous*, which means that two structures with similar efficiency do not necessarily share any evident common pattern. This structural heterogeneity prevents a straightforward identifications of the geometrical features that are compatible with efficient transport. To uncover these features, we apply the protocol based on complex networks described in Methods. All the clusters we identify are shown in Supp. Fig. \[fig:supp:VMDrepr1\],\[fig:supp:VMDrepr2\],\[fig:supp:VMDrepr3\]. A sketch of the most relevant ones is depicted in Fig. \[fig:N4cl12\]. According to the network analysis, systems with $N=4$ and $N=5$ are quite homogeneous, and few geometries are compatible with efficient transport. The configurations obtained are shown in Fig. \[fig:N4cl12\]. In both cases the clusterization algorithm gives two clusters, where the intermediate sites of the less populated cluster are more strongly aligned than the others. For $N=4$ we label the two clusters $\tau_a$ and $\tau_b$ (first row of Fig. \[fig:N4cl12\]); they represent 63.1% and 36.9% of the total population, respectively. In both cases the four sites are equidistant from each other. However, in the latter case, the two intermediate sites are arranged along the input-output axis while in the former case they are slightly offset. For $N=5$, the situation is similar, but there is an extra intermediate site. Two clusters, named $\pi_a$ and $\pi_b$ (second row of Fig. \[fig:N4cl12\]), represent 83.5% and 15.8% of the total population (the remaining 0.7% is noise). In this case there is a slight deviation from an equidistant distribution of the inter-site distances. In the cluster $\pi_b$ the 3 intermediate sites are aligned along an axis which is rotated with respect to the in-out one; in $\pi_a$ these three sites form a triangle. {width="99.00000%"} Interestingly, the structures found for $N=4$ and $N=5$ constitute the building blocks for the higher dimensional cases (see Supp. Fig. \[fig:supp:VMDrepr1\],\[fig:supp:VMDrepr2\],\[fig:supp:VMDrepr3\]). In fact, systems with $N>5$ present a higher degree of heterogeneity and a prototypical modular structure. The first module is comprised by four/five sites approximately lined up along the in-out axis; this defines a structural backbone. In all cases this module is compatible with either $\tau_a$/$\tau_b$ or $\pi_a$/$\pi_b$. The second module is essentially formed by the remaining two to four sites, organized in tightly packed *pairs* or *triplets*. Backbone sites are approximately equally spaced between input and output with typical inter-site distances of around $0.50-0.60 \ r_0$, depending on the specific case. Pair/triplet sites instead are always very close to each other with a inter-site distance of around $0.25 \ r_0$, depending on the particular organization. It is worth noting that the backbone arrangement of all mentioned systems is symmetric under input output inversion [@Mostarda2013; @Walschaers2013]. The organization of $\tau_b$ constitutes the backbone of the most populated cluster of $N=6$, $\tau_6$ (75.1%, third row of Fig. \[fig:N4cl12\]). In this cluster, we identify a pair whose position is less well defined than the position of the backbone sites. A backbone of four sites that resembles $\tau_b$ is present also in the second cluster for $N=7$, $\tau_7$ (46.9%, fourth row of Fig. \[fig:N4cl12\], on the right), where the remaining three sites lie close together at comparable reciprocal distances, [*i.e.*]{} they form a *triplet*. This triplet is located more heterogeneously than the intermediate sites, in a region comparable to the one occupied by the pair module in $\tau_6$. Lastly, the most populated cluster for $N=8$, $\tau_8$ (58.4%), is formed by a backbone of four sites as in $\tau_6$ and $\tau_7$, with the remaining four sites organized into two pairs. At an increase of the granularity parameter to the value $p=1.7$, this cluster breaks into two smaller clusters that differ only in the location of the pairs: the more populated cluster $\tau_8^a$ (47.1%) has the two pairs on the same side of the backbone, where they form a triangle with the two intermediate sites of the backbone (Fig. \[fig:N4cl12\], fifth row right); the two pairs of the smaller cluster $\tau_8^b$ (10.3%) are instead diametrically opposed with respect to the backbone axis. Backbones composed of five sites emerge for $N=7$. The most populated cluster $\pi_7$ (53,1%, fourth row of Fig. \[fig:N4cl12\], on the left), is in fact formed by five sites organized in a backbone geometry similar to $\pi_a$ with an additional pair similar to the case of $\tau_6$. With this choice for the granularity parameters, the remaining clusters are less well defined. The second cluster $v_6$ for $N=6$ (24.8%), is composed of heterogeneous structures which are hard to reconduct to a single structural motif. In this case, increasing $p$ to $1.4$ separates this cluster into $7$ smaller more homogeneous clusters, with sites either disposed on a line or in a sparse manner. A more detailed discussion of the case $N=6$ can be found in [@Mostarda2013]. Also the second cluster $v_8$ for $N=8$ (36.9% of the population) is poorly identifiable. Cluster $v_8$ is in fact composed by a well defined backbone-like module with five sites and the remaining three sites in a sparse configuration. With the chosen granularity value $p=1.5$, the latter module is not compatible with a triplet. Subgroups at higher $p$, but with a very small population, present a triplet in a similar manner as in $\tau_7$. Altogether these results provide evidence for the presence of a modular arrangement in the geometries of efficient structures. Inverse participation ratio --------------------------- So far, we have constructed and clusterized a complex network of efficient structures on purely geometrical grounds. Now we move to investigate the dynamics of these structures, to better understand whether the common geometrical features identified correspond to dynamical similarities. To quantitatively characterize the dynamical behavior of the identified structures, the inverse participation ratio (IPR, see Methods) is calculated at every instant of time for each structure. In Fig. \[fig:inverseparticip\] the distributions of the maxima of the IPR within $t\in(0,\mathcal{T})$ from $N=4$ to $N=8$ are shown, divided into clusters. Remarkably, the maxima of the IPR spontaneously group into two well defined distributions. The cases $N=4$ and $N=5$ are basically homogeneous, with negligible differences between the two clusters in both systems. The corresponding values lie around $3.3$ and $4.2$, respectively, which means that the excitation is shared between approximately four or five sites. The two values are prototypical for the IPR distributions of clusters with bigger number of sites. In fact, structures in cluster $\tau_6$ have IPR maxima values similar to those in $\tau_a$ and $\tau_b$, while values for structures in $v_6$ have values close to those in $\pi_a$ and $\pi_b$. The distributions for triplet cluster in $N=7$ and the double pair clusters in $N=8$ (red curves) correspond to those for $\tau_a$ and $\tau_b$, while the pair cluster in $N=7$ and the sparse cluster in $N=8$ (blue curves) have the same IPR distribution as $\pi_a$ and $\pi_b$. Strikingly, the distribution of IPR supports that indeed the backbones of the $\tau_X$ and $\pi_X$ clusters (where $X$ stands for any $N$) for $N>5$ correspond to $\tau_a$/$\tau_b$ and $\pi_a$/$\pi_b$ respectively, not only from a geometric point of view, but also dynamically. This is evidenced by the excellent overlap of the distributions for different $N$ (bottom right panel of Fig. \[fig:inverseparticip\]). Inactive sites enhanced transport --------------------------------- The distributions of the IPR maxima reveal that for $N>5$ only a subset of the sites is substantially excited at the same time. In fact, the sites arranged in a backbone and those forming a pair or triplet possess a different dynamical role; while the former carry the excitation actively, sites closely packed together are never significantly populated by the excitation. Such a behavior emerges systematically for all system sizes, such that the pairs identified before for the case $N=6$ [@Mostarda2013] are just one example. In the following sections, we will thus refer to the backbone and to the pairs/triplets as to the [*active*]{} and [*inactive*]{} modules of the clusters. ![Efficiency loss upon removal of inactive modules as a function of the original efficiency $\epsilon$ for the clusters with $N=7$ and $N=8$. Error bars are calculated according to the standard deviation. All cases but $v_8$ are compatible with the pair effect found for $\tau_6$.[]{data-label="fig:efflossinactivemod"}](efficiencylossvertical_6.pdf){width="44.00000%"} Removal of the inactive module results in a systematic efficiency loss. This is shown in Fig. \[fig:efflossinactivemod\], where all clusters, apart from $v_8$, behave similarly: the contribution of the inactive modules is particularly important for the most efficient realizations due to the sensitivity of perfect constructive interference. The loss upon pair removal typically ranges from $0.05$ to $0.15$, depending on the initial value of the efficiency. As we suggested from geometrical considerations, $v_8$ is a very noisy cluster and cannot be considered completely composed of a 5-sites backbone plus a triplet. In fact, removal of these three sites causes an efficiency drop up to 60%, which indicates that the triplet in $v_8$ cannot be considered an inactive module. While the triplets in the first cluster with $N=7$ play a role similar to the pair in $\tau_6$, it is not obvious whether the presence of two pairs in $\tau_{8}^a$ and $\tau_{8}^b$ is necessary or if only one of them is enough to obtain the same effect. In fact, the two pairs show a small degree of collectiveness, which means that one is dominant and the other one has a close-to-negligible effect (Fig. \[fig:efflossinactivemodonebyone\](c-d) in Supp.Mat.). This is confirmed by the fact that the efficiency loss upon removal of the two pairs at the same time is only slightly larger than the sum of efficiency losses upon removal either pair(Fig. \[fig:efflossinactivemodonebyone\](a-b) in Supp. Mat.). Inactive modules induce eigenvalue shift ---------------------------------------- The mechanism behind the influence of the inactive modules on the exciton dynamics can be understood from the distribution of the energy eigenvalues with and without the inactive sites as displayed in Fig. \[fig:lambdashift\]. Because only a single excitation is present in the system at any time, there are $N$ energy eigenvalues to study. Given the weak interaction between the backbone and $k$ pairs or a triplet there are $N-2k$ or $N-3$ eigenstates whose amplitudes are highly localized on the backbone. The amplitudes of the remaining $2k$ or $3$ eigenstates are instead localized on the inactive sites. The interaction between the backbone and the inactive sites results in a shift of the eigenfrequencies of the former $N-2k$ or $N-3$ eigenstates (denoted by $\mathit{\lambda}_i$ in Fig. \[fig:lambdashift\]), such that their differences are close to integer multiples of a fundamental frequency. With this shift, the excitation is transferred to the output site essentially perfectly after one period of this fundamental frequency. This is true for all the clusters, ranging from the pair and triplet clusters with $N=7$ ($\pi_7$ in Fig. \[fig:lambdashift\]) to the clusters with two pairs in the $N=8$ case ($\tau_{8}^a$ and $\tau_{8}^b$ in Fig. \[fig:lambdashift\]), what strongly suggests that this mechanism [@Mostarda2013] works independently of the system size. ![Shift of the first backbone eigenvalue upon removal of inactive modules. In systems with both $N=7$ (top row) and $N=8$ (bottom row) the removal of the inactive sites causes a shift in only the first eigenvalue. In the former case, the mechanism is similar for both the pair and the triplet.[]{data-label="fig:lambdashift"}](eigshift_7.pdf){width="44.00000%"} Robustness ---------- The analysis presented so far shows the emergence of two classes of geometrical and dynamical behavior, characterized by an arrangement into active and inactive modules. In the following, we explore this separation with respect to the robustness properties of the various clusters. {width="99.00000%"} Transport robustness is probed by random displacements of the individual sites of a structure. With displacements confined to a cube of side $0.05 \ r_0$ centered around the original position of the site, $\mathit{\Delta} \epsilon_{\text{rand}}$ is calculated for each structure as the difference between the original efficiency and the average efficiency obtained from 1000 site-randomizations. In this scheme, structures are kept rigid which corresponds to the assumption that the dynamics occurs on a much faster time scale than low-frequency fluctuations of the entire system (e.g. in the context of biological systems this would be equivalent to large-scale protein breathing). The distributions of $\mathit{\Delta} \epsilon_{\text{rand}}$ for $N=4-8$ are shown in Fig. \[fig:stabilitytot\]. For $N=4$, both $\tau_a$ and $\tau_b$ are very robust under random displacement, the former is slightly more stable than the latter with $\mathit{\Delta} \epsilon_{\text{rand}} = 0.041$ as compared to $\mathit{\Delta} \epsilon_{\text{rand}} = 0.044$. Also in the case of $N=5$, $\pi_a$ and $\pi_b$ present overall a quite similar pattern of robustness: $\mathit{\Delta} \epsilon_{\text{rand}}$ are $0.094$ and $0.107$ for the first and second cluster, respectively. It is however in the clusters obtained for system of size from $N=6$ to $N=8$ that we detect the largest separation in response to random displacements. In all these cases, the loss in efficiency for structures in $\tau_X$ clusters is roughly half of the efficiency of the losses in $\pi_X$ or $v_X$ clusters. Exact values can be found in the caption of Fig. \[fig:stabilitytot\], where can be also visually noticed that the two curves separate well from each other in all cases. Overall, the efficiency loss upon random displacement, which represents the robustness of our randomly generated structures, spontaneously group into two distributions, independently on the number of total sites. This is clearly shown by the overlap of all the curves into a single plot (bottom right panel of Fig. \[fig:stabilitytot\]). Two behaviors are present, depending on the number of sites that build the backbone. Clusters whose backbone is composed by four sites (red data in Fig. \[fig:stabilitytot\]) show good robustness under random displacement of the sites ($\mathit{\Delta} \epsilon_{\text{rand}}$ peaked around $0.06$), while the efficiency loss of backbones with a larger number of sites (typically $5$) peaks around $0.10$ in all cases (blue data). Poorly defined clusters $v_6$ and $v_8$ share the same response to noise as the $\pi_X$ clusters. Overall, this result suggests that the backbone size is already a good indicator on the robustness of a given efficient structure. In agreement with the IPR analysis, all the robustness distributions overlap very well (compare the bottom right panels of Fig. \[fig:inverseparticip\] for IPR and Fig. \[fig:stabilitytot\] for robustness). This provides strong evidence for a clear correlation between robustness, backbone size and inverse participation ratio. Conclusions =========== As shown here, the application of advanced statistical techniques from complex network analysis permit to find a geometrical characterization of efficient structures. The analysis of efficient transport in systems with a variable number of excitable sites from $N=4$ to $N=8$ highlights the emergence of clear structural signatures related to high efficiency, independently of system size. For growing $N$, a modular arrangement appears. The first is a backbone-like module, typically formed by four or five sites that actively carry the excitation. The remaining sites are arranged in one or more inactive modules composed by tightly packed sites whose function is to tune the eigenvalues of the backbone to realize constructive interference and enhance transport. This mechanism is statistically dominant: only the 2% of the structures with $N=7$ or $N=8$ does not possess any inactive module. Remarkably, common geometrical and dynamical features evidence the recursiveness of these modules. Efficient structures for smaller systems ($N=4,5$) are identified as building blocks for larger structures ($N\geq6$). The addition of inactive modules to these prototypical backbones seem to represent an effective general strategy for the construction of structures in which high efficient transport is achieved by means of constructive interference. The analysis presented so far has been performed for a purely coherent case, i.e. without any source of noise. This choice is consistent, because results would not change qualitatively in the presence of incoherent effects. It has been in fact emphazised before that as long as the interest lies in the characterization of [*fast*]{} transport, which is what motivates the current definition of efficiency equation (\[eq:emaxtau\]), environmental noise would decrease the efficiency of every structure with no specific distinction [@Mostarda2013], irrespective of the environment considered. The modularity identified here holds great promise for an explicit exploitation as design principle: the construction of large optimized system seems feasible if it can be decomposed into smaller, individually optimized units, whereas a simultaneous optimization over all degrees of freedom easily turns impractical. Existing aspects of such modularity in actual LHC’s underline the feasibility to obtain such optimal structures through evolutionary optimization. It should thus be expected that the features classified here pave a practical roadmap towards the design of systems that achieve highly efficient transport in a potentially robust fashion. [**SUPPLEMENTARY FIGURES**]{} {width="80mm"} {width="80mm"} {width="100mm"} {width="130mm"} {width="130mm"} {width="130mm"} [^1]: There are $(N-2)!$ permutations since input- and output-site are distinguished from the other sites.

--- abstract: 'A rattleback is a rigid, semi-elliptic toy which exhibits unintuitive behavior; when it is spun in one direction, it soon begins pitching and stops spinning, then it starts to spin in the opposite direction, but in the other direction, it seems to spin just steadily. This puzzling behavior results from the slight misalignment between the principal axes for the inertia and those for the curvature; the misalignment couples the spinning with the pitching and the rolling oscillations. It has been shown that under the no-slip condition and without dissipation the spin can reverse in both directions, and Garcia and Hubbard obtained the formula for the time required for the spin reversal $t_r$ \[Proc. R. Soc. Lond. A **418**, 165 (1988)\]. In this work, we reformulate the rattleback dynamics in a physically transparent way and reduce it to a three-variable dynamics for spinning, pitching, and rolling. We obtain an expression of the Garcia-Hubbard formula for $t_r$ by a simple product of four factors: (1) the misalignment angle, (2) the difference in the inverses of inertia moment for the two oscillations, (3) that in the radii for the two principal curvatures, and (4) the squared frequency of the oscillation. We perform extensive numerical simulations to examine validity and limitation of the formula, and find that (1) the Garcia-Hubbard formula is good for both spinning directions in the small spin and small oscillation regime, but (2) in the fast spin regime especially for the steady direction, the rattleback may not reverse and shows a rich variety of dynamics including steady spinning, spin wobbling, and chaotic behavior reminiscent of chaos in a dissipative system.' author: - Yoichiro - Hiizu title: Rattleback dynamics and its reversal time of rotation --- Introduction {#sec:introduction} ============ Spinning motions of rigid bodies have been studied for centuries and still are drawing interest in recent years, including the motions of Euler’s disks [@Moffatt2000], spinning eggs [@Moffatt2002], and rolling rings [@Jalali2015], to mention just a few. Also, macroscopic systems which convert vibrations to rotations have been studied in various context such as a circular granular ratchet [@Heckel2012], and bouncing dumbbells, which show a cascade of bifurcations [@Kubo2015]. Another interesting example of rigid body dynamics which involves such oscillation-rotation coupling is a rattleback, also called as a celt or wobble stone, which is a semi-elliptic spinning toy \[Fig. \[fig:notation\](a)\]. It spins smoothly when spun in one direction; however, when spun in the other direction, it soon starts wobbling or rattling about its short axis and stops spinning, then it starts to rotate in the opposite direction. One who has studied classical mechanics must be amazed by this reversal in spinning, because it apparently seems to violate the angular momentum conservation, and the chirality emerges from a seemingly symmetrical object. There are three requirements for a rattleback to show this reversal of rotation: (1) the two principal curvatures of the lower surface should be different, (2) the two horizontal principal moments of inertia should also be different, and (3) the principal axes of inertia should be misaligned to the principal directions of curvature. These characteristics induce the coupling between the spinning motion and the two oscillations: the pitching about the short horizontal axis and the rolling about the long horizontal axis. The coupling is asymmetric, i.e., the oscillations cause torque around the spin axis and the signs of the torque are opposite to each other. This also means that either the pitching or the rolling is excited depending on the direction of the spinning. We will see that the spinning couples with the pitching much stronger than that with the rolling; therefore, it takes much longer time for spin reversal in one direction than in the other direction, and that is why most rattlebacks reverse only for one way before they stop by dissipation. In the 1890s, a meteorologist, Walker, performed the first quantitative analysis of the rattleback motion [@Walker1896]. Under the assumptions that the rattleback does not slip at the contact point and that the rate of spinning speed changes much slower than other time scales, he linearized the equations of motion and showed that either the pitching or the rolling becomes unstable depending on the direction of the spin. More detailed analyses were performed by Bondi [@Bondi1986], and recently by Wakasugi [@WakasugiH23]. Case and Jalal [@Case2014] derived the growth rate of instability at slow spinning. Markeev [@Markeev1984], Pascal [@Pascal1983], and Blackowiak et al. [@Blackowiak1997] obtained the equations of the spin motion by extracting the slowly varying amplitudes of the fast oscillations of the pitching and the rolling. Moffatt and Tokieda [@MoffattTokieda2008] derived similar equations to those of Markeev [@Markeev1984] and Pascal [@Pascal1983], and pointed out the analogy to the dynamo theory. Garcia and Hubbard [@GarciaHubbard1988] obtained the expressions of the averaged torques generated by the pure pitching and the rolling, and derived the formula for spin reversal time. As the first numerical study, Kane and Levinson [@Kane1982] simulated the energy-conserving equations and showed that the rattleback changes its spinning direction indefinitely for certain parameter values and initial conditions. They also demonstrated the coupling between the oscillations and the spinning by showing that it starts to rotate when it begins with pure pitching or rolling, but the direction of the rotation is different between pitching and rolling. Similar simulations were performed by Lindberg and Longman independently [@Lindberg1983]. Nanda *et al.* simulated the spin resonance of the rattleback on a vibrating base [@Nanda2016]. Energy-conserving dynamical systems usually conserve the phase volume, but the present rattleback dynamics does not explore the whole phase volume with a given energy because of a non-holonomic constraint due to the no-slip condition. Therefore, the Liouville theorem does not hold, and such a system has been shown to behave much like dissipative systems. Borisov and Mamaev in fact reported the existence of “strange attractor” for certain parameter values in the present system [@Borisov2003]. The no-slip rattleback system has been actively studied in the context of chaotic dynamics during the last decade [@Borisov2006; @Borisov2014]. Effects of dissipation at the contact point have been investigated in several works. Magnus [@Magnus1974] and Karapetyan [@Karapetyan1981] incorporated a viscous type of friction force proportional to the velocity. Takano [@Takano2014] determined the conditions under which the reversal of rotation occurs with the viscous dissipation. Garcia and Hubbard [@GarciaHubbard1988] simulated equations with aerodynamic force, Coulomb friction in the spinning, and dissipation due to slippage, then they compared the results with a real rattleback. The dissipative rattleback models based on the contact mechanics with Coulomb friction have been developed by Zhuravlev and Klimov [@Zhuravlev2008] and Kudra and Awrejcewicz [@Awrejcewicz2012; @Kudra2013; @Kudra2015]. This paper is organized as follows. In the next section, we reformulate the rattleback dynamics under the no-slip and no dissipation condition in a physically transparent way. In the small-spin and small-oscillation approximation, the dynamics is reduced to a simplified three-variable dynamics. We then focus on the time required for reversal, or what we call *the time for reversal*, which is the most evident quantity that characterizes rattlebacks, and obtain a concise expression for the Garcia-Hubbard formula for the time for reversal [@GarciaHubbard1988]. In Sec. \[sec:simulation\], the results of the extensive numerical simulations are presented for various model parameters and initial conditions in order to examine the validity and the limitation of the theory. Discussions and conclusion are given in Sec. \[sec:discussion\] and Sec. \[sec:conclusion\], respectively. Theory {#sec:theory} ====== Equations of motion -------------------  A commercially available rattleback made of plastic. (b) Notations of the rattleback. (c) A schematic illustration of the shell-dumbbell model.](1.pdf){width="7cm"} We consider a rattleback as a rigid body, whose configuration can be represented by the position of the center of mass G and the Euler angles; both of them are obtained by integrating the velocity of the center of mass $\bm{v}$ and the angular velocity $\bm{\omega}$ around it [@Goldstein2002]. We investigate the rattleback motion on a horizontal plane, assuming that it is always in contact with the plane at a single point C without slipping. We ignore dissipation, then all the forces that act on the rattleback are the contact force $\bm{F}$ exerted by the plane at C and the gravitational force $-Mg\bm{u}$, where $\bm{u}$ represents the unit vertical vector pointing upward \[Fig. \[fig:notation\](b)\]. Therefore, the equations of motion are given by $$\begin{aligned} \frac{d(M\bm{v})}{dt} &= \bm{F} - Mg\bm{u},\label{eq:em-1}\\ \frac{d(\hat{I}\bm{\omega})}{dt} &= \bm{r} \times \bm{F}, \label{eq:em-2}\end{aligned}$$ where $M$ and $\hat{I}$ are the mass and the inertia tensor around G, respectively, and $\bm{r}$ is the vector from G to the contact point C. The contact force $\bm{F}$ is determined by the conditions of the contact point; our assumptions are that (1) the rattleback is always in contact at a point with the plane, and (2) there is no slip at the contact point. The second constraint is represented by the relation $$\bm{v} = \bm{r} \times \bm{\omega}.\label{eq:no-slip}$$ Before formulating the constraint (1), we specify the co-ordinate system. We employ the body-fixed co-ordinate with the origin being the center of mass G, and the axes being the principal axes of inertia; the $z$ axis is the one close to the spinning axis pointing downward, and the $x$ and $y$ axes are taken to be $I_{xx} > I_{yy}$ (Fig. \[fig:coordinate\]). In this co-ordinate, the lower surface function of the rattleback is assumed to be given by $$f(x,y,z) = 0,\label{eq:def-z}$$ where $$f(x,y,z) \equiv \frac{z}{a} - 1 + \frac{1}{2a^2}(x,\,y)\hat{R}(\xi)\hat{\Theta}\hat{R}^{-1}(\xi)\begin{pmatrix}x\\y\\\end{pmatrix},$$ with $$\begin{aligned} \hat{R}(\xi) \equiv \begin{pmatrix} \cos\xi, & -\sin\xi \\ \sin\xi, & \cos\xi \\ \end{pmatrix}, \quad \hat{\Theta} \equiv \begin{pmatrix} \theta, & 0 \\ 0 & \phi \\ \end{pmatrix}.\end{aligned}$$ Here $a$ is the distance between G and the surface at $x=y=0$, and $\xi$ is the *skew angle* by which the principal directions of curvature are rotated from the $x$-$y$ axes, which we choose as the principal axes of inertia (Fig. \[fig:coordinate\]). $\theta/a$ and $\phi/a$ are the principal curvatures at the bottom, namely at $(0,0,a)^{t}$. Now, we can formulate the contact point condition (1); the components of the contact point vector $\bm{r}$ should satisfy Eq. (\[eq:def-z\]), and the normal vector of the surface at C should be parallel to the vertical vector $\bm{u}$. Thus we have $$\bm{u} \parallel \nabla f,$$ which gives the relation $$\frac{\bm{r}_{\perp}}{a} = \frac{1}{u_{z}} \hat{R}(\xi)\hat{\Theta}^{-1}\hat{R}^{-1}(\xi)\bm{u}_{\perp} \label{eq:def-xy},$$ where $\bm{a}_{\perp}$ represents the $x$ and $y$ components of a vector $\bm{a}$ in the body-fixed co-ordinate. Before we proceed, we introduce a dotted derivative of a vector $\bm{a}$ defined as the time derivative of the vector components in the body-fixed co-ordinate. This is related to the time derivative by $$\frac{d\bm{a}}{dt} = \dot{\bm{a}} + \bm{\omega} \times \bm{a}.$$ Note that the vertical vector $\bm{u}$ does not depend on time, thus we have $$\frac{d\bm{u}}{dt} = \dot{\bm{u}} + \bm{\omega} \times \bm{u} = \bm{0}. \label{eq:diff-u}$$ These conditions, i.e., the no-slip condition (\[eq:no-slip\]), the conditions of the contact point (\[eq:def-z\]) and (\[eq:def-xy\]), and the vertical vector condition (\[eq:diff-u\]), close the equations of motion (\[eq:em-1\]) and (\[eq:em-2\]). Following Garcia and Hubbard [@GarciaHubbard1988], we describe the rattleback dynamics by $\bm{u}$ and $\bm{\omega}$. The evolution of $\bm{\omega}$ is obtained as $$\begin{gathered} \hat{I} \dot{\bm{\omega}} - M\bm{r} \times (\bm{r} \times \dot{\bm{\omega}}) = - \bm{\omega} \times (\hat{I}\bm{\omega}) \\ + M\bm{r}\times(\dot{\bm{r}}\times \bm{\omega} + \bm{\omega}\times (\bm{r} \times \bm{\omega})) + Mg\bm{r}\times\bm{u} \label{eq:diff-omega}\end{gathered}$$ by eliminating the contact force $\bm{F}$ from the equations of motion (\[eq:em-1\]) and (\[eq:em-2\]), and using the no-slip condition (\[eq:no-slip\]). The state variables $\bm{u}$ and $\bm{\omega}$ can be determined by Eqs. (\[eq:diff-u\]) and (\[eq:diff-omega\]) with the contact point conditions (\[eq:def-z\]) and (\[eq:def-xy\]). ![(color online) \[fig:coordinate\]A body-fixed co-ordinate viewed from below. The dashed lines indicate the principal directions of curvature, rotated by $\xi$ from the principal axes of inertia (the $x$-$y$ axes).](2r.pdf){width="8cm"} The rattleback is characterized by the inertial parameters $M$, $I_{xx}$, $I_{yy}$, $I_{zz}$, the geometrical parameters $\theta$, $\phi$, $a$, and the skew angle $\xi$. For the stability of the rattleback, both of the dimensionless curvatures $\theta$ and $\phi$ should be smaller than $1$; without loss of generality, we assume $$0 < \phi < \theta < 1,$$ then, it is enough to consider $$-\frac{\pi}{2} < \xi < 0,$$ for the range of the skew angle $\xi$. The positive $\xi$ case can be obtained by the reflection with respect to the $x$-$z$ plane. At this stage, we introduce the dimensionless inertial parameters $\alpha$, $\beta$, and $\gamma$ for later use after Bondi [@Bondi1986] as $$\alpha \equiv \frac{I_{xx}}{Ma^{2}}+1, \ \beta \equiv \frac{I_{yy}}{Ma^{2}}+1, \ \gamma \equiv \frac{I_{zz}}{Ma^2},\label{eq:def-abg}$$ which are dimensionless inertial moments around the contact point C. Note that $$\alpha > \beta > 1,$$ because we have assumed $I_{xx} > I_{yy}$. Small amplitude approximation of oscillations under $\omega_{z}=0$ {#subsec:linearization} ------------------------------------------------------------------ We consider the oscillation modes in the case of no spinning $\omega_{z} = 0$ in the small amplitude approximation, namely, in the linear approximation in $|\omega_{x}|,\,|\omega_{y}|\ll\sqrt{g/a}$, which leads to $|x|,\,|y| \ll a$, $|u_{x}|,\, |u_{y}| \ll 1 $, and $u_{z} \approx -1$. In this regime, the $x$ and $y$ components of Eq. (\[eq:diff-u\]) can be linearized as $$\dot{\bm{u}}_{\perp} \approx \hat{\varepsilon}\,\bm{\omega}_{\perp}, \quad \hat{\varepsilon} \equiv \begin{pmatrix} 0, & 1 \\ -1, & 0 \\ \end{pmatrix} = \hat{R}(-\pi/2).\label{eq:lin-u}$$ By using Eq. (\[eq:def-xy\]) with $u_{z} \approx -1$, Eq. (\[eq:diff-omega\]) can be linearized as $$\begin{aligned} \hat{J}\,\dot{\bm{\omega}}_{\perp} &\approx \frac{g}{a^2} (\bm{r}\times\bm{u})_{\perp}\notag\\ &= -\frac{g}{a}\hat{\varepsilon}\,[-\hat{R}(\xi)\hat{\Theta}^{-1}\hat{R}^{-1}(\xi) +1 ] \bm{u}_{\perp},\label{eq:lin-omg1}\end{aligned}$$ with the inertial matrix $$\hat{J} \equiv \begin{pmatrix} \alpha, & 0 \\ 0, & \beta \\ \end{pmatrix}.$$ From the linearized equations (\[eq:lin-u\]) and (\[eq:lin-omg1\]), we obtain $$\hat{J}\ddot{\bm{\omega}}_{\perp}= - \frac{g}{a}(\hat{\Gamma} -1) \bm{\omega}_{\perp},\label{eq:lin-omg2}$$ where $$\hat{\Gamma} \equiv \hat{R}(\xi +\pi/2) \hat{\Theta}^{-1}\hat{R}^{-1}(\xi+\pi/2).$$ At this point, it is convenient to introduce the bra-ket notation for the row and column vector of $\bm{\omega}_{\perp}$ as $\bra{\omega_{\perp}}$ and $\ket{\omega_{\perp}}$, respectively. With this notation, Eq. (\[eq:lin-omg2\]) can be put in the form of $$\begin{aligned} \ket{\ddot{\tilde{\omega}}_{\perp}}= -\hat{H}\ket{\tilde{\omega}_{\perp}}, \label{eq:lin-omg3}\end{aligned}$$ with $$\begin{aligned} \ket{\tilde{\omega}_{\perp}} \equiv \hat{J}^{1/2}\ket{\omega_{\perp}}, \quad \hat{H} \equiv \frac{g}{a}\hat{J}^{-1/2}(\hat{\Gamma} -1 )\hat{J}^{-1/2}, \label{eq:lin-omg4}\end{aligned}$$ where $\hat{H}$ is symmetric. The eigenvalue equation $$\begin{aligned} \hat{H} \ket{\tilde{\omega}_{j}}= \omega_{j}^2 \ket{\tilde{\omega}_{j}} \label{eq:def-omgpr}\end{aligned}$$ determines the two oscillation modes with $j=p$ or $r$, whose frequencies are given by $$\omega_{p,r}^2 = \frac{1}{2}\left[(H_{11}+H_{22}) \pm \sqrt{(H_{11}-H_{22})^2 + 4H_{12}^2}\right] \label{eq:def-omega_pr}$$ with $$\omega_{p} \ge \omega_{r}. \label{ineq:omega_pr}$$ Here, $H_{ij}$ denotes the $ij$ component of $\hat H$. The orthogonal condition for the eigenvectors $\ket{\tilde{\omega}_{p}}$ and $\ket{\tilde{\omega}_{r}}$ can be written using $\hat{\varepsilon}$ as $$\begin{aligned} \ket{\tilde{\omega}_{p}} &= \hat{\varepsilon} \ket{\tilde{\omega}_{r}},\quad \ket{\tilde{\omega}_{r}} = -\hat{\varepsilon} \ket{\tilde{\omega}_{p}}, \\ \bra{\tilde{\omega}_{r}} &= \bra{\tilde{\omega}_{p}}\hat{\varepsilon}, \quad \bra{\tilde{\omega}_{p}} = -\bra{\tilde{\omega}_{r}}\hat{\varepsilon}. \end{aligned}$$ In the case of zero skew angle, $\xi=0$, we have $$\begin{aligned} \omega_p^2 &=\left({g\over a}\right){1/\phi-1\over\alpha}\equiv \omega_{p0}^2, \label{def:omega_p0} \\ \omega_r^2 &=\left({g\over a}\right){1/\theta-1\over\beta}\equiv \omega_{r0}^2, \label{def:omega_r0}\end{aligned}$$ and the eigenvectors $\ket{\omega_{p}}$ and $\ket{\omega_{r}}$ are parallel to the $x$ and the $y$ axes, thus these modes correspond to the pitching and the rolling oscillations, respectively. This correspondence holds for $|\xi|\ll 1$ and $\omega_{p0}>\omega_{r0}$ as for a typical rattleback parameter, the case we will discuss mainly in the following [^1]. Garcia and Hubbard’s theory for the time for reversal {#subsec:GHtheory} ----------------------------------------------------- Based on our formalism, it is quite straightforward to derive Garcia and Hubbard’s formula for the reversal time of rotation. ### Asymmetric torque coefficients Due to the skewness, the pitching and the rolling are coupled with the spinning motion. We examine this coupling in the case of $\omega_{z} = 0$ by estimating the averaged torques around the vertical axis caused by the pitching and the rolling oscillations. From Eqs. (\[eq:em-1\]) and (\[eq:em-2\]) and the no-slip condition Eq. (\[eq:no-slip\]), the torque around $\bm{u}$ is given by $$\begin{aligned} T &\equiv \bm{u}\cdot(\bm{r} \times \bm{F}) \approx - Ma^2 [\dot{\bm{\omega}}_{\perp}\cdot\hat{\varepsilon}(\hat{\Gamma} - 1)\hat{\varepsilon} \,\bm{u}_{\perp}\,],\label{eq:ave-torque}\end{aligned}$$ within the linear approximation in $\bm{\omega}_{\perp}$, $\bm{u}_{\perp}$, and $\bm{r}_{\perp}$ discussed in Sec. \[subsec:linearization\]. We define the *asymmetric torque coefficients* $K_{p} $ and $K_{r}$ for each mode by $$-K_{p} \equiv \frac{\overline{T}_{p}}{\overline{E}_{p}}, \qquad K_{r} \equiv \frac{\overline{T}_{r}}{\overline{E}_{r}}, \label{eq:def-kpr}$$ where $\overline{T}_{j}\ (j=p\text{~or~}r)$ is the averaged torque over the oscillation period generated by each mode, and $\overline{E}_{j}$ is the corresponding averaged oscillation energy which can be estimated within the linear approximation as $$\begin{aligned} \overline{E} &\approx Ma^2 (\alpha \overline{\omega_{x}^2} + \beta \overline{\omega_{y}^2} ). \label{eq:ave-ene}\end{aligned}$$ The minus sign for the definition of $K_{p}$ is inserted in order that both $K_{p}$ and $K_{r}$ should be positive for typical rattleback parameters as can be seen below. Note that the asymmetric torque coefficients are dimensionless. From Eqs. (\[eq:ave-torque\]) and (\[eq:ave-ene\]), $-K_{p}$ is given by $$\begin{aligned} -K_{p} &= \frac{\braket{\omega_{p}|\,\hat{\varepsilon}(\hat{\Gamma} - 1)\hat{\varepsilon}\hat{\varepsilon}\,|\omega_{p}}}{\braket{ \omega_{p}|\hat{J}|\omega_{p}}}\notag\\ &= -\frac{(a/g)\braket{\tilde{\omega}_{p}|\,\hat{J}^{-1/2}\hat{\varepsilon}\hat{J}^{1/2}\hat{H}\,|\tilde{\omega}_{p}}}{\braket{\tilde{\omega}_{p}|\tilde{\omega}_{p}}}\label{eq:kp-1}\\ &= - \omega_{p}^2 \,\frac{(a/g)\braket{\tilde{\omega}_{p}|\,\hat{J}^{-1/2}\hat{\varepsilon}\hat{J}^{1/2}\,|\tilde{\omega}_{p}}}{\braket{\tilde{\omega}_{p}|\tilde{\omega}_{p}}}. \label{eq:kp-2}\end{aligned}$$ In the same way, $K_{r}$ is given by $$\begin{aligned} K_{r} &= -\frac{(a/g)\braket{\tilde{\omega}_{r}|\hat{J}^{-1/2}\hat{\varepsilon}\hat{J}^{1/2}\hat{H}|\tilde{\omega}_{r}}}{\braket{\tilde{\omega}_{r}|\tilde{\omega}_{r}}}\label{eq:kr-1}\\ &= \omega_{r}^2 \,\frac{(a/g)\braket{\tilde{\omega}_{p}|(\hat{J}^{-1/2}\hat{\varepsilon}\hat{J}^{1/2})^{\dagger}|\tilde{\omega}_{p}}}{\braket{\tilde{\omega}_{p}|\tilde{\omega}_{p}}}. \label{eq:kr-2}\end{aligned}$$ Equations (\[eq:kp-1\])–(\[eq:kr-2\]) yield simple relations for $K_{p}$ and $K_{r}$ as $$\begin{aligned} \frac{K_{p}}{K_{r}} = \frac{\omega_{p}^2}{\omega_{r}^{2}} \label{eq:k-rat}\end{aligned}$$ and $$\begin{aligned} K_{p} - K_{r}&= \frac{(a/g)}{\braket{\tilde{\omega}_{p}|\tilde{\omega}_{p}}} \mathrm{Tr}\left[\hat{J}^{-1/2}\hat{\varepsilon}\hat{J}^{-1/2}\hat{H}\right]\notag\\ & = -\frac{1}{2}\sin(2\xi)\left(\frac{1}{\beta} - \frac{1}{\alpha}\right)\left(\frac{1}{\phi} - \frac{1}{\theta}\right). \label{eq:k-diff}\end{aligned}$$ Equations (\[eq:k-rat\]) and (\[eq:k-diff\]) are enough to determine $$\begin{aligned} K_{p} = -\frac{1}{2}\sin(2\xi)\left(\frac{1}{\beta} - \frac{1}{\alpha}\right)\left(\frac{1}{\phi} - \frac{1}{\theta}\right) \frac{\omega_{p}^2}{\omega_{p}^2 - \omega_{r}^2},\label{eq:Kp}\\ K_{r} = -\frac{1}{2}\sin(2\xi)\left(\frac{1}{\beta} - \frac{1}{\alpha}\right)\left(\frac{1}{\phi} - \frac{1}{\theta}\right) \frac{\omega_{r}^2}{\omega_{p}^2 - \omega_{r}^2}\label{eq:Kr}.\end{aligned}$$ Note that Eqs. (\[eq:Kp\]) and (\[eq:Kr\]) are consistent with the three requirements of rattlebacks: $\xi \neq 0$, $\alpha \neq \beta$, and $\theta \neq \phi$. Equations (\[eq:Kp\]) and (\[eq:Kr\]) are shown to be equivalent to the corresponding expressions Eq. (42a,b) in Garcia and Hubbard [@GarciaHubbard1988] although their expressions look quite involved. These results also show that $$K_{p}K_{r} > 0 \quad \text{and hence} \quad \overline{T}_{p}\overline{T}_{r} <0,$$ namely, the torques generated by the pitching and the rolling always have opposite signs to each other. ### Typical rattleback parameters Typical rattleback parameters fall in the region that satisfies the following two conditions: (1) the skew angle is small, $$|\xi| \ll 1,$$ and (2) the pitch frequency is higher than the roll frequency. Under these conditions, the modes $p$ and $r$ of Eq. (\[eq:def-omgpr\]) correspond to the pitching and the rolling oscillations respectively, and $$\omega_{p}^2 \approx \omega_{p0}^2,\qquad \omega_{r}^2 \approx \omega_{r0}^2 \label{eq:omg_pr-approx-omgpr0} $$in accord with the inequality (\[ineq:omega\_pr\]) [@Note1]. From Eqs. (\[eq:def-kpr\]), (\[eq:Kp\]), and (\[eq:Kr\]), the signs of the asymmetric torque coefficients and the averaged torques for typical rattlebacks are given by $$K_{p}>0 \quad \text{and}\quad K_{r}>0, \label{eq:sign-typ-k}$$ and $$\overline{T}_{p}<0 \quad \text{and}\quad \overline{T}_{r}>0,$$ by noting $\xi<0$, $\alpha > \beta$, $\theta >\phi$. The fact that $\omega_{p0}>\omega_{r0}$ for a typical rattleback means that the shape factor, $1/\phi-1$ or $1/\theta-1$, contributes much more than the inertial factor, $1/\alpha$ or $1/\beta$, in Eqs. (\[def:omega\_p0\]) and (\[def:omega\_r0\]) although these two factors compete, i.e. $1/\phi-1>1/\theta-1$ and $1/\alpha<1/\beta$. This is a typical situation because the two curvatures of usual rattlebacks are markedly different, i.e., $\phi \ll \theta < 1$ as can be seen in Fig. \[fig:notation\](c). Moreover, we can show that the pitch frequency is always higher for an ellipsoid with a uniform mass density whose surface is given by $x^2/c^2 + y^2/b^2+ z^2/a^2 = 1\ (b^2 > c^2 > a^2)$. This also holds for a semi-ellipsoid for $b^2 > c^2 > (5/8)a^2$, where the co-ordinate system is the same as the ellipsoid. ### Time for reversal Now we study the time evolution of the *spin* $n$ defined as the vertical component of the angular velocity $$n \equiv \bm{u}\cdot\bm{\omega},$$ assuming that the expressions for the asymmetric torque coefficients, $K_p$ and $K_r$, obtained above are valid even when $\omega_{z}\ne 0$. We consider the quantities $\overline{n}$, $\overline{E}_p$, and $\overline{E}_r$, averaged over the time scale much longer than the oscillation periods, yet much shorter than the time scale for spin change. Then, these averaged quantities should follow the following evolution equations: $$\begin{aligned} I_{\textrm{eff}} \frac{d\overline{n}(t)}{dt} &= - K_{p} \overline{E}_{p}(t) + K_{r} \overline{E}_{r}(t) , \label{eq:diff-ne-1}\\ \frac{d \overline{E}_{p}(t)}{dt} & = K_{p} \overline{n}(t) \overline{E}_{p}(t),\label{eq:diff-ne-2}\\ \frac{d \overline{E}_{r}(t)}{dt} & = - K_{r} \overline{n}(t) \overline{E}_{r}(t) \label{eq:diff-ne-3}.\end{aligned}$$ Here, $I_{\textrm{eff}}$ is the effective moment of inertia around $\bm{u}$ under the existence of the oscillations, and is assumed to be constant; it should be close to $I_{zz}$. As can be seen easily, the total energy $E_{\textrm{tot}}$ defined by $$E_{\textrm{tot}} \equiv \frac{1}{2} I_{\textrm{eff}} \overline{n}(t)^2 + \overline{E}_{p}(t) + \overline E_{r}(t)$$ is conserved. It can be seen that there is another invariant, $$C_{I} \equiv \frac{1}{K_{p}}\ln \overline{E}_{p} + \frac{1}{K_{r}}\ln \overline{E}_{r},\label{eq:casimir}$$ which has been discussed in connection with a Casimir invariant [@MoffattTokieda2008; @Yoshida2016]. With these two conservatives, general solutions of the three-variable system (\[eq:diff-ne-1\])–(\[eq:diff-ne-3\]) should be periodic. Let us consider the case where the spin is positive at $t=0$ and the sum of the oscillation energies are small compared to the spinning energy: $$\overline{n}(0) \equiv n_i >0, \quad \overline{E}_{p}(0) + \overline{E}_{r}(0) \ll \frac{1}{2} I_{\textrm{eff}} n_{i}^2.$$ For a typical rattleback, the pitching develops and the rolling decays as long as $\overline n>0$ as can be seen from Eqs. (\[eq:sign-typ-k\]), (\[eq:diff-ne-2\]) and (\[eq:diff-ne-3\]). Thus the rolling is irrelevant and can be ignored, i.e., $\overline E_r(t) = 0$, to estimate the time for reversal. Then we can derive the equation $$\frac{d\overline{n}(t)}{dt} = -\frac{K_{p}}{2}\left(n_0^2- \overline{n}(t)^2\right) \label{eq:diff-n},$$ where the constant $n_0>0$ is defined by $$\frac{1}{2}I_{\textrm{eff}} n_{0}^2 \equiv E_{\textrm{tot}}.$$ This can be easily solved as $$\overline{n}(t) = n_{0}\frac{(n_{0} + n_{i})\exp(-n_{0}K_{p}t) - (n_{0} -n_{i}) }{(n_{0} + n_{i})\exp(-n_{0}K_{p}t) + (n_{0} -n_{i})}, \label{eq:gh-solution-p}$$ and we obtain the time for reversal $t_{rGH+}$ for the $n_{i} > 0$ case as $$t_{rGH+} = \frac{1}{n_0 K_p} \ln\left(\frac{n_{0}+n_{i}}{n_0 - n_i}\right), \label{eq:trgh-p}$$ by just setting $\overline{n}=0$ in Eq. (\[eq:gh-solution-p\]). Similarly, in the case of $n_i<0$, only the rolling develops and the pitching is irrelevant, thus we obtain $\overline{n}(t)$ and the time for reversal $t_{rGH-}$ as $$\overline{n}(t) = -n_{0}\frac{(n_{0} + |n_{i}|)\exp(-n_{0}K_{r}t) - (n_{0} - |n_{i}|) }{(n_{0} + |n_{i}|)\exp(-n_{0}K_{r}t) + (n_{0} - |n_{i}|)} \label{eq:gh-solution-m}$$ and $$t_{rGH-} =\frac{1}{n_0 K_r} \ln\left(\frac{n_0+|n_i|}{n_0 - |n_i|} \right). \label{eq:trgh-m}$$ Equations (\[eq:trgh-p\]) and (\[eq:trgh-m\]) are Garcia-Hubbard formulas for the times for reversal [@GarciaHubbard1988]. From the expressions of $K_{p}$ and $K_{r}$ given by Eqs. (\[eq:Kp\]) and (\[eq:Kr\]), we immediately notice that (1) the time for reversal is inversely proportional to the skew angle $\xi$ in the small skewness regime, and (2) the ratio of the time for reversal $t_{rGH-}/t_{rGH+}$ is simply given by the squared ratio of the pitch frequency to the roll frequency $\omega_{p}^2 / \omega_{r}^2$, provided initial values $n_{0}$ and $n_{i}$ are the same except their signs. For a typical rattleback, $\omega_{p}^2 \gg \omega_{r}^2$, thus $t_{rGH+} \ll t_{rGH-}$, i.e., the time for reversal is much shorter in the case of $n_{i}>0$ than in the case of $n_{i}<0$. Thus we call the spin direction of $n_{i} > 0$ the *unsteady direction* [@GarciaHubbard1988], and that of $n_i <0$ the *steady direction*. In the small skewness regime, this ratio of the squared frequencies is estimated as $$\begin{aligned} \frac{\omega_{p}^2}{\omega_{r}^2} \approx \frac{\omega_{p0}^2}{\omega_{r0}^2} = \frac{\beta}{\alpha}\frac{1/\phi - 1}{1/\theta -1}. \label{eq:ratomg}\end{aligned}$$ This becomes especially large as $\theta$ approaches $1$ or as $\phi$ approaches 0, namely, as the smaller radius of principal curvature approaches $a$, or as the larger radius of principal curvature becomes much larger than $a$. We remark that both of the inertial parameters $\alpha$ and $\beta$ are larger than $1$ by definition Eq. (\[eq:def-abg\]), and cannot be arbitrarily large for a typical rattleback. Let us consider these two limiting cases: $\phi\to 0$ and $\theta \to 1$ with $|\xi| \ll 1$. In the case of $\phi \to 0$, $$K_{p} \to \infty,\quad K_{r} \to (-\xi) \left(\frac{1}{\beta} - \frac{1}{\alpha}\right)\frac{\alpha}{\beta}\left(\frac{1}{\theta} - 1\right),$$ thus the time for reversal $t_{rGH-}$ remains constant while $t_{rGH+}$ approaches $0$. In the case of $\theta \to 1$, $$K_{p} \to (-\xi) \left(\frac{1}{\beta} - \frac{1}{\alpha}\right)\left(\frac{1}{\phi} - 1\right),\quad K_{r} \to 0,$$ and thus $t_{rGH+}$ remains constant while $t_{rGH-}$ diverges to infinity, i.e., the negative spin rotation never reverses. Simulation {#sec:simulation} ========== We perform numerical simulations for the times for the first spin reversal and compare them with Garcia-Hubbard formulas (\[eq:trgh-p\]) and (\[eq:trgh-m\]). Shell-dumbbell model -------------------- To consider a rattleback whose inertial and geometrical parameters can be set separately, we construct a simple model of the rattleback, or the *shell-dumbbell model*, which consists of a light shell and two dumbbells: the light shell defines the shape of the lower part of the rattleback and the dumbbells represent the masses and the moments of inertia. The shell is a paraboloid given by Eq. (\[eq:def-z\]). The dumbbells consist of couples of weights, $m_{x}/2$ and $m_{y}/2$, fixed at $(\pm r_{x},0,0)$ and $(0,\pm r_{y},0)$ in the body-fixed co-ordinate, respectively \[Fig. \[fig:notation\](c)\]. Then the total mass is $$M = m_{x} + m_{y}$$ and the inertia tensor is diagonal with its principal moments $$\begin{aligned} I_{xx} &= m_{y}r_{y}^{2},\quad I_{yy} = m_{x}r_{x}^{2}, \\ I_{zz} &= m_{y}r_{y}^{2} + m_{x}r_{x}^{2}.\end{aligned}$$ Note that the simple relation $$I_{zz} = I_{xx} + I_{yy}$$ holds for the shell-dumbbell model. We define $$f_{sd} \equiv I_{yy}/I_{zz},$$ then the dimensionless parameters $\alpha,\,\beta,$ and $\gamma$ defined by Eq. (\[eq:def-abg\]) are given by, $$\gamma = I_{zz}/Ma^{2}, \, \alpha = (1-f_{sd})\gamma + 1, \, \beta = f_{sd}\gamma + 1.$$ The parameter $f_{sd}$ satisfies $0<f_{sd}<0.5$, since we have assumed $\alpha > \beta$. The shell-dumbbell model makes it easier to visualize an actual object represented by the model with a set of parameters, and is used in the following simulations for determining the parameter ranges. Methods {#subsec:method} ------- $\gamma$ $f_{sd}$ $\alpha,\ \beta$ $\theta$ $\phi$ $-\xi$ (deg) ---- ---------- --------------- ------------------ -------------- -------------- -------------- -- GH $12.28$ — 13.04, 1.522 0.6429 0.0360 1.72 SD $[5,15]$ $[0.05,0.15]$ — $[0.6,0.95]$ \[0.01,0.1\] (0,6\]  (a) Cumulative distributions of the pitch and the roll frequencies for the parameter set SD in Table \[tab:parameters\]; $\omega_{p} \text{~and~} \omega_{r}$ of Eq. (\[eq:def-omgpr\]) and their zeroth order approximation $\omega_{p0}$ and $\omega_{r0}$ by Eqs. (\[def:omega\_p0\]) and (\[def:omega\_r0\])](3r.pdf){width="7.5cm"} are shown. The inset shows the cumulative distribution of $\omega_{p}/\omega_{r}$. The number of samples is $10^{6}$.  (a) Cumulative distributions of the asymmetric torque coefficients $K_{p}$ and $K_{r}$ for SD (Table \[tab:parameters\]). The number of samples is ${10}^{5}$. (b) A 2D color plot for the distribution of ($K_{p}$,$K_{r}$). The color code shown is in the logarithmic scale for the relative frequency $P(K_{p},\,K_{r})$, i.e., $-9 \le \log_{10}P(K_{p},\,K_{r}) \le 0$. The number of samples is $10^{8}$.](4.pdf){width="7.5cm"} The equations of motion (\[eq:diff-u\]) and (\[eq:diff-omega\]) with the contact point conditions (\[eq:def-z\]) and (\[eq:def-xy\]) are numerically integrated by the fourth-order Runge-Kutta method with an initial condition $\bm{\omega}(0)$ and $\bm{u}(0)$. In the simulations, we take $$\bm{u}(0) = (0,0,-1)^{t} \label{eq:sim-ic-1}$$ and specify $\bm{\omega}(0)$ as $$\bm{\omega}(0) = (|\omega_{xy0}|\cos\psi,\ |\omega_{xy0}|\sin\psi,\ -n_{i}) \label{eq:sim-ic-2}$$ in terms of $|\omega_{xy0}|$, $\psi$, and $n_{i}$. According to the simplified dynamics (\[eq:diff-ne-1\])–(\[eq:diff-ne-3\]), the irrelevant mode of oscillation does not affect the dynamics sensitively as long as the relevant mode exists and the initial spin energy is much larger than the initial oscillation energy. Thus we choose $\ket{\omega(0)}= (\omega_{x0}, \omega_{y0})^{t}$ in the direction of the relevant eigenmode, $$\psi = \psi_{p} \text{ for } n_{i}>0,\quad\text{and}\quad\psi = \psi_{r} \text{ for } n_{i}<0, \label{eq:sim-ic-3}$$ where $\psi_{p}$ and $\psi_{r}$ are the angles of the eigenvectors $\ket{\omega_{p}}$ and $\ket{\omega_{r}}$ from the $x$-axis, respectively. Numerical results are presented in the unit system where $M$, $a$, and $$\tilde{t} \equiv 1/\tilde{\omega} \equiv \sqrt{a/g}$$ as units of mass, length, and time. The size of the time step for the numerical integration is taken to be $0.002\,\tilde{t}$. In numerics, we determine the time for reversal $t_{r}$ by the time at which $n=\bm{\omega}\cdot\bm{u}$ becomes zero for the first time, and they are compared with Garcia-Hubbard formulas (\[eq:trgh-p\]) and (\[eq:trgh-m\]); $n_0$ is determined as $$\frac{\gamma n_{0}^{2}}{2} = \frac{1}{2}( \alpha\omega_{x0}^2 + \beta\omega_{y0}^2 + \gamma\omega_{z0}^2), \label{eq:def-n0}$$ assuming $I_{\mathrm{eff}} = I_{zz}$ at $t=0$. Here the potential energy $U(\bm{u})$ is set to zero where $\bm{u}(0) = (0,0,-1)^{t}$. The parameters used in the simulations are listed in Table \[tab:parameters\]. For the parameter set SD, the ranges are shown. When numerical results are plotted against $K_{p}$ or $K_{r}$, given by Eqs. (\[eq:Kp\]) or (\[eq:Kr\]), respectively, sets of parameters are chosen randomly from the ranges until resulting $K_{p}$ or $K_{r}$ falls within the range of $\pm0.1\%$ of a target value. The ranges of SD are chosen to meet the following two conditions: (1) $0<\phi \ll \theta <1$, $\beta < \alpha$, and $|\xi| \ll 1$ and (2) the pitch frequency should be higher than the roll frequency. As argued in Sec. \[subsec:GHtheory\], usual rattlebacks such as one in Fig. \[fig:notation\](a) satisfy these two conditions. Figure \[fig:omega-dist\] shows the cumulative distributions for the eigenfrequencies $\omega_{p}$ and $\omega_{r}$, and their approximate expressions $\omega_{p0}$ and $\omega_{r0}$ for the parameter set SD; it shows $(\omega_{p}/\omega_{r}) > 1.3$ in accordance with the condition (2). The parameter set GH gives $K_{p} = 0.553$ and $K_{r}=0.0967$, and the distributions of $K_{p}$ and $K_{r}$ for SD are shown in Fig. \[fig:k-dist\], where one can see $K_{p}\gg K_{r}$. From Eq. (\[eq:k-rat\]), this corresponds to $\omega_{p}^2 \gg \omega_{r}^2$, i.e., the pitch frequency is significantly faster than the roll frequency. Consequently, the time for reversal is much shorter for the unsteady direction $n_{i}>0$, where the pitching is induced, than for the steady direction $n_{i}<0$, where the rolling is induced. We denote the time for reversal for the unsteady direction as $t_{ru}$ and that for the steady direction as $t_{rs}$ when we consider a specific spinning direction. Results ------- ### General behavior for the parameter set GH ![\[fig:gh-t-n\] A typical spin evolution and the corresponding $\omega_{x}$ and $\omega_{y}$ for GH (Table \[tab:parameters\]). (a) The case of the initial spin in the unsteady direction. The initial condition is specified by Eqs. (\[eq:sim-ic-1\])–(\[eq:sim-ic-3\]) with $n_{i} = 0.1\,\tilde{\omega}$ and $|\omega_{xy0}|=0.01\,\tilde{\omega}$. (b) The case of the initial spin in the steady direction with $n_{i} = -0.1\,\tilde{\omega}$ and $|\omega_{xy0}|=0.01\,\tilde{\omega}$. The dashed lines in (a-1) and (b-1) show Garcia and Hubbard’s solution $\overline{n}(t)$ given by Eqs. (\[eq:gh-solution-p\]) and (\[eq:gh-solution-m\]), respectively.](5r.pdf){width="8cm"} In Fig. \[fig:gh-t-n\] we show a typical simulation result of the time evolution of the spin $n(t)$ along with the angular velocities $\omega_{x}(t)$ and $\omega_{y}(t)$ for the parameter set GH (Table \[tab:parameters\]) in the case of the unsteady direction $n_i > 0$ (a), and the steady direction $n_i < 0$ (b). Figure \[fig:gh-t-n\](a-1) shows that the spin $n$ changes its sign from positive to negative at $t_{ru}\approx 112\,\tilde{t}$, and Fig. \[fig:gh-t-n\](b-1) shows the spin $n$ changes its sign from negative to positive at $t_{rs} \approx 810\,\tilde{t}$. Garcia and Hubbard’s solutions $\overline{n}(t)$ of Eqs. (\[eq:gh-solution-p\]) and (\[eq:gh-solution-m\]) are shown by the dashed lines in Figs. \[fig:gh-t-n\](a-1) and (b-1), respectively; they are in good agreement with the numerical simulations. The angular velocities $\omega_{x}$ and $\omega_{y}$ oscillate in much shorter time scale, and their amplitudes evolve differently depending on the spin direction. In the case of Fig. \[fig:gh-t-n\](a), where the positive initial spin reverses to negative, the amplitude of $\omega_{x}$ becomes large and reaches its maximum around $t_{ru}$; the amplitude of $\omega_{y}$ also becomes large around both sides of $t_{ru}$ but shows the local minimum at $t_{ru}$. Both $\omega_{x}$ and $\omega_{y}$ oscillate at the pitch frequency $\omega_{p} \approx 1.44\,\tilde{\omega}$. In the case of Fig. \[fig:gh-t-n\](b) where the negative spin reverses to positive, the situation is similar but the amplitude of $\omega_{y}$ reaches its maximum around $t_{rs}$, and $\omega_{x}$ and $\omega_{y}$ oscillate at the roll frequency $\omega_{r} \approx 0.602\,\tilde{\omega}$. These features can be understood based on the analysis in the previous section as follows. The positive spin induces the pitching, which is mainly represented by $\omega_{x}$ because the eigenvector of the pitching $\ket{\omega_{p}}$ is nearly parallel to the $x$ axis, i.e., $\psi_{p} \approx -17^{\circ}$. Likewise, the negative spin induces the rolling, mainly represented by $\omega_{y}$, because $\psi_{r} \approx 88^{\circ}$. The local minima of the amplitude for $\omega_{y}$ in Fig. \[fig:gh-t-n\](a-3), or $\omega_{x}$ in Fig. \[fig:gh-t-n\](b-2), at the times for reversal are tricky; it might mean that the eigenvector of the pitching (rolling) deviates more from the $x$ axis ($y$ axis) for $\omega_{z} \neq 0$ than that for $\omega_{z} = 0$; as a result, the pitching (rolling) mode has a larger projection on the $y$ axis ($x$ axis) for $\omega_{z} \neq 0$. Note that for given $|n_{i}|$, the maximum value of $\omega_{y}$ in Fig. \[fig:gh-t-n\](b-3) is larger than that of $\omega_{x}$ in (a-2). This is due to $\alpha \gg \beta$; the oscillation energy around zero spin for the both cases should be the same, which gives $\alpha \overline{{\omega}_{x}^2}\approx \beta \overline{{\omega}_{y}^2}$ thus $\sqrt{\overline{\omega_{x}^2}} < \sqrt{\overline{\omega_{y}^2}}$. ### Simulations with the parameter set SD We present detailed results of the simulations for the ranges of the parameters given by SD in Table \[tab:parameters\]. #### Unsteady initial spin direction $(n_{i}>0)$. In this case, the system behaves basically as we expect from the Garcia-Hubbard formula unless the initial spin or oscillation is too large. Figure \[fig:kptr\] shows the time for reversal $t_{ru}$ as a function of $K_{p}$ when spun in the unsteady direction. The results are plotted against $K_{p}$ by the procedure described in Sec. \[subsec:method\]. When the initial spin $n_{i}$ is $n_{i} \lesssim 0.2\,\tilde{\omega}$ with $|\omega_{xy0}|=0.001\tilde{\omega},\, 0.01\tilde{\omega}$, $t_{ru}$ is in good agreement with the Garcia-Hubbard formula $t_{rGH+}$ of Eq. (\[eq:trgh-p\]), i.e., almost inversely proportional to $K_{p}$ with small scatter around the average. For a given $n_i$, as the initial oscillation amplitude $|\omega_{xy0}|$ becomes large, the standard deviations of $t_{ru}$ become large, and the average of $t_{ru}$ deviates upward from the Garcia-Hubbard formula $t_{rGH+}$, which is derived with the small amplitude approximation of $\omega_{x}$ and $\omega_{y}$. For larger $n_i$, $t_{rGH+}$ also underestimates $t_{ru}$, as already noted by Garcia and Hubbard [@GarciaHubbard1988] for the parameter set GH. The underestimation can be also seen in Fig. \[fig:gh-t-n\](a-1), where one can see that Garcia and Hubbard’s solution $\overline{n}(t)$ of Eq. (\[eq:gh-solution-p\]) changes its sign earlier than the simulation. For $n_i \gtrsim 0.4\,\tilde{\omega}$, $t_{ru}$ deviates notably upward from the Garcia-Hubbard formula $t_{rGH+}$. As $n_i$ increases, the average of $t_{ru}$ increases and the standard deviations become large. Figure \[fig:kptr\](b) shows a typical spin evolution with $n_{i} = 0.5\,\tilde{\omega}$. The spin oscillates widely at the pitch frequency, which is qualitatively different from typical spin behaviors at small $n_{i}$ and from Garcia and Hubbard’s solution $\overline{n}(t)$ of Eq. (\[eq:gh-solution-p\]) as in Fig. \[fig:gh-t-n\](a-1). In this region, the Garcia-Hubbard formula is no longer valid. ![(color online) \[fig:kptr\] (a) Time for reversal of the unsteady direction $t_{ru}$ for the parameter set SD (Table \[tab:parameters\]) as a function of the asymmetric torque coefficient $K_{p}$ in the logarithmic scale. The error bars indicate one standard deviation of $1000$ samples for each data point. The solid lines are $t_{rGH+}$ given by Eq. (\[eq:trgh-p\]), calculated using the mean values of $n_{0}$. (b) A typical spin evolution with $n_{i} = 0.5\,\tilde{\omega},\ |\omega_{xy0}|=0.01\,\tilde{\omega}$. The parameter set GH is used.](6r.pdf){width="8.4cm"} #### Steady initial spin direction $(n_{i}<0)$. {width="16.8cm"} ![(color online) \[fig:krtr-2\] Time for reversal $t_{rs}$ for the steady direction as a function of $K_{r}$ in the logarithmic scale. Each data point represents the average with the standard deviation of Type R samples out of $1000$ simulations from the parameter set SD (Table \[tab:parameters\]).](8r.pdf){width="8.4cm"} Much more complicated phenomena are observed when spun in the steady direction. When the initial spin $|n_i|$ is small enough, the spin simply reverses as shown in Fig. \[fig:gh-t-n\](b-1). We call this simple reversal behavior Type R. For larger $|n_i|$, however, there appear some cases where the spin never reverses; in such cases there are two types of behaviors: steady spinning at $n_{ss}$ (Type SS), and spin wobbling around $n_{w}$ ($n_{ss}<n_{w}<0$, Type SW). For Type SS samples, $n_{ss}$ is slightly less than $n_{i}$, i.e., $n_{ss}<n_{i}<0$, because small initial rolling decays and its energy is converted to the spin energy. Typical spin evolutions of a Type SS sample and a Type SW sample are shown in Figs. \[fig:krtr\](b-1) and (b-2). Figure \[fig:krtr\](a) shows the $K_{r}$ dependence of the fractions of Types R, SS, and SW for various initial conditions given by $n_{i}$ and $|\omega_{xy0}|$. For each sample, we wait up to $t = 5t_{rGH-}$; the spin evolution is classified as Type R if it reverses. If it does not, the spin evolution is classified as Type SS if the initial rolling amplitude decays monotonously, and classified as Type SW if the spin $n$ starts wobbling by the time $5t_{rGH-}$. The other samples, in which the rolling grows slowly yet shows no visible spin change by the time $5t_{rGH-}$, are labeled “unclassified" in Fig. \[fig:krtr\]. Such samples may show spin reversal or spin wobbling if we take a much longer simulation time. Type SS appears for $|n_i|\gtrsim 0.3\,\tilde{\omega}$ and its fraction increases as $|n_i|$ increases. The fraction is larger for smaller $K_{r}$ and smaller $|\omega_{xy0}|$, i.e., $|\omega_{xy0}|=0.001\,\tilde{\omega}$. Type SW appears for $|n_i| \gtrsim 0.1\,\tilde{\omega}$ and its fraction is also larger for smaller $K_{r}$, but stays around $0.2$ for $|n_i| \gtrsim 0.4\,\tilde{\omega}$. Figure \[fig:krtr-2\] shows the $K_{r}$ dependence of $t_{rs}$ only for the samples of Type R, which shows a spin reversal behavior. For small $|n_{i}| \lesssim 0.2\,\tilde{\omega}$ with $|\omega_{xy0}|=0.01\,\tilde{\omega}, 0.001\,\tilde{\omega}$, $t_{rs}$ is in good agreement with Garcia-Hubbard formula $t_{rGH-}$ of Eq. (\[eq:trgh-m\]), and the average of $t_{ru}$ is almost inversely proportional to $K_{r}$. As in the case of the unsteady direction, the standard deviations of $t_{rs}$ become large, and the average $t_{rs}$ deviates downward from $t_{rGH-}$ as initial oscillation amplitude $|\omega_{xy0}|$ becomes large. Note that $t_{rGH-}$ tends to overestimate $t_{rs}$, in contrast to the case of the unsteady direction, where $t_{rGH+}$ underestimates $t_{ru}$. This has also been noted by Garcia and Hubbard [@GarciaHubbard1988] for the parameter set GH, and can be seen by Garcia and Hubbard’s solution $\overline{n}(t)$ in Fig. \[fig:gh-t-n\](b-1). For $|n_{i}| \gtrsim 0.3\,\tilde{\omega}$, one may notice the standard deviations are large for $K_{r}\ll 0.1$. In these cases, we find that some samples appear to spin stably for quite a long time, i.e., several times of $t_{rGH-}$, and then abruptly starts to reverse its sign. During the time period $t<t_{rs}$, the rolling grows much more slowly than it should as predicted by the theory in Sec. \[sec:theory\]. Such samples make both the average and standard deviation large as Fig. \[fig:krtr-2\]. Next we consider the Type SS samples. There always exists a steady solution, $\bm{\omega}(0)=(0,0,\mathrm{const.})^{t}$ and $\bm{u}(0) = (0,0,-1)^{t}$, and Bondi [@Bondi1986] has shown that for the steady direction, this solution is linearly stable for $n < n_{c1}<0$, where $n_{c1} (<0)$ is given by $$n_{c1}^2 \equiv \frac{g}{a}\frac{-(1-\theta)(1-\phi)}{2-(\theta+\phi) - (\alpha + \beta - \gamma)(\theta + \phi - 2\theta\phi)}. \label{eq:def-nc1}$$ When the denominator of Eq. (\[eq:def-nc1\]) is positive, such a threshold does not actually exist, and the steady solution is always unstable. Note that $n_{c1}$ does not depend on $\xi$. In Fig. \[fig:krtr\], the filled triangles show the fraction of samples whose $|n_{c1}|$ is smaller than $|n_{i}|$, which should correspond with the ratio of Type SS. For $|\omega_{xy0}|=0.001\,\tilde{\omega}$, all samples whose $|n_{c1}|$ is smaller than $|n_{i}|$ actually show Type SS behaviors and vice versa. On the other hand, for $|\omega_{xy0}|=0.1\,\tilde{\omega}$, there are some samples whose $|n_{c1}|$ is smaller than $|n_{i}|$ yet do not show Type SS behavior; for $n_i = -0.3 \,\tilde{\omega}$, there are only several Type SS samples out of 8000 samples, which cannot be seen in Fig. \[fig:krtr\](a), and for $|n_i| \gtrsim 0.4\,\tilde{\omega}$, the fractions of Type SS for $|\omega_{xy0}|=0.1\,\tilde{\omega}$ are smaller than those for $|\omega_{xy0}|=0.001\,\tilde{\omega}$. This may be because $|\omega_{xy0}|=0.1\,\tilde{\omega}$ is not small perturbation, and the spin might have escaped from the basin of attractor of Type SS behavior. Last we consider the Type SW samples. The time when the spin starts to wobble roughly corresponds with $t_{rs}$ of Type R in Fig. \[fig:krtr-2\]; the center of wobbling $n_{w}$ and its amplitude vary from sample to sample. As in the case of Type R, there are some samples which start to wobble after several times of $t_{rGH-}$ where $K_{r} \ll 0.1$. Wobbling behaviors of such samples are similar to those which start wobbling around $t_{rGH-}$. We remark that there are two qualitatively different Type SW behaviors. When $|n_i| \lesssim 0.4\,\tilde{\omega}$, the spin of Type SW sample oscillates almost periodically. However, when $n_i = -0.5\,\tilde{\omega}$ and $K_{r} \ll 0.1$, we find some samples that show “chaotic" oscillations as an example shown in Fig \[fig:krtr\](b-3). Discussion {#sec:discussion} ========== In the present work, we study the minimal model for the rattleback dynamics, i.e., a spinning rigid body with a no-slip contact ignoring any form of dissipation. We have reduced the original dynamics to the simplified dynamics (\[eq:diff-ne-1\])–(\[eq:diff-ne-3\]) with the three variables. The assumptions and/or approximations used in the derivation are (1) the amplitudes of the oscillations are small, (2) the coupling between the spin and the oscillations does not depend on the spin, and (3) the time scale for the spin change is much longer than the oscillation periods. It is interesting to note that the last assumption is apparently analogous to that used in the derivation of an adiabatic invariant for some systems under slow change of an external parameter if the spin variable is regarded as a slow parameter. In the present case with this separation of time scales, the dynamics conserves the “Casimir invariant" $C_{I}$ of Eq. (\[eq:casimir\]). Our simplified dynamics can be compared with some previous works. Based on Bondi’s formulation [@Bondi1986], Case and Jalal obtained the growth rates $\delta_{x}$ and $\delta_{y}$ of the pitching and the rolling amplitudes around the $x$ and $y$ axes, respectively, at a small constant spin and small skewness [@Case2014]. Their results can be expressed as $$\delta_{x} = \frac{n}{2}K_{p},\label{eq:case-instab1}\quad \delta_{y} = -\frac{n}{2}K_{r},$$ using our notations. The factor $1/2$ comes from the choice of the variables; they chose the contact point co-ordinates, while we choose the oscillation energies, which are second order quantities of their variables. Moffatt and Tokieda [@MoffattTokieda2008] obtained equations for the oscillation amplitudes of pitching and rolling, $P$ and $R$, and the spinning $S$ for small spin and skewness as $$\frac{d}{d\tau} \begin{pmatrix} P \\ R \\ S\end{pmatrix} = \begin{pmatrix} R \\ \lambda P \\ 0\end{pmatrix}\times \begin{pmatrix} P \\ R \\ S\end{pmatrix} = \begin{pmatrix} \lambda PS \\ -RS \\ R^2 - \lambda P^2\end{pmatrix} , \label{eq:toki-prs}$$ where $\tau$ is rescaled time, and $\lambda$ is the squared ratio of the pitch frequency to the roll frequency. Equation (\[eq:toki-prs\]) is equivalent with Eqs. (\[eq:diff-ne-1\])–(\[eq:diff-ne-3\]); again the difference comes from choice of the variables. The mathematical structures of Eq. (\[eq:toki-prs\]) have been investigated recently in more detail by Yoshida *et al.* [@Yoshida2016] in connection with the Casimir invariant and chaotic behavior of the original dynamics. ![\[fig:periodic\] Three types of spin behaviors after the first reversal period in the small spin regime with $n_{i} = 0.1\tilde{\omega}$, $|\omega_{xy0}|=0.01\tilde{\omega}$. (a) A quasi-periodic behavior with the parameter set GH ($\theta = 0.6429$), (b) a chaotic behavior with $\theta = 0.82$, (c) a quasi-periodic behavior with a period shorter than the first one with $\theta = 0.9$. All the other parameters for (b) and (c) are the same as GH. The dashed lines show the spin evolutions for the corresponding simplified dynamics, where $\overline{E}_p(0) = Ma^2[\alpha (|\omega_{xy0}|\cos\psi_{p})^2 + \beta (|\omega_{xy0}|\sin\psi_{p})^2]/2$, $\overline{E}_r(0) = 3\times10^{-5}Ma^2\tilde{\omega}^2$, and $\overline{n}(0) = n_{i}$.](9r.pdf){width="7.5cm"} After the first round of spin reversals, our simplified dynamics (\[eq:diff-ne-1\])–(\[eq:diff-ne-3\]) repeats itself and shows periodic behavior as well as the dynamics studied by Moffatt and Tokieda Eq. (\[eq:toki-prs\]) because the system with only three variables has two conservatives, i.e., the total energy and the Casimir invariant. However, the Casimir invariant is an approximate one in the original dynamics, and invariant only under the approximations given at the beginning of this section. The Casimir “invariant” actually varies and the original system shows aperiodic behaviors. A few examples for longer time evolutions of spin $n(t)$ are given in Fig. \[fig:periodic\] for the system with the parameter set GH except for the curvature in the rolling direction $\theta=0.6429$ (a) for GH, $0.82$ (b), and $0.9$ (c) along with those by the corresponding simplified dynamics. The first example (a) almost shows a periodic spin reversal behavior as is expected by the simplified dynamics. It is, however, only quasi-periodic with fluctuating periodicity. The second example (b) does not show a periodic behavior; the initial spin reversal till $t/\tilde t\approx 100$ is nearly the same with (a), but after the time of the second spin reversal around $t/\tilde t\approx 3000$, it turns into chaotic, deviating from the simplified dynamics. The third example (c) may look similar to (a) but is peculiar; it shows a quasi-periodic behavior after the initial round of spin reversals, and its periodicity is much shorter than that by the simplified dynamics. The simplified dynamics seems to work reasonably well for the case of smaller $\theta$ in (a) but fails for larger $\theta$ close to $1$ in (b) and (c). This indicates that the approximations or assumptions used to derive the simplified dynamics are not valid for the larger curvature in the rolling direction $\theta$; as the radius of curvature $1/\theta$ becomes small and close to 1, i.e., the height of the center of mass, the restoration force for the rolling oscillation becomes weak. This should result in the rolling oscillation with larger amplitude and the slower frequency, thus the assumptions (1) and (3) given at the beginning of this section may not be good enough. The fact that the system shows a different behavior after the first round of spin reversals is reminiscent of the existence of attractors, which is normally prohibited in a conserving system by Liouville theorem. In the present system, however, the theorem is invalidated by the non-holonomic constraint due to the no-slip condition Eq. (\[eq:no-slip\]) [^2]. As mentioned already, the existence of strange attractors in an energy conserving system with a non-holonomic constraint has been studied by Borizov et al. [@Borisov2014], and chaotic behavior in the rattleback system has been discussed in connection with the Casimir invariant by Yoshida *et al.* [@Yoshida2016]. Summary and conclusion {#sec:conclusion} ====================== We have performed the theoretical analysis and numerical simulations on the minimal model of rattleback. By reformulating Garcia and Hubbard’s theory [@GarciaHubbard1988], we obtained the concise expressions for the asymmetric torque coefficients, Eqs. (\[eq:Kp\]) and (\[eq:Kr\]), gave the compact proof to the fact that the pitching and the rolling generate the torques with the opposite sign, and reduced the original dynamics to the three-variable dynamics by a physically transparent procedure. Our expressions for the asymmetric torque coefficients are equivalent to those by Garcia and Hubbard, but we explicitly elucidate that the ratio of the two coefficient for the pitching and the rolling oscillation is proportional to the squared ratio of those frequencies. Since the pitching frequency is significantly higher than that of the rolling for a typical rattleback, the time for reversal to one spin direction (or unsteady direction) is much shorter than that to the other direction (or steady direction); the spin reversal for the latter direction is not usually observed in a real rattleback due to dissipation. The simulations on the original dynamics for various parameter sets demonstrate that Garcia-Hubbard formulas for the first spin reversal time (\[eq:trgh-p\]) and (\[eq:trgh-m\]) are good in the case of small initial spin and small oscillation for both the unsteady and the steady directions. The deviation from the formula is especially large for the steady direction in the fast initial spin and small $K_r$ regime, where the rattleback may not reverse and shows a variety of dynamics, that includes steady spinning, periodic and chaotic wobbling. In conclusion, the rattleback is simple but shows fascinatingly rich dynamics, and keeps attracting physicists’ attention. [29]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , ****, (). , , , ****, (). , , , , ****, (). , , , , ****, (). , ****, (). , ****, (). , Master’s thesis, , (). , ****, (). , ****, (). , ****, (). , , , in **, paper DETC97/VIB-4103 (ASME, ). , ****, (). , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , , , ****, (). , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , , , **, 3rd ed. (, New York, ). Note1, , , , (). Note2, [^1]: Note that in the atypical case of $\omega_{p0}<\omega_{r0}$, i.e. the pitching is slower than the rolling, we have $\omega_p\approx\omega_{r0}$ and $\omega_r\approx\omega_{p0}$ for $|\xi|\ll 1$ because $\omega_p>\omega_r$ by Eq. (\[ineq:omega\_pr\]). [^2]: The no-slip condition should be violated in the situations where the ratio of the vertical and the inplane components of the contact force, i.e., $F_{\parallel}\equiv\bm F\cdot\bm u$ and $F_{\perp}\equiv|\bm{F} - (\bm{F}\cdot \bm{u})\bm u|$, exceeds the friction coefficient. The ratio $F_\perp/F_\parallel$ becomes large when the angular momentum around $\bm u$ changes. In the cases given in Fig. \[fig:periodic\], its largest value is around 0.2.

--- abstract: 'We present strong arguments that the deep structure of the quantum vacuum contains a web of microscopic wormholes or short-cuts. We develop the concept of wormhole spaces and show that this web of wormholes generate a peculiar array of long-range correlations in the patterns of vacuum fluctuations on the Planck scale. We conclude that this translocal structure represents the common cause for both the BH-entropy-area law, the more general holographic principle and the entanglement phenomena in quantum theory. In so far our approach exhibits a common structure which underlies both gravity and quantum theory on a microscopic scale. A central place in our analysis is occupied by a quantitative derivation of the distribution laws of microscopic wormholes in the quantum vacuum. This makes it possible to address a number of open questions and controversial topics in the field of quantum gravity.' --- 1.5cm 0.5 cm Institut für Theoretische Physik\ Universität Göttingen\ Friedrich-Hund-Platz 1\ 37077 Göttingen Germany\ (E-mail: requardt@theorie.physik.uni-goettingen.de) Introduction ============ In the following we want to give a new explanation of the area law of black hole (BH) entropy and the more general and stronger [ holographic principle]{}. Furthermore, we provide (in our view) convincing arguments that an important structural ingredient of the deep structure of our quantum vacuum is a network of microscopic wormholes. In contrast to e.g. [string theory]{} and [loop quantum gravity]{} (LQG), which both employ the quantum laws more or less unaltered all the way down to the remote [Planck scale]{}, we regard this as an at least debatable assumption. We rather view the holographic hypothesis as a means to understand how both quantum theory and gravitation do emerge as derived and secondary theories from a more fundamental theory living on a more microscopic scale. A central role in this enterprise is played by an analysis of the microscopic structure of the [quantum vacuum]{} which leads to the key concept of [wormhole spaces]{}. This important conceptual structure makes it possible to understand the holographic aspects of quantum gravity, on the one hand, and the (non-local) entanglement phenomena pervading ordinary quantum physics, on the other hand, in a relatively natural way. Furthermore we think that there exist links to the old ideas of e.g. Sakharov and Zeldovich, dubbed [induced gravity]{} (see for example [@Sa; @1],[@Sa; @2],[@Z1],[@J1]). Some words are in order regarding the relation of our investigation to the analysis of BH entropy in, say, string theory. Three scenarios are in our view in principle possible. Either, both approaches adress the same phenomena in different languages, or they deal with them on different scales of resolution of space-time. Be that as it may, we think that our observation that the true ground state of our quantum vacuum seems to be what we call a wormhole space (see section \[worm\]) is an aspect which is not apparent in the original string theory approach and may be helpful to fix the proper ground state in string theory. In [@Bekenstein2] Bekenstein remarked that the deeper meaning of black-hole entropy (BH-entropy) remains mysterious. He asks, is it similar to that of ordinary entropy, i.e. the log of a counting of internal BH-states, associated with a single BH-exterior? ([@Bekenstein1],[@Bekenstein3] or [@Hawking1]). Or, similarly, is it the log of the number of ways, in which the BH might be formed. Or is it the log of the number of horizon quantum states? ([@Hooft1],[@Susskind1]). Does it stand for information, lost in the transcendence of the hallowed principle of unitary evolution? ([@Hawking2],[@Giddings]). He then claims that the usefulness of any proposed interpretation of BH-entropy depends on how well it relates to the original “statistical” aspect of entropy as a measure of disorder, missing information, multiplicity of microstates compatible with a given macrostate, etc. Quite a few workers in the field argue that the peculiar dependence of BH-entropy on the area of the event horizon points to the fact that the degrees of freedom (DoF), responsible for BH-entropy, are situated near the event horizon. This seems to be further corroborated by the corresponding behavior of the so-called entanglement entropy, i.e. its (apparent) linear dependence on the area of the dividing surface (cf., just to mention a few sources, [@Hooft2],[@Sorkin1],[@Sorkin2] or the lively debate in [@Jacobson1], concerning entanglement entropy in a more general setting, [@Sorkin3],[@Srednicki]). This linear dependence does however not generally hold without further qualifications. It does in particular [not]{} hold for excited states (see [@Requ2])! That is, while some particular sort of entanglement certainly plays an important role in this context, the real question is in our view the scale of resolution of space-time where this entanglement becomes effective and the nature of the quantum vacuum on this level of resolution.\ Remark: We want to emphasize the in our view crucial (but frequently apparently not fully appreciated) point that the entropy content of a BH is maximal.\ We think, the usual version of entanglement, we observe on the scales of ordinary quantum theory, is only an [epiphenomenon]{}, representing rather the coarse-grained effect of a hidden structure which lives on a much more microscopic scale. I.e., we are sceptical whether on such a microscopic scale the quantum vacuum can still be treated in the way of an ordinary quantum field theory vacuum as suggested in some of the papers cited above. We think, the [maximum-entropy property]{} of the BH-interior suggests another interpretation. We will come back to this point in more detail in section \[4\] (cf. also the sceptical remarks in some of the review papers by Wald, e.g. [@Wald1] (see in particular sect.6, Open Issues), [@W2] (see in particular sect.4, Some unresolved Issues and Puzzles\],[@W3]). As BH-entropy is widely regarded as an observational window into the more hidden and primordial quantum underground of space-time, it should be expected that it can be naturally explained within the frameworks of the leading candidates of such a theory, i.e., to mention the most prominent, string theory or LQG. For certain extreme situations string theory manages to give an explanation of the BH-entropy-area law. Whether the explanation is really natural is perhaps debatable (it relies in fact on a number of assumptions and correspondences as e.g. peculiar intersections of various classes of p-branes). In a sense, it is rather a correspondence between BH-behavior and the configurational entropy of certain string states. To mention some representative papers, [@S1],[@S2],[@S3],[@S4],[@S5],[@S6]. In LQG, on the other hand, it is assumed from the outset (at least as far as we can see) that the corresponding DoF are sitting at the BH-horizon. Therefore the observed area dependence of BH-entropy is perhaps not so surprising (cf. e.g. [@L1],[@L2]). In the enumeration of the most promising candidates for a theory of quantum gravity one approach is usually left out which, we nevertheless think, has a certain potential. One may, for example, tentatively divide quantum gravity candidates into roughly three groups, the relativisation of quantum theory (with e.g. LQG and causal set theory as members), the quantisation of general relativity (string theory being a prominent candidate) or third, theories which underlie both general relativity [and]{} quantum theory but are in fact more fundamental and structurally different from both and contain these two pillars of modern physics as derived and perhaps merely effective sub-theories, living on coarser scales (cf. e.g. [@Isham]). In the following we want to develop such a model theory in more detail. As far as we can see, such a philosophy is also shared by ‘t Hooft who emphasized this point in quite a few papers (see e.g. [@H1],[@H2],[@H3],[@H4]). We quote from [@H2]: > […it may still be possible that the quantum mechanical nature of the phenomenological laws of nature at the atomic scale can be attributed to an underlying law that is deterministic at the Planck scale but with chaotic effects at all larger scales…Since, according to our philosophy, quantum states are identified with equivalence classes…]{} Furthermore: > […It is the author’s suspicion however, that these hidden variable theories failed because they were based far too much upon notions from everyday life and ‘ordinary physics’ and in particular because general relativistic effects have not been taken into account properly.]{} While ’t Hooft usually chooses his model theories from the cellular automaton (CA) class, we are adopting a point of view which is on the one hand more general and flexible but, on the other hand, technically more difficult and complex. Instead of a relatively rigid underlying geometric substratum in the case of CA (typically some fixed regular lattice) on which the CA are evolving according to a given fixed (typically local) CA-law, we are employing quite irregular, dynamic geometric structures called by us [cellular networks]{}, the main point being that connections ([edges]{} or [links]{}) between the respective [nodes]{} or [cells]{} can be created or annihilated according to a dynamical law which, in addition, determines the evolution of the local node- and edge-states. To put it briefly, the ‘matter distribution’ (i.e. the global pattern of node-states) acts on the geometry of the network (the global pattern of active edges) and vice versa. Thus, as in general relativity, the network is supposed to find both its internal geometry and its matter-energy distribution with the help of a generalized dynamical law which intertwines the two aspects (cf. e.g. [@R1] or [@R2] and further references given there). Technically, the geometric substructure can be modelled by large, usually quite irregular ([random]{}) [graphs]{}. To make our point clear, this approach should not be confused with e.g. the spin network approach in LQG or various forms of (dynamical) triangulations. Our networks are usually extremely irregular and wildly fluctuating on a microscopic scale, resembling rather Wheeler’s [space-time foam]{}, and smooth geometric structures (as e.g. dimensional notions) are hoped to emerge via some sort of a [geometric renormalisation process]{} (in fact a very particular organized form of [coarse-graining]{} steps). Some of the interesting deeper mathematical aspects can for example be looked up in [@R3]. In our dynamical network approach to quantum space-time physics the nodes are assumed to represent cells of some microscopic size (presumably Planck size), the internal details of which cannot be further resolved in principle or are ignored and averaged over for convenience and will be represented instead by a simple ansatz for a local (node) state. It can perhaps be compared with the many existing spin-models which are designed to implement certain characteristic features of complex solids. This is more or less the same philosophy as in the CA-framework. The elementary connections between the nodes (the edges in graph theory) are assumed to represent [elementary interactions]{} or [information channels]{} among the cells and also carry simple [edge-states]{}. We made a detailed numerical analysis of the behavior of such networks in [@R4].\ Remark: We would like to emphasize however, that our approach does not really rely on this particular framework. It rather serves as a means to illustrate the various steps in our analysis within a concrete model theory. The paper is organized as follows. In the next section we analyse the basic substratum, i.e. the microscopic patterns of vacuum fluctuations, in particular the [negative]{} energy fluctuations. In section \[worm\] we describe the three different roads which lead (in our view: inevitably) to the concept of [wormhole space]{}. The preparatory sections 2 and 3 are then amalgamated in section 4 into a detailed analysis of the microscopic distribution pattern of [short-cuts]{} or [wormholes]{} and their consequences for the number of [effective]{} DoF in a volume of space. We introduce a new type of dimension, the so-called [holographic dimension]{}. Furthermore, we explain the microscopic basis of the [holographic principle]{} in general and the [bulk-boundary correspondence]{} between the DoF in the interior of e.g. a BH and the DoF on the boundary. Some apparent counter examples concerning the area-scaling property (see e.g. [@Marolf]; Wheeler’s ’bag of gold’-spacetimes ) are very briefly addressed. In the last section we briefly comment on a number of immediate applications of our microscopic holographic approach and (open) problems which can be settled with the help of our framework. The Structure of the Vacuum Fluctuations on a Microscopic Scale =============================================================== A characteristic feature of the dynamical network models we investigated is their [undulatory]{} character. As a consequence of the [feedback]{} structure of the coupling between node (cell) states and [wiring diagram]{} of edges (i.e. the pattern of momentary elementary connections or interactions) the network never settles in a static, frozen final state. The network may of course end up in some [attracting]{} subset of phase space but typical are wild fluctuations on a small (microscopic) scale with possibly some macroscopic patterns emerging on a coarser scale forming some kind of [superstructure]{} (see e.g. [@R4]). It is in our view not sufficiently appreciated that, in contrast to most of the other systems being studied in physics, the quantum vacuum is in a state of eternal unrest on a microscopic scale, with, for all we know, short-lived excitations constantly popping up and being reabsorbed by the seething sea. It therefore seems reasonable to regard our above network model (investigated in e.g. [@R1] to [@R4]) as a (toy) model of the quantum vacuum with the energy-momentum fluctuations on short scales being associated with the fluctuations of the local node and edge states. In the following we adopt the working hypothesis of a parallelism of network behavior and microscopic behavior of the quantum vacuum. We now come to a detailed analysis of the microscopic pattern of vacuum fluctuations. In sect.4 of [@Requ1] we made a calculation which shows that, given the huge number of roughly Planck-size grains in a macroscopic piece of space and assuming that the individual grains are allowed to fluctuate almost independently, more precisely, some grain variable like e.g. the local energy, the total fluctuations in a macroscopic or mesoscopic piece of space of typical physical quantities are still so large (i.e. macroscopic) that they should be observable. Note that with the number of nodes of roughly Planck-size, $N_P$, in a macroscopic volume, $V$, being gigantic, its square root is still very large (for the details of the argument see [@Requ1]). More precisely, with $q_i$ some physical quantity belonging to a microscopic grain (e.g. energy, momentum, some charge etc. and taking for convenience $\langle q_i\rangle$=0) and $Q_V:=\sum_i q_i$ being the observable belonging to the volume $V$, the fluctuation of the latter behaves under the above assumption as $$\langle Q_VQ_V\rangle^{1/2}\sim (V/l_p^3)^{1/2}$$ with $N_P\sim V/l_p^3$ the number of grains in $V$. This is a consequence of the [central limit theorem]{}. As such large integrated fluctuations in a macroscopic region of the physical vacuum are not observed (they are in fact microscopic on macroscopic scales), we conclude: The individual grains or supposed elementary DoF do not fluctuate approximately independently. Remark: We note that this fact is also corroborated by other, independent observations. We can refine the result further (cf. [@Requ1]) by assuming that the fluctuations in the individual grains are in fact not independent but correlated over a certain distance or, more precisely, are [short-range correlated]{}. In mathematical form this is expressed as [integrable correlations]{}. This allows that “positive” and “negative” deviations from the mean value can compensate each other more effectively. Letting e.g. $q(x)$ be the density of a certain physical observable and $Q_V:=\int_V q(x)\,d^n\!x$ the integral over $V$. In order that $$\langle Q_VQ_V\rangle^{1/2}\ll V^{1/2}$$ we proved in [@Requ1] that it is necessary that $$\int_V \!d^n\!y\,\langle q(x)q(y)\rangle\approx 0 \label{fluc}$$ We made a more detailed analysis in [@Requ1] under what physical conditions property (\[fluc\]) can be achieved, arriving at the result: Nearly vanishing fluctuations in a macroscopic volume, $V$, together with short-range correlations imply that the fluctuations in the individual grains are anticorrelated in a fine-tuned and non-trivial way, i.e. positive and negative fluctuations strongly compensate each other which technically is expressed by property (\[fluc\]). Remark: In [@R5] we extended such a vacuum fluctuation analysis and applied it to measurement instruments, being designed to detect (possibly) microscopic fluctuations of distances due to passing gravitational waves. We hence infer that the fluctuation pattern of e.g. energy-momentum has to be strongly [anticorrelated]{}. But under the above assumption it is possible that the underlying compensation mechanism which balances e.g. the positive and negative energy fluctuations is of short-range type, viz., individual grain-energies may still fluctuate almost independently if their spatial distance is sufficiently large. We show in the following that the true significance of the so-called [holographic principle]{} is it, to enforce a very rigid and [long-ranged]{} anticorrelated fluctuation pattern in the quantum vacuum. As we are at the moment only interested in matters of principle, we assume the simplest case to prevail, called the [space-like]{} holographic principle (holding in contexts like e.g. quasi-static backgrounds or asymptotic Minkowski-space; see e.g. the beautiful review [@Bousso]). There exists a class of scenarios in which the maximal amount of information or entropy which can be stored in a spherical volume is proportional to the area of the bounding surface. The same holds then for the number of available DoF in $V$. This is the spacelike holographic principle. In a series of papers Brustein et al. developed a point of view that relates typical fluctuation results of quantum mechanical observables in quantum field theory with the area-law-like behavior of entanglement entropy and BH-entropy (cf. e.g. [@Bru]). We already made a brief remark to this approach in [@Requ1]. We note that we arrived at related results using different methods in another context (see e.g. [@CMP50] and [@JMP43]). As a more detailed comment would lead us too far astray, we plan to discuss this subject matter elsewhere. What we are going to show in the following is that the mechanism leading to the strange area-behavior of the entropy of an enclosed volume, $V$, is considerably subtler as usually envisaged. On the one hand, we will show that the number of elementary DoF contained in $V$ is in principle proportional to the volume. On the other hand we infer from observations on the macroscopic or mesoscopic scale that the fluctuations of e.g. the energy are strongly anticorrelated. However, as long as this compensation mechanism is short-ranged, we would still have a number of nearly independently fluctuating clusters of elementary DoF which again happens to be proportional to the volume as the cluster size is roughly equal to the correlation length. So the conclusion seems to be inescapable that the patterns of vacuum fluctuations must actually be [long-range correlated]{} on a microscopic scale. But we showed in [@Requ1] or [@R5] in quite some detail that even systems, displaying long-range correlations, will usually have an entropy which is proportional to the volume. A typical example is a (quantum) crystal ([@Requ1],[@R5]). It is certainly correct that below a phase transition point a system of particles in the crystal phase has a smaller entropy than in the liquid or gas phase, but still the entropy happens to be an extensive quantity. The reason is in our view that the system develops, as a result of the long-range correlations, new types of collective excitations (e.g. lattice phonons) which serve as new collective DoF. Approximately they may be treated as a gas of weakly interacting elementary modes with the usual extensive entropic behavior. That is, the holographic principle entails that the elementary DoF have to be long-range anticorrelated (cf. also the remarks in [@H4] or sect.7 of [@Susskind2]). But we see that this is only a necessary but not a sufficient property for an entropy-area law to hold. We hence arrive at the preliminary conclusion: From our preceding arguments and observations we conclude that the holographic principle implies that the fluctuation patterns in $V$ are long-range anticorrelated in a fine-tuned way on a microscopic scale and are essentially fixed by the state of the fluctuations on the bounding surface. The dynamical mechanism, which generates these long-range correlations must however, by necessity, have quite unusual properties (cf. subsection \[bulkboundary\]). Before we derive the wormhole structure of the quantum vacuum on a primordial scale in the next sections, we continue with the general analysis of the pattern of vacuum fluctuations and derive some useful properties of it. A particular role is usually played by the energy and its fluctuations. Furthermore, vacuum fluctuations are frequently discussed together with the so-called [zero-point energies]{}. While they are not exactly the same, they are closely related. Both occur also in connection with the [cosmological constant problem]{} (to mention only a few sources see e.g. [@Zinkernagel1],[@Zinkernagel2],[@Nernst],[@Enz],[@Boyer],[@Weinberg],[@Straumann]). In the simplest examples like e.g. the quantized harmonic oscillator or the electromagnetic field we have $$H=P^2/2m+m\omega/2\cdot Q^2$$ and with $$\langle P\rangle_0=\langle Q\rangle_0 =0$$ in the groundstate, $\psi_0$, we have $$\hbar\cdot\omega/2=\langle H\rangle_0=1/2m\cdot \langle (P-\langle P\rangle_0)^2\rangle_0+ m\omega/2\cdot\langle (Q-\langle Q\rangle_0)^2\rangle_0$$ with $$\langle (P-\langle P\rangle_0)^2\rangle_0\cdot \langle (Q-\langle Q\rangle_0)^2\rangle_0\geq \hbar^2/4$$ which follows from $[P,Q]=-i\hbar$. In the same way we have in (matter-free) QED: $$H=const\cdot ({\mathbf}{E}^2+{\mathbf}{B}^2)$$ with $$\langle {\mathbf}{E}\rangle_0=\langle {\mathbf}{B}\rangle_0 =0$$ so that again $\langle H\rangle_0$ is a sum over pure vacuum fluctuations of the non-commuting quantities ${\mathbf}{E}$ and ${\mathbf}{B}$. One should however note that in the quantum field context products of fields at the same space-time point have to be Wick-ordered (in order to be well-defined). It is, on the other hand, frequently argued that with gravity entering the stage, these eliminated zero-point energy fluctuations have to be taken into account again. In our view, this problem is not really settled. We now come to an important point. It is our impression that in some heuristic discussions (vacuum fluctuations as virtual particle-antiparticle pairs) the consequences of the fact that the vacuum state is an exact eigenstate of the Hamiltonian in a Hilbert space representation of some quantum field theory are not fully taken into account. I.e., we have $$H\,\Omega=0$$ (provided the ground state energy is for convenience normalized to zero; note however that this may be problematical in a theory containing gravity). Eigenstates, however, have the peculiar property that the standard deviation is necessarily zero, $$\Delta_{\Omega}\,H=\langle (H-\langle H\rangle_{\Omega})^2\rangle_{\Omega}^{1/2}=\langle H^2\rangle_{\Omega}^{1/2}=0$$ According to the standard interpretation of quantum theory combined with spectral theory, $H^2\geq 0$, this implies that in each individual observation process the total energy of the vacuum which is, according to conventional wisdom, the (hypothetical) sum or superposition of local (small scale) fluctuations, happens to be exactly zero. In other words, the elementary fluctuations have to exactly compensate each other in an apparently fine-tuned way. Put differently If there are positive local energy fluctuations, there have to be at the same time by necessity negative energy fluctuations of exactly the same order. That is, at each moment, the global pattern of energy fluctuations in the quantum vacuum is an array of rigidly correlated positive and negative local excitations. Remark: Note the similarity of this independent observation to what we have said above in connection with the holographic hypothesis.\ It should be mentioned that Hawking in [@Haw1] invoked exactly this picture of a particle pair excitation near the event horizon with the virtual particle, having negative energy, falling into the BH while the one with positive energy escapes to infinity. It would be useful to get more quantitative information on the spectral properties of the local observables, in particular estimates on negative fluctuations. One could try to make an explicit spectral resolution of these quantities, e.g. of the energy, contained in a finite volume, $V$, but this turns out to be difficult in general, even if one has given an explicit model theory in some Hilbert space. As we prefer a more general, model independent approach (not necessarily based on Hilbert space mathematics), we proceed by using (similar to Bell in his papers) a general probabilistic approach which rather exploits the statistics of individual measurement results. Unfortunately, we found that the standard estimates, known to us in this context (e.g. the Markov-Chebyshev-inequality), always go in the wrong direction (see e.g. [@Bauer] or [@Feller]). Therefore we present in the following our own estimate. The strategy is the following. We take an observable, $E_V$, localized in $V$ with, for convenience, discrete spectral values, $E_i$, and corresponding probabilities denoted by $p_i>0$. If we assume that the expectation of $E_V$ is zero (which can always be achieved by a simple shift) we have $$\sum\,p_i=1\quad , \quad \sum\,p_i\cdot E_i=0$$ Furthermore, we assume its standard deviation in e.g. the vacuum, $\Omega$, to be finite (which is automatically the case for bounded operators, but we want to include also more general statistical variables) $$\sum\,p_i\cdot E_i^2=(\Delta_{\Omega}E)^2<\infty$$ In a first step we make the simplifying assumption (taking e.g. a bounded function of the energy) $$|E_i|\leq\Lambda\quad\text{for all}\quad E_i$$ We are interested in the amount of negative (e.g. energy) fluctuations we will observe in measurements. A reasonable quantitative measure of it is $$\sum\,p_i^-\cdot |E_i^-|$$ with $E_i^-,p_i^-$ the negative spectral values and their corresponding probabilities. We then have (with $|E_i^-|/\Lambda\leq 1$) $$\sum\,p_i^-\cdot |E_i^-|/\Lambda\geq \sum\,p_i^-\cdot |E_i^-|^2/\Lambda^2$$ For the lhs we have $$\sum\,p_i^-\cdot |E_i^-|/\Lambda=\sum\,p_i^+\cdot |E_i^+|/\Lambda$$ as the expectation of $E$ was assumed to be zero. This yields $$\begin{gathered} \sum\,p_i^-\cdot |E_i^-|/\Lambda=1/2\cdot \left(\sum\,p_i^-\cdot |E_i^-|/\Lambda+\sum\,p_i^+\cdot |E_i^+|/\Lambda\right)\geq\\ 1/2\cdot \left(\sum\,p_i^-\cdot |E_i^-|^2/\Lambda^2+\sum\,p_i^+\cdot |E_i^+|^2/\Lambda^2\right) \end{gathered}$$ I.e. $$\sum\,p_i^-\cdot |E_i^-|\geq 1/2\cdot\sum\,p_i\cdot E_i^2/\Lambda=1/2\Lambda\cdot (\Delta_{\Omega}E)^2$$ On the other hand (Cauchy-Schwartz) $$\left(\sum\,p_i^-\cdot |E_i^-|\right)^2=1/4\cdot \left(\sum\,p_i^-\cdot |E_i^-|+\sum\,p_i^+\cdot |E_i^+|\right)^2\leq 1/4\cdot\sum\,p_i\cdot |E_i|^2$$ We hence arrive at the result If the observable, $E$, is bounded, so that its spectral values fulfill $|E_i|\leq\Lambda$, we have the estimate $$1/2\Lambda^{-1}\,(\Delta_{\Omega}E)^2\leq \left(\sum\,p_i^-\cdot |E_i^-|\right)\leq 1/2\, (\Delta_{\Omega}E)$$ with $p_i$ the probabilities that the negative spectral values $E_i$ occur in an observation. That is, we manage to bound a quantity, which is difficult to measure directly, by quantities, which are usually more easily accessible. We can generalize this result to situations where the $E_i$ are not exactly bounded by some $\Lambda$ but are bounded in at least an essential way. We assume that there exists some $\Lambda$ so that $$\sum_{|E_i|>\Lambda}\,p_i\cdot |E_i|^2<\varepsilon_{\Lambda}$$ We then have $$\begin{gathered} \sum\,p_i^-\cdot |E_i^-|/\Lambda=1/2\,\left(\sum\,p_i^-\cdot |E_i^-|/\Lambda+\sum\,p_j^+\cdot |E_j^+|/\Lambda\right)\geq\\ 1/2\,\left(\sum_{|E_i^-|\leq\Lambda}\,p_i^-\cdot |E_i^-|/\Lambda+\sum_{|E_j^+|\leq\Lambda}\,p_j^+\cdot |E_j^+|/\Lambda\right)\geq\\ 1/2\,\left(\sum_{|E_i^-|\leq\Lambda}\,p_i^-\cdot |E_i^-|^2/\Lambda^2+\sum_{|E_j^+|\leq\Lambda}\,p_j^+\cdot |E_j^+|^2/\Lambda^2\right)\geq\\ 1/2\,\left(\sum\,p_i\cdot |E_i|^2/\Lambda^2-\varepsilon_{\Lambda}/\Lambda^2\right)\end{gathered}$$ Under the above assumption of an essentially bounded $E$ we have $$\sum\,p_i^-\cdot |E_i^-|\geq 1/2\Lambda\,\left((\Delta_{\Omega}E)^2-\varepsilon_{\Lambda}\right)$$ Another, rigorous, but not quantitative, argument can be derived from axiomatic quantum field theory (see e.g. [@Wigh1]). It follows from the so-called [Reeh-Schlieder theorem]{} that there are no [local observables]{} or [fields]{} which can annihilate the vacuum (where by local we mean that the objects commute for space-like separation). I.e., we have for any local $A$ (with $A=A^*$) $$A\,\Omega\neq 0\quad\Rightarrow\quad (A\Omega|A\Omega)=(\Omega|A^2\Omega)\neq 0$$ We take now as local observable the energy density integrated over a certain spatial region, $V$, $$H_V:=\int_V\,h_{00}({\mathbf}{x},0)\,d^3\!x$$ One usually normalizes $h_{00}(x)$ to $$(\Omega|h_{00}(x)\,\Omega)=0\quad\Rightarrow\quad (\Omega|\int_V\,h_{00}({\mathbf}{x},0)d^3\!x\,\Omega)=0$$ The classical expression of the energy density, being derived in Lagrangian field theory, is positive. The corresponding quantized expression, after a necessary [Wick-ordering]{} (see e.g. [@Bjo] or [@Itz]) is however no longer positive definite as an operator (density). This can be seen as follows. If the quantized energy density were still positive, one can take the square root (via the spectral theorem) of e.g. the positive operator $H_V$ and get: $$0= (\Omega|H_V\Omega)= (H_V^{1/2}\Omega|H_V^{1/2}\Omega)$$ hence $$H_V^{1/2}\Omega=0$$ As $H_V^{1/2}$ is also a local observable this is a contradiction due to the Reeh-Schlieder theorem. $H_V$ is not a positive operator, hence its spectrum contains negative spectral values. It is then easy to construct Hilbert-space vectors, $\psi$, so that the measurement of $H_V$ in $\psi$ yields negative values for the local energy. Remark: We recently learned that this argument is originally attributed to Epstein (unpublished;[@Reeh] or see [@Summers1]), while the derivation which can be found in [@Gla] is a completely different one. The important message (in our view) of all this is that, perhaps in contrast to naive expectation, the quantum vacuum contains a lot of negative energy excitations which globally exactly balance the positive excitations. One may now speculate about the possibility of making use of this observation. \[worm\]Wormhole Spaces ======================= In this section we want to describe (very) briefly and sketchily the three different lines of reasoning which lead us to the concept of [wormhole spaces]{}. The first line originated from our investigation of the structure and dynamical behavior of the networks we described above. In e.g. [@R1] we analyzed in some quantitative detail the unfolding of the network structure and the various network epochs under the inscribed microscopic dynamical laws and developed the two-level concept of the network structure (or, rather, a multi-scale structure), which, under the right conditions, is relatively smooth on a sufficiently coarse-grained level (level 2) with, among other things, a distant measure (metric) of the more ordinary type and (hopefully) an integer-valued geometric dimension, while on a more microscopic scale (level 1) the network structure is expected to be very erratic with possibly a lot of links (elementary interactions or information channels) connecting regions which may be far apart with respect to the metric on level 2. The association of these links with microscopic wormholes thus suggests itself (cf. in particular observation 4.27 in [@R1]). Note furthermore that our network dynamics implies that these translocal connections are dynamically switched on or off. Compare this observation with the point of view expounded in e.g. [@DeWitt] > […But if a wormhole can fluctuate out of existence when its entrances are far apart …then, by the principle of microscopic reversibility, the fluctuation [into]{} existence of a wormhole having widely separated entrances ought to occur equally readily. This means that every region of space must, through the quantum principle, be potentially “close” to every other region, something that is certainly not obvious from the operator field equations which, like their classical counterparts, are strictly local.…It is difficult to imagine any way in which widely separated regions of space can be “potentially close” to each other unless space-time itself is embedded in a convoluted way in a higher-dimensional manifold. Additionally, a dynamical agency in that higher-dimensional manifold must exist which can transmit a sense of that closeness.]{} The quantitative network calculations in the mentioned papers have mainly been performed within the framework of [random graphs]{}. Important mathematical tools for the network analysis in the transition from microscopic, strongly fluctuating and geometrically irregular scales to coarse-grained and, by the same token, smoother scales have been the concepts of [cliques]{} of nodes, the [clique-graph]{} of a graph and an important network parameter which we dubbed [intrinsic scaling dimension]{} (we later learned, [@R3], that this concept plays also an important role in [geometric group theory]{} or [Cayley-graphs]{} where it is called the [growth degree]{}). To give a better feeling what is actually implied, we give the definitions of clique, clique-graph and internal scaling dimension (more about graph theory can e.g. be found in [@Bollo], [@R2], notions and properties of graph dimension were studied in e.g. [@ReqDim]). A simplex in a graph is a subset of vertices (nodes) with each pair of nodes in this subset being connected by an edge. In graph theory it is also called a complete subgraph. The maximal members in this class are called cliques. The clique graph, $C(G)$, of a graph, $G$, is built in the following way. Its set of nodes is given by the cliques of $G$, an edge is drawn between too of its nodes if the respective cliques have a non-empty overlap with respect to their set of nodes. Graphs carry a natural neighborhood structure and notion of distance. The neighborhood $U_n(x)$ of a node $x$ is the set of nodes $y$ which can be reached, starting at $x$ in $\leq n$ consecutive steps, i.e. there exists a path of $\leq n$ consecutive edges connecting the nodes $x$ and $y$. \[dist\]The canonical network or graph metric is given by $$d(x,y):=\min_{\gamma}\{l(\gamma)\,|\,\gamma\; \text{a path connecting}\; x\; \text{and}\; y\}$$ Here $l(\gamma)$ is the number of consecutive edges of the path. The above definition fulfills all properties of a metric. Thus graphs and networks are examples of [metric spaces]{}. \[Dim\] Let $x$ be an arbitrary node of $G$. Let $\#(U_n(x))$ denote the number of nodes in $U_n(x)$.We consider the sequence of real numbers $D_n(x):= \frac{\ln(\#(U_n(x))}{\ln(n)}$. We say $\underline{D}_S(x):= \liminf_{n \rightarrow \infty} D_n(x)$ is the [*lower*]{} and $\overline{D}_S(x):= \limsup_{n \rightarrow \infty} D_n(x)$ the [*upper internal scaling dimension*]{} of G starting from $x$. If $\underline{D}_S(x)= \overline{D}_S(x)=: D_S(x)$ we say $G$ has internal scaling dimension $D_S(x)$ starting from $x$. Finally, if $D_S(x)= D_S$ $\forall x$, we simply say $G$ has [ *internal scaling dimension $D_S$*]{}. We proved in [@ReqDim] (among other things) that this quantity does not depend on the choice of the base point for most classes of graphs. It turns out that this geometric notion is a very effective characteristic of the large-scale structure of graphs and networks. This topic was further studied in greater generality in e.g. [@R3]. In [@R2] we developed what we called the [geometric renormalization group]{}, to extract important geometric coarse grained, that is, large scale information from the microscopically quite chaotically looking network and its dynamics. The idea is, at least in principle, similar to the [block spin transformation]{} in statistical mechanics. That is, certain characteristic properties of the system are distilled from the microscopically wildly fluctuating statistical system by means of a series of algorithmic renormalization steps (i.e. coarse-graining plus purification). The central aim is it to arrive in the end at a system which resembles, on the surface, a classical space-time, or, on the other hand, to describe the criteria a network has to fulfill in order that it actually has such a [classical fixed point]{}. In the course of this analysis we observed (cf. section VIII of [@R2]) that the so-called [critical network geometries]{}, i.e. the microscopic network geometries which are expected to play a relevant role in the analysis, are necessarily in a very specific way [geometrically non-local]{}, put differently, they have to contain a very peculiar structure of non-local links, or [short-cuts]{}, that is, in other words, the kind of [wormhole structure]{}, we already described above. Relations to [non-commutative geometry]{} were established and studied in [@Connes]. We mention in particular section 7.2 “Microscopic Wormholes and Wheeler’s Space-Time Foam” and section 8 “Quantum Entanglement and Quantum Non-Locality”. The possible relevance for quantum theory is in fact quite apparent (as has also been emphasized in the papers by ’t Hooft), as these microscopic wormholes may be the origin of the ubiquitous entanglement phenomena in quantum theory. The following figures describe pictorially the nested structure of the cliques of nodes in consecutive renormalization steps and overlapping cliques of nodes, defining the local [near-order]{} of [physical points]{} together with shortcuts which connect distant parts of the coarse-grained surface structure. A second complex of (related) phenomena emerges in the field of [small world networks]{}. This is a particular class of networks of apparently quite a universal character (described and reviewed in some detail, for the first time, in [@Watts]) with applications in many fields of modern science. They consist essentially of an ordinary local network with its own local notion of distance superimposed by a typically very sparse network of so-called [short-cuts]{} living on the same set of nodes and playing a structural role similar to the microscopic wormholes described above. A typical example (with dimension of the underlying lattice $k=1$) is given in the following figure. Some further (in fact very few) references, taken from quite diverse fields are e.g. [@Strogatz],[@Grano],[@Lochmann]. Its, in our view, crucial characteristic is the existence of two metrics over the same network or graph. The first, $d_1(x,y)$, is defined (cf. definition \[dist\]) by taking into account the full set of edges (i.e., including the short-cuts) and a second (local) metric, $d_2(x,y)$, taking into account only the edges of the underlying local network. It hence holds $$d_1(x,y)\leq d_2(x,y)$$ Remark: The metric $d_2(x,y)$ may then be associated (after some renormalisation or coarse-graining steps) with an ordinary macroscopic metric defined on a smooth space (without wormholes) like our classical space-time. $d_1(x,y)$, on the other hand, should be regarded as a microscopic distance concept which employs the existence of wormholes. While, on the surface, the origin of this concept of small world networks seems to be quite independent of the wormholes in general relativity, it is the more surprising that on a conceptual meta level various subtle ties do emerge. To mention only one (in our view) important observation. In [@Schuster] it is for example shown, that a sparse network of shortcuts superimposed upon an underlying local network, has the propensity to stabilize the overall frequency pattern ([phase locking]{}) of so-called [phase-oscillators]{} which represent the nodes of the networks, the links representing the couplings. The oscillators are assumed to oscillate with (to a certain degree) independent frequencies. If we relate these local frequencies with some local notion of time (or clocks), we may infer that (microscopic) wormholes create or stabilize some global notion of time! We now come to the third strand, viz. the real wormholes of general relativity or quantum gravity. We mainly concentrate on the wormholes in true, i.e. Lorentzian space-time. Euclidean wormholes also (may) play an important role and have been discussed extensively in the context of the (nearly) vanishing value of the [cosmological constant]{} (see e.g. [@HawW1],[@ColW],[@KlebaW], [@PresW],[@UnrW],[@HawW2]). Of particular relevance in the Lorentzian context are the so-called [traversable]{} wormholes. Their study started (as far as we know) with two seminal papers by Thorne and coworkers (see [@Mo]). The geometric construction of such solutions is in fact not so difficult if performed by the so-called [g-method]{}. That is, one constructs a geometric wormhole, e.g. of the static type, and, in a second step, computes the energy-momentum tensor being consistent with this solution. Giving a rough outline, this can be done in following way. Two open balls are removed from two different pieces of e.g. approximately flat 3-space. Their boundaries are glued together with the junction being smoothed. As a consequence of the smoothing process a tube emerges interpolating between the two spheres (see e.g. [@Kras]). It is a remarkable fact that in this process the [weak energy condition]{} (WEC) has to be violated, the latter implying that $$T_{00}\geq 0\quad ,\quad T_{00}+T_{ii}\geq 0\quad\text{for}\quad i=1,2,3$$ that is, the matter-energy density is positive in any reference system. Put differently, In order to get a traversable wormhole, one has to violate the WEC. The WEC is always satisfied by classical matter. Therefore quantum effects are needed. The kind of negative energy needed is also called exotic matter. We showed in quite some detail in the preceding section that the quantum vacuum abounds with negative energy fluctuations. Therefore the speculation in section H of the first paper in [@Mo] does not seem to be too far-fetched. In a next step one can study networks of such traversable wormholes. In [@Ho1] it is speculated that such a network, existing in the early universe, may solve the [horizon problem]{}. The same situation was discussed from the point of view of our network approach in section 4.1 (The Embryonic Epoch) of [@R1]. All this comes already quite near the general picture we envoked in the beginning of this section. Furthermore one can envisage solutions combining black and white holes. This corresponds to some of our networks where the orientation (direction) of the links connecting two nodes can change under the dynamics. A review of Lorentzian wormholes can be found in the book by Visser ([@Vi]). Some other references are e.g. [@Red] and [@Few]. The above picture of a hypothetical network of wormholes sitting in the deep structure of the quantum vacuum is beautifully complemented by an approach (see e.g. [@Prep],[@Gara]) which investigates within a (semi)classical approximation the energy of a quantum vacuum state containing such an array of wormholes (or, rather, a gas of such wormholes) and compare it with a vacuum state which in zeroth order is flat Minkowski space. It comes out (apparently being a kind of Casimir effect) that the quantum vacuum containing the wormhole gas has in this semiclassical approximation a lower energy compared to the state, being a perturbation of Minkowski space. One should note however that this is a first order quantum effect! Anyhow, this observation seems to corroborate the space-time foam picture of e.g. Wheeler and we conclude this section with From our analysis in this and the preceding section emerges a model of the ground state of some preliminary version of quantum gravity which contains as an essential ingredient a network of microscopic wormholes. These wormholes can be created and annihilated and are in our picture the carriers of information between distant parts of classical space-time. We call such a physical structure a wormhole space and regard our cellular or small world networks, discussed above, as models, encoding and representing the typical characteristics of such systems. The typical characteristic is the existence of two types of distance, a microscopic one and an ordinary local one, being similar to ordinary macroscopic metrics on smooth spaces. \[4\]Wormhole Spaces as the Common Cause of the Holographic Principle and the Entropy-Area Law ============================================================================================== We learned in the preceding sections that two (presumably crucial) properties govern the behavior of the quantum vacuum on a microscopic scale. First, the vacuum fluctuations are strongly long-range anticorrelated on a microscopic scale, i.e. there exists a fine-tuned pattern of positive and negative (energy) fluctuations. Second, a quantum mechanical stability analysis seems to show that the quantum vacuum is pervaded by a network of microscopic wormholes. We argued above that these two features are not independent phenomena but rather are the two sides of the same medal. Furthermore, the presumed wormhole structure has been supported by observations coming from other fields of research like e.g. cellular or small-world networks. In this (central) section we will now combine these observations and show that they underlie (among other things) the [holographic principle]{} and the [entropy-area law]{} of BH-thermodynamics. In the following we will use (for convenience) the language of our networks with the nodes of the network representing microscopic grains of space (or space-time) of roughly Planck-size. Leaving out other details we treat our quantum vacuum as a wormhole space, i.e. as a (small world) network consisting of an ordinary local network structure being superimposed by a (presumably) sparse random network with edges consisting of short-cuts, i.e. links, connecting regions of space or space-time, which may be quite a distance apart with respect to the metric, belonging to the underlying local network. These short-cuts represent the wormholes of ordinary space-time. The crucial characteristic, from which everything is expected to follow, is the pattern and distribution of these short-cuts being immersed in the underlying local network. That is, we randomly select a node $x$ in the network $G$ ($G$ standing for graph) and study the distribution of short-cuts connecting $x$ with nodes $y$ on spheres of radius $R$ around $x$ (measured with respect to some macroscopic metric or the natural metric of the underlying [local]{} network). We expect that the precise distribution law will depend on the concrete type of space-time we are dealing with. This holds in particular if the space-time is not static. That is, our microscopic approach to holography makes it possible to understand how holography may depend on the concretely given type of space-time (cf. e.g. the covariant entropy bound of Bousso, [@Bousso]). Remark: We emphasize that the network or the quantum vacuum it is representing, is basically a statistical system with all local DoF fluctuating. That means, most of our statements in the following are about mean values or averages over finer statistical details. The Distribution of Short-Cuts or Wormholes ------------------------------------------- One can arrive at the law, describing the distribution of short-cuts or wormholes around some arbitrary but fixed generic node (viz. some fixed place in space-time) in roughly two ways. One can e.g. motivate the distribution law by appealing to certain fundamental principles like e.g. [scale-freeness]{} or absence of a particular and in some sense unnatural length scale on a fundamental level. Alternatively, one can show that a reasonable choice leads to far-reaching consequences and corroborates the findings and observations made on a more macroscopic level. To keep the discussion as briefly as possible we adopt in this section the second point of view. In the following we want to concentrate, for the sake of brevity, on a simple type of quantum vacuum, that is, the vacuum belonging to ordinary Minkowski space or a space-time which is asymptotically flat (e.g. a Schwarzschild space-time). We postpone the analysis of more general space-times as they occur in general relativity. We make the following conjecture: On the average the number of short-cuts from a central node $x$ to nodes $y$, sitting on the sphere, $S_R(x)$ about $x$ is independent of $R$. Denoting this number by $N_{S_R}(x)$, we hence have $$N_{S_R}(x)=N_0$$ Remark: As this number is a statistical average, it need not be an integer.\ The situation is depicted in the following picture. \[bb\] We will show in subsection \[bulkboundary\] in a detailed quantitative analysis that this result approximately holds as well for nodes, not sitting exactly in the center of the spheres $S_R$ (see the following picture). We denote the cluster of nodes in the ball $B_R$ being connected to an $x$ by short-cuts by $C_{B_R}(x)$. We previously introduced the internal scaling dimension of a network (see definition \[Dim\]). It roughly describes how fast the network is growing with respect to some base node. As this [growth degree]{} is to a large degree independent of the base node (see e.g. [@ReqDim]) it is a global characteristic of a given network, in fact of a whole class of similar networks ([@R3]). It is well known that the generalization of the concept of dimension away from smooth geometric structures is not unique. The above type of dimension has the tendency to grow if additional short-cuts are inserted into a given network geometry. We now introduce another dimensional concept which catches other important network properties being more closely related to the phenomena we want to analyze in this paper. It uses in an essential way the two metrics, $d_1,d_2$, introduced above. From the above we infer that the number of nodes in the cluster $C_{B_R}(x)$ is approximately equal to $N_0\cdot R$. Furthermore, if the network of short-cuts is very sparse, the clusters $C_{B_R}(x_i),C_{B_R}(x_j)$ with $x_i\neq x_j$ are essentially disjoint (the overlap is empty or very small). This is the phenomenon called [*spreading*]{} in the theory of random graphs. Hence, the following concept is reasonable. We define a [holographic dimension]{}, $D_H$, of a network in the following way. We take some ball $B_R$ with macroscopic radius $R$ around some fixed but arbitrary node $x$ with respect to the local metric $d_2$. We then form the $U_1^{(1)}(y)$-neighborhoods around the nodes $y\in B_R$ with respect to the microscopic metric $d_1$ . We construct a [minimal cover]{} of $B_R$ by such $U_1^{(1)}(y_i)$, i.e. a minimal selection of such $y_i$ s.t. $$\bigcup_i\,U_1^{(1)}(y_i)\supset B_R$$ The cardinality of such a minimal set we denote by $N_C(B_R)$. We take the limit $R$ large or $R\to\infty$ (in an infinite network) and define We call $$D_H:=\lim_{R\to\infty}\,\ln\,N_C(B_R)/\ln\,R$$ the holographic dimension of the graph (network), provided the limit exists. In the more general situation we can, as in definition \[Dim\], define upper and lower dimensions etc. As for the previously defined graph dimension, the limit is independent of the selected base point , $x$, if the network or graph is homogeneous on the average or in the large. Due to the sparseness of the embedded subgraph of short-cuts, which yields the spreading property mentioned above, the number $N_C(B_R)$ scales for the wormhole spaces or small-world networks as $$N_C(B_R)\sim R^{n-1}$$ with $n$ the dimension of the local network or its coarse-grained continuum limit space. Proof: The $U_1^{(1)}(y)$-neighborhoods consist of nodes lying in the neighborhoods with respect to the local metric, $d_2$, $U_1^{(2)}(y)$, plus the vertices connected by short-cuts with $y$. The cardinality of $U_1^{(2)}(y)$ is independent of $R$ and typically (at least in our models) a small number. For $R\to\infty$ $U_1^{(1)}(y)\cap B_R$ will therefore consist mainly of nodes connected to $y$ by short-cuts. Sparseness of the short-cut graph and spreading yield the result. [$ \hfill \Box $]{} For the type of wormhole spaces or small-world networks, defined above, we then have $$D_H=\lim_{R\to\infty}\,\ln(V(B_R)/R)/\ln\,R=n-1$$ That is, in this case we have the important result $$D_H= dim\,S_R= n-1$$ We now come to the holographic principle and the BH-entropy area law. As already mentioned, we discuss in this paper only the example of 4-dim. asymptotically flat (Minkowski) space-time. In Planck units a macroscopic ball, $B_R$, contains approximately $$|V(B_R)|:= V(B_R)/l_p^3$$ DoF or grains of Planck size. The typical cluster size is $$|C_{B_R}(x_i)|\approx N_0\cdot R/l_p$$ Due to the mentioned spreading property the number of (effectively) independent cluster in the above minimal cover is approximately $$\begin{gathered} N_C(B_R)\approx ((4/3)\pi\cdot R^3/N_0\cdot R)\cdot l_p^{-2}=(3N_0)^{-1}\cdot 4\pi R^2/l_p^2=\\(3N_0)^{-1}\cdot A(S_R)/l_p^2=:(3N_0)^{-1}\cdot |A(S_R)| \end{gathered}$$ with $A(S_R)$ denoting the area of $S_R$. The number of effectively independent clusters, $C_{B_R}(x_i)$ in $B_R$ is $$N_C(B_R)\approx (3N_0)^{-1}\cdot|A(S_R)|= (3N_0)^{-1}\cdot A(S_R)/l_p^2$$ with the typical cluster size $$|C_{B_R}(x_i)|\approx N_0\cdot R/l_p$$ To show now that the number of effective DoF in a generic volume (where by generic we mean a region in space with the diameter in all directions being roughly of the same order) is proportional to the surface area, $A(V)$, of its boundary, we employ a general observation, made e.g. in statistical mechanics. An important tool for the analysis of systems in statistical mechanics are correlation functions. Correlations decay usually for large separation of the respective DoF, but what is on the other hand certainly the case is, that nearest neighbors are strongly correlated (near order versus far order). We expect that the DoF in each of the $U_1^{(1)}(x)$ are strongly correlated. We hence take it for granted, that they act effectively as a single collective DoF. Remark: It may be possible, that this near order in the immediate neighborhood of the grains can be finally destroyed by the insertion of a huge amount of localized energy, but this does not seem possible with present means. Due to the existence of wormholes or short-cuts, distributed in space-time, the number of effective DoF (affiliated with the respective clusters $C_{B_R}(x_i)$) in e.g. a ball $B_R$ equals $N_C(B_R)$, that is $$\#(\text{DoF in}\;B_R)\approx (3N_0)^{-1}\cdot |A(S_R)|=(3N_0)^{-1}\cdot A(S_R)/l_p^2$$ This is the area-law behavior of entropy or number of DoF in a volume of space found in e.g. BH-entropy. We note however, that this law, in our formulation, is essentially a statement about the collective behavior of the elementary DoF in (the interior of) a volume of space. I.e., the respective DoF are [not]{} really sitting on the boundary of $V$. As to the details of the [bulk-boundary correspondence]{} see the following subsection. If we adopt the entropy-area law of BH-thermodynamics, which is, expressed in Planck units, $$S=1/4\cdot |A|$$ we have the possibility to fix our parameter $N_0$, which gives the number of wormholes connecting a central grain of space with the grains on a surrounding sphere $S_R$ for any $R$. However, entropy is not exactly identical to number of DoF. To relate the two, we have to make a simple model assumption. One frequently makes the assumption of [Boolean DoF]{}, i.e. the DoF on an elementary scale are [two-valued]{}. With this assumption we have the relation $$S=N\cdot \ln\,2\quad\text{i.e.}\quad N=|A|/4\cdot\ln\,2$$ with $S$ the entropy, $N$ the number of DoF. With the help of this identification we get $$N_0=4/3\cdot\ln\,2$$ which can in qualitative arguments be approximated by one! That is, in Planck units, there exists roughly one short-cut between a central vertex and a surrounding sphere of radius $R$. This shows that on an extremely microscopic scale, the network of short-cuts is indeed very sparse. However the picture changes considerably if we go over to more accessible length scales. If we use, for example an atomic length-scale of e.g. $l_a:=10^{-10}m$, we have approximately $$(10^{-10})^3/(10^{-35})^3=10^{75}$$ grains of Planck-size in a volume element of diameter $l_a$. If we then choose, instead of a sphere $S_R$, a spherical shell of radius $R$ and thickness $l_a$ we have approximately The number of wormholes or short-cuts between a central volume element of size $l_a^3$ and a corresponding spherical shell of radius $R$ is approximately $$\#(\text{short-cuts})\approx 10^{75}\cdot 10^{25}=10^{100}$$ which is quite a large number. If we choose for example $R=1m$, we see that roughly $10^{96}$ grains in the shell are the endpoints of about $10^{100}$ short-cuts coming from the central volume element of size $l_a^3$. If we replace $R$ by the approximate diameter of the universe, i.e. $R_0\approx 10^{10}$ ly, we get (with $1\,ly\approx 10^{17}m$): $$R_0\approx 10^{27}\,m$$ and for the number of Planck-size grains in a spherical shell of this radius: $$\#\,(\text{grains in shell of radius}\,R_0)\approx 10^{149}$$ with still $10^{100}$ short-cuts ending there. That is, only one in $10^{49}$ grains is the endpoint of a respective short-cut. But if we select a volume element of size $l_a^3$ in this shell, we have still The number of wormholes (short-cuts) between two volume elements of size $l_a^3$ being a distance $R_0$ apart, is still the large number $$\#\,(\text{short-cuts})\approx 10^{100}\cdot 10^{-149}\cdot 10^{75}= 10^{-49}\cdot 10^{75}=10^{26}$$ that is, even over such a large distance there exist still a substantial number of wormholes connecting the two volume elements. But nevertheless, the network is sparse, viewed at Planck-scale resolution. \[bulkboundary\]The Bulk-Boundary Correspondence ------------------------------------------------- We now come to the last point of this section. From what we have learned above, it is intuitively clear, that the DoF sitting on the boundary $S_R$ of e.g. a ball $B_R$ should fix (or slave) the DoF in the interior. But we note that in order that this can hold, we have to verify our statement made in observation \[bb\]. Furthermore, it is of tantamount importance to understand in more quantitative detail the influence of different shapes of the region under discussion and the effect of different space-time geometries. The prerequisites for this enterprise will be derived in the following. As an example we employ, as we already did above, the simple geometry of the spacelike holographic bound. For reasons of simplicity we place the center of the ball in the origin, i.e. ${\mathbf}{x_0}={\mathbf}{0}$. It is of great help if we can transform the problem into a problem of ordinary continuous analysis. To this end we introduce the probability that a node in the interior of $B_R$ and an arbitrary node on the boundary $S_R$ are connected by a short-cut. With ${\mathbf}{y}\in S_R$ and ${\mathbf}{x}\in B_R$ there spatial euclidean distance in three dimensions is $$|{\mathbf}{y}-{\mathbf}{x}|=\left(\sum_{i=1}^3\,(y_i-x_i)^2\right)^{1/2}$$ The edge probability is given by $$p(|{\mathbf}{y}-{\mathbf}{x}|)=N_0/|A(S_{|{\mathbf}{y}-{\mathbf}{x}|})|=(N_0\cdot l_p^2/4\pi)\cdot |{\mathbf}{y}-{\mathbf}{x}|^{-2}$$ Here $|A(S_{|{\mathbf}{y}-{\mathbf}{x}|}|$ is the number of nodes (or Planck-scale grains) on the sphere around ${\mathbf}{x}$ with radius $|{\mathbf}{y}-{\mathbf}{x}|$.This follows directly from what we have learned in the previous sections. What we are actually doing in the following is the calculation of the average number of short-cuts between an arbitrary node ${\mathbf}{x}$ in the interior of $B_R$ and the nodes on the boundary $S_R$. This will be done within the framework of [random graphs]{}. The above $p$ is the so-called [edge probability]{} (for the technical details see [@Bollo] or [@R1],[@R2]). The sample space is the space of graphs with [node set]{} comprising the node in ${\mathbf}{x}$ and all the nodes sitting on the boundary $S_R$ and [edge set]{} all possible different sets of short-cuts connecting $x$ with the nodes on $S_R$. The probability of each graph in the sample space is calculated with the help of the above elementary edge probability $p$ and its dual $q:=1-p$. We choose ${\mathbf}{x}$ arbitrary but fixed in $B_R({\mathbf}{0})$ and let ${\mathbf}{y}$ vary over the sphere $S_R({\mathbf}{0})$. The integral over $S_R({\mathbf}{0})$ will then give the mean number of short-cuts between ${\mathbf}{x}$ and the grains on $S_R({\mathbf}{0})$. The guiding idea is that the DoF in the interior are fixed by the DoF on the boundary if this integral is essentially $\gtrsim 1$, as according to our philosophy, developed previously, in that case every node in the interior has on average at least one partner on the boundary as nearest neighbor with respect to the microscopic metric $d_1$. To make the integration easier we choose, without loss of generality, $${\mathbf}{x}=\begin{pmatrix}0 \\ 0 \\ z \end{pmatrix}\quad , \quad z:=k\cdot R$$ with $0\leq k\leq 1$. A straightforward calculation (using polar coordinates and appropriate variable transformations) yields for the average number of short-cuts, $N_{S_R}({\mathbf}{x})$, $$\begin{gathered} N_{S_R}({\mathbf}{x})= (N_0\,l_p^2/4\pi)\cdot l_p^{-2}\cdot\int_{S_R}\,|{\mathbf}{y}-{\mathbf}{x}|^{-2}\,d\!o=\\ \left(N_0/4\pi\cdot R^2\right)\cdot 2\pi\,R^{-2}\cdot\int_{-1}^{+1}\,d\!u\,((1+k^2)-2ku)^{-1}=\\N_0/2\cdot\int_{-1}^{+1}\,d\!u\,((1+k^2)-2ku)^{-1} \end{gathered}$$ Note that the integrand $((1+k^2)-2ku)^{-1}$ is always positive. Furthermore, our choice of a Coulomb-like law (in three dimensions) for the distribution of short-cuts in the previous subsection, i.e. $p\sim R^{-2}$, makes the above integral independent of $R$. We can find a closed expression for the definite integral, i.e. $$I:= \int_{-1}^{+1}\,d\!u\,((1+k^2)-2ku)^{-1}=-1/2k\cdot\ln\,((1-k)^2/(1+k)^2)> 0$$ Note that the position of the point ${\mathbf}{x}$ relative to the center and the boundary can be regulated by the value of the parameter $0\leq k\leq 1$. We have tabulated the integral for $k$ from $0$ to $0.9$ in the following table.\ $k$ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ------- --- ----- ----- ------ ------ ------ ------ ------ ------ ------ $I_k$ 2 2 2 2.04 2.11 2.19 2.29 2.45 2.71 3.23 \ We see that the number of short-cuts is almost constant through the whole interior of $B_R$ apart from a thin shell near the boundary. But this is not really surprising because there the main contribution comes from the near side of the boundary and is no longer of a true short-cut character. Taking into account the additional prefactor, $N_0/2$, in front of the integral which is $\approx 1/2$ we have The number of short-cuts from an arbitrary node ${\mathbf}{x}$ in $B_R$ to the boundary $S_R$ is approximately $$p({\mathbf}{x})\gtrsim 1$$ for most of the nodes. Furthermore for our Coulomb-like distribution law it is independent of the radius of the sphere and is therefore consistent with the expected holographic behavior for this geometry. It is instructive to evaluate the above formula for $k>1$, i.e., the influence via short-cuts of the sphere $S_R$ on a DoF in the exterior of $S_R$. For $k$ large, the integral is dominated by the first term in the integrand, viz. for $k$ large we have $$I\approx \int_{-1}^1d\!u\,(1+k^2)^{-1}\sim k^{-2}$$ For nodes, $x$, lying outside of $S_R$, the effect of the short-cut connections between $x$ and $S_R$ decays like a Coulomb-law. That is, the DoF in the exterior are no longer fixed by the DoF on $S_R$. What remains instead is a statistical influence in form of a correlation which decays with increasing distance. By the same token, there cannot be an entropy-area law for the exterior of the sphere relative to its internal boundary. Anyhow, this example does not really contradict the correctness of the spatial holographic principle as being presented in this paper. It would be interesting to relate our findings to the covariant holographic principle of e.g. Bousso, [@Bousso] This simple observation has an important consequence for arguments being sometimes invoked against the general nature of the spatial holographic principle (cf. e.g. [@Bousso]). While we do not intend to discuss the holographic principle for more general space-times in this paper, we mention one counter-example which one finds frequently in the literature, i.e. a universe containing a closed spatial slice, $S$ with a small inner subregion, $S_2$ (see the following picture). The area-law in the usual form applies for the subregion $S_2$ relative to its boundary. However, according to our (microscopic) version of spatial holography, the DoF on the inner boundary cannot slave the DoF in the large region $S_1$ if the inner boundary becomes too small. They only establish some kind of correlation in the exterior. The quantitative details are given by integrating our Coulomb-like influence law over the inner surface. Another, related, class of interesting (but perhaps pathological) apparent counter examples (which we plan to address in greater detail elsewhere) is discussed in e.g. [@Marolf], i.e. spacetimes which are called by Marolf ’bag-of-gold spacetimes’. An essential ingredient is some FRW-spacetime hidden in the interior of a region which resembles an ordinary BH. The innner FRW-universe has of course an entropy which is proportional to its volume while from the outside the whole configuration looks like a BH. This seeming contradiction can be easily understood with the help of our microsopic holographic law as the FRW-spacetime is actually only weakly coupled with the exterior of the BH via wormholes. The technical arguments are the same as above. Commentary ========== In the preceding sections we developed only the groundwork of our approach. To keep the paper within reasonable size, we had to postpone a more detailed discussion of the many consequences and immediate applications. In this final section we at least undertake to briefly comment on a number of important points. It is however obvious that a more detailed discussion of each point would require a paper of its own.\ i) The possible connections to the ubiquituous phenomenon of entanglement in ordinary quantum theory are obvious. Interesting in this respect is e.g. the well-known tension in quantum theory between the locality and causality principle of special relativity and the instantaneous state reduction, accompanying the measurement process (cf. the respective sections in e.g. [@Aharonov]). We think, similar to e.g. ’t Hooft, that (the microscopic form of) holography (we developed in this paper) is the common basis which may unite quantum theory and gravitation.\ ii) The consequences of the BH-entropy being maximal, which is quite uncharacteristic for the ground state entanglement entropy in say ordinary quantum theory, should be further analysed.\ iii) The ADS-CFT-correspondence is regarded in string theory as the paradigm for bulk-boundary correspondence (we mention only the review [@Maldacena1] and the popular account [@Maldacena2]). In it two, at first glance, fundamentally different theories are related to each other, the one living in the bulk, the other living on the [boundary at infinity]{}. We must however say that the concrete physical epistemology of this latter notion is not entirely clear to us. The use of boundaries at infinity is wide spread in holography and is mathematically well-defined, in particular for certain well-adapted coordinate systems being in use in [hyperbolic geometry]{}. But in general it is rather an asymptotic property and not a concrete place. Note that in our approach full information about the interior of a (spatial) region is distributed essentially everywhere in the exterior of the region via wormholes, but usually not in the form of another field theory!\ iv) A virulent problem (the [unitarity problem]{}) in BH-thermodynamics is the question whether a pure state goes over into a mixed state or not, that is, if the laws of ordinary quantum theory are possibly violated in BH-thermodynamics (instead of the many published papers we mention only the reviews by Wald, cited above). This is a quite intricate epistomological problem somewhat similar to the quantum measurement problem. We think, part of the problem is that frequently pure states and mixtures are regarded as complete opposites. But this is not really correct. It is here not the place to go into more details. But in some respect it lies rather in the eye of the beholder. That is, it is the problem of dealing with the complete microscopic information of a state, or rather with some coarse-grained form. Note that in our approach microscopic information is widely scattered via short-cuts or wormholes over essentially the whole space. I.e., it is not fully accessible to a local observer. We recommend the study of some older classics on the [ergodic theorem]{} in quantum statistical mechanics ([@Neumann],[@Pauli],[@Kampen]).\ v) Our analysis should be extended to more general space-times where possibly different distribution laws may show up. 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--- abstract: 'Auto-encoder is an important architecture to understand point clouds in an encoding and decoding procedure of self reconstruction. Current auto-encoder mainly focuses on the learning of global structure by global shape reconstruction, while ignoring the learning of local structures. To resolve this issue, we propose Local-to-Global auto-encoder (L2G-AE) to simultaneously learn the local and global structure of point clouds by local to global reconstruction. Specifically, L2G-AE employs an encoder to encode the geometry information of multiple scales in a local region at the same time. In addition, we introduce a novel hierarchical self-attention mechanism to highlight the important points, scales and regions at different levels in the information aggregation of the encoder. Simultaneously, L2G-AE employs a recurrent neural network (RNN) as decoder to reconstruct a sequence of scales in a local region, based on which the global point cloud is incrementally reconstructed. Our outperforming results in shape classification, retrieval and upsampling show that L2G-AE can understand point clouds better than state-of-the-art methods.' author: - Xinhai Liu - Zhizhong Han - Xin Wen - 'Yu-Shen Liu' - Matthias Zwicker bibliography: - 'reference.bib' title: 'L2G Auto-encoder: Understanding Point Clouds by Local-to-Global Reconstruction with Hierarchical Self-Attention' --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010178.10010224&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Computer vision&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010178.10010224.10010240.10010242&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Shape representations&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10002951.10003317&lt;/concept\_id&gt; &lt;concept\_desc&gt;Information systems Information retrieval&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; {height="5cm" width="16cm"} Introduction ============ In recent years, point clouds have attracted increasing attention due to the popularity of various depth sensors in different applications. Not only the traditional methods, deep neural networks have also been applied to point cloud analysis and understanding. However, it remains a challenge to directly learn from point clouds. Different from 2D images, point cloud is an irregular 3D data which makes it difficult to directly use traditional deep learning framework, e.g., traditional convolution neural network (CNN). The traditional CNN usually requires some fixed spatial distribution around each pixel so as to facilitate the convolution. One way to alleviate the problem is to voxelize a point cloud into voxels and then apply 3D Cov-Nets. However, because of the sparsity of point clouds, it leads to resolution-loss and explosive computation complexity, which sacrifices the representation accuracy. To address above challenges, PointNet [@qi2016pointnet] has been proposed to directly learn shape representations from raw point sets. Along with the availability of directly learning from point clouds by deep learning models, auto-encoder (AE) has become an vital architecture of the involved neural networks. Current AE focuses on the learning of the global structure of point clouds in the encoding and decoding procedure. However, current AE structure is still limited by learning the local structure of point clouds, which tends to be an important piece of information for point cloud understanding. To simultaneously learn global and local structure of point clouds, we propose a novel auto-encoder called Local-to-Global auto-encoder (L2G-AE). Different from traditional auto-encoder, L2G-AE leverages a local region reconstruction to learn the local structure of a point cloud, based on which the global shape is incrementally reconstructed for the learning of the global structure. Specifically, the encoder of L2G-AE can hierarchically encode the information at point, scale and region levels, where a novel hierarchical self-attention is introduced to highlight the important elements in each level. The encoder further aggregates all the information extracted from the point cloud into a global feature. In addition, L2G-AE employs a RNN-based decoder to decode the learned global feature into a sequence of scales in each local region. And based on scale features, the global point cloud is incrementally reconstructed. L2G-AE leverages this local to global reconstruction to facilitate the point cloud understanding, which finally enables local and global reconstruction losses to train L2G-AE. Our key contributions are summarized as follows. - We propose L2G-AE to enable the learning of global and local structures of point clouds in an auto-encoder architecture, where the local structure is very important in learning highly discriminative representations of point clouds. - We propose hierarchical self-attention to highlight important elements in point, scale and region levels by learning the correlations among the elements in the same level. - We introduce RNN as decoding layer in an auto-encoder architecture to employ more detailed self supervision, where the RNN takes the advantage of the ordered multi-scale areas in each local region. Related Work ============ Point clouds is a fundamental type of 3D data format which is very close to the raw data of various 3D sensors. Recently, applications of learning directly on point clouds have received extensive attention, including shape completion [@Stutz2018CVPR], autonomous driving [@qi2017frustum], 3D object detection [@simon2018complex; @Yang2018pixor; @zhou2017voxelnet], recognition and classification [@qi2016pointnet; @NIPS2017_7095; @golovinskiy2009shape; @li2018pointcnn; @wang2018dynamic; @xu2018spidercnn; @shen2018mining; @xie2018attentional; @li2018so; @you2018pvnet], scene labeling [@NIPS2011_4226], upsampling [@yu2018pu; @2019arXiv181111286Y], dense labeling and segmentation [@Wang2018pointseg] , etc. Due to the irregular property of point cloud and the inspiring performances of 2D CNNs on large-scale image repositories such as ImageNet [@deng2009imagenet], it is intuitive to rasterize point clouds into 3D voxels and then apply 3D CNNs. Some studies [@qi2017frustum; @zhou2017voxelnet; @HanCyber17a] represent each voxel with a binary value which indicates the occupation of this location in space. The main problem of voxel-based methods is the fast growth of neural network size and computation complexity with the increasing of spatial resolution. To alleviate this problem, some improvements [@li2016fpnn] have been proposed to explore the data sparsity of point clouds. However, when dealing with point clouds with huge number of points, the complexity of the neural network is still unacceptable. Recently, deep neural networks work quite effectively on the raw 3D point clouds. Different from learning from readered views [@han2018seqviews2seqlabels; @han20193d2seqviews; @han20192seq2seq; @han2019view; @han2019parts4feature; @han20193dviewgraph] 2D meshes [@Zhizhong2016] or 3D voxels [@Zhizhong2016b; @han2017boscc; @han2018deep], PointNet [@qi2016pointnet] is the pioneer study which directly learns the representation for point clouds by computing features for each point individually and aggregating these features with max-pool operation. To capture the contextual information of local patterns inside point clouds, PointNet++ [@NIPS2017_7095] uses sampling and grouping operations to extract features from point clusters hierarchically. Similarly, several recent studies [@riegler2017octnet; @klokov2017escape] explores indexing structures, which divides the input point cloud into leaves, and then aggregates node features from leaves to the root. Inspired by the convolution operation, recent methods [@li2018pointcnn; @wang2018dynamic; @xu2018spidercnn] investigate well-designed CNN-like operations to aggregate points in local regions by building local connections with k-neareat-neighbors (kNN). Capturing the context information inside local regions is very important for the discriminative ability of the learned point cloud representations. KC-Net [@shen2018mining] employs a kernel correlation layer and a graph pooling layer to capture the local patterns of point clouds. ShapeContextNet [@xie2018attentional] extends 2D Shape Context [@belongie2001shape] to the 3D, which divides a local region into small bins and aggregates the bin features. Point2Seqeuce [@liu2019point2sequence] employs an attention-based sequence to sequence architecture to encode the multi-scale area features inside local regions. In order to alleviate the dependence on the labeled data, some studies have performed unsupervised learning for point clouds. FoldingNet [@yang2018foldingnet] proposes a folding operation to deform a canonical 2D grid onto the suface of a point cloud. 3D-PointCapsNet [@zhao2019pcn] employs a dynamic routing scheme in the reconstruction of input point clouds. However, it is difficult for these methods to capture the local patterns of point clouds. Similar to FoldingNet, PPF-FoldNet [@Deng2018ppffoldnet] also learns local descriptors on point cloud with a folding operation. LGAN [@Achlioptas2017latent] proposes an auto-encoder based on PointNet and extends the decoder module to the point cloud generation application with GAN. In this work, we propose a novel auto-encoder architecture to learn representations for point clouds. On the encoder side, an hierarchical self-attention mechanism is applied to embedding the correlation among features in each level. And on the decoder side, an interpolation layer and a RNN decoding layer are engaged to reconstruct multi-scale areas inside local regions. After building local areas, the global point cloud is generated by a fully-connected (FC) layer which acts as a down sampling function. Method ====== Now we introduce the L2G-AE in detail, where the structure is illustrated in Figure \[fig:architecture\]. The input of the encoder is an unordered point set $\bm{P} = \{p_1, p_2, \cdots, p_N \}$ with $N$ ($N=1024$) points. Each point in the point set is composed of a 3D coordinate $(x,y,z)$. L2G-AE first establishes multi-scale areas $\bm{A}_t$ $(t \in [1,T])$ in each local region around the sampled points. Then, a hierarchical feature abstraction is enforced to obtain the global features of input point clouds with self-attentions. In the decoder, we simultaneously reconstruct local scale areas and global point clouds by hierarchical feature decoding. The output of L2G-AE is the reconstructed local areas $\bm{A}^{'}_t$ and the reconstructed $\bm{P}^{'}$ with same number of points to $\bm{P}$. ![A multi-scale example inside a local region of an airplane point cloud, where there are four scales areas $[\bm{A}_1, \bm{A}_2, \bm{A}_3, \bm{A}_4]$ with different colors around the centroid point (red).[]{data-label="fig:multiscale"}](ms.pdf){height="3.5cm" width="5cm"} Multi-scale Establishment ------------------------- To capture fine-grained local patterns of point clouds, we first establish multi-scale areas in each local region, which is similar to PointNet++ [@NIPS2017_7095] and Point2Sequence [@liu2019point2sequence]. Firstly, a subset $\{p_{i_1}, p_{i_2},\\ \cdots, p_{i_M}\}$ of the input points is selected as the centroid of local regions by iterative farthest point sampling (FPS). The latest point $p_{i_j}$ is always the farthest one from the rest points $\{p_{i_1}, p_{i_2}, \cdots, p_{i_{j-1}}\}$. Compared to other sampling method, such as random sampling, FPS can achieve a better coverage of the entire point cloud with the given same number of centroids. As shown in Figure \[fig:multiscale\], around each sampled centroid, $T$ different scale local areas are established continuously by kNN searching with $\{K_1, K_2, \cdots, K_T\}$ nearest points, respectively. An alternative searching method is ball query [@NIPS2017_7095] which selects all points with a radius around the centroid. However, it is difficult for ball query to ensure the information inside local regions, which is sensitive to the sparsity of the input point clouds. ![Self-attention module. The input of this module is a $D_1 \times D_2$ feature map and the output is another $D_1 \times (D_2 + C)$ feature map, where $C$ is a parameter.[]{data-label="fig:c_e"}](CE.pdf){height="4cm" width="8cm"} Hierarchical Self-attention --------------------------- In current work of learning on point clouds, Multi-Layer-Perceptron (MLP) layer is widely applied to integrate multiple features. Traditional MLP layer first abstracts each feature into higher dimension individually and then aggregates these features by a concise max pooling operation. However, these two simple operations can hardly encode the correlation between feature vectors in the feature space. Inspired by the self-attention machanism in [@zhang2018sa], the attention machanism is suitable for improving the traditional MLP by learning the correlation between features. In this work, we propose a self-attention module to make up the defects of the MLP layer with an attention mechanism. Here, self-attention refers to learn the correlation among features in the same level. Different from the raw self-attention, we enforce a hierarchical feature extraction architecture with hierarchical self-attention in the encoder. There are three different levels inside the encoder, including point level, scale level, and region level. At each level, we introduce a self-attention module to learn self-attention weights by mining the correlations among the corresponding feature elements. Consequently, three self-attention modules are designed to propagate features from the lower level to the higher level. Supposed the input of the self-attention module is a feature map $\bm{x} \in \mathbb{R}^{D_1 \times D_2}$, where $D_1$, $D_2$ are the dimensions of the feature map. Therefore, $D_1$, $D_2$ are equal to $K_t$, $3$ in the point level, equal to $T$, $D$ in the scale level and equal to $M$, $D$ in the region level, respectively. As depicted in Figure \[fig:c\_e\], the feature map $\bm{x}$ is first transformed into two feature spaces $\bm{f}$ and $\bm{g}$ to calculate the attention below, where $\bm{f}(\bm{x}) = \bm{W_f x}$, $\bm{g}(\bm{x}) = \bm{W_g x}$, $$\beta_{j,i} = \frac{exp(s_{ij})}{\sum_{i=1}^{D_1} exp(s_{ij})}, \text{where } s_{ij} = \bm{f}(\bm{x_i})^T\bm{g}(\bm{x_j}),$$ and $\beta_{j,i}$ evaluates the attention degree which the model pays to the $i^{th}$ location when synthesizing the $j^{th}$ feature vector. Then the attention result is $\bm{r} = (\bm{r_1}, \bm{r_2}, \cdots, \bm{r_j}, \cdots, \bm{r_{D_1}}) \in \mathbb{R}^{D_1 \times D_2}$, where $$\bm{r_j} = \sum_{i=1}^{D_1} \beta_{j,i}\bm{h}(\bm{x_i}), \text{where }\bm{h}(\bm{x_i}) = \bm{W_h x_i}.$$ In above formulation, $\bm{W_f},\bm{W_g},\bm{W_h} \in \mathbb{R}^{D_2 \times C}$ are learned weight matrices, which are implemented as $1\times1$ convolutions. We use $C = M / 8 $ in the experiments. In addition, inspired by the skip link operation in ResNet[@he2016deep] and DenseNet [@huang2017densely], we further concatenate the result of the attention mechanism with the input feature matrix. Therefore, the final output of the self-attention module is given by $$\bm{o_i} = \bm{x_i} \oplus \bm{r_i},$$ where $\oplus$ is the concatenation operation. This allows the network to rely on the cues among the feature vectors. To aggregate the features with correlation information, a MLP layer and a max pooling operation are employed to integrate the multiple features. In particular, the first self-attention module aggregates the points in a scale to a D-dimensional feature vector. The second one encodes the multi-scale features in a region into a D-dimensional feature. The final one integrates features of all local regions on a point cloud into a 1024-dimensional global feature. Therefore, the encoder hierarchically abstracts point features from the levels of point, scale and region to a global representation of the input point cloud. Interpolation Layer ------------------- The target of the decoder is to generate the points of the local areas and entire points. Previous approaches [@Achlioptas2017latent; @yang2018foldingnet; @Deng2018ppffoldnet] usually use simple fully-connected (FC) layers or MLP layers to build the decoder. However, the expressive ability of the decoder is largely limited without considering the relationship among features. In this work, we propose a progressive decoding way which can be regarded as a reverse process of the encoding. The first step is to generate local region features from the global feature. To propagate the global feature $\bm{g}$ to region features, a simple interpolation operation is first engaged in the decoder. The local region feature $\bm{l}_i$ is calculated by $$\bm{l}_i = \frac{c}{(p_i-p_0)^2}\bm{g}, i \in [1,M],$$ where $c$ ($c={10}^{-10}$) is a constant. Here, $p_0 = (0,0,0)$ is the centroid of the input point cloud after the normalization processing. And $p_i$ is the centroid point of the corresponding local region. By the simple interpolation operation, the spatial distribution information of local region can be integrated to facilitate the feature decoding. The interpolated local region features are then concatenated with skip linked local region features from the encoder. The concatenated features are passed through another MLP layer into a $M \times D$ feature matrix. ![The decoding process of the RNN layer.[]{data-label="fig:rnn"}](RNN.pdf){height="2cm" width="8cm"} RNN Layer --------- Given the feature of local regions, we want to decode the scale level features. Due to the multi-scale setting, the features of different scales in a local region can be regarded as a feature sequence with length $T$. As we all know that recurrent neural network [@hochreiter1997long] has shown excellent performances in processing sequential data. Thus, a RNN decoding layer is employed to generate the multi-scale area features. The decoding process is shown in Figure \[fig:rnn\]. We first replicate the local region feature $\bm{l}_i$ for $T$ times, and the replicated local region features are feed into the RNN layer by $$\bm{h}_t = f(\bm{h}_{t-1}, \bm{l}_i^t), t \in [1,T],$$ where $f$ is a non-linear activation function and $t$ is the index of RNN step. Therefore, the predicted $t^{th}$ area feature $\bm{a}_t$ can be calculated by $$\bm{a}_t = \bm{W}_\theta \bm{h}_t.$$ Here, $\bm{W}_d$ is a learnable weight matrix. To generate the points inside each local area, several FC layers are adopted to reconstruct the points. The local area $\bm{A}^{'}_t$ is reconstructed by $$\bm{A}^{'}_t = \bm{W}_{\theta_t}\bm{a}_t + b_{\theta_t},$$ where $\bm{W}_{\theta_t}$, $b_{\theta_t}$ are weights of the FC layer. Based on the reconstructed local areas, another FC layer is applied to incrementally reconstruct the entire point cloud. All reconstructed areas are concatenated and then passed through the FC layer by $$\bm{P} = \bm{W} [\bm{A}^{'}_1 \oplus \bm{A}^{'}_2 \oplus \cdots \oplus \bm{A}^{'}_T] + b.$$ Here, $\oplus$ represents the concatenation operation. Loss Function ------------- We propose a new loss function to train the network in an end-to-end fashion. There are two parts in the loss function, local scale reconstruction and global point cloud reconstruction, respectively. As mentioned earlier, we should encourage accurate reconstruction of local areas and the global point cloud at the same time. Suppose $\bm{A}_t$ is the $t^{th}$ scale area in the multi-scale establishment subsection, then, the local reconstruction error for $\bm{A}^{'}_t$ is measured by the well-known Chamfer distance, $$\begin{split} L_{local} = d_{CH}(\bm{A}_t, \bm{A}^{'}_t) = \sum_{t=1}^T (\frac{1}{|\bm{A}_t|}\sum_{p_i \in \bm{A}_t} \min_{p^{'}_i \in \bm{A}^{'}_t} \lVert p_i - p^{'}_i \rVert_2 \\ + \frac{1}{|\bm{A}^{'}_t|}\sum_{p^{'}_i \in \bm{A}^{'}_t} \min_{p_i \in \bm{A}_t} \lVert p_i - p^{'}_i \rVert_2), \end{split}$$ Similarly, let the input point set be $\bm{P}$ and the reconstructed point set be $\bm{P}^{'}$. The global reconstruction error can be denoted by $$\begin{split} L_{global} = d_{CH}(\bm{P}, \bm{P}^{'}) = \frac{1}{|\bm{P}|}\sum_{p_i \in \bm{P}} \min_{p^{'}_i \in \bm{P}^{'}} \lVert p_i - p^{'}_i \rVert_2 \\ + \frac{1}{|\bm{P}^{'}|}\sum_{p^{'}_i \in \bm{P}^{'}} \min_{p_i \in \bm{P}} \lVert p_i - p^{'}_i \rVert_2. \end{split}$$ Altogether, the network is trained end-to-end by minimizing the following joint loss function $$L = L_{local} + \gamma L_{global},$$ where $\gamma$ ($\gamma=1$) is the proportion of two part errors. Experiments =========== In this section, we first investigate how some key parameters affect the performance of L2G-AE in the shape classification task on ModelNet10 [@wu20153d]. Then, an ablation study is done to show the effectiveness of each module in L2G-AE. Finally, we further evaluate the performances of L2G-AE in multiple applications including 3D shape classification, 3D shape retrieval and point cloud upsampling. Network Configuration --------------------- In L2G-AE, we first sample $M = 256$ points as the centroids of local regions by FPS. Then, around each centroid, a kNN searching algorithm selects $T=4$ scale areas with $[K_1=16, K_2=32, K_3=64, K_4=128]$ points inside each area. In the multi-level feature propagation process, we initialize the feature dimension $C = M/8 = 32$ and $D = 256$. The encoder learns a 1024-dimension global feature for the input point cloud through hierarchical feature extraction. Similarly, the decoder hierarchically reconstructs local scales and global point cloud. In the RNN decoding layer, we adopt LSTM as the default RNN cell with hidden state dimension $h = D = 256$. In the experiment, we train our network on a NVIDIA GTX 1080Ti GPU using ADAM optimizer with the initial learning rate of 0.0001 and batch size of 8. The learning rate is decreased by 0.3 for every 20 epochs. Parameters ---------- All experiments on parameter comparison are evaluated under ModelNet10. ModelNet10 contains 4899 CAD models from 10 categories and is split into 3991 for training and 908 for testing. For each model, we adopt 1024 points which are uniformly sampled from mesh faces and are normalized into a unit ball before being fed into the network. During the training process, the loss function keeps decreasing and stabilizes around the 180th epoch. To acquire the accuracies on ModelNet10, we train a linear SVM from the global features obtained by the auto-encoder. Specifically, the OneVsRest strategy is adopted with the linearSVM function as the kernel. We first explore the number of sampled points $M$ which determines the distribution of local regions inside point clouds. In the experiment, we keep the network settings as depicted in the network configuration and vary the number of sampled points $M$ from 128 to 320. $M$ 128 192 256 320 --------- ------- ------- ----------- ------- -- Acc (%) 93.83 94.38 **95.37** 93.94 : The effects of the number of sampled points $M$ under ModelNet10.[]{data-label="table:sampled_points"} The results are shown in Table \[table:sampled\_points\], where the instance accuracies on the benchmark of ModelNet10 have a tendency to rise first and then fall. This comparison implies that L2G-AE can effectively extract the contextual information in point clouds by multi-level feature propagation and $M=256$ is an optimal choice which can well cover input point clouds without excessive redundant. ![The reconstructed results with different sampled points, where the CD represents the Chamfer distance between ground-truth and the reconstructed point cloud.[]{data-label="fig:sample_re"}](sample.pdf){height="1.5cm" width="8cm"} To learn the reconstructed results intuitively, Figure \[fig:sample\_re\] shows the reconstructed point clouds with different sampled points. According to Chamfer distances, L2G-AE can also reconstruct the input point cloud with the varying of sampled points. With keeping the sampled points $M = 384$, we investigate the key parameter dimension $C$ inside the self-attention modules. To unify the parameter in self-attention module, we keep the same dimension $C$ in different semantic levels. We change the default $C = 32$ to 16 and 64, respectively. In Table \[table:c\], L2G-AE achieves the best performance when the feature dimension $C$ is 32. $M$ 16 32 64 --------- ------- ----------- ------- -- Acc (%) 93.94 **95.37** 94.16 : The effects of the feature dimension $C$ of the self-attention module under ModelNet10.[]{data-label="table:c"} Finally, we show the effects of feature dimension of local areas $D$ and the global feature $D_{global}$. The dimension is varied as shown in Table \[table:d\] and Table \[table:dglobal\]. Neither the biggest nor the smallest, L2G-AE gets better performances when $D$, $D_{global}$ are set to 256 and 1024 respectively. There is a trade-off between the network complexity and the expressive ability of our L2G-AE. $D$ 128 256 512 --------- ------- ----------- ------- -- Acc (%) 93.72 **95.37** 93.28 : The effects of the local feature dimension $D$ on ModelNet10.[]{data-label="table:d"} $D_{global}$ 512 1024 2048 -------------- ------- ----------- ------- -- Acc (%) 94.16 **95.37** 93.94 : The effects of the global feature dimension $D_{global}$ under ModelNet10.[]{data-label="table:dglobal"} Ablation Study -------------- To quantitatively evaluate the effect of the self-attention module, we show the performances of L2G-AE under four settings: with point level self-attention module only (PL), with area level self-attention module only (AL), with region level self-attention module only (RL), remove all self-attention modules (NSA) and with all self-attention modules (ASA). As shown in Table \[table:rsa\], the self-attention module is effective in learning highly discriminative representations of point clouds by capturing the correlation among feature vectors. The results with only one self-attention module outperform the results without any self-attention module. And we achieve the best performance when three self-attention modules work together. The performance of self-attentions is affected by the discriminative ability of features. At the area level, the features of areas in the same region are similar, since there are only four areas, which makes the self-attention at area level contribute the least among all three self-attentions. In contrast, at the point level and the region level, the features of points or regions change a lot, so these self-attentions contribute more. From our observation, the results of PL and RL are coincidentally equal in the experiments. Metric PL AL RL NSA ASA --------- ------- ------- ------- ------- ----------- Acc (%) 94.16 94.05 94.16 93.72 **95.37** : The effects of the self-attention module on ModelNet10.[]{data-label="table:rsa"} ![The reconstruction results of L2G-AE with only the local loss and only the global loss.[]{data-label="fig:loss_compare"}](loss_compare.pdf){width="8cm" height="2cm"} After exploring the self-attention module, we also discuss the contributions of the two loss functions $L_{local}$ and $L_{global}$. In Table \[table:loss\], the results with local loss only (Local), global loss only (Global) and two losses together (Local + Global) are listed. The local loss function is very important in capturing local patterns of point clouds. And the two loss functions together can further enhance the classification performances of our neural network. In addition, Figure \[fig:loss\_compare\] shows the reconstruction results of our L2G-AE with only local loss and only global loss, respectively. From the results of the reconstructed point clouds, L2G-AE can reconstruct the input point cloud with only part of the joint loss function. In particular, the local reconstructed result in Figure \[fig:loss\_compare\] is a dense point cloud. Metric Local Global Local+Global --------- ------- -------- -------------- Acc (%) 94.71 92.84 **95.37** : The effects of the two loss functions $L_{local}$ and $L_{global}$ on ModelNet10.[]{data-label="table:loss"} Methods Supervised MN40 MN10 ------------------ ------------ ----------- ----------- -- PointNet Yes 89.20 - PointNet++ Yes 90.70 - ShapeContextNet Yes 90.00 - Kd-Net Yes 91.80 94.00 KC-Net Yes 91.00 94.4 PointCNN Yes 92.20 - DGCNN Yes 92.20 - SO-Net Yes 90.90 94.1 Point2Sequence Yes 92.60 95.30 MAP-VAE No 90.15 94.82 LGAN No 85.70 95.30 LGAN(MN40) No 87.27 92.18 FoldingNet No 88.40 94.40 FoldingNet(MN40) No 84.36 91.85 Our No **90.64** **95.37** : The comparison of classification accuracy (%) under ModelNet10 and ModelNet40.[]{data-label="table:compare"} Classification -------------- In this subsection, we evaluate the performance of L2G-AE under ModelNet10 and ModelNet40 benchmarks, where ModelNet40 contains 12, 311 CAD models which is split into 9, 843 for training and 2, 468 for testing. Table \[table:compare\] compares L2G-AE with state-of-the-art methods in the shape classification task on ModelNet10 and ModelNet40. The compared methods include PointNet [@qi2016pointnet], PointNet++ [@NIPS2017_7095], ShapeContextNet [@xie2018attentional], KD-Net [@klokov2017escape], KC-Net [@shen2018mining], PointCNN [@li2018pointcnn], DGCNN [@wang2018dynamic], SO-Net [@li2018so], Point2Sequence [@liu2019point2sequence], MAP-VAE [@Zhizhong2019mapvae], LGAN [@Achlioptas2017latent] and FoldingNet [@yang2018foldingnet]. L2G-AE significantly outperforms all the unsupervised competitors under ModelNet10 and ModelNet40, respectively. In particular, L2G-AE achieves accuracy $95.37\%$ which is even higher than other methods of supervision under ModelNet10. Although the results of LGAN [@Achlioptas2017latent] and FoldingNet [@yang2018foldingnet] also show good performances under ModelNet10 and ModelNet40. This is because these methods are trained under a version of ShapeNet55 that contains more than 57,000 3D shapes. However, this version of ShapeNet55 dataset is not avaiable for public download from the official website. Therefore, we train all these methods under ModelNet40 for the fair comparison. Methods LGAN FoldingNet Our --------- ------- ------------ ----------- Acc (%) 49.94 53.42 **67.81** : The comparison of retrieval in terms of under ModelNet10.[]{data-label="table:retrieval"} ![The comparison of PR curves for retrieval under ModelNet10.[]{data-label="fig:pr"}](pr.pdf){height="3cm" width="4cm"} $10^{-3}$ bathtub bed chair desk dresser monitor n.stand sofa table toilet ----------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- -- PU **1.01** **1.12** **0.82** **1.22** 1.55 **1.19** **1.77** **1.13** **0.69** **1.39** EC 1.43 1.81 1.80 1.30 1.43 2.04 1.88 1.79 1.00 1.72 Our 1.74 1.46 1.58 2.08 **1.40** 1.61 1.86 1.67 1.86 2.10 Retrieval --------- L2G-AE is further evaluated in the shape retrieval task under ModelNet10 and compared with some other unsupervised methods of learning on point clouds. The compared results include two state-of-the-art unsupervised methods for point clouds, i.e., LGAN [@Achlioptas2017latent] and FoldingNet [@yang2018foldingnet]. The target of shape retrieval is to obtain the relevant information of a inquiry from a collection. In these experiments, the 3D shapes in the test set are used as quires to retrieve the rest shapes in the same set, and mean Average Precision (mAP) is used as a metric. As shown in Table \[table:retrieval\], our results outperform all the compared results under ModelNet10. It shows that L2G-AE can be effect in improving the performance of unsupervised shape retrieval on point clouds. Their PR curves under ModelNet10 are also compared in Figure \[fig:pr\] which intuitively shows the performances of these three methods. Unsupervised Upsampling for Point Clouds ---------------------------------------- Benefit from the design of local to global reconstruction, it is competent for our L2G-AE to be applied in the unsupervised point cloud upsampling application. In the local reconstruction, a dense point cloud is obtained by reconstructing multiple local scales with overlapping. Therefore, it is convenient to produce the upsampling results by downsampling from the dense local reconstructed results using some unsupervised methods, such as random sampling or farthest point sampling. As far as we know, L2G-AE is the first method which performs point cloud upsampling with deep neural networks in an unsupervised manner. To evaluate the performance of L2G-AE, We compare our method on relatively sparse (625 points) inputs with state-of-the-art supervised point cloud upsampling methods, including PU-Net [@yu2018pu] and EC-Net [@yu2018ecnet]. The target of upsampling is to generate a dense point clouds with 10000 points. For PU-Net and EC-Net, the $16 \times$ results (10000 points) are obtained from inputs (625 points) in a supervised manner. Differently, L2G-AE first obtains the local reconstruction results and then downsamples them to 10000 points. ![Some upsampled results of L2G-AE.[]{data-label="fig:upsampled"}](upsampling.pdf){width="8cm" height="6cm"} As shown in Table \[table:upsampling\], mean Chamfer Distance (mCD) is used as a metric for quantitative comparison with PU-Net (PU) and EC-Net (EC) under ModelNet10. Although the results of PU-Net and EC-Net are better than “Our” in some classes under ModelNet10, the most likely reason is that the ground-truth is not visible to L2G-AE in the training. In addition, the input point cloud with 625 points contains very limited information. Figure \[fig:upsampled\] shows some upsamled results of our L2G-AE. ![Some reconstructed examples of L2G-AE.[]{data-label="fig:reconstruct"}](reconstruct.pdf){height="7cm" width="7cm"} ![Some examples of the attention in the region level, where each subfigure represents a 3D object.[]{data-label="fig:attetnion"}](attention_rl.pdf){height="10cm" width="7cm"} Visualization ------------- In this section,we will show some important visualization results of L2G-AE. Firstly, some reconstructed point clouds by L2G-AE are listed with the ground-truths as shown in Figure \[fig:reconstruct\]. From the results, the reconstructed point clouds of L2G-AE are consistent with the ground-truths. Then, some visualizations of the attention map inside self-attention modules are engaged to show the effect of attentions in the hierarchical feature abstraction. There are three self-attention modules in the encoder, and we first visualize the attention map inside the local region level. For intuitively understanding, we directly attach the attention values to the centroids of local regions and then show these centroids. By summing attention map by column in the region level, the attention value of each centroid is caculated. For example, a $256 \times 256$ attention map is translated to a 256-dimension attention vector, when the number of sampled centroids is 256. Then, both the size and the color of centroids are associated with the attention values. Therefore, the centroids with lighter colors and larger sizes indicate larger attention values. As depicted in Figure \[fig:attetnion\], we show some examples of the region level attention. Figure \[fig:attetnion\] shows that the self-attention in the region level tends to on special local regions at conspicuous locations such as edges, corners or protruding parts. ![Some examples of the attention in the scale level. The abscissa represents the 4 scales $[s_1,s_2,s_3,s_4]$ around each centroid in a point cloud and the ordinate indicates the index of 256 centroids, where each subfigure represents a 3D object.[]{data-label="fig:sl_attention"}](sl_attention.pdf){width="7cm" height="6cm"} ![Some examples of the attention in the point level, where the four subfigures in each row represent the four scales of a local region.[]{data-label="fig:pl_attention"}](pl_attention.pdf){width="7cm" height="6cm"} Similarly, we also show some examples of the scale level attention in Figure \[fig:sl\_attention\] and the point level attention in Figure \[fig:pl\_attention\]. In Figure \[fig:sl\_attention\], each image shows the 4 scale attention values around 256 sampled centroids of a point cloud. And the color indicates the value of attention, where large attention value corresponds to a bright color such as yellow. The results indicate that the network tends to focus on the $4^{th}$ scale which contains more information of local structures. In Figure \[fig:pl\_attention\], each row represents the 4 scale areas around a centroid. In different scale areas, the network concern on different points inside the areas to capture the local patterns in the local region. Conclusions =========== In this paper, we propose a novel local to global Auto-encoder framework for point cloud understanding in the shape classification, retrieval and point cloud upsampling tasks. In the encoder, a self-attention mechanism is employed to explore the correlation among features in the same level. In contrast, an interpolation layer and RNN decoding layer successfully reconstruct local scales and global point clouds hierarchically. Experimental results show that our method achieves competitive performances with state-of-the-art methods.

--- abstract: 'The pants graph has proved to be influential in understanding 3-manifolds concretely. This stems from a quasi-isometry between the pants graph and the Teichmüller space with the Weil-Petersson metric. Currently, all estimates on the quasi-isometry constants are dependent on the surface in an undiscovered way. This paper starts effectivising some constants which begins the understanding how relevant constants change based on the surface. We do this by studying the hyperbolicity constant of the pants graph for the five-punctured sphere and the twice punctured torus. The hyperbolicity constant of the relative pants graph for complexity 3 surfaces is also calculated. Note, for higher complexity surfaces, the pants graph is not hyperbolic or even strongly relatively hyperbolic.' author: - Ashley Weber bibliography: - 'mybib.bib' date: title: Hyperbolicity constants for pants and relative pants graphs --- Introduction ============ The pants graph has been instrumental in understanding Teichmüller space. This is because the pants graph is quasi-isometric to Teichmüller space equipped with the Weil-Petersson metric [@Brock-WPtoPants]. Brock and Margalit used pants graphs to show that all isometries of Teichmüller space with the Weil-Petersson metric arise from the mapping class group of the surface [@BM-WPisom]. This relationship was also used to classify for which surfaces the associated Teichmüller space is hyperbolic. The relationship between the pants graph and Teichmüller space has been used to study volumes of 3-manifolds [@Brock-WPtoPants; @Brock-WPtrans]. In particular, it has been used to relate volumes of the convex core of a hyperbolic 3-manifold to the distance of two points in Teichmüller space. It has also related the volume of a hyperbolic 3-manifold arising from a psuedo-Anosov element in the mapping class group to the translation length of the psuedo-Anosov element as applied to the pants graph. Both of these relations have constants which depend on the surface; this paper is the start of effectivising those constants. Notice Aougab, Taylor, and Webb have some effective bounds on the quasi-isometry bounds, however even these still depend on the surface in a way that is unknown [@ATW]. Let $S_{g,p}$ be a surface with genus $g$ and $p$ punctures. We define the complexity of a surface to be $\xi(S_{g,p}) =3g + p - 3 $. Brock and Farb have shown that the pants graph is hyperbolic if and only if the complexity of the surface is less than or equal to $2$ [@BF]. Brock and Masur showed that in a few cases the pants graph is strongly relatively hyperbolic, specifically when $\xi(S) = 3$ [@BM]. Even though hyperbolicity is well studied for the pants graph, the hyperbolicity constants associated with the pants graph or the relative pants graph is not. In addition to having a further understanding of the quasi-isometry mentioned above and all of its applications, actual hyperbolicity constants are useful in answering questions about asymptotic time complexity of certain algorithms, especially those involving the mapping class group. More speculatively, estimates on hyperbolicity constants may be crucial to effectively understand the virtual fibering conjecture, which relates the geometry of the fiber to the geometry of the base surface. The focus of this paper is to find hyperbolicity constants for the pants graph and relative pants graph, when these graphs are hyperbolic. For a surface $S = S_{0,5}, S_{1,2}$, ${\mathcal{P}}(S)$ is $2,691,437$-thin hyperbolic. Computing the asymptotic translation lengths of an element in the mapping class group on ${\mathcal{P}}(S)$ is a question explored by Irmer [@Irmer]. Bell and Webb have an algorithm that answers this question for the curve graph [@BellWebb]. Combining the works of Irmer, and Bell and Webb, one could conceivably come up with an algorithm for asymptotic translation lengths on ${\mathcal{P}}(S)$. In this case, the above Theorem would put a bound on the run-time of the algorithm in the cases that $S = S_{0,5}, S_{1,2}$. We now turn our attention to the relatively hyperbolic cases. For a surface $S = S_{3,0}, S_{1,3}, S_{0,6}$, ${\mathcal{P}}_{rel}(S)$ is $2,606,810,489$-thin hyperbolic. To show both of our main theorems, we construct a family of paths that is very closely related to hierarchies, introduced in [@MMII]. We show that this family of paths satisfies the thin triangle condition which, by a theorem of Bowditch, allows us to conclude the whole space is hyperbolic [@Bow]. A key tool used throughout is the Bounded Geodesic Image Theorem [@MMII]. This theorem allows us to control the length of geodesics in subspaces. This method cannot be made to generalize to pants graphs in general since any pants graph of a surface with complexity higher than $3$ is not strongly relatively hyperbolic [@BM]. Although, this method may be able to be used for other graphs which are variants on the pants graph. One might consider approaching this problem by finding the sectional curvature of Teichmüller space and using the quasi-isometry to inform on the hyperbolicity constant of the pants graph. If the sectional curvature is bounded away from zero, one can relate the curvature of the space to the hyperbolicity constant of the space. However, the sectional curvature of Teichmüller sapce is not bounded away from zero [@Huang]. Therefore, this technique cannot be used. **Acknowledgments:** I would like to thank my advisor, Jeff Brock, for suggesting this problem, support, and helpful conversations. I’d also like to thank Tarik Aougab and Peihong Jiang for helpful conversations. Preliminaries ============= Hyperbolicity ------------- Assume $\Gamma$ is a connected graph which we equip with the metric where each edge has length 1. We give two definitions of a graph being hyperbolic. A triangle in $\Gamma$ is $k$-*centered* if there exists a vertex $c \in \Gamma$ such that $c$ is distance $\leq k$ from each of its three sides. $\Gamma$ is $k$-*centered hyperbolic* if all geodesic triangles (triangles whose edges are geodesics) are $k$-centered. We say a triangle in $\Gamma$ is $\delta$-*thin* if each side of the triangle is contained in the $\delta$-neighborhood of the other two sides for some $\delta \in {\mathbb{R}}$. A graph is $\delta$-*thin hyperbolic* if all geodesic triangles are $\delta$-thin. Note that $\delta$-thin hyperbolic and $k$-centered hyperbolic are equivalent up to a linear factor [@ABC]. \[centered to thin\] If $\Gamma$ is $k$-centered hyperbolic then $\Gamma$ is $4k$-thin hyperbolic. The following proof is very similar to the proof of an existence of a global minsize of triangles implies slim triangles in [@ABC] (Proposition 2.1). We denote $[a,b]$ as a geodesic between $a$ and $b$; if $c \in [a,b]$ then $[a, c]$ or $[c,b]$ refers to the subpath of $[a,b]$ with $c$ as one of the endpoints. Consider the triangle $xyz$ and assume it is $k$-centered. Let $p$ be the centered point and $x'$ be the point on the edge $[y,z]$ closest to $p$. Similarly define $y'$ and $z'$. Suppose there is a point $t \in [x,z']$ such that $d(t, [x, y']) > 2k$. Let $u$ be the point in $[t, z']$ nearest to $t$ such that $d(u, u') = 2k$ for some point $u' \in [x, y']$, see Figure \[center to thin figure\]. Consider the geodesic triangle $uu'x$. There exists points $a$, $b$, and $c$ on the three sides of $uu'x$ that are less than or equal to $k$ away from some point $q$, see Figure \[center to thin figure\]. Since $a \in [x, u]$, by assumption $a$ does not lie in $[t, u]$ and $d(u, a) \leq 4k$. So $d(t, u') \leq 4k$ or $d(t, c) \leq 4k$, making the triangle $xyz$ $4k$-thin. Bowditch shows, in [@Bow] Proposition 3.1, that we don’t always have to work with geodesic triangles to show hyperbolicity of a graph. \[subset hyperbolic\] Given $h \geq 0$, there exists $\delta \geq 0$ with the following property. Suppose that $G$ is a connected graph, and that for each $x, y \in V(G)$, we have associated a connected subgraph, ${\mathcal{L}}(x,y) \subset G$, with $x, y \in {\mathcal{L}}(x,y)$. Suppose that: 1. for all $x, y, z \in V(G)$, $${\mathcal{L}}(x,y) \subset N_h({\mathcal{L}}(x,z) \cup {\mathcal{L}}(z, y))$$ and 2. for any $x, y \in V(G)$ with $d(x,y) \leq 1$, the diameter of ${\mathcal{L}}(x,y)$ in $G$ is at most $h$. Then $G$ is $\delta$-thin hyperbolic. In fact, we can take any $\delta \geq (3m-10h)/2$, where $m$ is any positive real number satisfying $$2h(6 + \log_2(m+2)) \leq m.$$ Graphs ------ Let $S = S_{g,p}$ be a surface where $g$ is the genus and $p$ is the number of punctures. We define $\xi(S_{g,p}) = 3g + p -3$ and refer to $\xi(S_{g,p})$ as the complexity of $S_{g,p}$. When $\xi(S) > 1$ the curve graph of $S$, ${\mathcal{C}}(S)$, originally introduced by Harvey in [@Harvey], is a graph whose vertices are homotopy classes of essential simple closed curves on $S$ and there is an edge between two vertices if the curves can be realized disjointly, up to isotopy. From here on when we talk about curves we really mean a representative of the homotopy class of an essential, non-peripheral, simple closed curve. When $\xi(S) = 1$, the definition of the curve graph is slightly altered in order to have a non-trivial graph: the vertices have the same definition, but there is an edge between two curves if they have minimal intersection number. We can similarly define the *arc and curve graph*, ${\mathcal{A}}{\mathcal{C}}(S)$, where a vertex is either a homotopy class of curves or homotopy class of arcs and the edges represent disjointness. This definition is the same for all surfaces such that $\xi(S) > 0$. A related graph associated to a surface is the pants graph. We call a maximal set of disjoint curves on a surface a *pants decomposition*. For $\xi(S) \geq 1$ the *pants graph*, denoted ${\mathcal{P}}(S)$, of a surface $S$ is a graph whose vertices are homotopy classes of pants decompositions and there exists an edge between two pants decompositions if they are related by an elementary move. Pants decompositions $\alpha$ and $\beta$ differ by an elementary move if one curve, $c$, from $\alpha$ can be deleted and replaced by a curve that intersects $c$ minimally to obtain $\beta$, see Figure \[elementary moves\]. We equip both graphs with the metric where each edge is length 1. Then ${\mathcal{C}}(S)$ and ${\mathcal{P}}(S)$ are complete geodesic metric spaces. The hyperbolicity of these graphs have been studied before. \[curve hyp\] For any hyperbolic surface $S$, ${\mathcal{C}}(S)$ is $17$-centered hyperbolic. Brock and Farb showed: For any hyperbolic surface $S$, ${\mathcal{P}}(S)$ is hyperbolic if and only if $\xi(S) \leq 2$. Relative graphs --------------- Let $S$ be a hyperbolic surface such that $\xi(S) \geq 3$. We say that a curve $c \in {\mathcal{C}}(S)$ is *domain separating* if $S \backslash c$ has two components of positive complexity. Each domain separating curve $c$ determines a set in ${\mathcal{P}}(S)$, $X_c = \{\alpha \in {\mathcal{P}}(S) | c \in \alpha \}$. To form the *relative pants graph*, denoted ${\mathcal{P}}_{rel}(S)$, we add a point $p_c$ for each domain separating curve and an edge from $p_c$ to each vertex in $X_c$, where each edge has length $1$. Effectively, we have made the set $X_c$ have diameter $2$ in the relative pants graph. Brock and Masur have shown: For $S$ such that $\xi(S) = 3$, ${\mathcal{P}}_{rel}(S)$ is hyperbolic. Paths in the Pants Graph ------------------------ Here we describe how we will get a path in ${\mathcal{P}}(S)$ if $\xi(S) =2$ or ${\mathcal{P}}_{rel}(S)$ if $\xi(S) = 3$. The paths for ${\mathcal{P}}(S)$ are hierarchies and were originally introduced by Masur and Minsky in [@MMII] (in more generality than we will use here); the paths in ${\mathcal{P}}_{rel}(S)$ are motivated by hierarchies. Take two pants decompositions, $\alpha = \{ \alpha_0, \alpha_1\}$ and $\beta = \{ \beta_0, \beta_1\}$, in ${\mathcal{P}}(S)$ where $S = S_{0,5}$ or $S_{1,2}$. To create a hierarchy between $\alpha$ and $\beta$ first connect $\alpha_0$ and $\beta_0$ with a geodesic path in ${\mathcal{C}}(S)$. This geodesic is referred to as the *main geodesic*, $g_{\alpha\beta} = \{ \alpha_0 = g_0, \ldots, g_n = \beta_0\}$. For each $g_i$, $0 \leq i \leq n$, connect $g_{i-1}$ to $g_{i+1}$ by a geodesic, $\gamma_i$, in ${\mathcal{C}}(S\backslash g_i)$, where $g_{-1} = \alpha_1$ and $g_{n+1} = \beta_1$. The collection of all of these geodesics is a *hierarchy* between $\alpha$ and $\beta$, generally pictured as in Figure \[Hierarchy picture\]. We often refer to the geodesic $\gamma_i$ as the geodesics whose domain is ${\mathcal{C}}(S \backslash g_i)$ or the geodesic connecting $g_{i-1}$ and $g_{i+1}$. We can turn a hierarchy into a path in ${\mathcal{P}}(S)$ by looked at all edges in turn, as pictured in Figure \[Hierarchy picture\]. We will often blur the line between the hierarchy being a path in the pants graph or a collection of geodesics - and refer to both as the hierarchy between $\alpha$ and $\beta$. Let $\xi(S) =3$. We make a path in ${\mathcal{P}}_{rel}(S)$ using a similar technique. Take two pants decompositions in ${\mathcal{P}}_{rel}(S)$, $\alpha = \{\alpha_0, \alpha_1, \alpha_2\}$ and $\beta = \{\beta_0, \beta_1, \beta_2\}$. Connect $\alpha_0$ to $\beta_0$ with a geodesic $g_{\alpha\beta}$ in ${\mathcal{C}}(S)$, we still refer to this as the main geodesic. For every non-domain separating curve $w \in g$, connect $w^{-1}$ to $w^{+1}$ with a geodesic, $h$, in ${\mathcal{C}}(S \backslash w)$ where $w^{-1}$ and $w^{+1}$ are the curves before and after $w$ in $g$. If $w = \alpha_0$ then $w^{-1} = \alpha_1$ and if $w = \beta_0$ then $w^{+} = \beta_1$. Now for each non-domain separating curve $z \in h$ connect $z^{-1}$ to $z^{+1}$ with a geodesic in ${\mathcal{C}}(S \backslash (w \cup z))$, where $z^{-1}$ and $z^{+1}$ are the curves before and after $z$ in $h$. If $z = w^{-1}$ then $z^{-1}$ is the curve preceding $w$ in the geodesic whose domain is ${\mathcal{C}}(S \backslash w^{-1})$. If $z = w^{+1}$ then $z^{+1}$ is the curve following $w$ in the geodesic whose domain is ${\mathcal{C}}(S \backslash w^{+1})$ (see Figure \[general hierarchy\] (top)). We can get a path in ${\mathcal{P}}_{rel}(S)$ by a similar process as before - going along each of the edges. Whenever we come across a domain separating curve, $c$, where $c$ is in the main geodesic or in a geodesic whose domain is ${\mathcal{C}}(S \backslash w)$ where $w$ is in the main geodesic, we add in the point $p_c$ into the path before moving on. For an example see Figure \[general hierarchy\]. These paths are *relative 3-archies*. As before, we will blur the line between the collection of geodesics and the path of a relative 3-archy. When discussing hierarchies (or relative 3-archies), subsurface projections of curves or geodesics are involved. The following maps are to define what is meant by subsurface projections [@MMII]. An *essential subsurface* is a subsurface where each boundary component is essential. Let ${\mathscr{P}}(X)$ be the set of subsets of $X$. For a set $A$ we define $f(A) = \cup_{a \in A}f(a)$, for any map $f$. Take an essential, non-annular subsurface $Y \subset S$. We define a map $$\phi_Y: {\mathcal{C}}(S) {\longrightarrow}{\mathscr{P}}({\mathcal{A}}{\mathcal{C}}(Y))$$ such that $\phi_Y(a)$ is the set of arcs and curves obtained from $a \cap Y$ when $\partial Y$ and $a$ are in minimal position. Define another map $$\psi_Y : {\mathscr{P}}({\mathcal{A}}{\mathcal{C}}(Y)) {\longrightarrow}{\mathscr{P}}({\mathcal{C}}(Y))$$ such that if $a$ is a curve, then $\psi_Y(a) = a$, and if $b$ is an arc, then $\psi_Y(b)$ is the union of the non-trivial components of the regular neighborhood of $(b\cap Y) \cup \partial Y$ (see Figure \[nbhd\]). Composing these two maps we define the map $$\begin{aligned} \pi_Y: {\mathcal{C}}(S) &{\longrightarrow}{\mathscr{P}}({\mathcal{C}}(Y)) \\ c &\longmapsto \psi_Y(\phi_Y(c))\end{aligned}$$ We use this map to define distances in a subsurface: for any two sets $A$ and $B$ in ${\mathcal{C}}(S)$, $$d_Y(A, B) = d_Y(\pi_Y(A), \pi_Y(B)).$$ We often refer to this as the distance in the subsurface $Y$. The relationship between hierarchies and these maps give rise to some useful properties including the Bounded Geodesic Image Theorem which was originally proven by Masur-Minsky [@MMII]. \[bounded geodesic image\] Let $Y$ be a subsurface of $S$ with $\xi(Y) \neq 3$ and let $g$ be a geodesic segment, ray, or biinfinite line in ${\mathcal{C}}(S)$, such that $\pi_Y(v) \neq \emptyset$ for every vertex of $v$ of $g$. There is a constant $M$ depending only on $\xi(S)$ such that $${\mathrm{diam}}_Y(g) \leq M.$$ It can be shown that $M$ is at most $100$ for all surfaces [@Webb]. Hyperbolicity of Pants Graph for Complexity 2 ============================================= In this section we explore the hyperbolicity constant for the pants graph of surfaces with complexity $2$. Before we state any results, some notation must be discussed. Throughout the paper we denote $[a, b]_\Sigma$ as a geodesic in ${\mathcal{C}}(\Sigma)$ connecting $a$ to $b$, for any surface $\Sigma$. If a geodesic satisfying this is contained in a hierarchy (or relative 3-archy, in later sections) being discussed, $[a,b]_\Sigma$ denotes the geodesic in the hierarchy. \[hierarchy k-centered\] For $S = S_{0,5}, S_{1,2}$, hierarchy triangles in ${\mathcal{P}}(S)$ are $8,900$-centered. Let $S = S_{0,5}$ or $S_{1,2}$. Take three pants decompositions $\alpha = \{\alpha_0, \alpha_1\}$, $\beta = \{\beta_0, \beta_1\}$, and $\gamma = \{\gamma_0, \gamma_1\}$ in $S$. Consider the triangle $\alpha\beta\gamma$ in ${\mathcal{P}}(S)$ where the edges are taken to be hierarchies instead of geodesics. There are three cases: 1. All three main geodesics have a curve in common. 2. Any two of the main geodesics share a curve, but not the third. 3. None of the main geodesics have common curves. In all three cases we will find a pants decomposition such that the hierarchy connecting this pants decomposition to each edge in $\alpha\beta\gamma$ is less than $8,900$. **Case 1**: Assume the main geodesics of all three edges share the curve $v \in {\mathcal{C}}(S)$. Define $v_{\alpha \beta}^{-1}$ to be the curve on $g_{\alpha \beta}$ preceding $v$ and $v_{\alpha \beta}^{+1}$ the curve on $g_{\alpha \beta}$ following $v$ when viewing $g_{\alpha\beta}$ going from $\alpha_0$ to $\beta_0$. Similarly define $v_{\alpha \gamma}^{-1}$, $v_{\alpha \gamma}^{+1}$, $v_{\beta \gamma}^{-1}$, and $v_{\beta \gamma}^{+1}$. See Figure \[Case 1\]. We want to show the geodesics connecting $v_*^{-1}$ to $v_*^{+1}$ in ${\mathcal{C}}(S \backslash v)$ are not too far apart in ${\mathcal{C}}(S \backslash v)$. Connect $v_{\alpha\beta}^{-1}$ to $v_{\alpha\gamma}^{-1}$, $v_{\alpha\gamma}^{+1}$ to $v_{\beta \gamma}^{+1}$ and $v_{\beta\gamma}^{-1}$ to $v_{\alpha\beta}^{+1}$ by geodesics in ${\mathcal{C}}(S \backslash v)$. We now have a loop in ${\mathcal{C}}(S\backslash v)$. Since all curves besides $v$ in $S$ intersect the subsurface $S \backslash v$ non-trivially we can apply the Bounded Geodesic Image Theorem on $[v_{\alpha \beta}^{-1}, \alpha_1]_S$ and $[\alpha_1, v_{\alpha \gamma}^{-1}]_S$ to get $d_{{\mathcal{C}}(S\backslash v)}(v_{\alpha \beta}^{-1}, v_{\alpha \gamma}^{-1}) \leq 2M$. Similarly, $d_{{\mathcal{C}}(S\backslash v)}(v_{\alpha \gamma}^{+1}, v_{\beta \gamma}^{+1}) \leq 2M$ and $d_{{\mathcal{C}}(S\backslash v)}(v_{\beta \gamma}^{-1}, v_{\alpha\beta}^{+1}) \leq 2M$. Consider the geodesic triangle $v_{\alpha\beta}^{+1}v_{\alpha\gamma}^{-1}v_{\beta\gamma}^{+1}$ in ${\mathcal{C}}(S \backslash v)$. We now have the picture in ${\mathcal{C}}(S \backslash v)$ as in Figure \[2 links are thin\]. By Theorem \[curve hyp\], the inner triangle is $17$ centered, call this center $z$. Combining Theorem \[curve hyp\] and Lemma \[centered to thin\], the outer three triangles are $17*4$-thin. Therefore $z$ is at most $17*5 + 2M = 285$ away from each of the geodesics in the hierarchy triangle $\alpha\beta\gamma$ whose domain is ${\mathcal{C}}(S \backslash v)$. This all implies that $\alpha\beta\gamma$ is 285-centered at $\{v, z\}$. **Case 2**: Assume that at least two main geodesics share a common curve, but there is no point that all three main geodesics share the same curve. First assume there is only one such shared curve. Without loss of generality assume that $g_{\alpha\beta}$ and $g_{\alpha \gamma}$ share the curve $v$. Then we can consider a new triangle with the main geodesics forming the triangle $v\beta_1\gamma_1$, see Figure \[Case 2\]. This new triangle has no shared curves so is covered by Case 3. Now assume there is more than one shared curve between the main geodesics. By definition of a geodesic, for any two main geodesics that share multiple curves, those curves have to show up in each main geodesic in the same order from either end, therefore we can just take the inner triangle where the edges share no curves and apply Case 3. **Case 3**: The argument given for this case is similar to the short cut argument in [@MMII]. Assume none of the three main geodesics, $g_{\alpha \beta}, g_{\alpha \delta}$, and $g_{ \beta \delta}$ share a curve. By Theorem \[curve hyp\] there exists a curve $c \in {\mathcal{C}}(S)$ that is distance at most $17$ from $g_{\alpha \beta}, g_{\alpha \gamma}$, and $g_{ \beta \gamma}$; let $c$ be the curve that minimizes the distance from all three main geodesics. Define $v_{\alpha \beta}$ to be the vertex in $g_{\alpha \beta}$ which has the least distance to $c$, and similarly define $v_{\alpha \gamma}$ and $v_{\beta \gamma}$. Consider the geodesic $[v_{\alpha\beta}, c]_S$ and let $c_0$ be the curve adjacent to $c$ in this geodesic. Let $v_*^{-1}$ be the curve in $g_*$ that precedes $v_{*}$. Now connect $\{v_{\beta\gamma}, v_{\beta\gamma}^{-1}\}$ to $\{c, c_0 \}$ with a hierarchy. We denote the main geodesic of this hierarchy as $[c, v_{\beta\gamma}]_S$. Take a vertex $w \in [c, v_{ \beta \gamma}]_S$ where $w$ is not equal to $c$ or $v_{\beta\gamma}$ and let $w^{-1}$ and $w^{+1}$ denote the vertices directly before and after $w$ in $[c, v_{ \beta \gamma}]_S$. We want to show that the link connecting $w^{-1}$ to $w^{+1}$ in $S\backslash w$ is at most $5M$. Assume $d_{S \backslash w} (w^{-1}, w^{+1}) \geq 5M$. Consider the path $[w^{+1}, v_{\beta\gamma}]_S \cup [v_{ \beta \gamma}, \beta_0]_S \cup [\beta_0, v_{\alpha \beta}]_S \cup [v_{\alpha \beta}, c]_S \cup [c, w^{-1}]_S$, where geodesics are taken to be on $g_*$ where appropriate. The Bounded Geodesic Image Theorem, and our assumption that $d_{S \backslash w} (w^{-1}, w^{+1}) \geq 5M$, implies that $w$ must be somewhere on the path. $w$ cannot be in $[w^{+1}, v_{\beta\gamma}]_S$, $[v_{ \beta \gamma}, \beta_0]_S$, or $[c, w^{-1}]_S$ since that would contradict the fact that they are geodesics or the definition of how we chose $c$ and $v_{\beta\gamma}$. Therefore, $w$ is in $[\beta_0, v_{\alpha \beta}]_S$ or $[v_{\alpha \beta}, c]_S$. Without loss of generality assume $w \in [\beta_0, v_{\alpha \beta}]_S$. We can apply the same logic to the path $[w^{+1}, v_{\beta\gamma}]_S \cup [v_{ \beta \gamma}, \gamma_0]_S \cup [\gamma_0, v_{\alpha \gamma}]_S \cup [v_{\alpha \gamma}, c]_S \cup [c, w^{-1}]_S$. Now $w$ has to be in $[v_{\alpha \gamma}, c]_S$ so that it doesn’t contradict the fact that the three main geodesic of the triangle $\alpha\beta\gamma$ do not share any curves. However, now all three main geodesics are closer to $w$ than $c$, which contradicts our choice of $c$. Therefore, the length of $[w^{-1}, w^{+1}]_{S \backslash w}$ is at most $5M$. Using a similar argument we can show the geodesic in ${\mathcal{C}}(S \backslash v_{\beta\gamma})$ connecting $v_{\beta\gamma}^{-1}$ to the appropriate vertex in $[c, v_{\beta\gamma}]_S$ is $\leq 5M$. Now consider the geodesic in ${\mathcal{C}}(S \backslash c)$ connecting $c_0$ to the second vertex, $x$, of $[c, v_{\beta\gamma}]_S$. Consider the path $[x, v_{\beta\gamma}]_S \cup [v_{\beta\gamma}, \beta_0]_S \cup [\beta_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, c_0]_S$. $c$ cannot be in anywhere in this path, otherwise it would contradict how we chose $c$ or $v_*$. So we can apply the Bounded Geodesic Image Theorem and get that $d_{S \backslash c}(c', x) \leq 4M$. Therefore the path from $\{v_{\beta\gamma}, v_{\beta\gamma}^{-1}\}$ to $\{c, c_0\}$ in the pants graph is less than or equal to $16(5M) + 5M + 4M$. A similar argument can be made for the other two sides of the triangle $\alpha\beta\gamma$, so $\{c, c_0\}$ can be taken to be a center of the triangle. Since $M \leq 100$ the triangle $\alpha\beta\gamma$ is $8,900$-centered at $\{c, c_0\}$. \[main thm 1\] For a surface $S = S_{0,5}, S_{1,2}$, ${\mathcal{P}}(S)$ is $2,691,437$-thin hyperbolic. For $x, y \in {\mathcal{P}}(S)$ define ${\mathcal{L}}(x,y)$ to be the collection of hierarchy paths between $x$ and $y$. These are connected because each hierarchy path is connected and all contain $x$ and $y$. By Theorem \[hierarchy k-centered\] and Lemma \[centered to thin\] we have that for all $x, y, z \in {\mathcal{P}}(S)$ $${\mathcal{L}}(x, y) \subset N_{4*8,900}({\mathcal{L}}(x,z) \cup {\mathcal{L}}(z,y)).$$ If $d(x,y) \leq 1$ then any hierarchy between $x$ and $y$ is just the edge $\{ xy\}$, so ${\mathcal{L}}(x,y) = \{x, y\}$. Thus, both conditions of Proposition \[subset hyperbolic\] are satisfied. Therefore by applying Proposition \[subset hyperbolic\] we get ${\mathcal{P}}(S)$ is $2,691,437$-thin hyperbolic. Relative Hyperbolicity of Pants Graphs Complexity 3 =================================================== In this section we turn our attention to relative pants graphs and their hyperbolicity constant. \[relative hierarchy k-centered\] Take $S$ such that $\xi(S) = 3$. The relative 3-archy triangles in ${\mathcal{P}}_{rel}(S)$ are $6,191,300$-centered. Take three pants decompositions of $S$, say $\alpha = \{\alpha_0, \alpha_1, \alpha_2 \}$, $\beta = \{\beta_0, \beta_1, \beta_2\}$, and $\gamma = \{\gamma_0, \gamma_1, \gamma_2\}$. Form the triangle $\alpha\beta\gamma$ such that each edge in the triangle is a relative 3-archy in ${\mathcal{P}}_{rel}(S)$. Let $g_{\alpha\beta}$, $g_{\beta\gamma}$, and $g_{\alpha\gamma}$ be the three main geodesics that make up the triangle (which connects $\alpha_0$, $\beta_0$, and $\gamma_0$). As before in Theorem \[hierarchy k-centered\], there are three cases: 1. All three main geodesics have a curve in common. 2. Any two of the main geodesics share a curve, but not the third. 3. None of the main geodesics have common curves. For the rest of the proof, note that if $v \in {\mathcal{C}}(S)$ is a non-domain separating curve, then $S \backslash v$ has one connected component with positive complexity, so by abuse of notation, we denote this component as $S \backslash v$. This means that every curve in ${\mathcal{C}}(S)$ not equal to $v$ intersects $S \backslash v$ so we can use the Bounded Geodesic Image Theorem on any geodesic that doesn’t contain $v$. Take two non-domain separating curve $v,w \in {\mathcal{C}}(S)$ such that $v$ and $w$ are disjoint. Then, because $\xi(S) = 3$, $S \backslash (v \cup w)$ has one connected component with positive complexity, and again we denote this component as $S \backslash (v \cup w)$. Furthermore, every curve in ${\mathcal{C}}(S)$ not equal to $v$ or $w$ intersects $S \backslash (v \cup w)$, so we may use the Bounded Geodesic Image Theorem for any geodesic that doesn’t contain $v$ or $w$. Whenever a domain separating curve, $c$, shows up in a relative 3-archy in ${\mathcal{P}}_{rel}(S)$, the section of the relative 3-archy containing $c$ has length $2$. Therefore, when referring to a curve along a geodesic within a relative 3-archy we will assume it is non-domain separating since this type of curve adds the most length to the relative 3-archy. This also just makes the proof cleaner. **Case 1:** Let $v$ be a vertex where all three main geodesics intersect. If $v$ is a domain separating curve then each edge of the triangle $\alpha\beta\gamma$ contains the point $p_v$, so the triangle is $0$-centered. Now assume $v$ is not a domain separating curve. Let $v_{\alpha\beta}^{-1}$ and $v_{\alpha\beta}^{+1}$ be the curves that are directly before and after $v$ on $g_{\alpha\beta}$. Similarly define $v_{\alpha\gamma}^{-1}$, $v_{\alpha\gamma}^{+1}$, $v_{\beta\gamma}^{-1}$, and $v_{\beta\gamma}^{+1}$. Consider the geodesics associated with $v$ in each relative 3-archy edge; in other words, all geodesics in the relative 3-archy that contribute to defining the path where $v$ is a part of every pants decomposition. Let $x_{\alpha\beta}$ be the curve in $[v_{\alpha\beta}^{-1}, v_{\alpha\beta}^{+1}]_{S \backslash v}$ that is adjacent to $v_{\alpha\beta}^{-1}$; similarly define $x_{\alpha\gamma}$. Now connect $\{v_{\alpha\beta}^{-1}, x_{\alpha\beta} \}$ to $\{v_{\alpha\gamma}^{-1}, x_{\alpha\gamma}\}$ with a hierarchy in ${\mathcal{P}}(S \backslash v)$. Note, to make our notation cleaner, we will refer to this as the hierarchy between $v_{\alpha\beta}^{-1}$ and $v_{\alpha\gamma}^{-1}$; similarly later on we won’t necessarily specify the second curve. By the Bounded Geodesic Image Theorem the geodesic connecting $v_{\alpha\beta}^{-1}$ and $v_{\alpha\gamma}^{-1}$ in ${\mathcal{C}}(S\backslash v)$ has length at most $2M$. Now consider any curve, $w$, in the geodesic $[v_{\alpha\beta}^{-1}, v_{\alpha\gamma}^{-1}]_{S \backslash v}$ contained in the hierarchy connecting $\{v_{\alpha\beta}^{-1}, x_{\alpha\beta} \}$ to $\{v_{\alpha\gamma}^{-1}, x_{\alpha\gamma}\}$. Assume $w$ is not a domain separating curve in $S$ and let $w^{-1}$ and $w^{+1}$ be the two curves before and after $w$ on $[v_{\alpha\beta}^{-1}, v_{\alpha\gamma}^{-1}]_{S \backslash v}$. Then the geodesic connecting $w^{-1}$ to $w^{+1}$ in ${\mathcal{C}}(S \backslash (v \cup w))$ has length at most $4M$ by using the Bounded Geodesic Image Theorem on $[w^{-1}, v_{\alpha\beta}^{-1}]_{S \backslash v} \cup [v_{\alpha\beta}^{-1}, \alpha_0]_S \cup [\alpha_0, v_{\alpha\gamma}^{-1}]_S \cup [v_{\alpha\gamma}^{-1}, w^{+1}]_{S \backslash v}$; note $w$ cannot be on this path because $w$ is distance $1$ from $v$, so if it was anywhere in the path it would be violating the assumption that we have geodesics. Therefore the hierarchy between $v_{\alpha\beta}^{-1}$ and $v_{\alpha\gamma}^{-1}$ has length at most $8M^2$. Similarly the hierarchies between $v_{\alpha\gamma}^{+1}$ and $v_{\beta\gamma}^{+1}$, and $v_{\alpha\beta}^{+1}$ and $v_{\beta\gamma}^{-1}$ have length less than $8M^2$. Now, make a hierarchy triangle $v_{\alpha\beta}^{+1}v_{\alpha\gamma}^{-1}v_{\beta\gamma}^{+1}$ in ${\mathcal{P}}(S \backslash v)$, see Figure \[links are thin\] for how this fits in with above. By Theorem \[hierarchy k-centered\], $v_{\alpha\beta}^{+1}v_{\alpha\gamma}^{-1}v_{\beta\gamma}^{+1}$ in ${\mathcal{P}}(S \backslash v)$ is $8,900$ centered, call the point at the center $z$. Then by Theorem \[hierarchy k-centered\] and Lemma \[centered to thin\], the hierarchy triangles $v_{\alpha\beta}^{+1}v_{\alpha\beta}^{-1}v_{\alpha\gamma}^{-1}$, $v_{\beta\gamma}^{-1}v_{\beta\gamma}^{+1}v_{\alpha\gamma}^{+1}$, and $v_{\alpha\gamma}^{-1}v_{\alpha\gamma}^{+1}v_{\beta\gamma}^{+1}$ are $35,600$ thin. Therefore $z$ is at most $124,500$ away from each $[v_*^{+1}, v_{*}^{-1}]_{S \backslash v}$. This implies that $\{z, v\}$ is at most $124,500$-centered in the relative 3-archy triangle $\alpha\beta\gamma$. **Case 2:** For the same reasons as in Theorem \[hierarchy k-centered\] case 2, this case can be reduced to case 3. **Case 3:** This proceeds with the same strategy as in case 3 of Theorem \[hierarchy k-centered\]. By Theorem \[curve hyp\], we know the triangle of main geodesics, $g_{\alpha\beta}g_{\beta\gamma}g_{\alpha\gamma}$ in ${\mathcal{C}}(S)$ is $17$-centered. Let $c$ be the curve that is at the center of this triangle. Connect $c$ to $g_{\alpha\beta}$, $g_{\beta\gamma}$, and $g_{\alpha\gamma}$ with a geodesic in ${\mathcal{C}}(S)$. Define $v_{\alpha \beta}$ to be the vertex in $g_{\alpha \beta}$ which is the least distance to $c$, and similarly define $v_{\alpha \gamma}$ and $v_{\beta \gamma}$. Let $c_0$ be the curve directly preceding $c$ in $[v_{\alpha\beta}, c]_S$ and let $c^{-1}$ be the curve directly preceding $c_0$. Consider a geodesic in ${\mathcal{C}}(S \backslash c_0)$ which connects $c^{-1}$ to $c$, define $c_1$ to be the curve directly preceding $c$ in this geodesic. We will show $\{c, c_0, c_1\}$ is a center of our relative 3-archy triangle $\alpha\beta\gamma$. Let $v_{\beta\gamma}^{-1}$ be the curve before $v_{\beta\gamma}$ in $g_{\beta\gamma}$ and $v_{\beta\gamma}'$ be the curve adjacent to $v_{\beta\gamma}$ in the geodesic contained in the relative 3-archy connecting $\beta$ to $\gamma$ whose domain is ${\mathcal{C}}(S \backslash v_{\beta\gamma}^{-1})$. Now connect $\{v_{\beta\gamma}, v_{\beta\gamma}^{-1}, v_{\beta\gamma}'\}$ to $\{c, c_0, c_1 \}$ with a relative 3-archy, $H$. Our goal is to bound the length of $H$. Using the exact argument as in Theorem \[hierarchy k-centered\] case 3, for each $w \in [c, v_{\beta\gamma}]_S$ which is non-separating, the geodesic in $H$ whose domain is ${\mathcal{C}}(S\backslash w)$ has length no more than $5M$. Let $w^{-1}$ and $w^{+1}$ be the curves before and after $w$ in $[c, v_{\beta\gamma}]_S$ and let $[w^{-1}, w^{+1}]_{S \backslash w}$ be the geodesic coming from $H$. Take $z \in [w^{-1}, w^{+1}]_{S \backslash w}$ and consider the geodesic in $H$ with domain ${\mathcal{C}}(S \backslash (w \cup z))$. Define $z^{-1}$ and $z^{+1}$ to be the curves before and after $z$ on $[w^{-1}, w^{+1}]_{S \backslash w}$. We will show $[z^{-1}, z^{+1}]_{S \backslash (w \cup z)}$ has length at most $7M$. Assume towards a contradiction that the length of $[z^{-1}, z^{+1}]_{S \backslash (w \cup z)}$ is greater than $7M$. Then the path $[z^{+1}, w^{+1}]_{S \backslash w} \cup [w^{+1}, v_{\beta\gamma}]_{S} \cup [v_{\beta\gamma}, \gamma_0]_S \cup [\gamma_0, v_{\alpha\gamma}]_S \cup [v_{\alpha\gamma}, c]_S \cup [c, w^{-1}]_S \cup [w^{-1}, z^{-1}]_{S \backslash w}$ must contain $z$ or $w$ somewhere, otherwise by the Bounded Geodesic Image Theorem using this path we would get that the length of $[z^{-1}, z^{+1}]_{S \backslash (w \cup z)}$ is at most $7M$. Since $w$ and $z$ are distance $1$ apart, it doesn’t matter which one shows up in the path because we eventually will arise at the same contradiction. Thus, without loss of generality we assume $z$ is in the path (and all other paths considered for this argument). Then $z$ must be in $[\gamma_0, v_{\alpha\gamma}]_S$ or $[v_{\alpha\gamma}, c]_S$, otherwise there would be a contradiction with the definition of a geodesic or the definition of $c$ or $v_{\beta\gamma}$ Without loss of generality assume $z \in [\gamma_0, v_{\alpha\gamma}]_S$. Similarly the path $[z^{+1}, w^{+1}]_{S \backslash w} \cup [w^{+1}, v_{\beta\gamma}]_{S} \cup [v_{\beta\gamma}, \beta_0]_S \cup [\beta_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, c]_S \cup [c, w^{-1}]_S \cup [w^{-1}, z^{-1}]_{S \backslash w}$ must contain $z$. Again, the only place $z$ could be, without yielding a contradiction, is in $[v_{\alpha\beta}, c]_S$. However even here, since $z$ is adjacent to $w$, $w$ is strictly closer than $c$ to the three main geodesics of $\alpha\beta\gamma$ which contradicts our choice of $c$. Therefore, the length of $[z^{-1}, z^{+1}]_{S \backslash (w \cup z)}$ is at most $7M$. Now all that’s left to bound is the beginning and end geodesics, i.e. the ones associated to $c$ and $v_{\beta\gamma}$. Let $y$ be the curve adjacent to $v_{\beta\gamma}$ in $[c, v_{\beta\gamma}]_S$ and let $y'$ be the curve adjacent to $v_{\beta\gamma}$ in the geodesic contained in $H$ whose domain is ${\mathcal{C}}(S \backslash y)$. Then the very beginning part of $H$ is the hierarchy connecting $\{y, y' \}$ to $\{v_{\beta\gamma}^{-1}, v_{\beta\gamma}' \}$ in $S \backslash v_{\beta\gamma}$. We will first bound the length of the geodesic $[y, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}}$. Assume that the length is more than $5M$. Then the path $[v_{\beta\gamma}^{-1}, \beta_0]_S \cup [\beta_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, c]_S \cup [c, y]_S$ has to contain $v_{\beta\gamma}$. By our assumption that the main geodesics on the triangle $\alpha\beta\gamma$ don’t intersect, the only part of the path that $v_{\beta\gamma}$ could be on without forming a contraction would be $[v_{\alpha\gamma}, c]_S$. The same is true of the path $[v_{\beta\gamma}^{-1}, \beta_0]_S \cup [\beta_0, \alpha_0]_S \cup [\alpha_0, v_{\alpha\gamma}]_S \cup [v_{\alpha\gamma}, c]_S \cup [c, y]_S$, where $v_{\beta\gamma}$ would have to be in $[v_{\alpha\gamma}, c]_S$. However, then we could take $v_{\beta\gamma}$ to be the center of the main geodesic triangle which would give strictly smaller lengths to each of the sides, contradicting our choice of $c$. Therefore, $[y, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}}$ has length at most 5M. Now take $w \in [y, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}}$ and let $w^{-1}$ and $w^{+1}$ be the curves that come directly before and after $w$ in $[y, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}}$. We want to bound the length of $[w^{-1}, w^{+1}]_{S \backslash (v_{\beta\gamma} \cup w)}$. Assume the length is greater than $7M$. Then the path $[w^{+1}, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}} \cup [v_{\beta\gamma}^{-1}, \beta_0]_S \cup [\beta_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, c]_S \cup[c, y]_S \cup [y, w^{-1}]_{S \backslash v_{\beta\gamma}}$ must contain $w$ or $v_{\beta\gamma}$. The only two places this could happen without raising a contradiction is in $[\beta_0, v_{\alpha\beta}]_S$ or $[v_{\alpha\beta}, c]_S$. Again, whether we assume $w$ or $v_{\beta\gamma}$ is in the path doesn’t matter since we will arrive at the same contradiction, hence we can assume without loss of generality $w$ is always on the path. Therefore, assume $w \in [v_{\alpha\beta}, c]_S$. Similarly, $w$ is contained in the path $[w^{+1}, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}} \cup [v_{\beta\gamma}^{-1}, v_{\beta\gamma}^{+1}]_{S \backslash v_{\beta\gamma}} \cup [v_{\beta\gamma}^{+1}, \gamma_0] \cup [\gamma_0, v_{\alpha\gamma}]_S \cup [v_{\alpha\gamma}, c]_S \cup[c, y]_S \cup [y, w^{-1}]_{S \backslash v_{\beta\gamma}}$, where $w \in [\gamma_0, v_{\alpha\gamma}]_S$ since anywhere else in the path would lead to a contradiction as explained previously. Note if $w \in [v_{\alpha\gamma}, c]_S$ then since $w$ is disjoint from $v_{\beta\gamma}$ and that $w \in [v_{\alpha\beta}, c]_S$, we could make a shorter path to each of the three sides on the main geodesic triangle and then $v_{\beta\gamma}$ would be the center of the triangle, contradicting our choice of $c$. The path $[w^{+1}, v_{\beta\gamma}^{-1}]_{S \backslash v_{\beta\gamma}} \cup [v_{\beta\gamma}^{-1}, \beta_0]_S \cup [\beta_0, \alpha_0]_S \cup [\alpha_0, v_{\alpha\gamma}]_S \cup [v_{\alpha\gamma}, c]_S \cup[c, y]_S \cup [y, w^{-1}]_{S \backslash v_{\beta\gamma}}$ has to contain $w$ as well. No matter where $w$ is on this path is creates a contradiction - either with the definition of $c$, with the we have a geodesic, or with the assumption the main geodesics do not share any curves. Consequently, $[w^{-1}, w^{+1}]_{S \backslash (v_{\beta\gamma} \cup w)}$ must have length at most $7M$. Note that this argument also works when $w = y$ or $w = v_{\beta\gamma}^{-1}$, which gives a length bound on the geodesic in $H$ whose domain is ${\mathcal{C}}(S \backslash (v_{\beta\gamma} \cup y))$ or ${\mathcal{C}}(S \backslash (v_{\beta\gamma} \cup v_{\beta\gamma}^{-1}))$, respectively. Let $x$ be the curve adjacent to $c$ in $[v_{\beta\gamma}, c]_S$ and $x'$ be the last curve adjacent to $c$ in the geodesic from the hierarchy whose domain is ${\mathcal{C}}(S \backslash x)$. First, the geodesic $[c_0, x]_{S \backslash c}$ has length no more than $4M$ by the Bounded Geodesic Image Theorem applied to $[c_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, \beta_0]_S \cup [\beta_0, v_{\beta\gamma}]_S \cup [v_{\beta\gamma}, x]_S$, which doesn’t contain $c$ because if it did we would get a contradiction on the definition of $c$. Now take any curve $w \in [c_0, x]_{S \backslash c}$ and define $w^{-1}$ and $w^{+1}$ as before. Then the path $[w^{+1}, x]_{S \backslash c} \cup [x, v_{\beta \gamma}]_S \cup [v_{\beta\gamma}, \beta_0]_S \cup [\beta_0, v_{\alpha\beta}]_S \cup [v_{\alpha\beta}, c_0]_S \cup [c_0, w^{-1}]_{S \backslash c}$ cannot contain $w$ because $w$ is adjacent to $c$ so if any geodesic making up the path contained $w$ it would either contradict that it is a geodesic or that $c$ is minimal distance from the main geodesics of the triangle $\alpha\beta\gamma$. Hence, applying the Bounded Geodesic Image Theorem to the path we get that $[w^{-1}, w^{+1}]_{S \backslash (c \cup w)}$ has length no more than $6M$. This leaves bounding the lengths of the geodesics connecting $c_1$ to the second vertex of $[c_0, x]_{S \backslash c}$ and $x'$ to the penultimate vertex of $[c_0, x]_{S \backslash c}$. By a similar argument using the Bounded Geodesic Image Theorem each of these geodesics have length at most $6M$. Therefore, putting all the length bounds together we get that the relative 3-archy connecting $\{v_{\beta\gamma}, v_{\beta\gamma}^{-1}, v_{\beta\gamma}'\}$ to $\{c, c_0, c_1 \}$ has length at most $16*5M*7M + (4M-1)*6M+12M + (5M+1)*7M = 6,191,300$ Similarly $\{c, c_0, c_1\}$ is length at most $6,191,300$ from the other two sides of the triangle $\alpha\beta\gamma$. Therefore, the relative 3-archy triangle $\alpha\beta\gamma$ is $6,191,300$-centered. \[main thm 2\] For a surface $S$ such that $\xi(S) =3$, ${\mathcal{P}}_{rel}(S)$ is $1,607,425,314$-thin hyperbolic. For $x, y \in {\mathcal{P}}_{rel}(S)$ define ${\mathcal{L}}(x,y)$ to be the collection of relative 3-archy paths between $x$ and $y$. These are connected because each relative 3-archy path is connected and all the relative 3-archies in ${\mathcal{L}}(x, y)$ contain $x$ and $y$. By Theorem \[relative hierarchy k-centered\] and Lemma \[centered to thin\] we have that for all $x, y, z \in {\mathcal{P}}_{rel}(S)$ $${\mathcal{L}}(x, y) \subset N_{4*6,191,300}({\mathcal{L}}(x,z) \cup {\mathcal{L}}(z,y)).$$ If $d(x,y) \leq 1$ then any relative 3-archy between $x$ and $y$ is just the edge $\{ xy\}$, so ${\mathcal{L}}(x,y) = \{x, y\}$. We now have both conditions of Proposition \[subset hyperbolic\] satisfied. Therefore by applying Proposition \[subset hyperbolic\] we get that ${\mathcal{P}}_{rel}(S)$ is $1,607,425,314$-thin hyperbolic. [*Email:*]{}\ aweber@math.brown.edu

--- abstract: 'We present a multi-wavelength study of the young stellar population in the Cygnus-X DR15 region. We studied young stars forming or recently formed at and around the tip of a prominent molecular pillar and an infrared dark cloud. Using a combination of ground based near-infrared, space based infrared and X-ray data, we constructed a point source catalog from which we identified 226 young stellar sources, which we classified into evolutionary classes. We studied their spatial distribution across the molecular gas structures and identified several groups possibly belonging to distinct young star clusters. We obtained samples of these groups and constructed K-band luminosity functions that we compared with those of artificial clusters, allowing us to make first order estimates of the mean ages and age spreads of the groups. We used a $^{13}$CO(1-0) map to investigate the gas kinematics at the prominent gaseous envelope of the central cluster in DR15, and we infer that the removal of this envelope is relatively slow compared to other cluster regions, in which gas dispersal timescale could be similar or shorter than the circumstellar disk dissipation timescale. The presence of other groups with slightly older ages, associated with much less prominent gaseous structures may imply that the evolution of young clusters in this part of the complex proceeds in periods that last 3 to 5 Myr, perhaps after a slow dissipation of their dense molecular cloud birthplaces.' author: - 'S. Rivera-Gálvez, C. G. Román-Zúñiga, E. Jiménez-Bailón, J. E. Ybarra, J. F. Alves and Elizabeth A. Lada' title: 'The Young Stellar Population of the Cygnus-X DR15 Region' --- =1 Introduction \[intro\] ====================== The star forming complex of Cygnus-X region is one of the most prominent features in our Galaxy. Originally detected as an extended region with a thermal spectrum, @Piddington:1952aa named the region Cygnus-X, in order to distinguish it from the other nearby known radio source, Cygnus-A. Cygnus-X is composed of several OB associations, dozens of embedded stellar clusters, hundreds of HII regions and over 40 known massive protostars [see @Reipurth:2008aa for an extensive review]. The young star population in Cygnus-X is currently interacting with one of the most massive molecular cloud complexes in the Galaxy, with a total mass of $3\times 10^6\ \mathrm{M}_\odot$ [@Schneider:2006aa], as well as the X-ray emitting Cygnus Superbubble. It has been proposed that Cygnus-X could be the precursor of a globular cluster [@Knodlseder:2000aa]. To study the interaction between recently formed star clusters and their surrounding medium in Cygnus-X could provide very important clues about the present evolution of the complex. It is particularly important to focus on the numerous embedded cluster populations and to compare the properties of clusters across the region. For instance: how embedded clusters proceed from formation to emergence from their parental gas clumps in such a strong ionizing medium? Is cluster formation or evolution in Cygnus-X determined by the local environment? What are the time scales from formation to gas dispersal? Also, what happens after cluster emergence: are subsidiary clusters destined to disperse or could they end up swelling the ranks of the main association? In any case, we should expect that multiwavelength analysis of the young star clusters in Cygnus-X will help to fine tune current ideas about the formation and evolution of embedded stellar clusters or groups. In this paper we selected to study one of the most prominent embedded cluster populations in Cygnus-X: the region DR15, also listed as cluster 10 in the survey of @Dutra:2001aa. The DR15 region has been related to the HII region G79.306+0.282, the source IRAS 20306+4005/FIR-1 [@Campbell:1982aa], and sources IRS 1, 2 and 3 in the list of @Kleinmann:1979aa [see also @Odenwald:1990aa]. DR15 is located in the Cygnus-X South region, located at an estimated distance of 1.4 kpc [@Rygl:2012aa]. The cluster contains one prominent far-infrared source (FIR-1), which marks the location of a compact HII region formed by a pair of B type embedded sources [@Odenwald:1990aa; @Oka:2001aa; @Kurtz:1994aa], however the cluster hosts various intermediate to massive stars which create a rather complex structure of photodissociation regions, forming a nebulous envelope which glows brightly in infrared images (see Figure \[fig:RGB\]). DR15 sits at the tip of a long (about 10 pc), filamentary pillar that extends into the southern edge of the central OB from DR12, lying in projection about 1 degree south from the Cyg-X3 star. The structure of the pillar appears to be protruding from the DR12 ridge (see Figure \[fig:spitzer\]). The DR12-15 region in Cygnus-X possibly lies in front of the OB2 association, along our line of sight. A large filamentary infrared dark cloud (IRDC) with active star formation lies to the north and west of the cluster. This IRDC appears to be kinematically independent from DR15, as shown by @Schneider:2006aa analysis. Recently, the western segment of the IRDC, has been shown to host a young stellar cluster in interaction with the Luminous Blue Variable (LBV) source G79.29+0.46 [@Rizzo:2008aa; @Jimenez:2010aa]. Distance to G79.29+0.46 has been also estimated to be 1.4 kpc [@Rizzo:2014aa; @Palau:2014aa]. Our main goal in this study is to track the progression of the star formation in the DR15 cluster and its immediate surroundings, by looking at the properties and distributions of the young stellar population. DR15 is especially interesting for being at a very specific stage at which it is already emerging from its parental cloud (some members and parts of the reflection nebulosities are already detectable in visible wavelengths). Moreover, the interesting case it presents at being apparently formed at the tip of a filamentary structure and next to a much younger star forming spot in the IRDC, is worth of detailed attention. The paper is organized as follows: in Section \[obs\] we describe the datasets used in this work. Section \[analysis\] is dedicated to describe the procedures we followed to acomplish a description of the history of star formation in the region, a result which we discuss in Section \[discussion\]. Finally, a brief summary of our results can be found in Section \[discussion\]. Observations {#obs} ============ For the first part of our analysis, we combined high-quality deep near-IR observations with images and catalogs from the Spitzer Cygnus-X Legacy Survey [@Hora:2009aa hereafter CXLS]. We also made use of archival data from the Chandra X-Ray Observatory. By combining these datasets we prepared a multi-wavelength photometry catalog for the stellar population in DR15, which we use to reconstruct the history of star formation in the region. Near Infrared Observations {#obs:nir} -------------------------- Near-infrared images of the Cygnus-X DR15 region were obtained with the OMEGA 2000 camera at the 3.5m telescope of the Calar Alto Observatory, atop Sierra de los Filabres in Almería, Spain, during the nights of February 2nd and March 3rd, 2010, with excellent weather conditions. The dataset consists of 900 second co-added exposures in the $J$, $H$ and $K$ bands (1.209, 1.648 and 2.208 $\mu$m, respectively). The seeing values –measured directly from the average FWHM of stars in the final reduced mosaics– were 1.17, 1.13 and 0.98$\arcsec$ in $J$, $H$ and $K$, respectively. The reduction of the images and the extraction of Point Spread Function (PSF) photometry lists for all bands were performed with custom `IRAF` pipelines, and the `SExtractor` algorithm [@Bertin:1996aa], following a methodology equivalent to the one described in @Roman:2010aa. We constructed a near-IR catalog by matching individual filter photometry lists with the aid of [TOPCAT]{} [@Taylor:2005uq]. We replaced saturated sources and a small number of missing detections in the Calar Alto images with values from the 2MASS Point Source Catalog, obtained at the Infrared Processing and Analysis Center (IPAC, Caltech). The 10 percent photometric depth values –indicated by the average brightness at which photometric error reaches a value of 0.1 mag– are 21.5, 20.0 and 18.75 mag in $J$, $H$ and $K$, respectively, and are good indicators of the completeness of the data. These limits are enough to sample the young star population in DR15 down to 0.09 M$_\odot$ in regions of low to moderate extinction ($A_V<25.0$ mag). {width="6.0in"} Spitzer Cygnus-X Legacy Survey data {#obs:mir} ------------------------------------ The *Spitzer Space Telescope* has observed the DR15 cluster with the IRAC and MIPS detectors as part of the CXLS. We obtained archival enhanced product mosaics from the Spitzer Heritage Archive as well as a photometric catalog coincident with our region of interest directly from the CXLS Data Release 1 (DR1). The catalog contains calibrated magnitudes for sources detected with IRAC in its four cryogenic mission channels (3.6, 4.5, 5.8 and 8.0 $\mu$m), as well as in the 24 $\mu$m channel of MIPS. Using `TOPCAT` and IDL routines we combined our near-IR and the CXLS DR1 catalogs into a single infrared photometry list with a total of 46983 sources. {width="6.0in"} Chandra ACIS Observations {#obs:xray} ------------------------- The DR15 cluster was observed with the Imaging Array of the Chandra Advanced CCD Imaging Spectrometer (ACIS-I) on 2011 January 25 (ObsID 12390, P.I. Wright) with net exposure time of 39.875 ks. We processed the archival raw data using the routines from version 4.5 of the Chandra Interactive Analysis of Observations `CIAO` data analysis system [@Fruscione:2006aa]. We used the `chandra\_repro` reprocessing script to recalibrate our event data in order to ensure that consistent calibration updates were applied to the dataset. The X-ray image processing was performed as follows: first, we created a exposure-corrected image from our data in the broad band (0.5 to 7 keV with an effective energy of 2.3 keV) using `CIAO/fluximage` routine with a bin size of 1. Second, we applied the `wavedetect` tool to our broad band image in order to identify potential X-ray sources. We used wavelet scales from 1 to 16.5 pixels in s.pdf of $\sqrt{2}$ and a source significance threshold of $1\times10^{-6}$. We performed photometry on the list provided by `wavedetect` using the the `ACIS Extract` (AE) package [@Broos:2010aa]. AE permits an optimal determination of the local background and the best flux extraction apertures based on the PSF of the image. Photometry is then extracted on selected energy bands and a list of source properties, statistics and best fit spectral models is produced. We used three energy bands: Soft, from 0.2 to 2.0 keV, Medium, from 2.0 to 4.0 keV and Hard, from 4.0 to 7.0 keV. From the final list of sources produced by `ACIS Extract` we rejected those with a probability of 1$\%$ or higher of being a background fluctuation $P_B>0.01$). Our final lists contains a total of 131 X-ray sources. From these, a total of 109 (83.2%) sources have a counterpart in our IR catalog. In Figure \[fig:chandra\] we show a false color (RGB) map of the ACIS field for DR15 using images from the three energy bands, overlaid with contours of visual extinction from the NICEST extinction map ww constructed from near-IR data. In there, we see how sources with hard X-ray emission are preferently located in regions of high column density. This is because soft X-ray bands are prone to oscuration by dust. We see how most of the X-ray sources are associated with the molecular cloud and the embedded population, confirming that most of the point source X-ray emission in DR15 comes from young stars. {width="7.0in"} FCRAO Observations {#obs:others} ------------------- We made use of the Five College Radio Observatory (FCRAO) $^{13}$CO(1-0) molecular radio emission map of the South Cygnus-X region from the study of @Schneider:2011aa. The map is a RA-Dec-radial velocity cube, from which we extracted molecular gas properties using standard tools from the `MIRIAD` [@Sault:1995aa] package. Analysis ========= We limited our region of study to the area covered with OMEGA 2000, defined as $\mathrm{[RA,\ Dec]}=[307.969040,40.150383]\rightarrow[308.305620,40.405865]$. The analysis described below correspond to sources falling within that area only. Identification of YSOs {#analysis:ysoid} ----------------------- We identified Young Stellar Objects (YSOs) in the DR15 region, by applying color and brightness criteria to our multiwavelength catalog. For sources that were detected in the Spitzer IRAC and MIPS 24$\mu$m bands, we classified Class I/0 (embedded protostars) and Class II (Classic T Tauri) sources using the criteria by @Ybarra:2013aa, we also required that these sources had photometric uncertainty values less than 0.25 mag in each band. Our color criteria @Ybarra:2013aa are essencially based in the color criteria of @Gutermuth:2008aa and @Kryukova:2012aa, but as explained by @Ybarra:2013aa we add an additional \[5.8\]-\[8.0\] criteria for objects that do not have a MIPS 24 $\mu m$ detection. In addition, our use of $JHK$ photometry allows us to identify additional less bright candidates and with the use of X-ray data we are able to identify Class III sources that do not have an infrared excess. Class III candidate sources were selected from a list of sources in the Chandra catalog that match in position with an infrared point source, after removing those corresponding to Class I/0 and Class II candidates. We added the requirement that $$J-H\ge0$$, because sources with $J-H<0$ are most likely either galaxies or spurious sources associated with diffraction spikes of bright stars. We further depurated this first counterpart list by keeping only those sources that had no evidence of a prominent circumstellar disk (i.e., we only selected stars with mostly photospheric emission). For this last criteria we used the $J-H$ vs $K-[4.5]$ color space: $$J-H>1.97(K-[4.5]).$$ and for those sources that do not have a detection in \[4.5\] we used: $$J-H>1.74(H-K).$$ The remaining list of Chandra-NIR counterparts are sources with evidence of a disk that were not previously selected as Class 0/I or II sources. We used the additional near-IR criteria of @Gutermuth:2008aa to select a few more young sources from this group. If a Chandra counterpart had photometry in the first three IRAC bands, and photometric errors less than 0.1 mag in at least \[3.6\] and \[4.5\], then it was classified as a Class 0/I candidate if: $$[4.5]-[5.8]>0.5 \mathrm{\ and\ } [3.6]-[4.5]>0.7,$$ while for Class II candidate identification, we used: $$0.2<[3.6]-[4.5]<0.7 \mathrm{\ and\ } 0.5<[4.5]-[5.0]<1.5.$$ A total of 20 sources could not be classified with these criteria if they did lack a detection in one or more bands, but if they fall to the right of the reddening band in the $J-H$ vs $K-[4.5]$ color space, they could be bonafide young sources. For 13 of these sources we determined their class (I or II) by inspecting their spectral energy distrubutions (SED), which we constructed using the SED fitting tool of @Robitaille:2007aa. Four additional sources were identified as possible AGN galaxies. The 3 remaining sources did not have enough IR bands to permit a clear classification and were discarded from the list. In total, we identified in our selected region a total of 226 YSOs, distributed as follows: 26 Class I/0 candidates, 155 Class II candidates and 45 Class III candidates (11, 69 and 20 percent of the total, respectively). We list all identified YSOs in the tables of the Appendix \[app:ysos\]. We matched our Class I, Class II and Class III catalogs against the catalog of @Kryukova:2014aa, out of 23 sources coinciding with our region of study, 17 were also identified as YSO candidates in this study. The 6 remaining sources lack emission is the \[4.5\] band and could not be confirmed as YSOs using our criteria. In Figure \[fig:diagrams\] we show the distribution of the classified YSOs in two different color-color diagrams. In the $J-H$ vs. $K-[4.5]$ diagram we can see that the most of Class I sources lie to the right of the reddening band indicating the presence of intense excess emission at IR wavelegths due to the stellar radiation in the dusty material of their envelopes or circumstellar disks. Class II sources also present infrared excess emission, although in a lesser way due the dust clearing within their inner disks. Then, the Class III sources lie within the reddening band or along the dwarf main sequence and lack significant infrared excess. The three groups of sources appear well separated in the \[3.6\]-\[4.5\] vs. \[4.5\]-\[8.0\] two-color diagram. ![Color-color diagrams for YSO sources in the DR15 region, Class I sources are marked with a red dot symbol; Class II sources are marked with a green dot symbol; Class III sources are indicated with a blue dot symbol. Sources marked with gray dot symbols are sources in the field without a YSO designation and photometric errors less than 0.1 mag in all bands. a) $J-H$ vs. $K-[4.5]$ two-color diagram. The solid curve is a model for dwarf main sequence population from the Dartmouth [@Dotter:2008aa] grid; the solid line at its right side is the Classic T-Tauri star locus [@Meyer:1997aa] adapted to this color space as in @Teixeira:2012aa. b) \[3.6\]-\[4.5\] vs. \[4.5\]-\[8.0\] two-color diagram, showing a clear separation between evolutionary classes.[]{data-label="fig:diagrams"}](f4a.pdf "fig:"){width="3.0in"} ![Color-color diagrams for YSO sources in the DR15 region, Class I sources are marked with a red dot symbol; Class II sources are marked with a green dot symbol; Class III sources are indicated with a blue dot symbol. Sources marked with gray dot symbols are sources in the field without a YSO designation and photometric errors less than 0.1 mag in all bands. a) $J-H$ vs. $K-[4.5]$ two-color diagram. The solid curve is a model for dwarf main sequence population from the Dartmouth [@Dotter:2008aa] grid; the solid line at its right side is the Classic T-Tauri star locus [@Meyer:1997aa] adapted to this color space as in @Teixeira:2012aa. b) \[3.6\]-\[4.5\] vs. \[4.5\]-\[8.0\] two-color diagram, showing a clear separation between evolutionary classes.[]{data-label="fig:diagrams"}](f4b.pdf "fig:"){width="3.0in"} Dust Extinction and the Spatial Distribution of YSOs \[analysis:extmap\] ------------------------------------------------------------------------ In Figure \[fig:ysodist\] we plot the spatial distribution of the YSOs on a dust extinction map. This dust extinction (A$_V$) map for the DR15 region was constructed with the near-IR catalog an optimized version of the near-infrared excess (NICER) algorithm of @Lombardi:2001aa, which estimates extinction using the dust-reddened colors of stars in the background of the cloud. In order to construct the map, we removed all sources with colors indicative of intrinsic infrared excess, which would bias the estimated extinction towards higher values. We used a sigma-clipping scheme to remove outlier values from the final weighted averages at each position. The map was constructed using Nyquist sampling on a equally spaced equatorial grid, smoothing the individual extinction estimates of background stars with a Gaussian filter of 30$\arcsec$ FWHM (this implies a resolution near 0.2 pc at a distance of 1.4 kpc). In Figure \[fig:ysodist\] we see how the map clearly resolves the filamentary structure of the IRDC in the northern part of the field and the morphology of the dust pillar on which DR15 is located. The extinction contours are limited to a maximum level of $A_V$=30 mag and further smoothed on the figure with a factor 3 boxcar, in order to remove some spurious features at the IRDC region. The resultant map shown in Figure \[fig:ysodist\] is in good agreement with a column density map constructed from 250 to 500 micron dust emission images from Herschel/SPIRE[^1], which also allows a much larger dynamic range (up to $A_V\approx$150 mag; Schneider et al., private comm.) Clearly, the regions with higher column density values are those hosting a majority of the youngest stars. Using our extinction map we determined how the Class I sources are distributed in the highest density regions. Using contours of constant extinction in s.pdf of 1.0 mag, we counted the number of Class I sources above each level, and found that 23 out of 24 sources in our region of study lie above A$_V$=13.0 mag, and 20 out of 24 sources lie above $A_V$=15.0 mag. The fraction of the total of Class I sources, $N(>A_V)/N_{total}$, decreases steeply after that, with only 50 percent of the total number of sources remaining at levels above $A_V$=22.0 mag. It is also important to notice that given the filamentary morphology of the IRDC, the projected area of the map contained above each level decreases very steeply for $A_V>12.0$ mag. We also found that the surface density of Class I sources, $\Sigma_*(>A_V)$, defined as the number of sources divided by the area above a given level, deviates little from a power-law behavior with a slope $\beta \approx 2.9$ in the range $12<A_V<40.0$ mag. All of this is very consistent with a Schmidt type relation like it was found for a set of nearby Giant Molecular Clouds by @Lada:2013aa, except that in those clouds the linear regime appears to be set at a lower extinction interval. Unfortunately, our numbers are too small to attempt a Bayesian analysis like that of @Lada:2013aa, and possibly a comparison with other regions containing IRDCs would be more fair. However, the fact that we find a possible Schmidt-like behavior in a region like Cygnus-X DR15 is very interesting and worth of a dedicated comparative analysis with similar regions, which is beyond the scope of this paper.  Properties of the youngest stars in the DR15 region \[analysis:SED\] -------------------------------------------------------------------- Using our master infrared catalogs we were able to construct spectral energy distrubutions (SED) for 24 of the Class 0/I sources we identified using the methods previously described. In most cases these sources belong to the cluster population associated with the dark infrared cloud in the northern section of our region of study. In a few cases we were able to complement these SEDs with Herschel Space Telescope PACS mid/far infrared photometry from the catalog of @Ragan:2012aa and 850 $\mu$m photometry from the catalog of @Difrancesco:2008aa. Using the SED fitting tool of @Robitaille:2007aa, we estimated some basic properties for the sources. In Figure \[fig:C1disks\] we show histograms of star masses and the disk accretion rates. The mass distribution suggests that the sources we are able to detect in DR15 are mostly intermediate to massive (mostly solar type and above). The accretion rates are in good agreement with estimates of typical accretion disk for sources with masses above solar [@Fang:2013aa]. ![Left: Distribution of Class 0/I candidate sources mass estimates from SED model fits. Right: Distribution of Class 0/I candidate sources disk accretion rate estimates from SED model fits.[]{data-label="fig:C1disks"}](f6a.pdf "fig:"){width="2.5in"} ![Left: Distribution of Class 0/I candidate sources mass estimates from SED model fits. Right: Distribution of Class 0/I candidate sources disk accretion rate estimates from SED model fits.[]{data-label="fig:C1disks"}](f6b.pdf "fig:"){width="2.5in"} The Star Formation History of DR15 \[analysis:sfh\] --------------------------------------------------- ### Identification of the main stellar groups in DR15 {#analysis:sfh:clusterid} To further investigate the stellar formation history in the DR15 region we attempted to identify the individual stellar clusters present in our field of study. Two clusters are easy to identify, namely the cluster at the center of the field and the cluster embedded in the IRDC at the north. However, the map of Figure \[fig:ysodist\] shows a good number of sources located around the molecular gas filaments, which may or may not be part of previously formed stellar clusters or groups. In order to identify significant overdensities of stars in the DR15 region, we constructed surface density maps using the `Gather` algorithm of @Gladwin:1999id. This algorithm is based in turn on the *nearest neighbor* method, which assigns individual surface density values to points on a two-dimensional map based on the equivalent circular area defined by the distance to a $n$th. neighbor point (see @Casertano:1985uq for a description of this method in the particular case of stellar cluster identification; some examples of its use for embedded clusters can be found in @Roman-Zuniga:2008aa [@Gutermuth:2009aa] and @Roman:2015aa). The `gather` algorithm is adequate for identifying individual clusters in a relatively simple layout, optimizing the value of $n$ that defines the surface density measurement and the size of the smoothing kernel used to construct the surface density map. We made individual `gather` maps for each of the candidate YSO class lists. These maps are shown in Figure \[fig:gather\]. The maps show the concentration of Class I sources at the north IRDC region. Class II sources are distributed over the entire region of study, but still concentrate in a few clearly defined groups. This led us to define some populations of stars, which we use as samples for our analysis. We named the two known clusters as DR15-C and DR15-N because their location center and north of our region of study. We identified and named as well three other groups: DR15-W, DR15-SW and DR15-SE. The purpose of this selection is to compare the age and age spread of these populations with the age of DR15-C. {width="7.0in"} ### YSO population ages from the K-band Luminosity Function {#analysis:sfh:clusterages} One of the goals of this study is to reconstruct the history of star formation in the region, by estimating the mean age and age spread of a set of the clusters and groups we identified from the `gather` maps. For this purpose, we constructed the K-band luminosity functions (KLF) of the groups and clusters, and compared them to the KLF of artificial pre-main sequence populations with different age ranges and age spreads. These artificial KLFs were constructed using the pre-main sequence model interpolation code of @Muench:2000aa. Another example of this method applied to a young cluster population can be found in @Roman:2015aa We selected the samples for DR15-C,DR15-W, DR15-SW and DR15-SE for this analysis. The samples were defined as circular areas that covered the four overdensities. The radii of the circles was chosen as 0.03 deg for DR15-W,SW and SE, and as 0.0225 for DR15-C. The KLF was constructed by limiting the samples by extinction in the following manner: we restricted the sample to those sources that a) fall above an extinction vector corresponding $A_V$=20 mag that reaches the sensitivity limit in a $H$ vs. $J-K$ color-magnitude diagram, and b) fall between an unreddened and unreddened 3 Myr pre-main sequence isochrone, properly shifted to the estimated distance to DR15. The extinction-limited samples assure a minimum contamination from extragalactic sources (which mostly will fall in the area below the extinction vector and the sensitivity limit) and will minimize a bias due to the decrease in the number of detected sources as a function of extinction. In their study, @Muench:2000aa showed that the observed shape of the KLF (mostly defined by the peak value) of a pre-main sequence population is particularly sensitive to three intrinsic parameters: the underlying Initial Mass Function (IMF), the mean age of the population, and to a lesser extent, the age spread or period of formation. Our analysis consists of constructing a grid of artificial KLF for each of our samples, and compare them with our observed function. We constructed this grid by assuming a fixed IMF and let the time parameters, mean age and age spread, to run free. For each population (DR15-C,DR15-W, DR15-SW and DR15-SE) we the used code of @Muench:2000aa to simulate artificial KLFs for clusters with the same number of sources and the same distributions for extinction and disk fractions as a function of color ($H-K$). We drew the artificial population from a broken power law IMF using the parameters of the one obtained for the IC 348 cluster (2 Myr old) in the paper of @Muench:2003aa. To draw the artificial populations we used the pre-main sequence stellar evolution model by [@DAntona:1997aa], with $[D/H]=2\times 10^{-5}$ to draw the samples for the artificial clusters . We ran the models using a grid of ages in which we varied the mean age of the cluster between 0.5 and 10 Myr, in s.pdf of 0.5 Myr. For each case, we simulated five age spreads between 1.0 to 5.0 Myr. For each case we simulated 500 clusters. The method could not be applied succesfully for the DR15-N sample, due to the large extinction variations in that cluster, which does not allow us to obtain a sufficient detection rate in the K band at the highest column density regions. In the remaining clusters, we were able to isolate extinction limited samples satisfactorily. We determined which age/age spread set fits each of the observed KLFs by averaging the KLF over all simulations for each point in the grid and comparing to the observed KLF, obtaining for each case a reduced $\chi^2$ estimation. In Figures \[fig:nchi1\] and \[fig:nchi2\] we show, for each cluster a contour plot of the reduced $\chi^2$ in a mean age vs age spread plane, indicating the parameter region with the best adjustments. ![The panels show contour maps of the normalized $\chi ^2$ values for the age vs. age spread estimation of clusters DR15-C and DR15-W using pre-main sequence models of the KLF. The most likely values fall within regions with purple and dark blue colors. The white contours indicate 68 and 95 percent confidence limits.The limits of the model grid are indicated by a thin, red line.[]{data-label="fig:nchi1"}](f8a.pdf "fig:"){width="3.0in"} ![The panels show contour maps of the normalized $\chi ^2$ values for the age vs. age spread estimation of clusters DR15-C and DR15-W using pre-main sequence models of the KLF. The most likely values fall within regions with purple and dark blue colors. The white contours indicate 68 and 95 percent confidence limits.The limits of the model grid are indicated by a thin, red line.[]{data-label="fig:nchi1"}](f8b.pdf "fig:"){width="3.0in"} ![Same as in Figure \[fig:nchi1\], but for clusters DR15-SW and DR15-SE[]{data-label="fig:nchi2"}](f9a.pdf "fig:"){width="3.0in"} ![Same as in Figure \[fig:nchi1\], but for clusters DR15-SW and DR15-SE[]{data-label="fig:nchi2"}](f9b.pdf "fig:"){width="3.0in"} In Table \[tab:results\] we list the results of the artificial KLF modeling analysis for the cluster samples in DR15. For each cluster we also include the number of Class I, Class II and Class III YSOs found in each population. The more embedded clusters DR15-C and DR15-W contain a population that according to our analysis, could have a mean age of 3.0 Myr. The populations flanking the main molecular pillar, DR15-SE and DR15-SW, appear to have mean ages as large as 3.5 and 4.5 Myr, respectively. The contour plots for these two samples, however, appear to have two local minima, suggesting age spreads as short as 3.0-3.5 Myr or as large as 4.5-5.0 Myr. The adjustment to an older age is in agreement with the less embedded status of these two samples. Notice that the confidence ranges in all cases indicate that our method cannot really constrain the age spreads satisfactorily (the 95 percent confidence range in DR15-C is the only one that suggest a constraint towards an age spread of 3.5 to 5 Myr). Still, given the embedded nature of these young star populations, we think it is little plausible that age spreads can be significantly larger than 5 Myr. [lccccc]{} DR15-C & 3.0\[2.5,3.5\] & 4.5\[3.5,5.0\] & 1 & 8 & 4\ DR15-W & 3.5\[2.5,4.5\] & 4.5\[1.0,5.0\] & 2 & 16 & 5\ DR15-SE & 3.5\[3.0,4.5\] & 3.0\[1.0,5.0\] & 0 & 5 & 0\ DR15-SW & 4.5\[4.0,5.0\] & 3.0\[1.0,5.0\] & 0 & 11 & 2\ DR15-N & – & – & 15 & 21 & 6\ DR15-NE & – & – & 0 & 11 & 3\ \[tab:results\] The slow removal of the DR15-C cluster molecular envelope {#analysis:sfh:radio_envelope} --------------------------------------------------------- At 3 Myr of age, we could expect that the DR15-C cluster have removed a significant fraction of its molecular envelope, as it occurs in clusters of a similar age, e.g. IC-348 [@Muench:2003aa], IC 1795 [e.g. @Oey:2005ly  Román-Zúñiga et al 2015, in rev.]. In fact our images show how other groups adjacent to DR15-C, like DR15-SE and DR15-SW, which our analysis suggests have similar ages, are associated with much less prominent molecular cloud features. Still, the molecular envelope of DR15-C appears as a well defined structure, both dense and compact, surrounding the cluster atop a dense molecular pillar. Using the $^{13}$CO(1-0) map of @Schneider:2011aa we made two position-velocity (PV) cuts across the envelope of DR15-C. We used our extinction map and a zero moment (integrated intensity) integration of the $^{13}$CO(1-0) map as a guide. The first cut (L2) runs across the observable structure of the pillar in the region observed. The second cut (L4) runs almost perpendicular to L2. The L2 cut shows a component related to the neck of the pillar, almost 2 pc long with a radial velocity about 4 km/s away from the Cygnus OB2 system velocity (0 km/s). Near the center of the cut, coincidental with the envelope of DR15-C, the PV plot shows a hint of an elliptical shell structure, with a red component moving slightly above 4 km/s and a blue component near 0 km/s. The latter merges into the dark infrared cloud, which shows a very smooth gradient from 2 to 0 km/s. The L4 cut shows for the most part, gas with velocities near 0 km/s but also a much more clear elliptical shell structure at the envelope, which opens from 0 to 5 km/s. The elliptical shell structure corresponds to the expanding envelope of DR15-C. ![Top: Zero moment (integrated intensity) $^{13}$CO(1-0) map of the DR15 region, constructed from the survey of @Schneider:2006aa, the grayscale is indicated in units of km/s. Contours indicate visual extinction, similar to Figure \[fig:ysodist\]. The two red arrow lines labeled as L2 and L4 indicate cuts along which position-velocity (P-V) plots were obtained (see text). Center, Bottom: P-V plots along L2 and L4 cuts, as indicated in top panel; position along the length of each cut is indicated both in arcseconds and in parsecs, asumming a distance of 1.4 kpc.[]{data-label="fig:pvcuts"}](f10a.pdf "fig:"){width="3.0in"}\ ![Top: Zero moment (integrated intensity) $^{13}$CO(1-0) map of the DR15 region, constructed from the survey of @Schneider:2006aa, the grayscale is indicated in units of km/s. Contours indicate visual extinction, similar to Figure \[fig:ysodist\]. The two red arrow lines labeled as L2 and L4 indicate cuts along which position-velocity (P-V) plots were obtained (see text). Center, Bottom: P-V plots along L2 and L4 cuts, as indicated in top panel; position along the length of each cut is indicated both in arcseconds and in parsecs, asumming a distance of 1.4 kpc.[]{data-label="fig:pvcuts"}](f10b.pdf "fig:"){width="3.0in"}\ ![Top: Zero moment (integrated intensity) $^{13}$CO(1-0) map of the DR15 region, constructed from the survey of @Schneider:2006aa, the grayscale is indicated in units of km/s. Contours indicate visual extinction, similar to Figure \[fig:ysodist\]. The two red arrow lines labeled as L2 and L4 indicate cuts along which position-velocity (P-V) plots were obtained (see text). Center, Bottom: P-V plots along L2 and L4 cuts, as indicated in top panel; position along the length of each cut is indicated both in arcseconds and in parsecs, asumming a distance of 1.4 kpc.[]{data-label="fig:pvcuts"}](f10c.pdf "fig:"){width="3.0in"} We defined the area of the shell as a rectangle of 5.25$\arcmin \times$3.75$\arcmin$ around the defined center of DR15C. Following the prescription by @Estalella:1999aa, we estimated the column density, N(H$_2$) and the total mass of expanding gas within $3<v_r<6$ km/s, as $M_{out}=$103.5 M$_\odot$. Then we used the method described by [@Qiu:2009aa] to estimate the dynamical time of the component, $t_{dyn}=2.9\times 10^5$ yr. This implies a total mass loss rate of $\dot{M}_{out}=M_{out}/t_{dyn}=360 \mathrm{\ M}_\odot \mathrm{\ Myr}^{-1}$. At this rate, it would take about 3 Myr to remove the total mass of gas in the shell, which we estimate to be 1023 M$_\odot$, using our extinction map, and a distance to the pillar of 1.4 kpc. This result appears to be consistent with estimates of the mass loss rates in other molecular pillars. For instance, using integral field unit spectroscopy, @Westmoquette:2013aa estimated a mass loss rate of 300 $\mathrm{\ M}_\odot \mathrm{\ Myr}^{-1}$ for the pillars of NGC 3603, which contain a total of about 700 M$_\odot$ of gas. For the “Pillars of creation" in M16, an estimate by @Mcleod:2015aa is of 70 $\mathrm{\ M}_\odot \mathrm{\ Myr}^{-1}$. In both cases, the removal timescales are of the order of 2-3 Myr. However, we need to point out that those estimates are based on models of photoevaporation, while our estimate comes from an estimation of a gas outflow rate from a zero moment map of molecular gas emission. The shell mass could be overestimated, due mainly to a large column density in the line of sight towards Cygnus-X, in which case we would be more consistent with the optical spectroscopy studies. However, what we consider important to note is that the removal of the envelope of DR15-C is relatively slow compared to other embedded cluster regions, where gas dispersal timescales are similar or shorter than the T Tauri timescale, i.e. less than 2 Myr . Discussion =========== The main goal of this paper is to reconstruct the history of star formation in the Cygnus-X DR15 region. For this purpose, we made a) an analysis of the spatial distribution of YSO candidate sources classified by evolutionary classes, and b) a comparison of the observed KLF of several young star population samples in our region of study with those of artificial cluster samples drawn from pre-main sequence stellar evolution models. The classification of YSO candidates and their spatial distribution reveals that several populations of young stars are present in the field of study. At the north/northeast part of our field we identified a very young cluster, DR15-N, hosting 15 Class 0/I sources. The cluster is forming within an infrared dark cloud that runs in the east-west direction at an estimated distance of 1400 pc [@Rizzo:2014aa; @Palau:2014aa]. The cluster also contains a relatively large number (21) of Class II sources and thus it may host the youngest population in the region of study. For this cluster we were not able to construct an unbiased extinction-limited sample to construct a KLF. Instead, we constructed SEDs for all the Class I sources we identified within it, and we compared them to YSO models. We found that in most cases, the models with the best fits correspond to intermediate to high mass YSOs with ages of $\sim 1$ Myr. The mass of the IRDC, estimated from our NICEST extinction map is of $\sim 2400\mathrm{\ M}_\odot$, with an equivalent radii of 5.2 pc. These parameters are in good agreement with massive star forming IRDCs as defined by the analysis of @Kauffmann:2010aa. The revised distance to Cygnus-X from the study of @Rygl:2012aa is 1.4 kpc, in agreement with the estimations of @Rizzo:2014aa and @Palau:2014aa. From our position-velocity cuts, we note that the emission along the structure of the dark infrared cloud has a null relative velocity compared to the Cygnus-X systemic velocity, same as the DR12-15 structure. It would be difficult to determine if the dark infrared cloud is located at the same distance than the DR15 pillar based only on a radial velocity difference argument. Also, it is important to notice that @Rygl:2012aa could not confirm that Cygnus-X South structures are at the same distance as those in Cygnus-X North where most of their measurements were made. We also lack enough evidence to claim interaction between the DR15 pillar and the IRDC. Using purely morphological arguments we like to comment that the formation of molecular pillars like DR15 is thought to be the result of the interaction between the molecular cloud and the photoionization region formed by the ionizing radiation of the central cluster (Cygnus OB2 in our case), while the IRDC appears to be a highly dense and coherent structure, apparently less affected by the HII regions. Also, our images do not show any obvious pillar structures coincident with the IRDC. These arguments could work in favor of a scenario in which the IRDC could be located at a slightly different distance than the DR15 pillar. According to our observations and our KLF simulation results, DR15-C is a young cluster with an age of about $~3$ Myrs, but it has a low number of young sources, with only one identified Class I, 8 Class II and 4 Class III sources, all of this despite the presence of the thick, nebulous envelope evident in extinction and $^{13}$CO(1-0) integrated intensity maps. We know the DR15 pillar hosts an embedded cluster, as evidenced by well known far-infrared sources associated with the photodissociation region [@Odenwald:1990aa] and also from our KLF analysis, which indicates that 200 sources could be present within DR15-C, after background and foreground contamination corrections. We do not think the contamination by sources from the DR15-N cluster is too high, as most the density peak of that cluster is located to the northeast and it is more embedded. However, it cannot be fully discarded that some sources in DR15-C are actually sources from DR15-N (and viceversa). Also, there seems to be another YSO group in the filament, south of DR15-C, which reinforces the idea of star formation being active in the pillar. Our evidence points to a scenario in which a) the star formation episode in DR15-C has probably reached its end or it is near its end and b) the parental gas has dissipated relatively slow, or at least slower that the dispersion time of circumstellar disks of its member stars. Our analysis of the $^{13}$CO(1-0) zero moment map indicates it would take up to 4 Myr to remove the cluster gaseous envelope. However, our observations in other cluster forming regions like the Rosette Molecular Cloud and W3 [e.g. @Roman-Zuniga:2008aa; @Ybarra:2013aa; @Roman:2015aa] suggest that clusters hosting intermediate to massive stars may remove their gas envelopes in periods shorter than the T Tauri timescale. Therefore, our analysis of DR15-C suggests that clusters forming in this kind of pillar structures, could dissipate their envelopes at slower rates. As shown in recent studies like those of @Westmoquette:2013aa and @Mcleod:2015aa the mass loss rates in pillars due to photoevaporation are of the order of $\sim 10^2\mathrm{\ M}_\odot \mathrm{\ Myr}^{-1}$, not too different from our estimations in the expanding gas shell. Clearly, we are comparing very different methods, that refer to very different processes (shell expansion vs. shell photo-erosion). It is important to notice that the DR15 pillar may not be the same kind of dusty pillar as those in M16, because extinction is too high to allow delineating the structure in DR15 using optical images. The molecular fragment located west of DR15 shows as a shadow with a pillar morphology in optical images, which make us think that DR15 could be a pillar too. Even so, the fact that the mass loss estimates for DR15 coincide in the order of magnitude of the effect with respect to M16 is interesting, and motivates further investigation. A detailed study of the removal of the DR15-C envelope is out of the scope of this paper, but it is the main topic in a close following study (Román-Zúñiga et al., in prep). DR15-W, DR15-SE and DR15-SW appear to be slightly older populations ($3.5-4.75$ Myrs) with much less gas and dust observed. However, we find evidence of remains of a structure that could have been similar to the pillar associated with DR15-C. It may be possible that the DR15-SE and DR15-SW groups belong to clusters formed before DR15-C, but their ages may not be much older, as evident from the presence of Class II and Class III sources. The estimated ages of the DR15-SW and SE samples suggest that the cluster evolution period, from formation to gas dispersal in the region could be around 5 Myr. Summary ======= The Cygnus-X DR15 region presents a prominent gaseous pillar as well as an IRDC, both hosting clusters of young stars. In this investigation, we made a multi-wavelength study of the young stellar population in the region. For this purpose, we processed and analyzed deep, high quality near-IR images of the region, as well as X-ray images from the Chandra Observatory. Using these datasets we obtained photometry catalogs for all point sources we were able to detect. We combined these catalogs with the 3.6 to 24 $\mu$m photometry catalog of the Cygnus-X Spitzer Legacy Survey, resulting in a master catalog containing almost 47 thousand individual sources. From our master catalog, we identified 226 young stellar sources, which we classified according to their evolutionary class related to the prominence of their circumstellar disks. We found that the young sources distribute into 26 Class 0/I, 156 Class II and 45 Class III sources. From our near-IR we constructed an extinction map, which we used to study the spatial distribution of the young sources in the molecular cloud structure present in the region. We found that the youngest population of this region is currently forming at the IRDC, at the regions of highest column density. Combining this with maps of YSO surface density, we were able to identify several groups, possibly associated to distinct stellar clusters. We obtained extinction-limited samples of these groups in order to construct their K band luminosity functions (KLF). We compared the observed KLFs with those of artificial young cluster populations sampled from interpolation of pre-main sequence models. This allowed us to make first order estimations of the mean ages and age spreads of the cluster population samples. We constructed SEDs for all Class I sources identified within the IRDC region, which allowed us to estimate their mass and disk accretion rates. These estimations are consistent with the formation of an intermediate mass star cluster, indicating that structures of this kind in Cygnus-X are possibly birth places of massive clusters. Using the FCRAO $^{13}$CO(1-0) from the study of @Schneider:2011aa we estimated the radial velocity distribution along the IRDC and the DR15 pillar, and we found that the nebulous envelope of the DR15-C cluster at the tip of the pillar is consistent with an expanding shell morphology. We estimated the mass loss rate of gas in this expanding shell and we deduced that the dissipation process of the DR15-C cluster gas envelope is relatively slow, compared to what we found in other studies of cluster forming regions, where the gas dispersal process is shorter than the T Tauri timescale. The mass loss rate we estimated is in the same order of magnitude as mass loss rates by photodissociation found in studies of other gas pillars using optical spectroscopy. This suggests that clusters forming in gas pillars like DR15-C could have a different evolution process than clusters forming at dense clumps in other giant molecular clouds. The presence of other populations containing Class II and Class III sources at the regions flanking the DR15 pillar, and projected near less dense but still noticeable gaseous structures, with estimated ages of 4 to 5 Myr is suggestive of a process of cluster forming processes that take about that long to form an dissipate in the molecular cloud complex that surrounds the Cygnus-X HII region. **Acknowledgements:** We thank the referee, Nicola Schneider for providing a comprehensive and constructive review that greatly improved the content of our manuscript. We thank Nicola Schneider for kindly providing us with a copy of the FCRAO $^{13}$CO(1-0) map for our study. CRZ acknowledges support from CONACYT project CB2010-152160, Mexico and programs UNAM-DGAPA-PAPIIT IN103014 and IN116315. EAL acknowledges support from the National Science Foundation through NSF Grant AST-1109679 to the University of Florida. This study is based on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC). We acknowledge the staff at Calar Alto for top of the line queued observations at the 3.5m with OMEGA 2000. We acknowledge use of data products from the 2MASS, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Centre/California Institute of Technology (funded by the USA National Aeronautics and Space Administration and National Science Foundation). This work is partly based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. 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The remaining sources are listed with 2MASS or CXLS depending if they were detected in those surveys. [lllllllllllllllllllll]{} CAHA\_20322208\_402017 & 20:32:22.08 & +40:20:17.2 & 18.827 & 9.999 & 18.101 & 0.02 & 12.026 & 0.02 & 7.411 & 0.015 & 6.021 & 0.015 & 5.066 & 0.015 & 4.136 & 0.015 & 0.395 & 0.018 & — & —\ CAHA\_20322856\_401941 & 20:32:28.56 & +40:19:41.6 & 14.827 & 0.003 & 11.397 & 0.019 & 8.943 & 0.016 & 7.372 & 0.015 & 6.304 & 0.015 & 5.349 & 0.015 & 4.427 & 0.015 & 0.981 & 0.017 & — & —\ CAHA\_20321777\_401408 & 20:32:17.77 & +40:14:08.2 & 15.139 & 0.015 & 12.744 & 0.014 & 10.647 & 0.014 & 8.162 & 0.015 & 7.323 & 0.015 & 6.455 & 0.015 & 4.864 & 0.015 & 1.412 & 0.016 & — & —\ SSTCYGX\_J203222.99\_402021.4 & 20:32:22.99 & +40:20:21.4 & — & — & — & — & — & — & 13.706 & 0.17 & 11.445 & 0.106 & 9.627 & 0.041 & 7.858 & 0.033 & 1.662 & 0.019 & — & —\ CAHA\_20322060\_401950 & 20:32:20.60 & +40:19:50.1 & — & — & — & — & 16.815 & 0.02 & 10.233 & 0.021 & 8.048 & 0.016 & 6.613 & 0.015 & 5.777 & 0.017 & 2.145 & 0.038 & — & —\ CAHA\_20323152\_401352 & 20:32:31.52 & +40:13:53.0 & 20.457 & 0.042 & 14.909 & 0.008 & 14.374 & 0.089 & 8.519 & 0.016 & 7.417 & 0.015 & 6.565 & 0.017 & 5.96 & 0.03 & 2.229 & 0.111 & — & —\ CAHA\_20322113\_402025 & 20:32:21.13 & +40:20:25.6 & 17.814 & 9.999 & 16.994 & 0.013 & 13.068 & 0.009 & 9.281 & 0.016 & 8.148 & 0.016 & 7.201 & 0.016 & 6.35 & 0.017 & 2.316 & 0.021 & 2.694 & 5.880E-15\ CAHA\_20322194\_401937 & 20:32:21.94 & +40:19:37.9 & — & — & 20.053 & 0.075 & 15.564 & 0.017 & 11.613 & 0.021 & 10.495 & 0.02 & 7.246 & 0.016 & 5.487 & 0.021 & 2.668 & 0.056 & — & —\ CAHA\_20322126\_401601 & 20:32:21.26 & +40:16:01.8 & 18.458 & 0.023 & 14.213 & 0.007 & 11.565 & 0.018 & 8.827 & 0.015 & 7.851 & 0.015 & 7.005 & 0.015 & 6.26 & 0.017 & 3.036 & 0.18 & — & —\ CAHA\_20320254\_401838 & 20:32:02.54 & +40:18:39.0 & — & — & — & — & 17.492 & 0.073 & 11.914 & 0.017 & 10.031 & 0.016 & 8.879 & 0.017 & 8.241 & 0.033 & 3.077 & 0.046 & — & —\ CAHA\_20322282\_401940 & 20:32:22.82 & +40:19:40.9 & — & — & — & — & 16.245 & 0.038 & 12.171 & 0.03 & 10.362 & 0.019 & 9.161 & 0.021 & 8.26 & 0.059 & 3.583 & 0.118 & — & —\ CAHA\_20315797\_401835 & 20:31:57.97 & +40:18:35.8 & 21.759 & 0.22 & — & — & — & — & 13.975 & 0.028 & 12.189 & 0.027 & 11.35 & 0.036 & 10.973 & 0.112 & 3.677 & 0.035 & — & —\ SSTCYGX\_J203153.84\_401833.9 & 20:31:53.84 & +40:18:33.9 & — & — & — & — & — & — & 13.039 & 0.018 & 11.039 & 0.016 & 9.802 & 0.019 & 8.87 & 0.028 & 5.383 & 0.092 & — & —\ CAHA\_20322222\_401955 & 20:32:22.22 & +40:19:56.0 & 18.432 & 9.999 & 16.975 & 0.008 & 13.395 & 0.005 & 10.72 & 0.016 & 9.534 & 0.016 & 8.519 & 0.018 & 7.538 & 0.025 & — & — & — & —\ CAHA\_20322014\_401953 & 20:32:20.14 & +40:19:53.6 & 17.771 & 9.999 & 17.345 & 9.999 & 14.847 & 0.006 & 9.945 & 0.016 & 8.612 & 0.016 & 7.726 & 0.017 & 7.577 & 0.026 & — & — & — & —\ CAHA\_20322784\_401942 & 20:32:27.84 & +40:19:42.4 & 19.796 & 0.037 & 15.484 & 0.007 & 12.902 & 0.005 & 10.559 & 0.017 & 9.733 & 0.018 & 8.976 & 0.021 & 8.266 & 0.062 & — & — & — & —\ CAHA\_20322033\_402001 & 20:32:20.33 & +40:20:01.5 & 16.173 & 0.003 & 15.529 & 0.007 & 15.118 & 0.012 & 13.54 & 0.118 & 11.84 & 0.066 & 10.647 & 0.059 & 9.852 & 0.075 & — & — & — & —\ CAHA\_20322899\_401821 & 20:32:28.99 & +40:18:21.2 & — & — & — & — & 16.45 & 0.03 & 14.072 & 0.062 & 13.326 & 0.036 & 12.328 & 0.117 & 11.479 & 0.24 & — & — & — & —\ CAHA\_20322111\_402001 & 20:32:21.11 & +40:20:01.1 & — & — & — & — & 16.686 & 0.047 & 11.988 & 0.033 & 10.318 & 0.023 & 9.153 & 0.019 & 8.38 & 0.027 & — & — & — & —\ CAHA\_20322781\_402032 & 20:32:27.81 & +40:20:32.7 & — & — & — & — & 18.385 & 0.1 & 14.168 & 0.022 & 12.909 & 0.018 & 12.075 & 0.042 & 11.14 & 0.069 & — & — & — & —\ SSTCYGX\_J203226.39\_401847.4 & 20:32:26.39 & +40:18:47.4 & — & — & — & — & — & — & 14.761 & 0.051 & 13.612 & 0.028 & 12.252 & 0.112 & 11.323 & 0.185 & — & — & — & —\ CAHA\_20320404\_401856 & 20:32:04.04 & +40:18:56.6 & 18.559 & 0.009 & 15.162 & 0.01 & 13.261 & 0.01 & 11.352 & 0.017 & 10.546 & 0.016 & 9.895 & 0.03 & 9.09 & 0.064 & 5.954 & 0.42 & 4.636 & 1.717E-14\ CAHA\_20323480\_401629 & 20:32:34.80 & +40:16:29.2 & 17.797 & 0.023 & 14.591 & 0.016 & 12.513 & 0.009 & 10.185 & 0.05 & 9.455 & 0.048 & 8.048 & 0.145 & — & — & — & — & 3.906 & 1.154E-14\ CAHA\_20322210\_401800 & 20:32:22.10 & +40:18:00.3 & 13.681 & 0.003 & 12.608 & 0.006 & 12.447 & 0.023 & 11.337 & 0.021 & 11.004 & 0.022 & 10.653 & 0.153 & 9.602 & 0.323 & — & — & 1.438 & 6.480E-15\ CAHA\_20322358\_401729 & 20:32:23.58 & +40:17:29.9 & 14.661 & 0.003 & 13.556 & 0.005 & 13.146 & 0.005 & 12.243 & 0.08 & 12.081 & 0.112 & 9.757 & 0.146 & — & — & — & — & 1.204 & 4.064E-15\ CAHA\_20325161\_401945 & 20:32:51.61 & +40:19:45.6 & 16.385 & 0.005 & 14.341 & 0.007 & 13.271 & 0.005 & 12.165 & 0.017 & 11.689 & 0.017 & 11.375 & 0.024 & 11.162 & 0.054 & — & — & 3.351 & 1.147E-14\ \[tab:ClassI\] [lllllllllllllllllllll]{} CAHA\_20324212\_401726 & 20:32:42.12 & +40:17:27.0 & 12.732 & 0.005 & 11.778 & 0.03 & 11.309 & 0.023 & 11.001 & 0.018 & 10.834 & 0.018 & 10.6 & 0.103 & 9.503 & 0.22 & 2.175 & 0.175 & — & —\ TWOM\_20322301\_4019226 & 20:32:23.02 & +40:19:22.7 & 12.204 & 0.025 & 10.916 & 0.021 & 10.071 & 0.014 & 8.789 & 0.015 & 8.154 & 0.015 & 7.561 & 0.015 & 6.622 & 0.016 & 2.94 & 0.037 & — & —\ CAHA\_20323244\_402105 & 20:32:32.44 & +40:21:05.1 & 13.473 & 0.003 & 11.578 & 0.018 & 10.661 & 0.017 & 10.131 & 0.015 & 9.936 & 0.015 & 9.751 & 0.016 & 9.249 & 0.024 & 3.358 & 0.043 & — & —\ CAHA\_20321111\_401916 & 20:32:11.11 & +40:19:16.7 & 19.311 & 0.033 & 15.549 & 0.016 & 13.14 & 0.007 & 11.057 & 0.03 & 9.915 & 0.023 & 9.281 & 0.096 & 8.539 & 0.248 & 3.658 & 0.235 & — & —\ CAHA\_20321736\_401949 & 20:32:17.36 & +40:19:49.5 & 19.194 & 0.02 & 15.036 & 0.006 & 12.555 & 0.004 & 10.287 & 0.017 & 9.536 & 0.016 & 8.877 & 0.063 & 8.238 & 0.205 & 3.748 & 0.159 & — & —\ CAHA\_20320319\_402215 & 20:32:03.19 & +40:22:15.9 & 14.418 & 0.004 & 12.749 & 0.008 & 12.095 & 0.02 & 9.921 & 0.016 & 9.256 & 0.015 & 8.651 & 0.017 & 7.743 & 0.038 & 4.189 & 0.086 & — & —\ CAHA\_20324852\_402104 & 20:32:48.52 & +40:21:04.8 & 13.849 & 0.003 & 12.805 & 0.005 & 12.1 & 0.021 & 10.642 & 0.015 & 9.731 & 0.015 & 9.054 & 0.016 & 8.221 & 0.021 & 4.79 & 0.046 & — & —\ CAHA\_20325574\_402220 & 20:32:55.74 & +40:22:20.6 & — & — & 19.872 & 0.06 & 16.277 & 0.041 & 11.28 & 0.016 & 9.469 & 0.015 & 8.207 & 0.015 & 7.367 & 0.015 & 4.826 & 0.036 & — & —\ CAHA\_20315532\_402216 & 20:31:55.32 & +40:22:16.8 & 14.671 & 0.003 & 13.231 & 0.008 & 12.382 & 0.006 & 11.501 & 0.016 & 11.015 & 0.016 & 10.594 & 0.019 & 9.544 & 0.026 & 4.841 & 0.196 & — & —\ CAHA\_20321148\_401807 & 20:32:11.71 & +40:18:05.1 & 13.322 & 0.039 & 12.351 & 0.042 & 11.517 & 9.999 & 10.704 & 0.019 & 10.358 & 0.018 & 10.261 & 0.051 & 9.379 & 0.117 & 4.949 & 0.169 & — & —\ CAHA\_20320833\_401604 & 20:32:08.33 & +40:16:04.7 & 13.914 & 0.004 & 12.784 & 0.005 & 12.356 & 0.023 & 11.52 & 0.019 & 11.226 & 0.017 & 10.55 & 0.061 & 9.419 & 0.111 & 5.031 & 0.162 & 1.774 & 1.365E-14\ CAHA\_20321289\_401257 & 20:32:12.89 & +40:12:58.0 & 14.167 & 0.011 & 12.895 & 0.021 & 12.215 & 0.02 & 11.591 & 0.016 & 11.367 & 0.016 & 11.211 & 0.04 & 10.694 & 0.089 & 5.4 & 0.076 & 2.069 & 9.625E-15\ CAHA\_20324359\_402121 & 20:32:43.59 & +40:21:21.3 & 14.853 & 0.003 & 12.584 & 0.007 & 11.295 & 0.017 & 9.688 & 0.015 & 9.134 & 0.015 & 8.61 & 0.015 & 7.939 & 0.016 & 5.42 & 0.07 & — & —\ TWOM\_20315270\_4019059 & 20:31:52.71 & +40:19:06.0 & 17.14 & 0.217 & 14.15 & 0.042 & 12.799 & 0.029 & 12.079 & 0.019 & 11.8 & 0.026 & 11.499 & 0.045 & 10.641 & 0.084 & 5.502 & 0.245 & 2.373 & 9.074E-15\ CAHA\_20315661\_401614 & 20:31:56.61 & +40:16:14.0 & 12.979 & 0.005 & 12.093 & 0.019 & 11.357 & 0.016 & 10.733 & 0.016 & 10.412 & 0.016 & 10.187 & 0.034 & 9.427 & 0.073 & 5.586 & 0.221 & 1.511 & 1.315E-14\ CAHA\_20320053\_402026 & 20:32:00.53 & +40:20:27.0 & 15.688 & 0.003 & 13.881 & 0.005 & 12.739 & 0.005 & 11.66 & 0.017 & 10.921 & 0.016 & 10.346 & 0.027 & 9.495 & 0.049 & 5.696 & 0.133 & — & —\ CAHA\_20325540\_402150 & 20:32:55.40 & +40:21:50.7 & 15.923 & 0.005 & 13.857 & 0.007 & 12.355 & 0.007 & 10.89 & 0.016 & 10.376 & 0.016 & 9.785 & 0.016 & 9.012 & 0.018 & 5.954 & 0.056 & — & —\ CAHA\_20330082\_401800 & 20:33:00.82 & +40:18:00.2 & 13.978 & 0.007 & 12.929 & 0.012 & 12.615 & 0.026 & 11.539 & 0.016 & 11.08 & 0.016 & 10.769 & 0.018 & 9.75 & 0.022 & 6.142 & 0.09 & — & —\ CAHA\_20322395\_402218 & 20:32:23.95 & +40:22:18.3 & 15.943 & 0.037 & 14.013 & 0.035 & 12.994 & 0.033 & 11.783 & 0.016 & 11.241 & 0.016 & 10.787 & 0.021 & 10.031 & 0.034 & 6.152 & 0.158 & — & —\ CAHA\_20321650\_402141 & 20:32:16.50 & +40:21:41.1 & 18.196 & 0.012 & 15.382 & 0.016 & 13.564 & 0.021 & 12.04 & 0.017 & 11.356 & 0.016 & 10.867 & 0.029 & 10.155 & 0.057 & 6.186 & 0.177 & — & —\ CAHA\_20320880\_402054 & 20:32:08.80 & +40:20:54.9 & 15.698 & 0.003 & 13.914 & 0.006 & 12.866 & 0.004 & 11.565 & 0.016 & 10.84 & 0.016 & 10.039 & 0.025 & 8.887 & 0.039 & 6.378 & 0.098 & — & —\ CAHA\_20324730\_402217 & 20:32:47.30 & +40:22:17.9 & 14.143 & 0.004 & 13.131 & 0.006 & 12.514 & 0.006 & 11.697 & 0.016 & 11.226 & 0.016 & 10.764 & 0.019 & 10.016 & 0.021 & 6.462 & 0.123 & — & —\ CAHA\_20321184\_401733 & 20:32:11.84 & +40:17:33.8 & 14.27 & 0.002 & 12.991 & 0.005 & 12.4 & 0.009 & 11.543 & 0.017 & 11.161 & 0.017 & 10.727 & 0.055 & 10.004 & 0.086 & 6.559 & 0.163 & — & —\ CAHA\_20325680\_401223 & 20:32:56.80 & +40:12:23.3 & 15.594 & 0.015 & 13.946 & 0.017 & 12.806 & 0.017 & 11.997 & 0.016 & 11.55 & 0.016 & 11.297 & 0.021 & 10.649 & 0.029 & 6.835 & 0.132 & — & —\ CAHA\_20330112\_401449 & 20:33:01.12 & +40:14:49.2 & 14.081 & 0.011 & 13.502 & 0.036 & 12.493 & 0.025 & 11.214 & 0.016 & 10.878 & 0.016 & 10.53 & 0.018 & 9.727 & 0.02 & 6.93 & 0.237 & — & —\ CAHA\_20330982\_401154 & 20:33:09.82 & +40:11:54.6 & 14.388 & 0.01 & 13.128 & 0.012 & 12.538 & 0.024 & 11.265 & 0.016 & 10.82 & 0.016 & 10.411 & 0.017 & 9.723 & 0.017 & 6.99 & 0.199 & — & —\ CAHA\_20325220\_402406 & 20:32:52.20 & +40:24:06.2 & 15.228 & 0.006 & 13.653 & 0.008 & 12.688 & 0.009 & 11.626 & 0.016 & 11.181 & 0.016 & 10.784 & 0.018 & 10.062 & 0.029 & 7.023 & 0.064 & — & —\ CAHA\_20324795\_402214 & 20:32:47.95 & +40:22:14.2 & 18.103 & 0.007 & 15.928 & 0.007 & 14.661 & 0.007 & 13.505 & 0.038 & 12.873 & 0.032 & 12.302 & 0.053 & 11.42 & 0.07 & 7.027 & 0.246 & — & —\ CAHA\_20325444\_401949 & 20:32:54.44 & +40:19:49.5 & 14.826 & 0.004 & 13.712 & 0.007 & 13.046 & 0.005 & 12.011 & 0.016 & 11.721 & 0.016 & 11.454 & 0.025 & 11.034 & 0.037 & 7.044 & 0.183 & — & —\ CAHA\_20320189\_401007 & 20:32:01.89 & +40:10:07.8 & 16.296 & 0.009 & 14.641 & 0.011 & 13.655 & 0.049 & 12.911 & 0.02 & 12.625 & 0.02 & 12.386 & 0.049 & 11.962 & 0.109 & 7.045 & 0.217 & — & —\ CAHA\_20330546\_401215 & 20:33:05.46 & +40:12:15.7 & 14.32 & 0.036 & 13.247 & 0.038 & 12.594 & 0.055 & 11.881 & 0.016 & 11.534 & 0.016 & 11.129 & 0.021 & 10.243 & 0.035 & 7.084 & 0.19 & 1.701 & 1.217E-14\ CAHA\_20330798\_401915 & 20:33:07.98 & +40:19:15.5 & 13.951 & 0.007 & 12.724 & 0.01 & 12.251 & 0.024 & 11.21 & 0.016 & 10.83 & 0.016 & 10.462 & 0.017 & 9.941 & 0.022 & 7.182 & 0.209 & — & —\ CAHA\_20321078\_402356 & 20:32:10.78 & +40:23:56.0 & 16.728 & 0.005 & 14.42 & 0.011 & 12.906 & 0.005 & 11.646 & 0.016 & 10.951 & 0.016 & 10.453 & 0.017 & 9.679 & 0.019 & 7.496 & 0.154 & — & —\ CAHA\_20331260\_402247 & 20:33:12.60 & +40:22:47.2 & 17.066 & 0.011 & 15.378 & 0.096 & 14.455 & 0.096 & 13.466 & 0.024 & 13.22 & 0.021 & 12.702 & 0.059 & 11.862 & 0.087 & 7.898 & 0.246 & — & —\ CAHA\_20321598\_401023 & 20:32:15.98 & +40:10:23.5 & 14.683 & 0.013 & 13.6 & 0.014 & 12.856 & 0.033 & 12.218 & 0.016 & 11.902 & 0.016 & 11.536 & 0.023 & 11.097 & 0.04 & — & — & 1.467 & 1.295E-14\ CAHA\_20323581\_400914 & 20:32:35.81 & +40:09:14.3 & 16.045 & 9.999 & 14.927 & 0.018 & 13.921 & 0.023 & 13.471 & 0.021 & 13.326 & 0.022 & 12.951 & 0.121 & 12.124 & 0.243 & — & — & — & —\ CAHA\_20325027\_401053 & 20:32:50.27 & +40:10:53.5 & 16.775 & 0.023 & 15.134 & 0.023 & 13.962 & 0.025 & 13.741 & 0.034 & 13.334 & 0.032 & 12.821 & 0.084 & 11.766 & 0.129 & — & — & — & —\ CAHA\_20320910\_401441 & 20:32:09.10 & +40:14:41.7 & 15.85 & 0.006 & 13.621 & 0.006 & 12.556 & 0.021 & 11.916 & 0.019 & 11.779 & 0.018 & 11.381 & 0.052 & 10.657 & 0.113 & — & — & — & —\ CAHA\_20321210\_401240 & 20:32:12.10 & +40:12:41.0 & 15.8 & 0.01 & 14.19 & 0.009 & 13.448 & 0.03 & 12.56 & 0.018 & 12.053 & 0.017 & 11.504 & 0.035 & 10.608 & 0.052 & — & — & — & —\ CAHA\_20320975\_401131 & 20:32:09.75 & +40:11:31.3 & 17.666 & 0.01 & 15.457 & 0.011 & 14.374 & 0.035 & 13.816 & 0.027 & 13.654 & 0.033 & 13.254 & 0.131 & 12.599 & 0.17 & — & — & — & —\ CAHA\_20320721\_401330 & 20:32:07.21 & +40:13:30.5 & 15.47 & 0.016 & 14.657 & 0.025 & 14.144 & 0.026 & 13.503 & 0.028 & 13.251 & 0.027 & 12.661 & 0.125 & 11.52 & 0.207 & — & — & — & —\ CAHA\_20320583\_401139 & 20:32:05.83 & +40:11:39.8 & 17.571 & 0.012 & 15.648 & 0.011 & 14.458 & 0.035 & 13.53 & 0.022 & 13.079 & 0.02 & 12.704 & 0.065 & 12.004 & 0.105 & — & — & — & —\ CAHA\_20321307\_401250 & 20:32:13.07 & +40:12:50.0 & 16.731 & 0.009 & 15.006 & 0.01 & 13.919 & 0.027 & 12.918 & 0.024 & 12.355 & 0.021 & 12.049 & 0.051 & 11.509 & 0.141 & — & — & — & —\ CAHA\_20321094\_401157 & 20:32:10.94 & +40:11:57.3 & 14.802 & 0.01 & 13.906 & 0.009 & 13.395 & 0.03 & 13.259 & 0.022 & 13.186 & 0.025 & 12.688 & 0.096 & 11.88 & 0.237 & — & — & — & —\ CAHA\_20321434\_401357 & 20:32:14.34 & +40:13:57.8 & 16.7 & 0.152 & 14.895 & 0.062 & 14.25 & 0.024 & 12.878 & 0.017 & 12.516 & 0.019 & 12.323 & 0.056 & 11.361 & 0.082 & — & — & — & —\ CAHA\_20320296\_401350 & 20:32:02.96 & +40:13:50.9 & 17.226 & 0.009 & 14.281 & 0.007 & 12.922 & 0.024 & 12.062 & 0.018 & 11.914 & 0.019 & 11.574 & 0.078 & 10.965 & 0.228 & — & — & — & —\ CAHA\_20320488\_401316 & 20:32:04.88 & +40:13:16.0 & 16.034 & 0.007 & 14.24 & 0.007 & 13.273 & 0.026 & 12.802 & 0.022 & 12.554 & 0.022 & 11.825 & 0.082 & 10.472 & 0.128 & — & — & — & —\ CAHA\_20320511\_401231 & 20:32:05.11 & +40:12:31.7 & 19.14 & 0.027 & 15.766 & 0.01 & 14.088 & 0.03 & 13.316 & 0.027 & 13.068 & 0.026 & 12.75 & 0.106 & 11.833 & 0.237 & — & — & — & —\ CAHA\_20321626\_401254 & 20:32:16.26 & +40:12:54.2 & 17.864 & 0.009 & 15.707 & 0.01 & 14.282 & 0.023 & 12.711 & 0.017 & 12.039 & 0.017 & 11.581 & 0.049 & 11.111 & 0.103 & — & — & — & —\ CAHA\_20323916\_401223 & 20:32:39.16 & +40:12:23.2 & 14.6 & 0.018 & 12.759 & 0.022 & 12.049 & 0.022 & 11.536 & 0.017 & 11.475 & 0.019 & 11.083 & 0.083 & 10.366 & 0.216 & — & — & — & —\ CAHA\_20323223\_401347 & 20:32:32.23 & +40:13:47.1 & 17.797 & 0.01 & 15.102 & 0.01 & 13.512 & 0.012 & 11.859 & 0.046 & 11.264 & 0.046 & 9.956 & 0.147 & 8.584 & 0.184 & — & — & — & —\ CAHA\_20323296\_401334 & 20:32:32.96 & +40:13:34.5 & 14.659 & 0.008 & 12.915 & 0.021 & 12.118 & 0.022 & 10.675 & 0.024 & 10.098 & 0.021 & 9.251 & 0.097 & 8.062 & 0.161 & — & — & 1.920 & 1.205E-14\ CAHA\_20323055\_401356 & 20:32:30.55 & +40:13:56.2 & 18.796 & 0.015 & 15.275 & 0.01 & 13.144 & 0.017 & 11.272 & 0.046 & 10.48 & 0.033 & 9.671 & 0.127 & 8.665 & 0.228 & — & — & 3.059 & 1.572E-15\ CAHA\_20323042\_401221 & 20:32:30.42 & +40:12:21.1 & 17.321 & 0.015 & 15.434 & 0.015 & 14.292 & 0.022 & 13.077 & 0.045 & 12.658 & 0.028 & 11.364 & 0.121 & 10.082 & 0.182 & — & — & — & —\ CAHA\_20323549\_401308 & 20:32:35.49 & +40:13:08.9 & 17.794 & 0.012 & 14.456 & 0.012 & 12.912 & 0.023 & 11.898 & 0.024 & 11.703 & 0.023 & 10.857 & 0.111 & 9.75 & 0.194 & — & — & — & —\ CAHA\_20325130\_401150 & 20:32:51.30 & +40:11:50.8 & 17.119 & 0.016 & 15.336 & 0.016 & 14.289 & 0.016 & 13.299 & 0.03 & 12.742 & 0.021 & 12.286 & 0.049 & 11.36 & 0.058 & — & — & — & —\ CAHA\_20325783\_401208 & 20:32:57.83 & +40:12:08.4 & 15.826 & 0.015 & 14.792 & 0.017 & 14.297 & 0.021 & 14.107 & 0.027 & 13.942 & 0.029 & 13.696 & 0.095 & 12.871 & 0.135 & — & — & — & —\ CAHA\_20330832\_401317 & 20:33:08.32 & +40:13:17.2 & 15.914 & 0.01 & 15.062 & 0.016 & 14.513 & 0.029 & 14.045 & 0.026 & 13.567 & 0.027 & 13.146 & 0.07 & 12.456 & 0.114 & — & — & — & —\ CAHA\_20330731\_401428 & 20:33:07.31 & +40:14:28.4 & 15.442 & 0.009 & 14.386 & 0.016 & 13.868 & 0.021 & 13.555 & 0.03 & 13.252 & 0.029 & 12.772 & 0.049 & 12.064 & 0.09 & — & — & — & —\ CAHA\_20330608\_401211 & 20:33:06.08 & +40:12:11.6 & 14.922 & 0.01 & 13.97 & 0.015 & 13.365 & 0.036 & 13.081 & 0.023 & 12.807 & 0.023 & 12.479 & 0.048 & 11.86 & 0.058 & — & — & — & —\ CAHA\_20315908\_401647 & 20:31:59.08 & +40:16:47.9 & 14.435 & 0.006 & 13.58 & 0.008 & 13.221 & 0.013 & 12.687 & 0.021 & 12.142 & 0.019 & 11.856 & 0.057 & 10.977 & 0.139 & — & — & 1.570 & 7.604E-15\ CAHA\_20320234\_401726 & 20:32:02.34 & +40:17:26.9 & 18.316 & 0.011 & 15.371 & 0.007 & 13.777 & 0.008 & 12.528 & 0.023 & 11.981 & 0.02 & 11.466 & 0.051 & 10.813 & 0.088 & — & — & — & —\ CAHA\_20320067\_401621 & 20:32:00.67 & +40:16:22.0 & 14.559 & 0.004 & 13.399 & 0.006 & 12.684 & 0.014 & 12.3 & 0.024 & 12.007 & 0.023 & 11.448 & 0.053 & 10.343 & 0.062 & — & — & — & —\ CAHA\_20321121\_401703 & 20:32:11.21 & +40:17:03.5 & 16.9 & 0.004 & 15.195 & 0.006 & 14.244 & 0.011 & 12.913 & 0.022 & 12.358 & 0.02 & 11.972 & 0.076 & 11.223 & 0.171 & — & — & — & —\ CAHA\_20321075\_401624 & 20:32:10.75 & +40:16:24.8 & 13.638 & 0.004 & 12.659 & 0.006 & 11.911 & 0.018 & 11.199 & 0.016 & 10.843 & 0.016 & 10.509 & 0.019 & 10.005 & 0.057 & — & — & — & —\ CAHA\_20320290\_401724 & 20:32:02.90 & +40:17:24.4 & 16.272 & 0.005 & 15.33 & 0.027 & 14.234 & 0.021 & 12.75 & 0.04 & 12.243 & 0.033 & 11.751 & 0.074 & 11.132 & 0.178 & — & — & — & —\ CAHA\_20320046\_401506 & 20:32:00.46 & +40:15:06.1 & 15.626 & 0.006 & 13.109 & 0.009 & 12.109 & 0.018 & 11.339 & 0.023 & 11.244 & 0.023 & 10.875 & 0.043 & 10.259 & 0.107 & — & — & — & —\ CAHA\_20320665\_401519 & 20:32:06.65 & +40:15:20.0 & 14.918 & 0.005 & 12.888 & 0.007 & 11.834 & 0.02 & 10.679 & 0.016 & 10.308 & 0.016 & 9.932 & 0.038 & 9.529 & 0.117 & — & — & — & —\ CAHA\_20320783\_401636 & 20:32:07.83 & +40:16:36.3 & 15.209 & 0.004 & 14.078 & 0.007 & 13.417 & 0.014 & 12.87 & 0.019 & 12.576 & 0.019 & 12.233 & 0.058 & 11.23 & 0.176 & — & — & — & —\ CAHA\_20320403\_401828 & 20:32:04.03 & +40:18:28.1 & 20.924 & 0.058 & 17.029 & 0.012 & 14.745 & 0.008 & 13.107 & 0.033 & 12.324 & 0.023 & 11.886 & 0.098 & 10.963 & 0.229 & — & — & — & —\ CAHA\_20320309\_401738 & 20:32:03.09 & +40:17:38.4 & 19.014 & 0.018 & 15.72 & 0.007 & 13.95 & 0.009 & 12.491 & 0.021 & 11.803 & 0.02 & 11.231 & 0.048 & 10.465 & 0.089 & — & — & — & —\ CAHA\_20320807\_401715 & 20:32:08.07 & +40:17:15.7 & 14.595 & 0.003 & 13.256 & 0.006 & 12.477 & 0.014 & 11.741 & 0.016 & 11.405 & 0.016 & 11.075 & 0.033 & 10.171 & 0.121 & — & — & — & —\ CAHA\_20320848\_401550 & 20:32:08.48 & +40:15:50.7 & 16.506 & 0.003 & 15.376 & 0.006 & 14.944 & 0.015 & 14.359 & 0.061 & 14.224 & 0.063 & 12.389 & 0.102 & 10.463 & 0.09 & — & — & — & —\ CAHA\_20322058\_401825 & 20:32:20.58 & +40:18:25.4 & 18.092 & 0.007 & 15.837 & 0.012 & 14.668 & 0.023 & 12.748 & 0.023 & 12.0 & 0.02 & 11.443 & 0.074 & 10.774 & 0.114 & — & — & — & —\ CAHA\_20322443\_401818 & 20:32:24.43 & +40:18:18.4 & 16.859 & 0.024 & 14.573 & 0.024 & 13.207 & 0.024 & 11.911 & 0.021 & 11.259 & 0.02 & 10.694 & 0.083 & 9.749 & 0.196 & — & — & 2.913 & 1.306E-15\ CAHA\_20322935\_401844 & 20:32:29.35 & +40:18:44.7 & 14.703 & 0.003 & 13.689 & 0.005 & 13.172 & 0.004 & 12.806 & 0.021 & 12.618 & 0.024 & 12.336 & 0.124 & 11.746 & 0.24 & — & — & — & —\ CAHA\_20321655\_401808 & 20:32:16.55 & +40:18:08.8 & 15.139 & 0.003 & 13.688 & 0.007 & 12.914 & 0.005 & 12.245 & 0.022 & 11.862 & 0.019 & 11.021 & 0.072 & 9.779 & 0.069 & — & — & — & —\ CAHA\_20323555\_401840 & 20:32:35.55 & +40:18:40.3 & 18.715 & 0.015 & 15.218 & 0.005 & 13.545 & 0.006 & 12.638 & 0.022 & 12.426 & 0.023 & 12.034 & 0.112 & 11.721 & 0.239 & — & — & — & —\ CAHA\_20323489\_401810 & 20:32:34.89 & +40:18:11.0 & 13.619 & 0.004 & 12.464 & 0.005 & 11.896 & 0.021 & 10.95 & 0.016 & 10.578 & 0.016 & 10.286 & 0.043 & 9.551 & 0.082 & — & — & 1.219 & 1.186E-15\ CAHA\_20323249\_401602 & 20:32:32.49 & +40:16:02.9 & 17.053 & 0.027 & 13.156 & 0.007 & 10.815 & 0.017 & 8.458 & 0.017 & 7.656 & 0.017 & 7.065 & 0.034 & 6.613 & 0.093 & — & — & — & —\ CAHA\_20323176\_401616 & 20:32:31.76 & +40:16:16.5 & 14.715 & 0.005 & 11.98 & 0.018 & 10.456 & 0.017 & 9.25 & 0.031 & 8.842 & 0.054 & 7.789 & 0.142 & 5.961 & 0.131 & — & — & — & —\ CAHA\_20324131\_401807 & 20:32:41.31 & +40:18:07.2 & 17.244 & 0.005 & 15.14 & 0.007 & 14.169 & 0.007 & 13.383 & 0.021 & 12.98 & 0.022 & 12.605 & 0.101 & 11.901 & 0.207 & — & — & — & —\ CAHA\_20323108\_401608 & 20:32:31.08 & +40:16:08.6 & 16.236 & 0.009 & 13.524 & 0.005 & 12.042 & 0.032 & 10.022 & 0.044 & 9.314 & 0.046 & 8.55 & 0.133 & 6.872 & 0.167 & — & — & — & —\ CAHA\_20324014\_401812 & 20:32:40.14 & +40:18:12.7 & 16.204 & 0.004 & 14.726 & 0.007 & 13.956 & 0.005 & 13.485 & 0.029 & 13.178 & 0.035 & 12.759 & 0.13 & 11.857 & 0.213 & — & — & — & —\ CAHA\_20323095\_401649 & 20:32:30.95 & +40:16:49.6 & 13.021 & 0.004 & 10.756 & 0.022 & 9.763 & 0.017 & 9.074 & 0.032 & 8.531 & 0.031 & 7.958 & 0.127 & 6.609 & 0.208 & — & — & 2.708 & 6.967E-15\ CAHA\_20323708\_401737 & 20:32:37.08 & +40:17:37.5 & 13.577 & 0.004 & 11.519 & 0.031 & 10.529 & 0.026 & 9.853 & 0.023 & 9.483 & 0.027 & 8.839 & 0.084 & 7.252 & 0.093 & — & — & 2.110 & 3.729E-15\ CAHA\_20323555\_401605 & 20:32:35.55 & +40:16:05.8 & 13.195 & 0.007 & 10.399 & 0.022 & 8.493 & 0.017 & 6.322 & 0.016 & 5.666 & 0.015 & 5.036 & 0.021 & 4.051 & 0.038 & — & — & — & —\ CAHA\_20330586\_401552 & 20:33:05.86 & +40:15:52.9 & 17.201 & 0.014 & 15.822 & 0.028 & 15.098 & 0.043 & 14.265 & 0.027 & 13.921 & 0.024 & 13.669 & 0.099 & 13.24 & 0.204 & — & — & — & —\ CAHA\_20320906\_402043 & 20:32:09.06 & +40:20:43.0 & 14.099 & 0.002 & 12.956 & 0.006 & 12.338 & 0.004 & 12.017 & 0.017 & 11.927 & 0.018 & 11.576 & 0.078 & 10.78 & 0.15 & — & — & — & —\ CAHA\_20321381\_401908 & 20:32:13.81 & +40:19:08.6 & 20.069 & 0.057 & 16.556 & 0.007 & 14.37 & 0.006 & 12.346 & 0.092 & 11.718 & 0.054 & 10.147 & 0.178 & 8.759 & 0.227 & — & — & — & —\ CAHA\_20322602\_401904 & 20:32:26.02 & +40:19:04.3 & 18.818 & 9.999 & 16.463 & 0.008 & 14.207 & 0.005 & 12.653 & 0.017 & 12.093 & 0.017 & 11.66 & 0.049 & 11.05 & 0.116 & — & — & — & —\ CAHA\_20322626\_402216 & 20:32:26.26 & +40:22:16.8 & 15.245 & 0.003 & 13.693 & 0.007 & 12.927 & 0.004 & 12.298 & 0.017 & 11.832 & 0.016 & 11.483 & 0.035 & 10.776 & 0.072 & — & — & — & —\ CAHA\_20322761\_401914 & 20:32:27.61 & +40:19:14.5 & 20.543 & 0.057 & 16.125 & 0.006 & 13.617 & 0.005 & 12.102 & 0.017 & 11.45 & 0.017 & 11.021 & 0.045 & 10.301 & 0.058 & — & — & — & —\ CAHA\_20322942\_401917 & 20:32:29.42 & +40:19:17.2 & 19.044 & 0.015 & 16.167 & 0.005 & 14.449 & 0.009 & 13.061 & 0.02 & 12.395 & 0.021 & 12.073 & 0.054 & 11.34 & 0.107 & — & — & — & —\ CAHA\_20322974\_401906 & 20:32:29.74 & +40:19:06.3 & 18.843 & 0.012 & 16.098 & 0.007 & 14.651 & 0.007 & 13.529 & 0.025 & 12.964 & 0.023 & 12.437 & 0.067 & 11.733 & 0.081 & — & — & — & —\ CAHA\_20322536\_402054 & 20:32:25.36 & +40:20:54.5 & 14.601 & 0.003 & 12.966 & 0.006 & 12.168 & 0.004 & 11.392 & 0.017 & 11.041 & 0.016 & 10.586 & 0.019 & 9.925 & 0.055 & — & — & — & —\ CAHA\_20322352\_402035 & 20:32:23.52 & +40:20:35.2 & 19.808 & 0.038 & 16.256 & 0.007 & 14.394 & 0.007 & 13.064 & 0.023 & 12.391 & 0.02 & 11.636 & 0.059 & 10.866 & 0.08 & — & — & — & —\ CAHA\_20322337\_402212 & 20:32:23.37 & +40:22:12.2 & 15.341 & 0.003 & 13.874 & 0.008 & 13.182 & 0.004 & 12.554 & 0.017 & 12.186 & 0.017 & 11.763 & 0.028 & 10.759 & 0.04 & — & — & — & —\ CAHA\_20321759\_402226 & 20:32:17.59 & +40:22:26.4 & 17.119 & 0.006 & 15.531 & 0.008 & 14.738 & 0.006 & 13.952 & 0.028 & 13.531 & 0.024 & 13.111 & 0.081 & 12.349 & 0.072 & — & — & — & —\ CAHA\_20321778\_401905 & 20:32:17.78 & +40:19:05.8 & 15.883 & 0.003 & 13.723 & 0.006 & 12.612 & 0.005 & 11.801 & 0.035 & 11.492 & 0.029 & 10.374 & 0.156 & 9.075 & 0.23 & — & — & — & —\ CAHA\_20321874\_402003 & 20:32:18.74 & +40:20:03.2 & 19.097 & 0.014 & 16.012 & 0.007 & 14.24 & 0.007 & 12.801 & 0.018 & 12.146 & 0.023 & 11.508 & 0.063 & 10.417 & 0.126 & — & — & — & —\ CAHA\_20321559\_402131 & 20:32:15.59 & +40:21:31.2 & 17.573 & 0.005 & 15.115 & 0.007 & 13.785 & 0.004 & 13.103 & 0.019 & 12.853 & 0.02 & 12.689 & 0.113 & 11.855 & 0.246 & — & — & — & —\ CAHA\_20323947\_402116 & 20:32:39.47 & +40:21:16.5 & 15.868 & 0.003 & 14.169 & 0.006 & 13.308 & 0.005 & 12.546 & 0.018 & 12.045 & 0.017 & 11.555 & 0.034 & 10.922 & 0.063 & — & — & — & —\ CAHA\_20324423\_401939 & 20:32:44.23 & +40:19:39.4 & 16.834 & 0.004 & 15.494 & 0.007 & 14.667 & 0.009 & 13.761 & 0.023 & 13.168 & 0.022 & 12.481 & 0.054 & 11.586 & 0.071 & — & — & — & —\ CAHA\_20323153\_402048 & 20:32:31.53 & +40:20:49.0 & 18.767 & 0.018 & 15.671 & 0.006 & 14.18 & 0.009 & 13.323 & 0.02 & 13.097 & 0.021 & 12.761 & 0.058 & 12.403 & 0.122 & — & — & — & —\ CAHA\_20323570\_402141 & 20:32:35.70 & +40:21:41.9 & 18.206 & 0.012 & 15.383 & 0.006 & 14.069 & 0.006 & 13.276 & 0.019 & 13.112 & 0.02 & 12.83 & 0.071 & 12.158 & 0.155 & — & — & — & —\ CAHA\_20323809\_401943 & 20:32:38.09 & +40:19:43.7 & 18.638 & 0.015 & 15.449 & 0.006 & 13.945 & 0.007 & 13.14 & 0.02 & 12.916 & 0.02 & 12.593 & 0.078 & 12.069 & 0.157 & — & — & — & —\ CAHA\_20323774\_401916 & 20:32:37.74 & +40:19:16.4 & 20.18 & 0.044 & 16.788 & 0.011 & 14.861 & 0.006 & 13.415 & 0.021 & 12.829 & 0.021 & 12.384 & 0.095 & 11.399 & 0.107 & — & — & — & —\ CAHA\_20325212\_401914 & 20:32:52.12 & +40:19:14.8 & 17.619 & 0.004 & 15.916 & 0.008 & 15.151 & 0.012 & 14.45 & 0.043 & 13.974 & 0.038 & 13.495 & 0.118 & 12.946 & 0.175 & — & — & — & —\ CAHA\_20325635\_402129 & 20:32:56.35 & +40:21:29.0 & 17.837 & 0.008 & 15.611 & 0.007 & 14.432 & 0.009 & 13.374 & 0.019 & 12.82 & 0.017 & 12.432 & 0.035 & 11.756 & 0.056 & — & — & — & —\ CAHA\_20315935\_402411 & 20:31:59.35 & +40:24:11.4 & 15.12 & 0.006 & 13.917 & 0.012 & 13.485 & 0.043 & 12.841 & 0.018 & 12.476 & 0.017 & 12.116 & 0.032 & 11.572 & 0.049 & — & — & — & —\ CAHA\_20315954\_402231 & 20:31:59.54 & +40:22:31.8 & 16.638 & 0.004 & 15.201 & 0.008 & 14.36 & 0.006 & 13.77 & 0.024 & 13.34 & 0.024 & 13.058 & 0.063 & 12.413 & 0.135 & — & — & — & —\ CAHA\_20320915\_402301 & 20:32:09.15 & +40:23:01.9 & 16.366 & 0.004 & 14.708 & 0.009 & 13.754 & 0.004 & 12.628 & 0.022 & 12.111 & 0.019 & 11.678 & 0.047 & 10.876 & 0.095 & — & — & — & —\ CAHA\_20320939\_402250 & 20:32:09.39 & +40:22:50.6 & 13.076 & 0.004 & 11.106 & 0.023 & 10.307 & 0.017 & 9.56 & 0.016 & 9.078 & 0.016 & 8.597 & 0.016 & 7.961 & 0.017 & — & — & 1.920 & 3.140E-14\ CAHA\_20320112\_402313 & 20:32:01.12 & +40:23:13.2 & 17.417 & 0.019 & 15.762 & 0.03 & 14.796 & 0.039 & 13.528 & 0.019 & 12.909 & 0.017 & 12.462 & 0.047 & 11.808 & 0.077 & — & — & — & —\ CAHA\_20322465\_402329 & 20:32:24.65 & +40:23:29.2 & 17.824 & 0.008 & 15.397 & 0.009 & 14.258 & 0.006 & 13.65 & 0.023 & 13.479 & 0.023 & 13.182 & 0.062 & 12.632 & 0.166 & — & — & — & —\ CAHA\_20324555\_402331 & 20:32:45.55 & +40:23:31.7 & 17.385 & 0.006 & 15.647 & 0.007 & 14.723 & 0.005 & 14.097 & 0.024 & 13.687 & 0.023 & 13.609 & 0.115 & 12.844 & 0.202 & — & — & — & —\ CAHA\_20324550\_402231 & 20:32:45.50 & +40:22:31.7 & 16.875 & 0.004 & 15.837 & 0.007 & 15.174 & 0.005 & 14.419 & 0.038 & 13.977 & 0.033 & 13.046 & 0.059 & 12.014 & 0.091 & — & — & — & —\ CAHA\_20330316\_402332 & 20:33:03.16 & +40:23:33.0 & 18.123 & 0.031 & 15.717 & 0.026 & 14.347 & 0.018 & 12.893 & 0.019 & 12.421 & 0.017 & 12.128 & 0.046 & 11.701 & 0.117 & — & — & — & —\ TWOM\_20324634\_4009048 & 20:32:46.35 & +40:09:04.9 & 18.49 & 9.999 & 14.906 & 0.059 & 13.206 & 0.036 & 12.13 & 0.017 & 11.948 & 0.017 & 11.546 & 0.038 & 11.182 & 0.062 & — & — & — & —\ TWOM\_20315282\_4012188 & 20:31:52.82 & +40:12:18.8 & 13.994 & 0.027 & 13.068 & 0.029 & 12.612 & 0.026 & 12.05 & 0.017 & 11.775 & 0.017 & 11.533 & 0.027 & 11.024 & 0.046 & — & — & — & —\ TWOM\_20322814\_4017148 & 20:32:28.14 & +40:17:14.9 & 10.966 & 0.035 & 9.926 & 0.038 & 9.336 & 0.024 & 8.985 & 0.034 & 8.541 & 0.046 & 7.94 & 0.172 & 6.448 & 0.242 & — & — & — & —\ TWOM\_20331278\_4018418 & 20:33:12.79 & +40:18:41.8 & 18.61 & 9.999 & 15.771 & 0.13 & 14.561 & 0.103 & 13.589 & 0.029 & 13.368 & 0.024 & 13.103 & 0.098 & 12.477 & 0.185 & — & — & — & —\ CAHA\_20325255\_401152 & 20:32:52.55 & +40:11:52.0 & 19.187 & 0.034 & 17.342 & 0.018 & 16.09 & 0.017 & 14.485 & 0.027 & 13.904 & 0.024 & 13.684 & 0.111 & 12.616 & 0.157 & — & — & — & —\ CAHA\_20321844\_401724 & 20:32:18.23 & +40:17:27.7 & 18.294 & 0.033 & 16.583 & 0.015 & 15.157 & 0.007 & 13.024 & 0.026 & 12.173 & 0.019 & 11.537 & 0.046 & 10.547 & 0.045 & — & — & — & —\ CAHA\_20322275\_401739 & 20:32:22.75 & +40:17:39.6 & 20.249 & 0.058 & 18.616 & 0.064 & 17.053 & 0.089 & 12.84 & 0.153 & 12.698 & 0.187 & 9.895 & 0.171 & 8.093 & 0.184 & — & — & — & —\ CAHA\_20324785\_401817 & 20:32:47.85 & +40:18:17.7 & 20.327 & 0.059 & 16.924 & 0.008 & 15.097 & 0.008 & 14.101 & 0.04 & 13.779 & 0.046 & 13.599 & 0.16 & 12.532 & 0.219 & — & — & — & —\ CAHA\_20324105\_401849 & 20:32:41.05 & +40:18:49.3 & 18.836 & 0.017 & 16.75 & 0.007 & 15.68 & 0.012 & 14.572 & 0.041 & 14.287 & 0.038 & 13.87 & 0.173 & 12.149 & 0.14 & — & — & — & —\ CAHA\_20325040\_401850 & 20:32:50.40 & +40:18:50.4 & 17.912 & 0.009 & 16.288 & 0.01 & 15.384 & 0.01 & 14.69 & 0.056 & 14.384 & 0.051 & 13.923 & 0.118 & 13.129 & 0.199 & — & — & — & —\ CAHA\_20322563\_401850 & 20:32:25.63 & +40:18:50.8 & 20.371 & 0.064 & 17.687 & 0.01 & 16.093 & 0.02 & 14.339 & 0.037 & 13.47 & 0.029 & 13.026 & 0.19 & 11.419 & 0.218 & — & — & — & —\ CAHA\_20330273\_401903 & 20:33:02.73 & +40:19:03.2 & 18.929 & 0.019 & 16.783 & 0.015 & 15.583 & 0.011 & 14.568 & 0.054 & 14.16 & 0.054 & 13.824 & 0.121 & 12.891 & 0.146 & — & — & — & —\ CAHA\_20323802\_401934 & 20:32:38.02 & +40:19:34.4 & 20.414 & 0.051 & 17.149 & 0.012 & 15.2 & 0.009 & 13.652 & 0.023 & 13.145 & 0.021 & 12.602 & 0.075 & 11.952 & 0.115 & — & — & — & —\ CAHA\_20320973\_402253 & 20:32:09.73 & +40:22:53.4 & 13.802 & 0.004 & 12.505 & 0.009 & — & — & 10.709 & 0.017 & 10.351 & 0.017 & 9.975 & 0.02 & 9.154 & 0.025 & — & — & — & —\ CAHA\_20315905\_401755 & 20:31:59.05 & +40:17:56.0 & — & — & 18.606 & 0.03 & 16.205 & 0.017 & 14.226 & 0.057 & 13.447 & 0.055 & 12.447 & 0.145 & 11.06 & 0.236 & — & — & — & —\ CAHA\_20322723\_401923 & 20:32:27.23 & +40:19:23.1 & — & — & 17.445 & 0.011 & 14.791 & 0.008 & 12.988 & 0.029 & 12.078 & 0.03 & 11.483 & 0.095 & 10.825 & 0.189 & — & — & — & —\ CAHA\_20321904\_401942 & 20:32:19.04 & +40:19:42.4 & — & — & 17.846 & 0.011 & 14.913 & 0.009 & 12.756 & 0.017 & 12.021 & 0.018 & 11.126 & 0.068 & 9.878 & 0.125 & — & — & — & —\ CAHA\_20321823\_402039 & 20:32:18.23 & +40:20:39.4 & — & — & 19.331 & 0.053 & 15.653 & 0.02 & 12.337 & 0.017 & 11.325 & 0.016 & 10.716 & 0.061 & 10.249 & 0.167 & — & — & — & —\ CAHA\_20321538\_402356 & 20:32:15.38 & +40:23:56.9 & — & — & 17.495 & 0.023 & 15.832 & 0.034 & 14.414 & 0.024 & 13.942 & 0.025 & 13.44 & 0.08 & 12.978 & 0.241 & — & — & — & —\ CAHA\_20330081\_401020 & 20:33:00.81 & +40:10:20.1 & — & — & — & — & 18.342 & 0.121 & 15.457 & 0.055 & 14.702 & 0.061 & 13.348 & 0.082 & 12.192 & 0.091 & — & — & — & —\ CAHA\_20322687\_401910 & 20:32:26.87 & +40:19:10.3 & — & — & — & — & 12.997 & 0.006 & 11.497 & 0.016 & 10.941 & 0.016 & 10.605 & 0.031 & 10.164 & 0.069 & — & — & 1.832 & 3.097E-15\ CAHA\_20322007\_401933 & 20:32:20.07 & +40:19:33.6 & — & — & — & — & 16.563 & 0.02 & 13.016 & 0.041 & 11.857 & 0.027 & 11.221 & 0.13 & 9.785 & 0.223 & — & — & — & —\ SSTCYGX\_J203152.69\_401840.5 & 20:31:52.68 & +40:18:40.5 & — & — & — & — & — & — & 13.805 & 0.022 & 12.985 & 0.021 & 12.558 & 0.048 & 12.053 & 0.175 & — & — & — & —\ CAHA\_20321015\_401847 & 20:32:10.15 & +40:18:48.0 & — & — & 17.561 & 0.014 & 14.716 & 0.008 & 13.155 & 0.074 & 12.549 & 0.065 & 11.844 & 0.44 & 10.831 & 0.807 & — & — & 2.431 & 2.387E-15\ CAHA\_20321798\_401922 & 20:32:17.98 & +40:19:22.3 & 19.178 & 0.018 & 15.984 & 0.006 & 14.341 & 0.006 & 12.932 & 0.077 & 12.54 & 0.056 & 11.309 & 0.362 & — & — & — & — & 3.409 & 1.067E-14\ CAHA\_20321061\_401904 & 20:32:10.61 & +40:19:04.8 & — & — & — & — & 17.616 & 0.05 & 14.017 & 0.067 & 13.097 & 0.081 & 11.677 & 0.173 & — & — & — & — & 3.176 & 1.357E-14\ CAHA\_20315867\_401915 & 20:31:58.67 & +40:19:15.1 & 17.58 & 0.006 & 14.666 & 0.005 & 12.948 & 0.008 & 11.306 & 0.016 & 10.714 & 0.016 & 10.313 & 0.019 & 9.896 & 0.045 & 7.262 & 0.41 & 1.862 & 9.616E-15\ CAHA\_20320503\_401715 & 20:32:05.03 & +40:17:16.0 & 12.833 & 0.005 & 11.862 & 0.019 & 11.351 & 0.016 & 10.564 & 0.016 & 10.174 & 0.015 & 9.762 & 0.02 & 8.992 & 0.046 & 6.837 & 0.749 & 1.380 & 6.313E-15\ CAHA\_20320542\_402126 & 20:32:05.42 & +40:21:26.2 & 14.416 & 0.002 & 13.22 & 0.006 & 12.679 & 0.005 & 12.127 & 0.023 & 11.936 & 0.021 & 11.7 & 0.133 & 11.008 & 0.356 & 6.932 & 0.505 & 1.920 & 1.509E-14\ CAHA\_20322109\_401848 & 20:32:21.09 & +40:18:48.1 & 17.17 & 0.005 & 14.563 & 0.005 & 13.189 & 0.004 & 11.819 & 0.031 & 11.28 & 0.023 & 10.886 & 0.187 & 10.03 & 0.32 & — & — & 2.314 & 2.492E-15\ CAHA\_20322641\_401515 & 20:32:26.41 & +40:15:15.6 & 15.269 & 0.007 & 12.963 & 0.007 & 11.691 & 0.018 & 10.435 & 0.025 & 10.028 & 0.025 & 9.808 & 0.219 & 9.412 & 0.829 & — & — & 1.920 & 1.220E-15\ CAHA\_20323239\_401643 & 20:32:32.39 & +40:16:43.3 & 15.99 & 0.005 & 13.31 & 0.005 & 11.504 & 0.026 & 10.049 & 0.071 & 9.374 & 0.043 & 9.051 & 0.477 & — & — & — & — & 3.059 & 7.560E-15\ CAHA\_20323582\_401745 & 20:32:35.82 & +40:17:45.0 & 13.798 & 0.004 & 12.611 & 0.008 & 12.108 & 0.033 & 11.136 & 0.049 & 10.972 & 0.043 & 9.44 & 0.243 & — & — & — & — & 2.781 & 6.038E-15\ CAHA\_20324487\_401834 & 20:32:44.87 & +40:18:34.1 & 15.417 & 0.005 & 13.309 & 0.008 & 12.165 & 0.021 & 11.274 & 0.016 & 10.967 & 0.016 & 10.707 & 0.022 & 10.187 & 0.048 & 7.054 & 0.322 & 2.256 & 8.162E-15\ CAHA\_20325504\_401617 & 20:32:55.04 & +40:16:17.3 & 13.48 & 0.009 & 12.375 & 0.021 & 11.68 & 0.019 & 10.542 & 0.015 & 10.282 & 0.015 & 10.11 & 0.017 & 9.77 & 0.017 & 6.073 & 0.163 & 1.526 & 1.505E-14\ CAHA\_20330681\_401337 & 20:33:06.81 & +40:13:37.9 & 13.464 & 0.017 & 12.831 & 0.024 & 12.525 & 0.027 & 11.991 & 0.016 & 11.836 & 0.017 & 11.689 & 0.027 & 11.2 & 0.044 & 7.413 & 0.258 & 1.613 & 1.026E-14\ \[tab:ClassII\] [lllllllllllllllllllll]{} CAHA\_20322152\_401104 & 20:32:21.52 & +40:11:04.6 & 13.842 & 0.016 & 13.002 & 0.015 & 12.596 & 0.028 & 12.559 & 0.019 & 12.548 & 0.021 & 12.395 & 0.078 & 11.608 & 0.169 & — & — & 2.548 & 6.699E-15\ CAHA\_20323285\_401056 & 20:32:32.85 & +40:10:56.5 & 16.605 & 0.017 & 15.081 & 0.012 & 14.321 & 0.02 & 14.065 & 0.232 & 14.028 & 0.171 & — & — & — & — & — & — & 3.438 & 6.162E-15\ CAHA\_20320491\_401359 & 20:32:04.91 & +40:13:59.3 & 13.227 & 0.008 & 12.717 & 0.023 & 12.476 & 0.024 & 12.571 & 0.021 & 12.592 & 0.021 & 12.509 & 0.099 & — & — & — & — & 1.190 & 5.738E-15\ CAHA\_20321916\_401318 & 20:32:19.16 & +40:13:18.2 & 15.919 & 0.011 & 14.424 & 0.01 & 13.758 & 0.025 & 13.357 & 0.055 & 13.247 & 0.067 & 13.21 & 0.55 & — & — & — & — & 2.489 & 3.258E-15\ CAHA\_20323784\_401352 & 20:32:37.84 & +40:13:52.7 & 15.577 & 0.014 & 14.011 & 0.027 & 13.419 & 0.021 & 13.344 & 0.233 & 13.284 & 0.25 & — & — & — & — & — & — & 3.263 & 1.022E-14\ CAHA\_20324208\_401220 & 20:32:42.08 & +40:12:20.3 & 14.24 & 0.014 & 13.071 & 0.017 & 12.52 & 0.022 & 12.381 & 0.019 & 12.255 & 0.02 & 12.494 & 0.123 & — & — & — & — & 1.905 & 7.653E-15\ CAHA\_20323075\_401419 & 20:32:30.75 & +40:14:19.5 & 15.255 & 0.028 & 12.881 & 0.044 & 11.66 & 0.025 & 10.803 & 0.072 & 10.597 & 0.071 & — & — & — & — & — & — & 2.694 & 1.592E-14\ CAHA\_20325387\_401420 & 20:32:53.87 & +40:14:20.4 & 13.2 & 0.015 & 12.069 & 0.019 & 11.614 & 0.018 & 11.383 & 0.016 & 11.305 & 0.016 & 11.215 & 0.024 & 11.149 & 0.149 & — & — & 1.818 & 2.580E-14\ CAHA\_20325533\_401440 & 20:32:55.33 & +40:14:40.3 & 15.64 & 0.014 & 14.432 & 0.015 & 13.914 & 0.014 & 13.712 & 0.023 & 13.654 & 0.027 & 13.552 & 0.093 & — & — & — & — & 2.212 & 8.545E-15\ CAHA\_20320256\_401740 & 20:32:02.56 & +40:17:40.5 & 13.948 & 0.005 & 13.094 & 0.006 & 12.814 & 0.01 & 12.755 & 0.029 & 12.633 & 0.034 & 12.802 & 0.247 & — & — & — & — & 1.467 & 3.822E-15\ CAHA\_20320237\_401701 & 20:32:02.37 & +40:17:01.1 & 13.188 & 0.003 & 12.005 & 0.039 & 11.556 & 0.028 & 11.155 & 0.016 & 11.008 & 0.016 & 10.808 & 0.034 & 10.338 & 0.081 & — & — & 1.511 & 2.660E-14\ CAHA\_20321032\_401752 & 20:32:10.32 & +40:17:53.0 & 14.269 & 0.003 & 13.309 & 0.006 & 12.949 & 0.009 & 12.646 & 0.03 & 12.564 & 0.043 & 12.658 & 0.378 & 12.546 & 1.721 & — & — & 1.073 & 3.459E-15\ CAHA\_20321962\_401812 & 20:32:19.62 & +40:18:12.4 & 13.391 & 0.004 & 12.21 & 0.027 & 11.83 & 0.021 & 11.669 & 0.017 & 11.53 & 0.017 & 11.415 & 0.064 & 11.042 & 0.159 & — & — & 1.292 & 3.995E-15\ CAHA\_20322560\_401805 & 20:32:25.60 & +40:18:05.3 & 19.153 & 0.02 & 15.839 & 0.005 & 14.207 & 0.004 & 13.319 & 0.063 & 13.104 & 0.09 & 13.092 & 0.671 & — & — & — & — & 3.570 & 2.168E-13\ CAHA\_20323482\_401818 & 20:32:34.82 & +40:18:18.4 & 14.571 & 0.021 & 13.428 & 0.028 & 12.995 & 0.029 & 12.655 & 0.027 & 12.56 & 0.037 & 12.213 & 0.136 & — & — & — & — & 1.438 & 4.581E-15\ CAHA\_20324203\_401510 & 20:32:42.03 & +40:15:10.2 & 14.755 & 0.009 & 13.799 & 0.011 & 13.423 & 0.013 & 13.169 & 0.111 & 13.281 & 0.12 & — & — & — & — & — & — & 1.175 & 3.020E-15\ CAHA\_20323083\_401706 & 20:32:30.83 & +40:17:06.1 & 15.738 & 0.006 & 13.446 & 0.007 & 12.33 & 0.009 & 11.896 & 0.078 & 11.932 & 0.113 & — & — & — & — & — & — & 3.380 & 5.798E-14\ CAHA\_20323043\_401643 & 20:32:30.43 & +40:16:43.2 & 15.525 & 0.003 & 13.049 & 0.006 & 11.933 & 0.026 & 11.182 & 0.055 & 11.031 & 0.106 & — & — & — & — & — & — & 2.716 & 7.643E-15\ CAHA\_20325412\_401547 & 20:32:54.12 & +40:15:47.9 & 17.073 & 0.009 & 15.642 & 0.013 & 15.02 & 0.012 & 14.742 & 0.041 & 14.508 & 0.052 & — & — & — & — & — & — & 1.978 & 1.2521E-14\ CAHA\_20325095\_401520 & 20:32:50.95 & +40:15:20.2 & 14.535 & 0.01 & 13.575 & 0.013 & 13.178 & 0.008 & 12.989 & 0.022 & 12.938 & 0.028 & 12.74 & 0.112 & — & — & — & — & 1.219 & 8.092E-15\ CAHA\_20315441\_402153 & 20:31:54.41 & +40:21:53.2 & 13.824 & 0.005 & 12.53 & 0.009 & 12.024 & 0.018 & 11.721 & 0.016 & 11.667 & 0.016 & 11.605 & 0.027 & 11.665 & 0.093 & — & — & 1.745 & 8.670E-15\ CAHA\_20315532\_402127 & 20:31:55.32 & +40:21:27.6 & 14.938 & 0.002 & 13.572 & 0.008 & 12.999 & 0.005 & 12.716 & 0.02 & 12.601 & 0.021 & 12.687 & 0.105 & 12.069 & 0.322 & — & — & 1.657 & 3.869E-15\ CAHA\_20320065\_401931 & 20:32:00.65 & +40:19:31.8 & 17.586 & 0.005 & 15.52 & 0.006 & 14.493 & 0.008 & 13.831 & 0.039 & 13.648 & 0.036 & 13.508 & 0.404 & — & — & — & — & 2.022 & 5.032E-15\ CAHA\_20320051\_402020 & 20:32:00.51 & +40:20:20.2 & 13.538 & 0.002 & 12.756 & 0.006 & 12.468 & 0.005 & 12.438 & 0.033 & 12.417 & 0.044 & 12.317 & 0.12 & 11.485 & 0.299 & — & — & 1.365 & 5.296E-15\ CAHA\_20320956\_401901 & 20:32:09.56 & +40:19:01.3 & 18.248 & 0.009 & 13.633 & 0.006 & 11.067 & 0.016 & 9.439 & 0.017 & 8.995 & 0.017 & 8.7 & 0.039 & 8.721 & 0.137 & — & — & 2.533 & 4.274E-15\ CAHA\_20320899\_401937 & 20:32:08.99 & +40:19:37.3 & 13.226 & 0.003 & 12.039 & 0.021 & 11.586 & 0.016 & 11.278 & 0.021 & 11.204 & 0.029 & 11.309 & 0.203 & 12.854 & 3.984 & — & — & 1.803 & 3.219E-14\ CAHA\_20321374\_401856 & 20:32:13.74 & +40:18:57.0 & 18.195 & 0.007 & 14.942 & 0.007 & 13.283 & 0.005 & 12.483 & 0.081 & 12.185 & 0.058 & 11.035 & 0.227 & — & — & — & — & 2.548 & 1.6081E-14\ CAHA\_20322030\_401901 & 20:32:20.30 & +40:19:01.8 & 15.208 & 0.003 & 13.298 & 0.005 & 12.42 & 0.005 & 11.744 & 0.034 & 11.482 & 0.027 & 11.051 & 0.239 & — & — & — & — & 2.621 & 8.350E-15\ CAHA\_20322943\_401858 & 20:32:29.43 & +40:18:58.9 & 15.257 & 0.011 & 13.795 & 0.007 & 13.134 & 0.006 & 12.803 & 0.023 & 12.699 & 0.025 & 12.67 & 0.107 & 12.967 & 0.732 & — & — & 1.584 & 2.927E-15\ CAHA\_20322803\_401927 & 20:32:28.03 & +40:19:27.6 & 15.742 & 0.012 & 13.726 & 0.017 & 12.758 & 0.024 & 12.166 & 0.021 & 11.895 & 0.042 & 11.74 & 0.115 & — & — & — & — & 1.584 & 4.456E-15\ CAHA\_20321720\_401910 & 20:32:17.20 & +40:19:10.0 & 15.663 & 0.003 & 15.001 & 0.006 & 14.695 & 0.006 & 14.39 & 0.175 & 14.528 & 0.198 & — & — & — & — & — & — & 1.584 & 3.054E-15\ CAHA\_20324072\_402146 & 20:32:40.72 & +40:21:46.2 & 14.345 & 0.003 & 12.705 & 0.006 & 12.007 & 0.022 & 11.619 & 0.016 & 11.456 & 0.016 & 11.386 & 0.027 & 11.445 & 0.059 & — & — & 2.227 & 1.829E-14\ CAHA\_20323427\_401851 & 20:32:34.27 & +40:18:51.1 & 15.769 & 0.005 & 13.771 & 0.005 & 12.887 & 0.006 & 12.478 & 0.017 & 12.353 & 0.019 & 12.112 & 0.08 & 12.208 & 0.269 & — & — & 1.716 & 1.399E-15\ CAHA\_20320116\_402233 & 20:32:01.16 & +40:22:33.9 & 15.79 & 0.002 & 14.135 & 0.008 & 13.341 & 0.006 & 12.809 & 0.02 & 12.754 & 0.021 & 12.698 & 0.059 & 13.085 & 0.224 & — & — & 1.978 & 3.314E-15\ TWOM\_20324184\_4014001 & 20:32:41.84 & +40:14:00.2 & 11.113 & 0.021 & 10.838 & 0.018 & 10.789 & 0.017 & 10.793 & 0.019 & 10.816 & 0.021 & 10.639 & 0.116 & 10.248 & 0.302 & — & — & 0.883 & 4.458E-15\ TWOM\_20330733\_4014348 & 20:33:07.33 & +40:14:34.8 & 11.646 & 0.022 & 11.18 & 0.019 & 11.041 & 0.018 & 10.981 & 0.016 & 10.968 & 0.016 & 10.935 & 0.021 & 10.923 & 0.044 & — & — & 1.131 & 2.840E-14\ TWOM\_20323631\_4020142 & 20:32:36.32 & +40:20:14.3 & 11.918 & 9.999 & 11.508 & 0.027 & 11.443 & 0.026 & 11.432 & 0.016 & 11.48 & 0.017 & 11.409 & 0.073 & 11.268 & 0.231 & — & — & 0.986 & 3.822E-15\ TWOM\_20330655\_4022485 & 20:33:06.55 & +40:22:48.6 & 10.945 & 0.022 & 10.767 & 0.019 & 10.719 & 0.018 & 10.708 & 0.016 & 10.721 & 0.016 & 10.73 & 0.019 & 10.804 & 0.04 & — & — & 0.898 & 3.0744E-15\ CAHA\_20321512\_401714 & 20:32:15.12 & +40:17:14.8 & 14.779 & 0.003 & 13.287 & 0.005 & 12.676 & 0.008 & 12.291 & 0.019 & 12.159 & 0.022 & 12.094 & 0.135 & — & — & — & — & 1.789 & 5.472E-15\ CAHA\_20320777\_401759 & 20:32:07.77 & +40:17:59.4 & 16.206 & 0.092 & 13.979 & 0.038 & 13.316 & 0.009 & — & — & — & — & — & — & — & — & — & — & 4.417 & 2.468E-15\ CAHA\_20323293\_401511 & 20:32:32.93 & +40:15:11.6 & 17.384 & 0.011 & 15.979 & 0.01 & 15.275 & 0.011 & — & — & — & — & — & — & — & — & — & — & 2.650 & 1.288E-13\ CAHA\_20322779\_401512 & 20:32:27.79 & +40:15:12.4 & 18.761 & 0.011 & 16.498 & 0.009 & 15.323 & 0.011 & — & — & — & — & — & — & — & — & — & — & 5.351 & 4.285E-15\ CAHA\_20330360\_401707 & 20:33:03.60 & +40:17:07.4 & 20.784 & 0.065 & 19.02 & 0.044 & 18.445 & 0.094 & — & — & — & — & — & — & — & — & — & — & 2.562 & 7.871E-15\ CAHA\_20315779\_401717 & 20:31:57.79 & +40:17:17.9 & 15.757 & 0.004 & 14.392 & 0.006 & 13.699 & 0.015 & — & — & — & — & — & — & — & — & — & — & 1.920 & 3.300E-15\ CAHA\_20324085\_401304 & 20:32:40.85 & +40:13:04.7 & 20.26 & 0.069 & 17.919 & 0.041 & 17.023 & 0.049 & — & — & — & — & — & — & — & — & — & — & 1.949 & 2.980E-15\ \[tab:ClassIII\] [^1]: Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.

--- abstract: 'We study the 1D dynamics of dark-dark solitons in the miscible regime of two density-coupled Bose-Einstein condensates having repulsive interparticle interactions within each condensate ($g>0$). By using an adiabatic perturbation theory in the parameter $g_{12}/{g}$, we show that, contrary to the case of two solitons in scalar condensates, the interactions between solitons are attractive when the interparticle interactions between condensates are repulsive $g_{12}>0$. As a result, the relative motion of dark solitons with equal chemical potential $\mu$ is well approximated by harmonic oscillations of angular frequency $w_r=(\mu/\hbar)\sqrt{({8}/{15}){g_{12}}/{g}}$. We also show that in finite systems, the resonance of this anomalous excitation mode with the spin density mode of lowest energy gives rise to alternating dynamical instability and stability fringes as a function of the perturbative parameter. In the presence of harmonic trapping (with angular frequency $\Omega$) the solitons are driven by the superposition of two harmonic motions at a frequency given by $w^2=(\Omega/\sqrt{2})^2+w_r^2$. When $g_{12}<0$, these two oscillators compete to give rise to an overall effective potential that can be either single well or double well through a pitchfork bifurcation. All our theoretical results are compared with numerical solutions of the Gross-Pitaevskii equation for the dynamics and the Bogoliubov equations for the linear stability. A good agreement is found between them.' author: - 'I. Morera' - 'A. Muñoz Mateo' - 'A. Polls' - 'B. Juliá-Díaz' title: 'Dark-dark-soliton dynamics in two density-coupled Bose-Einstein condensates' --- Introduction ============ Over the last two decades, Bose-Einstein condensation (BEC) has enabled the study of numerous physical concepts [@Pitaevskii2003; @Pethick2008]. Among them, an interesting scenario is the connection between the non-linear waves and atomic systems, that leads to the so-called matter-wave solitons [@Kevrekidis2008; @Kevrekidis2015]. These structures emerge from the balance between linear dispersion and interactions, which are accounted for from non-linear terms in the equations of motion. Specifically, mean field descriptions of BECs, as provided by the Gross-Pitaevskii equation (GP), incorporate a non-linear term proportional to the interatomic interaction strength $g$, which is measured by the s-wave scattering length $a$. Depending on the sign and magnitude of the latter and the dimensionality of the system, a large number of structures can be found: dark [@Frantzeskakis2010] and bright [@Strecker2003] solitons, vortices [@Fetter2001], etc. Dark solitons in BECs are localized non-linear excitations that present a notch in the condensate density and a phase step across its center. They have been observed in numerous experiments using different techniques [@Burger1999; @Becker2008; @Weller2008; @Lamporesi2013; @Anderson1999] and this has inspired a large number of theoretical works (see [@Frantzeskakis2010] and references therein). Specifically, the problem of dark solitons in BECs confined by parabolic external traps has been extensively studied [@William1997; @Muryshev1999; @Busch2000; @Pelinovsky2005]. Another interesting aspect of BECs is the study of multi-component systems, which can be described by a set of coupled GP equations. These systems were soon realized in ultracold-gas experiments by coupling two condensates made of either different atomic species [@Cornell1998] or different hyperfine states of the same atomic species [@Myatt1997]. This fact inspired the study of matter-wave solitons in these settings, and new families of solitonic structures have been found: dark-dark [@Ohberg2001; @Hoefer2011; @Yan2012], dark-bright [@Middelkamp2011; @Yan2011; @Achilleos2011], dark-antidark [@Danaila2016], etc. Particular attention has been given to the so-called Manakov limit in 1D settings, where the intra- and inter-condensate particle interactions match [@Yan2012]. The aim of this work is to study the dynamics of dark-dark solitons in a one-dimensional setting of two density-coupled condensates out of the Manakov limit, for varying coupling between condensates. Specifically, we consider a mean-field description of BECs composed by two hyperfine states of the same alkali species, and inspect both the case without any axial trap, termed untrapped, and the case with a harmonic trap in the axial direction. Within the Hamiltonian approach of the perturbation theory for solitons [@Uzunov1993; @Kivshar1994; @Frantzeskakis2002], we obtain analytical expressions for the adiabatic evolution of the dark-dark soliton. For the untrapped case we find that when the interaction between components is repulsive the dark-dark soliton can be seen as a bound state of two dark solitons performing a relative harmonic motion. For attractive interactions such a bound state can not exist because the dark solitons repel each other. An equivalent study is also performed for the confined case, where the competition between soliton interactions and harmonic trapping is found to produce sizable changes on the dynamics, leading for example to bound states even with attractive interactions. Our analytical results are tested with different numerical techniques: direct simulations of the Gross-Pitaevskii equations, Fourier analysis to extract characteristic frequencies of the motion, and numerical solution of the Bogoliubov equations for the linear excitations of stationary states. In what follows, we first introduce the theoretical model in section II, and next we show our numerical results in section III. To sum up, we present our conclusions in section IV. Theoretical model ================= The dynamics of a BEC at zero temperature can be accurately described within the mean field approach in terms of a wave-function $\psi(\mathbf{r},t)$. In a one dimensional setting, the wave functions of two trapped, density-coupled BECs are governed by corresponding Gross-Pitaevskii equations $$\begin{aligned} \begin{aligned} &i\hbar \frac{\partial}{\partial t} \psi_1 = \left(\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial z^2}+\frac{m\Omega_z^2 z^2}{2}+ g |\psi_{1}|^2 + g_{12} |\psi_{2}|^2 \right)\psi_{1} \\ &i\hbar \frac{\partial}{\partial t} \psi_2 = \left(\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial z^2}+\frac{m\Omega_z^2 z^2}{2}+ g |\psi_{2}|^2 + g_{12} |\psi_{1}|^2 \right)\psi_{2}\, , \end{aligned} \label{eq:ebmr_dim}\end{aligned}$$ where $\Omega_z$ is the angular frequency of the axial harmonic trapping, $g=2\hbar \Omega_\perp a$ is the reduced 1D strength of the interaction between particles of the same condensate, which is proportional to the energy of the transverse trapping $\hbar \Omega_\perp$ and to the scattering length $a$, and $g_{12}=2\hbar \Omega_\perp a_{12}$ is the strength of the density coupling between condensates, proportional to the scattering length between particles of different condensates $a_{12}$. A stationary soliton solution to Eq. (\[eq:ebmr\_dim\]) $\psi_i(z,t)$, with $i=1,2$, can be written as the product of a time-independent (and real function) background component $\phi_i$ and the soliton excitation $v_i$, so that $$\begin{aligned} \psi_i(z,t)=\phi_i(z) v_i(z,t) e^{-i \mu_i t/\hbar}, \label{DS}\end{aligned}$$ where $\mu_i$ is the chemical potential. It is important to remark that both $\psi_i$ and $\phi_i$ satisfy the coupled GP equations for the same chemical potential: $$\begin{aligned} &\mu_i \phi_i = \left(-\frac{1}{2}\frac{\partial^2}{\partial z^2}+\frac{1}{2}\Omega^2 z^2+ g \phi_{i}^2 + g_{12} \phi_{j}^2 \right)\phi_{i} \label{eq:ebmr20} \\ &i \frac{\partial}{\partial t} \psi_i = \left(-\frac{1}{2}\frac{\partial^2}{\partial z^2}+\frac{1}{2}\Omega^2 z^2+ g |\psi_{i}|^2 + g_{12} |\psi_{j}|^2 \right)\psi_{i} \,, \label{eq:ebmr2}\end{aligned}$$ where $\Omega=\Omega_z/\Omega_\perp$ is the trap aspect ratio, and we have written the equations in dimensionless form by using transverse trap units, $a_\perp=\sqrt{\hbar/m\Omega_\perp}$ as unit length and $t_\perp=\Omega_\perp^{-1}$ as unit time, and for the sake of a simple notation have kept the same letters for the dimensionless couplings ${g}$ and ${g}_{12}$. So, from Eqs. (\[eq:ebmr20\])-(\[eq:ebmr2\]) we can obtain corresponding equations for the soliton wave functions $v_i(z,t)$ $$\begin{aligned} &\left(i \frac{\partial}{\partial t} + \frac{1}{2}\frac{\partial^2 }{\partial z^2}- g\phi^2_{i} \left(|v_{i}|^2-1\right) - g_{12}\phi^2_j\left(|v_{j}|^2-1\right)\right)v_i= \nonumber\\ &\hspace{.5cm} =-\frac{\partial v_i}{\partial z}\,\frac{\partial \ln \phi_i}{\partial z}\,, \label{eq:v}\end{aligned}$$ where the external trap does not appear explicitly, although its information is encoded in the background wavefunctions $\phi_i$. Untrapped case -------------- Here we set $\Omega=0$, hence the ground state solutions to Eq. (\[eq:ebmr20\]) are the constant density $|\phi_i|^2=n_i$ states $$n_i=\frac{g\mu_i-g_{12}\mu_j}{g^2-g^2_{12}}. \label{eq:n_i}$$ The equations of motion for soliton states Eq. (\[eq:v\]) can be written as $$\begin{aligned} \left(i \frac{\partial }{\partial t} + \frac{1}{2}\frac{\partial^2 }{\partial z^2} - g\,n_i\, \left(|v_{i}|^2-1\right) - g_{12}\,n_j\, \left(|v_{j}|^2-1\right)\right)v_i=0. \label{eq:v2}\end{aligned}$$ As we said before, we are going to deal with the coupling between condensates in a perturbative way. Then, assuming that $g_{12}<<g$, the background densities Eq. (\[eq:n\_i\]) become $n_i\approx \mu_i/g-(g_{12}/g)\,\mu_j/g$, and neglecting terms proportional to $g_{12}^2$ in Eq. (\[eq:v2\]) we get $$\begin{aligned} \begin{aligned} &i \frac{\partial v_i}{\partial t} + \frac{1}{2}\frac{\partial^2 v_i}{\partial z^2}- \mu_i \left(|v_{i}|^2-1\right)v_i = \\ &=- \frac{g_{12}}{g}\mu_j \left(|v_{i}|^2-|v_{j}|^2\right)v_i\equiv \frac{g_{12}}{g}\mu_j P(v_i,v_j). \end{aligned} \label{eq:v3}\end{aligned}$$ On the left hand side of this equality we have the well known non-linear Schrödinger equation for the wave function $v_i$, whereas on the right hand side we get a perturbative term in the parameter $g_{12}/g$. In order to obtain analytical expressions for the interactions between solitons, from now on we will focus on the case of equal backgrounds $\phi_1=\phi_2=\phi$, and then $\mu_1=\mu_2=\mu=(g+g_{12})n$. So that, by dividing the whole equation by $\mu$, we set new units for space $\hbar/\sqrt{m\mu}$ and time $\hbar/\mu$. In these units we introduce the general dark soliton solution $$\begin{aligned} \begin{aligned} &v_i(z,t)=\cos\varphi_i \tanh \zeta_i + i \sin\varphi_i, \end{aligned} \label{soliton}\end{aligned}$$ where $\zeta_i=\cos\varphi_i(z-t\sin\varphi_i )$, and $\varphi_i$ parametrizes the soliton darkness, that is $|v_i|^2=(1-\cos^2\varphi_i \, \mbox{sech}^2 \, \zeta_i)$, and the soliton velocity $\dot z_0=\sin \varphi_i$, that is the time derivative of the soliton position $z_0$, taken at minimum density. In the absence of perturbation, $P(v_i,v_j)=0$, the solution Eq. (\[soliton\]) describes overlapping solitons moving at constant velocity, and hence $\ddot z_0=\dot\varphi_i=0$. However, the effect of the perturbation $P(v_i,v_j)$ leads to the adiabatic time evolution of the soliton parameters, $(\varphi, z_0)\rightarrow (\varphi(t), z_0(t))$. This evolution can be described by the equation [@Kivshar1994] $$\begin{aligned} \dot\varphi_i=\frac{g_{12}/g}{2 \cos^2 \varphi_i \sin \varphi_i} \Re\left[\int_{-\infty}^{\infty} P(v_i,v_j)\frac{\partial v_i^*}{\partial t} dz\right]. \label{pert}\end{aligned}$$ Assuming solitons with high density depletions $\varphi_i<<1$ and separated by a relative distance $2z_0$, such that $\zeta_1=\cos \varphi_1 \left(z-z_0\right)$ and $\zeta_2=\cos \varphi_2 \left(z+z_0\right)$, Eq. (\[pert\]) gives $$\begin{aligned} \dot\varphi_1=-\frac{g_{12}}{g}f(z_0)\equiv \, F_{\rm eff}^{(i)}(z_0), \label{eq:phi}\end{aligned}$$ where $f(z_0)=8\sinh(2z_0)e^{6z_0}(3-3e^{8z_0}+(1+4e^{4z_0}+e^{8z_0})4z_0)/(e^{4z_0} -1)^5$. As can be seen, for $z_0=0$, that is for overlapped solitons, $\dot \varphi_1=0$ and the solitons evolve without relative motion. Since, in this limit ($\varphi<<1$) $\dot z_0\approx \varphi$, Eq. (\[eq:phi\]) is also an equation for $\ddot z_0$, with the right hand side playing the role of a classical force $F_{\rm eff}^{(i)}(z_0)$ derived from an effective potential $U_{\rm eff}^{(i)}(z_0)$ : $$\begin{aligned} \ddot z_0=-\frac{d U_{\rm eff}^{(i)}(z_0)}{dz_0}. \label{eq:rel}\end{aligned}$$ This potential accounts for the interactions between the solitons and is the source of their relative motion. After integration of the force Eq. (\[eq:phi\]) over $z_0$, we get $$\begin{aligned} U_{\rm eff}^{(i)}(z_0)=\frac{g_{12}}{g}\,\frac{2(1+e^{4z_0})(1-2z_0)-4} { e^{-4z_0}\,(e^{4z_0}-1)^3}. \label{eq:int_pot}\end{aligned}$$ As it has been anticipated, it presents a minimum at $z_0=0$, allowing for the existence of a bound state made of overlapped solitons, and tends exponentially to zero for finite values of the inter-soliton distance $2z_0$ (see Fig. \[fig:pert\]). For small separation $z_0\rightarrow 0$, the Taylor expansion reads $$\begin{aligned} U_{\rm eff}^{(i)}(z_0)\sim \frac{g_{12}}{g}\left(-\frac{1}{6}+\frac{4}{15} z^2_0 \right) +O[z^4_0]\,, $$ ![Top panel: Effective force experienced by a dark soliton forming a dark-dark soliton system for $g_{12}/g=0.01$ in the absence of trapping. Bottom panel: Soliton interaction potential Eq. (\[eq:int\_pot\]) from which the above effective force is derived. The solitons oscillate in a relative motion around the minimum at $z_0=0$.[]{data-label="fig:pert"}](Pot_g12_pos_a_001b){width="\columnwidth"} and the equation of relative motion Eq. (\[eq:rel\]) simplifies to that of a harmonic oscillator, $\ddot z_0+w_r^2 \,z_0=0$, with angular frequency (for generic chemical potential $\mu$) $$w_r=\mu\sqrt{\frac{8}{15}\frac{g_{12}}{g}}. \label{eq:freq_rel}$$ This is the frequency of small oscillations of the two solitons around their center of mass, moving at constant velocity $v_{CM}=(\sin \varphi_1+\sin \varphi_2)/2$. It is also interesting to note that this model predicts an instability (imaginary frequency) when $g_{12}<0$, due to the fact that the effective potential $U_{\rm eff}^{(i)}$ presents a maximum for attractive interparticle interactions between particles of different condensates, and as a consequence, it prevents the existence of a bound solitonic state in this case. Trapped case ------------ In the presence of axial harmonic trapping $V(z)=\frac{1}{2}\Omega^2 z^2$, we rely on the Thomas-Fermi (TF) approximation to study the strong (interparticle) interacting regime ($\mu_i>>\Omega$) [@Pitaevskii2003]. There the inhomogeneous ground state densities, $n_i(z)=|\phi_i|^2$, are given by $$\begin{aligned} n_i(z)=\frac{g\mu_i(z)-g_{12}\mu_j(z)}{g^2-g^2_{12}}, \label{eq:TF}\end{aligned}$$ where $\mu_i(z)=\mu_i-V(z)$ are the local chemical potentials. In the limit of $g_{12}<<g$, we get $$\begin{aligned} g\,n_i(z)={\mu_i(z)}-\frac{g_{12}}{g}{\mu_j(z)},\end{aligned}$$ and, as a result, the equations of motion for soliton solutions Eq. (\[eq:v\]) become $$\begin{aligned} &i \frac{\partial v_i}{\partial t} + \frac{1}{2}\frac{\partial^2 v_i}{\partial z^2}- \mu_i(z) \left(|v_{i}|^2-1\right)v_i = \nonumber \\ &=- \frac{g_{12}}{g}\mu_j(z) \left(|v_{i}|^2-|v_{j}|^2\right)v_i -\frac{\partial v_i}{\partial z}\,\frac{d\ln \phi_i}{dz} .\end{aligned}$$ The local chemical potentials and the last term in the right hand side of this equation are the main differences with respect to the untrapped case Eq. (\[eq:v3\]). In spite of these differences, and due to the fact that the background densities change slowly inside the TF regime ($dV/dz\rightarrow 0$), an analogue perturbative approach can still be followed, and the governing equation for the dark soliton $v_i$ is $$\begin{aligned} i \frac{\partial v_i}{\partial t} + \frac{1}{2}\frac{\partial^2 v_i}{\partial z^2}-\mu_i(z) v_i \left(|v_{i}|^2-1\right)=P(v_i,v_j;V(z)), \label{eq:v5}\end{aligned}$$ where the perturbation $P(v_i,v_j;V)$ is given by $$\begin{aligned} &P(v_i,v_j;V)\approx - \frac{g_{12}}{g} \mu_j(z) \left(|v_{i}|^2-|v_{j}|^2\right)v_i+ \nonumber\\ &\hspace{.5cm} +\frac{1}{2\mu(z)}\frac{dV}{dz}\frac{\partial v_i}{\partial z},\end{aligned}$$ which, along with the term associated to the interaction between dark solitons, contains a new term introduced by the external trap [@Frantzeskakis2010]. Again, we focus on the symmetric case $\mu_1=\mu_2=\mu$, where the local chemical potential reads $\mu(z)=\mu-V(z)=(g+g_{12})\,n(z)$. In the particular case of motion around the center of the harmonic potential $\mu(z)\rightarrow \mu$, and by following the same procedure as in the untrapped case (with $\hbar/\sqrt{m\mu}$ and $\hbar/\mu$ as space and time units respectively) we can obtain a particle-like evolution for the soliton position $z_0$ ![(Color online) Forces (top panel) and total, single-well potential (bottom panel) experienced by each dark soliton forming a dark-dark soliton system with $\mu=1$ and $g_{12}>0$ in the presence of a harmonic trap with $\Omega=0.1$. The dashed (red) line represents the force caused by the trap, whereas the continuous lines represent the effective force due to the interaction with the other dark soliton.[]{data-label="fig:potg12posa001"}](Pot_g12_pos_a_001){width="\linewidth"} $$\begin{aligned} \ddot{z}_0=-\frac{\Omega^2}{2 \mu^2} z_0- F_{\rm eff}^{(i)} , \label{eq:freq_trap}\end{aligned}$$ where the right hand side is the total effective force acting on the soliton $F_{\rm eff}=-dU_{\rm eff}/d\,z_0$, and embraces the superposition of the action of the external trap and the soliton interaction. For small-amplitude oscillations $z_0\approx 0$, we get $$\begin{aligned} \ddot{z}_0=-\left(\frac{\Omega^2}{2\mu^2}+\frac{8}{15}\frac{g_{12}}{g} \right)z_0+\frac{64}{63}\frac{g_{12}}{g}z_0^3+O(z_0^5), \label{eq:cl2}\end{aligned}$$ where the balance between the two harmonic force terms (inside the parenthesis) leads to different scenarios depending on the sign of the coupling interaction $g_{12}$. ![(Color online) Same as Fig. \[fig:potg12posa001\] for $g_{12}<0$. Three cases are depicted for different $g_{12}/g$ values, below (blue), at (orange) and above (green) the threshold $g_c$ indicating the change in the shape of the total effective potential from single well $g_{12}/g< c_c$ to double well $g_{12}/g>g_c$ (see text). []{data-label="fig:potg12nega001"}](Pot_g12_neg_a_001){width="\columnwidth"} ### Repulsive coupling: ${g_{12}>0}$ Figure \[fig:potg12posa001\] shows the effective force and effective potential defined in Eq. (\[eq:freq\_trap\]) for varying values of $g_{12}/g$ at a given trapping $\Omega=0.1$. As can be seen, the different parameters do not produce qualitative changes in the potential, which presents a single minimum capable to bound the coupled solitons. As in the untrapped case, our theory predicts that the dark solitons will perform small-amplitude oscillations around $z_0=0$ with angular frequency (in transverse oscillator units) $$w=\left(\frac{\Omega^2}{2}+\frac{8}{15}\frac{g_{12}}{g} \mu^2\right)^\frac { 1 } { 2 }. \label{eq:fre}$$ As we will see later, our numerical results demonstrate that the minimum supports a bound state that exists and is stable for small $g_{12}/g$. However, for increasing values of this parameter both stable and unstable cases can be found depending on the particular valued of the chemical potential of the system. ### Attractive coupling: ${g_{12}<0}$ The phenomenology is richer for attractive interactions between particles of different condensates. In contrast to the untrapped case, the presence of the harmonic potential allows for the generation of local minima in the total effective potential that can support stationary states made of two solitons. As shown in Fig. \[fig:potg12nega001\], by increasing the parameter $|g_{12}/g|$, the total effective potential modifies from a single well (an also a unique fixed point $z_0^*=0$ in the equation of motion) to a double well potential (with three fixed points). Specifically, the system shows a pitchfork bifurcation at $g_c=|g_{12}/g|={15}(\Omega/\mu)^2/16$, hence for $|g_{12}/g|>g_c$ the fixed point at $z_0^*=0$ loses its stability and two new off-center, stable fixed points appear. From Eq. (\[eq:cl2\]) we get their position at $$z_{0_\pm}^{*} = \pm \frac{1}{\sqrt{\mu}}\sqrt{\frac{21}{40}- \frac{63}{128} \left|\frac{g}{g_{12}}\right| \left(\frac{\Omega}{\mu}\right)^2 }. \label{eq:fixed_double}$$ ![(Color online) Stationary state made of two solitons situated at the fixed points $z_{0_\pm}^*$ of the effective potential (see text) for $\mu=10\,\Omega$ and $g_{12}/g=-0.1$.[]{data-label="fig:separated"}](separated_solitons){width="\columnwidth"} As can be noted, the separation between minima increases with $\mu/\, \Omega$ for given $g_{12}/g$, and saturates at a distance of $2z_0^*=1.44$. Fig. \[fig:separated\] shows an example of a stationary state with two separated solitons occupying the two minima of the effective potential at $\mu=10 \, \Omega$ and $g_{12}/g=-0.1<g_c$. For small distances $\epsilon(t)$ around the fixed points $z_{0_\pm}^*$, from the substitution of $z_0(t)=z_0^*+\epsilon(t)$ in Eq. (\[eq:cl2\]), the solitons oscillate according to $$\ddot{\epsilon}=-\left( w_1(z_{0_\pm}^*)^2 +w^2 \right) \epsilon \equiv -w_{\rm eff}^2 \epsilon , \label{eq:epsilon}$$ up to linear terms in the perturbation $\epsilon(t)$, where $w$ is the angular frequency given by , and $ w_1(z_0)^2=-2 f(z_0) \left( \mbox{sech}(z_0)- {5}/{\sinh(2z_0)} + 3\right)-$ $\left( \left(1+4e^{4z_0} +e^{ 8z_0}\right)+ \left(-12e^{8z_0}+\left(16e^{4z_0}+8e^{8z_0}\right) z_0 \right) \right)$ $ 64e^{6z_0} / \left(e^{4z_0}-1\right)^5 $. ![(Color online) Bogoliubov modes in 1D rings of different size (untrapped case). The upper panel depicts the real part of the frequencies and the lower panel the imaginary parts. The solid and dashed lines correspond to Eqs. (\[eq:freq\_rel\]) and (\[eq:spin\]), respectively, the crossing of which indicates the appearance of instabilities.[]{data-label="fig:bog_notrap"}](modes_mu05_Z10.pdf){width="\columnwidth"} Numerical results ================= In what follows, in order to test our analytical predictions on the dark-dark-soliton dynamics, we first numerically solve the GP equation to obtain these stationary states for varying chemical potentials and interactions strengths. Afterwards, the soliton stability is monitored both in the nonlinear regime (by simulating the real time evolution with the GP equation) and by linear analysis around the equilibrium states. The latter is performed by solving the Bogoliubov equations for the linear excitations $[u(z,t),v(z,t)]$ around the dark-dark soliton states $\psi(z,t)=\exp(-i \mu t)[\psi(z) +\sum_\omega(u \,e^{-i\omega t}+ v^* e^{i\omega t})]$. These equations are: $$\begin{aligned} B \left(\begin{array}{c} u_1 \\ v_1 \\ u_2 \\ v_2 \end{array}\right) = \omega \left(\begin{array}{c} u_1 \\ v_1 \\ u_2 \\ v_2 \end{array}\right), \label{eq:BdG}\end{aligned}$$ where: $$\begin{aligned} B= \left(\begin{array}{cccc} h_1 & g\psi_1^2 & g_{12}\psi_ 1 \psi_2^* & g_{12}\psi_1 \psi_2 \\ -g\psi_1^{*2} & -h_1 & -g_{12}\psi_1^* \psi_2^* & -g_{12}\psi_1^* \psi_2 \\ g_{12}\psi_1^* \psi_2 & g_{12}\psi_1 \psi_2 & h_2 & g\psi_2^2 \\ -g_{12}\psi_1^* \psi_2^* & - g_{12}\psi_1 \psi_2^* &- g\psi_2^{*2} & -h_2 \end{array}\right).\end{aligned}$$ Here $h_i=-\frac{1}{2} \frac{\partial^2}{\partial z^2} + \frac{1}{2}\Omega^2z^2 + 2g |\psi_i|^2 + g_{12} |\psi_{j\neq i}|^2 - \mu_i$. They can be seen as an eigenvalue problem with a non-trivial solution given by ${\rm det}|B- \omega I|=0$. Apart from the perturbative case $g_{12}<<g$, we also explore numerically the stability of overlapped solitons in the whole miscible regime $g_{12}<g$, and show that the finite size of the system determines the stability properties. Untrapped case -------------- ![(Color online) Spatio-temporal evolution of the density of two untrapped dark-dark solitons with $\mu=1$, $g_{12}=0.015$ and $g=1$ obtained from the numerical solution of the GP Eq. . The top and down panel correspond to component 1 and 2 respectively. The continuous (colour) lines represent the evolution of the dark solitons positions given by Eq. (\[eq:freq\_rel\]).[]{data-label="fig:osci003"}](untrap_osci_0015){width="\columnwidth"} ![Same as Fig. \[fig:osci003\] but with $g_{12}=0.1$. []{data-label="fig:osci01"}](untrap_osci_01){width="\columnwidth"} ![(Color online) Soliton relative motion frequency as a function of the interparticle strength ratio in the absence of external trap. The continuous (blue) line represents our theoretical prediction Eq. (\[eq:freq\_rel\]) and the (orange) dots are values extracted by means of Fourier analysis from the numerical solution of the time dependent GP equation.[]{data-label="fig:fourier"}](Fourier2){width="\columnwidth"} The excitation frequencies of the stationary states can be readily extracted by solving the Bogoliubov equations Eq. (\[eq:BdG\]). In the Appendix we analytically show that there are two zero modes of excitation at $g_{12}/g=0$, and $1$, and hence that the system of overlapped dark solitons is expected to be unstable in between. However such analysis assume an infinite system, and the situation is quite different in systems of finite size, where ranges of dynamical stability can be found. Fig. \[fig:bog\_notrap\] shows two examples of this phenomenon, where the Bogoliubov modes are computed for overlapped solitons with the same chemical potential $\mu=0.5$ in 1D rings of different sizes $L=20,\; 40$. The instability emerge from a Hopf bifurcation, which occurs due to the collision of two excitation modes (see the discussion about this collision in the next section): one mode associated to the oscillations around the minimum of the effective potential of soliton interactions, given by expression (\[eq:freq\_rel\]) and represented in Fig. \[fig:bog\_notrap\] by the solid red curve, and the background spin density mode of lowest energy (represented by the dashed curve for the longer ring), which is given by the analytical expression (in full units) [@Abad2013]: $$\hbar\omega=\sqrt{\frac{\hbar^2 k^2}{2m} \left( \frac{\hbar^2 k^2}{2m} + 2(g-g_{12})n\right)}. \label{eq:spin}$$ The small disagreement between the crossing of these analytical curves and the beginning of instabilities in the numerical results arises from the curvature of the modes near the bifurcation point, and decreases for longer rings. This first instability triggers new collisions between modes and, as a consequence, more instability regions. The longer the ring the higher the number of instability regions in the system, approaching the prediction for the infinite case. It is worth remarking that the bound state mode predicted by Eq. (\[eq:freq\_rel\]), in excellent agreement with the numerics, does not change with the size of the ring. To analyze the dynamics of dark-dark solitons in the bound state allowed by the repulsive interparticle interactions $g_{12}>0$, we excite the relative motion of the solitons by imposing the initial ansatz $$\begin{aligned} \begin{aligned} &\psi_1=\sqrt{n}\tanh\left(z-z_0\right) \\ &\psi_2=\sqrt{n}\tanh\left(z+z_0\right), \end{aligned}\end{aligned}$$ and we fix $\mu=1$. Figures \[fig:osci003\]–\[fig:osci01\] show the comparison between the subsequent motion of the solitons from the numerical solution of GP Eq. , and the analytical prediction by Eq. (\[eq:freq\_rel\]) fitted to $y(t)=z_0\cos{wt}$. As can be seen, it provides a reasonable good estimate for small $g_{12}$ (Fig. \[fig:osci003\]) but fails for larger $g_{12}$ (Fig. \[fig:osci01\]) or also long times. In order to obtain a more quantitative comparison we have run different real time evolutions for varying $g_{12}$ and equal chemical potential $\mu=1$. By tracking the position of the solitons (at minimum density), we have computed their characteristic frequency from a Fourier analysis in time. The numerical results are presented in Fig. \[fig:fourier\], and show a very good agreement with our analytical prediction for small values of $g_{12}/g$. Trapped case ------------ ### Attractive interaction between condensates: ${g_{12}<0}$ ![(Color online) Frequency of the dark soliton relative motion in a harmonic trap with $\Omega=0.1$. The real (top panel) and imaginary (bottom) parts of this frequency are shown according to our analytical prediction Eq. (continuous orange line) and the numerical solution of the excitation spectrum from Eq. for the out-of-phase anomalous mode (dash-dotted lines).[]{data-label="fig:freqg12nega00175"}](Freq_g12_neg_a_00175.pdf){width="\columnwidth"} As anticipated, in this case the configuration of overlapped solitons at $z_0*=0$ is unstable for $|g_{12}/g|>g_c$. This instability can also be detected by the appearance of an imaginary frequency in the excitation spectrum. Fig. \[fig:freqg12nega00175\] shows our results for the linear excitations of such a stationary state with $\mu=10\, \Omega$ from the solution of the Bogoliubov equations (dash-dotted lines). The frequency of the out-of-phase anomalous mode (see a discussion of this mode in next section) takes real values for $|g_{12}/g|<g_c$ and pure imaginary for $|g_{12}/g|>g_c$. This critical point $g_c$ is associated to the change of the total effective potential, from single-well to double-well. Along with the change of the total effective potential two new stable fixed points appear in the system. In Fig. \[fig:eqpos2\] we compare these points (red dots), obtained by extracting the mean position of the soliton oscillations around the equilibrium positions, with our analytical approach Eq. (orange line).The latter fails for increasing values of $|g/g_{12}|$. However the direct numerical computation of Eq. (blue line) provides a very good agreement for the regime of interest, and shows that the distance between equilibrium points increases with $|g_{12}/g|$ instead of being saturated at $2 z_0^*=1.44$. ![(Color online) Fixed points of the effective double well potential given by Eq. (continuous blue line) as a function of $g_{12}/g$. The continuous (orange) line represents the analytical result and the (red) dots are the mean position of the soliton oscillations extracted from the numerical solution of GP equation. []{data-label="fig:eqpos2"}](eq_pos2){width="1.\columnwidth"} ![Spatio-temporal evolution of a dark-dark-soliton density obtained from the numerical solution of the GP Eq. for $g_{12}/g=-0.046$ and harmonic trapping $\Omega=0.1$. Each panel corresponds to a different condensate. The dashed (red) lines represent the position of the equilibrium points given by . []{data-label="fig:doublewell0046"}](insta_outcenter_0046){width="\columnwidth"} We have numerically solved the GP Eq. (\[eq:ebmr\_dim\]) for the real-time evolution of  dark-dark solitons with values of $g_{12}$ below and above the bifurcation point $g_c$ for the change of stability. First, we have computed a case (see Fig. \[fig:doublewell0046\]) with overlapped solitons situated at $z_0=0$ and $g_{12}/g=-0.046<g_c$, hence unstable according to the linear prediction. The initial stationary state has been perturbed with white-noise of $1\%$ amplitude. As expected the system is unstable, and eventually the solitons separate by moving towards the minima of the effective double well potential Eq. (\[eq:fixed\_double\]). In this case, each dark soliton has enough energy to pass through the energy barrier created at the trap center $z_0=0$, so that collisions between them are observed to cause a shift in their trajectories. If instead the two dark solitons are initially situated at different locations, close to the positions of the fixed points Eq. (\[eq:fixed\_double\]) (see Fig. \[fig:osci\_outcenter\_00125\]), the solitons oscillate symmetrically around such points. Their time evolution can be fitted by $x(t)=z_0^*+\left(-z_0^*+z_0 \right)\cos(w_{\rm eff}t)$, where $w_{\rm eff}$ is given by , and provides a good approximation to the real time evolution obtained from the GP equation. ![Same as Fig. \[fig:doublewell0046\] with $g_{12}/g=-0.0125$. In this spatio-temporal evolution each dark soliton is situated off center ($z_0=\pm0.3$, near the equilibrium points given by ). The continuous line is fitted according to our analytical results (see text).[]{data-label="fig:osci_outcenter_00125"}](osci_outcenter_00125198){width="\columnwidth"} ### Repulsive interaction between condensates: ${g_{12}>0}$ ![(Color online) Numerical results from the Bogoliubov spectrum of a dark-dark soliton with $\mu=10 \,\Omega$ in a trap. The real (top panel) and imaginary (bottom panel) parts of the excitation frequencies are shown against the interaction strength ratio $g_{12}/g$. The continuous (orange) line represents the analytical values from Eq. .[]{data-label="fig:spectrumrealimag"}](spectrum_trap01_mu1.pdf){width="1.0\columnwidth"} Figure \[fig:spectrumrealimag\] represents the excitation spectrum of overlapped dark-dark solitons at $z_0^*=0$, in the range $g_{12}/g\in[0,1]$, for $\mu=10\, \Omega$, within the Thomas Fermi regime of the axial harmonic oscillator. At $g_{12}=0$, corresponding to uncoupled solitons, the linear modes have double degeneracy, and the lowest energy excitations are the anomalous mode, with frequency $\omega_0=\Omega/\sqrt{2}$, and the hydro-dynamical excitations $\omega_n=\sqrt{{n(n+1)}/{2}}\,\Omega$, with $n=1,2,...$ [@Stringari1996; @Busch2000; @Kevrekidis2015; @Frantzeskakis2010]. The degeneracy is broken for non null $g_{12}$ and gives rise to two branches of in-phase and out-of-phase modes. In the hydro-dynamical case, the in-phase modes are associated to excitations in the total density of the background, and remain constant for varying $g_{12}$. On the other hand, the out-of-phase or spin modes account for variations in the difference of the background densities [@Abad2013], and decrease their energy for increasing values of $g_{12}$. ![Real time evolution of three dark-dark solitons with varying coupling out of and inside the first instability region of Fig. \[fig:spectrumrealimag\], showing corresponding stable and unstable dynamics. []{data-label="fig:instab_trap"}](abc_evolution.pdf "fig:"){width="\columnwidth"}\ ![Real time evolution of three dark-dark solitons with varying coupling out of and inside the first instability region of Fig. \[fig:spectrumrealimag\], showing corresponding stable and unstable dynamics. []{data-label="fig:instab_trap"}](imaginary_freq_mu_1.pdf "fig:"){width="0.9\columnwidth"} The anomalous modes are characterized by the negative value of the quantity $norm \times energy$ [@MacKay1987], and in scalar condensates their frequency coincide with the oscillation frequency of the solitons in the trap [@Busch2000]. There are also two different anomalous modes for non null $g_{12}$, an in-phase one with constant energy, and an out-of-phase mode whose energy increases with $g_{12}$. The in-phase anomalous mode is associated with the small amplitude, abreast oscillations of the solitons in the trap, hence it is the same as in decoupled condensates: $\omega_0=\Omega/\sqrt{2}$. However, the out-of-phase anomalous mode is associated with the relative motion of the solitons. This mode, given by the analytical expression Eq. , is depicted (orange line) in the upper panel of Fig. \[fig:spectrumrealimag\], in good agreement with the numerical results for small values of $g_{12}/g$. The lower panel of Fig. \[fig:spectrumrealimag\] present our numerical results for the imaginary part of the spectrum. As previously commented in the untrapped case, these instabilities are characterized by the collision of the out-of-phase anomalous mode with the spin mode associated with the background. These collisions produce Hamiltonian-Hopf bifurcations where a complex frequency quartet appears in the excitation spectrum. It is interesting to see that the out-of-phase anomalous mode only collides with odd spin modes. Regions of stability and instability alternates up to a value of the coupling with $g_{12}/g$ close to 1, but interestingly inside the immiscible regime. The higher the chemical potential, the closer is this value to $g_{12}/g=1$, according with the Bogoliubov analysis for the untrapped case. Such a value can be well approximated within the Thomas-Fermi regime by Eq. (\[eq:spin\]) evaluated at maximum density $n=\mu/(g+g_{12})$. Again the crossing of this analytical frequency with the function for small oscillations Eq. (\[eq:fre\]) provides a good estimate for the beginning of the first instability region in the range $g_{12}/g\in[0,1]$. It is worth noticing that such instability is not captured by Eq. alone, due to the fact that for its derivation the soliton motion was assumed to be decoupled from the background. Figure \[fig:instab\_trap\] shows examples of stable and unstable dark-dark solitons near the first instability region of Fig. \[fig:spectrumrealimag\]. The evolution of the nonlinear systems develops according the linear stability analysis and demonstrate the existence of dynamically stable coupled solitons, cases (a) and (c), that could be experimentally realized. Conclusions =========== The dynamics of dark-dark soliton states in two density coupled BECs has been studied within GP theory. By performing a perturbation analysis in the parameter $g_{12}/g$ for solitons with equal chemical potential, we have derived analytical expressions describing their relative motion both in harmonic traps and untrapped systems. Contrary to the case of solitons in scalar condensates, our theoretical model predicts that the interaction between dark solitons excited in different condensates is attractive (repulsive) for repulsive (attractive) interparticle interactions $g_{12}$. In harmonically trapped systems, the scenario is specially interesting for negative $g_{12}$, where the effective potential felt by the solitons modifies its shape as a function of $g_{12}$ (through a pitchfork bifurcation) from a single-well potential to a double-well potential, then allowing for stationary states made of solitons located at different positions. The theoretical analytical predictions have been shown to be in good agreement with the numerical solutions of the Gross Pitaevskii equation for the real time evolution, and with the Bogoliubov equations for the linear excitations of the dark-dark solitons. In particular, we have demonstrated that the resonance of two out-of-phase modes, the anomalous one giving the frequency of the relative motion between solitons, and the lowest energy mode associated to the spin density excitation of the background, give rise to instabilities (Hopf bifurcations) that produce the decay of dark-dark solitons. This fact translate in finite systems, either harmonically trapped condensates or ring geometries, into alternating regions of dynamical stability and instability. The existence of dynamically stable dark-dark solitons open up the way for their experimental realization. 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The dark soliton solutions healing to the ground state with density $n$ are given by $$\psi_{1,2}(x)=\psi_d(x)e^{-i\mu\,t/\hbar}=\sqrt{\frac{\mu}{g+g_{12}}}\, \tanh\left(\frac{x}{\xi}\right) e^{-i\mu\,t/\hbar}\,, \label{eq:dark}$$ where $\xi=\sqrt{\hbar^2/m \mu}$ is the soliton healing length. As in the scalar case, it can be seen that the soliton state depends on both, the chemical potential $\mu$ and the (sum of the) interaction strength $g+g_{12}$. We check the stability of Eq. (\[eq:dark\]) by solving the Bogoliubov equations $$\begin{aligned} H_0 \, u_1+ \psi_d^2 \left[(2g+g_{12})u_1+g v_1 +g_{12}(u_2+v_2) \right] = \omega \, u_1 \nonumber\\ -H_0 \, v_1 - \psi_d^2 \left[(2g+g_{12})v_1+g u_1 +g_{12}(u_2+v_2) \right] = \omega \, v_1 \nonumber\\ H_0 \, u_2+ \psi_d^2 \left[(2g+g_{12})u_2+g v_2 +g_{12}(u_1+v_1) \right] = \omega \, u_2 \nonumber\\ -H_0 \, v_2 - \psi_d^2 \left[(2g+g_{12})v_2+g u_2 +g_{12}(u_1+v_1) \right] = \omega \, v_2 , \label{eq:bog0}\end{aligned}$$ where $H_0 = -(\hbar^2/2m)\partial_{xx} -\mu$, and $[u_{1}(x), v_{1}(x), u_{2}(x) ,v_{2}(x)]$ are the linear modes with energy $\omega$. By adding and substracting the the two first previous equations, on the one hand, and the two last, on the other hand, we obtain new equations for the linear combinations $f_{j\pm}=u_{j}(x) \pm v_{j}(x)$ $$\begin{aligned} \left(\frac{-\hbar^2}{2m}\partial_{xx}+ \, (g+g_{12}) \, \psi_d^2-\mu\right) f_{1-}= \omega f_{1+} \nonumber\\ \left(\frac{-\hbar^2}{2m}\partial_{xx}+ \, (g+g_{12}) \, \psi_d^2-\mu\right) f_{2-}= \omega f_{2+}, \label{eq:bog1-} \\ \left( \frac{-\hbar^2}{2m}\partial_{xx} + (3g+g_{12}) \psi_d^2 \, -\mu \right) f_{1+} +2g_{12}\psi_d^2 f_{2+}=\omega f_{1-} \, \nonumber\\ \left( \frac{-\hbar^2}{2m}\partial_{xx} + (3g+g_{12}) \psi_d^2 \, -\mu \right) f_{2+} +2g_{12}\psi_d^2 f_{1+}=\omega f_{2-} \,. \label{eq:bog1+}\end{aligned}$$ The first two equations Eqs. (\[eq:bog1-\]) are already decoupled for the modes $1$ and $2$, and (for $\omega=0$) contains zero energy excitations associated to the $U(1)$ symmetry presented within each condensate, so that a global phase can be arbitrarily picked in the soliton solutions Eq. (\[eq:dark\]). However, Eqs. (\[eq:bog1+\]) are still coupling the modes $1$ and $2$. In order to decouple them, we make the symmetric $f_{s\pm}=f_{1\pm}(x) + f_{2\pm}(x)$ and antisymmetric $f_{a\pm}=f_{1\pm}(x) - f_{2\pm}(x)$ linear combinarions to get $$\begin{aligned} \left(\frac{-\hbar^2}{2m}\partial_{xx}+ \, (g+g_{12}) \, \psi_d^2-\mu\right) f_{s-}= \omega f_{s+} \nonumber\\ \left( \frac{-\hbar^2}{2m}\partial_{xx} + 3(g+g_{12}) \psi_d^2 \, -\mu \right) f_{s+} =\omega f_{s-} \,, \label{eq:bog_s}\end{aligned}$$ and $$\begin{aligned} \left(\frac{-\hbar^2}{2m}\partial_{xx}+ \, (g+g_{12}) \, \psi_d^2-\mu\right) f_{a-}= \omega f_{a+} \, \nonumber\\ \left( \frac{-\hbar^2}{2m}\partial_{xx} + (3g-g_{12}) \psi_d^2 \, -\mu \right) f_{a+} =\omega f_{a-} \,. \label{eq:bog_a}\end{aligned}$$ Equations (\[eq:bog\_s\]) for the symmetric combinations $f_{s\pm}=f_{1\pm}(x) + f_{2\pm}(x)$ are equivalent to the Bog equations of a scalar condensate in a dark soliton state for an interaction strength $g+g_{12}$. As a result they do not present unstable modes, and contain two Goldstone modes (with $\omega=0$) reflecting the mentioned $U(1)$ symmetry and the translational invariance of the system. On the other hand, the Eqs. (\[eq:bog\_a\]) for the antisymmetric combinations $f_{a\pm}=f_{1\pm}(x) - f_{2\pm}(x)$ are relevant for the unstable modes having complex frequencies $\omega$. These modes can first appear at $\omega=0$ for particular values of the system parameters $\{\mu, g, g_{12}\}$ (for a given particle species of mass $m$), indicating a bifurcation of the soliton solutions giving place to a new stationary state. However, as we have seen in the text, the instabilities can also appear from a couple of nonzero real frequencies which become complex, indicating a Hopf bifurcation with a subsequent oscillatory dynamics. Below, we analyze the first case associated to the existence of zero modes. At $\omega=0$, since the first equation (\[eq:bog\_s\]) provides the commented Goldstone modes, we only have to look for solutions to $$\begin{aligned} \left( \frac{-\hbar^2}{2m}\partial_{xx} + \mu \frac{3g-g_{12}}{g+g_{12}}\, \tanh^2\left(\frac{x}{\xi} \, \right)-\mu \right)f_{a+} =0\,,\end{aligned}$$ that in units of the healing length gives $$\begin{aligned} -\frac{1}{2}\partial_{xx} f_{a+}- (1+\delta g) \, \mbox{sech}^2(x) f_{a+} = -\delta g \, f_{a+}, \label{eq:zero}\end{aligned}$$ where $\delta g=2(g-g_{12})/({g+g_{12}})=2(g-g_{12})n/\mu$ is the relevant parameter for the emergence of instabilities. Eq. (\[eq:zero\]) is a Schrödinger equation for the asymmetric wavefunction $f_{a+}$ in the potential well $\mbox{sech}^2(x)$ with depth $(1+\delta g)$. This well has bound states with energy $-\delta g$ whenever [@Rosen; @Sophie2017] $$\sqrt{2 \,\delta g+\frac{9}{4}}-\sqrt{2\,\delta g}-\frac{1}{2}=n \, ,$$ with $n=0,1,2,\dots$ and $n \leq \sqrt{2\delta g+9/4}-1/2$. The latter condition ensures that $g_{12}<g$ and saturates with $g=g_{12}$ at the starting point of the inmiscible regime.

--- abstract: 'We investigate the nonlinear scattering theory for quantum systems with strong Seebeck and Peltier effects, and consider their use as heat-engines and refrigerators with finite power outputs. This article gives detailed derivations of the results summarized in Phys. Rev. Lett. [**112**]{}, 130601 (2014). It shows how to use the scattering theory to find (i) the quantum thermoelectric with maximum possible power output, and (ii) the quantum thermoelectric with maximum efficiency at given power output. The latter corresponds to a minimal entropy production at that power output. These quantities are of quantum origin since they depend on system size over electronic wavelength, and so have no analogue in classical thermodynamics. The maximal efficiency coincides with Carnot efficiency at zero power output, but decreases with increasing power output. This gives a fundamental lower bound on entropy production, which means that reversibility (in the thermodynamic sense) is impossible for finite power output. The suppression of efficiency by (nonlinear) phonon and photon effects is addressed in detail; when these effects are strong, maximum efficiency coincides with maximum power. Finally, we show in particular limits (typically without magnetic fields) that relaxation within the quantum system does not allow the system to exceed the bounds derived for relaxation-free systems, however, a general proof of this remains elusive.' author: - 'Robert S. Whitney' date: 'March 16, 2015' title: | Finding the quantum thermoelectric with maximal efficiency\ and minimal entropy production at given power output --- Introduction ============ Thermoelectric effects in nanostructures [@Pekola-reviews; @Casati-review; @Sothmann-Sanchez-Jordan-review; @Haupt-review] and molecules [@Paulsson-Datta2003; @Reddy2007] are of great current interest. They might enable efficient electricity generation and refrigeration [@books; @DiSalvo-review; @Shakouri-reviews], and could also lead to new types of sub-Kelvin refrigeration, cooling electrons in solid-state samples to lower temperatures than with conventional cryostats [@Pekola-reviews], or cooling fermionic atomic gases [@Grenier2012; @Brantut-Grenier-et-al2013; @Grenier2014]. However, they are also extremely interesting at the level of fundamental physics, since they allow one to construct the simplest possible quantum machine that converts heat flows into useful work (electrical power in this case) or vice versa. This makes them an ideal case study for [*quantum thermodynamics*]{}, i.e. the thermodynamics of quantum systems [@QuantumThermodyn-book]. ![\[Fig:thermocouple\] (a) The simplest heat-engine is a thermocouple circuit made of two thermoelectrics (filled and open circles). The filled and open circles are quantum systems with opposite thermoelectric responses, an example could be that in (b). For a heat-engine, we assume $T_L > T_R$, so heat flows as shown, generating a current $I$, which provides power to a load (battery charger, motor, etc.) that converts the electrical power into some other form of work. The same thermocouple circuit can act as a refrigerator; if one replaces the load with a power supply that generates the current $I$. This induces the heat flow out of Reservoir $L$, which thereby refrigerates Reservoir $L$, so $T_L < T_R$. Note that in both cases the circuit works because the two thermoelectrics are electrically in series but thermally in parallel. In (b), $N$ indicates the number of transverse modes in the narrowest part of the quantum system. ](figure1.pdf){width="\columnwidth"} The simplest heat-engine is a thermocouple circuit, as shown in Fig. \[Fig:thermocouple\]. It consists of a pair of thermoelectrics with opposite thermoelectric responses (filled and open circles) and a load, connected in a ring. Between each such circuit element is a big reservoir of electrons, the reservoir on the left ($L$) is hotter than the others, $T_L > T_R$, so heat flows from left to right. One thermoelectric’s response causes an electric current to flow in the opposite direction to the heat flow (filled circle), while the other’s causes an electric current to flow in the same direction as the heat flow (open circle). Thus, the two thermoelectrics turn heat energy into electrical work; a current flow $I$ through the load. The load is assumed to be a device that turns the electrical work into some other form of work; it could be a battery-charger (turning electrical work into chemical work) or a motor (turning electrical work into mechanical work). The same thermocouple circuit can be made into a refrigerator simply by replacing the load with a power supply. The power supply does work to establish the current $I$ around the circuit, and this current through the thermoelectrics can “drag” heat out of reservoir $L$. In other words, the electrical current and heat flow are the same as for the heat-engine, but now the former causes the latter rather than vice versa. Thus, the refrigerator cools reservoir $L$, so $T_L < T_R$. The laws of classical thermodynamics inform us that entropy production can never be negative, and maximal efficiency occurs when a system operates reversibly (zero entropy production). Thus, it places fundamental bounds on heat-engine and refrigerator efficiencies, known as Carnot efficiencies. In both cases, the efficiency is defined as the power output divided by the power input. For the heat-engine, the power input is the heat current out of the hotter reservoir (reservoir $L$), $J_L$, and the power output is the electrical power generated $P_{\rm gen}$. Thus, the heat-engine (eng) efficiency is $$\begin{aligned} \eta_{\rm eng} = P_{\rm gen}\big/ J_L. \label{Eq:eff-eng}\end{aligned}$$ This efficiency can never exceed Carnot’s limit, $$\begin{aligned} \eta_{\rm eng}^{\rm Carnot} &=& 1-T_R/ T_L, \label{Eq:Carnot-eng}\end{aligned}$$ where we recall that we have $T_L > T_R$. For the refrigerator the situation is reversed, the load is replaced by a power supply, and the power input is the electrical power that the circuit absorbs from the power supply, $P_{\rm abs}$. The power output is the heat current out of the colder reservoir (reservoir $L$), $J_L$. This is called the cooling power, because it is the rate at which the circuit removes heat energy from reservoir $L$. Thus, the refrigerator (fri) efficiency is $$\begin{aligned} \eta_{\rm fri} = J_L \big/ P_{\rm abs}. \label{Eq:eff-fri}\end{aligned}$$ This efficiency is often called the coefficient of performance or COP. This efficiency can never exceed Carnot’s limit, $$\begin{aligned} \eta_{\rm fri}^{\rm Carnot} &=& (T_R/T_L -1)^{-1}, \ \ \label{Eq:Carnot-fri}\end{aligned}$$ where we recall that $T_L < T_R$ (opposite of heat-engine). Strangely, the laws of classical thermodynamics do not appear to place a fundamental bound on the power output associated with reversible (Carnot efficient) operation. Most textbooks say that reversibility requires “small” power output, but rarely define what “small” means. The central objective of Ref. \[\] was to find the meaning of “small”, and find a fundamental upper bound on the efficiency of an irreversible system in which the power output was [*not*]{} small. Ref. \[\] did this for the class of quantum thermoelectrics that are well modelled by a scattering theory, which enables one to straightforwardly treat quantum and thermodynamic effects on an equal footing. It summarized two principal results absent from classical thermodynamics. Firstly, there is a quantum bound (qb) on the power output, and no quantum system can exceed this bound (open circles in Fig. \[Fig:summary\]). Secondly, there is a upper bound on the efficiency at any given power output less than this bound (thick black curves in Fig. \[Fig:summary\]). The efficiency at given power output can only reach Carnot efficiency when the power output is very small compared to the quantum bound on power output. The upper bound on efficiency then decays monotonically as one increases the power output towards the quantum bound. The objective of this article is to explain in detail the methods used to derive these results, along with the other results that were summarized in Ref. \[\]. ![\[Fig:summary\] The thick black curves are qualitative sketches of the maximum efficiency as a function of heat-engine power output (main plot), or refrigerator cooling power (inset), with the shaded regions being forbidden. Precise plot of such curves for different temperature ratios, $T_R/T_L$, are shown in Fig. \[Fig:allpowers\]. The colored loops (red, grey and blue) are typical sketches of the efficiency versus power of [*individual*]{} heat-engines as we increase the load resistance (direction of arrows on loop). The power output $P_{\rm gen}=IV$ vanishes when the load resistance is zero (for which $V=0$) or infinite (for which $I=0$), with a maximum at an intermediate resistance (open square). The curves have a characteristic loop form [@Casati-review], however the exact shape of the loop depends on many system specific details, such as charging effects. The dashed blue loop is for a typical non-optimal system (always well below the upper bound), while the solid red and grey loops are for systems which achieve the upper bound for a particular value of the load. The star marks the Curzon-Ahlborn efficiency. ](figure2.pdf){width="\columnwidth"} Contents of this article ------------------------ This article provides detailed derivations of the results in Ref. \[\]. The first part of this article is an extended introduction. Section \[Sect:literature\] is a short review of the relevant literature. Section \[Sect:Unique\] discusses how we define temperature, heat and entropy. Section \[Sect:entropy-prod\] recalls the connection between efficiency and entropy production in any thermodynamic machine. Section \[Sect:ScatteringTheory\] reviews the nonlinear scattering theory, which section \[Sect:over-estimates\] uses to make very simple over-estimates of a quantum system’s maximum power output. The second part of this article considers how to optimize a system which is free of relaxation and has no phonons or photons. Section \[Sect:guess-heat\] gives a hand-waving explanation of the optimal heat engine, while Section \[Sect:eng\] gives the full derivation. Section \[Sect:guess-fri\] gives a hand-waving explanation of the optimal refrigerator, while Section \[Sect:fri\] gives the full derivation. Section \[Sect:chain\] proposes a system which could in principle come arbitrarily close to the optimal properties given in sections \[Sect:eng\] and \[Sect:fri\]. Section \[Sect:in-parallel\] considers many quantum thermoelectrics in parallel. The third part of this article considers certain effects neglected in the above idealized system. Section \[Sect:ph\] adds the parasitic effect of phonon or photon carrying heat in parallel to the electrons. Section \[Sect:Relax\] treats relaxation within the quantum system. Comments on existing literature {#Sect:literature} =============================== There is much interest in using thermoelectric effects to cool fermionic atomic gases [@Grenier2012; @Brantut-Grenier-et-al2013; @Grenier2014], which are hard to cool via other methods. This physics is extremely similar to that in this work, but there is a crucial difference. For the electronic systems that we consider, we can assume the temperatures to be much less than the reservoir’s Fermi energy, and so take all electrons to have the same Fermi wavelength. In contrast, fermionic atomic gases have temperatures of order the Fermi energy, so the high-energy particles in a reservoir have a different wavelength from the low-energy ones. Thus, our results do not apply to atomic gases, although our methodology does[@Grenier2014]. Nonlinear systems and the figure of merit $ZT$ {#Sect:nonlinear+ZT} ---------------------------------------------- Engineers commonly state that wide-ranging applications for thermoelectrics would require them to have a dimensionless figure of merit, $ZT$, greater than three. This dimensionless figure of merit is a dimensionless combination of the linear-response coefficients [@books] $ZT= TGS^2/\Theta$, for temperature $T$, Seebeck coefficient $S$, electrical conductance $G$, and thermal conductance $\Theta$ . Yet for us, $ZT$ is just a way to characterize the efficiency, via $$\begin{aligned} \eta_{\rm eng} = \eta_{\rm eng}^{\rm carnot} {\sqrt{ZT+1} -1 \over \sqrt{ZT+1} +1}, \nonumber\end{aligned}$$ with a similar relationship for refrigerators. Thus, someone asking for a device with a $ZT > 3$, actually requires one with an efficiency of more than one third of Carnot efficiency. This is crucial, because the efficiency is a physical quantity in linear and nonlinear situations, while $ZT$ is only meaningful in the linear-response regime [@Zebarjadi2007; @Grifoni2011; @2012w-pointcont; @Meair-Jacquod2013; @Michelini2014; @Azema-Lombardo-Dare2014]. Linear-response theory rarely fails for bulk semiconductors, even when $T_L$ and $T_R$ are very different. Yet it is completely [*inadequate*]{} for the quantum systems that we consider here. Linear-response theory requires the temperature drop on the scale of the electron relaxation length $l_{\rm rel}$ (distance travelled before thermalizing) to be much less than the average temperature. For a typical millimetre-thick bulk thermoelectric between a diesel motor’s exhaust system ($T_L\simeq 700$K) and its surroundings ($T_R\simeq 280$K), the relaxation length (inelastic scattering length) is of order the mean free path; typically 1-100nm. The temperature drop on this scale is tens of thousands of times smaller than the temperature drop across the whole thermoelectric. This is absolutely tiny compared with the average temperature, so linear-response [@Mahan-Sofo1996] works well, even though $(T_L-T_R)/T_L$ is of order one. In contrast, for quantum systems ($L \ll l_{\rm rel}$), the whole temperature drop occurs on the scale of a few nanometres or less, and so linear-response theory is inapplicable whenever $(T_L-T_R)/T_L$ is not small. Carnot efficiency {#Sect:Carnot} ----------------- A system must be reversible (create no entropy) to have Carnot efficiency; proposals exist to achieve this in bulk [@Mahan-Sofo1996] or quantum [@Humphrey-Linke2005; @Kim-Datta-Lundstrom2009; @Jordan-Sothmann-Sanchez-Buttiker2013] thermoelectric. It requires that electrons only pass between reservoirs L and R at the energy where the occupation probabilities are identical in the two reservoirs [@Humphrey-Linke2005]. Thus, a thermoelectric requires two things to be reversible. Firstly, it must have a $\delta$-function-like transmission [@Mahan-Sofo1996; @Humphrey-Linke2005; @Kim-Datta-Lundstrom2009; @Jordan-Sothmann-Sanchez-Buttiker2013; @Sothmann-Sanchez-Jordan-Buttiker2013], which only lets electrons through at energy ${\epsilon}_0$. Secondly,[@Humphrey-Linke2005] the load’s resistance must be such that ${ e^{\operatorname{-}} }V = {\epsilon}_0 (1-T_R/T_L)$, so the reservoirs’ occupations are equal at ${\epsilon}_0$, see Fig. \[Fig:Fermi\]. By definition this means the current vanishes, and thus so does the power output, $P_{\rm gen}$. However, one can see how $P_{\rm gen}$ vanishes by considering a quantum system which lets electrons through in a tiny energy window $\Delta$ from ${\epsilon}_0$ to ${\epsilon}_0+\Delta$, see Fig \[Fig:tophat-width\]. When we take $\Delta\big/({k_{\rm B}}T_{L,R}) \to 0$, one has Carnot efficiency, however we will see (leading order term in Eq. (\[Eq:Pgen-eng-lowpower\])) that $$\begin{aligned} P_{\rm gen} \propto {1 \over \hbar} \Delta^2, \label{Eq:power-for-Delta-to-zero}\end{aligned}$$ which vanishes as $\Delta\big/({k_{\rm B}}T_{L,R}) \to 0$. Heat-engine efficiency at finite power output and Curzon-Ahlborn efficiency {#Sect:eff-CA} --------------------------------------------------------------------------- To increase the power output beyond that of a reversible system, one has to consider irreversible machines which generate a finite amount of entropy per unit of work generated. Curzon and Ahlborn[@Curzon-Ahlborn1975] popularized the idea of studying the efficiency of a heat-engine running at its maximum power output. For classical pumps, this efficiency is $\eta_{\rm eng}^{\rm CA} = 1- \sqrt{T_L/T_R}$, which is now called the Curzon-Ahlborn efficiency, although already given in Refs. \[\]. As refrigerators, these pumps have an efficiency at maximum cooling power of zero, although Refs. \[\] discuss ways around this. The response of a given heat-engine is typically a “loop” of efficiency versus power (see Fig. \[Fig:summary\]) as one varies the load on the system[@Casati-review]. For a peaked transmission function with width $\Delta$ (see e.g. Fig. \[Fig:tophat-width\]), the loop moves to the left as one reduces $\Delta$. In the limit $\Delta \to 0$, the whole loop is squashed onto the $P_{\rm gen}=0$ axis. In linear-response language, this machine has $ZT \to \infty$. In this limit, the efficiency at maximum power can be very close to that of Curzon and Ahlborn [@Esposito2009-thermoelec] (the star in Fig. \[Fig:summary\]), just as its maximum efficiency can be that of Carnot[@Humphrey-Linke2005] (see previous section). However, its maximum power output is $\propto { e^{\operatorname{-}} }V\Delta/\hbar$ for small $\Delta$ (where $V$ is finite, chosen to ensure maximum power), which vanishes for $\Delta \to 0$, although it is much larger than Eq. (\[Eq:power-for-Delta-to-zero\]). Fig. \[Fig:summary\] shows that a system with larger $\Delta$ (such as the red curve) operating near its maximum efficiency will have both higher efficiency and higher power output than the one with small $\Delta$ (left most grey curve) operating at maximum power. This article shows how to derive the thick black curve in Fig. \[Fig:summary\], thereby showing that there is a fundamental trade-off between efficiency and power output in optimal thermodynamic machines made from thermoelectrics [@footnote:casati-review]. As such, our work overturns the idea that maximizing efficiency at maximum power is the best route to machines with both high efficiency and high power. It also overturns the idea that systems with the narrowest transmission distributions (the largest $ZT$ in linear-response) are automatically the best thermoelectrics. At this point we mention that other works[@Nakpathomkun-Xu-Linke2010; @Leijnse2010; @Meden2013; @Hershfield2013] have studied efficiencies for various systems with finite width transmission functions, for which power outputs can be finite. In particular, Ref. \[\] considered a boxcar transmission function, which is the form of transmission function that we have shown can be made optimal [@2014w-prl]. Pendry’s quantum bound on heat-flow {#Sect:Pendry} ----------------------------------- An essential ingredient in this work is Pendry’s upper bound [@Pendry1983] on the heat-flow through a quantum system between two reservoirs of fermions. He found this bound using a scattering theory of the type discussed in Section \[Sect:ScatteringTheory\] below. It is a concrete example of a general principle due to Bekenstein [@Bekenstein], and the same bound applies in the presence of thermoelectric effects [@2012w-2ndlaw]. The bound on the heat flow out of reservoir $L$ is achieved when all the electrons and holes arriving at the quantum system from reservoir $L$ escape into reservoir $R$ without impediment, while there is no back-flow of electrons or holes from reservoir $R$ to L. The easiest way to achieve this is to couple reservoir $L$ through the quantum system to a reservoir $R$ at zero temperature, and then ensure the quantum system does not reflect any particles. In this case the heat current equals $$\begin{aligned} J^{\rm qb}_L = {\pi^2 \over 6h} N {k_{\rm B}}^2 T_L^2, \label{Eq:Jqb}\end{aligned}$$ where $N$ is the number of transverse modes in the quantum system. We refer to this as the quantum bound (qb) on heat flow, because it depends on the quantum wave nature of the electrons; it depends on $N$, which is given by the cross-sectional area of the quantum system divided by $\lambda_{\rm F}^2$, where $\lambda_{\rm F}$ is the electron’s Fermi wavelength. As such $J_L^{\rm qb}$ is ill-defined within classical thermodynamics. Uniquely defining temperature, heat and entropy {#Sect:Unique} =============================================== ![\[Fig:Unique\] To implement the procedure in Section \[Sect:Unique\], one starts with the circuit unconnected, as in (a), one then connects the circuit, as in (b). After a long time $t_{\rm expt}$, one disconnects the circuit, returning to (a). The circles are the quantum thermoelectrics, as in Fig. \[Fig:thermocouple\]. ](figure3.pdf){width="\columnwidth"} Works on classical thermodynamics have shown that the definition of heat and entropy flows can be fraught with difficulties. For example, the rate of change of entropy cannot always be uniquely defined in classical continuum thermodynamics[@Kern1975; @Day1977; @book:irreversible-thermodyn]. Here the situation is even more difficult, since the electrons within the quantum systems (circles in Fig. \[Fig:thermocouple\]) are not at equilibrium, and so their temperature cannot be defined. Thus, it is crucial to specify the logic which leads to our definitions of temperature, heat flow and entropy flow. Our definition of heat flow originated in Refs. \[\], the rate of change of entropy is then found using the Clausius relation [@Footnote:Sivan-Imry] (see below). To explain these quantities and show they are unambiguous, we consider the following three step procedure for a heat engine. An analogue procedure works for a refrigerator. - [**Step 1.**]{} Reservoir $L$ is initially decoupled from the rest of the circuit (see Fig. \[Fig:Unique\]a), has internal heat energy $Q_L^{(0)}$, and is in internal equilibrium at temperature $T_L^{(0)}$. The rest of the circuit is in equilibrium at temperature $T_R^{(0)}$ with internal heat energy $Q_R^{(0)}$. The internal heat energy is the total energy of the reservoir’s electron gas minus the energy which that gas would have in its ground-state. As such, the internal energy can be written as a sum over electrons and holes, with an electron at energy ${\epsilon}$ above the reservoir’s chemical potential (or a hole at energy ${\epsilon}$ below that chemical potential) contributing ${\epsilon}$ to this internal heat energy. The initial entropies are then $S^{(0)}_i = Q^{(0)}_i \big/ T^{(0)}_i$ for $i=L,R$. - [**Step 2.**]{} We connect reservoir $L$ to the rest of the circuit ( (see Fig. \[Fig:Unique\]b) and leave it connected for a long time $t_{\rm expt}$. While we assume $t_{\rm expt}$ is long, we also assume that the reservoirs are all large enough that the energy distributions within them change very little during time $t_{\rm expt}$. Upon connecting the circuit elements, we assume a transient response during a time $t_{\rm trans}$, after which the circuit achieves a steady-state. We ensure that $t_{\rm expt}\gg t_{\rm trans}$, so the physics is dominated by this steady-state. Even then the flow will be noisy [@Blanter-Buttiker] due to the fact electrons are discrete with probabilistic dynamics. So we also ensure that $t_{\rm expt}$ is much longer than the noise correlation time, so that the noise in the currents is negligible compared to the average currents. - [**Step 3.**]{} After the time $t_{\rm expt}$, we disconnect reservoir $L$ from the rest of the circuit. Again, there will be a transient response, however we assume that a weak relaxation mechanism within the reservoirs will cause the two parts of the circuit to each relax to internal equilibrium (see Fig. \[Fig:Unique\]a). After this one can unambiguously identify the temperature, $T_i$, internal energy $Q_i$ and Clausius entropy $S_i=Q_i\big/ T_i$ of the two parts of the circuit (for $i=L,R$). Since the reservoirs are large, we assume $T_i = T_i^{(0)}$. Thus, we can unambiguously say that the heat-current out of reservoir $i$ [*averaged*]{} over the time $t_{\rm expt}$ is $$\begin{aligned} \langle J_i \rangle = \big (Q^{(0)}_i - Q_i \big) \big/ t_{\rm expt}.\end{aligned}$$ For the above thermocouple, we treat the currents for each thermoelectric separately, writing the heat current out of reservoir $L$ as $J_L+J_{L'}$, where $J_L$ is the heat current from reservoir $L$ into the lower thermoelectric in Fig. \[Fig:thermocouple\] (the filled circle), and $J_{L'}$ is the heat current from reservoir $L$ into the upper thermoelectric in Fig. \[Fig:thermocouple\] (the open circle). Treating each thermoelectric separately is convenient, and also allows one to generalize the results to “thermopiles”, which contain hundreds of thermoelectrics arranged so that they are electrically in series, but thermally in parallel. The average rate of change of entropy in the circuit is $\langle \dot S_{\rm circuit} \rangle = \langle \dot S \rangle +\langle \dot S' \rangle$, where $\langle \dot S \rangle$ is the average rate of change of entropy associated with the lower thermoelectric in Fig. (\[Fig:thermocouple\]), while $\langle \dot S' \rangle$ is that for the upper thermoelectric. Then $$\begin{aligned} \langle \dot S\rangle = \langle \dot S_L \rangle + \langle \dot S_R \rangle = -{\langle J_{\rm L} \rangle \big/ T_L} \,-\, {\langle J_{\rm R} \rangle \big/ T_R}\,, \label{Eq:average-dotS-def}\end{aligned}$$ while $\langle \dot S'\rangle$ is the same with $J_L,J_R,T_R$ replaced by $J_{L'},J_{R'},T_{R'}$. We neglect the entropy of the thermoelectrics and load, by assuming their initial and final state are the same. This will be the case if they are small compared to the reservoirs, so their initial and final states a simply given by the temperature $T_R$. The nonlinear scattering theory in Ref. \[\] captures long-time average currents (usually called the DC response in electronics), such as electrical current $\langle I_i \rangle$ and heat current $\langle J_i \rangle$, see references in Section \[Sect:ScatteringTheory\]. It is believed to be exact for non-interacting particles, and also applies when interactions can be treated in a mean-field approximation (see again section \[Sect:ScatteringTheory\]). A crucial aspect of the scattering theory is that we do not need to describe the non-equilibrium state of the quantum system during step 2. Instead, we need that quantum system’s transmission function, defined in section \[Sect:ScatteringTheory\]. In this article we will [*only*]{} discuss the long-time average of the rates of flows (not the noisy instantaneous flows), and thus will not explicitly indicate the average; so $I_i$, $J_i$ and $\dot S_i$ should be interpreted as $\langle I_i \rangle$, $\langle J_i \rangle$ and $\langle\dot S_i \rangle$. Entropy production {#Sect:entropy-prod} ================== There are little known universal relations between efficiency, power and and entropy production, which follow trivially from the laws of thermodynamics [@Cleuren2012]. Consider the lower thermoelectric in Fig. \[Fig:thermocouple\]a (filled circle), with $J_L$ and $J_R$ being steady-state heat currents into it from reservoir $L$ and R. Then the first law of thermodynamics is $$\begin{aligned} J_R + J_L=P_{\rm gen}, \label{Eq:firstlaw}\end{aligned}$$ where $P_{\rm gen}$ is the electrical power generated. The Clausius relation for the rate of change of total entropy averaged over long times as in Eq. (\[Eq:average-dotS-def\]), is $$\begin{aligned} \dot S = -{J_L \over T_L} + {J_L - P_{\rm gen} \over T_R}, \label{Eq:secondlaw}\end{aligned}$$ where we have used Eq. (\[Eq:firstlaw\]) to eliminate $J_R$. For a heat engine, we take $J_L$ to be positive, which means $T_L > T_R$ and $J_R$ is negative. We use Eq. (\[Eq:eff-eng\]) to replace $J_L$ with $P_{\rm gen}/\eta_{\rm eng}$ in Eq. (\[Eq:secondlaw\]). Then, the rate of entropy production by a heat-engine with efficiency $\eta_{\rm eng}(P_{\rm gen})$ at power output $P_{\rm gen}$ is $$\begin{aligned} \dot S (P_{\rm gen}) &=& {P_{\rm gen} \over T_R} \left({\eta_{\rm eng}^{\rm carnot} \over \eta_{\rm eng}(P_{\rm gen})} -1 \right), \label{Eq:dotS-eng}\end{aligned}$$ where the Carnot efficiency, $\eta_{\rm eng}^{\rm carnot}$, is given in Eq. (\[Eq:Carnot-eng\]). Hence, knowing the efficiency at power $P_{\rm gen}$, tells us the entropy production at that power. Maximizing the former minimizes the latter. For refrigeration, the load in Fig. \[Fig:thermocouple\] is replaced by a power supply, the thermoelectric thus absorbs a power $P_{\rm abs}$ to extract heat from the cold reservoir. We take reservoir $L$ as cold ($T_L < T_R$) , so $J_L$ is positive. We replace $P_{\rm gen}$ by $-P_{\rm abs}$ in Eqs. (\[Eq:firstlaw\],\[Eq:secondlaw\]). We then use Eq. (\[Eq:eff-fri\]) to replace $P_{\rm abs}$ by $J_L/\eta_{\rm fri}$. Then the rate of entropy production by a refrigerator at cooling power $J_L$ is $$\begin{aligned} \dot S (J_L) &=& {J_L \over T_R} \left({1\over \eta_{\rm fri}(J_L)} -{1 \over \eta_{\rm fri}^{\rm carnot}} \right), \label{Eq:dotS-fri}\end{aligned}$$ where the Carnot efficiency, $\eta_{\rm fri}^{\rm carnot}$, is given in Eq. (\[Eq:Carnot-fri\]). Hence knowing a refrigerator’s efficiency at cooling power $J_L$ gives us its entropy production, and we see that maximizing the former minimizes the latter. Eqs. (\[Eq:dotS-eng\],\[Eq:dotS-fri\]) hold for systems modelled by scattering theory, because this theory satisfies the laws of thermodynamics [@Bruneau2012]$^,$[@2012w-2ndlaw]. The rate of entropy production is zero when the efficiency is that of Carnot, but becomes increasingly positive as the efficiency reduces. In this article, we calculate the maximum efficiency for given power output, and then use Eqs. (\[Eq:dotS-eng\],\[Eq:dotS-fri\]) to get the minimum rate of entropy production at that power output. Nonlinear Scattering Theory {#Sect:ScatteringTheory} =========================== This work uses Christen and Büttiker’s nonlinear scattering theory [@Christen-ButtikerEPL96], which treats electron-electron interactions as mean-field charging effects. Refs. \[\] added thermoelectric effects by following works on linear-response [@Engquist-Anderson1981; @Sivan-Imry1986; @Butcher1990]. Particle and heat flows are given by the transmission function, ${\cal T}_{RL}({\epsilon})$, for electrons to go from left ($L$) to right ($R$) at energy ${\epsilon}$, where ${\cal T}_{RL}({\epsilon})$ is a [*self-consistently*]{} determined function of $T_L$, $T_R$ and $V$. In short, this self-consistency condition originates from the fact that electrons injected from the leads change the charge distribution in the quantum system, which in turn changes the behaviour of those injected electrons (via electron-electron interactions). The transmission function can be determined self-consistently with the charge distribution, if the latter is treated in a time-independent mean-field manner (neglecting single electron effects). We note that the same nonlinear scattering theory was also derived for resonant level models [@Humphrey-Linke2005; @Nakpathomkun-Xu-Linke2010] using functional RG to treat single-electron charging effects [@Meden2013]. The scattering theory for the heat current is based on the observation that an electron leaving reservoir $i$ at energy ${\epsilon}$ is carrying heat ${\epsilon}- \mu_i$ out of that reservoir [@Butcher1990], where $\mu_i$ is the reservoir’s chemical potential. Thus, a reservoir is cooled by removing an electron above the Fermi surface, but heated by removing a electron below the Fermi surface. It is convenient to treat empty states below a reference chemical potential (which we define as ${\epsilon}=0$), as “holes”. Then we do not need to keep track of a full Fermi sea of electrons, but only the holes in that Fermi sea. Then the heat-currents out of reservoirs L and R and into the quantum system are $$\begin{aligned} J_L \! &=& \! {1 \over h} \sum_\mu \int_0^\infty {\rm d}{\epsilon}\, ({\epsilon}- \mu{ e^{\operatorname{-}} }V_L) \, {\cal T}^{\mu\mu}_{RL}({\epsilon}) \, \big[f_L^\mu ({\epsilon}) - f_R^\mu ({\epsilon})\big], \nonumber \\ \label{Eq:JL} \\ J_R \! &=& \! {1 \over h} \sum_\mu \int_0^\infty {\rm d}{\epsilon}\, ({\epsilon}- \mu{ e^{\operatorname{-}} }V_R) \, {\cal T}^{\mu\mu}_{RL}({\epsilon}) \, \big[f_R^\mu ({\epsilon}) - f_L^\mu ({\epsilon})\big], \nonumber \\ \label{Eq:JR}\end{aligned}$$ where ${ e^{\operatorname{-}} }$ is the electron charge (${ e^{\operatorname{-}} }<0$), so ${ e^{\operatorname{-}} }V_i$ is the chemical potential of reservoir $i$ measured from the reference chemical potential (${\epsilon}=0$). The sum is over $\mu=1$ for “electron” states (full states above the reference chemical potential), and $\mu=-1$ for “hole” states (empty states below that chemical potential). The Fermi function for particles entering from reservoir $j$, is $$\begin{aligned} f_j^\mu({\epsilon}) = \left(1+\exp\left[({\epsilon}- \mu { e^{\operatorname{-}} }V_j)\big/ ({k_{\rm B}}T_j) \right] \right)^{-1}. \label{Eq:Fermi}\end{aligned}$$ The transmission function, ${\cal T}^{\nu\mu}_{ij}({\epsilon})$, is the probability that a particle $\mu$ with energy ${\epsilon}$ entering the quantum system from reservoir $j$ will exit into reservoir $i$ as a particle $\nu$ with energy ${\epsilon}$. We only allow $\nu=\mu$ here, since we do not consider electron to hole scattering within the quantum system (only common when superconductors are present). Interactions mean that ${\cal T}^{\mu\mu}_{RL}({\epsilon})$, is a [*self-consistently*]{} determined function of $T_L$, $T_R$ $V_L$ and $V_R$. The system generates power $P_{\rm gen} = (V_R-V_L) I_L$, so $$\begin{aligned} P_{\rm gen} \! &=& \! {1\over h} \sum_\mu \int_0^\infty {\rm d}{\epsilon}\ \mu{ e^{\operatorname{-}} }(V_R-V_L)\, {\cal T}^{\mu\mu}_{RL}({\epsilon}) \, \big[f_L^\mu ({\epsilon}) - f_R^\mu ({\epsilon})\big], \nonumber \\ \label{Eq:Pgen}\end{aligned}$$ It is easy to verify that Eqs. (\[Eq:JL\]-\[Eq:Pgen\]) satisfy the first law of thermodynamics, Eq. (\[Eq:firstlaw\]). This theory assumes the quantum system to be relaxation-free, although decoherence is allowed as it does not change the structure of Eqs. (\[Eq:JL\]-\[Eq:Pgen\]). Relaxation is discussed in Section \[Sect:Relax\]. We define the voltage drop as $V=V_R-V_L$. Without loss of generality we take the reference chemical potential to be that of reservoir $L$, so $$\begin{aligned} V_L=0, \qquad V_R=V, \label{Eq:def-V}\end{aligned}$$ then $J_L$ and $P_{\rm gen}$ coincide with Eqs. (8,9) in Ref. \[\]. Numerous works have found the properties of thermoelectric systems from their transmission functions, ${\cal T}_{RL}({\epsilon})$. Linear-response examples include Refs. \[\], while nonlinear responses were considered in Refs. \[\], see Refs. \[\] for recent reviews. However, here we do not ask what is the efficiency of a given system, we ask what is the system that would achieve the highest efficiency, and what is this efficiency? This is similar in spirit to Ref. \[\], except that we maximize the efficiency for given power output. We need to answer this question in the context of the mean-field treatment of electron-electron interactions[@Christen-ButtikerEPL96], in which the transmission function for any given system is the solution of the above mentioned self-consistency procedure. Despite this complexity, any transmission function (including all mean-field interactions) must obey $$\begin{aligned} 0\leq{\cal T}^{\mu\mu}_{RL}({\epsilon})\leq N \ \ \hbox{ for all }{\epsilon}, \label{Eq:basic-limits-on-transmisson}\end{aligned}$$ where $N$ is the number of transverse modes at the narrowest point in the nanostructure, see Fig. \[Fig:thermocouple\]. Let us assume that this is the [*only*]{} constraint on the transmission function. Let us assume that for any given $T_L$, $T_R$ and $V$, a clever physicist could engineer any desired transmission function, so long as it obeys Eq. (\[Eq:basic-limits-on-transmisson\]). Presumably they could do this either by solving the self-consistency equations for ${\cal T}^{\mu\mu}_{RL}({\epsilon})$, or by experimental trial and error. Thus, in this work, we find the ${\cal T}^{\mu\mu}_{RL}({\epsilon})$ which maximizes the efficiency given solely the constraint in Eq. (\[Eq:basic-limits-on-transmisson\]), and get this maximum efficiency. We then rely on future physicists to find a way to construct a system with this ${\cal T}^{\mu\mu}_{RL}({\epsilon})$ (although some hints are given in Section \[Sect:chain\]). From thermoelectric optimization to thermocouple optimization {#Sect:transforming-from-full-to-open} ============================================================= The rest of this article considers optimizing a single thermoelectric. However, an optimal thermocouple heat engine (or refrigerator) consists of two systems with opposite thermoelectric responses (full and open circles in Fig. \[Fig:thermocouple\]). So here we explain how to get the optimal thermocouple from the optimal thermoelectric. Suppose the optimal system between $L$ and $R$ (the full circle) has a given transmission function ${\cal T}_{RL}^{\mu,\mu} ({\epsilon})$, which we will find in Section \[Sect:eng\]. This system generates an electron flow parallel to heat flow (so electric current is anti-parallel to heat flow, implying a negative Peltier coefficient). The system between $L$ and $R'$ (the open circle) must have the opposite response. For this we interchange the role played by electrons and holes compared with ${\cal T}_{RL}^{\mu,\mu} ({\epsilon})$, so the optimal system between $L$ and $R'$ has $$\begin{aligned} {\cal T}_{R'L}^{\mu,\mu} ({\epsilon}) &=& {\cal T}_{RL}^{-\mu,-\mu} ({\epsilon}).\end{aligned}$$ If the optimal bias for the system between $L$ and $R$ is $V$ (which we will also find in Section \[Sect:eng\]), then the optimal bias for the system between $L$ and $R'$ is $-V$. Then the heat flow from reservoir $L$ into $R'$ equals that from $L$ into $R$, while the electrical current from $L$ into $R'$ is opposite to that from $L$ into $R$, and so $P_{\rm gen}$ is the same for each thermoelectric. The load across the thermocouple (the two thermoelectrics) must be chosen such that the bias across the thermocouple is $2V$. The condition that the charge current out of $L$ equals that into $L$ will then ensure that both thermoelectrics are at their optimal bias. In the rest of this article we discuss power output and heat input [*per thermoelectric*]{}. For a thermocouple, one simply needs to multiply these by two, so the efficiency is unchanged but the power output is doubled. Simple estimate of bounds on power output {#Sect:over-estimates} ========================================= One of the principal results of Ref. \[\] is the quantum bounds on the power output of heat-engines and refrigerators. The exact derivation of these bounds is given in Sections \[Sect:qb-eng\] and \[Sect:qb-fri\]. Here, we give simple arguments for their basic form based on Pendry’s limit of heat flow discussed in Section \[Sect:Pendry\] above. For a refrigerator, it is natural to argue that the upper bound on cooling power will be closely related to Pendry’s bound, Eq. (\[Eq:Jqb\]). We will show in Section \[Sect:qb-fri\] that this is the case. A two-lead thermoelectric can extract as much as half of $J^{\rm qb}_L$. In other words, the cooling power of any refrigerator must obey $$\begin{aligned} J_L &\leq& {1 \over 2} J^{\rm qb}_L \ =\ {\pi^2 \over 12h} N {k_{\rm B}}^2 T_L^2.\end{aligned}$$ Now let us turn to a heat-engine operating between a hot reservoir $L$ and cold reservoir $R$. Following Pendry’s logic, we can expect that the heat current into the quantum system from reservoir $L$ cannot be more than $J_L^{\hbox{\scriptsize over-estimate}} ={\pi^2 \over 6h} N {k_{\rm B}}^2 (T_L^2-T_R^2)$. Similarly, no heat engine can exceed Carnot’s efficiency, Eq. (\[Eq:Carnot-eng\]). Thus, we can safely assume any system’s power output is less than $$\begin{aligned} P_{\rm gen}^{\hbox{\scriptsize over-estimate}} &=& \eta_{\rm eng}^{\rm carnot} J_L^{\hbox{\scriptsize over-estimate}} \nonumber \\ &=& {\pi^2 N {k_{\rm B}}^2 (T_L+T_R) (T_L-T_R)^2 \over 6h \ T_L} .\end{aligned}$$ We know this is a significant over-estimate, because maximal heat flow cannot coincide with Carnot efficiency. Maximum heat flow requires ${\cal T}^{\mu\mu}_{RL}({\epsilon})$ is maximal for all ${\epsilon}$ and $\mu$, while Carnot efficiency requires a ${\cal T}^{\mu\mu}_{RL}({\epsilon})$ with a $\delta$-function-like dependence on ${\epsilon}$ (see Section \[Sect:Carnot\]). None the less, the full calculation in Section \[Sect:qb-eng\] shows that the true quantum bound on power output is such that [@footnote:qb2] $$\begin{aligned} P_{\rm gen} &\leq& P_{\rm gen}^{\rm qb2} \,\equiv\, A_0\, {\pi^2 \over h} N {k_{\rm B}}^2 \big(T_L-T_R\big)^2, \quad \quad $$ where $A_0 \simeq 0.0321$. Thus, the simple over-estimate of the bound, $P_{\rm gen}^{\hbox{\scriptsize over-estimate}}$, differs from the true bound $P_{\rm gen}^{\rm qb2}$ by a factor of $(1+T_R/T_L)/(6A_0)$. In other words it over estimates the quantum bound by a factor between 5.19 and 10.38 (that is 5.19 when $T_R=0$ and 10.38 when $T_R=T_L$). This is not bad for such a simple estimate. ![\[Fig:Fermi\] Sketch of Fermi functions $f_L^\mu({\epsilon})$ and $f_L^\mu({\epsilon})$ in Eq. (\[Eq:Fermi\]), when $\mu { e^{\operatorname{-}} }V$ is positive, and $T_L > T_R$. Eq. (\[Eq:Eps0-guess\]) gives the point where the two curves cross, ${\epsilon}_0$.](figure4.pdf){width="0.9\columnwidth"} Guessing the optimal transmission for a heat-engine {#Sect:guess-heat} =================================================== Here we use simple arguments to guess the transmission function which will maximize a heat-engine’s efficiency for a given power output. We consider the flow of electrons from reservoir $L$ to reservoir $R$ (the filled circle Fig. \[Fig:thermocouple\]a, remembering ${ e^{\operatorname{-}} }<0$, so electron flow is in the opposite direction to $I$). To produce power, the electrical current must flow against a bias, so we require ${ e^{\operatorname{-}} }V$ to be positive, with $V$ as in Eq. (\[Eq:def-V\]). Inspection of the integrand of Eq. (\[Eq:Pgen\]) shows that it only gives positive contributions to the power output, $P_{\rm gen}$, when $\mu \big(f^\mu_L({\epsilon}) - f^\mu_R({\epsilon})\big) >0$. From Eq. (\[Eq:Fermi\]), one can show that $f^\mu_L({\epsilon})$ and $f^\mu_R({\epsilon})$ cross at $$\begin{aligned} {\epsilon}_0 = \mu { e^{\operatorname{-}} }V \big/ (1-T_R/T_L), \label{Eq:Eps0-guess}\end{aligned}$$ see Fig. \[Fig:Fermi\]. Since ${ e^{\operatorname{-}} }V$ is positive, we maximize the power output by blocking the transmission of those electrons ($\mu=1$) which have ${\epsilon}< {\epsilon}_0$, and blocking the transmission all holes ($\mu=-1$). For $\mu=1$, all energies above ${\epsilon}_0$ add to the power output. Hence, maximizing transmission for all ${\epsilon}> {\epsilon}_0$ will maximize the power output, giving $P_{\rm gen}=P_{\rm gen}^{\rm qb}$. However, a detailed calculation, such as that in Section \[Sect:eng\], is required to find the $V$ which will maximize $P_{\rm gen}$; remembering that $P_{\rm gen}$ depends directly on $V$ as well as indirectly (via the above choice of ${\epsilon}_0$). Now we consider maximizing the efficiency at a given power output $P_{\rm gen}$, where $P_{\rm gen} < P_{\rm gen}^{\rm qb}$. Comparing the integrands in Eqs. (\[Eq:JL\],\[Eq:Pgen\]), we see that $J_L$ contains an extra factor of energy ${\epsilon}$ compared to $P_{\rm gen}$. As a result, the transmission of electrons ($\mu=1$) with large ${\epsilon}$ enhances the heat current much more than it enhances the power output. This means that the higher an electron’s ${\epsilon}$ is, the less efficiently it contributes to power production. Thus, one would guess that it is optimal to have an upper cut-off on transmission, ${\epsilon}_1$, which would be just high enough to ensure the desired power output $P_{\rm gen}$, but no higher. Then the transmission function will look like a “band-pass filter” (the “boxcar” form in Fig \[Fig:tophat-width\]), with ${\epsilon}_0$ and ${\epsilon}_1$ further apart for higher power outputs. This guess is correct, however the choice of $V$ affects both ${\epsilon}_0$ and ${\epsilon}_1$, so the calculation in Section \[Sect:eng\] is necessary to find the $V$, ${\epsilon}_0$ and ${\epsilon}_1$ which maximize the efficiency for given $P_{\rm gen}$. Maximizing heat-engine efficiency for given power output {#Sect:eng} ======================================================== ![\[Fig:tophat-width\] How the optimal “boxcar” transmission changes with increasing required power output. At maximum power output, a heat engine has ${\epsilon}_1 = \infty$ while ${\epsilon}_0$ remains finite. At maximum cooling power, a refrigerator has ${\epsilon}_1 = \infty$ and ${\epsilon}_0=0$. The qualitative features follow this sketch for all $T_R/T_L$, however the details depend on $T_R/T_L$, see Fig. \[Fig:Delta+V\]. ](figure5.pdf){width="\columnwidth"} Now we present the central calculations of this article, finding the maximum efficiency of a quantum thermoelectric with [*given*]{} power output. In this section we consider heat-engines, while Section \[Sect:fri\] addresses refrigerators. For a heat engine, our objective is to find the transmission function, ${\cal T}^{\mu\mu}_{RL}({\epsilon})$, and bias, $V$, that maximize the efficiency $\eta_{\rm eng}(P_{\rm gen})$ for given power output $P_{\rm gen}$. To do this we treat ${\cal T}^{\mu\mu}_{RL}({\epsilon})$ as a set of many slices each of width $\delta \to 0$, see the sketch in Fig. \[Fig:T-functions\]a. We define $\tau^{\mu}_\gamma$ as the height of the $\gamma$th slice, which is at energy ${\epsilon}_\gamma \equiv \gamma\delta$. Our objective is to find the optimal value of $\tau^{\mu}_\gamma$ for each $\mu,\gamma$, and optimal values of the bias, $V$; all under the constraint of fixed $P_{\rm gen}$. Often such optimization problems are formidable, however this one is fairly straightforward. The efficiency is maximum for a fixed power, $P_{\rm gen}$, if $J_L$ is minimum for that $P_{\rm gen}$. If we make an infinitesimal change of $\tau^{\mu}_\gamma$ and $V$, we note that $$\begin{aligned} \delta P_{\rm gen} &=& \left. {\partial P_{\rm gen} \over \partial \tau^{\mu}_\gamma} \right|_V \delta\tau^{\mu}_\gamma \ +\ P'_{\rm gen} \,\delta V, \label{Eq:deltaPgen} \\ \delta J_L &=& \left. {\partial J_L \over \partial \tau^{\mu}_\gamma} \right|_V \delta\tau^{\mu}_\gamma \ +\ J'_L \,\delta V, \label{Eq:deltaJL}\end{aligned}$$ where $|_x$ indicates that the derivative is taken at constant $x$, and the primed indicates $\partial/\partial V$ for fixed transmission functions. If we want to fix $P_{\rm gen}$ as we change $\tau^{\mu}_\gamma$, we must change the bias $V$ to compensate. For this, we set $\delta P_{\rm gen}=0$ in Eq. (\[Eq:deltaJL\]) and substitute the result for $\delta V$ into Eq. (\[Eq:deltaPgen\]). Then $J_L$ decreases (increasing efficiency) for an infinitesimal increase of $\tau^{\mu}_\gamma$ at fixed $P_{\rm gen}$, if $$\begin{aligned} \left.{\partial J_L \over \partial \tau^{\mu}_\gamma} \right|_{P_{\rm gen}} &=& \left.{\partial J_L \over \partial \tau^{\mu}_\gamma} \right|_V - {J'_L \over P'_{\rm gen}} \left.{\partial P_{\rm gen} \over \partial \tau^{\mu}_\gamma }\right|_V \ <\ 0. \qquad \label{Eq:eng-condition}\end{aligned}$$ Comparing Eq. (\[Eq:JL\]) and Eq. (\[Eq:Pgen\]), one sees that $$\begin{aligned} \left.{\partial J_L\over \partial \tau^{\mu}_\gamma }\right|_V &=& {{\epsilon}_\gamma\over \mu { e^{\operatorname{-}} }V}\, \left.{\partial P_{\rm gen} \over \partial \tau^{\mu}_\gamma }\right|_V . \label{Eq:change-JL-to-change-Pgen}\end{aligned}$$ where $V$ is given in Eq. (\[Eq:def-V\]). Thus, the efficiency $\eta_{\rm eng}(P_{\rm gen})$ grows with a small increase of $\tau^{\mu}_\gamma$ if $$\begin{aligned} \left({\epsilon}_\gamma - \mu { e^{\operatorname{-}} }V {J'_L \over P'_{\rm gen}} \right) \times \left.{\partial P_{\rm gen} \over \partial \tau^{\mu}_\gamma }\right|_V \ <\ 0, \label{Eq:eng-condition2}\end{aligned}$$ where $P_{\rm gen}$, $P'_{\rm gen}$, $J_L$, $J'_L$ and ${ e^{\operatorname{-}} }V$ are positive. ![\[Fig:T-functions\] A completely arbitrary transmission function $ {\cal T}_{RL}^{\mu\mu} ({\epsilon})$ (see Section \[Sect:eng\]). We take it to have infinitely many slices of width $\delta \to 0$, so slice $\gamma$ has energy ${\epsilon}_\gamma \equiv \gamma \delta$ and height $\tau^{\mu}_\gamma$. We find the optimal height for each slice. ](figure6.pdf){width="0.85\columnwidth"} For what follows, let us define two energies $$\begin{aligned} {\epsilon}_0 &=& { e^{\operatorname{-}} }V \big/ (1-T_R/T_L), \label{Eq:eng-bounds-eps0} \\ {\epsilon}_1 &=& { e^{\operatorname{-}} }V \, J'_L / P'_{\rm gen}. \label{Eq:eng-bounds-eps1}\end{aligned}$$ One can see that $ \left.\left({\partial P_{\rm gen}/\partial\tau^{\mu}_\gamma }\right)\right|_V >0$ when both $\mu=1$ and ${\epsilon}> {\epsilon}_0$, and is negative otherwise. Thus, for $\mu=1$, Eq. (\[Eq:eng-condition2\]) is satisfied when ${\epsilon}_\gamma$ is between ${\epsilon}_0$ and ${\epsilon}_1$. For $\mu=-1$, Eq. (\[Eq:eng-condition2\]) is never satisfied. A heat-engine is only useful if $P_{\rm gen}>0$, and this is only true for ${\epsilon}_0 <{\epsilon}_1$. Hence, if $\mu=1$ and ${\epsilon}_0 <{\epsilon}<{\epsilon}_1$, then $\eta_{\rm eng}(P_{\rm gen})$ is maximum for $\tau^{\mu}_\gamma$ at its maximum value, $\tau^{\mu}_\gamma=N$. For all other $\mu$ and ${\epsilon}_\gamma$, $\eta_{\rm eng}(P_{\rm gen})$ is maximum for $\tau^{\mu}_\gamma$ at its minimum value, $\tau^{\mu}_\gamma=0$. Since the left-hand-side of Eq. (\[Eq:eng-condition2\]) is not zero for any ${\epsilon}_\gamma\neq {\epsilon}_0,{\epsilon}_1$, there are no stationary points, which is why $\tau^{\mu}_\gamma$ never takes a value between its maximum and minimum values. Thus, the optimal ${\cal T}^{\mu\mu}_{RL}({\epsilon})$ is a “boxcar” or “top-hat” function, $$\begin{aligned} {\cal T}^{\mu\mu}_{RL}({\epsilon}) \! &=& \! \left\{ \! \begin{array}{cl} N & \hbox{ for } \mu=1 \ \hbox{ \& } \ \ {\epsilon}_0 \! <\! {\epsilon}\! <\! {\epsilon}_1 \phantom{\big|} \\ 0 & \hbox{ otherwise } \phantom{\big|} \end{array} \right. \quad \label{Eq:top-hat}\end{aligned}$$ see Fig. \[Fig:T-functions\]b. It hence acts as a band-pass filter, only allowing flow between L and R for electrons ($\mu=1$) in the energy window between ${\epsilon}_0$ to ${\epsilon}_1$. Substituting a boxcar transmission function with arbitrary ${\epsilon}_0$ and ${\epsilon}_1$ into Eqs. (\[Eq:JL\],\[Eq:Pgen\]) gives $$\begin{aligned} J_L &=& N \, \big[F_L({\epsilon}_0)-F_R({\epsilon}_0)-F_L({\epsilon}_1)+F_R({\epsilon}_1) \big], \label{Eq:JL-eng} \\ P_{\rm gen} \!\! &=& \!N{ e^{\operatorname{-}} }V \, \big[G_L({\epsilon}_0)-G_R({\epsilon}_0) -G_L({\epsilon}_1)+G_R({\epsilon}_1) \big], \qquad \label{Eq:Pgen-eng}\end{aligned}$$ where we define $$\begin{aligned} F_j({\epsilon}) = {1 \over h} \int_{\epsilon}^\infty { x \ {{\rm d}}x \over 1+ \exp\big[(x-{ e^{\operatorname{-}} }V_j)\big/({k_{\rm B}}T_j)\big] }, \label{Eq:Fintegral} \\ G_j({\epsilon}) = {1 \over h}\int_{\epsilon}^\infty {{{\rm d}}x \over 1+ \exp\big[(x-{ e^{\operatorname{-}} }V_j)\big/({k_{\rm B}}T_j)\big] }, \label{Eq:Gintegral}\end{aligned}$$ which are both positive for any ${\epsilon}>0$. Remembering that we took $V_L=0$ and $V_R=V$, these integrals are $$\begin{aligned} F_L({\epsilon}) &=& {\epsilon}G_L({\epsilon}) -{({k_{\rm B}}T_L)^2 \over h} {\rm Li}_2\big[-{{\rm e}}^{-{\epsilon}/({k_{\rm B}}T_L)}\big], \qquad \\ F_R({\epsilon}) &=& {\epsilon}G_R({\epsilon}) -{({k_{\rm B}}T_R)^2 \over h} {\rm Li}_2\big[-{{\rm e}}^{-( {\epsilon}-{ e^{\operatorname{-}} }V)/({k_{\rm B}}T_R)}\big], \qquad \\ G_L({\epsilon}) &=& {{k_{\rm B}}T_L\over h}\ln\big[1+{{\rm e}}^{-{\epsilon}/({k_{\rm B}}T_L)}\big], \\ G_R({\epsilon}) &=& {{k_{\rm B}}T_R\over h}\ln\big[1+{{\rm e}}^{-({\epsilon}-{ e^{\operatorname{-}} }V)/({k_{\rm B}}T_R)}\big],\end{aligned}$$ for dilogarithm function, ${\rm Li}_2(z)= \int_0^\infty t \, dt \big/({{\rm e}}^t/z -1)$. We are only interested in cases where ${\epsilon}_0$ fulfills the condition in Eq. (\[Eq:eng-bounds-eps0\]), in this case $({\epsilon}_0-{ e^{\operatorname{-}} }V)/({k_{\rm B}}T_R) = {\epsilon}_0/({k_{\rm B}}T_L)$, which means $G_R({\epsilon}_0)$ and $F_R({\epsilon}_0)$ are related to $G_L({\epsilon}_0)$ and $F_L({\epsilon}_0)$ by $$\begin{aligned} G_R({\epsilon}_0)&=&{T_R \over T_L}\, G_L({\epsilon}_0), \label{Eq:G_R} \\ F_R({\epsilon}_0)-{\epsilon}_0 G_R({\epsilon}_0)&=&{T_R^2\over T_L^2}\, \left( F_L({\epsilon}_0) -{\epsilon}_0 G_L({\epsilon}_0) \right). \quad\end{aligned}$$ ![\[Fig:bounds\] Solutions of the transcendental equations giving optimal ${\epsilon}_1$ (heat-engine) or ${\epsilon}_0$ (refrigerator). In (a), the red curve is the optimal ${\epsilon}_1(V)$ for ${\epsilon}_1> {\epsilon}_0$, and the thick black line is ${\epsilon}_0$ in Eq. (\[Eq:eng-bounds-eps0\]). The red circle and red arrow indicate the low and high power limits discussed in the text. In (b), the red curve is the optimal ${\epsilon}_0(V)$ for ${\epsilon}_0<{\epsilon}_1$, and the thick black line is ${\epsilon}_1$ in Eq. (\[Eq:fri-bounds-eps1\]). ](figure7.pdf){width="\columnwidth"} Eq. (\[Eq:eng-bounds-eps1\]) tells us that ${\epsilon}_1$ depends on $J_L$ and $P_{\rm gen}$, but that these depend in-turn on ${\epsilon}_1$. Hence to find ${\epsilon}_1$, we substitutes Eqs. (\[Eq:JL-eng\],\[Eq:Pgen-eng\]) into Eq. (\[Eq:eng-bounds-eps1\]) to get a transcendental equation for ${\epsilon}_1$ as a function of $V$ for given $T_R/T_L$. This equation is too hard to solve analytically (except in the high and low power limits, discussed in Sections \[Sect:qb-eng\] and \[Sect:eff-at-given-power\] respectively). The red curve in Fig. \[Fig:bounds\]a is a numerical solution for $T_R/T_L=0.2$. Having found ${\epsilon}_1$ as a function of $V$ for given $T_R/T_L$, we can use Eqs. (\[Eq:JL-eng\],\[Eq:Pgen-eng\]) to get $J_L(V)$ and $P_{\rm gen}(V)$. We can then invert the second relation to get $V(P_{\rm gen})$. At this point we can find $J_L(P_{\rm gen})$, and then use Eq. (\[Eq:eff-eng\]) to get the quantity that we desire — the maximum efficiency at given power output, $\eta_{\rm eng}(P_{\rm gen})$. In Section \[Sect:qb-eng\], we do this procedure analytically for high power ($P_{\rm gen} =P_{\rm gen}^{\rm qb2}$), and in Section \[Sect:eff-at-given-power\], we do this procedure analytically for low power ($P_{\rm gen} \ll P_{\rm gen}^{\rm qb2}$). For other cases, we only have a numerical solution for the transcendental equation for ${\epsilon}_1$ as a function of $V,T_R/T_L$, so we must do everything numerically. ![\[Fig:Delta+V\] (a) Plots of optimal $\Delta$ (left) and ${ e^{\operatorname{-}} }V$ (right) for a heat-engine with given power output, $P_{\rm gen}$, for $T_R/T_L=$ 0.05, 0.1, 0.2, 0.4, 0.6 and 0.8. We get ${\epsilon}_0$ from ${ e^{\operatorname{-}} }V$ by using Eq. (\[Eq:eng-bounds-eps0\]). (b) Plots of optimal $\Delta$ (left) and ${ e^{\operatorname{-}} }V$ (right) for a refrigerator with a given cooling power output, $J_L$, for $T_R/T_L=$ 1.05, 1.2, 1.5, 2, 4 and 10. We get ${\epsilon}_1$ from ${ e^{\operatorname{-}} }V$ by using Eq. (\[Eq:fri-bounds-eps1\]). ](figure8.pdf){width="\columnwidth"} ![\[Fig:allpowers\] Efficiencies of (a) heat-engines and (b) refrigerators. In (a) the curves are the maximum allowed heat-engine efficiency as a function of power outputs for $T_R/T_L= 0.05,0.2,0.4,0.6,0.8$ (from top to bottom). In (b) the curves are the maximum allowed refrigerator efficiency as a function of cooling power for $T_R/T_L= 1.05,1.2,1.5,2,4$ (from top to bottom). In both (a) and (b) the horizontal black lines indicate Carnot efficiency for each $T_R/T_L$, while the dashed black curves are the analytic theory for small cooling power, given in Eq. (\[Eq:eta-eng-small-Pgen\]) or Eq. (\[Eq:eta-fri-smallJ\]). The circles mark the analytic result for maximum power output. ](figure9.pdf){width="\columnwidth"} Fig. \[Fig:Delta+V\]a gives the values of $\Delta=({\epsilon}_1-{\epsilon}_0)$ and ${ e^{\operatorname{-}} }V$ which result from solving the transcendental equation numerically for a variety of different $T_R/T_L$. Eq. (\[Eq:eng-bounds-eps0\]) then relates ${\epsilon}_0$ to ${ e^{\operatorname{-}} }V$. The qualitative behaviour of the resulting boxcar transmission function is shown in Fig. \[Fig:tophat-width\]. This numerical evaluation enables us to find the efficiency as a function of $P_{\rm gen}$ and $T_R/T_L$, which we plot in Fig. \[Fig:allpowers\]a. Quantum bound on heat engine power output {#Sect:qb-eng} ----------------------------------------- Here we want to find the highest possible power output of the heat-engine. In the previous section, we had the power as a function of two independent parameters, $V$ and ${\epsilon}_1$, with ${\epsilon}_0$ given by Eq. (\[Eq:eng-bounds-eps0\]). However, we know that Eq. (\[Eq:eng-bounds-eps1\]) will then determine a line in this two-dimensional parameter space (Fig. \[Fig:bounds\]a), which we can parametrize by the parameter $V$. The maximum possible power corresponds to $P_{\rm gen}'=0$, where we recall $P_{\rm gen}' \equiv {{\rm d}}P_{\rm gen} \big/ {{\rm d}}V$. This has two consequences, the first is that from Eq. (\[Eq:eng-bounds-eps0\]), we see that $P_{\rm gen}'=0$ means that ${\epsilon}_1 \to \infty$. Thus, the transmission function ${\cal T}_{RL}^{\mu\mu}({\epsilon})$, taking the form of a Heaviside step function, $\theta({\epsilon}-{\epsilon}_0)$, where ${\epsilon}_0$ is given in Eq. (\[Eq:eng-bounds-eps0\]). Taking Eq. (\[Eq:Pgen-eng\]) combined with Eq. (\[Eq:G\_R\]) for ${\epsilon}_1 \to \infty$, gives $$\begin{aligned} P_{\rm gen}\big({\epsilon}_1\to\infty\big) &=& N{ e^{\operatorname{-}} }V \, \left(1-{T_R \over T_L}\right)\ G_L\left( {{ e^{\operatorname{-}} }V \over 1-T_R/T_L}\right). \nonumber\end{aligned}$$ The second consequence of $P_{\rm gen}'=0$, is that the $V$-derivative of this expression must be zero. This gives us the condition that $$\begin{aligned} (1+B_0)\ln[1+B_0] +B_0\ln[B_0] =0\end{aligned}$$ where we define $B_0= \exp[-{ e^{\operatorname{-}} }V/({k_{\rm B}}T_L-{k_{\rm B}}T_R)] = \exp[-{\epsilon}_0/({k_{\rm B}}T_L)]$. Numerically solving this equation gives $B_0 \simeq 0.318$. Eq. (\[Eq:eng-bounds-eps0\]) means that this corresponds to ${ e^{\operatorname{-}} }V = -{k_{\rm B}}(T_L-T_R) \ln[0.318]= 1.146 \,{k_{\rm B}}(T_L-T_R)$, indicated by the red arrow in Fig. \[Fig:bounds\]a. Substituting this back into $P_{\rm gen}\big({\epsilon}_1\to\infty\big)$ gives the maximum achievable value of $P_{\rm gen}$, $$\begin{aligned} P_{\rm gen}^{\rm qb2} = A_0\, {\pi^2 \over h} N {k_{\rm B}}^2 \big(T_L-T_R\big)^2 \quad \quad \label{Eq:P-qb2}\end{aligned}$$ with $$\begin{aligned} A_0 \equiv B_0\ln^2[B_0]\big/\big[\pi^2(1+B_0)\big] \simeq 0.0321.\end{aligned}$$ We refer to this as the quantum bound (qb) on power output[@footnote:qb2], because of its origin in the Fermi wavelength of the electrons, $\lambda_{\rm F}$. We see this in the fact that $P_{\rm gen}^{\rm qb2}$ is proportional to the number of transverse modes in the quantum system, $N$, which is given by the cross-sectional area of the quantum system divided by $\lambda_{\rm F}^2$. This quantity has no analogue in classical thermodynamics. The efficiency at this maximum power, $P_{\rm gen}^{\rm qb2}$, is $$\begin{aligned} \eta_{\rm eng} (P_{\rm gen}^{\rm qb2}) &=& \eta_{\rm eng}^{\rm Carnot}\big/ \big( 1+C_0 (1+T_R/T_L) \big), \label{Eq:Eff-at-Pqb2}\end{aligned}$$ with $$\begin{aligned} C_0=-(1+B_0){\rm Li}_2(-B_0)\big/\big(B_0\ln^2[B_0]\big) \simeq 0.936.\end{aligned}$$ As such, it varies with $T_R/T_L$, but is always more than $0.3\,\eta_{\rm eng}^{\rm Carnot}$. This efficiency is less than Curzon and Ahlborn’s efficiency for all $T_R/T_L$ (although not much less). However, the power output here is infinitely larger than the maximum power output of systems that achieve Curzon and Ahlborn’s efficiency, see Section \[Sect:eff-CA\]. The form of Eq. (\[Eq:Eff-at-Pqb2\]) is very different from Curzon and Ahlborn’s efficiency. However, we note in passing that Eq. (\[Eq:Eff-at-Pqb2\]) can easily be written as $\eta_{\rm eng} (P_{\rm gen}^{\rm qb2}) = \eta_{\rm eng}^{\rm carnot} \big/ \left[(1+2C_0) -C_0\eta_{\rm eng}^{\rm carnot}\right]$, which is reminiscent of the efficiency at maximum power found for very different systems (certain classical stochastic heat-engines) in Eq. (31) of Ref. \[\]. Optimal heat-engine at low power output {#Sect:eff-at-given-power} --------------------------------------- Now we turn to the opposite limit, that of low power output, $P_{\rm gen}\ll P_{\rm gen}^{\rm qb2}$, where we expect the maximum efficiency to be close to Carnot efficiency. In this limit, ${\epsilon}_1$ is close to ${\epsilon}_0$. Defining $\Delta= {\epsilon}_1- {\epsilon}_0$, we expand Eqs. (\[Eq:JL-eng\],\[Eq:Pgen-eng\]) in small $\Delta$ up to order $\Delta^3$. This gives $$\begin{aligned} J_L &=& {P_{\rm gen} \over 1-T_R/T_L} \ +\ {N\,\Delta^3\, (1-T_R/T_L) \over 3h\,{k_{\rm B}}T_R} g\!\left(x_0\right), \qquad \label{Eq:JL-eng-lowpower} \\ P_{\rm gen} &=& {N\,{\epsilon}_0 \,\Delta^2 \,(1-T_R/T_L)^2 \over 2h\,{k_{\rm B}}T_R} \nonumber \\ & & \times \bigg[ g\!\left(x_0\right) + {\Delta \, (1+T_R/T_L) \over 3 \, {k_{\rm B}}T_R} \,{{\rm d} g(x_0)\over {\rm d} x_0} \bigg], \label{Eq:Pgen-eng-lowpower}\end{aligned}$$ where Eq. (\[Eq:eng-bounds-eps0\]) was used to write ${ e^{\operatorname{-}} }V$ in terms of ${\epsilon}_0$, and we defined $x_0={\epsilon}_0/({k_{\rm B}}T_L)$, and $g(x)={{\rm e}}^x/(1+{{\rm e}}^x)^2$. Thus, for small $\Delta$ we find that, $$\begin{aligned} \eta_{\rm eng}(\Delta) = \eta_{\rm eng}^{\rm Carnot} \left(1-{2\Delta \over 3x_0{k_{\rm B}}T_L} + \cdots\right). \label{Eq:Efficiency-in-terms-of-Delta}\end{aligned}$$ Eq. (\[Eq:eng-bounds-eps1\]) gives a transcendental equation for $x_0$ and $\Delta$. However, $\Delta$ drops out when it is small, and the transcendental equation reduces to $$\begin{aligned} x_0 \tanh[x_0/2]=3, \label{Eq:transcendental-small-Delta}\end{aligned}$$ for which $x_0 \equiv {\epsilon}_0/({k_{\rm B}}T_L) \simeq 3.24$. Eq. (\[Eq:eng-bounds-eps0\]) means that this corresponds to ${ e^{\operatorname{-}} }V =3.24 \,{k_{\rm B}}(T_L-T_R)$, indicated by the circle in Fig. \[Fig:bounds\]a. Now we can use Eq. (\[Eq:Pgen-eng-lowpower\]) to lowest order in $\Delta$, to rewrite Eq. (\[Eq:Efficiency-in-terms-of-Delta\]) in terms of $P_{\rm gen}$. This gives the efficiency for small $P_{\rm gen}$ as, $$\begin{aligned} \eta_{\rm eng} \big(P_{\rm gen}\big) = \eta_{\rm eng}^{\rm Carnot} \left(1- 0.478 \sqrt{ {T_R \over T_L} \ {P_{\rm gen} \over P_{\rm gen}^{\rm qb2}}} \,+ \cdots \right)\!, \quad \label{Eq:eta-eng-small-Pgen}\end{aligned}$$ where the dots indicate terms of order $(P_{\rm gen} /P_{\rm gen}^{\rm qb2})$ or higher. Eq. (\[Eq:dotS-eng\]) then gives the minimum rate of entropy production at power output $P_{\rm gen}$, $$\begin{aligned} \dot S \big(P_{\rm gen}\big) = 0.478{P_{\rm gen}^{\rm qb2} \over \sqrt{T_LT_R} } \left( {P_{\rm gen} \over P_{\rm gen}^{\rm qb2}}\right)^{3/2} \,+ {\cal O}[P_{\rm gen}^2] , \quad \label{Eq:dotS-eng-small-Pgen}\end{aligned}$$ Thus, the maximal efficiency at small $P_{\rm gen}$ is that of Carnot minus a term that grows like $P_{\rm gen}^{1/2}$ (dashed curves in Fig. \[Fig:allpowers\]a), and the associated minimal rate of entropy production goes like $P_{\rm gen}^{3/2}$. Note that Eq. (\[Eq:Efficiency-in-terms-of-Delta\]), shows that Carnot efficiency occurs at any $x_0$ (i.e. any ${\epsilon}_0$) when $\Delta$ is strictly zero (and so $P_{\rm gen}$ is strictly zero). However, for arbitrary $x_0$ the factor 0.478 in Eq. (\[Eq:eta-eng-small-Pgen\]) is replaced by $\sqrt{ 8\pi^2 A_0/[9 x_0^3 g(x_0)]}$. The value of $x_0$ that satisfied Eq. (\[Eq:transcendental-small-Delta\]) is exactly the one which minimizes this prefactor (its minimum being 0.478), and thus maximizes the efficiency for any small but finite $P_{\rm gen}$. Guessing the optimal transmission for a refrigerator {#Sect:guess-fri} ==================================================== Here we use simple arguments to guess the transmission function which maximizes a refrigerator’s efficiency for given cooling power. The arguments are similar to those for heat-engines (Section \[Sect:guess-heat\]), although some crucial differences will appear. We consider the flow of electrons from reservoir $L$ to reservoir $R$ (the filled circle in Fig. \[Fig:thermocouple\]a, remembering ${ e^{\operatorname{-}} }<0$ so electrons flow in the opposite direction to $I$). To refrigerate, the thermoelectric must absorb power, so the electrical current must be due to a bias, this requires ${ e^{\operatorname{-}} }V$ to be negative, with $V$ as in Eq. (\[Eq:def-V\]). Inspection of the integrand of Eq. (\[Eq:JL\]) shows that it only gives positive contributions to the cooling power output, $J_L$, when $\big(f^\mu_L({\epsilon}) - f^\mu_R({\epsilon})\big) >0$. Since $T_L< T_R$ and ${ e^{\operatorname{-}} }V<0$, we can use Eq. (\[Eq:Fermi\]) to show that this is never true for holes ($\mu=-1$), and is only true for electrons ($\mu=1$) with energies ${\epsilon}< {\epsilon}_1$, where $$\begin{aligned} {\epsilon}_1 = -{ e^{\operatorname{-}} }V \big/ (T_R/T_L-1). \label{Eq:Eps1-guess}\end{aligned}$$ Thus, it is counter-productive to allow the transmission of electrons with ${\epsilon}> {\epsilon}_1$, or the transmission of any holes. Note that this argument gives us an [*upper*]{} cut-off on electron transmission energies, despite the fact it gave a [*lower*]{} cut-off for the heat engine (see Eq. (\[Eq:Eps0-guess\]) and the text around it). All electron ($\mu=1$) energies from zero to ${\epsilon}_1$ contribute positively to the cooling power $J_L$. To maximize the cooling power, one needs to maximize $\big(f^\mu_L({\epsilon}) - f^\mu_R({\epsilon})\big)$, this is done by taking ${ e^{\operatorname{-}} }V \to -\infty$ , for which ${\epsilon}_1 \to \infty$. This logic gives the maximum cooling power, which Section \[Sect:fri\] will show equals ${{\textstyle{\frac{1}{2}}}}J_L^{\rm qb}$. Now we consider maximizing the efficiency at a given cooling output $J_L$, when $J_L <{{\textstyle{\frac{1}{2}}}}J_L^{\rm qb}$. Comparing the integrands in Eqs. (\[Eq:JL\],\[Eq:Pgen\]), we see that the extra factor of ${\epsilon}$ in $J_L$, means that allowing the transmission of electrons at low energies has a small effect on cooling power, while costing a similar electrical power as higher energies. Thus, it would seem to be optimal to have a lower cut-off on transmission, ${\epsilon}_0$, which would be just low enough to ensure the desired cooling power $J_L$, but no lower. Then the transmission function will acts as a “band-pass filter” (the “box-car” in Fig \[Fig:tophat-width\]), with ${\epsilon}_0$ and ${\epsilon}_1$ further apart for higher cooling power. This is correct, however the choice of $V$ affects ${\epsilon}_0$ and ${\epsilon}_1$, so the calculation in Section \[Sect:fri\] is necessary to find the $V$, ${\epsilon}_0$ and ${\epsilon}_1$ which maximize the efficiency for cooling power $J_L$. Maximizing refrigerator efficiency for given cooling power {#Sect:fri} ========================================================== Here we find the maximum refrigerator efficiency, also called the coefficient of performance (COP), for given cooling power $J_L$. The method is very similar to that for heat-engines, and here we mainly summarize the differences. The refrigerator efficiency increases for a fixed cooling power, $J_L$, if the electrical power absorbed $P_{\rm abs}=-P_{\rm gen}$ decreases for fixed $J_L$. This is so if $$\begin{aligned} \left.{\partial P_{\rm abs} \over \partial \tau^{\mu}_\gamma }\right|_{J_L} &=& \left.{\partial P_{\rm abs} \over \partial \tau^{\mu}_\gamma }\right|_V - {P'_{\rm abs} \over J'_L} \left.{\partial J_L \over \partial \tau^{\mu}_\gamma }\right|_V \ <\ 0. \qquad \label{Eq:fri-condition}\end{aligned}$$ where we recall that the primed means $({{\rm d}}/ {{\rm d}}V)$. This is nothing but Eq. (\[Eq:eng-condition\]) with $J_L \to P_{\rm abs}$ and $P_{\rm gen} \to J_L$. Using Eq. (\[Eq:change-JL-to-change-Pgen\]), we see that $\eta_{\rm fri}(J_L)$ grows with $\tau^{\mu}_\gamma$ for $$\begin{aligned} \left( {-\mu { e^{\operatorname{-}} }V \over {\epsilon}_\gamma} - {P'_{\rm abs} \over J'_L} \right) \times \left.{\partial J_L \over \partial \tau^{\mu}_\gamma }\right|_V \ <\ 0, \label{Eq:fri-condition2}\end{aligned}$$ where $P_{\rm abs}$, $P'_{\rm abs}$, $J_L$, $J'_L$ and $-{ e^{\operatorname{-}} }V$ are all positive. To proceed we define the following energies $$\begin{aligned} {\epsilon}_0 &=& -{ e^{\operatorname{-}} }V \,J'_L / P'_{\rm abs}, \label{Eq:fri-bounds-eps0} \\ {\epsilon}_1 &=& {-{ e^{\operatorname{-}} }V \big/ (T_R/T_L-1)}. \label{Eq:fri-bounds-eps1}\end{aligned}$$ Then one can see that $ \left.\left({\partial J_L/\partial \tau^{\mu}_\gamma}\right)\right|_V$ is positive when both $\mu=1$ and ${\epsilon}< {\epsilon}_1^{\rm fri}$, and is negative otherwise. Thus, for $\mu=-1$, Eq. (\[Eq:fri-condition2\]) is never satisfied. For $\mu=1$, Eq. (\[Eq:fri-condition2\]) is satisfied when ${\epsilon}_\gamma$ is between ${\epsilon}_0^{\rm fri}$ and ${\epsilon}_1^{\rm fri}$. A refrigerator is only useful if $J_L>0$ (i.e. it removes heat from the cold reservoir), and this is only true for ${\epsilon}_0^{\rm fri} <{\epsilon}_1^{\rm fri}$. Hence, if $\mu=1$ and ${\epsilon}_0^{\rm fri} <{\epsilon}<{\epsilon}_1^{\rm fri}$, then $\eta_{\rm fri}(J_L)$ grows upon increasing $\tau^{\mu}_\gamma$. Thus, the optimum is when such $\tau^{\mu}_\gamma=N$. For all other $\mu$ and ${\epsilon}_\gamma$, $\eta_{\rm fri}(J_L)$ grows upon decreasing $\tau^{\mu}_\gamma$. Thus, the optimum is when such $\tau^{\mu}_\gamma=0$. This gives the boxcar transmission function in Eq. (\[Eq:top-hat\]), with ${\epsilon}_0$ and ${\epsilon}_1$ given by Eqs. (\[Eq:fri-bounds-eps0\],\[Eq:fri-bounds-eps1\]). Comparing with Eqs. (\[Eq:eng-bounds-eps0\],\[Eq:eng-bounds-eps1\]), we see these energies are the opposite way around for a refrigerator compared to a heat-engine (up to a minus sign). Substituting Eqs. (\[Eq:JL-eng\],\[Eq:Pgen-eng\]) into Eq. (\[Eq:fri-bounds-eps0\]), one gets a transcendental equation for ${\epsilon}_0$ as a function of $V$ for given $T_R/T_L$. This equation is too hard to solve analytically (except in the high and low power limits, discussed in Sections \[Sect:qb-fri\] and \[Sect:lowpower-fri\]). The red curve in Fig. \[Fig:bounds\]b is a numerical solution for $T_R/T_L=1.5$. Having found ${\epsilon}_0$ as a function of $V$ for given $T_R/T_L$, we can use Eqs. (\[Eq:JL-eng\],\[Eq:Pgen-eng\]) to get $J_L(V)$ and $P_{\rm abs}(V)=-P_{\rm gen}(V)$. We can then invert the first relation to get $V(J_L)$. Now, we can find $P_{\rm abs}(J_L)$, and then use Eq. (\[Eq:eff-fri\]) to get the quantity that we desire — the maximum efficiency (or COP), $\eta_{\rm fri}(J_L)$, at cooling power $J_L$. Fig. \[Fig:Delta+V\]b gives the values of $\Delta=({\epsilon}_1-{\epsilon}_0)$ and ${ e^{\operatorname{-}} }V$ which result from solving the transcendental equation numerically. As noted, ${\epsilon}_1$ is related to ${ e^{\operatorname{-}} }V$ by Eq. (\[Eq:fri-bounds-eps1\]). The qualitative behaviour of the resulting boxcar transmission function is sketched in Fig. \[Fig:tophat-width\]. This numerical evaluation enables us to find efficiency as a function of $J_L$ and $T_R/T_L$, which we plot in Fig. \[Fig:allpowers\]b. Quantum bound on refrigerator cooling power {#Sect:qb-fri} ------------------------------------------- To find the maximum allowed cooling power, $J_L$, we look for the place where $J'_L=0$. From Eq. (\[Eq:fri-bounds-eps0\]) we see that this immediately implies ${\epsilon}_0 =0$. Taking Eq. (\[Eq:JL-eng\]) with ${\epsilon}_0=0$, we note by using Eq. (\[Eq:Fintegral\]) that $F_L(0)-F_R(0)$ grows monotonically as one takes $-{ e^{\operatorname{-}} }V \to \infty$. Similarly, for ${\epsilon}_1$ given by Eq. (\[Eq:eng-bounds-eps1\]), we note by using Eq. (\[Eq:Fintegral\]) and $T_R > T_L$ that $F_R({\epsilon}_1)-F_L({\epsilon}_1)$ grows monotonically as one takes $-{ e^{\operatorname{-}} }V \to \infty$. Thus, we can conclude that $J_L$ is maximal for $-{ e^{\operatorname{-}} }V \to \infty$, which implies ${\epsilon}_1 \to \infty$ via Eq. (\[Eq:fri-bounds-eps1\]). Physically, this corresponds to all electrons arriving at the quantum system from reservoir $L$ being transmitted into reservoir $R$, but all holes arriving from reservoir $L$ being reflected back into reservoir $L$. At the same time, reservoir $R$ is so strongly biased that it has no electrons with ${\epsilon}>0$ (i.e. no electrons above reservoir $L$’s chemical potential) to carry heat from R to L. In this limit, $F_L({\epsilon}_1)=F_L({\epsilon}_1)=F_R({\epsilon}_0)=0$, so the maximal refrigerator cooling power is $$\begin{aligned} J_L = {\pi^2 \over 12 h} N {k_{\rm B}}^2 T_L^2 , \label{Eq:J-qb-fri}\end{aligned}$$ where we used the fact that ${\rm Li}_2[1] = \pi^2/12$. This is exactly half the quantum bound on heat current that can flow out of reservoir $L$ given in Eq. (\[Eq:Jqb\]). The quantum bound is achieved by coupling reservoir $L$ to another reservoir with a temperature of absolute zero, through an contact with $N$ transverse mode. By definition a refrigerator is cooling reservoir $L$ below the temperature of the other reservoirs around it. In doing so, we show its cooling power is always less than or equal to $J_L^{\rm qb}/2$. However, it is intriguing that the maximum cooling power is independent of the temperature of the environment, $T_R$, of the reservoir being cooled (reservoir $L$). In short, the best refrigerator can remove all electrons (or all holes) that reach it from reservoir $L$, but it cannot remove all electrons [*and*]{} all holes at the same time. It is easy to see that the efficiency of the refrigerator (COP) at this maximum possible cooling power is zero, simply because $|V| \to \infty$, so the power absorbed $P_{\rm abs} \to \infty$. However, one gets exponentially close to this limit for $-{ e^{\operatorname{-}} }V \gg {k_{\rm B}}T_R$, for which $P_{\rm abs}$ is large but finite, and so $\eta_{\rm fri}(J_L)$ remains finite (see Fig. \[Fig:allpowers\]b). Optimal refrigerator at low cooling power {#Sect:lowpower-fri} ----------------------------------------- Now we turn to the opposite limit, that of low cooling power output, $J_L\ll J_L^{\rm qb}$, where we expect the maximum efficiency to be close to Carnot efficiency. In this limit, ${\epsilon}_0$ is close to ${\epsilon}_1$. Defining $\Delta= {\epsilon}_1- {\epsilon}_0$, we expand Eqs. (\[Eq:JL-eng\],\[Eq:Pgen-eng\]) in small $\Delta$ up to order $\Delta^3$. This gives $$\begin{aligned} J_L &=& {P_{\rm abs} \over T_R/T_L-1} \ -\ {N\,\Delta^3 \,(T_R/T_L-1) \over 3h\,{k_{\rm B}}T_R} g\!\left(x_1\right), \qquad \label{Eq:JL-fri-lowpower} \\ P_{\rm abs}&=& {N\,{\epsilon}_1 \,\Delta^2 \,(T_R/T_L-1)^2 \over 2h\,{k_{\rm B}}T_R} \nonumber \\ & & \times \bigg[ g\!\left(x_1\right) - {\Delta \, (T_R/T_L+1) \over 3 \, {k_{\rm B}}T_R} \,{{\rm d} g(x_1)\over {\rm d} x_1} \bigg], \label{Eq:Pabs-fri-lowpower}\end{aligned}$$ where Eq. (\[Eq:fri-bounds-eps1\]) was used to write ${ e^{\operatorname{-}} }V$ in terms of ${\epsilon}_1$, and we define $x_1={\epsilon}_1/({k_{\rm B}}T_L)$, and $g(x)={{\rm e}}^x/(1+{{\rm e}}^x)^2$. Thus, for small $\Delta$ we find that the efficiency is $$\begin{aligned} \eta_{\rm fri}(\Delta) = \eta_{\rm fri}^{\rm Carnot} \left(1-{2\Delta \over 3x_1{k_{\rm B}}T_L} + \cdots\right). \label{Eq:COP-in-terms-of-Delta}\end{aligned}$$ Note that this is the same Eq. (\[Eq:Efficiency-in-terms-of-Delta\]) for the heat-engine at low power output, except that $x_0$ is replaced by $x_1$, and the Carnot efficiency is that of the refrigerator rather than that of the heat-engine. Eq. (\[Eq:fri-bounds-eps0\]) gives a transcendental equation for $x_1$ and $\Delta$. However, $\Delta$ drops out when it is small, and the transcendental equation reduces to $$\begin{aligned} x_1 \tanh [x_1/2]=3, \label{Eq:condition-fri-smallJ}\end{aligned}$$ for which $x_1\equiv {\epsilon}_1/({k_{\rm B}}T_L) = 3.2436\cdots$. Again this is the same as for a heat-engine, Eq. (\[Eq:transcendental-small-Delta\]), but with $x_1$ replacing $x_0$. Eq. (\[Eq:fri-bounds-eps1\]) means that this corresponds to $-{ e^{\operatorname{-}} }V =3.2436 \,{k_{\rm B}}(T_R-T_L)$, indicated by the circle in Fig. \[Fig:bounds\]b. Now we can use Eq. (\[Eq:JL-fri-lowpower\]) to lowest order in $\Delta$, to rewrite Eq. (\[Eq:COP-in-terms-of-Delta\]) in terms of $J_L$. This gives the efficiency (or coefficient of performance, COP) for small $J_L$ as, $$\begin{aligned} \eta_{\rm fri}(J_L) = \eta_{\rm fri}^{\rm Carnot} \left(1- 1.09 \sqrt{ \,{T_R \over T_R-T_L}\ {J_L \over J_L^{\rm qb}} }\, + \cdots \right)\!, \nonumber \\ \label{Eq:eta-fri-smallJ}\end{aligned}$$ where the dots indicate terms of order $(J_L/J_L^{\rm qb})$ or higher. Eq. (\[Eq:dotS-fri\]) gives the minimum rate of entropy generation at cooling power output $J_L$, as $$\begin{aligned} \dot S \big(J_L\big) = 1.09 {J^{\rm qb}_L \over T_L}\sqrt{1-{T_L\over T_R}} \left({J_L \over J_L^{\rm qb}} \right)^{3/2}\, + {\cal O}[J_L^2], \nonumber \\ \label{Eq:dotS-fri-smallJ}\end{aligned}$$ Thus, we conclude that the maximum efficiency at small $J_L$ is that of Carnot minus a term that grows like $J_L^{1/2}$ (dashed curves in Fig. \[Fig:allpowers\]b), while the associated minimum entropy production goes like $J_L^{3/2}$. We note that Carnot efficiency occurs at $J_L =0$ at any $x_1={\epsilon}_1/({k_{\rm B}}T_L)$. However, then the 1.09 factor in Eq. (\[Eq:eta-fri-smallJ\]) becomes $\sqrt{ 4\pi^2/[27 x_1^3 g(x_1)]}$. The condition in Eq. (\[Eq:condition-fri-smallJ\]) minimizes this factor (the minimum being 1.09), and thereby maximizes the efficiency for given $J_L$. Implementation with a chain of quantum systems {#Sect:chain} ============================================== ![\[Fig:band\] (a) A chain of single level quantum dots with their energy levels aligned at energy $E_0$. (b) Transmission function when all hoppings are equal (note the strong oscillations). (c) Transmission function when all hoppings are carefully chosen (see text). To aid comparison all bandwidths in the plots have been normalized. ](figure10.pdf){width="\columnwidth"} The previous sections have shown that maximum efficiency (at given power output) occurs when the thermoelectric system has a boxcar transmission function with the right position and width. In the limit of maximum power, the boxcar becomes a Heaviside step-function. Here, we give a detailed recipe for engineering such transmission functions for non-interacting electrons, and then discuss how to include mean-field interaction effects. A Heaviside step-function is easily implemented with point-contact, whose transmission function is [@Buttiker-pointcont], $$\begin{aligned} {\cal T}_{\rm L,isl}({\epsilon}) = \left(1+ \exp \left[- {{\epsilon}-E(V) \over D_{\rm tunnel} }\right] \right)^{-1} \label{Eq:transmission-pc}\end{aligned}$$ where $E(V)$ is the height of the energy barrier induced by the point contact, and $D_{\rm tunnel}$ is a measure of tunnelling through the point contact. A sufficiently long point contact exhibits negligible tunnelling, $D_{\rm tunnel} \to 0$, so the transmission function simplifies to the desired Heaviside step-function, $\theta[{\epsilon}-E(V)]$. For a potential implementation of a boxcar function we consider a chain of sites (quantum dots or molecules) with one level per site, as sketched in Fig. \[Fig:band\]a. The objective is that the hoppings between sites, $\{t_i\}$, will cause the states to hybridize to form a band centred at $E_0$, with a width given by the hopping[@Buttiker-private-comm]. Neglecting electron-electron interactions, the hopping Hamiltonian for five sites in the chain ($k=5$) can be written as $$\begin{aligned} {\cal H}_{\rm chain} = \left(\begin{array}{ccccc} -{{\rm i}}a_0 /2 \ & t_1 & 0 & 0 & 0 \\ t_1 & 0 & t_2 & 0 & 0 \\ 0 & t_2 & 0 & t_3 & 0 \\ 0 & 0 & t_3 & 0 & t_4 \\ 0 & 0 & 0 & t_4 & \ -{{\rm i}}a_0/2 \end{array}\right).\end{aligned}$$ This is easily generalized to arbitrary chain length, $k$. Here we treat $a_0$ as a phenomenological parameter, however in reality it would be given by $|t_0|^2$ multiplied by the density of states in the reservoir. The fact that particles escape from the chain into the reservoirs, means the wavefunction for any given particle in the chain will decay with time. To model this, the Hamiltonian must be non-Hermitian, with the non-Hermiticity entering in the matrix elements for coupling to the reservoirs (top-left and bottom right matrix elements). These induce an imaginary contribution to each eigenstate’s energy $E_i$, with the wavefunction of any eigenstate decaying at a rate given by the imaginary part of $E_i$. The non-Hermiticity of $ {\cal H}_{\rm chain}$ also means that its left and right eigenvectors are different, defining $\big | \psi_i^{\rm (r)}\big \rangle$ as the $i$th right eigenvector of the matrix ${\cal H}_{\rm chain}$, and $\big\langle\psi_i^{\rm (l)} \big|$ as the $i$th left eigenvector, we have $\big\langle\psi_i^{\rm (l)} \big | \psi_j^{\rm (r)}\big \rangle = \delta_{ij}$ and $\big\langle\psi_i^{\rm (l)} \big | {\cal H}_{\rm chain} \big | \psi_i^{\rm (r)}\big \rangle = E_i$. The resolution of unity is $\sum_{i} \big | \psi_i^{\rm (r)}\big \rangle \, \big\langle\psi_i^{\rm (l)} \big | = {\bm 1}$, where ${\bm 1}$ is the $k$-by-$k$ unit matrix. We define $|1\rangle$ as the vector whose first element is one while all its other elements are zero, and $|k\rangle$ as the vector whose last element (the $k$th element) is one while all its other elements are zero. Then the transmission probability at energy ${\epsilon}$ is given by $$\begin{aligned} {\cal T}_{RL}({\epsilon}) &=& \left| \big\langle k \big| \ \left[{\epsilon}-{\cal H}_{\rm chain}\right]^{-1} \big| 1 \big\rangle \right|^2 \ a_0 \, ,\end{aligned}$$ where $[\cdots]^{-1}$ is a matrix inverse. To evaluate this matrix inverse, we introduce a resolution of unity to the left and right of $\left[{\epsilon}-{\cal H}_{\rm chain}\right]^{-1}$. This gives $$\begin{aligned} {\cal T}_{RL} &=& \sum_i \ \left|{\big\langle k \big | \psi_i^{\rm (r)}\big\rangle \ \big\langle\psi_i^{\rm (l)} \big| 1 \big\rangle \over {\epsilon}-E_i} \right|^2 \ a_0. \label{Eq:T-for-chain}\end{aligned}$$ For any given set of hoppings $a_0, t_1,\cdots t_k$, one can easily use a suitable eigenvector finder (we used Mathematica) to evaluate this equation numerically, while an analytic solution is straight-forward[@Grenier-private] for $k\leq 3$. When all hoppings in the chain are equal, there is a mismatch between the electron’s hopping dynamics in the chain and their free motion in the reservoirs. This causes resonances in the transmission, giving the Fabry-Perot-type oscillations in Fig. \[Fig:band\]b for $k=5$. However, we can carefully tune the hoppings (to be smallest in the middle of the chain and increasing towards the ends) to get the smooth transmission functions in Fig. \[Fig:band\]c. The $k=5$ curve in Fig. \[Fig:band\]c has $t_1=t_4= 0.39a_0$ and $t_2=t_3= 0.28 a_0$, and we choose $a_0= 1.91$ to normalize the band width to 1. As the number of sites in the chain, $k$, increases, the transmission function tends to the desired boxcar function. The above logic assumes no electron-electron interactions. When we include interaction effects at the mean-field level, things get more complicated. If the states in the chain are all at the same energy $E_0$ when the chain is unbiased, they will not be aligned when there is a bias between the the reservoirs, because the reservoirs also act as gates on the chain states. To engineer a chain where the energies are aligned at the optimal bias, one must adjust the confinement potential of the dots in the chain (or adjust the chemistry of the molecules in the chain) so that their energies are sufficiently out of alignment at zero bias that they all align at optimal bias. In principle, we have the control to do this. However, in practice it would require a great deal of trial-and-error experimental fine tuning. We do not enter further into such practical issues here. Rather, we use the above example to show that there is no [*fundamental*]{} reason that the bound on efficiency cannot be achieved. Many quantum systems in parallel {#Sect:in-parallel} ================================ To increase the efficiency at given power output, one must increase the number of transverse modes, $N$. This is because the efficiency decays with the power output divided by the quantum bounds in Eqs. (\[Eq:P-qb2\],\[Eq:J-qb-fri\]), and these bounds go like $N$. However, a strong thermoelectric response requires a transmission function that is highly energy dependent, this typically only occurs when the quantum system (point-contact, quantum dot or molecule) has dimensions of about a wavelength, which implies that $N$ is of order one. Crucial exceptions (beyond the scope of this work) are systems containing superconductors, either SNS structures[@Pekola-reviews] or Andreev interferometers [@Chandra98] (see also Ref. \[\] and references therein), where strong thermoelectric effects occur for large $N$. In the absence of a superconductor, the only way to get large $N$ is to construct a device consisting of many $N=1$ systems in parallel, such as a surface covered with a certain density of such systems [@Jordan-Sothmann-Sanchez-Buttiker2013; @Sothmann-Sanchez-Jordan-Buttiker2013]. In this case $P_{\rm gen}^{\rm qb2}$ and $J_{\rm L}^{\rm qb}$ in Eqs. (\[Eq:P-qb2\],\[Eq:J-qb-fri\]) become bounds on the power per unit area, with $N$ being replaced by the number of transverse modes per unit area. With this one modification, all calculations and results in this article can be applied directly to such a situation. Carnot efficiency is achieved for a large enough surface area that the power per unit area is much less than $P_{\rm gen}^{\rm qb2}$ and $J_{\rm L}^{\rm qb}$. It is worth noting that the number of modes per unit area cannot exceed $\lambda_{\rm F}^{-2}$, for Fermi wavelength $\lambda_{\rm F}$. From this we can get a feeling for the magnitude of the bounds discussed in this article. Take a typical semiconductor thermoelectric (with $\lambda_{\rm F}\sim 10^{-8}$m), placed between reservoirs at 700 K and 300 K (typical temperatures for a thermoelectric recovering electricity from the heat in the exhaust gases of a diesel engined car). Eq. (\[Eq:P-qb2\]) tells us that to get 100 W of power output from a semiconductor thermoelectric one needs a cross section of at least 4 mm$^2$. Then Eq. (\[Eq:eta-eng-small-Pgen\]) tells us that to get this power at 90% of Carnot efficiency, one needs a cross section of at least 0.4 cm$^2$. Remarkably, it is [*quantum mechanics*]{} which gives these bounds, even though the cross sections in question are macroscopic. Phonons and photons carrying heat in parallel with electrons {#Sect:ph} ============================================================ ![\[Fig:thermocouple-phonons\] The thermocouple heat-engine in Fig. \[Fig:thermocouple\], showing the heat flow due to phonon and photons, which carry heat from hot to cold by all possible routes (in parallel with the heat carried by the electrons). This always reduces the efficiency, so it should be minimized with suitable thermal insulation.](figure11.pdf){width="\columnwidth"} Any charge-less excitation (such as phonons or photons) will carry heat from hot to cold, irrespective of the thermoelectric properties of the system. While some of the phonons and photons will flow through the thermoelectric quantum system, most will flow via other routes, see Fig. \[Fig:thermocouple-phonons\]. A number of theories for these phonon or photon heat currents take the form $$\begin{aligned} J_{\rm ph}= \alpha (T_L^\kappa-T_R^\kappa), \label{Eq:J_ph}\end{aligned}$$ where $J_{\rm ph}$ is the heat flow out of the L reservoir due to phonons or photons. The textbook example of such a theory is that of black-body radiation between the two reservoirs, then $\kappa=4$ and $\alpha$ is the Stefan-Boltzmann constant. An example relevant to suspended sub-Kelvin nanostructures is a situation where a finite number $N_{\rm ph}$ of phonon or photon modes carry heat between the two reservoirs [@Pendry1983; @photons; @phonons; @2012w-pointcont] then $\kappa=2$ and $\alpha \leq N_{\rm ph}\pi^2 {k_{\rm B}}^2/(6h)$. One of the biggest practical challenges for quantum thermoelectrics is that phonons and photons will often carry much more heat than the electrons. This is simply because the hot reservoir can typically radiate heat in all directions as phonons or photons, while electrons only carry heat through the few nanostructures connected to that reservoir. Thus, in many cases the phonon or photon heat flow will dominate over the electronic one. However, progress is being made in blocking phonon and photon flow, by suspending the nanostructure to minimize phonon flow [@phonons] and engineering the electromagnetic environment to minimize photon flow [@photons], and it can be hoped that phonon and phonon effects will be greatly reduced in the future. Hence, here we consider the full range from weak to strong phonon or photon heat flows. For compactness in what follows we will only refer to phonon heat flows (usually the dominant parasitic effect). However, strictly one should consider $J_{\rm ph}$ as the sum of the heat flow carried by phonons, photons and any more exotic charge-less excitations that might exist in a given circuit (mechanical oscillations, spin-waves, etc.). ![\[Fig:allpowers-phonons\] Plots of the maximum efficiency allowed when the there is a phonon heat flow, $J_{\rm ph}$, in parallel with the heat carried by the electrons. The curves in (a) are for $T_R/T_L=0.2$, with $J_{\rm ph} = 0, 0.01, 0.1, 1$ (from top to bottom); the curves come from Eq. (\[Eq:eng-e+ph\]) with $\eta_{\rm eng}(P_{\rm gen})$ given in Fig. \[Fig:allpowers\]a. The curves in (b) are for $T_R/T_L=1.5$, with $J_{\rm ph} = 0, 0.02, 0.1, 0.4$ (from top to bottom); the curves come from Eq. (\[Eq:fri-e+ph\]) with $\eta_{\rm eng}(P_{\rm gen})$ given in Fig. \[Fig:allpowers\]b. The maximum cooling power (open circles) is $({{\textstyle{\frac{1}{2}}}}J^{\rm qb}_L-J_{\rm ph})$. ](figure12.pdf){width="\columnwidth"} Heat-engine with phonons ------------------------ For heat-engines, the phonon heat-flow is in parallel with electronic heat-flow, so the heat-flow for a given $P_{\rm gen}$ is $(J_L+J_{\rm ph})$, rather than just $J_L$ (as it was in the absence of phonons). Thus, the efficiency in the presence of the phonons is $$\begin{aligned} \eta^{\rm e+ph}_{\rm eng}(P_{\rm gen})={P_{\rm gen} \over J_L(P_{\rm gen})+J_{\rm ph}}.\end{aligned}$$ Writing this in terms of the efficiency, we get $$\begin{aligned} \eta_{\rm eng}^{\rm e+ph} (P_{\rm gen}) &=& \big[ \eta_{\rm eng}^{-1} (P_{\rm gen}) +J_{\rm ph} / P_{\rm gen} \big]^{-1}, \label{Eq:eng-e+ph}\end{aligned}$$ where $\eta_{\rm eng} (P_{\rm gen})$ is the efficiency for $J_{\rm ph}=0$. Given the maximum efficiency at given power in the absence of phonons, we can use this result to find the maximum efficiency for a given phonon heat flow, $J_{\rm ph}$. An example of this is shown in Fig. \[Fig:allpowers-phonons\]a. It shows that for finite $J_{\rm ph}$, Carnot efficiency is not possible at any power output. Phonons have a huge effect on the efficiency at small power output. Whenever $J_{\rm ph}$ is non-zero, the efficiency vanishes at zero power output, with $$\begin{aligned} \eta^{\rm e+ph}_{\rm eng}(P_{\rm gen})=P_{\rm gen}\big/J_{\rm ph} \ \ \ \hbox{ for } \ P_{\rm gen} \ll J_{\rm ph}. \label{Eq:eta-with-phonons-smallP}\end{aligned}$$ As $J_{\rm ph}$ increases, the range of applicability of this small $P_{\rm gen}$ approximation (shown as dashed lines in Fig. \[Fig:allpowers-phonons\]) grows towards the maximum power $P_{\rm eng}^{\rm qb}$ (open circles). In contrast, phonon heat flows have little effect on the efficiency near the maximum power output, until these flows become strong enough that $J_{\rm ph} \sim P_{\rm gen}$. For strong phonon flow, where $J_{\rm ph} \gg P_{\rm gen}$, Eq. (\[Eq:eta-with-phonons-smallP\]) applies at all powers up to the maximum, $P_{\rm gen}^{\rm qb2}$. Then, the efficiency is maximal when the power is maximal, where maximal power is the quantum bound given in Eq. (\[Eq:P-qb2\]). Thus, the system with both maximal power and maximal efficiency is that with a Heaviside step transmission function (see section \[Sect:chain\]). Refrigerator with phonons ------------------------- For a refrigerator to extract heat from a reservoir at rate $J$ in the presence of phonons carrying a back flow of heat $J_{\rm ph}$, that refrigerator must extract heat at a rate $J_L=J+J_{\rm ph}$. Note that for clarity, in this section we take $J_{\rm ph}$ to be positive when $T_L< T_R$ (opposite sign of that in Eq. (\[Eq:J\_ph\])). Thus, the efficiency, or COP, in the presence of phonons, is the heat current extracted, $J$, divided by the electrical power required to extract heat at the rate $J_L=(J+J_{\rm ph})$. This means that $$\begin{aligned} \eta_{\rm fri}^{\rm e+ph}(J) &=& {J \, \eta_{\rm fri}(J+J_{\rm ph}) \over J+J_{\rm ph}} , \label{Eq:fri-e+ph}\end{aligned}$$ where $\eta_{\rm fri} (J)$ is the efficiency for $J_{\rm ph}=0$. We can use this result to find the maximum efficiency for a given phonon heat flow, $J_{\rm ph}$. An example is shown in Fig. \[Fig:allpowers-phonons\]b. Eq. (\[Eq:fri-e+ph\]) means that the phonon flow suppresses the maximum cooling power, so $J$ must now obey $$\begin{aligned} J&\leq& {{\textstyle{\frac{1}{2}}}}J_L^{\rm qb} -J_{\rm ph} \label{Eq:Jqb-phonons}\end{aligned}$$ with $J_L^{\rm qb}$ given in Eq. (\[Eq:Jqb\]). Thus, the upper bound (open circles) in Fig. \[Fig:allpowers-phonons\]b move to the left as $J_{\rm ph}$ increases. When the reservoir being refrigerated (reservoir $L$) is at ambient temperature, $T_R$, then $J_{\rm ph}=0$ while $J_L^{\rm qb}$ is finite. However, as reservoir $L$ is refrigerated (reducing $T_L$), $J_{\rm ph}$ grows, while $J_L^{\rm qb}$ shrinks. As a result, at some point (before $T_L$ gets to zero) one arrives at $J_{\rm ph} = {{\textstyle{\frac{1}{2}}}}J_L^{\rm qb}$, and further cooling of reservoir $L$ is impossible. Thus, given the $T_L$ of $J_{\rm ph}$ for a given system, one can easily find the lowest temperature that reservoir $L$ can be refrigerated to, by solving the equation $J_{\rm ph} = {{\textstyle{\frac{1}{2}}}}J_L^{\rm qb}$ for $T_L$ To achieve this temperature, one needs the refrigerator with the maximum cooling power (rather than the most efficient one), this is a system with a Heaviside step transmission function (see section \[Sect:chain\]). Such a system’s refrigeration capacities were discussed in Ref. \[\]. We also note that, as with the heat-engine, phonons have a huge effect on the efficiency at small cooling power, as can be seen in Fig. \[Fig:allpowers-phonons\]b. Whenever $0< J_{\rm ph}<{{\textstyle{\frac{1}{2}}}}J_L^{\rm qb}$, the efficiency vanishes for small cooling power, with $$\begin{aligned} \eta^{\rm e+ph}_{\rm fri}(J)=J \ {\eta_{\rm fri}(J_{\rm ph}) \over J_{\rm ph}}\ \ \ \ \hbox{ for } \ J \ll J_{\rm ph}.\end{aligned}$$ Relaxation in a quantum system without B-field {#Sect:Relax} ============================================== Elsewhere in this article, we neglected relaxation in the quantum system. In other words, we assumed that electrons traverse the system in a time much less than the time for inelastic scattering from phonons, photons or other electrons. We now consider systems in which there is such relaxation, and ask if this relaxation could enable a system to exceed the bounds found above for relaxationless systems. To make progress, we restrict our interest to systems with negligible external magnetic field (B-field) [@Footnote:Error-my-PRL]. As yet, we have not been able to consider the rich interplay of relaxation and B-field [@Casati2011; @Sanchez-Serra2011; @Entin-Wohlman2012]. We use the voltage-probe model [@voltage-probe] shown in Fig. \[Fig:relax\]a. A system with relaxation is modeled as a phase-coherent scatterer coupled to a fictitious reservoir $M$ (a region in which relaxation occurs instantaneously). The rate of the relaxation is controlled by the transmission of the lead coupling to reservoir $M$. We then separate the phase-coherent scatterer into scatterers 1,2 and 3, as shown in Fig. \[Fig:relax\]b, each with their own transmission functions ${\cal T}_{ij}({\epsilon})$ with $i,j \in L,M,R$. We assume that the transmission is unchanged under reversal of direction, so ${\cal T}_{ij}({\epsilon})={\cal T}_{ji}({\epsilon})$ for all ${\epsilon}$ and $i,j$. This condition is guaranteed by time-reversal symmetry whenever the B-field has a negligible effect on the electron and hole dynamics. However, it also applies for any B-field when all particles relax as they traverse the quantum system (then ${\cal T}_{LR}({\epsilon})={\cal T}_{RL}({\epsilon})=0$, which is sufficient to force ${\cal T}_{ij}({\epsilon})={\cal T}_{ji}({\epsilon})$ for all $i,j$). ![\[Fig:relax\] (a) A quantum system in which relaxation occurs is modelled phenomenologically by a coherent quantum system coupled to a third fictitious reservoir $M$ in which the relaxation occurs. (b) The same model after we have separated the system’s scattering matrix into three components. The dashed arrows are the exchange of phonons or photons. The arm containing scatterers 1 and 2 is shown in (c) for a heat-engine, and in (d) for a refrigerator. ](figure13.pdf){width="\columnwidth"} If the relaxation involves electron-phonon or electron-photon interactions (typically any system which is not sub-Kelvin), the phonons or photons with which the electrons interact usually flow easily between the system and the reservoirs. Thus, these phonons or photons can carry heat current between the fictitious reservoir $M$ and reservoirs $L,R$ (dashed arrows in Fig. \[Fig:relax\]). The total electrical and heat currents into reservoir $M$ must be zero, and this constraint determines reservoir $M$’s bias, $V_M$, and temperature, $T_M$. Method of over-estimation ------------------------- The optimal choice of ${\cal T}_{ML}$ and ${\cal T}_{RM}$ depends on $T_M$, while $T_M$ depend on the heat current, and thus on ${\cal T}_{ML}$ and ${\cal T}_{RM}$. The solution and optimization of this self-consistency problem has been beyond our ability to resolve, even though we have restricted ourselves to a simple model of relaxation in a system with negligible B-field. Instead, we make a simplification which leads to an [*over-estimate*]{} of the efficiency. We assume $V_M,T_M$ are free parameters (not determined from ${\cal T}_{ML}$ and ${\cal T}_{RM}$), with $T_M$ between $T_L$ and $T_R$. If we find the optimal ${\cal T}_{ML}$ and ${\cal T}_{RM}$ for given $T_M$, and then find the optimal $T_M$ (irrespective of whether it is consistent with ${\cal T}_{ML}$ and ${\cal T}_{RM}$ or not), we have an over-estimate of the maximal efficiency. Even with this simplification, we have only been able to address the low-power and high-power limits. However, we show below that this over-estimate is sufficient to prove the following. - At low power, relaxation cannot make the system’s efficiency exceed that of the optimal relaxation-free system with $N_{\rm max}$ modes. - Relaxation cannot make a system’s power exceed that of the maximum possible power of a relaxation-free system with $N_{\rm max}$ modes. Defining $N_L$ and $N_R$ as the number of transverse modes in the system to the left and right of the region where relaxation occurs, $$\begin{aligned} N_{\rm max}={\rm max}[N_L,N_R], \label{Eq:Nmax}\end{aligned}$$ Efficiency of heat-engine with relaxation {#Sect:eng-eff-relax} ----------------------------------------- To get the efficiency for our model of a quantum system with relaxation, we must find the efficiency for the system in Fig. \[Fig:relax\]b. This system has two “arms”. One arm contains scatterers 1 and 2, and we define its efficiency as $\eta_{\rm eng}^{(1\&2)}$. The other arm contains scatterer 3, and we define its efficiency as $\eta_{\rm eng}^{(3)}$. The efficiency of the full system, $ \eta_{\rm eng}^{\rm total}(P_{\rm gen})$, is given by $$\begin{aligned} {1 \over \eta_{\rm eng}^{\rm total}(P_{\rm gen}) } = {p_{\rm rel} \over \eta_{\rm eng}^{(1\&2)} (p_{\rm rel}P_{\rm gen}) } +{q_{\rm rel} \over \eta_{\rm eng}^{(3)} ( q_{\rm rel}P_{\rm gen}) }, \label{Eq:heatengines-in-parallel}\end{aligned}$$ Here $p_{\rm rel}$ is the proportion of transmitted electrons that have passed through the arm containing scatterers 1 and 2, while $q_{\rm rel}=(1-p_{\rm rel})$ is the proportion that have passed through the arm containing scatterer 3. Physically, $p_{\rm rel}$ is the probability that an electron entering the quantum system relaxes before transmitting, while $q_{\rm rel}$ is the probability that it transmits before relaxing. One sees from Eq. (\[Eq:heatengines-in-parallel\]) that the maximal efficiency for a given $p_{\rm rel}$ occurs when both $\eta_{\rm eng}^{(1\&2)}$ and $\eta_{\rm eng}^{(3)}$ are maximal. The upper-bound on $\eta_{\rm eng}^{(3)}$ is that given in section \[Sect:eng\] with $q_{\rm rel}N_L$ modes to the left and $q_{\rm rel}N_R$ modes to the right. Our objective now is to find the maximum $\eta_{\rm eng}^{(1\&2)}$ with $N_1=p_{\rm rel}N_L$ modes on the left and $N_2=p_{\rm rel}N_R$ modes on the right. More precisely our objective is to find an [*over-estimate*]{} of this maximum. For the heat flows indicated in Fig. \[Fig:relax\]c, the efficiency is $$\begin{aligned} \eta_{\rm eng}^{(1\&2)} &\equiv& P_{\rm gen}^{(1\&2)}\big/J \nonumber \\ &=&\!\! {1 \over J}\left[ P_{\rm gen}^{(1)}(J_1;T_M,T_L) + P_{\rm gen}^{(2)}(J_2;T_R,T_M)\right]\! , \qquad \ \end{aligned}$$ where $J_1=J-J_1^{\rm ph}-J^{\rm ph}$ and $J_2 =J-J_2^{\rm ph}-J^{\rm ph}-P_{\rm gen}^{(1)}$. One sees that $\eta_{\rm eng}^{(1\&2)}$ is maximal for given $T_M$ when $J^{\rm ph}=J_1^{\rm ph}=J_2^{\rm ph}=0$ (these heat currents cannot be negative because $T_L > T_M> T_R$). Thus, to get our over-estimate of the maximal efficiency for given $T_M$, we assume these phonon and photon heat-currents are zero. Then, with a little algebra, one finds that $$\begin{aligned} 1-\eta_{\rm eng}^{(1\&2)}\big(P_{\rm gen}^{(1\&2)}\big) = \left(1-\eta_{\rm eng}^{(1)}\big(P_{\rm gen}^{(1)}\big) \right)\left(1-\eta_{\rm eng}^{(2)}\big(P_{\rm gen}^{(2)}\big) \right), \nonumber\end{aligned}$$ where $P_{\rm gen}^{(1)}$ and $P_{\rm gen}^{(1)}$ are related to $P_{\rm gen}^{(1\&2)}$ by $$\begin{aligned} P_{\rm gen}^{(\mu)} = P_{\rm gen}^{(1\&2)} \eta_{\rm eng}^{(\mu)} \big/ \eta_{\rm eng}^{(1\& 2)}, \label{Eq:P1-or-2}\end{aligned}$$ for $\mu=1,2$. For given $T_M$, one maximizes $\eta_{\rm eng}^{(1\&2)}$ by independently maximizing $\eta_{\rm eng}^{(1)}$ and $\eta_{\rm eng}^{(2)}$. For low powers, Eq. (\[Eq:eta-eng-small-Pgen\]) with $P,N,T_R\to P_1, N_1, T_M$ gives $\eta_{\rm eng}^{(1)}$, while with $P,N,T_L\to P_2, N_2, T_M$ gives $\eta_{\rm eng}^{(2)}$. In this limit, we can treat efficiencies in Eq. (\[Eq:P1-or-2\]) to zeroth order in $P_{\rm gen}^{(1\&2)}$, taking them to be Carnot efficiencies, so $$\begin{aligned} P_{\rm gen}^{(1)} \simeq {T_L-T_M \over T_L-T_R}P_{\rm gen}^{(1\&2)}, \quad P_{\rm gen}^{(1)} \simeq {T_M-T_R \over T_L-T_R}P_{\rm gen}^{(1\&2)}. \nonumber\end{aligned}$$ Then some algebra gives the over-estimate of efficiency at low powers for given $T_M$, to be $$\begin{aligned} \eta_{\rm eng}^{(1\&2)} \leq \eta_{\rm eng}^{\rm Carnot} \left(1- 0.478 \sqrt{ {T_R \over T_L} \ {P_{\rm gen} \ K_{\rm rel}\over P_{\rm gen}^{\rm qb2}(N=1)} } \right)\! , \quad\end{aligned}$$ with $P_{\rm gen}^{\rm qb2}(N=1)$ given by Eq. (\[Eq:P-qb2\]) with $N=1$, and $$\begin{aligned} K_{\rm rel} = \sqrt{{1 \over N_1}\,{T_R(T_L-T_M) \over T_M(T_L-T_R)}} + \sqrt{{1 \over N_2}\,{T_L(T_M-T_R) \over T_M(T_L-T_R)}},\quad \label{Eq:Krelax}\end{aligned}$$ where $N_1= p_{\rm rel} N_L$ and $N_2=p_{\rm rel} N_L$ are respectively the number of transmission modes in scattering matrices 1 and 2. The over-estimate of $\eta_{\rm eng}^{(1\&2)}$ is maximal when $T_M$ is chosen to minimize $K_{\rm rel}$. The two minima of $K_{\rm rel}$ are at $T_M=T_R$ and $T_M=T_L$, for which the values of $K_{\rm rel}$ are $1/\sqrt{N_1}$ and $1/\sqrt{N_2}$ respectively. Thus, we have $$\begin{aligned} K_{\rm rel} \geq 1/\sqrt{p_{\rm rel}N_{\rm max}}\ , \label{Eq:Krelax-limit} \end{aligned}$$ with $N_{\rm max}$ in Eq. (\[Eq:Nmax\]). Thus, whatever $T_M$ may be, $$\begin{aligned} \eta_{\rm eng}^{(1\&2)} \left(P_{\rm gen}^{(1\&2)}\right) &\leq& \eta_{\rm eng}^{\rm Carnot} \nonumber \\ & & \times \left(\! 1- 0.478 \sqrt{ {T_R \over T_L} {P_{\rm gen}^{(1\&2)} \over P_{\rm gen}^{\rm qb2}(p_{\rm rel}N_{\rm max}) } } \right)\! . \nonumber \\ \label{Eq:eta1&2-bound}\end{aligned}$$ Since $P_{\rm gen}^{(1\&2)} = p_{\rm rel} P_{\rm gen}$, we can simplify Eq. (\[Eq:eta1&2-bound\]) by noting that $$\begin{aligned} {P_{\rm gen}^{(1\&2)} \over P_{\rm gen}^{\rm qb2}(p_{\rm rel}N_{\rm max}) } = {P_{\rm gen} \over P_{\rm gen}^{\rm qb2}(N_{\rm max}) } \end{aligned}$$ where $P_{\rm gen}$ is the total power generated by the combined system made of scatterers 1,2 and 3. Then substituting the result into Eq. (\[Eq:heatengines-in-parallel\]), we get an over-estimate of the efficiency at power output $P_{\rm gen}$ which is equal to the upper bound we found in the absence of relaxation, Eq. (\[Eq:eta-eng-small-Pgen\]). Thus, we can conclude that for small power outputs, no quantum system with relaxation within it can exceed the upper-bound on efficiency found for a [*relaxation-free*]{} system with $N_{\rm max}$ transverse modes. Since the proof is based on an over-estimate of the efficiency for a system with relaxation, we cannot say if a system with finite relaxation can approach the bound in Eq. (\[Eq:eta-eng-small-Pgen\]). Unlike in the relaxation-free case, we cannot say what properties the quantum system with relaxation (as given in terms of the properties of the effective scatterers 1, 2 and 3) are necessary to maximize the efficiency at given power output. We simply know that it cannot exceed Eq. (\[Eq:eta-eng-small-Pgen\]). Refrigerator with relaxation ---------------------------- Our objective is to find an over-estimate of the maximal efficiency of a refrigerator that is made of quantum systems in which relaxation occurs. The efficiency of the system with relaxation, $ \eta_{\rm fri}^{\rm total}(P_{\rm gen})$, is given by $$\begin{aligned} \eta_{\rm fri}^{\rm total}(J_L) = p_{\rm rel} \eta_{\rm fri}^{(1\&2)} (p_{\rm rel}J_L) +q_{\rm rel} \eta_{\rm fri}^{(3)} ( q_{\rm rel}J_L), \label{Eq:fridges-in-parallel}\end{aligned}$$ thus we need to find an upper bound on $\eta_{\rm fri}^{(1\&2)}$. We make an over-estimate of this efficiency by taking $T_M$ to be a free parameter between $T_L$ and $T_R$. For given $T_M$, the efficiency of the combined systems 1 and 2 is $$\begin{aligned} \eta_{\rm fri}^{(1\&2)} (J)= J \Big/ \big[ P_{\rm abs}^{(1)}(J_1) + P_{\rm abs}^{(2)}(J_2) \big], \end{aligned}$$ where $J_1=J+J_1^{\rm ph}+J^{\rm ph}$ and $J_2=J+J_2^{\rm ph} +J^{\rm ph}+P_{\rm abs}^{(1)}$, see Fig. \[Fig:relax\]d. This efficiency is maximized when $J_1^{\rm ph},J_2^{\rm ph},J^{\rm ph} = 0$ (since $T_L<T_M<T_R$ means these currents are not negative). Then a little algebra gives $$\begin{aligned} 1+{1 \over \eta_{\rm fri}^{(1\&2)}(J)} = \left[ 1+{1 \over \eta_{\rm fri}^{(1)}(J)} \right]\left[ 1+{1 \over \eta_{\rm fri}^{(2)}\big(J_2\big)} \right] \! , \qquad\end{aligned}$$ where $J_2= J+P_{\rm abs}^{(1)}= J\big[1+1/\eta_{\rm fri}^{(1)}(J)\big]$. Thus, to maximize $\eta_{\rm fri}^{(1\&2)}(J)$ for given $T_M$, one must maximize both $\eta_{\rm fri}^{(1)}$ and $\eta_{\rm fri}^{(2)}$. For low power, this can be done using Eq. (\[Eq:eta-fri-smallJ\]) (much as for the heat-engine in Section \[Sect:eng-eff-relax\] above) giving $$\begin{aligned} \eta_{\rm fri}^{(1\&2)} \leq \eta_{\rm fri}^{\rm Carnot} \! \left(\! 1- 1.09 \sqrt{ {T_R \over T_R-T_L}{J_L K_{\rm rel}\over J_L^{\rm qb}(N=1)}}\,\right) \! , \ \ \end{aligned}$$ where $K_{\rm rel}$ is given in Eq. (\[Eq:Krelax\]), and $J_L^{\rm qb}(N=1)$ is given by Eq. (\[Eq:J-qb-fri\]) with $N=1$. The over-estimate of $\eta_{\rm fri}^{(1\&2)}$ is maximal when $K_{\rm rel}$ is minimal, see Eq. (\[Eq:Krelax-limit\]). Substituting this into Eq. (\[Eq:fridges-in-parallel\]), we see that the efficiency with relaxation does not exceed the result in Eq. (\[Eq:eta-fri-smallJ\]) for a [*relaxation-free*]{} system with $N_{\rm max}$ transverse modes. Quantum bounds on power with relaxation --------------------------------------- For a heat-engine, the arm with scatterers 1 and 2, has a maximum power, $$\begin{aligned} P_{\rm gen}^{(1\&2)} \leq A_0\, {\pi^2 \over h} {k_{\rm B}}^2 \left[ N_1\big(T_L-T_M\big)^2+ N_2\big(T_M-T_R\big)^2 \right], \nonumber\end{aligned}$$ Since $(T_L-T_M)^2 +(T_M-T_R)^2 \leq (T_L-T_R)^2$, the power of the full system cannot exceed the maximum power of a relaxation-less system, Eq. (\[Eq:P-qb2\]), with $N_{\rm max}$ modes. For a refrigerator, the arm containing scatterers 1 and 2 has a maximum cooling power, $$\begin{aligned} J \leq \left\{ \begin{array}{l} \pi^2 N_1 {k_{\rm B}}^2 T_L^2 \big/(12 h) \\ \pi^2 N_2 {k_{\rm B}}^2 T_M^2 \big/(12 h) -P_{\rm abs}^{(1)} \ , \end{array}\right. \end{aligned}$$ where $P_{\rm abs}^{(1)}$ is the electrical power absorbed by scatter 1. The upper (lower) term is the limit on the heat-flow into scatterer 1 (scatterer 2), noting that the heat-flow into scatterer 2 is $J+P_{\rm abs}^{(1)}$. Unless $N_2 \gg N_1$, the lower limit is the more restrictive one. In any case, the cooling power of the full system can never exceed the maximum power of a relaxation-less system, Eq. (\[Eq:J-qb-fri\]), with $N_{\rm max} $ modes. Conclusions {#Sect:conclusions} =========== The upper bound on efficiency at zero power (i.e. Carnot efficiency) is classical, since it is independent of wavelike nature of the electrons. However, this work on thermoelectrics shows that the upper bound on efficiency at finite power is quantum, depending on the ratio of the thermoelectric’s cross-section to the electrons’ Fermi wavelength. If one thought that electrons were classical (strictly zero wavelength), one would believe that Carnot efficiency was achievable at any power output. Quantum mechanics appears to tell us that this is not so. However, a crucial point for future work is to discover how universal our bounds on efficiency at finite power are. Our bounds currently rely on the quantum system being (a) well modelled by the nonlinear scattering theory with its mean-field treatment of electron-electron interactions, (b) coupled to only two reservoirs (hot and cold), and (c) relaxation free. Under certain conditions we have also shown that they apply when there is relaxation in the quantum system. We cannot yet prove that our results are as general as Pendry’s bound on heat flow[@Pendry1983], which applies for arbitrary relaxation and for more than two reservoirs [@2012w-2ndlaw], as well as for electronic Luttinger liquids[@Kane-Fisher] and bosons[@Pendry1983]. It also remains to be seen if our bound occurs in systems with strong electron-electron interactions (Coulomb blockade, Kondo physics, etc.). More generally, we wonder whether similar bounds apply to those thermodynamic machines that do not rely on thermoelectric effects, such as Carnot heat engines. Acknowledgements ================ I am very grateful to M. Büttiker for the suggestion which led to the implementation in Section \[Sect:chain\]. I thank P. Hänggi for questions on entropy flow which led to section \[Sect:Unique\]. I thank L. Correa for questions which led to a great improvement of section \[Sect:eff-CA\]. I thank C. Grenier for an analytic solution of Eq. (\[Eq:T-for-chain\]) for $k=3$. 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--- abstract: 'The [Near-Bipartiteness]{} problem is that of deciding whether or not the vertices of a graph can be partitioned into sets $A$ and $B$, where $A$ is an independent set and $B$ induces a forest. The set $A$ in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that [Near-Bipartiteness]{} is polynomial-time solvable for graphs of diameter $2$ and [[NP]{}]{}-complete for graphs of diameter $4$. We show that [Near-Bipartiteness]{} is [[NP]{}]{}-complete for graphs of diameter $3$, resolving their open problem. We also generalise their result for diameter $2$ by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter $2$.' author: - Marthe Bonamy - 'Konrad K. Dabrowski' - Carl Feghali - | \ Matthew Johnson - Daniël Paulusma bibliography: - 'mybib.bib' title: | Independent Feedback Vertex Sets for\ Graphs of Bounded Diameter[^1] --- Introduction ============ A graph is [*near-bipartite*]{} if its vertex set can be partitioned into sets $A$ and $B$, where $A$ is an independent set and $B$ induces a forest. The set $A$ is said to be an [*independent feedback vertex set*]{} and the pair $(A,B)$ is said to be a [*near-bipartite decomposition*]{}. This leads to the following two related decision problems. [.99]{} <span style="font-variant:small-caps;">[Near-Bipartiteness]{}</span>\ ----------------- ---------------------------------------------------------------------------------------- *    Instance:* [a graph $G$.]{} *Question:* [is $G$ near-bipartite (that is, does $G$ have an independent feedback vertex set)?]{} ----------------- ---------------------------------------------------------------------------------------- [.99]{} <span style="font-variant:small-caps;">[Independent Feedback Vertex Set]{}</span>\ ----------------- --------------------------------------------------------------------------- *    Instance:* [a graph $G$ and an integer $k\geq 0$.]{} *Question:* [does $G$ have an independent feedback vertex set of size at most $k$?]{} ----------------- --------------------------------------------------------------------------- Setting $k=n$ shows that the latter problem is more general than the first problem. Thus, if [Near-Bipartiteness]{} is [[NP]{}]{}-complete for some graph class, then so is [Independent Feedback Vertex Set]{}, and if [Independent Feedback Vertex Set]{} is polynomial-time solvable for some graph class, then so is [Near-Bipartiteness]{}. Note that every near-bipartite graph is $3$-colourable, that is, its vertices can be coloured with at most three colours such that no two adjacent vertices are coloured alike. The problems [$3$-Colouring]{} [@Lo73] and [Near-Bipartiteness]{} [@BLS98] (and thus [Independent Feedback Vertex Set]{}) are [[NP]{}]{}-complete. However, their complexities do not necessarily coincide on special graph classes. Grötschel, Lovász and Schrijver [@GLS84] proved that [Colouring]{} is polynomial-time solvable for perfect graphs even if the permitted number of colours $k$ is part of the input. However, Brandstädt et al. [@BBKNP13] proved that [Near-Bipartiteness]{} remains [[NP]{}]{}-complete for perfect graphs. The same authors also showed that [Near-Bipartiteness]{} is polynomial-time solvable for $P_4$-free graphs. Yang and Yuan [@YY06] proved that [Near-Bipartiteness]{} also remains [[NP]{}]{}-complete for graphs of maximum degree $4$. To complement their hardness result, Yang and Yuan [@YY06] showed that every connected graph of maximum degree at most $3$ is near-bipartite except the complete graph $K_4$ on four vertices. This also follows from a more general result of Catlin and Lai [@CL95]. Recently we gave a linear-time algorithm for finding an independent feedback vertex set in a graph of maximum degree at most $3$ [@BDFJP17], and also proved that [Near-Bipartiteness]{} is [[NP]{}]{}-complete even for line graphs of maximum degree $4$ [@BDFJP17b]. It is also known that [Near-Bipartiteness]{} is [[NP]{}]{}-complete for planar graphs; this follows from a result of Dross, Montassier and Pinlou [@DMP16]; see the arXiv version of [@BDFJP17] for details. Tamura, Ito and Zhou [@TIZ15] proved that [Independent Feedback Vertex Set]{} is [[NP]{}]{}-complete for planar bipartite graphs of maximum degree $4$ (note that [Near-Bipartiteness]{} is trivial for bipartite graphs). They also proved that [Independent Feedback Vertex Set]{} is linear-time solvable for graphs of bounded treewidth, chordal graphs and $P_4$-free graphs (the latter result generalising the result of [@BBKNP13] for [Near-Bipartiteness]{} on $P_4$-free graphs). In [@BDFJP17b] we proved that finding a minimum independent feedback vertex set is polynomial-time solvable even for $P_5$-free graphs. We refer to [@AGSS16; @MPRS12] for FPT algorithms with parameter $k$ for finding an independent feedback vertex set of size at most $k$. The [*distance*]{} between two vertices $u$ and $v$ in a graph $G$ is the length (number of edges) of a shortest path between $u$ and $v$. The [*diameter*]{} of a graph $G$ is the maximum distance between any two vertices in $G$. In addition to their results for graphs of bounded maximum degree, Yang and Yuan [@YY06] proved that [Near-Bipartiteness]{} is polynomial-time solvable for graphs of diameter at most $2$ and [[NP]{}]{}-complete for graphs of diameter at most $4$. They asked the following question, which was also posed by Brandstädt et al. [@BBKNP13]: [*What is the complexity of [Near-Bipartiteness]{} for graphs of diameter $3$?*]{} [**Our Results.**]{} We complete the complexity classifications of [Near-Bipartiteness]{} and [Independent Feedback Vertex Set]{} for graphs of bounded diameter. In particular, we prove that [Near-Bipartiteness]{} is [[NP]{}]{}-complete for graphs of diameter $3$, which answers the above question. We also prove that [Independent Feedback Vertex Set]{} is polynomial-time solvable for graphs of diameter $2$. This generalises the result of Yang and Yuan [@YY06] for [Near-Bipartiteness]{} restricted to graphs of diameter $2$. \[t-main\] Let $k\geq 0$ be an integer. (i) If $k\leq 2$, then [Independent Feedback Vertex Set]{} (and thus [Near-Bipartiteness]{}) is polynomial-time solvable for graphs of diameter $k$.\ (ii) If $k\geq 3$, then [Near-Bipartiteness]{} (and thus [Independent Feedback Vertex Set]{}) is [[NP]{}]{}-complete for graphs of diameter $k$. We prove Theorem \[t-main\] (i) in Section \[s-poly\]. Yang and Yuan [@YY06] proved their result for [Near-Bipartiteness]{} by giving a polynomial-time verifiable characterisation of the class of near-bipartite graphs of diameter $2$. We use their characterisation as the starting point for our algorithm for [Independent Feedback Vertex Set]{}. In fact our algorithm not only solves the decision problem but even finds a minimum independent feedback vertex set in a graph of diameter $2$. We prove Theorem \[t-main\] (ii) in Section \[s-diam3\] by using a construction of Mertzios and Spirakis [@MS16], which they used to prove that [$3$-Colouring]{} is [[NP]{}]{}-complete for graphs of diameter $3$. The outline of their proof is straightforward: a reduction from [$3$-Satisfiability]{} that constructs, for any instance $\phi$, a graph $H_\phi$ that is $3$-colourable if and only if $\phi$ is satisfiable. We reduce [$3$-Satisfiability]{} to [Near-Bipartiteness]{} for graphs of diameter $3$ using the same construction, that is, we show that $H_\phi$ is near-bipartite if and only if $\phi$ is satisfiable. As such, our result is an observation about the proof of Mertzios and Spirakis, but, owing to the intricacy of $H_\phi$, this observation is non-trivial to verify. In Section \[s-diam3\] we therefore repeat the construction and describe our reduction in detail, though we rely on [@MS16] where possible in the proof. Independent Feedback Vertex Set for Diameter $2$ {#s-poly} ================================================ In this section we show how to compute a minimum independent feedback vertex set of a graph of diameter $2$ in polynomial time. As mentioned, our proof relies on a known characterisation of near-bipartite graphs of diameter $2$ [@YY06]. In order to explain this characterisation, we first need to introduce some terminology. Let $G=(V,E)$ be a graph and let $X\subseteq V$. Then the [*$2$-neighbour set*]{} of $X$, denoted by $A_X$, is the set that consists of all vertices in $V\setminus X$ that have at least two neighbours in $X$. A set $I\subseteq V$ is [*independent*]{} if no two vertices of $I$ are adjacent. For $u\in V$, we let $G-u$ denote the graph obtained from $G$ after deleting the vertex $u$ (and its incident edges). A graph is [*complete bipartite*]{} if its vertex set can be partitioned into two independent sets $S$ and $T$ such that there is an edge between every vertex of $S$ and every vertex of $T$. If $S$ or $T$ has size $1$, the graph is also called a [*star*]{}. \[t-yy06\] A graph $G=(V,E)$ of diameter $2$ is near-bipartite if and only if one of the following two conditions holds: (i) \[cond:i\]there exists a vertex $u$ such that $G-u$ is bipartite; or (ii) \[cond:ii\]there exists a set $X$, $4\leq |X|\leq 5$, such that $(A_X,V\setminus A_X)$ is a near-bipartite decomposition. As noted in [@YY06], Theorem \[t-yy06\] can be used to solve [Near-Bipartiteness]{} in polynomial time for graphs of diameter $2$, as conditions \[cond:i\] and \[cond:ii\] can be checked in polynomial time. However, Theorem \[t-yy06\] does not tell us how to determine the size of a minimum independent feedback vertex set. In order to find a minimum independent feedback vertex set, we will distinguish between the two cases of Theorem \[t-yy06\]. This leads to two corresponding lemmas. \[l-1\] Let $G=(V,E)$ be a near-bipartite graph of diameter $2$ that contains a vertex $u$ such that $G-u$ is bipartite. Then it is possible to find a minimum independent feedback vertex set of $G$ in polynomial time. We can partition $V\setminus\{u\}$ into four independent sets $S_1$, $S_2$, $T_1$, $T_2$ (some of which might be empty) such that (i) \[prop:i\]$S_1\cup S_2$ and $T_1\cup T_2$ form bipartition classes of $G-u$; (ii) \[prop:ii\]$u$ is adjacent to every vertex of $S_1\cup T_1$; and (iii) \[prop:iii\]$u$ is non-adjacent to every vertex of $S_2\cup T_2$. Moreover, as $G$ has diameter $2$, it follows that given a vertex of $S_2$ (respectively, $T_2$) and a vertex of $T_1\cup T_2$ (respectively, $S_1\cup S_2$), these two vertices must either be adjacent or have a common neighbour. As the latter is not possible, we deduce that (i) \[prop:iv\]every vertex of $S_2$ is adjacent to every vertex of $T_1\cup T_2$, and every vertex of $T_2$ is adjacent to every vertex of $S_1\cup S_2$ (see also  \[fig:STu\]). =\[circle,draw=black, fill=black, minimum size=5pt, inner sep=1pt\] =\[draw,black\] ///in [ (0,-2)/T/1/below, (0,2)/S/1/above, (5,-2)/T/2/below, (5,2)/S/2/above]{} (0.6,-0.4) rectangle (3.4,0.4); (1) at (1,0) ; (2) at (2,0) ; (3) at (3,0) ; at (2,0) [$\name_\number$]{}; \(u) at (0,0) ; /in [1/1,1/2,2/1,2/2,1/3,2/3,3/3,3/2,3/1]{} [ (S1) – (T2) – (S2) – (T1); ]{} in [1,2,3]{} [ (S1) – (u) – (T1); (S2) – (u) – (T2); ]{} at (u) [$u$]{}; A (not necessarily proper) $2$-colouring of the vertices of a graph is *good* if the vertices coloured $1$ form an independent set and the vertices coloured $2$ induce a forest. The set of vertices coloured $1$ in a good $2$-colouring is said to be a *$1$-set* and is, by definition, an independent feedback vertex of $G$. A good $2$-colouring of $G$ is [*optimal*]{} if its $1$-set is of minimum possible size among all good $2$-colourings. Our algorithm colours vertices one by one with colour $1$ or $2$ to obtain a number of good $2$-colourings. We will establish that our approach ensures that at least one of our good $2$-colourings is optimal. Therefore, as our algorithm finds different good $2$-colourings, it only needs to remember the smallest $1$-set seen so far. We note that $G$ certainly has good $2$-colourings as, for example, we can let either $S_1\cup S_2$ or $T_1\cup T_2$ be the set of vertices coloured $1$. We say that an edge is a [*$1$-edge*]{} if both its end-points have colour $1$ and say that a cycle of $G$ is a [*$2$-cycle*]{} if all its vertices have colour $2$. Our algorithm will consist of a number of branches depending on the way we will colour the vertices of $G$. Whenever we detect a $1$-edge or a $2$-cycle in a branch, we can discard the branch as we know that we are not going to generate a good $2$-colouring. Before we describe our algorithm, we first prove the following claim. Here, we say that an independent set $I$ is a [*twin-set*]{} if every vertex of $I$ has the same neighbourhood. [[[*Claim .* ]{}]{}[*\[clm:1\]Let $I$ be a twin-set. In every optimal $2$-colouring, at least $|I|-1$ vertices of $I$ obtain the same colour.*]{}\ ]{} We prove Claim \[clm:1\] as follows. If $|I|=1$, the claim is trivial. Suppose $|I|\geq 2$ and let $J$ be the neighbourhood of the vertices of $I$. Note that $J$ is non-empty since $|I|\geq 2$ and $G$ is connected. Let $c$ be an optimal $2$-colouring of $G$. If $c$ gives colour $1$ to a vertex of $J$, then every vertex of $I$ must receive colour $2$. Now suppose that $c$ gives colour $2$ to every vertex of $J$. If $|J|=1$, then $c$ colours every vertex of $I$ with colour $2$, as doing this will not create a $2$-cycle. If $|J|\geq 2$ then, in order to avoid a $2$-cycle, at least $|I|-1$ vertices of $I$ must be coloured $1$. This proves Claim \[clm:1\]. By \[prop:i\], \[prop:iii\], \[prop:iv\], we find that $S_2$ and $T_2$ are twin-sets. Let $Z$ be the set of isolated vertices in the subgraph of $G$ induced by $S_1\cup T_1$. Then by \[prop:i\], \[prop:ii\], \[prop:iv\], the neighbourhood of every vertex in $Z\cap S_1$ (respectively, $Z\cap T_1$) is $T_2\cup \{u\}$ (respectively, $S_2\cup \{u\}$). So $Z\cap S_1$ and $Z\cap T_1$ are twin-sets. We choose one vertex from each non-empty set in $\{S_2,T_2,Z\cap S_1,Z\cap T_1\}$ and let $W$ be the set of chosen vertices. Note that the choice of the vertices in $W$ can be done arbitrarily, since all four of these sets are twin-sets. We now branch by giving all vertices in $S_2 \setminus W$ the same colour, all vertices in $T_2 \setminus W$ the same colour, all vertices in $(Z\cap S_1)\setminus W$ the same colour and all vertices in $(Z\cap T_1)\setminus W$ the same colour. We then branch by colouring the at most four vertices of $W$ with every possible combination of colours. Hence the total number of branches is at most $2^8$. We discard any branch that yields a $1$-edge or $2$-cycle. Let $S_1' = S_1 \setminus Z$ and $T_1'=T_1 \setminus Z$. For each remaining branch we try to colour the remaining vertices of $G$, which are all in $S_1'\cup T_1'\cup \{u\}$, and keep track of any minimum $1$-set found. In the end we return a $1$-set of minimum size (recall that $G$ has at least two $1$-sets). For any remaining branch we do as follows. We first give colour $1$ to $u$. Then every vertex of $S_1'\cup T_1'$ must get colour $2$. If this does not yield a $1$-edge or $2$-cycle, we obtain a $1$-set, which we remember if it is the smallest one found so far. We now give colour $2$ to $u$. If $u$ was the only remaining vertex, we check for the presence of a $1$-edge or a $2$-cycle, and if none is present, we remember the $1$-set found if it is the smallest one found so far. Otherwise, we let $D_1,\ldots,D_r$ for some integer $r\geq 1$ be the connected components of the (bipartite) graph induced by $S_1'\cup T_1'$. As these vertices do not belong to $Z$, each $D_i$ contains at least one edge. Moreover, each $D_i$ is bipartite. For $i \in \{1,\ldots,r\}$, we denote the two non-empty bipartition classes of $D_i$ by $D_i^1$ and $D_i^2$ such that $|V(D_i^1)|\leq |V(D_i^2)|$. The following claim is crucial. [[[*Claim .* ]{}]{}[*\[clm:2\]For $i\in\{1,\ldots,r\}$, we must either colour all vertices of $D_i^1$ with colour $1$ and all vertices of $D_i^2$ with colour $2$, or vice versa.*]{}\ ]{} We prove Claim \[clm:2\] as follows. Suppose that $D_i^1$ contains a vertex with the same colour as a vertex of $D_i^2$. As $D_i$ is connected and bipartite, this means that $D_i$ contains an edge $vw$ whose end-vertices are either both coloured $1$ or coloured $2$. In the first case, we obtain a $1$-edge. In the second case the vertices $u$, $v$ and $w$ form a $2$-cycle in $G$. Hence we must use colours $1$ and $2$ for different partition classes of $D_i$. This proves Claim \[clm:2\]. We now proceed as follows. First suppose that $S_2 \cup T_2$ is non-empty. If we coloured a vertex in $S_2$ (respectively $T_2$) with colour $1$, then every vertex in $T_1'$ (respectively $S_1'$) must be coloured $2$ and therefore every vertex in $S_1'$ (respectively $T_1'$) must be coloured $1$ by Claim \[clm:2\]. Again, in this case we discard the branch if a $1$-edge or $2$-cycle is found; otherwise we remember the corresponding $1$-set if it is the best set found so far. In every other case, we must have coloured every vertex of non-empty set $S_2\cup T_2$ with colour $2$. Without loss of generality, assume that there is a vertex $s \in S_2$ that is coloured $2$. Then at most one vertex of $T_1'$ may have colour $2$, as otherwise we obtain a $2$-cycle by involving the vertices $s$ and $u$. We branch by guessing this vertex and then colouring it either $1$ or $2$, while assigning colour $1$ to all other vertices of $T_1'$. Then the only vertices with no colour yet are in $S_1'$, but their colour is determined by the colours of the vertices in $T_1'$ due to Claim \[clm:2\]. We are left to deal with the case where $S_2\cup T_2=\emptyset$. Claim \[clm:2\] tells us that we must either give every vertex of $D_i^1$ colour $1$ and every vertex of $D_i^2$ colour $2$, or vice versa. For $i \in \{1,\ldots,r\}$ we give colour $1$ to every vertex of every $D_i^1$; as $|V(D_i^1)|\leq |V(D_i^2)|$, this is the best possible good $2$-colouring for this branch. The correctness of our algorithm follows from the fact that we distinguish all possible cases and find a best possible good $2$-colouring (if one exists) in each case. Note that it takes polynomial time to find the sets $S_1$, $S_2$, $T_1$ and $T_2$. Moreover, the number of branches is $O(n)$ and each branch can be processed in polynomial time, as we only need to search for a $1$-edge or $2$-cycle. Hence our algorithm runs in polynomial time. \[l-2\] Let $G=(V,E)$ be a near-bipartite graph of diameter $2$ that contains no vertex $u$ such that $G-u$ is bipartite. Then it is possible to find a minimum independent feedback vertex of $G$ in polynomial time. As $G$ is near-bipartite, it has an independent feedback vertex set. Let $A$ be a minimum independent feedback vertex set. We claim that $G$ contains a set of vertices $X$ of size $4\leq |X|\leq 5$ such that $A_X=A$. This would immediately give us a polynomial-time algorithm. Indeed, it would suffice to check, for every set $X$ of size $4\leq |X|\leq 5$, whether $(A_X,V\setminus A_X)$ is a near-bipartite decomposition and to return a set $A_X$ of minimum size that satisfies this condition. This takes polynomial time. To prove the above claim we will follow the same line of reasoning as in the proof of Theorem \[t-yy06\] However, our arguments are slightly different, as we need to prove a stronger statement. Let $B=V\setminus A$ and let $F$ be the subgraph of $G$ induced by $B$. By definition, $F$ is a forest, so all of its connected components are trees. We will first consider the case where $F$ has a connected component $T$ of diameter at least $3$. Let $P$ be a longest path in the tree $T$ on vertices $v_1,\ldots,v_p$ in that order. As $T$ has diameter $3$, we find that $p\geq 4$. If $p\leq 5$, then we let $X=\{v_1,\ldots,v_p\}$. If $p\geq 6$, then we let $X=\{v_1,v_2,v_{p-1},v_p\}$. We will show that $A=A_X$. Let $u\in A$. As $G$ has diameter $2$ and $A$ is an independent set, $u$ is adjacent to $v_1$ or to a neighbour $v^*$ of $v_1$ in $B$. In the latter case, if $v^*\neq v_2$ then $v^*$ must have a neighbour in $\{v_2,\ldots,v_p\}$, otherwise we have found a path that is longer than $P$, but in this case $B$ contains a cycle, a contradiction. Hence, $u$ has at least one neighbour in $\{v_1,v_2\}$, and similarly, $u$ has at least one neighbour in $\{v_{p-1},v_p\}$. So $A\subseteq A_X$. Now suppose $u\in A_X$. Note that $u\neq v_3$ due to our choice of $X$. Then the subgraph of $G$ induced by $V(P)\cup \{u\}$ contains a cycle. Hence $u$ must belong to $A$. So $A_X\subseteq A$. We conclude that $A=A_X$. We now consider the case where every connected component of $F$ has diameter at most $2$. Such components are either isolated vertices or stars (we say that the latter components are [*star-components*]{} and that their non-leaf vertex is the [*star-centre*]{}; if such a component consists of a single edge, we arbitrarily choose one of them to be the star-centre). If $F$ contains no star-components, then $G$ is bipartite and therefore $G-u$ is bipartite for every vertex $u$, a contradiction. If $F$ contains exactly one star-component, then by choosing $u$ to be the star-centre we again find that $G-u$ is bipartite. Hence $F$ contains at least two star-components $D_1$ and $D_2$. For $i=1,2$, let $v_i$ be the star-centre and let $w_i$ be a leaf in $D_i$. We choose $X=\{v_1,v_2,w_1,w_2\}$ and show that $A=A_X$. Let $u\in A$. As $G$ has diameter $2$ and $A$ is an independent set, $u$ is either adjacent to $w_1$ or to a neighbour of $w_1$ in $B$. If this neighbour is not $v_1$, then $D_1$ is not a star-component, a contradiction. Hence, $u$ has at least one neighbour in $\{v_1,w_1\}$, and similarly, $u$ has at least one neighbour in $\{v_2,w_2\}$. So $A\subseteq A_X$. Now suppose $u\in A_X$. Then $X\cup \{u\}$ induces either a connected subgraph of $G$ that contains both $D_1$ and $D_2$ (and is therefore not a star-component) or a subgraph with a cycle. Hence $u$ must belong to $A$. So $A_X\subseteq A$. We conclude that $A=A_X$. This completes the proof of our claim and thus the proof of the lemma. We are now ready to prove the main result of this section. \[t-p\] The problem of finding a minimum independent feedback vertex set of a graph of diameter $2$ can be solved in polynomial time. Let $G$ be an $n$-vertex graph of diameter $2$. We first check in polynomial time whether $G$ contains a vertex $u$ such that $G-u$ is bipartite. If so, then we apply Lemma \[l-1\]. If not, then we check in polynomial time whether $G$ contains a set $X$ of size $4\leq |X|\leq 5$ such that $(A_X,V\setminus A_X)$ is a near-bipartite decomposition. If so, then $G$ is near-bipartite and we apply Lemma \[l-2\]. If not, then $G$ is not near-bipartite due to Theorem \[t-yy06\]. We note that the running time of the algorithm in Theorem \[t-p\] is determined by the time it takes to find and process each set $X$ of size $4\leq |X|\leq 5$. This takes $O(n^7)$ time, as checking the existence of a set $X$ takes $O(n^5)$ time using brute force, determining the $2$-neighbour set $A_X$ takes $O(n)$ time and checking if $(A_X,V\setminus A_X)$ is a near-bipartite decomposition takes $O(n^2)$ time. Near-Bipartiteness for Diameter $3$ {#s-diam3} =================================== In this section we prove that [Near-Bipartiteness]{} is [[NP]{}]{}-complete for graphs of diameter $3$. In order to prove this, we use a construction of Mertzios and Spirakis [@MS16]. To introduce this construction, we first consider the constraint graph $J$ defined in Figure \[fig:constraint-graph\]. \[lem:constraint-graph\] Let $X$ be a subset of $\{X_1, X_2, X_3\} \subset V(J)$ containing at most two vertices. Then there exists a near-bipartite decomposition $(A,B)$ of $J$ such that, for $1 \leq p \leq 3$, $X_p \in A$ if and only if $X_p \in X$. Noting the automorphic equivalence of $X_2$ and $X_3$, it is sufficient to consider the following two cases. If $X$ is a subset of $\{X_1, X_2\}$, let $A=X \cup \{Y_6,Y_7\}$. If $X = \{X_2,X_3\}$, let $A= \{X_2,X_3,Y_4\}$. Notice that there is no near-bipartite decomposition of $J$ with $\{X_1,X_2,X_3\} \subseteq A$. Combined with the above lemma, this gives an idea of how this will be used later. The vertices $X_1$, $X_2$ and $X_3$ will represent literals in a clause of an instance of [$3$-Sat]{} and membership of $A$ will indicate that a literal is false: thus $A$ can be extended to a near-bipartite decomposition except when every literal is false. (In [@MS16], a weaker result was shown: one can always find a $3$-colouring of $J$ such that members of a chosen *proper* subset of $\{X_1, X_2, X_3\}$ belong to the same class and excluded members do not belong to that class.) Let $\phi$ be an instance of [$3$-Sat]{} with $m$ clauses $C_1, \ldots, C_m$ and $n$ variables $v_1,\ldots,v_n$. We may assume that each clause has three distinct literals. For a clause $C_k$ in $\phi$, we describe a *clause graph* ${\mathcal C}^k$, illustrated within Figure \[fig:hphi\]. We think of ${\mathcal C}^k$ as an array of $n+5m+1$ rows and eight columns. In each row except the last, every (row,column) position contains exactly two vertices, which we refer to as the *true vertex* and the *false vertex*, and we say that these two vertices are *mates*. The first $n$ rows form the *variable block* of the graph and we think of row $i$ as representing the variable $v_i$. The next $5m$ rows are made up of $m$ *clause blocks* ${\mathcal C}^{k,1}, {\mathcal C}^{k,2}, \ldots, {\mathcal C}^{k,m}$, each of five rows. Every true vertex of the variable and clause blocks is joined by an edge to every false vertex in the same row except its mate. Hence the vertices of each row induce a complete bipartite graph minus a matching. In the final row, each column contains a single vertex, and each of these vertices is joined by an edge to every other vertex in the same column. We call this row the *dominating block*. We complete the definition of the clause graph by describing how we add further edges so that it contains the constraint graph $J$ as an induced subgraph. Let the literals of $C_k$ be $x_{\ell_1},x_{\ell_2},x_{\ell_3}$. We choose vertices from the first three columns of the variable block of ${\mathcal C}^k$ that we will denote $X_1^k, X_2^k, X_3^k$ to represent the literals. If $x_{\ell_p}$ is the variable $v_i$, then we choose as $X_p^k$ a vertex from row $i$ and column $p$, and choose the true vertex if the literal is positive and the false vertex if the literal is a negated variable. For $p \in \{4,\ldots,8\}$, let $Y_p^k$ be the true vertex from the $(p-3)$th row and $p$th column of the clause block ${\mathcal C}^{k,k}$. Finally add the ten edges $\{ X_1^kY_4^k, X_2^kY_5^k, X_2^kY_8^k, X_3^kY_6^k, X_3^kY_7^k, Y_4^kY_5^k, Y_4^kY_6^k, Y_5^kY_7^k, Y_6^kY_8^k, Y_7^kY_8^k\}$ so that $\{X_1^k,X_2^k,X_3^k, Y_4^k, Y_5^k,Y_6^k,Y_7^k,Y_6^k\}$ induces the constraint graph $J$. =\[circle,draw=black, fill=black, minimum size=5pt, inner sep=1pt\] =\[draw,black\] We now define the graph $H_\phi$. It contains: - the disjoint union of clause graphs ${\mathcal C}^k$, $1 \leq k \leq m$ (we think of the clause graphs as being arranged side-by-side, so that they form an array of $n+5m+1$ rows and $8m$ columns), - edges from each true vertex of each clause graph to each false vertex in the same row of other clause graphs, and - an additional vertex $v_0$ joined to each vertex in the dominating block of each clause graph. Note that each column of $H_\phi$ contains exactly one vertex that is in a constraint graph $J$ and the only rows that contain more than one such vertex are those in the variable block. For an instance $\phi$ of [$3$-Sat]{}, $H_\phi$ has diameter $3$. Note that in [@MS16], Lemma 2 proves the bound on the diameter for a graph that is a spanning subgraph of $H_\phi$ which is, of course, sufficient for an upper bound for the diameter of $H_\phi$ and it is easy to see that the diameter is not less than $3$. We note also that $H_\phi$ does not contain any triangles or any vertices that are siblings (two vertices are siblings if the neighbourhood of one is a subset of the neighbourhood of the other) so [Near-Bipartiteness]{} is also [[NP]{}]{}-complete for such instances. \[thm:diam3\] [Near-Bipartiteness]{} is [[NP]{}]{}-complete for graphs of diameter at most $3$. We prove that [$3$-Sat]{} can be polynomially-reduced to [Near-Bipartiteness]{} by showing that $\phi$ is satisfiable if and only if $H_\phi$ has a near-bipartite decomposition $(A,B)$. ($\Rightarrow$) Suppose that $\phi$ has a satisfying assignment. Let $v_0$ be in $A$, and let the vertices of all the dominating blocks be in $B$. If the variable $v_i$ is true, then let $B$ contain all the true vertices of row $i$ of the variable blocks of each clause graph. Otherwise let $B$ contain the false vertices. In each case, let $A$ contain the mates of these vertices. Consider the constraint graph that is an induced subgraph of each clause graph. The vertices $X_1$, $X_2$ and $X_3$ have been assigned to either $A$ or $B$ with at most two, representing false literals, belonging to $A$. By Lemma \[lem:constraint-graph\], we can assign the remaining vertices of the subgraph (which are all true vertices of clause blocks) to $A$ and $B$ such that on the subgraph they form a near-bipartite decomposition. When we assign a true vertex of a clause block to $A$ or $B$, we assign all other true vertices in the same row of $H_\phi$ to the same set and assign their mates to the other set. As each row of the clause blocks contains only one vertex in a constraint graph, this process assigns every vertex in $H_\phi$ to exactly one of $A$ and $B$, and we have assigned every vertex of $H_\phi$ to $A$ or $B$. It is immediately clear that $A$ is an independent set. We must show that $B$ contains no cycles. We know that $B$ contains all the vertices of the dominating blocks and, in each row, either all the true vertices or all the false vertices. Thus if $B$ contains a cycle then all the vertices of the cycle belong to the same clause graph (the only edges going between distinct clause graphs are those joining true vertices to false vertices in the same row). Let $G_B$ be a subgraph of a clause graph induced by vertices of $B$. Then each true and false vertex not in the constraint graph has degree $1$ (due to the edge joining it to the dominating block), and each vertex in the dominating block has at most one neighbour with degree more than $1$ (since it only has one neighbour in the constraint graph). Thus if $G_B$ contains a cycle then it belongs to the constraint graph, contradicting how $A$ and $B$ were chosen. ($\Leftarrow$) Suppose $A$ and $B$ form a near-bipartite decomposition of $H_\phi$. Then $B$ can be decomposed into two independent sets, and these, along with $A$, can be considered a $3$-colouring. In [@MS16 Theorem 5], it is shown that if $H_\phi$ has a $3$-colouring, then $\phi$ is satisfiable. Conclusions {#s-con} =========== We completed the computational complexity classifications of [Near-Bipartiteness]{} and [Independent Feedback Vertex Set]{} for graphs of diameter $k$ for every integer $k\geq 0$. We showed that the complexity of both problems jumps from being polynomial-time solvable to [[NP]{}]{}-complete when $k$ changes from $2$ to $3$. We recall that near-bipartite graphs are $3$-colourable. Interestingly, the complexity of [$3$-Colouring]{} for graphs of diameter $k$ has not yet been settled, as there is one remaining case left, namely when $k=2$. This is a notorious open problem, which has been frequently posed in the literature (see, for example, [@BKM12; @BFGP13; @MS16; @Pa15]). We note that the approach of solving [Near-Bipartiteness]{} and [Independent Feedback Vertex Set]{} for graphs of diameter $2$ does not work for [$3$-Colouring]{}. For instance, we cannot bound the size of the set $X$ in Lemma \[l-2\] if we drop the condition that the union of two colour classes must induce a forest. [^1]: This paper received support from EPSRC (EP/K025090/1), London Mathematical Society (41536), the Leverhulme Trust (RPG-2016-258) and Fondation Sciences Mathématiques de Paris. The hardness result (Theorem \[thm:diam3\]) of this paper has been announced in an extended abstract of the Proceedings of MFCS 2017 [@BDFJP17].

--- abstract: 'We present a class of 2D systems which shows a counterintuitive property that contradicts a semi classical intuition: A 2D quantum particle “prefers” tunneling through a barrier rather than traveling above it. Viewing the one particle 2D system as the system of two 1D particles, it is demonstrated that this effect occurs due to a specific symmetry of the barrier that forces excitations of the interparticle degree of freedom that, in turn, leads to the appearance of an effective potential barrier even though there is no “real” barrier. This phenomenon cannot exist in 1D.' author: - 'Denys I. Bondar' - 'Wing-Ki Liu' - 'Misha Yu. Ivanov' bibliography: - '2D\_Tunnelling.bib' title: Enhancement and suppression of tunneling by controlling symmetries of a potential barrier --- Introduction ============ Quantum tunneling has been one of the most important problems in quantum mechanics since its foundation. The simplest problems of tunneling are one-dimensional, which is where our intuition on tunneling comes from. The extension of 1D tunneling to many dimensions is not straightforward. There are many peculiarities that appear in many dimensional cases that do not exist in 1D (for systematic studies of such differences see, e.g., Refs. [@Chabanov1999; @Chabanov2000; @Zakhariev2008]). Quite often many dimensional tunneling is equated to the tunneling of complex (i.e., many particle) systems. Key aspects of the quantum mechanical tunneling of complex systems were analyzed by Zakhariev [*et al.*]{} [@Zakhariev1964; @Amirkhanov1966] in the mid-1960s; nevertheless, this problem has become an area of active research only in the past few decades (see, e.g., Refs. [@Tomsovic1998; @Takagi2002; @Zakhariev2002; @Razavy2003a; @Ankerhold2007] and references therein). Tunneling of a diatomic molecule has been studied in Refs. [@Goodvin2005; @Goodvin2005a; @Lee2006; @Hnybida2008; @Shegelski2008; @Kavka2010]. Mechanisms of single and double proton transfer have been modelled by multidimensional tunneling [@Smedarchina1995; @Smedarchina2007; @Smedarchina2008]. Time-dependent numerical study of tunneling dynamics of a two-particle quantum system with an internal degree of freedom has been analyzed in Ref. [@Volkova2006], and an enhancement of the tunneling probability due to the formation of a long-lived resonant state of the system in the barrier region has been discovered (similar analytical studies have been done in Ref. [@Yamamoto1996]). It has also been suggested that collective tunneling of electrons may have an important contribution to multiple ionization of atoms in a superstrong laser field [@Kornev2009]. Quantum tunneling of complex systems is not only of theoretical interest. Recent experiments where this phenomenon is observed directly include tunneling of a singe hydrogen atom [@Lauhon2000], resonant tunneling of Cooper pairs [@Toppari2007], and a bosonic Josephson junction consisting of two weakly coupled Bose-Einstein condensates in a macroscopic double-well potential [@Albiez2005]. We present a class of 2D systems which has a counterintuitive property that contradicts the semi classical intuition: A 2D quantum particle “prefers” tunneling through to flying above a barrier. According to our analysis, such “paradoxical” dynamics is caused by a peculiar symmetry of the barrier that leads to excitations of an interparticle degree of freedom. There is no 1D counterpart of such systems. The rest of the article is organized as follows: In Sec. \[Sec2\], we present the systems and describe the counterintuitive effect. The observed “paradox” is explained in Sec. \[Sec3\]. Connections between the phenomenon and classical physics are discussed in Sec. \[Sec4\]. Concluding remarks and a possible application of the effect to quantum control are presented in the last section. Formulation of the “paradox” {#Sec2} ============================ Let us consider a particle moving in 2D (coordinates $x_1$ and $x_2$) toward a barrier located at the origin $x_1=x_2=0$. The initial velocity of the particle is chosen to be directed along the diagonal $x_1=x_2$, incident on the barrier from the third (where $x_1 < 0$ and $x_2 < 0$) to the first quadrant (where $x_1 > 0$ and $x_2 > 0$); see Fig. \[Fig\_potentials\]. While the numerical calculations are done for a specific Hamiltonian, the analytical analysis that follows relies exclusively on the symmetry properties of the 2D potential, making our conclusions, drawn from the numerical analysis, general. The model Hamiltonian for our system is chosen as (atomic units are used throughout) $$\begin{aligned} && \hat{H}_N (\alpha) = -\frac 1{2} \left( \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2} \right) + \Omega_N (\alpha; x_1, x_2), \label{HamiltonianCartesianDef} \\ && \Omega_N (\alpha; x_1, x_2) = \alpha V(x_1) + 3V(x_2) + U_N(x_2 - x_1). \end{aligned}$$ where $N=1,2,4$ and $\alpha$ being an arbitrary real parameter. The potentials $V(x_1)$ and $V(x_2)$ describe the potential barriers near the origin, for the motion along each of the two coordinates. The parameter $\alpha$ allows us to vary the relative height of the barriers. We have chosen $$\begin{aligned} V(x) = x \exp(-x^2),\end{aligned}$$ which corresponds to a potential barrier preceded by a potential well. The potential $U_N(\rho)$ describes the coupling between the two degrees of freedom. In the absence of $U_N(\rho)$ \[i.e., for $U_N(\rho)=0$\], the 2D dynamics breaks into two uncoupled 1D motions. Nontrivial features in tunneling appear as the result of nonzero coupling of the two degrees of freedom. Before we describe the choice of $U_N(\rho)$ in our model, let us introduce the center of mass ($R$) and relative ($\rho$) coordinates $$\begin{aligned} \label{SimpleCMandRelativeCoord} R = (x_1 + x_2)/2, \qquad \rho = x_2 -x_1.\end{aligned}$$ The Hamiltonian (\[HamiltonianCartesianDef\]) in these new coordinates reads $$\begin{aligned} \label{Hamiltonian_NewCoord} \hat{H}_N(\alpha) &=& \frac{-1}{2M}\frac{\partial^2}{\partial R^2} + \frac{-1}{2\mu}\frac{\partial^2}{\partial \rho^2} + \Omega_N (\alpha; \rho, R), \nonumber\\ \Omega_N (\alpha; R, \rho) &=& \alpha V\left( R - \rho/2\right) + 3V\left( R + \rho/2\right) \nonumber\\ &&+ U_N(\rho), \label{PotentialDeff}\end{aligned}$$ where $\mu = 1/2$ and $M = 2$. Now we can specify the potential that couples the two degrees of freedom and see its role in the problem. If $U_N(\rho)$ is attractive, as it is in our calculations, it may support bound states. These bound states, and their symmetries, play a key role. Here, we set $U_N$ to describe a short-range attraction, $$\begin{aligned} \label{Un_potentials} U_N (\rho) = -A\exp\left( -\rho^2/r_N^2 \right).\end{aligned}$$ Varying the parameter $r_N$, we change the number of bound states supported by the attracting potential. In the calculations, we use $A=2$ and $r_1 = 1$, $r_2 = 1.961$, $r_4 =3.162$ corresponding to one, two, and four bound states supported by the Hamiltonian $-1/(2\mu) \partial^2/\partial \rho^2 + U_N(\rho)$. The energies of these states are $-0.955$ for $U_1$; $-1.377$ and $-0.372$ for $U_2$; and, finally, $-1.590$, $-0.856$, $-0.308$, and $-0.012$ for $U_4$. Following Ref. [@Volkova2006], we study the tunneling within the time-dependent approach solving the time-dependent Schrödinger equation, $$\begin{aligned} \label{TimDepSchEq} \left[ i\partial/\partial t - \hat{H}_N(\alpha)\right] \Psi_N(\alpha; t, x_1, x_2) = 0,\end{aligned}$$ with the initial condition at $t=0$ that reads in the coordinates (\[SimpleCMandRelativeCoord\]) as $$\begin{aligned} \label{InitialCondition} \Psi_N(\alpha; 0, R, \rho) = C \phi_g(\rho) e^{ -(R-\bar{R})^2/\left(2\sigma_R^2\right) + i\sqrt{2M E_{cm}}R }.\end{aligned}$$ Here $C$ is a normalization constant and $\phi_g(\rho)$ is the ground state of the interparticle Hamiltonian, $-1/(2\mu) \partial^2/\partial \rho^2 + U_N(\rho)$. In all our studies, we set $m=1$, $\bar{R} = -55$, $\sigma_R = 3$, and $E_{cm} = 1$ (all values are in atomic units). The initial wave function (\[InitialCondition\]) is localized in the third quadrant, and we calculate the probability of finding the particle in the first quadrant, i.e., the probability of tunneling at later time $\tau$. The reason for using $\phi_g(\rho)$ as the relative coordinate part of the initial wave function is that we wanted to avoid spreading of the wave packet along $\rho$ before it reached the potential barrier. We also present the initial expectation value of energy $$\begin{aligned} \label{AvEDef} \bar{E}_N = {\left\langle{\Psi_N(\alpha; 0, x_1, x_2) }\right|}\hat{H}_N(\alpha) {\left| \Psi_N(\alpha; 0, x_1, x_2) \right\rangle}, \\ \bar{E}_1 = 0.05911, \quad \bar{E}_2 = -0.3631, \quad \bar{E}_4 = -0.5766 \nonumber\end{aligned}$$ (all values are in atomic units). Rigorously speaking, $\bar{E}_N$ depends on $\alpha$; however, this dependence is very weak because the initial wave function (\[InitialCondition\]), independent of $\alpha$, is mostly localized in the region where the potential barrier, $\alpha V(x_1) + 3V(x_2)$, vanishes. The probabilities of tunneling, disintegration (see below for the clarification of this term), and reflection are defined as follows: $$\begin{aligned} P_T^{(N)} (\alpha, \tau) &=& \int_0^{\infty} dx_1 \int_0^{\infty} dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2, \label{Prob_T_deff}\\ P_D^{(N)} (\alpha, \tau) &=& \int_{-\infty}^0 dx_1 \int_0^{\infty} dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2 \nonumber\\ &+& \int_0^{\infty} dx_1 \int_{-\infty}^0 dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2, \label{Prob_D_deff}\\ P_R^{(N)} (\alpha, \tau) &=& \int_{-\infty}^0 dx_1 \int_{-\infty}^0 dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2. \label{Prob_R_deff}\end{aligned}$$ However, since the potential barrier, $\alpha V(x_1) + 3V(x_2)$, has “well” and “hill” regions, we also employ the corresponding “shifted” probabilities to exclude regions were the potential barrier is localized $$\begin{aligned} p_t^{(N)} (\alpha, \tau) &=& \int_3^{\infty} dx_1 \int_3^{\infty} dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2, \label{prob_t_deff} \\ p_d^{(N)} (\alpha, \tau) &=& \int_{-\infty}^{-3} dx_1 \int_3^{\infty} dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2 \nonumber\\ &+& \int_3^{\infty} dx_1 \int_{-\infty}^{-3} dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2, \label{prob_d_deff}\\ p_r^{(N)} (\alpha, \tau) &=& \int_{-\infty}^{-3} dx_1 \int_{-\infty}^{-3} dx_2 \, \left| \Psi_N(\alpha; \tau, x_1, x_2) \right|^2, \label{prob_r_deff}\\ p_s^{(N)} (\alpha, \tau) &=& 1 - p_t^{(N)} - p_d^{(N)} - p_r^{(N)},\end{aligned}$$ $p_s^{(N)}$ is the probability that a particle is trapped in the neighborhood of the potential barrier (in fact, mostly in the well region of the potential barrier). We introduce these quantities to verify that our conclusions are not due to variations of the probability density in a neighborhood of the potential barrier (see Ref. [@EPAPS_Animations]). The Hamiltonian (\[HamiltonianCartesianDef\]) can be viewed as the Hamiltonian of two 1D particles, where $x_{1,2}$ are coordinates of the first and second particles, respectively. This interpretation is crucial to explain the observed effect. Utilizing such a point of view, quantities $P_D^{(N)}$ \[Eq. (\[Prob\_D\_deff\])\] and $p_d^{(N)}$ \[Eq. (\[prob\_d\_deff\])\] can be indeed labeled as the probabilities of disintegration because the particles are flying apart (i.e., the two particle system is disintegrating) if after sufficiently long time $\tau$ either $x_1>0$ and $x_2 <0$ or $x_1<0$ and $x_2>0$. ![(Color online) Plots of the potential $\Omega_N (\alpha; R, \rho)$ \[Eq. (\[PotentialDeff\])\] for different $\alpha$ and $N$. Roman numerals in plot (a) label quadrants. Black solid lines denote the level sets of the expectation value of the initial energy, $\bar{E}_N$ \[Eq. (\[AvEDef\])\], i.e., the boundary between the classically allowed and classically forbidden regions. (a) $N=1$ and $\alpha=3$; (b) $N=1$ and $\alpha = -3$; (c) $N=2$ and $\alpha=3$; (d) $N=2$ and $\alpha=-3$; (e) $N=4$ and $\alpha=3$; (f) $N=4$ and $\alpha=-3$. []{data-label="Fig_potentials"}](Potentials.pdf) Before stating the results of numerical calculations, let us qualitatively analyze dynamics of the system within a semiclassical consideration. Figure \[Fig\_potentials\] presents the plots of the potentials (\[PotentialDeff\]). According to the initial condition \[see Eqs. (\[InitialCondition\]) and (\[AvEDef\])\], the particle is located on the axis $\rho=0$ and its initial velocity is directed along this axis toward the first quadrant, and the amplitude of the velocity is chosen such that the total energy of the particle equals $\bar{E}_N$; hence, the boundaries between the classically allowed and classically forbidden regions are drawn by solid black lines in Fig. \[Fig\_potentials\]. Now compare Fig. \[Fig\_potentials\](a) with Fig. \[Fig\_potentials\](b). Since the semi classical counterpart of our quantum particle “experiences” the barrier in Fig. \[Fig\_potentials\](a) (penetration though a barrier is of exponentially small probability) and does not “feel” any barrier in Fig. \[Fig\_potentials\](b) (the particle moves solely in the classically allowed region) while traveling along the axis $\rho=0$, then one would intuitively conclude that the probability of finding the particle in the first quadrant in Fig. \[Fig\_potentials\](a) ought to be smaller than in Fig. \[Fig\_potentials\](b). By the same token, the very same probabilities in Figs. \[Fig\_potentials\](c) and \[Fig\_potentials\](e) should be smaller than in Figs. \[Fig\_potentials\](d) and \[Fig\_potentials\](f), respectively. Further discussions of the phenomenon from the point of view of classical trajectories are presented in Sec. \[Sec4\]. The results presented in Figs. \[Fig\_tunneling\_prob\]–\[Fig\_probability\_tunnel\_time\] are obtained from the numerical solution of the time-dependent Schrödinger equation (\[TimDepSchEq\]) by means of the split-operator method with an absorbing boundary condition. Figures \[Fig\_tunneling\_prob\]–\[Fig\_reflection\_prob\] show the dependence of the probabilities of tunneling, disintegration, and reflection as function of the parameter $\alpha$ that characterizes the asymmetry of the potential barrier. Dynamics of tunneling processes occurring in Figs. \[Fig\_potentials\](a)–(f) are visualized as animations, which are available for viewing in Ref. [@EPAPS_Animations]. Remarkably, while our qualitative conclusion reached regarding Figs. \[Fig\_potentials\](a) and \[Fig\_potentials\](b) is indeed correct (see Fig. \[Fig\_probability\_tunnel\_time\]), the conclusions regarding the comparison of Figs. \[Fig\_potentials\](c) and \[Fig\_potentials\](d) and Figs. \[Fig\_potentials\](e) and \[Fig\_potentials\](f) turn out to be completely wrong. In other words, the particle does prefer to “go” through the barrier \[Figs. \[Fig\_potentials\](c) and \[Fig\_potentials\](e)\] rather than flying above the barrier \[Figs. \[Fig\_potentials\](d) and \[Fig\_potentials\](e)\]. Furthermore, even though the potentials $\Omega_1 (\pm 3; R, \rho)$ look similar to $\Omega_{2,4} (\pm 3; R, \rho)$, the particle favors motion above the barrier \[Fig. \[Fig\_potentials\](b)\] rather than penetration through the barrier \[Fig. \[Fig\_potentials\](a)\] for the former pair of the potentials. This “paradox” is resolved in the next section. ![(Color online) Probabilities of tunneling as a function of the height of the barrier ($\alpha$). (a) $P_T^{(N)}(\alpha, 150)$ \[Eq. (\[Prob\_T\_deff\])\]; (b) $p_t^{(N)}(\alpha, 150)$ \[Eq. (\[prob\_t\_deff\])\]. []{data-label="Fig_tunneling_prob"}](TunnellingProb.pdf) ![(Color online) Probabilities of disintegration as a function of the height of the barrier ($\alpha$). (a) $P_D^{(N)}(\alpha, 150)$ \[Eq. (\[Prob\_D\_deff\])\]; (b) $p_d^{(N)}(\alpha, 150)$ \[Eq. (\[prob\_d\_deff\])\]. []{data-label="Fig_disintegr_prob"}](DesintegrProb.pdf) ![(Color online) Probabilities of reflection as a function of the height of the barrier ($\alpha$). (a) $P_R^{(N)}(\alpha, 150)$ \[Eq. (\[Prob\_R\_deff\])\]; (b) $p_r^{(N)}(\alpha, 150)$ \[Eq. (\[prob\_r\_deff\])\]. []{data-label="Fig_reflection_prob"}](ReflectionProb.pdf) ![(Color online) The probabilities of tunneling $P_T^{(N)} (\alpha, \tau)$ \[Eq. (\[Prob\_T\_deff\])\] and $p_t^{(N)} (\alpha, \tau)$ \[Eq. (\[prob\_t\_deff\])\] as functions of time $\tau$. (a) Comparison of Fig. \[Fig\_potentials\](a) vs. Fig. \[Fig\_potentials\](b); (b) comparison of Fig. \[Fig\_potentials\](c) vs. Fig. \[Fig\_potentials\](d); (c) comparison of Fig. \[Fig\_potentials\](e) vs. Fig. \[Fig\_potentials\](f).[]{data-label="Fig_probability_tunnel_time"}](ProbTunnelingTime.pdf) Explanation of the effect {#Sec3} ========================= The “paradox” posed in Sec. \[Sec2\] is resolved in this section by analyzing a perturbation theory solution of the Schrödinger equation. In this section, we shall study a general two-1D-particles system given by the Hamiltonian $$\begin{aligned} \label{HamiltonianR_RHO} \hat{H} &=& \frac{-1}{2M}\frac{\partial^2}{\partial R^2} + \frac{-1}{2\mu}\frac{\partial^2}{\partial \rho^2} + \nonumber\\ && U(\rho) + V_1\left( R - \frac{\mu}{m_1} \rho\right) + V_2\left( R + \frac{\mu}{m_2} \rho\right),\end{aligned}$$ which is already written in the (general) center of mass ($R$) and relative ($\rho$) coordinates, $$\begin{aligned} && R = (m_1 x_1 + m_2 x_2)/(m_1 + m_2), \quad \rho = x_2 -x_1, \nonumber\\ && \mu = m_1 m_2 /M, \quad M = m_1 + m_2, \nonumber\end{aligned}$$ where $x_{1,2}$ ($m_{1,2}$), as previously, being the coordinates (masses) of the first and second particles, respectively. Let us introduce the following notation $$\begin{aligned} \hat{U}(t_f, t_i) &=& \hat{T} \exp\left[ -i \hat{H}(t_f - t_i)\right], \\ \hat{U}_0(t_f, t_i) &=& \hat{T} \exp\left[ -i( \hat{H} - V_1 - V_2)(t_f - t_i) \right],\end{aligned}$$ for total and unperturbed propagators, respectively. The sum of the potential barriers, $V_1 + V_2$, shall be considered as a perturbation. The eigenfunctions, ${\left| n \right\rangle}$, and eigenvalues, $E_n$, of the internal motion are the solutions of the problem $$\begin{aligned} \label{InternalMotionEigenstates} \left[ \frac{-1}{2\mu}\frac{d^2}{d\rho^2} + U(\rho) \right]{\left| n \right\rangle} = E_n{\left| n \right\rangle},\end{aligned}$$ where the index $n$ denotes bound and continuous states. Introducing ${\left| n k \right\rangle} \equiv {\left| n \right\rangle} \otimes {\left| k \right\rangle}$, where ${\left| k \right\rangle}$ is an eigenfunction of the free motion of the center of mass $\langle R {\left| k \right\rangle} \equiv \exp(ikR)/\sqrt{2\pi}$ – a plane wave and ${\left| n \right\rangle}$ is an eigenstate of the internal motion, the unperturbed propagator reads $$\begin{aligned} \hat{U}_0(t_f, t_i) = { \mathop{\hbox to4pt{ $\sum$ \hss}{\displaystyle\int}} }_n \int dk e^{-i\left(E_n + \frac{k^2}{2M}\right)(t_f - t_i)}{\left| n k \right\rangle}{\left\langle{k n}\right|}\end{aligned}$$ The total propagator is a solution of the Lippmann-Schwinger equation written in the “post” form $$\begin{aligned} \hat{U}(t_f, t_i) &=& \hat{U}_0(t_f, t_i) \nonumber\\ &-& i \int_{t_i}^{t_f} dt \hat{U}_0(t_f, t) [V_1 + V_2] e^{\epsilon t} \hat{U}(t, t_i),\end{aligned}$$ where we set $\epsilon \to 0$. Assuming that the initial condition ${\left| \Psi(t_i) \right\rangle} \equiv {\left| n \right\rangle}\otimes{\left| \psi_{in} \right\rangle}$, where ${\left| n \right\rangle}$ is one of the eigenstates (\[InternalMotionEigenstates\]) and ${\left| \psi_{in} \right\rangle}$ is a wave packet localized before the barriers \[see, e.g., Eq. (\[InitialCondition\])\], we obtain $$\begin{aligned} {\left| \Psi(+\infty) \right\rangle} &\approx& \hat{U}_0(+\infty, -\infty){\left| \Psi(-\infty) \right\rangle} + {\left| \Psi_1 \right\rangle} + {\left| \Psi_2 \right\rangle}, \label{Psi_at_plus_infty}\\ {\left| \Psi_1 \right\rangle} &=& -2\pi i{ \mathop{\hbox to4pt{ $\sum$ \hss}{\displaystyle\int}} }_{n'} \int dk dk' \delta\left( E_{n'} + \frac{k'^2}{2M} - E_n - \frac{k^2}{2M} \right) \nonumber\\ && \times {\left| n' k' \right\rangle} W_{n n'}(k-k') \langle k{\left| \psi_{in} \right\rangle}, \\ {\left| \Psi_2 \right\rangle} &=& -2\pi i{ \mathop{\hbox to4pt{ $\sum$ \hss}{\displaystyle\int}} }_{n'', n'} \int dkdk'dk'' {\left| n'' k'' \right\rangle}\langle k {\left| \psi_{in} \right\rangle} \nonumber\\ && \times \delta\left( E_{n''} + k''^2 /[2M] - E_n - k^2/[2M]\right) \nonumber\\ &&\times \frac{ W_{n' n''}(k'-k'') W_{n n'}(k-k')}{E_n + k^2/(2M) - E_{n'} - k'^2/(2M) + i0},\end{aligned}$$ where $W_{n n'}(k - k') = {\left\langle{k' n'}\right|} V_1 + V_2 {\left| n k \right\rangle}$ and ${\left| \Psi_{1,2} \right\rangle}$ are the first- and second-order corrections, respectively. Higher-order corrections can be derived in a similar manner, but what is important for our further analysis is that they are functions of $W$. We simplify the matrix element $W$ by representing it as follows $$\begin{aligned} \label{W_initial_expression} W_{n n'}(k - k') &=& \int \frac{dR d\rho dq}{2\pi} e^{i(k-k')R}\phi_{n'}^*(\rho)\phi_n(\rho) \nonumber\\ && \times [ V_1(q)\delta(R - \mu\rho/m_1 - q) \nonumber\\ && + V_2(q)\delta(R + \mu\rho/m_2 - q)],\end{aligned}$$ where $\phi_n(\rho) = \langle \rho {\left| n \right\rangle}$. After trivial integration over $R$, we obtain $$\begin{aligned} \label{Simplified_W_general} && W_{n n'}(k - k') = \mathrsfs{F}_{n n'}\left( \frac{\mu}{m_1}[k-k']\right)\int \frac{dq}{2\pi} e^{i(k-k')q} V_1(q) \nonumber\\ && \quad + \mathrsfs{F}_{n n'}\left( \frac{\mu}{m_2}[k'-k]\right)\int \frac{dq}{2\pi} e^{i(k-k')q} V_2(q),\end{aligned}$$ where the quantity, $$\begin{aligned} \label{Formfactor_Def} \mathrsfs{F}_{n n'}(p) = \int d\rho e^{ip\rho} \phi_{n'}^*(\rho)\phi_n(\rho),\end{aligned}$$ is called the form factor, and it is well known in the scattering theory. Its physical interpretation is the probability amplitude of transferring a momentum $p$ from the center of mass to the interparticle degree of freedom by making the transition $n\to n'$. Now we consider the case of identical particles: $m_1 = m_2 = m$ and $U(-\rho) = U(\rho)$. Then, there are two types of the eigenstates of the internal motion: even ($+$), $\phi_n(-\rho) = \phi_n(\rho)$, and odd ($-$), $\phi_n(-\rho) = -\phi_n(\rho)$. Since $ \mathrsfs{F}_{n n'}(-p) = \mathrsfs{F}_{n n'}(p)$ \[$\mathrsfs{F}_{n n'}(-p) = -\mathrsfs{F}_{n n'}(p)$\] in the case of $\phi_n$ and $\phi_{n'}$ being of the same (different) parity, Eq. (\[Simplified\_W\_general\]) takes the form $$\begin{aligned} \label{W_equivalent_particles} W_{n n'}(k - k') &=& \left\{ \begin{array}{ccl} \mathrsfs{F}_{n n'}\left( [k - k']/2\right) \int \frac{dq}{2\pi}e^{i(k-k')q} [V_1(q) + V_2(q)], &\mbox{if}& \mbox{$\phi_n$ and $\phi_{n'}$ have the same parity} \\ \mathrsfs{F}_{n n'}\left( [k - k']/2\right) \int \frac{dq}{2\pi}e^{i(k-k')q} [V_1(q) - V_2(q)], &\mbox{if}& \mbox{$\phi_n$ and $\phi_{n'}$ have different parities}, \end{array} \right.\end{aligned}$$ which is the product of the form factor and the Fourier transform of either the sum of the barriers or the difference of the barriers, depending on the parities of the initial and final states. Two conclusions can be readily drawn from Eq. (\[W\_equivalent\_particles\]): First, considering tunneling within the time-independent picture, Amirkhanov and Zakhariev [@Amirkhanov1966] have discovered the violation of the barrier penetration symmetry for complex particles, i.e., the penetration of composite particles through asymmetric barriers in opposite directions may differ. \[Note that the rates of tunneling of an elementary (structureless) particle are exactly the same in both the directions within the time-independent approach.\] The situations when the system approaches the barrier from the left and from the right differ only by inversion of the sign of the momentum of the center of mass. The only part of the wave function (\[Psi\_at\_plus\_infty\]) that maintains the dependence on the sign of the momentum is the matrix element (\[Simplified\_W\_general\]). Thus, the discussed phenomenon of tunneling asymmetry is manifested in our consideration as a physical consequence of the property $$\begin{aligned} \label{SymmetryW} W_{n n'}(k'-k) = W_{n n'}(k-k') \Longleftrightarrow V_1(q) = V_2(-q).\end{aligned}$$ Equation (\[SymmetryW\]) is not only an alternative and perhaps faster way of achieving the main result of Ref. [@Amirkhanov1966] but also the generalization of their conclusion for the case of nonidentical barriers ($V_1 \neq V_2$). Second, Eq. (\[W\_equivalent\_particles\]) basically provides an explanation of the observed anomalies related to the potentials $\Omega_n (\alpha; x_1, x_2)$ \[Eq. (\[PotentialDeff\])\] pictured in Fig. \[Fig\_potentials\], if we recall that tunneling of a 2D particle in the potential $\Omega_n (\alpha; x_1, x_2)$ is equivalent to collective tunneling of two equal 1D particles through the potential barriers $V_1(x_1) = \alpha V(x_1)$ and $V_2(x_2) = 3 V(x_2)$. Hence, Eq. (\[SymmetryW\]) determines the selection rule for transitions between states of the internal degree of freedom induced by (collective) tunneling. The key point is that the probability of collective tunneling strongly depends on whether an excitation of the internal degree of freedom is possible. If a system is initially in the ground state and the excitations are allowed, then by going to an excited state, the center of mass of the system lowers its kinetic energy (i.e., increasing the width of the barrier), consequently reducing the probability of tunneling (see Refs. [@Zakhariev2002; @Goodvin2005; @Goodvin2005a; @Hnybida2008; @Shegelski2008] and reference therein). We recall that the parity of the ground state is even, the first excited state – odd, the second excited state – even, etc.; therefore, according to Eq. (\[W\_equivalent\_particles\]), if $$\begin{aligned} \label{Condition_not_forcing_transitions} V_1(q) = V_2(q),\end{aligned}$$ then the transition from the ground state to the first excited state is forbidden, but if $$\begin{aligned} \label{Condition_forcing_transitions} V_1(q) = -V_2(q),\end{aligned}$$ then the transition is allowed. Paraphrasing, we note that if condition (\[Condition\_not\_forcing\_transitions\]) takes place then the interparticle degree of freedom may stay the same while the center of mass traverses the barriers, but if condition (\[Condition\_forcing\_transitions\]) holds then the state of the interparticle degree of freedom can change. In essence, this is the core of the observed phenomenon in Sec. \[Sec2\]. Indeed, condition (\[Condition\_forcing\_transitions\]) is satisfied for Figs. \[Fig\_potentials\](d) and \[Fig\_potentials\](f). Hence, the tunneling probability is less in these cases than in cases Figs. \[Fig\_potentials\](c) and \[Fig\_potentials\](e) for which equality (\[Condition\_not\_forcing\_transitions\]) takes place. On the whole, the same conclusion is valid as long as the potential $U_N(\rho)$ can have at lest two bound states. In Figs. \[Fig\_potentials\](a) and \[Fig\_potentials\](b), when there is only a single bound state supported by the intraparticle interaction, the simple intuitive picture holds. Why is this the case? After all, there are also continuum states of the intraparticle motion available for the excitation. To answer this question, let us look at collective tunneling from the point of view of the (time-independent) multichannel formalism, which is the most common method employed to the problem at hand (see, e.g., Refs. [@Zakhariev1964; @Amirkhanov1966; @Zakhariev2002; @Saito1994; @Penkov2000; @Chabanov2000; @Penkov2000a; @Razavy2003a; @Goodvin2005; @Goodvin2005a; @Lee2006; @Hnybida2008; @Shegelski2008; @Zakhariev2008]). According to the multi-channel approach, using the expansion $\Psi(R, \rho) = { \mathop{\hbox to4pt{ $\sum$ \hss}{\displaystyle\int}} }_n \phi_n(\rho)\chi_n (R)$, the stationary Schrödinger equation, $\hat{H}\Psi(R, \rho) = E\Psi(R, \rho)$, is reduced to the following system of ordinary differential equations for unknown functions $\chi_n(R)$: $$\begin{aligned} \nonumber \frac{1}{2M} \frac{d^2\chi_n(R)}{dR^2} - { \mathop{\hbox to4pt{ $\sum$ \hss}{\displaystyle\int}} }_{n'} Z_{nn'}(R) \chi_{n'}(R) = (E_n-E)\chi_n(R),\end{aligned}$$ where $Z_{nn'}(R)$ being effective potentials, $$\begin{aligned} \label{Effective_Potential_Def} Z_{nn'}(R) &=& \int d\rho \, \phi^*_n(\rho)\left[ V_1\left( R - \frac{\mu}{m_1} \rho\right) + \right. \nonumber\\ && \left. + V_2\left( R + \frac{\mu}{m_2} \rho\right) \right]\phi_{n'}(\rho).\end{aligned}$$ Such an effective potential may be interpreted as the potential barrier that the center of mass encounters while incident in the state $n'$ and reflected or transmitted in the state $n$. Let $g$ denote the ground as well as the single bound state of the potential $U_1(\rho)$ \[Eq. (\[Un\_potentials\])\], and $c$ a low-lying odd (unbound) state of the continuum spectrum, which is normalized to the delta function. Then, $Z_{cg}(R) \equiv Z_{gc}(R) \equiv 0$ since the $g\to c$ transition is forbidden in Fig. \[Fig\_potentials\](a) ; respectively, $Z_{gg}(R) \equiv 0$ in Fig. \[Fig\_potentials\](b). The first nonzero effective potentials in Figs. \[Fig\_potentials\](a) and \[Fig\_potentials\](b) are $Z_{gg}(R)$ and $Z_{cg}(R)$, respectively, and they are plotted in Fig. \[Fig\_effective\_potentials\]. Taking into account that $E_g - \bar{E}_1 \approx E_{cm} = 1$ (a.u.) ($E_g$ being the ground state energy) is the kinetic energy of the center of mass, we may qualitatively conclude from Fig. \[Fig\_effective\_potentials\] that the center of mass needs to tunnel through the barrier \[$Z_{gg}(R)$\] in Fig. \[Fig\_potentials\](a) and flies above the barrier \[$Z_{cg}(R)$\] in Fig. \[Fig\_potentials\](b); thus, the probability of finding the particle in the first quadrant in Fig. \[Fig\_potentials\](b) prevails over the probability of tunneling in Fig. \[Fig\_potentials\](a). Finally, we note that the first nonzero effective potentials in Figs. \[Fig\_potentials\](c) and \[Fig\_potentials\](d) are of the same order; the same statement is valid in cases Figs. \[Fig\_potentials\](e) and \[Fig\_potentials\](f). ![(Color online) Plots of effective potentials $Z_{nn'}(R)$ \[Eq. (\[Effective\_Potential\_Def\])\]. The solid line is $Z_{gg}(R)$ for case (a) of Fig. \[Fig\_potentials\]. The dashed line represents $Z_{cg}(R)$ for case (b) of Fig. \[Fig\_potentials\]. []{data-label="Fig_effective_potentials"}](EffectivePotentials.pdf) Traces of the forced excitations, which occur in Figs. \[Fig\_potentials\](d) and \[Fig\_potentials\](f), can be directly observed in obtained numerical data; these are steplike structures in $P^{(2)}_T(-3, \tau)$, $p^{(2)}_t (-3, \tau)$, $P^{(4)}_T(-3, \tau)$, and $p^{(4)}_t (-3, \tau)$ \[see Figs. \[Fig\_probability\_tunnel\_time\](b) and \[Fig\_probability\_tunnel\_time\](c)\] and a snakelike shape of the wave function $\Psi_{2,4}(-3; t, R, \rho)$ that emerges from the barrier (see Ref. [@EPAPS_Animations]). Nevertheless, the data regarding Fig. \[Fig\_potentials\](b) \[see Fig. \[Fig\_probability\_tunnel\_time\](a)\] seems not to reveal similar jumps at first sight. It is due to the fact that the coupling between bound states \[the form factor (\[Formfactor\_Def\]), more precisely\] is bigger than the coupling of a bound state to a state of the continuum spectrum. Regardless of smallness, these transitions show up in the observation that the probability of disintegration in Fig. \[Fig\_potentials\](a) is less than in Fig. \[Fig\_potentials\](b) \[see Fig. \[Fig\_disintegr\_prob\] that $P_D^{(1)} (3, 150) < P_D^{(1)} (-3, 150)$ as well as $p_d^{(1)} (3, 150) < p_d^{(1)} (-3, 150)$\]. Concluding this section, we list pivotal factors in explaining the “paradox” reported in Sec. \[Sec2\]. (i) This effect cannot exist in one dimension, it requires at least two dimensions. (ii) Our explanation of the effect relies on a natural isomorphism between systems of one 2D particle and two 1D particles of the same mass. The essence of the effect lies in possibility of tuning the potential barriers such that the intraparticle degree of freedom is excited. (iii) Dynamics of tunneling crucially depends on whether the intraparticle potential supports one or more bound states (an exact number is irrelevant for the qualitative description). Classical Physics and The “paradox” {#Sec4} =================================== A classical counterpart of the quantum system at hand \[Eq. (\[HamiltonianCartesianDef\])\] is a mechanical system with the Hamiltonian $$\begin{aligned} \label{ClassicalHamiltonianCartesian} \mathrsfs{H}(p_1, p_2; x_1, x_2) = \left( p_1^2 + p_2^2 \right)/2 + \Omega_N (\alpha; x_1, x_2),\end{aligned}$$ where $p_{1,2}$ and $x_{1,2}$ are canonically conjugate variables. One may perform the canonical transformation to rewrite the Hamiltonian (\[ClassicalHamiltonianCartesian\]) in terms of the new canonical variables $P_R$, $P_{\rho}$ and $R$, $\rho$, where the latter pair being the center of mass and relative coordinates \[Eq. (\[SimpleCMandRelativeCoord\])\], $$\begin{aligned} \label{ClassicalHamiltonianCM_Relative} \mathrsfs{H}\left( P_R, P_{\rho}; R, \rho\right) = P_R^2 /4 + P_{\rho}^2 + \tilde{\Omega}_N (\alpha; R, \rho).\end{aligned}$$ The connection between new and old canonical momenta reads: $ p_1 = P_R/2 - P_{\rho}, \quad p_2 = P_R/2 + P_{\rho}. $ The initial condition for the classical counterpart that corresponds to the initial condition (\[InitialCondition\]) is $x_1(0) = x_2(0) = \bar{R}$ and $p_1(0) = p_1(0) = \left[ \bar{E}_N -\Omega_N (\alpha; \bar{R}, \bar{R}) \right]^{1/2}$. Having calculated classical trajectories with this initial condition, we observe that the classical particle does not reach the first quadrant in Figs. \[Fig\_potentials\](b), \[Fig\_potentials\](d), and \[Fig\_potentials\](f); it is reflected back to the third quadrant. The same conclusion can be reach qualitatively by calculating the force that acts on the classical counterpart, $$\begin{aligned} F_{1,2}(x_1, x_2) = -\partial \Omega_N (\alpha; x_1, x_2) /\partial x_{1,2}.\end{aligned}$$ Since $$\begin{aligned} \nonumber F_1(x,x) = (2\alpha x^2 -\alpha) e^{-x^2}, \quad F_2(x,x) = (6x^2 - 3)e^{-x^2},\end{aligned}$$ the classical particle experiences the force that deflects it from moving along the diagonal, $x_1 = x_2$ (which coincides with the axis $\rho=0$), and pushes it toward a knee-like barrier located in the second quadrant (see Fig. \[Fig\_potentials\]); hence, the particle eventually bounces off the barrier back to the third quadrant. It is noteworthy to mention a peculiarity of numerical calculations. We have found that it is advantageous to employ a (fourth-order) symplectic integrator [@Forest1990; @Yoshida1990; @Candy1991] for solving Hamilton’s equations in this section due to the following reason: A sharp and localized shape of the kneelike barrier leads to an unstable motion of the classical particle. If one employs nonsymplectic integrators (e.g., the Runge-Kutta methods), a very tiny time step must be chosen in order to properly account for the influence of the kneelike barrier; this, in fact, often leads to instability of the numerical scheme for a long time propagation. A physical reason of such an instability lies in the fact that nonsymplectic integrators do not explicitly conserve energy while the symplectic integrators always do; hence, they give a proper long-time evolution of any chaotic Hamiltonian system. The observation of this behavior of the classical counterpart casts doubt on the quantum nature of the “paradox.” More precisely, is it possible that an ensemble of classical particles, which corresponds (in some sense) to the initial wave function of the system at hand (\[InitialCondition\]), would mimic the observed phenomenon? As shown below, the answer turns out to be negative. Furthermore, since it is well known that the application of the semi classical approximation, as a mediator between classical and quantum mechanics, to tunneling often is very fruitful in shading light on the physical nature of the studied process (see, e.g., Refs. [@Maitra1997; @Spanner2003]), the addressed question is important as the first step toward the usage of semiclassical methods for the interpretation of the “paradox.” Shirokov has proposed the unified formalism for quantum and classical mechanics [@Shirokov1979d]–a reformulation of both the theories in terms of the same physical and mathematical concepts. Crudely speaking, this formalism is based on the well-known fact that observables of quantum mechanics can be converted from operators to functions (i.e., to a very similar form as in classical physics) by means of the Weyl representation [@Weyl1950]. In these terms, the probability density of quantum states is represented by the Wigner quasiprobability density distribution function [@Wigner1932]. Hence, in order to construct a classical ensemble that corresponds to the quantum particle, we shall calculate the Wigner function $W(P_R, P_{\rho}; R, \rho)$ for the initial condition (\[InitialCondition\]), $$\begin{aligned} \label{WingerFunctionForInitialState} && W(P_R, P_{\rho}; R, \rho) = (2\pi)^2 \int \Psi\left( R-R'/2, \rho-\rho'/2\right)\\ && \qquad\quad \times \Psi^*\left( R+R'/2, \rho+\rho'/2\right) e^{i\left( P_R R'\ + P_{\rho} \rho' \right)} d\rho' dR', \nonumber\end{aligned}$$ where $\Psi (R, \rho) \equiv \Psi_N(\alpha; 0, R, \rho)$. Since $\phi_g(\rho)$ has no zeros and decays exponentially at infinity, we shall approximate it by a Gaussian $$\begin{aligned} \label{GaussApproxGroundState} \phi_g(\rho) \propto \exp\left[ -\rho^2 / \left(2\sigma_{\rho}^2\right) \right],\end{aligned}$$ where we set $\sigma_{\rho} = 1.5$ (a.u.). Substituting Eqs. (\[GaussApproxGroundState\]) and (\[InitialCondition\]) into Eq. (\[WingerFunctionForInitialState\]), one readily obtains $$\begin{aligned} W(P_R, P_{\rho}; R, \rho) &\propto& \exp\left\{ -(R-\bar{R})^2 / \sigma_R^2 -\rho^2 / \sigma_{\rho}^2 \right. \\ && \left. - \sigma_R^2 \left( P_R - \sqrt{2M E_{cm}}\right)^2 -\sigma_{\rho}^2 P_{\rho}^2 \right\}. \nonumber\end{aligned}$$ Therefore, our quantum system corresponds to an ensemble of classical particles with the Hamiltonian (\[ClassicalHamiltonianCM\_Relative\]), and the initial state of the ensemble that corresponds to the initial condition (\[InitialCondition\]) can be generated by considering $R$, $\rho$, $P_R$, and $P_{\rho}$ as independent normal random variables with means $\bar{R}$, 0, $\sqrt{2M E_{cm}}$, 0 and with standard deviations $\sigma_R/\sqrt{2}$, $\sigma_{\rho}/\sqrt{2}$, $\left[\sqrt{2}\sigma_R\right]^{-1}$, $\left[\sqrt{2}\sigma_{\rho}\right]^{-1}$, respectively.  Results of classical simulations of dynamics of an ensemble of $10^6$ particles are compared with [*ab initio*]{} quantum simulations in Fig. \[CvsQ\]. Foremost, one may notice how well the classical simulations reproduced quantum behavior in the classically allowed regions in almost all the cases. One observes a qualitative agreement between classical and quantum results in the cases of tunneling \[Figs. \[CvsQ\](a), \[CvsQ\](c), and \[CvsQ\](e)\]. Nevertheless, in the cases of the over barrier motion \[Figs. \[CvsQ\](b), \[CvsQ\](d), and \[CvsQ\](f)\], classical mechanics gives more asymmetric probability distributions than quantum mechanics. This can be explained by means of a simple observation that there are the kneelike potential barriers in Figs. \[Fig\_potentials\](b), \[Fig\_potentials\](d), and \[Fig\_potentials\](f), which force a majority of classical particles to go to the fourth quadrant. Having calculated the classical probability density distributions, we may introduce the probability of tunneling as well as the shifted probability of tunneling analogously to the corresponding quantum quantities \[Eqs. (\[Prob\_T\_deff\]) and (\[prob\_t\_deff\])\]. The classical probabilities of tunneling in the above-barrier cases are an order of magnitude larger than the corresponding classical probabilities in the under-barrier cases. This conclusion contradicts the results of the quantum calculations (Fig. \[Fig\_probability\_tunnel\_time\]). In other words, there is no “paradox” in classical physics. It is natural since the ensemble of classical particles should “prefer” going above than “tunneling through” the barrier. Hence, we have confirmed that the reported effect is genuinely quantum mechanical. Conclusions and discussions =========================== In Sec. \[Sec2\], we presented the 2D systems, whose potentials are plotted in Fig. \[Fig\_potentials\], which hold the unexpected property that the probability of tunneling through a barrier is larger than the probability of flying above a barrier. As it was clarified in Sec. \[Sec3\], this phenomenon occurs due to a specific symmetry of the potential \[Eq. (\[Condition\_forcing\_transitions\])\] that forces excitations of an interparticle degree of freedom, thus lowering the probability of tunneling. This effect is overlooked by the intuitive conclusion which uses the language of trajectories within the quasi classical approximation that the tunneling is an “exponentially harder” process than flying above a barrier. First and foremost, we note that the quasi classical approximation, being an elegant and insightful approach in 1D, is in fact very cumbersome and quite often impractical in 2D. Hence, in most situations of interest different modifications of the original quasi classical approximation that make additional assumptions on the wave function are employed (see, e.g., Ref. [@Razavy2003a] and references therein). From this point of view, we conclude that a quasiclassical model capable of explaining the reported “paradox” must not only rely on the language of trajectories but also include the quantum transitions that are at the core of the effect. An important undiscussed issue is the dependence of the reported effect on the initial condition (\[InitialCondition\]). If we substitute $\phi_g(\rho)$ in Eq. (\[InitialCondition\]) by the wave function of the first excited state of the interparticle Hamiltonian in the cases of $N=2$ and $N=4$ (note that $E_{cm}$ must be appropriately decreased such that it would be possible to talk about tunneling), then one may expect that the “paradox” should disappear, and one would observe a conventional situation: the probability of tunneling through the barrier \[Figs. \[Fig\_potentials\](c) and \[Fig\_potentials\](e)\] would be smaller than the probability of flying above the barrier \[Figs. \[Fig\_potentials\](d) and \[Fig\_potentials\](f)\]. Indeed, since the transition from the first excited state to the ground state is allowed because condition (\[Condition\_forcing\_transitions\]) is satisfied in Figs. \[Fig\_potentials\](d) and \[Fig\_potentials\](f), then after making such a jump, the center of mass gains the energy difference; hence, it can more easily tunnel in Figs. \[Fig\_potentials\](d) and \[Fig\_potentials\](f) than in Figs. \[Fig\_potentials\](c) and \[Fig\_potentials\](e) where this transition is forbidden. As far as applications of the effect to quantum control are considered, consider a system of two neutral atoms that interact through the dipole-dipole interaction and are trapped, e.g., by a dipole trap. The magnitudes of the atomic dipoles depend on internal states occupied by the atoms. The internal states of the atoms can be changed for each atom independently by means of a laser with an appropriately tuned frequency, assuming that the atoms have different spectra. Performing such excitations, we may be able to switch between the cases where either condition (\[Condition\_not\_forcing\_transitions\]) or condition (\[Condition\_forcing\_transitions\]) is valid. Hence, we may allow or forbid the two atomic system to tunnel through the trap. A generalization of Eq. (\[W\_equivalent\_particles\]) as well as Eqs. (\[SymmetryW\]), (\[Condition\_not\_forcing\_transitions\]), and (\[Condition\_forcing\_transitions\]) to the case of $n$ ($n\geqslant 3$) particles is a nontrivial question that should be addressed in the future. One might expect that such a generalization of the effect may reveal many new varieties of the phenomenon, which could be interesting from the point of view of quantum control of tunneling of complex systems. The authors thank Michael Spanner for fruitful comments. D.I.B. acknowledges the Ontario Graduate Scholarship program for financial support. M.Yu.I. and W.K.L. acknowledge support of NSERC discovery grants.

--- abstract: 'We study the problem of anonymizing tables containing personal information before releasing them for public use. One of the formulations considered in this context is the $k$-anonymization problem: given a table, suppress a minimum number of cells so that in the transformed table, each row is identical to atleast $k-1$ other rows. The problem is known to be NP-hard and MAXSNP-hard; but in the known reductions, the number of columns in the constructed tables is arbitrarily large. However, in practical settings the number of columns is much smaller. So, we study the complexity of the practical setting in which the number of columns $m$ is small. We show that the problem is NP-hard, even when the number of columns $m$ is a constant ($m=3$). We also prove MAXSNP-hardness for this restricted version and derive that the problem cannot be approximated within a factor of $\frac{6238}{6237}$. Our reduction uses alphabets $\Sigma$ of arbitrarily large size. A natural question is whether the problem remains NP-hard when both $m$ and $|\Sigma|$ are small. We prove that the $k$-anonymization problem is in $P$ when both $m$ and $|\Sigma|$ are constants.' author: - 'Venkatesan T. Chakaravarthy, Vinayaka Pandit, Yogish Sabharwal' bibliography: - 'anon.bib' date: | IBM Research - India, New Delhi and Bengaluru.\ [*{vechakra, pvinayak, ysabharwal}@in.ibm.com*]{} title: ' On the Complexity of the $k$-Anonymization Problem ' --- Introduction {#sec:intro} ============ Various organization such as hospitals and insurance companies collect massive amount of personal data. These need to be released publicly for the purpose of scientific data mining; for instance, data collected by hospitals could be mined to infer epidemics. However, a major risk in releasing personal data is that they can be used to infer sensitive information about individuals. A natural idea for protecting privacy is to remove obvious personal identifiers such as social security number, name and driving license number. However, Sweeney [@Swe02] showed that such a deidentified database can be joined with other publicly available databases (such as voter lists) to reidentify individuals. For instance, she showed that 87% of the population of the United States can be uniquely identified on the basis of gender, date of birth and zipcode. In the literature, such an identity leaking attribute combination is called a quasi-identifier. It is important to recognize quasi-identifiers and apply protective measures to eliminate the risk of identity disclosure via join attacks. Samaratti and Sweeyney [@Sam-Swe; @Swe02] introduced the notion of $k$-anonymity, which aims to preserve privacy either by suppressing or generalizing some of the sensitive data values. In this paper, we consider the basic $k$-anonymity problem with only suppression allowed. Suppose we have a table with $n$ rows and $m$ columns. In order to achieve anonymity, one is allowed to suppress the entries of the table so that in the modified table, every row is identical to at least $k-1$ other rows. The goal is to minimize the number of cells suppressed. This is called the [*$k$-anonymization*]{} problem. The motivation for the problem formulation are twofold: (i) any join attack would return groups of at least $k$ rows, thus preserving privacy with a parameter of $k$; (ii) lesser the number of entries suppressed, better is the value of the modified table for data mining. [**Example:** ]{}We now illustrate the problem definition with an example. An example input table and its anonymized output, for $k=2$, are shown in Figure \[fig:example\]. The number of rows is $n=4$ and number of columns is $m=3$. The suppressed cells are shown by “$*$”. We see that in the anonymized output table, the first and the third rows are identical, and the second and the fourth rows are identical. Thus the table on the right is $2$-anonymized. The cost of this anonymization is $4$, since $4$ cells are suppressed. This is an optimal solution. [ccccc]{} $x$ $a$ $b$ ----- ----- ----- $z$ $c$ $d$ $y$ $a$ $b$ $z$ $c$ $e$ & & & & $*$ $a$ $b$ ----- ----- ----- $z$ $c$ $*$ $*$ $a$ $b$ $z$ $c$ $*$ \ &&&&\ Original table & & & & $2$-Anonymized table [**Known and New Results:** ]{} Meyerson and Williams [@MW] proved the NP-hardness of the $k$-anonymization problem. Aggarwal et al. [@ICDT] improved the result by showing that the problem remains NP-hard even when the alphabet $\Sigma$ from which the symbols of the table are drawn is fixed to be ternary. Bonizzoni et al [@APX-hard] proved MAXSNP-hardness (and NP-hardness) even when the alphabet is binary. The value of the privacy parameter $k$ is a fixed constant in all the above results ($k=3$). On the algorithmic front, Meyerson and Williams gave a $O(k\log k)$-approximation algorithm. This was improved by Aggarwal et al. [@ICDT], who devised a $O(k)$-approximation algorithm. Park and Shim [@SIGMOD] presented an approximation algorithm with a ratio of $O(\log{k})$; however, we observe that the running time of their algorithm is exponential in the number of columns $m$ (but, polynomial in the number of rows $n$). We make the following observations regarding the previously known results. Firstly, the known NP-hardness reductions produce tables in which the number of columns is arbitrarily large. This is not satisfactory as the number of columns in practical settings is not large. Secondly, the algorithm of Park and Shim [@SIGMOD] is a polynomial time $O(\log k)$-approximation algorithm when the number of columns $m$ is small ($m=O(\log n)$). These observations raise a natural question: Does the $k$-anonymization problem remain NP-Complete even when the number of columns $m$ is small ($\log n$ or a constant)? We show that the $k$-anonymization problem remains NP-hard, even when the number columns $m$ is fixed to be a constant ($m=3$). In fact, we also show that the above restricted version is MAXSNP-hard, thus ruling out polynomial time approximation schemes. We also derive that the problem cannot be approximated within a factor of $\frac{6238}{6237}$. Even though our inapproximability bound is mild, it is the first explicit inapproximability bound proved for the $k$-anonymization problem. All our hardness results hold even when the privacy parameter $k$ is a constant ($k=7$). As we noted, the previous constructions ensured that the alphabet size is a fixed constant; but, in our constructions, the alphabet size is not a fixed constant, but it is arbitrarily large. However, this is not a serious issue; in most settings, tables have large number of unique entries (for example, a zipcode column takes a large number of distinct values). In the wake of previous results and our results mentioned above, a natural question is whether the problem is NP-hard when both the number of columns $m$ and the alphabet size $|\Sigma|$ are small. We show that the problem can be solved optimally in polynomial time when both $m$ and $|\Sigma|$ are fixed constants. Below, we summarize the new results proved in this paper. - The $k$-anonymization problem is NP-hard even when the number of columns $m$ and the privacy parameter $k$ are constants ($m=3$, $k=7$). We show that the restricted version is MAXSNP-hard and cannot be approximated within a factor of $\frac{6238}{6237}$. - When both the number of columns and the alphabet size are fixed constants, the problem can be solved optimally in polynomial time. The result is true, even when the privacy parameter $k$ is arbitrarily large. The known and new results put together provide a complete picture on the complexity of the problem for the four cases of the number of columns $m$ and the alphabet size $\Sigma$ being fixed constants or arbitrarily large. This is shown in Figure \[fig:summary\]. -- ------------------------------------------------------------------------------------------------------------------------------------------------- --     $\mathbf{|\Sigma|}$**[=constant]{} &     $\mathbf{|\Sigma|}$**[=arbitrary]{}\ & &\ & &\ $\mathbf{m}$**[=constant]{} & Solvable in P (even when $k$ is arbitrary) \[This paper\] & NP-hard (even when $k$ is a constant) \[This paper\]\ & &\ & &\ $\mathbf{m}$**[=arbitrary]{} & NP-hard (even when $k$ is a constant) [@ICDT] & NP-hard (even when $k$ is a constant) [@ICDT]\ & &\ ******** -- ------------------------------------------------------------------------------------------------------------------------------------------------- -- Problem Definition ================== The input to the $k$-anonymization problem is a $n\times m$ table $T$ having $n$ rows and $m$ columns, with symbols of the table drawn from an alphabet $\Sigma$. The input also includes a [*privacy parameter*]{} $k$. A feasible solution $\sigma$ transforms the given table $T$ to a new table $T'$ by suppressing some of the cells of $T$; namely, it replaces some of the cells of $T$ with “$*$”. In the transformed table $T'$, for any row $t$, there should exist $k-1$ other rows that are identical to $t$. The cost of the solution, denoted ${{\rm Cost}}(\sigma)$, is the number of suppressed cells. The goal is to find a solution having the minimum cost. Consider a solution $\sigma$. For a row $t$, we denote by ${{\rm Cost}}(t)$ the number of suppressed cells in $t$ and say that $t$ pays this cost. Thus, ${{\rm Cost}}(\sigma)$ is the sum of costs paid by all the rows. There is an equivalent way to view a solution in terms of partitioning the given table. Consider a subset of rows $S$. We say that a column is [*good*]{} with respect to $S$, if all the rows in $S$ take identical values on the column. A column is said to be [*bad*]{}, if it is not good; meaning, some two rows in $S$ have different values on the given column. Denote by $a(S)$ the number of bad columns in $S$. Then, our goal is to a find a partition of rows $\Pi={S_1, S_2, \ldots, S_{\ell}}$ such that each set $S_i$ is of size $|S_i|\geq k$. Each row $t$ in $S_i$ pays a cost of $a(S_i)$. The total cost of the solution is the sum of costs paid by all rows. Equivalently, the cost of the solution is given by $\sum_{i=1}^{\ell} |S_i|\cdot a(S_i)$. We shall interchangeably use either of the two descriptions in our discussions. Hardness Results with Three Columns =================================== In this section, we present results on the complexity of $k$-anonymization problem when both the number of columns and the privacy parameter are constants. NP-Hardness ----------- \[thm:AAA\] The $k$-anonymization problem is NP-hard even when the number of columns $m$ is 3 and the privacy parameter is fixed as $k=7$. [[*Proof:* ]{}]{}We give a reduction from the vertex cover problem on 3-regular graphs, which is known to be NP-hard (see [@GJ]). Recall that a vertex cover of a graph refers to a subset of vertices such that each edge has at least one endpoint in the subset and that a graph is said to be 3-regular, if every vertex has degree exactly 3. Let $G=(V,E)$ be the input 3-regular graph having $r$ vertices. The alphabet of the output table is as follows: For each vertex $u \in V$, we add a symbol $u$. Next, we have additional symbols ‘$0$’ and ‘$Z$’. Further, we need a number of [*special*]{} symbols. A special symbol appears only once in the whole of the table. The exposition becomes somewhat clumsy, if we explicitly introduce these special symbols. Instead, we use the generic symbol ‘$?$’ to mean the special symbols. The symbol ‘$?$’ is not a single symbol, but a general placeholder to mean a special symbol. We maintain a running list of special symbols (say $s_1, s_2, \ldots$) and whenever a new row containing ‘$?$’ is added to the table, we actually get a new symbol from the list and replace ‘$?$’ by the new symbol. For instance, suppose $\langle ?, u, u\rangle$ and $\langle ?, v, ?\rangle$ are the first two rows added to the table. Then, the actual rows added are $\langle s_1, u, u\rangle$ and $\langle s_2, u, s_3\rangle$. With the above discussion in mind, notice that two different instances of ‘$?$’ do not match with each other. The output table $T$ is constructed as follows. 1. For each vertex $u\in V$, add the following 20 rows. These are said to be rows corresponding to $u$. 1. Add the following row six times: $\langle 0, u, u\rangle$. \[type:AAA\] 2. Add a row $\langle ?, u, u\rangle$. It is called the [*critical*]{} row of $u$ and it plays a vital role in the construction. \[type:BBB\] 3. Add seven rows: $\langle ?, u, ?\rangle$. \[type:CCC\] 4. Add the following row 3 times: $\langle 0, u, Z\rangle$. \[type:DDD\] 5. Add the following row 3 times: $\langle 0, Z, u\rangle$. \[type:EEE\] 2. For each edge $(x, y)$, add two rows $\langle 0, x, y\rangle$ and $\langle 0, y, x\rangle$. These are called [*edge rows*]{}. 3. Add the following two sets of [*dummy rows*]{}: 1. Add seven rows as below: $\langle 0, ?, Z\rangle$. \[type:DDD-A\] 2. Add seven rows as below: $\langle 0, Z, ?\rangle$. \[type:DDD-B\] This completes the construction of the table. The privacy parameter is set as $k=7$. Consider any $k$-anonymization solution to the constructed table. For any row of the table, we can derive a lowerbound on the cost paid by the row; we refer to the lowerbound as the [*base cost*]{}. The base costs are derived as follows, for the various types of rows. Consider any vertex $u\in V$. First consider rows of type \[type:AAA\]. These rows are of the form $\langle 0, u, u\rangle$ and there are exactly six of them. Since $k=7$, these rows must be participating in a cluster having a different row. Hence, each of these rows must pay a cost of at least 1. We set the base cost for each of these rows to be 1. Now, consider the critical row of type \[type:BBB\]. This row is of the form $\langle ?, u, u\rangle$ and it must pay a base cost of 1, since it involves a special symbol. The base cost of the critical row is deemed to be 1. A type \[type:CCC\] row (of the form $\langle ?, u, ?\rangle$) must pay cost of at least two, since it has two special symbols. The base cost of such a row is deemed to be 2. By similar arguments, we see that any other type of row must pay a base cost of 1. To summarize, every row of type \[type:CCC\] (of the form $\langle ?, u, ?\rangle$) pays a base cost of 2, whereas any row of any other type pays a base cost of 1. For each vertex $u$, the total base cost across the 20 rows can be calculated as follows: (i) The six (type \[type:AAA\]) rows of the form $\langle 0, u, u\rangle$ pay a cost of $6$ in total; (ii) The critical row (of type \[type:BBB\]) pays a cost of $1$; (iii) The seven (type \[type:CCC\]) rows of the form $\langle ?, u, ?\rangle$ pay a cost of $2$ each, totaling $14$; (iv) The three (type \[type:DDD\]) rows of the form $\langle 0, u, Z\rangle$ pay cost of $3$ in total; (v) The three (type \[type:EEE\]) rows of the form $\langle 0, Z, u \rangle$ pay cost of $3$ in total. Thus, the total base cost for each vertex $u$ is $27$. Then, each edge has a base cost of 2, coming from the two rows corresponding to it. The two blocks dummy rows (of type \[type:DDD-A\] and type \[type:DDD-B\]) contribute a base cost of $7$ each, summing up to $14$. Thus, the [*aggregated base cost*]{} is $ABC = 27r + 2|E|+14$. For any row, the difference between the actual cost paid and the base cost is denoted as [*extra cost*]{}. Similarly, the total extra cost is the sum of extra costs over all the rows. Notice that the cost of the solution is the sum of $ABC$ and the total extra cost. We claim that the given graph has a vertex cover of size $\leq t$, if and only if there exists a $k$-anonymization solution with an extra cost $\leq t$. It would follow that the graph has a vertex cover of size $\leq t$, if and only if there exists a $k$-anonymization solution of cost $\leq ABC+t$. This would prove the required NP-hardness. We next proceed to prove the above claim. We split the proof into two parts. First, we shall argue that if the given graph has a vertex cover of size $\leq t$, then there exists a $k$-anonymization solution with extra cost $\leq t$. Suppose $C$ is a vertex cover of size $\leq t$. We shall construct a $k$-anonymization solution $\sigma$ in which the critical rows corresponding to the vertices in the cover $C$ pay an extra cost of 1 and every other row pays no extra cost. For each edge $(x, y)$, if $x\in C$, then [*attach*]{} the edge to $x$, else attach it to $y$. (If both the endpoints of the edge are in the cover, the edge can be attached arbitrarily to any one of the two vertices). Without loss of generality, assume that each vertex in the cover has at least one edge attached to it. Otherwise, the vertex can be safely removed from $C$, yielding a smaller cover. -------------------- ---------------- Case: $u\not\in C$ Case: $u\in C$ -------------------- ---------------- Form a $k$-anonymization solution $\sigma$ as follows. See Figure \[fig:hard\] for an illustration. - Form two clusters combining the dummy rows. - Form a cluster by combining the seven (type \[type:DDD-A\]) dummy rows of the form $\langle 0, ?, Z\rangle$; call this cluster $D_1$. - Form a cluster by combining the seven (type \[type:DDD-B\]) dummy rows of the form $\langle 0, Z, ?\rangle$; call this cluster $D_2$. - Consider each vertex $u$ not in the cover $C$ (i.e., $u\not\in C$). - Form a cluster by adding the six (type \[type:AAA\]) rows of the form $\langle 0, u, u\rangle$ and the critical row $\langle ?, u, u\rangle$. Each row in the cluster pays a cost of 1, and hence the extra cost is 0 for all these rows. - Form a cluster by adding the seven (type \[type:CCC\]) rows of the form $\langle ?, u, ?\rangle$. Each row in the cluster pays a cost of 2, and hence the extra cost is 0 for all these rows. - Add the three (type \[type:DDD\]) rows of the form $\langle 0, u, Z\rangle$ to $D_1$. Add the three (type \[type:EEE\]) rows of the form $\langle 0, Z, u\rangle$ to $D_2$. Thus, each row in $D_1$ and $D_2$ pays a cost of 1, and hence their extra costs are 0. - Consider each vertex $u$ in the cover $C$ (i.e., $u\in C$). - Form a cluster $A_u$ by adding three of the (type \[type:AAA\]) rows of the form $\langle 0, u, u\rangle$ - Form a cluster $B_u$ by adding the remaining three (type \[type:AAA\]) rows of the form $\langle 0, u, u\rangle$ - Consider each edge attached to $u$, say $(u, x)$ for some $x\in V$. Add the edge row $\langle 0, u, x\rangle$ to $A_u$ and add the edge row $\langle 0, x, u\rangle$ to $B_u$. - Add the three (type \[type:DDD\]) rows of the form $\langle 0, u, Z\rangle$ to $A_u$ and add the three (type \[type:EEE\]) rows of the form $\langle 0, Z, u\rangle$ to $B_u$. Notice that both $A_u$ and $B_u$ have at least seven rows each, since each vertex has at least one edge attached to it. Every row in these two clusters pays a cost of $1$ (thus, the extra cost paid by these rows is 0). - Form a cluster by adding the seven (type \[type:CCC\]) rows of the form $\langle ?, u, ?\rangle$. Add the critical row $\langle ?, u, u\rangle$ to this cluster. Notice that the seven rows each pay a cost of $2$, and hence their extra cost is $0$. [*The critical row pays a cost of $2$, and hence, its extra cost is $1$*]{}. Observe that all the rows of the table have been assigned to some cluster and each cluster has size at least $7$. From the above discussion, we see that the only rows having non-zero extra cost are the critical rows corresponding to the vertices in the cover $C$ and they pay an extra cost of $1$ each. We conclude that the total extra cost is $|C|$. We have proved the following claim: [Claim 1: ]{} If the given graph has a vertex cover of size $\leq t$, then there exists a $k$-anonymization solution with extra cost $\leq t$. We next proceed to prove the reverse direction: if there exists a $k$-anonymization solution $\sigma$ of extra cost $\leq t$, then there exists a vertex cover of size $\leq t$. Consider such a solution $\sigma$. We first make the following claim. [Claim 2: ]{} Consider a vertex $u$. Suppose the critical row $\langle ?, u, u\rangle$ pays an extra cost of 0. Then, the only cluster in which it can participate is the one obtained by combining the critical row with the six (type \[type:AAA\]) rows of the form $\langle 0, u, u\rangle$.\ [[*Proof:* ]{}]{}Clearly, the critical row must pay a cost of $1$, since it has a special symbol. If it pays no extra cost, then the rows it is combined with should have the symbol ’$u$’ in their second and third columns. There are exactly six such rows available and these are the (type \[type:AAA\]) rows of the form $\langle 0, u, u\rangle$. [$\Box$]{} We say that a vertex is [*perfect*]{}, if all the 20 rows corresponding to it pay an extra cost of 0. A vertex is said to be [*imperfect*]{}, if at least one of its 20 rows pay an extra cost of at least 1. [Claim 3: ]{}Consider an edge $(x, y)$. If both $x$ and $y$ are perfect, then at least one of the two edge rows corresponding to the edge pays a cost of at least 2.\ [[*Proof:* ]{}]{}Consider the edge row $\langle 0, x, y\rangle$, corresponding to the given edge. Let $S$ be the cluster to which this row belongs. Since $k=7$, we have $|S|\geq 7$. Recall that a column is said to be good with respect to $S$, if the rows of $S$ have identical values on the column; a column is said to be bad with respect to $S$, otherwise. We shall argue that at least two of the three columns are bad with respect to $S$. Let us consider the three possible choices for two-column subsets out of the three columns. - Clearly, both the second and the third columns cannot be good with respect to $S$, since there are no other rows that contain $x$ in their second column and $y$ in their third column. - Next, we argue that both the first and the second column cannot be good with respect to $S$. Since, both $x$ and $y$ are perfect, their critical rows do not pay any extra cost. By Claim 2, the six (type \[type:AAA\]) rows of the form $\langle 0, x, x\rangle$ have gone to some cluster other than $S$. Similarly, the six (type \[type:AAA\]) rows of the form $\langle 0, y, y\rangle$ have also gone to some other cluster. These rows cannot be part of $S$. Now, since the graph is 3-regular, there are only two other edge rows that have ‘$0$’ in their first column and ‘$x$’ in their second column; these correspond to the two other edges incident on $x$. There are three other rows (corresponding to the vertex $x$ and of type \[type:DDD\]) that have ‘$0$’ in their first column and ‘$x$’ in their second column. Thus, totally there are only 5 other rows that that have the above property. Since $|S|\geq 7$, it follows that both the first and the second column cannot be good with respect to $S$. - A similar argument shows that there are only 5 other rows that have ‘$0$’ in their first column and ‘$y$’ in their third column. This means that both the first column and the third column cannot be good in $S$. We conclude at least two of the three columns are bad respect to $S$. Thus, the concerned edge row $\langle 0, x, y\rangle$ must pay a cost of at least 2. [$\Box$]{} Let $V'$ be the set of all imperfect vertices. Let $E'$ be the set edges whose both endpoints are perfect. Each imperfect vertex (by definition) contributes at least 1 to the extra cost. By Claim 3, each edge in $E'$ pays an extra cost of at least 1. Therefore, $$\mbox{total extra cost of $\sigma$} \geq |V'| + |E'|.$$ Construct a vertex cover $C$ as follows. Add every imperfect vertex to $C$. For each edge in $E'$, add one of its endpoints (arbitrarily) to $C$. Clearly, $C$ is a vertex cover. So, $$|C| \leq |V'|+|E'| \leq \mbox{extra cost of $\sigma$}$$ We have proved the following claim: [Claim 4: ]{} If there exists a $k$-anonymization solution $\sigma$ of extra cost $\leq t$, then there exists a vertex cover of size $\leq t$. [$\Box$]{} We observed that the cost of a $k$-anonymous solution is the sum of $ABC$ and the extra cost of the solution. Now, by combining Claim 1 and Claim 4, we get the following: there exists a vertex cover of size $\leq t$, if and only if there exists a $k$-anonymization solution of cost $\leq ABC+t$. This completes the NP-hardness proof. [$\Box$]{} It is easy to show that our reduction is an $L$-reduction (see [@Papa] for a discussion on $L$-reductions). As the vertex cover problem on 3-regular graphs is MAXSNP-hard [@Alimonti], it follows that, \[thm:BBB\] The $k$-anonymization problem is MAXSNP-hard, even when the number of columns in 3 and the privacy parameter is fixed as $k=7$. Moreover, [[Chlebík]{}]{} and [[Chlebíkov[á]{}]{}]{} [@Chlebik] showed that the vertex cover problem on 3-regular graphs cannot be approximated within a factor of $\frac{100}{99}$. Now, taking the parameters of the $L$-reduction of our construction, and based on the result of [[Chlebík]{}]{} and [[Chlebíkov[á]{}]{}]{}, we can show that, The $k$-anonymization problem cannot be approximated within a factor of $\frac{6238}{6237}$, even when the number of columns in 3 and the privacy parameter is fixed as $k=7$. MAXSNP-Hardness and Inapproximability Bound ------------------------------------------- We show that the $k$-anonymization problem is MAXSNP-hard even when the number of columns is 3; meaning it cannot be approximated within a factor of $1+\epsilon$, for some $\epsilon>0$. \[thm:BBB\] The $k$-anonymization problem is MAXSNP-hard, even when the number of columns in 3 and the privacy parameter is fixed as $k=7$. [[*Proof:* ]{}]{}The vertex cover problem on 3-regular graphs is known to be MAXSNP-hard [@Alimonti]. We shall argue that the reduction given in Theorem \[thm:AAA\] is an $L$-reduction (see [@Papa] for a discussion on $L$-reductions); it would follow that the $k$-anonymization problem is MAXSNP-hard. Recall that an $L$-reduction from a minimization problem $P_1$ to a minimization problem $P_2$ consists of two polynomial time algorithms $R$ and $S$. The algorithm $R$ takes an instance $I_1$ of the problem $P_1$ as input and outputs an instance $I_2$ of the problem $P_2$. The algorithm $S$ takes as input a solution $S_2$ of the instance $I_2$ and produces a solution $S_1$ for the instance $I_1$. The following two requirements are to be met, for some constants $\alpha$ and $\beta$: (i) ${{\rm OPT}}(I_2)\leq \alpha {{\rm OPT}}(I_1)$; (ii) ${{\rm Cost}}(S_1) - {{\rm OPT}}(I_1) \leq \beta ({{\rm Cost}}(S_2) - {{\rm OPT}}(I_2))$. Let $G=(V,E)$ be the input 3-regular graph over $r$ vertices. The reduction given in Theorem \[thm:AAA\] produces a table $T$; this constitutes the required algorithm $R$. Let $C^*$ be the optimal vertex cover of $G$. In Theorem \[thm:AAA\], we showed (in Claim 1) that given a cover of size at most $t$, we can construct a $k$-anonymization solution $\sigma$ of cost at most $ABC+t$. It follows that the optimal $k$-anonymization solution satisfies ${{\rm Cost}}(\sigma^*)\leq ABC+|C^*|$. Recall that $ABC = 27r+2|E|+14$. Since the graph is 3-regular, number of edges is $|E|=3r/2$. Thus, $ABC=30r+14$. For sufficiently large $r$, we have $ABC\leq 31r$. Since $G$ is 3-regular, a vertex can cover at most three edges and so, $|C^*|\geq r/2$. It follows that $ABC \leq 62r$ and that ${{\rm Cost}}(\sigma^*)\leq 63|C^*|$. Thus, we have $\alpha = 63$. In Theorem \[thm:AAA\], we showed how to construct a vertex cover $C$, from a given $k$-anonymization solution $\sigma$ such that $|C|$ is at most the extra cost of $\sigma$; this construction is our algorithm $S$. It follows that $|C|\leq {{\rm Cost}}(\sigma)-ABC$. We already saw that ${{\rm Cost}}(\sigma^*) \leq ABC + |C^*|$. Combining the two inequalities, we get that $|C|-|C^*|\leq {{\rm Cost}}(\sigma)-{{\rm Cost}}(\sigma^*)$. Thus, we have $\beta = 1$. We have provided an $L$-reduction with $\alpha = 63$ and $\beta=1$. [$\Box$]{} We next extend the Theorem \[thm:BBB\] to get a mild inapproximability bound. The $k$-anonymization problem cannot be approximated within a factor of $\frac{6238}{6237}$, even when the number of columns in 3 and the privacy parameter is fixed as $k=7$. [[*Proof:* ]{}]{}Suppose we have an algorithm for solving the $k$-anonymization problem with an approximation ratio of $a$. Using the $L$-reduction given in Theorem \[thm:BBB\], we can obtain an algorithm for the vertex cover problem on 3-regular graphs with an approximation ratio of $(1+\alpha\beta (a-1))$. [[Chlebík]{}]{} and [[Chlebíkov[á]{}]{}]{} [@Chlebik] showed that the vertex cover problem on 3-regular graphs cannot be approximated within a factor of $\frac{100}{99}$. It follows that the $k$-anonymization problem cannot be approximated within a factor of $\frac{6238}{6237}$. [$\Box$]{} Special case: $m$ and $|\Sigma|$ are constants ============================================== As our NP-hardness reduction utilizes alphabets of arbitrarily large size, a natural question is whether the problem remains NP-hard when both the number of columns and the alphabet size are fixed constants. Here, we show that this case can be solved optimally in polynomial time. In the problem definition, let $m$, the number of columns, to be a constant and let the size of $\Sigma$ be a constant $s$ and let $k$ be the privacy parameter. By a [row pattern]{}, we mean a vector over $m$ columns whose entries belong to $\Sigma$. Let ${{\cal R}}$ denote the set of all row patterns; $|{{\cal R}}| = |\Sigma|^{m}$ is a constant. By an [*anonymization pattern*]{}, we mean a vector over $m$ columns whose each entry is either a symbol from $\Sigma$ or the suppression symbol ‘$*$’. Let ${{\cal P}}$ denote the set of all anonymization patterns; $|{{\cal P}}| = (|\Sigma|+1)^{m+1}$ is a constant . We say that a row pattern $t$ [*matches*]{} an anonymization pattern $p$, if $p$ and $t$ agree on all columns, except the columns suppressed in $p$. We use “$t\sim p$" as a shorthand to mean that $t$ matches $p$. Consider the optimal $k$-anonymization solution $\sigma^*$. For each row pattern $t$, the solution $\sigma^*$ chooses an anonymization pattern $p$ matching $t$ and applies $p$ to $t$. If a pattern $p\in {{\cal P}}$ is applied to a row pattern $t$, we say that $t$ is [*attached*]{} to $p$. The solution satisfies the property that, for each anonymization pattern $p \in {{\cal P}}$, number of row patterns attached to it is either zero or at least $k$. If no row pattern is attached to $p$, then we say that $p$ is [*closed*]{}; on the other hand, if at least $k$ row pattern are attached to $p$, we say that $p$ is [*open*]{}. Thus, the optimal solution $\sigma^*$ opens up some subset of patterns from ${{\cal P}}$. Of course, we do not know which patterns are open and which are closed. But, we can guess the set of open patterns by iterating over all possible subsets of ${{\cal P}}$. For each subset $P\subseteq {{\cal P}}$, our goal is to compute the optimal solution whose set of open patterns is exactly equal to $P$. The number of such subsets is $2^{|{{\cal P}}|}$, which is a constant since $|{{\cal P}}|$ is a constant. Then, we take the minimum of the over these solutions. Consider a subset of patterns $P$. Our goal is to find the optimal solution in which the set of open patterns is exactly equal to $P$. Notice that there may not exist any feasible solution for the subset $P$; we also need to determine, if this is the case. This can be formulated as the following integer linear program. For each row pattern $t \in {{\cal R}}$, $s(t)$ denotes the number of copies (i.e., tuples) of the row pattern in the input table ($s(t) = 0$ if the row pattern $t$ does not occur in the table). For each pair $(p \in {{\cal P}},t\in {{\cal R}})$ such that the row pattern $t$ matches the pattern $p$, we introduce an integer variable $x_{p,t}$. This variable captures the number of copies of the row pattern $t$ attached to the anonymization pattern $p$. For a pattern $p$, let ${{\rm Cost}}(p)$ denote that number of suppressed cells in $p$; this is the cost each copy of a row pattern $t$ would pay, if $t$ is attached to $p$. $$\begin{aligned} \notag \min \sum_{(p,t):t\sim p} {{\rm Cost}}(p) x_{p,t} & &\\ \notag \mbox{subject to:} &\\ \sum_{t:t\sim p} x_{p,t} \geq k & & \mbox{for all } p\in P\\ \label{eqn:AAA} \sum_{p:t\sim p} x_{p,t} = s(t) & & \mbox{for all } t\in {{\cal R}}\\ \label{eqn:BBB} x_{p, t}\in \mathbb{N}_0 & & \mbox{for all } (p,t):t\sim p\end{aligned}$$ Note that this integer linear program has a constant number of variable as the number of $x_{p,t}$ variables is bounded by $|{{\cal P}}|\cdot|{{\cal R}}| \leq m^{2|\Sigma|+1}$. By the famous result of Lenstra [@Lenstra], an integer linear program on constant number of variables can be solved in polynomial time. We relax the integrality constraints to get the following linear program. The equality constraints (\[eqn:AAA\]) are split into two inequalities and relax the integrality constraints (\[eqn:BBB\]). [^1]. $$\begin{aligned} \notag \min \sum_{(p,t):t\sim p} {{\rm Cost}}(p) x_{p,t} & &\\ \notag \mbox{subject to:} & \\ \label{eqn:CCC} \sum_{t:t\sim p} x_{p,t} \geq k & & \mbox{for all } p\in P\\ \label{eqn:DDD} \sum_{p:t\sim p} x_{p,t} \geq 1 & & \mbox{for all } t\in T\\ \label{eqn:EEE} -\sum_{p:t\sim p} x_{p,t} \geq -1 & & \mbox{for all } t\in T\\ \label{eqn:FFF} x_{p, t}\geq 0 & & \mbox{for all } (p,t):t\sim p\end{aligned}$$ Let $A$ denote the constraint matrix where each row is a constraint and each column is a variable. Our main claim is that the matrix $A$ is totally unimodular (see [@opt-book] for a discussion on totally unimodular matrices). Recall that a square matrix is unimodular, if its determinant is -1, 0 or +1 and that a matrix is totally unimodular, if all its square submatrices are unimodular. It is well known that if the constraint matrix is totally unimodular and the right hand side of the constraints are all integral, then the optimal solution is integral. This means that in the optimal solution returned by the linear program, the values of all the variables $x_{p,t}$ will be either 0 or 1. Then, our constraints imply that for each row $t$, $x_{p,t}=1$ for exactly one pattern $p$ that $t$ matches to. It is now easy to convert the linear program solution into a $k$-anonymization solution of same cost. We shall show that $A^T$ (transpose of $A$) is totally unimodular. Taking transpose preserves total unimodularity, and thus we would get that $A$ is also totally unimodular. Let $X$ be a $a\times b$ matrix and $Y$ be a $a\times c$ matrix. Let $X||Y$ denote the $a\times (b+c)$ matrix obtained by concatenating $X$ and $Y$ along the column-side (namely, the first $b$ columns of $X||Y$ are taken from $X$ and the remaining $c$ columns are taken from $Y$). The matrix $A^T$ can be decomposed into four parts given as $A^T = A_1 || A_2 || \overline{A}_2 || I$. The fours parts $A_1, A_2, \overline{A}_2$ and $I$, come from the four sets of constraints, \[eqn:CCC\], \[eqn:DDD\], \[eqn:EEE\] and \[eqn:FFF\], respectively. The matrix $A_1$ has a row corresponding to each variable $x_{p,t}$ and a column corresponding to each pattern $p'$, and a $1$ in the corresponding cell if $p=p'$ and $0$, otherwise. Similarly, the matrix $A_2$ has a row corresponding to each variable $x_{p,t}$ and a column corresponding to each row $t'$, and a $1$ in the corresponding cell if $t=t'$ and $0$, otherwise. The matrix $\overline{A}_2$ is given by $\overline{A}_2 = (-1)A_2$ and $I$ is an identity matrix. We can multiply each column of $A_2$ by $(-1)$; this may only change the sign of the determinants of the submatrices of $A$. Thus, it suffices to show that $A' = A_1 || A_2 || A_2 || I$ is totally unimodular. Notice that $A_1 || A_2$ is the incidence matrix of the bipartite graph $G$ defined below: the patterns in ${{\cal P}}$ form one side of the graph and the rows of the table $T$ form the other side; an edge is drawn between a pattern $p$ and a row $t$, if $t$ matches $p$. Now the claim that $A'$ is totally unimodular follows from the following three simple propositions.  [@opt-book] The incidence matrix of any bipartite graph is totally unimodular. If $X||Y$ is totally unimodular then $X||Y||Y$ is also totally unimodular. [[*Proof:* ]{}]{}Write $Z=X||Y||Y$. The matrix $Z$ has three parts to it. Consider any submatrix $S$ of $Z$. If $S$ includes only columns from the first two parts, then it is a submatrix of $X||Y$, which is guaranteed to be unimodular. Now, suppose $S$ includes a column from the second part and the corresponding column from the third part. Then, these two columns can be canceled out yielding a submatrix with a column full of 0’s and hence, having a determinant of $0$. The remaining case is that $S$ does not include any pair of corresponding columns. In this case $S$ is also a submatrix of $X||Y$ and hence, it is unimodular. [$\Box$]{}  [@opt-book] If $X$ is totally unimodular then $X||I$ is totally unimodular, where $I$ is an identity matrix. This approach, when applied to the practical case of $m=O(\log n)$ and $|\Sigma|$ being arbitrarily large, leads to a variant of facility location problem. The patterns ($n2^m=n^{O(1)}$ in number) can be viewed as facilities with a connection cost equal to the number of suppressed cells. The rows can be viewed as clients who can be serviced by any pattern that they match to. The goal is to open a subset of the facilities and attach the clients to the facilities such that every open facility has at least $k$ clients attached to it. Objective is to minimize the total connection cost of all the clients. Note that the distances here are non-metric. No approximation algorithms are known for this variant. Designing approximation algorithms for this facility location problem that can in turn yield approximation algorithm for the above case of anonymization problem would be interesting. Open Problems ============= For the general $k$-anonymization problem, the best known approximation algorithm, due to Aggarwal et al. [@ICDT], achieves a ratio of $O(k)$. Their algorithm is based on a natural graph theoretic framework. They showed that any poly-time algorithm that uses their framework cannot achieve a factor better than $O(k)$. Breaking the $O(k)$-approximation barrier seems to be a challenging open problem. Improving the $O(\log k)$ approximation ratio, due to Park and Shim [@SIGMOD], for the practical special case when $m=O(\log n)$ is an interesting open problem. For the case where $m$ is constant, a trivial constant factor approximation algorithm exists: suppressing all cells yields an $O(m)$ approximation ratio. However, it is challenging to design an algorithm that, for all constants $m$, guarantees a fixed constant approximation ratio (say, $2$); notice that such an algorithm is allowed to run in time $2^{2^m}$. Getting a hardness of approximation better than $\frac{6238}{6237}$ would be of interest. [^1]: We do not need to enforce the constraint “$x_{p,t}\leq 1$”, since this is automatically taken are by third set of constraints.

--- abstract: 'The ensemble of Euclidean gluon field configurations represented by the domain wall network is considered. A single domain wall is given by the sine-Gordon kink for the angle between chromomagnetic and chromoelectric components of the gauge field. The domain wall separates the regions with self-dual and anti-self-dual fields. The network of the domain wall defects is introduced as a combination of multiplicative and additive superpositions of kinks. The character of the spectrum and eigenmodes of color-charged fluctuations in the presence of the domain wall network is discussed. The concept of the confinement-deconfinement transition in terms of the ensemble of domain wall networks is outlined. Conditions for the formation of a stable thick domain wall junction (the chromomagnetic trap) during heavy ion collisions are discussed, and the spectrum of color charged quasiparticles inside the trap is evaluated. An important observation is the existence of the critical size $L_c$ of the trap stable against gluon tachyonic modes, which means that deconfinement can occur only in a finite region of space-time in principle. The size $L_c$ is related to the value of gluon condensate $\langle g^2F^2\rangle$.' author: - 'Sergei N. Nedelko[^1], Vladimir E. Voronin[^2]' title: Domain wall network as QCD vacuum and the chromomagnetic trap formation under extreme conditions --- Introduction ============ In general, diffusion of the relativized versions of ideas born in condensed matter and solid state physics to the quantum field theory has been proven to be extremely fruitful. It was realised long time ago that a complex of problems associated with investigation of the QCD vacuum structure appeared as particularly suitable object in this respect. This paper is focused on the further development of approach to QCD vacuum as a medium describable in terms of statistical ensemble of domain wall networks. This concept plays important role in description of condensed matter systems with rival order and disorder but has been insufficiently explored in application to QCD vacuum. The identification of the properties of nonperturbative gauge field configurations relevant to a coherent resolution of confinement, chiral symmetry breaking, $U_{\rm A}(1)$ and strong CP problems is an overall task pursued by most approaches to the QCD vacuum structure. As a rule, analytical as well as Lattice QCD studies of QCD vacuum structure are focused on localized topological configurations (instantons, monopoles and dyons, vortices) which *via* condensation could be seen as appropriate gauge field configurations responsible for confinement of static color charges and other nonperturbative features of strong interactions. In recent years, three-dimensional configurations akin to domain walls became popular as well [@Ilgenfritz:2007xu; @Moran:2008xq; @Moran:2007nc; @deForcrand:2008aw; @deForcrand:2006my; @Zhitn]. First of all, these are the $Z(3)$ domain walls related to the center symmetry of the pure Yang-Mills theory [@deForcrand:2008aw] and double-layer domain wall structures in topological charge density [@Zhitn]. Lattice QCD serves as a main source of motivation and verification tool for these studies in pair with the theoretically appealing scenario of static quark confinement in the spirit of the dual Meissner effect equipped with the Wilson and Polyakov loop criteria. The localized configurations are characterized by the vanishing ratio of the action to the 4-volume in the infinite volume limit. In this sense, instantons, monopoles, vortices and double-layer domain walls are localized configurations. A complementary treatment of the above mentioned overall task is based on the investigation of the properties of quantum effective action of QCD. As in other quantum systems with infinitely many degrees of freedom, the global minima of the effective action define the phase structure of QCD. The identification of global minima in different regimes (high energy density, high baryon density, strong external electromagnetic fields) has highest priority for understanding the phase transformations in hadronic matter. In general, a nontrivial global minimum corresponds to a gauge field with the strength not vanishing at space-time infinity and, hence, extensive action proportional to the four dimensional space-time volume of the system, unlike the localized configurations. A variety of essentially equivalent statements of the problem in the context of QCD can be found, for instance, in [@Minkowski; @Pagels; @Mink; @Leutwyler; @NK1; @Faddeev0]. Global minima related by discrete symmetry transformations like CP, Weyl symmetry in the root space of $su(N_{\rm c})$, center symmetry in particular, is a reason to look for field configurations interpolating between them. First of all, these are domain wall configurations, but also lower dimensional topological defects. There is among others one difference between this treatment and approaches based on localized objects: the last one intends to merge the initially isolated objects (e.g., instanton gas or liquid) while the former collects defects in an initially homogeneous background. At first sight, both ways seem to lead to a similar outcome - a class of nonperturbative gluon field configurations with a self-consistent balance of order and disorder which can be characterized, in particular, by nonzero gluon condensate and topological charge density. However, essential disparity can arise since a superposition of localized objects inherits the properties of isolated objects while the superposition of defects in the initially homogeneous ordered background brings some disorder and merely refines the overall properties of the background. For instance, the superposition of infinitely many instantons and anti-instantons is not a configuration with a finite classical action but it maintains the property to have integer-valued topological charge. On the contrary, the configuration obtained by implanting infinitely many domain wall defects into the Abelian covariantly constant (anti-)self-dual field can have any real value of the mean topological charge density as well as any real value of topological charge fraction per domain [@NK3]. Both configurations can be seen as lumps of the topological charge density distributed in the Euclidean space-time like in Fig.\[Fig:kink\_network\] or in the lattice configurations [@Moran:2008xq; @Ilgenfritz:2007xu]. However, in the instanton picture each lump carries an integer charge while in the treatment of global minima the charge is any real, irrational, for instance, number. This can have dramatic consequences for the fate of $\theta$ parameter in QCD and the natural resolution of the strong CP-problem [@NK3]. In the Euclidean formulation, the statement of the problem starts with the very basic symbol of the functional integral $$\begin{aligned} &&Z=N\int\limits_{{\cal F}} DA \exp\{-S[A]\}, \end{aligned}$$ where the functional space ${\cal F}$ is subject to the condition $$\begin{aligned} \label{cond0} {\cal F}=\{A: \lim_{V\to \infty} \frac{1}{V}\int\limits_V d^4xg^2F^a_{\mu\nu} (x)F^a_{\mu\nu}(x) =B_{\mathrm vac}^2\}.\end{aligned}$$ The constant $B_{\mathrm vac}$ is not equal to zero in the general case, which is equivalent to nonzero gluon condensate $\langle g^2F^2 \rangle$. Condition (\[cond0\]) singles out fields $B_\mu^a$ with the strength which is constant almost everywhere in $R^4$. It is a necessary requirement to allow gluon condensate to be nonzero. It does not forbid also fields with a finite action since the case $B_{\mathrm vac}=0$ has to be also studied. The dynamics chooses the value of $B_{\mathrm vac}$. However, the phenomenology of strong interactions has already required nonzero gluon and quark condensates. Hence, they must be allowed in the QCD functional integral from the very beginning. Separation of the long range modes $B_\mu^a$ responsible for gluon condensate and the local fluctuations $Q_\mu^a$ in the background $B_\mu^a$, must be supplemented by the gauge fixing condition. The background gauge condition for fluctuations $D(B)Q=0$ is the most natural choice. Further steps include integration over the fluctuation fields resulting in the effective action for the long-range fields and identification of the minima of this effective action (for more details see [@NK1; @NK3; @NG2011-1]) which dominate over the integral in the limit $V\to \infty$ and define the phase structure of the system. As soon as minima are identified, this setup defines a principal scheme for self-consistent identification of the class of gauge fields which almost everywhere coincide with the global minima of the quantum effective action. A treatment of these “vacuum fields” in the functional integral $$\begin{aligned} Z &=&N'\int\limits_{{\cal B}}DB \int\limits_{{\cal Q}} DQ \det[D(B)D(B+Q)] \\ &&\times\delta[D(B)Q] \exp\{-S[B+Q]+S[B]\}.\end{aligned}$$ must be nonperturbative. The fields $B_\mu^a\in{\cal B}$ are subject to condition (\[cond0\]) with the fixed vacuum value of the condensate $B_{\mathrm vac}^2$. The condensate plays the role of the scale parameter of QCD to be identified from the hadron phenomenology. The fluctuations $Q$ in the background of the vacuum fields can be seen as perturbations. The homogeneous fields with the domain wall defects are the most natural and simplest example of gluon configurations which are homogeneous almost everywhere in $R^4$ and satisfy the basic condition Eq.(\[cond0\]). Basic argumentation in favour of the Abelian (anti-)self-dual homogeneous fields as global minima of the effective action originates from papers [@Minkowski; @Leutwyler; @Pagels; @Woloshin; @Pawlowski; @NG2011]. Within the Ginzburg-Landau approach to the effective action the domain wall is described simply by the sine-Gordon kink for the angle between chromomagnetic and chromoelectric components of the gluon field [@NG2011]. This kink configuration can be seen as either Bloch or Néel domain wall separating the regions with self-dual and anti-self-dual Abelian gauge fields. On the domain wall the gluon field is Abelian with orthogonal to each other chromomagnetic and chromoelectric fields. We shall not repeat here arguments leading to this conclusion but just refer to papers [@NG2011; @NG2011-1] where a more detailed discussion can be found. Group theoretical analysis of the Weyl symmetry and subgroup embeddings behind the domain wall formation in the effective gauge theories is given in a recent paper [@George:2012sb]. It should be also mentioned that the model of confinement, chiral symmetry breaking and hadronization based on the dominance of the gluon fields which are (anti-)self-dual Abelian almost everywhere demonstrated high phenomenological performance [@EN; @NK1; @NK3; @NK4]. The purpose of the present paper is to evolve the approach outlined in article [@NG2011] in two respects: explicit analytical construction of the domain wall network in $R^4$ through a combination of additive and multiplicative superpositions of kinks, and refining the spectrum and eigenmodes of the color charged scalar, spinor and vector fields in the background of a domain wall. In particular the spectrum of quasiparticles inside the thick domain wall junction is evaluated. It is shown that the standard methods of the sine-Gordon model [@Vachaspati] allow one to generate various domain wall networks. The eigenvalues and eigenfunctions are found for the Laplace operator in the background of a single infinitely thin domain wall. In this case, the eigenvalue problem has to be solved separately in the bulk of $R^4$ and on the 3-dimensional hyperplane of the wall. The continuity of the charge current through the wall is required together with the square integrability of the bulk eigenfunctions. For the infinitely thin wall the bulk eigenfunction possesses the purely discrete spectrum which coincides with the spectrum for the case of a homogeneous (anti-)self-dual field without a kink defect. The eigenfunctions differ in a certain way but have the same harmonic oscillator type as the ones in the absence of the kink defect. These modes describe confined color charged fields. The eigenfunctions localized on the wall have continuous spectrum with the dispersion relation of charged quasi-particles. This confirms qualitative conjectures of [@NG2011; @NG2011-1]. It is argued that thick domain wall junction may be formed during heavy ion collisions and play the role of a trap for charged quasi-particles. Confinement is lost inside the trap of a finite size. There exists a critical size of the stable trap, beyond which the emerging tachyonic gluon modes destroy it. The paper is organized as follows. Section II is devoted to the domain wall network construction. The spectrum of scalar color charged field in the background of infinitely thin domain wall is discussed in section III. In the fourth section we discuss the chromomagnetic trap formation and evaluate the spectrum and eigenmodes of the color charged scalar, vector and spinor quasiparticles inside the trap. Nonzero gluon condensate $\langle g^2F^2\rangle$ and domain wall network in QCD vacuum ====================================================================================== The calculation of the effective quantum action for the Abelian (anti-)self-dual homogeneous gluon field within the functional renormalization group approach [@Pawlowski] has indicated that this configuration is a serious candidate for the role of global minimum of QCD effective action and has enhanced the older one-loop results [@Minkowski; @Leutwyler; @Pagels]. The functional RG result also supported conclusions of [@NK1; @NG2011] based on the Ginzburg-Landau type effective Lagrangian of the form $$\begin{aligned} \mathcal{L}_{\mathrm{eff}} &=& - \frac{1}{4\Lambda^2}\left(D^{ab}_\nu F^b_{\rho\mu} D^{ac}_\nu F^c_{\rho\mu} + D^{ab}_\mu F^b_{\mu\nu} D^{ac}_\rho F^c_{\rho\nu }\right) \nonumber\\ &-&U_{\mathrm{eff}} \nonumber \\ U_{\mathrm{eff}}&=&\frac{\Lambda^4}{12} {\rm Tr}\left(C_1\breve{ f}^2 + \frac{4}{3}C_2\breve{ f}^4 - \frac{16}{9}C_3\breve{ f}^6\right), \label{ueff}\end{aligned}$$ where $\Lambda$ is a scale of QCD related to gluon condensate, $\breve f=\breve F/\Lambda^2$, and $$\begin{aligned} && D^{ab}_\mu = \delta^{ab} \partial_\mu - i\breve{ A}^{ab}_\mu = \partial_\mu - iA^c_\mu {(T^c)^{ab}}, \\ && F^a_{\mu\nu} = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu - if^{abc} A^b_\mu A^c_\nu, \\ && \breve{ F}_{\mu\nu} = F^a_{\mu\nu} T^a,\ \ \ T^a_{bc} = -if^{abc} \\ && {\rm Tr}\left(\breve{ F}^2\right) = \breve{ F}^{ab}_{\mu\nu}\breve{ F}^{ba}_{\nu\mu} = -3 F^a_{\mu\nu}F^a_{\mu\nu} \leq 0, \\ && C_1>0, \ C_2>0, \ C_3 > 0.\end{aligned}$$ Detailed discussion of this expression can be found in [@NG2011]. Here it should be noted that all symmetries of QCD are respected and the signs of the constants are chosen so that the action is bounded from below and its minimum corresponds to the fields with nonzero strength, i.e. $F^2\not=0$ at the minimum. Thus, an important input is the existence of the nonzero gluon condensate. By inspection, one gets as an output twelve (for $SU(3)$) global degenerate discrete minima. The minima are achieved for covariantly constant Abelian (anti-)self-dual fields $$\begin{aligned} \breve{ A}_{\mu} = -\frac{1}{2}\breve{ n}_k F_{\mu\nu}x_\nu, \, \tilde F_{\mu\nu}=\pm F_{\mu\nu}\end{aligned}$$ where the matrix $\breve{n}_k$ belongs to the Cartan subalgebra of $su(3)$ $$\begin{aligned} \breve n_k &=& T^3\ \cos\left(\xi_k\right) + T^8\ \sin\left(\xi_k\right), \nonumber\\ \xi_k&=&\frac{2k+1}{6}\pi, \, k=0,1,\dots,5. \label{HLxik}\end{aligned}$$ The values $\xi_k$ correspond to the boundaries of the Weyl chambers in the root space of $su(3)$. The minima are connected by the discrete parity and Weyl transformations, which indicates that the system is prone to existence of solitons (in real space-time) and kink configurations (in Euclidean space). Below we shall concentrate on the simplest configuration – kink interpolating between self-dual and anti-self-dual Abelian vacua. If the angle $\omega$ between chromoelectric and chromomagnetic fields is allowed to deviate from the constant vacuum value and all other parameters are fixed to the vacuum values, then the Lagrangian takes the form $$\begin{aligned} \label{SG} \mathcal{L}_{\textrm{eff}} &=& -\frac{1 }{2}\Lambda^2 b_{\textrm{vac}}^2 \partial_\mu \omega \partial_\mu \omega \\ &-& b_{\textrm{vac}}^4 \Lambda^4 \left(C_2+3C_3b_{\textrm{vac}}^2 \right){\sin^2\omega},\end{aligned}$$ with the corresponding sine-Gordon equation $$\begin{aligned} \partial^2\omega = m_\omega^2 \sin 2\omega,\ \ m_\omega^2 = b_{\textrm{vac}}^2 \Lambda^2\left(C_2+3C_3b_{\textrm{vac}}^2 \right),\end{aligned}$$ and the standard kink solution $$\omega(x_\mu) = 2\ {\rm arctg} \left(\exp(\mu x_\mu)\right) \label{sakink}$$ interpolating between $0$ and $\pi$. Here $x_\mu$ stays for one of the four Euclidean coordinates. The kink describes a planar domain wall between the regions with almost homogeneous Abelian self-dual and anti-self-dual gluon fields. Chromomagnetic and chromoelectric fields are orthogonal to each other on the wall, see Fig.\[Fig:single\_kink\]. Far from the wall, the topological charge density is constant, its absolute value is equal to the value of the gluon condensate. The topological charge density vanishes on the wall. The upper plot shows the profiles of the components of the chromomagnetic and chromoelectric fields corresponding to the Bloch domain wall – the chromomagnetic field flips in the direction parallel to the wall plane. ![Kink profile in terms of the components of the chromomagnetic and chromoelectric field strengths (upper plot), and a two-dimensional slice for the topological charge density in the presence of a single kink measured in units of $g^2F^b_{\alpha\beta}F^b_{\alpha\beta}$ (lower plot). Here $\omega$ is the angle between the chromomagnetic and chromoelectric fields, $\cos\omega=F^a_{\mu\nu}\tilde F^a_{\mu\nu}/F^b_{\alpha\beta}F^b_{\alpha\beta}$. The three-dimensional planar domain wall separates the four-dimensional regions filled with the self-dual (blue color) and anti-self-dual (red color) Abelian covariantly constant gluon fields. The chromomagnetic and chromoelectric fields are orthogonal to each other inside the wall (green color). []{data-label="Fig:single_kink"}](kink){width="75mm"} ![Kink profile in terms of the components of the chromomagnetic and chromoelectric field strengths (upper plot), and a two-dimensional slice for the topological charge density in the presence of a single kink measured in units of $g^2F^b_{\alpha\beta}F^b_{\alpha\beta}$ (lower plot). Here $\omega$ is the angle between the chromomagnetic and chromoelectric fields, $\cos\omega=F^a_{\mu\nu}\tilde F^a_{\mu\nu}/F^b_{\alpha\beta}F^b_{\alpha\beta}$. The three-dimensional planar domain wall separates the four-dimensional regions filled with the self-dual (blue color) and anti-self-dual (red color) Abelian covariantly constant gluon fields. The chromomagnetic and chromoelectric fields are orthogonal to each other inside the wall (green color). []{data-label="Fig:single_kink"}](single_kink){width="70mm"} {width="70mm"} {width="75mm"} ![A two-dimensional slice of the four-dimensional lump of anti-self-dual field in the background of the self-dual configuration. The domain wall surrounding the lump in the four-dimensional space is given by the multiplicative superposition of eight kinks as it is defined by Eq.(\[asd\_lump\]). []{data-label="Fig:asd_lump"}](asd_lump){width="70mm"} The domain wall network can be now constructed by the standard methods [@Vachaspati]. Let us denote the general kink configuration as $$\zeta(\mu_i,\eta_\nu^{i}x_\nu-q^{i})=\frac{2}{\pi}\arctan\exp(\mu_i(\eta_\nu^{i}x_\nu-q^{i})),$$ where $\mu_i$ is the inverse width of the kink, $\eta_\nu^{i}$ is a normal vector to the plane of the wall, $q^{i}=\eta_\nu^{i}x^{i}_\nu$ with $x^{i}_\nu$ - coordinates of the wall. The topological charge density for the multiplicative superposition of two kinks with the normal vectors anti-parallel to each other $$\omega(x_1)=\pi\zeta(\mu_1,x_1-a_1)\zeta(\mu_2,-x_1-a_2)$$ is shown in Fig.\[multsupkink\]. The additive superposition of infinitely many pairs $$\label{layered} \omega(x_1)=\pi\sum\limits_{j=1}^{\infty}\zeta(\mu_j,x_1-a_j)\zeta(\mu_{j+1},-x_1-a_{j+1})$$ gives a layered topological charge structure in $R^4$, Fig.\[Fig:layered\]. Formally, one may try to go further and consider the product $$\label{asd_lump} \omega(x)=\pi\prod_{i=1}^k \zeta(\mu_i,\eta_\nu^{i}x_\nu-q^{i}).$$ For an appropriate choice of normal vectors $\eta^i$ this superposition represents a lump of anti-self-dual field in the background of the self-dual one, in two, three and four dimensions for $k=4,6,8$, respectively. The case $k=8$ is illustrated in Fig.\[Fig:asd\_lump\]. The general kink network is then given by the additive superposition of lumps (\[asd\_lump\]) $$\label{kink_network} \omega=\pi\sum_{j=1}^{\infty}\prod_{i=1}^k \zeta(\mu_{ij},\eta_\nu^{ij}x_\nu-q^{ij}).$$ The correponding topological charge density is shown in Fig. \[Fig:kink\_network\]. This figure as well as the LHS of Fig. \[Fig:layered\] represents the configuration with infinitely thin domain wall defects, that is the Abelian homogeneous (anti-)self-dual field almost everywhere in $R^4$ characterized by the nonzero absolute value of the topological charge density which is constant and proportional to the value of the action density almost everywhere. The most RHS plots in Figs. \[Fig:layered\] and \[Fig:kink\_network\] show the opposite case of the network composed of very thick kinks. Green color corresponds to the gauge field with an infinitesimally small topological charge density. Study of the spectrum of colorless and color charged fluctuations indicates that the LHS configuration is expected to be confining (only colorless hadrons can be excited as particles) while the RHS one (crossed orthogonal field) supports the color charged quasiparticles as the elementary excitations. It is expected that the RHS configuration can be triggered by external electromagnetic fields  [@NG2011-1; @D'Elia:2012zw; @Bali:2013esa]. Strong electromagnetic fields emerge in relativistic heavy ion collisions [@Skokov:2009qp; @toneev; @Warringa]. Even after switching off the external electromagnetic field the nearly pure chromomagnetic vacuum configuration (RHS Fig.\[Fig:kink\_network\]) can support strong anisotropies [@Tuchin:2013ie] and, in particular, influence the chiral symmetry realization in the collision region [@Fukushima:2012kc]. More detailed consideration of the spectrum of elementary color charged excitations at the domain wall junctions (the green regions) is given in the section \[trap\]. A comment on representation of the domain wall network in terms of the vector potential is in order. The domain wall network constructed in this section relies on the separation of the Abelian part from the general gauge field. The vector potential representation can be easily realized for the planar Bloch domain wall and their layered superposition, Fig. \[Fig:layered\]. The same is true also for the interior of a thick domain wall junction, where field is almost homogeneous. The description of the domain walls in the general network Fig. \[Fig:kink\_network\] in terms of the vector potential requires application of the gauge field parametrization suggested in a series of papers by Y.M. Cho [@Cho1; @Cho2], S. Shabanov [@shabanov1; @shabanov2], L.D. Faddeev and A. J. Niemi [@Faddeev] and, recently, by K.-I. Kondo [@Kondo]. In this parameterization the Abelian part ${\hat V}_\mu (x)$ of the gauge field ${\hat A}_\mu (x)$ is separated manifestly, $$\begin{aligned} \label{eqnsFaddeevNiemKondo} {\hat A}_\mu (x) &=& {\hat V}_\mu (x) + {\hat X}_\mu (x), \, {\hat V}_\mu (x) = {\hat B}_\mu (x) + {\hat C}_\mu (x), \\ {\hat B}_\mu (x) &=& [n^aA^a_\mu (x)]\hat{n} (x)=B_\mu(x)\hat{n}(x), \nonumber \\ {\hat C}_\mu (x) &=& g^{-1}\partial_\mu \hat{n}(x)\times \hat{n}(x), \nonumber\\ {\hat X}_\mu (x) &=& g^{-1}{\hat n}(x) \times \left( \partial_\mu {\hat n}(x) + g {\hat A}_\mu (x) \times {\hat n}(x) \right), \nonumber\end{aligned}$$ where ${\hat A}_\mu (x) = A^a_\mu (x) t^a$, ${\hat n} (x) = n_a (x) t^a$, $n^a n^a = 1$, and $$\begin{aligned} {\partial_\mu\hat n}\times {\hat n} = i f^{abc}\partial_\mu n^a n^b t^c,\, \, [t^a,t^b]=if^{abc}t^c.\end{aligned}$$ The field ${\hat V}_\mu $ is seen as the Abelian field in the sense that $[{\hat V}_\mu (x),{\hat V}_\nu (x)]=0$. The color vector field $n^a(x)$ may be used for detailed description of the thin domain wall junctions in general case. This issue is beyond the scope of the present paper and will be considered elsewhere. [![Three-dimensional slices of the kink network - additive superposition of numerous four-dimensional lumps as it is given by Eq. . The correspondence of colors to the character of the configuration is the same as in Fig. \[Fig:layered\]. []{data-label="Fig:kink_network"}](cube1 "fig:"){width="25mm"}]{} [![Three-dimensional slices of the kink network - additive superposition of numerous four-dimensional lumps as it is given by Eq. . The correspondence of colors to the character of the configuration is the same as in Fig. \[Fig:layered\]. []{data-label="Fig:kink_network"}](cube3 "fig:"){width="25mm"}]{} [![Three-dimensional slices of the kink network - additive superposition of numerous four-dimensional lumps as it is given by Eq. . The correspondence of colors to the character of the configuration is the same as in Fig. \[Fig:layered\]. []{data-label="Fig:kink_network"}](cube4 "fig:"){width="25mm"}]{} Charged field fluctuations in the background of a planar domain wall ==================================================================== Boundary condition ------------------ In this section we study the spectrum of color charged field fluctuations in the background of a single planar domain wall of the Bloch type. The best thing to do would be to solve the eigenvalue problem for the kink of the finite width. However, the problem turns out to be not that simple. Let us consider the problem for the scalar field in the adjoint representation, that is just the Faddeev-Popov ghost field in the background gauge. The quadratic part of the action for the scalar field in the background field of a planar kink with the finite width placed at $x_1=0$ looks like $$\begin{aligned} \label{action_sc} S[\Phi]&=&-\int d^4x (D_\mu\Phi)^\dagger(x)D_\mu\Phi(x) \\ \nonumber &=&\int d^4x \Phi^\dagger(x)D^2\Phi(x), \\ \nonumber D_\mu&=&\partial_\mu+i\breve B_\mu, \, \breve B_\mu=-\breve n B_\mu(x).\end{aligned}$$ Here $\breve n$ is the constant color matrix, $B_\mu$ is the vector potential for the planar Bloch domain wall. For our purposes the most convenient gauge for $B_\mu$ is $$\begin{aligned} \label{gauge1} && B_1=H_2(x_1)x_3+H_3(x_1)x_2, \\ \nonumber && B_2=B_3=0,\quad B_4=-Bx_3, \\ \nonumber &&H_2=B\sin\omega(x_1), \,H_3=-B\cos\omega(x_1), \\ \nonumber &&\omega(x_1)=2\ {\rm arctg} \exp\mu x_1.\end{aligned}$$ A kink with the finite width is a regular everywhere in $R^4$ function, the scalar field is assumed to be a continuous square integrable function. Integration by parts in Eq.(\[action\_sc\]) does not generate surface terms either at infinity or at the location of the kink. However, there is a peculiarity related to the chosen gauge of the background field. According to Eq.(\[gauge1\]), $$\begin{aligned} D^2&=& \tilde D^2+i\partial_\mu\breve B_\mu, \label{covder1}\\ \tilde D^2&=& \partial^2+2i\breve B_\mu\partial_\mu -i\breve B_\mu \breve B_\mu \nonumber\\ &=& (\partial_1-i\breve n H_2(x_1)x_3-i\breve nH_3(x_1)x_2)^2 \nonumber\\ && + \partial_2^2 +\partial_3^2 + (\partial_4+i\breve n Bx_3)^2 -i\partial_1B_1 \nonumber\\ \partial_\mu\breve B_\mu&=&-\breve n H^\prime_2(x_1)x_3-\breve n H^\prime_3(x_1)x_2. \label{sing_gauge}\end{aligned}$$ The action can be written as $$\begin{aligned} S[\Phi]&=&\int d^4x \Phi^\dagger(x)\tilde D^2\Phi(x) \label{surf_gauge}\\ &-&i\int d^4x \Phi^\dagger(x)\breve n\Phi(x) \left[H^\prime_2(x_1)x_3+H^\prime_3(x_1)x_2\right] . \nonumber\end{aligned}$$ It should be noted that the integral in the second line is equal to zero if $\Phi^\dagger(x)\breve n\Phi(x)$ is an even function of $x_2$ and $x_3$. The structure of $D^2$ in Eq.(\[covder1\]) is quite complicated. In the eigenvalue problem the variables can hardly be separated in the case of the finite width of the kink. The problem becomes much simpler and tractable in the limit of the infinitely thin domain wall $\mu\to\infty$. This limit brings discontinuity into the background field and thus creates a sharp boundary – the hyperplane of the domain wall. In such a situation one has to solve the problem in the bulk and on the wall and match the solutions according to some appropriate conditions. For our choice of the kink location there are three regions to be studied: $x_1<0$ with the self-dual field $B_\mu$, $x_1>0$ with the anti-self-dual field, and $x_1=0$ with the chromomagnetic and chromoelectric fields orthogonal to each other. Conditions imposed onto the eigenmodes of color charged fields on the sharp wall can be obtained from the requirement of preservation of the properties of eigenmodes for finite $\mu$ as far as they can be identified. The continuity of the normal to the wall component of the total (through the whole hypersurface of the wall) charged current offers a reliable guiding principle for identification of the matching conditions. Continuity of the total current means that the surface terms do not appear under integration by parts in the action, $$\begin{aligned} \label{current_cont_sc} &&\lim_{\varepsilon\to 0}\left[J_1(\varepsilon)-J_1(-\varepsilon)\right]=0, \\ \nonumber &&J_\mu(x_1)=\int d^3x \Phi^\dagger(x)D_\mu\Phi(x), \\ \nonumber &&d^3x=dx_2dx_3dx_4.\end{aligned}$$ Moreover, this requirement restricts the form of the eigenfunctions in such a way that the surface terms associated with the gauge dependent delta-function singularuties in $\partial_\mu \breve B_\mu$, Eq.(\[sing\_gauge\]) vanish as well. ![Derivatives of the components of the chromomagnetic field are plotted for two values of the width parameter $\mu/\sqrt{B}=3,10$.The coordinate $x_1$ is given in units of $1/\sqrt{B}$. In the limit of the infinitely thin domain wall ($\mu/\sqrt{B}\to\infty$) the derivatives develop the delta-function singularities at the location of the wall. []{data-label="Fig:der_single_kink"}](derivH.eps){width="75mm"} Confined fluctuations in the bulk --------------------------------- Let us consider the eigenvalue problem $$\begin{aligned} \label{scalar1} &&-\tilde D^2\Phi=\lambda\Phi.\end{aligned}$$ for the functions square integrable in $R^4$ and satisfying the condition (\[current\_cont\_sc\]). For all $x_1\not=0$ the operator $\tilde D^2$ takes the form $$\begin{aligned} \label{covder2} \tilde D^2&=& (\partial_1 \pm i\breve n B x_2)^2 \nonumber\\ && + \partial_2^2 +\partial_3^2 + (\partial_4+i\breve n Bx_3)^2 \nonumber\end{aligned}$$ where plus corresponds to the anti-self-dual configuration and minus is for the self-dual one. By inspection one can see that the eigenfunctions satisfy the relation $$\begin{aligned} \label{conditions} \Phi^{(+)}(x_1,x_\perp)=\Phi^{(-)}(-x_1,x_\perp),\end{aligned}$$ where $(\pm)$ denotes the duality of the background field for a given $x_1$. Respectively, the square integrable solutions are $$\begin{aligned} \label{eigenf_sc} \Phi^{(\pm)}_{kl}(x)&=&\phi^{(\pm)}_k(x_1,x_2)\chi_l(x_3,x_4) \\ \phi^{(\pm)}_k(x_1,x_2)&=&\int dp_1 f(p_1)e^{\pm ip_1x_1-\frac{1}{2}|\breve n|B(x_2+p_1/|\breve n|B)^2} \nonumber\\ &\times& H_k\left(\sqrt{|\breve n|B}\left[x_2+\frac{p_1}{|\breve n|B}\right]\right) \nonumber\\ \chi_k(x_3,x_4)&=&\int dp_4 g(p_4)e^{ip_4x_4-\frac{1}{2}|\breve n|B(x_3+p_4/|\breve n|B)^2 } \nonumber\\ &\times& H_l\left(\sqrt{|\breve n|B}\left[x_3+\frac{p_4}{|\breve n|B}\right]\right), \nonumber\end{aligned}$$ where $H_m$ are the Hermite polynomials. The eigenvalues are $$\begin{aligned} \label{eigenv_sc} \lambda_{kl}=2|\breve n|B(k+l+1), \, \, k,l=0,1,\dots.\end{aligned}$$ The amplitudes $f(p_1)$ and $g(p_4)$ have to provide square integrability of the eigenfunctions in $x_1$ and $x_4$. In order to satisfy condition (\[current\_cont\_sc\]) one has to restrict the amplitude $f(p_1)$ additionally. The integral current through the domain wall is continuous if both $f$ and $H_k$ are odd or even functions simultaneously under the combined change $p_1\to-p_1$ and $x_2\to-x_2$ $$\begin{aligned} \label{cont_cond_sc} f(-p_1)H_k(-z)=f(p_1)H_k(z).\end{aligned}$$ This property also guarantees the absence of the gauge specific contribution to the action related to the derivative of $H_3$ in Eqs.(\[sing\_gauge\],\[surf\_gauge\]). A combination of (\[cont\_cond\_sc\]) and (\[conditions\]) obviously leads to the relation $$\begin{aligned} \label{relations} \phi_{k}^{(\pm)}(x_1,x_2)=\phi_{k}^{(\pm)}(-x_1,-x_2),\end{aligned}$$ where $(\pm)$ denotes the duality of the background field for a given $x_1$. This identity allows one to show that the eigenfunctions $$\Phi_{kl}(x) = \left\{ \begin{array}{l} \Phi^{(+)}_{kl}(x), \, \, x_1\in L_+\\ \Phi^{(-)}_{kl}(x), \, \, x_1\in L_- \end{array} \right., \, k,l=0,1\dots$$ form a complete orthogonal set in the space of square integrable functions which are even with respect to simultaneous reflection $x_1\to -x_1$ and $x_2\to-x_2$. The eigenfunctions are of the bound state type with the purely discrete spectrum. Field fluctuations of this type can be seen as confined. It should be noted that the eigenvalues coincide with those for the purely homogeneous (anti-)self-dual Abelian field. In this sense, the domain wall defect does not destroy dynamical confinement of color charged fields. The eigenfunctions are restricted by the correlated evenness condition (\[cont\_cond\_sc\]), while in the case of the homogeneous field the properties of the amplitude $f(p_1)$ and the polynomial $H_k$ are mutually independent. Color charged quasiparticles on the wall ---------------------------------------- Let us now consider the eigenvalue problem on the domain wall, i.e. for the region $x_1=0$. On the wall the chromomagnetic and chromoelectric fields are orthogonal to each other (see Fig.\[Fig:single\_kink\]). In conformity with (\[current\_cont\_sc\]) the absence of the charged current off the infinitely thin domain wall requires $$\begin{aligned} \partial_1\Phi|_{x_1=0}=0,\end{aligned}$$ and the eigenvalue problem on the wall takes the form $$\begin{aligned} \label{covder_wall} \left[- \partial_2^2 -\partial_3^2 +\breve n^2 B^2 x_3^2 + (i\partial_4-\breve n Bx_3)^2\right]\Phi=\lambda\Phi \nonumber\end{aligned}$$ with the solution $$\begin{aligned} \label{on_wall_sol_sc} \Phi_k(x_2,x_3,x_4)&=&e^{ip_2x_2+ip_4x_4}e^{-\frac{|\breve n|B}{\sqrt{2}}\left(x_3-\frac{p_4}{2|\breve n|B}\right)^2} \nonumber\\ &\times& H_k\left[\sqrt{\sqrt{2}|\breve n|B}\left(x_3-\frac{p_4}{2|\breve n|B}\right)\right], \nonumber\\ \lambda_k(p^2_2,p_4^2)&=&\sqrt{2}|\breve n|B(2k+1)+\frac{p_4^2}{2}+\frac{p_2^2}{2}, \nonumber\\ &&k=0,1,2,\dots \nonumber\end{aligned}$$ The spectrum of the eigenmodes on the wall is continuous, it depends on the momentum $p_2$ longitudinal to the chromomagnetic field and Euclidean energy $p_4$, the corresponding eigenfunctions are oscillating in $x_2$ and $x_4$. In the direction $x_3$ transverse to the chromomagnetic field the eigenfunctions are bounded and the eigenvalues display the Landau level structure. The continuation $p_4^2=-p_0^2$ leads to the dispersion relation $$p_0^2=p_2^2+\mu^2_k,\quad \mu^2_k=2\sqrt2(2k+1)|\breve n|B,\, \, k=0,1,2,\dots .$$ This can be treated as the lack of confinement - the color charged quasiparticles with masses $\mu_k$ and momentum $\mathbf{p}$ parallel to the chromomagnetic field $\mathbf{H}$ can be excited on the wall. The case of the planar domain wall configuration (two infinite parts of the space-time separated by a three-dimentional hypersurface like in Fig.\[Fig:single\_kink\]) is rather artificial. Its weight in the whole ensemble of the gluon field configurations with the constant scalar condensate $\langle g^2F_{\mu\nu}F_{\mu\nu}\rangle$ and the lumpy structured distribution of the topological charge density $\langle g^2\tilde F_{\mu\nu}F_{\mu\nu}\rangle$ is negligible. The entropy-energy balance implies that the typical configuration should be highly disordered (see Fig.\[Fig:kink\_network\]). Moreover, in the case of the planar domain wall the eigenvalue problem for the square integrable vector gauge fields $$\begin{aligned} \label{vector} \left[-D^2\delta_{\mu\nu}+2i\breve F_{\mu\nu}\right]Q_\nu=\lambda Q_\mu\end{aligned}$$ leads to the negative eigenvalues and corresponding tachyonic modes on the wall where $\tilde F_{\mu\nu}F_{\mu\nu}=0$. This is a well-known instability of the Nielsen-Olesen type [@Nielsen:1978rm]. The presence of the tachyonic mode is due to the three infinite dimensions of the planar domain wall hypersurface. One can expect that finite size of boundaries between lumps in the typical kink network configuration, Fig.\[Fig:kink\_network\], removes the tachyonic modes. This is manifestly exemplified in the next section where the color charged field eigenvalues and modes are studied for thick cylindrical domain wall junction. The relatively stable defect of this type can occur in the ensemble of confining gluon fields due to the influence of the strong electromagnetic fields on the QCD vacuum structure. The spectrum of color charged quasiparticles trapped in a thick domain wall junction {#trap} ==================================================================================== Heavy ion collisions: the strong electromagnetic field as a trigger for deconfinement ------------------------------------------------------------------------------------- It has been observed that the strong electromagnetic fields generated in relativistic heavy ion collisions can play the role of a trigger for deconfinement [@NG2011-1]. The mechanism discussed in [@NG2011-1] is as follows. The electric $\mathbf E_{\rm el}$ and magnetic $\mathbf H_{\rm el}$ fields are practically orthogonal to each other [@toneev; @Skokov:2009qp]: $\mathbf E_{\rm el}\mathbf H_{\rm el}\approx 0 $. For this configuration of the external electromagnetic field the one-loop quark contribution to the QCD effective potential for the homogeneous Abelian gluon fields is minimal for the chromoelectric and chromomagnetic fields directed along the electric and magnetic fields respectively. The orthogonal chromo-fields are not confining: color charged quasiparticles can move along the chromomagnetic field. It has been noted also that this mechanism assumes the strong azimuthal anisotropy in momentum distribution of color charged quasiparticles. Deconfined quarks as well as gluons will move preferably along the direction of the magnetic field but this will happen due to the gluon field configuration even after switching the electromagnetic field off. A detailed and systematic analytical one-loop calculation of the QCD effective potential for the pure chromomagnetic field was performed recently in [@Ozaki:2013sfa] and confirmed the result that the chromomagnetic field prefers to be parallel (or anti-parallel) to the external magnetic field. Another important source of verification of the basic observations of paper [@NG2011-1] is due to the recent Lattice QCD studies of the response of the QCD vacuum to external electromagnetic fields [@D'Elia:2012zw; @Bali:2013esa; @Bali:2013owa; @Bonati:2013qra]. In particular, in qualitative agreement with [@NG2011-1] Lattice QCD study [@Bali:2013esa] has demonstrated that in the presence of external magnetic field the gluonic action develops an anisotropy: the chromomagnetic field parallel to the external field is enhanced, while the chromo-electric field in this direction is suppressed. The results of [@Bali:2013owa] indicated that the magnetic field can affect the azimuthal structure of the expansion of the system during heavy ion collisions. ![ Examples of two-dimensional slice of the cylindrical thick domain wall junctions. The correspondence of colors is the same as in Fig.\[Fig:layered\]. Blue and red regions represent self-dual and anti-self-dual lumps. Confinement is lost in the green region where $g^2\tilde F_{\mu\nu}(x)F_{\mu\nu}(x)=0$. The scalar condensate density $g^2F_{\mu\nu}(x)F_{\mu\nu}(x)$ is nonzero and homogeneous everywhere.[]{data-label="Fig:chromo_bag"}](trap_2d_1 "fig:"){width="38mm"}![ Examples of two-dimensional slice of the cylindrical thick domain wall junctions. The correspondence of colors is the same as in Fig.\[Fig:layered\]. Blue and red regions represent self-dual and anti-self-dual lumps. Confinement is lost in the green region where $g^2\tilde F_{\mu\nu}(x)F_{\mu\nu}(x)=0$. The scalar condensate density $g^2F_{\mu\nu}(x)F_{\mu\nu}(x)$ is nonzero and homogeneous everywhere.[]{data-label="Fig:chromo_bag"}](trap_2d_2 "fig:"){width="38mm"} Within the context of the confining domain wall network these observations mean that a flash of the strong electromagnetic field during heavy ion collisions produces a kind of defect in the form of the thick domain wall junction in the confining gluon background exactly in the region where collision occurs (see Fig.\[Fig:chromo\_bag\]). The electromagnetic flash can act as one of the preconditions for conversion of the high energy density and baryon density to the thermodynamics of color charged degrees of freedom.\ Cylindrical trap ----------------- ### Scalar field eigenmodes Since topological charge density is zero in the interior of the trap ($g^2\tilde F_{\mu\nu}(x)F_{\mu\nu}(x)=0$) there exists a specific reference frame where one can use the pure chromomagnetic field for description of the gluon background inside the trap. For simplicity we take cylindrical geometry of the trap and study the properties of scalar and vector (gluon) color charged field eigenmodes. Extension of the present consideration to more realistic form of the trap is straightforward. Consider the eigenvalue problem for the massless scalar field $\Phi^a$ $$\begin{aligned} \label{cylinder_s} -\left(\partial_\mu-i\breve B_\mu\right)^2\Phi(x)=\lambda^2\Phi(x)\end{aligned}$$ in the cylindrical region $$\begin{aligned} x\in\mathcal{T}=\left\{x_1^2+x_2^2<R^2, \ (x_3,x_4)\in \mathrm{R^2} \right\}\end{aligned}$$ with the homogeneous Dirichlet condition at the boundary $$\begin{aligned} \label{HDC} &&\Phi(x)=0, \ x\in \partial\mathcal{T} \\ &&\partial\mathcal{T} =\left\{x_1^2+x_2^2=R^2, \ (x_3,x_4)\in \mathrm{R^2} \right\}. \nonumber\end{aligned}$$ Here $\breve B_\mu$ stays for adjoiunt representaion of the homogeneous chromomagnetic field $H^a_i=\delta_{i3}n^a H$ with the vector potential taken in the symmetric gauge $$\begin{gathered} \breve B_\mu=-\frac{1}{2} \breve n B_{\mu\nu}x_\nu, \label{purechrmag}\\ \breve B_4=\breve B_3=0, \ B_{12}=-B_{21}=H, \nonumber\\ \breve n=T_3\cos(\xi)+T_8\sin(\xi). \nonumber\end{gathered}$$ The eigenvalues of the matrix $\breve n$ are $$\begin{aligned} \label{n_diag} \breve v=\mathrm{diag}\left[\cos\left(\xi\right),-\cos\left(\xi\right),0,\cos\left(\xi-\frac{\pi}{3}\right), -\cos\left(\xi-\frac{\pi}{3}\right), \cos\left(\xi+\frac{\pi}{3}\right),-\cos\left(\xi+\frac{\pi}{3}\right),0\right]. \label{vadj}\end{aligned}$$ For any value of the angle $\xi$ there are two zero eigenvalues $\breve{v}_3=\breve{v}_8=0$. Two additional zero elements occur in $\breve{v}$ if the angle takes values $\xi_k$ (see Eq. ) minimizing the effective potential and corresponding to the boundaries of the Weyl chambers. By inspection one can check that nonzero eigenvalues $\breve{v}$ take values $\pm v$ with $v=\sqrt{3}/2$. Below we use notation $$\begin{aligned} \nonumber \breve v^a=v\kappa^a.\end{aligned}$$ For example, if $\xi=\xi_0=\pi/6$ then the nonzero values of $v^a$ correspond to $a=1,2,4,5$ and $$\begin{aligned} \nonumber \kappa_1=1, \kappa_2=-1, \kappa_4=1, \ \kappa_5=-1.\end{aligned}$$ It has to be noted that the effective Lagrangian leads to the kink configuration (for details see [@NG2011]) $$\begin{aligned} \xi_{k}(x_i) =\frac{1}{3} \arctan\left[\sinh(m_\xi x_i)\right]+\frac{\pi k}{3}, \ k=0,\dots,5,\end{aligned}$$ interpolating between boundaries $\xi_k$ and $\xi_{k+1}$ of the $k$-th Weyl chamber. Superposition of these “color” domain walls can be arranged in a complete analogy with the “duality” domain walls. The only new feature of the “color” domains is that there are six different types interrelated by the Weyl reflections instead of two types as in the case of duality domains. Solution of the problem is straightforward. We give it below just for completeness. It is convenient to introduce dimensionless variables using the strength of the chromomagnetic field as a basic scale. Below all quantities are assumed to be measured in terms of this scale, for instance $$\sqrt{H}x_\mu \equiv x_\mu,\quad \frac{\lambda}{\sqrt{H}}\equiv \lambda.$$ After diagonalization with respect to color indices and transformation to the cylindrical coordinates Eq.  takes the form $$\begin{aligned} \label{HDC1} -\left[\partial_4^2+\partial_3^2 +\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \vartheta^2} -i\kappa^a v \frac{\partial}{\partial \vartheta} \right. \nonumber\\ \left. -\frac{1}{4} v^2r^2\right]\Phi^a=\lambda^2\Phi^a,\ \ \\end{aligned}$$ where it has been used that $$\begin{aligned} &&x_1=r\cos\vartheta,\ x_2=r\sin\vartheta,\\ \\ &&\frac{\partial}{\partial x_1}=\cos\vartheta\frac{\partial}{\partial r}-\frac{\sin\vartheta}{r}\frac{\partial}{\partial \vartheta}, \\ &&\frac{\partial}{\partial x_2}=\sin\vartheta\frac{\partial}{\partial r}+\frac{\cos\vartheta}{r}\frac{\partial}{\partial \vartheta}, \\ &&\partial_1^2+\partial_2^2=\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \vartheta^2}.\end{aligned}$$ The variables in Eq.  are separated by substitution $$\begin{aligned} \Phi^a=\phi^a(r)e^{il\vartheta}\exp\left(ip_3x_3+ip_4x_4\right).\end{aligned}$$ Periodicity of the solution in angle $\vartheta\in[0,2\pi]$ requires integer values of parameter $l$. The radial part $\phi(r)$ should satisfy equation $$\label{HDCR} -\left[\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}-\frac{1}{r^2}\left(\frac{1}{2}\breve v r^2-l\right)^2 \right]\phi=\mu^2\phi,$$ where $\mu$ is related to the original eigenvalue $\lambda$, $$\begin{aligned} \lambda^2=p_4^2+p_3^2+\mu^2.\end{aligned}$$ By means of the substitution $$\phi=r^le^{-\frac{1}{4}\breve{v}r^2}\chi,$$ one arrives at the Kummer equation ($z=\breve{v}r^2/2$) $$\left[z\frac{d^2}{dz^2}+(l+1-z)\frac{d}{dz}-\frac{\breve{v}-\mu^2}{2\breve{v}}\right]\chi=0.$$ The complete solution can be chosen in the form $$\chi(z)=C_1M\left(\frac{\breve{v}-\mu^2}{2\breve{v}},1+l,z\right)+C_2z^{-l}M\left(\frac{\breve{v}-\mu^2}{2\breve{v}}-l,1-l,z\right),$$ where $M(a,b,z)$ is Kummer function. General solution of equation takes the form $$\phi_{l}(r)= e^{-\frac{1}{4}\breve{v}r^2}\left[C_1r^lM\left(\frac{\breve{v}-\mu^2}{2\breve{v}},1+l,\frac12\breve{v}r^2\right) +C_2r^{-l}M\left(\frac{\breve{v}-\mu^2}{2\breve{v}}-l,1-l,\frac12\breve{v}r^2\right)\right]$$ The first term is regular at $r=0$ provided $l\geqslant 0$ while the second one is well-defined for $l\leqslant 0$. Therefore, the solution regular inside the cylinder is $$\begin{aligned} \phi_{al}&=&e^{-\frac{1}{4}\breve{v}_ar^2}r^lM\left(\frac{\breve{v}_a-\mu^2}{2\breve{v}_a},1+l,\frac12\breve{v}_ar^2\right), \ \ l\geqslant 0, \\ \phi_{al}&=&e^{-\frac{1}{4}\breve{v}_ar^2}r^{-l}M\left(\frac{\breve{v}_a-\mu^2}{2\breve{v}_a}-l,1-l,\frac12\breve{v}_ar^2\right), \ \ l< 0, \label{scalarphi}\end{aligned}$$ where the color index $a$ has been explicitly indicated. The color matrix elements $\breve{v}_a$ can be negative. In this case one has to apply Kummer transformation [@AS] $$M(a,b,z)=e^zM(b-a,b,-z).$$ Dirichlet boundary condition defines the infinite discrete set of eigenvalues as the solutions $\mu^2_{alk}$ ($k=0,1\dots\infty$) of the equations $$\begin{aligned} \label{eigenvaluesscalar1} M\left(\frac{\hat{v}_a-\mu^2}{2\hat{v}_a},1+l,\frac12\hat{v}_aR^2\right)=0,\quad l\geqslant 0, \\ M\left(\frac{\hat{v}_a-\mu^2}{2\hat{v}_a}-l,1-l,\frac12\hat{v}_aR^2\right)=0,\quad l< 0. \label{eigenvaluesscalar2}\end{aligned}$$ If $\mu^2_{alk}$ satisfies equation , than $\tilde\mu^2_{alk}=\mu^2_{alk}-2\hat{v}_al$ is a solution of . Finally the complete orthogonal set of eigenfunctions for the problem and reads $$\begin{aligned} &&\Phi_{alk}(p_3, p_4|r,\vartheta,x_3,x_4)=e^{ip_3x_3+ip_4x_4} e^{il\vartheta}\phi_{alk}(r), \nonumber \\ &&\lambda_{alk}^2=p_4^2+p_3^2+\mu_{akl}^2, \label{eigenv_eucl} \\ && k=0,1,\dots,\infty, \ \ l=-\infty\dots\infty, \nonumber\end{aligned}$$ where functions $\phi_{alk}$ are defined by with $\mu^2=\mu_{akl}^2$ solving the boundary condition . Unlike Landau levels in the infinite space the eigenvalues $\mu_{akl}^2$ are not equidistant in $k$ and non-degenerate in $l$ as it is illustrated in Fig.\[Fig:Eigen\_set1\]). The dependence of several low-lying eigenvalues $\mu_{akl}^2$ on the dimensionless size parameter $\sqrt{H}R$ is shown in Fig.\[Fig:Eigen\_flow\]. ### Vector field eigenmodes For pure chromomagnetic field the adjoint representation vector field Eq.  takes the form $$\begin{aligned} \label{vector1} \left[-\breve D^2\delta_{\mu\nu}+2i\breve n B_{\mu\nu}\right]Q_\nu=\lambda Q_\mu,\end{aligned}$$ and the boundary conditions are $$\begin{aligned} &&\breve n Q_{\mu}(x)=0, \ x\in \partial\mathcal{T} \nonumber\\ &&\partial\mathcal{T} =\left\{x_1^2+x_2^2=R^2, \ (x_3,x_4)\in \mathrm{R^2} \right\} \label{HDCV}\end{aligned}$$ In terms of the eigenvectors $\breve{Q}_\mu^a$ of matrices $B_{\mu\nu}$ and $\breve n$ Eqs.  and take the form $$\begin{aligned} \label{vector2} \left[-\breve D^2+2s_{\mu}\breve v H\right]^a\breve{Q}^{a}_\mu =\lambda_{a\mu}\breve{Q}^{a}_\mu, \\\nonumber \breve v \breve{Q}_{\mu}(x)=0, \ x\in \partial\mathcal{T}.\end{aligned}$$ Omitting obvious well-known details we just note that equation describes sixteen charged with respect to $\breve n$ spin-color polarizations of the gluon fluctuations with $(s_1=1, s_2=-1, s_3=s_4=0)$ and $\breve v^a\not=0$ as well as sixteen “color neutral” with respect to $\breve{n}$ modes $$\begin{aligned} \label{vector4} &&-\partial^2\breve{Q}_\mu^{(0)}=p^2 \breve{Q}_\mu^{(0)}. \label{Q0}\end{aligned}$$ Neutral mode $\breve{Q}_\mu^{(0)}$ is a zero mode of $\breve n$, and it is insensitive to the boundary condition . We shall briefly discuss the possible role of the neutral modes in the last section. Equations for the color charged modes have the same form as the scalar field equation in the previous subsection. The only essential difference is that the eigenvalues $\lambda_{alk\nu}$ for nonzero $v^a$ have an addition $\pm 2vH$ to the eigenvalues $\mu^2_{akl}$ of the scalar case: $$\begin{aligned} &&\lambda^2_{alk\nu}=p_4^2+p_3^2+\mu_{alk}^2+ 2s_\nu\kappa_a v, \label{eigenv_eucl_pm} \\ && k=0,1,\dots,\infty, \ \ l\in Z, \nonumber\\ && s_1=1, \ s_2=-1, \ s_3=s_4=0, \ \kappa_a=\pm 1. \nonumber\end{aligned}$$ where $\mu^2_{akl}$ are the same as in the scalar case. If we were considering the square integrable solutions in $R^4$ then the lowest mode $\lambda^2_{a00\nu}$ with $s_\nu\kappa_a=-1$ would be tachyonic. In the finite trap the lowest eigenvalue is $$\begin{aligned} \label{lambda00} \lambda^2_{a00\nu}=p_4^2+p_3^2+\mu_{a00}^2-2v, \ s_\nu\kappa_a=-1.\end{aligned}$$ The dependence of $\mu_{a00}^2$ on dimensionless size parameter $\sqrt{H}R$ is strongly nonlinear. Few lowest eigenvalues $\mu_{akl}^2$ as functions of $\sqrt{H}R$ are shown in Fig. \[Fig:Eigen\_flow\]. One concludes that if the dimensionless size $\sqrt{H}R$ of the trap is sufficiently small $$\begin{aligned} \label{dmlsc} \sqrt{H}R<\sqrt{H}R_{\rm c}\approx 1.91,\end{aligned}$$ then there are no unstable tachyonic modes in the spectrum of color charged vector fields. To estimate the critical size one may use the mean phenomenological value of the gluon condensate (gauge coupling constant $g$ is included into the field strength tensor) $$\begin{aligned} \langle F^a_{\mu\nu}F^{a\mu\nu}\rangle = 2H^2\approx 0.5{\rm GeV}^4.\end{aligned}$$ Equation  leads to the critical radius $$\begin{aligned} R_{\rm c}\approx 0.51 \ {\rm fm} \ (2R_{\rm c}\approx 1 \ {\rm fm}). \label{rc}\end{aligned}$$ Thus the tachyonic mode is absent if the diameter of the cylindrical trap is less or equal to $1 \ {\rm fm}$. ![Eigenvalues $\mu^2_{alk}$ for the scalar field problem, $l=-2,-1,0,1,2$ and $k=0,1,2$, for $\sqrt{H}R=1.6$. Eigenvalues are denoted by asterisks in the case of positive $v_a$ and by circles in the case of negative $v_a$. []{data-label="Fig:Eigen_set1"}](dirichlet_eigenvalues_set1){width="75mm"} ![The lowest eigenvalues corresponding to positive color orientation $\kappa^a=1$ as functions of $\sqrt{H}R$. The critical radius $R_{\rm c}$ corresponds to $\mu_{a00}^2=2v=\sqrt{3}$. For large $\sqrt{H}R$ eigenvalues approach correct Landau levels, the degeneracy in $l$ is restored.[]{data-label="Fig:Eigen_flow"}](dirichlet_eigenvalues_flow){width="75mm"} ### Quark field eigen modes In this subsection we address the eigenvalue problem for Dirac operator in the cylindrical region in the presence of chromomagnetic background field (\[purechrmag\]) $$\begin{aligned} &&\!\not\!\!D\psi(x)=\lambda\psi(x), \label{dirac}\\ &&D_\mu=\partial_\mu+\frac{i}{2}\hat n B_{\mu\nu}x_\nu, \nonumber\\ &&\hat n= t_3\cos\xi +t_8\sin\xi \label{nfund}\\ &&=\frac{1}{2}{\rm diag}\left(\cos\xi+\frac{\sin\xi}{\sqrt{3}}, -\cos\xi+\frac{\sin\xi}{\sqrt{3}}, -\frac{2\sin\xi}{\sqrt{3}}\right). \nonumber\end{aligned}$$ Euclidean Dirac matrices are taken in the anti-hermitian representation. The angle $\xi$ is assumed to take one of the vacuum values $\xi_k$, and according to Eq.  the following forms of the matrix $\hat n$ can occur $$\begin{aligned} &&\hat n=\left\{ \pm \frac{1}{\sqrt{3}}{\rm diag}\left(1, -\frac{1}{2}, -\frac{1}{2}\right), \right. \label{HLxikfund}\\ &&\left. \pm \frac{1}{\sqrt{3}}{\rm diag}\left(\frac{1}{2}, \frac{1}{2},-1\right), \pm \frac{1}{\sqrt{3}}{\rm diag}\left(-\frac{1}{2},1, -\frac{1}{2}\right) \right\}. \nonumber\end{aligned}$$ Below we use notation $$\begin{aligned} \hat{n}_{ij}=\delta_{ij}\hat{u}_j. \nonumber\\\end{aligned}$$ The boundary conditions are $$\begin{aligned} i\!\not\!\eta(x)e^{i\theta\gamma_5}\hat n\psi(x) = \hat n\psi(x), \ x\in\partial\mathcal{T}, \nonumber\\ \bar\psi(x)e^{i\theta\gamma_5} \hat n i \! \!\not\!\eta(x) = -\bar\psi(x) \hat n, \ x\in\partial\mathcal{T}, \label{bagbc}\end{aligned}$$ where $\eta_\mu$ is a unit vector normal to the cylinder surface $\partial\mathcal{T}$, see Eq. . These are simply the bag boundary conditions. This choice appears to be rather natural. Indeed, inside the thick domain wall junction one expects an existence of the color charged quasiparticles (quarks) being the carriers of the color current, but outside the junction gluon configurations are confining (see Fig. \[Fig:chromo\_bag\])) and the current has to vanish at the boundary. Unlike the adjoint representation of color matrix the matrix $\hat n$ in fundamental representation has no zero eigenvalues for any value of the angle $\xi$ corresponding to the boundaries of Weyl chambers, see Eq. . Boundary condition restricts all three color components of the quark field. Substitution $$\label{psirelation} \psi=\left(\not{\hspace*{-0.3em}D}+\lambda\right)\varphi$$ leads to the equation $$\begin{aligned} \label{squareD} -\left(D^2+\hat{u}H\Sigma_3\right)\varphi=\lambda^2\varphi,\end{aligned}$$ where it has been used that in the pure chromomagnetic field $$\begin{aligned} \frac12\sigma_{\mu\nu}\hat{B}_{\mu\nu}=\Sigma_3H\hat{u}, \ \Sigma_i=\frac12\varepsilon_{ijk}\sigma_{jk}.\end{aligned}$$ Equation is essentially the same as . Its solution in cylindrical coordinates ($2\pi$-periodic in $\vartheta$ and regular at $r=0$) is given by four independent components $\varphi_l^\alpha$ ($\alpha=1,\dots,4$, $l\in Z$): $$\begin{aligned} \varphi_l^\alpha=e^{-ip_3x_3-ip_4x_4}e^{il\vartheta}\phi_l^\alpha(r)\end{aligned}$$ with $$\label{philp} \phi_{l}^\alpha=e^{-\frac14\hat{u}r^2}r^{l} M\left(\frac{1+s_\alpha}{2}-\frac{\mu^2}{2\hat{u}},1+l,\frac{\hat{u}r^2}{2}\right)$$ for the case $l\geqslant 0$ and $$\label{philn*} \phi_{l}^{\alpha}=e^{-\frac14\hat{u}r^2}r^{-l} M\left(\frac{1+s_{\alpha}}{2}-\frac{\mu^2}{2\hat{u}}-l,1-l,\frac{\hat{u}r^2}{2}\right)$$ for $l<0$. Here $$s_\alpha=(-1)^\alpha, \ \alpha=1,\dots,4$$ denotes the sign of the quark spin projection on the direction of chromomagnetic field, and therefore $$\begin{aligned} \label{philarrows} \phi_l^3=\phi_l^1=\Phi_l^{\uparrow\uparrow}(r), \ \phi_l^4=\phi_l^2=\Phi_l^{\uparrow\downarrow}(r).\end{aligned}$$ The variable $\mu$ is related to the Dirac eigenvalues as $$\mu^2=\lambda^2-p_3^2-p_4^2. \label{mu}$$ Finally the Dirac operator eigenfunction $\psi$ can be obtained by means of relation with $$\not{\hspace*{-0.3em}D}+\lambda=\left( \begin{array}{cccc} \lambda&0&i\partial_4+\partial_3&D_1-iD_2\\ 0&\lambda&D_1+iD_2&i\partial_4-\partial_3\\ i\partial_4-\partial_3&-D_1+iD_2&\lambda&0\\ -D_1-iD_2&i\partial_4+\partial_3&0&\lambda \end{array} \right), \nonumber$$ where $$\begin{gathered} D_1+iD_2=e^{i\vartheta}\left(\frac{\partial}{\partial r}+\frac{i}{r}\frac{\partial}{\partial \vartheta}+\frac12\hat{u}r\right),\displaybreak[0] \\ D_1-iD_2=e^{-i\vartheta}\left(\frac{\partial}{\partial r}-\frac{i}{r}\frac{\partial}{\partial \vartheta}-\frac12\hat{u}r\right).\end{gathered}$$ Four solutions are for $l\geqslant 0$ $$\psi_l^{(1)}=e^{-ip_3x_3-ip_4x_4}\left( \begin{array}{c} \lambda \Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ 0\\ (p_4+ip_3)\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ \frac{\mu^2}{2(l+1)}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta} \end{array} \right)$$ $$\psi_l^{(2)}=e^{-ip_3x_3-ip_4x_4}\left( \begin{array}{c} 0\\ \lambda \Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ -2(l+1)\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ (p_4-ip_3)\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta} \end{array} \right)$$ $$\psi^{(3)}_l=e^{-ip_3x_3-ip_4x_4}\left( \begin{array}{c} (p_4-ip_3)\Phi_{l}^{\uparrow\uparrow}(r)e^{il\vartheta}\\ -\frac{\mu^2}{2(1+l)}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ \lambda \Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ 0 \end{array} \right)$$ $$\psi^{(4)}_l=e^{-ip_3x_3-ip_4x_4}\left( \begin{array}{c} 2(l+1)\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ (p_4+ip_3)\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ 0\\ \lambda \Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ \end{array} \right),$$ and for $l<0$ $$\psi_l^{(1)}=e^{-ip_3x_3-ip_4x_4}\left( \begin{array}{c} \lambda \Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ 0\\ (p_4+ip_3)\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ 2l\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta} \end{array} \right)$$ $$\psi_l^{(2)}=e^{-ip_3x_3-ip_4x_4}\left( \begin{array}{c} 0\\ \lambda \Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ -\frac{\mu^2}{2l}\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ (p_4-ip_3)\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta} \end{array} \right)$$ $$\psi^{(3)}_l=e^{-ip_3x_3-ip_4x_4}\left( \begin{array}{c} (p_4-ip_3)\Phi_{l}^{\uparrow\uparrow}(r)e^{il\vartheta}\\ -2l\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ \lambda \Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ 0 \end{array} \right)$$ $$\psi^{(4)}_l=e^{-ip_3x_3-ip_4x_4}\left( \begin{array}{c} \frac{\mu^2}{2l}\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ (p_4+ip_3)\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ 0\\ \lambda \Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ \end{array} \right).$$ All four spinors are eigenfunctions, $$\begin{aligned} J_3\psi_l^{(m)}=\left(l+\frac{1}{2}\right)\psi_l^{(m)},\end{aligned}$$ of the total momentum projection operator onto $x_3$. $$\begin{gathered} J_i=L_i+S_i,\quad L_i=-i\varepsilon_{ijk}x_j\partial_k,\quad S_i=\frac12\Sigma_i,\\ J_3=-i\frac{\partial}{\partial \vartheta}+\frac{1}{2}\left(\begin{array}{cc} \sigma_3&0\\ 0&\sigma_3 \end{array}\right).\end{gathered}$$ Only two of these solutions at given $l$ are linearly independent. We select $$\psi_l=A\psi^{(1)}_l+B\psi^{(4)}_l$$ as a general solution to equation for the reason that $\psi^{(1)}_l$ and $\psi^{(4)}_l$ remain linearly independent in the limit $\lambda\to 0$. The limit will be used in the next section for the solving the Dirac equation in Minkowski space-time. Boundary condition with $\theta=\pi/2$ leads to the equation defining the values of the parameter $\mu$ as well as the ratio of $A$ and $B$. For $l\geqslant 0$ one gets $$\begin{aligned} && A\left(\frac{\mu^2}{2(1+l)}\Phi^{\uparrow\downarrow}_{l+1}(R)+\lambda \Phi^{\uparrow\uparrow}_{l}(R)\right)+B\left(\lambda \Phi^{\uparrow\downarrow}_{l+1}(R)+2(l+1)\Phi^{\uparrow\uparrow}_{l}(R)\right)=0, \nonumber\\ &&A\Phi^{\uparrow\uparrow}_l(R)+B \Phi^{\uparrow\downarrow}_{l+1}(R)=0. \nonumber\end{aligned}$$ This system has a nontrivial solution for $A$ and $B$ if the determinant of the matrix composed of the coefficients in front of them is equal to zero $$\begin{gathered} \label{muquark} \left[\Phi^{\uparrow\uparrow}_l(R)\right]^2=\left[\frac{\mu}{2(1+l)}\Phi^{\uparrow\downarrow}_ {l+1}(R)\right]^2.\end{gathered}$$ This equation defines the spectrum of $\mu^2$. States with definite spin orientation with respect to the chromomagnetic field are mixed in the boundary condition, and the spin projection onto the direction of the field is not a good quantum number unlike the projection of the total momentum $j_3$ as it is taken into account in Fig.\[Fig:Q\_eigenv\]. As is illustrated in Fig.\[Fig:Q\_eigenv\] there is a discrete set of solutions $\mu_{ilk}>0$ which depend also on the color orientation $\hat u_i$ ($j_3=(2l+1)/2$ with $l\in Z$, $k\in N$, $j=1,2,3$). As a rule one can omit the color index $j$ assuming that $\mu_{lk}$ is a diagonal color matrix for any $l,k$. The values $\mu^2_{lk}$ has to be used to find the relation between $A$ and $B$ $$\begin{gathered} \frac{B_{lk}}{A_{lk}}=-\left.\frac{\Phi^{\uparrow\uparrow}_l(R)}{\Phi^{\uparrow\downarrow}_{l+1}(R)}\right|_{\mu^2=\mu^2_{lk}}= (-1)^{k+1}\frac{\mu_{lk}}{2(l+1)}, \label{lambdapm}\\ \lambda_{lk}=\pm\sqrt{\mu_{lk}^2+p_3^2+p_4^2}=\pm |\lambda_{lk}|. \nonumber\end{gathered}$$ Here $\mu_{lk}$ is taken to be positive, and $\lambda_{lk}$ takes both positive and negative values. Equation has been used in combination with observation (by inspection) that the sign of the ratio $B_{lk}$ and $A_{lk}$ depends on $k\in N$ as it is indicated in irrespectively to $l$ and color orientation. Analogous consideration for the case $l<0$ leads to the equation for $\mu$ $$\begin{gathered} \left[\Phi^{\uparrow\downarrow}_{l+1}(R)\right]^2=\left[\frac{\mu}{2l}\Phi^{\uparrow\uparrow}_ {l}(R)\right]^2,\end{gathered}$$ and for the ratio of coefficients $$\begin{gathered} \frac{A_{lk}}{B_{lk}}=-\left.\frac{\Phi^{\uparrow\uparrow}_l(R)}{\Phi^{\uparrow\downarrow}_{l+1}(R)}\right|_{\mu^2=\mu^2_{lk}}= (-1)^{k}\frac{\mu_{lk}}{2l}, \label{lumpm}\\ \lambda_{lk}=\pm\sqrt{\mu_{lk}^2+p_3^2+p_4^2}=\pm |\lambda_{lk}|. \nonumber\end{gathered}$$ The orthogonal normalized set of solutions has the form for $l\geqslant 0$ $$\psi_{lk}^{(\pm)}=\frac{A_{lk}}{(2\pi)^\frac32\sqrt{2|\lambda_{lk}|}}\left( \begin{array}{c} \frac{\pm|\lambda_{lk}|+(-1)^{k+1}\mu_{lk}}{\sqrt{p_4+ip_3}} \Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ (-1)^{k+1}\frac{\mu_{lk}\sqrt{p_4+ip_3}}{2(l+1)}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ \sqrt{p_4+ip_3}\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ \frac{\mu_{lk}(\mu_{lk}\pm(-1)^{k+1}|\lambda_{lk}|)}{2(l+1)\sqrt{p_4+ip_3}}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta} \end{array} \right)e^{-ip_3x_3-ip_4x_4}, \label{psi_eucl1}$$ and for $l<0$ $$\psi_{lk}^{(\pm)}=\frac{B_{lk}}{(2\pi)^\frac32\sqrt{2|\lambda_{lk}|}}\left( \begin{array}{c} \frac{\mu_{lk}(\mu_{lk}\pm(-1)^k|\lambda_{lk}|)}{2l\sqrt{p_4+ip_3}} \Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ \sqrt{p_4+ip_3}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ (-1)^k\frac{\mu_{lk}\sqrt{p_4+ip_3}}{2l}\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ \frac{\pm|\lambda_{lk}|+(-1)^k\mu_{lk}}{\sqrt{p_4+ip_3}}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta} \end{array} \right)e^{-ip_3x_3-ip_4x_4}. \nonumber$$ The spinors $\psi_{lk}^{(+)}$ and $\psi_{lk}^{(-)}$ correspond to the positive and negative eigenvalues $\lambda_{lk}$ in respectively, they are eigenfunctions of $J_3$ with $j_3=l+1/2$. Normalization constants are $$\begin{aligned} A^{-2}_{jlk}(R)&=&\int_0^R drr\left[\left(\frac{\mu_{jlk}}{2(l+1)}\Phi_{l+1}^{\uparrow\downarrow}(r)\right)^2+\left(\Phi_{l+1}^{\uparrow\uparrow}(r)\right)^2\right] \nonumber\\ B^{-2}_{jlk}(R)&=&\int_0^R drr\left[\left(\frac{\mu_{jlk}}{2l}\Phi_{l+1}^{\uparrow\uparrow}(r)\right)^2+\left(\Phi_{l+1}^{\uparrow\downarrow}(r)\right)^2\right] \label{normconst}\end{aligned}$$ The same procedure applied to the equation $$\bar\psi(x)\stackrel{\leftarrow}{\!\not\!\!D}=\lambda\bar\psi(x)$$ leads to the solutions for $l\geqslant 0$ $$\bar\psi_{lk}^{(\pm)}=\frac{A_{lk}}{(2\pi)^\frac32\sqrt{2|\lambda_{lk}|}}\left( \begin{array}{c} \pm\sqrt{p_4+ip_3}\Phi^{\uparrow\uparrow}_l(r)e^{-il\vartheta}\\ \frac{\mu_{lk}(\mp\mu_{lk}+(-1)^{k+1}|\lambda_{lk}|)}{2(1+l)\sqrt{p_4+ip_3}} \Phi^{\uparrow\downarrow}_{l+1}(r)e^{-i(l+1)\vartheta}\\ \frac{\pm(-1)^{k}\mu_{lk}+|\lambda_{lk}|}{\sqrt{p_4+ip_3}}\Phi^{\uparrow\uparrow}_l(r)e^{-il\vartheta}\\ \mp(-1)^{k}\frac{\mu_{lk}\sqrt{p_4+ip_3}}{2(1+l)}\Phi^{\uparrow\downarrow}_{l+1}(r)e^{-i(l+1)\vartheta} \end{array} \right)^\textrm{T}e^{ip_3x_3+ip_4x_4}, \label{psi_eucl}$$ and for $l<0$ $$\bar\psi_{lk}^{(\pm)}=\frac{B_{lk}}{(2\pi)^\frac32\sqrt{2|\lambda_{lk}|}}\left( \begin{array}{c} \pm(-1)^{k}\frac{\mu_{lk}\sqrt{p_4+ip_3}}{2l}\Phi^{\uparrow\uparrow}_l(r)e^{-il\vartheta}\\ \frac{|\lambda_{lk}|\mp(-1)^k\mu_{lk}}{\sqrt{p_4+ip_3}} \Phi^{\uparrow\downarrow}_{l+1}(r)e^{-i(l+1)\vartheta}\\ \frac{\mu_{lk}(\mp\mu_{lk}+(-1)^k|\lambda_{lk}|)}{2l\sqrt{p_4+ip_3}}\Phi^{\uparrow\uparrow}_l(r)e^{-il\vartheta}\\ \pm\sqrt{p_4+ip_3}\Phi^{\uparrow\downarrow}_{l+1}(r)e^{-i(l+1)\vartheta}\\ \end{array} \right)^\textrm{T}e^{ip_3x_3+ip_4x_4} . \nonumber$$ ![ The lowest values of $\mu$ solving Eq.  for $\sqrt{H}R=1.6$. Here $j_3=l+1/2$ is the projection of the total momentum on the direction of the chromomagnetic field. Eigenvalues are denoted by asterisks in the case of positive $u_j$ and by circles in the case of negative $u_j$.[]{data-label="Fig:Q_eigenv"}](Quark_eigen_val-6){width="80mm"} Quasiparticles --------------- To get insight into the physical treatment of above-considered Euclidean eigenmodes one has to solve the Minkowski space Klein-Gordon and Dirac equations in the presence of chromomagnetic field inside the cylinder with the bag-like boundary conditions. Solutions describe the elementary quasiparticle excitations inside the thick cylindrical domain wall junction. Quite detailed analysis of the notion of quasiparticles in relativistic quantum field theory can be found in [@Arteaga:2008ux]. Unlike the fundamental elementary and composite particles the characteristic properties of quasiparticles (for instance the specific form of the dispersion relation) need not be necessarily Lorentz invariant or even gauge invariant. The overall statement of the problem under consideration necessarily assumes that space direction along the chromomagnetic field is singled out by underlining experimental setup as it coincides with the direction of the strong magnetic field generated for short time in heavy ion collision. In generic relativistic frame both chromoelectric and chromomagnetic fields are present inside the domain wall junction. However since the topological charge density vanishes in the region (see Fig.\[Fig:chromo\_bag\]) there exists specific frame where chromoelectric field is absent. This frame is the most convenient for our purposes. ### Adjoint representation: color charged bosons In Minkowski space-time the problem and turns to the wave equation $$\begin{aligned} \label{cylinder_s_m} -\left(\partial_\mu-i\breve B_\mu\right)^2\phi(x)=0\end{aligned}$$ for color charged adjoint spin zero field inside a cylindrical wave guide. As it follows from and futher discussion the charged components of the adjoint field of the color matrix $\breve n$ comes in complex conjugate pairs. For instance if $\xi=\pi/6$ then there are two pairs $\phi_1\pm i\phi_2$ and $\phi_4\pm i\phi_5$. Thus $\phi^a$ is a complex scalar field, the corresponding solution of satisfying boundary condition takes the form $$\begin{aligned} &&\phi^a(x)=\sum_{lk}\int\limits_{-\infty}^{+\infty}\frac{dp_3}{2\pi}\frac{1}{\sqrt{2\omega_{alk}}} \left[a^{+}_{akl}(p_3)e^{ix_0\omega_{akl}-ip_3x_3} +b_{akl}(p_3)e^{-ix_0\omega_{akl}+ip_3x_3}\right]e^{il\vartheta}\phi_{alk}(r), \label{seigenf_mink} \\ &&\phi^{a\dagger}(x)=\sum_{lk}\int\limits_{-\infty}^{+\infty}\frac{dp_3}{2\pi}\frac{1}{\sqrt{2\omega_{alk}}} \left[b^{+}_{akl}(p_3)e^{-ix_0\omega_{akl}+ip_3x_3} +a_{akl}(p_3)e^{ix_0\omega_{akl}-ip_3x_3}\right]e^{-il\vartheta}\phi_{alk}(r), \nonumber \\ &&p_0^2=p_3^2+\mu_{akl}^2, \nonumber\\ &&p_0=\pm\omega_{akl}(p_3), \ \omega_{akl}=\sqrt{p_3^2+\mu_{akl}^2}, \label{mass_mink} \\ && k=0,1,\dots,\infty, \ \ l\in Z, \nonumber\end{aligned}$$ with $\phi_{alk}(r)$ defined in but here it is assumed to be normalized $$\begin{aligned} \int\limits_0^\infty dr r\int\limits_0^{2\pi} d\vartheta e^{i(l-l')\vartheta} \phi_{alk}(r) \phi_{al'k'}(r)=\delta_{ll'}\delta_{kk'}. \end{aligned}$$ Equation can be treated as the dispersion relation between energy $p_0$ and momentum $p_3$ for the quasiparticles with masses $\mu_{akl}$. These quasiparticles are extended in $x_1$ and $x_2$ directions and are classified by the quantum numbers $l,k$. The orthogonality, normalization and completeness of the set of functions $e^{il\vartheta}\phi_{alk}(r)$ guarantees the standard canonical commutation relations for the field $\phi^a$ and its canonically conjugated momentum if $a^\dagger_{akl}(p_3)$, $a_{akl}(p_3)$, $b^\dagger_{akl}(p_3)$ and $b_{akl}(p_3)$ are assumed to satisfy the standard commutation relations for creation and annihilation operators. The Fock space of states for the quasiparticles with masses $\mu_{akl}$ can be constructed by means of the standard QFT methods. This treatment provides one with a suitable terminology and formalism for discussion of the confining properties of various gluon field configurations in the context of QFT: unlike the chromomagnetic field the (anti-)self-dual fields characteristic for the bulk of domain network configuration (see the LHS plot in Fig. \[Fig:kink\_network\]) lead to purely discrete spectrum of eigenmodes in Euclidean space and do not possess any quasiparticle treatment in terms of dispersion relation between energy and momentum for elementary color charged excitations. If there is a reason for long-lived defect in the form of thick domain wall junction then its boundary defines a shape and a size for the space region which can be populated by color charged quasiparticles. The vector adjoint field can be elaborated in the similar to the scalar case way. A modification relates just to the inclusion of polarization vectors. As it has already been mentioned the most important feature is the absence of tachyonic mode of the vector color charged field if $R<R_{\mathrm c}$. Disappearance of the tachyonic mode for subcritical size of the trap is one of the most important observations of this paper. ### Fundamental representation: color charged fermions Neither the background field nor the boundary condition involve the time coordinate. The solution of the Dirac equation $$\begin{aligned} &&i\!\not\!\!D\psi(x)=0, \label{dirac1}\end{aligned}$$ satisfying condition can be obtained from Euclidean solutions (unnormalized solutions have to be used) by the analytical continuation $p_4\to ip_0$, $x_4\to ix_0$ and the requirement $\lambda_{lk}=0$, which leads to the energy-momentum relation for the solutions with definite $j_3$, $k$ and color $j$ $$\begin{gathered} p_0^2=p_3^2+\mu_{jlk}^2 ,\ \ p_0=\pm\omega_{jlk}(p_3), \\ \omega_{jlk}=\sqrt{p_3^2+\mu_{jlk}^2}.\end{gathered}$$ Finally the solution of the Dirac equation takes the form $$\begin{aligned} \psi^{j}(x)=\sum_{lk}\int\limits_{-\infty}^{+\infty}\frac{dp_3}{2\pi}\frac{1}{\sqrt{2\omega_{jlk}}} \left[a^{\dagger}_{jlk}(p_3)\chi_{jlk}(p_3|r,\vartheta) e^{ix_0\omega_{jlk}-ix_3p_3}+b_{jlk}(p_3)\upsilon_{jlk}(p_3|r,\vartheta)e^{-ix_0\omega_{jlk}+ix_3p_3}\right], \\ \bar\psi^{j}(x)=\sum_{lk}\int\limits_{-\infty}^{+\infty}\frac{dp_3}{2\pi}\frac{1}{\sqrt{2\omega_{jlk}}} \left[b^{\dagger}_{jlk}(p_3)\bar\chi_{jlk}(p_3|r,\vartheta) e^{-ix_0\omega_{jlk}+ix_3p_3}+a_{jlk}(p_3){\bar\upsilon }_{jlk}(p_3|r,\vartheta)e^{ix_0\omega_{jlk}-ix_3p_3}\right].\end{aligned}$$ Here the pair of spinors for positive $\chi_{lk}$ and negative $\upsilon_{lk}$ energy solutions are $$\begin{aligned} \chi_{lk}=A_{lk}\left( \begin{array}{c} (-1)^{k+1}\frac{\mu_{lk}}{\sqrt{\omega_{lk}+p_3}} \Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ i(-1)^{k+1}\frac{\mu_{lk}\sqrt{\omega_{lk}+p_3}}{2(l+1)}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ i\sqrt{\omega_{lk}+p_3}\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ \frac{\mu^2_{lk}}{2(l+1)\sqrt{\omega_{lk}+p_3}}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta} \end{array} \right), \ \ \ \ \upsilon_{lk}=A_{lk}\left( \begin{array}{c} (-1)^{k+1}\frac{\mu_{lk}}{\sqrt{\omega_{lk}+p_3}} \Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ i(-1)^{k}\frac{\mu_{lk}\sqrt{\omega_{lk}+p_3}}{2(l+1)}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ -i\sqrt{\omega_{lk}+p_3}\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ \frac{\mu^2_{lk}}{2(l+1)\sqrt{\omega_{lk}+p_3}}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta} \end{array} \right), \label{barchi_mink}\end{aligned}$$ for $l\geqslant0$ and $$\begin{aligned} \chi_{lk}=B_{lk}\left( \begin{array}{c} \frac{\mu^2_{lk}}{2l\sqrt{\omega_{lk}+p_3}} \Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ i\sqrt{\omega_{lk}+p_3}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ i(-1)^k\frac{\mu_{lk}\sqrt{\omega_{lk}+p_3}}{2l}\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ (-1)^k\frac{\mu_{lk}}{\sqrt{\omega_{lk}+p_3}}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta} \end{array} \right), \ \ \ \ \upsilon_{lk}=B_{lk}\left( \begin{array}{c} \frac{\mu^2_{lk}}{2l\sqrt{\omega_{lk}+p_3}} \Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ -i\sqrt{\omega_{lk}+p_3}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta}\\ i(-1)^{k+1}\frac{\mu_{lk}\sqrt{\omega_{lk}+p_3}}{2l}\Phi_l^{\uparrow\uparrow}(r)e^{il\vartheta}\\ (-1)^k\frac{\mu_{lk}}{\sqrt{\omega_{lk}+p_3}}\Phi_{l+1}^{\uparrow\downarrow}(r)e^{i(l+1)\vartheta} \end{array} \right) \label{barchi_mink_ln}\end{aligned}$$ for $l<0$. The spinors are normalized as $$\begin{aligned} \int\limits_{0}^{2\pi}d\vartheta\int\limits_0^R dr r \chi^\dagger_{jlk}(p_3|r,\vartheta)\chi_{jlk}(p_3|r,\vartheta)= \int\limits_{0}^{2\pi}d\vartheta\int\limits_0^R dr r \upsilon^\dagger_{jlk}(p_3|r,\vartheta)\upsilon_{jlk}(p_3|r,\vartheta)= 2\omega_{jlk}\end{aligned}$$ The Dirac conjugated spinors are $$\bar\psi^{j}(x)=\psi^{j\dagger}(x)\gamma_0$$ as usual. The Fock space can be constructed by means of the creation and annihilation operators $\left\{ a^{\dagger}_{jlk}(p_3), a_{jlk}(p_3), b^{\dagger}_{jlk}(p_3), b_{jlk}(p_3)\right\}$ satisfying the standard anticommutation relations. The one-particle state is characterized by a color orientation $j$, momentum $p_3$, projection $j_3=(l+1/2)$ of the total angular momentum and the energy $\omega_{jlk}=\sqrt{p_3^2+\mu^2_{jlk}}$. Since the boundary condition mixes the states with spin parallel and anti-parallel to the chromomagnetic field the spin projection is not a good quantum number unlike the half-integer valued projection of the total angular momentum $j_3$. Discussion ========== An ensemble of confining gluon configurations has been constructed explicitly as a domain wall networks representing the almost everywhere homogeneous Abelian (anti-)self-dual gluon fields. Confinement is understood here as the absence of the color charged wave-like elementary excitations. The dynamical quark confinement occurs in the (four-dimensional) bulk of the domain wall network. Inside the (three-dimensional) domain walls topological charge density vanishes and the color charged quasiparticles can be excited. Under extreme conditions, in particular under the influence of the strong electromagnetic field specific for relativistic heavy ion collisions, a relatively stable defect in the confining ensemble, a thick domain wall junction, can be formed. Though the scalar gluon condensate is nonzero everywhere $\langle g^2F^2\rangle\not=0$, the region of defect is characterized by the vanishing topological charge density $\langle |g^2\tilde FF|\rangle$=0 unlike the rest of the space, which indicates the lack of confinement in the junction. The quark field excitations inside the junction are represented by the color charged quasiparticles. The spectrum of gluon excitations besides the trapped color charged modes contains also the color neutral with respect to the background field modes. Almost obvious but important observation is that there exists a critical size $L_{\rm c}$ of the junction beyond which the tachyonic gluon modes emerge in the excitation spectrum and destabilize the defect. The critical size can be related to the value of the gluon condensate $\langle g^2F^2\rangle$ and in the case of the considered in the paper cylindrical trap $L_{\rm c}\approx 1$fm for the standard value of the condensate, see . The specific value of the critical size depends on the geometry of the trap but its very existence and its commensurability with a distance of order of $1$fm is a generic feature. This observation underlines the generic necessity of accounting for the essentially finite size of the space-time region in which deconfinement may occur. The reason is that thermodynamic limit does not exist as the system under consideration disappears as soon as the typical size of the space volume exceeds the critical value. Excess of the internal pressure of the trap filled by many charged quasiparticles leads to its expansion and breakdown of stability followed by its disintegration to many smaller traps (or bags), which is reminiscent of the heterophase fluctuations studied in [@Yukalov:2013yj] as well as the dynamics and statistical mechnaics of bags with a surface tension [@Bugaev]. The dynamics of the color charged quasiparticles as it is described above is strictly one-dimentional in space. This feature can be a source of the azimuthal asymmetries in heavy ion collisions, similarly to the approach of paper [@Tuchin:2013ie] upto a substitution of the magnetic field by the Abelian chromomagnetic field (see also [@Bali:2013owa]). However it should be noted that the one-dimensional dynamics is a property of the zero-th order approximation based on the quadratic part of the action. 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--- author: - 'Alex Garivaltis[^1]' title: '[**Cover’s Rebalancing Option with Discrete Hindsight Optimization**]{}' --- **JEL Classification:** C44, D80, D81, G11 Introduction ============ The main alternative to the Markowitz (1952) mean-variance theory of portfolio selection was popularized by Kelly (1956) who sought to optimize a gambler’s asymptotic continuously-compounded capital growth rate in repeated bets on horse races in the presence of partial inside information. His reasoning is in fact applicable to all gambling, insurance, and investment problems. Rather than optimize the static reward per unit of risk, the Kelly Criterion (Poundstone 2010) is equivalent to the prescription that one should act each round so as to maximize the expected log of his capital. Breiman (1961) showed that the Kelly Criterion constitutes asymptotically dominant behavior: a Kelly gambler will almost surely beat any other gambler in the long run by an exponential factor, and he has the shortest expected hitting time for a distant wealth goal. With probability approaching 1 as time goes on, the Kelly gambler’s bankroll will (amusingly) overtake that of a mean-variance investor, who has a smooth ride but ultimately cannot “eat his Sharpe ratio.” The books by Cover and Thomas (2006) and Luenberger (1998) are excellent primers of the theory of asymptotic capital growth in discrete and continuous time, respectively. Thorp (cf. his 2017 biography) demonstrated the practical effectiveness of the Kelly Criterion when he used it to size his Blackjack bets in certain favorable situations that are identifiable via his trademark (1966) theory of card counting. In this connection, the correct behavior is to bet the fraction $b^*:=p-q$ of your net worth on a given hand for which $p$ is the chance of winning and $q$ is the chance of losing. For growth opportunities in the stock market, the analog of Kelly’s fixed fraction betting scheme is a certain constant-rebalanced portfolio $b^*$ that trades continuously so as to maintain a target growth-optimal fraction of wealth in each risk asset. For instance, rather than bet $b:=2\%$ of wealth on a (favorable) hand of Blackjack, one could bet $2\%$ of wealth (or even $b:=200\%$ of wealth) on the S&P $500$ index. In theory, if stock market returns are *iid* across (discrete) time then one can calculate the corresponding log-optimal portfolio directly from the return distribution. But in practice, equity investors must get along without complete knowledge of the return distribution. Thus, a real-world investor cannot measure the exact regret of his portfolio relative to the Kelly bet for the simple reason that he does not know the Kelly bet. The way out of this conundrum was discovered by information theorist Thomas Cover (1938-2012), who formulated the *individual sequence approach* to investment. For a given observed sequence of asset prices, one can look back and determine which constant-rebalanced asset allocation would have yielded the greatest final wealth *for that particular sequence*. By definition, a Kelly gambler (who knows the distribution of returns but not the individual sequence that will occur in the future) will achieve a final wealth that is no greater than that of the best constant-rebalanced portfolio determined in hindsight for the actual sequence of returns. Thus began Cover’s important *universal portfolio theory* that formulated various on-line investment schemes (1986, 1991, 1996, 1998) that guarantee to achieve a high percentage of the final wealth of the best constant unlevered rebalancing rule (of any kind) in hindsight. Of course, any such scheme would then also guarantee to achieve a high percentage of the Kelly final wealth in *iid* stock markets. Contribution ------------ One can consider Cover’s performance benchmark to be a financial derivative (“Cover’s rebalancing option”) whose final payoff is equal to the wealth that would have accrued to a $\$1$ deposit into the best rebalancing rule (or fixed-fraction betting scheme) determined in hindsight. Ordentlich and Cover (1998) began the work of pricing this option in the Black-Scholes (1973) market at time-$0$ for unlevered hindsight optimization over a single underlying risk asset. Garivaltis (2018) priced and replicated the rebalancing option at any time $t$ for levered hindsight optimization over an arbitrary number of correlated stocks in geometric Brownian motion. That paper obtained the elegant result that for completely relaxed (levered) hindsight optimization, the corresponding delta-hedging strategy simply looks back over the observed price history $[0,t]$, computes the best rebalancing rule in hindsight $b(S_t,t)$, and bets that fraction of wealth over the next differential time step $[t,t+dt]$. The present paper studies Cover’s rebalancing option with hindsight optimization over a discrete set $\mathbb{B}:=\{b_1,...,b_n\}$ of rebalancing rules. Apart from the scientific obligation to extend Ordentlich and Cover’s incisive (1998) chain of reasoning, our approach has some interesting advantages relative to hindsight-optimization over all possible rebalancing rules. In our world, the (delta-hedging) practitioner is now free to express any of his institutional constraints or beliefs about future returns through a judicious choice of the set $\mathbb{B}$. Our newly austere mode of hindsight optimization yields a rock-bottom option price and correspondingly better guarantees of relative performance at the end of the planning period, whose shortened length is now well within a human life span. Say, for robust betting on the S&P 500 index, the author himself is inclined to use $\mathbb{B}:=\{0,0.5,1,1.5,2\}$, which amounts to the following five (continuously-rebalanced) asset allocations: 1. $0\%$ stocks, $100\%$ cash 2. $50\%$ stocks, $50\%$ cash 3. $100\%$ stocks, $0\%$ cash 4. $150\%$ stocks, $-50\%$ cash (margin loans) 5. $200\%$ stocks, $-100\%$ cash (margin loans) In this example, the author would like to avoid paying the full *Cost of Achieving the Best \[Rebalancing Rule\] in Hindsight* that would correspond (Garivaltis 2018) to $\mathbb{B}:=\mathbb{R}$ or even $\mathbb{B}:=[0,2]$. The paper is organized as follows. Section \[notation\] explains our basic notation and terminology. Section \[two\] develops our main techniques in the context of hindsight optimization over a pair $b>c$ of rebalancing rules and a single underlying risk asset. We price and replicate both the horizon-$T$ and perpetual versions of the rebalancing option, and give performance simulations that illustrate the general behavior of the replicating strategy. Section \[several\] extends the methodology to general discrete sets of asset allocations. We show how the rebalancing option can be interpreted as a certain portfolio of Margrabe-Fischer (1978) exchange options, and derive the general replicating strategy, which is a time- and state-varying convex combination of the $b_i$. We close the paper by proving that American-style rebalancing options (with general exercise price $K$) are always “worth more alive than dead” in equilibrium. Definitions and Notation {#notation} ======================== We start in the Black-Scholes (1973) market with a single underlying stock whose price $S_t$ follows the geometric Brownian motion $$\frac{dS_t}{S_t}=\mu\,dt+\sigma dW_t,$$where $\mu$ is the drift, $\sigma$ is the volatility, and $W_t$ is a standard Brownian motion. There is a risk-free bond whose price $B_t:=e^{rt}$ follows $$\frac{dB_t}{B_t}=r\,dt.$$A constant rebalancing rule $b\in(-\infty,+\infty)$ is a fixed-fraction betting scheme that continuously maintains the fraction $b$ of wealth in the stock and the fraction $1-b$ of wealth in bonds. We let $V_t(b)$ denote the wealth at $t$ that accrues to a $\$1$ deposit into the rebalancing rule $b$. Thus, the trader holds $\Delta:=bV_t(b)/S_t$ shares of the stock at time $t$, and his remaining $(1-b)V_t$ dollars are invested in bonds. Maintenance of the target asset allocation generally requires continuous trading. If $0<b<1$, the trader must sell a precise number of shares on every uptick (more precisely, whenever $dS_t/S_t\geq r\,dt$) to restore the target allocation. Similarly, when the risk asset underperforms cash over $[t,t+dt]$ (i.e. when $dS_t/S_t\leq r\,dt$) the trader must buy additional shares to restore the balance. This amounts to a volatility harvesting scheme (cf. Luenberger 1998) that “lives off the fluctuations” of the underlying. For $b=1$ the trader just buys the stock and holds it; for $b>1$ he carries a margin (debit) balance of $(b-1)V_t(b)$ dollars at time $t$. A levered rebalancing rule $b>1$ must continuously maintain a fixed debt-to-assets ratio of $1-1/b$. Thus, when the stock rises (and debt is now a smaller percentage of assets) the trader will borrow against his new wealth to buy additional shares. Similarly, when the stock falls he must sell some shares to reduce the loan-to-value ratio. This “buy high, sell low” strategy is only appropriate for stocks with relatively high drift and low volatility. Finally, for low quality underlyings one can hold all cash ($b=0$) or a continuously-rebalanced short position $b<0$. We now imagine a trader who starts with $\$1$ and has two favored rebalancing rules $b>c$, who wants to perform well relative to the best of $\mathbb{B}:=\{b,c\}$ in hindsight. Accordingly, we create for him the financial derivative whose final payoff at $T$ is $$\boxed{V_T^*:=\max\{V_T(b),V_T(c)\}}.$$ Ordentlich and Cover (1998) investigated the best unlevered rebalancing rule in hindsight, with payoff $V_T^*:=\max\limits_{0\leq b\leq1}{V_T(b)}$. They found the time-0 price of this contingent claim to be $$C_0=1+\sigma\sqrt{\frac{T}{2\pi}}.$$The owner of this rebalancing option (cf. Garivaltis 2018) will compound his money at the same asymptotic rate as the best unlevered rebalancing rule in hindsight. Indeed, the final excess continuously-compounded growth rate of the best rebalancing rule in hindsight over that of the replicating strategy is $\log\big\{1+\sigma\sqrt{T/(2\pi)}\big\}/T$, which tends to 0 as $T\to\infty$. This growth rate spread obtains deterministically, regardless of the realized price path $(S_t)_{0\leq t\leq T}$. Garivaltis (2018) extended the Ordentlich-Cover (1998) analysis by computing the general time-$t$ price $C(S,t)$ of Cover’s rebalancing option for both levered and unlevered hindsight optimization. For levered hindsight optimization (with payoff $V_T^*:=\max\limits_{b\in\mathbb{R}}{V_T(b)}$), Garivaltis (2018) found the general pricing formula $$C(S,t)=\sqrt{\frac{T}{t}}\exp\{rt+z_t^2/2\},$$ where $$z_t:=\frac{\log(S_t/S_0)-(r-\sigma^2/2)t}{\sigma\sqrt{t}}$$is an auxiliary variable that is distributed unit normal with respect to the equivalent martingale measure $\mathbb{Q}$. More generally, for a Black-Scholes market with $d$ correlated stocks in geometric Brownian motion, Garivaltis (2018) found that $$C(S,t)=\bigg(\frac{T}{t}\bigg)^{d/2}\cdot\exp\{rt+z_t'R^{-1}z_t/2\},$$where $R:=[\rho_{ij}]_{d\times d}$ is the correlation matrix of instantaneous returns, $$z_{it}:=\frac{\log(S_{it}/S_{i0})-(r-\sigma_i^2/2)t}{\sigma_i\sqrt{t}},$$are auxiliary variables, and $\sigma_i$ is the volatility of stock $i$. When we relax the hindsight optimization to include all levered rebalancing rules $b\in\mathbb{R}^d$, replication becomes especially simple. At time $t$, one just looks back at the observed price history $[0,t]$, finds the best ($d$-dimensional) rebalancing rule $b(S,t)$ in hindsight, and bets the fraction $b_i(S,t)$ of wealth on stock $i$ over $[t,t+dt]$. The relation $C(S,t;T)\propto T^{d/2}$ matches the model-independent $\mathcal{O}(T^{d/2})$ super-replicating price calculated by Cover $\&$ Company. In what follows, we work toward reducing the option price $\sqrt{T/t}\cdot\exp\{rt+z_t^2/2\}$ by replacing $\mathbb{B}=\mathbb{R}$ with $\mathbb{B}:=\{b,c\}$. In order to get the payoff $\max\{V_T(b),V_T(c)\}$ into a more practical form, we note that $V_t(b)$ is a geometric Brownian motion, since $$\label{eq:wealth} \frac{dV_t(b)}{V_t(b)}=b\frac{dS_t}{S_t}+(1-b)\frac{dB_t}{B_t}=[r+(\mu-r)b]dt+b\sigma dW_t.$$ Solving this stochastic differential equation, we obtain (cf. Wilmott 1998, 2001) $$V_t(b)=\exp\{[r+(\mu-r)b-\sigma^2b^2/2]t+b\sigma W_t\}.$$ In order to get the payoff in terms of the observable variable $S_t$ (rather than the Wiener process $W_t$), we start with the equation $$S_t=S_0\exp\{(\mu-\sigma^2/2)t+\sigma W_t\},$$ and solve for $\sigma W_t$ in terms of $S_t$. Substituting the resulting expression into \[eq:wealth\], we get $$V_t(b)=\exp\{(r-\sigma^2b^2/2)t+b[\log(S_t/S_0)-(r-\sigma^2/2)t]\}.$$ We thus have $$\label{fortune} \boxed{V_t(b)=\exp\{(r-\sigma^2b^2/2)t+b\sigma\sqrt{t}\cdot z_t\}},$$where $$\label{auxiliary} \boxed{z_t:=\frac{\log(S_t/S_0)-(r-\sigma^2/2)t}{\sigma\sqrt{t}}}$$is distributed unit normal with respect to the equivalent martingale measure $\mathbb{Q}$. Note that the drift $\mu$ (which is difficult to estimate) does not appear in this formula. The final wealth of the rebalancing rule $b$ is now expressed solely in terms of $z_t$, the risk-free rate $r$, the time $t$, and the volatility $\sigma$, which is easily estimated from high-frequency price data. The Best of Two Asset Allocations in Hindsight {#two} ============================================== Before we can price the rebalancing option with payoff $\max\{V_T(b),V_T(c)\}$, we must characterize the random outcomes under which $b$ will turn out to outperform $c$ over the interval $[0,t]$. Accordingly, we compare the exponents of $V_t(b)$ and $V_t(c)$ to obtain For two given rebalancing rules $b>c$, b outperforms $c$ over $[0,t]$ if and only if $$\boxed{z_t\geq\frac{b+c}{2}\sigma\sqrt{t}}.$$ The best rebalancing rule (of any kind) in hindsight over $[0,t]$, denoted $b(S,t)$, is $$\boxed{b(S,t):=\arg\max_{b\in\mathbb{R}}V_t(b)=\frac{z(S,t)}{\sigma\sqrt{t}}}.$$Given any closed set $\mathbb{B}$ of rebalancing rules, the best performer in hindsight is the $b\in\mathbb{B}$ that is nearest to $b(S,t)=z(S,t)/(\sigma\sqrt{t})$. We compute the abscissa of vertex of the parabola $b\mapsto\log\,V_t(b)$. This yields $$b(S,t)=\arg\max_{b\in\mathbb{R}}\log V_t(b)=\frac{-\sigma\sqrt{t}\cdot z_t}{2(-\sigma^2t/2)}=\frac{z_t}{\sigma\sqrt{t}}.$$Because the graph of a parabola is symmetric about its vertex, the $b\in\mathbb{B}$ that maximizes the height of this parabola is whichever element of $\mathbb{B}$ is nearest to the vertex $b(S,t)$. We proceed to compute the cost of achieving the best of two rebalancing rules in hindsight, by finding the expected present value of $\max\{V_T(b),V_T(c)\}$ at time-0 with respect to the equivalent martingale measure $\mathbb{Q}$. This cost is the sum of two integrals $I_1+I_2$, where $$I_1:=\frac{\exp(-\sigma^2b^2T/2)}{\sqrt{2\pi}}\int_{\frac{b+c}{2}\sigma\sqrt{T}}^\infty\exp(-z^2/2+b\sigma\sqrt{T}\cdot z)dz$$and $$I_2:=\frac{\exp(-\sigma^2c^2T/2)}{\sqrt{2\pi}}\int_{-\infty}^{\frac{b+c}{2}\sigma\sqrt{T}}\exp(-z^2/2+c\sigma\sqrt{T}\cdot z)dz.$$ In the sequel, we will often use the following general formula (i.e. the appendix to Reiner and Rubinstein 1992): $$\boxed{ \int_{A}^Be^{-\alpha y^2+\beta y}dy=\sqrt{\frac{\pi}{\alpha}}\exp\bigg(\frac{\beta^2}{4\alpha}\bigg)\bigg[N\bigg(B\sqrt{2\alpha}-\frac{\beta}{\sqrt{2\alpha}}\bigg)-N\bigg(A\sqrt{2\alpha}-\frac{\beta}{\sqrt{2\alpha}}\bigg)\bigg]},$$where $\alpha>0$ and $N(\bullet)$ is the cumulative normal distribution function. Simplifying the two integrals, we get $$I_1=I_2=N\bigg(\frac{b-c}{2}\sigma\sqrt{T}\bigg).$$ The time-$0$ cost of achieving the best of two rebalancing rules $\{b,c\}$ in hindsight is $$\boxed{C_0(\delta,\sigma,T)=2N\bigg(\frac{\delta}{2}\sigma\sqrt{T}\bigg)}.$$where $\delta:=|b-c|$ is the distance between the two rebalancing rules. The equilibrium price at $t=0$ of a perpetual option ($T:=\infty$) on the best of two rebalancing rules $\{b,c\}$ in hindsight is $C_0(\delta,\sigma,\infty)=\$2$. Note that the horizon-$T$ price is independent of the interest rate $r$, and it is translation invariant in the sense that it depends only on the distance $\delta=|b-c|$ between the two rebalancing rules. We always have $1\leq C_0(\delta,\sigma,T)\leq2$; besides the perpetual version of the option, the maximum $\$2$ price also obtains if $\sigma=\infty$ or $\delta=\infty$. The minimum $\$1$ price obtains if any of the parameters $\delta, \sigma, T$ tends to 0. Since the increasing function $N(\bullet)$ is concave over $[0,\infty)$, we see that the option price is increasing and concave separately in each of the parameters $\delta, \sigma, T$. Given two rebalancing rules $b>c$ with distance $\delta=|b-c|$, an initial $\$1$ deposit into the horizon-$T$ replicating strategy achieves at $T$ a compound growth-rate that is exactly $$\frac{100}{T}\log\bigg\{2N\bigg(\frac{\delta}{2}\sigma\sqrt{T}\bigg)\bigg\}$$percent lower than that of the best of $\{b,c\}$ in hindsight. A $\$1$ deposit into the corresponding horizon-free strategy (that replicates the perpetual version of the option) achieves a compound-growth rate at $T$ that is at most $100\log(2)/T$ percent lower than that of the best of $\{b,c\}$ in hindsight. The trader’s initial ($\$1$) deposit into the replicating strategy buys him $1/C_0$ units of the option at $t=0$. For the horizon-$T$ option, his wealth at expiration will be $\max\{V_T(b),V_T(c)\}/C_0$, and hence the excess continuously-compounded growth rate will be $$\frac{1}{T}\log[\max\{V_T(b),V_T(c)\}]-\frac{1}{T}\log[\max\{V_T(b),V_T(c)\}/C_0]=\frac{\log C_0(\delta,\sigma,T)}{T}.$$For the horizon-free option, the trader’s initial dollar buys him half a unit of the option at $t=0$. Thus, his wealth at $T$ will be at least half the exercise value of the option, which is $\max\{V_T(b),V_T(c)\}$. Hence, the excess continuously-compounded growth rate of the hindsight-optimized rule at $T$ is at most $$\frac{1}{T}\log[\max\{V_T(b),V_T(c)\}]-\frac{1}{T}\log[\max\{V_T(b),V_T(c)\}/2]=\frac{\log 2}{T}.$$ Consider the following robust scheme for $T:=25$ years of leveraged bets on the S&P 500 index. We put $b:=2$ and $c:=1$ (e.g. buy-and-hold), with $\sigma:=0.15$. We get $C_0=\$1.29$ and $\log(C_0)/T=1\%$, so the replicating strategy is guaranteed to achieve a final compound-growth rate that is $1\%$ lower than the best of $\{b,c\}$ in hindsight. If $b=2$ happens to outperform the index by more than $1\%$ per year, then the trader will beat the market over $t\in[0,25]$. If $b=2$ underperforms the index (or outperforms by less than $1\%$ a year), then the trader’s compound-growth rate will have lagged the market by at most $1\%$ a year. Note that the corresponding horizon-free strategy (that replicates the perpetual version of the option) can only guarantee to get within $\log(2)/T=2.8\%$ of the hindsight-optmized growth rate at $T=25$ We construct a robust $T:=25$ year scheme for long-run stock market investment that guarantees preservation of capital. We put $b:=1$ ($100\%$ stocks) and $c:=0$ (all cash). Assuming that $\sigma:=0.15$, the practitioner can rest easy, safe in the knowledge that his foray into risk assets will ultimately not cause him to lag the risk-free rate by more than $1\%$ a year. If $r>0.01$, then he is guaranteed not to lose money if he sticks to the Plan for $T=25$ years. At the same time, if stocks go through the roof, his strategy will earn the long-run market growth rate minus a $1\%$ “universality cost.” Would-be practitioners who enjoyed these example can use Figure \[fig:regret\] to inform the choice of horizon: it plots the excess continuously-compounded growth rate for different volatilities and maturities with $\delta:=1$. ![The excess percent growth rate of the best of two rebalancing rules over the replicating strategy, for different horizons and volatilities, with $\delta:=1$.[]{data-label="fig:regret"}](excess2.png){width="375px"} General Formulas for Pricing and Replication -------------------------------------------- Before we can put our on-line schemes for robust asset allocation into actual practice, we must derive general time-$t$ formulas for pricing and replication of the rebalancing option under discrete hindsight optimization. Thus, we proceed to extend the above integration technique to the general situation. To simplify the notation, we let $\tau:=T-t$ denote the remaining life of the option at time $t$. Inspired by Garivaltis (2018), we start with the decomposition $$\boxed{z_T=\sqrt{\frac{t}{T}}\cdot z_t+\sqrt{\frac{\tau}{T}}\cdot y},$$where $$\boxed{y:=\frac{\log(S_T/S_t)-(r-\sigma^2/2)\tau}{\sigma\sqrt{\tau}}}$$is distributed unit normal with respect to the equivalent martingale measure and the information available at $t$. Conditional on the values of time-$t$ variables, $b$ outperforms $c$ at $T$ if and only if $$\boxed{y\geq\frac{\frac{b+c}{2}\sigma T-\sqrt{t}\cdot z_t}{\sqrt{\tau}}}.$$ Thus, the general price $C(S,t)$ is equal to the sum of two integrals $I_1+I_2$, where $$I_1:=\frac{\exp(rt-\sigma^2b^2T/2+b\sigma\sqrt{t}\cdot z_t)}{\sqrt{2\pi}}\int_{[(b+c)\sigma T/2-\sqrt{t}\cdot z_t]/\sqrt{\tau}}^\infty\exp(-y^2/2+b\sigma\sqrt{\tau}\cdot y)dy$$and $$I_2:=\frac{\exp(rt-\sigma^2c^2T/2+c\sigma\sqrt{t}\cdot z_t)}{\sqrt{2\pi}}\int_{-\infty}^{[(b+c)\sigma T/2-\sqrt{t}\cdot z_t]/\sqrt{\tau}}\exp(-y^2/2+c\sigma\sqrt{\tau}\cdot y)dy.$$ These integrals simplify to $$I_1=N\bigg(\frac{[b-c]\sigma T/2+\sqrt{t}\cdot z_t-b\sigma t}{\sqrt{\tau}}\bigg)V_t(b)$$ and $$I_2=N\bigg(\frac{[b-c]\sigma T/2-\sqrt{t}\cdot z_t+c\sigma t}{\sqrt{\tau}}\bigg)V_t(c).$$ The general cost $C(S,t)$ of achieving the best of two rebalancing rules $b>c$ in hindsight is $$\label{eq:cost2} \boxed{C=N(d_1)V_t(b)+N(d_2)V_t(c)},$$where $$\label{d1} \boxed{d_1:=\frac{(b-c)\sigma T/2+\sqrt{t}\cdot z_t-b\sigma t}{\sqrt{\tau}}}$$ and $$\boxed{d_2:=(b-c)\sigma\sqrt{\tau}-d_1=\frac{(b-c)\sigma T/2-\sqrt{t}\cdot z_t+c\sigma t}{\sqrt{\tau}}}.$$ A perpetual option ($T:=\infty$) on the best of two rebalancing rules $b>c$ costs $C(S,t)=V_t(b)+V_t(c)$ in state $(S_t,t)$. To delta-hedge the perpetual option, one holds $$\boxed{\Delta=\frac{bV_t(b)+cV_t(c)}{S_t}}$$shares of the underlying in state $(S_t,t)$, and therefore bets the fraction $$\boxed{\hat{b}(S_t,t):=\frac{\Delta S}{C}=\frac{bV_t(b)+cV_t(c)}{V_t(b)+V_t(c)}}$$of wealth on the underlying at $t$. As $T\to\infty$, we see that $d_1,d_2\to+\infty$ and the option price converges to $V_t(b)+V_t(c)$. Next, one can verify by direct calculation from (\[fortune\]) and (\[auxiliary\]) that $$\frac{\partial V(b)}{\partial S}=\frac{\partial V(b)}{\partial z}\frac{\partial z}{\partial S}=\frac{bV_t(b)}{S_t}.$$Alternately, one can observe that the rebalancing rule $b$ keeps (by definition) $bV_t(b)$ dollars in the stock at time $t$, which amounts to $bV_t(b)/S_t$ shares. Thus, to replicate the sum $V_t(b)+V_t(c)$ we must own a total of $\Delta=bV_t(b)/S_t+cV_t(c)/S_t$ shares of the underlying. We should note that our general pricing formulas could have been obtained differently, by applying the theory of “exchange options” that was bequeathed to us in sumultaneous papers by Margrabe (1978) and Fischer (1978). Rather than the single underlying $S_t$, one could view the (perfectly correlated) geometric Brownian motions $U_1(t):=V_t(b)$ and $U_2(t):=V_t(c)$ as underlyings of a multi-asset option with payoff $$\max\{U_1,U_2\}=\max\{U_1-U_2,0\}+U_2.$$ This amounts to a $\$1$ deposit into the rebalancing rule $c$, plus the option to exchange the final wealth of $c$ for the final wealth of $b$ at $T$. Substituting the aggregate volatility $\sigma_a:=(b-c)\sigma$ into Margrabe’s Formula (cf. Zhang 1998) yields the same result $$\label{margrabe} \boxed{C(U_1,U_2,t)=N(d_1)U_1+N(\sigma_a\sqrt{\tau}-d_1)U_2},$$where $$\label{margrabe2} \boxed{d_1:=\frac{\log(U_1/U_2)}{\sigma_a\sqrt{\tau}}+\frac{\sigma_a\sqrt{\tau}}{2}},$$is in agreement with (\[d1\]). Figure \[fig:iv\] plots the price and intrinsic value of the rebalancing option for different values of $S$ under the parameters $r:=0.03, T:=10, S_0:=100, t:=5, \sigma:=0.7, b:=1.5,$ and $c:=0.5$. The horizon-$T$ replicating strategy for the best of two rebalancing rules $b>c$ in hindsight holds $$\boxed{\Delta=\frac{N(d_1)bV_t(b)+N(d_2)cV_t(c)}{S_t}}$$shares of the stock in state $(S_t,t)$, which amounts to betting the fraction $$\boxed{\hat{b}(S_t,t)=\frac{\Delta S}{C}=\frac{N(d_1)bV_t(b)+N(d_2)cV_t(c)}{N(d_1)V_t(b)+N(d_2)V_t(c)}}.$$ of wealth on the stock at $t$. Thus, the on-line fraction of wealth bet on the stock is a time- and state-varying convex combination of $b$ and $c$. First, we note the standard relations $\partial C/\partial U_1=N(d_1)$ and $\partial C/\partial U_2=N(d_2)$, which follow by direct calculation from (\[margrabe\]), (\[margrabe2\]), and the fact that $U_1\phi(d_1)=U_2\phi(d_2)$, where $\phi(\bullet)$ is the standard normal density function. Differentiating the option price, we get $$\frac{\partial C}{\partial S}=\frac{\partial C}{\partial U_1}\frac{\partial U_1}{\partial S}+\frac{\partial C}{\partial U_2}\frac{\partial U_2}{\partial S}=N(d_1)\frac{bU_1}{S}+N(d_2)\frac{cU_2}{S},$$which is the desired result. Thus, even if the best rebalancing rule (of any kind) in hindsight $b(S,t)=z(S,t)/(\sigma\sqrt{t})$ happens to lie between $b$ and $c$, the replicating strategy will not generally bet the hindsight-optimized fraction $\arg\max\limits_{b\in\mathbb{R}}V_t(b)$ of wealth on the stock. This phenomenon is illustrated in Figure \[fig:comparison\]. Instead, the relative weighting of $b$ and $c$ (which is initially $50/50$ at time-$0$) evolves with the observed performances $V_t(b), V_t(c)$ and the remaining life $\tau:=T-t$ of the option. For a fixed time $t$, if $U_1\to\infty$ or $U_2\to\infty$ then the on-line portfolio weight will converge to $b$ or $c$ accordingly. As $\tau\to 0$, $d_1$ tends to $\pm\infty$ and $d_2$ tends to $\mp\infty$ according as to whether $b$ or $c$ has outperformed over the known price history. Thus, small differences in the observed performances $V_t(b)$ and $V_t(c)$ get amplified in the on-line portfolio weight as $\tau\to0$. Figures \[fig:sim\] and \[fig:sim2\] simulate the performance of the replicating strategy for different parameter values over a $T:=30$ year horizon. ![The fraction of wealth bet by the replicating strategy for different stock prices, $r:=0.03, T:=10, S_0:=100, t:=5, \sigma:=0.7, b:=1.5, c:=0.5$.[]{data-label="fig:comparison"}](comparison.png){width="325px"} ![Option price and intrinsic value for different stock prices, $r:=0.03, T:=10, S_0:=100, t:=5, \sigma:=0.7, b:=1.5, c:=0.5$.[]{data-label="fig:iv"}](iv.png){width="300px"} ![Performance simulation over $T:=30$ years for the parameters $S_0:=1, b:=2, c:=0.5, r:=0.03, \sigma:=0.15, \nu:=0.1, \mu=\nu+\sigma^2/2.$[]{data-label="fig:sim"}](sim1.png){width="425px"} ![Performance simulation over $T:=30$ years for the parameters $S_0:=1, b:=2, c:=0.5, r:=0.03, \sigma:=0.7, \nu:=0.07, \mu=\nu+\sigma^2/2.$[]{data-label="fig:sim2"}](sim2.png){width="425px"} The General Discrete Set of Asset Allocations {#several} ============================================= We carry on with the general discrete set $\mathbb{B}:=\{b_1,...,b_n\}\subset\mathbb{R}$ of asset allocations, where the $b_i$ are arranged in increasing order: $b_1<b_2<\cdot\cdot\cdot<b_n$. Thus, we now deal with the payoff $\ V_t^*:=\max\limits_{1\leq i\leq n}V_t(b_i)$. For notational convenience, we will also write $b_0:=-\infty$ and $b_{n+1}:=+\infty$. We let $\Delta b_i:=b_{i+1}-b_{i}$ for $0\leq i\leq n$, and thus $\Delta b_0=\Delta b_{n}=+\infty$. For a given rebalancing rule $b_i\,(1\leq i\leq n)$, the final payoff of the option is equal to $V_T(b_i)$ if and only if $$\frac{b_{i-1}+b_{i}}{2}\sigma\sqrt{T}\leq z_T\leq \frac{b_i+b_{i+1}}{2}\sigma\sqrt{T}.$$ Thus, conditional on the values of time-$t$ variables, $b_i$ will turn out to be the best performer over $[0,T]$ if and only if $$\frac{(b_{i-1}+b_{i})\sigma T/2-\sqrt{t}\cdot z_t}{\sqrt{\tau}}\leq y\leq \frac{(b_i+b_{i+1})\sigma T/2-\sqrt{t}\cdot z_t}{\sqrt{\tau}}.$$ For simplicity, we will write this interval as $y\in[A_{i-1},A_i]$. Thus $A_0=-\infty$ and $A_n=+\infty$. The expected present value of the final payoff with respect to $\mathbb{Q}$ and the information available at $t$ is equal a sum of integrals $I_1+\cdot\cdot\cdot+I_n$, where $$I_i:=\frac{\exp(rt-\sigma^2b_i^2T/2+b_i\sigma\sqrt{t}\cdot z_t)}{\sqrt{2\pi}}\int_{A_{i-1}}^{A_i}\exp(-y^2/2+b_i\sigma\sqrt{\tau}\cdot y)dy.$$ Evaluating these integrals, we obtain the general pricing formula $$\boxed{C(S,t)=\sum\limits_{i=1}^n\{N(A_i-\beta_i)-N(A_{i-1}-\beta_i)\}V_t(b_i)},$$where $$\boxed{A_i:=\frac{(b_i+b_{i+1})\sigma T/2-\sqrt{t}\cdot z_t}{\sqrt{\tau}}}.$$and $$\boxed{\beta_i:=b_i\sigma\sqrt{\tau}}.$$ Bearing in mind that $A_0=-\infty$ and $A_n=+\infty$, we can also write $$\begin{gathered} \label{eq:genprice} C(S,t)=N(A_1-\beta_1)V_t(b_1)+\sum\limits_{i=2}^{n-1}\{N(A_i-\beta_i)-N(A_{i-1}-\beta_i)\}V_t(b_i)\\ +N(\beta_n-A_{n-1})V_t(b_n).\\\end{gathered}$$The general option price could again have been obtained differently, by an interesting application of Margrabe’s theory of exchange options. Indeed, we could consider the wealth processes $(V_t(b_i))_{i=1}^n$ as separate underlyings $U_i:=V_t(b_i)$ of a multi-asset option whose final payoff is equal to $\max\{U_1,U_2,...,U_n\}$. First of all, we remark that at any given time the ordered sequence of numbers $U_1(t),...,U_n(t)$ is unimodal, or single-peaked. This happens because the $(\log U_i)_{i=1}^n$ trace out a sequence of heights on the parabola $b\mapsto\log V_t(b)$ as we move from left to right over the abscissae $b_1<b_2<\cdot\cdot\cdot<b_n$. The peak occurs for the index $$i^*:=\arg\min\limits_{1\leq i\leq n}|b_i-b(S_t,t)|=\arg\max\limits_{1\leq i\leq n}V_t(b_i).$$Thus, $U_i$ is increasing in $i$ for $i<i^*$ and decreasing in $i$ for $i\geq i^*$. This unimodality in hand, we now have the identity $$\max\{U_1,...,U_n\}=U_1+(U_2-U_1)^++(U_3-U_2)^++\cdot\cdot\cdot+(U_n-U_{n-1})^+,$$where $x^+:=\max\{x,0\}$ denotes the positive part of $x$. Hence, the payoff $\max\limits_{1\leq i\leq n}U_i$ is equivalent to a portfolio consisting of one unit of $U_1$, plus an option to exchange $U_1$ for $U_2$, plus an option to exchange $U_2$ for $U_3$, $\cdot\cdot\cdot$ , plus an option to exchange $U_{n-1}$ for $U_n$. At expiration, the trader keeps exchanging $U_{i}$ for $U_{i+1}$ until the maximum $U_i^*$ is reached. Applying the Margrabe Formula (cf. Zhang 1998) in conjunction with linear pricing, we find that the no-arbitrage price of this portfolio (consisting of a unit of $U_1$ plus $n-1$ exchange options) is $$\label{easy} \boxed{C(U_1,...,U_n,t)=U_1+\sum\limits_{i=1}^{n-1}\{N(d_{1i})U_{i+1}-N(d_{2i})U_i\}},$$where $$\boxed{d_{1i}:=\frac{\log(U_{i+1}/U_i)}{\sigma_{ai}\sqrt{\tau}}+\frac{\sigma_{ai}\sqrt{\tau}}{2}},$$$d_{2i}:=d_{1i}-\sigma_{ai}\sqrt{\tau}$, and $\sigma_{ai}:=\Delta b_i\sigma$ is the aggregate volatility in a two-asset market consisting of $U_i$ and $U_{i+1}$. Collecting terms, we get the linear combination $$\label{combo} \boxed{C(U_1,...,U_n,t)=N(-d_{21})U_1+\sum_{i=2}^{n-1}[N(d_{1,i-1})-N(d_{2i})]U_i+N(d_{1,n-1})U_n},$$which agrees with equation (\[eq:genprice\]) above. Figure \[payoff2\] plots the option price and intrinsic value for different stock prices under the parameters $r:=0.03, T:=10, S_0:=100, t:=5, \sigma:=0.7,$ and $\mathbb{B}:=\{0,0.5,1,1.5,2\}.$ In general there will be several implied volatilities $\sigma$ that could rationalize an observed price of the rebalancing option. Figure \[vol\] plots the option price for different volatilities under the parameters $r:=0.03, T:=10, S_0:=100, t:=5, S_t:=200,$ and $\mathbb{B}:=\{0,0.5,1.5\}$. ![Option price and intrinsic value for different stock prices, $r:=0.03, T:=10, S_0:=100, t:=5, \sigma:=0.7, \mathbb{B}:=\{0,0.5,1,1.5,2\} $.[]{data-label="payoff2"}](generaliv.png){width="300px"} ![Option prices for different volatilities, $r:=0.03, T:=10, S_0:=100, t:=5, S_t:=200, \mathbb{B}:=\{0,0.5,1.5\} $.[]{data-label="vol"}](vol.png){width="300px"} In specializing the pricing formula for $t:=0$ and simplifying (remembering that $V_0(b_i):=1$), we get For hindsight optimization over $n$ discrete rebalancing rules $b_1<\cdot\cdot\cdot<b_n$, the time-0 cost of achieving the best $b_i$ in hindsight is $$\boxed{C_0=2-n+2\sum_{i=1}^{n-1}N(\Delta b_i\sigma\sqrt{T}/2)},$$where $\Delta b_i:=b_{i+1}-b_i$. If $\delta:=\max\limits_{1\leq i\leq n-1}\Delta b_i$, then $$\boxed{C_0\leq 2-n+2(n-1)N(\delta\sigma\sqrt{T}/2)}.$$ A perpetual option on the best of any $n$ distinct rebalancing rules in hindsight is worth $C_0=n$ dollars at time-0. Thus, we see that the time-0 price of the general horizon-$T$ rebalancing option is independent of the interest rate, and it is increasing and concave separately in the parameters $\Delta b_i, \sigma, T$. We again observe that horizontal translations of the point set $\{b_1,...,b_n\}$ do not alter the option price. We always have the relation $1\leq C_0\leq n$; the maximum $n$ dollar price obtains if any of the parameters tends to infinity and the minimum $\$1$ price obtains if any of the parameters tends to zero. For a $T:=25$ year planning horizon, we cherry pick five favored asset allocations $\mathbb{B}:=\{0,0.5,1,1.5,2\}$. Assuming stock market volatility of $\sigma:=0.15$ going forward, we get $C_0=\$1.59$, and the excess growth rate of the hindsight-optimized asset allocation will be exactly $\log(C_0)/T=1.87\%$. Assuming that the risk-free rate is greater than $1.87\%$, the replicating strategy is guaranteed not to lose money if the practitioner sticks to the Plan for the next $T=25$ years. The horizon-$T$ replicating strategy for the best of the rebalancing rules $b_1<b_2<\cdot\cdot\cdot<b_n$ in hindsight holds $$\boxed{\Delta=N(-d_{21})\frac{b_1V_t(b_1)}{S_t}+\sum\limits_{i=2}^{n-1}[N(d_{1,i-1})-N(d_{2i})]\frac{b_iV_t(b_i)}{S_t}+N(d_{1,n-1})\frac{b_nV_t(b_n)}{S_t}}$$shares of the stock in state $(S_t,t)$, thereby betting the fraction $\hat{b}(S,t)=\Delta S/C$ of its bankroll on the stock. This amounts to a time- and state-varying convex combination of the $b_i$. As $\tau\to0$, the option price converges to $U_{i^*}:=\max\limits_{1\leq i\leq n}U_i$ and the fraction of wealth bet by the replicating strategy converges to $\arg\max\limits_{b\in\mathbb{B}}V_t(b)$ if this set is a singleton; if $\arg\max\limits_{b\in\mathbb{B}}V_t(b)=U_{i^*}=U_{i^*+1}$ has two distinct points, then $\hat{b}$ converges to the midpoint $(b_{i^*}+b_{i^*+1})/2$ as $\tau\to0$. The horizon-free replicating strategy (corresponding to the perpetual version of the option) bets the performance-weighted average $$\hat{b}(S_t,t)=\frac{\sum\limits_{i=1}^nb_iV_t(b_i)}{\sum\limits_{i=1}^nV_t(b_i)}$$of the rebalancing rules $b_i$, which converges almost surely to $$\arg\max_{b\in\mathbb{B}}\bigg\{(\mu-r)b-\frac{\sigma^2b^2}{2}\bigg\}=\arg\min_{b\in\mathbb{B}}\bigg|b-\frac{\mu-r}{\sigma^2}\bigg|,$$i.e. it converges to whichever element of $\mathbb{B}$ is closest to the continuous time Kelly rule (cf. Luenberger 1998). Note that the pricing formula (\[combo\]) is a linearly homogeneous function of the underlyings $(U_1,...,U_n)$. By Euler’s theorem for homogeneous functions, we therefore have the relation $$C=\sum\limits_{i=1}^n\frac{\partial C}{\partial U_i}U_i.$$Accordingly, by direct calculation on (\[combo\]) one can (carefully) verify the partial derivatives $$\frac{\partial C}{\partial U_1}=N(-d_{21}),$$ $$\frac{\partial C}{\partial U_i}=N(d_{1,i-1})-N(d_{2i})\text{ for } 2\leq i\leq n-1,$$ $$\frac{\partial C}{\partial U_n}=N(d_{1,n-1}).$$To verify these partials easily, one needs the identity$$\frac{\phi(d_{2i})}{\phi(d_{1i})}=\frac{U_{i+1}}{U_i}\text{ for }1\leq i\leq n-1,$$where $\phi(\bullet)$ is the standard normal density function. Observe that $U_i$ generally appears in the terms of (\[combo\]) that correspond to the indices $i-1, i, \text{and } i+1$. $U_1$ appears in the first two terms and $U_n$ appears in the last two terms. This being done, the delta-hedging strategy now obtains from the chain rule $$\frac{\partial C}{\partial S}=\sum\limits_{i=1}^n\frac{\partial C}{\partial U_i}\frac{\partial U_i}{\partial S}$$in conjunction with the fact that $\partial U_i/\partial S=b_iV_t(b_i)/S$. To get the horizon-free result, we just observe that $d_{1i}\to+\infty$ and $d_{2i}\to-\infty$ as $T\to\infty$. Finally, consider what happens when $\tau\to0$. The numbers $d_{1i},d_{2i}$ will converge to the same limit $\pm\infty$ according as $U_{i+1}$ is greater or less than $U_i$. In the event that $U_{i+1}=U_i$ then $d_{1i}$ and $d_{2i}$ both converge to zero. The numbers $(U_i)_{i=1}^n$ will typically have a unique mode $U_{i^*}$, e.g. $U_1<\cdot\cdot\cdot<U_{i^*-1}<U_{i^*}>U_{i^*+1}>\cdot\cdot\cdot>U_n$. In this case, all coefficients in the linear combination (\[combo\]) converge to zero except the one corresponding to $i=i^*$, which converges to $1$. If there are two modes $U_{i^*}=U_{i^*+1}$, then the corresponding coefficients in (\[combo\]) both converge to $1/2$, and the result follows. Note that for $n>2$, the initial weighting of the $b_i$ at time-$0$ is not uniform, even if the $b_i$ themselves are equally spaced. The endpoints $b_1$ and $b_n$ have initial weights $N(\Delta b_1\sigma\sqrt{T}/2)/C_0$ and $N(\Delta b_{n-1}\sigma\sqrt{T}/2)/C_0$, respectively, and the rest of the $b_i$ have initial weights $[N(\Delta b_{i-1}\sigma\sqrt{T}/2)-N(-\Delta b_{i}\sigma\sqrt{T}/2)]/C_0$ for $2\leq i\leq n-1$. If the $b_i$ are equally spaced, then each of the intermediate points ($2\leq i\leq n-1$) gets initial weight $[2N(\delta\sigma\sqrt{T}/2)-1]/C_0$, but the endpoints $b_1,b_n$ get the higher initial weight $N(\delta\sigma\sqrt{T}/2)/C_0$. For any closed set $\mathbb{B}\subseteq\mathbb{R}$ of rebalancing rules (finite or infinite), the American-style version of Cover’s rebalancing option (with exercise price $K$ and payoff $\max\{\max\limits_{b\in\mathbb{B}}V_t(b)-K,0\})$ will never be exercised early in equilibrium, and its price $C_a(S_t,t)$ equals the price $C_e(S_t,t)$ of the corresponding European-style option. For simplicity, let $V_t^*:=\max\limits_{b\in\mathbb{B}}V_t(b)$ denote the hindsight-optimized wealth over $[0,t]$, and let $b^*_t:=\arg\max\limits_{b\in\mathbb{B}}V_t(b)$ denote the best (allowable) rebalancing rule in hindsight over $[0,t]$. Consider, from the standpoint of time $t$, the following two trading strategies: - Invest $Ke^{-r\tau}$ dollars in the risk-free bond and buy $1$ unit of Cover’s (European-style) rebalancing option at a price of $C_e(S_t,t)$. - Invest $V_t^*$ dollars into the rebalancing rule $b_t^*$. That is, take the best rebalancing rule in hindsight over $[0,t]$, and adhere to that same (constant) continuously-rebalanced portfolio over $[t,T]$. Observe that Strategy 1 has a final payoff of $\max\{V_T^*,K\}$ and Strategy 2 has a final payoff of $V_T(b_t^*)$. Since the payoff at $T$ of Strategy 1 is guaranteed to be at least as great as that of Strategy 2, the initial investment of $Ke^{-r\tau}+C_e(S_t,t)$ dollars into Strategy 1 must be greater or equal to the investment $V_t^*$ that is required for Strategy 2. Thus, we have the inequalities $$C_a(S_t,t)\geq C_e(S_t,t)\geq V_t^*-Ke^{-r\tau}\geq V_t^*-K.$$Hence, since the price of an American rebalancing option always exceeds the exercise value, the option “is worth more alive than dead” and will never be exercised in equilibrium. On account of the fact that early exercise rights are worthless anyhow, we must therefore have $C_a(S_t,t)=C_e(S_t,t)$. We remark that this is a general model-independent result that applies equally well to rebalancing rules $b\in\mathbb{B}\subseteq\mathbb{R}^d$ over arbitrary $d$-dimensional stock markets. The dominance argument only requires the market (and the set $\mathbb{B}$) to admit a well-defined hindsight-optimized rebalancing rule $b_t^*:=\arg\max\limits_{b\in\mathbb{B}}V_t(b)$. For $\mathbb{B}:=\{1\}$ the best rebalancing rule in hindsight is just $b_t^*=1$ and we get $V_t^*=S_t$; this recovers the proof given by Merton (1973, 1990) of the no-exercise theorem for vanilla call options. The special cases $\mathbb{B}:=\mathbb{R}^d$ and $\mathbb{B}:=[0,1]$ were observed by Garivaltis (2018) for a continuous-time Black-Scholes market with $K:=0$. Conclusion ========== This paper studied Cover’s rebalancing option with discrete hindsight optimization. In the context of a single risk asset, a constant (perhaps levered) rebalancing rule is a simple trading strategy that continuously maintains some fixed fraction of wealth in the underlying asset. Cover’s discrete-time universal portfolio theory derives robust on-line trading strategies that are guaranteed to achieve an acceptable percentage of the final wealth of the best rebalancing rule (of any kind) in hindsight. Working in continuous time, we formulated the less aggressive benchmark of the best rebalancing rule in hindsight that hails from some finite set $\mathbb{B}:=\{b_1,...,b_n\}$. This approach allows the (delta-hedging) practitioner to cherry pick a small number of favored rebalancing rules that could embody institutional leverage constraints or the trader’s own speculative beliefs as to the future pattern of returns in the stock market. Accordingly, we priced and replicated the financial option whose final payoff is equal to the wealth $V_T^*:=\max\limits_{1\leq i\leq n}V_T(b_i)$ that would have accrued to a $\$1$ deposit into the best $b_i$ in hindsight. We found that a perpetual option (with zero exercise price) on the best of $n$ distinct rebalancing rules costs $n$ dollars at $t=0$. The corresponding (horizon-free) replicating strategy amounts to depositing a dollar into each $b_i$ and “letting it ride.” If the option expires at some fixed date $T$ the price is lower; it is concavely increasing in $T$ and in the volatility $\sigma$ of the underlying risk asset. From the standpoint of $t=0$, the cost $C_0$ of achieving the best of the $b_i$ is translation invariant: it increases monotonically with the distances $\Delta b_i$ between adjacent rebalancing rules, but it does not otherwise depend on their precise location. In this connection, the replicating strategy amounts to a time- and state-varying convex combination of the $b_i$ that dynamically considers both the observed performances $V_t(b_i)$ and the remaining life $\tau:=T-t$ of the option. No-arbitrage considerations dictate that American-style rebalancing options (for general exercise price $K$) will never be exercised early in equilibrium. This model-independent result holds for arbitrary closed sets $\mathbb{B}\subseteq\mathbb{R}^d$ of rebalancing rules over any $d-$dimensional stock market that admits a well-defined best rebalancing rule in hindsight. Toward the end of the investment horizon (as it becomes more and more obvious which $b_i$ is likely to be the best in hindsight), even small differences in observed performance will cause the replicating strategy to dramatically over- or under-weight the various $b_i$. Any practitioner of the horizon-$T$ delta-hedging strategy is guaranteed to achieve at $T$ the *deterministic* fraction $1/C_0$ of the final wealth of the best $b_i$ in hindsight. The excess compound-growth rate at $T$ of the best $b_i$ (over and above the trader) is $\log(C_0)/T$, which tends to $0$ as $T\to\infty$. The replicating strategy will asymptotically beat the underlying (i.e. an S&P 500 ETF) if any of the $b_i$ turns out to achieve a compound-growth rate that is higher than $b=1$. If there is no such $b_i\in\mathbb{B}$, but the trader had the good sense to put $1\in\mathbb{B}$, then the trader’s compound-annual growth rate will lag the underlying risk asset by at most $100\log(C_0)/T$ percent at $T$. If we have $0\in\mathbb{B}$, then the trader also guarantees that he will ultimately not lose money over $[0,T]$ if the condition $\log(C_0)/T<r$ is satisfied. Hence, our trading strategy is in a sense the most conservative attempt at detecting on-the-fly whether any of the rebalancing rules in some finite set is capable of beating the underlying over a given investment horizon. We have therefore obtained a universal continuous-time asset allocation scheme that is computationally pleasant as well as feasible for the human life span. The on-line behavior is Markovian in the sense that the relevant state vector is just $(S_t,t,(V_t(b_i))_{i=1}^n)$. The algorithm requires no prior knowledge of the (hard-to-estimate) drift parameter $\mu$ of the stock market. Apart from the finite-dimensional state vector, the trader’s behavior depends only on the known parameters $r, \sigma, T,$ and $\mathbb{B}$. In just 25 years, say, our method guarantees to achieve within $1.87\%$ of the compound-annual growth rate of whichever turns to be the most profitable asset allocation among $\mathbb{B}:=\{0,0.5,1,1.5,2\}$. [9]{} **Black, F. and Scholes, M., 1973**. 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--- abstract: 'The antiferromagnetic (AF) model is generalized for the quasielectron system composed of identical ionic-covalent dimers. The density-fluctuation and covalent-correlation operators are constructed based on the extended AF density matrices, and the quasielectron system is decomposed into 4-level subsystems for the electron ionization and affinity. By considering the nearest-neighbor hopping near the covalent limit, we can see the importance of the bonding coefficients to the effective mass of the excited carrier in the crystal of the zincblende structure.' author: - 'C. F. Huang' title: 'An extended antiferromagnetic model for quasielectrons in identical ionic-covalent bonds' --- Introduction ============ Different quasiparticles are taken into account for many-electron systems. Based on the Bardeen Cooper Schrieffer (BCS) theory [@James], the Bogoliubov-BCS quasiparticles are constructed under the superconducting (SC) order by considering Bogoliubov-deGennes or Hartree-Bogoliubov equations [@Ghosal; @Barankov; @Paar]. The antiferromagnetic (AF) quasielectrons are introduced when the orbital antiferromagnetism or the d-density wave (DDW) order becomes important [@Gerami; @Chakravarty], and the quasiparticles under multiple orders are discussed in the literature [@Bena; @Kee; @Huang1; @Laughlin; @Ramshaw]. The Hubbard model [@Mahan] may help us to clarify the SC and AF behaviors, and the 4-level Hubbard dimer [@Matlak1; @Matlak2] is constructed by including the up- and down-spin orbitals at the two sites. The ionic-covalent chemical bond can be approximated by such a dimer when the two sites correspond to the atomic orbitals, and we have the Heitler-London state [@Fulde; @Soos], which is denoted by $| \Psi _{c} \rangle$ in this manuscript, for the covalent limit. The ionic state $| \Psi _{i} \rangle$ may become dominant in the hetero-diatom bond, in which we shall consider different on-site energies introduced in the ionic Hubbard model [@Kampf; @Buzatu]. The Bloch states [@Fulde; @Grosso; @Grafenstein] are important to construct the quasiparticle orbitals, including the plane-wave ones for the nearly free carriers in crystals. Such states can be introduced based on the mean-field methods such as the Hartree-Fock (HF) approach, which yields the self-consistent-field (SCF) solutions [@Fulde; @Grafenstein; @Stoyanova]. It has been discussed in the literature how to include the correlation energy beyond the HF approach by considering the coupled-cluster corrections such as those due to coupled-cluster doubles (CCDs) [@Fulde; @Grafenstein; @Stoyanova; @Doll; @Talukdar]. In addition, my group [@Huang1] has used the AF part of the extended Bogoliubov-BCS quasiparticles to construct the energy form of the one-bond system by considering [@Prasad] $$\begin{aligned} | \Psi _{b} \rangle = \alpha _{i} | \Psi _{i} \rangle + \alpha _{c} | \Psi _{c} \rangle.\end{aligned}$$ Here $| \Psi _{b} \rangle$ denotes the bonding wavefunction of the half-filled ionic-covalent bond, and the complex numbers $ \alpha _{i}$ and $\alpha _{c}$ are the bonding coefficients satisfying $| \alpha _{i}| ^{2} +|\alpha _{c}| ^{2} =1$. In this manuscript, the AF model is generalized for the compound in which the chemical bonds are identical to the red one in Fig. 1 (a). To include the bonding correlation, the extended AF density matrix $$\begin{aligned} \rho _{ea} = \left[ \begin{array}{c} \rho \text{ \ \ } \Delta \\ \Delta \text{ \ \ } \rho \end{array} \right] \end{aligned}$$ is introduced to construct the correlation operators $$\begin{aligned} \begin{cases} d ^{(1)} = ( \sqrt{2} -1) [ \rho ^{(1)} \rho ^{(2)} ( I - \rho ^{(1)}) + ( I - \rho ^{(1)}) \rho ^{(2)} \rho ^{(1)} ] \\ d ^{(2)} = ( I - \rho ^{(1)}) \rho ^{(2)} ( I - \rho ^{(1)}) \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \end{cases}\end{aligned}$$ for the electron pair in the bonding region $\Omega$. Here $\rho$ and $\Delta$ are self-adjoint operators, $I$ denotes the identity operator, and the operators $\rho ^{(1)} = \rho + \Delta$ and $\rho ^{(2)} = \rho - \Delta$ can serve as the density matrices for the quasielectrons in Fig. 2 (a), which shows the 4-level dimer corresponding to the one-bond system. The correlation matrices $d ^{(1)}$ and $d ^{(2)}$ are denoted as the density-fluctuation and covalent-correlation operators because they represent the fluctuating charge and the correlation due to the covalent component, respectively. For convenience, the background is mentioned in section II, and the operators $d ^{(1)} $ and $d ^{(2)}$ are introduced in subsection III-A by considering the non-interacting and interacting parts of the one-bond Hamiltonian. The assumptions about the one-bond dimer are discussed in Appendix A, and an orbital transformation is mentioned in Appendix B to improve my model. The ionized and affinitive processes for the one-bond system are taken into account in subsection IV-A. The quasielectron model for the binary compound is constructed in section III-B by considering the identical chemical bonds connecting the anions and cations. Figure 1 (b) shows such bonds in the zincblende-structure crystal as an example. The density-fluctuation and covalent-correlation operators are extended as $d ^{( \text{I} )}$ and $d ^{( \text{II} )}$ for the bonding correlation in the quasielectron system of the considered compound. I decompose such a system into the 4-level subsystems shown in Fig. 3 (a), and the energy difference to excite a carrier is obtained in section IV-B by considering the ionized/affinitive process. When the compound forms an ideal crystal following the periodic boundary condition, each subsystem may correspond to a Bloch-type function. Near the ionic limit, as shown in Appendix C, my model can be supported by the coupled-cluster theory. On the other hand, we can see the importance of the bonding coefficients to the excited carrier near the covalent limit under the strong e-e repulsive strength, which is responsible for the Mott insulating behaviors in some AF systems [@Lee1; @Liu]. The Hamiltonian family, which can correspond to the random Schr$\ddot{o}$dinger operators in the random-matrix theory [@Kirsc; @Erdos; @Lee2; @Huang2], is discussed in section V. Actually we may generalize Eq. (2) to include a set of Hamiltonians by constructing the multiple-component quasielectrons. I note that the multiple-component functions can be used to introduce the vector bundles [@Bohm; @Friedman; @Banks] for the gauge theory. The compound system composed of different ionic-covalent dimers are discussed in Appendix D. The summary is made in section VI. Background ========== The DDW Hamiltonian $H _{ddw} = \sum _{ {\bf k} \sigma } \chi _{ {\bf k} \sigma } ^\dag B _{ {\bf k} } \chi _{ {\bf k} \sigma } $ [@Gerami] has been introduced for the AF quasielectrons with $$\begin{aligned} B _{ {\bf k} } = \left[ \begin{array}{c} \varepsilon _{ {\bf k} } - \mu \text{ \ \ \ \ \ \ \ } \Delta _{ {\bf k} } \text{ \ \ \ \ } \\ \text{ \ \ \ } \Delta _{ {\bf k} } ^{\ast} \text{ \ \ \ \ } \varepsilon _{ {\bf k} + {\bf Q} } - \mu \end{array} \right]\end{aligned}$$ and $\chi _{ {\bf k} \sigma } ^{\dag} = ( c _{ {\bf k} \text{ } \sigma } ^{\dag} , - i c _{ {\bf k}+{\bf Q} \text{ } \sigma } ^{\dag} )$. Here $\mu$ denotes the chemical potential, the wave-vector [**k**]{} belongs to the reduced Brillouin zone (RBZ), ${\bf Q }$ equals the DDW ordering wavevector, $\sigma$ denotes the spin orientation $\uparrow$ or $\downarrow$, $\varepsilon _{ {\bf k} }$ and $ \varepsilon _{ {\bf k} + {\bf Q} }$ are the energy coefficients, $ \Delta _{ {\bf k} }$ is the DDW order parameter, and $ c _{ {\bf k} \text{ } \sigma }$ and $ c _{ {\bf k}+{\bf Q} \text{ } \sigma }$ are to annihilate electrons at $({\bf k} , \sigma )$ and $({\bf k} +{\bf Q}, \sigma )$, respectively. At half-filling, the zero-temperature chemical potential may lie in the gap between the two DDW energy bands, under which the gapless quasielectrons appear at the nodal points [@Gerami; @Chakravarty]. The lower band is filled with the AF quasielectrons while the upper one is empty. Each filled eigenket $\varphi_{f}$ can be denoted by the two-component wavefunction $$\begin{aligned} | \varphi _{ f } \rangle = \left[ \begin{array}{c} \varphi ^{\prime} _{ f } \\ \varphi ^{\prime \prime} _{ f } \end{array} \right]\end{aligned}$$ with the plane waves $\varphi ^{\prime} _{ f } = a _{f} ^{\prime} exp (i {\bf k}_{f} \cdot {\bf r})$ and $\varphi ^{\prime \prime} _{ f } = a _{f} ^{ \prime \prime} exp (i ({\bf k}_{f} + {\bf Q} ) \cdot {\bf r})$. Here ${\bf k}_{f} \in $ RBZ, and the coefficients $ a _{f} ^{\prime}$ and $ a _{f} ^{ \prime \prime}$ satisfying $( a _{f} ^{ \prime \ast }$, $ -i a _{f} ^{ \prime \prime \ast} ) B _{ {\bf k} } \propto ( a _{f} ^{ \prime \ast}$, $ -i a _{f} ^{ \prime \prime \ast} )$ are normalized such that $\langle \varphi _{f} | \varphi _{f} \rangle =1$. We may exchange the components in Eq. (5) to obtain $$\begin{aligned} T | \varphi _{ f } \rangle = \left[ \begin{array}{c} \varphi ^{\prime \prime} _{ f } \\ \varphi ^{\prime} _{ f } \end{array} \right],\end{aligned}$$ which also represents the ket $\varphi _{ f }$, by the operator $$\begin{aligned} T = \left[ \begin{array}{c} 0 \text{ \ \ } I \\ I \text{ \ \ } 0 \end{array} \right]. \end{aligned}$$ The extended AF density matrix $ \sum ( | \varphi _{ f } \rangle \langle \varphi _{ f } | + T | \varphi _{ f } \rangle \langle \varphi _{ f } | T ^{\dag} ) $ is of the form of $\rho _{ea}$ given by Eq. (2) because it commutes with $T$. We can decompose $\rho _{ea}$ as $$\begin{aligned} \rho _{ea} = \rho ^{(1)} \otimes e _{1} e _{1} ^{\dag} + \rho ^{(2)} \otimes e _{2} e _{2} ^{\dag}. \end{aligned}$$ such that $\rho ^{(1)} = \rho + \Delta = e _{1} ^{\dag} \rho _{ea} e _{1} $ and $\rho ^{(2)} = \rho - \Delta = e _{2} ^{\dag} \rho _{ea} e _{2} $, where $e _{1} ^{\dag} = \frac{1}{ \sqrt{2} } (1,1) $ and $e _{2} ^{\dag} = \frac{1}{ \sqrt{2} } (1,-1) $. The effective Hamiltonian $\hat{H_{D}}$ given by Eq. (49) in Ref. [@Huang1] can be obtained by generalizing $H _{ddw}$ for the extended AF density matrix. In addition to the orbital antiferromagnetism, the order parameter $\Delta _{ {\bf k} }$ is taken into account for d-wave superconductivity [@Chakravarty; @Bena; @Kee]. Moreover, we may use $\rho ^{(1)}$ and $\rho ^{(2)}$ to represent the quasielectrons in the one-bond system [@Huang1]. The 4-level dimer composed of the two spatial sites with up- and down-spin orientations has been introduced by considering the dimer Hamiltonian $H _{t-U} = \sum _{\sigma} (t _{AB} c _{A \sigma} ^{\dag} c _{B \sigma} +t _{AB} ^{*} c _{B \sigma} ^{\dag} c _{A \sigma}) + \hat{U}$ [@Matlak1]. Here $A$ and $B$ represent the two spatial sites, $t _{AB}$ is the hopping coefficient, the annihilators $c _{A \sigma}$ and $c _{B \sigma}$ follow $\{ c _{B \sigma} ,c _{B \sigma ^{\prime} } \} = \{ c _{A \sigma} ,c _{A \sigma ^{\prime} } \} = \{ c _{A \sigma} ,c _{B \sigma ^{\prime} } \} = \{ c _{A \sigma} ,c _{B \sigma ^{\prime} } ^{\dag} \} =0$ and $\{ c _{A \sigma} ,c _{A \sigma ^{\prime} } ^{\dag} \} = \{ c _{B \sigma} ,c _{B \sigma ^{\prime} } ^{\dag} \} = \delta _{ \sigma \sigma ^{\prime} } $, the factor $\sum _{\sigma} (t _{AB} c _{A \sigma} ^{\dag} c _{B \sigma} +t _{AB} ^{*} c _{B \sigma} ^{\dag} c _{A \sigma})$ is responsible for the intra-bond hopping, and $\hat{U}$ denotes effective e-e interaction potential. To model the AF states, we shall note that such states occur as the up- and down-spin electrons repel each other when they are at the same sites. We can take $$\begin{aligned} \hat{U} = \int d ^{3} r ^{\prime} d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{ \prime} - {\bf r} ^{ \prime \prime} | ) \psi _{ \uparrow } ^{\dag} ({\bf r} ^{\prime}) \psi _{ \uparrow } ( {\bf r} ^{\prime} ) \psi _{ \downarrow } ^{\dag} ({\bf r} ^{ \prime \prime}) \psi _{ \downarrow } ({\bf r} ^{ \prime \prime})\end{aligned}$$ as the short-range repulsive potential to reduce the double occupancy [@Lee1] such that the electrons prefer $c _{A \uparrow} ^{\dag} c _{B \downarrow} ^{\dag} | 0 \rangle $ and $c _{B \uparrow} ^{\dag} c _{A \downarrow} ^{\dag} | 0 \rangle $ at half filling. Here $| 0 \rangle $ denotes the vacuum state, the nonnegative function $U(|{\bf r}|)$ equals zero as $|{\bf r}|$ exceeds the short interacting length $a$, and $ \psi _{ \sigma } ( {\bf r} )$ is to annihilate the electron with the spin orientation $\sigma$ at position ${\bf r}$. Such a dimer can be used to model the ionic-covalent chemical bond, which separates atom $A$ from atom $B$ in Fig. 1 (a). Assume that the bonding electrons almost concentrate in $\Omega$ such that we can consider the approximation $ \langle {\bf r} | A \rangle = \langle {\bf r} | B \rangle =0$ as $ {\bf r} \notin \Omega$, where the $A$- and $B$-site orbitals $| A \rangle$ and $|B \rangle$ denote the normalized spatial parts of $c _{A \sigma} ^{\dag} | 0 \rangle$ and $c _{ B \sigma} ^{\dag} | 0 \rangle$, respectively. In this manuscript, I assume that $| A \rangle$ and $|B \rangle$ are localized near the corresponding atoms in the considered bond, as shown in Fig. 4. We can construct these site orbitals by considering the linear combinations of the atomic orbitals after truncating the tails outside the bonding region $\Omega$ and performing suitable orthogonalization to have the zero overlap integral [@Levine]. Let $ \mathbb{C} ^{2} _{\Omega} $ be the vector space composed of the linear combinations of $| A \rangle$ and $|B \rangle$. The space $ \mathbb{C} ^{2} _{\Omega} $ is isomorphic to $ \mathbb{C} ^{2} $, and is a subspace of the Hilbert space $ L^{2} _{\Omega} $ composed of the square-integrable functions which equal zero outside $\Omega$. The Heitler-London correlated state $| \Psi _{c} \rangle = \frac{1}{ \sqrt{2} } ( c_{A \uparrow} ^{\dag} c_{B \downarrow} ^{\dag} + c_{B \uparrow} ^{\dag} c_{A \downarrow} ^{\dag}) | 0 \rangle $, a linear combination of the AF states $c_{A \uparrow} ^{\dag} c_{B \downarrow} ^{\dag} | 0 \rangle $ and $c_{B \uparrow} ^{\dag} c_{A \downarrow} ^{\dag} | 0 \rangle $, has been taken into account in the covalent limit [@Fulde; @Soos] when the 4-level bond is half-filled. To include the ionic part, we may consider the ionic Hubbard model [@Kampf; @Buzatu] and modify $H _{t-U}$ as the one-bond Hamiltonian $$\begin{aligned} H _{b} = \sum _{\sigma} (t _{AB} c _{A \sigma} ^{\dag} c _{B \sigma} + t _{AB} ^{*} c _{B \sigma} ^{\dag} c _{A \sigma} + \varepsilon _{A} c _{A \sigma} ^{\dag} c _{A \sigma} + \varepsilon _{B} c _{B \sigma} ^{\dag} c _{B \sigma}) + \hat{U},\end{aligned}$$ which is suitable to model the polar molecule [@Prasad], to introduce the on-site energies $\varepsilon _{A}$ and $\varepsilon _{B}$. In the above equation, $\sum _{\sigma} (t _{AB} c _{A \sigma} ^{\dag} c _{B \sigma} + t _{AB} ^{*} c _{B \sigma} ^{\dag} c _{A \sigma} + \varepsilon _{A} c _{A \sigma} ^{\dag} c _{A \sigma} + \varepsilon _{B} c _{B \sigma} ^{\dag} c _{B \sigma}) = H _{b} - \hat{U} $ is the non-interacting part of $H _{b}$ while the e-e potential $\hat{U}$ serves as the interacting part. In the following, assume that $\varepsilon _{B} > \varepsilon _{A}$ such that atom $A$ has the higher electronegativity. The two-electron wavefunction becomes the uncorrelated state $ | \Psi _{i} \rangle = c _{A \uparrow} ^{\dag} c _{A \downarrow} ^{\dag} | 0 \rangle$ in the ionic limit, so we shall take both $ | \Psi _{i} \rangle$ and $ | \Psi _{c} \rangle$ in general and consider the bonding wavefunction given by Eq. (1) at half filling. The assumptions about Eq. (10) are discussed in Appendix A. While there exists another state $ | \Psi _{BB} \rangle = c ^{\dagger} _{B \uparrow} c ^{\dagger} _{B \downarrow} |0 \rangle$, as mentioned in Appendix B, its contribution is small and we can include it just by performing an orbital transformation. For convenience, I introduce my model without considering $ | \Psi _{BB} \rangle$ in the main text, and include its contribution in Appendix B by using such a transformation to preserve the form of Eq. (1). Because the bonding electrons in the covalent limit are described by the linear combination of the AF states $ c_{A \uparrow} ^{\dag} c_{B \downarrow} ^{\dag} | 0 \rangle$ and $ c_{B \uparrow} ^{\dag} c_{A \downarrow} ^{\dag} | 0 \rangle$, it is natural to try the extended AF density matrix $\rho _{ea}$ to construct the quasielectron orbitals of the one-bond system in Fig. 1 (a). In each AF state one electron is located at $ | A \rangle$ while the other one is located at $ |B \rangle$, so we shall take the matrices $\rho ^{(1)}$ and $\rho ^{(2)}$ in Eq. (8) as $| A \rangle \langle A |$ and $| B \rangle \langle B |$ in the covalent limit. On the other hand, both electrons occupy $ | A \rangle$ in the ionic limit and thus these two matrices should equal $| A \rangle \langle A |$ as $| \Psi _{i} \rangle$ is dominated. By taking [@Huang1] $$\begin{aligned} \rho ^{(1)} = | A \rangle \langle A | \text{ and } \rho ^{(2)} = | L \rangle \langle L | \end{aligned}$$ with $ | L \rangle = \alpha _ {i} | A \rangle + \alpha _{c} | B \rangle$ in $\mathbb{C} ^{2} _{ \Omega }$, the matrix $\rho ^{(1)}$ just represents the quasielectron occupying $| A \rangle$ under both limits. In addition, the matrix $\rho ^{(2)}$ corresponds to the other one jumping to $| A \rangle$ from $| B \rangle$ as $| \Psi _{i} \rangle$ becomes significant. The operator $\rho _{sb} = \frac{1}{2} (\rho ^{(1)} + \rho ^{(2)} + d ^{(1)} )$ satisfies $\langle {\bf r} ^{\prime} | \rho _{sb} | {\bf r} ^{ \prime \prime } \rangle = \langle \Psi _{b} | \psi _{ \uparrow } ^{ \dagger } ({\bf r} ^{ \prime } ) \psi _{ \uparrow } ({\bf r} ^{\prime \prime} ) | \Psi _{b} \rangle = \langle \Psi _{b} | \psi _{ \downarrow } ^{ \dagger } ({\bf r} ^{ \prime } ) \psi _{ \downarrow } ({\bf r} ^{\prime \prime} ) | \Psi _{b} \rangle $ and thus serves as the one-electron density matrix, which is spin-degenerate because $\langle \Psi _{b} | \psi _{ \sigma } ^{ \dagger } ({\bf r} ^{\prime}) \psi _{ - \sigma } ({\bf r}^{\prime \prime} ) | \Psi _{b} \rangle=0$ for $\sigma = \uparrow$ and $\downarrow$. Quasielectrons in ionic-covalent bonds ====================================== In this section, the energy forms are constructed for the quasielectrons in the half-filled bonding systems. For convenience, I discuss the one-bond system based on the 4-level dimer [@Matlak1] in subsection III-A, and extend the results to the compound system composed of identical dimers [@Matlak2] in subsection III-B. III-A One-bond system {#iii-a-one-bond-system .unnumbered} ===================== Consider the two uncorrelated states $| \Phi _{b} \rangle = c _{A \uparrow} ^{\dag} c _{ L \downarrow} ^{\dag} | 0 \rangle = \alpha _{i} | \Psi _{i} \rangle + \alpha _{c} c _{ A \uparrow} ^{\dag} c _{B \downarrow} ^{\dag} | 0 \rangle $ and $| \Phi _{b} ^{\prime} \rangle = c _{L \uparrow} ^{\dag} c _{ A \downarrow} ^{\dag} | 0 \rangle = \alpha _{i} | \Psi _{i} \rangle + \alpha _{c} c _{ B \uparrow} ^{\dag} c _{A \downarrow} ^{\dag} | 0 \rangle $, which can be obtained from $| \Psi _{b} \rangle$ by substituting the AF states $ c _{ A \uparrow} ^{\dag} c _{B \downarrow} ^{\dag} | 0 \rangle $ and $c _{ B \uparrow} ^{\dag} c _{A \downarrow} ^{\dag} | 0 \rangle$, respectively, for the covalent component $| \Psi _{c} \rangle$ in Eq. (1). Here $c _{L \sigma}$ is the operator to annihilate the electron with the spin orientation $\sigma$ at $| L \rangle$ in the bonding region $\Omega$ in Fig. 1 (a). The matrices $\rho ^{(1)}$ and $\rho ^{(2)}$ in Eq. (11) correspond to the up- and down-spin electrons in $| \Phi _{b} \rangle$, and correspond to the down- and up-spin ones in $ | \Phi _{b} ^{\prime} \rangle$. So $\rho ^{(1)}$ and $\rho ^{(2)}$ are for the quasielectrons with the opposite spin orientations in the one-bond system, and can represent the occupied levels of the half-filled 4-level dimer in Fig. 2 (a) if $| 1 \rangle \otimes | \sigma \rangle = |A \rangle \otimes | \sigma \rangle $ and $| 2 \rangle \otimes | - \sigma \rangle = | L \rangle \otimes | - \sigma \rangle $. The two unoccupied levels $| \bar{1} \rangle \otimes | \sigma \rangle$ and $| \bar{2} \rangle \otimes | - \sigma \rangle$ in Fig. 2(a) serve as $ | B \rangle \otimes | \sigma \rangle$ and $| \bar{L} \rangle \otimes | - \sigma \rangle$ in such a one-bond system, where $| \bar{L} \rangle = \alpha _{c} ^{*} | A \rangle - \alpha _{i} ^{*} | B \rangle \bot | L \rangle$ is a ket in $\mathbb{C} ^{2} _{\Omega}$. The uncorrelated energy $\langle \Phi _{b} | H _{b} | \Phi _{b} \rangle = \langle \Phi ^{\prime} _{b} | H _{b} | \Phi _{b} ^{\prime} \rangle$ equals $$\begin{aligned} tr (\rho ^{(1)} + \rho ^{(2)}) H_{sb} + \int _{ {\bf r} ^{ \prime },{\bf r} ^{ \prime \prime} \in \Omega} d ^{3} r ^{\prime} d ^{3} r ^{ \prime \prime } U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime} \rangle \langle {\bf r} ^{\prime \prime} | \rho ^{(2)} | {\bf r} ^{\prime \prime} \rangle,\end{aligned}$$ where $H_{sb} = t _{AB} | A \rangle \langle B | + t _{AB} ^{*} | B \rangle \langle A | + \varepsilon _{A} | A \rangle \langle A | + \varepsilon _{B} | B \rangle \langle B |$ results from the non-interacting part of $H_{b}$. The factors $t _{AB} | B \rangle \langle A | + t _{AB} ^{*} | A \rangle \langle B |$ and $\varepsilon _{A} | A \rangle \langle A | + \varepsilon _{B} | B \rangle \langle B |$ of $H _{sb}$ are responsible for the intra-bond hopping and on-site energy difference, respectively. While we may construct the qausielectron density matrices by Eq. (11), the wavefunction $| \Psi _{b} \rangle$ in Eq. (1) is different from the uncorrelated functions $| \Phi _{b} \rangle$ and $| \Phi _{b} ^{\prime} \rangle$ when $ \alpha _{c} \neq 0$. To obtain the bonding energy $E _{b} = \langle \Psi _{b} | H _{b} | \Psi _{b} \rangle$, therefore, we shall include the correlation contributions corresponding to the blue dash curve in Fig. 2 (a) by calculating the difference $\langle \Psi _{b} | H _{b} | \Psi _{b} \rangle - \langle \Phi _{b} | H _{b} | \Phi _{b} \rangle = \langle \Psi _{b} | H _{b} | \Psi _{b} \rangle - \langle \Phi ^{\prime} _{b} | H _{b} | \Phi _{b} ^{\prime} \rangle$. The non-interacting part of $H_{b}$ induces the correlation energy $ tr H_{sb} d ^{(1)} = \langle \Psi _{b} | (H _{b} - \hat{U}) | \Psi _{b} \rangle - \langle \Phi _{b} | (H _{b} - \hat{U}) | \Phi _{b} \rangle = \langle \Psi _{b} | (H _{b} - \hat{U}) | \Psi _{b} \rangle - \langle \Phi ^{\prime} _{b} | (H _{b} - \hat{U}) | \Phi ^{\prime} _{b} \rangle $. We can take $I = | A \rangle \langle A| + | B \rangle \langle B | $, which is the identity operator on $\mathbb{C} ^{2} _{\Omega}$, in Eq. (3) to introduce the density-fluctuation operator $ d ^{(1)} $. The operator $d ^{(1)}$ results from the factor $\alpha _{i} \alpha _{c} ^{*} \langle \Psi _{c} | (H _{b} - \hat{U}) | \Psi _{i} \rangle + \alpha _{c} \alpha _{i} ^{*} \langle \Psi _{i} | (H _{b} - \hat{U}) | \Psi _{c} \rangle $ when we calculate $\langle \Psi _{b} | (H _{b} - \hat{U}) | \Psi _{b} \rangle $, and becomes zero if the ionic and covalent components do not coexist. The up- or down-spin density $\langle {\bf r} | \rho _{sb} | {\bf r} \rangle $ at position ${\bf r}$ includes the factor $\langle {\bf r} | d ^{(1)} | {\bf r} \rangle$, which follows $\int _{ {\bf r} \in \Omega} d ^{3} r \langle {\bf r} | d ^{(1)} | {\bf r} \rangle=tr d ^{(1)} = ( \sqrt{2} -1) tr [ ( I - \rho ^{(1)}) \rho ^{(1)} \rho ^{(2)} + \rho ^{(2)} \rho ^{(1)} ( I - \rho ^{(1)}) ]=0 $ because $( I - \rho ^{(1)}) \rho ^{(1)}= \rho ^{(1)} ( I - \rho ^{(1)})=0$ under Eq. (11). Therefore, $\langle {\bf r} | d ^{(1)} | {\bf r} \rangle$ has no contribution to the total charge $2 \int _{ {\bf r} \in \Omega} d ^{3} r \langle {\bf r} | \rho _{sb} | {\bf r} \rangle $, and represents the fluctuating charge density due to the coexistence of the ionic and covalent components. I note that $\delta \rho$, the deviation from the average density [@Zhang] due to the correlation in the quantum Hall effect [@DHLee; @You], also has no contribution to the total charge. To include the correlation energy due to $\hat{U}$ in the quasielectron space, we can further introduce $d ^{(2)}$ to rewrite $ \langle \Psi _{b} | \hat{U} | \Psi _{b} \rangle - \langle \Phi _{b} | \hat{U} | \Phi _{b} \rangle$ or $\langle \Psi _{b} | \hat{U} | \Psi _{b} \rangle - \langle \Phi ^{\prime} _{b} | \hat{U} | \Phi _{b} ^{\prime} \rangle$ as $\int _{ {\bf r} ^{\prime},{\bf r} ^{\prime \prime} \in \Omega} d ^{3} r ^{\prime} d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) [ \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime \prime } \rangle \langle {\bf r} ^{\prime \prime} | d ^{(2)} | {\bf r} ^{ \prime } \rangle + \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime } \rangle \langle {\bf r} ^{\prime \prime} | d ^{(1)} | {\bf r} ^{\prime \prime } \rangle] $. We can take $I= |A\rangle \langle A| + | B \rangle \langle B| $ in Eq. (3) to introduce $d ^{(2)}$ on $\mathbb{C} ^{2} _{ \Omega} $. The operator $d ^{(2)}$ comes from the factor $ | \alpha _{c} | ^{2} \langle \Psi _{c} | \hat{U} | \Psi _{c} \rangle$ when we calculate $\langle \Psi _{b} | \hat{U} | \Psi _{b} \rangle$, and becomes zero when $\alpha _{c} =0$. Hence $d ^{(2)}$ represents the covalent correlation, and we shall include both $d ^{(1)}$ and $d ^{(2)}$ for the blue dash curve in Fig. 2 (a). By including the correlation contributions, we have $$\begin{aligned} E_{b} = tr (\rho ^{(1)} + \rho ^{(2)}) H_{sb} + \int _{ {\bf r} ^{ \prime },{\bf r} ^{ \prime \prime } \in \Omega} d ^{3} r ^{ \prime } d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime} \rangle \langle {\bf r} ^{\prime \prime} | \rho ^{(2)} | {\bf r} ^{\prime \prime} \rangle +\end{aligned}$$ $$tr H_{sb} d ^{(1)} + \int _{ {\bf r} ^{\prime},{\bf r} ^{\prime \prime} \in \Omega} d ^{3} r ^{\prime} d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime} \rangle \langle {\bf r} ^{\prime \prime} | d ^{(1)} | {\bf r} ^{\prime \prime} \rangle + \text{ \ \ \ \ \ }$$ $$\int _{ {\bf r} ^{ \prime },{\bf r} ^{\prime \prime} \in \Omega} d ^{3} r ^{\prime} d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime \prime} \rangle \langle {\bf r} ^{\prime \prime} | d ^{(2)} | {\bf r} ^{\prime} \rangle \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$$ from Eq. (12) when the ionic-covalent chemical bond in Fig. 1 (a) is half-filled. In the above equation, the factors in the second and third lines are due to the bonding correlation. The above equation provides the energy form of the density matrix $\rho _{ea}$, which can be decomposed into $\rho ^{(1)} = e _{1} ^{\dag} \rho _{ea} e _{1}$ and $\rho ^{(2)} = e _{2} ^{\dag} \rho _{ea} e _{2}$, for the one-bond system. III-B Compound system {#iii-b-compound-system .unnumbered} ===================== Consider the binary compound where the anions and cations are connected by the identical ionic-covalent bonds, and assume that all the bonds are well-separated without overlap. Figure 1 (b) shows such bonds in the zincblende-structure crystal [@Grosso], in which each atom provides 4 site orbitals, as an example. For convenience, I parameterize these bonds by the integer parameter $j=1 \sim N$ and denote the bonding region of the $j$-th bond as $\Omega _{j}$, where $N$ is the total number of the bonds. Each bond in the compound is a 4-level dimer just as the one-bond system in Fig. 1 (a). Assume that there exists the one-to-one mapping ${\bf R} _{j}$ to relate any position ${\bf r} _{j} \in \Omega _{j}$ in the $j$-th bond to ${\bf r} \in \Omega$ in Fig. 1 (a) by ${\bf R} _{j} ({\bf r}) = {\bf r} _{j}$ and ${\bf R} _{j} ^{-1} ({\bf r} _{j}) = {\bf r}$ such that the distance $|{\bf r}_{1}-{\bf r} _{2}|$ between ${\bf r}_{1}$ and ${\bf r}_{2} \in \Omega$ equals $|{\bf R} _{j} ( {\bf r} _{1} )- {\bf R} _{j}({\bf r} _{2})| $ for all $j$. Therefore, every ionic-covalent chemical bond in the compound is identical to that in Fig. 1 (a). Let $| A, j \rangle $ and $| B, j \rangle $ as the kets mapped from $| A \rangle $ and $| B \rangle $ under ${\bf R}_{j}$, respectively. The space $\mathbb{C} ^{2} _{ \Omega _{j} }$ spanned by $| A, j \rangle $ and $| B, j \rangle $ is a subspace of the Hilbert space $L^{2} _{ \Omega _{j} }$ composed of square-integrable functions which equal zero outside $\Omega _{j}$, and any two kets in $L^{2} _{ \Omega _{l} }$ and $L^{2} _{ \Omega _{p} }$ are orthogonal to each other when $l \neq p$. We can introduce the space $\mathbb{C} ^{N} \otimes \mathbb{C} ^{2} _{\Omega} = \mathbb{C} ^{2} _{\Omega _{1} } \oplus \mathbb{C} ^{2} _{\Omega _{2} } \oplus \textellipsis \oplus \mathbb{C} ^{2} _{\Omega _{N} }$ for the compound, and choose a set of orthonormal basis $\{ | w _{j} \rangle \}$ in $\mathbb{C} ^{N}$ to represent $| A, j \rangle $ and $| B, j \rangle $ by $|w _{j} \rangle \otimes | A \rangle $ and $|w _{j} \rangle \otimes | B \rangle $, respectively. The space $\mathbb{C} ^{N} \otimes \mathbb{C} ^{2} _{\Omega}$ is a subspace of $\mathbb{C} ^{N} \otimes L ^{2} _{\Omega} = L ^{2} _{\Omega _{1} } \oplus L ^{2} _{\Omega _{2} } \oplus \textellipsis \oplus L ^{2} _{\Omega _{N} }$, and the position ket $| {\bf r} _{j} \rangle$ corresponding to the position ${\bf r} _{j} \in \Omega _{j} $ is taken as $|w _{j} \rangle \otimes | {\bf R} _{j} ^{-1} ({\bf r} _{j}) \rangle $ to perform the integral in $\mathbb{C} ^{N} \otimes L ^{2} _{\Omega}$. For any function $F _{\Omega}$ defined on $\Omega$ and the position ${\bf r} _{j _{1}} \in {\Omega}_{j _{1}} $, we have $(\langle {\bf r} _{j _{1} } |)(| w _{ j _{2} } \rangle \otimes | F _{\Omega} \rangle) = ( \langle w _{ j _{1} } | \otimes \langle {\bf R} _{ j _{1} } ^{ -1 } ( {\bf r} _{ j _{1} } ) | )( | w _{ j _{2} } \rangle \otimes | F _{\Omega} \rangle ) = \delta _{ j _{1} , j _{2} } \langle {\bf R} _{ j _{1} } ^{ -1 } ( {\bf r} _{ j _{1} } ) | F _{\Omega} \rangle = \delta _{ j _{1} , j _{2} } F _{\Omega} ({\bf R} _{ j _{1} } ^{ -1 } ( {\bf r} _{ j _{1} } ) )$. By mapping the one-bond system in Fig. 1 (a) to the identical bonds in the compound, at half filling we can transfer $ \rho ^{(1)}$ and $ \rho ^{(2)}$ in Eq. (11) to the $j$-th bond and obtain $ \rho _{ w _{j} } ^{(1)} = | w _{j} \rangle \langle w _{j} | \otimes |A \rangle \langle A |$ and $\rho _{ w _{j} } ^{(2)} = | w _{j} \rangle \langle w _{j} | \otimes |L \rangle \langle L|$ for the quasielectrons at $ | A, j \rangle$ and $ | L, j \rangle $. Here $ | L, j \rangle = \alpha _{i} | A, j \rangle + \alpha _{c} | B, j \rangle$. The matrices $$\begin{aligned} \rho ^{( \text{I} )} = \sum _{j=1} ^{N} \rho _{ w _{j} } ^{(1)} \text{ \ \ and \ \ } \rho ^{( \text{II} )} = \sum _{j=1} ^{N} \rho _{ w _{j} } ^{(2)}\end{aligned}$$ for the half-filled compound are of the opposite spin orientations just as $ \rho ^{(1)}$ and $ \rho ^{(2)}$ in the one-bond system, and the corresponding extended AF density matrix is $\rho ^{( \text{I} )} e _{1} e _{1} ^{\dag} + \rho ^{( \text{II} )} e _{2} e _{2} ^{\dag}$. In fact, we can take $I _{ \mathbb{C} ^{N} } $ as the identity operator on $\mathbb{C} ^{N}$ and rewrite Eq. (14) by $\rho ^{( \text{I} )} = I _{ \mathbb{C} ^{N} } \otimes \rho ^{(1)} $ and $\rho ^{( \text{II} )} = I _{ \mathbb{C} ^{N} } \otimes \rho ^{(2)} $ to see that $\rho ^{( \text{I} )}$ and $\rho ^{( \text{II} )}$ are the natural extensions of $ \rho ^{(1)}$ and $ \rho ^{(2)}$, respectively. To include the bonding correlation due to the $j$-th bond, we shall substitute $ d _{ w _{j} } ^{(1)} = | w _{j} \rangle \langle w _{j} | \otimes d ^{(1)}$ and $ d _{ w _{j} } ^{(2)} = | w _{j} \rangle \langle w _{j} | \otimes d ^{(2)}$ for the operators in Eq. (3). The correlation operators $d ^{( \text{I} )}= \sum _{j} d _{ w _{j} } ^{(1)} = I _{ \mathbb{C} ^{N} } \otimes d ^{(1)}$ and $d ^{( \text{II} )}= \sum _{j} d _{ w _{j} } ^{(2)} = I _{ \mathbb{C} ^{N} } \otimes d ^{(2)}$ satisfy $$\begin{aligned} \begin{cases} d ^{( \text{I} )} = ( \sqrt{2} -1) [ \rho ^{( \text{I} )} \rho ^{( \text{II} )} ( I - \rho ^{( \text{I} )}) + ( I - \rho ^{( \text{I} )}) \rho ^{( \text{II} )} \rho ^{( \text{I} )} ] \\ d ^{( \text{II} )} = ( I - \rho ^{( \text{I} )}) \rho ^{( \text{II} )} ( I - \rho ^{( \text{I} )}) \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \end{cases},\end{aligned}$$ and the matrix $\rho _{sC} = \frac{1}{2}(\rho ^{( \text{I} )} + \rho ^{( \text{II} )} +d ^{( \text{I} )} ) = I _{ \mathbb{C} ^{N} } \otimes \rho _{sb}$ yields the density $Q({\bf r} _{j} ) = 2 \langle {\bf r} _{j} | \rho _{sC} | {\bf r} _{j} \rangle = 2 \langle {\bf R} _{j} ^{-1} ({\bf r} _{j}) | \rho _{sb} | {\bf R} _{j} ^{-1} ({\bf r} _{j}) \rangle $ at ${\bf r} _{j} \in \Omega _{j}$. In the above equation, the matrix $I$ denotes the identity operator on $\mathbb{C} ^{N} \otimes \mathbb{C} ^{2} _{\Omega} $, and we can interpret $d ^{( \text{I} )}$ and $d ^{( \text{II} )}$ as the density-fluctuation and covalent-correlation operators of the quasielectron system in the compound because they serve as $d ^{(1)}$ and $d ^{(2)}$. For the ideal crystal, we may discuss the correlation under the crystal symmetry imposed on $ \rho ^{( \text{I} )}$ and $ \rho ^{( \text{II} )}$ based on Eq. (15). When the distances between different bonds are larger than $a$, the interacting length of $\hat{U}$, there is no inter-bond e-e interaction in the half-filled compound. Therefore, the two quasielectrons in a specific bond only interact with each other just as those in the one-bond system in Fig. 1 (a), and the e-e energy term is composed of $$\text{(i) \ \ } \sum _{j=1} ^{N} \int _{ {\bf r} ^{\prime} _{ j },{\bf r} ^{\prime \prime} _{ j } \in \Omega _{j} } d ^{3} r ^{\prime} _{ j } d ^{3} r ^{\prime \prime} _{ j } U ( | {\bf r} ^{\prime} _{ j } - {\bf r} ^{\prime \prime} _{ j } | ) \langle {\bf r} ^{\prime} _{ j } | \rho ^{( \text{I} )} | {\bf r} ^{\prime} _{ j } \rangle \langle {\bf r} ^{\prime \prime} _{ j } | \rho ^{( \text{II} )} | {\bf r} ^{\prime \prime} _{ j } \rangle$$ $$\begin{aligned} \text{(ii) \ } \sum _{j=1} ^{N} \int _{ {\bf r} ^{\prime} _{ j },{\bf r} ^{\prime \prime} _{ j } \in \Omega _{j} } d ^{3} r ^{\prime} _{ j } d ^{3} r ^{\prime \prime} _{ j } U ( | {\bf r} ^{\prime} _{ j } - {\bf r} ^{\prime \prime} _{ j } | ) \langle {\bf r} ^{\prime} _{ j } | \rho ^{( \text{I} )} | {\bf r} ^{\prime} _{ j } \rangle \langle {\bf r} ^{\prime \prime} _{ j } | d ^{( \text{I} )} |{\bf r} ^{\prime \prime} _{ j } \rangle \end{aligned}$$ $$\text{(iii) } \sum _{j=1} ^{N} \int _{ {\bf r} ^{\prime} _{ j },{\bf r} ^{\prime \prime} _{ j } \in \Omega _{j} } d ^{3} r ^{\prime} _{ j } d ^{3} r ^{\prime \prime} _{ j } U ( | {\bf r} ^{\prime} _{ j } - {\bf r} ^{\prime \prime} _{ j } | ) \langle {\bf r} ^{\prime} _{ j } | \rho ^{( \text{I} )} | {\bf r} ^{\prime \prime } _{ j } \rangle \langle {\bf r} ^{\prime \prime} _{ j } | d ^{( \text{II} )} | {\bf r} ^{\prime} _{ j } \rangle. \text{ \ }$$ Equation (16)-(i) corresponds to the second term in Eq. (12) and can be obtained without considering the bonding correlation. On the other hand, Eqs. (16)-(ii) and (16)-(iii) correspond to the last two terms in Eq. (13) and provide the correlation contributions. To include the energy resulting from the non-interacting term $H _{sb}$, we shall consider the factor $2 tr \rho _{sC} ( \sum _{j=1} ^{N} | w _{j} \rangle \langle w _{j} | \otimes H_{sb} ) = 2 tr \rho _{sb} H _{sb} \times N$ in the compound system. In addition to the intra-bond hopping in $H _{sb}$, the inter-bond hopping $$\begin{aligned} H _{hop} = \sum _{ j \neq j ^{\prime} } t _{ j \xi , j ^{\prime} \xi ^{\prime} } | w _{ j } \rangle \langle w _{j ^{\prime} } | \otimes | \xi \rangle \langle \xi ^{\prime} |\end{aligned}$$ should be taken into account to relate different chemical bonds [@Matlak2]. Here $ | \xi \rangle \text{ and } | \xi ^{\prime} \rangle \in \{ | A \rangle , |B \rangle \} $, and each coefficient $t _{ j \xi , j ^{\prime} \xi ^{\prime} } = t^{*} _{j ^{\prime} \xi ^{\prime}, j \xi } $ is for the jump from $| w _{ j ^{\prime} } \rangle \otimes | \xi ^{\prime} \rangle$ to $| w _{ j } \rangle \otimes | \xi \rangle$. Therefore, we shall introduce the non-interacting Hamiltonian $H _{sC} = H _{hop}+ \sum _{j=1} ^{N} | w _{j} \rangle \langle w _{j} | \otimes H_{sb} $ and include the energy $2 tr \rho _{sC} H _{sC}$. In this manuscript I consider the short-range hopping, so $t _{ j \xi , j ^{\prime} \xi ^{\prime} }=0$ if the distance between the $j$-th and $j^{\prime}$-th bonds is longer than a specific length. The energy for the half-filled compound system is $$E_{Cr} = tr ( \rho ^{ ( \text{I} ) } + \rho ^{ ( \text{II} ) } ) H _{sC} + \sum _{j=1} ^{N} \int _{ {\bf r} ^{\prime} _{ j },{\bf r} ^{\prime \prime} _{ j } \in \Omega _{j} } d ^{3} r ^{\prime} _{ j } d ^{3} r ^{\prime \prime} _{ j } U ( | {\bf r} ^{\prime} _{ j } - {\bf r} ^{\prime \prime} _{ j } | ) \langle {\bf r} ^{\prime} _{ j } | \rho ^{( \text{I} )} | {\bf r} ^{\prime} _{ j } \rangle \langle {\bf r} ^{\prime \prime} _{ j } | \rho ^{( \text{II} )} | {\bf r} ^{\prime \prime} _{ j } \rangle +$$ $$\begin{aligned} trH _{sC} d^{ ( \text{I} )} + \sum _{j=1} ^{N} \int _{ {\bf r} ^{\prime} _{ j },{\bf r} ^{\prime \prime} _{ j } \in \Omega _{j} } d ^{3} r ^{\prime} _{ j } d ^{3} r ^{\prime \prime} _{ j } U ( | {\bf r} ^{\prime} _{ j } - {\bf r} ^{\prime \prime} _{ j } | ) \langle {\bf r} ^{\prime} _{ j } | \rho ^{( \text{I} )} | {\bf r} ^{\prime} _{ j } \rangle \langle {\bf r} ^{\prime \prime} _{ j } | d ^{( \text{I} )} |{\bf r} ^{\prime \prime} _{ j } \rangle +\end{aligned}$$ $$\sum _{j=1} ^{N} \int _{ {\bf r} ^{\prime} _{ j },{\bf r} ^{\prime \prime} _{ j } \in \Omega _{j} } d ^{3} r ^{\prime} _{ j } d ^{3} r ^{\prime \prime} _{ j } U ( | {\bf r} ^{\prime} _{ j } - {\bf r} ^{\prime \prime} _{ j } | ) \langle {\bf r} ^{\prime} _{ j } | \rho ^{( \text{I} )} | {\bf r} ^{\prime \prime } _{ j } \rangle \langle {\bf r} ^{\prime \prime} _{ j } | d ^{( \text{II} )} | {\bf r} ^{\prime} _{ j } \rangle. \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$$ In addition to $\{ | w _{j} \rangle \}$, we can choose another orthonormal complete set $ \{ | \eta _{j} \rangle \}$ in $ \mathbb{C} ^{N}$ to describe the quasielectron system of the compound. For an example, it is important to choose the set composed of the Bloch-type functions in $\mathbb{C} ^{N}$ when the considered compound is an ideal crystal following the periodic boundary condition. The matrices in Eqs. (14) and (15) can be rewritten as $\rho ^{( \text{I} )} = \sum _{j=1} ^{N} \rho _{ \eta _{j} } ^{(1)}$, $\rho ^{( \text{II} )} = \sum _{j=1} ^{N} \rho _{ \eta _{j} } ^{(2)}$, $d ^{( \text{I} )} = \sum _{j=1} ^{N} d_{ \eta _{j} } ^{(1)}$, and $d ^{( \text{II} )} = \sum _{j=1} ^{N} d_{ \eta _{j} } ^{(2)}$, where $\rho _{ \eta _{j} } ^{(1)} = | \eta _{j} \rangle \langle \eta _{j} | \otimes | A \rangle \langle A |$, $\rho _{ \eta _{j} } ^{(2)} = | \eta _{j} \rangle \langle \eta _{j} | \otimes | L \rangle \langle L |$, $d _{ \eta _{j} } ^{(1)} = | \eta _{j} \rangle \langle \eta _{j} | \otimes d ^{(1)}$, and $d _{ \eta _{j} } ^{(2)} = | \eta _{j} \rangle \langle \eta _{j} | \otimes d ^{(2)}$. Every $| \eta _{j} \rangle$ can correspond to the 4-level dimer in Fig. 2 (a) if we take $\rho _{ \eta _{j} } ^{(1)} $ and $\rho _{ \eta _{j} } ^{(2)}$ as the two quaielectrons at $| 1\rangle = | \eta _{j} \rangle \otimes | A \rangle $ and $| 2 \rangle = | \eta _{j} \rangle \otimes | L \rangle $. The spatial parts of the two empty orbitals are $| \bar{1} \rangle = | \eta _{j} \rangle \otimes |B \rangle $ and $ | \bar{2} \rangle = | \eta _{j} \rangle \otimes | \bar{L} \rangle $, respectively. The operators $d _{ \eta _{j} } ^{(1)}$ and $d _{ \eta _{j} } ^{(2)}$ are determined by $\rho _{ \eta _{j} } ^{(1)} $ and $\rho _{ \eta _{j} } ^{(2)}$ because $d _{ \eta _{j} } ^{(1)} = ( \sqrt{2} -1) [ \rho _{ \eta _{j} } ^{(1)} \rho _{ \eta _{j} } ^{(2)} ( I - \rho _{ \eta _{j} } ^{(1)}) + ( I - \rho _{ \eta _{j} } ^{(1)}) \rho _{ \eta _{j} } ^{(2)} \rho _{ \eta _{j} } ^{(1)} ]$ and $d _{ \eta _{j} } ^{(2)} = ( I- \rho _{ \eta _{j} } ^{(1)} )\rho _{ \eta _{j} } ^{(2)} ( I - \rho _{ \eta _{j} } ^{(1)}) $, and we can interpret $d _{ \eta _{j} } ^{(1)}$ and $d _{ \eta _{j} } ^{(2)}$ as the correlation contributions of the two quasielectrons in subsystem $\eta _{j}$. Therefore, the half-filled quasielectron system to model the compound are decomposed into the 4-level subsystems as shown in Fig. 3 (a), where the blue dash curves denote the corresponding correlation contributions. In addition, the two quasielectrons in each subsystem $\eta _{j}$ are correlated just as those described by $\rho ^{(1)}$ and $\rho ^{(2)}$ in the one-bond system. Electron affinity and ionization ================================ To model the ionized and affinitive processes, we shall consider how to remove and/or add one quasielectron to excite the carrier. For convenience, first I focus on the one-bond system in subsection IV-A. The assumptions about the one-bond Hamiltonian $H _{b}$ are discussed in Appendix A. Secondly, I consider only the change of the 4-level subsystem $\eta ^{\prime}$ in subsection IV-B to add and/or remove one quasielectron in the compound system, and discuss the excited carrier near the covalent limit to see the importance of bonding coefficients. IV-A Electron affinity and ionization of the one-bond system {#iv-a-electron-affinity-and-ionization-of-the-one-bond-system .unnumbered} ============================================================ The one-bond system in Fig. 1 (a) is taken as 4-level dimer to model the ionic-covalent bonding. When one quasielectron is removed from such a dimer, the remained one occupying $| 1 \rangle \otimes | \sigma \rangle $ in Fig. 2 (b) has no correlated partner. Because atom A has the higher electro-negativity, we can approximate the remained quasielectron by $\rho ^{(1)}$ in Eq. (11) and obtain the energy $E_{b} ^{(-)} = tr H _{sb} \rho ^{(1)}$ as $| \Psi _{b} \rangle$ becomes the uncorrelated one-electron state $| \Psi _{b,1, \uparrow } \rangle = c _{ A \uparrow } ^{\dag} | 0 \rangle$ or $| \Psi _{b,1, \downarrow } \rangle = c _{ A \downarrow } ^{\dag} | 0 \rangle$. On the other hand, there are three quasielectrons when we change $| \Psi _{b} \rangle$ to a three-electron state, which corresponds to the left-hand side of Fig. 2(c), by the affinitve process. We can see from Fig. 2 (c) that such a three-electron state is equivalent to the one-hole state because there are only 4 levels. The remained quasihole is located at $| \bar{1} \rangle \otimes | \sigma \rangle$, and its spatial part $| \bar{1} \rangle$ can be approximated by $ | B \rangle $ in $\mathbb{C}^{2} _{\Omega}$ because atom $B$ is of the lower electro-negativity. So we can take $| \Psi _{b,3, \sigma } \rangle = c _{ B \sigma} | Fb \rangle \propto c _{ B \underline{ \sigma } } ^{ \dag } c _{ A \sigma } ^{\dag} c _{ A \underline{\sigma} } ^{\dag} | 0 \rangle \propto c _{ L \underline{\sigma} } ^{\dag} c _{ A \sigma } ^{\dag} c _{ \bar{L} \underline{\sigma} } ^{\dag} | 0 \rangle$ as the wavefunction of the three-electron or one-hole state for $\sigma = \uparrow$ or $\downarrow$, and the added electron is located at the spatial ket $| \bar{L} \rangle \in \mathbb{C}^{2} _{\Omega}$. Here $ \underline{ \sigma } = - \sigma$, $c _{ \bar{L} \underline{ \sigma } } $ is to annihilate the electron at $ | \bar{L} \rangle \otimes | - \sigma \rangle $, and $| Fb \rangle$ denotes the four-electron state for the filled one-bond system. The state $| \Psi _{b,3, \sigma } \rangle$ is uncorrelated, and its density matrices for $\sigma$ and $- \sigma $ are $\rho ^{(1)} = | A \rangle \langle A |$ and $ \rho ^{(2)} _{+} = | A \rangle \langle A | + | B \rangle \langle B | = | L \rangle \langle L | + | \bar{L} \rangle \langle \bar{L} |$, respectively. Hence the energy for the one-bond system becomes $E ^{(+)} _{b} =tr (\rho ^{(1)} + \rho ^{(2)} _{+})H_{sb} + \int _{ {\bf r} ^{ \prime },{\bf r} ^{\prime \prime} \in \Omega} d ^{3} r ^{\prime} d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime} \rangle \langle {\bf r} ^{\prime \prime} | \rho ^{(2)} _{+} | {\bf r} ^{\prime \prime} \rangle$ after we add one electron. By taking $ \rho ^{(2)} _{-} = d ^{(1)} _{ \pm } = d ^{(2)} _{ \pm } =0$, we can rewrite $E^{( \pm )} _{b}$ as $$\begin{aligned} E_{b} ^{( \pm )}= tr (\rho ^{(1)} + \rho ^{(2)} _{ \pm }) H_{sb} + \int _{ {\bf r} ^{\prime},{\bf r} ^{\prime \prime} \in \Omega} d ^{3} r ^{\prime} d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime} \rangle \langle {\bf r} ^{\prime \prime} | \rho ^{(2)} _{ \pm } | {\bf r} ^{\prime \prime} \rangle +\end{aligned}$$ $$tr H_{sb} d ^{(1)} _{ \pm } + \int _{ {\bf r} ^{\prime},{\bf r} ^{\prime \prime} \in \Omega} d ^{3} r ^{\prime} d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime} \rangle \langle {\bf r} ^{\prime \prime} | d ^{(1)} _{ \pm } | {\bf r} ^{\prime \prime} \rangle + \text{ \ \ \ \ \ }$$ $$\int _{ {\bf r} ^{\prime},{\bf r} ^{\prime \prime} \in \Omega} d ^{3} r ^{\prime} d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime \prime} \rangle \langle {\bf r} ^{\prime \prime} | d ^{(2)} _{ \pm } | {\bf r} ^{\prime} \rangle. \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$$ The above equation can be obtained from Eq. (13) by substituting $ \rho ^{(2)} _{ \pm } $, $ d ^{(1)} _{ \pm }$, and $ d ^{(2)} _{ \pm } $ for $ \rho ^{(2)}$, $ d ^{(1)} $, and $ d ^{(2)} $, respectively. The meaning of $d ^{(1)} _{ \pm } = d ^{(2)} _{ \pm } =0$ is that there is no fluctuating charge or covalent correlation after the ionized and affinitive processes. The remained quasielectron in Fig. 2 (b) has no correlated partner, so it is natural that $d ^{(1)} _{ - } = d ^{(2)} _{ - } =0$. On the other hand, only one quasihole is left at the right-hand side of Fig. 2 (c), and the corresponding one-hole state should be similar to the one-electron state in Fig. 2 (b) based on the electron-hole symmetry. Hence it is reasonable that $d ^{(1)} _{ + } = d ^{(2)} _{ + } =0$. IV-B Electron affinity and ionization of the compound system {#iv-b-electron-affinity-and-ionization-of-the-compound-system .unnumbered} ============================================================ In subsection III-B, the quasielectron system to model the considered compound is decomposed into 4-level subsystems, as shown in Fig. 3 (a). To remove (add) one quasielectron from (to) the subsystem characterized by $| \eta ^{ \prime } \rangle \in \{ | \eta _{j} \rangle \}$, I note that $| \eta ^{\prime } \rangle \otimes | A \rangle$ and $| \eta ^{\prime } \rangle \otimes | B \rangle$ in subsystem $\eta ^{\prime}$ serve as $| A \rangle$ and $|B \rangle$ in the one-bond system, respectively. Therefore, we shall introduce $\rho ^{(2)} _{ \eta ^{ \prime} , - } =0$ and $\rho ^{(2)} _{ \eta ^{ \prime} , + } = | \eta ^{ \prime } \rangle \langle \eta ^{ \prime } | \otimes ( | A \rangle \langle A | + | B \rangle \langle B |) $ for the electron ionization and affinity just as how we introduce $\rho ^{(2)} _{ - }$ and $\rho ^{(2)} _{+} $ according to the electronegativities in subsection IV-A. The ionized quasielectron is removed from $| \eta ^{\prime } \rangle \otimes | L \rangle$ while one quasielectron enters $| \eta ^{\prime } \rangle \otimes | \bar{L} \rangle$ in the affinitive process, and the removed/added charge in the j-th chemical bond equals $| \langle w _{j} | \eta ^{\prime} \rangle | ^{2}$. Together with $\rho ^{(1)} _{ \eta ^{\prime}} $, the matrix $\rho ^{(2)} _{ \eta ^{ \prime} , - } $ corresponds to the one-electron state in Fig. 2 (b) while $\rho ^{(2)} _{ \eta ^{ \prime} , + } $ corresponds to the three-electron or one-hole state in Fig. 2 (c). Because the one- and three-electron states are both uncorrelated, we shall take $d ^{(1)} _{ \eta ^{\prime} , \pm } = d ^{(2)} _{ \eta ^{\prime} , \pm } =0$ to remove the correlation contribution of the subsystem $ \eta ^{\prime}$. If each subsystem characterized by $ \eta _{j} \neq \eta ^{\prime} $ remains unchanged, as shown in Fig. 3 (b), we shall replace $\rho ^{( \text{II} )}$, $d ^{( \text{I} )}$, and $d ^{( \text{II} )}$ by $ \rho ^{( \text{II} )} _{ \eta ^{\prime} , \pm } = \sum _{ \eta _{j} \neq \eta ^{\prime} } \rho ^{(2)} _{ \eta _{ j } } +\rho ^{(2)} _{ \eta ^{ \prime } , \pm } $, $d ^{( \text{I} )} _{ \eta ^{\prime} , \pm } = \sum _{ \eta _{j} \neq \eta ^{\prime} } d ^{(1)} _{ \eta _{ j } } +d ^{(1)} _{ \eta ^{ \prime} , \pm } $, and $d ^{( \text{II} )} _{ \eta _{\prime} , \pm } = \sum _{ \eta _{j} \neq \eta ^{\prime} } d ^{(2)} _{ \eta _{j} } +d ^{(2)} _{ \eta ^{ \prime} , \pm } $, respectively. The energy $E _{Cr} $ becomes $$E_{Cr} ^{ ( \eta ^{\prime} , \pm ) } = tr ( \rho ^{ ( \text{I} ) } + \rho ^{ ( \text{II} ) } _{ \eta ^{\prime}, \pm } ) H _{sC} + \sum _{j=1} ^{N} \int _{ {\bf r} ^{\prime} _{ j },{\bf r} ^{\prime \prime} _{ j } \in \Omega _{j} } d ^{3} r ^{\prime} _{ j } d ^{3} r ^{\prime \prime} _{ j } U ( | {\bf r} ^{\prime} _{ j } - {\bf r} ^{\prime \prime} _{ j } | ) \langle {\bf r} ^{\prime} _{ j } | \rho ^{( \text{I} )} | {\bf r} ^{\prime} _{ j } \rangle \langle {\bf r} ^{\prime \prime} _{ j } | \rho ^{( \text{II} )} _{ \eta ^{\prime}, \pm} | {\bf r} ^{\prime \prime} _{ j } \rangle +$$ $$\begin{aligned} trH _{sC} d^{ ( \text{I} )} _{ \eta ^{\prime}, \pm} + \sum _{j=1} ^{N} \int _{ {\bf r} ^{\prime} _{ j },{\bf r} ^{\prime \prime} _{ j } \in \Omega _{j} } d ^{3} r ^{\prime} _{ j } d ^{3} r ^{\prime \prime} _{ j } U ( | {\bf r} ^{\prime} _{ j } - {\bf r} ^{\prime \prime} _{ j } | ) \langle {\bf r} ^{\prime} _{ j } | \rho ^{( \text{I} )} | {\bf r} ^{\prime} _{ j } \rangle \langle {\bf r} ^{\prime \prime} _{ j } | d ^{( \text{I} )} _{ \eta ^{\prime}, \pm} |{\bf r} ^{\prime \prime} _{ j } \rangle +\end{aligned}$$ $$\sum _{j=1} ^{N} \int _{ {\bf r} ^{\prime} _{ j },{\bf r} ^{\prime \prime} _{ j } \in \Omega _{j} } d ^{3} r ^{\prime} _{ j } d ^{3} r ^{\prime \prime} _{ j } U ( | {\bf r} ^{\prime} _{ j } - {\bf r} ^{\prime \prime} _{ j } | ) \langle {\bf r} ^{\prime} _{ j } | \rho ^{( \text{I} )} | {\bf r} ^{\prime \prime } _{ j } \rangle \langle {\bf r} ^{\prime \prime} _{ j } | d ^{( \text{II} )} _{ \eta ^{\prime}, \pm} | {\bf r} ^{\prime} _{ j } \rangle \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$$ after we change the number of electrons in subsystem $\eta ^{\prime} $. An effective carrier is excited in the ionized/affinitive process, and we can obtain its excitation energy by calculating the difference $E_{Cr} ^{ ( \eta ^{\prime} , \pm ) } -E_{Cr}$ based on Eqs. (18) and (20). It is convenient to rewrite $\rho ^{( \text{II} )} _{ \eta ^{\prime} , \pm }$, $d ^{( \text{I} )} _{ \eta ^{\prime} ,\pm }$, and $d ^{( \text{II} )} _{ \eta ^{\prime} ,\pm }$ as $$\begin{aligned} \begin{cases} \rho ^{( \text{II} )} _{ \eta ^{\prime} , + } = \rho ^{( \text{II} )} + | \eta ^{ \prime } \rangle \langle \eta ^{ \prime } | \otimes | \bar{L} \rangle \langle \bar{L} | \\ \rho ^{( \text{II} )} _{ \eta ^{\prime} , - } = \rho ^{( \text{II} )} -\rho ^{(2)} _{ \eta ^{\prime} } \\ d ^{( \text{I} )} _{ \eta ^{\prime} ,\pm } = ( \sqrt{2} -1) [ \rho ^{( \text{I} )} \rho ^{( \text{II} )} _{ \eta ^{\prime} , -} ( I - \rho ^{( \text{I} )}) + ( I - \rho ^{( \text{I} )}) \rho ^{( \text{II} )} _{ \eta ^{\prime} , - } \rho ^{( \text{I} )} ] \\ d ^{( \text{II} )} _{ \eta ^{\prime} ,\pm } = ( I - \rho ^{( \text{I} )}) \rho ^{( \text{II} )} _{ \eta ^{\prime} , - } ( I - \rho ^{( \text{I} )}) \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \end{cases},\end{aligned}$$ in Eq. (20) to obtain the result irrelevant to $| \eta _{j} \rangle $ for all $ \eta _{j} \neq \eta ^{\prime}$. Direct calculation yields $$\begin{aligned} E_{Cr} ^{ ( \eta ^{\prime} , \pm ) } -E_{Cr}=\langle \eta ^{ \prime} | H _{\pm } | \eta ^{ \prime } \rangle + E _{ b} ^{ (\pm)} - E _{b} \text{ with } \begin{cases} H_{+} = tr ^{ \prime } H _{hop} (| \bar{L} \rangle \langle \bar{L} | - d ^{(1)}) \\ H_{-} = - tr ^{ \prime } H _{hop} (| L \rangle \langle L | + d ^{(1)}) \end{cases},\end{aligned}$$ where $tr ^{\prime} $ denotes the trace with respect to $\mathbb{C} ^{2} _{ \Omega}$. Near the ionic limit, it is shown in Appendix C that $E_{Cr} ^{ ( \eta ^{\prime} , \pm ) } -E_{Cr}$ can be close to the energy difference obtained by considering the coupled-cluster corrections after we improve my model based on Eq. (25). The operators $| \bar{L} \rangle \langle \bar{L} |$, $| L \rangle \langle L |$, and $ d ^{(1)}$ in Eq. (22) include the bonding coefficients $\alpha _{i}$ and $\alpha _{c}$, which depend on the repulsive strength of the e-e interaction potential $\hat{U}$ as mentioned in Appendix B. Therefore, we can obtain the interaction-dependent electron ionization and affinity for the excited carrier. To obtain the quantitative results, I consider the nearest-neighbor hopping in the zincblende-structure crystal, in which the atom $A$ located at ${\bf R} _{A} = n _{1} (l_{c}/2, l_{c}/2, 0) + n _{2} (0, l_{c}/2,l_{c}/2) + n _{3} (l_{c}/2, 0, l_{c}/2)$ is accompanied by the atom $B$ at ${\bf R} _{B} = {\bf R} _{A} + (l_{c}/4, l_{c}/4, l_{c}/4) $. Here $n _{1}$, $n _{2}$, and $n _{3}$ are integers, and $l _{c}$ is the length of the crystal lattice. For convenience, I denote ${\bf n} = (n _{1}, n _{2}, n _{3}) $ for the parameters $n _{1}$, $n _{2}$, and $n _{3}$ of ${\bf R} _{A}$, and take $ m =1 \sim 4$ to parameterize the 4 ionic-covalent bonds around the same atom $A$ such that each $| w _{j} \rangle$ can be re-parameterized as $| w _{ {\bf n}, m } \rangle$. Assume that the hopping coefficients equal $t _{A}$ and $t _{B}$ for the adjacent A- and B-site orbitals, respectively. By adding one quasielectron to the s-like Bloch-type orbital $ | \eta ^{\prime} \rangle =\frac{1}{2 \sqrt{N} } \sum _{ {\bf n} , m } e ^{ i {\bf k} \cdot {\bf R}_{A} } | w_{ {\bf n}, m } \rangle \in \mathbb{C} ^{N}$ near the covalent limit, we can obtain $$\begin{aligned} E_{Cr} ^{ ( \eta ^{\prime} , + ) } -E_{Cr}=3 t _{A} | \alpha _{c} | ^{2} + E _{b} ^{(+)} - E _{b} -\gamma \varepsilon _{dis} ( {\bf k}) \text{ with } \end{aligned}$$ $$\varepsilon _{dis} ( {\bf k}) = cos \frac{ l _{c} k _{x} }{2} cos \frac{ l _{c} k _{y} }{2} + cos \frac{ l _{c} k _{y} }{2} cos \frac{ l _{c} k _{z} }{2} + cos \frac{ l _{c} k _{z} }{2} cos \frac{ l _{c} k _{x} }{2}$$ as the energy dispersion curve [@Grosso] for the excited carrier in the tight-binding scheme. Here $ \gamma = - | \alpha _{i}| ^{2} t _{B} $ serves as the overlap energy [@Kittel], ${\bf k}= ( k _{x} , k _{y}, k _{z}) $ denotes the wavevector, and $\varepsilon _{dis} ( {\bf k})$ can be obtained by considering the twelve nearest-neighbor vectors [@Grosso]. The effective mass $m ^{*} ({\bf k})$ [@Grundmann] follows $ [ m ^{*-1}({\bf k}) ] _{i,j} = - \frac{ \gamma }{\hbar ^2} \frac{ \partial ^{2} }{\partial k_{i} \partial k _{j}} \varepsilon _{dis} ( {\bf k}) \propto |\alpha _{i}| ^{2} = 1- |\alpha _{c}| ^{2} $ at each ${\bf k}$, which reveals the importance of the bonding coefficients to the excited carrier. In addition, the carrier becomes immobile in the covalent limit because the bandwidth equals zero as $| \alpha _{i} | = 0$. The coefficients $\alpha _{i}$ and $\alpha _{c}$ are determined by the e-e repulsive strength as mentioned in Appendix B, so $m ^{*} ({\bf k})$ depends on the e-e interaction potential $\hat{U}$ in my model. The zero bandwidth in the covalent limit is due to the lack of the double occupancy at half filling under the strong repulsive strength of $\hat{U}$, which induces the Mott insulating behaviors in some AF systems [@Lee1; @Liu]. Discussion ========== In the last three sections, the non-negative function $U(| {\bf r} |)$ is taken into account to introduce $\hat{U}$ without considering the inter-bond e-e correction. Actually there should exist the inter-bond e-e energy $\frac{1}{2} \sum _{ j_{1} \neq j_{2} } \int _{ {\bf r} _{1} \in \Omega _{ j _{1} } } d ^{3} r _{1} \int _{ {\bf r} _{2} \in \Omega _{ j _{2} } } d ^{3} r _{2} U ^{\prime} (| {\bf r} _{1} - {\bf r} _{2} |) Q ({\bf r} _{1})Q({\bf r} _{2})$ in the quasielectron system to model the considered compound, where $U ^{\prime}$ represents the long-range e-e correction. Because the electron density $Q({\bf r} _{j} )= 2 \langle {\bf r} _{j} | \rho _{sC} | {\bf r} _{j} \rangle$ at ${\bf r} _{j} \in \Omega _{j}$ in the $j$-th bond and $\rho _{sC} = \frac{1}{2}( \rho ^{ ( \text{I} ) } + \rho ^{ ( \text{II} ) } + d ^{ ( \text{I} ) }) $, we shall include $$\text{(i) } \frac{1}{2} \sum _{ q = \text{I}, \text{II} } \text{ } \sum_{ q ^{\prime} = \text{I}, \text{II} } \text{ } \sum _{ j_{1} \neq j_{2} } \int _{ {\bf r} _{1} \in \Omega _{ j _{1} } } d ^{3} r _{1} \int _{ {\bf r} _{2} \in \Omega _{ j _{2} } } d ^{3} r _{2} U ^{\prime} (| {\bf r} _{1} - {\bf r} _{2} |) \langle {\bf r} _{1} | \rho ^{ (q) } | {\bf r} _{1} \rangle \langle {\bf r} _{2} | \rho ^{(q^{\prime})} | {\bf r} _{2} \rangle$$ $$\begin{aligned} \text{(ii) \ } \sum _{ q = \text{I}, \text{II} } \text{ } \sum _{ j_{1} \neq j_{2} } \int _{ {\bf r} _{1} \in \Omega _{ j _{1} } } d ^{3} r _{1} \int _{ {\bf r} _{2} \in \Omega _{ j _{2} } } d ^{3} r _{2} U ^{\prime} (| {\bf r} _{1} - {\bf r} _{2} |) \langle {\bf r} _{1} | \rho ^{ (q) } | {\bf r} _{1} \rangle \langle {\bf r} _{2} | d ^{ (I) } | {\bf r} _{2} \rangle \text{ \ \ \ \ \ \ } \end{aligned}$$ $$\text{(iii) } \frac{1}{2} \sum _{ j_{1} \neq j_{2} } \int _{ {\bf r} _{1} \in \Omega _{ j _{1} } } d ^{3} r _{1} \int _{ {\bf r} _{2} \in \Omega _{ j _{2} } } d ^{3} r _{2} U ^{\prime} (| {\bf r} _{1} - {\bf r} _{2} |) \langle {\bf r} _{1} | d^{( \text{I} )} | {\bf r} _{1} \rangle \langle {\bf r} _{2} | d^{( \text{I} )} | {\bf r} _{2} \rangle \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$$ for the inter-bond e-e correction. Since each chemical bond in the compound system is identical to the one-bond system discussed in subsection III-A, the charge fluctuation $\langle {\bf r} _{j} | d ^{( \text{I})} | {\bf r} _{j} \rangle$ at ${\bf r} _{j} \in \Omega _{j}$ for any $j$ has no contribution to the total electron charge just as $\langle {\bf r} | d ^{(1)} | {\bf r} \rangle$. I note that the deviation $\delta \rho$ responsible for the density-density interaction [@You] in the quantum Hall theory [@Zhang; @DHLee] also has no contribution to the total charge, and Eq. (24)-(iii) shows the universality of such interaction. The charge fluctuation due to $ d ^{( \text{I} )}$ may interact with the charge density given by $\rho ^{( \text{I} )}$ and $\rho ^{( \text{II} )}$, and Eq. (24)-(ii) just provides the corresponding energy together with Eq. (16)-(ii). Equations (16)-(i) and (24)-(i) yield the Hartree-potential energy resulting from $\rho ^{( \text{I} )}$ and $\rho ^{( \text{II} )}$, and we shall include the Fock-potential term [@Nelson] because of the lack of the self-interaction in Eq. (16)-(i). When the compound is an ideal crystal following the periodic boundary condition, it is important to consider the case that each $ | \eta _{j} \rangle \in \mathbb{C} ^{N}$ in Fig. 3 (a) corresponds to one Bloch-type function. The Bloch wavefunction, which consists of its Bloch-type part in $ \mathbb{C} ^{N}$ and the bonding part in $ \mathbb{C} ^{2} _{ \Omega}$, can be extended to the form $e ^{ i {\bf k} \cdot {\bf r} } u _{ {\bf k} } ({\bf r}) $ [@Kittel] to include the small density due to the electron tails in the shaded region in Fig. 1 (c) as $H _{ sC} $ is replaced by $\frac{{\bf p}^{2}}{2m_{0}} +V _{cr}({\bf r})$. Here $V_{cr}$ denotes the one-electron periodic potential in the crystal, $m _{0} $ is the electron mass in vacuum, ${\bf p}$ is for the momentum operators, and $u _{ {\bf k} } ( {\bf r})$ represents the periodic part of the corresponding Bloch state. To determine the hopping coefficients $t _{ j \xi, j^{\prime} \xi ^{\prime}}$ in Eq. (17), principally we can transform the Bloch wavefunctions to the Wannier ones [@Marzari], which serve as the localized atomic orbitals in the tight-binding model [@Mahan]. For the well-developed ionic-covalent bonds, in Fig. 1 (c) the electron density in the shaded region must be so low that the bonding electrons almost concentrate in the red region, where the high density induces the bonding correlation representing by Eq. (15). After introducing the inter-bond hopping, therefore, we may approximate the $j$-th bond’s Wannier function as zero outside $\Omega _{j} $ for all $j$ to calculate the e-e energy. Actually Eqs. (18) and (20) can be valid in the systems composed of different ionic-covalent dimers such as the those in the chalcopyrite-structure [@Grundmann] compound when all the bonds are well-separated, as shown in Appendix D. The Wannier basis, however, depends on the gauge freedom [@Marzari] and is not unique. When the ionic-covalent bonds in the compound are not identical, the decomposition in Fig. 3 (a) can become invalid because of the non-constant bonding coefficients. More studies are necessary to clarify how to exactly include the bonding correlation beyond the compound model developed in subsections III-B and IV-B. It is shown in subsection IV-B that bandwidth can become zero because of the strong e-e repulsive strength, which is responsible for the Mott insulator in some AF systems [@Lee1; @Liu]. It is known that the random fields [@Kirsc; @Erdos; @Lee2; @Huang2] modeled by a family of parameterized Hamiltonians can result in the disorder leading to different insulators, and both the disorder and e-e interaction effects have been observed in the quantum Hall systems [@TYH; @Wang]. The transition between insulating phases has been studied by considering the disordered interacting systems. [@Byczuk; @Braganc] To include a Hamiltonian family, we may replace $H_{sC}$ by the random-matrix set $H_{sC} ^{\omega}$ parameterized by $\omega$ and consider $\sum _{\omega} (\rho ^{( \text{I} )}_{\omega} e _{1,\omega} e _{1,\omega} ^{\dag} + \rho ^{( \text{II} )}_{\omega} e _{2,\omega} e _{2,\omega} ^{\dag})$. Here the set $\{ e _{1, \omega} , e _{2,\omega} \}$ is an orthonormal one in the corresponding vector space, and for each $\omega$ the matrices $\rho ^{( \text{I} )}_{\omega}$ and $\rho ^{( \text{II} )}_{\omega}$ serve as $\rho ^{( \text{I} )}$ and $\rho ^{( \text{II} )}$. If the one-bond system in Fig. 1 (a) is asymmetric with respect to the bonding axis, the rotation centered on such an axis is important to the mapping ${\bf R} _{j}$ in subsection III-B for each $j$ and we need to introduce the parameter $\omega$ for the rotation degrees of freedom [@Huang3]. In the Born-Oppenhemier method [@Bohm] (BOM), we also need to consider a family of Hamiltonians to determine the electron wavefunctions parameterized by the relative position of the nuclei. For any two $2^{n} \times 2^{n}$ matrices $\rho ^{\prime} _{ea}$ and $\rho ^{\prime \prime} _{ea}$, actually we can construct a $ 2 ^{n+1} \times 2 ^{n+1}$ matrix $\rho ^{\prime \prime \prime} _{ea} = \rho ^{\prime} _{ea} \otimes e _{1} e _{1} ^{\dag} + \rho ^{\prime \prime} _{ea} \otimes e _{2} e _{2} ^{\dag}$ and take Eq. (8) as the case for $n=0$, where the integer $n$ is non-negative. By this way we can construct AF-type quasielectrons with $2^{n+1}$ components for the Hamiltonian family parameterized by $\omega = 1 \sim 2 ^{n}$. The multiple-component orbitals can be used to include the multiple CCDs, which are briefly discussed in Appendix B after including $| \Psi _{BB} \rangle$, in the quasielectron space [@Huang3]. I note that the multiple-component functions are introduced to develope the vector bundles [@Bohm; @Friedman; @Banks]. While the hole components [@Huang1] do not appear in the density matrices in my ionic-covalent model, they may become important when both the particle-particle and particle-hole channels [@Yu] are taken into account for the Bogoliubov-BCS quasiparticles. By considering the fractal structures [@Schwalm] to extend such quasiparticles [@Huang1; @Huang4], in fact, we can obtain the form of Eq. (2) from the electron components of the extended ones. Summary ======= The extended AF quasielectrons are introduced for the compound where the ionic-covalent bonds are identical to the one-bond dimer. The density-fluctuation and covalent-correlation operators are constructed for the bonding correlation, and my quasielectron model shows the universality of the density-density interaction. The quasielectron system is decomposed into the 4-level subsystems for the electron ionization and affinity in the compound, and such a model can be supported by the coupled-cluster theory near the ionic limit. For the ideal crystal, each subsystem may correspond to one Bloch-type function. By considering the nearest-neighbor hopping in the zincblende-structure crystal, we can see the importance of the bonding coefficients to the effective mass near the covalent limit. Acknowledgment {#acknowledgment .unnumbered} ============== The author thanks Profs. I.-H. Tsai, Keh-Ning Huang, and Hrong-Tzer Yau for the valuable discussions about the vector bundles, coupled-cluster corrections, and random-matrix theory, respectively. Appendix A {#appendix-a .unnumbered} ========== For the one-bond system in Fig. 1 (a), I assume that the distance $\ell$ separating the peaks of $| \langle {\bf r } | A \rangle | ^{2}$ and $| \langle {\bf r } | B \rangle | ^{2}$ in Fig. 4 is longer than the interacting length $a$ of $\hat{U}$. In addition, assume that $\int _{ {\bf r} ^{\prime} \in \Omega} d ^{3} r ^{\prime} \int _{ {\bf r ^{\prime \prime} } \in \Omega} d^{3} r ^{ \prime \prime} U (|{\bf r} ^{\prime} -{\bf r} ^{ \prime \prime } |) | \langle {\bf r } ^{\prime} | A \rangle | ^{2} | \langle {\bf r ^{\prime \prime } } | A \rangle | ^{2} \sim \int _{ {\bf r} ^{\prime} \in \Omega} d ^{3} r ^{\prime} \int _{ {\bf r ^{\prime \prime } } \in \Omega} d^{3} r ^{ \prime \prime} U (|{\bf r} ^{\prime} -{\bf r} ^{ \prime \prime} |) | \langle {\bf r }^{\prime} | B \rangle | ^{2} | \langle {\bf r ^{\prime \prime} } | B \rangle | ^{2} \sim U _{0} >0$, where the positive parameter $U _{ 0} $ represents the repulsive strength of $\hat{U}$. Hence the effective e-e potential $\hat{U}$ in Eqs. (9) and (10) is dominated by $U _{0} c _{A \uparrow} ^{\dag} c _{A \uparrow} c _{A \downarrow} ^{\dag} c _{A \downarrow} + U _{0} c _{B \uparrow} ^{\dag} c _{B \uparrow} c _{B \downarrow} ^{\dag} c _{B \downarrow} \equiv \hat{U} _{0} $, which corresponds to the intrasite Coulomb repulsion [@Matlak1; @Matlak2] in the t-U model. Moreover, I assume that the difference $\varepsilon _{B} - \varepsilon_{A} $ is high enough for us to take $\sum _{\sigma} (t _{AB} c _{A \sigma} ^{\dag} c _{B \sigma} + t _{AB} ^{*} c _{B \sigma} ^{\dag} c _{A \sigma} ) + \hat{U}- \hat{U}_{0}$ as the perturbation part of $H _{b}$. Under the above assumptions, the remained quasielectron and quasihole in the one-bond system are roughly located at $| A \rangle$ and $| B \rangle$ as the one-bond wavefunction becomes the one- and three-electron ground states, respectively. Hence the spatial ket $| 1 \rangle $ of the occupied level in Fig. 2 (b) is close to $| A \rangle $ when the 4-level dimer in Fig. 2 represents such a one-bond system, and the ket $| \bar{1} \rangle$ for the quasihole at the right-hand side of Fig. 2 (c) can be approximated by $| B \rangle$. In addition, we may neglect the small change on $\rho ^{(1)}$ as one electron is removed/added. For the one- and three-electron states of subsystem $ \eta ^{\prime} $ in Fig. 3 (b), on the other hand, we shall take $| 1 \rangle \sim | \eta ^{\prime} \rangle \otimes | A \rangle $ and $| \bar{1} \rangle \sim | \eta ^{\prime} \rangle \otimes | B \rangle$ in Figs. 2 (b) and (c), respectively, and approximate $\rho ^{(1)} _{\eta ^{\prime} }$ as $( | \eta ^{\prime} \rangle \otimes | A \rangle)( \langle \eta ^{\prime} | \otimes \langle A |)$. We can tune $\rho ^{(1)}$ and $\rho ^{(1)} _{ \eta ^{\prime} }$ in section IV to modify the orbitals of the remained quasiparticles in the corresponding spaces $\mathbb{C} ^{2} _{\Omega}$ and $| \eta ^{\prime} \rangle \langle \eta ^{\prime} | \otimes \mathbb{C} ^{2} _{\Omega}$. The charge background due to the half-filled subsystems, which are characterized by $\eta _{j} \neq \eta ^{\prime}$ in Fig. 3 (b), should be taken into account to perform the modification for the compound system. Appendix B {#appendix-b .unnumbered} ========== In the one-bond system in Fig. 1 (a), the wavefunction $| \Psi _{b} ^{SCF} \rangle = ( \sqrt{1- | \lambda _{1} | ^{2} } c _{ A \uparrow} ^{\dag} + \lambda _{1} c _{ B \uparrow} ^{\dag} ) ( \sqrt{1- | \lambda _{1} | ^{2} } c _{ A \downarrow} ^{\dag} + \lambda _{1} c _{ B \downarrow} ^{\dag} ) | 0 \rangle$ can serve as the effective SCF state at half filling near the ionic limit if the small parameter $\lambda _{1}$ is determined by minimizing $\langle \Psi _{b} ^{SCF} | H _{b} | \Psi _{b} ^{SCF} \rangle$. The wavefunction $| \Psi _{b} ^{SCF} \rangle$ is a linear combination of $| \Psi _{i} \rangle$, $| \Psi _{c} \rangle$, and $| \Psi _{BB} \rangle$, so principally we should take $| \Psi _{BB} \rangle$ into account in addition to the ionic and covalent parts. The ket $| \Psi _{b} ^{CCD} \rangle = ( \sqrt{1- | \lambda _{1} | ^{2} } c _{ B \uparrow} ^{\dag} - \lambda _{1} ^{*} c _{ A \uparrow} ^{\dag} ) ( \sqrt{1- | \lambda _{1} | ^{2} } c _{ B \downarrow} ^{\dag} - \lambda _{1} ^{*} c _{ A \downarrow} ^{\dag} ) | 0 \rangle$ is the only allowed CCD for the bonding electrons. When Brillouin theorem [@Manninen; @Pople] is valid near the ionic limit, the coupled-cluster method is applicable and we may take $| \Psi _{b} ^{Br} \rangle = \sqrt{1- | \lambda _{2} | ^{2} } | \Psi _{b} ^{SCF} \rangle + \lambda _{2} | \Psi _{b} ^{CCD} \rangle $ as $| \Psi _{b} \rangle $ to model the ground state. Here the small parameter $\lambda _{2}$ is determined by minimizing $\langle \Psi _{b} ^{Br} | H _{b} | \Psi _{b} ^{Br} \rangle$. The single substitution [@Pople] is neglected in $| \Psi _{b} ^{Br} \rangle $. While Brillouin theorem may become invalid, we have the bonding wavefunction $ | \Psi _{b} \rangle = \tau _{i} | \Psi _{i} \rangle + \tau _{c} | \Psi _{c} \rangle+ \tau _{BB} | \Psi _{BB} \rangle $ [@Prasad; @Havenith] in general when the one-bond system is half-filled. Here $ \tau _{i} $, $ \tau _{c} $, and $ \tau _{BB} $ are the coefficients satisfying $ | \tau _{i} | ^{2} + | \tau _{c} | ^{2} + | \tau _{BB} | ^{2} =1$. The energies of $| \Psi _{i} \rangle$ and $| \Psi _{c} \rangle$ are close to $2 \varepsilon _{A} + U _{0}$ and $\varepsilon _{A}+\varepsilon _{B}$, respectively, and are both lower than the energy of $| \Psi _{BB} \rangle$ under the assumptions mentioned in Appendix A. Hence $| \tau_{BB} | $ should be small, and we can approximate $\tau _{i}$ and $\tau _{c}$ as the coefficients $\alpha _{i}$ and $\alpha _{c}$ in Eq. (1) if it is suitable to neglect the small contribution of $ | \Psi _{BB} \rangle$. The bonding wavefunction $| \Psi _{b} \rangle \rightarrow | \Psi _{i} \rangle$ near the ionic limit as the repulsive strength $ U_{0} << \varepsilon _{B}- \varepsilon _{A}$. With increasing the e-e repulsive strength, $| \alpha _{i} |$ decreases and $\alpha _{c}$ becomes significant. The wavefunction $|\Psi _{b} \rangle \rightarrow |\Psi _{c} \rangle$ near the covalent limit when $U _{0} >> \varepsilon _{B}- \varepsilon _{A} $, under which the double occupancy is forbidden. When it is inappropriate to neglect $| \Psi _{BB} \rangle$, we can perform the orbital transformation $$\begin{aligned} \begin{cases} c _{ A^{\prime} \sigma } =\sqrt{ 1 - | \lambda _{3} | ^{2} } c _{ A \sigma } + \lambda _{3} c _{ B \sigma } \\ c _{ B^{\prime} \sigma } = - \lambda _{3} ^{*} c _{ A \sigma } + \sqrt{ 1 - | \lambda _{3} | ^{2} } c _{ B \sigma } \end{cases},\end{aligned}$$ to rewrite $| \Psi _{b} \rangle$ as $\alpha_{i} ^{\prime} | \Psi _{i} ^{\prime} \rangle + \alpha_{c} ^{\prime} | \Psi _{c} ^{\prime} \rangle + \tau_{BB} ^{\prime} | \Psi _{BB} ^{\prime} \rangle $ with $ | \Psi _{i} ^{\prime} \rangle = c _{ A^{\prime} \uparrow } ^{\dag} c _{ A^{\prime} \downarrow } ^{\dag} | 0 \rangle $, $| \Psi _{c} ^{\prime} \rangle = \frac{1}{ \sqrt{2} } (c _{ A^{\prime} \uparrow } ^{\dag} c _{ B^{\prime} \downarrow } ^{\dag} + c _{ B^{\prime} \uparrow } ^{\dag} c _{ A^{\prime} \downarrow } ^{\dag}) | 0 \rangle$, and $ | \Psi _{BB} ^{\prime} \rangle = c _{ B^{\prime} \uparrow } ^{\dag} c _{ B^{\prime} \downarrow } ^{\dag} | 0 \rangle $. Here $\lambda _{3}$ is a complex number following $0 \leq | \lambda _{3} | \leq 1 $, and $\alpha_{i} ^{\prime}$, $\alpha_{c} ^{\prime}$, and $\tau_{BB} ^{\prime}$ are the coefficients determined by $\tau _{i}$, $\tau _{c}$, $\tau _{BB}$ and $\lambda _{3}$. The bonding wavefunction $| \Psi _{b} \rangle$ becomes $\alpha_{i} ^{\prime} | \Psi _{i} ^{\prime} \rangle + \alpha_{c} ^{\prime} | \Psi _{c} ^{\prime} \rangle$, the ionic-covalent form given by Eq. (1), if the parameter $\kappa \equiv \lambda ^{*} _{3} / \sqrt{1 - | \lambda ^{*} _{3} | ^{2} } $ follows $\tau _{i} \kappa ^{2} - \sqrt{2} \tau _{c} \kappa + \tau _{BB} =0$ such that $\tau_{BB} ^{\prime} =0 $. There are two solutions to $\kappa$, and we can choose the solution with the smaller absolute value while the other one is important to the spontaneous symmetry breaking [@Huang3]. To improve my model by including $| \Psi _{BB} \rangle$, we shall replace $|A\rangle$ and $| B \rangle $ by $| A ^{\prime} \rangle \equiv \sqrt{ 1 - | \lambda _{3} | ^{2} } | A \rangle + \lambda _{3} ^{*} | B \rangle $ and $| B ^{\prime} \rangle \equiv - \lambda _{3} | A \rangle + \sqrt{ 1 - | \lambda _{3} | ^{2} } | B \rangle$ such that $\rho ^{(1)} \rightarrow | A ^{\prime} \rangle \langle A ^{\prime} | $ and $\rho ^{(2)} \rightarrow | L ^{\prime} \rangle \langle L ^{\prime} | $ in Eq. (11), where $ | L ^{\prime} \rangle = \alpha _{i} ^{\prime} | A ^{\prime} \rangle + \alpha _{c} ^{\prime} | B ^{\prime} \rangle$. In addition, the ket $| \bar{L} \rangle$ in subsection III-A should be replaced by $ | \bar{L} ^{\prime} \rangle = \alpha _{c} ^{\prime \ast} | A ^{\prime} \rangle - \alpha _{i} ^{\prime \ast} | B ^{\prime} \rangle $. Based on the mapping ${\bf R}_{j}$, for Eq. (14) we have $ \rho _{ w _{j} } ^{(1)} = | w _{j} \rangle \langle w _{j} | \otimes |A ^{\prime} \rangle \langle A ^{\prime} |$ and $ \rho _{ w _{j} } ^{(2)} = | w _{j} \rangle \langle w _{j} | \otimes |L ^{\prime} \rangle \langle L ^{\prime} |$, under which $ \rho _{ \eta _{j} } ^{(1)} = | \eta _{j} \rangle \langle \eta _{j} | \otimes |A ^{\prime} \rangle \langle A ^{\prime} |$ and $ \rho _{ \eta _{j} } ^{(2)} = | \eta _{j} \rangle \langle \eta _{j} | \otimes |L ^{\prime} \rangle \langle L ^{\prime} |$. The matrix $\rho ^{(2)} _{+}$ in Eq. (19) and the operator $\rho ^{(2)} _{ \eta ^{\prime} +}$ for subsystem $\eta ^{\prime}$ in subsection IV-B both remain unchanged because $| A \rangle \langle A| + | B \rangle \langle B | = | A ^{\prime} \rangle \langle A ^{\prime} | + | B ^{\prime} \rangle \langle B ^{\prime} | $. The one- and three-electron states $| \Psi _{b,1,\sigma} \rangle$ and $| \Psi _{b,3,\sigma} \rangle$ in subsection IV-A should be replaced by $| \Psi ^{\prime} _{b,1,\sigma} \rangle \equiv c ^{\dagger} _{ A^{\prime} \sigma } | 0 \rangle$ and $ | \Psi ^{\prime} _{b,3,\sigma} \rangle \equiv c _{ B^{\prime} \sigma } | Fb \rangle $ if we neglect the small change on $\rho ^{(1)}$ when one electron is removed/added. In Eqs. (19) and (20), we can perform the modification discussed at the end of Appendix A by tuning $\rho ^{(1)}$ and $\rho ^{(1)} _{ \eta ^{\prime} }$. When we take $|\Psi ^{Br} _{b} \rangle $ as the bonding wavefunction according to the coupled-cluster method near the ionic limit, the coefficient $\tau _{c}$ can be small and become comparable with $\tau _{BB}$. The bonding wavefunction $| \Psi _{b} \rangle $ is dominated by $| \Psi _{i} \rangle$, but we cannot consider only the ionic part to probe the bonding correlation. Therefore, it is important to improve my model near such a limit by using Eq. (25) to include $| \Psi _{BB} \rangle$ in addition to the covalent part if we hope to exactly probe the bonding correlation. In the BOM, a family of Hamiltonians are taken into account by considering the variation on the positions of the nuclei. While $| \Psi _{b} ^{CCD} \rangle$ is the only allowed CCD in the 4-level dimer for the one-bond system in Fig. 1 (a), it depends on the positions of the nuclei and thus can generate a CCD family $ \{ | \Psi _{b} ^{CCD} (\omega) \rangle \}$. Here the parameter $\omega$ is to parameterize such a family. Multiple CCDs, in fact, can be incorporated in the quasiparticle space by considering the corresponding family [@Huang3], and we may extend the BOM to develop the quasiparticles including both the electron-correlation and nucleus-vibration effects. Appendix C {#appendix-c .unnumbered} ========== In subsection III-B, I consider the compound system where the identical bonds are parametrized by $j$. For convenience, let $c_{jA \sigma}$ and $c_{jB \sigma}$ as the annihilators to remove electrons with the spin orientation $\sigma$ in $| A, j \rangle$ and $| B, j \rangle$, respectively. The compound Hamiltonian ${\cal H}= \sum _{ j \neq j ^{ \prime}} H_{hop} ^{(j, j^{\prime})} + \sum _{ j } (H_{sb} ^{(j)} + \hat{U _{j} }) $, where $ H_{hop} ^{(j, j^{\prime})} = \sum _{\sigma} \sum _{ \xi , \xi ^{\prime} \in \{ A,B \} } t _{ j \xi , j ^{\prime} \xi ^{\prime} } c_{j \xi \sigma} ^{\dag} c_{ j ^{\prime} \xi ^{\prime} \sigma} $, $H_{sb} ^{(j)} = \sum _{\sigma} (t _{AB} c _{j A \sigma} ^{\dag} c _{ j B \sigma} + t _{AB} ^{*} c _{jB \sigma} ^{\dag} c _{jA \sigma} + \varepsilon _{A} c _{jA \sigma} ^{\dag} c _{jA \sigma} + \varepsilon _{B} c _{jB \sigma} ^{\dag} c _{jB \sigma})$, and $\hat{U _{j} } = \int _{ {\bf r} _{1},{\bf r} _{2} \in \Omega _{j} } d ^{3} r _{1} d ^{3} r _{2} U ( | {\bf r} _{1} - {\bf r} _{2} | ) \psi _{ \uparrow } ^{\dag} ({\bf r} _{1}) \psi _{ \uparrow } ({\bf r} _{1}) \psi _{ \downarrow } ^{\dag} ({\bf r} _{2}) \psi _{ \downarrow } ({\bf r} _{2})$. Assume that all the hopping coefficients $ t _{ j \xi , j ^{\prime} \xi ^{\prime} }$ are so small that every bond in the considered compound is almost independent and is mapped from the one-bond system in Fig. 1 (a). When $| \Psi ^{Br} _{b} \rangle$ is taken as the one-bond wavefunction near the ionic limit as mentioned in Appendix B, we shall take $ | \Psi ^{Br} _{j} \rangle= \sqrt{1- | \lambda _{2} | ^{2} } | \Psi _{j} ^{SCF} \rangle + \lambda _{2} | \Psi _{j} ^{CCD} \rangle$ with $| \Psi ^{SCF} _{j} \rangle=c _{j A^{\prime \prime} \uparrow } ^ {\dag} c _{j A^{\prime \prime} \downarrow } ^ {\dag} | 0 \rangle$ and $| \Psi ^{CCD} _{j} \rangle = c _{j B^{\prime \prime} \uparrow } ^ {\dag} c _{j B^{\prime \prime} \downarrow } ^ {\dag} | 0 \rangle$ for the $j$-th bond based on the mapping ${\bf R}_{j}$. Here $c _{j A^{\prime \prime} \sigma } ^ {\dag}= \sqrt{1- | \lambda _{1} | ^{2} } c _{ j A \sigma} ^{\dag} + \lambda _{1} c _{ j B \sigma} ^{\dagger} $ and $c _{j B^{\prime \prime} \sigma } ^ {\dag} = \sqrt{1- | \lambda _{1} | ^{2} } c _{ j B \sigma} ^{\dag} - \lambda _{1} ^{*} c _{ j A \sigma} ^{\dag} $. We may use Eq. (25) to rewrite $| \Psi ^{Br} _{b} \rangle$ and $ | \Psi ^{Br} _{j} \rangle$ by the ionic-covalent form, and approximate the one- and three-electron states of the one-bond system by $| \Psi ^{\prime} _{b,1,\sigma} \rangle $ and $ | \Psi ^{\prime} _{b,3,\sigma} \rangle $, which are introduced in Appendix B. Let $| \Psi ^{ \prime (j)} _{b,1,\sigma} \rangle = c _{j A^{\prime} \sigma } ^{\dagger} |0 \rangle$ and $| \Psi ^{\prime (j)} _{b,3,\sigma} \rangle = c _{j B^{\prime} \sigma } | Fb ^{(j)} \rangle $ as the $j$-th bond’s states mapped from $| \Psi ^{ \prime } _{b,1,\sigma} \rangle$ and $| \Psi ^{ \prime} _{b,3,\sigma} \rangle$. Here the 4-electron state $|Fb ^{(j)} \rangle$ describes the fully occupied $j$-th bond, and the annihilators $c _{j A^{\prime} \sigma } = \sqrt{ 1 - | \lambda _{3} | ^{2} } c _{ j A \sigma } + \lambda _{3} c _{ j B \sigma }$ and $c _{j B^{\prime} \sigma } = - \lambda _{3} ^{*} c _{ j A \sigma } + \sqrt{ 1 - | \lambda _{3} | ^{2} } c _{ j B \sigma }$. The effective SCF state for the compound is $| \Psi ^{SCF} _{Cr} \rangle = \prod _{j=1} ^{ N } c _{ j A^{\prime \prime} \uparrow } ^ {\dag} c _{j A^{\prime \prime} \downarrow } ^ {\dag} | 0 \rangle$, and $E _{Cr} ^{SCF} =\langle \Psi ^{SCF} _{Cr} | {\cal H} | \Psi ^{SCF} _{Cr} \rangle$ is the SCF value of $E _{Cr}$. For the electron affinity and ionization discussed in subsection IV-B, in the SCF calculation [@Stoyanova; @Grafenstein2] we shall calculate $ \langle \Psi ^{SCF} _{ \eta ^{\prime} \sigma, + } | {\cal H} | \Psi ^{SCF} _{ \eta ^{\prime} \sigma, + } \rangle$ and $\langle \Psi ^{SCF} _{ \eta ^{\prime} \sigma, - } | {\cal H} | \Psi ^{SCF} _{ \eta ^{\prime} \sigma, - } \rangle$ when the carrier excitations occur in the subsystem corresponding to $| \eta ^{\prime} \rangle$. Here $$\begin{aligned} | \Psi ^{SCF} _{ \eta ^{\prime} \sigma, \pm } \rangle= \begin{cases} \sum _{j=1} ^{N} \langle w _{j} | \eta ^{ \prime} \rangle c _{ j B^{ \prime \prime} \underline{ \sigma } } ^{\dag} | \Psi ^{SCF} _{Cr} \rangle \text{ \ \ for \ \ } + \\ \sum _{j=1} ^{N} \langle w _{j} | \eta ^{ \prime} \rangle ^{*} c _{ j A^{ \prime \prime} \underline{ \sigma } } | \Psi ^{SCF} _{Cr} \rangle \text{ \ for \ \ } - \end{cases}\end{aligned}$$ with $\underline{\sigma} = - \sigma$. We can denote the spin-independent values $ \langle \Psi ^{SCF} _{ \eta ^{\prime} \sigma, + } | {\cal H} | \Psi ^{SCF} _{ \eta ^{\prime} \sigma, + } \rangle$ and $\langle \Psi ^{SCF} _{ \eta ^{\prime} \sigma , - } | {\cal H} | \Psi ^{SCF} _{ \eta ^{\prime} \sigma , - } \rangle$ as $E ^{SCF} _{ \eta ^{\prime} , +} $ and $E ^{SCF} _{ \eta ^{\prime} , -} $, respectively, and choose $\sigma= \downarrow$ in Eq. (26) without loss of generality. In the SCF calculation, the added/removed charge in the $j$-th bond equals $| \langle w _{j} | \eta ^{\prime} \rangle |^{2}$ just as that in subsection IV-B, and $E ^{ ( \eta ^{\prime} ,\pm ) }_{Cr} - E _{Cr}$ is approximated as $$\begin{aligned} E ^{SCF} _{ \eta ^{\prime} , \pm} - E ^{SCF}_{Cr} = {\cal K} ^{SCF} _{ \eta ^{\prime} , \pm} + {\cal B}^{SCF} _{ \eta ^{\prime} , \pm} \text{ \ with } \end{aligned}$$ $$\begin{cases} {\cal B}^{SCF} _{ \eta ^{\prime} , \pm} = \sum _{j=1} ^{ N } | \langle w _{j} | \eta ^{\prime} \rangle |^{2} [ \langle j , \pm| (H_{sb} ^{(j)} + \hat{ U } _{j})| j , \pm \rangle - \langle \Psi ^{SCF}_{j} | (H_{sb} ^{(j)} + \hat{ U } _{j}) | \Psi ^{SCF} _{j} \rangle ] \\ {\cal K} ^{SCF} _{ \eta ^{\prime} , +} = \sum _{ j \neq j ^{\prime} } \sum _{ \xi \xi ^{\prime} } t _{ j \xi , j ^{\prime} \xi ^{\prime} } \langle w _{ j ^{\prime} } | \eta ^{\prime} \rangle \langle \eta ^{ \prime} | w _{ j } \rangle \langle j ,+| c _{j \xi \uparrow } ^{\dag} | \Psi ^{SCF} _{j} \rangle \langle \Psi ^{SCF} _{j ^{\prime} } | c _{j ^{\prime} \xi ^{\prime} \uparrow } | j ^{\prime} , + \rangle \\ {\cal K} ^{SCF} _{ \eta ^{\prime} , -} = - \sum _{ j \neq j ^{\prime} } \sum _{ \xi \xi ^{\prime} } t _{ j \xi , j ^{\prime} \xi ^{\prime} } \langle w _{ j ^{\prime} } | \eta ^{\prime} \rangle \langle \eta ^{ \prime} | w _{ j } \rangle \langle \Psi ^{SCF} _{j}| c _{j \xi \uparrow } ^{\dag} | j , - \rangle \langle j ^{\prime} , - | c _{j ^{\prime} \xi ^{\prime} \uparrow } |\Psi ^{SCF} _{j ^{\prime} } \rangle \end{cases}.$$ Here $| j , - \rangle = c _{j A^{\prime \prime} \uparrow } | \Psi ^{SCF} _{j} \rangle$ and $| j , + \rangle = c _{j B^{\prime \prime} \uparrow } ^{\dag} | \Psi ^{SCF} _{j} \rangle$. The energy factors ${\cal B}^{SCF} _{ \eta ^{\prime} , \pm}$ and ${\cal K} ^{SCF} _{ \eta ^{\prime} , \pm}$ are due to $\sum _{ j } (H_{sb} ^{(j)} + \hat{U _{j} })$ and $ \sum _{ j \neq j ^{ \prime}} H_{hop} ^{(j, j^{\prime})} $, respectively. To include the coupled-cluster corrections, in Eq. (27) we shall consider the two-particle excitation [@Stoyanova; @Grafenstein2] ${\cal S}^{(2)}_{J}= \frac{\lambda _{2}}{ \sqrt{ 1 - |\lambda _{2}| ^{2} } } c _{J B^{\prime \prime} \uparrow } ^ {\dag} c _{ J A^{\prime \prime} \uparrow } c _{J B^{\prime \prime} \downarrow } ^ {\dag} c _{J A^{\prime \prime} \downarrow } $ to replace $| \Psi ^{SCF} _{J} \rangle $ by $ | \Psi _{J} ^{Br} \rangle = \sqrt{ 1 - |\lambda _{2}| ^{2} } \times exp( {\cal S} ^{(2)} _{J}) | \Psi ^{SCF} _{J} \rangle $ for any $J=j$ or $j ^{\prime}$. In addition, we may introduce the one-particle excitation [@Stoyanova; @Grafenstein2] ${\cal S}^{(1)}_{J}= \frac{\lambda _{4}}{ \sqrt{ 1 - |\lambda _{4}| ^{2} } } c _{ J B ^{\prime \prime} \downarrow } ^{ \dag } c _{J A^{\prime \prime} \downarrow } $ to modify $| J, - \rangle$ and $| J, + \rangle$ as $| \Psi ^{\prime} _{b,1,\downarrow} \rangle = \sqrt{ 1 - |\lambda _{4}| ^{2} } \times exp ( {\cal S}^{(1)}_{J} ) | J , - \rangle $ and $| \Psi ^{\prime} _{b,3,\downarrow} \rangle= \sqrt{ 1 - |\lambda _{4}| ^{2} } \times exp( {\cal S}^{(1)}_{J} ) | J , + \rangle $, respectively, in the $J$-th bond. Here $\lambda _{4} = \lambda _{3} ^{\ast} \sqrt{ 1 - | \lambda _{1} | ^{2} } - \lambda _{1} \sqrt{ 1 - | \lambda _{3} | ^{2} } $. The generator for the ground-state correlation [@Grafenstein] equals $\sum _{J} {\cal S} ^{(2)} _{J}$, and the operator ${\cal S} ^{(1)} _{j} + \sum _{j \neq J} {\cal S} ^{(2)} _{J}$ generates the correlated states corresponding to $c _{ j B^{ \prime \prime} \uparrow } ^{\dag} | \Psi ^{SCF} _{Cr} \rangle$ and $c _{ j A^{ \prime \prime} \uparrow } | \Psi ^{SCF} _{Cr} \rangle$ [@Grafenstein; @Stoyanova; @Grafenstein2]. When the coupled-cluster method is applicable near the ionic limit, as mentioned in Appendix B, it is important to perform the orbital transformation in Eq. (25) to improve my model. Rewriting $| \Psi _{b} ^{Br} \rangle $ as the ionic-covalent form by performing such a transformation, in the improved model we can re-obtain the difference $ E _{b} ^{(\pm)} - E _{b}$ in Eq. (22) by using the coupled-cluster method to modify ${\cal B}^{SCF} _{ \eta ^{\prime} , \pm}$. The coupled-cluster corrections to $ {\cal K} ^{SCF} _{ \eta ^{\prime} , \pm}$, in fact, are small and we have $\langle \eta ^{ \prime} | H _{\pm } | \eta ^{ \prime } \rangle \simeq {\cal K} ^{SCF} _{ \eta ^{\prime} , \pm}$ under the assumption about the small hopping coefficients. So we can use the coupled-cluster method to correct $E ^{SCF} _{ \eta ^{\prime} , \pm} - E ^{SCF}_{Cr}$ and obtain the difference close to $E_{Cr} ^{ ( \eta ^{\prime} , \pm ) } -E_{Cr}$. In addition, the added/removed charge in the $j$-th bond equals $| \langle w _{j} | \eta ^{\prime} \rangle |^{2}$, which is the same as that in subsection IV-B, because the above one- and two-particle excitations do not change the number of electrons in each bond. Therefore, my improved model can be supported by the coupled-cluster theory near the ionic limit. Appendix D {#appendix-d .unnumbered} ========== The matrix $\rho ^{(2)} _{ \omega _{j} }$ in Eq. (14) represents the quasielectron at the orbital $ | \omega _{j} \rangle \otimes | L \rangle = \alpha _{i} | A, j \rangle + \alpha _{c} | B, j \rangle$. When the ionic-covalent bonds in the compound are not identical to each other, the bonding coefficients can depend on $j$ and we shall replace $\alpha _{i}$ and $\alpha _{c}$ by $\alpha _{i} ^{(j)}$ and $\alpha _{c} ^{(j)}$ in the $j$-th bond. Here the coefficients $\alpha _{i} ^{(j)}$ and $\alpha _{c} ^{(j)}$ satisfy $| \alpha _{i} ^{(j)} | ^{2} + | \alpha _{c} ^{(j)} | ^{2} =1$ for all $j=1 \sim N$. The matrix $\rho ^{(2)} _{ \omega _{j} }$ should be modified as $(\alpha _{i} ^{(j)} | A, j \rangle + \alpha _{c} ^{(j)} | B, j \rangle ) (\alpha _{i} ^{(j)} \langle A, j | + \alpha _{c} ^{(j)} \langle B, j |) $ and we can still take $\rho ^{(1)} _{ \omega _{j} } = | A, j \rangle \langle A, j |$. Equations (14) and (15) remain valid after the modification, and the energy $E _{Cr}$ can still be obtained based on Eq. (18). In subsection IV-B, the quasielectron at $| \eta ^{\prime } \rangle \otimes | L \rangle = \sum _{j} \langle \omega _{j} | \eta ^{\prime} \rangle (\alpha _{i} | A, j \rangle + \alpha _{c} | B, j \rangle)$ is ionized from subsystem $\eta ^{ \prime}$ while one quasielectron enters $| \eta ^{\prime } \rangle \otimes | \bar{L} \rangle = \sum _{j} \langle \omega _{j} | \eta ^{\prime} \rangle (\alpha _{c} ^{*} | A, j \rangle - \alpha _{i} ^{*} | B, j \rangle)$ in the affinitive process. When the coefficients for the ionic and covalent parts depend on $j$, the orbital of the quasielectron to be ionized should be modified as $ \sum _{j} \langle \omega _{j} | \eta ^{\prime} \rangle (\alpha _{i} ^{(j)} | A, j \rangle + \alpha _{c} ^{(j)} | B, j \rangle) \equiv| \eta ^{\prime} _{L} \rangle $. In addition, the orbital for the added quasielectron becomes $ \sum _{j} \langle \omega _{j} | \eta ^{\prime} \rangle (\alpha _{c} ^{(j) \ast} | A, j \rangle - \alpha _{i} ^{(j) \ast} | B, j \rangle) \equiv | \eta ^{\prime} _{ \bar{L} } \rangle$. So we need to modify the first two matrices in Eq. (21) as $\rho ^{( \text{II} )} _{ \eta ^{\prime} , + } = \rho ^{( \text{II} )} + | \eta ^{\prime} _{ \bar{L} } \rangle \langle \eta ^{\prime} _{ \bar{L} } |$ and $\rho ^{( \text{II} )} _{ \eta ^{\prime} , - } = \rho ^{( \text{II} )} - | \eta ^{\prime} _{L} \rangle \langle \eta ^{\prime} _{L} |$. The energy $E_{Cr} ^{ ( \eta ^{\prime} , \pm ) } $ can be calculated based on Eq. (20), where $d ^{( \text{I} )} _{ \eta ^{\prime} ,\pm } $ and $d ^{( \text{II} )} _{ \eta ^{\prime} ,\pm } $ can still be obtained from the last two lines in Eq. (21). In the compound composed of different chemical bonds, therefore, Eqs. 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--- abstract: 'It is well known that the three parameters that characterize the Kerr black hole (mass, angular momentum and horizon area) satisfy several important inequalities. Remarkably, some of these inequalities remain valid also for dynamical black holes. This kind of inequalities play an important role in the characterization of the gravitational collapse. They are closed related with the cosmic censorship conjecture. In this article recent results in this subject are reviewed.' author: - | Sergio Dain\ Facultad de Matemática, Astronomía y Física, FaMAF,\ Universidad Nacional de Córdoba,\ Instituto de Física Enrique Gaviola, IFEG, CONICET,\ Ciudad Universitaria (5000) Córdoba, Argentina. title: Geometric inequalities for black holes --- Geometric inequalities in General Relativity {#sec:geom-ineq} ============================================ A classical example of a geometric inequality is the isoperimetric inequality for closed plane curves given by $$\label{eq:54} L^2 \geq 4\pi A\quad (=\text{ circle}),$$ where $A$ is the area enclosed by a curve $C$ of length $L$. In (\[eq:54\]) equality holds if and only if $C$ is a circle, see figure \[fig:1\]. For a review on this subject see [@Osserman78]. {width="3cm"} {width="3cm"} \[fig:1\] The inequality (\[eq:54\]) applies to complicated geometric objects (i.e. arbitrary closed planar curves). The equality in (\[eq:54\]) is achieved only for an object of “optimal shape” (i.e. the circle) which is described by few parameters (in this case only one: the radius). Moreover, this object has a variational characterization: the circle is uniquely characterized by the property that among all simple closed plane curves of given length $L$, the circle of circumference $L$ encloses the maximum area. General Relativity is a geometric theory, hence it is not surprising that geometric inequalities appear naturally in it. Many of these inequalities are similar in spirit as the isoperimetric inequality (\[eq:54\]). In particular, all the geometric inequalities discussed in this article will have the same structure as (\[eq:54\]): the inequality applies for a rich class of objects and the equality only applies for an object of “optimal shape” (always indicated in parenthesis as in (\[eq:54\])). This object, like the circle, can be described by few parameters and it has also a variational characterization. However, General Relativity is also a physical theory. It is often the case that the quantities involved have a clear physical interpretation and the expected behavior of the gravitational and matter fields often suggests geometric inequalities which can be highly non-trivial from the mathematical point of view. The interplay between physics and geometry gives to geometric inequalities in General Relativity their distinguished character. These inequalities relate quantities that have both a physical interpretation and a geometrical definition. The plan of this article follows this interplay between physics and mathematics. In section \[sec:physical-picture\] we present the physical motivations for the black holes geometric inequalities. In section \[sec:theorems\] we summarize some theorems where these inequalities have been recently proved. Finally, in section \[sec:open-problems-recent\] we list relevant open problems and we also describe recent results on geometric inequalities for bodies. Physical picture {#sec:physical-picture} ================ An important example of a geometric inequality is the positive mass theorem. Let $m$ be the total ADM mass on an asymptotically flat complete initial data such that the dominant energy condition is satisfied. Then we have $$\label{eq:1} 0\leq m\quad (=\text{ Minkowski}).$$ The mass $m$ is a pure geometrical quantity [@Arnowitt62][@Bartnik86][@chrusciel86]. However, from the geometrical mass definition, without the physical picture, it would be very hard even to conjecture the inequality (\[eq:1\]). In fact the proof of the positive mass theorem turns out to be very subtle [@Schoen79b][@Schoen81][@witten81]. A key assumption in the positive mass theorem is that the matter fields should satisfy an energy condition. This condition is expected to hold for all physically realistic matter. This kind of general properties which do not depend very much on the details of the model are not easy to find for a macroscopic object. And hence it is difficult to obtain simple and general geometric inequalities among the parameters that characterize ordinary macroscopic objects. Black holes represent a unique class of very simple macroscopic objects and hence they are natural candidates for geometrical inequalities. Nevertheless, in section \[sec:open-problems-recent\] we will present also a geometric inequality valid for ordinary bodies. The black hole uniqueness theorem ensures that stationary black holes in vacuum are characterized by the Kerr exact solution of Einstein equations [^1]. For simplicity we will not consider the electromagnetic field in this article, however most of the results presented here can be generalized to include that case. It is somehow remarkable that the same family of solutions of Einstein equations that describe the unique stationary black hole (i.e. the Kerr metric) also describe naked singularities. In effect, the Kerr metric depends on two parameters: the mass $m$ and the angular momentum $J$. This metric is a solution of Einstein vacuum equations for any choice of the parameters $m$ and $J$. However, it represents a black hole if and only if the following remarkably inequality holds $$\label{eq:2} \sqrt{|J|}\leq m.$$ Otherwise the spacetime contains a naked singularity. Figure \[fig:2\] shows the parameter space of the Kerr solution. Extreme black holes are defined by the equality in (\[eq:2\]). These black holes lie at the boundary between naked singularities and black holes. For most of the inequalities discussed in this article, extreme black holes play the role of the circle in the isoperimetric inequality (\[eq:54\]): they reach the equality and they represent objects of “optimal shape”. ![A point in this graph is a Kerr solution with parameters $m$ and $J$. The horizontal axis where $m=0$ is Minkowski space. The Schwarzschild solution is given by the vertical axis where $J=0$. In the gray region the parameters satisfy the inequality (\[eq:2\]) and hence the Kerr solution describe a black hole. The boundary of this region is given by the equality in (\[eq:2\]), these solutions are called extreme black holes. In the white region, excluding the horizon axis, the Kerr solution contains a naked singularity. That includes also the negative mass region. []{data-label="fig:2"}](kerr-parametros-p.pdf){width="70.00000%"} The area of the horizon of the Kerr black hole is given by the simple but very important formula $$\label{eq:3} A=8\pi \left(m^2+ \sqrt{m^4-J^2} \right).$$ From equation (\[eq:3\]) we deduce that the following three geometric inequalities hold for a Kerr black hole $$\begin{aligned} \sqrt{\frac{A}{16\pi}} &\leq m &(=\text{Schwarzschild}),\label{eq:pen}\\ \sqrt{|J|} &\leq m &(= \text{Extreme Kerr}),\label{eq:mj}\\ 8\pi |J| &\leq A &(= \text{Extreme Kerr}).\label{eq:JA}\end{aligned}$$ As expected from the discussion above, the inequality (\[eq:mj\]) is needed to define the black hole horizon area in (\[eq:3\]): if (\[eq:mj\]) does not hold, then the expression (\[eq:3\]) is not a real number. We have listed this inequality again here to emphasize its connection with the other two in the following discussion. Inequalities (\[eq:pen\]) and (\[eq:JA\]) follow from (\[eq:mj\]) and (\[eq:3\]). Note that these inequalities relate the three relevant parameters of the Kerr black hole $(m,J,A)$. Let us discuss the physical meaning of the inequalities (\[eq:pen\]), (\[eq:mj\]) and (\[eq:JA\]). In the inequality (\[eq:pen\]), the difference $$\label{eq:4b} m-\sqrt{\frac{A}{16\pi}},$$ represents the rotational energy of the Kerr black hole. This is the maximum amount of energy that can be extracted from the black hole by the Penrose process [@Christodoulou70]. When the difference (\[eq:4b\]) is zero, the black hole has no angular momentum and hence it is the Schwarzschild black hole. From Newtonian considerations, we can interpret the inequality (\[eq:mj\]) as follows [@Wald71]. In a collapse the gravitational attraction ($\approx m^2/r^2$) at the horizon ($r \approx m $) dominates over the centrifugal repulsive forces ($\approx J^2/mr^3$). Finally, concerning the inequality (\[eq:JA\]), the black hole temperature is given by the following formula $$\label{eq:5c} \kappa= \frac{1}{4 m} \left(1-\frac{(8\pi J)^2 }{A^2} \right).$$ The temperature is positive if and only if the inequality (\[eq:JA\]) holds. Moreover the temperature is zero if and only if the equality in (\[eq:JA\]) holds and hence the black hole is extreme. There exists another relevant geometrical inequality which can be deduced from the formula (\[eq:3\]) $$\label{eq:10} 8\pi \left( m^2-\sqrt{m^4-J^2} \right) \leq A \quad (= \text{Extreme Kerr}).$$ Remarkably, as it was pointed out in [@Khuri:2013wha] for the case of the electric charge and in [@Dain:2013qia] for the present case of angular momentum, the inequality (\[eq:10\]) can be deduced purely from the inequalities (\[eq:mj\]) and (\[eq:JA\]) (i.e. without using the equality (\[eq:3\])) by simple algebra. Namely $$\begin{aligned} \label{eq:11} m^2 &= \sqrt{m^4-J^2+J^2},\\ & \leq |J| +\sqrt{m^4-J^2}, \label{eq:11b} \\ & \leq \frac{A}{8\pi}+ \sqrt{m^4-J^2},\label{eq:11c}\end{aligned}$$ where in the line (\[eq:11b\]) we have used (\[eq:mj\]) and in line (\[eq:11c\]) we have used (\[eq:JA\]). In that sense, the inequalities (\[eq:pen\]), (\[eq:mj\]) and (\[eq:JA\]) are more fundamental than (\[eq:10\]). However, the inequality (\[eq:10\]) is important by itself since it related with the Penrose inequality with angular momentum, see [@Khuri:2013wha] [@Dain:2013qia]. We have seen that for stationary black holes the inequalities (\[eq:pen\]), (\[eq:mj\]) and (\[eq:JA\]) are straightforward consequences of the area formula (\[eq:3\]). ![Schematic representation of an initial data for a non-stationary black hole. The black ring represents a trapped surface. Outside and inside the trapped surface the gravitational field is highly dynamical.[]{data-label="fig:3"}](dynamical-bh-p.pdf){width="6cm"} However, black holes are in general non stationary, see figure \[fig:3\]. Astrophysical phenomena like the formation of a black hole by gravitational collapse or a binary black hole collision are highly dynamical. For such systems, the black hole can not be characterized by few parameters as in the stationary case. In fact, even stationary but non-vacuum black holes have a complicated structure (for example black holes surrounded by a rotating ring of matter, see the numerical studies in [@Ansorg05]). Remarkably, inequalities (\[eq:pen\]), (\[eq:mj\]) and (\[eq:JA\]) extend (under appropriate assumptions) to the fully dynamical regime. Moreover, these inequalities are deeply connected with properties of the global evolution of Einstein equations, in particular with the cosmic censorship conjecture. To discuss the physical arguments that support these inequalities in the dynamical regime it is convenient to start with the inequality . For a dynamical black hole, the physical quantities that are well defined are the total ADM mass $m$ of the spacetime and the area $A$ of the black hole horizon. The total mass $m$ of the spacetime measures the sum of the black hole mass and the mass of the gravitational waves surrounding it. In the stationary case, the mass of the black hole is equal to the total mass of the spacetime, but this is no longer true for a dynamical black hole. The mass $m$ is a global quantity, it carries information on the whole spacetime. In contrast, the area of the horizon $A$ is a quasi-local quantity, it carries information on a bounded region of the spacetime. It is well known that the energy of the gravitational field cannot be represented by a local quantity (i.e. a scalar field). The best one can hope is to obtain a quasi-local expression. The same applies to the angular momentum. In general, it is difficult to find physically relevant quasi-local quantities like mass and angular momentum (see the review article [@Szabados04]). However, in axial symmetry, there is a well defined notion of quasi-local angular momentum: the Komar integral of the axial Killing vector. Moreover, the angular momentum is conserved in vacuum. That is, axially symmetric gravitational waves do not carry angular momentum. Then, for axially symmetric dynamical black holes we have two well defined quasi-local quantities: the area of the horizon $A$ and the angular momentum $J$. Note that the inequality relates only quasi-local quantities. Using $A$ and $J$ we can define the quasi-local mass for a dynamical black hole by the Kerr formula (\[eq:3\]), that is $$\label{eq:masa} {m_{bh}}= \sqrt{\frac{A}{16\pi}+\frac{4\pi J^2}{A}}.$$ This is, in principle, just a definition. Since ${m_{bh}}$ is given by the Kerr formula (\[eq:3\]) it automatically satisfies the inequalities (\[eq:mj\]) and (\[eq:pen\]). However, the relevant question is: does ${m_{bh}}$ describes the quasi-local mass of a non-stationary black hole? This question is closed related to the validity of the inequality (\[eq:JA\]) in the dynamical regime. In order to answer it let us analyze the evolution of ${m_{bh}}$. For a dynamical black hole, by the area theorem, we know that the horizon area $A$ increase with time, see figure \[fig:4\]. ![The area theorem. The horizon area of a dynamical black hole increase with time.[]{data-label="fig:4"}](area-th-p.pdf){width="6cm"} In general, the quasi-local mass of the black hole is not expected to be a monotonically increasing quantity. Energy can be extracted from a rotating black hole by the Penrose process. However, if we assume axial symmetry then the angular momentum will be conserved at the quasi-local level. On physical grounds, one would expect that in this situation the quasi-local mass of the black hole should increase with the area, since there is no mechanism at the classical level to extract mass from the black hole. In effect, the Penrose process involves an interchange of angular momentum between the black hole and the exterior. But the angular momentum transfer is forbidden in axial symmetry. Then, both the area $A$ and the quasi- local mass ${m_{bh}}$ should monotonically increase with time in axial symmetry. Let us take a time derivative of ${m_{bh}}$. To analyze this, it is illustrative to write down the complete differential, namely the first law of thermodynamics $$\label{eq:mq} \delta {m_{bh}}= \frac{\kappa}{8 \pi} \delta A + \Omega_H \delta J,$$ where $$\label{eq:7} \kappa= \frac{1}{4{m_{bh}}} \left(1-\frac{(8\pi J)^2 }{A^2} \right),\quad \Omega_H=\frac{4\pi J}{A \,{m_{bh}}}.$$ In equation (\[eq:mq\]) we have followed the standard notation for the formulation of the first law; we emphasize, however, that in our context this equation is a trivial consequence of . In axial symmetry $\delta J=0$ and hence we obtain $$\delta {m_{bh}}= \frac{\kappa}{8 \pi} \delta A.$$ By the area theorem we have $$\delta A \geq 0.$$ Then $\delta {m_{bh}}\geq 0$ if and only if $\kappa \geq 0$, that is $\delta {m_{bh}}\geq 0$ if and only if the inequality (\[eq:JA\]) holds. Then, it is natural to conjecture that this inequality should be satisfied for any axially symmetric black hole. If the horizon violates (\[eq:JA\]), then in the evolution the area will increase but the mass ${m_{bh}}$ will decrease. This will indicate that the quantity ${m_{bh}}$ does not have the desired physical meaning. Also, a rigidity statement is expected. Namely, the equality in (\[eq:JA\]) is reached only by the extreme Kerr black hole where $\kappa=0$. This inequality provides a remarkable quasi-local measure of how far a dynamical black hole is from the extreme case, namely an ‘extremality criteria’ in the spirit of [@Booth:2007wu], although restricted only to axial symmetry. In the article [@Dain:2007pk] it has been conjectured that, within axially symmetry, to prove the stability of a nearly extreme black hole is perhaps simpler than a Schwarzschild black hole. It is possible that this quasi-local extremality criteria will have relevant applications in this context. Note also that the inequality allows to define, at least formally, the positive temperature of a dynamical black hole $\kappa$ by the formula (\[eq:7\]) (see Refs. [@Ashtekar03] [@Ashtekar02] for a related discussion of the first law in dynamical horizons). If inequality holds, then ${m_{bh}}$ defines a non-trivial quantity that increase monotonically with time, like the black hole area $A$. It is important to emphasize that the physical arguments presented above in support of are certainly weaker in comparison with the ones behind the Penrose inequalities that support the inequalities (\[eq:pen\]) and (\[eq:mj\]) that we will discuss bellow. A counter example of any of these inequality will prove that the standard picture of the gravitational collapse is wrong. On the other hand, a counter example of will just prove that the quasi-local mass is not appropriate to describe the evolution of a non-stationary black hole. One can imagine other expressions for quasi-local mass, may be more involved, in axial symmetry. On the contrary, reversing the argument, a proof of will certainly suggest that the mass has physical meaning for non-stationary black holes as a natural quasi-local mass (at least in axial symmetry). Also, the inequality provide a non trivial control of the size of a black hole valid at any time. In a seminal article Penrose [@Penrose73] proposed a remarkably physical argument that connects global properties of the gravitational collapse with geometric inequalities on the initial conditions. That argument lead to the well known Penrose inequality (\[eq:pen\]) for dynamical black holes (without any symmetry assumption). In the following we review this argument imposing axial symmetry, where angular momentum is conserved. And, more important, we include a relevant new ingredient: we assume that the inequality (\[eq:JA\]) holds. We will assume that the following statements hold in a gravitational collapse: - Gravitational collapse results in a black hole (weak cosmic censorship). - The spacetime settles down to a stationary final state. We will further assume that at some finite time all the matter have fallen into the black hole and hence the exterior region is vacuum. Conjectures (i) and (ii) constitute the standard picture of the gravitational collapse. Relevant examples where this picture is confirmed (and where the role of angular momentum is analyzed) are the collapse of neutron stars studied numerically in [@Baiotti:2004wn] [@Giacomazzo:2011cv]. The black hole uniqueness theorem implies that the final stationary state postulated in (ii) is given by the Kerr black hole. Let us denote by $m_0, J_0, A_0$, respectively, the mass, angular momentum and horizon area of the remainder Kerr black hole. Penrose argument runs as follows. Take a Cauchy surface $S$ in the spacetime such that the collapse has already occurred. This is shown in figure \[fig:5\]. ![The Penrose diagram of a gravitational collapse. The initial Cauchy surface is denoted by $S$. The area $A$ increase along the event horizon. The mass $m$ decrease along null infinity. We have assumed axial symmetry and hence the angular momentum remains constant along null infinity $J=J_0$.[]{data-label="fig:5"}](penrose-ineq-js-p.pdf){width="50.00000%"} Let ${\Sigma}$ denotes the intersection of the event horizon with the Cauchy surface $S$ and let $A$ be its area. Let $(m, J)$ be the total mass and angular momentum at spacelike infinity. These quantities can be computed from the initial surface $S$. By the black hole area theorem we have that the area of the black hole increase with time and hence $$\label{eq:15} A_0\geq A.$$ Since gravitational waves carry positive energy, the total mass of the spacetime should be bigger than the final mass of the remainder Kerr black hole $$\label{eq:4} m\geq m_0.$$ The difference $m-m_0$ is the total amount of gravitational radiation emitted by the system. To related the initial angular momentum $J$ with the final angular momentum $J_0$ is much more complicated. Angular momentum is in general non-conserved. There exists no simple relation between the total angular momentum $J$ of the initial conditions and the angular momentum $J_0$ of the final black hole. For example, a system can have $J=0$ initially, but collapse to a black hole with final angular momentum $J_0\neq 0$. We can imagine that on the initial conditions there are two parts with opposite angular momentum, one of them falls in to the black hole and the other escape to infinity. Axially symmetric vacuum spacetimes constitute a remarkable exception because the angular momentum is conserved. In that case we have $$\label{eq:59} J=J_0.$$ For a discussion of this conservation law in detail see [@dain12] and reference therein. We have assumed that the inequality holds, then by the discussion above we have that the quasi-local mass ${m_{bh}}$ increase with time, that is $$\label{eq:8b} {m_{bh}}\leq m_0.$$ We emphasize that this inequality is highly non-trivial. The quantity ${m_{bh}}$ is computed on the initial surface $S$, in contrast to compute $m_0$ we need to known the whole spacetime. Using (\[eq:8b\]) and (\[eq:4\]) we finally obtain $$\label{eq:6} \sqrt{\frac{A}{16\pi}+\frac{4\pi J^2}{A}}= {m_{bh}}\leq m.$$ This inequality has the natural interpretation that the mass of the black hole ${m_{bh}}$ should always be smaller than the total mass of the spacetime $m$. The inequality (\[eq:6\]) represents a generalization of the Penrose inequality with angular momentum. This inequality implies $$\label{eq:mjd} \sqrt{|J|}\leq m.$$ In fact, the inequality can be deduced directly by the same heuristic argument without using the area theorem. It depends only on the following assumptions - Gravitational waves carry positive energy. - Angular momentum is conserved in axial symmetry. - In a gravitational collapse the spacetime settles down to a final Kerr black hole. Let us summarize the discussion of this section. For an axially symmetric, dynamical black hole, the following two geometrical inequalities are expected $$\begin{aligned} 8\pi |J| & \leq A \quad (=\text{Extreme Kerr horizon}), \label{eq:JAd}\\ \sqrt{\frac{A}{16\pi}+\frac{4\pi J^2}{A}} & \leq m \quad (=\text{Kerr black hole}). \label{eq:pendj} \end{aligned}$$ The inequality is quasi-local and the inequality is global. The global inequality implies the following two inequalities $$\begin{aligned} \sqrt{\frac{A}{16\pi}} &\leq m \quad (=\text{Schwarzschild}),\label{eq:penddd}\\ \sqrt{|J|} &\leq m. \quad (=\text{extreme Kerr black hole}).\label{eq:mjdd}\end{aligned}$$ That is: > *The three geometrical inequalities (\[eq:pen\]), (\[eq:mj\]) and (\[eq:JA\]) valid for the Kerr black holes are expected to hold also for axially symmetric, dynamical black holes.* The Penrose inequality is valid also without the axial symmetry assumption. It is important to emphasize that all the quantities involved in the geometrical inequalities above can be calculated on the initial surface. For simplicity, we have avoided the distinction between event horizon and apparent horizons (defined in terms of trapped surfaces) to calculate the area $A$. This point is important for the Penrose inequality (see the discussion in [@Mars:2009cj]) but not for the other inequalities which are the main subject of this review. In particular the horizon area $A$ in (\[eq:JAd\]) is the area of an appropriated defined trapped surface. A counter example of the global inequality (\[eq:pendj\]) will imply that cosmic censorship is not true. Conversely a proof of it gives indirect evidence of the validity of censorship, since it is very hard to understand why this highly nontrivial inequality should hold unless censorship can be thought of as providing the underlying physical reason behind it. The inequalities (\[eq:pen\]), (\[eq:mj\]) and (\[eq:JA\]) can be divided into two groups: 1. $\sqrt{\frac{A}{16\pi}}\leq m$: the area appears as lower bound. 2. $\sqrt{|J|} \leq m$ and $ 8\pi |J| \leq A $: the angular momentum appears as lower bound and the area appears as upper bound. The mathematical methods used to study these two groups are, up to now, very different. This review is mainly concerned with the second group. Finally, we mention that for the Kerr black hole there exists a remarkable equality of the form $(8\pi J)^2 =A^+A^-$, where $A^+$ and $A^-$ denote the areas of event and Cauchy horizon (see figure \[fig:kerr-diag\]). This equality has been proved for general stationary spacetimes in the following series of articles [@Ansorg:2009yi] [@Hennig:2009aa] [@Ansorg:2008bv]. It has recently received considerable attention in the string community (see [@Cvetic:2010mn] and [@Visser:2012wu] and references therein). The key property used in these studies is that the product of horizon areas is independent of the mass of the black hole. It is interesting to note that there exists, up to now, no generalization of this kind of equality (or a related inequality) to the dynamical regime. Theorems {#sec:theorems} ======== The Penrose inequality $$\label{eq:9} \sqrt{\frac{A}{16\pi}} \leq m \quad (=\text{Schwarzschild}),$$ has been intensively studied. It is a very relevant geometric inequality for black holes since it is valid without any symmetry assumption. For a comprehensive review on this subject see [@Mars:2009cj]. The most important results concerning this inequality are the proofs of Huisken-Ilmanen [@Huisken01] and Bray [@Bray01] for the Riemannian case. The general case remains open. Also, there is up to now no result concerning the Penrose inequality with angular momentum (\[eq:pendj\]) discussed in the previous section. In the following we present a sample of the main results concerning inequalities (\[eq:mjdd\]) and (\[eq:JAd\]) that have been recently proved. For the global inequality (\[eq:mjdd\]) we have the following theorem. \[t:1\] Consider an axially symmetric, vacuum, asymptotically flat and maximal initial data set with two asymptotics ends. Let $m$ and $J$ denote the total mass and angular momentum at one of the ends. Then, the following inequality holds $$\label{eq:60} \sqrt{|J|} \leq m \quad (= \text{Extreme Kerr}).$$ For the precise definitions, fall off conditions an assumptions on the initial data we refer to original articles cited bellow. The first proof of the global inequality (\[eq:60\]) was provided in a series of articles [@Dain05c], [@Dain05d], [@Dain05e] which end up in the global proof given in [@Dain06c]. The proof is based on a variational characterization of the extreme Kerr initial data. In [@Chrusciel:2007dd] and [@Chrusciel:2007ak] the result was generalized and the proof simplified. In [@Chrusciel:2009ki] [@Costa:2009hn] the charge was included. In [@Schoen:2012nh] relevant improvements on the rigidity statements were made. In particular in that article it was proved the first rigidity result including charge and a measure of the distance to extreme Kerr black hole was introduced. In [@zhou12] the result was proved with the maximal condition replaced by a small trace assumption for the second fundamental form of the initial data. Related results concerning the force between black holes were proved in [@Clement:2012np]. Finally, the mass formula and the variational techniques involved in the proof of the inequality (\[eq:60\]) were very recently used to study the linear stability of the extreme Kerr black hole [@Dain:2014iba]. Under the hypothesis of theorem \[t:1\] (namely, vacuum and axial symmetry) the angular momentum is defined as conserved quasi-local integral. In particular, if the topology of the manifold is trivial (i.e. ${\mathbb{R}^3}$), then the angular momentum is zero and hence theorem \[t:1\] reduces to the positive mass theorem. In order to have non-zero angular momentum we need to allow non-trivial topologies, for example manifolds with two asymptotic ends as it is the case in theorem \[t:1\]. An important initial data set that satisfies the hypothesis of the theorem is provided by an slice $t=constant$ in the Kerr black hole in the standard Boyer-Lindquist coordinates, see figures \[fig:kerr-diag\] and \[fig:kerr-extrem-diag\]. The non-extreme initial data have a different geometry as the extreme initial data. The former are asymptotically flat at both ends. In contrast, extreme initial data, which reach the equality in (\[eq:60\]), have one asymptotically flat end and one cylindrical end, see figure \[fig:non-extreme-id\]. That geometry represents the “optimal shape” with respect to the inequality (\[eq:60\]). Figure \[fig:non-extreme-id\] is the analog of figure \[fig:1\] for the geometrical inequality (\[eq:60\]). ![Conformal diagram of the non-extreme Kerr black hole. The points $i_0$ represent spacelike infinity. The surface $S$ have two identical asymptotically flat ends $i_0$.[]{data-label="fig:kerr-diag"}](kerr-color-p.pdf){width="6cm"} ![Conformal diagram of the extreme Kerr black hole. The point $i_0$ represents spacelike infinity, the point $i_c$ represent the cylindrical end. The surface $S$ has one asymptotically flat end $i_0$ and one cylindrical end $i_c$.[]{data-label="fig:kerr-extrem-diag"}](kerr-extremo-color-p.pdf){width="5cm"} ![On the left, an the initial data with two asymptotically flat ends, like the non-extreme Kerr black holes. For these data the strict inequality holds. On the right, the data of extreme Kerr black hole, with one asymptotically flat and one cylindrical end. For this data the equality holds.[]{data-label="fig:non-extreme-id"}](kerr-initial-data-iso-p.pdf "fig:"){width="30.00000%"} ![On the left, an the initial data with two asymptotically flat ends, like the non-extreme Kerr black holes. For these data the strict inequality holds. On the right, the data of extreme Kerr black hole, with one asymptotically flat and one cylindrical end. For this data the equality holds.[]{data-label="fig:non-extreme-id"}](kerr-ex-initial-data-iso-p.pdf "fig:"){width="35.00000%"} Regarding the quasi-local inequality (\[eq:JAd\]) we have the following result. \[t:2\] Given an axisymmetric closed marginally trapped and stable surface ${\Sigma}$, in a spacetime with non-negative cosmological constant and fulfilling the dominant energy condition, it holds the inequality $$\label{eq:JAt} 8\pi |J| \leq A \quad (= \text{Extreme Kerr throat}),$$ where $A$ and $J$ are the area and angular momentum of ${\Sigma}$. This is a pure spacetime and local result. That is, there is no mention of a three-dimensional initial hypersurface where the two-dimension surface ${\Sigma}$ is embedded. Axisymmetry is only imposed on ${\Sigma}$. Moreover, this theorem does not assume vacuum. The matter fields can have also angular momentum and it can be transferred to the black hole, however the inequality (\[eq:JAt\]) remains true even for that case. It is important to note that the angular momentum that appears in (\[eq:JAt\]) is the gravitational one (i.e. the Komar integral). In fact this inequality is non-trivial even for the Kerr-Newman black hole, see the discussion in [@dain12]. Theorem \[t:2\] has the following history. The quasi-local inequality (\[eq:JAt\]) was first conjectured to hold in stationary spacetimes surrounded by matter in [@Ansorg:2007fh]. In that article the extreme limit of this inequality was analyzed and also numerical evidences for the validity in the stationary case was presented (using the numerical method and code developed in [@Ansorg05]). In a series of articles [@hennig08] [@Hennig:2008zy] the inequality (\[eq:JAt\]) (including also the electromagnetic charge) was proved for that class of stationary black holes. See also the review article [@Ansorg:2010ru]. In the dynamical regime, the inequality (\[eq:JAt\]) was conjectured to hold in [@dain10d] based on the heuristic argument mentioned in section \[sec:physical-picture\]. In that article also the main relevant techniques for its proof were introduced, namely the mass functional on the surface and its connections with the area. A proof (but with technical restrictions) was obtained in [@Acena:2010ws] [@Clement:2011kz]. The first general and pure quasi-local result was proven in [@Dain:2011pi], where the relevant role of the stability condition for minimal surfaces was pointed out. The generalization to trapped surfaces and non-vacuum has been proved in [@Jaramillo:2011pg]. The electromagnetic charge was included in [@Clement:2011np] and [@Clement:2012vb]. This inequality has been extended to higher dimensions in [@Hollands:2011sy] and [@Paetz:2013rka]. In [@Yazadjiev:2012bx] [@Yazadjiev:2013hk] and [@Fajman:2013ffa] it has been also extended to Einstein-Maxwell dilaton gravity. In [@Reiris:2013jaa] related inequalities that involve the shape of the black hole were proved. ![Axially symmetric two-surface. The axial Killing vector $\eta$ is tangent to the surface. The null vectors $\ell^a$ and $k^a$ are normal to ${\cal S}$[]{data-label="fig:axial-2s"}](axial-two-surface.pdf){width="20.00000%"} To describe the concept of stable trapped surface (this condition was first introduced in [@andersson08]) used in theorem \[t:2\] let us consider an axially symmetric closed two-surface ${\Sigma}$ with the topology of a two-sphere. The surface ${\Sigma}$ is embedded in the spacetime. Let $\ell^a$ and $k^a$ be null vectors spanning the normal plane to ${\Sigma}$ and normalized as $\ell^a k_a = -1$, see figure \[fig:axial-2s\]. The expansion is defined by $\theta^{(\ell)}= \nabla_a\ell^a$, where $\nabla$ is the spacetime connection. The surface ${\Sigma}$ is marginally trapped if $\theta^{(\ell)}=0$. Given a closed marginally trapped surface ${\Sigma}$ we will refer to it as spacetime stably outermost if there exists an outgoing ($-k^a$-oriented) vector $X^a= \gamma \ell^a - \psi k^a$, with $\gamma\geq0$ and $\psi>0$, such that the variation of $\theta^{(\ell)}$ with respect to $X^a$ fulfills the condition $$\label{e:stability_condition} \delta_X \theta^{(\ell)} \geq 0.$$ Here $\delta$ denotes a variation operator associated with a deformation of the surface ${\Sigma}$ (c.f. for example [@Booth:2006bn] [@andersson08])). For maximal initial data the stability condition (\[e:stability\_condition\]) is closed related with the stability condition for minimal surfaces (see [@Dain:2011kb], [@Jaramillo:2011pg]). The stability of a minimal surface is the requirement that the area is a local minimum. The extreme throat geometry, with angular momentum $J$, was defined in [@dain10d] (see also [@Acena:2010ws] and [@Dain:2011pi]). This concept captures the local geometry near the horizon of an extreme Kerr black hole. The extreme throat is the asymptotic limit in the cylindrical end of an extreme Kerr black hole, see figure \[fig:throat-cd\] and \[fig:throat-id\]. Both the intrinsic and extrinsic geometry of this surface are fixed. For example, it has an intrinsic metric given by $$\label{eq:gamma0} |J| \left( (1+\cos^2\theta) d\theta^2+ \frac{4\sin^2\theta}{(1+\cos^2\theta)} d\phi^2 \right).$$ It is an oblate sphere with respect to the axis of rotation (see figure \[fig:arb-st\], on the right). ![Location of the extreme Kerr throat surface ${\Sigma}$ in the spacetime. []{data-label="fig:throat-cd"}](kerr-extremo-s-p.pdf){width="50.00000%"} ![Location of the extreme Kerr throat surface ${\Sigma}$ on the initial data. []{data-label="fig:throat-id"}](kerr-ex-initial-data-s-p.pdf){width="40.00000%"} ![On the left, an arbitrary axially symmetric stable two surface. For this kind of surface the strict inequality holds. On the right, the extreme throat sphere, where the equality holds.[]{data-label="fig:arb-st"}](axial-two-surface-iso-p.pdf "fig:"){width="30.00000%"} ![On the left, an arbitrary axially symmetric stable two surface. For this kind of surface the strict inequality holds. On the right, the extreme throat sphere, where the equality holds.[]{data-label="fig:arb-st"}](extreme-throat-sphere-p.pdf "fig:"){width="30.00000%"} The extreme Kerr throat achieve the equality in (\[eq:JAt\]), this surface has the “optimal shape” with respect this inequality. It has also a variational characterization. Figure \[fig:arb-st\] is the analog of figures \[fig:1\] and \[fig:non-extreme-id\] for inequality (\[eq:JAt\]). The results in theorem \[t:2\] has been used in a recent non-existence proof of stationary black holes binaries [@Neugebauer:2013ee] [@Neugebauer:2011qb] [@Chrusciel:2011iv]. The rigidity statement in theorem \[t:2\] (namely that the equality in implies that the surface is an extreme Kerr throat) has been proved in a different context: for extreme isolated horizon and near-horizon geometries of extremal black holes in [@Hajicek:1974oua], [@Lewandowski:2002ua] and [@Kunduri:2008rs], see also the review article [@lrr-2013-8] and reference therein. Open problems and recent results on bodies {#sec:open-problems-recent} ========================================== In this final section I would like to present the main open problems regarding the black holes geometrical inequalities discussed in the previous sections. My aim is to present open problems which are relevant (and probably involve the discovery of new techniques) and at the same time they appear feasible to solve. For more details see the review article [@dain12]. The open problem mentioned there regarding the inclusion of the electric charge in the quasi-local inequality (\[eq:JAt\]) have been solved [@Clement:2011np] [@Clement:2012vb]. For the global inequality (\[eq:60\]) there are two main open problems, which involve generalizations of the assumptions in theorem \[t:1\]: - Remove the maximal condition. - Generalization for asymptotic flat manifolds with multiple ends. Concerning the maximal condition, as we mention above, in a recent article [@zhou12] this assumption have been replaced by a small trace condition. See also the discussion in [@dain12]. The most relevant open problem is the second one. The physical heuristic argument presented in section \[sec:physical-picture\] applies to that case and hence there little doubt that the inequality holds. This problem is related with the uniqueness of the Kerr black hole with degenerate and disconnected horizons. It is probably a hard problem. There are very interesting partial results in [@Chrusciel:2007ak] and also numerical evidences in [@Dain:2009qb]. Probably the most important open problem for geometrical inequalities for axially symmetric black holes is the following: - Prove the Penrose inequality with angular momentum (\[eq:6\]). We mention in section \[sec:physical-picture\] that there is a clear physical connection between the global inequality (\[eq:60\]) and the Penrose inequality with angular momentum in axial symmetry (\[eq:6\]). However, the techniques used to prove the inequality (\[eq:60\]) are very different than the one used to prove the classical Penrose inequality (\[eq:9\]) (see the discussion in [@dain12]). For the quasi-local inequality (\[eq:JAt\]) the two main problems are the following: - A generalization of the inequality (\[eq:JAt\]) without axial symmetry. - A generalization of the inequality (\[eq:JAt\]) for ordinary bodies. The problem of finding versions of inequality (\[eq:JAt\]) without any symmetry assumption, in contrast with the other open problems presented above, is not a well-defined mathematical problem since there is no unique notion of quasi-local angular momentum in the general case. However, exploring the scope of the inequality in regions close to axial symmetry (in some appropriate sense) can perhaps provide such a notion. From the physical point of view, we do not see any reason why this inequality should only hold in axial symmetry. Note that the global inequality (\[eq:60\]) only holds in axial symmetry. This is clear from the physical point of view (see the discussion in [@dain12]) and in [@huang11] highly non-trivial counter examples have been constructed. Finally, concerning the second problem there have been recently some results in [@Dain:2013gma]. Consider a rotating body ${U}$ with angular momentum $J({U})$, see figure \[fig:body\]. Let ${\mathcal{R}}({U})$ be a measure (with units of length) of the size of the body. ![Axially symmetric rotating body.[]{data-label="fig:body"}](body3.pdf){width="2.6cm"} In [@Dain:2013gma], the following universal inequality for all bodies is conjectured $$\label{eq:22} {\mathcal{R}}^2({U}) \apprge \frac{G}{c^3} |J({U})|,$$ where $G$ is the gravitational constant and $c$ the speed of light. The symbol $\apprge$ is intended as an order of magnitude, the precise universal (i.e. independent of the body) constant will depend on the definition of ${\mathcal{R}}$. We have reintroduced in (\[eq:22\]) the fundamental constants in order to make more transparent the discussion bellow. The arguments in support of the inequality (\[eq:22\]) are based in the following three physical principles: - The speed of light $c$ is the maximum speed. - For bodies which are not contained in a black hole the following inequality holds $$\label{eq:2b} {\mathcal{R}}({U}) \apprge\frac{G}{c^2} m({U}),$$ where $m({U})$ is the mass of the body. - The inequality (\[eq:22\]) holds for black holes. Let us discuss these assumptions. Item (i) is clear. Item (ii) is called the *trapped surface conjecture* [@Seifert79]. Essentially, it says that if the reverse inequality as in (\[eq:2b\]) holds then a trapped surface should enclose ${U}$. That is: if matter is enclosed in a sufficiently small region, then the system should collapse to a black hole. This is related with the *hoop conjecture* [@thorne72] (see also [@Wald99] [@PhysRevD.44.2409] [@Malec:1992ap] ). The trapped surface conjecture has been proved in spherical symmetry [@Bizon:1989xm] [@Bizon:1988vv] [@Khuri:2009dt] and also for a relevant class of non-spherical initial data [@Malec:1991nf]. The general case remains open but it is expected that some version of this conjecture should hold. Concerning item (iii), the area $A$ is a measure of the size of a trapped surface, hence the inequality (\[eq:JAt\]) represents a version of for axially symmetric black holes. If we include the physical constants, this inequality has the form $$\label{eq:5} A\geq8\pi\frac{G}{c^3} |J|.$$ In fact the inequality (\[eq:5\]) was the inspiration for the inequality (\[eq:22\]). A possible generalization of (\[eq:5\]) for bodies is to take the area $A(\partial {U})$ of the boundary $\partial {U}$ of the body ${U}$ as measure of size. But unfortunately the area of the boundary is not a good measure of the size of a body in the presence of curvature. In particular, an inequality of the form $A(\partial {U}) \apprge G c^{-3} |J({U})| $ does not holds for bodies. The counter example is essentially given by a rotating torus in the weak field limit, with large major radius and small minor radius. The details of this calculation will be presented in [@Anglada13]. Using the three physical principles (i), (ii) and (iii) in [@Dain:2013gma] it is argued that the inequality (\[eq:22\]) should hold. One of the main difficulties in the study of inequalities of the form is the very definition of the measure of size. In fact, despite the intensive research on the subject, there is no know universal measure of size such that the trapped surface conjecture (or, more general, the hoop conjecture) holds (see the interesting discussions in [@Malec:1992ap] [@Gibbons:2012ac] [@Senovilla:2007dw] [@Reiris:2013jaa]). However, the remarkable point is that in order to find an appropriate measure of size ${\mathcal{R}}$ such that holds it is not necessary to prove first , and hence we do not need to find the relevant measure of mass $m({U})$ for the trapped surface conjecture. In [@Dain:2013gma] a size measure is proposed and for that measure the following version of the inequality has been proved for constant density bodies. This theorem is a consequence of the Schoen-Yau theorem [@schoen83d]. \[t:3\] Consider a maximal, axially symmetric, initial data set that satisfy the dominant energy condition. Let ${U}$ be an open set on the data. Assume that the energy density is constant on ${U}$. Then the following inequality holds $$\label{eq:7d} {\mathcal{R}}^2({U}) \geq \frac{24}{\pi^3}\frac{G}{c^3} |J({U})|.$$ The definition of the radius ${\mathcal{R}}$ in (\[eq:7d\]) is as follow. Let ${\mathcal{R}_{SY}}({U})$ be the Schoen-Yau radius defined in [@schoen83d]. This radius is expressed in terms of the largest torus that can be embedded in ${U}$. See figure \[fig:sy-torus\]. Consider a region ${U}$ with a Killing vector $\eta^i$ with norm $\lambda$, we define the radius ${\mathcal{R}}$ by $$\label{eq:8} {\mathcal{R}}({U}) = \frac{2}{\pi} \frac{\left(\int_{U}\lambda \right)^{1/2}}{{\mathcal{R}_{SY}}({U})}.$$ The definition of the radius (\[eq:8\]) is, no doubt, very involved. It is not expected to be the optimal size measure for a body. It should be considered, together with theorem \[t:3\], as an example where the conjecture (\[eq:22\]) can be proved with the current available mathematical techniques. For examples and further discussion on this radius we refer to [@Dain:2013gma]. ![On the left, the Schoen-Yau ${\mathcal{R}_{SY}}$ radius for a body is defined in terms of the biggest embedded torus. On the right, the same torus is showed on the plane orthogonal to the axial Killing vector. On that plane the torus is a circle, and the radius ${\mathcal{R}_{SY}}$ is related to the radius of the biggest embedded circle.[]{data-label="fig:sy-torus"}](toro-interno-p.pdf "fig:"){width="3.2cm"} ![On the left, the Schoen-Yau ${\mathcal{R}_{SY}}$ radius for a body is defined in terms of the biggest embedded torus. On the right, the same torus is showed on the plane orthogonal to the axial Killing vector. On that plane the torus is a circle, and the radius ${\mathcal{R}_{SY}}$ is related to the radius of the biggest embedded circle.[]{data-label="fig:sy-torus"}](sy-radius-as-2-p.pdf "fig:"){width="1.8cm"} This article is based on the longer review article [@dain12], we refer to that article for more details. The two main differences with respect to [@dain12] are the following. First, several new results appeared after the publication of [@dain12]. These results have been included here. 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--- abstract: 'We present three families of pairs of geometrically non-isomorphic curves whose Jacobians are isomorphic to one another as unpolarized abelian varieties. Each family is parametrized by an open subset of ${{\mathbb P}}^1$. The first family consists of pairs of genus-$2$ curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible Jacobians. The second family also consists of pairs of genus-$2$ curves, but generically the curves in this family have absolutely simple Jacobians. The third family consists of pairs of genus-$3$ curves, one member of each pair being a hyperelliptic curve and the other a plane quartic. Examples from these families show that in general it is impossible to tell from the Jacobian of a genus-$2$ curve over ${{\mathbb Q}}$ whether or not the curve has rational points — or indeed whether or not it has real points. Our constructions depend on earlier joint work with Franck Leprévost and Bjorn Poonen, and on Peter Bending’s explicit description of the curves of genus $2$ whose Jacobians have real multiplication by ${{\mathbb Z}}[\sqrt{2}]$.' address: 'Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1967, USA.' author: - 'Everett W. Howe' date: 12 May 2003 title: 'Infinite families of pairs of curves over ${{\mathbb Q}}$ with isomorphic Jacobians' --- Introduction {#S-intro} ============ Torelli’s theorem shows that a curve is completely determined by its polarized Jacobian variety, but it has been known since the late 1800s that distinct curves can have isomorphic unpolarized Jacobians. In particular, the unpolarized Jacobian of a curve may not reflect all of the curve’s geometric properties. Proving that a particular property of curves cannot always be determined from the Jacobian is equivalent to showing that there exist two curves, one with the given property and one without, whose Jacobians are isomorphic to one another. Thus, for example, the pairs of curves written down in [@Howe:PAMS] show that one cannot tell whether or not a curve of genus $3$ over the complex numbers is hyperelliptic simply by looking at its Jacobian. One would also like to find [*arithmetic*]{} properties of curves that are not determined by the Jacobian, but from an arithmetic perspective the heretofore-known explicit examples of distinct curves with isomorphic Jacobians (catalogued in the introduction to [@Howe:PAMS]) are not entirely satisfying. The primary complaint is that none of the examples involves curves that can be defined over ${{\mathbb Q}}$; in addition, for any given number field only finitely many of the examples can be defined over that field. Furthermore, all of the explicit examples in characteristic $0$ known before now involve curves with geometrically reducible Jacobians, and the arithmetic of such curves differs qualitatively from that of curves whose Jacobians are irreducible. In this paper we address these concerns by providing three new explicit families of pairs of non-isomorphic curves with isomorphic Jacobians. Each family is parametrized by an open subset of ${{\mathbb P}}^1$, so each family gives an infinite number of examples over ${{\mathbb Q}}$. Also, the Jacobians of the curves in one of the families are typically absolutely simple. Using examples from these families, we show that the Jacobian of a genus-$2$ curve over ${{\mathbb Q}}$ does not determine whether or not the curve has rational points, or indeed whether or not the curve has real points. Liu, Lorenzini, and Raynaud [@LLR] have used our results to show that the Jacobian of a genus-$2$ curve over ${{\mathbb Q}}$ does not determine the number of components on the reduction of a minimal model of the curve modulo a prime. Our first family of pairs of curves can be defined over an arbitrary field $K$ whose characteristic is not $2$. If $t$ is an element of $K$ with $t(t+1)(t^2+1)\neq 0$ then the equation $$(t + 1) y^2 = (2 x^2 - t) (4 t^2 x^4 + 4(t^2 + t + 1)x^2 + 1)$$ defines a curve of genus $2$ that we will denote $C(t)$. Clearly the quotient of $C(t)$ by the involution $(x,y)\mapsto (-x,y)$ is an elliptic curve, so the Jacobian of $C(t)$ splits over $K$. \[T-nonsimple2\] Let $K$ be a field of characteristic not $2$ and suppose $t$ is an element of $K$ such that $t(t^2-1)(t^2+1)$ is nonzero. Then $C(t)$ and $C(-t)$ are curves of genus $2$ over $K$ whose Jacobians are isomorphic over $K$. Furthermore, $C(t)$ and $C(-t)$ are geometrically non-isomorphic unless $K$ has characteristic $11$ and $t^2 \in \{-3, -4\}$. Our next family takes a little more effort to describe. In order to do so we must define the [*Richelot duals*]{} of a genus-$2$ curve over a field $K$ of characteristic not $2$ (see [@Cassels-Flynn Ch. 9], [@Bending §3]). Suppose $C$ is a genus-$2$ curve over $K$ defined by an equation $\delta y^2 = f$, where $\delta\in K^*$ and where $f$ is a monic separable polynomial in $K[x]$ of degree $6$. Let ${{K\llap{$\overline{\phantom{\rmK}}$}}}$ be a separable closure of $K$, and suppose $f$ can be factored as a product $g_1 g_2 g_3$ of three monic quadratic polynomials in ${{K\llap{$\overline{\phantom{\rmK}}$}}}[x]$ that are permuted by ${\operatorname{Gal}}({{K\llap{$\overline{\phantom{\rmK}}$}}}/K)$. For each $i$ write $g_i = x^2 - t_i x + n_i$ and suppose the determinant $$d = \left| \begin{matrix} 1 & t_1 & n_1 \\ 1 & t_2 & n_2 \\ 1 & t_3 & n_3 \end{matrix} \right| ,$$ which is an element of $K$, is nonzero. Define three new polynomials by setting $$\begin{aligned} h_1 &= g_3 \frac{dg_2}{dx} - g_2 \frac{dg_3}{dx}\\ h_2 &= g_1 \frac{dg_3}{dx} - g_3 \frac{dg_1}{dx}\\ h_3 &= g_2 \frac{dg_1}{dx} - g_1 \frac{dg_2}{dx}.\end{aligned}$$ Then the product $h_1 h_2 h_3$ is a separable polynomial in $K[x]$ of degree $5$ or $6$. The [*Richelot dual of $C$ associated to the factorization $f = g_1 g_2 g_3$*]{} is the genus-$2$ curve $D$ defined by $d \delta y^2 = h_1 h_2 h_3.$ \[T-simple2\] Let $K$ be a field of characteristic not $2$, let $v$ be an element of $K\setminus\{0,1,4\}$ such that $$(v^2 - v + 4) (v^2 + v + 2) (v^2 + 3 v + 4) (v^3 - 6 v^2 - 7 v - 4) (v^3 - 4 v^2 + 7 v + 4) \neq 0,$$ let $w$ be a square root of $v$ in ${{K\llap{$\overline{\phantom{\rmK}}$}}}$, and define numbers $\rho_1,\ldots,\rho_6$ by setting $$\begin{aligned} \rho_1 & = \frac{(-2+w)(1+w)}{2w^2} & \qquad \qquad \rho_2 & = \frac{(-2-w)(1-w)}{2w^2}\\ \rho_3 & = \frac{-2(2+w)}{(-2+w)(1+w)} & \qquad \qquad \rho_4 & = \frac{(-2-w)(1-w)}{(-w)(-1-w)} \\ \rho_5 & = \frac{(-2+w)(1+w)}{w(-1+w)} & \qquad \qquad \rho_6 & = \frac{-2(2-w)}{(-2-w)(1-w)}.\end{aligned}$$ The $\rho_i$ are distinct from one another, so that if we let $f = \prod(x-\rho_i)$ then the curve $D$ over $K$ defined by $y^2 = f$ has genus $2$. Set $$\begin{aligned} g_1 & = (x-\rho_1)(x-\rho_5) & \qquad \qquad g_1' & = (x-\rho_1)(x-\rho_3)\\ g_2 & = (x-\rho_2)(x-\rho_4) & \qquad \qquad g_2' & = (x-\rho_2)(x-\rho_6)\\ g_3 & = (x-\rho_3)(x-\rho_6) & \qquad \qquad g_3' & = (x-\rho_4)(x-\rho_5).\end{aligned}$$ The Richelot duals $C$ and $C'$ of $D$ with respect to the factorizations $f = g_1 g_2 g_3$ and $f = g_1' g_2' g_3'$ exist, and their Jacobians become isomorphic to one another over $K(\sqrt{v(v-4)}\,)$. The curves $C$ and $C'$ are geometrically non-isomorphic unless one of the following conditions holds[:]{} - ${\operatorname{char}}K = 3$ and $v^{10} - v^8 + v^7 - v^6 - v^5 + v + 1 = 0$[;]{} - ${\operatorname{char}}K = 19$ and $v + 1 = 0$[;]{} - ${\operatorname{char}}K = 89$ and $v + 36 = 0$[;]{} - ${\operatorname{char}}K = 1033$ and $v + 508 = 0$. Furthermore, if $K$ has characteristic $0$ and if $v$ is not an algebraic number, then the Jacobians of $C$ and $C'$ are absolutely simple. In fact, when $K$ has characteristic $0$ it is very easy to find [*algebraic*]{} numbers $v$ in $K$ for which the Jacobians in Theorem \[T-simple2\] are absolutely simple. For example, suppose $R$ is a subring of $K$ for which there is a homomorphism $\varphi$ to an extension of ${{\mathbb F}}_{13}$. We show in the proof of Theorem \[T-simple2\] that in this case the Jacobians of $D$ and $D'$ are geometrically irreducible whenever $v$ lies in $\varphi^{-1}(2)$ or $\varphi^{-1}(6)$. Theorem \[T-simple2\] gives a $1$-parameter family of pairs of non-isomorphic curves with isomorphic Jacobians. In fact, we shall see that there is a family of such pairs of curves parametrized by an elliptic surface; over ${{\mathbb Q}}$, this surface has positive rank. Our third family of pairs of curves with isomorphic Jacobians is again easy to write down. Suppose $K$ is a field of characteristic not $2$ and suppose $t$ is an element of $K$ with $t(t+1)(t^2+1)(t^2+t+1)\neq 0$. Let $H(t)$ be the genus-$3$ hyperelliptic curve defined by the homogeneous equations $$\begin{aligned} W^2 Z^2 &= - \frac{ (t^2+1) }{t(t+1)(t^2+t+1)}X^4 - \frac{4(t^2+1) }{ (t+1)(t^2+t+1)}Y^4 + \frac{ 1}{t }Z^4 \label{E-H1} \\ 0 &= - X^2 + 2 t Y^2 + (t+1) Z^2 \label{E-H2}\end{aligned}$$ and let $Q(t)$ be the plane quartic $$\begin{gathered} \label{E-Q} X^4 + 4t^2 Y^4 + (t+1)^2 Z^4 + (8t^2 + 4t + 8)X^2Y^2 \\ - (4t^2 + 2t + 2)X^2Z^2 + (4t^2 + 4t +8)Y^2Z^2 = 0.\end{gathered}$$ \[T-nonsimple3\] Let $K$ be a field of characteristic not $2$ and let $t$ be an element of $K$ such that $t(t+1)(t^2+1)(t^2+t+1)\neq 0$. Then the Jacobians of the two genus-$3$ curves $H(t)$ and $Q(t)$ are isomorphic to one another over $K$. In Section \[S-curves\] we mention some simple facts about abelian surfaces with two non-isomorphic principal polarizations and we show how Richelot isogenies can in principle be used to produce such surfaces from an abelian surface that has nontrivial automorphisms. In Section \[S-nonsimple\] we prove Theorem \[T-nonsimple2\]. In Section \[S-RM\] we review a result of Bending that shows how to obtain genus-$2$ curves over a given field $K$ whose Jacobians have real multiplication by $\sqrt{2}$ over $K$, and we show how to adapt Bending’s result to obtain curves over $K$ with real multiplication by $\sqrt{2}$ over a quadratic extension of $K$. In Section \[S-Galois\] we give some Galois restrictions on our generalization of Bending’s construction that ensure that the curves we construct have two rational Richelot isogenies to curves with isomorphic Jacobians. In Section \[S-application\] we show that there is a positive-rank elliptic surface whose points give rise to pairs of genus-$2$ curves with isomorphic Jacobians, and we prove Theorem \[T-simple2\]. In Section \[S-genus3\] we prove Theorem \[T-nonsimple3\]. Finally, in Section \[S-examples\] we provide some explicit examples of curves over ${{\mathbb Q}}$ produced by our theorems, and we show that the Jacobian of a curve over ${{\mathbb Q}}$ does not determine whether or not the curve has rational points, or even whether or not it has real points. We relied heavily on the computer algebra system Magma [@Magma] while working on this paper. Some of our Magma routines are available on the web: to find them, start at [http://alumni.caltech.edu/\~however/biblio.html]{} and follow the links related to this paper. Abelian surfaces with non-isomorphic polarizations {#S-curves} ================================================== Weil [@Weil] showed that an abelian surface with an indecomposable principal polarization is a Jacobian, so one of our goals in this paper is to write down abelian surfaces with two non-isomorphic principal polarizations. In this section we will make a few observations about such surfaces. Suppose $B$ is an abelian surface with two principal polarizations $\mu$ and $\mu'$, which we view as isogenies from $B$ to its dual variety $\hat{B}$. The polarized varieties $(B,\mu)$ and $(B,\mu')$ are isomorphic to one another if and only if there is an automorphism $\beta$ of $B$ such that $\mu' = \hat{\beta} \mu \beta$, where $\hat{\beta}$ is the dual of $\beta$. We would like to write down an abelian surface $B$ with two non-isomorphic principal polarizations $\mu$ and $\mu'$, so we would like to avoid the existence of such an automorphism $\beta$. We will accomplish this by obtaining $\mu'$ from $\mu$ through the use of an automorphism of a surface [*isogenous*]{} to $B$. Our main tool is the following well-known construction: Suppose $(A,\lambda)$ is a principally-polarized abelian surface over a field $K$, suppose $n$ is a positive integer, and suppose $G$ is a rank-$n^2$ subgroupscheme of the $n$-torsion $A[n]$ of $A$ that is isotropic with respect to the $\lambda$-Weil pairing on $A[n]$. Let $B$ be the quotient abelian surface $A/G$ and let $\varphi:A\to B$ be the natural map. Then there is a unique principal polarization $\mu$ of $B$ that makes the following diagram commute: $$\begin{matrix} A & {\mathop{{\relbar\joinrel\longrightarrow}}\limits^{n\lambda}} & \hat{A} \\ {\vbox{\vbox to 4pt{}\vbox{\hbox{ \Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle\varphi$}}$}}\vfill}}} & & {\vbox{\vbox to 4pt{}\vbox{\hbox{ \Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle\hat{\varphi}$}}$}}\vfill}}} \\ B & {\mathop{{\relbar\joinrel\longrightarrow}}\limits^{\mu}} & \hat{B} \end{matrix}$$ Now suppose that $A$ has an automorphism $\alpha$ such that $G':=\alpha(G)$ is also an isotropic subgroup of $A[n]$, and let $(B',\mu')$ be the principally polarized abelian surface obtained from $G'$ as above. \[P-basic\] The automorphism $\alpha$ of $A$ provides an isomorphism $B\to B'$. If we identify $B'$ with $B$ via this automorphism, then $\mu' = \hat{\beta} \mu \beta$, where $\beta$ is the image of $\alpha^{-1}$ in $({\operatorname{End}}B)\otimes {{\mathbb Q}}$. We have a commutative diagram with exact rows: $$\begin{matrix} 0 &{\longrightarrow}& G &{\longrightarrow}& A &{\mathop{{\relbar\joinrel\longrightarrow}}\limits^{\varphi}} &B &{\longrightarrow}& 0\\ & &{\vbox{\vbox to 4pt{}\vbox{\hbox{ \Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle\alpha$}}$}}\vfill}}} & &{\vbox{\vbox to 4pt{}\vbox{\hbox{ \Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle\alpha$}}$}}\vfill}}} & & & & \\ 0 &{\longrightarrow}& G' &{\longrightarrow}& A &{\mathop{{\relbar\joinrel\longrightarrow}}\limits^{\varphi'}}&B' &{\longrightarrow}& 0, \end{matrix}$$ where $\varphi$ and $\varphi'$ are the natural maps from $A$ to $B$ and $B'$, respectively. Completing the diagram, we find an isomorphism $B\to B'$. This proves the first statement of the theorem. The second statement follows by an easy diagram chase. This proposition leaves us some hope, because the $\beta$ in the proposition will not be an element of ${\operatorname{End}}B$ if $G\neq G'$. Also, if we consider the case $n=2$ and if the principally-polarized surface $(A,\lambda)$ is given to us explicitly as either a Jacobian or a product of polarized elliptic curves, then the theory of the Richelot isogeny will allow us to write down $(B,\mu)$ and $(B,\mu')$ explicitly as Jacobians, as we explain below. Thus, we would like to explicitly write down abelian surfaces $A$ with non-trivial automorphisms. In later sections we will consider two families of such explicitly-given surfaces: products of isogenous elliptic curves, and Jacobians with real multiplication by $\sqrt{2}$. We close this section with a comment about Richelot duals and maximal isotropic subgroups. Suppose $C$ is a genus-$2$ curve defined by an equation $\delta y^2 = f$ and suppose $D$ is the Richelot dual of $C$ corresponding to a factorization $f = g_1 g_2 g_3$. Then there is an isogeny from the Jacobian of $C$ to the Jacobian of $D$ whose kernel is the order-$4$ subgroup $G$ of ${\operatorname{Jac}}C$ containing the classes of the divisors $(a_i,0) - (b_i,0)$, where $a_i$ and $b_i$ are the roots of $g_i$ in ${{K\llap{$\overline{\phantom{\rmK}}$}}}$ (see [@Cassels-Flynn Ch. 9]). The subgroup $G$ is a maximal isotropic subgroup of the $2$-torsion of ${\operatorname{Jac}}C$ under the Weil pairing. Conversely, every $K$-defined maximal isotropic subgroup $G$ of $({\operatorname{Jac}}C)[2]$ arises in this way. Thus, given a $K$-defined maximal isotropic subgroup $G$, we can define the [*$G$-Richelot dual*]{} of $C$ to be the Richelot dual of $C$ with respect to the factorization $f = g_1 g_2 g_3$ that gives rise to $G$. Proof of Theorem \[T-nonsimple2\] {#S-nonsimple} ================================= In this section we will prove Theorem \[T-nonsimple2\] by following the outline given in Section \[S-curves\] in the case where $A$ is a split abelian surface. Let $E$ and $E'$ be elliptic curves over a field $K$ of characteristic not $2$ and suppose there is a $2$-isogeny $\psi$ from $E$ to $E'$. Let $Q$ be the nonzero element of $E[2](K)$ in the kernel of $\psi$, and let $P$ and $R$ be the other two geometric $2$-torsion points of $E$. Let $Q' = \psi(P) = \psi(R)$, so that $Q'$ is a nonzero element of $E'[2](K)$, and let $P'$ and $R'$ be the other geometric $2$-torsion points of $E'$. Suppose the discriminants of $E$ and $E'$ are equal up to squares, so that the fields $K(P)$ and $K(P')$ are the same. Let $A$ be the surface $E\times E'$ and let $\lambda$ be the product principal polarization on $A$. Let $\alpha$ be the automorphism of $A$ that sends a point $(U,V)$ to $(U, V+\psi(U))$. Let $G$ be the $K$-defined subgroup $$G = \{(O,O), (P, P'), (Q, Q'), (R, R')\}$$ of $A[2]$ and let $G' = \alpha(G)$, so that $G'$ is the $K$-defined subgroup $$G'= \{(O,O), (P, R'), (Q, Q'), (R, P')\}.$$ Let $(B,\mu)$ and $(B,\mu')$ be the principally-polarized surfaces obtained from $G$ and $G'$ as in Section \[S-curves\]. The polarizations $\mu$ and $\mu'$ will be indecomposable except in unusual circumstances, so there will usually be curves $C$ and $C'$ whose polarized Jacobians are isomorphic to $(B,\mu)$ and $(B,\mu')$, respectively. If $E$ and $E'$ are given to us by explicit equations, then $C$ and $C'$ can also be given by explicit equations — see [@HLP §3.2], where the unusual circumstances are also explained. To make this outline explicit and to prove Theorem \[T-nonsimple2\] we must start with an explicit $2$-isogeny $\psi:E\to E'$, where the discriminants of $E$ and $E'$ are equal up to squares. Let $t$ be an element of $K$ such that $t (t^2 + 1) (t^2 - 1)$ is nonzero, and let $E$ and $E'$ be the elliptic curves $$\begin{aligned} E: \quad y^2 & = x (x^2 - 4(t^2+1) x + 4(t^2+1))\\ E': \quad y^2 & = x (x^2 + 8(t^2+1) x + 16 t^2(t^2+1)).\end{aligned}$$ It is easy to check that the discriminants of $E$ and $E'$ are both equal to $t^2 + 1$, up to squares. Let $s$ be a square root of $t^2 + 1$ in an algebraic closure of $K$, so that the $2$-torsion points of $E$ are $$\begin{aligned} P &= (2t^2 + 2 + 2st, 0)\\ Q &= (0,0)\\ R &= (2t^2 + 2 - 2st,0)\\ \intertext{and the $2$-torsion points of $E'$ are} P' &= (-4t^2 - 4 + 4s, 0)\\ Q' &= (0,0)\\ R' &= (-4t^2 - 4 - 4s,0).\end{aligned}$$ It is easy to check that the map $$(x,y) \mapsto \left( \frac{y^2}{x^2}, \frac{(x^2 - 4(t^2+1))y}{x^2} \right)$$ defines a $2$-isogeny $\psi:E\to E'$ that kills $Q$ and that sends $P$ to $Q'$ (see [@Silverman Example III.4.5]). Let $G$ and $G'$ be the subgroups of $A[2]$ defined above and let $(B,\mu)$ and $(B,\mu')$ be the principally-polarized surfaces obtained from $G$ and $G'$ as above. If we apply [@HLP Prop. 4] we find that $(B,\mu)$ is isomorphic over $K$ to the polarized Jacobian of the curve $y^2 = h_t$, where $$h_t = 2^{38} t^6 (t+1)^3 (t^2+1)^{12} (2 x^2 - t) (4 t^2 x^4 + 4(t^2 + t + 1)x^2 + 1).$$ Furthermore, $(B,\mu')$ is isomorphic to the polarized Jacobians of the curve $y^2 = h_{-t}$. Scaling $h_t$ and $h_{-t}$ by squares in $K$, we find that $y^2 = h_t$ is isomorphic to the curve $C(t)$ of Theorem \[T-nonsimple2\] and that $y^2 = h_{-t}$ is isomorphic to the curve $C(-t)$. To complete the proof of Theorem \[T-nonsimple2\] we must show that $C(t)$ and $C(-t)$ are geometrically non-isomorphic, except for the special cases listed in the theorem. The simplest way to do this is to use Igusa invariants (see [@Igusa], [@Mestre]). Facilities for computing Igusa invariants are included in the computer algebra package Magma [@Magma]. Let us begin by working over the ring ${{\mathbb Z}}[t]$, where $t$ is an indeterminate. Let $J_2(t), J_4(t), J_6(t), J_8(t),$ and $J_{10}(t)$ be the Igusa invariants of the twist $y^2 = (2x^2 - t)(4x^4 + 4(t^2 + t + 1)x^2 + 1)$ of $C(t)$. (The invariants $J_{2i}(t)$ of this curve, scaled by $4^i$, can be computed in Magma using the function [ScaledIgusaInvariants]{}.) Let $$\begin{aligned} R_{2} &= \frac{J_4(t) J_2(-t)^2 - J_4(-t) J_2(t)^2} { t (t^2+1)^3 } \\ R_{3} &= \frac{J_6(t) J_2(-t)^3 - J_6(-t) J_2(t)^3} { t^3 (t^2+1)^3} \\ R_{5} &= \frac{J_{10}(t) J_2(-t)^5 - J_{10}(-t) J_2(t)^5} {t^3 (t^2+1)^7},\end{aligned}$$ all of which we view as elements of ${{\mathbb Z}}[t]$. If $C(t)$ and $C(-t)$ are isomorphic for a given value of $t$ in a given field $K$, then the polynomials $R_{2}, R_{3},$ and $R_{5}$ must all evaluate to $0$ at this value. But we compute that $$\gcd({\operatorname{resultant}}(R_{2},R_{3}), {\operatorname{resultant}}(R_{2},R_{5})) = 2^{980} 3^{48} 11^8,$$ so if the characteristic of $K$ is neither $3$ nor $11$ then the two curves $C(t)$ and $C(-t)$ are geometrically non-isomorphic for every value of $t$ in $K$ with $t(t^2+1)(t^2-1)\neq 0$. We repeat the above calculation in the ring ${{\mathbb F}}_3[t]$, only now we define $$\begin{aligned} R_{2} &= \frac{J_4(t) J_2(-t)^2 - J_4(-t) J_2(t)^2} {t (t^2-1)^2 (t^2+1)^7} \\ R_{3} &= \frac{J_6(t) J_2(-t)^3 - J_6(-t) J_2(t)^3} {t^3 (t^2+1)^9}. \end{aligned}$$ We find that $\gcd(R_{2},R_{3}) = 1$, so the two curves $C(t)$ and $C(-t)$ are geometrically non-isomorphic for every value of $t$ in characteristic $3$, as long as $t(t^2+1)(t^2-1)\neq 0$. Next we repeat the above calculation in the ring ${{\mathbb F}}_{11}[t]$, with $$\begin{aligned} R_{2} &= \frac{J_4(t) J_2(-t)^2 - J_4(-t) J_2(t)^2} { t (t^2+1)^3 }\\ R_{3} &= \frac{J_6(t) J_2(-t)^3 - J_6(-t) J_2(t)^3} { t^3 (t^2+1)^3}. \end{aligned}$$ We find that $\gcd(R_{2},R_{3}) = (t^2 + 3)(t^2 + 4)$, so that the two curves are geometrically non-isomorphic in characteristic $11$ except possibly when $t^2$ is $-3$ or $-4$. Finally we note that in characteristic $11$, when $t^2$ is $-3$ or $-4$ the curve $C(t)$ is geometrically isomorphic to the supersingular curve $y^2 = x^6 + x^4 + 4 x^2 + 7$. Thus, $C(t)$ and $C(-t)$ are geometrically isomorphic for these values of $t$. It was not critical in our construction that the isogeny $\psi:E\to E'$ have degree $2$. Similar constructions can be made with other kinds of isogenies. Jacobians with real multiplication by $\sqrt{2}$ {#S-RM} ================================================ In this section we review a construction of Bending [@Bending] that produces every genus-$2$ curve over a given field $K$ whose Jacobian has a $K$-rational endomorphism that is fixed by the Rosati involution and whose square is $2$. We will give a variant of Bending’s construction that produces curves over $K$ with a not-necessarily $K$-rational endomorphism that is fixed by Rosati and whose square is $2$. We do not claim that our construction will produce all such curves. First we recall Bending’s construction. Let $K$ be a field of characteristic not $2$ and let $A$, $P$, and $Q$ be elements of $K$ with $P$ nonzero. Define $$\begin{aligned} B &= (APQ - Q^2 + 4P^2 + 1)/P^2\\ C &= 4(AP - Q)/P\\ R &= 4P\end{aligned}$$ and let $\alpha_1$, $\alpha_2$, $\alpha_3$ be the roots of $T^3 + AT^2 + BT + C$ in a separable closure ${{K\llap{$\overline{\phantom{\rmK}}$}}}$ of $K$. For $i = 1,2,3$ let $$G_i = X^2 - \alpha_i X + P\alpha_i^2 + Q\alpha_i + R,$$ and suppose that the product $G_1 G_2 G_3 \in K[X]$ has nonzero discriminant. Let $D$ be a nonzero element of $K$. \[T-Bending\] The Jacobian of the genus-$2$ curve $D Y^2 = G_1 G_2 G_3$ has a $K$-rational endomorphism that is fixed by the Rosati involution and whose square is $2$. Furthermore, if $\#K>5$ then every curve over $K$ whose Jacobian has such an endomorphism is isomorphic to a curve that arises in this way from some choice of $A$, $P$, $Q$, and $D$ in $K$. See [@Bending Theorem 4.1]. Bending assumes that the base field $K$ has characteristic $0$, but his proof works over an arbitrary field $K$ of characteristic not $2$ so long as every genus-$2$ curve over $K$ can be written in the form $y^2 = \text{(sextic)}$. This is the case for every field with more than $5$ elements. The endomorphism of $D Y^2 = G_1 G_2 G_3$ whose existence is claimed by Theorem \[T-Bending\] is obtained by noting that the obvious Richelot dual of the curve is isomorphic over $K$ to the curve itself. Thus the degree-$4$ isogeny from the Jacobian of the curve to the Jacobian of its dual can be viewed as an endomorphism of the curve’s Jacobian, and this endomorphism has the properties claimed in the theorem. We will want to consider curves over $K$ whose Jacobians have real multiplication by $\sqrt{2}$ that is not necessarily defined over $K$. For this reason, we will require the following variant of Bending’s construction: Suppose $r$, $s$, and $t$ are elements of a field $K$ of characteristic not $2$, with $s\neq 0$, $s\neq 1$, and $t\neq 1$. Let $$\begin{aligned} c_2 &= r + 4t\\ c_1 &= 4 t (r + s^3 - s^2 t - 2 s^2 + 5 s + t)\\ c_0 &= 4 t (s - 1) (r s^2 - r s t - r s - r t - 8 s t)\end{aligned}$$ and suppose that the polynomial $T^3 - c_2 T^2 + c_1 T - c_0$ has three distinct roots $\beta_1$, $\beta_2$, $\beta_3$ in ${{K\llap{$\overline{\phantom{\rmK}}$}}}$. For $i = 1,2,3$ let $$g_i = x^2 - 2\beta_i x + (1-s)\beta_i^2 - 4 s (s - 1)^2 t (s - t - 1),$$ and suppose that the discriminant of the product $f = g_1 g_2 g_3$ is nonzero. Let ${{\mathcal C}}(r,s,t)$ be the curve over $K$ defined by $y^2 = f$. \[T-RM\] The Richelot dual of ${{\mathcal C}}(r,s,t)$ associated to the factorization $f = g_1 g_2 g_3$ is isomorphic over $K(\sqrt{st}\,)$ to ${{\mathcal C}}(r,s,t)$. The endomorphism of ${\operatorname{Jac}}{{\mathcal C}}(r,s,t)$ [(]{}over $K(\sqrt{st}\,)$[)]{} obtained by composing the Richelot isogeny with the natural isomorphism from the dual curve to ${{\mathcal C}}(r,s,t)$ is fixed by the Rosati involution, and its square is the multiplication-by-$2$ endomorphism. This theorem can be proven by direct calculation, but here we will prove it by relating the curve ${{\mathcal C}}(r,s,t)$ back to Bending’s construction. Let $Q$ be a square root of $st$ in ${{K\llap{$\overline{\phantom{\rmK}}$}}}$, let $P = (1 - s)/4$, let $A = (r + 6 s t - 2 t)/(4PQ)$, let $D=1$, and let ${{\mathcal C}}'$ be the curve over $K(Q)$ defined by using this $A$, $P$, $Q$, and $D$ in Bending’s construction. Then the $\alpha_i$ are related to the $\beta_i$ by $$Q\alpha_i = \beta_i/(s-1) + 2t,$$ and the curve $Y^2 = G_1 G_2 G_3$ is isomorphic $y^2 = g_1 g_2 g_3$ via the relation $x = 2(s-1)(QX + t)$. This shows that ${{\mathcal C}}(r,s,t)$ is isomorphic to its Richelot dual over $K(Q)$. The rest of the theorem follows from Bending’s theorem. Bending’s family of curves has three “geometric” parameters $A$, $P$, and $Q$ and one “arithmetic” parameter $D$ (which parametrizes quadratic twists of the curve determined by $A$, $P$, and $Q$). Since the moduli space ${{\mathcal M}}$ of genus-$2$ curves with real multiplication by $\sqrt{2}$ is a two-dimensional rational variety, one might hope to replace Bending’s three-geometric-parameter family with a two-parameter family. But there is an obstruction, which stems from the fact that ${{\mathcal M}}$ is a coarse moduli space and not a fine one: A $K$-rational point on ${{\mathcal M}}$ does not necessarily give rise to a curve over $K$. Indeed, Mestre [@Mestre] has shown that to every $K$-rational point $P$ on the moduli space of genus-$2$ curves there is naturally associated a genus-$0$ curve over $K$, and $P$ corresponds to a curve over $K$ if and only if the genus-$0$ curve has a $K$-rational point. Galois restrictions {#S-Galois} =================== In order to prove Theorem \[T-simple2\] we will apply the construction outlined in Section \[S-curves\] to a Jacobian with real multiplication by $\sqrt{2}$ that we will obtain from Theorem \[T-RM\]; we will take the automorphism $\alpha$ to be $1+\sqrt{2}$. The construction requires that we find a Galois-stable maximal isotropic subgroup $G$ of the $2$-torsion of the Jacobian such that $G' = (1+\sqrt{2})(G)$ is a maximal isotropic subgroup different from $G$. This requirement imposes some restrictions on the values of $r$, $s$, and $t$ that we will be able to use in Theorem \[T-RM\]. In this section we will make these restrictions explicit, and in Section \[S-application\] we will find an elliptic surface that parametrizes a subset of the allowable values of $r$, $s$, and $t$. Recall the basic outline of Theorem \[T-RM\]: Given three elements $r$, $s$, $t$ of our base field $K$, we define a polynomial $h = T^3 - c_2 T^2 + c_1 T - c_0$ in the polynomial ring $K[T]$, and we assume that $h$ is separable. We use the roots $\beta_1$, $\beta_2$, $\beta_3$ of $h$ to define three polynomials $g_1$, $g_2$, $g_3$ in the polynomial ring ${{K\llap{$\overline{\phantom{\rmK}}$}}}[x]$, we assume that the product $f = g_1 g_2 g_3$ is separable, and we define a curve $C$ by $y^2 = f$. Then we show that the Richelot dual of $C$ corresponding to the factorization $f = g_1 g_2 g_3$ is geometrically isomorphic to $C$ itself. Let $L$ be the quotient of the polynomial ring $K[T]$ by the ideal generated by the polynomial $h$ and let $\beta$ be the image of $T$ in $L$. Since $h$ is separable, the algebra $L$ is a product of fields. Let $g\in L[x]$ be the polynomial $$g = x^2 - 2\beta x + (1-s)\beta^2 - 4 s (s-1)^2 t (s - t - 1).$$ Let $\Delta\in K^*$ be the discriminant of $h$ and let $\Delta'\in L^*$ be the discriminant of $g$. \[T-Galois\] There are distinct Galois-stable maximal isotropic subgroups $G$ and $G'$ of $({\operatorname{Jac}}C)[2]$ with $G' = (1+\sqrt{2})(G)$ if and only if $\Delta\Delta'$ is a square in the algebra $L$. Let the roots of $g_1$ (respectively, $g_2$, $g_3$) be $r_1$ and $r_2$ (respectively, $r_3$ and $r_4$, $r_5$ and $r_6$). For each $i$ let $W_i$ be the Weierstraß point of $C$ corresponding to the root $r_i$ of $f = g_1 g_2 g_3$. The kernel $H$ of the Richelot isogeny multiplication-by-$\sqrt{2}$ on the Jacobian $J$ of $C$ is the order-$4$ subgroup containing the divisor classes $[W_1 - W_2]$, $[W_3 - W_4]$, and $[W_5 - W_6]$. Suppose there are distinct Galois-stable maximal isotropic subgroups $G$ and $G'$ of $({\operatorname{Jac}}C)[2]$ with $G' = (1+\sqrt{2})(G)$. Then clearly $G\neq H$, so $\#(G\cap H)$ is either $2$ or $1$. Suppose $\#(G\cap H)=2$. By renumbering the polynomials $g_i$ and by renumbering their roots, we may assume that $G$ is the order-$4$ subgroup $$G = \{ 0, [W_1 - W_2], [W_3 - W_5], [W_4 - W_6]\}.$$ Then $\sqrt{2}$ kills the first two elements of $G$ and sends the second two elements to $[W_1 - W_2]$, and it follows that $(1+\sqrt{2})(G) = G$, contradicting our assumption that $G$ and $G'$ are distinct. So now we know that $G\cap H = \{0\}$. By renumbering the polynomials $g_i$ and by renumbering their roots, we may assume that $G$ is the order-$4$ subgroup $$G = \{ 0, [W_1 - W_5], [W_2 - W_4], [W_3 - W_6]\}.$$ It is not hard to show that the automorphism $1 + \sqrt{2}$ of $J$ sends $[W_1 - W_5]$ to $[W_2 - W_6]$, $[W_2 - W_4]$ to $[W_1 - W_3]$, and $[W_3 - W_6]$ to $[W_4 - W_5]$, so we have $$G'= \{ 0, [W_1 - W_3], [W_2 - W_6], [W_4 - W_5]\}.$$ Suppose $\sigma$ is an element of the Galois group such that $r_1^\sigma = r_2$. Since $G$ is Galois stable, it follows that $\sigma$ sends $[W_1 - W_5]$ to $[W_2 - W_4]$, and therefore $r_5^\sigma = r_4$. But since $G'$ is Galois stable, we see that $\sigma$ must send $[W_4 - W_5]$ to itself, and it follows that $r_4^\sigma = r_5$. Continuing in this manner, we find that $r_2^\sigma = r_1$ and $r_6^\sigma = r_3$ and $r_3^\sigma = r_6$. Thus, $\sigma$ acts on the roots of $f$ according to the permutation $(1 2)(3 6)(4 5)$ of the subscripts. By considering the other choices for $r_1^\sigma$ and using the same reasoning as above, we find that the image of the absolute Galois group of $K$ in the symmetric group on the the roots of $f$ is contained in the subgroup $$S = \{ {\operatorname{Id}}, (1 2)(3 6)(4 5), (1 3)(2 4)(5 6), (1 4 6)(2 3 5), (1 5) (2 6) (3 4), (1 6 4) (2 5 3)\};$$ here of course we identify the root $r_i$ with the integer $i$. In particular, note that the action of $\sigma$ on the $r_i$ is determined by the action of $\sigma$ on the $\beta_i$. To show that $\Delta\Delta'$ is square in the algebra $L$, we will consider three cases, depending on the splitting of the polynomial $h$. *Case 1.* Suppose $h$ is irreducible. Then $L$ is a field, and the condition that $\Delta\Delta'$ be a square in $L$ is equivalent to saying that $g$ defines the Galois closure $M$ of $L$ over $K$. So suppose, to obtain a contradiction, that $g$ does not define $M$ over $L$. There are two ways that this can happen: either the roots of $g$ do not lie in $M$, or $M$ is a quadratic extension of $L$ and the roots of $g$ lie in $L$. Suppose that the roots of $g$ do not lie in $M$. Then there is an element $\sigma$ of the absolute Galois group of $K$ that fixes $M$ but that moves the roots of $g$. But this contradicts the fact that the the action of $\sigma$ on the roots of $g$ is determined by the action of $\sigma$ on the $\beta_i$. Suppose $M$ is a quadratic extension of $L$, so that the image of the absolute Galois group in the symmetric group on the $\beta_i$ is the full symmetric group. Then the image of the absolute Galois group in the symmetric group on the $r_i$ must be the entire group $S$ given above, which acts transitively on the $r_i$. But if the roots of $g$ lie in $L$, then the $r_i$ will form two orbits under the action of the absolute Galois group, giving a contradiction. *Case 2.* Suppose $h$ factors as a linear polynomial times an irreducible quadratic. Then one of the $\beta_i$, say $\beta_1$, lies in $K$, while $\beta_2$ and $\beta_3$ are conjugate elements in a quadratic extension $M$ of $K$. With the labelings we have chosen, this means that the image of the absolute Galois group in the symmetric group on the $r_i$ must be equal to the two-element group $$S' = \{ {\operatorname{Id}}, (1 2)(3 6)(4 5)\};$$ In particular, we see that $r_3$ and $r_6$ (and $r_4$ and $r_5$) are quadratic conjugates of one another, so they all must be elements of $M$. This means that the image of $g$ in $K[x]$ (obtained by sending $\beta$ to $\beta_1$) is an irreducible polynomial that defines $M$, while the image of $g$ in $M[x]$ (obtained by sending $\beta$ to $\beta_2$) splits into two linear factors. Thus, the discriminant $\Delta'$ of $g$ in $L = K\times M$ is equal to the discriminant of $M$ (up to squares) in the first component, and is a square in the second component. But $\Delta$ has this same property, so the product $\Delta\Delta'$ is a square in $L$. *Case 3.* Suppose $h$ splits over $K$ into three linear factors, so that $\Delta$ is a square in $K$. Then the absolute Galois group of $K$ acts trivially on the $\beta_i$, so it must act trivially on the $r_i$ as well. This means that the discriminant $\Delta'$ of $g$ must be a square in each factor of $L = K\times K\times K$, so $\Delta\Delta'$ is a square in $L$ as well. We see that if there are subgroups $G$ and $G'$ as in the statement of the theorem then $\Delta\Delta'$ must be a square in $L$. We leave the details of the proof of the converse statement to the reader; the point is that in each of the three cases above, the reasoning is reversible. Application of our construction to\ curves with real multiplication {#S-application} =================================== In this section we will follow the outline given in Section \[S-curves\] in the case where $A$ is a Jacobian with real multiplication by $\sqrt{2}$ that has appropriate Galois-stable subgroups. Theorem \[T-simple2\] will follow quickly from the result we obtain. We will continue to use the notation from previous sections: - $r$, $s$, and $t$ will be elements of a field $K$; - $h$ will be a polynomial in $K[T]$ defined in terms of $r$, $s$, and $t$; - $L$ will be the algebra $K[T]/(h)$; - $\beta$ will be the image of $T$ in $L$; - $g$ will be a polynomial in $L[x]$ defined in terms of $r$, $s$, $t$, and $\beta$; - $\Delta\in K^*$ will be the discriminant of $h$; and - $\Delta' \in L^*$ will be the discriminant of $g$. \[T-gensimple\] Let $K$ be a field of characteristic not $2$, suppose $r$, $s$, and $t$ are elements of $K$ that satisfy the hypotheses appearing before the statement of Theorem [\[T-RM\]]{}, and let $C$ be the curve ${{\mathcal C}}(r,s,t)$ from Theorem [\[T-RM\]]{}. Suppose further that the product $\Delta\Delta'$ is a square in $L^*$, so that there are Galois-stable subgroups $G$ and $G'$ of $({\operatorname{Jac}}C)[2]$ as in the statement of Theorem [\[T-Galois\]]{}. Then the Jacobian of the $G$-Richelot dual of $C$ is isomorphic over $K(\sqrt{st}\,)$ to the Jacobian of the $G'$-Richelot dual of $C$. Let $D$ be the $G$-Richelot dual of $C$ and let $D'$ be the $G'$-Richelot dual of $C$. Theorem \[T-RM\] and Theorem \[T-Galois\], combined with the argument in Section \[S-curves\], show that the Jacobian of $D$ becomes isomorphic to the Jacobian of $D'$ when the base field is extended to $K(\sqrt{st}\,)$. Since $D$ and $D'$ are defined over $K$, and since their Jacobians become isomorphic over $K(\sqrt{st}\,)$, it is tempting to think that ${\operatorname{Jac}}D$ must be isomorphic over $K$ to either the Jacobian of $D'$ or the Jacobian of the standard quadratic twist of $D'$ over $K(\sqrt{st}\,)$. But in fact this is not the case. It is true that ${\operatorname{Jac}}D'$ is a $K(\sqrt{st}\,)/K$-twist of ${\operatorname{Jac}}D$, but the twist is by an automorphism of ${\operatorname{Jac}}D$ that does not come from an automorphism of $D$. Indeed, generically the automorphism group of $D$ contains $2$ elements, while the automorphism group of ${\operatorname{Jac}}D$ is isomorphic to the unit group of ${{\mathbb Z}}[2\sqrt{2}]$. Suppose $t = s-1$ and let $u = r + 2$. Then $\Delta\Delta'$ is a square in $L$ if and only if $(u^2 + a)^2 + 8bu + 4c$ is a square in $K$, where $$\begin{aligned} a &= -4 s (s^2 + 11s - 11)\\ b &= -8s^2 (s-1) (4s-1) \\ c &= -16s^2 (s-1) (28s^2 - 19s + 1).\end{aligned}$$ When $t = s-1$ and $r = u - 2$, we find that the coefficients of the polynomial $h$ used to define the algebra $L$ are $$\begin{aligned} c_2 &= 4s + u - 6 \\ c_1 &= - 4 (s-1) (s^2 - 6s - u + 3) \\ c_0 &= 4 (s-1)^3 (-8s - u + 2),\end{aligned}$$ and we compute that $$\Delta = 16 s (s-1)^2 ((u^2 + a)^2 + 8bu + 4c),$$ where $a$, $b$, and $c$ are as in the statement of the proposition. Furthermore, the polynomial $g\in L[x]$ defined in Section \[S-Galois\] is $x^2 - 2\beta x + (1-s)\beta^2$, so that $\Delta' = 4 s \beta^2$. We see that $\Delta\Delta'$ is a square in $L$ if and only if the element $\delta = (u^2 + a)^2 + 8bu + 4c$ of $K$ is a square in $L$. If $L$ is a field then it is a cubic extension of $K$, and $\delta$ is a square in $L$ if and only if it is a square in $K$. If $L$ is not a field then it has $K$ as a factor, and again $\delta$ is a square in $L$ if and only if it is a square in $K$. \[P-ellipticsurface\] Let $K={{\mathbb Q}}(s)$ be the function field in the variable $s$ over ${{\mathbb Q}}$, let $$\begin{aligned} a &= -4 s (s^2 + 11s - 11)\\ b &= -8s^2 (s-1) (4s-1) \\ c &= -16s^2 (s-1) (28s^2 - 19s + 1),\end{aligned}$$ let $F$ be the curve over $K$ defined by $$z^2 = (u^2 + a)^2 + 8bu + 4c,$$ and let $E$ be the elliptic curve over $K$ defined by $$y^2 = x^3 - a x^2 - c x + b^2.$$ Then - the map $u = (y-b)/x$, $z = 2x - u^2 - a$ gives an isomorphism from $E$ to $F$, whose inverse is $x = (z + u^2 + a)/2$, $y = ux + b$[;]{} - the point $P = (0, b)$ on $E$ has infinite order[;]{} - the point $T = (4s^2(1-s), 0)$ on $E$ has order $2$[;]{} - the isomorphism in statement [(a)]{} takes the involution $(u,z)\mapsto (u,-z)$ on $F$ to the involution $Q \mapsto -Q-P$ on $E$[;]{} - the isomorphism in statement [(a)]{} takes $-P$ and the origin of $E$ to the two infinite points on $F$. An easy calculation shows that statement (a) is true; the particular values of $a$, $b$, and $c$ are irrelevant to the calculation. To show that the point $P$ has infinite order, it suffices to show that when we specialize $s$ to a particular value the specialized $P$ has infinite order. For example, if we set $s = 2$, then $E$ becomes the curve $$y^2 = x^3 + 120 x^2 + 4800 x + 50176$$ and $P$ becomes the point $(0, -224)$. Translating $x$ by $40$, we find a new equation for $E$: $y^2 = x^3 - 13824,$ where now $P = (40,-224)$. But $7$ divides $224$ and $7$ does not divide $13824$, so by the Lutz-Nagell theorem [@Silverman Cor. VIII.7.2] the point $P$ has infinite order. This proves statement (b). Statement (c) is clear. Let $R$ be a point $(u,z)$ on $F$ and let $\tilde{R} = (u,-z)$ be its involute. Let $Q$ and $\tilde{Q}$ be the images of $R$ and $\tilde{R}$ on $E$. Clearly $Q$ and $\tilde{Q}$ both lie on the line $y = ux + b$, and the third intersection point of this line with $E$ is easily seen to be $P$. Thus, the involution on $E$ satisfies $\tilde{Q} + Q = -P$, and this is statement (d). The equations for the isomorphism show that $-P$ is mapped to an infinite point on $F$, and statement (d) shows that $O_E$ gets mapped to an infinite point as well. If we view the curve $F\cong E$ as an elliptic surface $S$ over ${{\mathbb Q}}$, then the points $P$ and $T$ of Proposition \[P-ellipticsurface\] can be viewed as rational curves on $S$. By adding multiples of $P$ and $T$ together, we get a countable family of rational curves on $S$. But $S$ contains more rational curves than just the ones in this family. For example, we have the curves $$\begin{aligned} s &= 5/4 \\ u &= (4w^2 + 5w + 40)/(4w) \\ z &= (2w^4 + 5w^3 - 50w - 200)/(2w^2)\end{aligned}$$ and $$\begin{aligned} s &= -1 \\ u &= (2w^2 - 10w - 4)/w \\ z &= (4w^4 - 40w^3 - 80w - 16)/w^2,\end{aligned}$$ where $w$ is a parameter; the curve $$\begin{aligned} s &= (5-w^2)/4\\ u &= (-w^4 + w^3 + 7w^2 - 5w - 10)/(4w + 8) \\ z &= (w^8 + 9w^7 + 22w^6 - 18w^5 - 135w^4 - 135w^3)/(8w^2 + 32w + 32),\end{aligned}$$ which corresponds to a $3$-torsion point on $E$ defined over a genus-$0$ extension of the function field ${{\mathbb Q}}(s)$; five curves in which $u$ is a linear expression in $s$, for example $$\begin{aligned} s &= 4(w^2 + 9w + 19)/w\\ u &= -s\\ z &= 16(16w^6 + 283w^5 + 1555w^4 - 29545w^2 - 102163w - 109744)/w^3;\end{aligned}$$ and three curves in which $u$ is a quadratic expression in $s$, for example $$\begin{aligned} s &= (w^2 + 3w + 1)/w\\ u &= 4s^2 - 6s\\ z &= 4(4w^8 + 35w^7 + 105w^6 + 119w^5 - 119w^3 - 105w^2 - 35w - 4)/w^4.\end{aligned}$$ One can check that the image of the elliptic surface $S$ in the moduli space of genus-$2$ curves is $2$-dimensional. To check this, one need only write explicitly the Igusa invariants of the genus-$2$ curve obtained from a pair $(s,u)$ and verify that the rank of the Jacobian matrix of the mapping from $(s,u)$-pairs to Igusa invariants at some arbitrary point is $2$. We now have enough machinery available to prove Theorem \[T-simple2\]. Consider the point $P = (0,b)$ from statement (b) of Proposition \[P-ellipticsurface\]. The $u$-coördinate of its image on the curve $F$ is $$u = (28s^2 - 19s + 1)/(1-4s).$$ So given any $s\in K$, we will obtain a curve satisfying the conclusion of Theorem \[T-gensimple\] if we set $$\begin{aligned} t &= s - 1\\ r &= -2 + (28s^2 - 19s + 1)/(1-4s) \end{aligned}$$ and set $C = {{\mathcal C}}(r,s,t)$. Given a $v \in K\setminus\{0,1,4\}$, let us apply the preceding observation with $s = v/4$, so that $$\begin{aligned} s &= v/4\\ t &= (v - 4)/4\\ r &= (7 v^2 - 11 v - 4)/(4-4v).\end{aligned}$$ The coefficients of the polynomial $h$ used in the construction of Section \[S-RM\] are $$\begin{aligned} c_2 &= \frac{3v^2 + 9v - 20}{4(1 - v)} \\ c_1 &= \frac{(v-4)(v^3 + 3v^2 - 4v - 32)}{16(1-v)} \\ c_0 &= \frac{(v-4)^3 (v^2 + 3v + 4)}{64(1-v)},\end{aligned}$$ and over $K(w)$ the roots of $h$ are $$\begin{aligned} \beta_1 &= \frac{(2 + w)(2 - w)}{4} \\ \beta_2 &= \frac{-(2+w)^2(2 - w + w^2)}{4(1 + w)}\\ \beta_3 &= \frac{-(2-w)^2(2 + w + w^2)}{4(1 - w)}.\end{aligned}$$ Each polynomial $g_i$ is $x^2 - 2\beta_i x + (1-s)\beta_i^2$ and has roots $\beta_i (1 \pm w/2)$, so we calculate that the roots of $g_1$ are $$\begin{aligned} r_1 &= -(1/8) (2 - w)^2 (2 + w) \\ r_2 &= -(1/8) (2 - w) (2 + w)^2, \\ \intertext{the roots of $g_2$ are} r_3 &= -(1/8) (2 + w)^3 (2 - w + w^2) / (1 + w) \\ r_4 &= -(1/8) (2 - w) (2 + w)^2 (2 - w + w^2) / (1 + w),\\ \intertext{and the roots of $g_3$ are} r_5 &= -(1/8) (2 - w)^2 (2 + w) (2 + w + w^2) / (1 - w) \\ r_6 &= -(1/8) (2 - w)^3 (2 + w + w^2) / (1 - w).\end{aligned}$$ These roots are indexed in a manner consistent with the indexing of the roots in the proof of Theorem \[T-Galois\]. Note that the $r_i$ are related to the $\rho_i$ of Theorem \[T-simple2\] by the relation $$r_i = 4s(s-1)\rho_i - 2(s-1)^2,$$ so the $r_i$ are distinct exactly when the $\rho_i$ are distinct. It is easy to check that when $(v^2 + 3 v + 4)(v^2 - v + 4) (v^3 - 6 v^2 - 7 v - 4)\neq 0$ the $\rho_i$ are distinct, so in this case the curve $D$ of Theorem \[T-simple2\] has genus $2$. Furthermore, we see that $D$ is isomorphic to the curve ${{\mathcal C}}(r,s,t).$ For each $i$ let $W_i$ be the point $(\rho_i,0)$ of $D$. Let $G$ be the Galois-stable subgroup of the Jacobian of $D$ that consists of the divisor classes $$\{ [0], [W_1 - W_5], [W_2 - W_4], [W_3 - W_6]\}$$ and let $G'$ be the Galois-stable subgroup $$\{ [0], [W_1 - W_3], [W_2 - W_6], [W_4 - W_5]\}.$$ An easy computation shows that when $v^3 - 4 v^2 + 7 v + 4$ and $v^2 + v + 2$ are nonzero the $G$-Richelot dual of $D$ and the $G'$-Richelot dual of $D$ are defined (that is, the determinants mentioned in the definition of the two Richelot duals are nonzero). Then the results of Section \[S-RM\] show that the $G$-Richelot dual of $D$ and the $G'$-Richelot dual of $D$ become isomorphic over $K(\sqrt{st}\,) = K(\sqrt{v(v-4)}\,)$. The proof that these two Richelot duals of $D$ are geometrically non-isomorphic to one another (except in the special cases listed in the theorem) is a computation along the same lines as the proof of the corresponding statement of Theorem \[T-nonsimple2\]. We leave the details to the reader. Finally, suppose that $K$ has characteristic $0$ and suppose that there is a ring homomorphism from ${{\mathbb Z}}[v]$ to ${{\mathbb F}}_{13}$ that takes $v$ to either $2$ or $6$. Then the curve $D$ reduces modulo $13$ to one of the curves over ${{\mathbb F}}_{13}$ obtained when $v=2$ or $v=6$. One can compute that the characteristic polynomials of Frobenius for the Jacobians of these two curves are $t^4 - 4 t^3 + 22 t^2 - 52 t + 169$ and $t^4 + 4 t^3 + 22 t^2 + 52 t + 169$, respectively. Then [@Howe-Zhu Thm. 6] shows that the Jacobians are absolutely simple. Since $D$ modulo $13$ is absolutely simple, so is $D$ itself. And finally, since $C$ and $C'$ have Jacobians isogenous to that of $D$, we see that their Jacobians are absolutely simple too. The final statement of the theorem then follows from the observation that if $v$ is not algebraic, then there is a homomorphism ${{\mathbb Z}}[v]\to{{\mathbb F}}_{13}$ that sends $v$ to any given element. We used the field ${{\mathbb F}}_{13}$ at the end of the proof simply because it is the smallest prime field that contains values of $v$ that give rise to absolutely simple Jacobians. Other prime fields have a larger proportion of good values of $v$. For example, there are $341$ values of $v$ in ${{\mathbb F}}_{769}$ that give rise to absolutely simple Jacobians. For three-digit primes $p$ the number of good $v$-values is typically greater than $0.3 p$. This implies that for a “randomly chosen” rational number $v$, it is almost certainly the case that $v$ will give rise to an absolutely simple Jacobian. Proof of Theorem [\[T-nonsimple3\]]{} {#S-genus3} ===================================== The proof of Theorem \[T-nonsimple3\] is very much like the proof of Theorem \[T-nonsimple2\]: We will produce three elliptic curves $E_1$, $E_2$, $E_3$, two maximal isotropic subgroups $G$, $G'$ of the $2$-torsion of $A = E_1\times E_2\times E_3$, and an automorphism $\alpha$ of $A$ that takes $G$ to $G'$. Then we will produce a hyperelliptic curve whose Jacobian is $A/G$ and a plane quartic whose Jacobian is $A/G'$. To produce these curves we will use the results of [@HLP §4]. Our notation will be chosen to match that of [@HLP], except that we will continue to call our base field $K$, instead of $k$. Let $K$ be an arbitrary field of characteristic not $2$ and let $t$ be an element of $K$ with $t (t + 1) (t^2 + 1) (t^2 + t + 1) \neq 0$. Let $s = -(t^2 + t + 1)$ and let $r$ be a square root of $t^2 + 1$ in an algebraic closure of $K$. Let ---------------------------- --------------------------- -- $A_1 = -2(t^2+1)s$ $B_1 = (t^2+1)s^2$ $A_2 = 4(t^2+1)s$ $B_2 = 4t^2(t^2+1)s^2$ $A_3 = -2(t^2+t+1)s\qquad$ $B_3 = (t+1)^2(t^2+1)s^2$ ---------------------------- --------------------------- -- and for each $i$ let $$\begin{aligned} \Delta_1 &= A_1^2 - 4B_1 = 4t^2(t^2+1)s^2 \\ \Delta_2 &= A_2^2 - 4B_2 = 16(t^2+1)s^2 \\ \Delta_3 &= A_3^2 - 4B_3 = 4t^2s^2.\end{aligned}$$ Note that the $\Delta_i$ and the $B_i$ are nonzero, so we may define for each $i$ an elliptic curve $E_i$ by $$y^2 = x(x^2 + A_i x + B_i).$$ We define $2$-torsion points $P_i$ on the $E_i$ by setting $$\begin{aligned} P_1 &= \bigl((t^2+1)s - rts, 0\bigr)\\ P_2 &= \bigl( -2(t^2+1)s - 2rs, 0\bigr)\\ P_3 &= \bigl((t^2+t+1)s - ts, 0\bigr)\end{aligned}$$ and for each $i$ we let $Q_i$ be the $2$-torsion point $(0,0)$ on $E_i$ and we let $R_i = P_i + Q_i$. Let $A = E_1\times E_2\times E_3$ and let $G$ be the subgroup of $A[2]$ generated by $(P_1,P_2,P_3)$, $(Q_1,Q_2,0)$, and $(Q_1,0,Q_3)$. Associated to these choices of $A$ and $G$ there is a quantity called the [*twisting factor*]{} $T$ (see [@HLP §4]). Using the formula in [@HLP §4] we find that for our $A$ and $G$ the twisting factor is $0$, so we may apply [@HLP Prop. 14] to find a hyperelliptic genus-$3$ curve whose Jacobian is isomorphic over $K$ to $A/G$. The curve given by [@HLP Prop. 14] is defined by two equations in ${{\mathbb P}}^3$, namely $$\begin{aligned} W^2 Z^2 &= a X^4 + b Y^4 + c Z^4 \label{E-oldH1}\\ 0 &= d X^2 + e Y^2 + f Z^2 \label{E-oldH2}\end{aligned}$$ where $$\begin{aligned} a &= 4 t (t+1) (t^2+1)^3 s^5 \\ b &= 16 t^2 (t+1) (t^2+1)^3 s^5 \\ c &= 4 t (t+1)^2 (t^2+1)^2 s^6 \\ 1/d &= -2 t (t+1) (t^2+1) s^2 \\ 1/e &= (t+1) (t^2+1) s^2 \\ 1/f &= 2 t (t^2+1) s^2.\end{aligned}$$ If we replace $W$ by $2t(t+1)(t^2+1)s^3 W$ in Equation \[E-oldH1\] and divide out common factors, we get Equation \[E-H1\], and if we multiply Equation \[E-oldH2\] by $2 t (t+1) (t^2+1) s^2$ we get Equation \[E-H2\]. This shows that the Jacobian of $H(t)$ is isomorphic to $A/G$. Now let $G'$ be the subgroup of $A[2]$ generated by $(P_1,P_2,R_3)$, $(Q_1,Q_2,0)$, and $(Q_1,0,Q_3)$, and let $T'$ be the twisting factor associated to $A$ and $G'$. The formula in [@HLP §4] shows that $$T' = -64(t^2+1)^2(t^2+t+1)s^3 = 64(t^2+1)^2s^4,$$ so the twisting factor is a nonzero square. Then [@HLP Prop. 15] shows that there is a plane quartic whose Jacobian is isomorphic (over $K$) to $A/G'$. The plane quartic is given by $$\label{E-oldQ} B_1 X^4 + B_2 Y^4 + B_3 Z^4 + d' X^2Y^2 + e' X^2Z^2 + f'Y^2Z^2 = 0$$ where $$\begin{aligned} d' &= 4 (t^2 + 1) (2t^2 + t + 2) s^2\\ e' &= -2 (t^2 + 1) (2t^2 + t + 1) s^2\\ f' &= 4 (t^2 + 1) ( t^2 + t + 2) s^2.\end{aligned}$$ Dividing Equation \[E-oldQ\] by $(t^2+1)s^2$ gives Equation \[E-Q\], so the Jacobian of $Q(t)$ is isomorphic to $A/G'$. To complete the proof we must show that $A/G \cong A/G'$. Note that there is a $2$-isogeny $\psi$ from $E_1$ to $E_2$ that kills $Q_1$ and that takes $P_1$ and $R_1$ to $Q_2$ (see [@Silverman Example III.4.5]). Consider the automorphism $\alpha$ of $A$ that sends a point $(S_1,S_2,S_3)$ to $(S_1,S_2+\psi(S_1),S_3)$. It is easy to check that $\alpha(G) = G'$, and it follows that $A/G\cong A/G'$, as desired, so the Jacobians of $H(t)$ and $Q(t)$ are isomorphic over $K$. Examples {#S-examples} ======== The curves $$3y^2 = (x^2 - 4) (x^4 + 7x^2 + 1)$$ and $$- y^2 = (x^2 + 4) (x^4 + 3x^2 + 1)$$ over ${{\mathbb Q}}$ are geometrically non-isomorphic, and yet their Jacobians are isomorphic to one another over ${{\mathbb Q}}$. If we take the two curves obtained by taking $t = 2$ in Theorem \[T-nonsimple2\], replace $x$ by $x/2$ in each equation, and twist both curves by $2$, we get the two curves given above. The curves $$5 y^2 = - 6 x^6 - 64 x^5 - 113 x^4 + 262 x^3 - 331 x^2 + 584 x + 232$$ and $$2 y^2 = - 21 x^6 - 236 x^5 + 45 x^4 - 440 x^3 - 615 x^2 - 76 x - 553$$ are geometrically non-isomorphic, but their Jacobians become isomorphic to one another over ${{\mathbb Q}}(\sqrt{-1})$. Furthermore, their Jacobians are absolutely simple. Take $v = 2$ in Theorem \[T-simple2\]. We find that $\rho_1 = -w/4$, $\rho_2 = w/4$, $\rho_3 = 2w + 2$, $\rho_4 = w - 1$, $\rho_5 = -w - 1$, and $\rho_6 = -2w + 2$, where $w = \sqrt{2}$. The curves $C$ and $C'$ in the theorem are $y^2 = f_1$ and $y^2 = f_2$, where $$\begin{aligned} f_1 &= -(30625/32) x^6 - (67375/16) x^5 - (305025/64) x^4 \\ & \qquad - (23765/16) x^3 + (28665/16) x^2 + (1715/2) x - (735/2)\end{aligned}$$ and $$f_2 = -(553/2) x^6 + 38 x^5 - (615/2) x^4 + 220 x^3 + (45/2) x^2 + 118 x - (21/2).$$ Replacing $x$ with $-2/(x+1)$ in $f_1$ and multiplying the result by $(1/5)(2/7)^2 (x+1)^6$ gives rise to the first curve given in the example. Replacing $x$ with $-1/x$ in $f_2$ and multiplying the result by $2 x^6$ gives rise to the second curve. Thus the Jacobians of the two curves become isomorphic to one another over ${{\mathbb Q}}(\sqrt{v(v-4)}\,) = {{\mathbb Q}}(\sqrt{-1})$. The Jacobians are simple because we chose our $v$ to be $2$ modulo $13$. \[EX-realpoints\] The curves $$y^2 + (x^3 + x^2 + x) y = 31 x^6 - 38 x^5 - 217 x^4 - 380 x^3 + 304 x^2 + 501 x - 366$$ and $$11 y^2 = - 49 x^6 - 378 x^5 - 755 x^4 + 110 x^3 - 2285 x^2 + 732 x - 1368$$ are geometrically non-isomorphic, but their Jacobians are isomorphic to one another over ${{\mathbb Q}}$. Furthermore, their Jacobians are absolutely simple. Take $v = -4/3$ in Theorem \[T-simple2\]. The curves $C$ and $C'$ we obtain are $y^2 = f_1$ and $y^2 = f_2$, where $$\begin{aligned} f_1 &= (28125/268912) x^6 - (11250/16807) x^5 \\ & \qquad + (3154875/1882384) x^4 - (812325/470596) x^3 \\ & \qquad\qquad - (57675/470596) x^2 + (26325/16807) x - (2025/2401)\end{aligned}$$ and $$\begin{aligned} f_2 &= -(131769/38416) x^6 + (11979/343) x^5 \\ & \qquad - (5595645/38416) x^4 + (62535/196) x^3 \\ & \qquad\qquad - (3735435/9604) x^2 + (86229/343) x - (23199/343).\end{aligned}$$ If we replace $x$ with $(2x+2)/(x+2)$ in $f_1$, multiply the result by $(343/5)^2 (x+2)^6$, and twist by $3$, we get the curve $$y^2 = 125 x^6 - 150 x^5 - 865 x^4 - 1518 x^3 + 1217 x^2 + 2004 x - 1464;$$ replacing $y$ with $2y + (x^3 + x^2 + x)$ gives the first curve in the example. If we replace $x$ with $(x+2)/(x+1)$ in $f_2$, multiply the result by $(1/11) 196^2 (x+1)^6$, and twist by $3$, we get the second curve in the example. The Jacobians of the two curves become isomorphic to one another over ${{\mathbb Q}}(\sqrt{v(v-4)}\,) = {{\mathbb Q}}$. The Jacobians are absolutely simple because their reductions modulo $17$ are absolutely simple. It is easy to see that the first curve in Example \[EX-realpoints\] has real-valued points, while the second curve does not. It follows that the real topology of a curve over ${{\mathbb Q}}$ is not determined by its Jacobian. Furthermore, suppose we choose a positive integer $d$ such that the quadratic twist of the second curve by $d$ has rational points. The quadratic twist of the first curve by $d$ will still not have any real points, let alone any rational points, so we see that the existence of rational points on a genus-$2$ curve over ${{\mathbb Q}}$ is not determined by its Jacobian, even if the Jacobian is absolutely simple. There are also triples $(r,s,t)$ that satisfy the hypotheses of Theorem \[T-gensimple\] but that do not lie on the elliptic surface discussed in Section \[S-application\]. \[EX-twiceseen\] The curves $$y^2 = x^6 - 24 x^4 + 80 x^3 - 63 x^2 - 24 x - 2$$ and $$y^2 = -2 x^6 + 6 x^5 + 9 x^4 - 48 x^3 + 162 x - 171$$ are geometrically non-isomorphic, but their Jacobians become isomorphic to one another over ${{\mathbb Q}}(\sqrt{2})$. Furthermore, their Jacobians are absolutely simple. We take $r = -7/4$ and $s = 1/2$ and $t = 1/4$ in Theorem \[T-gensimple\]. Let $\xi$ be a root of the irreducible polynomial $$x^6 + 6x^4 + 9x^2 + 16$$ and let $K$ be the number field generated by $\xi$. The polynomial $h$ of Section \[S-Galois\] is $$h = T^3 + 3/4 T^2 + 9/16 T + 3/64$$ and its roots are the elements $$\begin{aligned} \beta_1 &= ( -\xi^4 -7\xi^2 -12)/16 \\ \beta_2 &= ( - \xi^5 +\xi^4 -5\xi^3 +7\xi^2 -10\xi )/32 \\ \beta_3 &= (\phantom{-}\xi^5 +\xi^4 +5\xi^3 +7\xi^2 +10\xi )/32\end{aligned}$$ of $K$. The polynomials $g_i$ are given by $$g_i = x^2 - 2\beta_i x + \beta_i^2 / 2 + 3/32,$$ and their roots (indexed in accordance with the proof of Theorem \[T-Galois\]) are $$\begin{aligned} r_1 &=( - \xi^4 - \xi^3 - 7\xi^2 - 7\xi - 12)/16 \\ r_2 &=( - \xi^4 + \xi^3 - 7\xi^2 + 7\xi - 12)/16 \\ r_3 &=( - \xi^5 - \xi^4 - 3\xi^3 - 3\xi^2 - 8\xi - 8)/32 \\ r_4 &=( - \xi^5 + 3\xi^4 - 7\xi^3 + 17\xi^2 - 12\xi + 8)/32 \\ r_5 &=(\phantom{-}\xi^5 + 3\xi^4 + 7\xi^3 + 17\xi^2 + 12\xi + 8)/32 \\ r_6 &=(\phantom{-}\xi^5 - \xi^4 + 3\xi^3 - 3\xi^2 + 8\xi - 8)/32\end{aligned}$$ We note that the two subgroups $G$ and $G'$ that appear in the proof of Theorem \[T-Galois\] are indeed Galois stable. We compute that the two Richelot duals are $y^2 = f_1 $ and $y^2 = f_2$, where $$\begin{aligned} f_1 &= - (81/512) x^6 - (1215/1024) x^5 - (21141/8192) x^4 - (8991/8192) x^3 \\ & \qquad - (19683/131072) x^2 - (2187/262144) x + (729/2097152)\\ f_2 &= - (1863/256) x^6 - (3159/512) x^5 - (26973/4096) x^4 - (11421/4096) x^3 \\ & \qquad - (76545/65536) x^2 - (28431/131072) x - (13851/1048576).\end{aligned}$$ Evaluating $f_1$ at $(2-x)/(4x)$ and multiplying the result by $(256/9)^2 x^6$ gives the first curve mentioned in the example; evaluating $f_2$ at $-x/(4x-8)$ and multiplying the result by $(128/9)^2 (x-2)^6$ gives the second curve. These curves are geometrically non-isomorphic, and their Jacobians become isomorphic over ${{\mathbb Q}}(\sqrt{st}\,) = {{\mathbb Q}}(\sqrt{2})$. Furthermore, their Jacobians are absolutely simple because their reductions modulo $7$ are absolutely simple. We obtained Example \[EX-twiceseen\] from a triple $(r,s,t)$ that does not lie on the elliptic surface from Section \[S-application\], but the same example can be obtained from the triple $(r,s,t) = (-10,-1,-2)$, which *does* lie on the surface. We computed all triples $(r,s,t)$ of naïve height at most $20$ for which the curve ${{\mathcal C}}(r,s,t)$ has two ${{\mathbb Q}}$-rational Richelot duals whose Jacobians are absolutely simple and isomorphic over ${{\mathbb Q}}$. Of all the examples we found, the triple $(r,s,t) = (-19/3, -6, -1/6)$ gave rise to the curves with the smallest coefficients: \[EX-smallest\] The curves $$y^2 = -9 x^6 + 6 x^5 - 47 x^4 - 14 x^3 - 5 x^2 - 36 x - 72$$ and $$y^2 = 8 x^6 - 60 x^5 + 235 x^4 - 186 x^3 - 239 x^2 - 30 x - 1$$ are geometrically non-isomorphic, but their Jacobians are isomorphic to one another over ${{\mathbb Q}}$. Furthermore, their Jacobians are absolutely simple. \[EX-g3\] The Jacobian of the hyperelliptic curve $$3 v^2 = - 17 u^8 + 56 u^7 - 84 u^6 + 56 u^5 - 70 u^4 - 56 u^3 - 84 u^2 - 56 u - 17$$ and the Jacobian of the plane quartic $$x^4 + 4 y^4 + 4 z^4 + 20 x^2 y^2 - 8 x^2 z^2 + 16 y^2 z^2 = 0$$ are isomorphic to one another over ${{\mathbb Q}}$. We take $t=1$ in Theorem \[T-nonsimple3\]. The plane quartic $Q(1)$ from the theorem is the plane quartic given in the example. The hyperelliptic curve from the theorem is given by the pair of homogeneous equations $$\begin{aligned} W^2 Z^2 &= -(1/3) X^4 - (4/3) Y^4 + Z^4 \label{E-exH1}\\ 0 &= -X^2 + 2Y^2 + 2Z^2. \label{E-exH2}\end{aligned}$$ We dehomogenize the equations by setting $Z=1$, and we parametrize the conic given by Equation \[E-exH2\] by setting $$\begin{aligned} X &= 2(u^2+1)/(u^2 + 2u - 1)\\ Y &= (u^2 - 2u - 1)/(u^2 + 2u - 1).\end{aligned}$$ Taking $W = v / (u^2 + 2u - 1)^4$ then gives us the hyperelliptic curve in our example. We note that the discriminant of the degree-$8$ polynomial used to define the hyperelliptic curve in Example \[EX-g3\] is $2^{94}$! [99]{} Curves of genus two with $\sqrt{2}$ multiplication, [http://front.math.ucdavis.edu/ANT/0213/]{}. : The Magma algebra system. I. The user language, [*J. Symbolic Comput.*]{} [**24**]{} (1997) 235–265. , London Math. Soc. Lecture Note Ser. [**230**]{}, Cambridge Univ. Press, 1996. : Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian, [*Proc. Amer. Math. Soc.*]{} [**129**]{} (2001) 1647-1657. [arXiv:math.AG/9805121]{}. : Large torsion subgroups of split Jacobians of curves of genus two or three, [*Forum Math.*]{} [**12**]{} (2000) 315–364. [http://front.math.ucdavis.edu/ANT/0057/]{}. : On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field, [*J. Number Theory*]{} [**92**]{} (2002) 139–163. [arXiv:math.AG/0002205]{}. : Arithmetic variety of moduli for genus two, [*Ann. of Math. (2)*]{} [**72**]{} (1960) 612–649. : Neron models, Lie algebras, and reduction of curves of genus one, in preparation. : Construction de courbes de genre 2 à partir de leurs modules, pp. 313-334 in: [*Effective methods in algebraic geometry*]{} (T. Mora and C. Traverso, eds.), Progr. Math. [**94**]{}, Birkhäuser, Boston, 1991. : [*The Arithmetic of Elliptic Curves*]{}, Grad. Texts in Math. [**106**]{}, Springer-Verlag, New York, 1986. : Zum Beweis des Torellischen Satzes, [*Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa*]{} (1957) 33–53.

--- abstract: 'The role of the off-shell dependence of $\rho-\omega$ mixing in the charge symmetry breaking nucleon-nucleon potential is discussed. It is shown that models describing the off-shell dependence of $\rho-\omega$ mixing are not sufficient to determine the charge symmetry breaking nucleon-nucleon potential.' address: 'Department of Physics$^a$ FM-15 and Institute For Nuclear Theory$^b$ NK-12, University of Washington, Seattle, Washington 98195, USA' author: - 'Thomas D. Cohen$^{a,b}$[^1], and Gerald A. Miller$^a$' title: 'Rho-Omega Mixing Off Shell and Charge Symmetry Breaking In the N-N Potential ' --- Introduction ============ Charge symmetry breaking has been studied for a long time, see e.g. the reviews[@NS69; @H69; @SHLOMO; @HM79; @MNS90; @NMS92; @MVO94] and especially the references therein. We follow ref. [@MVO94] in summarizing a few main features. Charge independence and charge symmetry breaking is caused by the $d$-$u$ quark mass difference $m_d-m_u >0$, and electromagnetic effects. The general goal of this area of research is to find small but observable effects of the breaking of charge independence and charge symmetry. This provides significant insight into strong interaction dynamics since the underlying origin of the breaking is understood. Over the years there has been substantial experimental and theoretical progress. First, we recall the old idea that $m_d-m_u>0$ along with electromagnetic effects accounts for the observed mass differences between members of hadronic isospin multiplets. This mass difference also leads to the notion that the physical $\rho$ and $\omega$ mesons are isospin mixed superpositions of bare states of good isospin. Indeed, substantial effects of $\rho-\omega$ mixing have been observed in the $e^+e^- \rightarrow \pi^+\pi^-$ cross section at $q^2\approx m^2_\omega$ [@Q78; @B85]. These results allow an extraction of the strong contribution to the $\rho-\omega$ mixing matrix element $<\rho|H_{str}|\omega>\approx$ - 5200 MeV$^2$. [@M94; @MVO94] Two nucleons may exchange a mixed $\rho-\omega$ meson. If one uses $<\rho|H_{str}|\omega>\approx$ - 5200 MeV$^2$ one obtains a nucleon-nucleon interaction which is consistent with the experimental value $\Delta a_{CSB} = a^N_{pp}-a^N_{nn} = 1.5\pm 0.5$ fm [@CB87]. Such a force can also consistently account most of the strong interaction contribution to the $^3H$-$^3He$ binding energy difference [@CB87] and for much of the Nolen-Schiffer anomaly[@BI87]. The TRIUMF (477 MeV [@A86] and 350 MeV[@GV369]) and IUCF (183 MeV) [@K90] experiments have compared analyzing powers of $\vec np$ and $n\vec p$ scattering and observe charge symmetry breaking at the level expected from $\pi,\gamma$ and $\rho$-$\omega$ exchange effects. The latter effects are important at 183 MeV. Thus $\rho$-$\omega$ mixing seems to describe most of the observed features or charge symmetry breaking in nuclear physics. While it is certainly true that other mechanism cannot be ruled out, $\rho$-$\omega$ mixing appears to give a consistent description of the bulk of the experimental data. Recently, this success has been called into question. The momentum dependence of the $\rho-\omega$ mixing amplitude has been calculated using several different models[@GHT92; @PW92; @KTW93; @MTRC94]. While these models are based on quite different physical assumptions, they all share one important quality: the $\rho-\omega$ mixing at spacelike momenta in all of these models is quite different from its value at the $\omega$ pole—generally of the opposite sign and significantly reduced in magnitude. Indeed it has been shown that for a wide class of models[@OCPTW94] the mixing must go to zero at $q^2=0$ implying that amplitude changes sign. Moreover, a QCD sum-rule calculation, also apparently gives a similarly large momentum dependence of the coupling[@HHKM94]. Since the N-N potential probes the spacelike region, this appears to imply that the vector meson exchange part of the charge symmetry breaking nucleon-nucleon NN potential is very different from one based on the on-mass-shell mixing. Indeed, NN potentials have been constructed based on these momentum dependent mixing amplitudes and these are quite different from the ones used in the successful phenomenology[@GHT92]-[@MTRC94],[@IN94]. The purpose of the present paper is to study the general role of the off-shell meson propagator in NN potentials. We find that knowledge of the off-shell meson propagator is not sufficient to determine the potential. In particular, one needs the vertex functions computed from the same theory that supplied the propagator. None of the present treatments of the off-shell propagator deals with the issue of the necessary vertex functions. It is not our intent to compute these functions. Rather, we wish to clarify issues of principle. Accordingly we have included a number of simple illustrative examples. We do show, however, that the CSB induced by the $\rho-\omega$ exchange potential can account for the existing data even if the the $q^2$ dependence is exactly as specified in any of the references [@GHT92]-[@MTRC94],[@IN94],[@HHKM94]. This is done by using CSB vertex functions We turn to an outline of this paper. In Sect. II we discuss the problem that in hadronic field theoretic models there is never a unique choice for fields, even in a renormalizable theory[@Haag], [@Ruelle], [@Borchers] and [@Coleman]. This means that the propagator and the vertex functions are not unique. We argue generally and with two explicit examples that while the propagator depends on the choice of field variables, the observables do not. Thus, knowledge of the off-shell meson propagator by itself gives no information unless one knows which field is used. One may be able to deduce which definition of the field has been used from a complete theory by studying the interactions with the other degrees of freedom in the problem. However, if the theory is incomplete and the interactions of the field with all of the other degrees of freedom is unknown, knowledge of the off-shell propagator by itself is not physically meaningful. There is an even more serious problem. Modern meson exchange potentials are motivated by field theoretic concerns. However, there is no first principle method for obtaining the “correct” NN potential directly from either QCD, or from some hadron field theoretic model or from any experimentally accessible set of data of hadronic properties. Given this essential difficulty, we believe it is sensible to adopt the general approach used in the construction of meson exchange potentials to the case of charge symmetry breaking. This approach makes the pragmatic assumptions of including the long range features in the meson propagators and the short range features in the vertex functions. This separation is discussed in Sect. III. Such a separation may be questioned, but [*a priori*]{} these assumptions should be no worse for the case of CSB potentials then they are for the isospin conserving part of the interaction. Moreover given the lack of rigor in the construction of potential from the underlying field theory, some assumptions must be made in order to make make any connection between $\rho-\omega$ mixing and the CSB potential. Given this, it is highly desirable to make sure that the assumptions are consistent with those made elsewhere in the problem. It is worth stressing at the outset, that in conventional treatments of meson exchange potentials the off-shell propagator plays no role. This is discussed in Sect. IV where realistic boson-exchange charge symmetric potentials are defined to be those that are consistent with the separation discussed in Sect.III. We show that for models with realistic spectral functions the momentum dependence of the meson propagator can be absorbed into that of the vertex function. An example of an unrealistic momentum dependent $\omega$ self energy is presented. The ideas of the Sects. II-IV are applied to the CSB potential caused by $\rho-\omega $ exchange in Sect. V. We show that the influence of the momentum dependence of the $\rho-\omega $ mixing matrix element can be included by allowing the $\rho$-nucleon coupling constant to violate charge symmetry. In particular, if the model of Ref.[@HHKM94] is used one needs CSB coupling constants that are 0.8% of the standard coupling constants to reproduce the results of a potential obtained without momentum dependence in the $\rho-\omega $ mixing matrix element and without CSB in the coupling constants. We summarize the analysis in Sect. VI. Field Redefinitions And Off-Shell Propagators \[fr\] ==================================================== It has been known for quite some time that value of an off shell propagator is completely dependent on the choice of field. This is an example of a general theorem proved by Haag[@Haag], Ruelle[@Ruelle], and Borchers[@Borchers] which has been discussed by Coleman, Wess and Zumino[@Coleman]. The off-shell propagators depend on the choice of interpolating fields, whereas all S-matrix elements are independent of this choice. Thus an off-shell propagator, taken in isolation, can have no physical meaning. To illustrate why this is so, let us consider the simplest possible case, the field corresponding to a stable scalar particle in some nontrivial interacting field theory. The equation of motion for this system may be written as $$\Box \phi(x) + m^2 \phi(x) = -j(x).$$ This equation of motion is determined from a Lagrangean density ${\cal L}(\phi,j)$. Furthermore, let us insist on studying the renormalized field, mass and current. This means that the correlation function for $\phi$ will have a pole with residue of unity at the physical mass, $m$: $$\langle \phi, \vec{p} | \phi(x) | {\rm vac} \rangle = e^{ip\cdot x}$$ which implies that $$\lim_{q^2 \rightarrow m^2}\, (q^2 - m^2) \, \int {\rm d}^4 x \, e^{i q \cdot x} \, \langle {\rm vac}| {\rm T}[\phi(x) \phi(0)] |{\rm vac} \rangle \, = \, i . \label{renorm}$$ We are concentrating on the renormalized quantities because un-renormalized properties are not observable and depend on the details of the renormalization procedure. Ultimately we will be interested in the spectral decomposition of the propagator in terms of the physical states of the system and this is directly related to the [*renormalized*]{} fields and sources. It is worth noting that the renormalization conditions put quite stringent constraints on matrix elements of the renormalized source. In particular they imply that source does not connect the vacuum to a one particle state $$\langle \phi , \vec{p} | j(0) | {\rm vac} \rangle \, = \, 0 . \label{cond}$$ This can be seen simply: $$\langle \phi , \vec{p} | j(x) | {\rm vac}\rangle = \langle \phi , \vec{p} | (\Box + m^2) \phi(x) | {\rm vac}\rangle = (-p^2 + m^2) \langle \phi , \vec{p} | \phi(x) | {\rm vac} \rangle$$ where $p^2$ is the square of the four momentum of the state which is $m^2$. Now we come to the crux of the issue. There is enormous freedom in the choice of field variables, and consequently the Green’s functions. In particular, we introduce a new renormalized field and a new source current according to: $$\begin{aligned} \phi^{\prime}(x) & = & \phi(x) + a(x), \label{def}\\ j^{\prime}(x)& = & j(x) + (\Box + m^2)\, a(x) \label{jpdef}\end{aligned}$$ where $a(x)$ is an operator such that $\langle {\rm vac} | a| \phi, \vec{p} \rangle = 0$. Thus, for example, $a(x)$ may be a multiple of the renormalized source $j(x)$ or $a(x)$ could have the form $a(x) = (\Box + m^2) b(x)$ where $b(x)$ is an arbitrary renormalized local composite operator. The new field and source satisfies an equation of motion with the same form as the original: $$(\Box + m^2) \phi^{\prime}(x) = j^{\prime}(x) \label{eomp}$$ It also satisfies the same renormalization conditions $$\begin{aligned} \langle \phi, \vec{p} | \phi^{\prime}(x) | {\rm vac} \rangle & = & e^{i p\cdot x} \\ \lim_{q^2 \rightarrow m^2} \, (q^2 - m^2) \, \int {\rm d}^4 x \, e^{i q \cdot x} \, \langle {\rm vac}|{\rm T}[\phi^{\prime}(x) \phi^{\prime}(0)]|{\rm vac} \rangle \, & = &\, i \label {renorm2}\end{aligned}$$ The field variable $\phi^{\prime}$ is as good a choice for the field variable as the original field $\phi$—its equation of motion is of the same form and it satisfies the correct renormalization conditions. It makes no difference to any [*physical*]{} amplitude whether one chooses to describe the physics in terms of the field $\phi$ or $\phi^{\prime}$. Thus, the masses of particles and possible bound states and S matrix elements for scattering states must be identical with either description. Going from one to the other amounts to nothing more than a change of variables. While the physics clearly does not depend on which field is chosen, the propagator depends strongly on this choice: $$\begin{aligned} \int {\rm d}^4 x \, e^{i q \cdot x} \, \langle {\rm vac}| {\rm T}[\phi^{\prime}(x) \phi^{\prime}(0)] |{\rm vac}\rangle \, \nonumber\\ = \int {\rm d}^4 x \, e^{i q \cdot x} \, \langle {\rm vac}|{\rm T}[\phi(x) \phi(0)] \rangle +\int {\rm d}^4 x \, e^{i q \cdot x} \, \langle {\rm T}[\phi(x) a(0)] + {\rm T}[a(x) \phi(0)] + {\rm T}[a(x) a(0)]|{\rm vac}\rangle\end{aligned}$$ Eq.(\[renorm2\]), which picks out the pole at $q^2 = m^2$, is obtained since by construction $a$ does not connect the vacuum to the one $\phi$ state. Clearly, this is necessary since the correlation functions for $\phi$ and $\phi^{\prime}$ satisfy the renormalization conditions in eqs. (\[renorm\]) and (\[renorm2\]). Off shell, however, there is no requirement that this term vanish and the two propagators will in general differ. Moreover, since the overall scale of $a$ is arbitrary it is clear that one can make the difference between the two descriptions arbitrarily large. Let us make these ideas explicit by considering two examples from a theory in which the current $j$ is a static external source. In this case the energy of the system is given by $$\begin{aligned} E=\int {\rm d}^3r {1\over 2} j(\vec r)\phi(\vec r)\end{aligned}$$ or $$\begin{aligned} E=\int {\rm d}^3r {1\over 2} j(\vec r) G(\vec r,\vec r^{\,\prime}) j(\vec r^{\,\prime}),\end{aligned}$$ where $G(\vec r,\vec r^{\,\prime})$ is the inverse of the operator $\nabla^2-m^2$. Let us first take $a(x)$ to be a simple function of $\vec x$, which is independent of $\phi$. Then $\langle \phi,\vec p|a(x)|{\rm vac}\rangle=0$ and the renormalization conditions of eqns. (\[renorm\],\[renorm2\]) are satisfied. One may determine a new Lagrangean density ${\cal L}^\prime$ and a new Hamiltonian ${\cal H}^\prime$ by starting with the original ${\cal L}$ and transforming the variables. Then the new energy $E^\prime$ is given by $$\begin{aligned} E^\prime=\int {\rm d}^3r {1\over 2} [\vec \nabla (\phi^\prime(\vec r)+a(\vec r)\cdot(\phi^\prime(\vec r)+a(\vec r)) + \nonumber\\ m^2(\phi^\prime(\vec r)+a(\vec r))^2 +2 j(\vec r)(\phi^\prime(\vec r) +a(\vec r))],\end{aligned}$$ and using the equation of motion (\[eomp\]) in the static limit leads to $$\begin{aligned} E^\prime=\int {\rm d}^3r {1\over 2}j(\vec r)(\phi^\prime(\vec r) +a(\vec r)).\end{aligned}$$ But Eq.(\[def\]) tells us that $E^\prime=E$. Even though the current $j^\prime$ of Eq.(\[jpdef\]) is different than $j$ the energy of the system does not depend on the choice of the function $a(\vec x)$. A more interesting example is obtained by letting $\phi=(1+f(\vec x))\phi^\prime $ (or $a(\vec x)= -{f(\vec x)\over 1+f(\vec x)} \phi(x)).$ We place the static source j at the origin and choose $f(\vec x)$ to vanish at large values of $|\vec x|$ faster than $e^{-m|\vec x|}/|\vec x|$. This maintains the original value of the renormalized coupling constant ( which is proportional to the asymptotic field) and therefore is the analog of our renormalization for problems with static sources. In this case the equation of motion is $$D\phi^\prime=-j^\prime \label {neom}$$ where $$D\equiv (1+f)^2 (-\nabla^2+m^2) -2(1+f) \partial_\mu f \partial^\mu -(1+f)\nabla^2f$$ and $$j^\prime\equiv j(1+f).$$ Clearly the Green’s function $G^\prime$ ( the inverse of $D$) and current $j^\prime$ are both fairly complicated. The use of the new Hamiltonian density ${\cal H}^\prime $ gives $$\begin{aligned} E^\prime=\int {\rm d}^3 r{1\over 2}[ (1+f)^2 \vec\nabla\phi^\prime\cdot\vec\nabla\phi^\prime \nonumber \\ +(\phi^\prime)^2 \vec\nabla f\cdot\vec\nabla f +2\phi^\prime(1+f)\vec\nabla f\cdot\vec\nabla\phi^\prime+ (m\phi^\prime)^2(1+f)^2+2j\phi^\prime(1+f)].\end{aligned}$$ Integration by parts and the equation of motion Eq.(\[neom\]) allows one to obtain $$E^\prime \, = \, {1\over 2}\int {\rm d}^3 r j(1+f)\phi^\prime \, =\,{1\over 2}\int {\rm d}^3 r j^\prime(\vec r)G^\prime(\vec r,\vec r^\prime)j^\prime(\vec r^\prime),$$ which is just the original energy since $(1+f)\phi^\prime=\phi$. Thus we have seen two explicit examples in which transformations of field variables change the equation of motion, the Green’s functions and the currents without changing, the physical observable, the energy of the system. These same arguments of Eqns. (5-10) can be used to show that the various n-point vertex functions of the also depend on the specific choice of field. The generalization of the argument to vector fields rather than scalars and to correlation functions of two different fields uses standard techniques. Again one finds that off-shell propagators and the vertex functions depend explicitly on the choice of field. It is clear what is going on here. Neither the off-shell propagators nor the vertex functions are directly observable. From a theoretical point of view, the values of these quantities depend explicitly on which arbitrary choice of field one makes. Various combinations of the propagators and the vertex functions correspond to physical quantities and it is only these combinations which can be measured. Choosing a particular field amounts to making a bookkeeping choice—it only determines whether some bit of the physics will be found in the vertex or in the propagator. The point we wish to stress is that knowledge, however precise, of the off-shell propagator contains no physical information unless one specifies the choice of the quantum field or equivalently unless one has knowledge of how the field couples to the rest of the system—[*i.e.*]{} knowledge of the vertex functions which arise from the same field choice. Thus, a model for the off-shell propagator in the absence of a [*consistent*]{} model for the vertex functions is not complete. The models of refs. [@GHT92]-[@MTRC94],[@IN94] present the mixed $\rho-\omega$ propagator off-shell, but do not give the necessary simultaneous consistent description of the CSB N-N-vector meson vertices. Philosophy Of Meson Exchange Potentials ======================================= The preceeding argument that off-shell meson propagators are not sufficient is entirely based on field theoretic considerations. Clearly, this does not help us to to compute observables, it does not address the question of how one can compute CSB (or any other) observables in nuclear physics. One typically constructs a nucleon-nucleon potential and then computes wavefunctions, hoping that the potentials capture the essential aspects of the underlying field theory. Nevertheless, there is no unambiguous way to construct potentials. Nontrivial assumptions must be made. Here we will assume that the assumptions underlying phenomenologically successful meson exchange models are reasonable. While one can construct equally successful purely phenomenological models, the meson exchange models make a connection to the spectral properties of the underlying theory. Moreover, the entire question we are investigating—the role of $\rho-\omega$ mixing in CSB effects in nuclear physics—can only be addressed in the context of a potential model which employs vector mesons. There is a definite philosophy underlying the construction of NN potentials from meson exchange. One principal idea is the need for a separation of momentum or length scales. One explicitly includes the exchange of mesons lighter (and hence more long-ranged) than some scale separation point. All short ranged effects are either incorporated in phenomenologically determined vertex functions or by some other purely phenomenological means. The physical picture underlying this philosophy is that the nucleon has a three-quark core which cannot be described efficiently in terms of mesons, while at longer distances the nucleon structure is dominated by a meson cloud. To some extent, the fact that short ranged effects are handled as pure phenomenology is of little importance in most low energy nuclear physics applications. Because of repulsion at short distances, nuclear wave functions have strong short distance correlations which prevent the system from feeling the truly short range part of the potential. Moreover, at very short distances the concept of an NN potential becomes particularly inappropriate. Typically, in meson exchange potentials this scale separation point, which we will call $\Lambda_s$, is taken to be of order 1 GeV so that $\rho$ and $\omega$ mesons are explicitly included while heavier vector mesons are not. It is worth observing, however, that this does not mean that the short distance physics does not have important long range consequences. In particular, the value of the meson-nucleon coupling constant, determined by short distanced physics, plays an essential role in the potentials at long and intermediate ranges. We believe that this general approach of treating the short range part of the NN interaction phenomenologically while explicitly including the effects of lighter mesons is reasonable. This general approach ought to be applicable to charge symmetry breaking effects. There is another important assumption which underlies these models. It is assumed that at except at short distances the vector part of the potential is dominated by the vector mesons. Thus it is assumed that continuum two pion vector-isovector and three pion vector-isoscalar exchange contributions are small— [*i.e.*]{} that the only substantial strength arising from the two pion vector-isovector exchange is sufficiently concentrated at the $\rho$ mass as to be well described by $\rho$ exchange and analogously for three pions and the $\omega$ exchange. We note that this assumption can be questioned. In its favor we note that in $e^+ e^- \rightarrow$ pions, the $\rho$ and $\omega$ peaks do, in fact, completely dominate the low lying spectral function. In our discussions we will adopt the Bonn potential [@BONN] strategy of incorporating all short range effects in vertex functions. In such a strategy the scale separation between long and short range is particularly easy to enforce: the phenomenological vertex functions are analytic for $q^2 < \Lambda_s^2$ while the propagators are analytic for $q^2 > \Lambda_s^2$, where $q^2$ is the square of the four momentum. Momentum-Dependent Self-Energies in Meson-Exchange Potentials \[csc\] ====================================================================== It is probably useful to discuss an analogous, and perhaps somewhat simpler problem before discussing charge symmetry breaking. The $\rho-\omega$ mixing matrix element is an off-diagonal mass term. Models which give momentum dependence to this off-diagonal mass can also be expected to give momentum dependence to the analogous diagonal mass terms—[*i.e.*]{} to the vector meson self-energies. It is clearly useful to understand the role of the momentum dependence of the $\rho$ and $\omega$ self energies in the charge symmetry preserving potential before taking on the challenge of understanding the the momentum dependence of the $\rho-\omega$ mixing. For simplicity we examine the one $\omega$ exchange contribution. First consider the traditional meson exchange model description with the scale separation as outlined above. The potential is given by $$V_{\omega} (q^2) = \frac{( g_\omega^{ \rm v}(q^2)\gamma^{(1)}_{\mu} \, + \, g_\omega^{ \rm t}(q^2) q^\alpha \sigma^{(1)}_{\alpha \mu} ) (g^{\mu \nu} - q^{\mu}q^{\nu}/m_{\omega}^2) (g_{\omega}^{\rm v}(q^2) \gamma^{(2)}_{\nu} + g_{\omega}^{\rm t}(q^2) \sigma_{\nu \beta} q^\beta ) }{q^2 - m_\omega^2} \label{oep}$$ where $g_{\omega}^{\rm v} (q^2)$ and $g_{\omega}^{\rm t}(q^2)$ are the vector and tensor couplings of the omega to the nucleons. The superscripts 1 and 2 label the nucleon. These couplings are analytic functions of $q^2$ for $q^2 < \Lambda_s^2$; the propagator is clearly analytic for $q^2 > \Lambda_s^2$. In principle, we could consider a more sophisticated model consistent with the philosophy outlined above. For example, one could explicitly include the exchange of three low-energy pions (with the quantum numbers of the rho) along with an omega self-energy due to its coupling with the three pion channel and a longer-range part of the $\omega$-N vertex due to three pion exchange. In practice, one expects such effects to be small: in part they serve to simply widen the omega pole by an amount of no practical significance to the potential; other effects of coupling to the three pion channel are small because they are weakly coupled. In any event, we will stick to the conventional assumptions underlying meson exchange models and neglect such effects. In the remainder of this paper we will ignore such effects. Let us now suppose that we had a detailed microscopic model of the $\omega$ meson which enables us to calculate a momentum dependent $\omega$ self energy, $\pi_{\omega}(q^2)$. As a matter of convention, we will include any effects of mass and wavefunction renormalizations of the $\omega$ in $\pi_{\omega}(q^2)$. This means that $\pi_{\omega}$ and its derivative vanishes at $q^2=m_\omega^2$. The omega exchange part of the N-N potential with such a model is given by $$V_{\omega}(q^2) = \frac{( \hat{g}_{\omega}^{\rm v}(q^2)\gamma^{(1)}_{\mu} \, + \, \hat{g}_{\omega}^{\rm t}(q^2) q^\alpha \sigma^{(1)}_{\alpha \mu} ) (g^{\mu \nu} - q^{\mu}q^{\nu}/m_{\omega}^2) (\hat{g}_{\omega}^{\rm v}(q^2) \gamma^{(2)}_{\nu} + \hat{g}_{\omega}^{\rm t}(q^2) \sigma_{\nu \beta} q^\beta ) }{q^2 - m_\omega^2 + \pi_\omega(q^2)}\;. \label{oep2}$$ We have written the couplings as $\hat{g}_{\omega}^{\rm v,t}$ and rather than $g_{\omega}^{\rm v,t}$ to make evident the fact that the vertex functions used in the model in eq. (\[oep2\]) need not be the same as the vertex functions used in the model in eq. (\[oep\]): [*these vertex functions are phenomenological and depend on how the rest of the problem is treated*]{}. Given that the vertex functions may differ between the two models, we note that the two models may be identical—[*i.e.*]{} they may be two equivalent ways of representing the same physics. One way for this to occur is if the vertex functions in the two models are related by $$g_\omega^{\rm v,t} \, = \, \hat{g}_\omega^{\rm v,t} \, \left (\frac{q^2 - m_\omega^2}{q^2 - m_\omega^2 + \pi_{\omega}(q^2)} \right )^{1/2}. \label{condition}$$ Note that the square root factor is unity for $q^2=m_\rho^2$ due to the renormalization of $ \pi_{\omega}(q^2)$. The result (\[condition\]) is not surprising in light of the formal analysis of Sect.II. Neither the propagator off-shell nor the vertex function are separately meaningful. Given that vertex functions are fit to some set of data, the only reason the condition in eq. (\[condition\]) would not be satisfied would be due to practical and philosophical limitations in the forms used in the fitting of the vertex functions. The practical limitation is that one must take some limited trial form for the phenomenological coupling. To the extent that meson exchange models make sense in the regime where they are used, the trial forms must be rich enough to describe the data with reasonable precision. Thus, apart from the philosophical concerns discussed below, eq. (\[condition\]) can be satisfied well enough so that any difference between the potentials of eqs. (\[oep\]) and (\[oep2\]) will have a small effect on the physics. The philosophical limitation is that the vertex functions are supposed to only contain effect of a range shorter than $\Lambda_s^{-1}$. Longer range effects are to be included by explicit dynamics of the lighter degrees of freedom in the problem. Thus, the issue of whether the two models are equivalent comes down to whether both $g_{\rm v,t}(q^2)$ and $\hat{g}_{\rm v,t}(q^2)$ can be analytic for $q^2<\Lambda_s^2$ while satisfying eq. (\[condition\]). In effect, the question is whether $$f(q^2) = \left (\frac{q^2 - m_\omega^2}{q^2 - m_\omega^2 + \pi_{\omega}(q^2)} \right )^{1/2}$$ is analytic for $q^2 < \Lambda_s^2$. Non-analyticity can occur when either the numerator or denominator vanishes or when $\pi_{\omega}$ is non-analytic. In fact, we should relax this restriction slightly—the non-analyticity associated with the $\omega$ coupling to three low energy pions which slightly broadens the pole and gives a small non-resonant contribution is, as discussed above, innocuous. In any event, this issue does not arise in the context of the models in refs. [@GHT92]-[@MTRC94]. Clearly, the analytic structure of $f(q^2)$ depends in detail on the choice of model. The simplest way to make the physics explicit is to make a spectral representation[@spectralref] for the propagator: $$\frac{1}{q^2 - m_\omega^2 + \pi_{\omega} (q^2)} = \int \, {\rm d}s \, \frac{\rho(s) }{q^2 - s + i \epsilon} + {\rm subtraction \quad terms}. \label{prop}$$ Different models will give rise to different spectral functions. However, if the model is realistic, the only substantial spectral strength for $q^2< \Lambda_s^2$ occurs at or near the omega pole. Accordingly any model which gives significant amounts of spectral strength below $\Lambda_s^2$ (apart from the $\omega$ pole), can be considered as unrealistic in our philosophy. If, however, all of the spectral strength is either at the $\omega$ pole or above $\Lambda_s^2$, then $f(q^2)$ is analytic for $q^2$ below $\Lambda_s^2$. The apparent non-analyticity due to the denominator vanishing at $q^2= m_\omega^2$ is precisely canceled by a vanishing numerator. (Recall, all renormalization effects are included in $\pi_\omega$ so that the position of the $\omega$ pole does not shift.) In this case, one may re-define the vertex functions according to eq. (\[condition\]). To see how the spectral representation constrains the allowable forms of the self energies consider consider the following simple example in which the self energy has the form $$\pi_{\omega} (q^2)=(q^2-m_\omega^2)^2 Bq^2. \label{exam}$$ This form is motivated by the renormalization requirements, that $\pi_\omega(q^2)$ and its derivative vanish at $q^2=0$. One determines the nucleon-nucleon potential generated by the propagator of Eq.(\[prop\]) by taking $q^2$ to be space-like $q^2=-Q^2<0$; the potential is proportional to the integral $$\int {\rm d}Q Q{\sin (Qr)\over r} \frac{1}{-Q^2 - m_\omega^2 -(Q^2+m_\omega^2)^2 BQ^2 }.$$ One does the contour integration by identifying the poles. There is always a pole at $Q^2=-m_\omega^2$ which is the standard term expected from the exchange of an $\omega^2$-meson. There are other poles at positions determined by the value of B. One finds that if $ m_\omega^4> m_\omega^4+4/B\ge0$ there will be poles with $Q^2$ real and negative. At least one of the poles must be at $|Q^2|<m_\omega^2$, which is unrealistic in our philosophy. If $m_\omega^4+4/B > m_\omega^4 $, the poles occur for $Q^2>0$ which are physically un-allowable tachyonic excitations. Similarly, if $m_\omega^4+4/B<0$, there are poles off the real axis, which violates the spectral representation and also renders the model for $ \pi_{\omega} (q^2)$ as useless. This analysis demonstrates that a spectral function of the form in Eq.(\[exam\]) is not viable. Let us now summarize the effects of the momentum dependence of the $\omega$ self energy on the meson exchange potential. In any realistic model, ([*i.e.*]{} any model without unphysical low $q^2$ spectral strength in the $\omega$ propagator) all of the effects of the momentum dependence of the self-energy can be re-absorbed into momentum dependence of the phenomenological vertex functions. Accordingly, there are no observable physical effects in the NN potential induced by such a momentum dependent self-energy. Moreover, including the short range part of the momentum dependence in the propagator of a meson exchange model violates the bookkeeping arrangement in which all of the short range effects are segregated into phenomenological vertices. The Charge Symmetry Breaking NN Potential And The Momentum Dependence of $\rho -\omega$ Mixing ============================================================================================== The preceeding section gives us a paradigm for what happens in the charge symmetry breaking part of the potential. We will show, for any realistic model of the momentum dependence of the mixing amplitude, that all of the effects of the momentum dependence can be absorbed into phenomenological short ranged charge-symmetry-breaking nucleon-vector meson couplings. Consider the charge-symmetry-breaking potential arising from vector meson exchange. Let us begin by implementing this according to the philosophy of scale separation discussed in the previous two sections. Assuming that only the meson exchanges we need to consider are the $\rho$ and $\omega$, the charge symmetry breaking interaction potential can be written as $$\begin{aligned} V_{\omega,\rho}^{\rm CSB} (q^2) \, = \, \frac{[g_\omega^{\rm v}(q^2) \gamma_{\mu}^{(1)} + g_\omega^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(1)}] [g^{\mu \nu} - q^{\mu}q^{\nu}/m_{\omega}^2] [g_\omega^{\rm v \, CSB}(q^2) \tau_3^{(2)} \gamma_{\nu}^{(2)} + g_\omega^{\rm t \, CSB}(q^2) \tau_3^{(2)} \sigma_{\nu \beta}^{(2)} q^\beta] }{q^2 - m_\omega^2} \nonumber \\ \nonumber \\ + \, \frac{ [g_\omega^{\rm v}(q^2) \gamma_{\mu}^{(2)} + g_\omega^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(2)}] [g^{\mu \nu} - q^{\mu}q^{\nu}/m_{\omega}^2] [g_\omega^{\rm v \, CSB}(q^2) \tau_3^{(1)} \gamma_{\nu}^{(1)} + g_\omega^{\rm t \, CSB}(q^2) \tau_3^{(1)} \sigma_{\nu \beta}^{(1)} q^\beta] }{q^2 - m_\omega^2} \nonumber \\ \nonumber \\ + \, \frac{ [g_\rho^{\rm v} (q^2) \gamma_{\mu}^{(1)} \tau_3^{(1)} + g_\rho^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(1)} \tau_3^{(1)}][g^{\mu \nu} - q^{\mu}q^{\nu}/m_{\rho}^2][g_\rho^{\rm v \, CSB}(q^2) \gamma_{\nu}^{(2)} + g_\rho^{\rm t \, CSB}(q^2) \sigma_{\nu \beta}^{(2)} q^\beta] }{q^2 - m_\rho^2} \nonumber \\ \nonumber \\ + \, \frac{ [g_\rho^{\rm v} (q^2) \tau_3^{(2)} \gamma_{\mu}^{(2)} + g_\rho^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(2)} \tau_3^{(2)}] [g^{\mu \nu} - q^{\mu}q^{\nu}/m_{\rho}^2][g_\rho^{\rm v \, CSB}(q^2) \gamma_{\nu}^{(1)} + g_\rho^{\rm t \, CSB}(q^2) \sigma_{\nu \beta}^{(1)} q^\beta] }{q^2 - m_\rho^2} \nonumber \\ \nonumber \\ \, + \, m^2_{\rho \omega} \, \, \frac{[g_\omega^{\rm v}(q^2) \gamma_{\mu}^{(1)} + g_\omega^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(1)}] [g^{\mu \gamma} - q^{\mu}q^{\gamma}/m_{\omega}^2] [g^{\gamma \nu} - q^{\gamma}q^{\nu}/m_{\rho}^2] [g_\rho^{\rm v}(q^2) \tau_3^{(2)}\gamma_{\nu}^{(2)} + g_\rho^{\rm t}(q^2) \tau_3^{(2)} \sigma_{\nu \beta}^{(2)} q^\beta] }{(q^2-m_\omega^2)(q^2-m_\rho^2)} \nonumber \\ \nonumber \\ + \, m^2_{\rho \omega} \, \, \frac{ [g_\omega^{\rm v}(q^2) \gamma_{\mu}^{(2)} + g_\omega^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(2)}] [g^{\mu \gamma} - q^{\mu}q^{\gamma}/m_{\omega}^2] [g^{\gamma \nu} - q^{\gamma}q^{\nu}/m_{\rho}^2] [g_\rho^{\rm v}(q^2) \tau_3^{(1)}\gamma_{\nu}^{(1)} + g_\rho^{\rm t}(q^2) \tau_3^{(1)} \sigma_{\nu \beta}^{(1)} q^\beta] } {(q^2-m_\omega^2)(q^2-m_\rho^2)} \nonumber\ \\ \label{CSB1}\end{aligned}$$ This form is rather general: in addition to $\rho-\omega$ mixing, it explicitly includes possible charge symmetry breaking couplings between the vector mesons and the nucleons arising from short distance effects: these couplings are labeled by the superscript CSB. The coefficient $m^2_{\rho \omega}$ is the mixing parameter which in this model is taken to be independent of $q^2$. It should be noted that the general form of eq. (\[CSB1\]) is consistent with the general philosophy of meson exchange used here. In particular all short-ranged effects are merely parameterized, while the long ranged effects are treated dynamically in terms of the mesons. It is for this reason, that we must include the $\rho-\omega$ mixing explicitly rather than including all of the effects in terms of the charge symmetry breaking couplings. The couplings $g_{\omega,\rho}^{\rm v,t } (q^2)$ are presumed to have been determined in fits to the charge symmetry conserving interactions. In principle, the coupling constants $g_{\omega,\rho}^{\rm v,t \, CSB} (q^2)$ must be determined phenomenologically from experimental data on charge symmetry breaking. In fact, in the treatments of CSB in refs. [@CB87; @BI87; @M94; @MVO94] these couplings were all taken to be zero. In that work, model assumptions and existing nucleon-nucleon and pion-nucleon scattering data were used to make [*a priori*]{} arguments that these couplings should be small and hence could be neglected. See, for example, Refs. [@MNS90] and [@MVO94] which reviews the charge-dependence of the couplings. The neglect of charge dependence in the meson-nucleon coupling constants is not invalidated by present data. In particular, descriptions of all known CSB effects do not require the inclusion of such terms. Had the data required the inclusion of such terms they could have been included without violating the spirit of a meson exchange potential model. Now suppose, we had a detailed model for the structure of the vector mesons in which the $\rho-\omega$ mixing amplitude has a nontrivial momentum dependence. The form for the CSB potential is very similar to the form above: $$\begin{aligned} V_{\omega,\rho}^{\rm CSB} (q^2) = \frac{[g_\omega^{\rm v}(q^2) \gamma_{\mu}^{(1)} + g_\omega^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(1)}] [g^{\mu \nu} - q^{\mu}q^{\nu}/m_{\omega}^2] [\hat{g}_\omega^{\rm v \, CSB}(q^2) \tau_3^{(2)} \gamma_{\nu}^{(2)} + \hat{g}_\omega^{\rm t \, CSB}(q^2) \tau_3^{(2)} \sigma_{\nu \beta}^{(2)} q^\beta] }{q^2 - m_\omega^2} \nonumber \\ \nonumber \\ + \frac{ [g_\omega^{\rm v}(q^2) \gamma_{\mu}^{(2)} + g_\omega^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(2)}] [g^{\mu \nu} - q^{\mu}q^{\nu}/m_{\omega}^2] [\hat{g}_\omega^{\rm v \, CSB}(q^2) \tau_3^{(1)} \gamma_{\nu}^{(1)} + \hat{g}_\omega^{\rm t \, CSB}(q^2) \tau_3^{(1)} \sigma_{\nu \beta}^{(1)} q^\beta] }{q^2 - m_\omega^2} \nonumber \\ \nonumber \\ + \frac{ [g_\rho^{\rm v} (q^2) \gamma_{\mu}^{(1)} \tau_3^{(1)} + g_\rho^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(1)} \tau_3^{(1)}][g^{\mu \nu} - q^{\mu}q^{\nu}/m_{\rho}^2][\hat{g}_\rho^{\rm v \, CSB}(q^2) \gamma_{\nu}^{(2)} + \hat{g} _\rho^{\rm t \, CSB}(q^2) \sigma_{\nu \beta}^{(2)} q^\beta] }{q^2 - m_\rho^2} \nonumber \\ \nonumber \\ + \frac{ [g_\rho^{\rm v} (q^2) \tau_3^{(2)} \gamma_{\mu}^{(2)} + g_\rho^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(2)} \tau_3^{(2)}] [g^{\mu \nu} - q^{\mu}q^{\nu}/m_{\rho}^2][\hat{g}_\rho^{\rm v \, CSB}(q^2) \gamma_{\nu}^{(1)} + \hat{g}_\rho^{\rm t \, CSB}(q^2) \sigma_{\nu \beta}^{(1)} q^\beta] }{q^2 - m_\rho^2} \nonumber \\ \nonumber \\ + \, m^2_{\rho \omega}(q^2) \, \, \frac{[g_\omega^{\rm v}(q^2) \gamma_{\mu}^{(1)} + g_\omega^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(1)}] [g^{\mu \gamma} - q^{\mu}q^{\gamma}/m_{\omega}^2] [g^{\gamma \nu} - q^{\gamma}q^{\nu}/m_{\rho}^2] [g_\rho^{\rm v}(q^2) \tau_3^{(2)}\gamma_{\nu}^{(2)} + g_\rho^{\rm t}(q^2) \tau_3^{(2)} \sigma_{\nu \beta}^{(2)} q^\beta] }{(q^2-m_\omega^2)(q^2-m_\rho^2)} \nonumber \\ \nonumber \\ + m^2_{\rho \omega}(q^2) \, \, \frac{ [g_\omega^{\rm v}(q^2) \gamma_{\mu}^{(2)} + g_\omega^{\rm t}(q^2) q^\alpha \sigma_{\alpha \mu}^{(2)}] [g^{\mu \gamma} - q^{\mu}q^{\gamma}/m_{\omega}^2] [g^{\gamma \nu} - q^{\gamma}q^{\nu}/m_{\rho}^2] [g_\rho^{\rm v}(q^2) \tau_3^{(1)}\gamma_{\nu}^{(1)} + g_\rho^{\rm t}(q^2) \tau_3^{(1)} \sigma_{\nu \beta}^{(1)} q^\beta] } {(q^2-m_\omega^2)(q^2-m_\rho^2)}\nonumber\end{aligned}$$ $$\label{CSB2}$$ We have labeled the CSB coupling as $\hat{g}_{\omega,\rho}^{\rm v,t \,CSB}$ rather than $g_{\omega,\rho}^{\rm v,t \, CSB}$ to make explicit the fact the CSB couplings in eq. (\[CSB2\]) may be different from the CSB couplings in eq. (\[CSB1\]). The question we wish to address is whether the model in eq. (\[CSB2\]) is equivalent to the model in eq. (\[CSB1\]). The issue comes down to whether the effects of the momentum dependence of the mixing can be entirely absorbed into differences between $\hat{g}_{\omega,\rho}^{\rm v,t \, CSB}$ and $g_{\omega,\rho}^{\rm v,t \, CSB}$ without introducing any unnaturally long range effects into the CSB couplings. We shall show that this can be done. The $\rho-\omega$ mixing is measured rather accurately at the pole at $q^2=m_\omega^2$. Accordingly, it is sensible to express $$m^2_{\rho \omega}(q^2) = m^2_{\rho \omega} + \delta m^2_{\rho \omega}(q^2)$$ with $\delta m^2_{\rho \omega}(m_{\omega}^2) = 0$. Thus, the expression $$\frac{ \delta m^2_{\rho \omega}(q^2)}{(q^2-m_\rho^2)(q^2-m_\omega^2)}$$ has no $\omega$ pole. All effects with this term are indistinguishable from terms arising due $\rho$ exchange with a CSB vertex. In particular, if $$\begin{aligned} \hat{g}_{\omega}^{\rm v,t CSB} & = & g_{\omega}^{\rm v,t CSB} \label{CSB3}\\ \hat{g}_{\rho}^{\rm v,t CSB} & = & g_{\rho}^{\rm v,t CSB} - \frac{ \delta m^2_{\rho \omega}(q^2)}{q^2-m_\omega^2} g_{\rho}^{\rm v,t} \label{CSB4}\end{aligned}$$ then the potential in eq. (\[CSB2\]) is identical with the one of eq. (\[CSB1\]). This result can also be obtained from Feynman diagrams. Let an $\omega$ be emitted from a nucleon and then be converted via $m^2_{\rho \omega}(q^2)$ into a $\rho$. One can draw a box which includes the strong vertex and $m^2_{\rho \omega}(q^2)$. This box is the charge-dependent $\rho$-nucleon coupling constant. Alternatively one can regard the $m^2_{\rho \omega}(q^2)$ as part of the propagator. Either way, the result is the same. We can do a specific calculation. For example, suppose $\delta m^2_{\rho \omega}(q^2)=m^2_{\rho\omega}/m_\omega^2 (q^2-m_\omega^2)$. This is a good approximation to the $m^2_{\rho \omega}(q^2)$ obtained in the sum rule work of Ref.[@HHKM94]. Then the difference between $\hat{g}_{\omega}^{\rm v,t CSB}$ and $ g_{\rho}^{\rm v,t CSB}$ is a simple constant $\approx -.008g_\rho^{v,t}$; if $ \hat{g}_{\rho}^{\rm v,t CSB}$ were chosen as the negative of that constant, one would obtain the standard form of the $\rho-\omega$ mixing contribution to the NN potential. See also Ref.[@G95]. Moreover, for any reasonable model of the momentum dependence of the mixing, eq. (\[CSB4\]) can be satisfied without introducing unnaturally long ranged effects into the meson-nucleon vertex functions. The issues are completely analogous to the ones raised in connection with the $\omega$ exchange potential discussed in the previous section. First, it should be noted that there is no $\omega$ pole singularity on the right hand side of eq. (\[CSB4\])—it is eliminated because $\delta m_{\rho \omega}^2$ vanishes at the $\omega$ pole. Thus, the only source of long range contamination of the couplings is in $\delta m_{\rho \omega}^2 (q^2)$ itself. Note, that by construction $\delta m_{\rho \omega}^2 (q^2)$ cannot have a singularity associated with either the $\rho$ or the $\omega$. Moreover, we know that the only substantial strength in the vector channels at $q^2 < \Lambda_s^2$ is through the $\rho$ and $\omega$ mesons. Thus, any model which yields long range effects in $\delta m_{\rho \omega}^2 (q^2)$ must be regarded as unrealistic according to our philosophy. Summary ======= We are working in the framework of boson exchange potentials. This means that in realistic boson-exchange models long range effects are included via boson exchanges and that short range effects are included in the vertex functions. For any such realistic model of the momentum dependence of the $\rho-\omega$ mixing parameter, there are no effects in the CSB breaking potential which cannot be absorbed into a redefinition of a short ranged CSB $\rho$-N vertex. Thus, a model which provides knowledge of the momentum dependence of the mixing parameter alone, without simultaneously giving a self-consistent model for the short ranged CSB vector meson-nucleon couplings gives no information about the CSB N-N potential. The work of refs. [@GHT92]-[@MTRC94] found major differences between the CSB potentials based on the on-shell $\rho-\omega$ mixing and models with a large momentum dependence. Our boson exchange model view is that this is because the short ranged CSB vector meson-nucleon coupling are assumed to be zero— just as in the models based on the on-shell $\rho-\omega$ mixing. However, there is no reason, [*a priori*]{} that this assumption is be true for the models under discussion. 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--- abstract: 'It is shown that the quasi-normal modes arise, in a natural way, when considering the oscillations in unbounded regions by imposing the radiation condition at spatial infinity with a complex wave vector $k$. Hence quasi-normal modes are not peculiarities of gravitation problems only (black holes and relativistic stars). It is proposed to consider the space form of the quasi-normal modes with allowance for their time dependence. As a result, the problem of their unbounded increase when $r\to \infty$ is not encountered more. The properties of quasi-normal modes of a compact dielectric sphere are discussed in detail. It is argued that the spatial form of these modes (especially so-called surface modes) should be taken into account, for example, when estimating the potential health hazards due to the use of portable telephones.' author: - 'V. V. Nesterenko' - 'A. Feoli' - 'G. Lambiase' - 'G. Scarpetta' title: | Quasi-normal modes of a dielectric sphere\ and some their implications --- Introduction ============ Quasi-normal modes (qnm) are widely used now in black hole physics and in relativistic theory of stellar structure (see, for example, Refs. [@Nollert; @FN; @KS]). The corresponding eigenfrequencies are complex numbers, however it is not due to the dissipative processes but it is a consequence of unbounded region occupied by the oscillating system. The latter naturally leads to the energy loss due to the wave emission (for example, gravity waves). In this paper we would like to show that the quasi-normal modes are not the peculiarities of the gravitational problems only. Actually they appear, in a natural way, when considering the oscillating systems unbounded in space. The necessary condition for emergence of such modes is imposing the radiation condition at spatial infinity on the field functions. It is this condition that leads to the characteristic behaviour of the quasi-normal modes, namely, these solutions to the relevant equations exponentially decay in time when $t\to \infty$ and simultaneously they exponentially rise at spatial infinity $r\to \infty$. An interesting and physically motivated example is provided here by the oscillations of electromagnetic field connected with a compact dielectric sphere placed in unrestricted homogeneous media with a different refraction index or in vacuum. Taking here the formal limit $\varepsilon \to \infty$ one passes to a perfectly conducting sphere ($\varepsilon $ is the refraction index of the sphere material). In this case the quasi-normal modes describing the electromagnetic oscillations outside the sphere are tractable analytically. We propose to consider the spatial form of a quasi-normal modes with allowance for their time dependence. Doing in this way one can escape the exponential rise of quasi-normal modes at spatial infinity. The eigenfrequencies of a dielectric sphere are complex $\omega =\omega'-i\, \omega''$, where $\omega '$ is the free oscillation (radian) frequency and $\omega ''$ is its relaxation time. These modes can be classified as the interior and exterior ones and, at the same time, as volume modes and surface modes. In physical applications the surface modes turn out to be important, for example, when estimating the health hazards due to the use of portable telephones. The point is the eigenfrequencies of a dielectric sphere with physical characteristics close to those of a human head lay in the GSM 400 MHz frequency band which has been used in a first generation of mobile phone systems and now is considered for using again. In this situation one can assume that the surface modes excited by a cellular phone will lead to higher heat generation in the tissues close to a head surface as compared with the predictions of routine calculations in this field. The layout of the paper is as follows. In Sec. II we show that the quasi-normal modes are the eigenfunctions of unbounded oscillating regions. As a simple example the quasi-normal modes of a perfectly conducting sphere are considered. The Sec.  III is devoted to the consideration of the quasi-normal modes of a dielectric sphere. The main features of these modes are revealed and their classification is presented. The implication of these quasi-normal modes for estimation of the health hazards of portable telephones is considered in Sec. IV. In Conclusion (Sec. V) the main results are formulated and their relation to the general theory of open systems is discussed. Quasi-normal modes as the eigenfunctions of unbounded oscillating regions ========================================================================= Here we show in the general case in what way complex frequencies and quasi-normal modes appear when considering harmonic oscillations in unbounded regions. Let a closed smooth surface $S$ divides the $d$-dimensional Euclidean space $\mathbb{R}^d$ into a compact internal region $D_{\text{ in}}$ and noncompact external region $D_{\text{ex}}$. We consider here a simple scalar wave equation $$\label{2-1} \left ( \Delta -\frac{1}{c^2} \frac{\partial ^2}{\partial t^2} \right )u(t, {\textbf x})=0\,{,}$$ where $c$ is the velocity of oscillation propagation and $\Delta $ is the Laplace operator in $\mathbb{R}^d$. For harmonic oscillations $$\label{2-2} u(t,{\textbf x})=e^{-i\omega t}u({\textbf x})$$ the wave equation (\[2-1\]) is reduced to the Helmholtz equation $$\label{2-3} (\Delta +k^2)\, u(\textbf{x}) =0, \quad k={\omega}/{c}\,{.}$$ The oscillations in the internal region $D_{\text{in}}$ are described by an infinite countable set of normal modes $$\label{2-4} u_n(t,\textbf{x})=e^{-i\omega _n t}u_n(\textbf{x}), \quad n=1,2,\ldots .$$ The spatial form of the normal modes (the functions $u_n(\textbf{x})$) is determined by the boundary conditions which are imposed upon the function $u(\textbf{x})$ on the internal side of the surface $S$. These conditions should fit the physical content of the problem under study. The set of normal modes is a complete one. Hence any solution of (\[2-3\]) obeying relevant boundary conditions can be expanded in terms of the normal modes $u_n(\textbf{x})$. When considering the oscillations in the external domain $D_{\rm ex }$ one imposes, in addition to the conditions on the compact surface $S$, a special requirement concerning the behavior of the function $u(\textbf{x})$ at large $r\equiv |\textbf{x}|$. In the classical mathematical physics [@RR] the radiation conditions, proposed by Sommerfeld [@Sommerfeld; @FM], are used here $$\label{2-5} \lim_{r\to \infty} r^{\frac{d-1}{2}}u(r)= \textrm{const}\,{,}\qquad \lim_{r\to\infty}r^{\frac{d-1}{2}}\left ( \frac{\partial u}{\partial r} -i ku\right )= 0\,{.}$$ For real values of the wave vector $k$ (for real frequencies $\omega$) the solution to Eq. (\[2-3\]), which obeys the radiation conditions (\[2-5\]) and reasonable boundary condition on a compact surface $S$, identically vanishes. In this case the Laplace operator entering the Helmholtz equation (\[2-3\]) has no eigenfunctions with real eigenvalues. The physical content of the radiation conditions is very clear. They select only the oscillations with real frequencies driven by external sources which are situated in a compact spatial area. From the mathematical standpoint, these conditions ensure the [*uniqueness*]{} of the solutions to [*inhomogeneous*]{} wave or Helmholtz equations with external sources on the right-hand side, when these solutions are considered in the external region $D_{\text{ex}}$ or in the whole space $D_{\text{in}}+D_{\text{ex}}$ in the case of compound media. Here the question arises, how to change minimally the conditions in the [*homogeneous*]{} problem at hand in order to get nonzero solutions, i.e., eigenfunctions in unbounded regions. The energy conservation law prompts a simple way to construct nonzero solutions to the homogeneous wave equation (\[2-1\]) or to the Helmholtz equation (\[2-3\]) describing the outgoing waves at spatial infinity, namely, one has to introduce [*complex*]{} frequencies $\omega = \omega'- i\,\omega '',\quad \omega ''>0$. We may hope that in this case the factor $e^{-\omega ''t}$ will describe the decay of the initial solutions in time accounting the fact that outgoing waves take away the energy. In other words, we are dealing here with the radiation of the energy with the amplitude decaying in time. Indeed, if we remove the requirement of reality of the wave vector $k$, then the homogeneous wave equation (2.1) and the Helmholtz equation (2.3) will have nonzero solutions with [*complex frequencies*]{}, these solutions obeying the radiation conditions (\[2-5\]) and a common boundary condition on a compact surface $S$ (for instance, Dirichlet or Neumann conditions). In quantum mechanics the radiation condition with a complex wave number $k$ is known as the Gamov condition which singles out the resonance states in the spectrum of the Hamiltonian [@Gamov; @G1; @G2]. When introducing the radiation conditions and proving the respective uniqueness theorem in the text books [@RR] only the real wave vector $k$ is considered. The possibility of existence of quasi-normal modes with complex frequencies satisfying the radiation conditions at spatial infinity with a complex $k$ is not mentioned usually. We are aware only of alone textbook where the eigenfunctions with complex frequencies are noted in this context. It is the article written by Sommerfeld in the book [@FM], where it is emphasized that the uniqueness in this problem is only up to the eigenfunctions with complex frequencies, i.e., up to the quasi-normal modes. Thus imposing the radiation conditions with real $k$ we remove the quasi-normal modes from our consideration only. However this cannot prevent the excitation of these modes in the real physical problem. Hence, when dealing with systems unbounded in space (open systems) one has always to investigate the consequences of qnm excitation. An example of such a problem will be considered in Sec. III. As a very simple and physically motivated example of quasi-normal modes we consider here the oscillations of electromagnetic field outside a perfectly conducting sphere of radius $a$. In this case the electric and magnetic fields are expressed in terms of two scalar functions $f_{kl}^{\rm{TE}}(r)$ and $f_{kl}^{\rm{TM}}(r)$ (Debye potentials [@Stratton]) which are the radial parts of the solutions to the scalar wave equation (\[2-1\]). Outside the perfectly conducting sphere placed in vacuum the solution to the Helmholtz equation (\[2-3\]) obeying the radiation conditions (\[2-5\]) has the form $(d=3)$ $$\label{2-6} f_{kl}(r)=C\, h^{(1)}_l\left ( \frac{\omega}{c}\,r \right ){,}\quad r>a\,{,}$$ where $ h^{(1)}_l(z)$ is the spherical Hankel function of the first kind [@AS]. At the surface of perfectly conducting sphere the tangential component of the electric field should vanish. This leads to the following frequency equation for TE-modes $$\label{2-7} h^{(1)}_l\left ( \frac{\omega}{c}\,a \right )=0{,}\quad l\geq 1$$ and for TM-modes $$\label{2-8} \frac{d}{d r}\left ( r \,h^{(1)}_l\left ( \frac{\omega}{c}\,r \right ) \right )=0, \quad r=a,\quad l\geq 1\,{.}$$ The spherical Hankel function $h^{(1)}_l(z)$ is $e ^{i z}$ multiplied by the polynomial in $1/z$ of a finite order [@AS]. Hence frequency equations (\[2-7\]) and (\[2-8\]) have a finite number of roots which are in the general case complex numbers. For $l=1$ (the lowest oscillations) Eqs.(\[2-7\]) and (\[2-8\]) assume the form $(z=a\,\omega/c)$ $$\begin{aligned} h^{(1)}_l(z)&=&-\frac{1}{z}\,e^{i z}\left ( 1+\frac{i}{z} \right )=0\quad (\text{TE modes}), \label{2-8a} \\ \frac{d }{d z}\left (z\, h^{(1)}_l(z)\right )&=&-\frac{i}{z^2}\,e^{i z}\left ( z^2+i z-1 \right )=0\quad (\text{TM modes})\,{.} \label{2-9}\end{aligned}$$ Thus the lowest eigenfrequencies are $$\begin{aligned} \frac{\omega}{c}&=&-\frac{i}{a}\quad (\mbox{TE modes})\,{,} \label{2-10} \\ \frac{\omega}{c}&=&-\frac{1}{2a}(i\pm\sqrt 3)\quad (\text{TM modes})\,{.} \label{2-11}\end{aligned}$$ The complex eigenfrequencies lead to a specific time and spatial dependence of the respective natural modes and ultimately of the electromagnetic fields. So, with allowance of (\[2-10\]), we obtain $$\label{2-12} e^{-i \omega t}f_{k1}^{\text{TE}}(r)=-i \,C\,\frac{a}{r}\,e^{(r-ct)/a}\left ( 1-\frac{a}{r} \right ), \quad r\geq a\,{.}$$ Thus, the eigenfunctions are exponentially going down in time and exponentially going up when $r$ increases. Such a time and spatial behaviour is a direct consequence of radiation conditions (\[2-5\]) and it is typical for eigenfunctions describing oscillations in external unbounded regions, the physical content and details of oscillation process being irrelevant. The eigenfunctions corresponding to complex eigenvalues are called [*quasi-normal modes*]{} keeping in mind their unusual properties [@Nollert]. The physical origin of such features is obvious, in fact we are dealing here with [*open systems*]{} in which the energy can be radiated to infinity. Therefore in open systems field cannot acquire a stationary state. Quasi-normal modes do not obey the standard completeness condition and the notion of norm cannot be defined for them [@Nollert]. Therefore these eigenfunctions cannot be used for expansion of the classical field with the aim to quantize it and to introduce the relevant Fock operators. The treatment of these problems can be found, for example, in [@Ching; @Leung; @Chang]. It is worthy to investigate the spatial form of the of quasi-normal modes with allowance for their time dependence. Indeed, these solutions have the character of propagating waves that are eventually going to spatial infinity. Let us take the point which is sufficiently far from the region with nontrivial dynamics in the system under study. Obviously, it has sense to say about the value of the quasi-normal mode at a given point only after arrival at this point of the wave described by this mode. The maximal value of the quasi-normal mode is observed just at the moment of its arrival at this point. At the later moments the quasi-normal mode is dumping due to its characteristic time dependence. Indeed, taking into account all this we obtain for the maximal observed value of the quasi-normal mode (\[2-12\]) the following physically acceptable expression $$\label{2-13}\left ( e^{-i \omega t}f_{k1}^{\text{TE}}(r)\right )_{\text{max-obs}}=-i \,C\,\frac{a}{r}\left ( 1-\frac{a}{r} \right ), \quad r\geq a\,{.}$$ Thus, in our consideration the problem of unbounded (exponential) rising of quasi-normal modes, when $r\to \infty$, does not arise. In order to associate with resonance phenomenon a single square integrable eigenfunction, rather sophisticated methods are used, for example, complex scaling [@scaling] (known also as the complex-coordinate method or as the complex-rotational method). The necessary condition for appearing the quasi-normal modes is imposing the radiation conditions at spatial infinity on the field functions. When other conditions are used at spatial infinity, the quasi-normal modes do not arise. For example, if we demand that the solution to the wave equation (\[2-1\]) becomes the sum of incoming and outgoing waves when $r\to \infty$ then the spectrum of the variable $k^2$ in the Helmholtz equation (\[2-3\]) will be positive and continuous. Quasi-normal modes of a dielectric sphere ========================================= The same situation, with regard to quasi-normal modes, takes place when we consider the oscillations of compound unbounded media. In this case in both the regions $D_{\text{in}}$ and $D_{\text{ex}}$ the wave equations are defined $$\begin{aligned} \left ( \Delta -\frac{1}{c_{\text{in}}^2}\frac{\partial }{\partial t^2} \right ) u_{\text{in}}(t,\mathbf{x}) &=& 0, \quad \mathbf{x}\in D_{\text{in}}\,{,} \label{2-14}\\ \left ( \Delta -\frac{1}{c_{\text{ex}}^2}\frac{\partial }{\partial t^2} \right ) u_{\text{ex} }(t,\mathbf{x}) &=& 0, \quad \mathbf{x}\in D_{\text{ ex}} \label{2-15}\end{aligned}$$ with the matching conditions at the interface $S$, for example, of the following kind $$\begin{aligned} u_{\text{ in}}(t,\mathbf{x})&=&u_{\text{ex}}(t, \mathbf{x}){,} \label{2-16} \\ \lambda_{\text{ in}}\frac{\partial u_{\text{in }}(t,\mathbf{x})}{\partial n_{\text{in }}(\mathbf{x})}&=&\lambda_{\text{ ex}}\frac{\partial u_{\rm ex}(t,\mathbf{x})}{\partial n_{\text{ ex }}(\mathbf{x})}, \quad \mathbf{x}\in S\,{,} \label{2-17}\end{aligned}$$ where $n_{\text{in}}(\textbf{x})$ and $n_{\text{ ex}}(\textbf{x})$ are the normals to the surface $S$ at the point $\textbf{x}$ for the regions $D_{\text{in}}$ and $D_{\text{ex}}$, respectively. The parameters $c_{\text{in}}$, $c_{\text{ex}}$, $\lambda _{\text{in}}$, and $\lambda _{\text{ex}}$ specify the material characteristics of the media. At the spatial infinity the solution $u_{\text{ex}}(t, \mathbf{x})$ should satisfy the radiation conditions (\[2-5\]). For real $k$ we again have only zero solution in this problem, both functions $u_{\text{in}}(t, \mathbf{x})$ and $u_{\text{ex}}(t, \mathbf{x})$ vanishing. However, the wave equations (\[2-14\]) and (\[2-15\]) have nonzero solutions with complex frequencies, i.e. quasi-normal modes, which satisfy the matching conditions (\[2-16\]) and (\[2-17\]) at the interface $S$ and radiation conditions (\[2-5\]) at spatial infinity. It is important, that the frequencies of oscillations in internal ($D_{\text{in}}$) and external ($D_{\text{ex}}$) regions are the same. A typical example here is the complex eigenfrequencies of a dielectric sphere. This problem has been investigated by Debye in his PhD thesis concerned with the light pressure on a material particles [@Debye]. Let us consider a sphere of radius $a$, consisting of a material which is characterized by permittivity $\varepsilon_1 $ and permeability $\mu_1$. The sphere is assumed to be placed in an infinite medium with permittivity $\varepsilon_2 $ and permeability $\mu_2$. In the case of spherical symmetry the solutions to Maxwell equations are expressed in terms of two scalar Debye potentials $\psi$ (see, for example, textbooks [@Stratton; @Jackson]): $$\begin{aligned} \mathbf{E}^{\text{TM}}_{lm}&=&\bm{\nabla} \times \bm{\nabla}\times(\mathbf{r}\psi^{\text{TM}}_{lm}),\quad \mathbf{H}^{\text{TM}}_{lm}=-i\,\omega \,\bm{\nabla} \times (\mathbf{r}\psi^{\text{TM}}_{lm})\quad (\text{E-modes}), \nonumber \\ \mathbf{E}^{\text{TE}}_{lm}&=&i\,\omega \,\bm{\nabla} \times (\mathbf{r}\psi^{\text{TE}}_{lm}),\quad \mathbf{H}^{\text{TE}}_{lm}=\bm{\nabla} \times \bm{\nabla}\times(\mathbf{r}\psi^{\text{TE}}_{lm})\quad (\text{H-modes})\,{.} \label{3-1}\end{aligned}$$ These potentials obey the Helmholtz equation and have the indicated angular dependence $$\label{3-2} \left ( \mathbf{\nabla}^2+k^2_i\right )\psi_{lm}=0,\quad k_i^2=\varepsilon_i\,\mu_i\,\frac{\omega^2}{c^2}, \quad i=1,2\quad (r\neq a); \quad \psi_{lm}(\mathbf{r})=f_l(r)Y_{lm}(\Omega)\,{.}$$ Equations (\[3-2\]) should be supplemented by the boundary conditions at the origin, at the sphere surface, and at infinity. In order for the fields to be finite at $r=0$ the Debye potentials should be regular here. At the spatial infinity we impose the radiation conditions with the goal to find the spectrum of eigenfunctions with complex frequencies (quasi-normal modes in the problem at hand). At the sphere surface the standard matching conditions for electric and magnetic fields should be satisfied [@Stratton]. In view of all this the Helmholtz equation (\[3-2\]) becomes now the spectral problem for the Laplace operator multiplied by the discontinuous factor $-1/(\varepsilon (r)\,\mu(r))$ $$\label{3-2a}-\, \frac{1}{\varepsilon (r)\,\mu(r)}\Delta \,\psi_{\omega l m}(r)=\frac{\omega^2}{c^2}\,\psi_{\omega l m}(r), \quad r\neq a\,{,}$$ where $$\varepsilon (r)\,\mu(r)= \begin{cases} \varepsilon_1\,\mu_1\,{,}&r<a \,{,}\\ \varepsilon_2\,\mu_2\,{,}&r>a \,{.} \end{cases}$$ In this problem the spectral parameter is $\omega^2/c^2$. In order to obey the boundary conditions at the origin and at spatial infinity formulated above, the solution to the spectral problem (\[3-2a\]) should have the form $$\label{3-2b} f_{\omega l}(r)=C_1\,j_l(k_1r)\,{,} \quad r<a,\quad f_{\omega l}(r)=C_2\,h^{(1)}_l(k_2r)\,{,} \quad r>a\,{,}$$ where $j_l(z)$ is the spherical Bessel function and $h_l^{(1)}(z)$ is the spherical Hankel function of the first kind [@AS], the latter obeys the radiation conditions (\[2-5\]). Now we address the matching conditions at the sphere surface. By making use of Eqs. (\[3-1\]) we can write, in an explicit form, the radial (r) and tangential (t) components of electric and magnetic fields in the case of spherical symmetry. For TE-modes these equations read $$\begin{aligned} E^{\text{TE}}_{klm,\text{r}}&=&0\,{,} \label{a-20}\\ E^{\text{TE}}_{klm,\text{t}}&=&a_{lm}(k)\,f_{kl}^{\text{TE}}(r)\,\mathbf{X}_{lm}\,{,} \label{a-21}\\ H^{\text{TE}}_{klm,\text{r}}&=&\frac{1}{kr}\left ( \frac{\varepsilon}{\mu} \right )^{1/2}\sqrt{l(l+1)}\,a_{lm}(k)f_{kl}^{\text{TE}}(r)\,Y_{lm} \,{,} \label{a-22}\\ H^{\text{TE}}_{klm,\text{t}}&=&\frac{i}{kr}\left ( \frac{\varepsilon}{\mu} \right )^{1/2}a_{lm}(k)\,\frac{d}{dr}\left( rf_{kl}^{\text{TE}}(r) \right)\,\mathbf{X}_{lm}^\perp\,{,} \label{a-23}\end{aligned}$$ and the same for the TM-modes $$\begin{aligned} E^{\text{TM}}_{klm,\text{r}}&=&-\frac{1}{kr}\left ( \frac{\mu}{\varepsilon} \right )^{1/2}\sqrt{l(l+1)}\,b_{lm}(k)f_{kl}^{\text{TM}}(r)\,Y_{lm} \,{,} \\ E^{\text{TM}}_{klm,\text{t}}&=&-\frac{i}{kr}\left ( \frac{\mu}{\varepsilon} \right )^{1/2}b_{lm}(k)\,\frac{d}{dr}\left( rf_{kl}^{\text{TM}}(r) \right)\,\mathbf{X}_{lm}^\perp\,{,} \label{a-25}\\ H^{\text{TM}}_{klm,\text{r}}&=&0\,{,} \label{a-26}\\ H^{\text{TM}}_{klm,\text{t}}&=&b_{lm}(k)\,f_{kl}^{\text{TM}}(r)\,\mathbf{X}_{lm}\,{.} \label{a-27}\end{aligned}$$ Here $\mathbf{X}_{lm}$ are the vector spherical harmonics [@Jackson] $$\label{vsh} \mathbf{X}_{lm}(\theta,\phi)=\frac{\mathbf{L}\,Y_{lm}(\theta,\phi)}{\sqrt{l(l+1)}}\,{,}\quad l\geq1\,{,}$$ where $\mathbf{L}$ is the angular momentum operator $$\mathbf{L}=-i\,(\mathbf{r}\times \bm{\nabla})\,{.}$$ The vector spherical harmonic $\mathbf{X}_{lm}^{\perp}$ is obtained from $\mathbf{X}_{lm}$ after rotation by the angle $\pi/2$ around the normal $\mathbf{n}=\mathbf{r}/r$. From Eqs.(\[a-20\]) – (\[a-27\]) it follows, in particular, that the tangential components of electric field in TE- and TM-modes are orthogonal each other and the same holds for the magnetic field. It implies that the matching conditions on the sphere surface do not couple TE- and TM-modes. At the sphere surface the tangential components of electric and magnetic fields are continuous (see Eqs. (\[a-20\]) – (\[a-27\])). As a result, the eigenfrequencies of electromagnetic field for this configuration are determined [@Stratton] by the frequency equation for the TE-modes $$\label{3-3} \Delta^{\text{TE}}_l(a\omega)\equiv \sqrt{\varepsilon_1\mu_2}\,\hat j_l'(k_1a)\,\hat h_l(k_2a)- \sqrt{\varepsilon_2\mu_1}\,\hat j_l(k_1a)\,{\hat h_l}'(k_2a)=0\,{}$$ and by the analogous equation for the TM-modes $$\label{3-4} \Delta^{\text{TE}}_l(a\omega)\equiv \sqrt{\varepsilon_2\mu_1}\,\hat j_l'(k_1a)\,\hat h_l(k_2a)- \sqrt{\varepsilon_1\mu_2}\,\hat j_l(k_1a)\,{\hat h_l}'(k_2a)=0\,{,}$$ where $k_i=\sqrt{\varepsilon_i\mu_i}\,\omega/c,\quad i=1,2$ are the wave numbers inside and outside the sphere, respectively, and $\hat j_l(z)$ and $\hat h_l(z)$ are the Riccati-Bessel functions [@AS] $$\label{3-5} \hat j_l(z)=z\,j_l(z)=\sqrt{\frac{\pi z}{2}}\,J_{l+1/2}(z)\,{,}\quad \hat h_l(z)=z\,h_l^{(1)}(z)=\sqrt{\frac{\pi z}{2}}\,H^{(1)}_{l+1/2}(z)\,{.}$$ In Eqs. (\[3-3\]) and (\[3-4\]) the orbital momentum $l$ assumes the values $1,2,\ldots $, and prime stands for the differentiation with respect of the arguments $k_1a$ and $k_2a$ of the Riccati-Bessel functions. The frequency equations for a dielectric sphere of permittivity $\varepsilon $ placed in vacuum follow from Eqs. (\[3-3\]) and (\[3-4\]) after putting there $$\label{3-12} \varepsilon_1=\varepsilon, \quad \varepsilon_2=\mu_1=\mu_2=1\,{.}$$ The roots of these equations have been studied in the Debye paper [@Debye] by making use of an approximate method. As the starting solution the eigenfrequencies of a perfectly conducting sphere were used. These frequencies are different for electromagnetic oscillations inside and outside sphere. Namely, inside sphere they are given by the roots of the following equations $(l\geq 1)$ $$\begin{aligned} j_l\left ( \frac{\omega}{c}\,a \right)&=&0 \quad (\text{TE-modes})\,{,} \label{3-13}\\ \frac{d}{dr} \left(r\,j_l\left ( \frac{\omega}{c}\,a \right)\right)&=&0\,{,} \quad r=a \quad (\text{TM-modes})\,{,} \label{3-14}\end{aligned}$$ while outside sphere they are determined by Eqs. (\[2-7\]) and (\[2-8\]). The frequency equations for perfectly conducting sphere (\[2-7\]), (\[2-8\]) and (\[3-13\]), (\[3-14\]) can be formally derived by substituting (\[3-12\]) into frequency equations (\[3-3\]) and (\[3-4\]) and taking there the limit $\varepsilon \to \infty$. Approximate calculation of the eigenfrequencies of a dielectric sphere without using computer [@Debye] didn’t allow one to reveal the characteristic features of the respective eigenfunctions (quasi-normal modes). The computer analysis of this spectral problem was accomplished in the work [@Gastine] where the experimental verification of the calculated frequencies was accomplished also by making use of radio engineering measurements. These studies enable one to separate all the dielectric sphere modes into the [*interior*]{} and [*exterior*]{} modes and, at the same time, into the [*volume*]{} and [*surface*]{} modes. It is worth noting that all the eigenfrequencies are complex $$\label{3-15} \omega=\omega '-i\,\omega''\,{.}$$ Thus we are dealing with “leaky modes”. The classification of the modes as the interior and exterior ones relies on the investigation of the behaviour of a given eigenfrequency in the limit $\varepsilon \to \infty$. The modes are called ”interior” when the product $k\,a= \sqrt {\varepsilon}\,\omega \,a/c$ remains finite in the limit $\varepsilon \to \infty$, provided the imaginary part of the frequency ($\omega ''$) tends to zero. The modes are referred to as ”exterior” when the product $k\,a/\sqrt{\varepsilon}= \omega\,a/c$ remains finite with growing $\omega''$. In the first case the frequency equations for a dielectric sphere (\[3-3\]) and (\[3-4\]) tend to Eqs. (\[3-13\]) and (\[3-14\]) and in the second case they tend to Eqs. (\[2-7\]) and (\[2-8\]). The order of the root obtained will be denoted by the index $r$ for interior modes and by $r'$ for exterior modes. Thus $\text{TE}_{\,lr}$ and $\text{TM}_{\,lr}$ denote the interior TE- and TM-modes, respectively, while $\text{TE}_{\,lr'}$ and $\text{TM}_{\,lr'}$ stand for the exterior TE- and TM-modes. For fixed $l$ the number of the modes of exterior type is limited because the frequency equations for exterior oscillations of a perfectly conducting sphere (\[2-7\]) and (\[2-8\]) have finite number of solutions (see the preceding Section). In view of this, the number of exterior TE- and TM-modes is given by the following rule. For even $l$ there are $l/2$ exterior TE-modes and $l/2$ exterior TM-modes, for odd $l$ the number of the modes $\text{TE}_{\,l\,r'}$ is $(l+1)/2$ and the number of the modes $\text{TM}_{\,l\,r'}$ equals $(l-1)/2$. An important parameter is the $Q$ factor $$\label{3-16} Q_{\text{rad}}=\frac{\omega'}{2\omega''}= 2\,\pi\,\frac{\text{stored energy}}{\text{radiated energy per cycle}}\,{.}$$ For exterior modes the value of $Q_{\text{rad}}$ is always less than 1, hence these modes can never be observed as sharp resonances. At the same time for $\varepsilon $ greater than 5, the $Q_{\text{rad}}$ for interior modes is greater than 10 and it can reach very high values when $\varepsilon \to \infty$. For physical implications more important is the classification in terms of [*volume*]{} or [*surface*]{} modes according to whether $r> l$ or $l> r$. For volume modes the electromagnetic energy is distributed in the whole volume of the sphere while in the case of surface modes the energy is concentrated in the proximity of the sphere surface. The exterior modes are the first roots of the characteristic equations and it can be shown that they are always surface modes. ![Electric energy density $r^2\,E_{\text{t}}^2$ for the surface (A) and volume (B) TE-modes of a dielectric sphere with $\varepsilon =40$ placed in vacuum.[]{data-label="Plot:s-v-modes"}](pic1.eps "fig:"){width="75mm"} ![Electric energy density $r^2\,E_{\text{t}}^2$ for the surface (A) and volume (B) TE-modes of a dielectric sphere with $\varepsilon =40$ placed in vacuum.[]{data-label="Plot:s-v-modes"}](pic2.eps "fig:"){width="75mm"} Figure 1 shows a typical spatial behaviour of the surface and volume modes of a dielectric sphere. Thus a substantial part of the sphere modes (about one half) belong to the interior surface modes. It is important that respective frequencies are the [*first*]{} roots of the characteristic equations. In order to escape the confusion, it is worth noting here that the surface modes in the problem in question obey the same boundary conditions at the sphere surface and when $r\to \infty$ as the volume modes do. Hence, these surface modes cannot be classified as the evanescent surface waves propagating along the interface between two media (propagating waves along dielectric waveguides [@Jackson], surface plasmon waves on the interface between metal bulk and adjacent vacuum [@Raether; @BPN] and so on). When describing the evanescent waves one imposes the requirement of their exponential decaying away from interface between two media. In this respect the evanescent surface wave differ from the modes in the bulk. Implication of QNM of a dielectric sphere for estimation of the health hazard of portable telephones ==================================================================================================== Here we shall argue that the features of the quasi-normal modes of a dielectric sphere (namely, existence of surface and volume modes) should be taken into account, in particular, when estimating the potential health hazards due to the use of the cellular phones. The safety guidelines in this field [@Health] are based on the findings from animal experiments that the biological hazards due to radio waves result mainly from the temperature rise in tissues[^1] and a whole-body-averaged specific absorption rate (SAR) below 0.4 W/kg is not hazardous to human health. This corresponds to a limits on the output from the cellular phones (0.6 W at 900 MHz frequency band and 0.27 W at 1.5 GHz frequency band). Obviously, the [*local*]{} absorption rate should be also considered especially in a human head [@WF]. In such studies the following point should be taken into account. The parts of human body (for example, head) posse the eigenfrequencies of electromagnetic oscillations like any compact body. In particular, one can anticipate that the eigenfrequencies of human head are close to those of a dielectric sphere with radius $a\approx 8$ cm and permittivity $\varepsilon \approx 40$ (for human brain $\varepsilon =44.1$ for 900 MHz and $\varepsilon =42.8$ for 1.5 GHz [@WF]). Certainly, our model is very rough, however for the evaluation of the order of the effect anticipated (see below) it is sufficient. By making use of the results of calculations conducted in the work [@Gastine] one can easily obtain the eigenfrequencies of a dialectic sphere with the parameters mentioned above. For $\text{TE}_{l1}$ modes with $l=1,2,3$ we have, respectively, the following frequencies: 280 MHz, 420 MHz, and 545 MHz. For $\text{TM}_{l1}$ modes with $l=1,2,3$ the resonance frequencies are 425 MHz, 540 MHz, and 665 MHz. The imaginary parts of these eigenfrequencies are very small so the $Q$ factor in Eq. (\[3-16\]) responsible for radiation is greater than 100. These eigenfrequencies belong to a new GSM 400 MHz frequency band which is now being standardized by the European Telecommunications Standards Institute. This band was primarily used in Nordic countries, Eastern Europe, and Russia in a first generation of mobile phone system prior to the introduction of GSM. Due to the Ohmic losses the resonances of a dielectric sphere in question are in fact broad, overlapping and, as the result, they cannot be manifested separately. Indeed, the electric conductance $\sigma$ of the human brain is rather substantial. According to the data presented in Ref. [@WF] $\sigma\simeq 1.0$ S/m. The eigenfrequencies of a dielectric dissipative sphere with allowance for a finite conductance $\sigma $ can be found in the following way. As known [@LL] the effects of $\sigma $ on electromagnetic processes in a media possessing a common real dielectric constant $\varepsilon$ are described by a complex dielectric constant $\varepsilon_{\text{diss}}$ depending on frequency $$\label{4-1} \varepsilon_{\text{diss}} =\varepsilon +i\frac{4\pi \sigma}{\omega}\,{.}$$ The eigenfrequencies $\omega$, calculated for a real $\varepsilon $, are related to eigenfrequencies $\omega_{\text{diss}}$ for $\varepsilon_{\text{diss}}$ by the formula [@LL] $$\label{4-2} \omega_{\text{diss}} =\frac{\omega}{\sqrt{\varepsilon_{\text{diss}} }}\simeq \omega -2\pi\, i \,\frac{\sigma}{\varepsilon}\,{.}$$ The corresponding factor $Q_{\text{diss}}$ is $$\label{4-3} Q_{\text{diss}}=\frac{\omega'_{\text{diss}}}{2 \omega''_{\text{diss}}}\simeq \frac{\varepsilon \, \omega}{4\pi\, \sigma}\,{.}$$ Substituting in this equation the values $\omega /2\pi =0.5\cdot 10^9\;\text{Hz}, \quad \varepsilon =40, \quad \sigma = 1\,\text{S/m}=9\cdot 10^9\; \text{s}^{-1}$ one finds $$\label{4-4} Q_{\text{diss}}\simeq\frac{20}{18}\simeq 1\,{.}$$ Thus the real spectrum of electromagnetic oscillations in the problem under study is practically a continuous band around the frequency 400 MHz. The radiation of the cellular telephone with frequency laying in this band, will excite (practically with the same amplitudes) all the neighbouring modes of a dissipative dielectric sphere. In order to get the upper bound for the anticipated effect (see below) we assume that the number of excited modes is sufficiently large so that the half of these modes are surface modes.[^2] Thus one can expect that the resulting spatial configuration of electric and magnetic fields inside a dielectric sphere with Ohm losses will follow, to some extent, the spatial behaviour of the relevant natural modes of the sphere, volume and surface ones. It is obvious that due to excitation of surface modes the maximum values of electric and magnetic fields inside dissipative sphere will be shifted to its outer part $r> a/2$. When assuming the total number of the surface modes to be the same as those for the volume modes and consequently it is equal to a half of all the dielectric sphere modes, then one can anticipate that the temperature rise in the head tissues close to head surface may be by a factor 1.5 higher in comparison with the standard calculations using the numerical methods without special allowance for the spatial behaviour of the relevant natural modes. However the numerical methods used for estimation of the temperature rise in human tissues due to the radio frequency irradiation do not take into account this effect. Indeed, such calculations (see, for example, paper [@WF]) are carried out in two steps. First the electric and magnetic fields inside the human body are calculated by solving the Maxwell equations with a given source (antenna of a portable telephone). The electric field gives rise to conduction currents with the energy dissipation rate $\sigma \,E^2/2$, where $\sigma $ is a conduction constant. In turn it leads to the temperature rising. The second step is the solution of the respective heat conduction equation (or more precisely, bioheat equation [@WF]) with found local heat sources $\sigma \,E^2/2$ and with allowance for all the possible heat currents. Hence, for this method the distribution of electric field inside the head is of primary importance. The spatial behaviour of the eigenfunctions characterize the system as a whole, and these properties cannot be taken into account by local methods for calculating the solution to partial differential equations (in our case, to the Maxwell equations). Conclusion ========== We have shown that such different, at first glance, notions as quasi-normal modes in black hole physics, Gamov states in quantum mechanics, and quasi-bound states in the theory of open electromagnetic resonators [@Vaynstayn] have the same origin, namely, all these are the eigenmodes of oscillating unbounded domains. By making use of a simple but physically motivated example of electromagnetic oscillations outside a perfectly conducting sphere, which admits analytical treatment, we have easily shown the main features of such oscillations, namely, their exponential decaying in time and, simultaneously, their exponential grows at spatial infinity. These properties are a direct consequence of the radiation condition which is met by qnm at spatial infinity. It is shown also that the exponential rising of qnm at infinity is not observable because the time dependence of qnm should be taken into account here. In the considered example the qnm are the outgoing spherical waves for $r>a$. This point is disregarded in all the attempts to treat qnm mathematically, in particular, to formulate the completeness condition and the relevant expansions in terms of qnm. We have considered the qnm modes in the problem without inhomogeneous potential (instead of the potential the boundary conditions at the surface of the sphere are introduced). It enables us to infer the conclusion stated above clear and easy. It is argued also that imposing the standard radiation conditions with a real wave vector $k=\omega /c$ does not prevent us from the necessity to investigate the eigenmodes with complex $\omega'$s, i.e., qnm in a given problem. The importance of this is demonstrated by investigating the role of the qnm in the problem of estimating the potential health hazards due to the use of portable telephones. The general analysis of the qnm spectra of a dielectric sphere with allowance for the dissipative processes enables us to estimate quantitatively the expected effect, namely, a possible temperature rise in the tissues laying in the outer part of the human head may be 1.5 times greater as compared with the inner part of the brain. The predicted effect is, in some sense, analogous to the usual (but weakly manifested here) skin-effect. It is not surprising because the substantial conductivity of the sphere material plays a principal part in our consideration. Due to this conductivity the individual resonances of a sphere become very overlapping and, being not observed separately, many of them (volume and surface ones) are excited by the cellular telephone irradiation with the frequency in this band. In view of different spatial behaviour, the surface and volume modes will lead to different spatial distributions of the net heat inside the sphere (and also inside a human head). When we have the quasi-normal modes instead the usual normal modes it implies that we are dealing with the open systems [@Open]. Open systems admit a dual description: on the one hand, they can be considered from the “inside” point of view, treating the coupling to the environment as a – not necessarily small – perturbation. From this point of view, one can study the (discreet) eigenvalues of the system, the width of resonances and the resulting decay properties [@Ching; @Leung]. On the other hand, open systems allow to take the “outside” point of view, considering the system as a perturbation of the environment. The typical quantity to be investigated from this point of view is the [*scattering matrix*]{} ($S$-matrix), i.e. the amplitude for passing from a given incoming field configuration to a certain outgoing configuration as a function of energy [@Vaynstayn]. The registration of the quasi-normal modes of a black hole is considered now as a possible way to detect this object [@Nollert; @KS]. In this connection it is surpassing that till now the quasi-normal modes have not been used in an analogous acoustic problem, i.e., for description of sound generation by ringing body [@Raylaygh; @Lamb; @Morse]. At the same time one can find in literature the statement (without proving) that we hear ringing quasi-normal modes of a bell when we hear the bell sound [@Moss]. This paper was completed during the visit of on of the authors (VVN) to Salerno University. It is his pleasant duty to thank G.Scarpetta and G. Lambiase for the kind hospitality extended to him. VVN was supported in part by the Russian Foundation for Basic Research (Grant No. 03-01-00025). The financial support of INFN is acknowledged. The authors are indebted to A.V. Nesterenko for preparing the figure. [99]{} H.-P. Nollert, Class. Quantum Grav. [**16**]{}, R159 (1999). 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I. G. Moss, [*Can you hear the shape of a bell?: Asymptotics of quasinormal modes,*]{} Nucl. Phys. B (Proc. Suppl.) [**104**]{}, 181 – 184 (2002). [^1]: In principle, non-ionizing radiation can lead also to other effects in biological tissues [@Sernelius]. [^2]: As it was shown in preceding Section this relation between the number of the surface and volume modes holds only for the spectra as a whole.

--- abstract: 'Density-wave instabilities have been observed and studied in a multitude of materials. Most recently, in the context of unconventional superconductors like the iron-based superconductors, they have excited considerable interest. We analyze the fluctuation corrections to the equation of state of the density-wave order parameter for commensurate charge-density waves and spin-density waves due to perfect nesting. For XY magnets, we find that contributions due to longitudinal and transverse fluctuations cancel each other, making the mean-field analysis of the problem controlled. This is consistent with the analysis of fluctuation corrections to the BCS theory of superconductivity \[Š. Kos, A. J. Millis, and A. I. Larkin, Phys. Rev. B **70**, 214531 (2004)\]. However, this cancellation does not occur in density-wave systems when the order parameter is a real $N$-component object with $N\neq2$. Then, the number of transverse fluctuating modes differs from the number of longitudinal fluctuating modes. Indeed, in the case of charge-density waves as well as spin-density waves with Heisenberg symmetry, we find that fluctuation corrections are not negligible, and hence mean-field theories are not justified. These singular fluctuations originate from the intermediate length-scale regime, with wavelengths between the lattice constant and the $T=0$ correlation length. The resulting logarithmic fluctuation contributions to the gap equation originate from the derivative of the anomalous polarization function, and the crucial process is an interaction of quasiparticles through the exchange of fluctuations.' author: - Mareike Hoyer - Jörg Schmalian title: 'Role of fluctuations for density-wave instabilities: Failure of the mean-field description' --- Introduction ============ In condensed matter systems, the interaction between electrons often gives rise to an instability of the Fermi-liquid ground state, resulting in the formation of a low-temperature ordered phase. Examples of such ground states include superconductivity, charge-density wave order and magnetically ordered states, and are expected in all materials at least in the clean limit. Often, the low-temperature ordered phase can be described in terms of an effective mean field that characterizes the novel ground state and serves as an order parameter (Fig. \[fig:visu-fluctuations\]). The poster child of such a mean-field theory is the celebrated BCS theory [@BardeenCooperSchrieffer-PR106.1957; @BardeenCooperSchrieffer-PR108.1957] of superconductivity, which offers arguably the single most successful mean-field description of an interacting many-body system. The principal reasons for the success of this theory are the following: 1. the instability towards the BCS state occurs for arbitrarily small interactions, 2. fluctuations beyond mean-field theory can be neglected for small interaction strength since the coherence length is large compared to the inverse Fermi momentum in conventional superconductors, $\xi\gg k_\mathrm{F}^{-1}$, resulting in a narrow Ginzburg regime. ![Visualization of the order parameter and fluctuations around its mean-field value (red dot) for three examples: (a) commensurate charge-density wave order, (b) superconductivity, and (c) commensurate spin-density wave order of Heisenberg spins. The longitudinal mode is indicated in green while the $N-1$ transverse modes are shown in blue. In case of (a) and (b), the manifold of degenerate ground states (red) is shown within the corresponding energy landscape (gray) below the phase transition, whereas for (c) only the former.[]{data-label="fig:visu-fluctuations"}](visu-fluct-OP){width="\columnwidth"} There are instabilities in the particle-hole channel with perfect nesting [@Peierls1955; @Overhauser-PR1962] which share the first characteristic with the BCS theory [@FeddersMartin-PR1966; @ChanHeine-JPhysF1973; @Gorkov-JETPLett1973; @Gorkov-JETP1974; @RiceScott-PRL1975; @HirschScalapino-PRL1986], and which are governed by the same self-consistent mean-field equation, $$\Delta_0={\rho_{\mathrm{F}}}V\int_{-\omega_0}^{\omega_0}\mathrm{d}\varepsilon_{{\boldsymbol{k}}}\,\frac{[1-2n_\mathrm{F}(E_{{\boldsymbol{k}}})]\Delta_0}{2E_{{\boldsymbol{k}}}}{~ ,}\label{eq:gap-equation}$$ which determines the order parameter $\Delta_0$. Here, we abbreviated $E_{{\boldsymbol{k}}}=\sqrt{\varepsilon_{{\boldsymbol{k}}}^2+\Delta_0^2}$, with the energy dispersion $\varepsilon_{{\boldsymbol{k}}}$. Furthermore, ${\rho_{\mathrm{F}}}V$ is the dimensionless interaction leading to long-range order, $n_\mathrm{F}(E_{{\boldsymbol{k}}})$ is the Fermi distribution, and $\omega_0$ refers to the energetic cut-off of the theory. The physical interpretation of $\Delta_0$ is the pairing gap in the case of superconductivity. In the case of charge-density wave (CDW) or spin-density wave (SDW) ordering, it corresponds to the amplitude of the modulation of the commensurate charge density or spin density, respectively. At zero temperature, the gap equation  can be integrated straightforwardly, resulting for ${\rho_{\mathrm{F}}}V\ll1$ in the well-known expression for the zero-temperature energy gap $$\Delta_0=2\omega_0\mathrm{e}^{-\frac{1}{{\rho_{\mathrm{F}}}V}}{~ .}\label{eq:gap}$$ Experimentally, deviations from mean-field behavior have been observed for density-wave systems in the quantum regime [@JaramilloEtAl-Nature2009; @SokolovEtAl-PRB2014; @FreitasEtAl-PRB2015; @FengEtAl-NPhys2015]. The aim of this paper is to theoretically clarify to what extent CDW and SDW systems fulfill the second characteristic of the BCS theory, namely a lack of fluctuation corrections as discovered in the case of superconductivity [@GeorgesYedidia-PRB1991; @vanDongen-PRL1991; @Martin-RoderoFlores-PRB1992; @KosMillisLarkin-PRB2004; @EberleinMetzner-PRB2013; @Eberlein-PRB2014; @FischerEtAl2017]. We find that the result depends on the order-parameter manifold (in particular, the number of components), and on whether longitudinal or transverse fluctuations, i.e., fluctuations of the amplitude or of the phase of the order parameter, dominate. In Fig. \[fig:visu-fluctuations\], the different types of fluctuating modes are visualized for three examples. In all cases, we show that the characteristic length scales of these fluctuations are within the coherence volume $\xi^d$, with coherence length $\xi\sim v_\mathrm{F}/\Delta_0$. Our analysis focuses on the stability of the ordered state at $T=0$. A complementary view is the investigation of the instability of the disordered state. For the density-wave systems discussed here, the latter can be performed using a renormalization group (RG) treatment [@Shankar-RevModPhys1994]. Consistent with our results, channel interference of the RG analysis suggests corrections to mean-field behavior. This is discussed briefly in Sec. \[sec:rg\] and Appendix \[app:two-band\]. The appeal of our approach is, however, that it offers a simple physical picture for the role of order-parameter fluctuations: amplitude variations suppress the order parameter compared to its mean-field value, whereas phase fluctuations enhance it. In this paper, we address the role of fluctuations for mean-field theories of commensurate density-wave order in systems with perfect nesting: We provide a self-consistent calculation of Gaussian fluctuation corrections to the zero-temperature gap equation. We come to the conclusion that, in contrast to BCS theory, these mean-field theories are not justified since sizable fluctuation corrections are inherent to these theories. The mean-field approach is then only valid if the number of amplitude and phase fluctuations is the same. Thus the mean-field approach is neither valid for Heisenberg SDWs nor for CDWs. The only exception is spin-density wave order of XY spins, where longitudinal and transverse fluctuation contributions cancel exactly – in accordance with the cancellation of amplitude and phase fluctuations in superconductors [@KosMillisLarkin-PRB2004; @FischerEtAl2017]. Mean-field theory for density-wave instabilities ================================================ Density-wave order arises naturally for systems in which the Fermi surface is perfectly nested, i.e., different parts of the Fermi surface are connected by the vector ${\boldsymbol{Q}}$ such that $\varepsilon_{{\boldsymbol{k}}+{\boldsymbol{Q}}}=-\varepsilon_{{\boldsymbol{k}}}$ holds for the energy dispersion. This nesting condition suggests we formulate mean-field theories for density-wave order in complete structural analogy to the BCS theory of superconductivity. The noninteracting part of the Hamiltonian $\mathcal{H}=\mathcal{H}_0+\mathcal{H}_\mathrm{int}$ is of the usual form $$\label{eq:H0} \mathcal{H}_0= \sum_{{\boldsymbol{k}},\sigma}\varepsilon_{{\boldsymbol{k}}}\psi^\dagger_{{\boldsymbol{k}},\sigma}\psi^{}_{{\boldsymbol{k}},\sigma}{~ ,}$$ where $\psi_{{\boldsymbol{k}},\sigma}^{(\dagger)}$ annihilates (creates) a fermionic state with momentum ${\boldsymbol{k}}$ and spin $\sigma$, where the nesting condition for the dispersion will be implied in the remainder. The most general $\mathrm{SU}(2)$-invariant interaction can be written as $$\begin{aligned} \mathcal{H}_\mathrm{int}&= \frac{1}{2}\sum\psi^\dagger_{{\boldsymbol{k}}_1,\sigma_1}\psi^\dagger_{{\boldsymbol{k}}_2,\sigma_2} U_{\sigma_1\sigma_2;\sigma_3\sigma_4}({\boldsymbol{k}}_1,{\boldsymbol{k}}_2;{\boldsymbol{k}}_3,{\boldsymbol{k}}_4) \nonumber \\ &\qquad \times \psi^{}_{{\boldsymbol{k}}_3,\sigma_3}\psi^{}_{{\boldsymbol{k}}_4,\sigma_4}\delta({\boldsymbol{k}}_1+{\boldsymbol{k}}_2-{\boldsymbol{k}}_3-{\boldsymbol{k}}_4){~ ,}\label{eq:H_interaction}\end{aligned}$$ where the summation is over momenta ${\boldsymbol{k}}_i$ and spins $\sigma_i$. The spin sector of the interaction can be decomposed into charge (ch) and spin (sp) channel according to $$U_{\sigma_1\sigma_2;\sigma_3\sigma_4} = U_\mathrm{ch}\,\delta_{\sigma_1\sigma_4}\delta_{\sigma_2\sigma_3} +U_\mathrm{sp}\,{\boldsymbol{\sigma}}_{\sigma_1\sigma_4}\cdot{\boldsymbol{\sigma}}_{\sigma_2\sigma_3}{~ ,}\label{eq:decomposition}$$ where we introduced ${\boldsymbol{\sigma}}=(\sigma_1,\sigma_2,\sigma_3)$ as the vector of Pauli matrices in spin space. Furthermore, we could allow for spin-orbit interaction and consider Ising spins or XY spins instead of the Heisenberg spins introduced in Eq.  by restricting ourselves to ${\boldsymbol{\sigma}}=(\sigma_1)$ or ${\boldsymbol{\sigma}}=(\sigma_1,\sigma_2)$, respectively. When formulating the mean-field theories for charge-density wave (CDW) order and spin-density wave (SDW) order, we consider these two channels separately. The interaction projected onto the respective channels takes the form \[eq:interaction\] $$\begin{aligned} \mathcal{H}_\mathrm{ch}&=-\frac{V_\mathrm{ch}}{8}\sum_{\sigma\sigma^\prime}\sum_{{\boldsymbol{k}}{\boldsymbol{k}}^\prime{\boldsymbol{q}}} (\psi^\dagger_{{\boldsymbol{k}},\sigma}\psi^{}_{{\boldsymbol{k}}+{\boldsymbol{q}},\sigma})(\psi^\dagger_{{\boldsymbol{k}}^\prime,\sigma^\prime}\psi^{}_{{\boldsymbol{k}}^\prime-{\boldsymbol{q}},\sigma^\prime}) {~ ,}\\ \mathcal{H}_\mathrm{sp}&=-\frac{V_\mathrm{sp}}{8}\sum_{\sigma_1^{}\sigma_2^{}}\sum_{\sigma_1^\prime\sigma_2^\prime}\sum_{{\boldsymbol{k}}{\boldsymbol{k}}^\prime{\boldsymbol{q}}} (\psi^\dagger_{{\boldsymbol{k}},\sigma_1}{\boldsymbol{\sigma}}_{\sigma_1\sigma_2}\psi^{}_{{\boldsymbol{k}}+{\boldsymbol{q}},\sigma_2}) \nonumber \\ &\qquad \qquad \cdot (\psi^\dagger_{{\boldsymbol{k}}^\prime,\sigma^\prime_1}{\boldsymbol{\sigma}}_{\sigma_1^\prime\sigma_2^\prime}\psi^{}_{{\boldsymbol{k}}^\prime-{\boldsymbol{q}},\sigma^\prime_2}){~ ,}\end{aligned}$$ where we assumed the couplings $U_{\mathrm{ch},\mathrm{sp}}$ to be independent of momenta and thus introduced $U_{\mathrm{ch},\mathrm{sp}}({\boldsymbol{k}}_1,{\boldsymbol{k}}_2;{\boldsymbol{k}}_3,{\boldsymbol{k}}_4)\equiv -V_{\mathrm{ch},\mathrm{sp}}/4$. The corresponding mean-field theories can be derived from the microscopic Hamiltonian by the usual procedure of introducing an effective bosonic field ${\boldsymbol{\Phi}}$ via a Hubbard-Stratonovich decoupling $$\begin{aligned} &\mathrm{e}^{\frac{V}{2}\sum_{{\boldsymbol{q}}}{\boldsymbol{b}}_{{\boldsymbol{q}}}\cdot{\boldsymbol{b}}_{-{\boldsymbol{q}}}}= \nonumber \\ &\quad \int\mathcal{D}{\boldsymbol{\Phi}}\,\mathrm{e}^{-\frac{1}{2V}\sum_{{\boldsymbol{q}}}{\boldsymbol{\Phi}}_{{\boldsymbol{q}}}\cdot{\boldsymbol{\Phi}}_{-{\boldsymbol{q}}}+\frac{1}{2}\sum_{{\boldsymbol{q}}}{\boldsymbol{\Phi}}_{{\boldsymbol{q}}}\cdot{\boldsymbol{b}}_{-{\boldsymbol{q}}}+\frac{1}{2}\sum_{{\boldsymbol{q}}}{\boldsymbol{b}}_{{\boldsymbol{q}}}\cdot{\boldsymbol{\Phi}}_{-{\boldsymbol{q}}}}\label{eq:decoupling}\end{aligned}$$ of the interaction  in the channel of interest and subsequently integrating out the fermions. This requires $V>0$ for the interaction in the respective channel, where we skip the subindex ch or sp in what follows. The effective field ${\boldsymbol{\Phi}}\in\mathds{R}^N$ plays the role of an order parameter and its dimensionality $N$ depends on the channel in which the decoupling of the interaction is performed. Formulated in the language of field integrals, the corresponding mean-field theory follows immediately at the level of the saddle-point approximation, i.e., assuming that the order parameter be temporally homogeneous and $\delta$-distributed in momentum space, which we denote ${\boldsymbol{\Phi}}_0$. The mean-field theories for density-wave order resulting from nesting bear the same structure as the BCS theory of superconductivity. The ground-state energy $$E_\mathrm{MF}(\Phi_0)=E_0-2{\rho_{\mathrm{F}}}L^d\Phi_0^2\bigg[\ln\bigg(\frac{2E_\mathrm{F}}{\Phi_0}\bigg)+\frac{1}{2}-\frac{1}{{\rho_{\mathrm{F}}}V}\bigg]$$ deep inside the ordered phase (where $\Phi_0\equiv|{\boldsymbol{\Phi}}_0|$ is the magnitude of the order parameter) is reduced as compared to the high-temperature ground-state energy $E_0$. For the instantaneous electronic interaction  considered here, the energy window of the attractive pairing goes up to the Fermi energy. That means, in contrast to the BCS theory where the energetic cut-off $\omega_0$ of the theory is given by the Debye energy, we consider $\omega_0\simeq E_\mathrm{F}$ in the remainder. Furthermore, ${\rho_{\mathrm{F}}}$ refers to the density of states at the Fermi level and $L^d$ denotes the system’s volume. This structure results in the characteristic logarithm appearing in the zero-temperature mean-field gap equation $$0=\frac{1}{{\rho_{\mathrm{F}}}L^d}\frac{{\mathrm{d}}E_\mathrm{MF}(\Phi_0)}{{\mathrm{d}}\Phi_0^2} \approx \frac{1}{{\rho_{\mathrm{F}}}V}-\ln\bigg(\frac{2E_\mathrm{F}}{\Phi_0}\bigg){~ ,}\label{eq:gap-equation2}$$ which self-consistently determines the magnitude of the mean-field value of the order parameter. Details on the derivation can be found in Appendix \[app:calculation\]. Note that Eq.  is equivalent to Eq.  at $T=0$. The logarithmic contribution arises from a relative sign provided by the nesting condition $\varepsilon_{{\boldsymbol{k}}+{\boldsymbol{Q}}}=-\varepsilon_{{\boldsymbol{k}}}$ in the case of density-wave order, whereas for superconductivity it results from the pairing of time-reversed states. Hence, the density-wave analog for the Nambu spinors is given by $$\Psi_{{\boldsymbol{k}}}=\begin{pmatrix} \psi^\dagger_{{\boldsymbol{k}},\uparrow} & \psi^\dagger_{{\boldsymbol{k}},\downarrow} & \psi^\dagger_{{\boldsymbol{k}}+{\boldsymbol{Q}},\uparrow} &\psi^\dagger_{{\boldsymbol{k}}+{\boldsymbol{Q}},\downarrow}\end{pmatrix}^T {~ ,}\label{eq:spinor}$$ allowing us to represent the mean-field ground-state energy as $E_\mathrm{MF}(\Phi_0)=2\Phi_0^2/V-\int_k\operatorname{tr}\ln[-(\mathcal{G}_k^\mathrm{MF})^{-1}]$. The corresponding matrix Green’s function in Matsubara representation is given by $$\begin{aligned} &\big(\mathcal{G}_k^{\mathrm{MF}}\big)^{-1}=\frac{1}{2}\left(\mathrm{i}\nu_n\tau_0\sigma_0-\varepsilon_{{\boldsymbol{k}}}\tau_3\sigma_0\right) \nonumber \\ &\qquad +\frac{1}{2}\tau_1\left\{\begin{matrix}\Phi_0\sigma_0 & \text{for CDW order} {~ ,}\\ {\boldsymbol{\Phi}}_0\cdot {\boldsymbol{\sigma}}& \text{for Heisenberg SDW order}{~ ,}\end{matrix}\right.\end{aligned}$$ where we abbreviated $k\equiv({\boldsymbol{k}},\nu_n)$ with the fermionic Matsubara frequency $\nu_n=(2n+1)\pi T$. The Pauli matrices $\sigma_i$ refer to the spin sector whereas the $\tau_i$ denote Pauli matrices in the band space that emerges due to a doubling of the unit cell. We assume $2{\boldsymbol{Q}}$ to be a reciprocal lattice vector. Fluctuation corrections to the gap equation =========================================== The validity of the mean-field approximation as discussed in the previous section relies on the assumption that deviations of the order parameter from its mean-field value lead to negligible corrections to the mean-field theory in the sense that fluctuation corrections to physical observables are small compared to their mean-field value. Of course, there are regimes in which the role of long-wavelength ($|{\boldsymbol{q}}|\ll2\xi^{-1}$) fluctuations is well-understood. Firstly, thermal fluctuations drive the phase transition. The Ginzburg regime in which these thermal fluctuations lead to sizable corrections to mean-field theory is restricted to the vicinity of the critical point. Secondly, in low-dimensional systems fluctuations are important down to lower temperatures, where they can lead to the breakdown of true long-range order [@MerminWagner-PRL1966; @Hohenberg-PR1967]. At zero temperature, deep inside the ordered phase, (quantum) fluctuations are expected to have severe consequences only for one-dimensional systems, while they are small in spatial dimensions $d\geq2$. Here we consider different fluctuations with characteristic length scales shorter than $\xi$. To self-consistently check the validity of a mean-field theory, we determine fluctuation corrections to the zero-temperature gap equation and compare the resulting contributions due to fluctuations to the mean-field terms. This approach was put forward in the context of superconductivity by Kos, Millis, and Larkin [@KosMillisLarkin-PRB2004]; and their self-consistent calculation of corrections to the BCS gap equation indeed showed that the BCS mean-field theory is justified. This analysis for $s$-wave superconductors can be extended to charged superconductors [@FischerEtAl2017] as well as anisotropic superconductors [@ParamekantiEtAl-PRB2000; @BarlasVarma-PRB2013; @Hoyer-Thesis2017], resulting in the conclusion that fluctuation corrections to the zero-temperature gap equation are generally negligible for superconductors. Remarkably, this is due to an exact cancellation of individually large contributions that can be assigned to fluctuations of the amplitude and the phase of the order parameter. Therefore, a natural question in the context of density-wave instabilities is whether an analogous mechanism of cancellation of longitudinal and transverse fluctuation corrections ensures the validity of the respective mean-field theories – or whether quantum fluctuations become sizable such that the mean-field description is not justified. This question will be addressed in the remainder of this paper. For density-wave instabilities, the order parameter governing the effective action is a real $N$-component vector, $\boldsymbol{\Phi}(\boldsymbol{r},\tau)\in\mathds{R}^N$, where $\tau$ refers to the imaginary time in Matsubara formalism. This order parameter can be split into the static and homogeneous mean-field value and spatial and temporal fluctuations around this mean-field value as $$\boldsymbol{\Phi}(\boldsymbol{r},\tau)=\Phi_0\hat{\boldsymbol{e}}+\delta\boldsymbol{\Phi}(\boldsymbol{r},\tau){~ .}$$ Here, we introduced $\hat{\boldsymbol{e}}$ as the unit vector along the direction of the mean-field order parameter and the magnitude of the mean-field order parameter $\Phi_0$ can be determined self-consistently from the gap equation. The fluctuations $\delta\boldsymbol{\Phi}$ of the order parameter around the mean-field configuration can be further split into one longitudinal mode $\parallel \hat{\boldsymbol{e}}$ and $N-1$ transverse modes $\perp \hat{\boldsymbol{e}}$. In the remainder of this paper, we restrict ourselves to the leading contribution and evaluate the fluctuation corrections for Gaussian fluctuations, i.e., taking into account contributions up to $\mathcal{O}[(\delta{\boldsymbol{\Phi}})^2]$. ![Diagrammatic representation of contributions to the gap equation. Straight lines represent the fermionic propagators while wiggly lines stand for the fluctuation propagator. The order parameter (indicated by a dashed line) is added for the sake of clarity here, however, it does not contribute to the derivative ${\mathrm{d}}E/{\mathrm{d}}\Phi_0^2$. The logarithmic contribution to the mean-field gap equation is shown in (a), while the structure of Gaussian fluctuation corrections to the gap equation is presented in (b).[]{data-label="fig:gap-equation"}](gap-equation){width="\columnwidth"} Including fluctuations around the saddle-point configuration results in additional contributions to the ground-state energy, $E(\Phi_0)=E_\mathrm{MF}(\Phi_0)+E_\mathrm{fluct}(\Phi_0)$, which are also reflected in the gap equation as $$\frac{1}{{\rho_{\mathrm{F}}}L^d}\frac{{\mathrm{d}}E(\Phi_0)}{{\mathrm{d}}\Phi_0^2}=\frac{1}{\lambda}-\ln\Big(\frac{2E_\mathrm{F}}{\Phi_0}\Big)+\frac{1}{{\rho_{\mathrm{F}}}L^d}\frac{{\mathrm{d}}E_\mathrm{fluct}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}=0{~ ,}\label{eq:gap-equation-fluct}$$ where we introduced the dimensionless interaction $\lambda={\rho_{\mathrm{F}}}V$. Unless the additional contribution stemming from fluctuations is smaller than the first two terms already appearing at mean-field level \[cf. Eq. \], the mean-field theory is not justified. The logarithmic contribution already appearing at mean-field level corresponds to the fermionic loop diagram depicted in Fig. \[fig:gap-equation\](a). The Gaussian fluctuation corrections studied here take the form presented in Fig. \[fig:gap-equation\](b). In the remainder, we show that such terms indeed give rise to a logarithmic contribution to the gap equation. The usual contributions due to critical fluctuations, important in the long-wavelength limit and for small frequencies, correspond to the limit where the fermionic triangle becomes structureless, cf. Fig. \[fig:long-wavelength\], but our analysis shows that the internal structure of the triangle diagram is indeed important. \] becomes structureless. This limit corresponds to the usual contributions known from the consideration of critical fluctuations. Note that in the opposite regime of short-wavelength fluctuations ($|{\boldsymbol{q}}|\gg2\xi^{-1}$), the inner structure of the fermionic triangle becomes crucial, cf. Fig. \[fig:triangles\].[]{data-label="fig:long-wavelength"}](long-wavelength){width="\columnwidth"} If the Gaussian fluctuation corrections are of the same order as the mean-field contribution to the zero-temperature gap equation, they can be effectively restated as a modification of the prefactor of the logarithmic contribution. Then the solution of Eq.  can be written in the familiar form  by introducing the effective interaction $$\lambda_\mathrm{eff}=\lambda\bigg[1-\frac{1}{{\rho_{\mathrm{F}}}L^{d}\ln\big(\frac{2 E_\mathrm{F}}{\Phi_0}\big)}\frac{{\mathrm{d}}E_\mathrm{fluct}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}\bigg]{~ .}$$ Hence the sign of the fluctuation contribution plays an interesting role as negative fluctuation corrections further enhance the gap as compared to its mean-field value whereas positive fluctuation contributions result in a reduction, i.e., weaken the ordered state. Let us now analyze the structure of fluctuation corrections in the case of density-wave instabilities due to nesting following the logic of Ref. , also building on the calculations presented in Ref. . More details specific to our derivation for both charge-density wave order and spin-density wave order are presented in Appendix \[app:calculation\]. The Gaussian fluctuation corrections can be expressed in terms of the fluctuation propagator $\boldsymbol{\mathcal{D}}_q$ as $E_\mathrm{fluct}(\Phi_0)=\frac{L^d}{2}\int_q\ln\det({\boldsymbol{\mathcal{D}}}^{-1}_q)$, where the dimensionality of the order parameter translates to the dimensionality of the fluctuation propagator. Accordingly, the Gaussian fluctuation corrections to the zero-temperature gap equation take the form $$\frac{1}{{\rho_{\mathrm{F}}}L^d}\frac{{\mathrm{d}}E_\mathrm{fluct}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}=\frac{1}{2{\rho_{\mathrm{F}}}}\int_q\,\frac{\frac{{\mathrm{d}}}{{\mathrm{d}}\Phi_0^2}\det(\boldsymbol{\mathcal{D}}^{-1}_q)}{\det(\boldsymbol{\mathcal{D}}^{-1}_q)}{~ ,}\label{eq:fluctuation-corrections}$$ where we introduced $q\equiv({\boldsymbol{q}},\omega)$ and the integration $\int_q\ldots=\int\frac{{\mathrm{d}}\omega}{2\pi}\int\frac{{\mathrm{d}}^d{\boldsymbol{q}}}{(2\pi)^d}\ldots$ runs over all external frequencies and momenta up to the cutoff. Consequently, an expansion of the fluctuation spectrum in small momenta and frequencies as often discussed in the context of collective modes is not sufficient here. Instead, the fluctuation propagator has to be determined for all frequencies and momenta, and in fact, the short-wavelength fluctuations with momenta $2\xi^{-1}=2\Delta_0/v_\mathrm{F}\ll|{\boldsymbol{q}}|\ll k_\mathrm{F}$ turn out to be crucial, as they give rise to an additional logarithmic contribution to the zero-temperature gap equation as discussed in Sec. \[sec:results\]. The inverse matrix of the fluctuation propagator for density-wave order can be stated in terms of the polarization matrix $\boldsymbol{\Pi}_q$ as $$\boldsymbol{\mathcal{D}}^{-1}_q=\frac{1}{V}\boldsymbol{\mathds{1}}-\boldsymbol{\Pi}_q{~ .}$$ For density-wave order, transverse and longitudinal fluctuations are not coupled and hence the polarization matrix is diagonal. The Gaussian fluctuation corrections can thus be expressed in terms of longitudinal and transverse contributions using $$E_\mathrm{fluct}(\Phi_0)=\frac{L^d}{2}\int_q\ln\Big[4^N\Big(\frac{1}{V}-\Pi^\perp_q\Big)^{N-1}\Big(\frac{1}{V}-\Pi^\parallel_q\Big)\Big]{~ .}$$ Both the longitudinal and the transverse contribution consist of a normal part $\Pi^\mathrm{n}_q=\frac{1}{2}\int_k(G_{k+q}G_{-k}+G_{-k}G_{k-q})$ which survives the limit $\Phi_0\rightarrow0$, and an anomalous contribution $\Pi^\mathrm{a}_q=-\int_kF_kF_{k+q}$ which vanishes in the high-temperature normal state, $$\begin{aligned} \Pi^\perp_q&=\Pi^\mathrm{n}_q-\Pi^\mathrm{a}_q{~ ,}\label{eq:Pi-perp} \\ \Pi^\parallel_q&=\Pi^\mathrm{n}_q+\Pi^\mathrm{a}_q{~ .}\label{eq:Pi-parallel}\end{aligned}$$ These integrals (see Appendix \[app:polarization-function\] for details) are of the same structure as those arising in the context of superconductivity, and hence we can build on the results obtained by previous studies [@KosMillisLarkin-PRB2004; @FischerEtAl2017] in the remainder. Results and Discussion {#sec:results} ====================== ![Diagrammatic representation of the terms contributing to the derivative of the polarization function. For the definition of the diagrammatic elements see Fig. \[fig:diagrammatic-elements\]. (a) and (b) Derivatives of the anomalous part, while (c) represents the derivative of the normal contribution. While in the long-wavelength regime, the fermionic triangle part depicted here becomes structureless (cf. Fig. \[fig:long-wavelength\]), it is of particular importance for the contributions stemming from the regime of short wavelengths. The diagram which yields the crucial contribution to the gap equation discussed in this paper is highlighted in gray.[]{data-label="fig:triangles"}](triangle-diagram-DW){width="\columnwidth"} We can straightforwardly adopt the formalism developed in the context of superconductivity [@KosMillisLarkin-PRB2004; @FischerEtAl2017] to calculate the leading-order corrections to the zero-temperature mean-field gap equation due to Gaussian fluctuations of the order parameter for density-wave order. We consider the regimes of small and large momenta/frequencies separately, which is possible [@VaksGalitskiiLarkin-JETP1962] since the integrals only depend on the combination $r=\sqrt{\omega^2+(v_\mathrm{F}|{\boldsymbol{q}}|\cos\theta)^2}/(2\Phi_0)$, where $\theta$ denotes the angle between fermionic momentum ${\boldsymbol{k}}$ and bosonic momentum ${\boldsymbol{q}}$. The crucial contribution to the fluctuation corrections, as given by Eq.  and diagrammatically represented in Fig. \[fig:gap-equation\](b), stems from the regime where $r\gg1$ and $v_\mathrm{F}|{\boldsymbol{q}}|>\omega$, whereas long-wavelength fluctuations lead to corrections that are negligible compared to the mean-field terms. This is due to the fact that the fermionic triangle (cf. Fig. \[fig:triangles\]) associated with the derivative of the polarization function becomes structureless in the limit $r\ll1$ and hence fluctuation corrections reduce to the simpler form shown in Fig. \[fig:long-wavelength\]. In the regime of interest, the fluctuation propagator is dominated by the normal part of the polarization function, while its derivative is largely determined by the anomalous part. Therefore, the leading contribution to the gap equation evaluates to \[eq:result\]$$\begin{aligned} &\frac{1}{{\rho_{\mathrm{F}}}L^d}\frac{{\mathrm{d}}E_\mathrm{fluct}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}\Big|_{r\gg1} \approx \frac{1}{2{\rho_{\mathrm{F}}}}\int_q\frac{[(N-1)-1]\frac{{\mathrm{d}}\Pi^\mathrm{a}_q}{{\mathrm{d}}\Phi_0^2}}{\frac{1}{V}-\Pi^\mathrm{n}_q} \\ &\qquad \qquad \approx -\frac{1}{2}\frac{(N-1)-1}{8\pi (d-1)}\ln\Big(\frac{E_\mathrm{F}}{\Phi_0}\Big){~ .}\end{aligned}$$ This reveals that Gaussian fluctuations indeed quite generically entail an additional logarithmic contribution to the gap equation for density-wave instabilities due to nesting. This logarithmic divergence can be traced back to the fermionic triangle depicted in Fig. \[fig:triangles\](a), which results from the derivative of the anomalous part of the polarization function that enters the numerator of ${\mathrm{d}}E_\mathrm{fluct}(\Phi_0)/{\mathrm{d}}\Phi_0^2$, while the two other contributions \[Figs. \[fig:triangles\](b) and \[fig:triangles\](c)\] do not give rise to additional logarithms. Thus, the precarious process that invalidates the mean-field approach in the case of density-wave order is the interaction of quasiparticles through fluctuations. fluctuations ------------------ -------------- ---------------- Heisenberg SDW ($N=3$) increase gap XY SDW ($N=2$) are negligible Ising SDW or CDW ($N=1$) decrease gap : Gaussian fluctuation corrections in $\mathrm{O}(N)$ models. Depending on the dimensionality of the order parameter, fluctuation corrections can increase or decrease the gap value compared to its mean-field value.[]{data-label="tab:summary-sdw-fluctuations"} The above result is in stark contrast to the insignificant role that fluctuations play in the context of superconductivity: Analogous contributions to the BCS gap equation are not only suppressed by the smallness of the Debye energy as compared to the Fermi energy, but the corresponding contributions stemming from fluctuations of phase and amplitude of the complex order parameter even cancel exactly. Nonetheless, the corrections due to the longitudinal mode and the $N-1$ transverse modes enter the result  with opposite signs, and hence the prefactor depends on the number of transverse modes. Only for XY spins ($N=2$), the large fluctuation corrections stemming from the regime $r\gg1$ cancel – which is consistent with previous results in the context of superconductivity [@KosMillisLarkin-PRB2004; @FischerEtAl2017]. In conclusion, the analysis of the role of fluctuation corrections in the case of density-wave order reveals that mean-field approaches are generally not justified in this situation, the only exception being spin-density wave order of XY spins. Furthermore, the sign of the fluctuation corrections allows us to judge whether the presence of fluctuations is advantageous or detrimental to the formation of density-wave order: The effective interaction $$\begin{aligned} \lambda_\mathrm{eff} &= \lambda\bigg[1+\frac{N-2}{16\pi(d-1)}\bigg]{~ ,}\label{eq:effective-interaction} $$ which governs the ordering in the presence of fluctuations, either decreases (${\mathrm{d}}E_\mathrm{fluct}/{\mathrm{d}}\Phi_0^2>0$) or increases (${\mathrm{d}}E_\mathrm{fluct}/{\mathrm{d}}\Phi_0^2<0$), and the same is true for the solution of the zero-temperature gap equation  in the presence of fluctuations. For $N=1$ or $N=3$ and $d=3$, the relative change in $\lambda$ is $1/(32\pi)\simeq0.01$, i.e., rather small. More important than the numerical value of this correction is the fact that there is no guarantee that even higher-order processes give rise to equally non-negligible corrections. Our analysis of fluctuation corrections shows that contributions stemming from the longitudinal mode lead to a decrease of the gap compared to its mean-field value, whereas transverse fluctuations increase the gap. Hence, if the latter dominate, fluctuations are favorable to the formation of density-wave order, as it is the case for spin-density wave order of Heisenberg spins. If, on the other hand, transverse fluctuations cannot compensate for the effect of the longitudinal mode, the ordered state is weakened. This applies to mean-field theories for charge-density wave order as well as for spin-density wave order of Ising spins, see also Table \[tab:summary-sdw-fluctuations\] for an overview of our results. Channel interference in the renormalization group approach {#sec:rg} ========================================================== {width="\textwidth"} Our analysis of the role of fluctuations for density-wave instabilities has been performed at zero temperature, i.e., deep inside the ordered phase. In doing so, we concentrated on a single channel and neglected the potential presence of competing instabilities. Another, complementary perspective is provided by a renormalization group (RG) analysis [@Shankar-RevModPhys1994], in which competing instabilities of the system can be treated on equal footing and thereby channel interference can be studied within this framework. However, within the RG scheme, the phase transition is approached coming from high energies, allowing us to determine the leading instability of the system, but this approach is less suited to explore the ordered state further. For example, the model defined by the Hamiltonian $\mathcal{H}=\mathcal{H}_0+\mathcal{H}_\mathrm{int}$ as stated in Eqs.  and  is in principle also unstable towards the formation of superconductivity, which is expected to interfere with SDW ordering. In fact, the complex phase diagrams of materials of current interest such as the iron-based superconductors can be understood as a result of the interplay of different competing instabilities: Already the simple two-band model allows for superconductivity as well as SDW order and CDW order [@ChubukovEfremovEremin-PRB2008; @PodolskyKeeKim-EPL2009; @KangTesanovic-PRB2011], owing to the nested nature of the Fermi surface. This model is essentially a band-basis translation of the model investigated in this paper. Hence the calculation of fluctuation corrections to the corresponding zero-temperature gap equations is straightforward, see Appendix \[app:calculation\]. What is more, we can use the RG equations derived for the same model [@ChubukovEfremovEremin-PRB2008] to analyze the effect of channel interference on the transition temperature towards a given ordered state. The resulting flow of the couplings in the density-wave channels as a function of $t=\log\frac{W}{E}$, where $W$ is the bandwidth and $E$ the running energy scale, takes the form \[eq:channel-interference-couplings\] $$\begin{aligned} \dot{\Gamma}_\mathrm{SDW}&= (\Gamma_\mathrm{SDW})^2\pm 2u_3(u_1-u_2-u_4){~ ,}\\ \dot{\Gamma}_\mathrm{CDW}&= (\Gamma_\mathrm{CDW})^2\mp 2u_3(u_1+u_2-u_4) {~ .}\end{aligned}$$ The couplings $u_i$ refer to different intraband and interband processes that are connected to the couplings in spin and charge channel according to Eq. . If the second term were zero, this would result in the usual logarithmic divergence $\Gamma=\Gamma_0/(1-\Gamma_0\log\frac{W}{E})$ of the coupling in the respective channel. Hence the presence of the second term implies corrections due to channel interference that are intrinsic to the model as long as the interband pair-hopping process which is associated with $u_3$, and involves a momentum transfer of $2{\boldsymbol{Q}}$, is effective. Motivated by the structure of Eq. , we used $u_3^{(0)}$ as a measure of channel interference in our brief analysis, where we tuned the bare couplings such that the instability in the channel under consideration is favored within the mean-field description, while the bare couplings in the competing channels were tuned to zero. We then kept the bare value of the coupling in a given channel fixed while increasing the channel interference strength. Details on our derivation can be found in Appendix \[app:two-band\]. We find that the energy scale at which the coupling diverges is affected by channel interference. Furthermore, the effect of channel interference on charge-density wave order and spin-density wave order is different: While increasing the channel-interference strength via $u_3^{(0)}$ results in higher transition temperatures for SDW order (since the divergence is pushed to lower energies), the opposite is true for CDW order. In Fig. \[fig:channel-interference\], we present our numerical solution of the RG equations  for SDW order characterized by a real order parameter and CDW order associated with a purely imaginary order parameter, since these are the two instabilities arising from the parameter range usually discussed in the context of iron-based superconductors. While our analysis and the perturbative RG investigation address rather different phenomena, we note that the trends we find from the RG flow are consistent with our analysis of fluctuation corrections: Channel interference is favorable for the formation of spin-density wave order as it leads to an increase of the corresponding transition temperature, whereas the transition towards charge-density wave order is hindered by channel interference in the sense that the corresponding transition temperature decreases. Summary ======= Mean-field theories for density-wave order resulting from a nesting of the Fermi surface can be derived from a microscopic model of interacting fermions in full analogy to the formulation of BCS theory in the language of field integrals. The BCS theory of superconductivity is the poster child of mean-field theories since neither thermal nor quantum fluctuations lead to sizable effects in conventional superconductors and hence can be neglected. In this paper, we showed that, in contrast to superconductivity (where the order parameter is a complex scalar), the impact of fluctuations is crucial in the case of commensurate density-wave order (characterized by a real $N$-component order parameter) as long as $N\neq2$. To be specific, we have investigated the role of fluctuations for charge-density wave order and spin-density wave order due to nesting of the Fermi surface. Our main finding is that, generally, fluctuation corrections to the zero-temperature gap equations for such density-wave instabilities are of the same order $\mathcal{O}[\ln(E_\mathrm{F}/\Phi_0)]$ as the terms already appearing at mean-field level. In conclusion, the mean-field theories for density-wave instabilities are not justified since the large fluctuation corrections imply that the respective mean-field theory cannot capture all relevant contributions. Of course, our analysis does not imply that the effect of fluctuations is merely to replace $\lambda$ by $\lambda_\mathrm{eff}$ of Eq. , as there is no guarantee that even higher-order processes will not give rise to corrections of the same order. Moreover, we find that the additional logarithmic contribution to the gap equation stemming from fluctuations originates from the derivative of the anomalous polarization function, and the crucial process is the interaction of quasiparticles through the exchange of fluctuations depicted in Fig. \[fig:triangles\](a). Interestingly, we find that the impact of longitudinal and transverse modes is quite different: Longitudinal fluctuations always yield $E_\mathrm{fluct}>0$ and are thus detrimental to the formation of long-range order. Transverse fluctuations, on the other hand, only yield $E_\mathrm{fluct}>0$ in the long-wavelength regime. In the opposite regime of transverse fluctuations on lengthscales smaller than the coherence length, we find that the respective fluctuation correction surprisingly lowers the energy. Furthermore, this contribution is the dominant one since it yields the additional logarithm to the gap equation in the case of perfect nesting that ultimately leads to an increase of the gap as compared to its mean-field value. It is due to the twist of the “phase” induced by the excitation of quasiparticles inside the coherence volume which enhances the kinetic energy of quasiparticle excitations. In contrast, longitudinal fluctuations can only lead to an increase of the energy since the potential energy is already minimized by the mean-field configuration. Because of the different nature of longitudinal and transverse fluctuations, the case $N=2$ is an interesting exception: The logarithmic contributions to the gap equation stemming from the longitudinal mode and the single transverse mode cancel exactly which renders the overall fluctuation corrections negligible. This cancellation legitimates the mean-field approach to density-wave order of XY spins. This is in accordance with the analysis of fluctuations in the context of superconductivity [@KosMillisLarkin-PRB2004; @FischerEtAl2017], where fluctuation corrections from phase and amplitude mode cancel analogously, providing the justification of the BCS mean-field theory. Acknowledgments =============== We thank M. Bard, A. V. Chubukov, S. Fischer, M. Hecker, N. Kainaris, and M. S. Scheurer for helpful discussions. Calculation of fluctuation corrections {#app:calculation} ====================================== This Appendix provides technical details on our calculation of fluctuation corrections, closely following the logic and notation introduced in the context of superconductivity in Ref.  and extended by Ref. . In what follows, we consider the partition function $\mathcal{Z}=\int\mathcal{D}[\bar{\psi},\psi]\,\exp(-\mathcal{S}[\bar{\psi},\psi])$ with the appropriate action $\mathcal{S}[\bar{\psi},\psi]=\int_0^\beta{\mathrm{d}}\tau\,(\sum_\sigma\int{\mathrm{d}}^d{\boldsymbol{x}}\,\bar{\psi}_\sigma({\boldsymbol{x}})\partial_\tau\psi_\sigma({\boldsymbol{x}})+\mathcal{H})$ corresponding to the Hamiltonian as stated in Eqs.  and . The effective theory in terms of the order parameter ${\boldsymbol{\Phi}}$ follows from the Hubbard-Stratonovich transformation  and successively integrating out the fermions, resulting in $\mathcal{Z}=\int\mathcal{D}{\boldsymbol{\Phi}}\,\exp(-\mathcal{S}_\mathrm{eff}[{\boldsymbol{\Phi}}])$. Since the derivation of fluctuation corrections to the zero-temperature mean-field gap equation for charge-density wave (CDW) order and spin-density wave (SDW) order follow the same logic and primarily differ in the dimensionality of the respective order parameter ${\boldsymbol{\Phi}}\in\mathds{R}^N$, we treat them simultaneously here. For CDW order, the order parameter (associated with the charge density $\rho$) introduced by the Hubbard-Stratonovich transformation  is a scalar, $$\rho_{{\boldsymbol{q}}}=\sum_{{\boldsymbol{k}},\sigma,\sigma^\prime}\big<\psi_{{\boldsymbol{k}},\sigma}^\dagger\delta_{\sigma\sigma^\prime}\psi_{{\boldsymbol{k}}+{\boldsymbol{Q}}+{\boldsymbol{q}},\sigma^\prime}^{}\pm\psi_{{\boldsymbol{k}}+{\boldsymbol{Q}},\sigma}^\dagger\delta_{\sigma\sigma^\prime}\psi_{{\boldsymbol{k}}+{\boldsymbol{q}},\sigma^\prime}^{}\big>{~ ,}$$ corresponding to $N=1$. The upper sign refers to CDW order characterized by a real (r) order parameter, whereas the lower sign refers to CDW order with an imaginary (i) order parameter. The latter follows from assuming $V<0$ in the respective channel. All derivations for iCDW order can be performed in complete analogy to those for rCDW order and since we come to the same conclusions in both cases, we concentrate on rCDW order in the following. For SDW order of Heisenberg spins, the order parameter (associated with the magnetization ${\boldsymbol{M}}$) is a three-component vectorial object, $${\boldsymbol{M}}_{{\boldsymbol{q}}}=\sum_{{\boldsymbol{k}},\sigma,\sigma^\prime}\big<\psi_{{\boldsymbol{k}},\sigma}^\dagger{\boldsymbol{\sigma}}_{\sigma\sigma^\prime}\psi_{{\boldsymbol{k}}+{\boldsymbol{Q}}+{\boldsymbol{q}},\sigma^\prime}^{}\pm\psi_{{\boldsymbol{k}}+{\boldsymbol{Q}},\sigma}^\dagger{\boldsymbol{\sigma}}_{\sigma\sigma^\prime}\psi_{{\boldsymbol{k}}+{\boldsymbol{q}},\sigma^\prime}^{}\big>{~ ,}$$ which corresponds to $N=3$. Again, for the sake of clarity, we only discuss rSDW order (upper sign) since, mutatis mutandis, the same results can be obtained for iSDW order (lower sign). For both types of density-wave order, the order parameter (which we denote ${\boldsymbol{\Phi}}$ henceforth) can be split into the static and homogeneous mean-field value $\Phi_0$ and fluctuations $\delta{\boldsymbol{\Phi}}$ around this mean-field value as $${\boldsymbol{\Phi}}_q=\Phi_0\hat{{\boldsymbol{e}}}\delta_{q,0}+\delta{\boldsymbol{\Phi}}_q{~ ,}$$ where we introduced $\hat{{\boldsymbol{e}}}$ as the unit vector along the direction of the mean-field order parameter. For CDW order, $\hat{{\boldsymbol{e}}}=1$, while for SDW order, we assume w.l.o.g. $\hat{{\boldsymbol{e}}}=\hat{{\boldsymbol{e}}}_3$ in the remainder. The usual procedure of integrating out the fermions after the decoupling  then leads to the effective action in terms of the fermionic Green’s function $$\mathcal{S}_\mathrm{eff}(\Phi_0)=\int_q\frac{2|{\boldsymbol{\Phi}}_q|^2}{V}-\int_{k,k^\prime}\operatorname{tr}\ln(\mathcal{G}_{kk^\prime}^{-1}) {~ ,}$$ which can be split into a mean-field part and fluctuations as well using $$\begin{aligned} &\operatorname{tr}\ln\big(-\mathcal{G}^{-1}_{kk^\prime}\big)=\operatorname{tr}\ln\big[-\big((\mathcal{G}^\mathrm{MF}_k)^{-1}\delta_{kk^\prime}+\eta_{kk^\prime}\big)\big]\\ &\quad =\operatorname{tr}\ln\big[-\big(\mathcal{G}^{\mathrm{MF}}_k)^{-1}\delta_{kk^\prime}\big)\big]-\frac{1}{2}\operatorname{tr}\big(\mathcal{G}^\mathrm{MF}_k\eta_{kk^\prime}\mathcal{G}^\mathrm{MF}_{k^\prime}\eta_{k^\prime k}\big) \nonumber \\ &\quad \qquad +\mathcal{O}\big[(\delta{\boldsymbol{\Phi}})^3\big] \end{aligned}$$ by expanding the fluctuations up to Gaussian order. Here, the mean-field part of the inverse matrix Green’s function in Matsubara representation is given by $$\begin{aligned} &\big(\mathcal{G}_k^{\mathrm{MF}})^{-1}=\tfrac{1}{2}\left(\mathrm{i}\nu_n\tau_0\sigma_0-\varepsilon_{{\boldsymbol{k}}}\tau_3\sigma_0\right) \nonumber \\ &\qquad +\tfrac{1}{2}\tau_1\left\{\begin{matrix}\Phi_0 \sigma_0 & \text{for CDW order} {~ ,}\\ {\boldsymbol{\Phi}}_0\cdot {\boldsymbol{\sigma}}& \text{for SDW order}{~ ,}\end{matrix}\right.\end{aligned}$$ while the fluctuation matrix is given by $$\eta_{kk^\prime}=\tfrac{1}{2}\tau_1\left\{\begin{matrix} \delta\Phi_{k-k^\prime}\sigma_0 & \text{for CDW order} {~ ,}\\ \delta{\boldsymbol{\Phi}}_{k-k^\prime}\cdot{\boldsymbol{\sigma}} & \text{for SDW order}{~ .}\end{matrix} \right.$$ Note that when considering density-wave order with an imaginary order parameter, the order parameter and its fluctuations are associated with $\tau_2$ rather than $\tau_1$. One easily finds that this change will not affect the conclusions of our analysis. Then the partition function can be expressed as $$\mathcal{Z}=\int\mathcal{D}[\bar{\Psi},\Psi]\, \mathrm{e}^{-(\mathcal{S}_0+\mathcal{S}_\mathrm{int})}\approx\mathrm{e}^{-(\mathcal{S}_\mathrm{MF}+\mathcal{S}_\mathrm{fluct})} {~ ,}$$ where the mean-field action takes the form $$\mathcal{S}_\mathrm{MF}(\Phi_0)=\frac{2\Phi_0^2}{V}-\int_k\operatorname{tr}\ln\Big[-\big(\mathcal{G}_k^\mathrm{MF}\big)^{-1}\Big]{~ ,}$$ resulting in the famous form of the gap equation $$\begin{aligned} 0&=\frac{{\mathrm{d}}\mathcal{S}_\mathrm{MF}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}=\frac{2}{V}-\int\frac{{\mathrm{d}}^d{\boldsymbol{k}}}{(2\pi)^d}\int_{-\infty}^\infty\frac{{\mathrm{d}}\nu}{2\pi}\,\frac{2}{\varepsilon_{{\boldsymbol{k}}}^2+\nu^2+\Phi_0^2} \nonumber \\ &\approx\frac{2}{V}-2{\rho_{\mathrm{F}}}\ln\Big(\frac{2 E_\mathrm{F}}{\Phi_0}\Big) \end{aligned}$$ at zero temperature. The Gaussian fluctuation part can be integrated exactly, $$\begin{aligned} \begin{split} \mathrm{e}^{-\mathcal{S}_\mathrm{fluct}}&=\int\mathcal{D}[\delta{\boldsymbol{\Phi}}]\,\mathrm{e}^{-\frac{1}{2}\int_q\delta{\boldsymbol{\Phi}}_q\cdot{\boldsymbol{\mathcal{D}}}_q^{-1}\cdot\delta{\boldsymbol{\Phi}}_{-q}} \\ &= \mathrm{e}^{-\frac{1}{2}\int_q\ln\det[{\boldsymbol{\mathcal{D}}}^{-1}_q]}{~ .}\end{split}\end{aligned}$$ Alternatively, the inverse fluctuation propagator matrix ${\boldsymbol{\mathcal{D}}}^{-1}_q$, and consequently the Gaussian fluctuation corrections to the action, can be expressed in terms of the polarization matrix ${\boldsymbol{\Pi}}_q$ as $${\boldsymbol{\mathcal{D}}}_q^{-1}= \frac{4}{V}{\boldsymbol{\mathds{1}}}-{\boldsymbol{\Pi}}_q$$ where ${\boldsymbol{\Pi}}_q$ is either given by $$\Pi^\mathrm{CDW}_q=-\frac{1}{4}\int_k\operatorname{tr}\big[\mathcal{G}^\mathrm{CDW}_{k+\frac{q}{2}}(\tau_1\sigma_0)\mathcal{G}^\mathrm{CDW}_{k-\frac{q}{2}}(\tau_1\sigma_0)\big] {~ ,}$$ or by $$\begin{aligned} ({\boldsymbol{\Pi}}^\mathrm{SDW}_q)_{ij}&=-\frac{1}{4}\int_k\operatorname{tr}\big[\mathcal{G}^\mathrm{SDW}_{k+\frac{q}{2}}(\tau_1\sigma_i)\mathcal{G}_{k-\frac{q}{2}}^\mathrm{SDW}(\tau_1\sigma_j)\big] \\ &=\begin{pmatrix} 4\Pi^\perp_q & 0 &0 \\ 0 & 4\Pi^\perp_q & 0\\ 0 & 0 & 4 \Pi^\parallel_q\end{pmatrix}\label{eq:Pi-SDW-matrix}{~ .}\end{aligned}$$ In the last line, we introduced longitudinal and transverse contributions $$\begin{aligned} \Pi^\perp_q&=\Pi^\mathrm{n}_q-\Pi^\mathrm{a}_q \\ \Pi^\parallel_q&=\Pi^\mathrm{n}_q+\Pi^\mathrm{a}_q \end{aligned}$$ in terms of the normal ($\Pi_q^\mathrm{n}$) and anomalous ($\Pi_q^\mathrm{a}$) part of the polarization function, which are discussed in more detail in Appendix \[app:polarization-function\]. The generalization to spin dimensionality $N$ is straightforward, and the resulting polarization matrix differs from Eq.  only in the number of transverse modes. The fluctuation corrections to the action then take the form $$\mathcal{S}_\mathrm{fluct}(\Phi_0)=\frac{1}{2}\int_q\ln\Big[4^N\Big(\frac{1}{V}-\Pi_q^\perp\Big)^{N-1}\Big(\frac{1}{V}-\Pi_q^\parallel\Big)\Big]$$ for both SDW order and CDW order, where the latter corresponds to $N=1$. Owing to the diagonal structure of ${\boldsymbol{\mathcal{D}}}^{-1}_q$, the Gaussian fluctuation corrections to the gap equation, $$\begin{aligned} \frac{{\mathrm{d}}\mathcal{S}_\mathrm{fluct}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}&=\frac{1}{2}\int_q\frac{\frac{{\mathrm{d}}}{{\mathrm{d}}\Phi_0^2}\det({\boldsymbol{\mathcal{D}}}_q^{-1})}{\det({\boldsymbol{\mathcal{D}}}_q^{-1})} \\ &= -\frac{1}{2}\int_q\Bigg[\frac{(N-1)\frac{{\mathrm{d}}\Pi^\perp_q}{{\mathrm{d}}\Phi_0^2}}{\frac{1}{V}-\Pi^\perp_q}+\frac{\frac{{\mathrm{d}}\Pi^\parallel_q}{{\mathrm{d}}\Phi_0^2}}{\frac{1}{V}-\Pi^\parallel_q}\Bigg]{~ ,}\nonumber\end{aligned}$$ can alternatively be written as $$\frac{{\mathrm{d}}\mathcal{S}_\mathrm{fluct}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}=-\frac{1}{2}\int_q \Big({\boldsymbol{\mathcal{D}}}_q\cdot\frac{{\mathrm{d}}{\boldsymbol{\Pi}}_q}{{\mathrm{d}}\Phi_0^2}\Big){~ ,}\label{eq:fluctuation-integral}$$ which is represented by the diagram in Fig. \[fig:gap-equation\](b), where the fermionic triangle (also see Fig. \[fig:triangles\]) represents the derivative of the polarization function, which is also evaluated in Appendix \[app:polarization-function\]. As shown in Refs.  and , the integral  is dominated by contributions from the regime where $r=\sqrt{\omega^2+(v_\mathrm{F}|{\boldsymbol{q}}|\cos\theta)^2}/(2\Phi_0)\gg1$ as well as $v_\mathrm{F}|{\boldsymbol{q}}|>\omega$, whereas corrections stemming from long-wavelength fluctuations are negligible. In that regime, the fluctuation propagator is dominated by the normal part of the polarization function, $|1/V-\Pi^\mathrm{n}_q|\gg|\Pi^\mathrm{a}_q|$, while the leading contribution to the derivative of the polarization function stems from the anomalous part, $|{\mathrm{d}}\Pi^\mathrm{a}_q/{\mathrm{d}}\Phi_0^2|\gg|{\mathrm{d}}\Pi^\mathrm{n}_q/{\mathrm{d}}\Phi_0^2|$. Consequently, the crucial correction terms to the gap equation due to fluctuations are given by $$\begin{aligned} \left.\frac{{\mathrm{d}}\mathcal{S}_\mathrm{fluct}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}\right|_{r\gg1}\approx \frac{1}{2}\int_q\frac{[(N-1)-1]\frac{{\mathrm{d}}\Pi_q^\mathrm{a}}{{\mathrm{d}}\Phi_0^2}}{\frac{1}{V}-\Pi_q^\mathrm{n}}{~ .}\end{aligned}$$ Then, depending on the number of transverse modes, i.e., the dimensionality of the order parameter, the fluctuation corrections are either positive ($N<2$), negative ($N>2$), or negligible ($N=2$). The polarization function and its derivatives {#app:polarization-function} ============================================= For the sake of completeness, we summarize the results for the polarization function and its derivatives obtained by previous studies [@KosMillisLarkin-PRB2004; @FischerEtAl2017]. To pinpoint the physical meaning of the individual terms eventually contributing to the gap equation, we introduce normal and anomalous fermionic Green’s functions as $$\begin{aligned} G_k&\equiv G({\boldsymbol{k}},\nu_n)= -\frac{\mathrm{i}\nu_n+\varepsilon_{{\boldsymbol{k}}}}{\varepsilon_{{\boldsymbol{k}}}^2+\nu_n^2+\Phi_0^2} \\ \text{and} \quad F_k&\equiv F({\boldsymbol{k}},\nu_n)= \frac{\Phi_0}{\varepsilon_{{\boldsymbol{k}}}^2+\nu_n^2+\Phi_0^2}{~ ,}\end{aligned}$$ respectively. It holds that $F_k=F_{-k}$, and furthermore, nesting implies that $G({\boldsymbol{k}}+{\boldsymbol{Q}},\nu_n)=-G(-{\boldsymbol{k}},-\nu_n)\equiv -G_{-k}$ as well as $F({\boldsymbol{k}}+{\boldsymbol{Q}},\nu_n)=F({\boldsymbol{k}},\nu_n)\equiv F_k$. Using these definitions, the matrix Green’s functions can be rewritten as $$\begin{aligned} \mathcal{G}_k^\mathrm{CDW}&= 2\begin{pmatrix} G_k\sigma_0 & F_k\sigma_0 \\ F_k\sigma_0 & -G_{-k}\sigma_0 \end{pmatrix} \\ \text{and} \quad \mathcal{G}_k^\mathrm{SDW} &= 2\begin{pmatrix} G_k \sigma_0 & F_k \sigma_3 \\ F_k\sigma_3 & -G_{-k}\sigma_0\end{pmatrix} {~ ,}\end{aligned}$$ which makes the structural congruence with the BCS theory of superconductivity obvious. The normal part of the polarization function is then solely determined by normal Green’s functions. It corresponds to $\Pi^\mathrm{n}_q=\frac{1}{2}(\Pi_{11,q}+\Pi_{22,q})$ in the notation of Ref.  and reads $$\begin{aligned} \Pi^\mathrm{n}_q&= \frac{1}{2}\int_k\big[G_{k+\frac{q}{2}}G_{-(k-\frac{q}{2})}+G_{-(k+\frac{q}{2})}G_{k-\frac{q}{2}}\big] \\ &=\int_k\frac{\nu_+\nu_-+\varepsilon_+\varepsilon_-}{(\varepsilon_+^2+\nu_+^2+\Phi_0^2)(\varepsilon_-^2+\nu_-^2+\Phi_0^2)} \\ &={\rho_{\mathrm{F}}}\ln\bigg(\frac{2 E_\mathrm{F}}{\Phi_0}\bigg)-{\rho_{\mathrm{F}}}\int_\Omega\frac{(2r^2+1)\operatorname{arsinh}(r)}{2r\sqrt{r^2+1}}\label{eq:normal-Pi} {~ ,}\end{aligned}$$ where we abbreviated $\int_k\ldots={\rho_{\mathrm{F}}}\int{\mathrm{d}}\varepsilon\int\frac{{\mathrm{d}}\nu}{2\pi}\int_\Omega\ldots$ and the integration $\int_\Omega\dots =\frac{1}{\Omega_d}\int{\mathrm{d}}\Omega\dots$ refers to the integration over the direction of the fermionic momentum with $\Omega_d$ being the volume of a $d$-dimensional sphere. Since we consider the spectrum to be isotropic, the integrand only depends on the momentum direction via $\theta=\sphericalangle({\boldsymbol{k}},{\boldsymbol{q}})$. Furthermore, we introduced $\nu_\pm=\nu\pm\frac{\omega}{2}$ as well as ${\boldsymbol{k}}_\pm={\boldsymbol{k}}\pm\frac{1}{2}{\boldsymbol{q}}$, and linearized the dispersion $\varepsilon_\pm=\varepsilon\pm \frac{1}{2}v_\mathrm{F}|{\boldsymbol{q}}|\cos\theta$. For the evaluation of the polarization function, it is useful that the integrals discussed here depend on external momenta and frequency only via the combination $$r=\frac{\sqrt{\omega^2+(v_\mathrm{F} |{\boldsymbol{q}}|\cos\theta)^2}}{2\Phi_0}{~ ,}$$ see Refs.  and  for details. Evaluating the above expression at $|{\boldsymbol{q}}|=0$ and $\omega=2\Phi_0r$ then results in the last line, where the angular integration still remains to be done. Unfortunately, this cannot be performed for arbitrary values of $r$, and we resort to approximations in the regimes $r\ll1$ and $r\gg1$, cf. Refs.  and  for details. Analogously, the anomalous part of the polarization function can be expressed as $$\begin{aligned} \Pi^\mathrm{a}_q&= -\int_k F_{k+\frac{q}{2}}F_{k-\frac{q}{2}} \\ &= -\int_k\frac{\Phi_0^2}{(\varepsilon_+^2+\nu_+^2+\Phi_0^2)(\varepsilon_-^2+\nu_-^2+\Phi_0^2)} \\ &= -{\rho_{\mathrm{F}}}\int_\Omega\frac{\operatorname{arsinh}(r)}{2r\sqrt{r^2+1}} \label{eq:anomalous-Pi} {~ .}\end{aligned}$$ In the notation of Ref. , this corresponds to $\Pi^\mathrm{a}_q=\frac{1}{2}(\Pi_{11,q}-\Pi_{22,q})$. This part vanishes in the limit $\Phi_0\rightarrow0$, i.e., in the disordered high-temperature phase. Furthermore, it is obvious from the expressions  and  that $$\left|\tfrac{1}{V}-\Pi_q^\mathrm{n}\right|>\left|\Pi_q^\mathrm{a}\right|$$ holds for arbitrary $r$. The corresponding derivatives can also be expressed in terms of normal and anomalous Green’s function, and the individual terms are represented by the fermionic triangle diagrams shown in Fig. \[fig:triangles\]. All contributions to the derivative of the normal part, $\frac{{\mathrm{d}}\Pi_q^\mathrm{n}}{{\mathrm{d}}\Phi_0^2}=-\frac{1}{2}\Big(\frac{{\mathrm{d}}\mathcal{D}^{-1}_{11,q}}{{\mathrm{d}}\Phi_0^2}+\frac{{\mathrm{d}}\mathcal{D}^{-1}_{22,q}}{{\mathrm{d}}\Phi_0^2}\Big)$, have the structure presented in Fig. \[fig:triangles\](c), and the analytic expression is given by $$\begin{aligned} \frac{{\mathrm{d}}\Pi_q^\mathrm{n}}{{\mathrm{d}}\Phi_0^2}&=-\frac{1}{2\Phi_0}\int_k\big[G_{k+\frac{q}{2}}F_{k+\frac{q}{2}}G_{-(k-\frac{q}{2})}+G_{k+\frac{q}{2}}F_{-(k-\frac{q}{2})}G_{-(k-\frac{q}{2})}+G_{k-\frac{q}{2}}F_{k-\frac{q}{2}}G_{-(k+\frac{q}{2})}+G_{k-\frac{q}{2}}F_{-(k+\frac{q}{2})}G_{-(k+\frac{q}{2})}\big] \\ &=-\frac{{\rho_{\mathrm{F}}}}{4\Phi_0^2}\int_\Omega\bigg[\frac{\operatorname{arsinh}(r)}{r(r^2+1)^{3/2}}+\frac{1}{r^2+1}\bigg]{~ .}\end{aligned}$$ The derivation of $\Pi_q^\mathrm{a}$ w.r.t. $\Phi_0^2$ generates two different types of contributions – the first \[cf. Fig. \[fig:triangles\](a)\] comes from the exchange of fluctuations between quasiparticles while the second \[cf. Fig. \[fig:triangles\](b)\] is solely determined by anomalous propagators, $$\begin{aligned} \frac{{\mathrm{d}}\Pi_q^\mathrm{a}}{{\mathrm{d}}\Phi_0^2}&=-\frac{1}{2\Phi_0}\int_k\big[G_{k+\frac{q}{2}}G_{-(k+\frac{q}{2})}F_{k-\frac{q}{2}}+G_{k-\frac{q}{2}}G_{-(k-\frac{q}{2})}F_{k+\frac{q}{2}}-F_{k+\frac{q}{2}}F_{k+\frac{q}{2}}F_{k-\frac{q}{2}}-F_{k+\frac{q}{2}}F_{k-\frac{q}{2}}F_{k-\frac{q}{2}}\big]\\ &=-\frac{{\rho_{\mathrm{F}}}}{4\Phi_0^2}\int_\Omega\bigg[\frac{(2r^2+1)\operatorname{arsinh}(r)}{r(r^2+1)^{3/2}}-\frac{1}{r^2+1}\bigg]{~ .}\end{aligned}$$ The crucial contribution to $\frac{{\mathrm{d}}\Pi_q^\mathrm{a}}{{\mathrm{d}}\Phi_0^2}=-\frac{1}{2}\Big(\frac{{\mathrm{d}}\mathcal{D}^{-1}_{11,q}}{{\mathrm{d}}\Phi_0^2}-\frac{{\mathrm{d}}\mathcal{D}^{-1}_{22,q}}{{\mathrm{d}}\Phi_0^2}\Big)$ that ultimately generates the additional logarithm contributing to the gap equation stems from the term $\propto\int_\Omega \ln(2r)/r^2$ in ${\mathrm{d}}\Pi_q^\mathrm{a}/{\mathrm{d}}\Phi_0^2$ in the regime $r\gg1$. It can be related to the first term, i.e., the contribution depicted in Fig. \[fig:triangles\](a). The diagrammatic key elements used throughout the paper are introduced in Fig. \[fig:diagrammatic-elements\]. ![Diagrammatic elements. The normal and anomalous propagators are represented by straight lines, and the numbers refer to the matrix structure in the band space emerging as a consequence of the doubling of the unit cell, as introduced in Eq. . Furthermore, two types of vertices appear in the diagrammatic representation of the gap equation: the coupling to the order parameter as well as to the respective fluctuations.[]{data-label="fig:diagrammatic-elements"}](triangle-diagram-vertices){width="\columnwidth"} Effect of channel interference on density-wave instabilities {#app:two-band} ============================================================ In this appendix, we demonstrate how the presence of competing instabilities can affect the transition towards a new low-temperature ordered phase. We use the two-band model of iron-based superconductors as an example, which has been studied in great detail in the recent past [@FernandesChubukov-RepProgPhys2017]. In our brief analysis, we greatly build on the RG analysis of this model as presented in Ref. . We start with a brief summary of their results before using them to analyze the effect of channel interference on density-wave instabilities. The model analyzed in this appendix consists of two nested Fermi pockets: one with a hole-like dispersion centered around ${\boldsymbol{0}}$, and another one with an electron-like dispersion centered around ${\boldsymbol{Q}}$, which we assume to be perfectly nested. The notation used in this appendix then merely differs from the notation used in the main text in that we introduce the band index $j\in\{1,2\}$ and measure momenta ${\boldsymbol{k}}$ as deviations from ${\boldsymbol{0}}$ and ${\boldsymbol{Q}}$, respectively. The noninteracting part \[cf. Eq. \] of the Hamiltonian $\mathcal{H}=\mathcal{H}_0+\mathcal{H}_\mathrm{int}$ then reads $$\mathcal{H}_0= \sum_{j,{\boldsymbol{k}},\sigma}\varepsilon_{j,{\boldsymbol{k}}}\psi^\dagger_{j,{\boldsymbol{k}},\sigma}\psi_{j,{\boldsymbol{k}},\sigma}{~ ,}$$ and the nesting condition takes the form $\varepsilon_{1,{\boldsymbol{k}}}=-\varepsilon_{2,{\boldsymbol{k}}}$. Furthermore, the interaction  translated to band notation is given by $$\begin{aligned} \mathcal{H}_\mathrm{int}&= \frac{1}{2}\sum\psi^\dagger_{j_1,{\boldsymbol{k}}_1,\sigma_1}\psi^\dagger_{j_2,{\boldsymbol{k}}_2,\sigma_2} U^{\sigma_1\sigma_2,\sigma_3\sigma_4}_{j_1j_2,j_3j_4}({\boldsymbol{k}}_1,{\boldsymbol{k}}_2;{\boldsymbol{k}}_3,{\boldsymbol{k}}_4) \nonumber \\ &\qquad \times \psi^{}_{j_3,{\boldsymbol{k}}_3,\sigma_3}\psi^{}_{j_4,{\boldsymbol{k}}_4,\sigma_4}\delta({\boldsymbol{k}}_1+{\boldsymbol{k}}_2-{\boldsymbol{k}}_3-{\boldsymbol{k}}_4){~ ,}\label{eq:H_int}\end{aligned}$$ where the summation is over band indices $j_i$, momenta ${\boldsymbol{k}}_i$, and spins $\sigma_i$. In contrast to the discussion in the main text, we allow for a weak momentum dependence of the couplings in the sense that they still depend on band indices, i.e., on whether the momenta are close to ${\boldsymbol{0}}$ or close to ${\boldsymbol{Q}}$. This results in several coupling constants associated with the different intraband and interband processes. After decomposing the interaction into charge (ch) and spin (sp) channel according to $$\begin{aligned} U^{\sigma_1\sigma_2,\sigma_3\sigma_4}_{j_1j_2,j_3j_4}({\boldsymbol{k}}_1{\boldsymbol{k}}_2;{\boldsymbol{k}}_3,{\boldsymbol{k}}_4)= U_{j_1j_2;j_3j_4}^\mathrm{ch}\delta_{\sigma_1\sigma_4}\delta_{\sigma_2\sigma_3} \nonumber \\ + U_{j_1j_2;j_3j_4}^\mathrm{sp}{\boldsymbol{\sigma}}_{\sigma_1\sigma_4}\cdot{\boldsymbol{\sigma}}_{\sigma_2\sigma_3}{~ ,}\end{aligned}$$ the spin sums can be partially evaluated and the resulting expression for the interaction term contains five independent interaction constants $U_i^{(0)}$, $$\begin{aligned} \mathcal{H}^\prime_\mathrm{int}&= U_1^{(0)}\sum_{{\boldsymbol{k}}{\boldsymbol{k}}^\prime{\boldsymbol{q}}}\psi^\dagger_{1,{\boldsymbol{k}},\sigma}\psi^\dagger_{2,{\boldsymbol{k}}^\prime,\sigma^\prime}\psi^{}_{2,{\boldsymbol{k}}^\prime-{\boldsymbol{q}},\sigma^\prime}\psi^{}_{1,{\boldsymbol{k}}+{\boldsymbol{q}},\sigma} \nonumber \\ &\quad + U_2^{(0)}\sum_{{\boldsymbol{k}}{\boldsymbol{k}}^\prime{\boldsymbol{q}}}\psi^\dagger_{2,{\boldsymbol{k}},\sigma}\psi^\dagger_{1,{\boldsymbol{k}}^\prime,\sigma^\prime}\psi^{}_{2,{\boldsymbol{k}}^\prime-{\boldsymbol{q}},\sigma^\prime}\psi^{}_{1,{\boldsymbol{k}}+{\boldsymbol{q}},\sigma} \nonumber \\ &\quad + \frac{U_3^{(0)}}{2}\sum_{{\boldsymbol{k}}{\boldsymbol{k}}^\prime{\boldsymbol{q}}}\big[\psi^\dagger_{2,{\boldsymbol{k}},\sigma}\psi^\dagger_{2,{\boldsymbol{k}}^\prime,\sigma^\prime}\psi_{1,{\boldsymbol{k}}^\prime-{\boldsymbol{q}},\sigma^\prime}\psi_{1,{\boldsymbol{k}}+{\boldsymbol{q}},\sigma}+\mathrm{H.\,c.}\big] \nonumber \\ &\quad + \frac{U_4^{(0)}}{2}\sum_{{\boldsymbol{k}}{\boldsymbol{k}}^\prime{\boldsymbol{q}}}\psi^\dagger_{2,{\boldsymbol{k}},\sigma}\psi^\dagger_{2,{\boldsymbol{k}}^\prime,\sigma^\prime}\psi^{}_{2,{\boldsymbol{k}}^\prime-{\boldsymbol{q}},\sigma^\prime}\psi^{}_{2,{\boldsymbol{k}}+{\boldsymbol{q}},\sigma} \nonumber \\ &\quad + \frac{U_5^{(0)}}{2}\sum_{{\boldsymbol{k}}{\boldsymbol{k}}^\prime{\boldsymbol{q}}}\psi^\dagger_{1,{\boldsymbol{k}},\sigma}\psi^\dagger_{1,{\boldsymbol{k}}^\prime,\sigma^\prime}\psi^{}_{1,{\boldsymbol{k}}^\prime-{\boldsymbol{q}},\sigma^\prime}\psi^{}_{1,{\boldsymbol{k}}+{\boldsymbol{q}},\sigma}{~ ,}\label{eq:H_int2}\end{aligned}$$ which we labeled in accordance with Ref. . Here the index $0$ indicates that these are the bare couplings. Note that the interband pair-hopping process associated with $U_3^{(0)}$ is only allowed if $2{\boldsymbol{Q}}$ is from the reciprocal lattice, as it is the case for iron-based superconductors. $U_4^{(0)}$ and $U_5^{(0)}$ are the intraband couplings in the two bands, and as we consider the system at perfect nesting, i.e., particle-hole symmetry, it holds that $U_4^{(0)}=U_5^{(0)}$. $U_1^{(0)}$ and $U_2^{(0)}$ refer to interband processes with a momentum transfer of ${\boldsymbol{0}}$ and ${\boldsymbol{Q}}$, respectively. The couplings $U_i^{(0)}$ are connected to the couplings in spin and charge channel by \[eq:couplings-connection\] $$\begin{aligned} U_{11,11}^\mathrm{ch}&= \frac{U_5^{(0)}}{4} {~ ,}& U_{11,11}^\mathrm{sp}&=-\frac{U_5^{(0)}}{4} {~ ,}\\ U_{22,22}^\mathrm{ch}&= \frac{U_4^{(0)}}{4} {~ ,}& U_{22,22}^\mathrm{sp}&=-\frac{U_4^{(0)}}{4} {~ ,}\\ U_{11,22}^\mathrm{ch}&= \frac{U_3^{(0)}}{4} {~ ,}& U_{11,22}^\mathrm{sp}&=-\frac{U_3^{(0)}}{4} {~ ,}\\ U_{12,12}^\mathrm{ch}&= -\frac{U_1^{(0)}}{4}+\frac{U_2^{(0)}}{2} {~ ,}& U_{12,12}^\mathrm{sp}&=-\frac{U_1^{(0)}}{4} {~ ,}\\ U_{12,21}^\mathrm{ch}&= \frac{U_1^{(0)}}{2}-\frac{U_2^{(0)}}{4} {~ ,}& U_{12,21}^\mathrm{sp}&=-\frac{U_2^{(0)}}{4} {~ .}\end{aligned}$$ In the remainder, we will work with dimensionless quantities, and to this end, we introduce dimensionless couplings via $u_i={\rho_{\mathrm{F}}}U_i$, where ${\rho_{\mathrm{F}}}$ is the density of states at the Fermi level. The presence of the interaction  implies three different types of instabilities: towards the formation of charge-density wave (CDW) order and spin-density wave (SDW) order, both with momentum ${\boldsymbol{Q}}$, as well as a superconducting (SC) instability resulting from Cooper pairing either in the conventional $s^{++}$-wave channel or in the sign-changing $s^{+-}$ channel. The couplings $\Gamma$ in the respective channels \[cf. the couplings $V_\mathrm{ch}$ and $V_\mathrm{sp}$ as introduced in Eq.  in the main text\] are given by the combinations \[eq:couplings\] $$\begin{aligned} \Gamma_\mathrm{rSDW}&= u_1+ u_3 {~ ,}& \Gamma_\mathrm{iSDW}&= u_1-u_3 {~ ,}\\ \Gamma_\mathrm{iCDW}&= u_1+u_3-2u_2 {~ ,}& \Gamma_\mathrm{rCDW}&= u_1-u_3-2u_2 {~ ,}\\ \Gamma_{s^{+-}}&=u_4-u_3 {~ ,}& \Gamma_{s^{++}}&= u_4+u_3 {~ .}\end{aligned}$$ Here, the labels r and i for density-wave instabilities refer to density-wave order characterized by a purely real and a purely imaginary order parameter, respectively. What is more, the interaction  possesses an $\mathrm{SO}(6)$ symmetry provided that $u_2^{(0)}=0$ and $u_4^{(0)}=-u_1^{(0)}$, as discussed by Ref. . Then, three out of the six states of emerging order are degenerate in energy – namely rSDW, iCDW, and $s^{+-}$ SC for repulsive interband pair hopping ($u_3^{(0)}>0$), whereas attractive interband pair hopping ($u_3^{(0)}<0$) may result in iSDW, rCDW, and $s^{++}$ SC. Real materials only approximately exhibit this enhanced symmetry, meaning that one of the instabilities wins. In the parent compounds of iron-based superconductors, for instance, spin-density wave order is realized and hence the coupling in the rSDW channel is the one which diverges first upon successively integrating out high-energy modes in a renormalization group (RG) analysis of the model. However, since other candidates for low-energy ordered phases are close in energy, they are competing for the same electrons, implying that phase competition is important in such systems. In the remainder, we analyze the effect of channel interference on the energy scale at which the instability towards density-wave order occurs. The RG flow of the couplings $u_i$ as functions of $t=\log\tfrac{W}{E}$, where $W$ is the bandwidth and $E$ the running energy scale, is governed by the coupled differential equations \[eq:RG-equations\] $$\begin{aligned} \dot{u}_1&= u_1^2+u_3^2 {~ ,}\\ \dot{u}_2&= 2u_2(u_1-u_2) {~ ,}\\ \dot{u}_3&= 2u_3(2u_1-u_2-u_4) {~ ,}\\ \dot{u}_4&= -u_3^2-u_4^2 {~ .}\end{aligned}$$ For the derivation and a detailed discussion, we refer to Ref. . Analogous results have been obtained by Refs.  for a related model without the pair-hopping process. Consequently, the flow of the couplings in the density-wave channels takes the form \[eq:channel-interference\] $$\begin{aligned} \dot{\Gamma}_\mathrm{SDW}&= (\Gamma_\mathrm{SDW})^2\pm 2u_3(u_1-u_2-u_4){~ ,}\\ \dot{\Gamma}_\mathrm{CDW}&= (\Gamma_\mathrm{CDW})^2\mp 2u_3(u_1+u_2-u_4) {~ ,}\end{aligned}$$ where the upper sign refers to rSDW and rCDW, while the lower sign refers to iSDW and iCDW. If the second term were zero, this would result in the usual logarithmic divergence $\Gamma=\Gamma_0/(1-\Gamma_0\log\frac{W}{E})$ of the coupling in the respective channel. Hence the presence of the second term implies corrections due to channel interference, meaning that these effects are intrinsic to the model. They vanish only for $u_3^{(0)}=0$, which we cannot generically assume for the iron-based superconductors as $2{\boldsymbol{Q}}$ is a reciprocal lattice vector here. Let us further note here that the second term combines the effect of competing density-wave and superconducting instabilities, even though we cannot differentiate between the effect of different channels on a given instability within this approach. Motivated by Eq. , we use $u_3^{(0)}$ as a measure of channel-interference strength. We may then analyze whether channel interference is beneficial or detrimental to the formation of a certain type of order by numerically solving the RG equations  for different values of $u_3^{(0)}$. For notational convenience, let us concentrate on the parameter range appropriate to describe the physics of iron-based superconductors here, i.e., repulsive interband pair hopping $u_3^{(0)}>0$ leading to the competition of rSDW order with iCDW order and $s^{+-}$ SC. We note here that the same trends are found mutatis mutandis for attractive $u_3^{(0)}<0$, i.e., for competing iSDW, rCDW, and $s^{++}$ SC instabilities. In order to analyze the effect of channel interference, we compare the flow of the coupling in a given channel for fixed $\Gamma^{(0)}$ upon varying $u_3^{(0)}$, which constitutes a measure of channel interference strength. Here, the bare parameters $u_i^{(0)}$ are chosen such that the ordering in the channel under consideration is favorable within the mean-field description, that is, $\Gamma^{(0)}\neq0$, while the bare couplings in the competing channels are tuned to zero. Although the couplings in the competing channels grow with the flow and are relevant as well, the leading instability remains the same as long as the channel interference does not change which of the couplings diverges first. However, the energy scale at which the couplings diverge turns out to be affected by the bare value of the interband pair hopping $u_3^{(0)}$, which can be used as a sign indicating whether channel interference promotes or hinders the formation of order in a given channel. In Fig. \[fig:channel-interference\](a), we exemplarily show the flow of $\Gamma_\mathrm{rSDW}$ obtained from solving Eqs.  for fixed bare interaction $\Gamma_\mathrm{rSDW}^{(0)}=0.5$. Upon increasing the channel interference strength via increasing $u_3^{(0)}$, the energy scale at which the interaction in the rSDW channel diverges is pushed to lower energies, that is, happens at higher transition temperatures. As a result, we find that channel interference is beneficial for the formation of SDW order. On the other hand, the energy scale at which the interaction $\Gamma_\mathrm{iCDW}$ in the iCDW channel diverges is pushed to higher energies upon increasing the channel interference strength $u_3^{(0)}$ while keeping $\Gamma_\mathrm{iCDW}^{(0)}=0.5$ fixed. Therefore, channel interference is detrimental to the formation of CDW order as the phase transition now happens at lower temperatures, as illustrated in Fig. \[fig:channel-interference\](b). 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--- abstract: 'In this paper we introduce a general fault-tolerant quantum error correction protocol using flag circuits for measuring stabilizers of arbitrary distance codes. In addition to extending flag error correction beyond distance-three codes for the first time, our protocol also applies to a broader class of distance-three codes than was previously known. Flag circuits use extra ancilla qubits to signal when errors resulting from $v$ faults in the circuit have weight greater than $v$. The flag error correction protocol is applicable to stabilizer codes of arbitrary distance which satisfy a set of conditions and uses fewer qubits than other schemes such as Shor, Steane and Knill error correction. We give examples of infinite code families which satisfy these conditions and analyze the behaviour of distance-three and -five examples numerically. Requiring fewer resources than Shor error correction, flag error correction could potentially be used in low-overhead fault-tolerant error correction protocols using low density parity check quantum codes of large code length.' author: - Christopher Chamberland - 'Michael E. Beverland' bibliography: - 'bibtex\_chamberland.bib' title: 'Flag fault-tolerant error correction with arbitrary distance codes' --- Introduction and formalism {#sec:Intro} ========================== Scalable quantum computers are expected to require some form of error correction (EC) to function reliably. Unfortunately, no practical model for a self-correcting quantum memory has been proposed to date, despite considerable effort [@BrownMemory16]. The models that come closest to this goal involve topological protection in the presence of physically imposed symmetries [@KitaevWire01; @KarzigScalable17], but even these are not expected to reduce error rates sufficiently for large computations. Therefore active protocols that require measuring the check operators of an error correcting code are probably necessary to realize scalable quantum computing. There are three general approaches of fault-tolerant error correction (FTEC) applicable to a wide range of stabilizer codes due to Shor [@Shor96], Steane [@Steane97], and Knill [@KL2005]. There are also a number of promising code-specific FTEC schemes, most notably the surface code with a minimum weight matching error correction scheme [@BK98; @DKLP02; @FMMC12]. This approach gives the best fault-tolerant thresholds to date and only requires geometrically local measurements. A high threshold [@Shor96; @AB97; @Preskill98; @KLZ98] implies that relatively imperfect hardware could be used to reliably implement long quantum computations. Despite this, the hardware and overhead requirements for the surface code are sufficiently demanding that it remains extremely challenging to implement in the lab. Fortunately, there are reasons to believe that there could be better alternatives to the surface code. For example, dramatically improved thresholds could be possible using concatenated codes if they enjoyed the same level of optimization as the surface code has in recent years [@Poulin06; @PhysRevA.83.020302]. Another enticing alternative is to find and use efficiently-decodable low density parity check (LDPC) codes with high rate [@Gallager1960; @LDPC13; @TZLDPC14] in a low-overhead FTEC protocol [@Gottesman13LDPC]. For these and other reasons, it is important to have general FTEC schemes applicable to a wide range of codes and to develop new schemes. Shor EC can be applied to any stabilizer code, but typically requires more syndrome measurement repetitions than Steane and Knill EC. Furthermore, all weight-$w$ stabilizer generators are measured sequentially using $w$-qubit verified cat states. On the other hand, Steane EC has higher thresholds than Shor EC and has the advantage that all Clifford gates are applied transversally during the protocol. However, Steane EC is only applicable to CSS [@CS96; @Steane97] codes and uses a verified logical ${|+\rangle}$ state encoded in the same code to simultaneously obtain all $X$-type syndromes, using transversal measurement (similarly for $Z$). Knill EC can also be applied to any stabilizer code but requires two additional ancilla code blocks (encoded in the same code that protects the data) prepared in a logical Bell state. The Bell state teleports the encoded information to one of the ancilla code blocks and the extra information from the transversal Bell measurement gives the error syndrome. Knill EC typically achieves higher thresholds than Shor and Steane EC but often uses more qubits [@Knill05; @Fern08KnillUpperBound]. It is noteworthy that for large LDPC codes, in which low weight generators are required be fault-tolerantly measured, Shor EC is much more favourable than Steane or Knill EC. Many improvements in these schemes have been made. For examples, in [@DA07], ancilla decoding was introduced to correct errors arising during state preparation in Shor and Steane EC rather than simply rejecting all states which fail the verification procedure. In this work, we build on a number of recent papers [@CR17v1; @CR17v2; @Yoder2017surfacecodetwist] that demonstrate flag error correction for particular distance-three and error detecting codes and present a general protocol for arbitrary distance codes. Flag error correction uses extra ancilla qubits to detect potentially problematic high weight errors that arise during the measurement of a stabilizer. We provide a set of requirements for a stabilizer code (along with the circuits used to measure the stabilizers) which, if satisfied, can be used for flag error correction. We are primarily concerned with extending the lifetime of encoded information using fault-tolerant error correction and defer the study of implementing gates fault-tolerantly to future work. Our approach can be applied to a broad class of codes (including but not limited to surface codes, color codes and quantum Reed-Muller codes). Of the three general schemes described above, flag EC has most in common with Shor EC. Further, flag EC does not require verified state preparation, and for all codes considered to date, requires fewer ancilla qubits. Lastly, we note that in order to satisfy the fault-tolerant error correction definition presented in \[subsec:Section0\], our protocol applied to distance-three codes differs from [@CR17v1]. We foresee a number of potential applications of these results. Firstly we believe it is advantageous to have new EC schemes with different properties that can be used in various settings. Secondly, flag EC involves small qubit overhead, hence possibly the schemes presented here and in other flag approaches [@CR17v1; @CR17v2; @Yoder2017surfacecodetwist] will find applications in early qubit-limited experiments. Thirdly, we expect the flag EC protocol presented here could potentially be useful for LDPC codes as described in [@Gottesman13LDPC]. In \[subsec:ReviewChaoReichardt,subsec:Distance5protocol\] we provide important definitions and introduce flag FTEC for distance-three and -five codes. In \[subsec:ApplicationProtocolColorCode\] we apply the protocol to two examples: the [$[\![19,1,5]\!]$]{} and [$[\![17,1,5]\!]$]{} color codes, which importantly have a variety of different weight stabilizers. The general flag FTEC protocol for arbitrary distance codes is given in \[subsec:GeneralProtocol\]. A proof that the general protocol satisfies the fault-tolerance criteria is given in \[app:ProtocolGeneralProof\]. In \[subsec:Remarks\] we provide examples of codes that satisfy the conditions that we required for flag FTEC. Flag circuit constructions for measuring stabilizers of the codes in \[subsec:Remarks\] are given \[app:GeneralTflaggedCircuitConstruction\]. We also provide a candidate circuit construction for measuring arbitrary weight stabilizers in \[App:GeneralwFlagCircuitConstruction\]. In \[sec:CircuitLevelNoiseFTEC\], we analyze numerically a number of flag EC schemes and compare with other FTEC schemes under various types of circuit level noise. We find that flag EC schemes, which have large numbers of idle qubit locations, behave best in error models in which idle qubit errors occur with a lower probability than CNOT errors. The remainder of this section is devoted to FTEC and noise model/simulation methods. Fault-tolerant error correction {#subsec:Section0} ------------------------------- Throughout this paper, we assume a simple depolarizing noise model in which idle qubits fail with probability $\tilde{p}$ and all other circuit operations (gates, preparations and measurements) fail with probability $p$, which recovers standard circuit noise when $\tilde{p}=p$. A detailed description is given in \[subsec:NoiseAndNumerics\]. The weight of a Pauli operator $E$ ($\text{wt}(E)$) is the number of qubits on which it has non-trivial support. We first make some definitions, $$\begin{aligned} \mathcal{E}_{t} = \{ E \in \mathcal{P}_{n} | \text{wt}(E) \le t \}, \end{aligned}$$ where $\mathcal{P}_{n}$ is the $n$-qubit Pauli group. \[Def:EpsilontSet\] Given a stabilizer group $\mathcal{S} = \langle g_{1}, \cdots, g_{m} \rangle$, we define the syndrome $s(E)$ to be a bit string, with i’th bit equal to zero if $g_i$ and $E$ commute, and one otherwise. Let $E_{\text{min}}(s)$ be a minimal weight correction $E$ where $s(E)=s$. We say operators $E$ and $E'$ are logically equivalent, written as $E \sim E'$, iff $E' \propto g E$ for $g \in \mathcal{S}$. \[Def:LogEquivDef\] An error correction protocol typically consists of a sequence of basic operations to infer syndrome measurements of a stabilizer code $C$, followed by the application of a Pauli operator (either directly or through Pauli frame tracking [@DA07; @Barbara15; @CIP17]) intended to correct errors in the system. Roughly speaking, a given protocol is fault-tolerant if for sufficiently weak noise, the effective noise on the logical qubits is even weaker. More precisely, we say that an error correction protocol is a $t$-FTEC if the following is satisfied: For $t = \lfloor (d-1)/2\rfloor$, an error correction protocol using a distance-$d$ stabilizer code $C$ is $t$-fault-tolerant if the following two conditions are satisfied: 1. For an input codeword with error of weight $s_{1}$, if $s_{2}$ faults occur during the protocol with $s_{1} + s_{2} \le t$, ideally decoding the output state gives the same codeword as ideally decoding the input state. 2. For $s$ faults during the protocol with $s \le t$, no matter how many errors are present in the input state, the output state differs from a codeword by an error of at most weight $s$. \[Def:FaultTolerantDef\] Here ideally decoding is equivalent to performing fault-free error correction. By codeword, we mean any state ${|\overline{\psi}\rangle} \in C$ such that $g{|\overline{\psi}\rangle} = {|\overline{\psi}\rangle} \thinspace \forall \thinspace g \in \mathcal{S}$ where $\mathcal{S}$ is the stabilizer group for the code $C$. Note that for the second criteria in \[Def:FaultTolerantDef\], the output and input codeword can differ by a logical operator. The first criteria in \[Def:FaultTolerantDef\] ensures that correctable errors don’t spread to uncorrectable errors during the error correction protocol. Note however that the first condition alone isn’t sufficient. For instance, the trivial protocol where no correction is ever applied at the end of the EC round also satisfies the first condition, but clearly is not fault-tolerant. The second condition is not always checked for protocols in the literature, but it is important as it ensures that errors do not accumulate uncontrollably in consecutive rounds of error correction (see [@AGP06] for a rigorous proof and [@CDT09] for an analysis of the role of input errors in an extended rectangle). To give further motivation as to why the second condition is important, consider a scenario with $s$ faults introduced during each round of error correction, and assume that $t/n<s<(2t+1)/3$ for some integer $n$ (see Fig. \[fig:ConditionTwoJustification\]). Consider an error correction protocol in which $r$ input errors and $s$ faults in an EC block leads to an output state with at most $r+s$ errors[^1]. Clearly condition 1 is satisfied. With the above considerations, an input state $E_{1}{|\bar{\psi}\rangle}$ with $\text{wt}(E_{1})\leq s$ is taken to $E_{2}{|\bar{\psi}\rangle}$, with $\text{wt}(E_{2})\leq 2s$ by one error correction round with $s$ faults. After the $j$th round, the state will be $E_{j}{|\bar{\psi}\rangle}$ with the first condition implying $\text{wt}(E_{j})\leq j \cdot s$ provided that $j \leq n$. However, when $j > n$, the requirement of the first condition is no longer satisfied so we cannot use it to upper bound $\text{wt}(E_{j})$. Now consider the same scenario but assuming both conditions hold. The second condition implies that after the first round, the input state $E_{1}{|\bar{\psi}\rangle}$ becomes $E'_{2}{|\bar{\phi}\rangle} = E_{2}{|\bar{\psi}\rangle}$, with $\text{wt}(E_{2}')\leq s$, and where ${|\bar{\phi}\rangle}$ is a codeword. Therefore the codewords are related by: ${|\bar{\phi}\rangle}= (E_{2}^{'\dagger} E_{2}) {|\bar{\psi}\rangle}$, with logical operator $(E_{2}^{'\dagger} E_{2})$ having weight at most $3s$, since $\text{wt}(E_{2})+\text{wt}(E_{2}') \leq 3s$. However, the minimum non-trivial logical operator of the code has weight $(2t+1)>3s$, implying that ${|\bar{\psi}\rangle} = {|\bar{\phi}\rangle}$, and therefore that $\text{wt}(E_{2}) = \text{wt}(E_{2}') \leq s$. Hence, for the $j$th round, $\text{wt}(E_{j}) \leq s$ for all $j$, i.e. the distance from the codeword is not increased by consecutive error correction rounds with $s$ faults, provided $s < (2t+1)/3$. ![An example showing the first fault tolerance condition alone in \[Def:FaultTolerantDef\] is not sufficient to imply a long lifetime. We represent $s$ faults occurring during a round of error correction with a vertical arrow, and a state a distance $r$ from the desired codeword with a horizontal arrow with $r$ above. The first condition alone allows errors to build up over time as in the top figure, which would quickly lead to a failure. However provided $s<(2t+1)/3$, both conditions together ensure that errors in consecutive error correction rounds do not build up, provided each error correction round introduces no more than $s$ faults, which could remain true for a long time.[]{data-label="fig:ConditionTwoJustification"}](ConditionTwoJustification.png){width="45.00000%"} Noise model and pseudo-threshold calculations {#subsec:NoiseAndNumerics} --------------------------------------------- In \[sec:CircuitLevelNoiseFTEC\], we perform a full circuit level noise analysis of various error correction protocols. Unless otherwise stated, we use the following depolarizing noise model: 1. With probability $p$, each two-qubit gate is followed by a two-qubit Pauli error drawn uniformly and independently from $\{I,X,Y,Z\}^{\otimes 2}\setminus \{I\otimes I\}$. 2. With probability $\frac{2p}{3}$, the preparation of the ${|0\rangle}$ state is replaced by ${|1\rangle}=X{|0\rangle}$. Similarly, with probability $\frac{2p}{3}$, the preparation of the ${|+\rangle}$ state is replaced by ${|-\rangle}=Z{|+\rangle}$. 3. With probability $\frac{2p}{3}$, any single qubit measurement has its outcome flipped. 4. Lastly, with probability $\tilde{p}$, each resting qubit location is followed by a Pauli error drawn uniformly and independently from $\{ X,Y,Z \}$. Some error correction schemes that we analyze contain a significant number of idle qubit locations. Consequently, most schemes will be analyzed using three ratios ($\tilde{p} = p$, $\tilde{p} = p/10$ and $\tilde{p} = p/100$) to highlight the impact of idle qubit locations on the logical failure rate. The two-qubit gates we consider are: CNOT, XNOT$ = H_1(\text{CNOT})H_1$, and CZ$ = H_2(\text{CNOT})H_2$. Logical failure rates are estimated using an $N$-run Monte Carlo simulation. During a particular run, errors are added at each location following the noise model described above. Once the error locations are fixed, the errors are propagated through a fault-tolerant error correction circuit and a recovery operation is applied. After performing a correction, the output is ideally decoded to verify if a logical fault occurred. For an error correction protocol implemented using a stabilizer code $C$ and a fixed value of $p$, we define the logical failure rate $$p_{\text{L}}^{(C)}(p) = \lim_{N \to\infty} \frac{N_{\text{fail}}^{(C)}(p)}{N} ,$$ where $N_{\text{fail}}^{(C)}(p)$ is the number of times a logical $X$ *or* logical $Z$ error occurred during the $N$ rounds. In practice we take $N$ sufficiently large to estimate $p_{\text{L}}^{(C)}(p)$, and provide error bars [@AliferisCross07; @CJL16b]. In this paper we are concerned with evaluating the performance of FTEC protocols (i.e. we do not consider performing logical gates fault-tolerantly). We define the pseudo-threshold of an error correction protocol to be the value of $p$ such that $$\begin{aligned} \tilde{p}(p) = p^{(C)}_{L}(p). \label{Def:PseudoThreshDef}\end{aligned}$$ Note that it is important to have $\tilde{p}$ on the left of \[Def:PseudoThreshDef\] instead of $p$ since we want an encoded qubit to have a lower logical failure rate than an unencoded idle qubit. From the above noise model, a resting qubit will fail with probability $\tilde{p}$. Flag error correction for small distance codes {#sec:Section1} ============================================== In this and the next section, we present a $t$-fault-tolerant flag error correction protocol with distance-$(2t+1)$ codes satisfying a certain condition. Our approach extends that introduced by Chao and Reichardt [@CR17v1] for distance three codes, which we first review using our terminology in \[subsec:ReviewChaoReichardt\]. In \[subsec:Distance5protocol\] we present the protocol for distance five CSS codes which contains most of the main ideas of the general case (which is provided in \[app:GeneralFTEC\]). Lastly, in section \[subsec:ApplicationProtocolColorCode\] we provide examples of how the protocol is applied to the [$[\![19,1,5]\!]$]{} and [$[\![17,1,5]\!]$]{} color codes. Definitions and Flag $1$-FTEC with distance-3 codes {#subsec:ReviewChaoReichardt} --------------------------------------------------- In what follows, we use the term location to refer to a gate, state preparation, measurement or idle qubit where a fault may occur. Note also that a two-qubit Pauli error $P_{1}\otimes P_{2}$ arising at a two-qubit gate location counts as a single fault. It is well known that with only a single measurement ancilla, a single fault in a blue CNOT of the stabilizer measurement circuit shown in \[fig:StabNonFT\] can result in a multi-weight error on the data block. This could cause a distance-$3$ code to fail, or more generally could cause a distance-$d$ code to fail due to fewer than $(d-1)/2$ total faults. We therefore say the blue CNOTs are *bad* according to the following definition: A circuit location in which a single fault can result in a Pauli error $E$ on the data block with $\mathrm{wt}(E) \ge 2$ will be referred to as a bad location. \[Def:BadErrorDef\] [0.25]{} ![Circuits for measuring the operator $ZZZZ$ (can be converted to any Pauli by single qubit Cliffords). (a) Non-fault-tolerant circuit. A single fault $IZ$ occurring on the third CNOT (from the left) results in the error $IIZZ$ on the data block. (b) Flag version of \[fig:StabNonFT\]. An ancilla (flag) qubit prepared in ${|+\rangle}$ and two extra CNOT gates signals when a weight two data error is caused by a single fault. Subsequent rounds of error correction may identify which error occurred. Consider an $IZ$ error on the second CNOT, in the non-flag circuit this would result in a weight two error, but in this case, this fault causes the circuit to flag. (c) An alternative flag circuit with lower depth than (b). All bad locations are illustrated in blue.[]{data-label="fig:ErrorPropSteane"}](StabNonFT.png "fig:"){width="\textwidth"} [0.3]{} ![Circuits for measuring the operator $ZZZZ$ (can be converted to any Pauli by single qubit Cliffords). (a) Non-fault-tolerant circuit. A single fault $IZ$ occurring on the third CNOT (from the left) results in the error $IIZZ$ on the data block. (b) Flag version of \[fig:StabNonFT\]. An ancilla (flag) qubit prepared in ${|+\rangle}$ and two extra CNOT gates signals when a weight two data error is caused by a single fault. Subsequent rounds of error correction may identify which error occurred. Consider an $IZ$ error on the second CNOT, in the non-flag circuit this would result in a weight two error, but in this case, this fault causes the circuit to flag. (c) An alternative flag circuit with lower depth than (b). All bad locations are illustrated in blue.[]{data-label="fig:ErrorPropSteane"}](Weight4Generator.png "fig:"){width="\textwidth"} [0.25]{} ![Circuits for measuring the operator $ZZZZ$ (can be converted to any Pauli by single qubit Cliffords). (a) Non-fault-tolerant circuit. A single fault $IZ$ occurring on the third CNOT (from the left) results in the error $IIZZ$ on the data block. (b) Flag version of \[fig:StabNonFT\]. An ancilla (flag) qubit prepared in ${|+\rangle}$ and two extra CNOT gates signals when a weight two data error is caused by a single fault. Subsequent rounds of error correction may identify which error occurred. Consider an $IZ$ error on the second CNOT, in the non-flag circuit this would result in a weight two error, but in this case, this fault causes the circuit to flag. (c) An alternative flag circuit with lower depth than (b). All bad locations are illustrated in blue.[]{data-label="fig:ErrorPropSteane"}](StabFTwithAncillaV2.png "fig:"){width="\textwidth"} As shown in \[fig:StabFTwithAncilla\], the circuit can be modified by including an additional ancilla (flag) qubit, and two extra CNOT gates. This modification leaves the bad locations and the fault-free action of the circuit unchanged. However, any single fault leading to an error $E$ with $\mathrm{wt}(E) \ge 2$ will also cause the measurement outcome of the flag qubit to flip [@CR17v1]. The following definitions will be useful: Consider a circuit for measuring a stabilizer generator that includes at least one flag ancilla. The ancilla used to infer the stabilizer outcome is referred to as the *measurement qubit*. We say the circuit has flagged if the eigenvalue of a flag qubit is measured as $-1$. If the eigenvalue of a measurement qubit is measured as $-1$, we will say that the measurement qubit flipped. \[Def:GlagMeasureQubitsDef\] The purpose of flag qubits is to signal when high weight data qubit errors result from few fault locations during a stabilizer measurement. Two key definitions are: A circuit[^2] $C(P)$ which, when fault-free, implements a projective measurement of a weight-$w$ Pauli $P$ without flagging is a $t$-flag circuit if the following holds: For any set of $v$ faults at up to $t$ locations in $C(P)$ resulting in an error $E$ with $\text{min}(\text{wt}(E),\text{wt}(E P)) > v$, the circuit flags. \[Def:tFlaggedCircuitDef\] Note that a $t$-flag circuit for measuring a weight-$t$ stabilizer $P$ is also a $k$-flag circuit for any $k>t$. In \[app:GeneralTflaggedCircuitConstruction\] we give constructions for some $t$-flag circuits. Let $\mathcal{E}(g_{i})$ be the set of all errors caused by one fault which caused the circuit $C(g_i)$ to flag. \[Def:FlagErrSetDef1\] Note that the flag error set can contain the identity as well as weight one errors. Suppose all errors in a flag error set $\mathcal{E}(g)$ for a 1-flag circuit $C(g)$ have distinct syndromes. As $C(g)$ is a 1-flag circuit, a single fault that leads to an error of weight greater than one will cause the circuit $C(g)$ to flag. Moreover, when a flag has occurred due to at most one fault, a complete set of fault-free stabilizer measurements will infer the resulting element of the flag error set which has been applied to the data qubits. In fact, one would only require distinct syndromes for errors in the flag error set that are logically inequivalent, as defined in \[Def:LogEquivDef\]. As an example, consider the 1-flag circuit in \[fig:StabFTwithAncilla\]. A single fault at any of the blue CNOT gates can lead to an error $E_{b}$ with $\text{wt}(E_{b}) \le 2$ on the data. The set $\mathcal{E}(Z^{\otimes 4})$ contains all errors $E_{b}$ which resulted from a fault at a blue CNOT gate causing the circuit $C(Z^{\otimes 4})$ of \[fig:StabFTwithAncilla\] to flag, i.e., $\mathcal{E}(g) = \{ I,Z_{q_{3}}Z_{q_{4}},X_{q_{2}}Z_{q_{3}}Z_{q_{4}},Z_{q_{1}}X_{q_{2}},Z_{q_{4}},$ $X_{q_{3}}Z_{q_{4}},Y_{q_{3}}Z_{q_{4}} \}$ with qubits $q_1$ to $q_4$. With the above definitions, we can construct a fault-tolerant flag error correction protocol for $d=3$ stabilizer codes satisfying the following condition. Consider a stabilizer code $\mathcal{S} = \langle g_{1},g_{2},\cdots , g_{r} \rangle$ and $1$-flag circuits $\{ C(g_{1}),C(g_{2}), \cdots , C(g_{r}) \}$. For every generator $g_{i}$, all pairs of elements $E,E'\in \mathcal{E}(g_{i})$ satisfy $s(E)\neq s(E')$ or $E \sim E'$. \[Def:Flag1FTECcondition\] In other words, we require that any two errors that arise when a circuit $C(g_{i})$ flags due to a single fault must be either distinguishable or logically equivalent. For the following protocol to satisfy the FTEC conditions in \[Def:FaultTolerantDef\], one can assume there is at most 1 fault. If the Flag $1$-FTEC condition is satisfied, the protocol is implemented as follows: ![Tree diagram illustrating the possible paths of the Flag $1$-FTEC Protocol. Numbers enclosed in red circles at the end of the edges indicate which step to implement in the Flag $1$-FTEC Protocol. A dashed line is followed when any of the 1-flag circuits $C(g_{i})$ flags. Solid squares indicate a syndrome measurement using 1-flag circuits whereas rings indicate a decision based on syndrome outcomes. Note that the syndrome measurement is repeated at most three times.[]{data-label="fig:TreeD3Diag"}](TreeD3.png){width="50.00000%"} [0.45]{}  after rotating the lattice such that the relevant face is on the bottom left. (b) For $g = Z_{q_{1}}Z_{q_{2}}Z_{q_{3}}Z_{q_{4}}$, the flag error set is $\mathcal{E}(g) = \{ I,Z_{q_{3}}Z_{q_{4}},X_{q_{2}}Z_{q_{3}}Z_{q_{4}},Z_{q_{1}}X_{q_{2}},Z_{q_{4}},$ $X_{q_{3}}Z_{q_{4}},X_{q_{3}}Z_{q_{3}}Z_{q_{4}} \}$ which contains all errors arising from a single fault that causes the stabilizer measurement circuit $C(g)$ to flag. Since the Steane code is a CSS code, the $X$ component of an error will be corrected independently allowing us to consider the $Z$-part of the flag error set $\mathcal{E}_Z(g)=\{I,Z_{q_1},Z_{q_4},Z_{q_3}Z_{q_4} \}$. As required, the elements of $\mathcal{E}_Z(g)$ all have distinct syndromes (with satisfied stabilizers represented by a plus). []{data-label="fig:Steane"}](SteaneCodeColor.png "fig:"){width="\textwidth"} [0.45]{}  after rotating the lattice such that the relevant face is on the bottom left. (b) For $g = Z_{q_{1}}Z_{q_{2}}Z_{q_{3}}Z_{q_{4}}$, the flag error set is $\mathcal{E}(g) = \{ I,Z_{q_{3}}Z_{q_{4}},X_{q_{2}}Z_{q_{3}}Z_{q_{4}},Z_{q_{1}}X_{q_{2}},Z_{q_{4}},$ $X_{q_{3}}Z_{q_{4}},X_{q_{3}}Z_{q_{3}}Z_{q_{4}} \}$ which contains all errors arising from a single fault that causes the stabilizer measurement circuit $C(g)$ to flag. Since the Steane code is a CSS code, the $X$ component of an error will be corrected independently allowing us to consider the $Z$-part of the flag error set $\mathcal{E}_Z(g)=\{I,Z_{q_1},Z_{q_4},Z_{q_3}Z_{q_4} \}$. As required, the elements of $\mathcal{E}_Z(g)$ all have distinct syndromes (with satisfied stabilizers represented by a plus). []{data-label="fig:Steane"}](SteaneConditionSatisfied.png "fig:"){width="\textwidth"} A tree diagram for the flag $1$-FTEC Protocol is illustrated in \[fig:TreeD3Diag\]. We now outline the proof that the flag 1-FTEC protocol satisfies the fault-tolerance criteria of \[Def:FaultTolerantDef\] (a more rigorous proof of the general case is presented in \[app:ProtocolGeneralProof\]). To show that Flag $1$-FTEC Protocol satisfies the criteria of \[Def:FaultTolerantDef\], we can assume there is at most one fault during the protocol. If a single fault occurs in either the first or second round leading to a flag, repeating the syndrome measurement will correctly diagnose the error. If there are no flags and a fault occurs which causes the syndromes in the first two rounds to change, then the syndrome during the third round will correctly diagnose the error. There could also be a fault during either the first or second round that goes undetected. But since there were no flags it cannot spread to an error of weight-2. In this case applying a minimum weight correction based on the measured syndrome of the second round will guarantee that the output codeword differs from a valid codeword by an error of weight at most one. Note that the above argument applies irrespective of any errors on the input state, hence the second criteria of \[Def:FaultTolerantDef\] is satisfied. It is worth pointing out that up to three repetitions are required in order to guarantee that the second criteria of \[Def:FaultTolerantDef\] is satisfied (unless the code has the property that all states are at most a weight-one error away from a valid codeword, as in [@CR17v1]). The Steane code is an example which satisfies the Flag $1$-FTEC condition with a simple choice of circuits. To verify this, the representation of the Steane code given in \[fig:SteaneColor1\] is useful. There is an $X$- and a $Z$-type stabilizer generator supported on the four qubits of each of the three faces. First let us specify all six stabilizer measurement circuits. The circuit that measures $Z_{q_1}Z_{q_2}Z_{q_3} Z_{q_4}$ is specified by taking qubits $q_1$, $q_2$, $q_3$, and $q_4$ to be the four data qubits in descending order in the 1-flag circuit in \[fig:StabFTwithAncilla\]. The other two $Z$-stabilizer measurement circuits are obtained by first rotating \[fig:SteaneColor1\] by $120^{\circ}$ and $240^{\circ}$ and then using \[fig:StabFTwithAncilla\]. The $X$-stabilizer circuit for each face is the same as the $Z$-stabilizer circuit for that face, replacing CNOT gates acting on data qubits by XNOT gates. The $Z$ component of the flag error set of the circuit in \[fig:StabFTwithAncilla\] is $\mathcal{E}_Z(Z_{q_1}Z_{q_2}Z_{q_3}Z_{q_4}) = \{ I,Z_{q_1},Z_{q_4},Z_{q_3}Z_{q_4} \}$. As can be seen from \[fig:SteaneColor1\], each of these has a distinct syndrome, thus the measurement circuit for $Z_{q_1}Z_{q_2}Z_{q_3} Z_{q_4}$ satisfies the flag $1$-FTEC condition, as do the remaining five measurement circuits by symmetry. Flag $2$-FTEC with distance-5 codes {#subsec:Distance5protocol} ----------------------------------- Before explicitly describing the conditions and protocol, we discuss some of the complications that arise for codes with $d>3$. For distance-5 codes, we must ensure that if two faults occur during the error correction protocol, the output state will differ from a codeword by an error of at most weight-two. For instance, if two faults occur in a circuit for measuring a stabilizer of weight greater than four, the resulting error $E$ on the data should satisfy $\text{wt}(E) \le 2$ unless there is a flag. In other words, all stabilizer generators should be measured using 2-flag circuits. In another case, two faults could occur during the measurement of *different* stabilizer generators $g_{i}$ and $g_{j}$. If two bad locations fail and are both flagged, and assuming there are no more faults, the measured syndrome will correspond to the product of the error caused in each circuit (which could have weight greater than two). Consequently, one should modify \[Def:FlagErrSetDef1\] of the flag error set to include these types of errors. One then decodes based on the pair of errors that resulted in the measured syndrome, provided logically inequivalent errors have distinct syndromes. Before stating the protocol, we extend some definitions from \[subsec:ReviewChaoReichardt\]. Consider a stabilizer code $\mathcal{S} = \langle g_{1},g_{2},\cdots , g_{r} \rangle$ and $t$-flag circuits $C(g_{i})$ for measuring the generator $g_{i}$. Let $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}})$ be the set of all errors caused by precisely $m$ faults spread amongst the circuits $C(g_{i_{1}}),C(g_{i_{2}}), \cdots , C(g_{i_{k}})$ which all flagged. \[Def:FlagErrSetDef\] Note that there could be more than one fault in a single circuit $C(g_{i_{k}})$. Examples of flag error sets are given in \[tab:PossibleCorrelatedErrors\] where only contributions from $Z$ errors are included (since the considered code is a CSS code). We also define a general $t$-fault correction set: $$\begin{aligned} \tilde{E}_{t}^{m}(g_{i_{1}},\cdots , g_{i_{k}},s) = \begin{cases} \{ E \in \mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m} \\ \text{ such that } s(E) = s \} \\ \{ E_{\text{min}}(s) \} \text{ if above set empty. } \end{cases} \label{eq:GeneralLookupTable}\end{aligned}$$ By $E \in \mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m}$, we are considering the set consisting of products between errors caused by $k$ flags and any error of weight $t-m$. As will be seen below, the correction set will form a critical part of the protocol by specifying the correction applied based on the measured syndrome and flag outcomes over multiple syndrome measurement rounds. In the case where $k$ $t$-flag circuits flagged caused by $k \le m \le t$ faults, the correction applied to the data block will correspond to an element of $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m}$ if the measured syndrome corresponds to an element in this set (there could also be $t-m$ faults which did not give rise to a flag). However in practice, there could be more than $t$ faults and so the measured syndrome may not be consistent with any element of the set $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m}$. In this case, and for the error correction protocol to satisfy the second criteria of \[Def:FaultTolerantDef\], the correction will correspond to $E_{\text{min}}(s)$. These features are all included in the set $\tilde{E}_{t}^{m}(g_{i_{1}},\cdots , g_{i_{k}},s)$. Consider a stabilizer code $\mathcal{S} = \langle g_{1},g_{2},\cdots , g_{r} \rangle$ and $2$-flag circuits $\{ C(g_{1}),C(g_{2}), \cdots , C(g_{r}) \}$. For any choice of generators $\{ g_{i}, g_{j} \}$: 1. $E,E' \in \mathcal{E}_{2}(g_{i},g_{j}) \Rightarrow s(E)\neq s(E')$ or $E \sim E'$, 2. $E,E' \in \mathcal{E}_{2}(g_{i})\cup (\mathcal{E}_{1}(g_{i}) \times \mathcal{E}_{1}) \Rightarrow s(E) \neq s(E')$ or $E \sim E'$. \[Def:Flag2FTECcondition\] In order to state the protocol, we define an update rule given a sequence of syndrome measurements using $t$-flag circuits for the counters[^3] $n_{\text{diff}}$ and $n_{\text{same}}$ as follows: For the following protocol to satisfy \[Def:FaultTolerantDef\], one can assume there are at most 2 faults. If the Flag $2$-FTEC condition is satisfied, the protocol is implemented as follows: ![Tree diagram for the Flag $2$-FTEC protocol. Numbers encircled in red at the end of the edges indicate which step to implement in the Flag $2$-FTEC Protocol. A dashed line is followed when any of the 2-flag circuits $C(g_{i})$ flags. Solid squares indicate a syndrome measurement using 2-flag circuits whereas rings indicate a decision based on syndrome outcomes. Edges with different colors indicate the current value of $n_{\text{diff}}$ in the protocol. Note that the protocol is repeated at most 6 times.[]{data-label="fig:TreeDiagramD5"}](TreeD5.png){width="50.00000%"} Note that when computing the update rules, if a flag occurs during the $j$’th round of syndrome measurements, the syndrome is not recorded for that round since all stabilizers must be measured. Thus when computing $n_{\text{diff}}$ and $n_{\text{same}}$ using consecutive syndromes $s_k$ and $s_{k+1}$, we are assuming that no flags occurred during rounds $k$ and $k+1$. In each case of the protocol, the correction sets correspond to those data errors which could arise from up to two faults which are consistent with the conditions of the case. As the elements are logically equivalent (by \[eq:GeneralLookupTable,Def:Flag2FTECcondition\]), which element is applied is unimportant. The general protocol for codes of arbitrary distance is given in \[app:GeneralFTEC\]. Examples of flag 2-FTEC applied to $d=5$ codes {#subsec:ApplicationProtocolColorCode} ---------------------------------------------- [0.22]{} ![Graphical representation of (a) the 19-qubit 2D color code and (b) the 17-qubit 2D color code. The $X$ and $Z$ stabilizers of the code are symmetric, given by the vertices of each plaquette. Both codes have distance-5.[]{data-label="fig:ColorCodeLattices"}](19qubitColorLattice.png "fig:"){width="\textwidth"} [0.25]{} ![Graphical representation of (a) the 19-qubit 2D color code and (b) the 17-qubit 2D color code. The $X$ and $Z$ stabilizers of the code are symmetric, given by the vertices of each plaquette. Both codes have distance-5.[]{data-label="fig:ColorCodeLattices"}](17qubitColorLattice.png "fig:"){width="\textwidth"} In this section we give examples of the flag $2$-FTEC protocol applied to the 2-dimensional [$[\![19,1,5]\!]$]{} and [$[\![17,1,5]\!]$]{} color codes, (see \[fig:19qubitLatticeColor,fig:17qubitLatticeColor\]). We first find 2-flag circuits for all generators (weight-4 and -6 for the 19-qubit code and weight-4 and -8 for the 17-qubit code). We also show that the flag 2-FTEC condition is satisfied for both codes. -------------- --------------------- --------------------- ------------------------------------- 1-fault 2-faults 1-fault 2-faults $I$,$Z_{1}$ $I$,$Z_{1}$,$Z_{2}$ $I$,$Z_{1}$,$Z_{6}$ $I$,$Z_{1}$,$Z_{2}$ $Z_{4}$ $Z_{3}$,$Z_{4}$ $Z_{1}Z_{2}$ $Z_{3}$, $Z_{4}$,$Z_{5}$,$Z_{6}$ $Z_{3}Z_{4}$ $Z_{1}Z_{2}$ $Z_{5}Z_{6}$ $Z_{1}Z_{2}$,$Z_{1}Z_{3}$ $Z_{1}Z_{4}$ $Z_{4}Z_{5}Z_{6}$ $Z_{1}Z_{4}$,$Z_{1}Z_{5}$ $Z_{2}Z_{4}$ $Z_{1}Z_{6}$,$Z_{2}Z_{3}$ $Z_{2}Z_{6}$,$Z_{3}Z_{4}$ $Z_{3}Z_{6}$,$Z_{4}Z_{5}$ $Z_{4}Z_{6}$,$Z_{5}Z_{6}$ $Z_{1}Z_{2}Z_{3}$,$Z_{1}Z_{5}Z_{6}$ $Z_{2}Z_{5}Z_{6}$,$Z_{3}Z_{4}Z_{5}$ $Z_{3}Z_{4}Z_{6}$,$Z_{3}Z_{5}Z_{6}$ $Z_{4}Z_{5}Z_{6}$ -------------- --------------------- --------------------- ------------------------------------- : $Z$ part of the flag error set of \[Def:FlagErrSetDef\] for flag circuits used to measure the stabilizers $g_{1} = Z_{1}Z_{2}Z_{3}Z_{4}$ and $g_{3} = Z_{1}Z_{2}Z_{3}Z_{4}Z_{5}Z_{6}$ (we removed errors equivalent up to the stabilizer being measured).[]{data-label="tab:PossibleCorrelatedErrors"} For a 2-flag circuit, two faults leading to an error of weight greater or equal to 3 (up to multiplication by the stabilizer) must always cause at least one of the flag qubits to flag. As shown in \[app:GeneralTflaggedCircuitConstruction\], a 2-flag circuit satisfying these properties can always be constructed using at most four flag qubits. We show 2-flag circuits for measuring weight six and eight generators in \[fig:Flag2CircuitExamples\]. In \[subsec:Remarks\], it will be shown that the family of color codes with a hexagonal lattice satisfy a sufficient condition which guarantees that the flag 2-FTEC condition is satisfied. However, there are codes that do not satisfy the sufficient condition but which nonetheless satisfy the 2-Flag FTEC condition. For the 19-qubit and 17-qubit color codes, we verified that the flag 2-FTEC condition was satisfied by enumerating all errors as one would have to for a generic code. In particular, in the case where the 2-flag circuits $C(g_{i})$ and $C(g_{j})$ flag, the resulting errors belonging to the set $\mathcal{E}_{2}(g_{i},g_{j})$ must be logically equivalent or have distinct syndromes (which we verified to be true). If a single circuit $C(g_{i})$ flags, there could either have been two faults in the circuit or a single fault along with another error that did not cause a flag. If the same syndrome is measured twice in a row after a flag, then errors in the set $\mathcal{E}_{2}(g_{i})\cup (\mathcal{E}_{1}(g_{i}) \times \mathcal{E}_{1})$ must be logically equivalent or have distinct syndromes (which we verified). If there is a flag but two different syndromes are measured in a row, errors belonging to the set $\mathcal{E}_{1}(g_{i}) \times \mathcal{E}_{1}$ must be logically equivalent or have distinct syndromes (as was already checked). The flag error sets (see \[Def:FlagErrSetDef\]) for the 19-qubit code can be obtained using the Pauli’s shown in \[tab:PossibleCorrelatedErrors\]. [0.32]{} ![ Illustration of 2-flag circuits for measuring (a) $Z^{\otimes 6}$ requiring only two flag qubits and (b) $Z^{\otimes 8}$ requiring only three flag qubits. Flag qubits are prepared in the ${|+\rangle}$ state, and measurement qubits in the ${|0\rangle}$ state. []{data-label="fig:Flag2CircuitExamples"}](Weight6Generator.png "fig:"){width="\textwidth"} [0.37]{} ![ Illustration of 2-flag circuits for measuring (a) $Z^{\otimes 6}$ requiring only two flag qubits and (b) $Z^{\otimes 8}$ requiring only three flag qubits. Flag qubits are prepared in the ${|+\rangle}$ state, and measurement qubits in the ${|0\rangle}$ state. []{data-label="fig:Flag2CircuitExamples"}](Weight8Generator.png "fig:"){width="\textwidth"} Given that the flag 2-FTEC condition is satisfied, the flag 2-FTEC protocol can be implemented following the steps of \[subsec:Distance5protocol\] and the tree diagram illustrated in \[fig:TreeDiagramD5\]. Flag error correction protocol for arbitrary distance codes {#app:GeneralFTEC} =========================================================== In this section we first provide the general flag $t$-FTEC protocol in \[subsec:GeneralProtocol\]. In \[subsec:Remarks\] we give a sufficient condition for stabilizer codes that allow us to easily prove that flag FTEC can be applied to a number of infinite code families. We show that the families of surface codes, hexagonal lattice color codes and quantum Reed-Muller codes satisfy the sufficient condition. Lastly, in \[app:GeneralTflaggedCircuitConstruction\], we give general $t$-flag circuit constructions which are applicable to the code families described in \[subsec:Remarks\]. We assume the reader is familiar with all previous definitions. However, to make this section reasonably self contained, we repeat some key definitions below. [6]{} A circuit $C(P)$ which, when fault-free, implements a projective measurement of a weight-$w$ Pauli $P$ without flagging is a $t$-flag circuit if the following holds: For any set of $v$ faults at up to $t$ locations in $C(P)$ resulting in an error $E$ with $\text{min}(\text{wt}(E),\text{wt}(E P)) > v$, the circuit flags. [9]{} Let $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}})$ be the set of all errors caused by precisely $m$ faults spread amongst the circuits $C(g_{i_{1}}),C(g_{i_{2}}), \cdots , C(g_{i_{k}})$ which all flagged. We also remind the reader of the correction set $$\begin{aligned} \tilde{E}_{t}^{m}(g_{i_{1}},\cdots , g_{i_{k}},s) = \begin{cases} \{ E \in \mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m} \\ \text{ such that } s(E) = s \} \\ \{ E_{\text{min}}(s) \} \text{ if above set empty. } \end{cases} \label{eq:GeneralLookupTableV2}\end{aligned}$$ Conditions and protocol {#subsec:GeneralProtocol} ----------------------- In what follows we generalize the fault-tolerant error correction protocol presented in \[subsec:Distance5protocol\] to stabilizer codes of arbitrary distance. Consider a stabilizer code $\mathcal{S} = \langle g_{1},g_{2},\cdots , g_{r} \rangle$ and $t$-flag circuits $\{ C(g_{1}),C(g_{2}), \cdots , C(g_{r}) \}$. For any set of $m$ stabilizer generators $\{ g_{i_{1}},\cdots ,g_{i_{m}} \}$ such that $1 \le m \le t$, every pair of elements $E,E' \in \bigcup_{j=0}^{t-m}\mathcal{E}_{t-j}(g_{i_{1}},\cdots ,g_{i_{m}})\times \mathcal{E}_{j}$ either satisfy $s(E)\neq s(E')$ or $E \sim E'$. \[Def:FlagtFTECcondition\] The above conditions ensure that if there are at most $t=\lfloor (d-1)/2 \rfloor$ faults, the protocol described below will satisfy the fault-tolerance conditions of \[Def:FaultTolerantDef\]. In order to state the protocol, we define an update rule given a sequence of syndrome measurements using $t$-flag circuits for the counters $n_{\text{diff}}$ and $n_{\text{same}}$ as follows (see also \[subsec:Distance5protocol\] and the associated footnote): In each case of the protocol, the correction sets correspond to those data errors which could arise from up to $t$ faults which are consistent with the conditions of the case. As the elements are logically equivalent (by \[eq:GeneralLookupTableV2,Def:FlagtFTECcondition\]), which element is applied is unimportant. For the protocol to satisfy the fault-tolerance criteria, the syndrome measurement needs to be repeated a minimum of $t+1$ times. In the scenario where the most syndrome measurement rounds are performed, $t$ identical syndromes are obtained before a fault causes the $t+1$’th syndrome to change (in which case $n_{\text{diff}}$ would increase by one). Afterwords, one measures the same syndrome $t-1$ times in a row until another fault causes the syndrome to change. This continues until all of the $t$ possible faults have been exhausted. At this stage, $n_{\text{diff}}=t$ so an extra syndrome measurement round will be performed using non-flag circuits. Thus the maximum number of syndrome measurement rounds $n_{\text{max}}$ is given by $$\begin{aligned} n_{\text{max}} = \sum_{j=0}^{t-1}(t-j) + t+1 = \frac{1}{2}(t^{2}+3t+2). \label{Eq:Nmax}\end{aligned}$$ Note that a similar approach by repeating syndrome measurements is used for Shor error correction [@AGP06; @Gottesman2010]. However, our scheme requires fewer syndrome measurement repetitions than is often described for Shor error correction and does not require the preparation and verification of a $w$-qubit cat state when measuring a stabilizer of weight-$w$. [^4] For codes that satisfy the flag $t$-FTEC condition, we also show in \[app:StatePrepAndMeasure\] how to fault-tolerantly prepare and measure logical states using the flag $t$-FTEC protocol. Sufficient condition and satisfying code families {#subsec:Remarks} ------------------------------------------------- The general flag $t$-FTEC condition can be difficult to verify for a given code since it depends on precisely which $t$-flag circuits are used. A sufficient (but not necessary) condition that implies the flag $t$-FTEC condition is as follows: **Sufficient flag $t$-FTEC condition:** Given a stabilizer code with distance $d>1$, and $\mathcal{S} = \langle g_{1},g_{2},\cdots , g_{r} \rangle$, we require that for all $v = 0,1, \dots t$, all choices $Q_{t-v}$ of $2(t-v)$ qubits, and all subsets of $v$ stabilizer generators $\{ g_{i_1},\dots ,g_{i_v} \} \subset \{ g_{1},\cdots , g_{r} \}$, there is no logical operator $l \in N(\mathcal{S}) \setminus \mathcal{S}$ such that $$\begin{aligned} \text{supp}(l) \subset \text{supp}(g_{i_1}) \cup \cdots \cup \text{supp}(g_{i_v}) \cup Q_{t-v},\end{aligned}$$ where $N(\mathcal{S})$ is the normalizer of the stabilizer group. If this condition holds, then the flag $t$-FTEC condition is implied for any choice of $t$-flag circuits $\{ C(g_{1}),C(g_{2}), \cdots , C(g_{r}) \}$. To prove this, we must show that it implies that none of the sets appearing in the $t$-FTEC condition contain elements that differ by a logical operator. Consider the set $\bigcup_{j=0}^{t-m}\mathcal{E}_{t-j}(g_{i_{1}},\cdots ,g_{i_{m}})\times \mathcal{E}_{j}$ for some set of $m$ stabilizer generators $\{ g_{i_{1}},\cdots ,g_{i_{m}} \}$ with $1 \le m \le t$. An error $E$ from this set will have support in the union of the support of the $m$ stabilizer generators $\{ g_{i_{1}},\cdots ,g_{i_{m}} \}$, along with up to $t-m$ other single qubits. Another error $E'$ from this set will have support in the union of support of the same $m$ stabilizer generators $\{ g_{i_{1}},\cdots ,g_{i_{m}} \}$, along with up to $t-m$ other *potentially different* single qubits. If the sufficient condition holds, then $\text{supp}(E E')$ cannot contain a logical operator. The sufficient flag $t$-FTEC condition is straightforward to verify for a number of code families with a lot of structure in their stabilizer generators and logical operators. We briefly provide a few examples. **Surface codes flag $t$-FTEC:** The rotated surface code [@KITAEV97Surface; @TS14; @BK98; @PhysRevLett.90.016803] family [$[\![d^2,1,d]\!]$]{} for all odd $d=2t+1$ (see \[fig:surfacecodeproof\]) satisfies the flag $t$-FTEC condition using any 4-flag circuits. Firstly, by performing an exhaustive search, we verified that the circuit of \[fig:StabFTwithAncilla\] is a 4-flag circuit. As a CSS code, we can restrict our attention to purely $X$-type and $Z$-type logical operators. An $X$ type logical operator must have at least one qubit in each of the $2t+1$ rows of the lattice shown. However, each stabilizer only contains qubits in two different rows. Therefore, with $v$ stabilizer generators, at most $2v$ of the rows could have support. With an additional $2(t-v)$ qubits, at most $2t$ rows can be covered, which is fewer than the number of rows, and therefore no logical $X$ operator is supported on the union of the support of $v$ stabilizers and $2(t-v)$ qubits. An analogous argument holds for $Z$-type logical operators, therefore the sufficient $t$-FTEC condition is satisfied. ![ The $d=5$ rotated surface code. Qubits are represented by white circles, and $X$ and $Z$ stabilizer generators are represented by red and green faces. As in the example, any logical $X$ operator has $X$ operators acting on at least five qubits, with at least one in each row of the lattice, involving an even number in any green face. In this case, no two stabilizer generators can have qubits in five rows, and therefore cannot contain an $X$ type logical operator. The argument is analogous for logical $Z$ operators. []{data-label="fig:surfacecodeproof"}](sufacecodeproof.png){width="30.00000%"} **Color codes flag $t$-FTEC:** Here we show that any distance $d=(2t+1)$ self-dual CSS code with at most weight-6 stabilizer generators satisfies the flag $t$-FTEC condition using any 6-flag circuits (see \[fig:6flagCircuit\] for an example). Examples include the hexagonal color code [@Bombin06TopQuantDist] family [$[\![(3d^2+1)/4,1,d]\!]$]{} (see \[fig:19qubitLatticeColor\]). As a self-dual CSS code, $X$ and $Z$ type stabilizer generators are identically supported and we can consider a pure $X$-type logical operator without loss of generality. Consider an $X$ type logical operator $l$ such that $$\begin{aligned} \text{supp}(l) \subset \text{supp}(g_{i_1}) \cup \cdots \cup \text{supp}(g_{i_v}) \cup Q_{t-v}, \label{eq:SuppRepeated}\end{aligned}$$ for some set of $v$ stabilizer generators $\{ g_{i_1},\dots ,g_{i_v} \} \subset \{ g_{1},\cdots , g_{r} \}$ along with $2(t-v)$ other qubits $Q_{t-v}$. Restricted to the support of any of the $v$ stabilizers $g_i$, $l|_{g_i}$ must have weight 0, 2, 4, or 6 (otherwise it would anti-commute with the corresponding $Z$ type stabilizer). If the restricted weight is 4 or 6, we can produce an equivalent lower weight logical operator $l' = g_i l$, which still satisfies \[eq:SuppRepeated\]. Repeating this procedure until the weight of the logical operator can no longer be reduced yields a logical operator $l_{\text{min}}$ which has weight either 0 or 2 when restricted to the support of any of the $v$ stabilizer generators. The total weight of $l_{\text{min}}$ is then at most $2v+2(t-v) =2t$, which is less than the distance of the code, giving a contradiction which therefore implies that $l$ could not have been a logical operator. An analogous arguments holds for $Z$-type logical operators, therefore the sufficient $t$-FTEC condition is satisfied. This proof can be easily extended to show that any distance $d=(2t+1)$ self-dual CSS code with at most weight-$2 v$ stabilizer generators for some integer $v$ satisfies the flag $t'$-FTEC condition using any $(v-1)$-flag circuits, where $t'= t/\lfloor v/2\rfloor$. **Quantum Reed-Muller codes flag $1$-FTEC:** The [$[\![n=2^m-1,k=1,d=3]\!]$]{} quantum Reed-Muller code family for every integer $m\geq 3$ satisfies the flag 1-FTEC condition using any 1-flag circuits for the standard choice of generators. We use the following facts about the Quantum Reed-Muller code family (see \[app:QRMcodes\] and [@ADP14] for proofs of these facts): (1) The code is CSS, allowing us to restrict to pure $X$ type and pure $Z$ type logical operators, (2) all pure $X$ or $Z$ type logical operators have odd support, (3) every $X$-type stabilizer generator has the same support as some $Z$-type stabilizer generator, and (4) every $Z$-type stabilizer generator is contained within the support of an $X$ type generator. We only need to prove the sufficient condition for $v=0,1$ in this case. For $v=0$, no two qubits can support a logical operator, as any logical operator has weight at least three. For $v=1$, assume the support of an $X$-type stabilizer generator contains a logical operator $l$. That logical operator $l$ cannot be $Z$ type or it would anti-commute with the $X$-stabilizer due to its odd support. However, by fact (3), there is a $Z$ type stabilizer with the same support as the $X$ type stabilizer, therefore implying $l$ cannot be $X$ type either. Therefore, by contradiction we conclude that no logical operator can be contained in the support of an $X$ stabilizer generator. Since every other stabilizer generator is contained within the support of an $X$-type stabilizer generator, a logical operator cannot be contained in the support of any stabilizer generator. Note that the Hamming code family has a stabilizer group which is a proper subgroup of that of the quantum Reed-Muller codes described here. The $X$-type generators of each Hamming code are the same as for a quantum Reed-Muller code, and the Hamming codes are self-dual CSS codes. It is clear that the sufficient condition cannot be applied to the Hamming code since it has even-weight $Z$-type logical operators (which are stabilizers for the quantum Reed-Muller code) supported within the support of some stabilizer generators. **Codes which satisfy flag $t$-FTEC condition but not the sufficient flag $t$-FTEC condition:** [0.3]{} ![(a) A 1-flag circuit for measuring the stabilizer $Z_{8}Z_{9}Z_{10}Z_{11}Z_{12}Z_{13}Z_{14}Z_{15}$ of the [$[\![15,7,3]\!]$]{} Hamming code. However a single fault on the fourth or fifth CNOT can lead to the error $Z_{12}Z_{13}Z_{14}Z_{15}$ on the data which is a logical fault. With the CNOT gates permuted as shown in (b), the [$[\![15,7,3]\!]$]{} satisfies the general flag 1-FTEC condition.](CircuitReichardt1.png "fig:"){width="\textwidth"} [0.3]{} ![(a) A 1-flag circuit for measuring the stabilizer $Z_{8}Z_{9}Z_{10}Z_{11}Z_{12}Z_{13}Z_{14}Z_{15}$ of the [$[\![15,7,3]\!]$]{} Hamming code. However a single fault on the fourth or fifth CNOT can lead to the error $Z_{12}Z_{13}Z_{14}Z_{15}$ on the data which is a logical fault. With the CNOT gates permuted as shown in (b), the [$[\![15,7,3]\!]$]{} satisfies the general flag 1-FTEC condition.](CircuitReichardt2.png "fig:"){width="\textwidth"} Note that there are codes which satisfy the general flag $t$-FTEC condition but not the sufficient condition presented in this section. An example of such a code is the [$[\![5,1,3]\!]$]{} code (see \[tab:StabilizerGeneratorsLists\] for the codes stabilizer generators and logical operators). Another example includes the Hamming codes as was explained in the discussion on quantum Reed-Muller codes. For instance, consider the [$[\![15,7,3]\!]$]{} Hamming code. Using the 1-flag circuit shown in \[fig:CircuitReichardt1\], the [$[\![15,7,3]\!]$]{} will not satisfy the general flag 1-FTEC condition since a single fault can lead to a logical error on the data. As was shown in [@CR17v1], by permuting the CNOT gates resulting in the circuit illustrated in \[fig:CircuitReichardt2\], the flag 1-FTEC condition is satisfied. Circuits {#app:GeneralTflaggedCircuitConstruction} -------- [0.35]{} ![(a) Illustration of a w-flag circuit for measuring the operator $Z^{\otimes w}$ where $w=6$ using the smallest number of flag qubits. (b) Illustration of a 3-flag circuit for measuring $Z^{\otimes 8}$ using the smallest number of flag qubits.[]{data-label="fig:ExamplesOfLargeFlagCircuits"}](6FlagCircuit.png "fig:"){width="\textwidth"} [0.35]{} ![(a) Illustration of a w-flag circuit for measuring the operator $Z^{\otimes w}$ where $w=6$ using the smallest number of flag qubits. (b) Illustration of a 3-flag circuit for measuring $Z^{\otimes 8}$ using the smallest number of flag qubits.[]{data-label="fig:ExamplesOfLargeFlagCircuits"}](3flagWeight8.png "fig:"){width="\textwidth"} In \[subsec:Remarks\] we showed that the family of surface codes, color codes with a hexagonal lattice and quantum Reed-Muller codes satisfied a sufficient condition allowing them to be used in the flag $t$-FTEC protocol. Along with the general 1-flag circuit construction of \[fig:General1FlagCircuitSecCircuit\], the $6$-flag circuit for measuring $Z^{\otimes 6}$ of \[fig:6flagCircuit\] can be used as $t$-flag circuits for all of the codes in \[subsec:Remarks\]. Note that the circuit in \[fig:StabFTwithAncilla\] (which is a special case of \[fig:General1FlagCircuitSecCircuit\] when $w=4$) is a 4-flag circuit which is used for measuring $Z^{\otimes 4}$. Before describing general 1- and 2-flag circuit constructions, we give the following two definitions which we will frequently use: Any CNOT that couples a data qubit to the measurement qubit will be referred to as $\text{CNOT}_{dm}$ and any CNOT coupling a measurement qubit to a flag qubit will be referred to as $\text{CNOT}_{fm}$. In both cases the target qubit will always be the measurement qubit. **1- and 2-flag circuits for weight $w$ stabilizer measurements:** We provide 1- and 2-flag circuit constructions for measuring a weight-$w$ stabilizer. The 1-flag circuit requires a single flag qubit, and the 2-flag circuit requires at most four flag qubits. [0.35]{} ![(a) General 1-flag circuit for measuring the stabilizer $Z^{\otimes w}$. (b) Example of a 2-flag circuit for measuring $Z^{\otimes 12}$ using our general 2-flag circuit construction. (c) An equivalent circuit using fewer flag qubits by reusing a measured flag qubit and reinitializing it in the ${|+\rangle}$ state for use in another pair of $\text{CNOT}_{\text{fm}}$ gates.[]{data-label="fig:Generall1And2FlagCircuits"}](General1FlagCircuitSecCircuit.png "fig:"){width="\textwidth"} [0.43]{} ![(a) General 1-flag circuit for measuring the stabilizer $Z^{\otimes w}$. (b) Example of a 2-flag circuit for measuring $Z^{\otimes 12}$ using our general 2-flag circuit construction. (c) An equivalent circuit using fewer flag qubits by reusing a measured flag qubit and reinitializing it in the ${|+\rangle}$ state for use in another pair of $\text{CNOT}_{\text{fm}}$ gates.[]{data-label="fig:Generall1And2FlagCircuits"}](General2FlagCircuitSecCircuit.png "fig:"){width="\textwidth"} [0.43]{} ![(a) General 1-flag circuit for measuring the stabilizer $Z^{\otimes w}$. (b) Example of a 2-flag circuit for measuring $Z^{\otimes 12}$ using our general 2-flag circuit construction. (c) An equivalent circuit using fewer flag qubits by reusing a measured flag qubit and reinitializing it in the ${|+\rangle}$ state for use in another pair of $\text{CNOT}_{\text{fm}}$ gates.[]{data-label="fig:Generall1And2FlagCircuits"}](General2FlagCircuitSecCircuitCompressed.png "fig:"){width="\textwidth"} Without loss of generality, in proving that the circuit constructions described below are 1- and 2-flag circuits, we can assume that all faults occurred on CNOT gates. This is because any set of $v$ faults (including those at idle, preparation or measurement locations) will have the same output Pauli operator and flag measurement results as some set of at most $v$ faults on CNOT gates (since every qubit is involved in at least one CNOT). As was shown in Ref. [@CR17v1], \[fig:General1FlagCircuitSecCircuit\] illustrates a general 1-flag circuit construction for measuring the stabilizer $Z^{\otimes w}$ which requires only two $\text{CNOT}_{\text{fm}}$ gates. To see that the first construction is a 1-flag circuit, note that an $IZ$ error occurring on any CNOT will give rise to a flag unless it occurs on the first or last $\text{CNOT}_{\text{dm}}$ gates or the last $\text{CNOT}_{\text{fm}}$ gate. However, such a fault on any of these three gates can give rise to an error of weight at most one (after multiplying by the stabilizer $Z^{\otimes w}$). One can also verify that if there are no faults, the circuit in \[fig:General1FlagCircuitSecCircuit\] implements a projective measurement of $Z^{\otimes w}$ without flagging. Following the approach in [@LAR11], one simply needs to check that the circuit preserves the stabilizer group generated by $Z^{\otimes w}$ and $X$ on each ancilla prepared in the ${|+\rangle}$ state and $Z$ on each ancilla prepared in the ${|0\rangle}$ state. By using pairs of $\text{CNOT}_{\text{fm}}$ gates, this construction satisfies the requirement. We now give a general 2-flag circuit construction for measuring $Z^{\otimes w}$ for arbitrary $w$ (see \[fig:General2FlagCircuitSecCircuit\] for an example). The circuit consists of pairs of $\text{CNOT}_{\text{fm}}$ gates each connected to a different flag qubit prepared in the ${|+\rangle}$ state and measured in the $X$ basis. The general 2-flag circuit construction involves the following placement of $w/2-1$ pairs of $\text{CNOT}_{\text{fm}}$ gates: 1. Place a $\text{CNOT}_{\text{fm}}$ pair between the first and second last $\text{CNOT}_{\text{dm}}$ gates. 2. Place a $\text{CNOT}_{\text{fm}}$ pair between the second and last $\text{CNOT}_{\text{dm}}$ gates. 3. After the second $\text{CNOT}_{\text{fm}}$ gate, place the first $\text{CNOT}_{\text{fm}}$ gate of the remaining pairs after every two $\text{CNOT}_{\text{dm}}$ gates. The second $\text{CNOT}_{\text{fm}}$ gate of a pair is placed after every three $\text{CNOT}_{\text{dm}}$ gates. As shown in \[fig:General2FlagCircuitSecCircuitCompressed\], it is possible to reuse some flag qubits to measure multiple pairs of $\text{CNOT}_{\text{fm}}$ gates at the cost of introducing extra time steps into the circuit. For this reason, at most four flag qubits will be needed, however, if $w \le 8$, then $w/2-1$ flag qubits are sufficient. We now show that the above construction satisfies the requirements of a 2-flag circuit. If one CNOT gate fails, by an argument analogous to that used for the 1-flag circuit, there will be a flag or an error of at most weight-one on the data. If the first pair of $\text{CNOT}_{\text{fm}}$ gates fail causing no flag qubits to flag, after multiplying the data qubits by $Z^{\otimes w}$, the resulting error $E_{r}$ will have $\text{wt}(E_{r}) \le 2$. For any other pair of $\text{CNOT}_{\text{fm}}$ gates that fail causing an error of weight greater than two on the data, by construction there will always be another $\text{CNOT}_{\text{fm}}$ gate between the two that fail which will propagate a $Z$ error to a flag qubit causing it to flag. Similarly, if pairs of $\text{CNOT}_{\text{dm}}$ gates fail resulting in the data error $E_{r}$ with $\text{wt}(E_{r}) \ge 2$, by construction there will always be an odd number of $Z$ errors propagating to a flag qubit due to the $\text{CNOT}_{\text{fm}}$ gates in between the $\text{CNOT}_{\text{dm}}$ gates that failed causing a flag qubit to flag. The same argument applies if a failure occurs between a $\text{CNOT}_{\text{dm}}$ and $\text{CNOT}_{\text{fm}}$ gate. Lastly, a proposed general $w$-flag circuit construction for arbitrary $w$ is provided in \[App:GeneralwFlagCircuitConstruction\]. **Use of flag information:** As seen in \[fig:6flagCircuit,fig:3flagWeight8,fig:General2FlagCircuitSecCircuit,fig:General2FlagCircuitSecCircuitCompressed\], in general $t$-flag circuits require more than one flag qubit. Apart from their use in satisfying the $t$-flag circuit properties, the extra flag qubits could be used to reduce the size of the flag error sets (defined in \[Def:FlagErrSetDef\]) when verifying the Flag $t$-FTEC condition of \[app:GeneralFTEC\]. To do so, we first define $f$, where $f$ is a bit string of length $u$ (here $u$ is the number of flag qubits) with $f_{i} = 1$ if the i’th flag qubit flagged and 0 otherwise. In this case, the correction set of \[eq:GeneralLookupTableV2\] can be modified to include flag information as follows: $$\begin{aligned} &\tilde{E}_{t}^{m}(g_{i_{1}},\cdots , g_{i_{k}},s,f_{i_{1}},\cdots ,f_{i_{k}}) = \nonumber \\ &\begin{cases} \{ E \in \mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}},f_{i_{1}},\cdots ,f_{i_{k}}) \times \mathcal{E}_{t-m} \\ \text{ such that } s(E) = s \} \\ \{ E_{\text{min}}(s) \} \text{ if above set empty., } \end{cases} \label{eq:CorrectionSetWithFlagInfo}\end{aligned}$$ where $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}},f_{i_{1}},\cdots,f_{i_{k}})$ is the new flag error set containing only errors caused by precisely $m$ faults spread amongst the circuits $C(g_{i_{1}}),C(g_{i_{2}}), \cdots , C(g_{i_{k}})$ which each gave rise to the flag outcomes $f_{i_{1}},\cdots,f_{i_{k}}$. Hence only errors which result from the measured flag outcome would be stored in the correction set. With enough flag qubits, this could potentially broaden the family of codes which satisfy the Flag $t$-FTEC condition. Circuit level noise analysis {#sec:CircuitLevelNoiseFTEC} ============================ The purpose of this section is to demonstrate explicitly the flag 2-FTEC protocol, and to identify parameter regimes in which flag FTEC presented both here and in other works offers advantages over other existing FTEC schemes. In \[subsec:Numerics19\] we analyze the logical failure rates of the [$[\![19,1,5]\!]$]{} color code and compute it’s pseudo-threshold for the three choices of $\tilde{p}$. In \[subsec:CompareFlagecschemes\] we compare logical failure rates of several fault-tolerant error correction schemes applied to distance-three and distance-five stabilizer codes. The stabilizers for all of the studied codes are given in \[tab:StabilizerGeneratorsLists\]. Logical failure rates are computed using the full circuit level noise model and simulation methods described in \[subsec:NoiseAndNumerics\]. Numerical analysis of the [$[\![19,1,5]\!]$]{} color code {#subsec:Numerics19} --------------------------------------------------------- The full circuit-level noise analysis of the flag 2-FTEC protocol applied to the [$[\![19,1,5]\!]$]{} color code was performed using the stabilizer measurement circuits of \[fig:StabFTwithAncilla,fig:WeightSixGenerators\]. In the weight-six stabilizer measurement circuit of \[fig:WeightSixGenerators\], there are 10 CNOT gates, three measurement and state-preparation locations, and 230 resting qubit locations. When measuring all stabilizer generators using non-flag circuits, there are 42 CNOT and 42 XNOT gates, 18 measurement and state-preparation locations, and 2196 resting qubit locations. Consequently, we expect the error suppression capabilities of the flag EC scheme to depend strongly on the number of idle qubit locations. three-qubit flag EC pseudo-threshold ---------------------------------------------------- -------------------------------------------------------- [$[\![19,1,5]\!]$]{} and $\tilde{p}=p$ $p_{\mathrm{pseudo}} = (1.14 \pm 0.02) \times 10^{-5}$ [$[\![19,1,5]\!]$]{} and $\tilde{p}=\frac{p}{10}$ $p_{\mathrm{pseudo}} = (6.70 \pm 0.07) \times 10^{-5}$ [$[\![19,1,5]\!]$]{} and $\tilde{p}=\frac{p}{100}$ $p_{\mathrm{pseudo}} = (7.74 \pm 0.16) \times 10^{-5}$ : Table containing pseudo-threshold values for the flag 2-FTEC protocol applied to the [$[\![19,1,5]\!]$]{} color code for $\tilde{p}=p$, $\tilde{p}=p/10$ and $\tilde{p}=p/100$.[]{data-label="tab:PseudoThresholAllThree1915"} ![Logical failure rates of the [$[\![19,1,5]\!]$]{} color code after implementing the flag 2-FTEC protocol presented in \[subsec:Distance5protocol\] for the three noise models described in \[subsec:NoiseAndNumerics\]. The dashed curves represent the lines $\tilde{p}=p$, $\tilde{p}=p/10$ and $\tilde{p}=p/100$. The crossing point between $\tilde{p}$ and the curve corresponding to $p^{({\ensuremath{[\![19,1,5]\!]}})}_{L}(\tilde{p})$ in \[Def:PseudoThreshDef\] gives the pseudo-threshold.[]{data-label="fig:PseudoThreshPlots19ColorAllThree"}](CBplotsAllThreeCurves.png){width="50.00000%"} Pseudo-thresholds of the [$[\![19,1,5]\!]$]{} code were obtained using the methods of \[subsec:NoiseAndNumerics\]. Recall that for extending the lifetime of a qubit (when idle qubit locations fail with probability $\tilde{p}$), the probability of failure after implementing an FTEC protocol should be smaller than $\tilde{p}$. We calculated the pseudo-threshold using \[Def:PseudoThreshDef\] for the three cases were idle qubits failed with probability $\tilde{p}=p$, $\tilde{p}=p/10$ and $\tilde{p}=p/100$. The results are shown in \[tab:PseudoThresholAllThree1915\]. The logical failure rates for the three noise models are shown in \[fig:PseudoThreshPlots19ColorAllThree\]. It can be seen that when the probability of error on a resting qubit decreases from $p$ to $p/10$, the pseudo-threshold improves by nearly a factor of six showing the strong dependence of the scheme on the probability of failure of idle qubits. Comparison of flag 1- and 2-FTEC with other FTEC schemes {#subsec:CompareFlagecschemes} -------------------------------------------------------- [0.33]{} {width="\textwidth"} {width="\textwidth"} [0.33]{} {width="\textwidth"} {width="\textwidth"} [0.33]{} {width="\textwidth"} {width="\textwidth"} The most promising schemes for testing fault-tolerance in near term quantum devices are those which achieve high pseudo-thresholds while maintaining a low qubit overhead. The flag FTEC protocol presented in this paper uses fewer qubits compared to other well known fault-tolerance schemes but typically has increased circuit depth. In this section we apply the flag FTEC protocol of \[subsec:ReviewChaoReichardt,subsec:Distance5protocol\] to the [$[\![5,1,3]\!]$]{}, [$[\![7,1,3]\!]$]{} and [$[\![19,1,5]\!]$]{} codes. We compare logical failure rates for three values of $\tilde{p}$ with Steane error correction applied to the [$[\![7,1,3]\!]$]{} and [$[\![19,1,5]\!]$]{} codes and with the $d=3$ and $d=5$ rotated surface code. More details on Steane error correction and surface codes are provided in \[app:SurfaceECSection,app:SteaneECSection\]. Note that recent work by Goto has provided optimizations to prepare Steane ancillas [@Goto16]. However, our numerical results for Steane-EC were produced using the methods presented in \[app:SteaneECSection\]. Results of the logical failure rates for $\tilde{p}=p$, $\tilde{p}=p/10$ and $\tilde{p}=p/100$ are shown in \[fig:AllComparisonPlotsCombined\]. Various pseudo-thresholds and required time-steps for the considered fault-tolerant error correction methods are given in \[tab:PseudoThresholdsAllECschemesD3,tab:PseudoThresholdsAllECschemesD5\]. The circuits for measuring the stabilizers of the 5-qubit code were similar to the ones used in \[fig:StabFTwithAncilla\] (for an $X$ Pauli replace the CNOT by an XNOT). For flag-FTEC methods, it can be seen that the [$[\![5,1,3]\!]$]{} code always achieves lower logical failure rates compared to the [$[\![7,1,3]\!]$]{} code. However, when $\tilde{p}=p$, both the $d=3$ surface code as well as Steane-EC achieves lower logical failure rates (with Steane-EC achieving the best performance). For $\tilde{p}=p/10$, flag-EC applied to the [$[\![5,1,3]\!]$]{} code achieves nearly identical logical failure rates compared to the $d=3$ surface code. For $\tilde{p}=p/100$, flag 1-FTEC applied to the [$[\![5,1,3]\!]$]{} code achieves lower logical failure rates than the $d=3$ surface code but still has higher logical failure rates compared to Steane-EC. FTEC scheme Noise model Number of qubits Time steps ($T_{\mathrm{time}}$) Pseudo-threshold ------------------------------- --------------------- ------------------ ------------------------------------ -------------------------------------------------------- Flag-EC [$[\![5,1,3]\!]$]{} $\tilde{p} = p$ $7$ $64 \le T_{\mathrm{time}} \le 88$ $p_{\mathrm{pseudo}} = (7.09 \pm 0.03) \times 10^{-5}$ Flag-EC [$[\![7,1,3]\!]$]{} $9$ $72 \le T_{\mathrm{time}} \le 108$ $p_{\mathrm{pseudo}} = (3.39 \pm 0.10) \times 10^{-5}$ $d=3$ Surface code 17 $\ge 18$ $p_{\mathrm{pseudo}} = (3.29 \pm 0.16) \times 10^{-4}$ Steane-EC [$[\![7,1,3]\!]$]{} $\ge 35$ 15 $p_{\mathrm{pseudo}} = (6.29 \pm 0.13) \times 10^{-4}$ Flag-EC [$[\![5,1,3]\!]$]{} $\tilde{p} = p/10$ $7$ $64 \le T_{\mathrm{time}} \le 88$ $p_{\mathrm{pseudo}} = (1.11 \pm 0.02) \times 10^{-4}$ Flag-EC [$[\![7,1,3]\!]$]{} $9$ $72 \le T_{\mathrm{time}} \le 108$ $p_{\mathrm{pseudo}} = (8.68 \pm 0.15) \times 10^{-5}$ $d=3$ Surface code 17 $\ge 18$ $p_{\mathrm{pseudo}} = (1.04 \pm 0.02) \times 10^{-4}$ Steane-EC [$[\![7,1,3]\!]$]{} $\ge 35$ 15 $p_{\mathrm{pseudo}} = (3.08 \pm 0.01) \times 10^{-4}$ Flag-EC [$[\![5,1,3]\!]$]{} $\tilde{p} = p/100$ $7$ $64 \le T_{\mathrm{time}} \le 88$ $p_{\mathrm{pseudo}} = (2.32 \pm 0.03) \times 10^{-5}$ Flag-EC [$[\![7,1,3]\!]$]{} $9$ $72 \le T_{\mathrm{time}} \le 108$ $p_{\mathrm{pseudo}} = (1.41 \pm 0.05) \times 10^{-5}$ $d=3$ Surface code 17 $\ge 18$ $p_{\mathrm{pseudo}} = (1.37 \pm 0.03) \times 10^{-5}$ Steane-EC [$[\![7,1,3]\!]$]{} $\ge 35$ 15 $p_{\mathrm{pseudo}} = (3.84 \pm 0.01) \times 10^{-5}$ We also note that the pseudo-threshold increases when $\tilde{p}$ goes from $p$ to $p/10$ for both the [$[\![5,1,3]\!]$]{} and [$[\![7,1,3]\!]$]{} codes when implemented using the flag 1-FTEC protocol. This is primarily due to the large circuit depth in flag-EC protocols since idle qubits locations significantly outnumber other gate locations. For the surface code, the opposite behaviour is observed. As was shown in [@FMMC12], CNOT gate failures have the largest impact on the pseudo-threshold of the surface code. Thus, when idle qubits have lower failure probability, lower physical error rates will be required in order to achieve better logical failure rates. For instance, if idle qubits never failed, then performing error correction would be guaranteed to *increase* the probability of failure due to the non-zero failure probability of other types of locations (CNOT, measurements and state-preparation). Lastly, the pseudo-threshold for Steane-EC also decreases with lower idle qubit failure rates, but the change in pseudo-threshold is not as large as the surface code. This is primarily due to the fact that all CNOT gates are applied transversally in Steane-EC, so that the pseudo-threshold is not as sensitive to CNOT errors compared to the surface code. Furthermore, most high-weight errors arising during the state-preparation of the logical ancilla’s will be detected (see \[app:SteaneECSection\]). Hence, idle qubit errors play a larger role than in the surface code, but Steane-EC has fewer idle qubit locations compared to flag-EC (see \[tab:PseudoThresholdsAllECschemesD3\] for the circuit depths of all schemes). FTEC scheme Noise model Number of qubits Time steps ($T_{\mathrm{time}}$) Pseudo-threshold -------------------------------- --------------------- ------------------ ------------------------------------- -------------------------------------------------------- Flag-EC [$[\![19,1,5]\!]$]{} $\tilde{p} = p$ $22$ $504 \le T_{\mathrm{time}} \le 960$ $p_{\mathrm{pseudo}} = (1.14 \pm 0.02) \times 10^{-5}$ $d=5$ Surface code 49 $\ge 18$ $p_{\mathrm{pseudo}} = (9.41 \pm 0.17) \times 10^{-4}$ Steane-EC [$[\![19,1,5]\!]$]{} $\ge 95$ 15 $p_{\mathrm{pseudo}} = (1.18 \pm 0.02) \times 10^{-3}$ Flag-EC [$[\![19,1,5]\!]$]{} $\tilde{p} = p/10$ $22$ $504 \le T_{\mathrm{time}} \le 960$ $p_{\mathrm{pseudo}} = (6.70 \pm 0.07) \times 10^{-5}$ $d=5$ Surface code 49 $\ge 18$ $p_{\mathrm{pseudo}} = (7.38 \pm 0.22) \times 10^{-4}$ Steane-EC [$[\![19,1,5]\!]$]{} $\ge 95$ 15 $p_{\mathrm{pseudo}} = (4.42 \pm 0.27) \times 10^{-4}$ Flag-EC [$[\![19,1,5]\!]$]{} $\tilde{p} = p/100$ $22$ $504 \le T_{\mathrm{time}} \le 960$ $p_{\mathrm{pseudo}} = (7.74 \pm 0.16) \times 10^{-5}$ $d=5$ Surface code 49 $\ge 18$ $p_{\mathrm{pseudo}} = (2.63 \pm 0.18) \times 10^{-4}$ Steane-EC [$[\![19,1,5]\!]$]{} $\ge 95$ 15 $p_{\mathrm{pseudo}} = (5.60 \pm 0.43) \times 10^{-5}$ Although Steane-EC achieves the lowest logical failure rates compared to the other fault-tolerant error correction schemes, it requires a minimum of 35 qubits (more details are provided in \[app:SteaneECSection\]). In contrast, the $d=3$ surface code requires 17 qubits, and flag 1-FTEC applied to the [$[\![5,1,3]\!]$]{} code requires only 7 qubits. Therefore, if the probability of idle qubit errors is much lower than gate, state preparation and measurement errors, flag-FTEC methods could be good candidates for early fault-tolerant experiments. It is important to keep in mind that for the flag 1-FTEC protocol applied to the distance-three codes considered in this section, the same ancilla qubits are used to measure all stabilizers. A more parallelized version of flag-FTEC applied to the [$[\![7,1,3]\!]$]{} code using four ancilla qubits instead of two is considered in \[app:CompactRepFlagQubit\]. In computing the number of time steps required by the flag $t$-FTEC protocols, a lower bound is given in the case where there are no flags and the same syndrome is repeated $t+1$ times. In \[app:GeneralFTEC\] it was shown that the full syndrome measurement for flag-FTEC is repeated at most $\frac{1}{2}(t^{2} + 3t + 2)$ times where $t = \lfloor (d-1)/2 \rfloor$. An upper bound on the total number of required time steps is thus obtained from a worst case scenario where syndrome measurements are repeated $\frac{1}{2}(t^{2} + 3t + 2)$ times. For distance-five codes, the first thing to notice from \[fig:AllComparisonPlotsCombined\] is that the slopes of the logical failure rate curves of flag-EC applied to the [$[\![19,1,5]\!]$]{} code and $d=5$ surface code are different from the slopes of Steane-EC applied to the [$[\![19,1,5]\!]$]{} code. In particular, $p_{\text{L}} = cp^{3} + \mathcal{O}(p^{4})$ for flag-EC and the surface code whereas $p_{\text{L}} = c_{1}p^{2} + c_{2}p^{3} + \mathcal{O}(p^{4})$ for Steane-EC ($c$, $c_{1}$ and $c_{2}$ are constants that depend on the code and FTEC method). The reason that Steane-EC has non-zero $\mathcal{O}(p^{2})$ contributions to the logical failure rates is that there are instances where errors occurring at two different locations can lead to a logical fault. Consequently, the Steane-EC method that was used is not strictly fault-tolerant according to \[Def:FaultTolerantDef\]. In \[app:SteaneECSection\], more details on the fault tolerant properties of Steane-EC are provided and a fully fault-tolerant implementation of Steane-EC is analyzed (at the cost of using more qubits). For $d=5$, the surface code achieves significantly lower logical failure rates compared to all other distance 5 schemes but uses 49 qubits instead of 22 for the [$[\![19,1,5]\!]$]{} code. Furthermore, due the differences in the slopes of flag-2 FTEC protocol compared with Steane-EC applied to the [$[\![19,1,5]\!]$]{} code, there is a regime where flag-2 FTEC achieves lower logical failure rates compared to Steane-EC. For $\tilde{p}=p/100$, it can be seen in \[fig:AllComparisonPlotsCombined\] that this regime occurs when $p \lesssim 10^{-4}$. We also note that the pseudo-threshold of flag-EC applied to the [$[\![19,1,5]\!]$]{} color code increases for all noise models whereas the pseudo-threshold decreases for the other FTEC schemes. Again, this is due to the fact that flag-EC has a larger circuit depth compared to the other FTEC methods and is thus more sensitive to idle qubit errors. Comparing the flag 2-FTEC protocol (applied to the [$[\![19,1,5]\!]$]{} color code) to all the $d=3$ schemes that were considered in this section, due to the higher distance of the 19-qubit code, there will always be a parameter regime where the 19-qubit color code acheives lower logical failure rates than both the $d=3$ surface code and Steane-EC applied to the [$[\![7,1,3]\!]$]{} code. In the case where $\tilde{p}=p/100$ and with $p \lesssim 1.5 \times10^{-4}$, using flag error correction with only 22 qubits outperforms Steane error correction (which uses a minimum of 35 qubits) and the $d=3$ rotated surface code (which uses 17 qubits). Note the considerable number of time steps involved in a round of flag-EC, particularly in the $d=5$ case (see \[tab:PseudoThresholdsAllECschemesD5\]). For many applications, this is a major drawback, for example for quantum computation when the time of an error correction round dictates the time of a logical gate. However there are some cases in which having a larger number of time-steps in an EC round while holding the logical error rate fixed is advantageous as it corresponds to a longer physical lifetime of the encoded information. Such schemes could be useful for example in demonstrating that encoded logical quantum information can be stored for longer time scales in the lab using repeated rounds of FTEC. Conclusion {#sec:Conclusion} ========== Building on definitions and a new flag FTEC protocol applied to distance-three and -five codes presented in \[sec:Section1\], in \[subsec:GeneralProtocol\] we presented a general flag FTEC protocol, which we called flag $t$-FTEC, and which is applicable to stabilizer codes of distance $d = 2t+1$ that satisfy the flag $t$-FTEC condition. The protocol makes use of flag ancilla qubits which signal when $v$ faults lead to errors of weight greater than $v$ on the data when performing stabilizer measurements. In \[subsec:ApplicationProtocolColorCode,app:GeneralTflaggedCircuitConstruction\] we gave explicit circuit constructions, including those needed for distance 3 and 5 codes measuring stabilizers of weight 4, 6 and 8. In \[subsec:Remarks\] we gave a sufficient condition for codes to satisfy the requirements for flag $t$-FTEC. Quantum Reed-Muller codes, Surface codes and hexagonal lattice color codes were shown to be families of codes that satisfy the sufficient condition. The flag $t$-FTEC protocol could be useful for fault-tolerant experiments performed in near term quantum devices since it tends to use fewer qubits than other FTEC schemes such as Steane, Knill and Shor EC. In \[subsec:CompareFlagecschemes\] we provided numerical evidence that with only 22 qubits, the flag $2$-FTEC protocol applied to the [$[\![19,1,5]\!]$]{} color code can achieve lower logical failure rates than other codes using similar numbers of qubits such as the rotated distance-3 surface code and Steane-EC applied to the Steane code. A clear direction of future work would be to find optimal general constructions of $t$-flag circuits for stabilizers of arbitrary weight that improve upon the general construction given in \[App:GeneralwFlagCircuitConstruction\]. Of particular interest would be circuits using few flag qubits and CNOT gates while minimizing the probability of false-positives (i.e. when the circuit flags without a high-weight error occurring). Finding other families of stabilizer codes which satisfy the sufficient or more general condition for flag $t$-FTEC would also be of great interest. One could also envisage hybrid schemes combining flag EC with other FTEC approaches. Another direction of future research would be to find general circuit constructions for simultaneously measuring multiple stabilizers while minimizing the number of required ancilla qubits. Further, we believe performing a rigorous numerical analysis to understand the impact of more compact circuit constructions on the codes threshold is of great interest. Lastly, the decoding complexity (i.e. generating the flag error set lookup tables) is limited by the decoding complexity of the code. In some cases, for example concatenated codes, it may be possible to exploit some structure to generate the flag error sets more efficiently. In the case of concatenated code, the decoding complexity would be reduced to the decoding complexity of the codes used at every level. Finding other scalable constructions for efficient decoding schemes using flag error correction remains an open problem. Acknowledgements ================ The authors would like to thank Krysta Svore, Tomas Jochym-O’Connor, Nicolas Delfosse and Jeongwan Haah for useful discussions. We also thank Steve Weiss for providing the necessary computational tools that allowed us to complete our work in a timely manner. C. C. would like to acknowledge the support of QEII-GSST and thank Microsoft and the QuArC group for its hospitality where all of this work was completed. Proof that the flag $t$-FTEC protocol satisfies the fault-tolerance criteria of \[Def:FaultTolerantDef\] {#app:ProtocolGeneralProof} ======================================================================================================== Consider the flag $t$-FTEC protocol described in \[subsec:GeneralProtocol\]. If the flag $t$-FTEC condition is satisfied, then both fault-tolerance criteria of \[Def:FaultTolerantDef\] will be satisfied. First note that the protocol always terminates. As was shown in the arguments leading to \[Eq:Nmax\] presented in \[subsec:GeneralProtocol\], the maximum number of syndrome measurement rounds is $\frac{1}{2}(t^{2}+3t+2)$. To prove fault-tolerance, in what follows we assume that there are at most $t$-faults during the protocol. Also, we define a benign fault to be a fault that either leaves all syndrome measurements in the protocol unchanged. By repeating the syndrome measurement using $t$-flag circuits, the following cases exhaust all possible errors for the occurrence of at most $t$ faults. *: The same syndrome is measured $t-n_{\text{diff}}+1$ times in a row and there are no flags.* At any time during the protocol, if there are no flags, there can be at most $t-n_{\text{diff}}$ remaining faults that occur (since it is guaranteed that there were at least $n_{\text{diff}}$ faults). Therefore, if the same syndrome was measured $t-n_{\text{diff}}+1$ times in a row, at least one round (say $r$) had to have been fault-free yielding the correct syndrome corresponding to the data qubit errors present at that time. Applying $E_{\text{min}}(s)$ will remove those errors. Furthermore, since all syndrome measurements are identical and there are no flags, there can be at most $t-n_{\text{diff}}$ errors which are introduced on the data blocks from faults during the $t-n_{\text{diff}}+1$ syndrome measurement rounds (excluding round $r$). Since none of the errors change the syndrome, after applying the correction, the output state can differ from the input codeword by an error of weight at most $t-n_{\text{diff}}$ (if the total number of faults input errors was $t$). For input states afflicted by an error of arbitrary weight, the output state will differ from a valid codeword (but not necessarily the input codeword) by an error of weight at most $t-n_{\text{diff}}$. Thus both conditions of \[Def:FaultTolerantDef\] are satisfied. *: There are no flags and $n_{\text{diff}} = t$.* The only way that $n_{\text{diff}} = t$ is if there were $t$-faults that each changed the syndrome measurement outcome. Further since there were no flags, an error $E$ afflicting the data qubits must satisfy $\text{wt}(E) \le t$. Thus repeating the syndrome measurement using non-flag circuits will correctly identify and remove the error in the case where the number of input errors and faults is $t$ or project the system back to the code space (to a possibly differ codeword) if there were $t$ faults and the input state was afflicted by an error of arbitrary weight . *: A set of $t$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{t}}) \}$ flagged.* Since $t$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{t}}) \}$ flagged, then no other faults can occur during the protocol. Hence, when repeating the syndrome measurement using non-flag circuits, the measured syndrome will correspond to an error $E_{r} \in \tilde{E}_{t}^{t}(g_{i_{1}},\cdots g_{i_{t}},s)$. Since from the flag $t$-FTEC condition all elements of $\tilde{E}_{t}^{t}(g_{i_{1}},\cdots g_{i_{t}},s)$ are logically equivalent, the product of errors resulting from the flag circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{t}}) \}$ will be corrected. Note that for an input error $E_{\text{in}}$ of arbitrary weight and since the final round must be error free, applying a correction a correction from the set $\tilde{E}_{t}^{t}(g_{i_{1}},\cdots g_{i_{t}},s)$ is guaranteed to return the system to the codespace. Thus both conditions of \[Def:FaultTolerantDef\] are satisfied. *: The $m$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{m}}) \}$ flagged with $1 \le m < t$, $n_{\text{diff}} = t -m $.* Here we can assume that at any point during the protocol and after the $j$’th flag, the syndrome never repeated more than $t-j-n_{\text{diff}}$ times. Otherwise case 5 of the protocol would already have occurred. As $m$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{m}}) \}$ have flagged and $n_{\text{diff}}=t-m$, then there can be no more faults. The final syndrome measurement using non-flag circuits will yield a syndrome corresponding to an error in the set $\tilde{E}_{t}^{m}(g_{i_{1}},\cdots g_{i_{m}},s)$ (and all elements are logically equivalent from the flag $t$-FTEC condition). Applying a recovery operator chosen from this set will thus remove the errors afflicting the data. If the input state differs from a valid codeword by an error of arbitrary weight, by definition of $\tilde{E}_{t}^{m}(g_{i_{1}},\cdots g_{i_{m}},s)$ the output state will be a valid codeword. *: The $m$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{m}}) \}$ flagged with $1 \le m < t$, $n_{\text{same}} = t -m - n_{\text{diff}} + 1$.* Given that $m$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{m}}) \}$ flagged, there are $r$ remaining faults that don’t result in a flag with $n_{\text{diff}} \le r \le t-m$. In this case, after the $m$’th flag, the syndrome measurement was repeated using $t$-flag circuits $t-m- n_{\text{diff}}+1$ times in a row and all syndromes were the same. It is thus guaranteed that at least one of the syndrome measurements $s$ was fault-free and correctly identified the errors arising from the flags and errors causing the syndrome to change giving $n_{\text{diff}}$ (along with some error $E$ which did not cause the circuits to flag with $\text{wt}(E) \le t-m- n_{\text{diff}}$). Consequently, if there are no errors on the input state, the overall error on the data will be $EE_{r}$ with $E_{r} \in \bigcup_{j=0}^{t-m-n_{\text{diff}}}\tilde{E}_{t}^{t-j-n_{\text{diff}}}(g_{i_{1}},\cdots ,g_{i_{m}}, s)$. Since all elements in $\bigcup_{j=0}^{t-m-n_{\text{diff}}}\tilde{E}_{t}^{t-j-n_{\text{diff}}}(g_{i_{1}},\cdots ,g_{i_{m}}, s)$ are logically equivalent from the flag $t$-FTEC condition, by choosing a correction from this set, the output state can differ from the input codeword by an error of at most weight $t-m- n_{\text{diff}}$. If there is an input error of arbitrary weight, then again one of the $t-m- n_{\text{diff}}+1$ rounds will have the correct syndrome $s$. The actual state of the data qubits after the protocol can differ from the state which had the correct syndrome by an error of weight at most $t-m- n_{\text{diff}}$. Therefore applying any correction with syndrome $s$ will return the system to the code space up to an error of weight at most $t-m- n_{\text{diff}}$. Fault-tolerant state preparation and measurement using flag $t$-FTEC {#app:StatePrepAndMeasure} ==================================================================== In this section we show how to fault-tolerantly prepare a logical ${|\overline{0}\rangle}$ state and how to perform fault-tolerant measurements for codes that satisfy the flag $t$-FTEC condition of \[app:GeneralFTEC\]. Note that there are several methods that can be used for doing so. Here we follow a procedure similar to that shown in [@Gottesman2010] when performing Shor EC. However, compared to Shor EC, the flag $t$-FTEC protocol requires fewer qubits. Furthermore, postselection is not necessary. Consider an $n$-qubit stabilizer code $C$ with stabilizer group $\mathcal{S} = \langle g_{1},\cdots , g_{n-k} \rangle$ that can correct up to $t$ errors. Notice that the encoded ${|\overline{0}\rangle}$ state is a $+1$ eigenstate of the logical $\overline{Z}$ operator and all of the codes stabilizer generators. For $k$ encoded qubits, ${|\overline{0}\rangle}$ would be $+1$ eigenstate of $\{ \overline{Z}_{1}, \cdots \overline{Z}_{k} \}$ and all of the codes stabilizers. For notational simplicity, in what follows we assume $k=1$. The state ${|\overline{0}\rangle}$ is a stabilizer state completely specified by the full stabilizer generators of $\mathcal{S}$ and $\overline{Z}$. We can think of $\mathcal{S}' = \langle g_{1}, \cdots g_{n-1}, \overline{Z} \rangle$ as a stabilizer code with zero encoded qubits and a $2^{0}=1$ dimensional Hilbert space. Thus any state which is a $+1$ eigenstate of all operators in $\mathcal{S}'$ will correspond to the encoded ${|\overline{0}\rangle}$ state. Now, suppose we prepare ${|\overline{0}\rangle}_{\text{in}}$ using a non-fault-tolerant encoding and perform a round of flag $t$-FTEC using the extended stabilizers $\langle g_{1}, \cdots g_{n-1}, \overline{Z} \rangle$. Then by the second criteria of \[Def:FaultTolerantDef\], the output state ${|\overline{0}\rangle}_{\text{out}}$ is guaranteed to be a valid codeword with at most $t$ single-qubit errors. But for the extended stabilizers $\langle g_{1}, \cdots g_{n-1}, \overline{Z} \rangle$ there is only *one* valid codeword which corresponds to the encoded ${|\overline{0}\rangle}$ state. In fact, by the second criteria of \[Def:FaultTolerantDef\], any $n$-qubit input state prepared using non-fault-tolerant circuits is guaranteed to be an encoded ${|\overline{0}\rangle}$ state if there are no more than $t$ faults in the EC round. We point out that the flag $t$-FTEC condition of \[subsec:GeneralProtocol\] is trivially satisfied for $\mathcal{S}'$ since the codes logical operators are now stabilizers. In other words, if two errors belong to the set $\tilde{E}^{m}_{t}(g_{i_{1}}, \cdots, g_{i_{k}},s)$, then their product will always be a stabilizer. Therefore, the flag $t$-FTEC protocol can always be applied for the code $\mathcal{S}'$. To summarize, the encoded ${|\overline{0}\rangle}$ state can be prepared by first preparing any $n$-qubit state using non-fault-tolerant circuits followed by applying a round of flag $t$-FTEC using the extended stabilizers $\langle g_{1}, \cdots g_{n-1}, \overline{Z} \rangle$. This guarantees that the output state will be the encoded ${|\overline{0}\rangle}$ state with at most $t$ single-qubit errors. Now suppose we want to measure the eigenvalue of a logical operator $\overline{P}$ where $P$ is a Pauli. If $C$ is a CSS code and the logical operator being measured is $X$ or $Z$, one could measure the eigenvalue by performing the measurement transversally. So suppose $C$ is not a CSS code. From [@Gottesman2010] we require that performing a measurement with $s$ faults on an input state with $r$ errors ($r+s \le t$) is equivalent to correcting the $r$ errors and performing the measurement perfectly. The protocol for fault-tolerantly measuring the eigenvalue of $\overline{P}$ is described as follows: 1. Perform flag $t$-FTEC. 2. Use a $t$-flag circuit to measure the eigenvalue of $\overline{P}$. 3. Repeat steps 1 and 2 $2t+1$ times and take the majority of the eigenvalue of $\overline{P}$. Step 1 is used to remove input errors to the measurement procedure. However during error correction, a fault can occur which could cause a new error on the data. Thus by repeating the measurement without performing error correction, the wrong state would be measured each time if there were no more faults. But repeating the syndrome $2t+1$ times, it is guaranteed that at least $t+1$ of the syndrome measurements had no faults and that the correct eigenvalue of $\overline{P}$ was measured. Thus taking the majority of the measured eigenvalues will give the correct answer. Note that during the fault-tolerant measurement procedure, if there is a flag either during the error correction round or during the measurement of $\overline{P}$, when error correction is performed one corrects based on the possible set of errors resulting from the flag. Candidate general $w$-flag circuit construction {#App:GeneralwFlagCircuitConstruction} =============================================== {width="105.00000%"} In this section we provide a candidate general $w$-flag circuit construction for measuring the stabilizer $Z^{\otimes w}$. Although we do not provide a rigorous proof that our construction results in a $w$-flag circuit, we give several arguments as evidence that it satisfies all the criteria of a $w$-flag circuit. An illustration of the circuit construction (for $w=12$) is given in \[fig:GeneralWflagFig\] and the description for how the circuit is constructed for arbitrary $w$ is provided in the caption. In what follows, we can restrict our attention to the case in which all $v$ faults occur on CNOT gates in the circuit. The effect on the measurement outcomes and data qubits due to a set of $v$ faults that include faults at idle and measurement locations can always occur due to at most $v$ faults at CNOT locations only (as every qubit is involved in at least one CNOT). Moreover, we can assume that for $\text{CNOT}_{\text{fm}}$ gates, the faults belong to the set $\{ IZ, ZI, ZZ \}$ since $X$ errors would never propagate to the data or affect the measurement outcome of a flag qubit. For $\text{CNOT}_{\text{dm}}$ gates, we can assume that faults belong to the set $\{XZ,XI \}$. We only consider $Z$ errors on the target qubit of a $\text{CNOT}_{\text{dm}}$ for the same reason that was given for $\text{CNOT}_{\text{fm}}$ gates. For the control qubit, an X errors guarantees that the weight of the data qubit error increases even after the application of a satbilizer (since we are measuring $Z^{\otimes w}$). We will use the following useful terminology: we say that a single-qubit Pauli at a time step in the circuit propagates to a qubit at a particular time-step if it would do so in the fault-free circuit. Given a single-qubit Pauli at a time step in the circuit, we say that another qubit is affected by the Pauli if it propagates to that qubit in any time step. We now provide arguments for why the circuit is a $w$-flag circuit. First, note that every $\text{CNOT}_{\text{fm}}$ gate comes as part of a pair with the measurement qubit being the target qubit. This ensures that when the circuit is fault-free, it implements a projective measurement of $Z^{\otimes w}$ without flagging. Next, notice that apart from the last two $\text{CNOT}_{\text{dm}}$ gates, each $\text{CNOT}_{\text{dm}}$ gate is followed by two $\text{CNOT}_{\text{fm}}$ gates, one with its partnering $\text{CNOT}_{\text{fm}}$ located before the first $\text{CNOT}_{\text{dm}}$ and the other partner is located in between the last two $\text{CNOT}_{\text{dm}}$ gates. Thus if there is a single $Z$ error on the measurement qubit which propagates to any of the data qubits, the circuit will flag. In all circuits considered in this section, $s_{0}$ will correspond to the sequence of $\text{CNOT}_{\text{fm}}$ gates that come before the first $\text{CNOT}_{\text{dm}}$ gate. First consider the shorter circuit construction using only the first family of $\text{CNOT}_{\text{fm}}$ gates from the construction in \[fig:GeneralWflagFig\] (see the example in \[fig:ErrorPropLogic\]). We can separate the set of all locations into subsets including two $\text{CNOT}_{\text{fm}}$ gates and one $\text{CNOT}_{\text{dm}}$ gate as shown in \[fig:CNOTsectionAppendix\] (apart from the last $\text{CNOT}_{\text{dm}}$). This circuit segment can increase the weight of the data error by at most one. There are four cases with inputs on the measurement qubit before the first $\text{CNOT}_{\text{fm}}$ and $\text{CNOT}_{\text{dm}}$ being $\{(I,I),(I,Z),(Z,I),(Z,Z)\}$. Note that if the following property held for each segment, then the circuit would be $w$-flag: for all inputs to the segment, if the weight of the data error increases and there are no faults in the segment, the segment flags. Unfortunately, for the input $(Z,Z)$, this is not the case. Both input $Z$ must come from at least two faults. ![Example of five faults that lead to an error of weight six on the data without causing a flag when only the first family of $\text{CNOT}_{\text{fm}}$ gates are used in the construction of \[fig:GeneralWflagFig\] (here $w=10$). Errors arising from faults are shown in blue and the resulting errors after propagating through the CNOT gates are shown in red. []{data-label="fig:ErrorPropLogic"}](ErrorPropLogic.PNG){width="45.00000%"} Note that if $v$ faults results in a data qubit error of weight greater than $v$ without causing the circuit in \[fig:ErrorPropLogic\] to flag, there must be either an $IZ$ fault followed by no fault in a consecutive pair of $\text{CNOT}_{\text{fm}}$ gates belonging to $s_{0}$ or a $ZZ$ fault followed by two $\text{CNOT}_{\text{fm}}$ gates that don’t fail in $s_{0}$. Moreover, a poor choice of ordering of the $\text{CNOT}_{\text{fm}}$ gates in $s_{1},s_{2}$ and $s_{3}$ can result in four faults causing a weight $\frac{w}{2}+1$ error on the data without causing the circuit to flag. Therefore, the ordering of the $\text{CNOT}_{\text{fm}}$ gates in the sets $s_{1},s_{2}$ and $s_{3}$ is chosen such that most $Z$ errors in $s_{0}$ that first propagate to flag qubits connected to gates in $s_{1}$, will then propagate to flag qubits in $s_{3}$ and vice-versa. Typically, if a $Z$ error propagates through multiple $\text{CNOT}_{\text{dm}}$ gates in $s_{1}$, then unless $\text{CNOT}_{\text{fm}}$ gates in $s_{3}$ fail, the flag qubits affected by the $Z$ error would flag. Furthermore, the total number of required failures for gates in $s_{3}$ to cancel the $Z$ errors will typically be equal to the number of times the $Z$ error propagated to the data. There are however cases which don’t flag in which $v$ faults in the circuit construction presented in \[fig:ErrorPropLogic\] lead to more than $v$ errors on the data qubit, such as the example given in the figure. All such problematic cases that we found had a $Z$ error on the target qubit in one of the last few $\text{CNOT}_{\text{fm}}$ gates in $s_0$, followed by a $Z$ error on the target qubit in one of the first few $\text{CNOT}_{\text{dm}}$ gates in $s_1$. Then further $Z$ errors occur throughout the remainder of the circuit which propagate to the data while preventing the flag qubits affected by the previous errors from flagging. Further, a $Z$ error on the control qubit of the second $\text{CNOT}_{\text{fm}}$ in $s_{2}$ cancels the $Z$ which propagates to the flag qubit coupled to that $\text{CNOT}_{\text{fm}}$ gate. ![Illustration of a pair of $\text{CNOT}_{\text{fm}}$ gates as well as a segment of $\text{CNOT}_{\text{dm}}$ followed by $\text{CNOT}_{\text{fm}}$ gate. The first $\text{CNOT}_{\text{fm}}$ gate belongs to the sequence of $\text{CNOT}_{\text{fm}}$ gates that come before the first $\text{CNOT}_{\text{dm}}$ gate (see \[fig:GeneralWflagFig\]). []{data-label="fig:CNOTsectionAppendix"}](CNOTsectionAppendix.PNG){width="40.00000%"} This particular problematic fault pattern would lead to flags if it occurred within the full circuit construction of \[fig:GeneralWflagFig\] (if the additional locations of the larger circuit do not fail). As this was the only type of problematic fault pattern that we found, one would hope that all problematic fault patterns are rendered non problematic provided no additional locations fail. Since the additional $\text{CNOT}_{\text{fm}}$ gates always occur immediately after one of the original $\text{CNOT}_{\text{fm}}$ gates (or after the last $\text{CNOT}_{\text{fm}}$ gate), as far as the flag properties of the original circuit are concerned, no new problematic fault patterns are introduced. We conclude this section by noting that our candidate general $w$-flag circuit construction requires $w-1$ flag qubits and is implemented in $7w-8$ time steps. This is clearly not optimal in general since for example, as shown in \[fig:6flagCircuit\], a $w$-flag circuit was found (for $w=6$) which requires only three flag qubits instead of five and the circuit is implemented in 14 time steps instead of 34. It is thus still an open problem to find optimal $w$-flag circuits for arbitrary $w$. Quantum Reed-Muller codes {#app:QRMcodes} ========================= In this section we first describe how to construct the family of quantum Reed-Muller codes $\text{QRM}(m)$ with code parameters $[\![ 2^{m}-1,k=1,d=3 ]\!]$ following [@ADP14]. We then show that the family of $\text{QRM}(m)$ codes satisfy the sufficient flag 1-FTEC condition of \[subsec:Remarks\]. Reed-Muller codes of order $m$ ($\text{RM}(1,m)$) are defined recursively from the following generator matrices: First, $\text{RM}(1,1)$ has generator matrix $$\begin{aligned} G_{1}=\left( \begin{array}{cc} 1 & 1\\ 0 & 1\\ \end{array} \right), \label{eq:G1mat}\end{aligned}$$ and $\text{RM}(1,m+1)$ has generator matrix $$\begin{aligned} G_{m+1}=\left( \begin{array}{cc} G_{m} & G_{m}\\ 0 & 1\\ \end{array} \right), \label{eq:GMmat}\end{aligned}$$ where 0 and 1 are vectors of zeros and ones in \[eq:GMmat\]. The dual of $\text{RM}(1,m+1)$ is given by the higher order Reed-Muller code $\text{RM}(m-2,m)$. In general, the generator matrices for higher-order Reed-Muller codes $\text{RM}(r,m)$ are given by $$\begin{aligned} H_{r,m+1}=\left( \begin{array}{cc} H_{r,m} & H_{r,m}\\ 0 & H_{r-1,m}\\ \end{array} \right). \label{eq:GRMmat}\end{aligned}$$ with $$\begin{aligned} H_{2,1}=H_{1,1} = \left( \begin{array}{cc} 1 &1\\ 0 & 1\\ \end{array} \right), \label{eq:G21mat}\end{aligned}$$ The $X$ stabilizer generators of $\text{QRM}(m)$ are derived from shortened Reed-Muller codes where the first row and column of $G_{m}$ are deleted. We define the resulting generator matrix as $\overline{G}_{m}$. The $Z$ stabilizer generators are obtained by deleting the first row and column of $H_{m-2,m}$. Similarly, we define the resulting generator matrix as $\overline{H}_{m-2,m}$. As was shown in [@ADP14], $\text{rows}(\overline{G}_{m}) \subset \text{rows}(\overline{H}_{m-2,m})$ and each row has weight $2^{m-1}$. Therefore, all the $X$-type stabilizer generators of $\text{QRM}(m)$ have corresponding $Z$-type stabilizers. By construction, the remaining rows of $\overline{H}_{m-2,m}$ will have weight $2^{m-2}$. Furthermore, every weight $2^{m-2}$ row has support contained within some weight $2^{m-1}$ row of the generator matrix $\overline{H}_{m-2,m}$. Therefore, every $Z$-type stabilizer generator has support within the support of an $X$ generator. Implementation of Steane error correction {#app:SteaneECSection} ========================================= In this section we describe how to implement Steane error correction and discuss its fault-tolerant properties. We also provide a comparison of a version of Steane error correction with flag 2-FTEC protocol described in \[subsec:Distance5protocol\] applied to the [$[\![19,1,5]\!]$]{} code. Steane error correction is a fault-tolerant scheme that applies to the Calderbank-Shor-Steane (CSS) family of stabilizer codes [@Steane97]. In Steane error correction, the idea is to use encoded ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle} = ({|\overline{0}\rangle} + {|\overline{1}\rangle})/\sqrt{2}$ ancilla states to perform the syndrome extraction. The ancilla’s are encoded in the same error correcting code that is used to protect the data. The $X$ stabilizer generators are measured by preparing the encoded ${|\overline{0}\rangle}$ state and performing transversal CNOT gates between the ancilla and the data, with the ancilla acting as the control qubits and the data acting as the target qubits. After applying the transversal CNOT gates, the syndrome is obtained by measuring ${|\overline{0}\rangle}$ transversally in the $X$-basis. The code construction for CSS codes is what guarantees that the correct syndrome is obtained after applying a transversal measurement (see [@Gottesman2010] for more details). [0.25]{} ![(a) Fault-tolerant Steane error correction circuit for distance-three CSS codes. Each line represents an encoded qubit. The circuit uses only two encoded ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ ancilla states (encoded in the same error correcting code which protects the data) to ensure that faults in the preparation circuits of the ancilla’s don’t spread to the data block. (b) Fault-tolerant Steane error correction circuit which can be used for any distance-three CSS stabilizer code encoding the data. There are a total of eight encoded ancilla qubits instead of four. The dark bold lines represent resting qubits. Note that the circuit in \[fig:SteaneCSSforNonPerfect\] could in some cases be used for higher distance CSS codes with appropriately chosen circuits for ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ ancilla states (see [@PR12]).[]{data-label="fig:SteaneECcircuitsAll"}](SteaneECcircuitforPerfectCSS.png "fig:"){width="\textwidth"} [0.25]{} ![(a) Fault-tolerant Steane error correction circuit for distance-three CSS codes. Each line represents an encoded qubit. The circuit uses only two encoded ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ ancilla states (encoded in the same error correcting code which protects the data) to ensure that faults in the preparation circuits of the ancilla’s don’t spread to the data block. (b) Fault-tolerant Steane error correction circuit which can be used for any distance-three CSS stabilizer code encoding the data. There are a total of eight encoded ancilla qubits instead of four. The dark bold lines represent resting qubits. Note that the circuit in \[fig:SteaneCSSforNonPerfect\] could in some cases be used for higher distance CSS codes with appropriately chosen circuits for ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ ancilla states (see [@PR12]).[]{data-label="fig:SteaneECcircuitsAll"}](SteaneECcircuitforPerfectNonCSS.png "fig:"){width="\textwidth"} FTEC scheme Noise model Number of qubits Time steps ($T_{\mathrm{time}}$) Pseudo-threshold ------------------------------ --------------------- ------------------ ------------------------------------- -------------------------------------------------------- Full Steane-EC $\tilde{p} = p$ $\ge 171$ 15 $p_{\mathrm{pseudo}} = (3.50 \pm 0.14) \times 10^{-3}$ Full Steane-EC $\tilde{p} = p/100$ $\ge 171$ 15 $p_{\mathrm{pseudo}} = (1.05 \pm 0.04) \times 10^{-3}$ Flag-EC [$[\![19,1,5]\!]$]{} $\tilde{p} = p$ 22 $504 \le T_{\mathrm{time}} \le 960$ $p_{\mathrm{pseudo}} = (1.14 \pm 0.02) \times 10^{-5}$ Flag-EC [$[\![19,1,5]\!]$]{} $\tilde{p} = p/100$ 22 $504 \le T_{\mathrm{time}} \le 960$ $p_{\mathrm{pseudo}} = (7.74 \pm 0.16) \times 10^{-5}$ Similarly, the $Z$-stabilizer generators are measured by preparing the encoded ${|\overline{+}\rangle}$, applying CNOT gates transversally between the ancilla and the data with the data acting as the control qubits and the ancilla’s acting as the target qubits. The syndrome is then obtained by measuring ${|\overline{+}\rangle}$ transversally in the $Z$-basis. The above protocol as stated is not sufficient in order to be fault-tolerant. The reason is that in general the circuits for preparing the encoded ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ are not fault-tolerant in the sense that a single error can spread to a multi-weight error which could then spread to the data block when applying the transversal CNOT gates. To make the protocol fault-tolerant, extra ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ ancilla states (which we call verifier qubits) are needed to check for multi-weight errors at the output of the ancilla states. [0.3]{} ![Logical failure rate of the full fault-tolerant Steane error correction approach of \[fig:SteaneCSSforNonPerfect\] and the flag 2-FTEC protocol of \[subsec:Distance5protocol\] applied to the [$[\![19,1,5]\!]$]{} code. In (a) idle qubits are chosen to fail with a total probability $\tilde{p} = p$ while in (b) idle qubits fail with probability $\tilde{p} = p/100$. The intersection between the dashed curve and solid lines represent the pseudo-threshold of both error correction schemes. []{data-label="fig:ResultFullFaultTolerantSteanevsCB"}](FullFaulTolerantSteanep.png "fig:"){width="\textwidth"} [0.3]{} ![Logical failure rate of the full fault-tolerant Steane error correction approach of \[fig:SteaneCSSforNonPerfect\] and the flag 2-FTEC protocol of \[subsec:Distance5protocol\] applied to the [$[\![19,1,5]\!]$]{} code. In (a) idle qubits are chosen to fail with a total probability $\tilde{p} = p$ while in (b) idle qubits fail with probability $\tilde{p} = p/100$. The intersection between the dashed curve and solid lines represent the pseudo-threshold of both error correction schemes. []{data-label="fig:ResultFullFaultTolerantSteanevsCB"}](FullFaulTolerantSteanep100.png "fig:"){width="\textwidth"} For the ${|\overline{0}\rangle}$ ancilla, multiple $X$ errors can spread to the data if left unchecked. Therefore, another encoded ${|\overline{0}\rangle}$ ancilla is prepared and a transversal CNOT gate is applied between the two states with the ancilla acting as the control and the verifier state acting as target. Anytime $X$ errors are detected the state is rejected and the error correction protocol start over. Further, if the verifier qubit measures a $-1$ eigenvalue of the logical $Z$ operator, the ancilla qubit is also rejected. A similar technique is used for verifying the ${|\overline{+}\rangle}$ state (see \[fig:SteaneCSSforPerfect\]). For the [$[\![7,1,3]\!]$]{} Steane code, an error $E=Z_{i}Z_{j}$ can always be written as $E=\overline{Z}Z_{k}$ where $\overline{Z}$ is the logical $Z$ operator (this is not true for general CSS codes). But ${|\overline{0}\rangle}$ is a $+1$ eigenstate of $\overline{Z}$. Therefore, we don’t need to worry about $Z$ errors of weight greater than one occurring during the preparation of the ${|\overline{0}\rangle}$ state. In [@AGP06] it was shown that unlike for the [$[\![7,1,3]\!]$]{} code, for general CSS codes, the encoded ancilla states need to be verified for both $X$ and $Z$ errors in order for Steane error correction to satisfy the fault-tolerant properties of \[Def:FaultTolerantDef\]. We show the general distance-three fault-tolerant scheme in \[fig:SteaneCSSforNonPerfect\]. Note that the circuit in \[fig:SteaneCSSforPerfect\] will only satisfy the fault-tolerant criteria of \[Def:FaultTolerantDef\] for perfect distance-three CSS codes (see [@AGP06] for more details). In \[subsec:CompareFlagecschemes\] we computed logical failure rates for Steane error correction applied to the [$[\![19,1,5]\!]$]{} code using the circuit of figure \[fig:SteaneCSSforPerfect\] in order to minimize the number of physical qubits. However, since the [$[\![19,1,5]\!]$]{} code is not a perfect CSS code, only the circuit in \[fig:SteaneCSSforNonPerfect\] satisfies all the criteria of \[Def:FaultTolerantDef\]. This explains why the leading order contributions to the logical failure was of the form $p_{\text{L}} = c_{1}p^{2}+c_{2}p^{3} + \mathcal{O}(p^{4})$ instead of $p_{\text{L}} = cp^{3} + \mathcal{O}(p^{4})$ (which would be the case for a distance-5 code). In \[fig:ResultFullFaultTolerantSteanevsCB\] we applied Steane error correction using the circuit of \[fig:SteaneCSSforNonPerfect\] to achieve the full error correcting capabilities of the [$[\![19,1,5]\!]$]{} code. We used methods presented in [@PR12; @CJL16b] in order to obtain the encoded ${|\overline{0}\rangle}$ state (since the [$[\![19,1,5]\!]$]{} code is self-dual, the ${|\overline{+}\rangle}$ state is obtain by interchanging all physical ${|0\rangle}$ and ${|+\rangle}$ states and reversing the direction of the CNOT gates). Note that not all ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ circuits had the same sequence of CNOT gates. This was to ensure that a single fault in two different preparation circuits, i.e. for ${|\overline{0}\rangle}$ and for ${|\overline{+}\rangle}$, would not lead to uncorrectable $X$ or $Z$ errors that would go undetected by the verifier ancillas and at the same time propagate to the data block. The results are compared with the flag 2-FTEC protocol of \[subsec:Distance5protocol\] applied to the [$[\![19,1,5]\!]$]{} for the noise models where idle qubits fail with probability $\tilde{p}=p$ and $\tilde{p}=p/100$. In both cases the logical failure rates have a leading order $p^{3}$ contribution (which is determined from finding the best fit curve to the data). The pseudo-threshold results are given in \[tab:PseudoThreshFullSteane\]. As can be seen, the full Steane-EC protocol using the circuit of \[fig:SteaneCSSforNonPerfect\] achieves significantly lower logical failure rates compared to Steane-EC using the circuit in \[fig:SteaneCSSforPerfect\] at the cost of using a minimum of 171 qubits compared to a minimum of 95 qubits. In contrast, the flag 2-FTEC scheme of \[subsec:Distance5protocol\] has a pseudo-threshold that is one to two orders of magnitude lower than than the full Steane-EC scheme but requires only 22 qubits. Implementation of Surface code error correction {#app:SurfaceECSection} =============================================== [0.2]{} ![(a) The $d=3$ surface code, with data qubits represented by white circles. The $X$ ($Z$) stabilizer generators are measured with measurement ancillas (gray) in red (green) faces (b) For perfectg measurements, the graph $G_{\text{2D}}$ used to correct $X$ type errors (here for $d=5$) consists of a black node for each $Z$-stabilizer, and a black edge for each data qubit in the surface code. White boundary nodes and blue boundary edges are added. Black and blue edges are given weight one and zero respectively. In this example, a two qubit $X$ error has occurred causing three stabilizers to be violated (red nodes). A boundary node is also highlighted and a minimum weight correction (red edges) which terminates on highlighted nodes is found. The algorithm succeeds as the error plus correction is a stabilizer.[]{data-label="fig:Surface17Circuits"}](SurfaceCodeLattice.png "fig:"){width="\textwidth"} [0.2]{} ![(a) The $d=3$ surface code, with data qubits represented by white circles. The $X$ ($Z$) stabilizer generators are measured with measurement ancillas (gray) in red (green) faces (b) For perfectg measurements, the graph $G_{\text{2D}}$ used to correct $X$ type errors (here for $d=5$) consists of a black node for each $Z$-stabilizer, and a black edge for each data qubit in the surface code. White boundary nodes and blue boundary edges are added. Black and blue edges are given weight one and zero respectively. In this example, a two qubit $X$ error has occurred causing three stabilizers to be violated (red nodes). A boundary node is also highlighted and a minimum weight correction (red edges) which terminates on highlighted nodes is found. The algorithm succeeds as the error plus correction is a stabilizer.[]{data-label="fig:Surface17Circuits"}](Graphd5Planar.png "fig:"){width="\textwidth"} We consider the rotated surface code [@KITAEV97Surface; @TS14; @BK98; @DKLP02; @FMMC12; @PhysRevLett.90.016803] as shown in Fig. \[fig:Surface17Lattice\], which has $n=d^2$ data qubits for distance $d$. Although we are concerned with error correction under the circuit level noise model described in \[subsec:NoiseAndNumerics\], it is useful to build intuition by first considering the idealized noise model in which stabilizer measurements are perfect, and single qubit $X$ errors occur with probability $2p/3$ ($Z$ errors can be treated in the same way). An $X$ type error $E$ occurs with probability $\mathcal{O}(p^{\text{wt}(E)})$, and has syndrome $s(E)$. The minimum weight $X$-type correction can be found efficiently for the surface code in terms of the graph $G_{\text{2D}}$ shown in Fig. \[fig:Surfaced5Graph\]. The graph $G_{\text{2D}}$ has a bulk node (black circle) for each $Z$ stabilizer generator, and a bulk edge (black) for each data qubit. A bulk edge coming from a bulk node corresponds to the edge’s data qubit being in the support of the node’s stabilizer. The graph also contains boundary nodes (white boxes) and boundary edges (blue), which do not correspond to stabilizers or data qubits. Each bulk and boundary edge is assigned weight one and zero respectively. The minimum weight decoder is then implemented as follows. After the error $E$ is applied, the nodes corresponding to unsatisfied stabilizers are highlighted. If an odd number of stabilizers was unsatisfied, one of the boundary nodes is also highlighted. Highlighted nodes are then efficiently paired together by the minimum weight connections in the graph, by Edmonds’ algorithm [@Edmonds65; @Kolmogorov09]. The correction $C$ is applied to the edges in the connection. Note that any single $\mathcal{O}(p)$ fault in this noise model corresponds to a weight one edge on the graph. [0.3]{} ![Circuits for measuring (a) $Z$-type, and (b) $X$-type generators. Identity gates (black rectangles) are inserted in the $Z$-type stabilizer measurement circuits to ensure that all measurements are synchronized. Note that unlike in [@FMMC12], to be consistent with the other schemes in this paper, we assume that we can prepare and measure in both the $X$ and $Z$ basis.[]{data-label="fig:Surface17Circuits"}](ZstabMeasSurfaceCode.png "fig:"){width="\textwidth"} [0.3]{} ![Circuits for measuring (a) $Z$-type, and (b) $X$-type generators. Identity gates (black rectangles) are inserted in the $Z$-type stabilizer measurement circuits to ensure that all measurements are synchronized. Note that unlike in [@FMMC12], to be consistent with the other schemes in this paper, we assume that we can prepare and measure in both the $X$ and $Z$ basis.[]{data-label="fig:Surface17Circuits"}](XstabMeasSurfaceCode.png "fig:"){width="\textwidth"} For circuit noise, we introduce a measurement qubit for each stabilizer generator, as represented by gray circles in Fig. \[fig:Surface17Lattice\], and circuits must be specified to implement the measurements, such as those in Fig. \[fig:Surface17Circuits\]. The performance of the code is sensitive to the choice of circuit [@TS14], for example a poor choice could allow a single fault to cause a logical failure for $d=3$ for any choice of decoder. To implement the decoder, first construct a new three dimensional graph $G_{\text{3D}}$ by stacking $d$ copies of the planar graph $G_{\text{2D}}$ that was shown in Fig. \[fig:Surfaced5Graph\], and adding new bulk (boudnary) edges to connect bulk (boudnary) nodes in neighboring layers. We also add additional diagonal edges such that any single $\mathcal{O}(p)$ fault in the measurement circuits corresponds to a weight-one edge in $G_{\text{3D}}$ (see Fig. \[fig:LowerRightAndLowerLeft\]). For simplicity, we do not involve further possible optimizations such as setting edge weights based on precise probabilities and including $X$-$Z$ correlations [@PhysRevA.83.020302]. [0.11]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge1.png "fig:"){width="\textwidth"} [0.11]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge2.png "fig:"){width="\textwidth"} [0.11]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge3.png "fig:"){width="\textwidth"} [0.15]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge5.png "fig:"){width="\textwidth"} [0.15]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge6.png "fig:"){width="\textwidth"} [0.25]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge4.png "fig:"){width="\textwidth"} All simulations of the surface code are performed using the circuit noise model in \[subsec:NoiseAndNumerics\], with the graph $G_{\text{3D}}$ described above as follows (to correct $X$ errors): 1. : Stabilizer outcomes are stored over $d$ rounds of noisy error correction, followed by one round of perfect error correction. The net error $E$ applied over all $d$ rounds is recorded. 2. : Nodes in the graph $G_{\text{3D}}$ are highlighted if the corresponding $Z$-type stabilizer outcome changes in two consecutive rounds. [^5] 3. : Find a minimal edge set forming paths that terminate on highlighted nodes. Highlight the edge set. 4. : The highlighted edges in $G_{\text{3D}}$ are mapped edges in the planar graph $G_{\text{2D}}$, and are then added modulo 2. 5. : The $X$-type correction $C_X$ is applied to highlighted edges in $G_{\text{2D}}$. The $Z$ correction $C_Z$ is found analogously. Finally, if the residual Pauli $R = E C_X C_Z$ is a logical operator, we say the protocol succeeded, otherwise we say it failed. FTEC scheme Noise model Number of qubits Time steps ($T_{\mathrm{time}}$) Pseudo-threshold ----------------------------- ----------------- ------------------ ------------------------------------ -------------------------------------------------------- Flag-EC [$[\![7,1,3]\!]$]{} $\tilde{p} = p$ 9 $36 \le T_{\mathrm{time}} \le108$ $p_{\mathrm{pseudo}} = (3.39 \pm 0.10) \times 10^{-5}$ Flag-EC [$[\![7,1,3]\!]$]{} 11 $34 \le T_{\mathrm{time}} \le 104$ $p_{\mathrm{pseudo}} = (2.97 \pm 0.01) \times 10^{-5}$ Compact implementation of flag error correction {#app:CompactRepFlagQubit} =============================================== [$[\![5,1,3]\!]$]{} [$[\![7,1,3]\!]$]{} [$[\![19,1,5]\!]$]{} code [$[\![17,1,5]\!]$]{} code -------------------------------------------------------------------- ------------------------------------------------------------------- --------------------------------------------------------------------- ----------------------------------------------------------------- $X_{1}Z_{2}Z_{3}X_{4}$ $Z_{4}Z_{5}Z_{6}Z_{7}$ $Z_{1}Z_{2}Z_{3}Z_{4}$,  $X_{1}X_{2}X_{3}X_{4}$ $Z_{1}Z_{2}Z_{3}Z_{4}$,  $X_{1}X_{2}X_{3}X_{4}$ $X_{2}Z_{3}Z_{4}X_{5}$ $Z_{2}Z_{3}Z_{6}Z_{7}$ $Z_{1}Z_{3}Z_{5}Z_{7}$,  $X_{1}X_{3}X_{5}X_{7}$ $Z_{1}Z_{3}Z_{5}Z_{6}$,   $X_{1}X_{3}X_{5}X_{6}$ $X_{1}X_{3}Z_{4}Z_{5}$ $Z_{1}Z_{3}Z_{5}Z_{7}$ $Z_{12}Z_{13}Z_{14}Z_{15}$,  $X_{12}X_{13}X_{14}X_{15}$ $Z_{5}Z_{6}Z_{9}Z_{10}$,  $X_{5}X_{6}Z_{9}Z_{10}$ $Z_{1}X_{2}X_{4}Z_{5}$ $X_{4}X_{5}X_{6}X_{7}$ $Z_{1}Z_{2}Z_{5}Z_{6}Z_{8}Z_{9}$,  $X_{1}X_{2}X_{5}X_{6}X_{8}X_{9}$ $Z_{7}Z_{8}Z_{11}Z_{12}$,  $X_{7}X_{8}X_{11}X_{12}$ $X_{2}X_{3}X_{6}X_{7}$ $Z_{6}Z_{9}Z_{16}Z_{19}$,  $X_{6}X_{9}X_{16}X_{19}$ $Z_{9}Z_{10}Z_{13}Z_{14}$,  $X_{9}X_{10}X_{13}X_{14}$ $X_{1}X_{3}X_{5}X_{7}$ $Z_{16}Z_{17}Z_{18}Z_{19}$,  $X_{16}X_{17}X_{18}X_{19}$ $Z_{11}Z_{12}Z_{15}Z_{16}$,  $X_{11}X_{12}X_{15}X_{16}$ $Z_{10}Z_{11}Z_{12}Z_{15}$,   $X_{10}X_{11}X_{12}X_{15}$ $Z_{8}Z_{12}Z_{16}Z_{17}$,  $X_{8}X_{12}X_{16}X_{17}$ $Z_{8}Z_{9}Z_{10}Z_{11}Z_{16}Z_{17}$ $Z_{3}Z_{4}Z_{6}Z_{7}Z_{10}Z_{11}Z_{14}Z_{15}$ $Z_{5}Z_{7}Z_{8}Z_{11}Z_{12}Z_{13}$ $X_{3}X_{4}X_{6}X_{7}X_{10}X_{11}X_{14}X_{15}$ $X_{5}X_{7}X_{8}X_{11}X_{12}X_{13}$ $X_{8}X_{9}X_{10}X_{11}X_{16}X_{17}$ $\overline{X} = X^{\otimes 5}$,$\overline{Z} = Z^{\otimes 5}$ $\overline{X} = X^{\otimes 7}$,$\overline{Z} = Z^{\otimes 7}$ $\overline{X} = X^{\otimes 19}$,$\overline{Z} = Z^{\otimes 19}$ $\overline{X} = X^{\otimes 17}$,$\overline{Z} = Z^{\otimes 17}$ ![Circuit for measuring the $Z$ stabilizer generators of the [$[\![7,1,3]\!]$]{} code using one flag qubit and three measurement qubits. The circuit is constructed such that any single fault at a bad location leading to an error of weight greater than one will cause the circuit to flag. Moreover, any error that occurs when the circuit flags due to a single fault has a unique syndrome.[]{data-label="fig:CompactStabMeasure"}](SteaneCode4Ancilla.png){width="40.00000%"} ![Logical failure rates of the flag 1-FTEC protocols using two and four ancilla qubits applied to the [$[\![7,1,3]\!]$]{} Steane code.[]{data-label="fig:PseudoThreshCompareSteaneDepth"}](CompareCompactFullp.png){width="35.00000%"} In [@CR17v1], it was shown that by using extra ancilla qubits in the flag-EC protocol, it is possible to measure multiple stabilizer generators during one measurement cycle which could reduce the circuit depth. Note that for the Steane code, measuring the $Z$ stabilizers using \[fig:StabFTwithAncilla\] requires only one extra time step. In this section we compare logical failure rates of the [$[\![7,1,3]\!]$]{} code using the flag-EC method of \[subsec:ReviewChaoReichardt\] which requires only two ancilla qubits and a flag-EC method which uses four ancilla qubits but that can measure all $Z$ stabilizer generators in one cycle (see \[fig:CompactStabMeasure\]). All $X$ stabilizers are measured in a separate cycle. Logical failure rates for $\tilde{p}=p$ are shown in \[fig:PseudoThreshCompareSteaneDepth\]. Pseudo-thresholds and the number of time steps required to implement the protocols are given in \[tab:PseudoThreshFullSteane2\]. Note that measuring stabilizers using two ancilla’s requires at most two extra time steps. Furthermore, the extra ancilla’s for measuring multiple stabilizers result in more idle qubit locations compared to using only two ancilla qubits. With the added locations for errors to be introduced, the flag error correction protocol using only two ancilla’s achieves a *higher* pseudo-threshold compared to the protocol using more ancilla’s. Thus assuming that reinitializing qubits can be done without introducing many errors into the system, FTEC using fewer qubits could achieve lower logical failure rates compared to certain schemes using more qubits. Stabilizer generators of various codes. {#app:ListStabGenerator} ======================================= In \[tab:StabilizerGeneratorsLists\] we provide stabilizer generators for the [$[\![5,1,3]\!]$]{} code, [$[\![7,1,3]\!]$]{} Steane code, [$[\![19,1,5]\!]$]{} and [$[\![17,1,5]\!]$]{} color codes. [^1]: This is the case for Shor, Steane and Knill EC with appropriately verified ancilla states. However the surface code does not satisfy this due to hook errors but nonetheless still satisfies condition 1 of \[Def:FaultTolerantDef\]. [^2]: To avoid confusing the notation of $C(P)$ that represents a circuit and $C$ that represents a code space, we always include the measured Pauli in parenthesis unless clear from context. [^3]: $n_{\text{diff}}$ tracks the minimum number of faults that could have caused the observed syndrome outcomes. For example, if the sequence $s_{1},s_{2},s_{1}$ was measured, $n_{\text{diff}}$ would increase by one since a single measurement fault could give rise to the given sequence (for example, this could be caused by a single CNOT failure which resulted in a data qubit and measurement error). However for the sequence $s_{1},s_{2},s_{1},s_{2}$, $n_{\text{diff}}$ would increase by two. [^4]: One could also define update rules analogous to those for $n_{\text{diff}}$ and $n_{\text{same}}$ when implementing Shor-EC which would only require at most $\frac{1}{2}(t^{2}+3t+2)$ syndrome measurement repetitions as in the flag $t$-FTEC protocol. [^5]: For an odd number of highlighted vertices, highlight the boundary vertex.

--- abstract: 'This paper introduces a microscopic approach to model epidemics, which can explicitly consider the consequences of individual’s decisions on the spread of the disease. We first formulate a microscopic multi-agent epidemic model where every agent can choose its activity level that affects the spread of the disease. Then by minimizing agents’ cost functions, we solve for the optimal decisions for individual agents in the framework of game theory and multi-agent reinforcement learning. Given the optimal decisions of all agents, we can make predictions about the spread of the disease. We show that there are negative externalities in the sense that infected agents do not have enough incentives to protect others, which then necessitates external interventions to regulate agents’ behaviors. In the discussion section, future directions are pointed out to make the model more realistic.' author: - 'Changliu Liu[^1]' bibliography: - 'main.bib' date: title: 'A Microscopic Epidemic Model and Pandemic Prediction Using Multi-Agent Reinforcement Learning' --- Introduction ============ With the COVID-19 pandemic souring across the world, a reliable model is needed to describe the observed spread of the disease, make predictions about future, and guide public policy design to control the spread. #### Existing Epidemic Models There are many existing macroscopic epidemic models.[@daley2001epidemic] For example, the SI model describes the growth of infection rate as the product of the current infection rate and the current susceptible rate. The SIR model further incorporates the effect of recovery into the model, i.e., when the infected population turns into immune population after a certain period of time. The SIRS model considers the case that immunity is not for lifetime and that the immune population can become susceptible population again. In addition to these models, the SEIR model incorporates the incubation period into analysis. Incubation period refers to the duration before symptoms show up.[@yan2006seir] The most important factor in all those models is $R0$, the regeneration number, which tells how fast the disease can spread. $R0$ can be regressed from data. #### Limitations of Existing Models Although these models are useful in predicting the spread of epidemics, they lack the granularity needed for analyzing individual behaviors during an epidemic and understanding the relationship between individual decisions and the spread of the disease.[@barrett2009estimating] For example, many countries now announced “lock-down", “shelter-in-place", “stay-at-home", or similar orders. However, their effects are very different across different countries, or even across different counties in the same country. One factor that can possibly explain these differences is the cultural difference. In different cultures, individuals make different choices. For instance, in the west, people exhibit greater inertia to give up their working/life routines so that they do not follow the orders seriously. While in the east, people tend to obey the rules better. These different individual choices can result in significantly different outcomes in disease propagation that cannot be captured by a macroscopic model. #### A Microscopic Epidemic Model In this paper, we develop a microscopic epidemic model by explicitly considering individual decisions and the interaction among different individuals in the population, in the framework of multi-agent systems. The aforementioned cultural difference can be understood as a difference in agents’ cost functions, which then affect their behaviors when they are trying to minimize their cost functions. The details of the microscopic epidemic model will be explained in the next section, followed by the analysis of the dynamics of the multi-agent system, and the prediction of system trajectories using multi-agent reinforcement learning. The model is still in its preliminary form. In the discussion section, future directions are pointed out to make the model more realistic. Microscopic Epidemic Model ========================== Suppose there are $M$ agents in the environment. Initially, $m_0$ agents are infected. Agents are indexed from $1$ to $M$. Every agent has its own state and control input. The model is in discrete time. The time interval is set to be one day. The evolution of the infection rate for consecutive days depends on agents’ actions. The questions of interest are: How many agents will eventually be infected? How fast they will be infected? How can we slow down the growth of the infection rate? Agent Model ----------- We consider two state values for an agent, e.g., for agent $i$, $x_i = 0$ means healthy (susceptible), $x_i = 1$ means infected. Everyday, every agent $i$ decides its level of activities $u_i\in[0,1]$. The level of activities for agent $i$ can be understood as the expected percentage of other agents in the system that agent $i$ wants to meet. For example, $u_i = 1/M$ means agent $i$ expects to meet one other agent. The actual number of agents that agent $i$ meets depends not only on agent $i$’s activity level, but also on other agents’ activity level. For example, if all other agents choose an activity level $0$, then agent $i$ will not be able to meet any other agent no matter what $u_i$ it chooses. Mathematically, the chance for agent $i$ and agent $j$ to meet each other depends on the minimum of the activity levels of these two agents, i.e., $\min\{u_i, u_j\}$. In the extreme cases, if agent $i$ decides to meet everyone in the system by choosing $u_i = 1$, then the chance for agent $j$ to meet with agent $i$ is $u_j$. If agent $i$ decides to not meet anyone in the system by choosing $u_i = 0$, then the chance for agent $j$ to meet with agent $i$ is $0$. Before we derive the system dynamic model, the assumptions are listed below: 1. In the agent model, we only consider two states: healthy (susceptible) and infected. All healthy agents are susceptible to the disease. There is no recovery and no death for infected agents. There is no incubation period for infected agents, i.e., once infected, the agent can start to infect other healthy agents. To relax this assumption, we may introduce more states for every agent. 2. The interactions among agents are assumed to be uniform, although it is not true in the real world. In the real world, given a fixed activity level, agents are more likely to meet with close families, friends, colleagues than strangers on the street. To incorporate this non-uniformity into the model, we need to redefine the chance for agent $i$ and agent $j$ to meet each other to be $\beta_{i,j}\min\{u_i, u_j\}$, where $\beta_{i,j}\in[0,1]$ is a coefficient that encodes the proximity between agent $i$ and agent $j$ and will affect the chance for them to meet with each other. For simplicity, we assume that the interaction patterns are uniform in this paper. 3. Meeting with infected agents will result in immediate infection. To relax this assumption, we may introduce an infection probability to describe how likely it is for a healthy agent to be infected if it meets with an infected agent. System Dynamic Model -------------------- On day $k$, denote agent $i$’s state and control as $x_{i,k}\in\mathcal{X}$ and $u_{i,k}\in\mathcal{U}$. By definition, the agent state space is $\mathcal{X} = \{0,1\}$ and the agent control space is $\mathcal{U}=[0,1]$. The system state space is denoted $\mathcal{X}^M:= \mathcal{X}\times\cdots \times \mathcal{X}$. The system control space is denoted $\mathcal{U}^M:=\mathcal{U}\times\cdots \times \mathcal{U}$. Define $m_k = \sum_{i} x_{i,k}$ as the number of infected agents at time $k$. The set of infected agents is denoted: $$\mathcal{I}_k := \{i: x_{i,k}= 1\}.$$ The state transition probability for the multi-agent system is a mapping $$\mathbb{T}: \mathcal{X}^M \times \mathcal{U}^M \times \mathcal{X}^M \mapsto [0,1].$$ According to the assumptions, an infected agent will always remain infected. Hence the state transition probability for an infected agent $i$ does not depend on other agents’ states or any control. However, the state transition probability for a healthy agent $i$ depends on others. The chance for a healthy agent $i$ to not meet an infected agent $j\in\mathcal{I}_k$ is $1-\min\{u_i,u_j\}$. A healthy agent can stay healthy if and only if it does not meet any infected agent, the probability of which is $\Pi_{j\in\mathcal{I}_k} (1-\min\{u_i,u_j\})$. Then the probability for a healthy agent to be infected is $1-\Pi_{j\in\mathcal{I}_k} (1-\min\{u_i,u_j\})$. From the expression $\Pi_{j\in\mathcal{I}_k} (1-\min\{u_i,u_j\})$, we can infer that: the chance for a healthy agent $i$ to stay health is higher if - the agent $i$ limits its own activity by choosing a smaller $u_i$; - the number of infected agents is smaller; - the infected agents in $\mathcal{I}_k$ limit their activities. The state transition probability for an agent $i$ is summarized in \[table:dynamics\]. \[table:dynamics\] #### Example Consider a four-agent system shown in \[fig:example\]. Only agent $1$ is infected. And the agents choose the following activity levels: $u_1 = 0.1, u_2 = 0.2, u_3 = 0.3, u_4 = 0.4$. Then the chance $p_{i,j}$ for agents $i$ and $j$ to meet with each other is $p_{1,2} = p_{1,3} = p_{1,4} = 0.1$, $p_{2,3} = p_{2,4} = 0.2$, and $p_{3,4} = 0.3$. Note that $p_{i,j} = p_{j,i}$. The chance for agents $2$, $3$, and $4$ to stay healthy is $0.9$, although they have different activity levels. {width="3.5cm"} Case Study ---------- Before we start to derive the optimal strategies for individual agents and analyze the closed-loop multi-agent system, we first characterize the (open-loop) multi-agent system dynamics by Monte Carlo simulation according to the state transition probability in \[table:dynamics\]. Suppose we have $M=1000$ agents. At the beginning, only agent $1$ is infected. We consider two levels of activities: normal activity level $u$ and reduced activity level $u^*$. The two activity levels are assigned to different agents following different strategies as described below. In particular, we consider “no intervention" case where all agents continue to follow the normal activity level, “immediate isolation" case where the activity levels of infected agents immediately drop to the reduced level, “delayed isolation" case where the activity levels of infected agents drop to the reduced level after several days, and “lockdown" case where the activity levels of all agents drop to the reduced level immediately. {width="\linewidth"} For each case, we simulate 200 system trajectories and compute the average, maximum, and minimum $m_k$ (number of infected agents) versus $k$ from all trajectories. A system trajectory in the “no intervention" case is illustrated in \[fig:monte\_carlo\], where $u=1/M$ for all agents. The $m_k$ trajectories under different cases are shown in \[fig:open\_loop\], where the solid curves illustrate the average $m_k$ and the shaded area corresponds to the range from min $m_k$ to max $m_k$. The results are explained below. - Case 0: no intervention. All agents keep the normal activity level $u$. The scenarios for $u= 1/M$ and $u= 2/M$ are illustrated in \[fig:open\_loop\]. As expected, a higher activity level for all agents will lead to faster infection. The trajectory of $m_k$ has a $S$ shape, whose growth rate is relatively slow when either the infected population is small or the healthy population is small, and is maximized when $50\%$ agents are infected. It will be shown in the following discussion that (empirical) macroscopic models also generate $S$-curves. - Case 1: immediate isolation of infected agents. The activity levels of infected agents immediately drop to $u^*$, while others remain $u$. The scenario for $u = 1/M$ and $u^* = 0.1/M$ is illustrated in \[fig:open\_loop\]. Immediate isolation significantly slows down the growth of the infections rate. As expected, it has the best performance in terms of flattening the curve, same as the lockdown case. The trajectory also has a $S$ shape. - Case 2: delayed isolation of infected agents. The activity levels of infected agents drop to $u^*$ after $T$ days, while others remain $u$. In the simulation, $u = 1/M$ and $u^* = 0.1/M$. The scenarios for $T=1$ and $T=2$ are illustrated in \[fig:open\_loop\]. As expected, the longer the delay, the faster the infection rate grows, though the growth of the infection rate is still slower than the “no intervention" case. Moreover, the peak growth rate (when $50\%$ agents are infected) is higher when the delay is longer. - Case 3: lockdown. The activity levels of all agents drop to $u^*$. The scenario for $u^* = 0.1/M$ is illustrated in \[fig:open\_loop\]. As expected, it has the best performance in terms of flattening the curve, same as the immediate isolation case. Since the epidemic model is monotone, every agent will eventually be infected as long as the probability to meet infected agents does not drop to zero. Moreover, we have not discussed decision making by individual agents yet. The activity levels are just predefined in the simulation. #### Remark The model we introduced is microscopic, in the sense that interactions among individual agents are considered. The simulated open-loop trajectories are indeed similar to those from a macroscopic model. Since only susceptible and infected populations are considered in the proposed microscopic model, we then compare it with the macroscopic Susceptible-Infected (SI) model. Define the state $s\in[0,1]$ as the fraction of infected population. The growth of the infected population is proportional to the susceptible population and the infected population. Suppose the infection coefficient is $\beta$, the system dynamics in the SI model follow: $$\dot s = \beta s (1-s).$$ {width="\linewidth"} We simulate the system trajectory under different infection coefficients as shown in \[fig:SI\]. The trajectories also have S shapes, similar to the ones in the microscopic model. However, since this macroscopic SI model is deterministic, there is no “uncertainty" range as shown in the microscopic model. The infection coefficient $\beta$ depends on the agents’ choices of activity levels. However, there is not an explicit relationship yet. It is better to directly use the microscopic model to analyze the consequences of individual agents’ choices. Distributed Optimal Control =========================== This section tries to answer the following question: in the microscopic multi-agent epidemic model, what is the best control strategy for individual agents? To answer that, we need to first specify the knowledge and observation models as well as the cost (reward) functions for individual agents. Then we will derive the optimal choices of agents in a distributed manner. The resulting system dynamics correspond to a Nash Equilibrium of the system. Knowledge and Observation Model ------------------------------- A knowledge and observation model for agent $i$ includes two aspects: what does agent $i$ know about itself, and what does agent $i$ know about others? The knowledge about any agent $j$ includes the dynamic function of agent $j$ and the cost function of agent $j$. The observation corresponds to run-time measurements, i.e., the observation of any agent $j$ includes the run-time state $x_{j,k}$ and the run-time control $u_{j,k}$. In the following discussion, regarding the knowledge and observation model, we make the following assumptions: - An agent knows its own dynamics and cost function; - All agents are homogeneous in the sense that they share the same dynamics and cost functions. And agents know that all agents are homogeneous, hence they know others’ dynamics and cost functions; - At time $k$, agents can measure $x_{j,k}$ for all $j$. But they cannot measure $u_{j,k}$ until time $k+1$. Hence, the agents are playing a simultaneous game. They need to infer others’ decisions when making their own decisions at any time $k$. Cost Function ------------- We consider two conflicting interests for every agent: - Limit the activity level to minimize the chance to get infected; - Maintain a certain activity level for living. We define the run-time cost for agent $i$ at time $k$ as $$l_{i,k} = x_{i,k+1} + \alpha_i p(u_{i,k}),$$ where $x_{i,k+1}$ corresponds to the first interest, $p(u_{i,k})$ corresponds to the second interest, and $\alpha_i > 0$ adjusts the preference between the two interests. The function $p(u)$ is assumed to be smooth. Due to our homogeneity assumption on agents, they should have identical preferences, i.e., $\alpha_i = \alpha$ for all $i$. Agent $i$ chooses its action at time $k$ by minimizing the expected cumulative cost in the future: $$\label{eq: repeated game} u_{i,k} = \arg\min \mathbb{E}[\sum_{t=k}^{\infty} \gamma^{t-k} l_{i,k} ],$$ where $\gamma\in[0,1]$ is a discount factor. The objective function depends on all agents’ current and future actions. It is difficult to directly obtain an analytical solution of . Later we will use multi-agent reinforcement learning to obtain a numerical solution. In this section, to simplify the problem, we consider a single stage game where the agents have zero discount of the future, i.e., $\gamma = 0$. Hence the objective function is reduced to $$u_{i,k} = \arg\min \mathbb{E}[l_{i,k} ],$$ which only depends on the current actions of agents. According to the state transition probability in \[table:dynamics\], the expected cost is $$\label{eq: expected cost} \mathbb{E}[l_{i,k} ] = \left\{\begin{array}{ll} 1-\Pi_{j\in\mathcal{I}_k} (1-\min\{u_i,u_j\}) + \alpha_i p(u_{i,k}) & \text{if }x_{i,k} = 0\\ 1 + \alpha_i p(u_{i,k}) & \text{if }x_{i,k} = 1 \end{array}\right..$$ Nash Equilibrium ---------------- {width="\linewidth"} According to , the expect cost for an infected agent only depends on its own action. Hence the optimal choice for an infected agent is $u_{i,k} = \bar u := \arg\min_u p(u)$. Then the optimal choice for a healthy agent satisfies: $$\begin{aligned} u_{i,k} &=& \arg\min_u [1-\Pi_{j\in\mathcal{I}_k} (1-\min\{u,\bar u\}) + \alpha_i p(u)],\\ &=&\arg\min_u [1-(1-\min\{u,\bar u\})^{m_k} + \alpha_i p(u)].\label{eq: healthy agent cost}\end{aligned}$$ Note that the term $1-(1-\min\{u,\bar u\})^{m_k}$ is positive and is increasing for $u\in[0,\bar u]$ and then constant for $u\in[\bar u, 1]$. Hence, the optimal solution for should be smaller than $\bar u = \arg\min_u p(u)$. Then the objective in can be simplified as $1-(1-u)^{m_k} + \alpha_i p(u)$. In summary, the optimal actions for both the infected and the healthy agents in the Nash Equilibrium can be compactly written as $$\label{eq: Nash} u_{i,k} = \arg\min_u \{1-(1-u)^{m_k}(1-x_{i,k}) + \alpha_i p(u)\}, \forall i.$$ #### Example Consider the previous example with four agents shown in \[fig:example\]. Define $$\label{eq: cost on activity} p(u) = \exp(\frac{1}{u-1}),$$ which is a monotonically decreasing function as illustrated in \[fig: p\]. Then the optimal actions in the Nash Equilibrium for this specific problem satisfy: $$\label{eq: example Nash} u_{i,k} = \arg\min_u \{u+x_{i,k}-ux_{i,k} + \alpha_i \exp(\frac{1}{u-1})\}, \forall i.$$ {width="\linewidth"} Solving for , for infected agents, $u_{i,k} = 1$. For healthy agents, the choice also depends on $\alpha_i$ as illustrated in \[fig: alpha\]. We have assumed that $\alpha_i = \alpha$ which is identical for all agents. We further assume that $\alpha < 2$ such that the optimal solution for healthy agents should be $u_{i,k} = 0$. The optimal actions and the corresponding costs for all agents are listed in \[fig: example cost\]. In the Nash Equilibrium, no agent will meet each other, since all agents except agent $1$ reduce their activity levels to zero. The actual cost (received at the next time step) equals to the expected cost (computed at the current time step). Agent ID State $x_{i,k}$ Optimal $u_{i,k}$ Optimal $\mathbb{E}[l_{i,k}]$ Actual $l_{i,k}$ ---------- ----------------- ------------------- ------------------------------- ----------------------- 1 1 1 1 1 2,3,4 0 0 $\alpha \exp(-1)$ $\alpha \exp(-1)$ Total $1+ 3\alpha \exp(-1)$ : List of the agent decisions and associated costs in the Nash Equilibrium in the four-agent example. \[fig: example cost\] However, let us consider another situation where the infected agent chooses $0$ activity level and all other healthy agents choose $1$ activity level. The resulting costs are summarized in \[fig: example cost global\]. Obviously, the overall cost is reduced in the new situation. However, this better situation cannot be attained spontaneously by the agents, due to externality of the system which will be explained below. Agent ID State $x_{i,k}$ Optimal $u_{i,k}$ Optimal $\mathbb{E}[l_{i,k}]$ Actual $l_{i,k}$ ---------- ----------------- ------------------- ------------------------------- ---------------------- 1 1 0 1+$\alpha \exp(-1)$ 1+$\alpha \exp(-1)$ 2,3,4 0 1 0 0 Total $1+ \alpha \exp(-1)$ : List of the agent decisions and associated costs in a situation better than the Nash Equilibrium in the four-agent example. \[fig: example cost global\] Dealing with Externality ------------------------ For a multi-agent system, define the system cost as a summation of the individual costs: $$L_k := \sum_i l_{i,k}.$$ The system cost in the Nash Equilibrium is denoted $L^*_k$, which corresponds to the evaluation of $L_k$ under agent actions specified in . On the other hand, the optimal system cost is defined as $$\label{eq: system optimum} L^o_k := \min_{u_{i,k},\forall i} L_k.$$ The optimization problem is solved in a centralized manner, which is different from how the Nash Equilibrium is obtained. To obtain the Nash Equilibrium, all agents are solving their own optimization problems independently. Although their objective functions depend on other agents’ actions, they are not jointly make the decisions, but only “infer" what others will do. By definition, $L^o_k \leq L^*_k$. In the example above, $L^*_k = 1+3\alpha \exp(-1)$ and $L^o_k = 1+\alpha \exp(-1)$. The difference $L^*_k - L^o_k$ is called the **loss of social welfare**. In the epidemic model, the loss of social welfare is due to the fact that bad consequences (i.e., infecting others) are not penalized in the cost functions of the infected agents. Those unpenalized consequences are called *externality*. There can be both positive externality and negative externality. Under positive externality, agents are lacking motivations to do things that are good for the society. Under negative externality, agents are lacking motivations to prevent things that are bad for the society. In the epidemic model, there are negative externality with infected agents. To improve social welfare, we need to “internalize" externality, i.e., add penalty for “spreading" the disease. Now let us redefine agent $i$’s run-time cost as $$\label{eq: shaped cost} \tilde l_{i,k} = x_{i,k+1} + \alpha_i p(u_{i,k}) + x_{i,k} q(u_{i,k}),$$ where $q(\cdot)$ is a monotonically increasing function. The last term $x_{i,k}q(u_{i,k})$ does not affect healthy agents since $x_{i,k} = 0$, but adds a penalty for infected agents if they choose large activity level. One candidate function for $q(u)$ is $1-(1-u)^{m_k}$. In the real world, such “cost shaping" using $q$ can be achieved through social norms or government regulation. The expected cost becomes $$\label{eq: expected cost tilde} \mathbb{E}[\tilde l_{i,k} ] = \left\{\begin{array}{ll} 1-\Pi_{j\in\mathcal{I}_k} (1-\min\{u_i,u_j\}) + \alpha_i p(u_{i,k}) & \text{if }x_{i,k} = 0\\ 1 + \alpha_i p(u_{i,k}) + q(u_{i,k}) & \text{if }x_{i,k} = 1 \end{array}\right.$$ Suppose the function $q$ is well tuned such that the $\arg\min_u [ \alpha_i p(u) + q(u)] = 0$. Then although the expected costs for infected agents are still independent from others, their decision is considerate to healthy agents. When the infected agents choose $u =0$, then for healthy agents, the expected cost becomes $\alpha_i p(u_{i,k})$, meaning that they do not need to worry about getting infected. Let us now compute the resulting Nash Equilibrium under the shaped costs using the previous example. #### Example In the four-agent example, set $q(u) = u$. Then $\arg\min_u [ \alpha p(u) + u] = 0$. Hence agent 1 will choose $u_{1,k} = 0$. For agents $i= 2,3,4$, they will choose $u_{i,k}=1$ since they are only minimizing $p(u)$. The resulting costs are summarized in \[fig: example cost improved\]. With the shaped costs, the system enters into a better Nash Equilibrium which indeed aligns with the system optimum in . A few remarks: - Cost shaping did not increase the overall cost for the multi-agent system. - The system optimum remains the same before and after cost shaping. - Cost shaping helped agents to arrive at the system optimum without centralized optimization. Agent ID State $x_{i,k}$ Optimal $u_{i,k}$ Optimal $\mathbb{E}[\tilde l_{i,k}]$ Actual $\tilde l_{i,k}$ ---------- ----------------- ------------------- -------------------------------------- ------------------------- 1 1 0 1+$\alpha \exp(-1)$ 1+$\alpha \exp(-1)$ 2,3,4 0 1 0 0 Total $1+ \alpha \exp(-1)$ : List of the agent decisions and associated costs in the Nash Equilibrium with shaped cost functions in the four-agent example. \[fig: example cost improved\] Multi-Agent Reinforcement Learning ================================== We have shown how to compute the Nash Equilibrium of the multi-agent epidemic model in a single stage. However, it is analytically intractable to compute the Nash Equilibrium when we consider repeated games . The complexity will further grow when the number of agents increases and when there are information asymmetry. Nonetheless, we can apply multi-agent reinforcement learning[@bucsoniu2010multi] to numerically compute the Nash Equilibrium. Then the evolution of the pandemic can be predicted by simulating the system under the Nash Equilibrium. Q Learning ---------- As evident from , the optimal action for agent $i$ at time $k$ is a function of $x_{i,k}$ and $m_k$. Hence we can define a Q function (action value function) for agent $i$ as $$Q_i: x_{i,k}\times m_k\times u_{i,k}\mapsto \mathbb{R}.$$ According to the assumptions made in the observation model, all agents can observe $m_k$ at time $k$. For a single stage game, we have derived in that $Q_i (x,m,u)= 1-(1-u)^{m}(1-x) + \alpha_i p(u)$. For repeated games , we can learn the Q function using temporal different learning. At every time $k$, agent $i$ chooses its action as $$\label{eq: optimal action} u_{i,k} = \arg\min_u Q_i(x_{i,k}, m_k, u).$$ After taking the action $u_{i,k}$, agent $i$ observes $x_{i,k+1}$ and $m_{k+1}$ and receives the cost $l_{i,k}$ at time $k+1$. Then agent $i$ updates its Q function: $$\begin{aligned} &Q_i (x_{i,k},m_k,u_{i,k}) \leftarrow Q_i (x_{i,k},m_k,u_{i,k}) + \eta \delta_{i,k},\\ &\delta_{i,k} = l_{i,k} + \gamma \min_u Q_i(x_{i,k+1},m_{k+1},u) - Q_i(x_{i,k},m_k,u_{i,k}),\end{aligned}$$ where $\eta$ is the learning gain and $\delta_{i,k}$ is the temporal difference error. All agents can run the above algorithm to learn their $Q$ functions during the interaction with others. However, the algorithm introduced above has several problems: - Exploration and limited rationality. There is no exploration in . Indeed, Q-learning is usually applied together with $\epsilon$-greedy where with probability $1-\epsilon$, the action $u_{i,k}$ is chosen to be the optimal action in , and with probability $\epsilon$, the action is randomly chosen with a uniform distribution over the action space. The $\epsilon$-greedy approach is introduced mainly from an algorithmic perspective to improve convergence of the learning process. When applied to the epidemic model, it has a unique societal implication. When agents are randomly choosing their behaviors, it represents the fact that agents have only limited rationality. Hence in the learning process, we apply $\epsilon$-greedy as a way to incorporate exploration for faster convergence as well as to take into account limited rationality of agents. - Data efficiency and parameter sharing. Keeping separated Q functions for individual agents is not data efficient. An agent may not be able to collect enough samples to properly learn the desired Q function. Due to the homogeneity assumptions we made earlier about agents’ cost functions, it is more data efficient to share the Q function for all agents. Its societal implication is that agents are sharing information and knowledge with each other. Hence, we apply parameter sharing[@gupta2017cooperative] as a way to improve data efficiency as well as to consider information sharing among agents during the learning process. With the above modifications, the multi-agent Q learning algorithm[@hu2003nash] is summarized below. - For every time step $k$, agents choose their actions as: $$u_{i,k} = \left\{\begin{array}{ll} \arg\min_u Q(x_{i,k}, m_k, u) & \text{probability }1-\epsilon\\ \text{random} &\text{probability }\epsilon \end{array}\right. \forall i.$$ - At the next time step $k+1$, agents observe the new states $x_{i,k+1}$ and receive rewards $l_{i,k}$ for all $i$. Then the Q function is updated: $$\begin{aligned} &Q (x_{i,k},m_k,u_{i,k}) \leftarrow Q (x_{i,k},m_k,u_{i,k}) + \eta \delta_{i,k},\forall i,\\ &\delta_{i,k} = l_{i,k} + \gamma \min_u Q(x_{i,k+1},m_{k+1},u) - Q(x_{i,k},m_k,u_{i,k}).\end{aligned}$$ #### Example In this example, we consider $M=50$ agents in the system. Only one agent is infected in the beginning. The run-time cost is the same as in the example in the *distributed optimal control* section, i.e., $l_{i,k} = x_{i,k+1} + \alpha \exp (\frac{1}{u_{i,k}-1})$ where $\alpha$ is chosen to be $1$. For simplicity, the action space is discretized to be $\{0, 1/M, 10/M\}$, called as low, medium, and high. Hence the Q function can be stored as a $2\times M\times 3$ matrix. In the learning algorithm, the learning rate is set to $\eta = 1$. The exploration rate is set to decay in different episodes, i.e., $\epsilon = 0.5(1-E/\max E)$ where $E$ denotes the current episode and the maximum episode is $\max E = 200$. The Q function is initialized to be $10$ for all entries. Three different cases are considered. For each case, we illustrate the Q function learned after 200 episodes as well as the system trajectories for episodes $10, 20,\ldots, 200$, blue for earlier episodes and red for later episodes. The results are shown in \[fig: marl\]. - Case 1: discount $\gamma = 0$ with runtime cost $l_{i,k}$. With $\gamma = 0$, this case reduces to a single stage game as discussed in the *distributed optimal control* section. The result should align with the analytical Nash Equilibrium in . As shown in the left plot in \[fig: marl\](a), the optimal action for a healthy agent is always *low* (solid green), while the optimal action for an infected agent is always *high* (dashed magenta). The Q values for infected agents do not depend on $m_k$. The Q values for healthy agents increase when $m_k$ increases if the activity level is not zero, due to the fact that: for a fixed activity level, the chance to get infected is higher when there are more infected agents in the system. All these results align with our previous theoretical analysis. Moreover, as shown in the right plot in \[fig: marl\](a), the agents are learning to flatten the curve across different episodes. - Case 2: discount $\gamma = 0.5$ with runtime cost $l_{i,k}$. Since the agents are now computing cumulative costs as in , the corresponding Q values are higher than those in case 1. However, the optimal actions remain the same, *low* (solid green) for healthy agents, *high* (dashed magenta) for infected agents, as shown in the left plot in \[fig: marl\](b). The trends of the Q curves also remain the same: the Q values do not depend on $m_k$ for infected agents and for healthy agents whose activity levels are zero. However, as shown in the right plot in \[fig: marl\](b), the agents learned to flatten the curve faster than in case 1, mainly because healthy agents are more cautious (converge faster to low activity levels) when they start to consider cumulative costs. - Case 3: discount $\gamma = 0.5$ with shaped runtime cost $\tilde l_{i,k}$ in . The shaped cost changes the optimal actions for all agents as well as the resulting Q values. As shown in the left plot in \[fig: marl\](c), the optimal action for an infected agent is *low* (dashed green), while that for a healthy agent is *high* (solid magenta) when $m_k$ is small and *low* (solid green) when $m_k$ is big. Note that when $m_k$ is high, the healthy agents still prefer low activity level, though the optimal actions for infected agents are low. That is because: due to the randomization introduced in $\epsilon$-greedy, there is still chance for infected agents to have medium or high activity levels. When $m_k$ is high, the healthy agents would rather limit their own activity levels to avoid the risk to meet with infected agents that are taking random actions. This result captures the fact that agents understand others may have limited rationality and prefer more conservative behaviors. We observe the same trends for the Q curves as the previous two cases: the Q values do not depend on $m_k$ for infected agents and for healthy agents whose activity levels are not zero. In terms of absolute values, the Q values for infected agents are higher than those in case 2 due to the additional cost $q(u)$ in $\tilde l_{i,k}$. The Q values for healthy agents are smaller than those in case 2 for medium and high activity levels, since the chance to get infected is smaller as infected agents now prefer low activity levels. The Q values remain the same for healthy agents with zero activity levels. With shaped costs, the agents learned to flatten the curve even faster than in case 2, as shown in the right plot in \[fig: marl\](c), since the shaped cost encourages infected agents to lower their activity levels. Discussion and Future Work ========================== #### Agents vs humans The epidemic model can be used to analyze real-world societal problems. Nonetheless, it is important to understand the differences between agents and humans. We can directly design and shape the cost function for agents, but not for humans. For agents, their behavior is predictable once we fully specify the problem (i.e., cost, dynamics, measurement, etc). Hence we can optimize the design (i.e., the cost function) to get desired system trajectory. For humans, their behavior is not fully predictable due to limited rationality. We need to constantly modify the knowledge and observation model as well as the cost function to match the true human behavior. #### Future work The proposed model is in its preliminary form. Many future directions can be pursued. - Relaxation of assumptions. We may add more agent states to consider recovery, incubation period, and death. We may consider the fact that the interaction patterns among agents are not uniform. We may consider a wide variety of agents who are not homogeneous. For example, health providers and equipment suppliers are key parts in fighting the disease. They should receive lower cost (higher reward) for maintaining or even expanding their activity levels than ordinary people. Their services can then lead to higher recovery rate. In addition, we may relax the assumptions on agents’ knowledge and observation models, to consider information asymmetry as well as partial observation. For example, agents cannot get immediate measurement whether they are infected or not, or how many agents are infected in the system. - Realistic cost functions for agents. The cost functions for agents are currently hand-tuned. We may learn those cost functions from data through inverse reinforcement learning. Those cost functions can vary for agents from different countries, different age groups, and different occupations. Moreover, the cost functions carry important cultural, demographical, economical, and political information. A realistic cost function can help us understand why we observe significantly different outcomes of the pandemic around the world, as well as enable more realistic predictions into the future by fully considering those cultural, demographical, economical, and political factors. - Incorporation of public policies. For now, the only external intervention we introduced is cost shaping. We may consider a wider range of public policies that can change the closed-loop system dynamics. For example, shut-down of transportation, isolation of infected agents, contact tracing, antibody testing, etc. - Transient vs steady state system behaviors. We have focused on the steady state system behaviors in the Nash Equilibrium. However, as agents live in a highly dynamic world, it is not guaranteed that a Nash Equilibrium can always be attained. While agents are learning to deal with unforeseen situations, there are many interesting transient dynamics, some of which is captured in \[fig: marl\], i.e., agents may learn to flatten the curve at different rates. Methods to understand and predict transient dynamics may be developed in the future. - Validation against real world historical data. To use the proposed model for prediction in the real world, we need to validate its fidelity again the historical data. The validation can be performed on the $m_k$ trajectories, i.e., for the same initial condition, the predicted $m_k$ trajectories should align with the ground truth $m_k$ trajectories. Conclusion ========== This paper introduced a microscopic multi-agent epidemic model, which explicitly considered the consequences of individual’s decisions on the spread of the disease. In the model, every agent can choose its activity level to minimize its cost function consisting of two conflicting components: staying healthy by limiting activities and maintaining high activity levels for living. We solved for the optimal decisions for individual agents in the framework of game theory and multi-agent reinforcement learning. Given the optimal decisions of all agents, we can make predictions about the spread of the disease. The system had negative externality in the sense that infected agents did not have enough incentives to protect others, which then required external interventions such as cost shaping. We identified future directions were pointed out to make the model more realistic. [^1]: C. Liu is with Carnegie Mellon University, Pittsburgh PA 15213. Email: cliu6@andrew.cmu.edu

--- abstract: 'It is known that the greatest common divisor of two Fibonacci numbers is again a Fibonacci number. It is called the *strong divisibility property*. However, this property does not hold for every second order sequence. In this paper we study the generalized Fibonacci polynomials and classify them in two types depending on their Binet formula. We give a complete characterization for those polynomials that satisfy the strong divisibility property. We also give formulas to calculate the $\gcd$ of those polynomials that do not satisfy the strong divisibility property.' --- Rigoberto Flórez\ Department of Mathematics and Computer Science\ The Citadel\ Charleston, SC\ U.S.A.\ [rigo.florez@citadel.edu]{}\  \ Robinson A. Higuita\ Instituto de Matemáticas\ Universidad de Antioquia\ Medellín\ Colombia\ [robinson.higuita@udea.edu.co]{}\  \ Antara Mukherjee\ Department of Mathematics and Computer Science\ The Citadel\ Charleston, SC\ U.S.A.\ [antara.mukherjee@citadel.edu]{} .2 in Introduction ============ It is well known that the greatest common divisor ($\gcd$) of two Fibonacci numbers is a Fibonacci number [@koshy]. Thus, $\gcd(F_m,F_n)=F_{\gcd(m,n)}$. It is called the *strong divisibility property* or *Fibonacci gcd property*. In this paper we study divisibility properties of generalized Fibonacci polynomials (GFP) and in particular we give characterization of the strong divisibility property for these polynomials. We classify the GFP in two types, the Lucas type and the Fibonacci type, depending on their closed formulas or their Binet formulas (see for example, $L_n(x)$ (\[bineformulados\]) and $R_n(x)$ (\[bineformulauno\]), and the Table \[equivalent\]). That is, if after solving the characteristic polynomial of the GFP we obtain a closed formula that look like the Binet formula for Fibonacci (Lucas) numbers, it is called Fibonacci (Lucas) type polynomials. Familiar examples of Fibonacci type polynomials are: Fibonacci polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials of second kind, Jacobsthal polynomials and one type of Morgan-Voyce polynomials. Examples of Lucas type polynomials are: Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Chebyshev polynomials of first kind, Jacobsthal-Lucas polynomials and second type of Morgan-Voyce polynomials. In Theorem \[gcd:property:fibonacci\] we prove that a GFP satisfies the strong divisibility property if and only if it is of Fibonacci type. The Theorem \[second:main:thm\] shows that the Lucas type polynomials satisfy the strong divisibility property partially and also gives the $gcd$ for those cases in which the property is not satisfied. A Lucas type polynomial is equivalent to a Fibonacci type polynomial if they both have the same recurrence relation but different initial conditions (see also Flórez et al. [@florezHiguitaMuk]). The Theorem \[combine:gcd:Lucas:Fibobacci\] proves that two equivalent GFP satisfies the strong divisibility property partially and gives the $gcd$ for those cases in which the property is not satisfied. In 1969 Webb and Parberry [@Webb] extended the strong divisibility property to the Fibonacci Polynomials. In 1974 Hoggatt and Long [@HoggattLong] proved the strong divisibility property for one type of generalized Fibonacci polynomial. In 1978 Hoggatt and Bicknell-Johnson [@hoggatt] extended the result mentioned in [@HoggattLong] to some cases of Fibonacci type polynomials. However, they did not prove the necessary and sufficient condition and their paper does not cover the Lucas type case. In 2005 Rayes, Trevisan, and Wang [@Rayes] proved that the strong divisibility property holds partially for the Chebyshev polynomials (we prove the general result in Theorem \[second:main:thm\]). Over the years several other authors have been also interested in the divisibility properties of sequences, some papers are [@hall; @kimberling; @kimberlingStrong1; @kimberlingStrong2; @lucas; @mcdaniel; @norfleet; @ward]. Lucas [@lucas] proved the strong divisibility property (SDP) for Fibonacci numbers. However, the study of SDP for Lucas numbers took until 1991, when McDaniel [@mcdaniel] provided proofs that the Lucas numbers satisfy the SDP partially. In 1995 Hilton et al. [@hilton] gave some more precise results about this property. As mentioned above several authors have been interested in the divisibility properties for Fibonacci type polynomials. However, the Lucas type polynomials have been less studied. Here we complete three cases of the SDP. Indeed, we give a characterization for the SDP for Fibonacci type polynomials and study both the SDP for Lucas type polynomials and the SDP for the combinations of Lucas type polynomials and Fibonacci type polynomials. Finally we provide an open question for the most general case of combination of two polynomials. Generalized Fibonacci polynomials GFP {#General:Fibonacci:Polynomial} ===================================== In the literature there are several definitions of generalized Fibonacci polynomials. However, the definition that we introduce here is simpler and covers other definitions. The background given in this section is a summary of the background given in [@florezHiguitaMuk]. However, the definition of generalized Fibonacci polynomial here is not exactly the same as in [@florezHiguitaMuk]. The *generalized Fibonacci polynomial* sequence, denoted by GFP, is defined by the recurrence relation $$\label{Fibonacci;general} G_0(x)=p_0(x), \; G_1(x)= p_1(x),\; \text{and} \; G_{n}(x)= d(x) G_{n - 1}(x) + g(x) G_{n - 2}(x) \text{ for } n\ge 2$$ where $p_0(x)$ is a constant and $p_1(x)$, $d(x)$ and $g(x)$ are non-zero polynomials in $\mathbb{Z}[x]$ with $\gcd(d(x), g(x))=1$. For example, if we let $p_0(x)=0$, $p_1(x)=1$, $d(x)=x$, and $g(x)=1$ we obtain the regular Fibonacci polynomial sequence. Thus, $$F_0(x)= 0, \; F_1(x)= 1, \; \text{and} \; F_{n}(x)= x F_{n - 1}(x) + F_{n - 2}(x) \text{ for } n\ge 2.$$ Letting $x=1$ and choosing the correct values for $p_0(x)$, $p_1(x)$, $d(x)$ and $g(x)$, the generalized Fibonacci polynomial sequence gives rise to three classical numerical sequences, the Fibonacci sequence, the Lucas sequence and the generalized Fibonacci sequence. In Table \[familiarfibonacci\] there are more familiar examples of GFP (see [@florezHiguitaMuk; @Pell; @Fermat; @koshy]). Hoggatt and Bicknell-Johnson [@hoggatt] show that Schechter polynomials are another example of generalized Fibonacci polynomials. \[!ht\] Fibonacci type and Lucas type polynomials ----------------------------------------- If we impose some conditions on the Definition (\[Fibonacci;general\]) we obtain two type of distinguishable polynomials. We say that a sequences as in (\[Fibonacci;general\]) is *Lucas type* or *first type* if $2p_{1}(x)=p_{0}(x)d(x)$ with $p_{0}\ne 0$. We say that a sequences as in (\[Fibonacci;general\]) is *Fibonacci type* or *second type* if $p_{0}(x)=0$ with $p_{1}(x)$ a constant. If $d^2(x)+4g(x)> 0$, then the explicit formula for the recurrence relation (\[Fibonacci;general\]) is given by $$\label{solutionrecurrencerelationuno} G_{n}(x) = t_1 a^{n} + t_2 b^{n}$$ where $a(x)$ and $b(x)$ are the solutions of the quadratic equation associated to the second order recurrence relation $G_{n}(x)$. That is, $a(x)$ and $b(x)$ are the solutions of $z^2-d(x)z-g(x)=0$. The explicit formula for $G_{n}(x)$ given in (\[solutionrecurrencerelationuno\]) with $G_{0}(x)=p_{0}(x)$ and $G_{1}(x)=p_{1}(x)$ imply that $$\label{solutionrecurrencerelationdos} t_{1}=\dfrac{p_{1}(x)-p_{0}(x)b(x)}{a(x)-b(x)} \text{ and } t_{2}=\dfrac{-p_{1}(x)+p_{0}(x)a(x)}{a(x)-b(x)}$$ Using (\[solutionrecurrencerelationuno\]) and (\[solutionrecurrencerelationdos\]) we obtain the Binet formulas for the Generalized Fibonacci sequences of the Lucas type and Fibonacci type. Thus, substituting $2p_{1}(x)=p_{0}(x)d(x)$ in (\[solutionrecurrencerelationdos\]) we obtain that $t_{1}=t_{2}= p_{0}(x)/2$. Substituting this in (\[solutionrecurrencerelationuno\]) and letting $\alpha$ be $2/p_{0}(x)$ we obtain the Binet formula for Generalized Fibonacci sequence of *Lucas type* or *first type* $$\label{bineformulados} L_n(x)=\frac{a^{n}(x)+b^{n}(x)}{\alpha}.$$ We want $\alpha$ be an integer, therefore $|p_{0}(x)|=1 \text { or } 2$. Now, substituting $p_{0}(x)=0$ and the constant $p_{1}(x)$ in (\[solutionrecurrencerelationdos\]) we obtain that $t_{1}=t_{2}=p_{1}(x)$. Substituting this in (\[solutionrecurrencerelationuno\]) we obtain the Binet formula for Generalized Fibonacci sequence of *Fibonacci type* or *second type* $$\label{bineformulaunogeneral} R_n(x)=\frac{p_{1}(x)\left(a^{n}(x)-b^{n}(x)\right)}{a(x)-b(x)}.$$ In this paper we are interested only on $R_n(x)$ when $p_1(x)=1$. Therefore, the Binet formula $R_n(x)$ that we use here is $$\label{bineformulauno} R_n(x)=\frac{a^{n}(x)-b^{n}(x)}{a(x)-b(x)}.$$ Note that $a(x)+b(x)=d(x)$, $a(x)b(x)= -g(x)$ and $a(x)-b(x)=\sqrt{d^2(x)+4g(x)}$ where $d(x)$ and $g(x)$ are the polynomials defined on the generalized Fibonacci polynomials. A generalized Fibonacci polynomial which satisfies the Binet formula (\[bineformulados\]) is said to be of *first type* or *Lucas type* and it is of *Second type* or *Fibonacci type* if it satisfies the Binet formula (\[bineformulauno\]). Horadam [@horadam-synthesis] and André-Jeannin [@Richard] have studied these polynomials in detail. The sequence of polynomials that have Binet representations $R_n(x)$ or $L_n(x)$ depend only on $d(x)$ and $g(x)$ defined on the generalized Fibonacci polynomials. We say that a generalized Fibonacci sequence of Lucas (Fibonacci) type is *equivalent* to a sequence of the Fibonacci (Lucas) type, if their recursive sequences are determined by the same polynomials $d(x)$ and $g(x)$. Notice that two equivalent polynomials have the same $a(x)$ and $b(x)$ in their Binet representations. For example, the Lucas polynomial is a GFP of Lucas type, whereas the Fibonacci polynomial is a GFP of Fibonacci type. Lucas and Fibonacci polynomials are equivalent because $d(x)=x$ and $g(x)=1$ (see Table \[familiarfibonacci\]). Note that in their Binet representations they both have $a(x)= (x+\sqrt{x^2+4})/2$ and $b(x)=(x-\sqrt{x^2+4})/2$. The Table \[equivalent\] is based on information from the following papers [@Richard; @florezHiguitaMuk; @horadam-synthesis]. The leftmost polynomials in Table \[equivalent\] are of the Lucas type and their equivalent polynomials are in the third column on the same line. In the last two columns of Table \[equivalent\] we can see the $a(x)$ and $b(x)$ that the pairs of equivalent polynomials share. It is easy to obtain the characteristic equations of the sequences given in Table \[familiarfibonacci\], and the roots of the equations are $a(x)$ and $b(x)$. For the sake of simplicity throughout this paper we use $a$ in place of $a(x)$ and $b$ in place of $b(x)$ when they appear in the Binet formulas. We use the notation $G_n^{*}$ or $G_n^{'}$ for $G_n$ depending on when it satisfies the Binet formulas (\[bineformulados\]) or (\[bineformulauno\]), respectively, (see Section \[mainresults:section\]). For most of the proofs of GFP of Lucas type it is required that $\gcd(p_0(x), p_1(x))=1$, $\gcd(p_0(x), d(x))=1$ and $\gcd(d(x), g(x))=1$. It is easy to see that $\gcd(\alpha, G_{n}^{*}(x))=1$. Therefore, for the rest the paper we suppose that these four mentioned conditions hold for all generalized Fibonacci polynomial sequences of Lucas type treated here. We use $\rho$ to denote $\gcd(d(x),G_1(x))$. Notice that in the definition Pell-Lucas we have that $p_0(x)=2$ and $p_1(x)=2x$. Thus, the $\gcd(p_0(x),p_1(x))\ne 1$. Therefore, Pell-Lucas does not satisfy the extra conditions that were just imposed for Generalized Fibonacci polynomial. To solve this problem we define *Pell-Lucas-prime* as follows: $$Q_0^{\prime}(x)= 1, \; Q_1^{\prime}(x)= x, \; \text{and} \; Q_{n}^{\prime}(x)= 2x Q_{n - 1}^{\prime}(x) + Q_{n - 2}^{\prime}(x) \text{ for } n\ge 2.$$ It easy to see that $2Q_{n}^{'}(x)=Q_{n}(x)$ and that $\alpha=2$. Flórez, Junes and Higuita [@florezHiguitaJunes] worked similar problems for numerical sequences. **Note.** The definition of Generalized Fibonacci Polynomial in [@florezHiguitaMuk] differs with definition on this papers due to the initial conditions on the Fibonacci type polynomials. Thus, the initial conditions for the Fibonacci type polynomials in [@florezHiguitaMuk] is $G_{0}(x)=p_{0}(x)=1$ and so implicitly $G_{-1}(x)=0$. However, our definition for the Lucas type polynomials are the same in both papers. \[!ht\] Divisibility properties of Generalized Fibonacci Polynomials ============================================================ In this section we prove a few divisibility and $\gcd$ properties which are true for all GFP. These results will be used in a section later on to prove the main results of this paper. Proposition \[prop2;1\] parts (1) and (2) is a generalization of Proposition 2.2 in [@florezjunes]. The proof here is similar the proof in [@florezjunes] since both use properties of integral domains. The reader can therefore update the proof in the afore-mentioned paper to obtain the proof of this proposition. \[prop2;1\] Let $p(x), q(x), r(x),$ and $s(x)$ be polynomials. (1) If $\gcd(p(x),q(x))=\gcd(r(x),s(x))=1$, then $$\gcd(p(x)q(x),r(x)s(x))=\gcd(p(x),r(x))\gcd(p(x),s(x))\gcd(q(x),r(x))\gcd(q(x),s(x)).$$ (2) If $\gcd(p(x),r(x))=1\gcd(q(x),s(x))=1$, then $$\gcd(p(x)q(x),r(x)s(x))=\gcd(p(x),s(x))\gcd(q(x),r(x)).$$ (3) If $z_1(x)=\gcd(p(x),r(x))$ and $z_2(x)=\gcd(q(x),s(x))$, then $$\gcd(p(x)q(x),r(x)s(x))=\frac{\gcd( z_2(x)p(x),z_1(x)s(x))\gcd( z_1(x)q(x), z_2(x)r(x))}{z_1(x)z_2(x)}.$$ We prove part (3). Since $\gcd(p(x),r(x))=z_1(x) $ and $\gcd(q(x),s(x))=z_2(x)$, there are polynomials $P(x)$, $S(x)$, $Q(x)$ and $R(x)$ with $\gcd(P(x),R(x))=\gcd(Q(x),S(x))=1$, such that $ p(x) = z_1(x) P(x),\; s(x)= z_2(x) S(x),\; r(x) = z_1(x) R(x), \; q(x)= z_2(x) Q(x)$. So, $$\begin{aligned} \gcd(p(x)q(x),r(x)s(x)) &=&\gcd(z_1(x) P(x) z_2(x)Q(x), z_1(x) R(x) z_2(x)S(x))\\ &=&z_1(x)z_2(x)\gcd( P(x)Q(x), R(x)S(x)).\\\end{aligned}$$ From part (2) we know that $ \gcd( P(x)Q(x), R(x) S(x))=\gcd( P(x),S(x))\gcd( Q(x), R(x)). $ Now it is easy to see that $$\gcd(p(x)q(x),r(x)s(x))=\frac{\gcd( z_2(x)p(x),z_1(x)s(x))\gcd( z_1(x)q(x), z_2(x)r(x))}{z_1(x)z_2(x)}.$$ This proves part (3). We recall that $\rho=\gcd(d(x),G_1(x))$ and that for GFP of Lucas type it is required that $\gcd(p_0(x), p_1(x))=1$, $\gcd(p_0(x), d(x))=1$, $\gcd(p_0(x), g(x))=1$ and $\gcd(d(x), g(x))=1$. We also recall $p_0(x)=0$ and $p_1(x)=1$ for GFP of Fibonacci type. For the rest of the paper we use the notation $G_n^{*}$ if the generalized Fibonacci polynomial $G_n$ satisfies the Binet formula (\[bineformulados\]) and $G_n^{'}$ if the generalized Fibonacci polynomial $G_n$ satisfies the Binet formula (\[bineformulauno\]). We use $G_n$ if the result does not need the mentioned classification to be true. We recall that for Lucas type polynomials $|p_{0}(x)| =1 \text{ or } 2$ and for Fibonacci type polynomial $p_{1}(x) =1$. \[gcdlemmas\] If $\{ G_{n}(x)\}$ is a GFP of either Lucas or Fibonacci type, then (1) $\gcd(d(x), G_{2n+1}(x))=G_1(x)$ for every positive integer $n$, (2) If the GFP is of Lucas type, then $\gcd(d(x), G_{2n}^{*}(x))= 1$ and if the GFP is of Fibonacci type, then $\gcd(d(x), G_{2n}^{\prime}(x))= d(x)$ (3) $\gcd(g(x), G_n(x))=\gcd(g(x), G_{1}(x))=1,$ for every positive integer $n$. We prove part (1) by induction, the proof of part (2) is similar and we omit it. Let $\{G_n\}$ a GFP. Let $S(n)$ be the statement $$\rho=\gcd(d(x), G_{2n+1}(x))\text{ for }n\ge 1.$$ To prove $S(1)$ we suppose that $\gcd(d(x), G_{3}(x))=r$. Thus, $r$ divides any linear combination of $d(x)$ and $G_3(x)$. Therefore, $r$ divides $G_3(x)-d(x) G_2(x)$. This and given that $G_3(x)= d(x) G_{2}(x) + g(x) G_{1}(x)$, imply that $r\mid g(x)G_{1}(x)$. So, $r\mid \gcd(d(x),g(x)G_{1}(x))$. Since $\gcd(d(x), g(x))=1$, we have that $r\mid \rho$. It is easy to see that $\rho \mid r$. Thus, $r=\gcd(d(x),G_1(x))$. This proves $S(1)$. We suppose that $S(n)$ is true for $n=k-1$. That is, suppose that $\gcd(d(x), G_{2k-1}(x)) = \rho$. To prove $S(k)$ we suppose that $\gcd(d(x), G_{2k+1}(x))=r'$. Thus, $r'$ divides any linear combination of $d(x)$ and $G_{2k+1}(x)$. Therefore, $r'$ divides $G_{2k+1}(x)-d(x) G_{2k}(x)$. This and given that $G_{2k+1}(x)= d(x) G_{2k}(x) + g(x) G_{2k-1}(x)$, imply that $r'\mid g(x)G_{2k-1}(x)$. Therefore, $r'\mid \gcd(d(x),g(x)G_{2k-1}(x))$. Since $\gcd(d(x), g(x))=1$, we have that $r'\mid \gcd(d(x),G_{2k-1}(x))$. By the inductive hypothesis we know that $\gcd(d(x), G_{2k-1}(x)) = \rho$. Thus, $r'\mid \rho$. It is easy to see that $\gcd(d(x),G_{2k+1}(x))$ divides $r'$. So, $r'=\gcd(G_1(x), d(x))$. We now show that depending on the type of the sequence it holds that $\gcd(d(x),G_1(x))=G_1$. If $\{G_n(x)\}$ is a GFP of Fibonacci type, by definition of $p(x)$ we that $G_1(x)=1$ (see comments after Binet formula ). Suppose that $\{G_n(x)\}$ is a GFP of Lucas type. Recall that $2p_{1}(x)=p_{0}(x)d(x)$ and that $|p_{0}(x)|=1 \text { or } 2$. The conclusion is straightforward since $G_1(x)=(a(x)+b(x))/\alpha=d(x)/\alpha$. Proof of part (2). Let $S(n)$ be the statement $$\rho=\gcd(d(x), G_{2n}(x))\text{ for }n\ge 1.$$ To prove $S(2)$ we suppose that $\gcd(d(x), G_{4}(x))=r$. Thus, $r$ divides any linear combination of $d(x)$ and $G_4(x)$. Therefore, $r$ divides $G_4(x)-d(x) G_3(x)$. This and given that $G_4(x)= d(x) G_{3}(x) + g(x) G_{2}(x)$, imply that $r\mid g(x)G_{2}(x)$. Therefore, $r\mid \gcd(d(x),g(x)G_{2}(x))$. Since $\gcd(d(x), g(x))=1$, we have that $r\mid \rho$. It is easy to see that $\rho \mid r$. Thus, $r=\gcd(d(x),G_2(x))$. This proves $S(2)$. We suppose that $S(n)$ is true for $n=k-1$. That is, suppose that $\gcd(d(x), G_{2k-2}(x)) = \rho$. To prove $S(k)$ we suppose that $\gcd(d(x), G_{2k}(x))=r'$. Thus, $r'$ divides any linear combination of $d(x)$ and $G_{2k}(x)$. Therefore, $r'$ divides $G_{2k}(x)-d(x) G_{2k-1}(x)$. This and given that $G_{2k}(x)= d(x) G_{2k-1}(x) + g(x) G_{2k-2}(x)$, imply that $r'\mid g(x)G_{2k-2}(x)$. Therefore, $r'\mid \gcd(d(x),g(x)G_{2k-2}(x))$. Since $\gcd(d(x), g(x))=1$, we have that $r'\mid \gcd(d(x),G_{2k-2}(x))$. By the inductive hypothesis we know that $\gcd(d(x), G_{2k-2}(x)) = \rho$. Thus, $r'\mid \rho$. It is easy to see that $\gcd(d(x),G_{2k}(x))$ divides $r'$. So, $r'=\gcd(G_2(x), d(x))$. We observe that for a GFP of Fibonacci type it holds that $G_2^{\prime}(x)=a(x)+b(x)=d(x)$. So, it is clear that $\gcd(G_{2n}^{\prime}(x),d(x))=d(x)$. For a GFP of Lucas type it holds that $G_0^{*}(x)$ is a non-zero constant. Since $G_2^{*}(x)=d(x)G_1^{*}(x)+G_{0}^{*}g(x)$, and $\gcd(d(x), g(x))=1$, we have $$\gcd(d(x),G_2^{*}(x))=\gcd(d(x),d(x)G_1^{*}(x)+G_{0}^{*}g(x))=\gcd(d(x),G_0^{*}(x)g(x))=1.$$ Proof of part (3). We now prove that $\gcd(g(x), G_1(x))=1$ by cases. If $G_1(x)$ is of the Fibonacci type, the conclusion is straightforward. As a second case we suppose that $G_1(x)$ is of the Lucas type. That is $G_1(x)$ satisfies the Binet formula (\[bineformulados\]). Therefore, we have that $$\gcd(g(x), G_1(x))=\gcd(g(x), L_1(x)) = \gcd(g(x), [a+b]/\alpha)=\gcd(g(x), d(x)/\alpha).$$ Since $\gcd(g(x), d(x))=1$, we have that $\gcd(g(x), d(x)/\alpha)=1$. This completes the proof. \[gcdforproposition4\] If $\{G_{n}(x)\}$ is a GFP polynomial sequence, then for every positive integer $n$ it holds that (1) $\gcd(G_{n}(x), G_{n+1}(x))$ divides $\gcd(G_{n}(x),g(x)G_{n-1}(x))=\gcd(G_{n}(x),G_{n-1}(x))$, (2) $\gcd(G_{n}(x), G_{n+2}(x))$ divides $\gcd(G_{n}(x), d(x)G_{n+1}(x))$. We prove part (1), the proof of part (2) is similar and we omit it. If $r$ is equal to $ \gcd(G_{n}(x), G_{n+1}(x))$, then $r$ divides any linear combination of $G_{n}(x)$ and $G_{n+1}(x)$. Therefore, $r\mid (G_{n+1}(x)-d(x) G_{n}(x))$. This and the recursive definition of $G_{n+1}(x)$ imply that $r\mid g(x)G_{n-1}(x)$. Therefore, $r\mid \gcd(g(x)G_{n-1}(x),G_{n}(x))$. Since $\gcd(g(x),G_{n}(x))=1$, we have that $\gcd(g(x)G_{n-1}(x),G_{n}(x))=\gcd(G_{n-1}(x),G_{n}(x))$. This completes the proof. \[gcddistance1;2\] Let $m$ and $n$ be positive integers with $0<|m-n|\le 2$. (1) If $\{G_{t}^{*}(x)\}$ is a GFP of Lucas type, then $$\gcd(G_m^{*}(x),G_n^{*}(x))= \begin{cases} G_{1}^{*}(x), & \mbox{if $m$ and $n$ are both odd;} \\ 1, & \mbox{otherwise. } \end{cases}$$ (2) If $\{G_{t}^{\prime}(x)\}$ is a GFP of Fibonacci type, then $$\gcd(G_m^{\prime}(x),G_n^{\prime}(x))= \begin{cases} G_{2}^{\prime}(x) & \mbox{if $m$ and $n$ are both even;} \\ 1, & \mbox{otherwise. } \end{cases}$$ We prove part (1) using several cases based on the values of $m$ and $n$. The proof of part (2) is similar and we omit. We first provide the proof for the case when $m$ and $n$ are consecutive integers using induction on $m$. Let $S(m)$ be the statement $$\gcd(G_{m}^{*}(x), G_{m+1}^{*}(x))=1\text{ for } m\ge 1.$$ First we prove $S(1)$. From Lemma \[gcdforproposition4\] part (1), we know that $$\label{formula1:for:gcddistance1;2} \gcd(G_{1}^{*}(x),G_{2}^{*}(x)) \text{ divides } \gcd(G_{1}^{*}(x), g(x)G_{0}^{*}(x)).$$ Since $$\gcd(G_{0}^{*}(x),G_{1}^{*}(x))=\gcd(p_{0}(x),p_{1}(x))=1,$$ we have that $$\gcd(G_{1}^{*}(x),g(x)G_{0}^{*}(x))=\gcd(G_{1}^{*}(x),g(x)).$$ This, (\[formula1:for:gcddistance1;2\]) and Lemma \[gcdlemmas\] part (3) imply that $\gcd(G_{1}^{*}(x),G_{2}^{*}(x))=1$. We suppose that $S(m)$ is true for $m=k-1$. Thus, suppose that $\gcd(G_{k-1}^{*}(x),G_{k}^{*}(x))=1$. We prove that $S(k)$ is true. From Lemma \[gcdforproposition4\] part (1), we know that $$\label{formula2:for:gcddistance1;2} \gcd(G_{k}^{*}(x),G_{k+1}^{*}(x)) \text{ divides } \gcd(G_{k}^{*}(x), g(x)G_{k-1}^{*}(x)).$$ From Lemma \[gcdlemmas\] part (3) we know that $\gcd(G_{k}^{*}(x),g(x))=1$. So, $$\gcd(G_{k}^{*}(x), g(x)G_{k-1}^{*}(x))= \gcd(G_{k}^{*}(x), G_{k-1}^{*}(x)).$$ This, (\[formula2:for:gcddistance1;2\]) and the inductive hypothesis imply that $\gcd(G_{k}^{*}(x),G_{k+1}^{*}(x)=1$. We now prove the proposition for consecutive even integers (this proof is actually a direct consequence of the previous proof). From Lemma \[gcdforproposition4\] part (2), we have $\gcd(G_{2k}^{*}(x), G_{2k+2}^{*}(x))$ divides $\gcd(G_{2k}^{*}(x),d(x)G_{2k+1}^{*}(x))$. From Lemma \[gcdlemmas\] part (2) we know that $\gcd(d(x),G_{2k}^{*}(x))=1$. This implies that $\gcd(G_{2k}^{*}(x),d(x)G_{2k+1}^{*}(x))= \gcd(G_{2k}^{*}(x),G_{2k+1}^{*}(x))$. From the previous part –that is, the case when $m$ and $n$ are consecutive integers– of this proof we conclude that $\gcd(G_{2k}^{*}(x),G_{2k+1}^{*}(x))=1$. This proves that $\gcd(G_{2k}^{*}(x), G_{2k+2}^{*}(x))=1$. Finally we prove the proposition for consecutive odd integers. From the recursive definition of GFP we have that $\gcd(G_{2k+1}^{*}(x),G_{2k-1}^{*}(x))$ equals $$\gcd(d(x)G_{2k}^{*}(x)+g(x)G_{2k-1}^{*}(x),G_{2k-1}^{*}(x))=\gcd(d(x)G_{2k}^{*}(x),G_{2k-1}^{*}(x))$$ From the first case in this proof we know that $\gcd(G_{2k}^{*}(x),G_{2k-1}^{*}(x))=1$. This implies that $\gcd(G_{2k+1}^{*}(x),G_{2k-1}^{*}(x))=\gcd(d(x),G_{2k-1}^{*}(x))$. This and Lemma \[gcdlemmas\] imply that $$\gcd(G_{2k+1}^{*}(x),G_{2k-1}^{*}(x))=\gcd(d(x),G_{2k-1}^{*}(x))=G_1^{*}(x).$$ This completes the proof of part (1). Identities and other properties of Generalized Fibonacci Polynomials {#mainresults:section} ==================================================================== In this section we present some identities that the GFP satisfy. These identities are required for the proofs of certain divisibility properties of the GFP. The results in this section are proved using the Binet formulas (see Section \[General:Fibonacci:Polynomial\]). Proposition \[divisity:Hogat:property1\] part (1) is a variation of a result proved by Hoggatt and Long [@HoggattLong], similarly Proposition \[divisity:property:fibonacci\] is a variation of a divisibility property proved by them in the same paper. In 1963 Ruggles [@koshy] proved that $F_{n+m}= F_{n} L_{m}-(-1)^m F_{n-m}$. Proposition \[divisity:Hogat:property1\] parts (2) and (3) are a generalization of this numerical identity. In 1972 Hansen [@hansen] proved that $ 5 F_{m +n - 1}=L_{m} L_{n} + L_{m - 1} L_{n -1} $. Proposition \[divisity:Hogat:property2\] part (1) is a generalization of this numerical identity. \[divisity:Hogat:property1\] If $\{ G_{n}^{*}(x)\}$ and $\{ G_{n}^{'}(x)\}$ are equivalent GFP sequences, then (1) $G_{m+n+1}^{'}(x)= G_{m+1}^{'}(x)G_{n+1}^{'}(x)+g(x)G_{m}^{'}(x)G_{n}^{'}(x)$, (2) if $n\ge m$, then $G_{n+m}^{'}(x)= \alpha G_{n}^{'}(x) G_{m}^{*}(x)-(-g(x))^{m} G_{n-m}^{'}(x)$, (3) if $n\ge m$, then $G_{n+m}^{'}(x)= \alpha G_{m}^{'}(x)G_{n}^{*}(x) +(-g(x))^{m} G_{n-m}^{'}(x)$. We prove part (1). We know that $\{ G_{n}^{'}(x)\}$ satisfies the Binet formula (\[bineformulauno\]). That is, $R_n(x)= (a^{n}-b^{n})/(a-b)$ . (Recall that we use $a:=a(x)$ and $b:=b(x)$.) Therefore, $G_{m+1}^{'}(x)G_{n+1}^{'}(x)+g(x)G_{m}^{'}(x)G_{n}^{'}(x)$ is equal to, $$\left[(a^{m+1}-b^{m+1})(a^{n+1}-b^{n+1})+g(x)(a^{m}-b^{m})(a^{n}-b^{n})\right]/(a-b)^{2}.$$ Simplifying and factoring terms we obtain, $$\left[\left(a^{n+m}(a^2+g(x))+b^{n+m}(b^2+g(x))\right)-(a^nb^m+b^na^m)(ab+g(x))\right]/(a-b)^{2}.$$ Next, since $ab=-g(x)$, we see that the above expression is equal to, $$\left[a^{n+m}(a^2+g(x))+b^{n+m}(b^2+g(x))\right]/(a-b)^{2}.$$ This, with the facts that, $(a^2+g(x))=a(a-b)$ and $(b^2+g(x))=-b(a-b)$, shows that the above expression is equal to $$\left(a^{n+m+1}-b^{n+m+1}\right) /(a-b)=R_{n+m+1}(x).$$ We prove part (2) the proof of part (3) is identical and we omit it. Suppose that $G_{k}^{*}(x)$ is equivalent to $G_{k}^{'}(x)$ and that $G_{k}^{*}(x)$ is of the Lucas type for all $k$. For simplicity let us suppose that $\alpha=1$ (the proof when $\alpha\neq 1$ is similar, so we omit it). Using the Binet formulas (\[bineformulados\]) and (\[bineformulauno\]) we obtain that $G_{n}^{'}(x) G_{m}^{*}(x)-(-g(x))^{m} G_{n-m}^{'}(x)$ equals $$\displaystyle{\frac{(a^{n}-b^{n})(a^{m}+b^{m})-(-g(x))^{m}(a^{n-m}-b^{n-m})}{(a-b)}}.$$ After performing the indicated multiplication and simplifying we obtain that this expression is equal to $$\left[\frac{a^{n+m}-b^{n+m}}{a-b} \right]+ \left[\frac{a^{n}b^{m}-a^{m}b^{n}-(-g(x))^m a^{n-m}+(-g(x))^mb^{n-m}}{a-b}\right].$$ Since $-g(x)=ab$, it is easy to see that the expression in the right bracket is equal to zero. Thus, $(a^{n+m}-b^{n+m})/(a-b)= G_{n+m}^{'}(x)$. This completes the proof of part (2). \[divisity:Hogat:property2\] Let $\{ G_{n}^{*}(x)\}$ and $\{ G_{n}^{'}(x)\}$ be equivalent GFP sequences. If $m\ge 0$ and $n\ge 0$, then (1) $(a-b)^2G^{'}_{m+n+1}(x) = \alpha^2 G_{m+1}^{*}(x)G_{n+1}^{*}(x)+\alpha^2 g(x)G_{m}^{*}(x)G_{n}^{*}(x),$ (2) $G_{m+n+2}^{*}(x) = \alpha G_{m+1}^{*}(x)G_{n+1}^{*}(x)+g(x)[\alpha G_{m}^{*}(x)G_{n}^{*}(x)-G_{m+n}^{*}(x)].$ In this proof we use $\alpha=1$, the proof when $\alpha\neq 1$ is similar, so we omit it. (Recall, once again, that we use $a:=a(x)$ and $b:=b(x)$.) Proof of part (1). Since $ G_{n}^{*}(x)$ is a Fibonacci polynomial of the Lucas type, we have that $G_{n}^{*}(x)$ satisfies the Binet formula $L_n(x)= (a^{n}+b^{n})/\alpha$ given in (\[bineformulados\]). Therefore, $$\label{divisity:Hogat:property2:formula1} G_{m+1}^{*}(x)G_{n+1}^{*}(x)+g(x)G_{m}^{*}(x)G_{n}^{*}(x)$$ is equal to, $$\left[a^{n+1}+b^{n+1}\right]\left[a^{m+1}+b^{m+1}\right]+g(x)\left[a^{n}+b^{n}\right]\left[a^{m}+b^{m}\right].$$ Simplifying and factoring we see that this expression is equal to $$a^{m+n}\left[a^2+g(x)\right]+b^{m+n}\left[b^2+g(x)\right]+(ab+g(x))\left[a^mb^n+a^nb^m\right].$$ Since $$ab=-g(x),\; a^2+g(x)=a(a-b),\text{ and } b^2+g(x)=-b(a-b),$$ we have that the expression in (\[divisity:Hogat:property2:formula1\]) is equal to $(a-b)(a^{m+n+1}-b^{m+n+1})$. We recall that $G_{m+n+1}^{'}(x)$ is equivalent to $G_{m+n+1}^{*}(x)$. Thus, $G^{'}_{m+n+1}(x) =(a^{m+n+1}-b^{m+n+1})/(a-b)$. Therefore, $(a-b)^2G^{'}_{m+n+1}(x) =(a-b)\left[a^{m+n+1}-b^{m+n+1}\right]$. This completes the proof of part (1). Proof of part (2). From the proof of part (1) we know that $$(a-b)^2G^{'}_{m+n+1}(x)=(a-b)[a^{m+n+1}-b^{m+n+1}].$$ Simplifying the right side of the previous equality we have that $$(a-b)^2G^{'}_{m+n+1}(x)=a^{m+n+2}-ba^{m+n+1}- ab^{m+n+1} + b^{m+n+2}.$$ So, $(a-b)^2G^{'}_{m+n+1}(x)=a^{m+n+2}+ b^{m+n+2} -ab[a^{m+n}+ b^{m+n}]$. We recall that $ab=-g(x)$. Thus, $$(a-b)^2G^{'}_{m+n+1}(x)=a^{m+n+2}+ b^{m+n+2} + g(x)[a^{m+n}+ b^{m+n}].$$ This and the Binet formula (\[bineformulados\]), imply that $$(a-b)^2G^{'}_{m+n+1}(x)=G_{m+n+2}^{*}(x) + g(x)G_{m+n}^{*}(x).$$ So, the proof follows from part (1) of this Proposition. \[Dic2\] Let $\{ G_{n}^{*}(x)\}$ be a GFP sequence of the Lucas type. If $m$, $n$, $r$, and $q$ are positive integers, then (1) if $m\leq n$, then $G_{m+n}^{*}(x)=\alpha G_{m}^{*}(x)G_{n}^{*}(x)+(-1)^{m+1}(g(x))^mG_{n-m}^{*}(x)$, (2) if $r<m$, then there is a polynomial $T(x)$ such that $$G_{mq+r}^{*}(x)= \left\{ \begin{array}{ll} G_{m}^{*}(x)T(x)+(-1)^{m(t-1)+t+r}(g(x))^{(t-1)m+r} G_{m-r}^{*}(x), & \hbox{ if } $q$ \hbox{ is odd;} \\ G_{m}^{*}(x)T(x)+(-1)^{(m+1)t}(g(x))^{mt} G_{r}^{*}(x), & \hbox{ if } $q$ \hbox{ is even} \end{array} \right.$$ where $t=\left\lceil \frac{q}{2} \right\rceil$, (3) if $n>1$, then there is a polynomial $T_n(x)$ such that $$G_{2^nr}^{*}(x)= G_{r}^{*}(x)T_n(x)+(2/\alpha)(g(x))^{2^{n-1}r}.$$ We prove part (1). Since $G_{m}^{*}(x)$ and $G_{n}^{*}(x)$ are of the Lucas type, they both satisfy the Binet formula (\[bineformulados\]). Thus, $$G_{m}^{*}(x)G_{n}^{*}(x)=\left(\frac{a^m+b^m}{\alpha}\right)\left(\frac{a^n+b^n}{\alpha}\right) =\frac{a^{m+n}+b^{m+n}}{\alpha^2}+\frac{(ab)^{m}\left(a^{n-m}+b^{n-m}\right)}{\alpha^2}.$$ So, $G_{m}^{*}(x)G_{n}^{*}(x)=\left[G_{n+m}^{*}(x)+(ab)^mG_{n-m}^{*}(x)\right]/\alpha.$ This and $ab=-g(x)$, imply that $$G_{m+n}^{*}(x)=\alpha G_{m}^{*}(x)G_{n}^{*}(x)-(-g(x))^{m} G_{n-m}^{*}(x).$$ This completes the proof of part (1). We first prove part (2) when $q$ is odd by induction. Suppose $q=2t-1$, and let $S(t)$ be the following statement. For every positive integer $t$ there is a polynomial $T_t(x)$ such that $$G_{m(2t-1)+r}^{*}(x)=G_{m}^{*}(x)T_t(x)+(-1)^{m(t-1)+t+r}(g(x))^{(t-1)m+r} G_{m-r}^{*}(x).$$ From part (1), taking $T_1(x)=\alpha G_{r}^{*}(x)$, it is easy to see that $S(t)$ is true if $t=1$. We suppose that $S(k)$ is true. That is, suppose that there is a polynomial $T_{k}(x)$ such that $$\label{formula1:for:Dic2} G_{m(2k-1)+r}^{*}(x)=G_{m}^{*}(x)T_k(x)+(-1)^{m(k-1)+t+r}(g(x))^{(k-1)m+r} G_{m-r}^{*}(x).$$ We prove that $S(k+1)$ is true. Notice that $G_{m(2k+1)+r}^{*}(x)=G_{(2km+r)+m}^{*}(x)$. Therefore, from part (1) we have $$G_{m(2k+1)+r}^{*}(x)=\alpha G_{m}^{*}(x)G_{2km+r}^{*}(x)+(-1)^{m+1}g^m(x)G_{m(2k-1)+r}^{*}(x).$$ This and $S(k)$ (see (\[formula1:for:Dic2\])), imply that $G_{m(2(k+1)+r}^{*}(x)$ equals $$\alpha G_{m}^{*}(x)G_{2km+r}^{*}(x)+(-1)^{m+1}(g(x))^{m}G_{m}^{*}(x)T_k (x)+(-1)^{km+(t+1)+r}(g(x))^{km+r} G_{m-r}^{*}(x).$$ Therefore, $G_{m(2(k+1)+r}^{*}(x)$ equals $$G_{m}^{*}(x)[\alpha G_{2km+r}^{*}(x)+(-1)^{m+1}(g(x))^{m}T_k (x)]+(-1)^{km+(t+1)+r}(g(x))^{km+r} G_{m-r}^{*}(x).$$ This, with $T_{k+1}(x):=\alpha G_{2km+r}^{*}(x)+(-1)^{m+1}(g(x))^mT_k (x)$, implies $S(k+1)$. This complete the proof when $q$ is odd. We now prove part (2) when $q$ is even by induction –this proof is similar to the case when $q$ is odd–. Suppose $q=2t$, and let $H(t)$ be the following statement. For every positive integer number there is a polynomial $T_t(x)$ such that $$G_{m(2t)+r}^{*}(x)=G_{m}^{*}(x)T_t(x)+(-1)^{(m+1)t}(g(x))^{mt} G_{r}^{*}(x).$$ From part (1), taking $T_1(x)=\alpha G_{r}^{*}(x)$, it is easy to see that $H(t)$ is true if $t=1$. We suppose that $H(k)$ is true. That is, suppose that there is a polynomial $T_{k}(x)$ such that $$\label{formula2:for:Dic2} G_{m(2k)+r}^{*}(x)=G_{m}^{*}(x)T_k(x)+(-1)^{(m+1)k}(g(x))^{mk} G_{r}^{*}(x).$$ We prove that $H(k+1)$ is true. Notice that $G_{2m(k+1)+r}^{*}(x)=G_{((2k+1)m+r)+m}^{*}(x)$. Therefore, from part (1) we have that $$G_{2m(k+1)+r}^{*}(x)=\alpha G_{m}^{*}(x)G_{(2k+1)m+r}^{*}(x)+(-1)^{m+1}(g(x))^m G_{2mk+r}^{*}(x).$$ This and $H(k)$ (see (\[formula2:for:Dic2\])), imply that $G_{m(2(k+1))+r}^{*}(x)$ equals $$\alpha G_{m}^{*}(x)G_{(2k+1)m+r}^{*}(x)+(-1)^{m+1}(g(x))^mG_{m}^{*}(x)T_{k} (x)+(-1)^{(k+1)(m+1)}(g(x))^{m(k+1)}G_{r}^{*}(x).$$ Therefore, $G_{m(2(k+1))+r}^{*}(x)$ equals $$G_{m}^{*}(x) \left[\alpha G_{(2k+1)m+r}^{*}(x)+(-1)^{m+1}(g(x))^mT_{k} (x)\right]+(-1)^{(m+1)(k+1)}(g(x))^{m(k+1)} G_{r}^{*}(x).$$ This with $T_{k+1}(x):=\alpha G_{(2k+2)m+r}^{*}(x)+(-1)^{m}(g(x))^mT_{k} (x)$, implies $H(k+1)$. We finally prove part (3) by induction. Since $G_{n}^{*}(x)$ is of the Lucas type, by the Binet formula it is easy to see that $G_0(x)=2/\alpha$. Let $S(n)$ be the statement: for every positive integer $n$ there is a polynomial $T_n(x)$ such that $G_{2^nr}^{*}(x)= G_{r}^{*}(x)T_n(x)+(2/\alpha)g^{2^{n-1}r}(x)$. Proof of $S(2)$. From part (1) we have $G_{2^2r}^{*}(x)= G_{2r+2r}^{*}(x)=\alpha(G_{2r}^{*}(x))^2-(2/\alpha)(g(x))^{2r}$. Applying again the result in part (1) for $G_{2r}^{*}(x)$ (and simplifying) we obtain $$\begin{aligned} G_{2^2r}^{*}(x)&=&\alpha[\alpha(G_{r}^{*}(x))^2-\frac{2}{\alpha}(-g(x))^{r}]^2-\frac{2}{\alpha}(g(x))^{2r}\\ &=&G_{r}^{*}(x)[\alpha^3(G_{r}^{*}(x))^3+(-1)^{r+1}4\alpha G_r^{*}(x)(g(x))^{r}]+\frac{4}{\alpha}(g(x))^{2r} -\frac{2}{\alpha}(g(x))^{2r}\\ &=&G_{r}^{*}(x)T_2(x)+\frac{2}{\alpha}(g(x))^{2r}.\end{aligned}$$ where $T_2(x)=\alpha^3(G_{r}^{*}(x))^3+(-1)^{r+1}4\alpha G_{r}^{*}(x)(g(x))^{r}$. This proves $S(2)$. We suppose that $S(k)$ is true for $k>2$, and we prove $S(k+1)$ is true. That is, we suppose that for a fixed $k$ there is a polynomial $T_k(x)$ such that $$G_{2^kr}^{*}(x)= G_{r}^{*}(x)T_k(x)+(2/\alpha)g^{2^{k-1}r}(x).$$ From part (1) we have $G_{2^{k+1}r}^{*}(x)= G_{2^k r+2^k r}^{*}(x)=\alpha(G_{2^k r}^{*}(x))^2-(2/\alpha)(g(x))^{2^k r}$. Using the result from the inductive hypothesis $S(k)$ and simplifying, we obtain $$\begin{aligned} G_{2^{k+1}r}^{*}(x)&=&\alpha[ G_{r}^{*}(x)T_k(x)+\frac{2}{\alpha}(g(x))^{2^{k-1} r} ]^2-\frac{2}{\alpha}(g(x))^{2^k r}\\ &=&G_{r}^{*}(x)[\alpha G_{r}^{*}(x)T_k ^2(x)+4T_k(x)(g(x))^{2^{k-1} r} ]+\frac{4}{\alpha}(g(x))^{2^k r} -\frac{2}{\alpha}(g(x))^{2^kr}\\ &=&G_{r}^{*}(x)T_{k+1}(x)+\frac{2}{\alpha}(g(x))^{2^kr},\end{aligned}$$ where $T_{k+1}(x)=\alpha G_{r}^{*}(x)T_k ^2(x)+4T_k(x)(g(x))^{2^{k-1} r} $. This completes the proof of part (3). In the following part of this section, we present two divisibility properties for the GFP. \[divisity:property:fibonacci\] If $\{ G_{n}^{'}(x)\}$ is a GFP sequence of the Fibonacci type, then $G_{m}^{'}(x)$ divides $G_{n}^{'}(x)$ if and only if $m$ divides $n$. We first prove the sufficiency. Based on the hypothesis that $m$ divides $n$, there is an integer $q\ge1$ such that $n=mq$. Then, using the Binet formula (\[bineformulauno\]), we have that, $$G_{m}^{'}(x)=(a^m-b^m)/(a-b)\,\, \text{ and }\,\, G_{mq}^{'}(x)=(a^{mq}-b^{mq})/(a-b).$$ It is easy to see –using induction on $q$– that $(a^m-b^m)$ divides $(a^{mq}-b^{mq})$ which implies that $G_{m}^{'}(x)$ divides $G_{mq}^{'}(x)$. This proves the sufficiency. We now prove the necessity. Suppose that $m$ does not divide $n$ and that $G_{m}^{'}(x)$ divides $G_{n}^{'}(x)$ for $m$ and $n$ greater than $1$. Therefore, there integers $q$ and $r$ with $0<r<n$ such that $n=mq+r$. Then by Proposition \[divisity:Hogat:property1\] part (1) $$\begin{aligned} G_{n}^{'}(x) &=& G_{mq+r}^{'}(x) \\ &=& G_{mq+1}^{'}(x)G_{r}^{'}(x)+g(x)G_{mq}^{'}(x)G_{r-1}^{'}(x)\\ &=& \left(d(x)G_{mq}^{'}(x)+g(x)G_{mq-1}^{'}(x)\right)G_{r}^{'}(x)+g(x)G_{mq}^{'}(x)G_{r-1}^{'}(x)\\ &=& d(x)(x) G_{mq}^{'}(x) G_{r}^{'}(x)+g(x)G_{mq-1}^{'}(x)G_{r}^{'}(x)+g(x)G_{mq-1}^{'}(x)G_{r}^{'}(x).\end{aligned}$$ Grouping terms and simplifying we obtain, $$G_{n}^{'}(x)=G_{mq}^{'}(x)G_{r+1}^{'}(x)+g(x)G_{mq-1}^{'}(x)G_{r}^{'}(x).$$ This and the fact that $G_{m}^{'}(x)\mid G_{n}^{'}(x)$ and $G_{m}^{'}(x) \mid G_{mq}^{'}(x)$ imply that $$G_{m}^{'}(x)\mid g(x)G_{mq-1}^{'}(x)G_{r}^{'}(x).$$ From Lemma \[gcdlemmas\] part (3) and Proposition \[gcddistance1;2\] we know that $\gcd(G_{mq}^{'}(x),g(x))=1$ and $\gcd(G_{mq-1}^{'}(x), G_{mq}^{'}(x))=1$, respectively. This implies that $G_{m}^{'}(x)\mid G_{r}^{'}(x).$ That is a contradiction since degree ($G_{r-1}^{'}(x))<$ degree $(G_{m-1}^{'}(x))$. This completes the proof. \[divisibity:property:first:type\] Let $m$ be a positive integer that is not a power of two. If $G_{m}^{*}(x)$ is a GFP of Lucas type, then for all odd divisors $q$ of $m$, it holds that $G_{m/q}^{*}(x)$ divides $G_{m}^{*}(x)$. More over $G_{m/q}^{*}(x)$ is of the Lucas type. Let $q$ be an odd divisor of $m$. If $q=1$ the result is obvious. Let us suppose that $q\not = 1$. Therefore, there is an integer $d>1$ such that $m=dq$. Using the Binet formula (\[bineformulados\]), where $a:=a(x)$ and $b:=b(x)$, we have $G_{m}^{*}(x)=G_{dq}^{*}(x)=(a^{dq}+b^{dq})/\alpha$. Let $X=a^{d}$ and $Y=b^{d}$. Using induction it is possible to prove that $X+Y$ divides $X^q+Y^q$. This implies that there is a polynomial $Q(x)$ such that $(X^q+Y^q)/\alpha=Q(x)(X+Y)/\alpha$. Therefore, $$G_{m}^{*}(x)=G_{dq}^{*}(x)=(a^{dq}+b^{dq})/\alpha=Q(x)(a^{d}+b^{d})/\alpha.$$ This and the Binet formula (\[bineformulados\]) imply that $G_{m}^{*}(x)=G_{d}^{*}(x) Q(x)$. Characterization of the strong divisibility property {#gcd:characterization} ==================================================== In this section we prove the main results of this paper. Thus, we prove the necessary and sufficient condition for the strong divisibility property for generalized Fibonacci polynomial of the Fibonacci type. We also prove that the strong divisibility property holds partially for generalized Fibonacci polynomial of the Lucas type. The other important result in this section is that the strong divisibility property holds partially for a generalized Fibonacci polynomial and its equivalent. The results here therefore provide a complete characterization of the strong divisibility property satisfied by the GFP of Fibonacci type. We note that if $G_{m}^{*}(x)$ and $G_{n}^{\prime}(x)$ are two equivalent polynomial from Table \[equivalent\], then $\gcd(G_{m}^{*}(x), G_{n}^{'}(x))$ is either one or $G_{\gcd(m,n)}^{*}(x)$. However, it is not true in general. Here we give an example of a couple GFP polynomials that do not satisfies this property. Firstly we define a Fibonacci type polynomial $$G_0^{\prime}(x)=0, \; G_1^{\prime}(x)= 1,\; \text{and} \; G_{n}^{\prime}(x)= (2x+1) G_{n - 1}^{\prime}(x) + G_{n - 2}^{\prime}(x) \text{ for } n\ge 2.$$ We now define the equivalent polynomial of the Lucas type $$G_0^{*}(x)=2, \; G_1^{*}(x)= 2x+1,\; \text{and} \; G_{n}^{*}(x)= (2x+1) G_{n - 1}^{*}(x) + G_{n - 2}^{*}(x) \text{ for } n\ge 2.$$ After some calculations we can see that $\gcd(G_{m}^{*}(x), G_{n}^{'}(x))$ is one, two or $G_{\gcd(m,n)}^{*}(x)$. Using the same polynomials we can see also that $\gcd(G_{m}^{*}(x), G_{n}^{*}(x))$ is one, two or $G_{\gcd(m,n)}^{*}(x)$. If we do the same calculations for numerical sequences (Fibonacci and Lucas numbers), we can see that they have the same behaviour. In this section we use the notation $E_{2}(n)$ to represent the *integer exponent base two* of a positive integer $n$ which is defined to be the largest integer $k$ such that $2^{k}\mid n$. \[Dic1\] If $R(x)$, $S(x)$ and $T(x)$ are polynomial in $\mathbb{Z}[x]$, then $$\gcd(R(x),T(x))=\gcd(R(x),R(x)S(x)-T(x)).$$ \[propiedadDivision\] Let $\{ G_{n}^{*}(x)\}$ be a GFP sequence of the Lucas type. If $m \mid n$ and $E_2(n)=E_{2}(m)$, then $ \gcd(G_{n}^{*}(x),G_{m}^{*}(x))=G_{m}^{*}(x). $ First of all we recall that $E_{2}(n)$ is the largest integer $k$ such that $2^{k}\mid n$. We suppose that $n=mq$ with $q\in\mathbb{N}$. Since $E_{2}(m)=E_{2}(n)=E_{2}(mq)$, we conclude that $q$ is odd. This, Lemma \[Dic1\] and Proposition $\ref{Dic2}$ part (2) imply that $$\begin{aligned} \gcd(G_{n}^{*}(x),G_{m}^{*}(x))&=&\gcd(G_{qm}^{*}(x),G_{m}^{*}(x))\\ &=&\gcd (G_{m}^{*}(x)T(x)+(-1)^{(n-1)m+n}(g(x))^{(n-1)m} G_{m}^{*}(x),G_{m}^{*}(x))\\ &=&G_{m}^{*}(x). \hspace{10.1cm}\end{aligned}$$ This proves the proposition. \[divisity:property:firstype\] Let $G_{m}^{*}(x)$ be a GFP of Lucas type. If $m>0$ is not a power of two, then for all odd divisors $q$ of $m$, it follows that $G_{m/q}^{*}(x)$ divides $G_{m}^{*}(x)$. More over $G_{m/q}^{*}(x)$ is of the Lucas type. It is easy to see that $E_2(m/q)=E_2(m)$. Therefore, the conclusion follows by Proposition \[propiedadDivision\]. \[propiedadDivisioncase2\] Let $d_{k}=\gcd(G_0 ^*(x),G_{k} ^{*}(x))$ where $G_{k}^{*}(x)$ is GFP of the Lucas type. Suppose that there is an integer $k'>0$ such that $d_{k'}=2$. If $m$ is the minimum positive integer such that $d_{m}=2$, then $m|n$ if and only if $d_{n}=2$. We suppose $m$ is the minimum positive integer such that $d_{m}=2$. Suppose that $m|n$, by Proposition \[propiedadDivision\] we know that $\gcd(G^* _{m}(x),G_{n}^{*}(x))=G_{m}^{*}(x)$ (we recall that $G^*_{0}(x)=p_0(x)$ and $|p_0(x)|=1$ or $2$). This and the fact that $2|G_{m} ^*(x)$, implies that $\gcd(G_{0} ^*(x),G_{n} ^*(x))=2$. This proves that $d_{n}=2$. Suppose that there is $n\in \mathbb{N}-\{m\}$ that satisfies that $d_{n}=2$ (note $2|\gcd(G^{*}_{m}(x),G_{n} ^*(x)$). From the division algorithms we have that there are integer $q$ and $r$ such that $n=mq+r$ where $0\le r< m$. This and Proposition \[Dic2\] part (2), imply that $$\gcd(G^{*} _{m}(x),G_{n} ^*(x))= \left\{ \begin{array}{ll} \gcd(G^* _{m}(x),(g(x))^{(t-1)m+r} G_{m-r}^{*}(x)) & \hbox{ if } $q$ \hbox{ is odd;} \\ \gcd(G^* _{m}(x),(g(x))^{mt} G_{r}^{*}(x)) & \hbox{ if } $q$ \hbox{ is even.} \end{array} \right.$$ This, Lemma \[gcdlemmas\] part (3), implies that $\gcd(G^* _{m}(x),G_{n} ^*(x))$ is either $\gcd(G^* _{m}(x),G_{m-r}^{*}(x))$ or $\gcd(G^* _{m}(x),G_{r}^{*}(x))$. From this and the fact that $2|\gcd(G^* _{m}(x),G_{n} ^*(x))$, we have that $\gcd(G^* _{r}(x),G^* _{0}(x))=2$ or $\gcd(G^* _{m-r}(x),G^*_{0}(x))=2$ . This holds only if $r=0$, due to definition of $m$. Therefore, $n=mq$. \[fundamental\] Let $G_{k}^{*}(x)$ be a GFP of Lucas type and let $n=mq+r$ where $m, q$ and $r$ are positive integers with $r<m$. If $m_1= m-r$ when $q$ is odd and $m_1=r$ when $q$ is even, then $ \gcd(G_{n}^{*}(x),G_{m}^{*}(x))=\gcd (G_{m_1}^{*}(x),G_{m}^{*}(x)). $ Let $$f(x)= \begin{cases} (-1)^{m(t-1)+t+r}(g(x))^{(t-1)m+r} , & \mbox{ if $q$ is odd;}\\ (-1)^{(m+1)t}(g(x))^{mt}, & \mbox{ if $q$ is even.} \end{cases}$$ This and Lemma \[gcdlemmas\] part (3) imply that $\gcd(G_{m}^{*}(x),f(x))=1$. Therefore, by Proposition \[Dic2\] part (2) it follows that $\gcd(G_{mq+r}^{*}(x),G_{m}^{*}(x))=\gcd (G_{m}^{*}(x)T(x)+f(x)G_{m_1}^{*}(x),G_{m}^{*})$. Now it is easy to see that $$\gcd (G_{m}^{*}(x)T(x)+f(x)G_{m_1}^{*}(x),G_{m}^{*})=\gcd (f(x) G_{m_1}^{*}(x),G_{m}^{*}(x)).$$ Since $\gcd(G_{m}^{*}(x),f(x))=1$, by Proposition \[prop2;1\] part (1) we have $\gcd (f(x) G_{m_1}^{*}(x),G_{m}^{*}(x))=\gcd (G_{m_1}^{*}(x),G_{m}^{*}(x))$. \[second:main:thm\] Let $\{ G_{n}^{*}(x)\}$ be a GFP of the Lucas type. If $m$ and $n$ are positive integers and $d=\gcd(m,n)$, then $$\gcd(G_{m}^{*}(x),G_{n}^{*}(x))= \begin{cases} G_{d}^{*}(x) & \mbox{ if }\; E_{2}(m)= E_{2}(n);\\ \gcd(G_{d}^*(x),G_{0}^*(x)) & \mbox{ otherwise}. \end{cases}$$ First of all we prove the result for $E_{2}(n)=E_{2}(m)$. From the Euclidean algorithm we know that there are non-negative integers $q$ and $r$ such that $n=mq+r$ with $r<m$. Let $d=\gcd(m,n)$. Clearly, if $r=0$, then $d=m$. Therefore, the result holds by Proposition \[propiedadDivision\]. We suppose that $r\neq 0$. If we take $m_1$ as in Lemma \[fundamental\], then $$\gcd (G_{n}^{*}(x),G_{m}^{*}(x))=\gcd(G_{mq+r}^{*}(x),G_{m}^{*}(x))=\gcd (G_{m_1}^{*}(x),G_{m}^{*}(x)).$$ Let $M_1=\{m,m_1 \}$. Notice that $\gcd(m_1,m)=d$, $E_2(m)=E_2(m_1)$ and that $m_1<m$. Therefore, there are non-negative integers $q_1$ and $r_1$ such that $m=m_1q_1+r_1$ with $r_1<m_1$. Again, if $r_1=0$, by Proposition \[propiedadDivision\] we obtain that $ \gcd(G_{n}^{*}(x),G_{m}^{*}(x))=\gcd(G_{m}^{*}(x),G_{m_1}^{*}(x))=G_{d}^{*} (x). $ If $r_1\neq 0$ we repeat the previous step and then we can guarantee that $$\gcd (G_{n}^{*}(x),G_{m}^{*}(x))=\gcd (G_{m_1}^{*}(x),G_{m}^{*}(x))=\gcd (G_{m_1}^{*}(x),G_{m_2}^{*}(x)),$$ where $$m_2=\begin{cases}m_1-r_1 & \mbox{ if $q$ is odd};\\ r_1 & \mbox{ if } \mbox{ if $q$ is even}. \end{cases}$$ We repeat this procedure $t$ times until we obtain the ordered decreasing sequence $m>m_1>m_2>\cdots >m_t\geq d$ such that $E_2(m)=E_2(m_t)$ and $\gcd(m_t,m_{t-1})=d$, where $$m_t=\begin{cases}m_{t-1}-r_{t-1} & \mbox{ if $q$ is odd};\\ r_{t-1} & \mbox{ if } \mbox{ if $q$ is even}. \end{cases}$$ Notice that $M_t=\{m, m_1, m_2, \ldots, m_t\} =M_{t-1}\cup\{m_t\}$ is an ordered set of natural numbers, therefore there is a minimum element. Since $M_t$ is constructed with a sequence of decreasing positive integer numbers, there must be an integer $k$ such that $M_{t}\subset M_k$ for all $t<k$ and $M_{k+1}$ is undefined. Thus, the procedure ends with $M_k$. Note that $m>m_1>m_2>\cdots >m_k \geq d$ such that $E_2(m)=E_2(m_k)$ and $\gcd(m_k,m_{k-1})=d$. [**Claim**]{}. The minimum element of $M_k$ is $m_k=d$ and $m_{k}\mid m_{k-1}$. Proof of claim. From the Euclidean algorithm we know that there are non-negative integers $q_k$ and $r_k$ such that $m_{k-1}=m_kq_k+r_k$ with $r_k<m_k$. If $r_k\neq 0$ we can repeat the procedure describe above to obtain a new set $M_{k+1}$ with $M_{k} \subset M_{k+1}$. That is a contradiction. Therefore, $r_k=0$. So, $m_{k-1}=m_kq_k$. This implies that $\gcd(m_k,m_{k-1})=d$. Thus, $m_k=d$. This proves the claim. The Claim and the Proposition \[propiedadDivision\] allow us to conclude that $$\gcd (G_{n}^{*}(x),G_{m}^{*}(x))=\gcd (G_{m}^{*}(x),G_{m_1}^{*}(x))=\cdots=\gcd (G_{m_{k-1}}^{*}(x),G_{m_{k}}^{*}(x))=G_{d}^{*}.$$ We now prove by cases that $\gcd (G_{n}^{*}(x),G_{m}^{*}(x))=\gcd (G_{d}^{*}(x),G_{0}^{*}(x))$ if $E_{2}(n)\neq E_{2}(m)$ and $d=\gcd(n,m)$. **Case 1.** Suppose that $m<n$ and that $E_2(n)<E_2(m)$. From the Euclidean algorithm there are two non-negative integers $q$ and $r$ such that $n=mq+r$ with $r<m$. Let $m_1= m-r$ when $q$ is odd and $m_1=r$ when $q$ is even (as defined as in Lemma \[fundamental\]). Since $n=mq+r$ and $E_2(n)<E_2(m)$, we have that $r\neq 0$. It is easy to see that $E_2(n)=E_2(r)$, and therefore $E_2(n)=E_2(m_1)$. This and Lemma \[fundamental\] imply that $ \gcd (G_{n}^{*}(x),G_{m}^{*}(x))=\gcd (G_{m}^{*}(x),G_{m_1}^{*}(x)). $ Since $E_2(m_1)=E_2(n)<E_2(m)$ and $m_1<m$, the criteria for Case 2 are satisfied here, so the proof of this case may be completed as we are going to do in Case 2 below. **Case 2.** Suppose that $E_2(m)<E_2(n)$ and that $m<n$. From the Euclidean algorithm we know that there are two non-negative integers $r$ and $q$ such that $n=mq+r$ with $ r<m$. If $r=0$, then $q$ must be even (because $E_2(m)<E_2(n)$). Let $k=E_2(q)$ and we consider two subcases on $k$. **Subcase 1.** If $k=1$, then $q=2t$ where $t$ is odd. Therefore, by the Proposition \[Dic2\] part (1) we have that $ G_{n}^{*}(x)=G_{2mt}^{*}(x)=\alpha (G_{mt}^{*}(x))^2+(-1)^{mt+1}(G_0 ^*(x))(-g(x))^{mt}. $ This, Proposition \[propiedadDivision\], Lemma \[gcdlemmas\] part (3) and Lemma \[Dic1\] imply that $$\begin{aligned} \gcd (G_{n}^{*}(x),G_{m}^{*}(x))&=& \gcd\left(\alpha (G_{mt}^{*}(x))^2+(-1)^{mt+1}G_0 ^*(x)(-g(x))^{mt}, G_{m}^{*}(x)\right) \\ &=& \gcd((-1)^{mt+1}G_0 ^*(x)(-g(x))^{mt},G_{m}^{*}(x)) \\ &=& \gcd(G_0 ^*(x),G_{m}^{*}(x))\\ &=& \gcd(G_0 ^*(x),G_{d}^{*}(x)). \end{aligned}$$ **Subcase 2.** If $k>1$, then $q=2^kt$ where $t$ is odd. Therefore, by the Proposition \[Dic2\] part (3), there is a polynomial $T_k(x)$ such that $ G_{n}^{*}=G_{2^kmt}^{*}=G_{mt}^{*}(x)T_k(x)+G_0 ^*(x)g^{2^{k-1}mt}(x). $ This, Proposition \[propiedadDivision\], Lemma \[gcdlemmas\] part (3) and Lemma \[Dic1\] imply $$\begin{aligned} \gcd (G_{n}^{*}(x),G_{m}^{*}(x))&=& \gcd(G_{mt}^{*}(x)T_k(x)+G_0 ^*(x) g^{2^{k-1}mt}(x) , G_{m}^{*}(x)) \\ &=& \gcd(G_0 ^*(x) g^{2^{k-1}mt}(x) ,G_{m}^{*}(x)) \\ &=&\gcd(G_0 ^*(x) ,G_{m}^{*}(x))\\ &=& \gcd(G_0 ^*(x),G_{d}^{*}(x)). \end{aligned}$$ Let us now suppose that $r\neq 0$. This and Lemma \[fundamental\], imply that $ \gcd (G_{n}^{*}(x),G_{m}^{*}(x))=\gcd (G_{m}^{*}(x),G_{m_1}^{*}(x)), $ where $m_1= m-r$ when $q$ is odd and $m_1=r$ when $q$ is even (as defined as in Lemma \[fundamental\]). Therefore, $m_1<m<n$ and $\gcd(m,n)=\gcd(m,m_1)=d$. We analyze both, the case in which $m_1\mid m$ and the case in which $m_1 \nmid m$. Suppose that $m=m_1 q_2$ and we consider two cases for $q_2$. **Subcase $q_2$ is odd.** If $q_2$ is odd we have that $E_2(m_1)=E_2(m)$. Therefore, by Proposition \[propiedadDivision\] we obtain that $$\gcd(G_{m}^{*}(x),G_{n}^{*}(x))=\gcd(G_{m}^{*}(x),G_{m_1}^{*}(x))=G_{d}^{*}(x)\; \text{ and }\; E_{2}(G_{d}^{*}(x))<E_{2}(G_{n}^{*}(x)).$$ This imply that $\gcd(G_{m}^{*}(x),G_{n}^{*}(x))=\gcd(G_{d}^{*}(x),G_{0}^{*}(x))$. **Subcase $q_2$ is even.** If $q_2$ is even, then $E_2(m_1)<E_{2}(m)$. Now it is easy to see that $\gcd(G_{m}^{*}(x),G_{n}^{*}(x))=\gcd(G_{m_1}^{*}(x),G_{0}^{*}(x))=\gcd(G_{d}^{*}(x),G_{0}^{*}(x))$. Now suppose that $m_1 \nmid m$. Therefore there are two non-negative integers $r_2$ and $q_2$ such that $m=m_1q_2+r_2$ where $0<r_2<m_1$. From Lemma \[fundamental\] we guarantee that we can find $m_2$ such that $m_2<m_1$, $\gcd(m_1,m_2)=d$ and $\gcd(G_{m_1}^{*}(x),G_{m_2}^{*}(x))=\gcd(G_{m_1}^{*}(x),G_{m}^{*}(x))$. In this form we construct a set $M_t= \{n, m, m_1, m_2, \ldots, m_t\}$ where $n> m>m_1>\cdots >m_t$ such that $\gcd(m_j,m_{j-1})=d$ and $$\gcd(G_{n}^{*}(x),G_{m}^{*}(x))=\gcd(G_{m_1}^{*}(x),G_{m}^{*}(x))=\cdots= \gcd(G_{m_j}^{*}(x),G_{m_{j-1}}^{*}(x)).$$ From Lemma \[fundamental\] we know that $n> m>m_1>\cdots >m_j$ ends only if $r_j=0$. Since $M_j= \{n, m, m_1, m_2, \ldots, m_j\}$ is an ordered sequence of natural numbers, it has a minimum element $m_j$. Therefore, $m_{j}\mid m_{j-1}$. It is easy to see that $ \gcd(G_{n}^{*}(x),G_{m}^{*}(x))=\gcd(G_{m_j}^{*}(x),G_{m_{j-1}}^{*}(x))$. This is equivalent to $\gcd(G_{n}^{*}(x),G_{m}^{*}(x))=\gcd(G_{d}^{*}(x),G_{0}^{*}(x)). $ This completes the proof. Let $d_{k}=\gcd(G_0 ^*(x),G_{k} ^{*}(x))$ where $G_{k}^{*}(x)$ is GFP of the Lucas type. If $m$ and $n$ are positive integers such that $E_{2}(n)\ne E_{2}(n)$, then (1) Suppose that there is an integer $k'>0$ such that $d_{k'}=2$. If $r$ is the minimum positive integer such that $d_{r}=2$, then $$\gcd(G_{m}^{*}(x),G_{n}^{*}(x))= \begin{cases} 2 & \mbox{ if }\; r|\gcd(m,n);\\ 1 & \mbox{ otherwise}. \end{cases}$$ (2) If $d_{k}\ne 2$ for every positive integer $k$, then $\gcd(G_{m}^{*}(x),G_{n}^{*}(x))=1$. It straightforward from Proposition (\[propiedadDivisioncase2\]). \[gcd:fibonaccilucas:samegcdlucaslucas\] Let $\{ G_{n}^{*}(x)\}$ and $\{ G_{n}^{'}(x)\}$ be equivalent GFP. If $m$ and $n$ are positive integers, then (1) $\gcd(G^{'}_{m+n+1}(x),G_{n}^{*}(x))=\gcd(G_{m+1}^{*}(x),G_{n}^{*}(x)),$ (2) if $m>n$, then $\gcd(G^{'}_{m-n+1}(x),G_{n}^{*}(x))=\gcd(G_{m+1}^{*}(x),G_{n}^{*}(x)),$ (3) if $m<n$, then $\gcd(G^{'}_{n-m+1}(x),G_{n}^{*}(x))=\gcd(G_{m-1}^{*}(x),G_{n}^{*}(x)).$ We first prove part (1) by induction. Let $S(n)$ be the statement (recall that $a-b=a(x)-b(x)$): for every $n\ge 1$ $\gcd((a-b)^2,G_{n}^{*}(x))=1$. Recall that in a GFP of Lucas type holds that $\gcd(p_0(x), p_1(x))=\gcd(p_0(x), d(x))=1$ and that $2p_1(x)=p_0(x)d_(x)$. From this and using Proposition \[divisity:Hogat:property2\] part (1) with $m=n=0$, it is easy to see that $\gcd((a-b)^2,G_{1}^{*}(x))=1$. We now prove that $S(2)$ is also true. It is easy to see that $$\begin{aligned} \gcd((a-b)^2,G_{2}^{*}(x))&=&\gcd(a^2(x)+b^2(x)-2ab,G_2^{*}(x))\\ &=&\gcd(G_{2}^{*}(x)+2g(x),G_{2}^{*}(x))\\ &=&\gcd(2g(x),G_{2}^{*}(x)).\end{aligned}$$ From Lemma (\[gcdlemmas\]) part (3) we know that $\gcd(g(x),G_{2}^{*}(x))=1$. This implies that either $$\gcd((a-b)^2,G_{2}^{*}(x))=1 \; \text{ or } \; \gcd((a-b)^2,G_{2}^{*}(x))=2.$$ If $\gcd((a-b)^2,G_{2}^{*}(x))=2$, then $2\mid \left(d^2(x)+4g(x)\right)$ and $2\mid G_{2}^{*}(x)$. So, $2\mid d^2(x)$ and $2\mid G_{2}^{*}(x)$. From Lemma (\[gcdlemmas\]) part (2) we know that $\gcd(d(x),G_{2}^{*}(x))=1$. This implies that $2 \mid 1$. Therefore, $\gcd((a-b)^2,G_{2}^{*}(x))=1$. This proves $S(2)$. Suppose that that $S(n)$ is true for some $k$. Thus, suppose that $\gcd((a-b)^2,G_{k}^{*}(x))=1$. We prove that $S(k+1)$ is true. Suppose that $\gcd((a-b)^2,G_{k+1}^{*}(x))=r(x)$. Therefore, $r(x)\mid(a-b)^2$ and $r(x)\mid G_{k+1}^{*}(x)$. So, $r(x) \mid [(a-b)^2G^{'} _{2k+1}(x)-\alpha^2(G^*_{k+1}(x))^2]$. From Proposition \[divisity:Hogat:property2\] part (1) we know that if $m=n=k$, then $$(a-b)^2G^{'} _{k+k+1}(x)=\alpha^2 G_{k+1}^{*}(x)G_{k+1}^{*}(x)+\alpha^2g(x)G_{k}^{*}(x)G_{k}^{*}(x).$$ Thus, $ (a-b)^2G^{'} _{2k+1}(x)-\alpha^2(G^{*}_{k+1}(x))^2=\alpha^2g(x)(G^{*}_k (x))^2. $ This implies that $r(x)$ divides $ \alpha^2 g(x)(G^{*} _k(x))^2$. Since $|\alpha| = 1$ or $2$, from the definition of GFP and Proposition \[gcddistance1;2\] is easy to see that $\gcd(\alpha,g(x))=1$. We know that $\gcd(\alpha, G_n)=1$ for every $n$. So, $\gcd(\alpha, r(x))=1$. We recall that from Lemma (\[gcdlemmas\]) part (3), that $\gcd(g(x),G_{k+1}^{*}(x))=1$. This and $r(x)\mid G_{k+1}^{*}(x)$ imply that $\gcd(r(x),g(x))=1$. Now it is easy to see that $r(x)\mid (G^{*} _{k}(x))^2$. Since $r(x)\mid (a-b)^2$ and $\gcd((a-b)^2,G_{k}^{*}(x))=1$, we have that $r(x)=1$. This proves that $S(k+1)$ is true. That is, $\gcd((a-b)^2,G_{n}^{*}(x))=1$. We now prove that $\gcd(G^{'}_{m+n+1}(x),G_{n}^{*}(x))=\gcd(G_{m+1}^{*}(x),G_{n}^{*}(x)).$ Proposition \[divisity:Hogat:property2\] part (1), implies that $$\begin{aligned} \gcd((a-b)^2G^{'}_{m+n+1}(x),G_{n}^{*}(x))&=&\gcd(\alpha^{2}G_{m+1}^{*}(x)G_{n+1}^{*}(x)+\alpha^{2}g(x)G_{m}^{*}(x)G_{n}^{*}(x),G_{n}^{*}(x))\\ &=&\gcd(\alpha^{2} G_{m+1}^{*}(x)G_{n+1}^{*}(x),G_{n}^{*}(x)).\end{aligned}$$ From Proposition \[gcddistance1;2\] and $\gcd (\alpha, G_{n+1})=1$ we know that $\gcd(\alpha^{2} G_{n+1}^{*}(x),G_{n}^{*}(x))=1$. Therefore, by Proposition \[prop2;1\] part (2) we have $\gcd(G_{m+1}^{*}(x)G_{n+1}^{*}(x),G_{n}^{*}(x))=\gcd(G_{m+1}^{*}(x),G_{n}^{*}(x))$. This implies that $$\gcd((a-b)^2G^{'}_{m+n+1}(x),G_{n}^{*}(x))=\gcd(G_{m+1}^{*}(x),G_{n}^{*}(x)).$$ This and $\gcd((a-b)^2,G_{n}^{*}(x))=1$ imply that $$\gcd(G^{'}_{m+n+1}(x),G_{n}^{*}(x))=\gcd(G_{m+1}^{*}(x),G_{n}^{*}(x)).$$ Proof of part (2). From Lemma \[gcdlemmas\] part (3) it is easy to see that $\gcd(G_{m-n+1}^{'}(x), G_{n}^{*}(x))$ is equal to $\gcd((g(x))^n G_{m+1-n}^{'}(x), G_{n}^{*}(x))$. This and Proposition \[divisity:Hogat:property1\] part (2) (after interchanging the roles of $m$ and $n$), imply that $\gcd(G_{m-n+1}^{'}(x), G_{n}^{*}(x))$ equals $$\gcd(\alpha G_{m+1}^{'}(x)G_{n}^{*}(x)-G_{m+1+n}^{'}(x),G_{n}^{*}(x))=\gcd(G_{m+n+1}^{'}(x), G_{n}^{*}(x)).$$ The conclusion follows from part (1). Proof of part (3). From Lemma \[gcdlemmas\] part (3) it is easy to see that $$\gcd(G_{n-m+1}^{'}(x), G_{n}^{*}(x)) =\gcd((-g(x))^{m-1} G_{n-(m-1)}^{'}(x), G_{n}^{*}(x)).$$ This and Proposition \[divisity:Hogat:property1\] part (3), imply that $\gcd(G_{m-n+1}^{'}(x), G_{n}^{*}(x))$ equals $$\gcd(G_{n+m-1}^{'}(x)-\alpha G_{m-1}^{'}(x)G_{n}^{*}(x),G_{n}^{*}(x))=\gcd(G_{n+(m-2)+1}^{'}(x), G_{n}^{*}(x)).$$ The conclusion follows from part (1). \[combine:gcd:Lucas:Fibobacci\] Let $\{ G_{n}^{*}(x)\}$ and $\{ G_{n}^{'}(x)\}$ be equivalent GFP. If $m$ and $n$ are positive integers and $\gcd(m,n)=d$, then $$\gcd(G^{'}_{m}(x),G_{n}^{*}(x))= \begin{cases} G_{d}^{*}(x) & \mbox{ if } E_2(m)>E_2(n); \\ \gcd(G_{ d }^{*}(x),G_{0}^{*}(x)) & \mbox{ otherwise. } \end{cases}$$ We suppose that $E_2(m)>E_2(n)$. We prove this part of the Theorem by cases. [**Case**]{} $m>n$. Since $m>n$, there is a positive integer $l$ such that $m=n+l$. Therefore, $\gcd(G^{'}_{m}(x),G_{n}^{*}(x))=\gcd(G^{'}_{l-1+n+1}(x),G_{n}^{*}(x))$. This and Proposition \[gcd:fibonaccilucas:samegcdlucaslucas\] part (1) imply that $\gcd(G^{'}_{m}(x),G_{n}^{*}(x))=\gcd(G_{l}^{*}(x),G_{n}^{*}(x))$. Since $E_2(m)>E_2(n)$ and $m=n+l$, we have that $E_2(l)=E_2(n)$. This and Theorem \[second:main:thm\] imply that $\gcd(G_{l}^{*}(x),G_{n}^{*}(x))=G_{\gcd(l,n)}^{*}(x)$. From Lemma \[Dic1\] it is easy to see that $\gcd(l,n)=\gcd(m,n)$. Thus, $\gcd(G_{l}^{*}(x),G_{n}^{*}(x))=G_{\gcd(m,n)}^{*}(x)$. So, $\gcd(G^{'}_{m}(x),G_{n}^{*}(x))=G_{\gcd(m,n)}^{*}(x)$. [**Case**]{} $m<n$. The proof of this case is similar to the proof of Case $m>n$. It is enough to replace $m$ by $n-(l+1)$ in $\gcd(G^{'}_{m}(x),G_{n}^{*}(x))$, and then use Proposition \[gcd:fibonaccilucas:samegcdlucaslucas\] part (3). We now suppose that $E_2(m)\le E_2(n)$. We prove this part of the Theorem by cases. [**Case**]{} $m>n$. So, there is a positive integer $l$ such that $m= l+n$. Therefore by Proposition \[gcd:fibonaccilucas:samegcdlucaslucas\] part (1) we have $$\gcd(G^{'}_{m}(x),G_{n}^{*}(x))=\gcd(G^{'}_{n+(l-1)+1}(x),G_{n}^{*}(x))=\gcd(G_{l}^{*}(x),G_{n}^{*}(x)).$$ Note that if $m=n+l$ and $E_2(m)\le E_2(n)$, then there are integers $k_1$, $k_2$, $q_1$ and $q_2$ with $k_1\leq k_2$ such that $m=2^{k_1} q_1$ and $n=2^{k_2} q_2$. Since $m=n+l$, we see that $E_2(l)\neq E_2(n)$. This and Theorem \[second:main:thm\] imply that $\gcd(G_{l}^{*}(x),G_{n}^{*}(x))=\gcd(G_{0}^{*}(x),G_{\gcd(n,l)}^{*}(x))$. Thus, $\gcd(G^{'}_{m}(x),G_{n}^{*}(x))=\gcd(G_{0}^{*}(x),G_{\gcd(m,n)}^{*}(x))$. [**Case**]{} $m<n$. The proof of this case is similar to the proof of Case $m>n$. It is enough to replace $m$ by $n-(l+1)+1$ in $\gcd(G^{'}_{m}(x),G_{n}^{*}(x))$, and then use Proposition \[gcd:fibonaccilucas:samegcdlucaslucas\] part (3). [**Case**]{} $m=n$. Since $n=(2n-1)-n+1$, taking $m=2n-1$ in Proposition \[gcd:fibonaccilucas:samegcdlucaslucas\] part (2) and using Theorem \[second:main:thm\] we obtain that $$\gcd(G'_{n}(x),G^{*} _{n}(x))=\gcd(G'_{(2n-1)-n+1}(x),G^{*} _{n}(x))=\gcd(G^{*} _{2n}(x),G^{*} _n(x))=\gcd(G_{0}^{*}(x),G_{n}^{*}(x)).$$ This completes the proof. Hoggatt and Bicknell-Johnson [@hoggatt Thm. 3.4] proved that Fibonacci polynomials, Chebyshev polynomials of second kind, Morgan-Voyce polynomial, and Schechter polynomial satisfy the strong divisibility property. Theorem \[gcd:property:fibonacci\] proves the necessary and sufficient condition for the polynomials in a generalized Fibonacci polynomial sequence to satisfy the strong divisibility property. Norfleet [@norfleet] also prove the some strong divisibility property for GFP of Fibonacci type. \[gcd:property:fibonacci\] Let $\{ G_{k}(x)\}$ be a GFP of either Fibonacci type or Lucas type. For any two positive integers $m$ and $n$ it holds that $\gcd(G_{m}(x),G_{n}(x))=G_{\gcd(m,n)}(x)$ if and only if $\{ G_{k}(x)\}$ is a sequence of GFP of the Fibonacci type. Let $\{ G_{n}^{'}(x)\}$ be a generalized Fibonacci polynomial sequence of the Fibonacci type, we now show that $\gcd(G_{m}^{'}(x),G_{n}^{'}(x))$ divides $G_{\gcd(m,n)}^{'}(x)$ for $m>0$, $n>0$ and vice versa. If $G_{n}^{'}$ is of Fibonacci type, by Proposition \[divisity:property:fibonacci\], it is clear that $G_{\gcd(m,n)}^{'} \mid \gcd(G_{m}^{'}(x),G_{n}^{'}(x))$. Next we show that $\gcd(G_{m}^{'}(x),G_{n}^{'}(x))$ divides $G_{\gcd(m,n)}^{'}$. Let $k=\gcd(m,n)$ and suppose without lost of generality that $k$ is neither equal $n$ nor equal $m$. The Bézout identity implies that there are two positive integers $r$ and $s$ such that $k=rm-sn$. So, $rm=k+sn$ and $G_{rm}^{'}(x) =G_{k+sn}^{'}(x)$ This, Proposition \[divisity:Hogat:property1\] part (1), and the fact that $k+sn=(k+(sn-1))+1$, imply that $$G_{rm}^{'}(x)= G_{k+1}^{'}(x)G_{s'n}^{'}(x)+g(x)G_{k}^{'}(x)G_{sn-1}^{'}(x).$$ We note that, by Proposition \[divisity:property:fibonacci\], $G_{m}^{'}(x)$ divides $G_{rm}^{'}(x)$ and $G_{n}^{'}(x)$ divides $G_{sn}^{'}(x)$. Since $\gcd(G_{m}^{'}(x),G_{n}^{'}(x)) \mid G_{m}^{'}(x)$ and $\gcd(G_{m}^{'}(x),G_{n}^{'}(x)) \mid G_{n}^{'}(x)$, and $G_{m}^{'}(x)\mid G_{rm}^{'}(x)$ and $G_{n}^{'}(x)\mid G_{s'n}^{'}(x)$, we have that $\gcd(G_{m}^{'}(x),G_{n}^{'}(x))$ divides $G_{rm}^{'}(x)$ and $G_{s'n}^{'}(x)$. This together with Lemma \[gcdlemmas\] part (3) and the fact that $\gcd(G_{m}^{'}(x),G_{n}^{'}(x))$ does not divide $G_{s'n-1}^{'}(x)$, implies that $\gcd(G_{m}^{'}(x),G_{n}^{'}(x))$ divides $G_{k}^{'}(x)$. Conversely, suppose that $\{ G_{n}(x)\}$ is a generalized Fibonacci polynomial sequence such that the strong divisibility property holds, or $\gcd(G_{m}(x),G_{n}(x))=G_{\gcd(m,n)}(x)$ for any two positive integers $m$ and $n$, we now show that both $G_{m}(x)$ and $G_{n}(x)$ are GFP of the Fibonacci type. We prove this using the method of contradiction. If $G_{m}(x)$ and $G_{n}(x)$ are in $\{ G_{n}(x)\}$ such that they are both GFP of the Lucas type, then by Theorem \[second:main:thm\] we obtain a contradiction. This completes the proof. The gcd properties of Familiar GFP and questions ================================================ In this section we formulate a general question and present three tables which are corollaries of the main results in Section \[gcd:characterization\]. These tables give us the strong divisibility property of the familiar polynomials which satisfy the Binet formulas (\[bineformulados\]) and (\[bineformulauno\]). Table \[corollary\_Fibonacci\] gives the gcd’s for Fibonacci polynomials, Pell polynomials, Fermat polynomials, Jacobsthal polynomials, Chebyshev’s second kind polynomials and one type of Morgan-Voyce $B_n$ polynomials. Table \[corollary\_lucas\] gives the strong divisibility property of the Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Jacobsthal-Lucas polynomials, Chebyshev’s first kind polynomials and Morgan-Voyce $C_n$ polynomials, while Table \[corollary\_lucas\_2\] gives the $\gcd$ of a polynomial of the Lucas type and its equivalent. We should note here that in the case of Table \[corollary\_lucas\], the strong divisibility property is partially satisfied since it only holds when the largest powers of 2 that divides $m$ and the largest powers of 2 that divides $n$ are both equal. (That is, $E_{2}(m)= E_{2}(n)$.) Similarly the strong divisibility property only holds in Table \[corollary\_lucas\_2\] when $E_{2}(n)<E_{2}(m)$. \[!ht\] \[!ht\] \[!ht\] Questions --------- 1. Let $\{G_{n}^{*}(x)\}$ and $\{ S_{n}(x)\}$ be generalized Fibonacci polynomial sequences of Lucas type and Fibonacci type, respectively. If $S_{n}(x) $ is not the equivalent of $G_{n}^{*}(x)$, what is the $\gcd(G_{k}^{*}(x),S_{m}(x))$? We believe that the answer is: $1$ or $x$. 2. Let $\{G_{n}(x)\}$ and $\{S_{n}(x)\}$ be two different Fibonacci polynomial sequences of the same type, then do they satisfy the strong divisibility property? 3. ([**Conjecture.**]{}) The GFP $T_{n}$ and $S_{m}$ satisfy the strong divisibility property if and only if $T_{n}$ and $S_{m}$ are both of Fibonacci type and they belong to the same generalized Fibonacci polynomial sequence. Theorems \[combine:gcd:Lucas:Fibobacci\] and \[gcd:property:fibonacci\] are evidence that the conjecture is true. 4. Let $\mathcal{R}$ be a set of recursive functions. If $\mathcal{F} : \mathbb{N} \to \mathcal{R}$, $\mathcal{G} : \mathcal{R} \times \mathcal{R} \to \mathcal{R}$ and $g: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, under what conditions $\mathcal{G} \circ (\mathcal{F} \times \mathcal{F} )= \mathcal{F} \circ g$ for all $\mathcal{F}\in \mathcal{R}$ and a fix $g$? Acknowledgement =============== The first and last authors were partially supported by The Citadel Foundation. [99]{} R. André-Jeannin, Differential properties of a general class of polynomials, *Fibonacci Quart.* **33** (1995), 453–458. R. André-Jeannin, A generalization of Morgan-Voyce polynomials, *Fibonacci Quart.* **32** (1994), 228–231. R. Flórez, R. Higuita and A. Mukherjee, Alternating sums in the Hosoya polynomial triangle, *J. Integer Seq.*, **17** (2014), Article 14.9.5. R. Flórez, R. Higuita and L. Junes, 9-modularity and gcd properties of generalized Fibonacci numbers, *Integers*, **14** (2014), Paper No. A55, 14pp. R. Flórez and L. Junes, gcd properties in Hosoya’s triangle, *Fibonacci Quart.* **50** (2012), 163–174. M. Hall, Divisibility Sequences of Third Order, *Amer. J. Math.* **58** (1936), 577-584. R. T. Hansen, Generating Identities for Fibonacci and Lucas Triples , *Fibonacci Quart.* **10** (1972), 571–578. P. Hilton, P. J. Pedersen, and L. Vrancken, On certain arithmetic properties of Fibonacci and Lucas numbers, *Fibonacci Quart.* **3** (1995), 211–217. V. E. Hoggatt,  Jr., and C. T. Long, Divisibility properties of generalized Fibonacci polynomials, *Fibonacci Quart.* **12** (1974), 113–120. V. E. Hoggatt,  Jr., and M. Bicknell-Johnson, Divisibility properties of polynomials in Pascal’s triangle, *Fibonacci Quart.* **16** (1978), 501–513. A. F. Horadam, A synthesis of certain polynomial sequences, *Applications of Fibonacci numbers*, Vol. 6 (Pullman, WA, 1994), 215–229, Kluwer Acad. Publ., Dordrecht, 1996. C. Kimberling, , Greatest common divisors of sums and differences of Fibonacci, Lucas, and Chebyshev polynomials, *Fibonacci Quart.* **17** (1979), 18–22. C. Kimberling, Strong divisibility sequences and some conjectures, *Fibonacci Quart.* **17** (1979), 13–17. C. Kimberling, Strong divisibility sequences with nonzero initial term, *Fibonacci Quart.* **16** (1978), 541–544. E. Lucas, Théorie des fonctions numériques simplement périodiques, *Amer. J. Math.* 1 (1878), 184-240, 289-321. A. F. Horadam and J. M. Mahon, Pell and Pell-Lucas polynomials, *Fibonacci Quart.* **23** (1985), 7–20. A. F. Horadam, Chebyshev and Fermat polynomials for diagonal functions, *Fibonacci Quart.* **17** (1979), 328–333. T. Koshy, *Fibonacci and Lucas Numbers with Applications*, John Wiley, New York, 2001. J. Lahr, Fibonacci and Lucas numbers and the Morgan-Voyce polynomials in ladder networks and in electric line theory, *Fibonacci numbers and their applications*, (Patras, 1984), 141–161. W. McDaniel, The g.c.d. in Lucas sequences and Lehmer number sequences, *Fibonacci Quart.* **29** (1991), 24–29. A. M. Morgan-Voyce, Ladder network analysis using Fibonacci numbers, *IRE Trans. Circuit Th.* **CT-6** (1959), 321–322. M. Norfleet, Characterization of second-order strong divisibility sequences of polynomials, *Fibonacci Quart.* **43** (2005), 166–169. M. O. Rayes, V. Trevisan, and P. S. Wang, Factorization properties of Chebyshev polynomials, *Comput. Math. Appl.* **50** (2005), 1231–1240. M. N. S. Swamy, Properties of the polynomials defined by Morgan-Voyce, *Fibonacci Quart.* **4** (1966a), 73–81. M. Ward, Note on divisibility sequences, *Bull. Amer. Math. Soc.* **42** (1936), 843-845. W. A. Webb and E. A. Parberry, Divisibility properties of Fibonacci polynomials, *Fibonacci Quart.* **7** (1969), 457–463. ------------------------------------------------------------------------ MSC 2010: Primary 11B39; Secondary 11B83. *Keywords:* Greatest common divisor, strong divisibility property, generalized Fibonacci polynomial, Fibonacci polynomial, Lucas polynomial.

--- author: - 'Esteban F. E. Morales' - Friedrich Wyrowski - Frederic Schuller - 'Karl M. Menten' bibliography: - 'references.bib' date: 'Received 3 April 2013 / Accepted 16 September 2013' title: 'Stellar clusters in the inner Galaxy and their correlation with cold dust emission[^1]' --- [Stars are born within dense clumps of giant molecular clouds, constituting young stellar agglomerates known as embedded clusters, which only evolve into bound open clusters under special conditions.]{} [We statistically study all embedded clusters (ECs) and open clusters (OCs) known so far in the inner Galaxy, investigating particularly their interaction with the surrounding molecular environment and the differences in their evolution.]{} [We first compiled a merged list of 3904 clusters from optical and infrared clusters catalogs in the literature, including 75 new (mostly embedded) clusters discovered by us in the GLIMPSE survey. From this list, 695 clusters are within the Galactic range $|\ell| \le 60\degr$ and $|b| \le 1.5\degr$ covered by the ATLASGAL survey, which was used to search for correlations with submm dust continuum emission tracing dense molecular gas. We defined an evolutionary sequence of five morphological types: deeply embedded cluster (EC1), partially embedded cluster (EC2), emerging open cluster (OC0), OC still associated with a submm clump in the vicinity (OC1), and OC without correlation with ATLASGAL emission (OC2). Together with this process, we performed a thorough literature survey of these 695 clusters, compiling a considerable number of physical and observational properties in a catalog that is publicly available.]{} [We found that an OC defined observationally as OC0, OC1, or OC2 and confirmed as a real cluster is equivalent to the physical concept of OC (a bound exposed cluster) for ages in excess of $\sim 16$ Myr. Some observed OCs younger than this limit can actually be unbound associations. We found that our OC and EC samples are roughly complete up to $\sim 1$ kpc and $\sim 1.8$ kpc from the Sun, respectively, beyond which the completeness decays exponentially. Using available age estimates for a few ECs, we derived an upper limit of 3 Myr for the duration of the embedded phase. Combined with the OC age distribution within 3 kpc of the Sun, which shows an excess of young exposed clusters compared to a theoretical fit that considers classical disruption mechanisms, we computed an embedded and young cluster dissolution fraction of $88 \pm 8\%$. This high fraction is thought to be produced by several factors and not only by the classical paradigm of fast gas expulsion.]{} Introduction {#sec:introduction} ============ Stars form by gravitational collapse of high-density fluctuations in the interstellar molecular gas, which are generated by supersonic turbulent motions [e.g., @Klessen2011-lectures]. Following the nomenclature of @Williams2000, star formation takes place in dense ($n \gtrsim 10^4$ ) *clumps*, which are in turn fragmented into denser ($n \gtrsim 10^5$ ) *cores*, in which individual stars or small multiple systems are born. Given this nature of the star formation process, stars are born correlated in space and time, with typical scales of 1 pc and 1 Myr, respectively [see @Kroupa2011], constituting young stellar agglomerates known as *embedded clusters* (ECs). @Bressert2010 studied the spatial distribution of star formation within 500 pc from the Sun and found that, in fact, most of the young stellar objects (YSOs) in their sample are found in regions with number densities greater than $\sim 2\,{\rm pc}^{-3}$, which is more than an order of magnitude higher than the density of field stars in the Galactic disk, $0.13\,{\rm pc}^{-3}$ [@Chabrier2001]. Many of the ECs defined in this way, however, are not gravitationally bound and will not become classical open clusters (OCs), i.e., bound stellar agglomerates that are free of gas and have lifetimes on the order of 100 Myr. It is very important to make the distinction from the start because there is often some confusion about this in the literature. In the definition used throughout this work (see Section \[sec:cluster-definition\]), ECs are *not* necessarily the direct progenitors of bound OCs, but just the natural outcome of the star formation process, which is “clustered” with respect to the field stars. The dynamical evolution of an EC is quite complex and can progress in several possible ways, depending on both the characteristics of the recently born stellar population and the physical properties of the parent molecular cloud. A gravitationally unbound molecular cloud or an unbound region of a molecular complex might still be able to form stars in subregions that are locally bound [e.g., @Bonnell2011], but the resulting EC born there is globally unbound and quickly disperses into the field. On the other hand, within a molecular complex, especially in bound regions, many ECs might merge and form a few large entities [@Maschberger2010]. If a certain EC (once born or after merging) manages to remain gravitationally bound in the gas potential, at some point the effect of stellar feedback starts to influence the parent molecular material in the vicinity. These feedback mechanisms include protostellar outflows, evaporation driven by non-ionizing ultraviolet radiation, photoionization and subsequent region expansion, stellar winds, radiation pressure and, eventually, supernovae. Again, the relative importance of a certain dissipation process is determined by the physical conditions of the system and the environment [@Fall2010]. The energy and momentum introduced by stellar feedback eventually disrupts the clump and sweeps up the residual gas out of the cluster volume. The stars of this emerging cluster are now tied to each other uniquely by the stellar gravitational potential, which might not be sufficient to keep the stars together, so that the cluster dissolves. This is the classical “infant mortality” paradigm established by @LadaLada2003. However, @Kruijssen2011 argue that this effect is only important in low-density regions, and by analyzing the dynamical state of the ECs arising from star formation hydrodynamic simulations, they find that in dense regions the formed clusters are actually bound and even close to virial equilibrium. They propose that those clusters are instead destroyed via tidal shocks from the surrounding dense gas. An alternative disruption mechanism for small-$N$ systems or larger clusters with a hierarchical substructure has recently been studied by @Moeckel2012, who find through $N$-body simulations that those clusters undergo a quick expansion owing fast internal relaxation. Bound exposed clusters are therefore the few survivors of all these processes and represent the remnants of originally more massive ECs. The observational study of ECs is fundamental to account for most of the newly formed stellar population in the Galaxy and to investigate the interaction with its parent molecular material through stellar feedback. In the past decade, thanks to the development of all-sky infrared imaging surveys, such as 2MASS and GLIMPSE (see Section \[sec:galactic-surveys\]), many new ECs have been discovered in the Galaxy [e.g., @Dutra2003-2mass; @Bica2003-2mass; @Mercer2005; @Borissova2011], significantly increasing the number of known systems. However, so far there have only been a few systematic studies of the whole current sample of ECs and OCs in a significant fraction of the Galactic plane [e.g., @BonattoBica2011; @Kharchenko2012], and none of these studies has distinguished clearly the embedded population from the OC sample (see below). The main goal of this paper is to fill this gap. Here, we statistically study all OCs and ECs known so far in the inner Galaxy from different cluster catalogs in the literature, after compiling a considerable number of physical and observational properties of these objects, particularly their degree of correlation with the surrounding molecular environment, if present. We take advantage of the recently completed ATLASGAL submm continuum survey (see Section \[sec:galactic-surveys\]), which provides a spatially unbiased view of the distribution of the dense molecular material in the Milky Way. While the distinction of ECs from OCs in these catalogs has primarily been made via correlations with known regions or nebulae seen in the infrared, the ATLASGAL survey allows us to objectively tell[^2] whether or not these objects are associated with dense molecular gas, as well as to possibly detect the presence of stellar feedback via simple morphological criteria. This paper is organized as follows. In the remainder of this introduction, we shortly present the main observational data and the nomenclature used throughout this work (Sections \[sec:galactic-surveys\] and \[sec:cluster-definition\], respectively). In Section \[sec:catalogs-summary\], we describe the literature compilation of a merged list of Galactic OCs and ECs, including a new search for ECs we conducted on the GLIMPSE survey; more details about the literature cluster lists used here are given in Appendix \[sec:catalogs-long\]. Section \[sec:huge-table\] summarizes the construction of an extensive catalog for the cluster sample within the Galactic range covered by ATLASGAL, with many pieces of information, including: characteristics of the submm and mid-infrared emission, correlation with known objects, distances (kinematic and/or stellar), ages, and membership in big molecular complexes. A more detailed description of all the assumptions and procedures made when organizing this information in the catalog is given in Appendix \[sec:huge-table-details\]. In Section \[sec:analysis\], we report the results of a statistical analysis performed on this catalog, in which we delineate a morphological evolutionary sequence with decreasing correlation with ATLASGAL emission, classify the sample in ECs and OCs, and separately study their distance distribution, completeness, and age distribution. Finally, Section \[sec:conclusions\] summarizes the main conclusions of this paper. Observations: Galactic surveys {#sec:galactic-surveys} ------------------------------ The APEX Telescope Large Area Survey of the Galaxy [ATLASGAL, @Schuller2009] is the first unbiased submm continuum survey of the whole inner Galactic disk, covering a total of 360 square degrees of the sky with Galactic coordinates in the range $|\ell| \le 60\degr$ and $|b| \le 1.5\degr$. The observations were carried out at 870  using the Large APEX Bolometer Camera [LABOCA; @Siringo2009] of the APEX telescope [@Guensten2006], located on Llano de Chajnantor, Chile, at 5100 m of altitude. With an antenna diameter of 12 m, the observations reach an angular resolution[^3] of $19.2''$ at this wavelength. The submm continuum emission mainly represents thermal radiation from cool dust, which is generally optically thin and, therefore, an excellent tracer of the amount of interstellar material on the line of sight. The ATLASGAL survey reaches an average rms noise level of $\sim 50$ mJy/beam, which translates in a $3\sigma$ detection limit of $\sim 4~M_{\sun}$ of total molecular mass (for a nominal distance of 2 kpc and a dust temperature of $T_\rd = 20$ K). In the infrared, we primarily use two large scale surveys that cover the inner Galactic plane: The Two Micron All Sky Survey [2MASS, @Skrutskie2006] which provides near-infrared (NIR) images of the whole sky, in the $J$ (1.25 ), $H$ (1.65 ), and $K_s$ (2.16 ) filters, with an angular resolution of $\sim 2.5''$; and the Galactic Legacy Infrared Mid-Plane Survey Extraordinaire [GLIMPSE, @Benjamin2003; @Churchwell2009], which is a set of various mid-infrared (MIR) surveys of the Galactic plane carried out with the InfraRed Array Camera [IRAC, @Fazio2004], on board of the *Spitzer Space Telescope* [@Werner2004]. Here we use the GLIMPSE I and II surveys which cover the $(\ell,b)$ ranges: $5\degr < |\ell| \le 65\degr$ and $|b| \le 1\degr$; $2\degr < |\ell| \le 5\degr$ and $|b| \le 1.5\degr$; $|\ell| \le 2\degr$ and $|b| \le 2\degr$, comprising a total of 274 square degrees. The IRAC camera provides images at four filters centered at wavelengths 3.6, 4.5, 5.6, and 8.0 , with an angular resolution of $\sim 2''$. The GLIMPSE surveys have revealed very peculiar structures in star-forming regions [a summary is provided in Section 2 of @Churchwell2009]. The 8.0  filter is particularly useful to detect the presence of bright fluorescent emission from polycyclic aromatic hydrocarbons (PAHs), which are excited by the stellar far ultraviolet (UV) field, but are destroyed by the harder UV radiation present within ionized gas regions. Thus, PAH emission is often observed from *IR bubbles*, which appear projected as ring-like structures and in many cases are tracing molecular material swept up by the expansion of regions created by the ionizing radiation from massive stars [@Deharveng2010]. On the other hand, *infrared dark clouds* (IRDCs), already found in previous MIR surveys, are seen as extinction features against the bright and diffuse mid-infrared Galactic background. They represent the densest and coldest condensations within giant molecular clouds and are the most likely sites of future star formation. For a few regions within the ATLASGAL Galactic range not covered by the GLIMPSE survey, we use data from the Wide-field Infrared Survey Explorer [WISE, @Wright2010], which mapped the entire sky in four infrared bands centered at 3.4, 4.6, 12, and 22 , with an angular resolution of $\sim 6''$ in the first three bands. Despite the lower sensitivity and coarser resolution as compared with GLIMPSE, bright PAH emission and prominent IRDCs can still be identified in the WISE images, specially at 12  (see Section \[sec:MIR-morphology\]). “Stellar cluster” definitions {#sec:cluster-definition} ----------------------------- In this paper, we define: - an *embedded cluster* (EC) as any stellar group recently born and still containing an important fraction of residual gas within and surrounding its volume, keeping in mind that it may never become a bound open cluster on its own. Since star formation takes place in molecular clouds, this definition is equivalent to the concept of a *correlated star formation event* introduced by @Kroupa2011; we keep the term “cluster” in order to match older designations in the literature. - an *open cluster* (OC) as any agglomerate of spatially correlated stars, and relatively free of the remaining gas. We use this observational definition of OC (see also Section \[sec:classification-oc-ec\]) in order to account for those objects that observationally appear like classical OCs, but whose dynamical state is unknown, in some cases they can actually be gravitationally unbound. - a *physical OC* as a gravitationally bound OC (i.e., a classical OC). - an *association* as an unbound OC. In this work, we sometimes use the term “star clusters” generically for all the classes defined above, especially when concerning observations. Bound, exposed star clusters, however, will be always be referred to explicitly as *physical OCs*. Compilation of cluster lists {#sec:catalogs-summary} ============================ Although the number of known OCs and ECs in the Galaxy has considerably increased over the last years, the current cluster sample is still far from being complete. As we discuss in Section \[sec:completeness\], the detection of a stellar cluster in the inner Galactic plane is particularly difficult, due to the high extinction and the crowded stellar background, making the cluster sample severely incomplete for distances larger than a few kpc from the Sun. If we are able to quantify this incompleteness, however, all the statistical results can properly be corrected, as we do in this work. Of course, the more complete the cluster sample, the smaller the corresponding uncertainties. We thus performed an extensive compilation of all Galactic star cluster catalogs from the literature. For completeness, this compilation was initially not restricted to the ATLASGAL Galactic range; we only did it afterwards for the comparison with ATLASGAL emission and all the subsequent analysis. The catalogs are listed in the first three columns of Table \[tab:catalogs\], where we give, respectively, an ID used throughout this work, the corresponding reference, and its category according to the wavelength at which the clusters are detected: *optical*, *NIR* or *MIR*. Optical clusters are taken mostly from the current version (3.1, from November, 2010) of the catalog by @Dias2002. NIR cluster catalogs are compilations, or lists from visual and automated searches mainly performed on the 2MASS survey. MIR clusters represent the objects detected by @Mercer2005 in the GLIMPSE data, and the new clusters discovered by us using a different search method on the same survey, which were missing in the @Mercer2005 list (see Section \[sec:newglimpse\]). In our total sample, we also included individual star clusters from the literature not listed in the previous catalogs (referred to as “Not cataloged clusters” in Table \[tab:catalogs\]). A more detailed description of the diverse catalogs and references used to construct our cluster sample is given in Appendix \[sec:catalogs-long\]. This literature compilation has been updated till August, 2011. Since we are dealing with different cluster catalogs which were constructed independently, a specific object can be present in more than one list. We therefore implemented a simple merging procedure to finally have an unique sample of stellar clusters. The first condition to identify one repetition, i.e., the same object in two different catalogs, was that the angular distance between the two given center positions were less than both listed angular diameters. We checked all merged objects under this criterion looking for the corresponding cluster names, when available, and confirmed a repetition when the names coincided. Otherwise (names not available or different), two clusters were considered the same object when the angular distance was less than both angular radii, which were also required to agree within a factor of 5. The last condition was imposed to account for the case when a compact infrared cluster shares the same field of view of a (different) optical cluster with a large angular size. This cross-identification process was not intended to be perfect, but good enough to not affect the statistical results of the whole cluster sample. Within the ATLASGAL Galactic range, a much more thorough revision was done (see Section \[sec:huge-table\]), further refining the cross-identifications, and even recognizing a few duplications and spurious clusters which were excluded from the final sample (see Section \[sec:spurious\]). ---- ------------------------- ----------- -------------- ---------------- -------------- ---------------- -------------- ---------------- Type ID Reference $N_{\rm cl}$ $N_{\rm cl}^*$ $N_{\rm cl}$ $N_{\rm cl}^*$ $N_{\rm cl}$ $N_{\rm cl}^*$ 01 @Dias2002 [ver. 3.1] *Optical* 2117 2117 216 216 29 29 02 @Kronberger2006 *Optical* 239 130 29 11 5 4 03 @DutraBica2000 *NIR* 22 8 18 8 8 2 04 @Bica2003-lit *NIR* 275 264 30 28 28 26 05 @Dutra2003-2mass *NIR* 174 167 81 80 78 77 06 @Bica2003-2mass *NIR* 163 155 69 68 63 62 07 @LadaLada2003 *NIR* 76 12 4 0 4 0 08 @Porras2003 *NIR* 73 21 0 0 0 0 09 @Mercer2005 *MIR* 90 86 83 81 55 54 10 @Kumar2006 *NIR* 54 20 0 0 0 0 11 @Froebrich2007 *NIR* 998 676 44 21 2 0 12 @Faustini2009 *NIR* 23 16 9 9 9 9 13 @Glushkova2010 *NIR* 194 32 12 4 1 0 14 @Borissova2011 *NIR* 96 96 85 85 65 65 15 Not cataloged (NIR) *NIR* 26 26 12 12 10 10 16 Not cataloged (MIR) *MIR* 3 3 3 3 0 0 17 New GLIMPSE (this work) *MIR* 111 75 103 69 94 67 Total *Optical* 2247 2247 227 227 33 33 Total *NIR* 1950 1493 356 315 265 251 Total *MIR* 197 164 182 153 144 121 ---- ------------------------- ----------- -------------- ---------------- -------------- ---------------- -------------- ---------------- In Table \[tab:catalogs\], for a given reference, we represent as $N_{\rm cl}$ the absolute (original) number of clusters in the catalog, whereas $N_{\rm cl}^*$ is the number of different entries with respect to all catalogs listed before it (i.e., after merging). The optical catalogs were put first, so that any cluster visible in the optical is considered an *optical* cluster. The infrared lists (including the *NIR* and *MIR* clusters) were positioned afterwards in chronological order, and therefore following roughly the discovery time. Absolute and after-merging numbers are presented for the total sky range of every list, the ATLASGAL Galactic range ($|\ell| \le 60\degr$ and $|b| \le 1.5\degr$), and finally for only those associated with ATLASGAL emission according to the criterion explained in Section \[sec:evolutionary-sequence\]. We warn that the number of clusters given there are after removing a few spurious objects and globular clusters (listed in Table \[tab:spurious\]). After cross-identifications, we ended up with a final sample of 3904 stellar clusters, of which 2247 are *optical*, 1493 *NIR*, and 164 *MIR* clusters. Taking into account the repetitions within each category, but not between them, the numbers of objects are 2247 for *optical*, 1950 for *NIR*, and 197 for *MIR*. Note that the low number of *MIR* clusters is due to the confined Galactic range of the GLIMPSE survey; actually, when only considering the ATLASGAL range, which is similar to the GLIMPSE range, the numbers of objects are of the same order for the different categories: 227 *optical*, 315 *NIR*, and 153 *MIR* clusters, after merging. As argued in Section \[sec:spurious\], for ECs (as defined in this work) we expect a minimal contamination by spurious detections, whereas for OCs that have not been confirmed by follow-up studies, we estimate a spurious contamination rate of $\sim 50\%$, following @Froebrich2007. New search for ECs in GLIMPSE {#sec:newglimpse} ----------------------------- The GLIMPSE on-line viewer[^4] from the Space Science Institute represents a very useful tool to quickly examine color images constructed from data collected in the four 3.6, 4.5, 5.8 and 8.0  IRAC filters, of the whole survey. By inspecting some specific regions with this viewer, we noticed that some heavily ECs are still missing in the @Mercer2005 list. An EC consists mostly of YSOs, which are intrinsically redder than field stars due to thermal emission from circumstellar dust, so that they are distinguished from background/foreground stars mainly by their red colors. Such a cluster would therefore produce a clearer spatial overdensity of stars in a point source catalog previously filtered by a red-color criterion, and would be more likely missed in a search of overdensities considering the totality of point sources, due the high number of field stars. We believe that this is the principal reason which would explain the incompleteness of the @Mercer2005 catalog. We then implemented a very simple automated algorithm using the GLIMPSE point source catalog to find the locations of EC candidates. First, we selected all point sources satisfying a red-color criterion: $[4.5] - [8.0] \ge 1$, following @Robitaille2008, who applied this condition to create their catalog of GLIMPSE intrinsically red sources. As already explained in that work, the use of these specific IRAC bands is supported by the fact that the interstellar extinction law is approximately flat between 4.5 and 8.0 , and therefore the contamination by extinguished field stars in this selection is reduced compared to other red-color criteria. By applying this condition to the entire GLIMPSE catalog, 268513 sources were selected. We did not impose the additional brightness and quality restrictions used by @Robitaille2008 because we favor the number of sources (and therefore higher sensitivity to possible YSO overdensities) rather than strict completeness and photometric reliability, which are not needed to only detect the locations of potential ECs. With the 268513 selected sources, a stellar surface density map was constructed by counting the number of sources within boxes of 0.01$\degr$ ($=36\arcsec$), in steps of 0.002$\degr$ ($=7.2\arcsec$). This significant oversampling was adopted in order to detect density enhancements that would have fallen into two or more boxes if we had used not overlapping bins. The bin size correspond to the typical angular dimension of some ECs serendipitously found using the on-line GLIMPSE viewer. To account for larger overdensities, a second stellar density map was produced with a bin size of 0.018$\degr$ ($=64.8\arcsec$), using the same step size of 0.002$\degr$. The red-source density maps were checked in a test field, and we found that thresholds of 5 sources for the small bin, and 7 sources for the large bin, are needed to detect the positions of all clusters which can be identified by-eye using the GLIMPSE on-line viewer within that area, although at the same time these low thresholds yield the detection of many spurious red-source overdensities that do not contain clusters. We decided to keep these thresholds in order not to miss any real cluster that might have a low number of members listed in the point source catalog, and perform a visual inspection of the images after the automated search to filter all spurious detections. It was also noticed that using the GLIMPSE point source archive instead of the catalog is roughly equivalent to utilizing the catalog with a lower threshold, so as long as we choose a correct threshold, the use of the more reliable GLIMPSE catalog (with respect to the archive) is justified. Within the whole GLIMPSE area, we detected 702 independent positions of overdensities (bins containing not-intersecting subsets of red sources), corresponding to 172 bins of 36$\arcsec$ with densities $\geq 5$ sources/bin, 195 bins of 64.8$\arcsec$ with densities $\geq 7$ sources/bin, and 335 locations satisfying the thresholds for both bin sizes. It should be noted that since the red-color criterion produced density maps with low crowding and therefore the local background density is always close to zero, a more sophisticated algorithm is not needed. In fact, the red-source density maps have a mean and a standard deviation of 0.039 and 0.21 sources/bin for the small bin, and 0.13 and 0.43 sources/bin for the large bin, which means that the used thresholds are above the $15\sigma$ level. Again, we emphasize that the automated search was only used to find possible locations of ECs; we did not intend to catch the complete YSO population for a given cluster in this process. [lrrccrrcl]{} \ G3CC & $\ell$ & $b$ & $\alpha$ & $\delta$ & Diam. & $N_{\rm circ}$ & Det. & Flags\ & ($\degr$)& ($\degr$)& (J2000) & (J2000) & ($\arcsec$) & & &\ (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9)\ \ G3CC & $l$ & $b$ & $\alpha$ & $\delta$ & Diam. & $N_{\rm circ}$ & Det. & Flags\ & ($\degr$)& ($\degr$)& (J2000) & (J2000) & ($\arcsec$) & & &\ (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9)\ 1 & 295.151 & $-$0.587 & 11:43:24.9 & $-$62:25:36 & 98 & 16 & A & C8,E8,S\ 2 & 299.014 & 0.128 & 12:17:24.9 & $-$62:29:04 & 60 & 4 & V & B,E8\ 3 & 299.051 & 0.181 & 12:17:47.9 & $-$62:26:12 & 81 & 14 & A & C8\ 4 & 299.337 & $-$0.319 & 12:19:43.1 & $-$62:58:08 & 51 & 9 & A & BR,E8\ 5 & 300.913 & 0.887 & 12:34:16.2 & $-$61:55:04 & 76 & 10 & A & C8,E8\ 6 & 301.643 & $-$0.240 & 12:40:02.6 & $-$63:05:01 & 67 & 9 & A & DC,E8,S\ 7 & 301.947 & 0.313 & 12:42:53.7 & $-$62:32:32 & 65 & 12 & A & E8\ 8 & 303.927 & $-$0.687 & 13:00:22.2 & $-$63:32:30 & 107 & 14 & A & C8,E8\ 9 & 304.002 & 0.464 & 13:00:40.3 & $-$62:23:17 & 82 & $\cdots$ & A & BR,E8,S\ 10 & 304.887 & 0.635 & 13:08:12.3 & $-$62:10:23 & 41 & 7 & A & DC,E4\ 11 & 307.083 & 0.528 & 13:26:58.8 & $-$62:03:25 & 71 & 8 & A & C8,DC,E8,S\ 12 & 309.421 & $-$0.621 & 13:48:38.1 & $-$62:46:11 & 48 & 10 & A & DC\ 13 & 309.537 & $-$0.742 & 13:49:51.6 & $-$62:51:42 & 38 & 7 & A & C8,DC,E8\ 14 & 309.968 & 0.302 & 13:51:25.6 & $-$61:44:51 & 40 & 6 & A & DC,E8\ 15 & 309.996 & 0.507 & 13:51:15.8 & $-$61:32:30 & 88 & 8 & A & E8,DC\ 16 & 313.762 & $-$0.860 & 14:24:58.6 & $-$61:44:56 & 80 & 15 & A & BR,C8,DC,E4,E8,U8\ 17 & 314.203 & 0.213 & 14:25:15.4 & $-$60:35:22 & 86 & 12 & A & C8,E8,U8\ 18 & 314.269 & 0.092 & 14:26:06.6 & $-$60:40:43 & 87 & 8 & A & C8,DC,E8,S,V2\ 19 & 317.466 & $-$0.401 & 14:51:19.3 & $-$59:50:46 & 45 & 7 & A & DC,E4,E8\ 20 & 317.884 & $-$0.253 & 14:53:45.6 & $-$59:31:34 & 74 & 15 & A & DC,E4,E8\ 21 & 318.049 & 0.088 & 14:53:42.2 & $-$59:08:49 & 88 & 20 & A & C8,DC,U8\ 22 & 318.777 & $-$0.144 & 14:59:33.5 & $-$59:00:59 & 105 & 8 & A & B,E8,V2\ 23 & 319.336 & 0.912 & 14:59:31.0 & $-$57:49:18 & 65 & 12 & A &\ 24 & 321.937 & $-$0.006 & 15:19:43.2 & $-$57:18:04 & 33 & 9 & A & C8,DC,E8\ 25 & 321.952 & 0.014 & 15:19:44.6 & $-$57:16:35 & 37 & 10 & A & E8\ 26 & 326.476 & 0.699 & 15:43:18.0 & $-$54:07:23 & 81 & 12 & A & C8,DC,E4,U8\ 27 & 326.796 & 0.385 & 15:46:20.3 & $-$54:10:35 & 54 & 10 & A & DC,E4\ 28 & 328.165 & 0.587 & 15:52:42.6 & $-$53:09:48 & 31 & 6 & A & E4,U8\ 29 & 328.252 & $-$0.531 & 15:57:58.9 & $-$53:58:02 & 58 & 9 & A & C8,DC,E4,E8\ 30 & 328.809 & 0.635 & 15:55:48.4 & $-$52:43:00 & 82 & 9 & V & C8,DC,E4\ 31 & 329.184 & $-$0.313 & 16:01:47.0 & $-$53:11:40 & 73 & 8 & A & DC,E4,U8\ 32 & 330.031 & 1.043 & 16:00:09.4 & $-$51:36:52 & 56 & 6 & A & DC,E8,S\ 33 & 335.061 & $-$0.428 & 16:29:23.5 & $-$49:12:25 & 63 & 6 & A & C8,DC,E4\ 34 & 337.153 & $-$0.393 & 16:37:48.5 & $-$47:38:53 & 49 & 4 & A & DC,U8,V2\ 35 & 338.396 & $-$0.406 & 16:42:43.2 & $-$46:43:36 & 65 & 8 & A & C8,DC,E4\ 36 & 338.922 & 0.390 & 16:41:15.7 & $-$45:48:23 & 97 & 11 & A & C8,E8,S\ 37 & 338.930 & $-$0.495 & 16:45:08.6 & $-$46:22:50 & 80 & 11 & A & C8,DC,E8,U8\ 38 & 339.584 & $-$0.127 & 16:45:59.1 & $-$45:38:44 & 53 & 9 & A & DC,E4,E8\ 39 & 344.221 & $-$0.569 & 17:04:06.6 & $-$42:18:57 & 51 & 11 & A & BR,E4,E8\ 40 & 344.996 & $-$0.224 & 17:05:09.7 & $-$41:29:26 & 75 & 15 & A & DC,E4,U8\ 41 & 347.883 & $-$0.291 & 17:14:27.3 & $-$39:12:35 & 62 & 6 & V & C8,E8\ 42 & 348.180 & 0.483 & 17:12:08.1 & $-$38:30:54 & 38 & 7 & A & E8\ 43 & 348.584 & $-$0.920 & 17:19:11.6 & $-$39:00:08 & 52 & 10 & A & C8,E4\ 44 & 350.105 & 0.085 & 17:19:26.7 & $-$37:10:48 & 167 & 25 & A & C8,E8,V2\ 45 & 350.930 & 0.753 & 17:19:04.7 & $-$36:07:16 & 90 & 14 & A & C8,DC,E8,S\ 46 & 351.776 & $-$0.538 & 17:26:43.1 & $-$36:09:18 & 93 & 14 & A & C8,DC,E4,E8\ 47 & 352.489 & 0.797 & 17:23:15.6 & $-$34:48:53 & 84 & 7 & A & C8,E8\ 48 & 358.386 & $-$0.482 & 17:43:37.5 & $-$30:33:51 & 57 & 5 & A & C8,DC,E4,E8,V2\ 49 & 0.675 & $-$0.046 & 17:47:23.7 & $-$28:22:59 & 140 & 23 & A & C8,E8,S\ 50 & 4.001 & 0.335 & 17:53:34.5 & $-$25:19:57 & 56 & 12 & A & BR,C8,E8\ 51 & 5.636 & 0.239 & 17:57:33.9 & $-$23:58:05 & 65 & 7 & A & C8,DC,E8\ 52 & 6.797 & $-$0.256 & 18:01:57.6 & $-$23:12:26 & 50 & 11 & A & C8,DC,E4,U8\ 53 & 8.492 & $-$0.633 & 18:06:59.3 & $-$21:54:55 & 126 & 28 & A & DC,S\ 54 & 9.221 & 0.166 & 18:05:31.3 & $-$20:53:21 & 42 & 7 & A & DC,E8\ 55 & 14.113 & $-$0.571 & 18:18:12.4 & $-$16:57:18 & 57 & 9 & A & DC,E8\ 56 & 14.341 & $-$0.642 & 18:18:55.2 & $-$16:47:15 & 124 & 15 & A & C8,DC,E4,E8\ 57 & 17.168 & 0.815 & 18:19:08.4 & $-$13:36:29 & 61 & 12 & A & DC\ 58 & 25.297 & 0.309 & 18:36:20.5 & $-$06:38:57 & 39 & 8 & A & E8\ 59 & 26.507 & 0.284 & 18:38:40.0 & $-$05:35:06 & 49 & 7 & A & C8,DC\ 60 & 31.158 & 0.047 & 18:48:02.1 & $-$01:33:26 & 50 & 8 & A & E8\ 61 & 34.403 & 0.229 & 18:53:18.4 & 01:24:47 & 91 & 8 & A & DC,E4\ 62 & 39.497 & $-$0.993 & 19:07:00.0 & 05:23:05 & 53 & 7 & A & C8,V2\ 63 & 43.040 & $-$0.451 & 19:11:38.7 & 08:46:40 & 52 & 6 & A & C8,E4,E8\ 64 & 43.893 & $-$0.785 & 19:14:26.8 & 09:22:44 & 63 & 7 & A & C8,E8\ 65 & 47.874 & 0.309 & 19:18:04.1 & 13:24:41 & 68 & 11 & A & C8,E8\ 66 & 49.912 & 0.369 & 19:21:47.7 & 15:14:20 & 55 & 11 & V & BR,C8,E8\ 67 & 50.053 & 0.064 & 19:23:11.3 & 15:13:10 & 107 & 14 & A & DC,S\ 68 & 52.570 & $-$0.955 & 19:31:54.7 & 16:56:44 & 44 & 9 & A & E4,E8\ 69 & 53.147 & 0.071 & 19:29:18.0 & 17:56:41 & 119 & 13 & A & C8,DC,S\ 70 & 53.237 & 0.056 & 19:29:32.3 & 18:00:57 & 76 & 19 & A & DC,S\ 71 & 56.961 & $-$0.234 & 19:38:16.7 & 21:08:02 & 58 & 8 & A & C8,E8\ 72 & 58.471 & 0.432 & 19:38:58.4 & 22:46:32 & 73 & 10 & A & C8,E8\ 73 & 59.783 & 0.071 & 19:43:09.9 & 23:44:14 & 120 & 11 & A & C8,E4,E8,V2\ 74 & 62.379 & 0.298 & 19:48:02.4 & 26:05:51 & 47 & 7 & A &\ 75 & 64.272 & $-$0.425 & 19:55:09.4 & 27:21:18 & 55 & 10 & A & BR\ As pointed out above, a subsequent visual selection was performed by examining the GLIMPSE images, based on a series of criteria which are explained in the following. Because the GLIMPSE on-line viewer has limited pixel resolution and is not efficient to inspect a high number of specific locations, we downloaded original GLIMPSE cutouts around these 702 positions and constructed three-color images using the 3.6 (blue), 4.5 (green) and 8.0  (red) IRAC bands. This by-eye inspection led us to finally select 88 overdensities as locations of clusters, 17 of which are identified as known clusters from our literature compilation presented before. The remaining 71 new objects are listed in Table \[tab:newglimpse\]. The adopted identification is a record number (column 1) preceded by the acronym “G3CC” (GLIMPSE 3-Color Cluster[^5]). The final coordinates and the angular diameter (column 6) were estimated by eye on the GLIMPSE three-color images fitting circles interactively with the display software *SAO Image DS9*[^6]. The visual criteria applied to select the 88 overdensities are identified for each new object as flags in the last column of Table \[tab:newglimpse\]. Figure \[fig:newglimpse-examples\] shows GLIMPSE three-color images of 6 clusters, illustrating these different criteria. An almost ubiquitous characteristic of the selected clusters (present in 82 cases) is their association with typical mid-infrared star formation signposts (see Section \[sec:galactic-surveys\]), namely: extended 8.0  emission in the immediate surroundings (flag E8, see Fig. \[fig:newglimpse-examples\](a,b,c,d,f)), likely representing radiation from UV-excited PAHs or warm dust; more localized extended 4.5  emission within the cluster area (flag E4, Fig. \[fig:newglimpse-examples\](a)), which might trace shocked gas by outflowing activity from protostars [see @Cyganowski2008 and references therein]; and presence of an infrared dark cloud in which the cluster is embedded (flag DC, Fig. \[fig:newglimpse-examples\](a,e)). We also indicate whether a cluster appears to have more stellar members than those identified by the red-color criterion, including the following situations: cluster composed of red sources and additional bright normal (not reddened) stars (flag BR, Fig. \[fig:newglimpse-examples\](d)), suggesting that the cluster is in a more evolved phase, probably emerging from the molecular cloud; cluster exclusively composed of bright normal stars (flag B, but only two cases, in conjunction with flag V2, see below); and presence of additional probable YSOs within the cluster, identified as sources uniquely detected at 8.0  (flag U8, representing extreme cases of red color), or compact 8.0  objects not listed in the point source catalog or archive (flag C8, Fig. \[fig:newglimpse-examples\](b,c,d,f)), due to the bright and variable extended emission at this wavelength, saturation for bright sources, or localized diffuse emission around a particular source which makes its apparent size larger than a point-source. The other flags indicate when the cluster shows up as a sparse, not centrally condensed set of sources (flag S, Fig. \[fig:newglimpse-examples\](b)), or if the cluster was noticed by-eye on the GLIMPSE images in a nearby location of an automatically detected overdensity, but not exactly at the same position (flag V2). {width="90.00000%"} The remaining positions were rejected as clusters, and typically correspond to background stars extinguished by dark clouds or seen behind foreground 8.0  diffuse emission, producing a red-source density enhancement by chance, sometimes together in the same line of sight with a couple of intrinsically red sources (YSOs) which however do not represent a cluster by their own. Quantitatively, we found that, in general, most of the rejected positions are overdensities with fewer elements than the ones selected as clusters. In fact, if we choose stricter thresholds of 8 sources for the small bin, and 10 sources for the large bin, instead of the originally used 5 and 7, respectively, the total set of overdensities decreases from 702 to just 87 independent positions, 37 of which represent our clusters. This would mean an improved “success” rate of $37/87 = 43\%$ for the automated method rather than the original $88/702 = 13\%$. Furthermore, if we consider the *effective* number of elements in the 88 bins originally selected as being locations of clusters, i.e., summing possible additional stellar members (flags BR,C8,U8) within the bins, we find that 61 of our clusters satisfy the new threshold. We emphasize, however, that the additional stellar members of each cluster were recognized after detailed inspection of the GLIMPSE images, so that the use of low density thresholds in the automated method was necessary to identify the initial cluster locations, despite of the consequent detection of many spurious red-source overdensities. If we had used from the beginning the stricter thresholds, we would have missed $88-37=51$ clusters. Column 7 of Table \[tab:newglimpse\] lists for every cluster the estimated number of stellar members within the assumed radius, $N_{\rm circ}$, counting the YSOs selected by the red-color criterion and the additional members identified in the images (flags BR,C8,U8). Note that this number represents a lower limit, especially in distant clusters, since lower mass members could still be undetected due to the limited angular resolution and sensitivity of the GLIMPSE data. We note that, because our simple automated method to find YSO overdensities is based on the GLIMPSE point source catalog, it is unavoidably biased towards young ECs that are not yet associated with very bright extended emission, which would hide many of the cluster members from the point source detection algorithm. Fortunately, it is quite likely that those bright nebulae were already looked for the presence of clusters by previous by-eye searches (see Section \[sec:completeness\]), so probably a few of them are really missing in our total compiled sample. We tried anyway to complete our list of new clusters by performing a systematic visual inspection with the on-line viewer over the entire area surveyed by GLIMPSE, including also fully exposed clusters that appear bright at 3.6  (equivalent to flag ‘B’). We found from this process 23 additional clusters, of which, however, only 4 are new discoveries with respect to our literature compilation. They are marked in column 8 of Table \[tab:newglimpse\] with a ‘V’, while the ones detected by the automated method are indicated with an ‘A’. We remark that, of the 17 known clusters we rediscovered from the red-source overdensities, only 3 are from the @Mercer2005 list. This practically null overlap between the two detection methods demonstrates that our search is fully complementary and particularly useful to detect ECs, confirming the ideas we presented at the beginning of this Section. Although our literature compilation of clusters is up to date until August, 2011, it is interesting to cross-check our list of new GLIMPSE clusters with the ECs recently discovered by @Majaess2013, who applied a combination of color and spectral index criteria to find YSO candidates using the WISE and 2MASS catalogs, and then looked for clusters by visually inspecting the YSOs spatial distribution. We found that only 5 new GLIMPSE clusters (they are indicated in Table \[tab:newglimpse\]) are associated with objects from the published list by @Majaess2013, in particular these 5 clusters are *contained* within the corresponding objects identified by @Majaess2013, which cover a much larger area. Due to the coarser angular resolution of WISE data with respect to GLIMPSE data, the typical stellar densities in our ECs are probably too high to make all the individual members detectable at the WISE resolution, and consequently they are hidden in the @Majaess2013 YSO selection. Properties of the cluster sample {#sec:huge-table} ================================ The next step of this work was to characterize the ATLASGAL emission, if present, at the positions of the star clusters compiled in Section \[sec:catalogs-summary\], and to compare this emission with NIR and MIR images. Hereafter, our study is naturally restricted to the ATLASGAL Galactic range ($|\ell| \le 60\degr$ and $|b| \le 1.5\degr$), and we refer to the list of the 695 stellar clusters within that range as the “whole cluster sample” (or simply as the “cluster sample”), unless noted. Together with this process, we performed a critical literature revision in order to add and update distances and ages for an important fraction of the sample, as well as to look for connections with known regions, IRDCs, and IR bubbles. We organize all this information in an unique catalog, whose construction is summarized in the following, and described in more detail in Appendix \[sec:huge-table-details\]. The catalog is only available in electronic form at the CDS, together with a companion list of all the references with the corresponding identification numbers used throughout the table. For illustration, an excerpt of the catalog is given in Appendix \[sec:catalog-excerpt\]. ATLASGAL and MIR emission {#sec:atlasgal-and-mir} ------------------------- In order to search for submm dust continuum emission tracing molecular gas likely associated with the clusters, we examined the ATLASGAL emission around the cluster positions. The column `Morph` is a text flag that gives information about the morphology of the detected ATLASGAL emission versus the IR emission. It is composed of two parts separated by a period. The first part tells about how the ATLASGAL emission is distributed throughout the immediate star cluster area, including the following cases: - `emb`: cluster fully embedded, with its center matching the submm clump peak (Fig. \[fig:EC-examples\], *top*). - `p-emb`: cluster partially embedded, whose area is not completely covered, or the submm clump peak is significantly shifted from the (proto-) stars locations (Fig. \[fig:EC-examples\], *bottom*). - `surr`: possibly associated submm emission surrounding the cluster or close to its boundaries (Fig. \[fig:OC-examples\], *top*). - `few`: one or a few ATLASGAL clumps within the cluster area (mostly for optical clusters having a large angular size), not necessarily physically related with the cluster. - `few*`: the same morphology as before, but now the clump(s) is (are) likely associated with the star cluster according to previous studies in the literature, or because the kinematic distance derived from molecular lines agrees with the stellar distance. See Section \[sec:distance-and-ages\] for a brief description of the distance determinations. - `exp`: exposed cluster, without ATLASGAL emission in immediate surroundings (Fig. \[fig:OC-examples\], *middle* and *bottom*). - `exp*`: cluster that is physically exposed, but presents submm emission within the cluster area which appears in the same line of sight, but with a kinematic distance discrepant from the stellar distance (the cluster would be categorized as `few` or `surr` if no distance information were available). We indicate in the second part of the column `Morph` (after the period) details about the mid-infrared morphology of each cluster, after visually inspecting GLIMPSE three-color images made with the 3.6 (blue), 4.5 (green) and 8.0  (red) bands. For a few clusters with no coverage in the GLIMPSE survey (7% of the cluster sample), we instead examined WISE three-color images using the 3.4, 4.6 and 12  filters. This flag includes the following cases: - `bub-cen`: presence of an IR bubble which seems to be produced by the cluster through stellar feedback, and appears in the images centered near the cluster position (Fig. \[fig:OC-examples\], *top*). - `bub-cen-trig`: the same situation than before, together with the presence of possible YSOs at the periphery of the bubble identified by their reddened appearance in the images, suggesting triggered star formation generated by the cluster (see also Fig. \[fig:OC-examples\], *top*). - `bub-edge`: in this case, the cluster itself appears at the edge of an IR bubble, suggesting that it was probably formed by triggering from an independent cluster or massive star. - `pah`: presence of bright and irregular emission at 8.0  (12  for WISE) which seems to be produced by the cluster through stellar radiative feedback (Fig. \[fig:EC-examples\], *bottom*); it is attributed to radiation from UV excited PAHs or warm dust, but is not clearly identified as an IR bubble (though it sometimes shows bubble-like borders)[^7]. ![Examples of the two morphological types defined for ECs (see Section \[sec:evolutionary-sequence\]): The cluster G3CC 38 of type EC1 (top panels), and the cluster $[$DBS2003$]$ 113 of type EC2 (bottom panels). The left panels show *Spitzer*-IRAC three-color images made with the 3.6 (blue), 4.5 (green) and 8.0  (red) bands. The right panels present 2MASS three-color images of the same field of view, constructed with the $J$ (blue), $H$ (green), and $K_s$ (red) bands. The overlaid contours on the 2MASS images represent ATLASGAL emission (870 ); the contour levels are $\{5,8.8,15,25,46,88,170\}\times \sigma$, where $\sigma$ is the local rms noise level ($\sigma = 45$ mJy/beam for G3CC 38, and $\sigma = 42$ mJy/beam for $[$DBS2003$]$ 113). The images are in Galactic coordinates and the given offsets are with respect to the cluster center, indicated in the left panels below the cluster name. The dashed circles represent the estimated angular sizes from the original cluster catalogs (see Section \[sec:basic-information\]). The 1 pc scale-bar was estimated using the corresponding distance adopted in our catalog.[]{data-label="fig:EC-examples"}](figures/morph-sequence-ec.eps){width="49.00000%"} ![Examples of the three morphological types defined for OCs (see Section \[sec:evolutionary-sequence\]): The cluster $[$DBS2003$]$ 176 of type OC0 (top panels), the cluster NGC 6823 of type OC1 (middle panels), and the cluster BH 222 of type OC2 (bottom panels). The local rms noise level of the ATLASGAL emission is, respectively, 36, 46, and 29 mJy/beam. See caption of Figure \[fig:EC-examples\] for more details of the images.[]{data-label="fig:OC-examples"}](figures/morph-sequence-oc.eps){width="49.00000%"} Correlation with known objects {#sec:known-objects} ------------------------------ Associated IR bubbles that are listed in the catalogs by @Churchwell2006 [@Churchwell2007] are identified in the table column `Bub`. On the GLIMPSE three-color images and on the 8.0  images (WISE three-color and 12  images when GLIMPSE data were not available), we also identified the presence of an infrared dark cloud in which the cluster appears to be embedded (column `IRDC`; see Fig. \[fig:EC-examples\], *top*), and we give the designation from the catalogs by @Simon2006 or @PerettoFuller2009 when the object is listed there. Finally, we searched in the literature for associated regions (column `HII_reg`), and we flagged the sources that have been classified in the literature as ultra compact (UC) regions. Distance and age {#sec:distance-and-ages} ---------------- An important part of this work was to assign distances to as many clusters as possible. In this regard, we took advantage of the fact that many of the ATLASGAL clumps at the locations or in the vicinity of the stellar clusters have measurements of molecular line LSR velocities [e.g., @Wienen2012; @Bronfman1996; @Urquhart2008]. Using these velocities and a combined rotation curve based on the models by @BrandBlitz1993 and @Levine2008, we computed kinematic distances for the clumps (column `KDist`) and, therefore, for the corresponding clusters when they were assumed to be physically associated. The kinematic distance ambiguity (KDA) was disentangled mainly by searching for previous resolutions in the literature [e.g. @CaswellHaynes1987; @Faundez2004; @AndersonBania2009; @Roman-Duval2009], for the clumps themselves or nearby regions in the phase space. A total of 424 clusters have kinematic distance estimates for the ATLASGAL clumps, 92% of which have available KDA solutions. The uncertainties (column `e_KDist`) have been determined by shifting the LSR velocities by $\pm 7$  to account for random motions, following @Reid2009, who suggest this value as the typical virial velocity dispersion of a massive star-forming region. We also compiled values for the stellar distance (column `SDist`) and age (column `Age`), estimated from studies of the stellar population of the clusters. These data were obtained from the original cluster catalogs or from new references found in SIMBAD. To prevent underestimation of the uncertainties (provided in columns `e_SDist` and `e_Age`), we imposed minimum errors depending on the computation method for the stellar distance, and on the range for the age [the latter following @BonattoBica2011]. Stellar distances are available for 222 clusters (32% of the sample), and ages for 209 clusters (30% of the sample). The most common method for stellar distance and age determination is isochrone fitting [e.g., @Loktin2001], which implies that these parameters are available mainly for exposed clusters (see Section \[sec:age-distribution\]). The final adopted distance for each cluster (column `Dist`) was chosen to be the available distance estimate with the lowest uncertainty. In some cases, we adopted independent distance estimates from the literature if they were more accurate than `SDist` and `KDist` [e.g., from maser parallax measurements; see @Reid2009 and references therein]. Clusters within a particular complex (identified in the column `Complex`) were assumed to be all located at the same distance, determined from the literature, or kinematically from an average position and velocity. In total, there are distance determinations (`Dist`) for 538 clusters, i.e., for 77% of our sample. Naturally, there is a dichotomy in the distance estimation method depending on whether or not the cluster is associated with an ATLASGAL source with available velocity, so that most exposed clusters uniquely have stellar distances, whereas the distances for ECs are mainly kinematic or from associations with complexes. However, it is still possible to compare stellar and kinematic determinations for a subsample of 38 clusters (mostly embedded) which have distances available from both methods. This comparison is shown in Figure \[fig:dist-comparison\], where plus symbols mean agreement between stellar and kinematic distances within the corresponding uncertainties, and circles are the cases in which there is a discrepancy between both techniques; the color indicates which distance estimate was finally adopted in our catalog: stellar (*red*), kinematic (*blue*), and other (*black*). The plot reveals that in our cluster sample, both methods are quite consistent with each other, with a 84% of agreement (32 out of 38 objects). We note that among the discrepant cases, there are two ECs (points $(2.16, 4.30)$ kpc and $(5.05, 1.30)$ kpc in the plot) whose method for age and (stellar) distance estimation was found to be particularly inaccurate (see Section \[sec:age-embedded\]). The rms between the stellar and kinematic distances compared in Figure \[fig:dist-comparison\] is 1.28 kpc, which represents the combined error, for this particular subsample, of both stellar and kinematic distances added in quadrature. If we compute this error from the estimated uncertainties `e_KDist` and `e_SDist` averaged over the subsample, we obtain a value of 1.59 kpc, which means that we slightly overestimated some of the uncertainties, probably because we were quite conservative in determining the minimum errors for the stellar distances (see Section \[sec:physical-parameters\]). The average uncertainties are $\langle\verb|e_KDist|\rangle = 0.67$ kpc and $\langle\verb|e_SDist|\rangle = 1.45$ kpc for the subsample of the 38 clusters used for comparison, and $\langle\verb|e_KDist|\rangle = 0.68$ kpc and $\langle\verb|e_SDist|\rangle = 0.58$ kpc for the whole sample. The high average error for the stellar distance in the subsample with respect to the whole sample is due to the fact that most of these clusters have stellar distances estimated from the spectrophotometric method, which is more inaccurate than, e.g., main sequence or isochrone fitting (see Section \[sec:physical-parameters\]). The average estimated uncertainty in the adopted distance is $\langle\verb|e_Dist|\rangle = 0.51$ kpc for the whole sample (and 0.52 kpc for the subsample). Analysis {#sec:analysis} ======== Morphological evolutionary sequence {#sec:evolutionary-sequence} ----------------------------------- Here, we use the characterization of the ATLASGAL emission found throughout each cluster’s area and/or environment (described in Section \[sec:atlasgal-and-mir\]) to define main morphological types and delineate an evolutionary sequence. First, in order to test our visual ATLASGAL morphological flags specified above (corresponding to the first part of the column `Morph`, and represented hereafter by `m`$_0$), we compared them against the more quantitative parameter $s \equiv \verb|Clump_sep|$ of our catalog, which is the projected distance of the nearest ATLASGAL emission pixel, normalized to the cluster angular radius. We found a reasonable correlation: $s = 0$ for all deeply ECs (`m`$_0$ = `emb`), $s < 0.42$ for partially ECs (`m`$_0$ = `p-emb`), $0.40 < s < 1.97$ for clusters surrounded by submm emission (`m`$_0$ = `surr`), and $s > 0.94$ for exposed clusters (`m`$_0$ = `exp`). Exposed clusters with $s < 1$ only comprise a few cases with a large angular size and very faint emission close to their borders. The remaining morphological flags are very specific and we do not expect any correlation with the quantity `Clump_sep`. ![Comparison of kinematic and stellar distances for the 38 clusters of our sample with both estimations available. Plus signs (+) indicate agreement within the errors, and circles mark the discrepant cases. Colors indicate which distance estimate was finally adopted in our catalog: stellar (*red*), kinematic (*blue*), and other (*black*). The dashed line is the identity.[]{data-label="fig:dist-comparison"}](figures/distance-comparison.eps){width="45.00000%"} Denoting by `Cf`$_0$ the first digit of the flag `Clump_flag` from our catalog (a value $>0$ means that the nearest ATLASGAL clump is likely associated with the cluster), and using the logical operators $\land$, $\lor$ and $\lnot$ (‘and’, ‘or’, and ‘not’, respectively), we define five morphological types as follows: - EC1: $\verb|m|_0 = \verb|emb|$ - EC2: $\verb|m|_0 = \verb|p-emb|$ - OC0: $\verb|m|_0 = \verb|surr| ~\lor~ \verb|m|_0 = \verb|few*| ~\lor~ (\verb|m|_0 = \verb|few| ~\land~ \verb|Cf|_0 > 0)$ - OC1: $\verb|m|_0 = \verb|exp| ~\land~ (\verb|Cf|_0 > 0 ~\lor~ \verb|KDist| \simeq \verb|SDist|)$ - OC2: ($\verb|m|_0 = \verb|exp| ~\lor~ \verb|m|_0 = \verb|exp*| ~\lor~ \verb|m|_0 = \verb|few|) ~\land~ \lnot(\rm{OC1} ~\lor~ \rm{OC2})$ The morphological type for each cluster is given in the column `Morph_type` of our catalog. Figures \[fig:EC-examples\] and \[fig:OC-examples\] present one example cluster for each morphological type, shown in GLIMPSE three-color images, and 2MASS three-color images overlaid with ATLASGAL contours. In simpler words, given that star clusters are expected to be less and less associated with molecular gas as time evolves, due to gas dispersal driven by stellar feedback, we have defined above a morphological evolutionary sequence, with decreasing correlation with ATLASGAL emission. EC1 are deeply ECs (Fig. \[fig:EC-examples\], *top*), EC2 are partially ECs (Fig. \[fig:EC-examples\], *bottom*), OC0 are emerging exposed clusters (Fig. \[fig:OC-examples\], *top*), and finally there are two kinds of totally exposed clusters: OC1 are still physically associated with molecular gas in their surrounding neighborhood (an ATLASGAL clump at a projected distance of `Clump_sep` times the cluster radius, see Fig. \[fig:OC-examples\], *middle*), whereas OC2 are all the remaining exposed clusters, which present no correlation with ATLASGAL emission (Fig. \[fig:OC-examples\], *bottom*). |ref\_Conf| ------ -------------- -------------------------- ------------------------------- ------------------------------------------ ----------------------------------------- Type $N_{\rm cl}$ $N_{\rm cl}$($D$ avail.) $N_{\rm cl}(\le D_{\rm rep})$ $N_{\rm cl}^{\rm conf}(\le D_{\rm rep})$ $N_{\rm cl}^{\rm tot}(\le D_{\rm rep})$ (1) (2) (3) (4) (5) (6) EC1 132 125 44 16 56 EC2 195 177 54 25 68 OC0 56 49 17 10 36 OC1 22 22 6 3 11 OC2 290 167 136 133 475 ------ -------------- -------------------------- ------------------------------- ------------------------------------------ ----------------------------------------- Note that, however, this classification is not perfect. For example, although the gas velocity and stellar distance data are quite extensive, they are not complete to identify all the $\verb|m|_0 = \verb|few*|$, $\verb|m|_0 = \verb|exp*|$ and $\verb|KDist| \simeq \verb|SDist|$ cases, so that some misclassification might occur in the type OC2. Similarly, the physical link between the submm emission and the ECs was based on the morphology seen in the images, and some chance alignments might still be present in a few cases (estimated to be about 5%, see Section \[sec:chance-alignments\]). Therefore, the defined morphological types should primarily be considered in a statistical way, and for individual objects they must be treated with caution. Column 2 of Table \[tab:morph-types\] lists how many objects fall in each morphological type for the whole cluster sample. Note that the low number of OC1 clusters could be partially due to the observational difficulty in identifying an exposed cluster physically associated with molecular gas in their surroundings, as remarked before. Column 3 gives the number of clusters with available distances, and the remaining columns will be described in Section \[sec:representative-sample\]. With this morphological classification, it is easy to determine (again, statistically) which clusters are associated with ATLASGAL emission: simply as those with types EC1,EC2,OC0 or OC1. These clusters are counted for every catalog in the last two columns of Table \[tab:catalogs\], as absolute and after-merging numbers of objects ($N_{\rm cl}$ and $N_{\rm cl}^*$, respectively). As expected, optical clusters are rarely associated with ATLASGAL emission (only $\sim 15\%$ of them, most of which are of type OC0 or OC1), since otherwise they would be barely visible at optical wavelengths due to dust extinction. On the other hand, the majority of the NIR and MIR clusters are physically related with submm dust radiation ($\sim 79\%$ and 74% of them, respectively). Although this is also expected because infrared emission is much less affected by dust extinction than visible light, these high percentages might partially be a consequence of the detection method of the infrared cluster catalogs, which in most cases tried to intentionally highlight the EC population. For example, the 2MASS by-eye searches by @Dutra2003-2mass and @Bica2003-2mass were done towards known radio/optical nebulae, and our new GLIMPSE cluster candidates were detected after applying a red-color criterion (see Section \[sec:newglimpse\]). In these particular catalogs, almost the totality of objects are associated with ATLASGAL emission. Chance alignments {#sec:chance-alignments} ----------------- We computed the probability of chance alignments of our stellar clusters with ATLASGAL clumps, and the different known objects looked for spatial correlation in our catalog (see Section \[sec:known-objects\]), in order to test the validity of the assumption of physical relation, when this is only based on the position of the objects on the sky. For a given sample of objects, this probability was estimated semi-analytically by assuming that the objects within $|b| \le 1\degr$ (where most sources are located for all samples used) and the longitude range originally covered, are uniformly distributed over that area, and that their angular sizes are distributed according to the observed sizes. We first calculated the probability of overlap of each cluster with one or more objects from this hypothetical sample, and then we averaged these probabilities over two different sets of clusters: morphological types EC1 and EC2 together (hereafter EC-); and types OC0, OC1 and OC2 together (hereafter OC-). For ATLASGAL clumps, we adopted a total number of 6451 objects within $330\degr \le \ell \le 21\degr$ and $|b| \le 1\degr$, from the compact source catalog by @Contreras2013, which, together with their estimated effective radii, gives an average chance alignment probability of 8.8% for clusters with types EC-, and 32% for clusters with types OC-. Considering that the submm and infrared morphologies of deeply ECs (type EC1) usually support the real physical relation with molecular gas (e.g., matching peaks of submm emission and stellar density), and that partially ECs (type EC2) are generally associated with more than one ATLASGAL clump, in practice the fraction of chance alignments of EC- clusters with ATLASGAL compact sources is likely below 5%, which is low enough to not affect the statistics of this work. Due to their larger angular sizes, clusters of types OC- are more prone to be aligned with ATLASGAL clumps by chance, and therefore our additional requirements to assume that an exposed cluster is associated with ATLASGAL emission are justified (morphological criteria or matching distances for types OC0 and OC1). For the known objects considered in our catalog, we assume that there are 4936 IR bubbles in the range $|\ell| \le 60\degr$ and $|b| \le 1\degr$ [@Simpson2012][^8], $17,364$ IRDCs within $10\degr \le |\ell| \le 60\degr$ and $|b| \le 1\degr$ [from the catalogs by @Simon2006; @PerettoFuller2009], and 944  regions in the range $343\degr \le \ell \le 60\degr$ and $|b| \le 1\degr$ [from the recently discovered and previously known regions listed in @Anderson2011]. In this case, to compute the chance alignment probability of each cluster with the objects of a given sample, we also required that the objects were larger than half the size of the cluster and that the distance between the object’s position and the cluster center were less than the sum of both radii divided by two, so that the alignment really mimics a physical relation misidentified by eye. The averaged probabilities are quite similar for clusters with types EC- and OC-, and they are all low: $\sim 2\%$ for IR bubbles, $\sim 3.5\%$ for IRDCs, and $\sim 0.3\%$ for regions. Observational classification of OCs and ECs {#sec:classification-oc-ec} ------------------------------------------- We can also use the morphological evolutionary sequence established in Section \[sec:evolutionary-sequence\] to observationally define in our sample the concepts of EC and OC. Since any stellar agglomerate that appears deeply or partially embedded in ATLASGAL emission would satisfy our physical definition of EC presented in Section \[sec:cluster-definition\], we simply use as observational definition the embedded morphological types: EC = EC1 $\lor$ EC2. We consider the remaining morphological types as OCs, but excluding those objects that have not been confirmed by follow-up studies, since we expect for them a high contamination rate by spurious candidates (see Section \[sec:spurious\]): OC = (OC0 $\lor$ OC1 $\lor$ OC2) $\land$ (`ref_Conf` not empty), where `ref_Conf` is the column in the catalog indicating the reference for cluster confirmation (see Section \[sec:physical-parameters\]). However, this observational definition of OC does not necessarily mean that the cluster is bound by its own gravity, and therefore, is not fully equivalent to the concept of *physical OC* defined in Section \[sec:cluster-definition\]. To investigate under which conditions both definitions agree, we can apply the empirical criterion proposed by @GielesPortegies2011 which distinguishes between physical OCs and associations by comparing the age of the object with its crossing time, $t_{\rm cross}$, computed as if it were in virial equilibrium. In useful physical units, Equation (1) of @GielesPortegies2011 becomes[^9] $$\label{eq:tcross-units-GP11} t_{\rm cross} = 9.33 \left(\frac{100\, M_{\sun}}{M}\right)^{1/2} \left(\frac{R_{\rm eff}}{\rm pc}\right)^{3/2} \, {\rm Myr},$$ where $M$ and $R_{\rm eff}$ are, respectively, the mass and the observed 2D projected half-light radius of the cluster. Unfortunately, mass estimates and accurate structural parameters are usually not directly available in the OC catalogs; in particular, there are no mass data in the @Dias2002 catalog, and the given sizes come from individual studies compiled there and are mostly derived from visual inspection. We therefore used the masses and radii determined by @Piskunov2007, who fitted a three-parameter King’s profile [@King1962] to the observed stellar surface density distribution of 236 objects taken from an homogeneous sample of 650 optical clusters in the solar neighborhood [@Kharchenko2005-known; @Kharchenko2005-new], which is a subset of the current version of the @Dias2002 catalog. @Piskunov2007 estimated the masses from the tidal radii, and the effective radius $R_{\rm eff}$ entering in Equation (\[eq:tcross-units-GP11\]) can be derived from both the core and tidal radius [we used Equation (B1) of @Wolf2010]. Because only 14 of the clusters analyzed by @Piskunov2007 are within the ATLASGAL sky coverage, in order to improve the statistics we applied the @GielesPortegies2011 criterion to the 236 studied objects, under the assumption that they are all OCs as observationally defined by us. This supposition is quite acceptable since they are optically-detected clusters and indeed within the ATLASGAL range almost all of them (13 out of 14) are classified as OCs. We computed the crossing times using Equation (\[eq:tcross-units-GP11\]), and in Figure \[fig:clusters-vs-associations\] they are plotted versus the corresponding ages available from the @Kharchenko2005-known [@Kharchenko2005-new] catalogs. The dashed line is the identity $t_{\rm cross} =$ Age, which divides the physical OCs ($t_{\rm cross} \leq$ Age) from associations ($t_{\rm cross} >$ Age). It can be seen in the plot that, because the resulting crossing times are relatively short ($\log(t_{\rm cross}/{\rm yr}) \lesssim 7.6$), the majority of the objects studied by @Piskunov2007 are physical OCs for ages in excess of 10 Myr. In fact, for $\log({\rm Age}/{\rm yr}) > 7.2$, which is the threshold above which the age distribution can uniquely be explained through classical cluster disruption mechanisms (see Section \[sec:age-distribution-fit\]), only 2.6% of the objects are formally associations. We thus conclude that our observational definition of OC agrees with the physical one provided by @GielesPortegies2011 [what we call a *physical OC*] for ages greater than $\sim 16$ Myr, which corresponds to the 74% of our OC sample within the ATLASGAL range. Younger OCs can be either associations, as a result of early dissolution, or already physical OCs. Spatial distribution {#sec:spatial-distribution} -------------------- ![Crossing time vs. age for an all-sky sample of 236 clusters [@Piskunov2006] taken from an homogeneous catalog of 650 optical clusters in the solar neighborhood [@Kharchenko2005-known; @Kharchenko2005-new]. The dashed line is the identity $t_{\rm cross} =$ Age, which divides the physical OCs ($t_{\rm cross} \leq$ Age) from associations ($t_{\rm cross} >$ Age) according to the criterion proposed by @GielesPortegies2011.[]{data-label="fig:clusters-vs-associations"}](figures/clusters-vs-associations.eps){width="48.00000%"} ![Galactic locations of (a) OCs and (b) ECs within the ATLASGAL range, superimposed over an artist’s conception of the Milky Way (R. Hurt from the *Spitzer* Science Center, in consultation with R. Benjamin), which was based on data obtained from the literature at radio, infrared, and visible wavelengths, and attempts to synthesize many of the key elements of the Galactic structure. The coordinate system is centered at the Sun position, indicated by the ‘$\sun$’ symbol, and we have scaled the image such that $R_0 = 8.23$ kpc [@Genzel2010]. The two diagonal lines represent the ATLASGAL range in Galactic longitude ($|\ell| \le 60\degr$). In panel (a), we indicate the names of the spiral arms.[]{data-label="fig:galactic-distribution"}](figures/galactic-distribution.eps){width="50.00000%"} In this Section, for the clusters in our sample with available distance estimates we study their spatial distribution in the Galaxy, and with respect to the Sun. Figure \[fig:galactic-distribution\] shows the Galactic distribution of the clusters separated in the (a) OC and (b) EC categories defined in the previous Section, on top of an artist’s conception of the Milky Way viewed from the north Galactic pole (R. Hurt from the *Spitzer* Science Center, in consultation with R. Benjamin). The image was constructed based on multiwavelength data obtained from the literature, and we have scaled it to $R_0 = 8.23$ kpc [@Genzel2010 see Section \[sec:kin-distance\]]. It is clear from the image that ECs probe deeper the inner Galaxy than the OC sample, which is concentrated within a few kpc from the Sun ($\lesssim 2$ kpc). This, of course is an observational effect mainly produced by the difficulty in detecting exposed clusters against the Galactic background, compared to ECs (see Section \[sec:completeness\]), and enhanced by the fact that some genuine OCs have no distance estimates and therefore cannot be included in the spatial distribution analysis (e.g., there are 123 clusters of type OC2 without available distance, half of which might be real). ECs are spread over larger distances from the Sun ($\lesssim 6$ kpc) and, although few of them can be detected beyond the Galactic center, a paucity of ECs is hinted within the Galactic bar, augmented by some apparent crowding close to both ends of the bar. The Galactic distribution of ECs is consistent with the spiral structure delineated on the background image; however, the large distance uncertainties ($\sim 0.5$ kpc on average, see Section \[sec:distance-and-ages\]), and the limited distance coverage, prevent the ECs from clearly defining the spiral arms by their own. To really quantify how deep our OC and EC samples reach into the inner Galaxy, and to estimate the completeness fraction at a given distance, we need to study the observed heliocentric distance distribution of the clusters, and compare it to what is expected from making some basic assumptions. In the following, we denote by $D$ the distance of the cluster from the Sun, projected on the Galactic plane[^10], and by $z$ the height of the cluster above the Galactic plane. For simplicity, we also define $Z \equiv z - z_0$, where $z_0$ is the displacement of the Sun above the plane; this is actually what we obtain directly[^11] from the cluster distance $d$ and its Galactic latitude $b$, $Z = d \sin b$. The observed $Z$- and $D$-distributions are shown, respectively, in Figures \[fig:Z-distribution\] and \[fig:D-distribution\], for our cluster sample separated in OC and EC categories. In the construction of the histograms, we used fixed bins of $\Delta Z = 10$ pc and $\Delta D = 0.4$ kpc, but since the distance uncertainties are quite nonuniform, we have fractionally spread the ranges determined by the central values and their uncertainties over the covered bins. In other words, for a cluster with distance and uncertainty $D \pm \sigma_D$, we considered all the bins overlapping with the range $[D-\sigma_D,D+\sigma_D]$ and in each bin we added the fraction (with respect to the total width of the range, $2\sigma_D$) comprised by the corresponding overlap. The total OC and EC distance distributions were obtained by repeating this procedure for all the clusters. The $Z$-distributions were constructed using the same method, and the fitted curves plotted in Figures  \[fig:Z-distribution\] and \[fig:D-distribution\] are explained in the following. ### Assumed model for the spatial distribution In general, we can assume that the spatial number-density of OCs or ECs in the Galaxy is described by a combination of two independent exponential-decay laws for the cylindrical coordinates $z$ and $R$, centered in the Galactic center: $\rho(R,z) = \rho_0 \,\varphi_R(R) \,\varphi_z(z)$, with $\varphi_R(R) = e^{-R/R_{\rm D}}$ and $\varphi_z(z) = e^{-|z|/z_{\rm h}}$. This is a common functional form used to characterize the Galactic distribution of stars [see Section 1.1.2 of @BinneyTremaine2008], and has already been applied in previous OC studies [@Bonatto2006; @Piskunov2006]. One might want to consider the imprint of spiral arm structure in the azimuthal distribution of ECs, since they are still embedded in molecular clouds, but here we are interested in the distance and height longitude-averaged distributions, for which azimuthal substructure is less important. Furthermore, as noted above, our EC distances are not accurate enough to constrain the location of the spiral arms. If we transform the density $\rho(R,z)$ to a coordinate system centered at the Sun, and assume that we are observing the *totality* of the clusters in the Galaxy within the ATLASGAL range ($|b| \le b_1$ and $|\ell| \le \ell_1$, with $b_1 \equiv 1.5\degr$ and $\ell_1 \equiv 60\degr$), the resulting density (not averaged in longitude $\ell$ yet) can be written as $$\label{eq:rho(D,l,Z)} \rho_{\rm tot}(D,\ell,Z) = \left\{ \begin{array}{ll} \rho_0 \, \varphi(D,\ell) \, \varphi_z(Z + Z_0) & \textrm{if}~~|Z| \le D \tan b_1 \\ 0 & \textrm{else~,} \end{array} \right.$$ where $$\label{eq:phi(D,l)} \varphi(D,\ell) \equiv \varphi_R\left(\sqrt{R_0^2 + D^2 - 2 R_0 D \cos \ell}\right)~.$$ Now we can derive an analytical expression for the $D$-distribution of an ideally complete sample: $$\begin{aligned} \Phi_D^{\rm tot}(D) & \equiv & \int_{-\infty}^{\,\infty} \int_{-\ell_1}^{\,\ell_1} \rho_{\rm tot}(D,\ell,Z)\, D\,\rd \ell\,\rd Z\label{eq:Phi(D)-definition}\\ & = & \Sigma_0\, f_{b_1}(D)\, D \int_{-\ell_1}^{\,\ell_1} \varphi(D,\ell) \,\rd \ell~, \label{eq:Phi(D)-general}\end{aligned}$$ where $\Sigma_0 \equiv 2 z_{\rm h} \rho_0$ is the surface number-density on the Galactic disk for $R=0$, and we have defined the function $f_{b_1}(D)$ as $$\label{eq:fb1(D)} f_{b_1}(D) \equiv \left\{ \begin{array}{ll} e^{-z_0/z_{\rm h}}\, \sinh(D \tan b_1/ z_{\rm h}) & \textrm{if}~~D \le z_0 / \tan b_1 \\ 1 - \cosh(z_0 / z_{\rm h})\, e^{-D \tan b_1 / z_{\rm h}} & \textrm{else~,} \end{array} \right.$$ which arises from the fact that the limited latitude coverage restricts the integration in $Z$ at each distance. ![Histogram of heights from the Galactic plane, as measured from the Sun ($Z = z -z_0$), for (a) OCs and (b) ECs, using a bin width of $\Delta Z = 10$ pc and Poisson uncertainties. The overplotted solid curve in each panel represents: (a) the fitted $Z$-distribution $\Phi_Z(Z)$ from Equation (\[eq:Phi(Z)\]) with best-fit parameters $z_0 = 14.7 \pm 3.7$ pc and $z_{\rm h} = 42.5 \pm 9.9$ pc; (b) the predicted $Z$-distribution from Equation (\[eq:Phi(Z)\]), using the parameters fitted for the OC sample. In panel (b), the darker shaded region is the $Z$-histogram for ECs with distances $D < 4$ kpc, whereas the dashed curve indicates the corresponding distribution as predicted from Equation (\[eq:Phi(Z)\]) and the same parameters $z_0$ and $z_{\rm h}$.[]{data-label="fig:Z-distribution"}](figures/Z-distributions.eps){width="48.00000%"} ![Histogram of heliocentric distances, $D$, for (a) OCs and (b) ECs, using a bin width of $\Delta D = 0.4$ kpc and Poisson uncertainties. In each panel, the solid curve represents the fitted $D$-distribution $\Phi_D(D)$ from Equation (\[eq:Phi(D)-observed\]), with the completeness distance $D_{\rm c}$ as free parameter (see Equation (\[eq:fc(D)\])); the dashed curve shows the fit with fixed $D_{\rm c} = 0$ (see text for details). The best-fit parameters are given in Table \[tab:parameters-ZD\].[]{data-label="fig:D-distribution"}](figures/D-distributions.eps){width="48.00000%"} ### Completeness fraction In practice, however, as already mentioned before and discussed in Section \[sec:completeness\], we are unable to detect the totality of the clusters within the ATLASGAL range, due to the difficulty in star cluster identification towards the inner Galaxy. Indeed, the $D$-distributions that we really observe for OCs and ECs (see Figure \[fig:D-distribution\]) do not increase with distance up to the Galactic center ($D = R_0$), as we would expect from Equation (\[eq:Phi(D)-general\]); instead, they reach a maximum at a nearby distance and then decay considerably, especially for optical clusters. The observed $D$-distributions are dominated by the high incompleteness at increasingly larger distances from the Sun, and therefore, are insensitive to large scale structure on the Galactic disk such as the scale length $R_D$. Attempts to include $R_D$ in the parametric fit to the distance distributions described below resulted in heavily degenerated output parameters and practically no constraint on their values. We then eliminated the dependence of the model on $R_D$ by making the rough approximation that the underlying radial distribution of clusters is uniform, i.e., $\varphi_R(R) = 1$. This is supported by the fact that, due to the incompleteness, most clusters in our sample are within a few kpc from the Sun, where the variations in $\varphi_R(R)$ can be considered small relative to the completeness decay. The constants $\rho_0$ and $\Sigma_0$ must now be interpreted as Solar neighborhood values, and from Equation (\[eq:Phi(D)-general\]) the complete $D$-distribution becomes $$\label{eq:Phi(D)-complete} \Phi_D^{\rm tot}(D) = 2 \ell_1 \,\Sigma_0 \,f_{b_1}(D) \,D~.$$ On the other hand, defining a fractional factor $f_{\rm c}(D)$ that quantifies the completeness of the cluster sample as a function of distance[^12], we can express the observed $D$-distribution $\Phi_D(D)$ as $$\label{eq:Phi(D)-observed} \Phi_D(D) = 2 \ell_1 \,\Sigma_0 \,f_{\rm c}(D) \,f_{b_1}(D) \,D~.$$ In order to assign a particular parametric shape to the completeness fraction, we chose an ansatz for $f_{\rm c}(D)$ based on previous statistical works of OCs in the whole sky. @Bonatto2006 studied the WEBDA database[^13] at that time and found, by completeness simulations, that their analyzed OC sample is highly incomplete in the inner Galaxy, even within what they called the “restricted zone”, defined as an annulus segment with Galactocentric distances $R$ in the range $[R_0 -1.3~{\rm kpc},R_0 +1.3~{\rm kpc}]$. The completeness fraction they determined decays almost immediately from $R = R_0$ to $R < R_0$ (see their Fig. 11; note that $R_0 = 8.0$ kpc in that work). However, @Piskunov2006 claim that the @Kharchenko2005-known [@Kharchenko2005-new] OC catalogs constitute a complete sample up to about 0.85 kpc from the Sun. This is nicely illustrated in their Fig. 1, where a flat distribution of surface number-density of clusters is exhibited up to that distance, after which the distribution starts to decrease considerably. If the completeness fraction of their sample in the inner Galaxy were similar to that obtained by @Bonatto2006, the surface density distribution would be a decreasing function immediately from $D=0$ kpc rather than from $D=0.85$ kpc[^14]. We think that this discrepancy is mainly caused by two effects: 1) the cluster sample studied by @Bonatto2006 (654 objects with known distances) is less complete than, e.g., the current version of the @Dias2002 catalog used in this work (1309 clusters with available distances), which is equivalent to the @Kharchenko2005-known [@Kharchenko2005-new] sample within 0.85 kpc; and 2) the “restricted zone” considered by @Bonatto2006 covers a larger area than the circle defined by the completeness limit of @Piskunov2006 (radius of 0.85 kpc centered at the Sun), and thus includes regions where the OC sample is indeed incomplete. In fact, we performed a quick test on the current @Dias2002 catalog by constructing the Galactocentric radii distribution of clusters within 1 kpc from the Sun, and we obtained a shape that is not incompatible with a exponential law in the whole range, as opposed to the distribution derived by @Bonatto2006 [their Fig. 9]. Based on the above discussion, the completeness fraction for our OC sample is likely $\sim 1$ up to a close distance from the Sun, $D_{\rm c}$, and then starts to decay significantly. We assume that the decay is exponential: $$\label{eq:fc(D)} f_{\rm c}(D) = \left\{ \begin{array}{ll} 1 & \textrm{if}~~D \le D_{\rm c}\\ e^{-(D-D_{\rm c})/s_0} & \textrm{else~.} \end{array} \right.$$ This parametrization allows us to investigate the possibility that the sample is always incomplete, as for @Bonatto2006, by just imposing $D_{\rm c} = 0$. We employ the same functional form for the completeness fraction of ECs, but of course varying the parameters $D_{\rm c}$ and $s_0$. ### Fit for the height distribution {#sec:height-distribution} Before proceeding to fit Equation (\[eq:Phi(D)-observed\]) to the observed $D$-distributions, we first need some estimates for $z_{\rm h}$ and $z_0$ which are used to compute the factor $f_{b_1}(D)$. We obtain those estimates from the $Z$-distribution, which can be analytically written as $$\label{eq:Phi(Z)} \Phi_Z(Z) = e^{-|Z + z_0|/z_{\rm h}} \int_{|Z|/\tan b_1}^{\infty} \,\frac{\Phi_D(D)}{2 z_{\rm h}\,f_{b_1}(D)} \,\rd D~.$$ The advantage in writing this equation explicitly in terms of $\Phi_D(D)$ is that we can directly use the observed $D$-distribution instead of its analytical expression (and compute the integral numerically), so that it is possible to fit the $Z$-distribution with only two free parameters, $z_0$ and $z_{\rm h}$, and independently of the fit for the distance distribution. All the fits were performed using the Levenberg-Marquardt least-squares minimization package `mpfit` [@Markwardt2009], implemented in IDL, and we have assumed Poisson uncertainties. The best fit of Equation (\[eq:Phi(Z)\]) to the observed $Z$-distribution of OCs is shown in Figure \[fig:Z-distribution\](a) as a solid curve, and the corresponding fitted parameters are $z_0 = 14.7 \pm 3.7$ pc and $z_{\rm h} = 42.5 \pm 9.9$ pc. These values are in excellent agreement with the ones derived by @Bonatto2006, if we consider their scale height $z_{\rm h}$ within the Solar circle (which is the case for almost the totality of our OC sample). The observed $Z$-distribution of ECs (Figure \[fig:Z-distribution\](b)) is much more irregular than that of OCs, and therefore a proper fit is not possible. This is likely due to the fact that ECs are spread over a larger area than OCs, and therefore, present lower statistics in the Solar neighborhood and larger average errors in $Z$ ($Z \propto D$). In addition, ECs are usually grouped in complexes, as we will see in Section \[sec:statistics\] and can already be noted in Figure \[fig:galactic-distribution\](b), where some particular locations appear crowded with many close objects, enhancing the non-uniformity of their spatial distribution. However, if we adopt the same parameters $z_0$ and $z_{\rm h}$ derived from the OC sample and compute the predicted distribution from Equation (\[eq:Phi(Z)\]) (naturally, using now the observed $\Phi_D(D)$ of ECs), the resulting curve is roughly consistent with the observed $Z$-distribution, as shown in Figure \[fig:Z-distribution\](b) (solid line). The most systematic discrepancy can be identified for $Z < -40$ pc, where there is a significant deficit of observed clusters with respect to the predicted distribution, probably due to the difficulty in detecting ECs below the Galactic disk for large distances. Indeed, Figure \[fig:Z-distribution\](b) also shows the observed $Z$-distribution for ECs with $D < 4$ kpc (darker inner histogram) and the corresponding prediction (dashed curve), and we can see that in this case the deficit of observed clusters below the Galactic plane is only marginal. Another explanation might be the fact that we have assumed that the $b=0$ plane is parallel to the Galactic disk, while in reality the combined effect of the offset of the Sun above the “true” Galactic plane, and of the Galactic center below the $b=0$ plane, slightly tilts the $b=0$ plane towards the south of the Galaxy (see Goodman et al., in preparation), so that clusters at large distances from the Sun and below the Galactic plane would appear at more negative values in the true $Z$-distribution. This could help to populate the bins in the range of the deficit of observed clusters, and would also explain why the deficit is less important for the distribution of clusters with $D < 4$ kpc. ### Fit for the distance distribution Using now values for $z_0$ and $z_{\rm h}$ obtained from the OC sample, which are also consistent with the EC height distribution, to compute the factor $f_{b_1}(D)$ defined in Equation (\[eq:fb1(D)\]), we fitted the analytical distribution $\Phi_D(D)$ from Equation (\[eq:Phi(D)-observed\]) to the observed $D$-distributions of OCs and ECs, with free parameters $\Sigma_0$, $D_{\rm c}$, $s_0$. The last two parameters are implicit in the completeness factor $f_{\rm c}(D)$ defined in Equation (\[eq:fc(D)\]). The best fits are overplotted as solid curves on the corresponding histograms of Figure \[fig:D-distribution\], and the fitted parameters are given in Table \[tab:parameters-ZD\]. As can be already noted in the plots and confirmed by the reduced $\chi^2$ values (0.90 for OCs, and 1.48 for ECs), the assumed form of the completeness fraction (Equation (\[eq:fc(D)\])) is a good representation of the overall detectability of star clusters in the inner Galaxy. The few outliers in the observed distribution with respect to the fitted analytical function for OCs with distances $D \gtrsim 6$ kpc mainly correspond to exposed clusters recently discovered at infrared wavelengths. A similar tendency is hinted for ECs with $D \gtrsim 11$ kpc, although in this case these outliers are also consistent with the irregular nature of the distribution in general, which slightly deviates (at one-sigma level) from the fitted curve at other distance bins. However, some problems with the resolution of the KDA, resulting in ECs incorrectly assigned to the far distance, cannot be ruled out. It is remarkable that, despite the lower statistics caused by restricting to the ATLASGAL range, the fitted completeness limit of our OC sample, $D_{\rm c} = 1.01 \pm 0.16$ kpc, is consistent with that derived by @Piskunov2006 for their all-sky sample in the Solar neighborhood[^15]. For ECs, both the completeness limit $D_{\rm c}$ and the completeness scale length $s_0$ are larger than the corresponding values of the OC distribution (see Table \[tab:parameters-ZD\]), quantitatively confirming that, from an observational point of view, the EC sample traces larger distances from the Sun than the ones traced by our OC sample. Parameter OC EC -------------------------- -------------- ------------- $z_0$ (pc) 14.7 (3.7) $z_{\rm h}$ (pc) 42.5 (9.9) $\Sigma_0$ (kpc$^{-2}$) 82.9 (12.9) 19.5 (3.1) $s_0$ (kpc) 0.72 (0.05) 1.81 (0.10) $D_{\rm c}$ (kpc) 1.01 (0.16) 1.84 (0.35) $\Sigma_0'$ (kpc$^{-2}$) 209.1 (33.3) 40.3 (5.0) $s_0'$ (kpc) 0.82 (0.04) 1.99 (0.09) : Best-fit parameters from the $Z$- and $D$-distributions of OCs and ECs.[]{data-label="tab:parameters-ZD"} The fitted completeness limits for OCs and ECs are significantly above zero, practically discarding the possibility that the cluster samples are always incomplete in the inner Galaxy, as suggested by @Bonatto2006 for OCs. To further test this option, we performed an alternative fit of Equation (\[eq:Phi(D)-observed\]) to the observed $D$-distributions, now fixing $D_{\rm c} = 0$. For each distribution in Figure \[fig:D-distribution\], the resulting best fit is shown as a dashed line, and we immediately notice that this alternative fit is poorer than the one with $D_{\rm c}$ as free parameter, specially for OCs. Indeed, we applied a Kolmogorov-Smirnov test to all the fitted distribution functions in a distance range free of far-distance outliers ($D \le 6$ kpc for OCs, $D \le 9$ kpc for ECs), and we found that the $D_{\rm c} = 0$ fit can be rejected with a significance level of 5% for OCs, and 6.5% for ECs. We thus conclude that the OC and EC samples in the inner Galaxy are roughly complete up to a distance of $\sim 1$ kpc and $\sim 1.8$ kpc, respectively, as derived from the free-$D_{\rm c}$ fits. Discussion on the completeness {#sec:completeness} ------------------------------ In general, the existence of a stellar cluster is observationally established by an excess surface density of stars over the background, so that its detectability depends on its richness, its angular size, the number of resolved individual members and their apparent brightness (which is directly related to the distance), the surface density of field stars, and the amount of extinction on the line of sight [@LadaLada2003]. Consequently, it is particularly difficult to identify a star cluster in the inner Galactic plane, where both the stellar background and the extinction are relatively high, or a very distant cluster, for which its members appear faint and could be confused as a few single stars due to limited angular resolution of the observations. In fact, we have shown in the previous Section that the current samples of OCs and ECs in the inner Galaxy are complete up to only a close distance from the Sun, and then the completeness heavily decreases as distance increases. We have also seen that incompleteness affects the OC sample more severely than the ECs, i.e., the latter have a higher completeness limit and a less drastic decay in the completeness fraction. At first glance, this might seem contradictory since ECs are, by definition, embedded in molecular clouds and thus subject to a high degree of in situ dust extinction. However, at infrared wavelengths, ECs become easier to detect than exposed clusters because it is easier to distinguish them from the field population. Since ECs are usually associated with illuminated interstellar material, they can be identified by eye towards the locations of known nebulae or star-forming regions [e.g., @Dutra2003-2mass; @Bica2003-2mass; @Borissova2011], even if the clusters are partially resolved or highly contaminated by extended emission. In other words, despite bright nebular emission can prevent young stars from being found by point source detection algorithms and therefore hide the host EC from automated searches, at the same time it can help to identify such a cluster when searched by eye against a high stellar background. For clusters with fainter or less irregular extended emission, automated searches can also take advantage of some distinctive characteristic of ECs (like the red-color criterion of our GLIMPSE search, see Section \[sec:newglimpse\]) to separate them from the background, which is in general not feasible for an evolved OC because its member stars present similar observational properties than the field population. It is interesting to compare our distance distribution of ECs (Figure \[fig:D-distribution\](b)) with that of individual *Spitzer*-detected YSOs [@Robitaille2008], as simulated by @RobitailleWhitney2010 using a population synthesis model. They show that the synthetic YSOs that would have been detected by *Spitzer* and included in the @Robitaille2008 catalog correspond to massive objects with a mass distribution that peaks at $\sim 8 M_{\sun}$. The corresponding distance distribution of this model is presented in Fig. 1 of @Beuther2012 for the $10\degr \le \ell \le 20\degr$ range. The plot reveals a high number of far YSOs up to distances of $\sim 14$ kpc, showing that, despite the high extinction, individual (massive) YSOs can be detected deep into the Galactic plane, as opposed to ECs. We therefore think that the low detectability of a far EC is mainly due to the faint apparent brightness of its low-mass population and confusion of its members, so that the whole cluster might be misidentified as an individual massive young star. At near-infrared wavelengths, however, extinction could still play an important role in hiding a far EC. Definition of a representative sample {#sec:representative-sample} ------------------------------------- We can quantify how many OCs and ECs we are missing within a certain distance from the Sun, using the analytical expressions for the observed distance distribution, $\Phi_D(D)$ (Equation (\[eq:Phi(D)-observed\])), and for the distance distribution that would be observed if we detected the totality of the clusters in the inner Galaxy, $\Phi_D^{\rm tot}(D)$ (Equation (\[eq:Phi(D)-complete\])), and using the fitted parameters given in Table \[tab:parameters-ZD\]. We define the cumulative completeness fraction, $F_{\rm c}(D)$, as the ratio of the number of observed clusters with distances $\le D$ to the number that would represent a complete sample within $D$: $$\label{eq:cumulative-completeness} F_{\rm c}(D) \equiv \frac{N_{\rm cl}(\le D)}{N_{\rm cl}^{\rm tot}(\le D)} = \frac{\displaystyle \int_0^D \Phi_D(D')\,\rd D'} {\displaystyle \int_0^D \Phi_D^{\rm tot}(D')\,\rd D'}~.$$ Now we can define a *representative* cluster sample as all objects with distances $D \le D_{\rm rep}$ for which the fraction $F_{\rm c}(D_{\rm rep})$ is above a certain threshold in both the OC and EC samples (this naturally places the restriction on the OC sample alone, since it is more incomplete). We chose a threshold of 0.25, for which the distance has to be $D \le 3.15$ kpc. For simplicity, we just adopt $D_{\rm rep} = 3.0$ kpc, where $F_{\rm c}(D_{\rm rep}) = 0.28$ and $F_{\rm c}(D_{\rm rep}) = 0.79$ for the OC and EC samples, respectively. Note that although the selection of the threshold is somewhat arbitrary, if we keep in mind the above fractions, we only need a certain distance limit $D_{\rm rep}$ where the samples are not too incomplete and at the same time have a reasonable absolute number of objects to perform a statistical analysis. In Column 4 of Table \[tab:morph-types\], we list the number of clusters with $D \le 3.0$ kpc for each morphological type; the total number of ECs in the representative sample is 98. To count the number of OCs, according to our definition we need that the clusters are also confirmed (`ref_Conf` not empty). The number of confirmed clusters with $D \le 3.0$ kpc is given in Column 5 for each morphological type, from which we obtain a total number of 146 OCs in the representative sample. With the fractions $F_{\rm c}(D_{\rm rep})$ computed before, it is also possible to estimate the number of clusters $N_{\rm cl}^{\rm tot}(\le D_{\rm rep})$ that we would observe within 3 kpc, if we had complete samples of OCs and ECs. The corresponding estimates are listed in Column 6, and were simply derived as $N_{\rm cl}(\le D_{\rm rep})/0.79$ for EC types, and $N_{\rm cl}^{\rm conf}(\le D_{\rm rep})/0.28$ for OC types. Note that the large number of OC2 clusters in this ideally complete sample is due to the fact that they cover a wide age range. The age distribution of our sample is analyzed in the next Section. Ages {#sec:age-distribution} ---- We would expect that the ages of the stellar clusters increase along the morphological evolutionary sequence defined in Section \[sec:evolutionary-sequence\]. By dividing the cluster sample in such morphological types, we indeed obtained an increasing tendency in the corresponding ages distributions. However, we were unable to estimate an average age or age ranges for each individual type, given the low number of clusters with available ages that fall within each category, except for OC2. In the whole sample, for types EC1, EC2, OC0 and OC1 there are, respectively, only 9, 16, 15 and 9 objects with age estimates, whereas for OC2 clusters there are 160. Note that for types OC0 and OC1, the total number of objects is also low (see Table \[tab:morph-types\]), so that the main reason for the small number of age estimates is the low absolute statistics. On the other hand, for the much more numerous EC1 and EC2 morphological types (and possibly also part of the OC0 type), the lack of age estimates may simply be caused by the difficulties involved in obtaining these values. It is still possible, however, to derive an upper limit for the ages of the ECs (EC1 and EC2 together), and also to study the age distribution of the whole OC population (OC0, OC1 and OC2 together), as described below. ### Upper limit age of ECs {#sec:age-embedded} The EC ages compiled from the literature were estimated using a variety of methods, including: comparison with theoretical isochrones on a Hertzsprung-Russell diagram constructed after spectroscopic classification in the near-infrared [e.g., @Furness2010], use of the relation between the circumstellar disk fraction in the cluster and its age [following @Haisch2001], and comparison with synthetic clusters constructed by Monte Carlo simulations [@SteadHoare2011], among others. We remark that from the 25 ECs with available age estimates, there are two objects that seem to be artificial outliers, with too old ages to be embedded, namely $7.5 \pm 2.6$ Myr and $25 \pm 7.5$ Myr [respectively, clusters VVV CL100 and VVV CL059 from @Borissova2011][^16]. These two objects are precisely the only ECs in our sample whose age was determined with the distance via isochrone fitting and the high uncertainty of this method for very young clusters is indeed acknowledged by the authors [@Borissova2011]. In a few other cases where isochrone fitting was used to derive the age of an EC, an independent measure of the distance was used as input in order to reduce the uncertainty [e.g., @Ojha2010]. Excluding these two outliers from our sample, we found that 90% (21 out of 23) of the ECs with available age estimates are younger than 3 Myr. Furthermore, given the high errors in this age range, even the remaining two clusters are consistent with being younger than 3 Myr, within the uncertainties: age of $3.3 \pm 2.1$ Myr for $[$BDS2003$]$ 139 [@SteadHoare2011], and $4.2 \pm 1.5$ Myr for $[$DBS2003$]$ 118 [@Roman2007]^\[fn:age-errors\]^. We therefore adopt an upper limit of 3 Myr for the embedded phase, which represents a better constraint than the 5 Myr limit often quoted in the literature [from @Leisawitz1989]. Since practically all available EC ages in our sample are $\lesssim 3$ Myr, the same result is obtained if we consider the representative sample ($D \leq D_{\rm rep} = 3$ kpc), despite the low statistics (10 out of 11 ECs are formally younger than 3 Myr, after removing one outlier). ### Age distribution of OCs {#sec:age-distribution-fit} ![Age distribution of OCs within the representative sample ($D \leq 3$ kpc), using a logarithmic bin width of $\Delta \log({\rm Age}/{\rm yr}) = 0.25$ and Poisson uncertainties. The solid curve corresponds to the fitted age distribution from Equation (\[eq:age-distribution\]), following @LamersGieles2006, with best-fit parameters ${\rm CFR} = 0.93 \pm 0.09$ Myr$^{-1}$ and $M_{\rm max} = (4.46 \pm 0.85)\times 10^4~M_{\sun}$.[]{data-label="fig:age-distribution"}](figures/age-distribution-oc.eps){width="48.00000%"} The much higher number of OCs with available age estimates allowed us to study their age distribution, which is shown in Figure \[fig:age-distribution\] for the representative sample (a total of 143 OCs). Assuming a constant cluster formation rate (CFR), the decreasing number of OCs as time evolves is due to the effect of different disruption processes. @LamersGieles2006 provide a theoretical parameterization of the survival time of initially bound OCs in the solar neighborhood, taking into account four main mechanisms: stellar evolution, tidal stripping by the Galactic gravitational field, shocking by spiral arms, and encounters with giant molecular clouds. They show that the observed age distribution $\Phi_a(a)$ for a constant CFR and a power-law cluster initial mass function with a slope of $-2$ can be written as $$\label{eq:age-distribution} \Phi_a(a) = C \left[\left(\frac{M_{\rm lim}(a)}{M_{\sun}}\right)^{-1} - \left(\frac{M_{\rm max}}{M_{\sun}}\right)^{-1} \right]~,$$ where $a$ is the age, $C$ is a constant, $M_{\rm lim}(a)$ is the initial mass of a cluster that, at an age $a$, reaches a mass equal to the detection limit (assumed to be 100 $M_{\sun}$), and $M_{\rm max}$ is the maximum initial mass of clusters that are formed. It can be shown that the cluster formation rate within the initial mass range $[100\,M_{\sun},M_{\rm max}]$ is related with the factor $C$ by $$\label{eq:cfr} {\rm CFR} = C \left[\frac{1}{100} - \left(\frac{M_{\rm max}}{M_{\sun}}\right)^{-1} \right]~.$$ We fitted $\Phi_a(a)$ from Equation (\[eq:age-distribution\]) to the observed age distribution of OCs in the representative sample, with free parameters $C$ and $M_{\rm max}$; the input function $M_{\rm lim}(a)$ was obtained by digitizing the dashed curve in Fig. 2 of @LamersGieles2006. We plot the resulting best fit as a solid curve in Figure \[fig:age-distribution\], corresponding to the parameters ${\rm CFR} = 0.93 \pm 0.09$ Myr$^{-1}$ and $M_{\rm max} = (4.46 \pm 0.85)\times 10^4~M_{\sun}$. It is clear from the figure that there is an excess of observed young OCs with respect to the fitted theoretical distribution, whereas for older ages the fit is a pretty good representation of the data. The observed excess of young OCs could be the result of two effects. First, young OCs dominate at larger distances because they contain more luminous stars, so that within an incomplete sample the proportion of young OCs is relatively higher than that of older clusters [@Piskunov2006]. Second, since the parameterization of @LamersGieles2006 considers the dissolution of initially bound OCs due to classical mechanisms, the observed over-population of young clusters might consists of associations, i.e., clusters which are already unbound due to disruption processes that are not accounted for by @LamersGieles2006. These associations will quickly dissolve into the field and, therefore, will not be able to populate the older age bins of the distribution in the future. While the age-dependent incompleteness is likely playing a role within our $D_{\rm rep} = 3$ kpc limit, it is interesting to investigate whether or not there is also a contribution from the presence of associations, for which we need to restrict the sample to smaller distances, where the incompleteness is not important. We found that the excess of observed young OCs still holds if we perform the fit for samples restricted to successively smaller distances, down to $D \leq 1.4$ kpc; nevertheless, the low statistics in the Solar neighborhood within the ATLASGAL range prevents us to perform this test on an even more restricted subsample of our catalog. We therefore fitted the model to all-sky samples of OCs, namely, the @Dias2002 catalog and the @Kharchenko2005-known [@Kharchenko2005-new] sample, restricted to a certain limit in projected distance, $D$. For clusters with $D \leq 0.6$ kpc, in both samples, we recovered the results from @LamersGieles2006[^17], whose observed age distribution practically does not show the excess of young OCs with respect to the fitted curve (see their Fig. 3). If we restrict the samples to $D \leq 1.4$ kpc, however, the age distribution for the @Dias2002 catalog presents a statistically significant over-population of young OCs, whereas for the @Kharchenko2005-known [@Kharchenko2005-new] sample the excess is only marginal. Given that the @Kharchenko2005-known [@Kharchenko2005-new] sample is a subset of the @Dias2002 catalog, this behavior means that the young excess in the sample with $D \leq 1.4$ kpc cannot *purely* be due to the age-dependent incompleteness, since otherwise we would obtain a more noticeable effect in the less complete sample. Then, there must necessarily be a contribution from presence of associations. The excess is less significant for the Kharchenko et al. catalog and not noticeable for clusters in both samples with $D \leq 0.6$ kpc probably because there is an observational limitation in detecting associations at very close distances, due to their larger sizes. In summary, we think that the excess of young clusters in our representative OC sample ($D \leq 3.0$ kpc) with respect to the theoretical description of @LamersGieles2006 is caused by a combination of age-dependent incompleteness and presence of associations. The age distribution shown in Figure \[fig:age-distribution\] was constructed using a bin width large enough to ensure good statistics over the whole age range, but we can refine the grid to constrain better a certain feature, as long as the presentation remains statistically significant. By constructing the age distribution with smaller bin widths and doing the fitting again, we found that the transition after which the theoretical description fits well the data occurs at an age of $\log(a/{\rm yr}) \simeq 7.2$, i.e., $\sim 16$ Myr. Consistently, we have seen in Section \[sec:classification-oc-ec\] that the $\sim 16$ Myr limit is roughly the age before which an observed OC might be either an association or a physical OC, whereas observed OCs older than that are practically always bound and therefore are disrupted through “classical” mechanisms over a longer timescale. ### Young cluster dissolution Similarly to the estimation of the cumulative completeness fraction (see Section \[sec:representative-sample\]), we can use the analytical expressions for the distance distributions from Section \[sec:spatial-distribution\] to transform the absolute CFR in the representative sample to an incompleteness-corrected cluster formation rate per unit area, $\dot{\Sigma}$, representative of the inner Galaxy close to the Sun. It can be easily shown that the conversion is $$\label{eq:cfr/area} \dot{\Sigma} = \frac{{\rm CFR(D \leq D_{\rm rep})}}{\ell_1 D_{\rm eff}^2(D_{\rm rep})}~,$$ where $$\label{eq:Deff} D_{\rm eff}^2(D) \equiv 2 \int_0^D f_{\rm c}(D') \,f_{b_1}(D') \,D'\,\rd D'~.$$ For the OC sample, $D_{\rm eff}(D_{\rm rep}) = 1.28$ kpc, which implies that the fitted cluster formation rate per unit area is $\dot{\Sigma}_{\rm fit} = 0.54 \pm 0.05$ Myr$^{-1}$ kpc$^{-2}$. This value can now be compared with the analogous parameter in the @LamersGieles2006 fit for a complete all-sky sample within 0.6 kpc from the Sun, $\dot{\Sigma}_{\rm LG06} = 0.63$ Myr$^{-1}$ kpc$^{-2}$. Together with the maximum mass of $M_{\rm max} = 3 \times 10^4~M_{\sun}$ they obtain, we can see that both fits are consistent within the uncertainties, assuming that their errors are similar to ours (theirs are not provided). On the other hand, from the observed number of OCs in our representative sample with ages $\log(a/{\rm yr}) < 7.2$, we derive $\dot{\Sigma}_{\rm obs} = 1.18 \pm 0.22$ Myr$^{-1}$ kpc$^{-2}$ (using Poisson errors), which sets an upper limit of $\sim 0.5$ to the fraction of observed young OCs that are actually associations. The observed cluster formation rate corrected by age-dependent incompleteness is some value between $\dot{\Sigma}_{\rm fit}$ and $\dot{\Sigma}_{\rm obs}$ that can be parametrized as $\dot{\Sigma}_{\rm obs}^{\rm corr} = \dot{\Sigma}_{\rm obs} - f_{\rm adi}(\dot{\Sigma}_{\rm obs} - \dot{\Sigma}_{\rm fit})$, where $f_{\rm adi}$ is a factor in the range $[0,1]$ ($f_{\rm adi} = 0$ for no age-dependent incompleteness, and $f_{\rm adi} = 1$ for no intrinsic young excess). To obtain a realistic estimate of the fraction of young clusters that will dissolve or merge with other(s) agglomerate(s), and therefore will not become physical OCs by their own, we also need an equivalent estimate for the formation rate of ECs. For that, we can simply take the local surface density $\Sigma_0$ obtained from fitting the distance distribution of ECs (Table \[tab:parameters-ZD\]), and divide it by their upper limit age of 3 Myr, resulting in $\dot{\Sigma}_{\rm EC} = 6.50 \pm 1.03$ Myr$^{-1}$ kpc$^{-2}$. This EC formation rate, however, is not directly comparable to that of OCs, since within 3 kpc from the Sun we are likely detecting ECs with masses below the detection limit of 100 $M_{\sun}$ adopted by @LamersGieles2006 for OCs, as shown, e.g., by @LadaLada2003, whose EC catalog includes objects with masses down to 20 $M_{\sun}$, with a large number of clusters with masses in the range $[50,100]~M_{\sun}$. Fortunately, we found that the uncertainty in the fraction of ECs with masses above 100 $M_{\sun}$, $f_{>100\,M_{\sun}}$, is not dominant and does not prevent us to compute a good estimate of the young dissolution fraction. If we assume that $f_{>100\,M_{\sun}}$ is in the range $[0.1,1]$, we obtain that the fraction of ECs and young exposed clusters, $f_{\rm diss}$, that will not become physical OCs is $$\label{eq:fdiss} f_{\rm diss} = 1 - \frac{\dot{\Sigma}_{\rm fit}}{\dot{\Sigma}_{\rm obs} - f_{\rm adi}(\dot{\Sigma}_{\rm obs} - \dot{\Sigma}_{\rm fit}) + f_{>100\,M_{\sun}}\dot{\Sigma}_{\rm EC}} = 88 \pm 8 \%~,$$ where the uncertainty has been numerically computed assuming Gaussian random variables, except for $f_{>100\,M_{\sun}}$ and $f_{\rm adi}$ which were drawn from uniform probability distributions in the corresponding domains ($[0,1]$ range for $f_{\rm adi}$, see above). The value is in excellent agreement with that obtained by @LadaLada2003. However, the explanation proposed by these authors, that this high fraction is produced by the dissolution of ECs after fast gas expulsion, has been modified (or extended) considerably in recent years. As we have reviewed in the Introduction, depending on the physical conditions of each individual system and its environment, several other phenomena can contribute to the high observed number of ECs relative to physical OCs, namely: dissolving associations from birth, merging of young subclusters, and young cluster dispersion due to tidal shocks from environment or due to fast relaxation for small-$N$ systems. Correlations {#sec:statistics} ------------ ------ ------------- ----------- --------- ---------- ----------- ----------- --------- Type PAH or Bub. Trigg. Edge IRDC UC Complex (1) (2) (3) (4) (5) (6) (7) (8) EC1 59 (8) 0 (0.8) 3 (1.5) 52 (13) 62 (9) 18 (4) 52 (8) EC2 87 (9) 8.2 (2.1) 0 (0.5) 11 (5) 69 (8) 5.6 (1.7) 63 (7) OC0 50 (12) 12 (5) 0 (1.8) 0 (5.9) 55 (12) 0 (1.8) 52 (12) OC1 50 (18) 9.1 (6.7) 0 (4.5) 0 (16.7) 59 (21) 0 (4.5) 45 (17) OC2 1.4 (0.7) 0 (0.3) 0 (0.3) 0 (0.7) 0.7 (0.5) 0 (0.3) 1 (0.6) ------ ------------- ----------- --------- ---------- ----------- ----------- --------- In this Section, we look for correlations between the morphological types defined in Section \[sec:evolutionary-sequence\] and other information compiled in our cluster catalog, such as the MIR morphology and association with known objects. The percentages of clusters that satisfy the studied criteria within each morphological type are presented in Table \[tab:statistics\]. Column 2 gives the percentage of clusters that appear to be exciting PAH emission through UV radiation from their stars, as traced by bright diffuse 8  emission (12  for WISE) or the presence of IR bubbles (MIR morphology `bub-cen`, `bub-cen-trig`, or `pah`, see Section \[sec:atlasgal-and-mir\]). Column 3 lists the fraction of clusters that seem to be triggering further star formation at the edge of the associated IR bubble (MIR morphology `bub-cen-trig` alone), whereas Column 4 indicates the fraction of clusters that are located at the edge of an IR bubble (MIR morphology `bub-cen-edge`). Columns 5, 6 and 7 give, respectively, the percentage of objects that are associated with IRDCs, regions of any type, and UC regions alone. Finally, Column 8 lists the fraction of clusters that are part of a complex of several clusters (see Section \[sec:complexes\]). In this table we present the statistics calculated for the whole cluster sample, because we obtained the same results for the representative sample, within the uncertainties (assumed to be Poisson errors). The only exception is the association with infrared dark clouds, for which we give the fractions within the representative sample. This is expected since an IRDC can only be identified at a relatively near distance because, to be detectable, it has to manifest itself as a dark extinction feature in front of the diffuse Galactic background. We also computed the statistics restricted to clusters with GLIMPSE data available, in order to minimize possible systematic errors arising from the lower resolution and sensitivity of the WISE images (see Section \[sec:MIR-morphology\]), but since only 7% of the clusters have no GLIMPSE data, we obtained identical results than those presented in Table \[tab:statistics\]. We note from the table that the presence of stellar feedback as traced by PAH emission and regions is very important in the first four stages of the evolutionary sequence. When excluding UC regions, we found that both indicators of feedback are roughly equivalent, i.e., the same clusters present both tracers. That a few clusters have PAH emission but no region is probably due to the incompleteness of the current sample of regions. Alternately, in some cases we might be dealing with lower mass clusters whose UV radiation is strong enough to excite the PAH molecules, but not to produce a detectable region of ionized gas [@Allen2007]. On the other hand, the few regions without PAH emission are probably more evolved, or UC regions not identified as such. However, it is remarkable that although the identification of an ultra compact region was only based on the literature, such objects are much more frequently associated with the first morphological type, which presumably covers the youngest clusters. The almost null correlation of OC2 clusters with indicators of stellar feedback is consistent with the fact that these clusters are mostly classical OCs and already gas-free. Concerning triggered star formation, we see that only EC2, OC0, and OC1 clusters are able to produce it, in roughly 10% of the cases. EC1 clusters are not able because they are too embedded and have not yet started to sweep up the surrounding material; in turn, their formation might be triggered itself by another cluster or massive star, but in only a very small fraction (see Column 4). We warn, however, that our diagnoses of triggered star formation are purely based on morphology, so that its real existence in these cases is definitely not conclusive. Infrared dark clouds are mostly associated with the first morphological type, confirming that they trace the earliest phases of star cluster formation. Interestingly, we found that the presence of IRDCs and PAH emission are almost mutually exclusive: within the representative sample, both tracers combined practically account for the totality of EC1 clusters, with almost null intersection. In other words, IRDCs and PAH emission trace, respectively, an earlier and later stage within the deeply embedded phase (type EC1). A simple interpretation for this behavior is that at some point IRDCs are “illuminated” by the radiation of the recently formed ECs, before their actual disruption, so that they become undetectable as extinction features in the mid-infrared but still prominent in the submm dust continuum emission traced by ATLASGAL. Although we have not identified the totality of complexes of physically related clusters in our sample, Table \[tab:statistics\] shows a clear tendency for ECs to be grouped in complexes. In contrast, OCs are much more isolated (the type OC2 dominates the OC population). Only those OCs that are still associated with some molecular gas (types OC0, OC1) present a similar degree of grouping with other clusters as ECs. This is consistent with the fact that star formation occurs in giant molecular cloud complexes with a hierarchical structure, in which star-forming regions with a relatively higher stellar density would be observationally identified as ECs. Many of them will dissolve, while others, if close enough, will undergo a merging process as a result of dynamical evolution, all in a timescale shorter than $\sim 15$ Myr (see Section \[sec:age-distribution\]). The final outcome, after the parent molecular cloud is destroyed, might therefore be very few or even an unique physical OC, which will appear relatively in isolation. Conclusions {#sec:conclusions} =========== We have statistically studied all ECs and OCs known so far in the inner Galactic plane and their correlation with dense molecular gas, taking particular advantage of the improved cluster sample over the past decade and the ATLASGAL submm continuum survey, which traces cold dust and dense molecular gas. The main results and conclusions presented in this paper are summarized as follows. 1. We compiled a merged full-sky list of 3904 ECs and OCs in the Galaxy, collected from several optical and infrared cluster catalogs in the literature, dealing properly with cross-identifications. 2. As part of the above compilation, we performed our own search for ECs on the mid-infrared GLIMPSE survey, complementing the catalog of 92 exposed and less-embedded clusters detected by @Mercer2005 on the same data. Our method basically consisted on visual inspection of three-color images around positions previously selected as potential YSO overdensities, which correspond to enhancements on a stellar density map of the GLIMPSE point source catalog filtered by a red color criterion. With this technique, we found 75 new clusters. 3. The sample of 695 ECs and OCs within the ATLASGAL Galactic range ($|\ell| \le 60\degr$ and $|b| \le 1.5\degr$) was studied in more detail, particularly regarding the correlation with submm emission. We constructed an extensive catalog (available in electronic form at the CDS) with all the relevant information on these objects, including: the characteristics of the submm and mid-infrared emission; correlation with IRDCs, IR bubbles, and regions; distances (kinematic and/or stellar) and ages; and membership in big molecular complexes. 4. Based on the morphology of the submm emission and, for exposed clusters, on the agreement of the clump kinematic distances and cluster stellar distances, we defined an evolutionary sequence with decreasing correlation with ATLASGAL emission: deeply embedded clusters (EC1), partially embedded clusters (EC2), emerging exposed clusters (OC0), totally exposed clusters still physically associated with molecular gas in their surrounding neighborhood (OC1), and all the remaining exposed clusters, with no correlation with ATLASGAL emission (OC2). 5. The morphological evolutionary sequence correlates well with other observational indicators of evolution. In particular, we found that IR bubbles/PAH emission and regions are both equivalently important in the first four stages of the evolutionary sequence, suggesting that ionization is one of the main feedback mechanisms in our cluster sample. IRDCs are significant mostly in the first type (EC1), tracing a very early phase prior to the stage in which the EC starts to “illuminate” the host molecular clump while still embedded (EC1 clusters with PAH emission). The presence of big complexes containing several clusters is, again, relevant in the first four morphological types, which is consistent with the fact that star formation occurs in giant molecular clouds, and that older OCs (OC2) are just the bound survivors of a very complex process of merging and dissolution of young agglomerates. 6. We observationally defined an EC as any cluster with morphological types EC1 or EC2; OCs were defined as all the remaining types, OC0, OC1, and OC2, but were required to be confirmed by follow-up studies, in order to minimize the contamination by spurious candidates. 7. We found that our observational definition of OC agrees with the physical one (a bound exposed cluster, referred to in this work as a *physical OC*) for ages greater than $\sim 16$ Myr. In our sample, some OCs younger than this limit can actually be associations. 8. By fitting the observed heliocentric distance distribution for OCs and ECs within the ATLASGAL range, we found that our OC and EC samples are roughly complete up to a distance of $\sim 1$ kpc and $\sim 1.8$ kpc, respectively. Beyond these limits, the completeness of the OC and EC samples decay exponentially with scale lengths of $\sim 0.7$ kpc and $\sim 1.8$ kpc, respectively. 9. We argued that ECs probe deeper the inner Galactic plane than OCs because, at infrared wavelengths, ECs can be more easily distinguished from the field population than OCs. On the other hand, a very distant EC is hardly detected due to the combined effect of extinction, the faint apparent brightness of its low-mass population and confusion of its members. 10. From a subsample of 23 ECs with available age estimates, we derived an upper limit of 3 Myr for the duration of the embedded phase. 11. We studied the OC age distribution within 3 kpc from the Sun, which was used to fit the theoretical parametrization of @LamersGieles2006 of different disruption mechanisms for bound OCs. We found an excess of observed young OCs with respect to the fit, thought to be a combined effect of age dependent incompleteness and presence of associations for ages $\lesssim 16$ Myr. 12. We derived formation rates of 0.54, 1.18, and 6.50 Myr$^{-1}$ kpc$^{-2}$ for bound OCs, all observed young OCs, and ECs, respectively, which translates into a EC dissolution fraction of $88 \pm 8\%$. This high fraction is thought to be produced by a combination of the following effects: dissolving associations from birth; merging of young subclusters; and young cluster dispersion due to fast gas expulsion, tidal shocks from environment, or fast relaxation for small-$N$ systems. The new generation of all-sky near-infrared surveys, such as the UKIDSS Galactic Plane Survey [@Lucas2008] and VISTA Variables in the Vía Láctea [VVV, @Minniti2010], will constitute valuable tools to discover new OCs and ECs in the Galactic plane and to start filling in the highly incomplete parts of the plane beyond 1 or 2 kpc from the Sun (for OCs and ECs, respectively). In the future, we plan to update our cluster database for the inner Galaxy to include the new discoveries. Furthermore, the improved sensitivity and resolution of these surveys relative to 2MASS will allow studies of the stellar population of ECs which appear too crowded and/or faint in the 2MASS data. Very importantly, this will increase the number of young clusters with available estimates of their physical properties, such as ages and masses. In particular, stellar masses can be combined with estimates of gas masses (e.g., from ATLASGAL) to derive star formation efficiencies and investigate possible trends with the age and the presence of feedback, placing important constraints on star formation theories. We thank the referee for making useful suggestions that improved the clarity of the paper, and Thomas Robitaille for reading the manuscript and providing helpful comments. We acknowledge the useful discussions with Pavel Kroupa, Maria Messineo (about the GLIMPSE search for ECs), and Marion Wienen (about kinematic distances). We also benefited from the email discussions with D. Froebrich (about its catalog of clusters), A. Moisés (about NIR spectrophotometric distances), and M. Gieles (about Equation (\[eq:tcross-units-GP11\])). This research is based on: data from the ATLASGAL project, which is a collaboration between the Max-Planck-Gesellschaft (MPIfR and MPIA), the European Southern Observatory and the Universidad de Chile; observations made with the *Spitzer Space Telescope*, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA; data products from the 2MASS, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation; and data products from the WISE, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This work has made use of the SIMBAD database, operated at CDS, Strasbourg, France, the NASA’s Astrophysics Data System, and the VizieR database of astronomical catalogs [@Ochsenbein2000]. This paper has made use of information from the Red MSX Source survey database at [www.ast.leeds.ac.uk/RMS](www.ast.leeds.ac.uk/RMS) which was constructed with support from the Science and Technology Facilities Council of the UK. E.F.E.M was supported for part of this research through a stipend from the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne. This was work partially carried out in the Max Planck Research Group *Star formation throughout the Milky Way Galaxy* at the Max Planck Institute for Astronomy (MPIA). Cluster lists in the literature {#sec:catalogs-long} =============================== In this appendix, we describe the diverse catalogs and references used for our cluster compilation, separated in three categories according to the wavelength at which the clusters are detected: optical, NIR and MIR clusters. Furthermore, we present a brief discussion of the contamination by false cluster candidates. Again, as for Table \[tab:catalogs\], the number of clusters quoted within the text represent values after removing these spurious objects and some globular clusters (listed in Table \[tab:spurious\]), unless explicitly mentioned. Optical clusters {#sec:optical} ---------------- @Dias2002 provide the most complete catalog of optically visible OCs and candidates, containing revised data compiled from old catalogs and from isolated papers recently published. The list is regularly updated on a dedicated webpage[^18], with additional clusters seen in the optical and revised fundamental parameters from new references. We used the version 3.1 (from November, 2010), which contains 2117 objects, of which 99.7% have estimated angular diameters, and 59.4% have simultaneous reddening, distance and age determinations. Kinematic information is also given for a fraction of clusters, 22.9% of the list have both radial velocity and proper motion data. It should be noted that this catalog aims at collecting not only the OCs first *detected* in the optical, but also most of (ideally, all) the clusters which were detected in the infrared and are *visible* in the optical. For example, 293 objects from the 998 2MASS-detected clusters of @Froebrich2007 were included in the last version of the catalog, based on by-eye inspection of the Digitized Sky Survey (DSS) images. We also included in our compilation the list of new galactic OC candidates by @Kronberger2006, who did a visual inspection of DSS and 2MASS images towards selected regions, and a subsequent analysis of the 2MASS color-magnitude diagrams of the candidates. The clusters were divided in different lists, some of them with fundamental parameters determined, and are all included in the @Dias2002 [ver. 3.1] catalog, except most of the stellar fields classified as *suspected* OC candidates (their Table 2e), which adds 130 objects to the optical cluster sample. NIR clusters ------------ Stellar clusters detected by NIR imaging, mainly from surveys of individual star-forming regions, are compiled from the literature by @Porras2003, @LadaLada2003, and @Bica2003-lit. The first two catalogs are exclusively limited to nearby regions (distances less than 1 kpc and $\simeq 2$ kpc, respectively); @Bica2003-lit did not use that restriction, but their list is only representative for nearby distances too ($\lesssim 2$ kpc). It is not surprising that the three compilations overlap considerably, as is shown in Table \[tab:catalogs\]. All together, these catalogs contribute 297 additional objects with respect to the optical cluster sample. However, most of the NIR clusters correspond to recent discoveries using the 2MASS survey. More than 300 new clusters were found by visual inspection of a huge number of 2MASS $J$, $H$, and specially $K_s$ images [@DutraBica2000; @DutraBica2001; @Bica2003-2mass; @Dutra2003-2mass]. In the pioneer work of @DutraBica2000, 58 star clusters and candidates were originally detected by doing a systematic visual search on a field of $5\degr \times 5\degr$ centered close to the Galactic Center, and towards the directions of regions and dark clouds for $|\ell| \le 4\degr$; though most of them were observed later at higher angular resolution, and 36 turned out to be spurious detections mainly due to the high contamination from field stars in this area (see Section \[sec:spurious\]). Additional 42 objects were discovered by @DutraBica2001, who searched for ECs around the central positions of optical and radio nebulae in the Cygnus X region and other specific regions of the sky [they are included in the literature compilation by @Bica2003-lit]. They extended the method for the whole Milky Way [@Dutra2003-2mass; @Bica2003-2mass southern and equatorial/northern Galaxy, respectively], inspecting a sample of 4450 nebulae collected from the literature, and they found a total of 337 new clusters. In addition to the visual inspection technique, a large number of 2MASS star clusters have been discovered by automated searches, which are based on the selection of enhancements on stellar surface density maps constructed with the point source catalog. The early works of @Ivanov2002 and @Borissova2003 led to 14 detections (the ones not present in any of the catalogs mentioned above are counted in the “Not cataloged (NIR)” row of Table \[tab:catalogs\]); similarly, @Kumar2006 found 54 ECs of which 20 are new detections, focusing the search around the positions of massive protostellar candidates. More recently, @Froebrich2007 searched for 2MASS clusters along the entire Galactic Plane with $|b| \le 20\degr$, automatically looking for star density enhancements, and manually selecting all remaining objects possessing the same visual appearance in the star density maps as known star clusters. They identified a total of 1788 star cluster candidates, 1021 of which resulted to be new discoveries and were presented as a catalog; an estimate of the contamination suggested that about half of these new candidates are real star clusters. A considerable number of objects from the @Froebrich2007 list have been analyzed in more detail by a variety of authors, and they were compiled by @Froebrich2008. For these objects and the ones recently studied by @Froebrich2010 (comprising a total of 68 clusters), we use the refined coordinates and diameters instead of the original ones. The follow-up studies compiled by @Froebrich2008 also unveil 22 spurious clusters and one globular cluster (see Table \[tab:spurious\]). A similar automatic 2MASS search done by @Glushkova2010 in the $|b| < 24\degr$ range, which includes the verification of the obtained star density enhancements by the analysis of color-magnitude diagrams and radial density distributions, produced a list of $\sim 100$ new clusters [most of them included in the last version of the catalog by @Dias2002], providing physical parameters for a total of 168 new and previously discovered objects. Expectations for the near future are that the new generation of all-sky NIR surveys, such as the United Kingdom Infrared Deep Sky Survey (UKIDSS) and VISTA Variables in the Vía Láctea (VVV), will give rise to the discovery of many more stellar clusters, thanks to their improved limiting magnitude and angular resolution compared to 2MASS. A cluster search using these data has already been performed by @Borissova2011, who found 96 previously unknown stellar clusters by visually inspecting multiwavelength NIR images of the VVV survey in the covered disk area ($295\degr \le \ell \le 350\degr$ and $|b| \le 2\degr$), towards directions of star formation signposts (masers, radio, and infrared sources). The objects listed in their catalog were required to present distinguishable sequences on the color-color and color-magnitude diagrams, after applying a field-star decontamination algorithm, in order to minimize the presence of false detections. Automated cluster searches in the UKIDSS and VVV surveys are being done by the corresponding teams.[^19] In our star cluster compilation, we also included recent NIR studies towards specific star-forming regions, or individual star clusters, which are not listed in the previous catalogs. In their NIR survey of 26 high-mass star-forming regions, @Faustini2009 identified the presence of 23 clusters, 16 of which are new discoveries. Additional individual new objects are counted as “Not cataloged clusters (NIR)” in Table \[tab:catalogs\]. MIR clusters ------------ As a result of the high sensitivity of the GLIMPSE mid-infrared survey, @Mercer2005 managed to find 92 new star clusters (2 of which are globular clusters) using an automated algorithm applied to the GLIMPSE point source catalog and archive, and a visual inspection of the image mosaics to search for ECs (the GLIMPSE Galactic range at that time was $10\degr \le |\ell| \le 65\degr$ and $|b| \le 1\degr$, excluding the inner part of the GLIMPSE II survey). The automatic detection method consisted of the construction of a renormalized star density map, which accounts for the varying background, the estimation of the clusters’ spatial parameters by fitting 2D Gaussians to the point sources with an expectation-maximization algorithm, and finally the removal of false detections by using a Bayesian criterion. This technique yielded 91 cluster candidates, 59 of which were new discoveries. Most of the clusters were detected applying a bright magnitude cut at 3.6  before the construction of the stellar density map. Additional 33 new ECs were identified by the visual inspection, which were missed by the automated method. However, simple by-eye examination of some GLIMPSE color images led us to conclude that there are still some ECs missing in the @Mercer2005 list. Because of this (and also to cover the GLIMPSE II area) we performed a new semi-automatic search in the whole GLIMPSE data, focused in the ECs, which resulted in increasing the number of MIR clusters to a total of 164 objects[^20]. The search is described in Section \[sec:newglimpse\]. Spurious cluster candidates {#sec:spurious} --------------------------- The majority of the new IR star cluster catalogs compiled here are based on algorithmic or by-eye detections of stellar density enhancements on images of IR Galactic surveys, and do not provide information whether the identified objects are really composed of physically related stars or are instead produced by chance alignments on the line of sight. Due to the patchy interstellar extinction, an apparent stellar overdensity can simply correspond to a low extinction region with high extinction surroundings. In addition, bright spatially extended emission might be incorrectly classified as unresolved star clusters embedded in nebulae. Confirmation of a real cluster can be achieved through deeper, high-resolution IR photometry or through spectroscopic observations of the candidate stellar members [e.g., @Dutra2003-ntt; @Borissova2005; @Borissova2006; @Messineo2009; @Hanson2010; @Davies2011-Mc81], which in some cases enables the estimation of physical parameters. Though an important number of such studies have been carried out during the past decade, they still cover a small fraction of the total sample of cluster candidates to be confirmed, mainly because these objects represent relatively new discoveries and the observations needed for a more detailed analysis are very time-consuming. Nevertheless, we can roughly estimate the contamination by spurious detections in our sample of cluster candidates in a statistical way. For example, by comparison of the basic characteristics (Galactic distribution, detection method and morphology) of the cluster candidates with those of known clusters rediscovered by their method, @Froebrich2007 found that about 50% of their catalog entries correspond to false clusters. Detailed follow-up studies of unbiased subsets of objects from this catalog, only restricted to certain areas, have determined similar contamination fractions [@Froebrich2008 and references therein]. Another example is the @DutraBica2000 catalog, where 52 (out of 58) candidates have been observed using higher resolution NIR imaging [@Dutra2003-ntt; @Borissova2005], resulting in 36 previously unresolved alignments of a few bright stars (probably in most cases unrelated) which resemble compact clusters at the 2MASS resolution. This would imply a $\sim 70\%$ contamination by spurious detections, but we note that, since this catalog is based on a systematic search for sources projected close to the Galactic center, it is particularly affected by a higher number of background/foreground stars and more intervening dust, which all help to mimic (or hide) star clusters. The subsequent 2MASS by-eye searches performed by this team [@DutraBica2001; @Dutra2003-2mass; @Bica2003-2mass] cover the whole Galactic plane and, furthermore, they are focused on radio/optical nebulae which generally correspond to regions, increasing the chance to find real stellar clusters. Typical spurious clusters associated with radio/optical nebulae represent one or a couple of bright stars plus extended emission [e.g., @Borissova2005]. We caution that, however, as the number of stars in these embedded multiple systems is larger, under the assumption that the stars are physically related, the consideration of a particular candidate as spurious or possible cluster is more dependent on how we define an EC. Under the definition used throughout this work (see Section \[sec:cluster-definition\]), since we do not impose any constraint on the number of members, we expect a minimal contamination by false detections for clusters associated with molecular gas[^21]. For exposed clusters, on the contrary, the probability that a cluster candidate consists of only unrelated stars on the same line of sight is much higher. Based on the above discussion, we estimate an overall spurious contamination rate of $\sim 50\%$ for exposed clusters that have not been confirmed by follow-up studies. In Table \[tab:spurious\] we list the spurious candidates within the compiled cluster catalogs that were not included in our final sample. This table comprises the false detections found by @Dutra2003-ntt and @Borissova2005, and the candidates from the @Froebrich2007 catalog listed as “not a cluster” by the literature compilation of follow-up studies by @Froebrich2008. The other objects are a few globular clusters and false clusters or duplications found in this work, primarily from the literature revision of the cluster sample in the ATLASGAL range (see Appendix \[sec:huge-table-details\]). [lclll]{} \ Name & Flag & Catalog & Ref. & Comments\ \ Name & Flag & Catalog & Ref. & Comments\ $[$DB2000$]$ 2 & S & 03 & 1 &\ $[$DB2000$]$ 3 & S & 03 & 1 &\ $[$DB2000$]$ 4 & S & 03 & 1 &\ $[$DB2000$]$ 7 & S & 01,03 & 2 &\ $[$DB2000$]$ 8 & S & 03 & 1,2 &\ $[$DB2000$]$ 9 & S & 03 & 1 &\ $[$DB2000$]$ 13 & S & 03 & 1 &\ $[$DB2000$]$ 14 & S & 03 & 1 &\ $[$DB2000$]$ 15 & S & 03 & 1 &\ $[$DB2000$]$ 16 & S & 03 & 1 &\ $[$DB2000$]$ 19 & S & 03 & 1 &\ $[$DB2000$]$ 20 & S & 03 & 1 &\ $[$DB2000$]$ 21 & S & 03 & 1 &\ $[$DB2000$]$ 22 & S & 03 & 1 &\ $[$DB2000$]$ 23 & S & 03 & 1 &\ $[$DB2000$]$ 24 & S & 03 & 1 &\ $[$DB2000$]$ 29 & S & 03 & 1 &\ $[$DB2000$]$ 30 & S & 03 & 1 &\ $[$DB2000$]$ 33 & S & 03 & 1 &\ $[$DB2000$]$ 34 & S & 03 & 1 &\ $[$DB2000$]$ 36 & S & 03 & 1 &\ $[$DB2000$]$ 37 & S & 03 & 1 &\ $[$DB2000$]$ 38 & S & 03 & 1 &\ $[$DB2000$]$ 39 & S & 03 & 1 &\ $[$DB2000$]$ 40 & S & 01,03 & 2 &\ $[$DB2000$]$ 41 & S & 03 & 2 &\ $[$DB2000$]$ 43 & S & 03 & 1 &\ $[$DB2000$]$ 44 & S & 03 & 1 &\ $[$DB2000$]$ 46 & S & 03 & 1 &\ $[$DB2000$]$ 47 & S & 03 & 1 &\ $[$DB2000$]$ 48 & S & 03 & 1 &\ $[$DB2000$]$ 53 & S & 03 & 1 &\ $[$DB2000$]$ 54 & S & 03 & 1 &\ $[$DB2000$]$ 56 & S & 03 & 2 &\ $[$DB2000$]$ 57 & S & 03 & 1 &\ $[$DB2000$]$ 58 & S & 01,03 & 2 &\ NGC 6334 VI & S & 04 & 3 &\ $[$DBS2003$]$ 83 & S & 05 & 2 &\ $[$DBS2003$]$ 84 & S & 05 & 2 &\ $[$DBS2003$]$ 95 & D & 05 & 4 &\ $[$DBS2003$]$ 170 & S & 05 & 2 &\ $[$DBS2003$]$ 172 & S & 05 & 5 &\ $[$BDS2003$]$ 101 & S & 06 & 2 &\ $[$BDS2003$]$ 103 & GC & 06 & 2 &\ $[$BDS2003$]$ 105 & S & 06 & 2 &\ $[$BDS2003$]$ 150 & D & 06 & 4 &\ $[$MCM2005b$]$ 3 & GC & 09 & 6,7 &\ $[$MCM2005b$]$ 5 & GC & 09 & 8 &\ $[$FSR2007$]$ 2 & S & 11 & 9 &\ $[$FSR2007$]$ 23 & S & 01,11 & 9 &\ $[$FSR2007$]$ 41 & S & 11 & 10 &\ $[$FSR2007$]$ 91 & S & 11 & 10 &\ $[$FSR2007$]$ 94 & S & 01,11 & 9 &\ $[$FSR2007$]$ 114 & S & 11 & 10 &\ $[$FSR2007$]$ 128 & S & 11 & 10 &\ $[$FSR2007$]$ 744 & S & 01,11 & 11 &\ $[$FSR2007$]$ 776 & S & 11 & 11 &\ $[$FSR2007$]$ 801 & S & 11 & 11 &\ $[$FSR2007$]$ 841 & S & 11 & 11 &\ $[$FSR2007$]$ 894 & S & 01,11 & 11 &\ $[$FSR2007$]$ 927 & S & 01,11 & 11 &\ $[$FSR2007$]$ 956 & S & 01,11 & 11 &\ $[$FSR2007$]$ 1527 & S & 11 & 9 &\ $[$FSR2007$]$ 1635 & S & 11 & 10 &\ $[$FSR2007$]$ 1647 & S & 11 & 10 &\ $[$FSR2007$]$ 1659 & S & 11 & 9 &\ $[$FSR2007$]$ 1685 & S & 11 & 10 &\ $[$FSR2007$]$ 1695 & S & 11 & 10 &\ $[$FSR2007$]$ 1754 & S & 11 & 10,9 &\ $[$FSR2007$]$ 1767 & S & 01,11 & 9 &\ $[$FSR2007$]$ 1735 & GC & 11 & 12,9 &\ Ruprecht 166 & S & 01 & 13 &\ Lynga 3 & S & 01 & 14 &\ NGC 6334 & S & 01 & 4 &\ NGC 6357 & D & 01 & 4 &\ SAI 24 & D & 01 & 4 &\ $[$FSR2007$]$ 101 & D & 01 & 4 &\ $[$FSR2007$]$ 124 & S & 01 & 4 &\ $[$FSR2007$]$ 178 & D & 01 & 4 &\ $[$FSR2007$]$ 198 & D & 01 & 4 &\ $[$FSR2007$]$ 869 & D & 01 & 4 &\ $[$FSR2007$]$ 923 & D & 01 & 4 &\ $[$FSR2007$]$ 974 & D & 01 & 4 &\ $[$FSR2007$]$ 1471 & D & 01 & 4 &\ Construction of the cluster catalog {#sec:huge-table-details} =================================== Here, we report in detail the construction of our cluster catalog within the ATLASGAL Galactic range ($|\ell| \le 60\degr$ and $|b| \le 1.5\degr$), including explanations for all the assumptions and procedures made when compiling the used information, as well as descriptions for all the columns provided in the catalog. The catalog and a list of cited references are electronically available at the CDS, and an excerpt is given in Appendix \[sec:catalog-excerpt\]. Designations, position and angular size {#sec:basic-information} --------------------------------------- The basic information for each cluster is directly obtained from the original cluster catalogs compiled (see Section \[sec:catalogs-summary\]). The column `ID` is a record number from 1 to 695 with the clusters sorted by Galactic longitude. The cluster designation, based on the original catalog, is listed in the column `Name`, which was chosen, in general, to be consistent with the SIMBAD database identifier. Other common names, or designations from other catalog(s) (for clusters originally present in more than one catalog), are given in the column `OName`. In the column `Cat`, we provide the original cluster catalog(s) from which each object was extracted, using the reference ID defined in Table \[tab:catalogs\]. The position of each object is based on the equatorial coordinates listed in the original catalog(s). For multiple catalogs, we averaged the listed positions and angular sizes to obtain the final values given here, ignoring in some cases certain references that were considered less accurate or redundant (which are listed between parentheses in the column `Cat`). The galactic coordinates are given in `GLON` and `GLAT`, whereas the equatorial coordinates (J2000.0) are listed in `RAJ2000` and `DECJ2000`. The column `Diam` is the angular diameter in arcseconds. ATLASGAL emission {#sec:clumpfind} ----------------- From the ATLASGAL survey images, we extracted submaps centered at the cluster locations and with a field of view of $\max\{30\arcmin, 2*\verb|Diam|\}$ to search for submm dust continuum emission tracing molecular gas likely associated with the clusters, and to then characterize its morphology. The first computation needed to determine the presence of real emission in those fields is a proper estimation of the local rms noise level, $\sigma$, for which we used an iterative sigma-clipping procedure[^22] with a threshold of $2\sigma$ and a convergence criterion of 1% (iteration stops when the non-sky pixels are a fraction lower than 1% of the total of sky pixels of the previous iteration). With these chosen parameters, the computed values of $\sigma$ agree well with quick estimates of the noise over emission-free regions identified by eye in some test fields. The average noise level is $\sigma = 45$ mJy/beam, and 95% of the total of fields have $\sigma$ in the range $[30,60]$ mJy/beam. Using the computed rms noise level of each field, we identified clumps of emission by applying the decomposition algorithm *Clumpfind* [@Williams1994] in its IDL implementation for 2D data, `clfind2d`. This routine requires only two input parameters: 1) the intensity threshold, which determines the minimum emission to be included in the decomposition; and 2) the stepsize which sets the contrast needed between two contiguous features to be identified as different clumps. We chose threshold = stepsize = $3\sigma$, after visualizing the decomposition on some test fields and requiring that the obtained clumps were roughly similar to those that would be identified by the human eye. We slightly modified the IDL code of `clfind2d` to improve the clump decomposition and to avoid false detections. Originally, the code developed by @Williams1994 deals with blended emission by splitting it into its corresponding clumps using a simple friends-of-friends method, but instead the current implementation breaks up the emission by assigning the blended pixels to the clump with the nearest peak, which produces some disconnected clumps, i.e., pixels of the same clump not connected by a continuous path. We thus changed the peak distance criterion by the *minimum distance to a clump* to assign blended emission to the existing clumps, which noticeably minimizes the effect of disconnected clumps and resembles the friends-of-friends method. A second modification to the code was to require that the clumps have angular sizes larger than the beam in both image directions, in order to reject “snake”-shaped clumps marginally above the threshold which correspond to minor image artifacts rather than real astronomical emission. The employed algorithm assigns into clumps all the emission above the given threshold and with an extent larger than the beam. We computed the angular distance from the cluster center of the nearest detected ATLASGAL emission pixel to have a quick first impression of the presence of molecular gas. Such values are listed in the column `Clump_sep`, normalized to the cluster angular radius (when no emission is detected in the whole ATLASGAL submap, a lower limit is given). We also performed a careful visual inspection of every ATLASGAL submap, using an IDL script to overplot the positions of all star clusters of our sample within the field, and the submm clumps detected before, as well as any interesting object, such as the positions of measured molecular line velocities (see Section \[sec:line-velocities\]). In another window, the script displays a smaller field of view ($\sim 10\arcmin$) with the cluster itself seen by whole set of IR images (2MASS and GLIMPSE, including three-color images) overlaid with ATLASGAL contours, in order to morphologically compare the IR and the submm emissions. The column `Clump_flag` is a two-digit flag which indicates whether or not the cluster appears physically related to the nearest submm clump detected by *Clumpfind*, as seen by the inspection of these images. The first digit of `Clump_flag` can take the values: 0, when the nearest ATLASGAL clump does not seem to be associated with the cluster; 1, when it does seem to be clearly associated, specially for the cases of star clusters deeply embedded within centrally condensed ATLASGAL clumps; and 2, when the physical connection is less clear but still likely, in most cases when the clump appears to belong to the same star-forming region than the stellar cluster, connected by some diffuse MIR emission. The second digit of `Clump_flag` provides information about the line velocity available for each object and will be described in Section \[sec:line-velocities\]. The column `Morph` is a text flag composed of two parts separated by a period. The first part gives further information about the morphology of the detected ATLASGAL emission throughout the immediate star cluster area, including the cases: `emb`, `p-emb`, `surr`, `few`, `few*`, `exp`, and `exp*`, which are explained in Section \[sec:atlasgal-and-mir\]. The second part indicates the MIR morphology and will be described in the next Section. MIR morphology and association with known objects {#sec:MIR-morphology} ------------------------------------------------- The mid-infrared morphology of a stellar cluster can also provide some clues about its evolutionary stage and presence of feedback, in particular the intensity and distribution of the 8.0  emission. We indicate in the second part of the column `Morph` (after the period) details about the 8.0  morphology of each cluster, after visually inspecting GLIMPSE three-color images made with the 3.6 (blue), 4.5 (green) and 8.0  (red) bands, as part of the process described in the previous Section. This flag includes the cases: `bub-cen`, `bub-cen-trig`, `bub-edge`, and `pah`, which are explained in Section \[sec:atlasgal-and-mir\]. All IR bubbles associated with star clusters and recognized in this work are identified in the table column `Bub`. We give the bubble names from the catalogs by @Churchwell2006 [@Churchwell2007] when the objects are listed there, otherwise an identifier based on the cluster `ID` is provided. We also list in this column IR bubbles that are located in the neighborhood of the clusters but that do not appear clearly associated with them or do not represent any of the scenarios defined above (e.g., bubble in the same star-forming region but not directly interacting with the cluster). Similarly, on the GLIMPSE three-color images and on the 8.0  images we identified the presence of an infrared dark cloud in which the cluster appears to be embedded (see Fig. \[fig:EC-examples\], *top*). These objects are listed in the column `IRDC` using a name based on the cluster `ID` when the IRDC has not been cataloged so far, or the designations from the catalogs by @Simon2006 and @PerettoFuller2009 if it was identified there before. Unlike the IR bubbles, since we do not provide information of the IRDCs within the `Morph` flag, we only list in the column `IRDC` those objects that exhibit possible physical connection with the cluster. Many of the IRDCs reported by @PerettoFuller2009 are only small dark fluctuations over a bright background and do not constitute cluster-forming clumps. We note that, since the ATLASGAL Galactic range is wider than the GLIMPSE coverage, 7% of the cluster sample have no GLIMPSE data available, and this is indicated in the column `no_GL` ($\verb|no_GL| = 1$ when there is no GLIMPSE data, otherwise $\verb|no_GL| = 0$). In those cases, we used WISE three-color images made with the 3.4 (blue), 4.6 (green) and 12  (red) filters, to identify all the features described above. Prominent PAH bands are covered by the 12  filter; indeed, by comparing both sets of 3-color images for clusters with GLIMPSE data available, we found that bright PAH 8.0  emission illuminated by the clusters is unambiguously detected at 12 . Similarly, most of the extended IRDCs identified at 8.0  can also be seen at 12 . However, because of saturation and the relatively low resolution, more detailed structures such as the presence of IR bubbles, smaller IRDCs, or possible triggered star formation are much harder to identify than in the GLIMPSE images. In addition, we searched in the literature for the presence of regions associated with the clusters, and they are listed in the column `HII_reg` with designations compatible with SIMBAD or common names used in the literature for large molecular complexes (see the references for complexes, `ref_Complex`, explained in Section \[sec:complexes\]). Particular designations used here which do not exist in SIMBAD and do not belong to complexes are those starting with: “HRDS”, indicating the regions discovered recently by @Anderson2011 using radio recombination line (RRL) observations; and “RMS”, which represent possible regions corresponding to radio continuum sources found by the RMS survey (see Section \[sec:line-velocities\] for a description of the on-line search we performed in such database; the objects listed here were taken from the “Radio Catalogue Search Results” section of the webpage of each individual RMS source investigated). It is worth noting that, for the regions primarily found using SIMBAD, we carefully checked their nature in the literature by requiring the presence of radio continuum emission or RRLs, since some sources are misclassified as regions in SIMBAD. Two important consulted references of RRL observations were @CaswellHaynes1987 (sources with prefix \[CH87\]) and @Lockman1989 (sources with prefix \[L89b\]). We also specified two flags at the end of some names to indicate two particular situations: the flag “(UC)”, when the source is classified as an ultra compact region in the literature; and the flag “(bub)”, when the region appears associated with the listed IR bubble, but not directly with the star cluster. However, we note that classification as an UC region may not be accurate, considering that detailed interferometric and large-scale observations are needed to really unveil the spatial distribution and evolutionary status of a particular region. Kinematic distance {#sec:kin-distance} ------------------ As stated in Section \[sec:distance-and-ages\], many of the ATLASGAL clumps at the locations or in the vicinity of the stellar clusters have measurements of molecular line LSR velocities. By assuming a Galactic rotation model, we can transform these velocities in kinematic distance estimates for the clumps and, therefore, for the corresponding clusters when they were assumed to be physically associated. ### Line velocities {#sec:line-velocities} We used four main references of line velocities, which were systematically searched on the ATLASGAL submaps (positions overlaid there), in the following priority order: 1) follow-up NH$_3\,(1,1)$ observations towards bright ATLASGAL sources [@Wienen2012 for northern sources; and Wienen et al., in preparation, for southern ones]; 2) similar targets observed in the N$_2$H$^+\,(1-0)$ line (Wyrowski et al., in preparation); 3) the CS$\,(2-1)$ Galactic survey by @Bronfman1996 towards *IRAS* sources with colors typical of compact regions; and 4) velocities of massive YSO candidates from the Red MSX Source (RMS) survey [@Urquhart2008] available on-line[^23], corresponding mainly to targeted observations in the $(1-0)$ and $(2-1)$ transitions of $^{13}$CO, or literature velocities compiled there. The priority sequence was primarily based on the number of ATLASGAL clumps available in each of the lists, in order to make the velocity assignments more uniform; the RMS survey was put at the end because the $^{13}$CO traces less dense gas than the other three molecules, which are unambiguously linked to the ATLASGAL emission. We note that, however, when the same clump is found in more than one list, the velocity differences are negligible compared to the error assumed for the computation of the kinematic distance (7 , see below). The adopted LSR velocity is listed in the column `Vlsr` (in ) of the catalog. We give the corresponding reference in the column `ref_Vlsr`, and the source name in `name_Vlsr` (SIMBAD compatible or the one used in the original paper). If no velocity was available from any of the four main lists mentioned before, additional velocity references were found by doing a coordinate query in SIMBAD. In some cases, we did not find any velocity for the closest detected ATLASGAL clump, but we did for another possibly associated clump or for the region. This information is indicated in the second digit of the flag `Clump_flag`, which can take the values: 0, when no velocity is available; 1, when the listed velocity is from the nearest ATLASGAL clump or from a clump directly adjacent to it; 2, when the clump with the velocity is not the nearest but is within the cluster area (used in cases of optical clusters with large angular size); 3, when the velocity is from an ATLASGAL clump which is apparently associated with the cluster as seen in the images, but is independent of the nearest one; and 4, when we list the RRL velocity of the related region. Considering the value of `Clump_flag` as an unique integer number, i.e., combining the first digit which gives information about the closest ATLASGAL clump (see Section \[sec:clumpfind\]) with the second digit explained here, the kinematic distance computed from `Vlsr` can be assigned to the star cluster if $\verb|Clump_flag| \geq 03$. ### Rotation curve {#sec:rotation-curve} Once all the available LSR velocities had been collected, kinematic distances were calculated using a Galactic rotation curve. The widely employed rotation curve fitted by @BrandBlitz1993 was based on a sample of regions and reflection nebulae with known stellar distances, and their associated molecular clouds, which have the velocity information. Most of these sources are located in the outer Galaxy, out to a Galactocentric radius $R$ of about 17 kpc. They added to the sample the tangent point velocities available at that time to cover the inner Galaxy, (i.e., for $R < R_0$, where $R_0 \sim 8$ kpc is the distance from the Sun of the Galactic center). However, since they used a global functional form to simultaneously fit the inner and the outer Galaxy, this curve does not properly match the data for $R < R_0$, as is shown, e.g., in Figures 6 and 7 of @Levine2008. These authors constructed an updated rotation curve for the inner Galaxy using recent high-resolution tangent point data. The linear function fitted by them to $R \leq 8$ kpc resulted to be steeper than the @BrandBlitz1993 curve in that range, and better reproduces the increase of the rotation velocity with increasing $R$. Given that most of our studied sources are within the solar circle ($R < R_0$), we decided to adopt the @Levine2008[^24] rotation curve for $R/R_0 \leq 0.78$, which is the point where it intersects the @BrandBlitz1993 curve. For $R/R_0 > 0.78$, we adopted the @BrandBlitz1993 curve to cover large Galactocentric radii. We used this intersection point instead of the whole range available in @Levine2008 to ensure continuity of the overall rotation curve assumed. It is worth mentioning that the fourth quadrant part of the same data used by @Levine2008 were previously analyzed by @McClureDickey2007 who fitted their own rotation curve. As already suspected by @Levine2008, the systematic shift of $\sim 7$  between the two curves (see their Figure 7) is due to the differences in determining the terminal velocities from the data. We note that the erfc fitting method [used by @McClureDickey2007] is conceptually equivalent to consider the half-power point of the tangent velocity profile. Fitting instead the theoretical function derived by @Celnik1979, which is a better approximation of the tangent velocity profile, it is found that the half-power point is shifted by $\sim 0.7\sigma_v$ from the real terminal velocity (where $\sigma_v$ is the typical velocity dispersion; see the proof in that paper). We thus favor the rotation curve by @Levine2008, since they fitted @Celnik1979 profiles to derive the tangent point velocities. We did not use the more recent rotation curve by @Reid2009 mainly because it is based on maser parallax distances of only 18 star-forming regions, which cover just the first and second quadrant, so that the obtained rotation curve is not fully representative of our Galactic range and, as the authors acknowledge, cannot conclusively be distinguished from a flat curve (which is their assumed form at the end). In addition, their recommended fit assumes that the massive star-forming gas orbits slower the Galaxy than expected for circular rotation, which has been questioned by some subsequent studies [@Baba2009; @McMillanBinney2010]. ### Derivation of the kinematic distances {#sec:derivation-kdistance} Both rotation curves used here [@BrandBlitz1993; @Levine2008] were originally constructed assuming the standard IAU values for the Galactocentric radius and the orbital velocity of the Sun, $R_0 = 8.5$ kpc and $\Theta_0 = 220$ , respectively. Nevertheless, it can be easily shown that the solution for $x = R/R_0$ derived by applying these curves and a particular LSR velocity is practically independent of the choice of $(R_0, \Theta_0)$ [fully independent for the case of a linear rotation curve constructed from tangent point velocities, as for @Levine2008], and that any scaling of the curve parameters to match updated values of $(R_0, \Theta_0)$ is equivalent to adopt the original parameters in all the parts of the equations. The only thing we need afterwards is an accurate value for $R_0$, to transform from the dimensionless solution $x$ to the physical Galactocentric radius $R$. Moreover, it can be also shown that the solution does not depend on the exact definition of the LSR, provided that the rotation curves and the input data use the same solar motion (generally standard in radiotelescopes), and that any possible correction is only important in the direction of the Galactic rotation, $V_{\sun}$ (which is also true; see Table 5 of @Reid2009, and @Schonrich2010), so that if applied it would be canceled out in the equations. We then applied the original rotation curves and the velocities `Vlsr` with no correction, to solve for $x = R/R_0$. To finally obtain $R$, we adopted $R_0 = 8.23$ ($\pm 0.20$) kpc from @Genzel2010, who computed the weighted mean of all recent *direct* estimations of the Galactic center distance from the Sun. We exclude from the kinematic distance estimation those sources with $R < 2.4$ kpc (only 2% of the cases), which is the point were the approaching and receding parts of the rotation curve constructed by @MarascoFraternali2011 [using coarser resolution data, but covering smaller $R$] start to show significant differences likely due to non-circular motions in the region of the Galactic bar. The @Levine2008 curve covers radii $R \geq 3$ kpc, which means that we implicitly extrapolated it to $R = 2.4$ kpc when we solved the equation for $x$. There is a simple geometrical relation between the obtained Galactocentric radius $R$ and the kinematic distance, but within the solar circle (in our sample, 99% of all kinematic distance estimations) an unique value of $R$ results in two possible distances equally spaced on either side of the tangent point, which are referred to as the near and far distances. This is known as the kinematic distance ambiguity (KDA) problem. Fortunately, as discussed in Section \[sec:KDA-resolution\], there exist a number of methods that have been applied in the literature for an important fraction of the sample to solve the KDA, which allowed us to assign an unique kinematic distance in the 92% of the cases. We list the 424 derived kinematic distances in the table column `KDist` (in kpc); when the KDA is not solved, both near and far distances are given separated by ‘/’. Uncertainties in these distances, provided in the column `e_KDist`, have been determined by shifting the LSR velocities by $\pm 7$  to account for random motions, following @Reid2009, who suggest this value as the typical virial velocity dispersion of a massive star-forming region. We acknowledge, however, that the error in the kinematic distance can be larger due to randomly oriented peculiar motions of up to 20 or 30  with respect to Galactic rotation, as shown, e.g., by the hydrodynamical simulations by @Baba2009. Similarly, such large systematic velocities have been found from maser parallax observations, leading to up to a factor 2 wrong kinematic distances [e.g., @Xu2006; @Kurayama2011]. However, in some such cases it has been found also that the star-forming region does follow circular rotation [e.g., @Sato2010-W51]. With the assumed velocity dispersion of $\sigma_v = 7$ , there are some critical cases where we can only assign an upper limit for the near distance ($|\verb|Vlsr|| < \sigma_v$), or a lower limit for the far distance (`Vlsr` within $\sigma_v$ from the forbidden velocity), and that are properly indicated in the table column `KDist`. ### Resolution of the kinematic distance ambiguity {#sec:KDA-resolution} The solutions for the distance ambiguity found in the literature are given in the table column `KDA`, which informs whether the source with available velocity (listed in `name_Vlsr`) is located on the near (`KDA` = `N`) or far side (`KDA` = `F`), or just at the tangent point (`KDA` = `T`). A companion question mark indicates a doubtful assignation, e.g., from low-quality flags in the original reference, but this happens for only 2% of the solutions. The most common methods for resolution of the distance ambiguity are (examples of references are given below): 1) radio recombination lines in conjunction with absorption toward regions, called the Emission/Absorption method ( E/A); and 2) self-absorption ( SA) and molecular line emission towards molecular clouds and massive YSOs. We considered any source with `Vlsr` within $\sigma_v = 7$  of the terminal velocity as consistent with being at the tangent point, and in general we assigned a `KDA` = `T`. However, for some of these sources, there still exist reliable[^25] KDA solutions that can further constrain the kinematic distance to a either the near (for which `KDA` = `NT`) or the far distance (`KDA` = `FT`). The following references for resolved KDAs were checked systematically (positions overplotted on the ATLASGAL submaps) : @CaswellHaynes1987 [presence/absence of optical counterparts + E/A for a few sources], @Faundez2004 [application of a spiral arms model of the IV quadrant], @AndersonBania2009 [ E/A + SA], @Roman-Duval2009 [ SA], and the RMS survey [@Urquhart2008]. For the RMS survey, which is an ongoing project, we took the KDA solutions from an on-line search we performed for every possibly associated source on “The RMS Database Server”[^26]; these solutions arise from dedicated application of absorption methods [@Urquhart2011; @Urquhart2012], from the literature, or from grouping of sources close in the phase space where there is at least one with resolved KDA. Additional KDA solutions were found through the SIMBAD coordinate query of each source, or from the reference from which the final cluster distance was adopted (e.g., a more accurate method such as maser parallax, see Section \[sec:complexes\]). All used references are listed as integer numbers in the column table `ref_KDA`. An ‘`*`’ following the number means that the source in the corresponding reference with resolved KDA is not located at the same position of the source from which we took the velocity, but is nearby in the phase space (close position and similar velocity) indicating that is likely connected. A reference between parentheses means that it contradicts the KDA solution adopted in this work (see below). Non-numeric flags in the column `ref_KDA` indicate complementary criteria used here to solve the distance ambiguity: - `C`: we adopt the KDA solution for the whole associated complex (see Section \[sec:complexes\]), or from a particular source in the complex. - `D`: source associated with an IRDC, favoring the near distance [see the arguments given by @Jackson2008] - `O`: out of the solar circle, i.e., no ambiguity in the kinematic distance. - `S`: adopted KDA solution consistent with the stellar distance (see Section \[sec:physical-parameters\]) - `z`: near distance adopted, since if located at the far distance the source would be too high above the Galactic plane. We adopted a height value of $|z| = 200$ pc to exclude the far distance, following @Blitz1991. If the assumption of two or more references or criteria delivered contradictory solutions for the KDA, in general we adopted the more recent, or the one using a more accurate method. Although this decision is somehow arbitrary, there are some reasonable guidelines that can be applied, e.g., we favor the consistency with stellar distance or with the complex (flags S and C), and we adopted the solution from the E/A method when conflicting with the SA method, since the first has been found to be more robust [@AndersonBania2009]. In any case, the KDA solutions from different references usually agree; discrepant ones are only the 12% of the total of resolutions and should not affect the statistical results of this work. Stellar distance and age {#sec:physical-parameters} ------------------------ A direct estimation of the distance to a cluster, i.e., from the member stars, is particularly useful when the accuracy is better than that of the kinematic distance from the gas (e.g., when a large sample of stars is used), or when the cluster is fully exposed and there is no nebula that can be associated to it. Using data from the original cluster catalogs and new references found in SIMBAD for each object, we compiled values for the stellar distance (in kpc; table column `SDist`) and its uncertainty (column `e_SDist`), as well as the age and its error (in Myr; columns `Age` and `e_Age`, respectively) computed by studies of the cluster stellar population. The corresponding references of the adopted parameters are listed in the columns `ref_SDist` and `ref_Age`. For the optical clusters in the @Dias2002 [see Section \[sec:optical\]] catalog, we generally used the original parameters given there, unless new estimates based on a better method (or data) provided a real improvement in accuracy. A more rigorous approach for multiple references of the same cluster would be similar to that taken in @PaunzenNetopil2006, and is beyond the scope of this work. However, these authors concluded that their literature-averaged parameters have the same statistical significance as the data from the @Dias2002 catalog, so that for the purposes of our work a correct estimation of the uncertainties (see below) is much more important than careful averaging. Out of the 216 clusters from the @Dias2002 catalog present in our sample, 131 objects come with determinations of both age and distance (+4 clusters with only the distance). We adopted these parameters for most of clusters (110 with original values, and 21 with new ones), and added parameters for 25 more. To keep track of all these changes, the original references used in the @Dias2002 catalog are listed in the column `ref_Dias`. The uncertainties in the cluster fundamental parameters are often ignored or underestimated in the literature; in particular, they are not provided in the @Dias2002 catalog. We therefore collected all available errors from the corresponding references and, to prevent underestimation, we imposed uniform *minimum* uncertainties in the derived parameters. We also assumed these values as errors when they were not given in the literature. For the stellar distance, the minimum uncertainty was carefully chosen depending on the method used to calculate it, in order to correctly compare it with the kinematic distance (e.g., to decide which of both distances is finally adopted, see Section \[sec:complexes\]). All most common methods for cluster distance determination use stellar photometry, so that the corresponding uncertainty is dominated by the errors from the absolute magnitude calibration and from the extinction estimation [e.g., @Pinheiro2010]. For the extinction, in addition to the statistical error intrinsic to the method, there is a systematic error produced by possible variations in the extinction law [e.g., @Fritz2011; @Moises2011], which is often not considered in the literature and might be particularly relevant in the NIR regime. In the optical, we can consider that the typical extinction law assumed ($R_V \simeq 3.1$, appropriate for diffuse local gas) is not subject to important variations, since the observed stars are relatively close to the Sun and not heavily embedded in the associated molecular clouds (if any), otherwise they would not be visible at these wavelengths. In the NIR, the extinction law can be described by a power law, $A_\lambda \propto \lambda^{-\beta}$, and the variations can be accounted for with different values for the exponent $\beta$. Using the typical spread in $\beta$ obtained by @Fritz2011 in their compilation, we found that the corresponding uncertainty in the $K$-band extinction is $\sigma(A_K) \simeq 0.2 \,A_K$. In the following, we list the main methods for stellar distance determinations of the used references, and the corresponding minimum uncertainties adopted in this work: - Optical main-sequence (MS) or isochrone fitting [e.g., @Kharchenko2005-known; @Loktin2001]: In this case, we follow @PhelpsJanes1994 who estimated an uncertainty in distance modulus of $\sigma(m-M) \sim 0.32$, from a detailed analysis of the typical error in fitting a template main sequence to the optical color-magnitude diagram. This is equivalent to an error of $\sim 15\%$ in distance. Due to the fact that, from the point of view of the distance uncertainty, fitting a MS is analogous to fitting an isochrone, we also adopted a minimum error of $\sim 15\%$ for the isochrone method. Furthermore, this is consistent with the spread in distance modulus found by @GrocholskiSarajedini2003 [see their Table 2] in their comparison of different isochrone models. - NIR isochrone fitting [e.g., @Tadross2008; @Glushkova2010]: We adopted the same minimum distance error as for optical isochrone fitting, 15%. Extinction law variations might be present, but since the type of clusters for which isochrone fitting is possible are not severely extinguished (they are generally not young), the corresponding uncertainty in $A_K$ due to these variations is also low (recall $\sigma(A_K) \simeq 0.2 \,A_K$). - Optical spectrophotometric distance [e.g., @Herbst1975]: Here, we assumed an absolute magnitude calibration uncertainty of $\sigma(M_V) \simeq 0.5$, consistent with the typical spread of massive OB star calibration scales [e.g., @Martins2005], and an error in spectral type determination of 1 subtype, equivalent to $\pm 0.3$ magnitudes in $M_V$ for the @Martins2005 calibration. Adding both contributions in quadrature gives an overall uncertainty of $\sim 0.58$ magnitudes in distance modulus, or $\sim 27\%$ in distance. - NIR spectrophotometric distance [e.g., @Moises2011]: For calibration and spectral type errors, we adopted the same overall uncertainty of $\sim 0.58$ magnitudes in distance modulus as for the optical method (absolute magnitudes are usually converted from the optical to the NIR using tabulated intrinsic colors with little error). We added in quadrature an uncertainty to account for possible extinction law variations: assuming a typical extinction of $A_K \simeq 1.5$, $\sigma(A_K) \simeq 0.2 \,A_K \simeq 0.3$. The final error in distance modulus is $\sim 0.66$ magnitudes, equivalent to $\sim 30\%$ in distance. - Average of spectrophotometric distances from many stars [e.g., @Moises2011; @Pinheiro2010]: Redefining the errors here would mean a complete re-computation of the average distance, since the minimum errors should be imposed in every individual star. Fortunately, in general the uncertainty of the average is dominated by the variance of the sample rather than by the individual errors. We thus kept the original quoted uncertainty in this case. - Kinematic distance from average stellar radial velocity [e.g., @Davies2008]: For consistency with gas kinematic distances, here we recomputed the stellar kinematic distance using the cluster LSR velocity, a velocity dispersion of 7  (in all cases higher than the quoted error in the cluster velocity) and the rotation curve as described in Section \[sec:kin-distance\]. This special case is indicated with the flag ‘(K)’ after the reference number in the column `ref_SDist`. - 10$^{\rm th}$ brightest star method [@Dutra2003-ntt; @Borissova2005]: We do not use the stellar distances derived by applying this technique, because they are very uncertain. The errors can easily reach a factor 10 or more in distance [@Borissova2005], which thus places no constraints on the cluster location at Galactic scales. For the cluster ages, we simply adopted uniform minimum errors based on the corresponding age range, following @BonattoBica2011: 35% for $\verb|Age| < 20$ Myr, 30% for 20 Myr $\leq \verb|Age| < 100$ Myr, 20% for 100 Myr $\leq \verb|Age| < 2$ Gyr, and 50% for $\verb|Age| \geq 2$ Gyr. The most common method for age determination is isochrone fitting [e.g., @Loktin2001]. For a few clusters with stars studied spectroscopically, the age can be estimated using the evolutionary types of the identified stars and knowledge about their typical ages and lifetimes [e.g., @Messineo2009]. For a total of 209 clusters age estimates can be found in the literature (30% of our sample). For some clusters of our sample for which no fundamental parameters are available, there are still some studies in the literature that present what can be considered as *confirmations* of the star cluster nature of the objects, i.e., the possibility of an erroneous identification as a cluster can be practically discarded. These references are given in the column `ref_Conf` of the catalog, and usually report higher resolution or/and sensitivity imaging NIR observations in which the star cluster is unequivocally revealed [e.g., @Dutra2003-ntt; @Borissova2005; @Kumar2004]. They also comprise detailed studies towards star-forming regions which are too young to really constrain the cluster physical parameters by isochrone fitting, but where it is still possible to recognize YSO candidates within the cluster as color excess sources in color-color and color-magnitude diagrams [e.g., @RomanAbraham2006]. The objects with both determined age and stellar distance can also be considered as confirmed stellar clusters, because the derivation of parameters usually requires the identification of the cluster sequence or stellar spectroscopy. We thus listed again the references for age and distance in the column `ref_Conf`, including in some cases additional references presenting further cluster analysis. Complexes, subclusters, and adopted distance {#sec:complexes} -------------------------------------------- Young star clusters are normally not found in isolation but within bigger complexes of gas, stars and other clusters, as a result of the fact that star formation occurs in giant molecular clouds with a hierarchical structure. If a group of stellar clusters in our sample was found to form a physically associated complex according to their positions and radial velocities, we identified it in the column `Complex` of the catalog. When the complex was identified in the literature, we here list its name [e.g., the giant molecular cloud W51; @Kang2010]. References for complex identification and analysis are provided in the column `ref_Complex`. Small complexes of clusters not previously established in the literature but whose morphology in the IR images (field of view of $\sim 10\arcmin$) suggests that they belong to the same star-forming region are indicated by `Complex` = MC-$i$, where $i$ is a record number. Bigger complexes of stellar clusters not found in the literature and visually identified within the ATLASGAL fields (of $\sim 30\arcmin$) through the proximity of their members in the phase-space are marked by `Complex` = KC-$j$, where $j$ is another record number. We warn that, however, since the complexes were recognized as part of the visual inspection of the maps, or were found in the literature, not all possible physical groupings of star clusters are provided here. For that, a subsequent statistical analysis is needed, which will be presented in a forthcoming paper. We also identified in the IR images a few cases where there is a pair of star clusters even closer, usually sharing part of their population, which can be considered as subclusters of an unique merging (or merged) entity. Those subclusters are indicated in the table column `SubCl` with an identical record number. For all the clusters of our sample, the final adopted distances with their corresponding errors are listed in the table columns `Dist` and `e_Dist` (in kpc), respectively, and were chosen to be the available distance estimate with the lowest uncertainty, corresponding in some cases to a determination from the literature which was more accurate than `SDist` and `KDist`. Clusters within a particular complex were assumed to be all located at the same distance. The origin of the adopted distance is properly indicated in the column `ref_Dist`, and can be one of the following: - `K`: kinematic distance adopted, $\verb|Dist|=\verb|KDist|$. - `S`: stellar distance adopted, $\verb|Dist|=\verb|SDist|$. - `Ref:`$n$: adopted distance from literature reference with identification number $n$. - `KC`: complex distance computed kinematically from an average position and velocity, using the values compiled here for all the clusters within the complex with available (and not repeated) `Vlsr`, and the rotation curve used in Section \[sec:kin-distance\]. - `SC`: complex distance computed by averaging the stellar distances (`SDist`) of the member clusters. - `C(Ref:`$n$`)`: distance for the whole complex adopted from literature reference with identification number $n$. - `CV(Ref:`$n$`)`: complex distance computed kinematically from an average position and velocity given by the reference with identification number $n$, and the rotation curve used in this work. - `C(ID:`$m$`)`: adopted for the whole complex the distance given for the cluster with $\verb|ID|=m$ (used when a particular cluster within a complex has a very accurate distance estimation). Additional comments {#sec:comments} ------------------- Specific comments about the stellar cluster itself, or its compiled fundamental parameters (stellar distance and age) are provided in column `Comments1`. We give additional remarks about the ATLASGAL emission, the associated complex or other objects, or about the finally adopted distance in column `Comments2`. For comments, the quoted literature is indicated by the code `Ref:`$n$, where $n$ is the identification number of the used reference. Excerpt of the cluster catalog {#sec:catalog-excerpt} ============================== This appendix gives an excerpt of the cluster catalog whose construction is explained in Appendix \[sec:huge-table-details\]. The totality of the catalog, together with a list of cited references, is electronically available at the CDS. Here, we present all the catalog columns (except columns `Comments1` and `Comments2` which are sometimes too wide for the paper version) for 50 (out of 695) stellar clusters. Only for presentation, here the columns are distributed in five tables (Tables \[tab:big-catalog1\] to \[tab:big-catalog5\]), but the on-line version of the catalog is a single table. The names of the columns are the same as defined in Appendix \[sec:huge-table-details\], and they are briefly described in the following (the corresponding Sections of the paper in which they are explained in more detail are given in parentheses): - `ID` : identification number (Section \[sec:basic-information\]) - `Name` : main name (Section \[sec:basic-information\]) - `OName` : other designation (Section \[sec:basic-information\]) - `Cat` : catalogs from which each cluster was extracted (Section \[sec:basic-information\]) - `GLON` : Galactic longitude (Section \[sec:basic-information\]) - `GLAT` : Galactic latitude (Section \[sec:basic-information\]) - `RAJ2000` : right ascension (Section \[sec:basic-information\]) - `DEC2000` : declination (Section \[sec:basic-information\]) - `Diam` : angular size (Section \[sec:basic-information\]) - `Dist` : adopted distance (Section \[sec:complexes\]) - `e_Dist` : distance error (Section \[sec:complexes\]) - `ref_Dist` : distance reference (Section \[sec:complexes\]) - `Age` : age (Section \[sec:physical-parameters\]) - `e_Age` : age error (Section \[sec:physical-parameters\]) - `ref_Age` : age reference (Section \[sec:physical-parameters\]) - `Morph_type` : morphological type (Section \[sec:evolutionary-sequence\]) - `Morph` : morphological flag (Section \[sec:atlasgal-and-mir\]) - `Clump_sep` : projected distance to the nearest ATLASGAL emission pixel (Section \[sec:clumpfind\]) - `Clump_flag` : gives information about the correlation with ATLASGAL and line velocity available (Sections \[sec:clumpfind\] and \[sec:line-velocities\]) - `name_Vlsr` : source name for line velocity (Section \[sec:line-velocities\]) - `Vlsr` : gas line velocity (Section \[sec:line-velocities\]) - `ref_Vlsr` : reference for line velocity (Section \[sec:line-velocities\]) - `KDist` : kinematic distance (Section \[sec:derivation-kdistance\]) - `e_KDist` : error in the kinematic distance (Section \[sec:derivation-kdistance\]) - `KDA` : solution of the kinematic distance ambiguity (Section \[sec:KDA-resolution\]) - `ref_KDA` : reference for the KDA solution (Section \[sec:KDA-resolution\]) - `SDist` : stellar distance (Section \[sec:physical-parameters\]) - `e_SDist` : error in the stellar distance (Section \[sec:physical-parameters\]) - `ref_Sdist` : reference for the stellar distance (Section \[sec:physical-parameters\]) - `ref_Dias` : reference for stellar parameters adopted in the @Dias2002 catalog (Section \[sec:physical-parameters\]) - `ref_Conf` : reference for cluster confirmation (as real cluster) or further studies (Section \[sec:physical-parameters\]) - `HII_reg` : associated region (Section \[sec:MIR-morphology\]) - `Bub` : associated infrared bubble (Section \[sec:MIR-morphology\]) - `IRDC` : associated infrared dark cloud (Section \[sec:MIR-morphology\]) - `no_GL` : indicates when there is no GLIMPSE data available (Section \[sec:MIR-morphology\]) - `SubCl` : groups subclusters (Section \[sec:complexes\]) - `Complex` : groups spatially associated clusters (Section \[sec:complexes\]) - `ref_Complex` : reference for complex identification (Section \[sec:complexes\]) [rlllrrcc]{} `ID` & `Name` & `OName` & `Cat` & `GLON` & `GLAT` & `RAJ2000` & `DEC2000`\ & & & & ($\degr$) & ($\degr$) & ( $^{\rm h}$: $^{\rm m}$: $^{\rm s}$) & ( $\degr$: $'$: $''$)\ [rrcclccccl]{} `ID` & `Diam` & `Dist` & `e_Dist` & `ref_Dist` & `Age` & `e_Age` & `ref_Age` & `Morph_type` & `Morph`\ & ($''$) & (kpc) & (kpc) & & (Myr) & (Myr) & & &\ [rrclccccc]{} `ID` & `Clump_sep` & `Clump_flag` & `name_Vlsr` & `Vlsr` & `ref_Vlsr` & `KDist` & `e_KDist` & `KDA`\ & (`Diam`/2) & & & () & & (kpc) & (kpc) &\ [rlcccccl]{} `ID` & `ref_KDA` & `SDist` & `e_SDist` & `ref_Sdist` & `ref_Dias` & `ref_Conf` & `HII_reg`\ & & (kpc) & (kpc) & & & &\ [rllcclc]{} `ID` & `Bub` & `IRDC` & `no_GL` & `SubCl` & `Complex` & `ref_Complex`\ \ [^1]: The full catalog of 695 stellar clusters within the ATLASGAL Galactic range is only available in electronic form at the CDS via anonymous ftp to `cdsarc.u-strasbg.fr (130.79.128.5)` or via <http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/> [^2]: In combination with distance information for cases of ambiguous physical relation. [^3]: Throughout this paper, we will refer as angular resolution to the full width at half-maximum of the point-spread function (or telescope beam). [^4]: <http://www.alienearths.org/glimpse/glimpse.php> [^5]: Referring to the fact that the clusters were finally selected on the GLIMPSE three-color images [^6]: <http://hea-www.harvard.edu/RD/ds9/> [^7]: This situation is conceptually different from the one indicated by the flag E8 for G3CC objects (see Section \[sec:newglimpse\]), where any extended 8.0  emission in the vicinity of the cluster is flagged. Here, the emission has to be located throughout most of the cluster area and appear as produced by the whole cluster. [^8]: This is a recent catalog of IR bubbles which is much more complete than the @Churchwell2006 [@Churchwell2007] catalogs, but was not used in this work because it was published after our cluster catalog was constructed. In any case, we searched for IR bubbles by eye at every cluster position to describe the MIR morphology (see Section \[sec:atlasgal-and-mir\]). [^9]: Before converting to physical units, we corrected a mistake in the original equation by @GielesPortegies2011: the transformation from virial radius to projected half-light radius is just $16/(3\pi)$ for a Plummer model, so that the constant in their equation is $[32/(3\pi)]^{3/2} = 6.26$ instead of 10. [^10]: In practice, we did not distinguish between the distance $d$ and the projected distance $D = d \cos b$. Since the maximum latitude within the ATLASGAL range is $|b| = 1.5\degr$, the difference is less than 0.03%, far below the distance uncertainties. [^11]: In this paper, for simplicity we have assumed that the $b=0$ plane is parallel to the “true” Galactic plane, although in reality this is not the case (Goodman et al., in preparation). While this has a negligible effect on the distance distribution and the completeness, it may distort the derived height distribution when considering clusters at large distances from the Sun (see Section \[sec:height-distribution\]). [^12]: Ideally, one should consider a completeness fraction dependent on Galactic longitude also, $f_{\rm c}(D,\ell)$, as we expect lower cluster detectability for low $|\ell|$, where the stellar background is higher. However, since we made the approximation $\varphi(D,\ell) = 1$, the integration in longitude would only affect the term $f_{\rm c}(D,\ell)$, and therefore the factor $f_{\rm c}(D)$ we used can be thought as a longitude-averaged completeness fraction. [^13]: WEBDA is an on-line OC database originally developed by @Mermilliod1996, and available on <http://www.univie.ac.at/webda/>; the clusters of this database are included in the @Dias2002 catalog. [^14]: We checked by numerical integration of $\Sigma(D) \propto \int_0^{2\pi} \varphi(D,\ell) \rd \ell$ that the raising of the surface density distribution in the inner Galaxy due to an exponential Galactic disk is practically imperceptible for $D < 1$ kpc, and therefore, a flat distribution cannot be the combined result of incompleteness and exponential disk structure. [^15]: Very recently, a significant effort in obtaining distances and other parameters of most of the known OCs and ECs has been published by @Kharchenko2013, who claim an overall completeness limit of 1.8 kpc. Since ECs are not dominant within a complete sample, the new limit represents an intrinsic improvement in the OC completeness. [^16]: Note that the quoted uncertainties are from our catalog, which might be larger than the values given in the original paper because we adopted minimum errors for the age estimates (see Section \[sec:distance-and-ages\])\[fn:age-errors\]. [^17]: This is totally expected for the Kharchenko et al. sample, since @LamersGieles2006 used basically the same clusters. The only difference is that they did not include the objects newly detected by @Kharchenko2005-new. On the other hand, the fact that for the @Dias2002 sample we obtain the same result implies that there are no systematic effects arising from differences between both samples, in particular regarding the age estimates. [^18]: <http://www.astro.iag.usp.br/~wilton/> [^19]: According to unpublished data, there seem to be more than 300 new clusters detected so far by the UKIDSS team. An independent automated search on UKIDSS, leading to the discovery of 167 additional clusters and multiple star forming regions, has already been published by @Solin2012, after the last update of our cluster compilation was done. [^20]: Including 3 additional GLIMPSE clusters from the literature counted as ‘Not cataloged clusters (MIR)’’ in Table \[tab:catalogs\] [^21]: For consistency with earlier studies, however, we anyway excluded from our sample a few EC candidates that have been considered spurious in the literature. [^22]: We use the routine `meanclip` from the IDL Astronomy User’s Library. [^23]: <http://www.ast.leeds.ac.uk/cgi-bin/RMS/RMS_SUMMARY_PAGE.cgi> [^24]: @Levine2008 provide a rotation curve as a function of both Galactocentric radius, $R$, and height off the Galactic plane, $z$. Here we $z$-averaged their rotation curve, so that it only depends on $R$. [^25]: Considering that the source is near the tangent point and some method/solution combinations are not longer valid. Examples of reliable solutions are: an associated stellar distance, a far solution from the E/A method, or a near solution from the SA method. [^26]: <http://www.ast.leeds.ac.uk/cgi-bin/RMS/RMS_DATABASE.cgi>; we did the search on August, 2011.

--- abstract: 'We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian. This distance reduces to the Grassmann distance when the subspaces are equidimensional and does not depend on any embedding into a larger ambient space. Furthermore, it has a concrete expression involving principal angles, and is efficiently computable in numerically stable ways. Our results are largely independent of the Grassmann distance — if desired, it may be substituted by any other common distances between subspaces. Our approach depends on a concrete algebraic geometric view of the Grassmannian that parallels the differential geometric perspective that is well-established in applied and computational mathematics.' address: - 'Department of Mathematics, University of Chicago, Chicago, IL 60637.' - 'Computational and Applied Mathematics Initiative, Department of Statistics, University of Chicago, Chicago, IL 60637.' author: - Ke Ye - 'Lek-Heng Lim' title: Schubert varieties and distances between subspaces of different dimensions --- Introduction {#sec:motivation} ============ Biological data (e.g. gene expression levels, metabolomic profile), image data (e.g.  <span style="font-variant:small-caps;">mri</span> tractographs, movie clips), text data (e.g. blogs, tweets), etc., often come in the form of a set of feature vectors $a_1,\dots,a_m \in \mathbb{R}^d$ and can be conveniently represented by a matrix $A \in \mathbb{R}^{m \times d}$ (e.g. gene-microarray matrices of gene expression levels, frame-pixel matrices of grey scale values, term-document matrices of term frequencies-inverse document frequencies). In modern applications, it is often the case that one will encounter an exceedingly large sample size $m$ (massive) or an exceedingly large number of variables $d$ (high-dimensional) or both. The raw data $A$ is usually less interesting and informative than the spaces it defines, e.g. its row and column spaces or its principal subspaces. Moreoever, it often happens that $A$ can be well-approximated by a subspace $\mathbf{A} \in \operatorname{Gr}(k,n)$ where $k \ll m$ and $n \ll d$. The process of getting from $A$ to $\mathbf{A}$ is well-studied, e.g. randomly sample a subset of representative landmarks or compute principal components. Subspace-valued data appears in a wide range of applications: computer vision [@MYDF; @VidMaSas2005], bioinformatics [@HH], machine learning [@HRS; @LZ], communication [@LHS; @ZT], coding theory [@AC; @BN; @CHS; @DHST], statistical classification [@HamLee2008], and system identification [@MYDF]. In computational mathematics, subspaces arise in the form of Krylov subspaces [@LS] and their variants [@C], as subspaces of structured matrices (e.g. Toeplitz, Hankel, banded), and in recent developments such as compressive sensing (e.g. Grassmannian dictionaries [@SH], online matrix completion [@BNR]). One of the most basic problems with subspaces is to define a notion of separation between them. The solution is well-known for subspaces of the same dimension $k$ in $\mathbb{R}^n$. These are points on the Grassmannian $\operatorname{Gr}(k,n)$, a Riemannian manifold, and the geodesic distance between them gives us an intrinsic distance. The *Grassmann distance* is independent of the choice of coordinates and can be readily related to principal angles and thus computed via the singular value decomposition (<span style="font-variant:small-caps;">svd</span>): For subspaces $\mathbf{A}, \mathbf{B} \in \operatorname{Gr}(k,n)$, form matrices $A, B \in \mathbb{R}^{n \times k}$ whose columns are their respective orthonormal bases, then $$\label{eq:grassdist} d(\mathbf{A},\mathbf{B}) = \Bigl(\sum\nolimits_{i=1}^k \theta_i^2\Bigr)^{1/2},$$ where $\theta_i = \cos^{-1} \bigl(\sigma_i(A^\mathsf{T} B)\bigr)$ is the $i$th principal angle between $\mathbf{A}$ and $\mathbf{B}$. This is the geodesic distance on the Grassmannian viewed as a Riemannian manifold. There are many other common distances defined on Grassmannians — Asimov, Binet–Cauchy, chordal, Fubini–Study, Martin, Procrustes, projection, spectral (see Table \[tab:distances\]). What if the subspaces are of different dimensions? In fact, if one examines the aforementioned applications, one invariably finds that the most general settings for each of them would fall under this situation. The restriction to equidimensional subspaces thus somewhat limits the utility of these applications. For example, the principal subspaces of two matrices $A$ and $B$ for a given noise level would typically be of different dimensions, since there is no reason to expect the number of singular values of $A$ above a given threshold to be the same as that of $B$. As such one may also find many applications that involve distances between subspaces of different dimensions: numerical linear algebra [@BES; @SS], information retrieval [@ref2a; @text], facial recognition [@ref2b; @face], image classification [@ref2a; @ref2b], motion segmentation [@motion; @ref2c; @ref2d], <span style="font-variant:small-caps;">eeg</span> signal analysis [@eeg], mechanical engineering [@mech], economics [@econ], network analysis [@network], blog spam detection [@blog], and decoding colored barcodes [@bar]. These applications are all based on two existing proposals for a distance between subspaces of different dimensions: The *containment gap* [@Kato pp. 197–199] and the *symmetric directional distance* [@SWF; @WWF]. They are however somewhat ad hoc and bear little relation to the natural geometry of subspaces. Also, it is not clear what they are suppose to measure and neither restricts to the Grassmann distance when the subspaces are of the same dimension. Our main objective is to show that there is an alternative definition that does generalize the Grassmann distance but our work will also shed light on these two distances. Main Contributions ------------------ Our main result (see Theorem \[thm1\]) can be stated in simple linear algebraic terms: Given any two subspaces in $\mathbb{R}^n$, $\mathbf{A}$ of dimension $k$ and $\mathbf{B}$ of dimension $l$, assuming $k< l$ without loss of generality, the distance from $\mathbf{A}$ to the nearest $k$-dimensional subspace contained in $\mathbf{B}$ *equals* the distance from $\mathbf{B}$ to the nearest $l$-dimensional subspace that contains $\mathbf{A}$. Their common value gives the distance between $\mathbf{A}$ and $\mathbf{B}$. Taking an algebraic geometric point-of-view: 1. \[quote\] *The distance between subspaces of different dimensions is the distance between a point and a certain Schubert variety within the Grassmannian.* This distance has the following properties, established in Section \[sec:main\]: 1. readily computable via <span style="font-variant:small-caps;">svd</span>; 2. restricts to the usual Grassmann distance for subspaces of the same dimension; 3. independent of the choice of local coordinates; 4. independent of the dimension of the ambient space (i.e., $n$); 5. \[other\] may be defined in conjunction with other common distances in Table \[tab:distances\]. We will see in Section \[sec:existing\] that the two existing notions of distance between subspaces of different dimensions are special cases of . Evidently, the word ‘distance’ in ($\ast$) is used in the sense of a distance of a point to a set. For example, if a subspace is contained in another, then the distance between them is zero, even if they are distinct subspaces. Thus the distance in ($\ast$) is not a metric[^1]. In Section \[sec:metric\], we define a metric on the set of subspaces of all dimensions using an analogue of our main result: Given any two subspaces in $\mathbb{R}^n$, $\mathbf{A}$ of dimension $k$ and $\mathbf{B}$ of dimension $l$ with $k< l$, the distance from $\mathbf{A}$ to the furthest $k$-dimensional subspace contained in $\mathbf{B}$ *equals* the distance from $\mathbf{B}$ to the furthest $l$-dimensional subspace that contains $\mathbf{A}$. Their common value gives a metric between $\mathbf{A}$ and $\mathbf{B}$. The most interesting metrics for subspaces of different dimensions can be found in Table \[tab:metrics\]. In Section \[sec:volume\], we obtain a volumetric analogue of our main result: Given two arbitrary subspaces in $\mathbb{R}^n$, $\mathbf{A}$ of dimension $k$ and $\mathbf{B}$ of dimension $l$ with $k < l$, we show that the probability a random $l$-dimensional subspace contains $\mathbf{A}$ *equals* the probability a random $k$-dimensional subspace is contained in $\mathbf{B}$. The far-reaching work [@EAS] popularized the basic *differential geometry* of Stiefel and Grassmannian manifolds by casting the discussions concretely in terms of matrices. Subsequent works, notably [@AMSbook; @AMS; @AMSV], have further enriched this concrete matrix-based approach. A secondary objective of our article is to do the same for the basic *algebraic geometry* of Grassmannians. In particular, we introduce some of the objects in Table \[tab:objects\] to an applied and computational mathematics readership. The proofs of our main results essentially use only the <span style="font-variant:small-caps;">svd</span>. Everything else is explained within the article and accessible to anyone willing to accept a small handful of unfamiliar terminologies and facts on faith. -------------------------------- --------------------------------------------- ------------------------------------------------------------------------------------------------------------- ------------------- *Grassmannian* $\operatorname{Gr}(k,n)$ models $k$-dimensional subspaces in $\mathbb{R}^n$ §\[sec:Grass\] *Infinite Grassmannian* $\operatorname{Gr}(k,\infty)$ models $k$-dimensional subspaces regardless of ambient space §\[sec:infty\] *Doubly-infinite Grassmannian* $\operatorname{Gr}(\infty,\infty)$ models subspaces of all dimensions regardless of ambient space §\[sec:dinfty\] *Flag variety* $\operatorname{Flag}(k_1,\dots,k_m,n)$ models nested sequences of subspaces in $\mathbb{R}^n$; $\operatorname{Flag}(k,n) = \operatorname{Gr}(k,n)$ §\[sec:schubert\] *Schubert variety* $\Omega(\mathbf{X}_1,\dots,\mathbf{X}_m,n)$ ‘linearly constrained’ subset of $\operatorname{Gr}(k,n)$ §\[sec:schubert\] -------------------------------- --------------------------------------------- ------------------------------------------------------------------------------------------------------------- ------------------- Grassmannian of linear subspaces {#sec:Grass} ================================ We will selectively review some basic properties of the Grassmannian. The differential geometric perspectives are drawn from [@Husemoller; @MS], the more concrete matrix-theoretic view from [@AMS; @EAS; @Wong], and the computational aspects from [@GVL]. We fix the ambient space $\mathbb{R}^n$. A *$k$-plane* is a $k$-dimensional subspace of $\mathbb{R}^n$. A *$k$-frame* is an ordered orthonormal basis of a $k$-plane, regarded as an $n \times k$ matrix whose columns $a_1,\dots, a_k$ are the orthonormal basis vectors. A *flag* is a strictly increasing sequence of nested subspaces, $\mathbf{X}_0\subset \mathbf{X}_1\subset \cdots \subset \mathbf{X}_m\subset \mathbb{R}^n$; it is *complete* if $m=n$. We write $\operatorname{Gr}(k,n)$ for the *Grassmannian* of $k$-planes in $\mathbb{R}^n$, $\operatorname{V}(k,n)$ for the *Stiefel manifold* of orthonormal $k$-frames, and $\operatorname{O}(n) \coloneqq \operatorname{V}(n,n)$ for the *orthogonal group* of $n \times n$ orthogonal matrices. $\operatorname{V}(k,n)$ may be regarded as a homogeneous space, $$\operatorname{V}(k,n) \cong \operatorname{O}(n)/\operatorname{O}(n-k),$$ or more concretely as the set of $n \times k$ matrices with orthonormal columns. There is a *right action* of $\operatorname{O}(k)$ on $\operatorname{V}(k,n)$: For $Q\in \operatorname{O}(k)$ and $A \in \operatorname{V}(k,n)$, the action yields $AQ \in \operatorname{V}(k,n)$ and the resulting homogeneous space is $\operatorname{Gr}(k,n)$, i.e., $$\label{eq:homo} \operatorname{Gr}(k,n) \cong \operatorname{V}(k,n)/\operatorname{O}(k) \cong \operatorname{O}(n)/\bigl(\operatorname{O}(n-k) \times \operatorname{O}(k)\bigr).$$ In this picture, a subspace $\mathbf{A} \in \operatorname{Gr}(k,n)$ is identified with an equivalence class comprising all its $k$-frames $\{ AQ \in \operatorname{V}(k,n): Q \in \operatorname{O}(k)\}$. Note that $\operatorname{span}(AQ) = \operatorname{span}(A)$ for all $Q \in \operatorname{O}(k)$. There is a *left action* of $\operatorname{O}(n)$ on $\operatorname{Gr}(k,n)$: For any $Q \in \operatorname{O}(n)$ and $\mathbf{A} = \operatorname{span}(A) \in \operatorname{Gr}(k,n)$ where $A$ is a $k$-frame of $\mathbf{A}$, the action yields $$\label{eq:action} Q \cdot \mathbf{A}\coloneqq \operatorname{span}(QA) \in \operatorname{Gr}(k,n).$$ This action is transitive as any $k$-plane can be rotated onto any other $k$-plane by some $Q \in \operatorname{O}(n)$. A $k$-plane $\mathbf{A} \in \operatorname{Gr}(k,n)$ will be denoted in boldface; the corresponding italized letter $A =[a_1,\dots,a_k] \in \operatorname{V}(k,n)$ will denote a $k$-frame of $\mathbf{A}$. $\operatorname{Gr}(k,n)$ and $\operatorname{V}(k,n)$ are smooth manifolds of dimensions $k(n-k)$ and $nk-k(k+1)/2$ respectively. As a set of $n\times k$ matrices, $\operatorname{V}(k,n)$ is a submanifold of $\mathbb{R}^{n\times k}$ and inherits a Riemannian metric from the Euclidean metric on $\mathbb{R}^{n\times k}$, i.e., given $A=[a_1,\dots, a_k]$ and $B=[b_1,\dots, b_k]$ in $T_X \operatorname{V}(k,n)$, the tangent space at $X \in \operatorname{V}(k,n)$, the Riemannian metric $g$ is defined by $g_X(A,B)=\sum_{i=1}^k a_i^\mathsf{T} b_i= \operatorname{tr}(A^\mathsf{T}B)$. As $g$ is invariant under the action of $\operatorname{O}(k)$, it descends to a Riemannian metric on $\operatorname{Gr}(k,n)$ and in turn induces a geodesic distance on $\operatorname{Gr}(k,n)$ which we define below. Let $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$ respectively. Let $r \coloneqq \min (k,l)$. The $i$th *principal vectors* $(p_{i},q_{i})$, $i=1,\dots,r$, are defined recursively as solutions to the optimization problem$$\begin{tabular} [c]{rcl}maximize & & $p^{\mathsf{T}}q$\\ subject to & & $p\in \mathbf{A},\; p^{\mathsf{T}}p_{1}=\dots=p^{\mathsf{T}}p_{i-1}=0,\;\lVert p\rVert=1,$\\ & & $q\in \mathbf{B}, \; q^{\mathsf{T}}q_{1}=\dots=q^{\mathsf{T}}q_{i-1}=0,\;\lVert q\rVert=1,$\end{tabular}$$ for $i=1,\dots,r$. The *principal angles* are then defined by$$\cos\theta_{i}=p_{i}^\mathsf{T}q_{i}, \quad i = 1,\dots,r.$$ Clearly $0\le \theta_{1}\leq\dots\leq\theta_r \le \pi/2$. We will let $\theta_i(\mathbf{A},\mathbf{B})$ denote the $i$th principal angle between $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$. Principal vectors and principal angles may be readily computed using <span style="font-variant:small-caps;">qr</span> and <span style="font-variant:small-caps;">svd</span> [@BG; @GVL]. Let $A = [a_1,\dots,a_k]$ and $B =[ b_1,\dots, b_l]$ be orthonormal bases and let $$\label{eq:svd} A^\mathsf{T} B = U\Sigma V^\mathsf{T}$$ be the full <span style="font-variant:small-caps;">svd</span> of $A^\mathsf{T} B$, i.e., $U \in \operatorname{O}(k)$, $V \in \operatorname{O}(l)$, $\Sigma = \left[\begin{smallmatrix}\Sigma_1 & 0 \\ 0 & 0 \end{smallmatrix}\right]\in \mathbb{R}^{k \times l}$ with $\Sigma_1 = \operatorname{diag}(\sigma_1,\dots,\sigma_r) \in \mathbb{R}^{r \times r}$ where $\sigma_1 \ge \dots \ge \sigma_{r} \ge 0$. The principal angles $\theta_{1}\leq\dots\leq\theta_{r}$ are given by $$\label{eq:angles} \theta_i = \cos^{-1} \sigma_i, \quad i = 1,\dots,r.$$ It is customary to write $A^\mathsf{T} B = U (\cos\Theta) V^\mathsf{T}$, where $\Theta = \operatorname{diag}(\theta_1,\dots,\theta_{r},1,\dots,1) \in \mathbb{R}^{k \times l}$ and $\Theta_1 = \operatorname{diag}(\theta_1,\dots,\theta_{r}) \in \mathbb{R}^{r \times r}$. Consider the column vectors, $$AU = [p_1,\dots,p_k], \quad BV = [q_1,\dots,q_l].$$ The principal vectors are given by $(p_1,q_1),\dots,(p_r, q_r)$. Strictly speaking, principal vectors come in pairs but we will also call the vectors $p_{r+1},\dots,p_k$ (if $r = l < k$) or $q_{r+1},\dots,q_l$ (if $r = k < l$) principal vectors for lack of a better term. We will use the following fact from [@GVL Theorem 6.4.2]. \[prop:angles\] Let $r = \min (k,l)$ and $\theta_1,\dots,\theta_{r}$ and $(p_1,q_1),\dots,(p_{r}, q_{r})$ be the principal angles and principal vectors between $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$ respectively. If $m < r$ is such that $1 = \cos \theta_1 = \dots = \cos \theta_m > \cos \theta_{m+1}$, then $$\mathbf{A} \cap \mathbf{B} = \operatorname{span} \{p_1,\dots, p_m \} = \operatorname{span} \{q_1,\dots, q_m \}.$$ If $k = l$, the geodesic distance between $\mathbf{A}$ and $\mathbf{B}$ in $\operatorname{Gr}(k,n)$ is called the *Grassmann distance* and is given by $$\label{eq:grassdist1} d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^k \theta_i^2\Bigr)^{1/2} = \lVert\cos^{-1}\Sigma\rVert_{F}.$$ An explicit expression for the geodesic [@AMS] connecting $\mathbf{A}$ to $\mathbf{B}$ on $\operatorname{Gr}(k,n)$ that minimizes the Grassmann distance is given by $\gamma : [0,1] \to \operatorname{Gr}(k,n)$, $$\label{eq:geodesic} \gamma(t) = \operatorname{span}( AU\cos t\Theta+ Q\sin t \Theta )$$ where $M= Q(\tan\Theta) U^{\mathsf{T}}$ is a condensed <span style="font-variant:small-caps;">svd</span> of the matrix $$M \coloneqq (I - AA^\mathsf{T})B(A^\mathsf{T} B)^{-1} \in \mathbb{R}^{n \times k}$$ and where $U \in \operatorname{O}(k)$ and $\Theta=\operatorname{diag}(\theta_1,\dots,\theta_k)\in \mathbb{R}^{k\times k}$ are as in and . Note that if $\cos\Theta=\Sigma$, then $\tan\Theta=(\Sigma^{-2}-I)^{1/2}$. Also, $\gamma(0) =\mathbf{A}$ and $\gamma(1) = \mathbf{B}$. Aside from the Grassmann distance, there are many well-known distances between subspaces [@BN; @DD; @DHST; @EAS; @HamLee2008]. We present some of these in Table \[tab:distances\]. -------------- --------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------- *Principal angles* *Orthonormal bases* Asimov $d^{\alpha}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \theta_k$ $\cos^{-1} \lVert A^\mathsf{T} B \rVert_2$ Binet–Cauchy $d^{\beta}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \left(1 - \prod\nolimits_{i=1}^k\cos^2\theta_i\right)^{1/2}$ $(1 - (\det A^\mathsf{T} B)^2)^{1/2}$ Chordal $d^{\kappa}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \left(\sum\nolimits_{i=1}^k\sin^2\theta_i\right)^{1/2}$ $\frac{1}{\sqrt{2}} \lVert AA^\mathsf{T} - BB^\mathsf{T} \rVert_F$ Fubini–Study $d^{\phi}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \cos^{-1} \left(\prod\nolimits_{i=1}^k\cos \theta_i\right)$ $\cos^{-1} \lvert \det A^\mathsf{T} B \rvert$ Martin $d^{\mu}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \left( \log \prod\nolimits_{i=1}^k 1/\cos^2 \theta_i \right)^{1/2}$ $(-2 \log \det A^\mathsf{T} B)^{1/2}$ Procrustes $d^{\rho}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = 2\left(\sum\nolimits_{i=1}^k\sin^2(\theta_i/2)\right)^{1/2}$ $\lVert AU - BV \rVert_F$ Projection $d^{\pi}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \sin \theta_k$ $\lVert AA^\mathsf{T} - BB^\mathsf{T} \rVert_2$ Spectral $d^{\sigma}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = 2\sin (\theta_k/2)$ $\lVert AU - BV \rVert_2$ -------------- --------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------- The value $\sin \theta_1$ is sometimes called the *max correlation distance* [@HamLee2008] or spectral distance [@DHST] but it is not a distance in the sense of a metric (can be zero for a pair of distinct subspaces) and thus not listed. The spectral distance $d^{\sigma}_{\operatorname{Gr}(k,n)}$ is also called chordal $2$-norm distance [@BN]. For each distance in Table \[tab:distances\] defined for equidimensional $\mathbf{A}$ and $\mathbf{B}$, Theorem \[thm:othermetrics\] provides a corresponding version for when $\dim \mathbf{A} \ne \dim \mathbf{B}$. The fact that all these distances in Table \[tab:distances\] depend on the principal angles is not a coincidence — the result [@Wong Theorem 3] implies the following. \[thm:wong\] Any notion of distance between $k$-dimensional subspaces in $\mathbb{R}^n$ that depends only on the relative positions of the subspaces, i.e., invariant under any rotation in $\operatorname{O}(n)$, must be a function of their principal angles. To be more specific, if a distance $d : \operatorname{Gr}(k,n) \times \operatorname{Gr}(k,n) \to [0, \infty)$ satisfies $$d(Q \cdot \mathbf{A}, Q \cdot \mathbf{B}) =d(\mathbf{A}, \mathbf{B}),$$ for all $\mathbf{A}, \mathbf{B} \in \operatorname{Gr}(k,n)$ and all $Q \in \operatorname{O}(n)$, where the action is as defined in , then $d$ must be a function of $\theta_i(\mathbf{A},\mathbf{B})$, $i=1,\dots,k$. We will next introduce the *infinite Grassmannian* $\operatorname{Gr}(k,\infty)$ to show that these distances between subspaces are independent of the dimension of their ambient space. The Infinite Grassmannian {#sec:infty} ========================= One way of defining a distance between $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B}\in \operatorname{Gr}(l,n)$ where $k \ne l$ is to first isometrically embed $\operatorname{Gr}(k,n)$ and $\operatorname{Gr}(l,n)$ into an ambient Riemannian manifold and then define the distance between $\mathbf{A}$ and $\mathbf{B}$ as their distance in the ambient space. This approach is taken in [@CHS; @SLLM], via an isometric embedding of $\operatorname{Gr}(0,n), \operatorname{Gr}(1,n), \dots, \operatorname{Gr}(n,n )$ into a sphere of dimension $(n-1)(n+2)/2$. Such a distance suffers from two shortcomings: It is not intrinsic to the Grassmannian and it depends on both the embedding and the ambient space. The distance that we propose in Section \[sec:main\] will depend only on the intrinsic distance of the Grassmannian and is independent of $n$, i.e., a $k$-plane $\mathbf{A}$ and an $l$-plane $\mathbf{B}$ in $\mathbb{R}^n$ will have the same distance if we regard them as subspaces in $\mathbb{R}^m$ for any $m \ge \min(k,l)$. We will first establish this for the special case $k = l$. Consider the inclusion map $\iota_n:\mathbb{R}^n\to \mathbb{R}^{n+1}$, $\iota_n(x_1,\dots, x_n)=(x_1,\dots, x_n,0)$. It is easy to see that $\iota_n$ induces a natural inclusion of $\operatorname{Gr}(k,n)$ into $\operatorname{Gr}(k,n+1)$ which we will also denote by $\iota_n$. For any $m > n$, composition of successive natural inclusions gives the inclusion $\iota_{nm} : \operatorname{Gr}(k,n) \to \operatorname{Gr}(k,m)$, where $\iota_{nm} \coloneqq \iota_n \circ \iota_{n+1} \circ \dots \circ \iota_{m-1}$. To be more concrete, if $A \in \mathbb{R}^{n \times k}$ has orthonormal columns, then $$\label{eq:iota} \iota_{nm} : \operatorname{Gr}(k,n) \to \operatorname{Gr}(k,m), \qquad \operatorname{span} (A) \mapsto \operatorname{span} \left( \begin{bmatrix}A \\ 0 \end{bmatrix} \right),$$ where the zero block matrix is $(m-n) \times k$ so that $\left[ \begin{smallmatrix}A \\ 0 \end{smallmatrix} \right] \in \mathbb{R}^{m \times k}$. For a fixed $k$, the family of Grassmannians $\{ \operatorname{Gr}(k,n): n\in \mathbb{N}, \; n \ge k \}$ together with the inclusion maps $\iota_{nm}:\operatorname{Gr}(k,n)\to \operatorname{Gr}(k,m)$ for $m > n$ form a direct system. The *infinite Grassmannian* of $k$-planes is defined to be the direct limit of this system in the category of topological spaces and denoted by $$\operatorname{Gr}(k,\infty)\coloneqq \varinjlim\operatorname{Gr}(k,n).$$ Those unfamiliar with the notion of direct limits may simply take $$\operatorname{Gr}(k, \infty) = \bigcup\nolimits_{n=k}^\infty \operatorname{Gr}(k,n),$$ where we regard $\operatorname{Gr}(k,n) \subset \operatorname{Gr}(k,n+1)$ by identifying $\operatorname{Gr}(k,n)$ with $\iota_n\bigl(\operatorname{Gr}(k,n)\bigr)$. With this identification, we no longer need to distinguish between $\mathbf{A} \in \operatorname{Gr}(k,n)$ and its image $\iota_{n}(\mathbf{A})\in \operatorname{Gr}(k,n+1)$ and may regard $\mathbf{A} \in \operatorname{Gr}(k,m)$ for all $m > n$. We now define a distance $d_{\operatorname{Gr}(k, \infty)}$ on $\operatorname{Gr}(k,\infty)$ that is consistent with the Grassmann distance on $\operatorname{Gr}(k,n)$ for all $n$ sufficiently large. \[lem:infty\] The natural inclusion $\iota_n:\operatorname{Gr}(k,n)\to \operatorname{Gr}(k,n+1)$ is isometric, i.e., $$\label{eq:iso} d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=d_{\operatorname{Gr}(k,n+1)}\bigl(\iota_n(\mathbf{A}),\iota_n(\mathbf{B})\bigr).$$ Repeated applications of yields $$\label{eq:iso1} d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=d_{\operatorname{Gr}(k,m)}\bigl(\iota_{nm}(\mathbf{A}),\iota_{nm}(\mathbf{B})\bigr)$$ for all $m > n$ and if we identify $\operatorname{Gr}(k,n)$ with $\iota_n\bigl(\operatorname{Gr}(k,n)\bigr)$, we may rewrite as $$d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=d_{\operatorname{Gr}(k,m)}(\mathbf{A},\mathbf{B})$$ for all $m > n$. If $a\in \mathbb{R}^n$, we write $\hat{a} = \left[\begin{smallmatrix} a \\ 0 \end{smallmatrix} \right]\in \mathbb{R}^{n+1}$. Let $A = [a_1,\dots, a_k]$ and $B = [b_1,\dots,b_k]$ be any orthonormal bases of $\mathbf{A}$ and $\mathbf{B}$ respectively. By the definition of $\iota_n$, $\iota_n(\mathbf{A})$ is the subspace in $\mathbb{R}^{n+1}$ spanned by an orthonormal basis that we will denote by $\iota_n(A) \coloneqq [\hat{a}_1,\dots, \hat{a}_k] \in \mathbb{R}^{(n+1) \times k}$. Hence we have $$\iota_n(A)^\mathsf{T} \iota_n(B) = \begin{bmatrix} A^\mathsf{T} B \\ 0 \end{bmatrix}.$$ By the expression for Grassmann distance in , we see that must hold. Since the inclusion of $\operatorname{Gr}(k,n)$ in $\operatorname{Gr}(k,n+1)$ is isometric, a geodesic in $\operatorname{Gr}(k,n)$ remains a geodesic in $\operatorname{Gr}(k,n+1)$. Given $\mathbf{A},\mathbf{B} \in \operatorname{Gr}(k, \infty)$, there must exist some $n$ sufficiently large so that both $\mathbf{A},\mathbf{B} \in \operatorname{Gr}(k, n)$ and in which case we define the distance between $\mathbf{A}$ and $\mathbf{B}$ in $ \operatorname{Gr}(k, \infty)$ to be $$d_{\operatorname{Gr}(k, \infty)}(\mathbf{A},\mathbf{B}) \coloneqq d_{\operatorname{Gr}(k, n)}(\mathbf{A},\mathbf{B}).$$ By Lemma \[lem:infty\], this value is independent of our choice of $n$ and is the same for all $m \ge n$. In particular, $d_{\operatorname{Gr}(k, \infty)}$ is well-defined and yields a distance on $\operatorname{Gr}(k,\infty)$. We summarize these observations below. \[cor:infty\] The Grassmann distance between two $k$-planes in $\operatorname{Gr}(k,n)$ is the geodesic distance in $\operatorname{Gr}(k,\infty)$ and is therefore independent of $n$. Also, the expression for a distance minimizing geodesic in $\operatorname{Gr}(k,n)$ extends to $\operatorname{Gr}(k,\infty)$. Lemma \[lem:infty\] also holds for other distances on $\operatorname{Gr}(k,n)$ in Table \[tab:distances\], allowing us to define them on $\operatorname{Gr}(k, \infty)$. \[lem:inclusion\] For all $m > n$, the inclusion $\iota_{nm}:\operatorname{Gr}(k,n)\to \operatorname{Gr}(k,m)$ is isometric when $\operatorname{Gr}(k,n)$ and $\operatorname{Gr}(k,m)$ are both equipped with the same distance in Table \[tab:distances\], i.e., $$d^*_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=d^*_{\operatorname{Gr}(k,m)}(\iota_{nm}(\mathbf{A}),\iota_{nm}\bigl(\mathbf{B})\bigr),$$ $* = \alpha, \beta, \kappa, \phi, \mu, \rho, \pi, \sigma$. Consequently $d^{\ast}_{\operatorname{Gr}(k,\infty)}$ is well-defined. $d^{\ast}_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})$ and $d^{\ast}_{\operatorname{Gr}(k,n+1)}\bigl(\iota_n(\mathbf{A}),\iota_n\bigl(\mathbf{B})\bigr)$ depend only on the principal angles between $\mathbf{A}$ and $\mathbf{B}$, so the distance remains unchanged under $\iota_n$. Repeated applications to $\iota_n \circ \iota_{n+1} \circ \dots \circ \iota_{m-1} = \iota_{nm}$ yield the required isometry. Distances between subspaces of different dimensions {#sec:main} =================================================== We now address our main problem. The proposed notion of distance will be that of a point $x \in X$ to a set $S \subset X$ in a metric space $(X,d)$. Recall that this is defined by $d(x, S) \coloneqq \inf\{d(x,y) : y \in S \}$. For us, $X$ is a Grassmannian, therefore compact, and so $d(x,S)$ is finite. Also, $S$ will be a closed subset and so we write $\min$ instead of $\inf$. We will introduce two possible candidates for $S$. \[def:Omega\] Let $k,l, n \in \mathbb{N}$ be such that $k\le l \le n$. For any $\mathbf{A}\in \operatorname{Gr}(k,n)$ and $\mathbf{B}\in \operatorname{Gr}(l,n)$, we define the subsets $$\Omega_{+}(\mathbf{A})\coloneqq\bigl\{\mathbf{X}\in \operatorname{Gr}(l,n) : \mathbf{A}\subseteq \mathbf{X}\bigr\},\quad \Omega_{-}(\mathbf{B})\coloneqq\bigl\{\mathbf{Y}\in \operatorname{Gr}(k,n) : \mathbf{Y} \subseteq \mathbf{B}\bigr\}.$$ We will call $\Omega_{+}(\mathbf{A})$ the *Schubert variety of $l$-planes containing $\mathbf{A}$* and $\Omega_{-}(\mathbf{B})$ the *Schubert variety of $k$-planes contained in $\mathbf{B}$*. As we will see in Section \[sec:schubert\], $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are indeed Schubert varieties and therefore closed subsets of $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively. Furthermore, $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are uniquely determined by $\mathbf{A}$ and $\mathbf{B}$ (see Proposition \[prop:Omega1\]) and may be regarded as ‘sub-Grassmannians’ of $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively (see Proposition \[prop:OmegaGrass\]). How could one define the distance between a subspace $\mathbf{A}$ of dimension $k$ and a subspace $\mathbf{B}$ of dimension $l$ in $\mathbb{R}^n$ when $k \ne l$? We may assume $k < l\le n$ without loss of generality. In which case a very natural solution is to define the required distance $\delta(\mathbf{A}, \mathbf{B}) $ as that between the $k$-plane $\mathbf{A}$ and the closest $k$-plane $\mathbf{Y}$ contained in $\mathbf{B}$, measured within $\operatorname{Gr}(k,n)$. In other words, we want the Grassmann distance from $\mathbf{A}$ to the closed subset $\Omega_{-}(\mathbf{B})$, $$\label{eq:minus} \delta(\mathbf{A}, \mathbf{B}) \coloneqq d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A}, \Omega_{-}\bigl(\mathbf{B})\bigr) = \min\bigl\{ d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{Y}) : \mathbf{Y}\in \Omega_{-}(\mathbf{B})\bigr\}.$$ This has the advantage of being intrinsic — the distance $\delta(\mathbf{A}, \mathbf{B}) $ is measured in $d_{\operatorname{Gr}(k,n)}$ and is defined wholly within $\operatorname{Gr}(k,n)$ without any embedding of $\operatorname{Gr}(k,n)$ into an arbitrary ambient space. Furthermore, by the property of $d_{\operatorname{Gr}(k,n)}$ in Corollary \[cor:infty\], $\delta(\mathbf{A}, \mathbf{B}) $ does not depend on $n$ and takes the same value for any $m \ge n$. We illustrate this in Figure \[fig:sphere\]: The sphere is intended to be a depiction of $\operatorname{Gr}(1,3)$ though to be accurate antipodal points on the sphere should be identified. (-4,0) circle\[radius=0.05\]; at (-4.08,0) [$\mathbf{A}$]{}; (0,4) arc (90:270:2cm and 4cm); (0,4) arc (90:-90:2cm and 4cm); (0,0) circle (4cm); (0,0) circle (4cm); at (3,-3) [$\operatorname{Gr}(1,3)$]{}; at (0,2.5) [$\Omega_{-}(\mathbf{B})$]{}; at (-3,-1.25) [$\gamma$]{}; (-4,0) arc (180:243:4cm and 2cm); (-1.8,-1.8) circle\[radius=0.05\]; at (-2,-1.9) [$\mathbf{X}$]{}; However, it is equally natural to define $\delta(\mathbf{A}, \mathbf{B}) $ as the distance between the $l$-plane $\mathbf{B}$ and the closest $l$-plane $\mathbf{Y}$ containing $\mathbf{A}$, measured within $\operatorname{Gr}(l,n)$. In other words, we could have instead defined it as the Grassmann distance from $\mathbf{B}$ to the closed subset $\Omega_{+}(\mathbf{A})$, $$\label{eq:plus} \delta(\mathbf{A}, \mathbf{B}) \coloneqq d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B}, \Omega_{+}\bigl(\mathbf{A})\bigr) = \min\bigl\{ d_{\operatorname{Gr}(l,n)}(\mathbf{B},\mathbf{X}) : \mathbf{X}\in \Omega_{+}(\mathbf{A})\bigr\}.$$ It will have the same desirable features as the one in except that the distance is now measured in $d_{\operatorname{Gr}(l,n)}$ and within $\operatorname{Gr}(l,n)$. It turns out that the two values in and are equal, allowing us to define $\delta(\mathbf{A}, \mathbf{B})$ as their common value. We will establish this equality and the properties of $\delta(\mathbf{A}, \mathbf{B})$ in the remainder of this section. The results are summarized in Theorem \[thm1\]. Our proof is constructive: In addition to showing the equality of and , it shows how one may explicitly find the closest points on Schubert varieties $\mathbf{X} \in \Omega_{-}(\mathbf{B})$ and $\mathbf{Y} \in \Omega_{+}(\mathbf{A})$ to any given point in the respective Grassmannians. \[thm1\] Let $\mathbf{A}$ be a subspace of dimension $k$ and $\mathbf{B}$ be a subspace of dimension $l$ in $\mathbb{R}^n$. Suppose $k \le l \le n$. Then $$\label{eq:equal} d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A}, \Omega_{-}\bigl(\mathbf{B})\bigr) = d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B}, \Omega_{+}\bigl(\mathbf{A})\bigr).$$ Their common value defines a distance $\delta(\mathbf{A}, \mathbf{B}) $ between the two subspaces with the following properties: 1. \[indp\] $\delta(\mathbf{A}, \mathbf{B}) $ is independent of the dimension of the ambient space $n$ and is the same for all $n \ge l+1$; 2. \[reduce\] $\delta(\mathbf{A}, \mathbf{B}) $ reduces to the Grassmann distance between $\mathbf{A}$ and $\mathbf{B}$ when $k = l$; 3. \[explicit\] $\delta(\mathbf{A}, \mathbf{B}) $ may be computed explicitly as $$\label{eq:grassdist2} \delta(\mathbf{A},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^{\min \{k,l\}}\theta_i(\mathbf{A},\mathbf{B})^2 \Bigr)^{1/2}$$ where $\theta_i(\mathbf{A},\mathbf{B})$ is the $i$th principal angle between $\mathbf{A}$ and $\mathbf{B}$, $i =1,\dots,\min(k,l)$. Rewriting as $$\min_{\mathbf{X}\in \Omega_{+}(\mathbf{A})}d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B}) = \min_{\mathbf{Y} \in \Omega_{-}(\mathbf{B})} d_{\operatorname{Gr}(k,n)}(\mathbf{Y},\mathbf{A}),$$ the equation says that the distance from $\mathbf{B}$ to the nearest $l$-dimensional subspace that contains $\mathbf{A}$ equals the distance from $\mathbf{A}$ to the nearest $k$-dimensional subspace contained in $\mathbf{B}$. This relation has several parallels. We will see that: 1. the Grassmann distance may be replaced by any of the distances in Table \[tab:distances\] (see Theorem \[thm:othermetrics\]); 2. ‘nearest’ may be replaced by ‘furthest’ and ‘min’ above replaced by ‘max’ when $n$ is sufficiently large (see Proposition \[prop:prob4\]); 3. ‘distance’ may be replaced by ‘volume’ with respect to the intrinsic uniform probability density on the Grassmannian (see Section \[sec:volume\]). We will prove Theorem \[thm1\] by way of the next two lemmas. \[lem:ineq1\] Let $k \le l \le n$ be positive integers. Let $\delta :\operatorname{Gr}(k,n)\times \operatorname{Gr}(l,n)\to [0,\infty)$ be the function defined by $$\delta(\mathbf{A},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^k\theta_i^2\Bigr)^{1/2}$$ where $\theta_i \coloneqq \theta_i(\mathbf{A},\mathbf{B})$, $i=1,\dots,k$. Then $$\delta(\mathbf{A},\mathbf{B})\ge d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr).$$ It suffices to find an $\mathbf{X}\in \Omega_{+}(\mathbf{A})$ such that $\delta(\mathbf{A},\mathbf{B})=d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B})$. Let $(p_1,q_1),\dots, (p_k,q_k)$ be the principal vectors between $\mathbf{A}$ and $\mathbf{B}$. We will extend $q_1,\dots,q_k$ into an orthonormal basis of $\mathbf{B}$ by appending appropriate orthonormal vectors $q_{k+1},\dots,q_l$. The principal angles are given by $\theta_i=\cos^{-1} p_i^\mathsf{T} q_i$, $\lVert p_i \rVert =\lVert q_i \rVert=1$. If we take $\mathbf{X} \in \operatorname{Gr}(l,n)$ to be the subspace spanned by $p_1,\dots,p_k,q_{k+1},\dots, q_l$, then $$\begin{aligned} \label{eq:ineq1} d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B}) &= [(\cos^{-1} p_1^\mathsf{T} q_1)^2 + \dots + (\cos^{-1} p_k^\mathsf{T} q_k)^2 \nonumber \\ &\qquad + (\cos^{-1} q_{k+1}^\mathsf{T} q_{k+1})^2 + \dots + (\cos^{-1} q_l^\mathsf{T} q_l)^2 ]^{1/2}\\ &= [ \theta_1^2 + \dots + \theta_k^2 + 0^2 + \dots + 0^2]^{1/2} = \delta(\mathbf{A},\mathbf{B}). \nonumber \end{aligned}$$ We state the following well-known fact [@Horn Corollary 3.1.3] for easy reference and deduce a corollary that will be useful for Lemma \[lem:ineq2\]. \[prop:compare\] Let $k\le l\le n$ be positive integers. Suppose $B \in \mathbb{R}^{n\times l}$ and $B_k \in \mathbb{R}^{n\times k}$ is a submatrix obtained by removing any $l-k$ columns from $B$. Then the $i$th singular values satisfy $\sigma_i(B_k)\le \sigma_i(B)$ for $i=1,\dots, k$. \[cor:compare\] Let $B$ and $B_k$ be as in Proposition \[prop:compare\] and $\mathbf{B}$ and $\mathbf{B}_k$ be subspaces of $\mathbb{R}^n$ spanned by the column vectors of $B$ and $B_k$ respectively. Then for any subspace $\mathbf{A}$ of $\mathbb{R}^n$, the principal angles between the respective subspaces satisfy $$\theta_i(\mathbf{A},\mathbf{B})\le \theta_i(\mathbf{A},\mathbf{B}_k)$$ for $i = 1,\dots,\min(\dim \mathbf{A}, \dim \mathbf{B}_k)$. By appropriate orthogonalization if necessary, we may assume that $B$ and its submatrix $B_k$ are orthonormal bases of $\mathbf{B}$ and $\mathbf{B}_k$. Let $A$ be an orthonormal basis of $\mathbf{A}$. Then $\sigma_i(A^\mathsf{T}B)$ and $\sigma_i(A^\mathsf{T}B_k)$ take values in $[0,1]$. Since $\theta_i(\mathbf{A},\mathbf{B})= \cos^{-1}(\sigma_i\bigl(A^\mathsf{T}B)\bigr)$ and $\cos^{-1}$ is monotone decreasing in $[0,1]$, the result follows from $\sigma_i(A^\mathsf{T}B)\ge \sigma_i(A^\mathsf{T}B_k)$, by Proposition \[prop:compare\] applied to the submatrix $A^\mathsf{T}B_k$ of $A^\mathsf{T}B$. \[lem:ineq2\] Let $\mathbf{A}$, $\mathbf{B}$ be as in Lemma \[lem:ineq1\]. Then $\delta(\mathbf{A}, \mathbf{B})\le d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}(\mathbf{B})\bigr)$. Let $\mathbf{Y} \in \Omega_{-}(\mathbf{B})$. Then $\mathbf{Y}$ is a $k$-dimensional subspace contained in $\mathbf{B}$ and in the notation of Corollary \[cor:compare\], we may write $\mathbf{Y} = \mathbf{B}_{k}$. By the same corollary we get $\theta_i(\mathbf{A},\mathbf{B})\le \theta_i(\mathbf{A},\mathbf{Y})$ for $i = 1, \dots, k$. Hence $$\label{eq:ineq2} \delta(\mathbf{A},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^{k}\theta_i(\mathbf{A},\mathbf{B})^2\Bigr)^{1/2} \le \Bigl(\sum\nolimits_{i=1}^{k}\theta_i(\mathbf{A},\mathbf{Y})^2\Bigr)^{1/2}=d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{Y}).$$ The desired inequality follows since this holds for arbitrary $\mathbf{Y} \in \Omega_{-}(\mathbf{B})$. Recall that Grassmannians satisfy an isomorphism $$\operatorname{Gr}(k,n)\cong \operatorname{Gr}(n-k,n)$$ that takes a $k$-plane $\mathbf{Y}$ to the $(n-k)$-plane $\mathbf{Y}^{\perp}$ of linear forms vanishing on $\mathbf{Y}$. It is easy to see that this isomorphism is an isometry. Using this isometric isomorphism, together with Lemma \[lem:ineq1\] and Lemma \[lem:ineq2\], we can immediately deduce that $$\delta(\mathbf{A}, \mathbf{B}) \le d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}\bigl(\mathbf{B})\bigr)=d_{\operatorname{Gr}(n-k,n)}\bigl(\mathbf{A^{\perp}},\Omega_{+}\bigl(\mathbf{B}^{\perp})\bigr)\le \delta(\mathbf{A}^{\perp},\mathbf{B}^{\perp}).$$ On the other hand, by results in [@Knyazev], we have $\delta(\mathbf{A}, \mathbf{B}) = \delta(\mathbf{A}^{\perp},\mathbf{B}^{\perp})$ and hence $$\delta(\mathbf{A}, \mathbf{B}) = d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}(\mathbf{B})\bigr).$$ Similarly we can obtain $$\delta(\mathbf{A}, \mathbf{B}) = d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr).$$ Hence we have the required equalities and in Theorem \[thm1\]. Property  is obvious from and Property  follows from Lemma \[lem:infty\]. The proof of Lemma \[lem:ineq1\] provides a simple way to find a point $\mathbf{X} \in \Omega_{+}(\mathbf{A})$ that realizes the distance $d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr)=\delta(\mathbf{A},\mathbf{B})$. Similarly we may explicitly determine a point $\mathbf{Y} \in \Omega_{-}(\mathbf{B})$ that realizes the distance $d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}(\mathbf{B})\bigr)=\delta(\mathbf{A},\mathbf{B})$. One might wonder whether or not Theorem \[thm1\] still holds if we replace $d_{\operatorname{Gr}(k,n)}$ by other distance functions described in Table \[tab:distances\]. The answer is yes. \[thm:othermetrics\] Let $k \le l \le n$. Let $\mathbf{A}\in \operatorname{Gr}(k,n)$ and $\mathbf{B}\in\operatorname{Gr}(l,n)$. Then $$d^*_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A}, \Omega_{-}(\mathbf{B})\bigr) = d^*_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B}, \Omega_{+}(\mathbf{A})\bigr),$$ for $* = \alpha, \beta, \kappa, \phi, \mu, \rho, \pi, \sigma$. Their common value $\delta^*(\mathbf{A},\mathbf{B})$ is given by: $$\begin{aligned} \delta^{\alpha}(\mathbf{A}, \mathbf{B}) &= \theta_k, & \delta^{\beta}(\mathbf{A},\mathbf{B}) &= \Bigl(1 - \prod\nolimits_{i=1}^k\cos^2\theta_i\Bigr)^{1/2},\\ \delta^{\kappa}(\mathbf{A}, \mathbf{B}) &= \Bigl(\sum\nolimits_{i=1}^k\sin^2\theta_i\Bigr)^{1/2}, & \delta^{\phi}(\mathbf{A}, \mathbf{B}) &= \cos^{-1}\bigl(\prod\nolimits_{i=1}^k\cos \theta_i\Bigr),\\ \delta^{\mu}(\mathbf{A}, \mathbf{B}) &= \Bigl( \log \prod\nolimits_{i=1}^k\frac{1}{\cos^2 \theta_i}\Bigr)^{1/2}, & \delta^{\rho}(\mathbf{A}, \mathbf{B}) &= \Bigl(2\sum\nolimits_{i=1}^k\sin^2(\theta_i/2)\Bigr)^{1/2},\\ \delta^{\pi}(\mathbf{A}, \mathbf{B}) &= \sin \theta_k, & \delta^{\sigma}(\mathbf{A}, \mathbf{B}) &= 2\sin (\theta_k/2),\end{aligned}$$ or more generally with $\min(k,l)$ in place of the index $k$ when we do not require $k \le l$. This follows from observing that our proof of Theorem \[thm1\] only involves principal angles between $\mathbf{A}$ and $\mathbf{B}$ and the diffeomorphism between $\operatorname{Gr}(k,n)$ and $\operatorname{Gr}(n-k,n)$ remains an isometry under these distances. In particular, both and still hold with any of these distances in place of the Grassmann distance. We will see in Section \[sec:existing\] that the projection distance $\delta^\pi$ in Theorem \[thm:othermetrics\] is equivalent to the containment gap, a measure of distance between subspaces of different dimensions originally proposed in operator theory [@Kato]. Grassmannian of subspaces of all dimensions {#sec:dinfty} =========================================== We view the equality of $d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}(\mathbf{B})\bigr)$ and $d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr)$ as the strongest evidence that their common value $\delta(\mathbf{A}, \mathbf{B})$ provides the most natural notion of distance between subspaces of different dimensions. As we pointed out earlier, $\delta$ is a distance in the sense of a distance from a point to a set, but not a distance in the sense of a metric on the set of all subspaces of all dimensions. For instance, $\delta$ does not satisfy the separation property: $\delta(\mathbf{A}, \mathbf{B}) =0$ for any $\mathbf{A} \subsetneq \mathbf{B}$. In fact, it is easy to observe the following. \[lem:zero\] Let $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$. Then $\delta(\mathbf{A},\mathbf{B}) = 0$ iff $\mathbf{A} \subseteq \mathbf{B}$ or $\mathbf{B} \subseteq \mathbf{A}$. $\delta$ also does not satisfy the triangle inequality: For a line $\mathbf{L}$ not contained in a subspace $\mathbf{A}$, the triangle inequality, if true, would imply $$\begin{aligned} \delta(\mathbf{L},\mathbf{A})&=\delta(\mathbf{L},\mathbf{A})+\delta(\mathbf{A},\mathbf{B})\ge \delta(\mathbf{L},\mathbf{B}),\\ \delta(\mathbf{L},\mathbf{B})&=\delta(\mathbf{L},\mathbf{B})+\delta(\mathbf{A},\mathbf{B})\ge \delta(\mathbf{L},\mathbf{A}),\end{aligned}$$ giving $\delta(\mathbf{L},\mathbf{A})=\delta(\mathbf{L},\mathbf{B})$ for any subspace $\mathbf{B}$, which is evidently false by Lemma \[lem:zero\] (e.g. take $\mathbf{B} = \mathbf{A} \oplus \mathbf{L}$). These observations also apply verbatim to all the other similarly-defined distances $\delta^*$ in Theorem \[thm:othermetrics\], i.e., none of them are metrics. The set of all subspaces of all dimensions is parameterized by $\operatorname{Gr}(\infty, \infty)$, the *doubly infinite Grassmannian* [@FH], which may be viewed informally as the disjoint union of all $k$-dimensional subspaces[^2] over all $k \in \mathbb{N}$, $$\operatorname{Gr}(\infty, \infty)=\coprod\nolimits_{k=1}^{\infty} \operatorname{Gr}(k,\infty).$$ To define a metric between any pair of subspaces of arbitrary dimensions is to define one on $\operatorname{Gr}(\infty,\infty)$. It is easy to define metrics on $\operatorname{Gr}(\infty,\infty)$ that bear little relation to the geometry of Grassmannian but we will propose one in Section \[sec:metric\] that is consistent with $\delta$ and with $d_{\operatorname{Gr}(k,n)}$ for all $k \le n$. We will require the formal definition of $\operatorname{Gr}(\infty, \infty)$, namely, it is the direct limit of the direct system of Grassmannians $\{ \operatorname{Gr}(k,n) : (k,n) \in \mathbb{N} \times \mathbb{N} \}$ with inclusion maps $\iota^{kl}_{nm} : \operatorname{Gr}(k,n) \to \operatorname{Gr}(l,m)$ for all $k \le l$ and $n \le m$ such that $l-k\le m-n$. For $A \in \mathbb{R}^{n \times k}$ with orthonormal columns, the embedding is given by $$\label{eq:epsilon} \iota^{kl}_{nm} :\operatorname{Gr}(k,n) \to \operatorname{Gr}(l,m), \qquad \operatorname{span} (A) \mapsto \operatorname{span} \left( \begin{bmatrix} A & 0\\ 0 & 0\\ 0 & I_{l-k} \end{bmatrix}\right),$$ where $I_{l-k} \in \mathbb{R}^{(l -k) \times (l -k)}$ is an identity matrix and we have $(m-n) - (l -k)$ zero rows in the middle so that the $3 \times 2$ block matrix is in $\mathbb{R}^{m \times l}$. Note that for a fixed $k$, $\iota^{kk}_{nm} $ reduces to $\iota_{nm}$ in . Since our distance $\delta(\mathbf{A},\mathbf{B})$ is defined for subspaces $\mathbf{A}$ and $\mathbf{B}$ of all dimensions, it defines a function $\delta:\operatorname{Gr}(\infty,\infty)\times \operatorname{Gr}(\infty,\infty)\to \mathbb{R}$ that is a *premetric* on $\operatorname{Gr}(\infty,\infty)$, i.e., $\delta(\mathbf{A},\mathbf{B})\ge 0$ and $\delta(\mathbf{A},\mathbf{A})=0$ for all $\mathbf{A},\mathbf{B} \in \operatorname{Gr}(\infty,\infty)$. This in turn defines a topology $\tau $ on $\operatorname{Gr}(\infty,\infty)$ in a standard way: The $\varepsilon$-ball centered at $\mathbf{A}$ is $$B_\varepsilon(\mathbf{A})\coloneqq\{ \mathbf{X}\in \operatorname{Gr}(\infty,\infty) : \delta(\mathbf{A},\mathbf{X})< \varepsilon \},$$ and $U \subseteq \operatorname{Gr}(\infty,\infty)$ is defined to be open if for any $\mathbf{A}\in U$, there is an $\varepsilon$-ball $B_\varepsilon(\mathbf{A}) \subseteq U$. The topology $\tau$ is consistent with the usual topology of Grassmannians (but it is not the disjoint union topology). If we restrict $\tau$ to $\operatorname{Gr}(k,\infty)$, then the subspace topology is the same as the topology induced by the metric $d_{\operatorname{Gr}(k,\infty)}$ on $\operatorname{Gr}(k,\infty)$ as defined in Section \[sec:infty\]. Nevertheless this apparently natural topology on $\operatorname{Gr}(\infty,\infty)$ turns out to be a strange one. \[prop:tau\] The topology $\tau$ on $\operatorname{Gr}(\infty,\infty)$ is non-Hausdorff and therefore non-metrizable. $\tau $ is not Hausdorff since it is not possible to separate $\mathbf{A}\subsetneq \mathbf{B}$ by open subsets, as we saw in Lemma \[lem:zero\]. Metrizable spaces are necessarily Hausdorff. Even though $\tau$ restricts to the metric space topology on $\operatorname{Gr}(k,\infty)$ induced by the Grassmann distance $d_{\operatorname{Gr}(k,\infty)}$ for every $k \in \mathbb{N}$, it is not itself a metric space topology. We view this as a consequence of a more general phenomenon, namely, the category $\boldsymbol{\mathsf{Met}}$ of metric spaces (objects) and continuous contractions (morphisms) has no coproduct, i.e., given a collection of metric spaces, there is in general no metric space that will behave like the disjoint union of the collection of metric spaces. To see this, take metric spaces $(X_1,d_1)$ and $(X_2,d_2)$ where $X_1 = \{ x_1\}$, $X_2 = \{x_2\}$. Suppose a coproduct $(X,d)$ of $(X_1,d_1)$ and $(X_2,d_2)$ exists. Let $Y=\{y_1,y_2\}$ and let $d_Y$ be the metric on $Y$ induced by $d_Y(y_1,y_2)=2 d(x_1,x_2)\ne 0$. Now define $\varphi_i: X_i\to Y$ by $\varphi_i(x_i)=y_i$, $i=1,2$. One sees that no morphism $\varphi:X\to Y$ in $\boldsymbol{\mathsf{Met}}$ is compatible with $\varphi_1$ and $\varphi_2$, contradicting the assumption that $X$ is the coproduct of $X_1$ and $X_2$. If we instead look at the category of metric spaces with continuous or uniformly continuous maps as morphisms, then coproducts always exist [@Helemskii]. In Section \[sec:metric\], we will relax our requirement and construct a metric $d_{\operatorname{Gr}(\infty,\infty)}$ on $\operatorname{Gr}(\infty,\infty)$ that restricts to $d_{\operatorname{Gr}(k,\infty)}$ for all $k \in \mathbb{N}$ but without requiring that it comes from a coproduct of $\{(\operatorname{Gr}(k,\infty), d_{\operatorname{Gr}(k,\infty)}) : k \in \mathbb{N}\}$ in $\boldsymbol{\mathsf{Met}}$. Metrics for subspaces of all dimensions {#sec:metric} ======================================= We will describe a simple recipe for turning the distances $\delta^*$ in Theorem \[thm:othermetrics\] into metrics on $\operatorname{Gr}(\infty,\infty)$. Suppose $k \le l$ and we have $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$. In this case there are $k$ principal angles between $\mathbf{A}$ and $\mathbf{B}$, $\theta_1,\dots, \theta_k$, as defined in . First we will set $\theta_{k+1} = \dots = \theta_l = \pi/2$. Then we take the Grassmann distance $\delta$ or any of the distances $\delta^*$ in Theorem \[thm:othermetrics\], replace the index $k$ by $l$, and call the resulting expressions $d_{\operatorname{Gr}(\infty, \infty)}(\mathbf{A}, \mathbf{B})$ (for Grassmann distance) and $d^*_{\operatorname{Gr}(\infty, \infty)}(\mathbf{A}, \mathbf{B})$ (for other distances) respectively. When $n$ is sufficiently large, setting $\theta_{k+1},\dots,\theta_l$ all equal to $\pi/2$ is equivalent to completing $\mathbf{A}$ to an $l$ dimensional subspace of $\mathbb{R}^n$, by adding $l-k$ vectors orthonormal to the subspace $\mathbf{B}$. Hence the distance between $\mathbf{A}$ and $\mathbf{B}$ is defined by the distance function on the Grassmannian $\operatorname{Gr}(l,n)$. We show in Proposition \[prop:metric\] that these expressions will indeed define metrics on $\operatorname{Gr}(\infty,\infty)$. Applying the above recipe to the Grassmann, chordal, and Procrustes distances yield the *Grassmann*, *chordal*, and *Procrustes metrics* on $\operatorname{Gr}(\infty,\infty)$ given in Table \[tab:metrics\]. ------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------- Grassmann metric $d_{\operatorname{Gr}(\infty, \infty)}(\mathbf{A}, \mathbf{B}) = \Bigl(\lvert k - l \rvert\pi^2/4 + \sum\nolimits_{i=1}^{\min(k,l)}\theta_i^2\Bigr)^{1/2}$ Chordal metric $d_{\operatorname{Gr}(\infty, \infty)}^{\kappa}(\mathbf{A}, \mathbf{B}) = \Bigl(\lvert k - l \rvert + \sum\nolimits_{i=1}^{\min(k,l)} \sin^2\theta_i\Bigr)^{1/2}$ Procrustes metric $d_{\operatorname{Gr}(\infty, \infty)}^{\rho}(\mathbf{A}, \mathbf{B}) =\Bigl(\lvert k - l \rvert + 2\sum\nolimits_{i=1}^{\min(k,l)}\sin ^2(\theta_i/2)\Bigr)^{1/2}$ ------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------- Evidently the metrics in Table \[tab:metrics\] are all of the form $$\label{eq:rms} d^*_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B}) = \sqrt{\delta^*(\mathbf{A}, \mathbf{B})^2 + c_*^2 \epsilon(\mathbf{A},\mathbf{B})^2},$$ where $\epsilon(\mathbf{A},\mathbf{B}) \coloneqq \lvert \dim \mathbf{A} - \dim \mathbf{B} \rvert^{1/2}$. On the other hand, applying the above recipe to other distances in Table \[tab:distances\] yield the *Asimov*, *Binet–Cauchy*, *Fubini–Study*, *Martin*, *projection*, and *spectral metrics* on $\operatorname{Gr}(\infty,\infty)$ given by $$\label{eq:indicator} d_{\operatorname{Gr}(\infty, \infty)}^{*}(\mathbf{A}, \mathbf{B}) \\ = \begin{cases} d_{\operatorname{Gr}(k,\infty)}^* (\mathbf{A}, \mathbf{B}) & \text{if } \dim \mathbf{A} = \dim \mathbf{B} = k,\\ c_* & \text{if } \dim \mathbf{A} \ne \dim \mathbf{B}, \end{cases}$$ for $ * = \alpha, \beta, \phi, \mu, \pi, \sigma$, respectively. The constants $c_* > 0$ can be seen to be $$c = c_\alpha = \pi/2, \quad c_\sigma = \sqrt{2}, \quad c_\mu = \infty,\quad c_\beta = c_\phi = c_\pi = c_\kappa = c_\rho = 1.$$ In all cases, for subspaces $\mathbf{A}$ and $\mathbf{B}$ of equal dimension $k$, these metrics on $\operatorname{Gr}(\infty,\infty)$ restrict to the corresponding ones on $\operatorname{Gr}(k,\infty)$, i.e., $$d^*_{\operatorname{Gr}(\infty, \infty)}(\mathbf{A}, \mathbf{B}) = d^*_{\operatorname{Gr}(k, \infty)}(\mathbf{A}, \mathbf{B}),$$ where the latter is as described in Corollary \[cor:infty\] and Lemma \[lem:inclusion\]. These metrics on $\operatorname{Gr}(\infty,\infty)$ are the amalgamation of two pieces of information, the distance $\delta^*(\mathbf{A},\mathbf{B})$ and the difference in dimensions $\lvert \dim \mathbf{A} - \dim \mathbf{B} \rvert$, either via a root mean square or an indicator function. The Grassmann metric has a natural interpretation (see Proposition \[prop:prob4\]): > *$d_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B})$ is the distance from $\mathbf{B}$ to the furthest $l$-dimensional subspace that contains $\mathbf{A}$, which equals the distance from $\mathbf{A}$ to the furthest $k$-dimensional subspace contained in $\mathbf{B}$.* The chordal metric in Table \[tab:metrics\] is equivalent to the symmetric directional distance, a metric on subspaces of different dimensions [@SWF; @WWF] popular in machine learning [@bar; @ref2a; @motion; @ref2b; @eeg; @mech; @blog; @ref2c; @econ; @network; @face; @ref2d; @text] (see Section \[sec:existing\]). \[prop:metric\] The expressions in Table \[tab:metrics\] and are metrics on $\operatorname{Gr}(\infty,\infty)$. It is trivial to see that the expression defined in yields a metric on $\operatorname{Gr}(\infty,\infty)$ for $ * = \alpha, \beta, \mu, \pi, \sigma, \phi$, and so we just need to check the remaining three cases that take the form in . Of the four defining properties of a metric, only the triangle inequality is not immediately clear from . Let $k = \dim \mathbf{A}$, $l = \dim \mathbf{B}$, and $m = \dim \mathbf{C}$. We may assume <span style="font-variant:small-caps;">wlog</span> that $k\le l \le m \le n$ where $n$ is chosen sufficiently large so that $\mathbf{A}, \mathbf{B}, \mathbf{C}$ are subspaces in $\mathbb{R}^n$. Let $A \in \mathbb{R}^{n\times k}$, $B \in \mathbb{R}^{n \times l}$, $C \in \mathbb{R}^{n \times m}$ be matrices whose columns are orthonormal bases of $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ respectively. Consider the following $(n+m-k)\times m$ matrices: $$A'=\begin{bmatrix} A & 0\\ 0 & I_{m-k} \end{bmatrix},\quad B'=\begin{bmatrix} B & 0\\ 0 & 0\\ 0 & I_{m-l} \end{bmatrix},\quad C'=\begin{bmatrix} C\\ 0 \end{bmatrix}.$$ and set $\mathbf{A}' = \operatorname{span}(A')$, $\mathbf{B}' = \operatorname{span}(B')$, $\mathbf{C}' = \operatorname{span}(C')$; note that these are just $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ embedded in $\operatorname{Graff}(m,n+m-k)$ via . The expressions in Table \[tab:metrics\] satisfy $$\begin{aligned} d^*_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B}) = d^*_{\operatorname{Gr}(m,n+m-k)}(\mathbf{A}',\mathbf{B}'),\\ d^*_{\operatorname{Gr}(\infty,\infty)} (\mathbf{B},\mathbf{C}) = d^*_{\operatorname{Gr}(m,n+m-k)}(\mathbf{B}',\mathbf{C}'),\\ d^*_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{C}) = d^*_{\operatorname{Gr}(m,n+m-k)}(\mathbf{A}',\mathbf{C}').\end{aligned}$$ Since $\mathbf{A}',\mathbf{B}',\mathbf{C}' \in \operatorname{Gr}(m,n+m-k)$, the triangle inequality for $d^*_{\operatorname{Gr}(m,n+m-k)}$ immediately yields the triangle inequality for $d^*_{\operatorname{Gr}(\infty,\infty)}$. The proof shows that for any $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$ where $k \le l \le n$, $$d^*_{\operatorname{Gr}(\infty,\infty)}(\mathbf{A},\mathbf{B}) =d^*_{\operatorname{Gr}(l,n+l-k)}\bigl(\iota^{k,l}_{n,n+l-k}(\mathbf{A}),\iota^{l,l}_{n,n+l-k}(\mathbf{B})\bigr).$$ The embeddings $\iota^{k,l}_{n,n+l-k}:\operatorname{Gr}(k,n)\to \operatorname{Gr}(l,n+l-k)$ and $\iota^{l,l}_{n,n+l-k}:\operatorname{Gr}(l,n)\to \operatorname{Gr}(l,n+l-k)$ are as defined in and are isometric for all $ k\le l \le n$. \[prop:prob4\] Let $k\le l \le n/2$ and $\mathbf{A} \in \operatorname{Gr}(k,n)$, $\mathbf{B}\in \operatorname{Gr}(l,n)$. Then $$\label{eq:equal2} \max_{\mathbf{X}\in \Omega_{+}(\mathbf{A})}d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B}) = \max_{\mathbf{Y} \in \Omega_{-}(\mathbf{B})} d_{\operatorname{Gr}(k,n)}(\mathbf{Y},\mathbf{A}) = d_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B}),$$ i.e., $d_{\operatorname{Gr}(\infty,\infty)}$ is the distance between *furthest* subspaces. We assume <span style="font-variant:small-caps;">wlog</span> that $\mathbf{A}\cap \mathbf{B}=\{0 \}$ by Proposition \[prop:angles\]. Since $$d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B})=\delta(\mathbf{X},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^l \theta_i(\mathbf{X},\mathbf{B})^2\Bigr)^{1/2},$$ and by Corollary \[cor:compare\], $\theta_i(\mathbf{X},\mathbf{B})\le \theta_i(\mathbf{A},\mathbf{B})$, $i =1,\dots,k$, we obtain $$d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B})\le \Bigl(\delta(\mathbf{A},\mathbf{B})^2+\sum\nolimits_{i=k+1}^l\theta_i(\mathbf{X},\mathbf{B})^2\Bigr)^{1/2}.$$ Let $(a_1,b_1),\dots, (a_k, b_k)$ be the principal vectors between $\mathbf{A}$ and $\mathbf{B}$. We extend $b_1,\dots, b_k$ to obtain an orthonormal basis $b_1,\dots, b_k,b_{k+1},\dots, b_l$ of $\mathbf{B}$. Let $\mathbf{X}\cap\mathbf{A}^\perp$ be the orthogonal complement of $\mathbf{A}$ in $\mathbf{X}$ and let $\mathbf{B}_0 \coloneqq\operatorname{span}\{b_{k+1},\dots, b_l\}$. Then $$\Bigl(\sum\nolimits_{i=k+1}^l\theta_i(\mathbf{X},\mathbf{B})^2 \Bigr)^{1/2} = \delta(\mathbf{X}\cap\mathbf{A}^\perp,\mathbf{B}_0),$$ and the last inequality becomes $$d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B}) \le \sqrt{\delta(\mathbf{A},\mathbf{B})^2+\delta(\mathbf{X}\cap\mathbf{A}^\perp,\mathbf{B}_0)^2}.$$ If $n\ge 2l$, then there exist $l-k$ vectors $c_1,\dots,c_{l-k}$ orthogonal to $\mathbf{A}$ and $\mathbf{B}$ simultaneously. Choosing $\mathbf{X} = \operatorname{span}\{a_1,\dots,a_k,c_1,\dots,c_{l-k}\}$, we attain the required maximum: $$d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B})=\sqrt{\delta(\mathbf{A},\mathbf{B})^2+(l-k)\pi^2/4} = d_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B}).$$ The second equality in follows from $d_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B}) = d_{\operatorname{Gr}(\infty,\infty)} (\mathbf{B},\mathbf{A})$, given that $d_{\operatorname{Gr}(\infty,\infty)}$ is a metric by Proposition \[prop:metric\]. The existence of the metrics $d^*_{\operatorname{Gr}(\infty,\infty)}$ as defined in and does not contradict our earlier discussion about the general nonexistence of coproduct in $\boldsymbol{\mathsf{Met}}$ as these metrics do not respect continuous *contractions*. Take the Grassmann metric on $\operatorname{Gr}(\infty,\infty)$ for instance. $(\operatorname{Gr}(\infty,\infty),d_{\operatorname{Gr}(\infty,\infty)})$ is an object of the category $\boldsymbol{\mathsf{Met}}$ but it is *not* the coproduct of $\{(\operatorname{Gr}(k,\infty), d_{\operatorname{Gr}(k,\infty)} ) : k \in \mathbb{N}\}$. Indeed, let $Y = \{y_1, y_2\}$ with metric defined by $d_Y(y_1, y_2) = 1$. Consider a family of maps $f_k:\operatorname{Gr}(k,\infty)\to Y$, $$f_k(\mathbf{A})= \begin{cases} y_1&\text{if }k=2,\\ y_2&\text{otherwise}. \end{cases}$$ Then $f_k$ is a continuous contraction between $\operatorname{Gr}(k,\infty)$ and $Y$. So $\{f_k :k \in \mathbb{N}\}$ is a family of morphisms in $\boldsymbol{\mathsf{Met}}$ compatible with $\{(\operatorname{Gr}(k,\infty), d_{\operatorname{Gr}(k,\infty)} ) : k \in \mathbb{N}\}$. If $(\operatorname{Gr}(\infty,\infty),d_{\operatorname{Gr}(\infty,\infty)})$ is the coproduct of this family, then there must be a continuous contraction $f:\operatorname{Gr}(\infty,\infty)\to Y$ such that $f\circ \iota_k=f_k$ with $\iota_k$ being the natural inclusion of $\operatorname{Gr}(k,\infty)$ into $\operatorname{Gr}(\infty,\infty)$. But taking $\mathbf{A}\in \operatorname{Gr}(2,\infty)$ and $\mathbf{B}\in \operatorname{Gr}(3,\infty)$, we see that $$d_{\operatorname{Gr}(\infty,\infty)}(\mathbf{A},\mathbf{B})\ge \frac{\pi}{2} > 1=d_Y\bigl(f(\mathbf{A}),f(\mathbf{B})\bigr),$$ contradicting the surmise that $f$ is a contraction. Similarly, one may show that $(\operatorname{Gr}(\infty,\infty),d_{\operatorname{Gr}(\infty,\infty)}^*)$ is not a coproduct in $\boldsymbol{\mathsf{Met}}$ for any $* = \alpha, \beta, \kappa, \mu, \pi, \rho, \sigma, \phi$. $(\operatorname{Gr}(\infty,\infty),d_{\operatorname{Gr}(\infty,\infty)})$ is also not the coproduct of $\{(\operatorname{Gr}(k,\infty), d_{\operatorname{Gr}(k,\infty)} ) : k \in \mathbb{N}\}$ in the category of metric spaces with continuous (or uniformly continuous) maps as morphisms. The coproduct in this category is simply $\operatorname{Gr}(\infty,\infty)$ with the metric induced by the disjoint union topology, which is too fine (in the sense of topology) to be interesting. In particular, such a metric is unrelated to the distance $\delta$. Comparison with existing works {#sec:existing} ============================== There are two existing proposals for a distance between subspaces of different dimensions — the containment gap and the symmetric directional distance. These turn out to be special cases of our distance in Section \[sec:main\] and our metric in Section \[sec:metric\]. Let $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$. The *containment gap* is defined as $$\gamma(\mathbf{A}, \mathbf{B}) \coloneqq \max_{a\in \mathbf{A}} \; \min_{b\in \mathbf{B}} \frac{\lVert a - b \rVert}{\lVert a \rVert}.$$ This was proposed in [@Kato pp. 197–199] and used in numerical linear algebra [@SS] for measuring separation between Krylov subspaces [@BES]. It is equivalent to our *projection distance* $\delta^\pi$ in Theorem \[thm:othermetrics\]. It was observed in [@BES p. 495] that $$\gamma(\mathbf{A}, \mathbf{B}) = \sin\bigl( \theta_k(\mathbf{A},\mathbf{Y}) \bigr)$$ where $\mathbf{Y} \in \Omega_{-}(\mathbf{B})$ is nearest to $\mathbf{A}$ in the projection distance $d^\pi_{\operatorname{Gr}(k,n)}$. By Theorem \[thm:othermetrics\], we deduce that it can also be realized as $$\gamma(\mathbf{A}, \mathbf{B}) = \sin\bigl( \theta_l(\mathbf{B},\mathbf{X}) \bigr)$$ where $\mathbf{X} \in \Omega_{+}(\mathbf{A})$ is nearest to $\mathbf{B}$ in the projection distance $d^\pi_{\operatorname{Gr}(l,n)}$, a fact about the containment gap that had not been observed before. Indeed, by Theorem \[thm:othermetrics\], we get $$\gamma(\mathbf{A}, \mathbf{B}) = \delta^\pi (\mathbf{A}, \mathbf{B})$$ for all $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$. The *symmetric directional distance* is defined as $$\label{eq:sdd} d_{\Delta}(\mathbf{A},\mathbf{B}) \coloneqq \Bigl( \max(k,l) - \sum\nolimits_{i,j=1}^{k,l} (a_i^\mathsf{T} b_j)^2 \Bigr)^{1/2}$$ where $A = [a_1,\dots,a_k]$ and $B = [b_1,\dots,b_l]$ are the respective orthonormal bases. This was proposed in [@SWF; @WWF], and is widely used [@bar; @ref2a; @motion; @ref2b; @eeg; @mech; @blog; @ref2c; @econ; @network; @face; @ref2d; @text]. The definition is equivalent to our *chordal metric* $d^\kappa_{\operatorname{Gr}(\infty,\infty)}$ in Table \[tab:metrics\], $$d_{\operatorname{Gr}(\infty,\infty)}^\kappa (\mathbf{A},\mathbf{B})^2 = \lvert k - l \rvert + \sum_{i=1}^{\min(k,l)} \sin^2 \theta_i = \max(k,l) - \sum_{i,j=1}^{k,l} (a_i^\mathsf{T} b_j)^{2} =d_\Delta (\mathbf{A}, \mathbf{B})^2,$$ since $ \lvert k - l \rvert = \max(k,l) - \min(k,l)$, and $$\sum\nolimits_{i,j=1}^{k,l} (a_i^\mathsf{T} b_j)^{2} = \lVert A^\mathsf{T} B \rVert_F^2 = \sum\nolimits_{i=1}^{\min(k,l)} \cos^2 \theta_i = \min(k,l) - \sum\nolimits_{i=1}^{\min(k,l)} \sin^2 \theta_i .$$ Geometry of $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ {#sec:schubert} ================================================================= Up to this point, $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$, as defined in Definition \[def:Omega\], are treated as mere subsets of $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively. We will see that $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ have rich geometric properties. Firstly, we will show that they are Schubert varieties, justifying their names. \[def:Schubert\] Let $\mathbf{X}_1\subset \mathbf{X}_2 \subset \dots \subset \mathbf{X}_k$ be a fixed flag in $\mathbb{R}^n$. The *Schubert variety* $\Omega(\mathbf{X}_1, \dots, \mathbf{X}_k , n)$ is the set of $k$-planes $\mathbf{Y}$ satisfying the Schubert conditions $\dim (\mathbf{Y}\cap \mathbf{X}_i)\ge i$, $i=1,\dots, k$, i.e., $$\Omega(\mathbf{X}_1, \dots, \mathbf{X}_k , n) = \{\mathbf{Y} \in \operatorname{Gr}(k,n): \dim (\mathbf{Y}\cap \mathbf{X}_i)\ge i, \; i=1,\dots, k \}.$$ Let $0 \eqqcolon k_0 < k_1<\cdots < k_{m+1} \coloneqq n$ be a sequence of increasing nonnegative integers. The associated *flag variety* is the set of flags satisfying the condition $\dim \mathbf{X}_i = k_i$, $i=0, 1,\dots, m+1$. We denote it by $\operatorname{Flag}(k_1,\dots, k_m, n)$, i.e., $$\{(\mathbf{X}_1,\dots,\mathbf{X}_{m}) \in \operatorname{Gr}(k_1,n) \times \dots \times \operatorname{Gr}(k_m,n) : \mathbf{X}_i\subset \mathbf{X}_{i+1}, \; i =1,\dots,m \}.$$ Observe that a Schubert variety depends on a specific increasing sequence of subspaces whereas a flag variety depends only on an increasing sequence of dimensions (of subspaces). Flag varieties may be viewed as a generalization of Grassmannians since if $m=1$, then $\operatorname{Flag}(k,n) = \operatorname{Gr}(k,n)$. Like Grassmannians, $\operatorname{Flag}(k_1,\dots, k_m, n)$ is a smooth manifold and sometimes called a *flag manifold*. The parallel goes further, $\operatorname{Flag}(k_1,\dots, k_m, n) $ is a homogeneous space, $$\label{eq:flag} \operatorname{Flag}(k_1,\dots, k_m, n) \cong \operatorname{O}(n)/\bigl(\operatorname{O}(d_1) \times \dots \times \operatorname{O}(d_{m+1})\bigr)$$ where $d_i = k_{i} - k_{i-1}$ for $i =1,\dots, m+1$, generalizing . Let $\mathbf{A}\in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$ with $k\le l$. Then $$\Omega_{+}(\mathbf{A}) = \Omega(\mathbf{A}_1, \dots, \mathbf{A}_l , n),\qquad \Omega_{-}(\mathbf{B}) = \Omega(\mathbf{B}_1, \dots, \mathbf{B}_{k} , n),$$ are Schubert varieties in $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively with the flags $$\begin{gathered} \{0\} \eqqcolon \mathbf{A}_0 \subset \mathbf{A}_1\subset \dots \subset \mathbf{A}_k\coloneqq\mathbf{A}\subset \mathbf{A}_{k+1}\dots \subset \mathbf{A}_{l}, \\ \{0\} \eqqcolon \mathbf{B}_0 \subset \mathbf{B}_1 \subset \dots \subset \mathbf{B}_{k} \coloneqq\mathbf{B}.\end{gathered}$$ where $\mathbf{A}_{k+i}$ is a subspace of $\mathbb{R}^n$ containing $\mathbf{A}$ of dimension $n-l+(k+i)$ for $1\le i\le l-k$. The isomorphism $\operatorname{Gr}(l,n) \cong \operatorname{Gr}(n-l,n)$ (resp. $\operatorname{Gr}(k,n) \cong \operatorname{Gr}(n-k,n)$) that sends $\mathbf{X}$ to $\mathbf{X}^{\perp}$ takes $\Omega_{+}(\mathbf{A})$ to $\Omega_{-}(\mathbf{A}^{\perp})$ (resp. $\Omega_{-}(\mathbf{B})$ to $\Omega_{+}(\mathbf{B}^{\perp})$). Thus $\Omega_{+}(\mathbf{A})$ (resp. $\Omega_{-}(\mathbf{B})$) may also be viewed as Schubert varieties in $\operatorname{Gr}(n-l,n)$ (resp. $\operatorname{Gr}(n-k,n)$). More importantly, this observation implies that $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$, despite superficial difference in their definitions, are essentially the same type of objects. \[prop:Omega\] For any $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$, we have $$\Omega_{+}(\mathbf{A}) \cong \Omega_{-}(\mathbf{A}^{\perp}) \quad\text{and}\quad \Omega_{-}(\mathbf{B}) \cong \Omega_{+}(\mathbf{B}^{\perp}).$$ Also, $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are uniquely determined by $\mathbf{A}$ and $\mathbf{B}$ respectively. \[prop:Omega1\] Let $\mathbf{A}, \mathbf{A}' \in \operatorname{Gr}(k,n)$ and $\mathbf{B},\mathbf{B}' \in \operatorname{Gr}(l,n)$. Then $$\begin{gathered} \Omega_{+}(\mathbf{A}) =\Omega_{+}(\mathbf{A'})\quad \text{if and only if} \quad \mathbf{A} = \mathbf{A}',\\ \Omega_{-}(\mathbf{B}) =\Omega_{-}(\mathbf{B}')\quad \text{if and only if} \quad \mathbf{B} = \mathbf{B}'.\end{gathered}$$ Suppose $\Omega_{+}(\mathbf{A})=\Omega_{+}(\mathbf{A'})$. Observe that the intersection of all $l$-planes containing $\mathbf{A}$ is exactly $\mathbf{A}$ and ditto for $\mathbf{A}'$. So $$\mathbf{A} = \bigcap\nolimits_{\mathbf{X} \in\Omega_{+}(\mathbf{A})} \mathbf{X} = \bigcap\nolimits_{\mathbf{X} \in\Omega_{+}(\mathbf{A}')} \mathbf{X} = \mathbf{A}'.$$ The converse is obvious. The statement for $\Omega_{-}$ then follows from Proposition \[prop:Omega\]. This observation allows us to treat subspaces of different dimensions on the same footing by regarding them as *subsets* in the same Grassmannian. If we have a collection of subspaces of dimensions $k\le k_1 < k_2 < \dots < k_m \le l$, the injective map $\mathbf{A} \mapsto \Omega_{+}(\mathbf{A})$ takes all of them into distinct subsets of $\operatorname{Gr}(l,n)$. Alternatively, the injective map $\mathbf{B} \mapsto \Omega_{-}(\mathbf{B})$ takes all of them into distinct subsets of $\operatorname{Gr}(k,n)$. The resemblance between $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ in Proposition \[prop:Omega\] goes further — we may view them as ‘sub-Grassmannians’. \[prop:OmegaGrass\] Let $k \le l \le n$ and $\mathbf{A} \in \operatorname{Gr}(k,n)$, $\mathbf{B} \in \operatorname{Gr}(l,n)$. Then $$\Omega_{+}(\mathbf{A}) \cong \operatorname{Gr}(l-k,n-k), \quad \Omega_{-}(\mathbf{B}) \cong \operatorname{Gr}(k,l),$$ isomorphic as algebraic varieties and diffeomorphic as smooth manifolds. Thus $$\dim \Omega_{+}(\mathbf{A})=(n-l)(l-k),\quad \dim \Omega_{-}(\mathbf{B}) = k(l-k).$$ The first isomorphism is the quotient map $\varphi: \Omega_{+}(\mathbf{A})\to \operatorname{Gr}_{l-k}(\mathbb{R}^n/\mathbf{A})$, $\mathbf{X} \mapsto \mathbf{X}/\mathbf{A} \subseteq \mathbb{R}^n/\mathbf{A}$, composed with the isomorphism $ \operatorname{Gr}_{l-k}(\mathbb{R}^n/\mathbf{A}) \cong \operatorname{Gr}(l-k,n-k)$. The second isomorphism is obtained by regarding a $k$-dimensional subspace $\mathbf{Y}$ of $\mathbb{R}^n$ in $\Omega_{-}(\mathbf{B}) $ as a $k$-dimensional subspace of $\mathbf{B}$, i.e., $\Omega_{-}(\mathbf{B}) = \operatorname{Gr}_k(\mathbf{B}) \cong \operatorname{Gr}(k,l)$. That $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are Grassmannians allows us to infer the following: 1. \[top\] as topological spaces, they are compact and path-connected; 2. as algebraic varieties, they are irreducible and nonsingular; 3. \[diff\] as differential manifolds, they are smooth and any two points on them can be connected by a length-minimizing geodesic. The topology in refers to the metric space topology, not Zariski topology. A consequence of compactness is that the distance $d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}(\mathbf{B})\bigr) = d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr)$ can be attained by points in $\Omega_{-}(\mathbf{B})$ and $\Omega_{+}(\mathbf{A})$ respectively. We constructed these closest points explicitly when we proved Theorem \[thm1\]. Many more topological and geometric properties of $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ follow from Proposition \[prop:OmegaGrass\] as they inherit everything that we know about Grassmannians (e.g. coordinate ring, cohomology ring, Plücker relations, etc.); in particular, $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are also flag varieties. The last property in requires a proof. The length-minimizing geodesic is not unique and so $\Omega_+(\mathbf{A})$ and $\Omega_-(\mathbf{B})$ are not *geodesically convex* [@Nicolaescu Definition 4.1.35]. Any two points in $\Omega_{-}(\mathbf{B})$ (resp. $\Omega_{+}(\mathbf{A})$) can be connected by a length-minimizing geodesic in $\operatorname{Gr}(k,n)$ (resp. $\operatorname{Gr}(l,n)$). By Proposition \[prop:Omega\], it suffices to show that any two points in $\Omega_{-}(\mathbf{B})$ can be connected by a geodesic curve in $\Omega_{-}(\mathbf{B})$. By Proposition \[prop:OmegaGrass\], $\Omega_{-}(\mathbf{B})$ is the image of $\operatorname{Gr}(k,l)$ embedded isometrically in $\operatorname{Gr}(k,n)$. So by Lemma \[lem:infty\], for any $\mathbf{X}_1,\mathbf{X}_2 \in \operatorname{Gr}(k,l)$, $d_{\operatorname{Gr}(k,n)}(\mathbf{X}_1,\mathbf{X}_2)= d_{\operatorname{Gr}(k,l)}(\mathbf{X}_1,\mathbf{X}_2) = d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$, where the last is the geodesic distance in $\Omega_{-}(\mathbf{B})$. Hence if $d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$ is realized by a geodesic curve $\gamma$ in $\Omega_{-}(\mathbf{B})$, then $\gamma$ must also be a geodesic curve in $\operatorname{Gr}(k,n)$. We have represented $\operatorname{Gr}(k,n)$ as a set of *equivalence classes* of matrices but it may also be represented as a set of *actual matrices* [@Nicolaescu Example 1.2.20], namely, idempotent symmetric matrices of trace $k$: $$\operatorname{Gr}(k,n)\cong \{ P \in \mathbb{R}^{n \times n} : P^{\mathsf{T}} = P^2 = P, \; \operatorname{tr}(P)=k \}.$$ The isomorphism maps each subspace $\mathbf{A}\in \operatorname{Gr}(k,n)$ to $P_{\mathbf{A}} \in \mathbb{R}^{n \times n}$, the unique orthogonal projection onto $\mathbf{A}$, and its inverse takes an orthogonal projection $P$ to the subspace $\operatorname{im}( P) \in \operatorname{Gr}(k,n)$. $P$ is an orthogonal projection iff it is symmetric and idempotent, i.e., $P^{\mathsf{T}} = P^2 = P$. The eigenvalues of an orthogonal projection onto a subspace of dimension $k$ are $1$’s and $0$’s with multiplicities $k$ and $n-k$, so $\operatorname{tr}(P) = k$ is equivalent to $\operatorname{rank}(P) = k$, ensuring $\operatorname{im}(P)$ has dimension $k$. In this representation, $$\begin{aligned} \Omega_{+}(\mathbf{A}) &\cong \{ P \in \mathbb{R}^{n \times n} : P^{\mathsf{T}} = P^2 = P, \; \operatorname{tr}(P)=l, \; \operatorname{im}(A) \subseteq \operatorname{im}(P) \},\\ \Omega_{-}(\mathbf{B}) &\cong \{ P \in \mathbb{R}^{n \times n} : P^{\mathsf{T}} = P^2 = P, \; \operatorname{tr}(P)=k, \; \operatorname{im}(P) \subseteq \operatorname{im}(B) \},\end{aligned}$$ allowing us to treat $\operatorname{Gr}(k,n)$, $\operatorname{Gr}(l,n)$, $\Omega_{+}(\mathbf{A})$, $\Omega_{-}(\mathbf{B})$ all as subvarieties of $\mathbb{R}^{n \times n}$. Probability density on the Grassmannian {#sec:volume} ======================================= We determine the relative volumes of $\Omega_{+}(\mathbf{A})$, $\Omega_{-}(\mathbf{B})$ and prove a volumetric analogue of in Theorem \[thm1\]: > *Given $k$-dimensional subspace $\mathbf{A}$ and $l$-dimensional subspace $\mathbf{B}$ in $\mathbb{R}^n$, the probability that a randomly chosen $l$-dimensional subspace in $\mathbb{R}^n$ contains $\mathbf{A}$ equals the probability that a randomly chosen $k$-dimensional subspace in $\mathbb{R}^n$ is contained in $\mathbf{B}$.* Every Riemannian metric on a Riemannian manifold yields a *volume density* that is consistent with the metric [@Nicolaescu Example 3.4.2]. The Riemannian metric[^3] on $\operatorname{Gr}(k,n)$ that gives us the Grassmann distance in and the geodesic in also gives a density $d\gamma_{k,n}$ on $\operatorname{Gr}(k,n)$. The volume of $\operatorname{Gr}(k,n)$ is then $$\label{eq:volume} \operatorname{Vol}\bigl(\operatorname{Gr}(k,n)\bigr)=\int_{\operatorname{Gr}(k,n)} |d\gamma_{k,n}| =\binom{n}{k}\frac{\prod_{j=1}^{n}{\omega_j}}{\bigl(\prod_{j=1}^k\omega_j \bigr)\bigl(\prod_{j=1}^{n-k}\omega_j\bigr)},$$ where $\omega_m \coloneqq \pi^{m/2}/\Gamma(1+m/2)$, volume of the unit ball in $\mathbb{R}^m$ [@Nicolaescu Proposition 9.1.12]. The normalized density $d\mu_{k,n} \coloneqq \operatorname{Vol}\bigl(\operatorname{Gr}(k,n)\bigr)^{-1} \lvert d\gamma_{k,n}\rvert$ defines a natural *uniform probability density* on $\operatorname{Gr}(k,n)$. With respect to this, the probability of landing on $\Omega_{+}(\mathbf{A})$ in $\operatorname{Gr}(l,n)$ equals the probability of landing on $\Omega_{-}(\mathbf{B})$ in $\operatorname{Gr}(k,n)$. \[cor:volume\] Let $k \le l \le n$ and $\mathbf{A} \in \operatorname{Gr}(k,n)$, $\mathbf{B} \in \operatorname{Gr}(l,n)$. The relative volumes of $\Omega_{+}(\mathbf{A})$ in $\operatorname{Gr}(l,n)$ and $\Omega_{-}(\mathbf{B})$ in $\operatorname{Gr}(k,n)$ are equal and their common value depends only on $k,l,n$, $$\mu_{l,n}\bigl(\Omega_{+}(\mathbf{A})\bigr) = \mu_{k,n}\bigl(\Omega_{-}(\mathbf{B})\bigr) = \frac{l!(n-k)!\prod_{j=l-k+1}^{l}{\omega_j}}{n!(l-k)!\prod_{j=n-k+1}^{n}\omega_j}.$$ By Proposition \[prop:OmegaGrass\], $\Omega_{+}(\mathbf{A})$ is isometric to $\operatorname{Gr}(n-l,n-k)$ and $\Omega_{-}(\mathbf{B})$ is isometric to $\operatorname{Gr}(k,l)$, so by their volumes are $$\binom{n-k}{n-l}\frac{\prod_{j=1}^{n-k}{\omega_j}}{\bigl(\prod_{j=1}^{n-l}\omega_j\bigr)\bigl(\prod_{j=1}^{l-k}\omega_j)}, \qquad \binom{l}{k}\frac{\prod_{j=1}^{l}{\omega_j}}{\bigl(\prod_{j=1}^k\omega_j\bigr)\bigl(\prod_{j=1}^{l-k}\omega_j)}$$ respectively. Now divide by the volumes of $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively. By definition, relative volume depends on the volume of ambient space and the dependence on $n$ is expected, a slight departure from Theorem \[thm1\]. Conclusions =========== We provided what we hope is a thorough study of subspace distances, a topic of wide-ranging interest. We investigated the topic from different angles and filled in the most glaring gap in our existing knowledge — defining distances and metrics for inequidimensional subspaces. We also developed simple geometric models for subspaces of all dimensions and enriched the existing differential geometric view of Grassmannians with algebraic geometric perspectives. We expect these to be of independent interest to applied and computational mathematicians. Most of the topics discussed in this article have been extended to affine subspaces in [@LWY]. Acknowledgment {#acknowledgment .unnumbered} -------------- We are very grateful to the two anonymous referees for their invaluable suggestions, both mathematical and stylistic. 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--- abstract: 'In this paper we study the synchronisation of three identical oscillators, i.e., clocks, hanging from the same hard support. We consider the case where each clock interacts with the other two clocks. The synchronisation is attained through the exchange of small impacts between each pair of oscillators. The fundamental result of this article is that the final locked state is at phase difference of $\frac{2\pi}{3}$ from successive clocks (clockwise or counter-clockwise). Moreover, the locked states attract a set whose closure is the global set of initial conditions. The methodology of our analysis consists in the construction a model, which is a non-linear discrete dynamical system, i.e. a non-linear difference equation. The results are extendable to any set of three oscillators under mutual symmetric interaction, despite the particular models of the oscillators.' author: - 'Emma D’Aniello and Henrique M. Oliveira' - Dipartimento di Matematica e Fisica - Università degli Studi della Campania Luigi Vanvitelli - 'Viale Lincoln n. 5 - 81100 Caserta, Italia' - | emma.daniello@unicampania.it\ Department of Mathematics\ Center for Mathematical Analysis, Geometry and Dynamical Systems,\ Instituto Superior Técnico, University of Lisbon,\ Av. Rovisco Pais, 1049-001, Lisboa, Portugal.\ holiv@math.tecnico.ulisboa.pt title: Huygens synchronisation of three clocks equidistant from each other --- Introduction ============ Synchronisation among oscillators with some form of coupling has been called universal [@strogatz2004] and is prevalent concept in Nature [@Pit]. In 1665 Huygens, the inventor of the pendulum clock, observed synchronisation between two pendulum clocks [@Huy] hanging from the same support. The first observation of Huygens was about the two clocks hanging in the same wall beam of his house when he was lying in bed with some indisposition. He observed both phase and phase opposition as a final state of the coupled system. The second observation was made later, when Huygens hung the two clocks on a board sitting on two chairs. The two systems observed by Huygens are quite different, therefore originating two lines of research completely separated. The later case has been studied in many papers [@Benn; @Col; @Frad; @Jova; @Col2; @Martens; @Oud; @Sen] by considering momentum conservation in the clocks-beam system, since the plank supporting the clocks is able to move. The system in that case is non-perturbative, there are three bodies that can move, i.e., three degrees of freedom: the pendula and the wood beam. The model is a classical mechanics paradigm, and in most of the works cited above the friction is considered viscous and not dry. The first case, when the pendulum clocks are suspended at a very rigid house beam, therefore not able to move, has been approached in the works [@Abr2; @Abr; @Vass]. In this model the interaction is considered perturbative. When one of the pendulums suffers the internal impact from the clock escape mechanism, a small travelling wave perturbs the second one and vice-versa. Recently, in [@OlMe] a theoretical model for this interaction was developed. In the same work, simulation and some experimental studies were carried on, that confirmed the validity of the proposed model when the support wall has infinite mass and, therefore, not moving. Naturally, the center of mass of the wall is not, in this case, a degree of freedom of the system and the coupling is, via very weak travelling waves, propagated in the rigid structure of the wall. In this article, we use as a conceptual starting point [@OlMe] presenting a mathematical model where the coupling is assumed to be attained through the exchange of impacts between three identical oscillators, where each one of the clocks interacts with the two other clocks. The model presents the advantage of being independent of the physical nature of the oscillators, and thus it can be used in other oscillator systems where synchronisation and phase locking has been observed [@Pit]. The ideas for the model presented here originated from the Andronov [@And; @OlMe] model of the phase-space limit cycle of isolated clocks, and assumes the exchange of single impacts (travelling solitons, for this system) between the oscillators at a specific point of the limit cycle. The fundamental hypotheses in this article are the existence of an asymptotically stable limit cycle for each oscillator and one very small interaction between each pair of clocks per cycle. We point out that in [@OlMe] the authors obtained phase opposition, which is in line with the original Huygens observations [@Huy]. In this paper we obtain a particularly symmetric asymptotic state at which all the clocks remain at a phase difference of $\frac{2\pi}{3}$ between each other. A natural step forward is to generalise the results obtained here to a larger set of oscillators and apply these results to bidimensional and tridimensional swarms of oscillators. The results of the present work can be used, namely, to study interacting insects or neuronal networks, that have been studied using the over-simplified integrate and fire models of Kuramoto [@Campbell1997; @Campbell1999; @Kuramoto1975; @Mirollo1990; @Strogatz2000]. The paper is organised in $5$ sections. In section $2$, we discuss the original model of the pendulum clock and we briefly recall the model for two identical clocks. In section $3$, we deduce the model for three identical clocks hanging at the same wall with mutual interactions. In section$\ 4$, we analyse the model, computing its symmetries and stabilities. In section $5 $, we draw conclusions and point out directions for future work. Model for the synchronisation of two oscillators ================================================ Some background --------------- For the sake of completeness we present here a very short theory of synchronisation for two oscillators exchanging small perturbations at each cycle. We consider identical oscillators. This theory can be applied to networks of identical oscillators, electronic oscillators and many other real world systems. We intend to consider, in future work, the case of slighly different oscillators, which give rise to regions of stability versus instability in the parameter space of these systems, i.e., Arnold Tongues [@boyland1986; @gilmore2011]. For basic, classical definitions and notions related to synchronisation, like phase and frequence, we follow and refer to [@Pit], and for concepts concerning general theory of dynamical systems like, for instance, limit cycle, we refer to [@arrowsmith1990]. In this paper, we always assume that an oscillator is a dynamical system having a limit cycle. We use the word clock when referring to a special type of oscillator described by the Andronov model [@OlMe]. Given a point $p_{0}$ in the limit cycle $\gamma$, the necessary time to return to $p_{0}$, after one round on the limit cycle, is the period $T_{0}$. A phase $\varphi$ is a real coordinate that describes the position of the representative point of the system on the limit cycle [@Nakao; @Pit]. Let $B_{\gamma}$ be the basin of attraction of the limit cycle. Consider the points outside the limit-cycle $\gamma$ but in $B_{\gamma}$. We extend the definition of phase to $B_{\gamma}$ as follows. We assign the same phase $\varphi$ to all points $p$ in $B_{\gamma}$ that converge to the same $p_{0}$ on the limit cycle $\gamma$ as $t\rightarrow\infty$, being the phase of $p_{0}$ precisely $\varphi$ [@gu1975]. The set of points $p$ that share the same phase is an *isochron curve*. If the oscillator’s states are on the same isochron at a given point in time, they continue to be on the same isochron in time [@gu1975; @Nakao]. When each clock suffers a perturbation, its state can go slightly off the limit cycle and generically jump to another isochron. Moreover, we assume that the limit cycles are structurally stable under small perturbations. When we consider two oscillators, $1$ and $2$, with orbits on the limit-cycle or sufficiently near the limit cycle, each one has a particular phase, respectively $\varphi$ and $\psi$. The study of the synchronisation of these two oscillators consists in establishing a dynamical system for the phase difference of the two oscillators. We have two possible lines of research [@Pit]. The first is to consider the phase difference along continuous time, i.e., to look at the function $\phi\left( t\right) =\psi\left( t\right) -\varphi\left( t\right) $ for $t\in\left[ 0,+\infty\right[ $. The second line of research, that we adopt in this paper, is to consider the phase difference $\phi_{n}=\psi_{n}% -\varphi_{n}$ taken at discrete instants $n=0,1,2,\ldots$. In this paper, we consider exclusively this last approach. There is phase synchronisation when the phase differences between the oscillators tend to a specific attractor. When this attractor is an isolated point, then there is phase locking. Naturally, richer coupled states can occur [@Martens]. The main goal for any theory of synchronisation is to obtain this phase difference dynamics and to establish the existence and nature of the attractor. In the case of Huygens observations, the attractor was the point $0$ or the point $\pi$ and the phase dynamics was unidimensional. The Andronov model for an isolated clock ---------------------------------------- We recall here the model for the sake of completeness of this article. Assuming that dry friction predominates in the internal metal pieces of the clock and the viscous damping is not predominant, using the angular coordinate $q$, the differential equation governing the isolated pendulum clock is$$\ddot{q}+\mu\text{ }\operatorname*{sign}\dot{q}+q=0,$$ where $\mu>0$ is the dry friction coefficient and $\operatorname*{sign}\left( x\right) $ is the classical function taking the value $-1$ at $x<0$ and $1$ at $x>0$. In [@And] it was considered that, in each cycle, the escape mechanism gives to the pendulum a fixed amount of normalized kinetic energy $\frac{h^{2}}{2}$ so to compensate the loss of kinetic energy occurred because of the dry friction in each complete cycle. This transfer of kinetic energy is called a *kick*. The origin is fixed so that the kick is given precisely when $q=-\mu$. The phase portrait is shown in Fig. $1$. \[h\] [limitcycle.eps]{} As in [@OlMe], with initial conditions $q\left( t=0\right) =-\mu$ and $\dot{q}\left( t=0\right) =v_{0}$, a Poincaré section [@Bir] is represented vol. II, page 268) as the half line $q=-\mu^{+}$ and $\dot{q}>0$ [@And]. The symbol $+$ means that we are considering that the section is taken immediately after the kick. Due to friction during a complete cycle, a loss of velocity $-4\mu$ occurs. By considering the velocity, $v_{n}=\dot {q}\left( 2n\pi^{+}\right) $, at the Poincaré section in each cycle, the non-linear discrete dynamical system [@And] is obtained $$v_{n+1}=\sqrt{\left( v_{n}-4\mu\right) ^{2}+h^{2}}\text{.} \label{recu}%$$ This equation has the asymptotically stable fixed point $$v_{f}=\frac{h^{2}}{8\mu}+2\mu\text{.}%$$ Any initial condition $v_{0}\in\left( 4\mu,+\infty\right) $ is attracted to $v_{f}$. Each cycle corresponds to a phase increment of $2\pi$ and the phase $\varphi$ is linear with respect to $t$, precisely $$\varphi=2\pi t.$$ As already mentioned, the nature of limit cycle is not of fundamental importance when we consider the interaction of three identical clocks, as we shall see in the sequel. We have presented here the basis of our reasonings in the non-usual case when the computations of the limit cycle are explicit and the usual angular phase is a linear function of $t$. Two interacting oscillators --------------------------- We present briefly the conclusions of considering two pendulum clocks suspended at the same wall, in a simplified version of [@OlMe], where the clocks are assumed to have natural angular frequencies near each other but different. Here, we assume that the two clocks have the same angular frequency. When one clock receives the kick, the impact propagates in the wall slightly perturbing the second clock. The perturbation is assumed to be instantaneous since the time of travel of sound in the wall between the clocks is assumed very small compared to the period. Consider two oscillators and index them by $i=1,2$. Each oscillator satisfies the differential equation $$\ddot{q}_{i}+\mu_{i}\text{ sign\ }\dot{q}_{i}+q_{i}=-\alpha_{i}\digamma\left( q_{j}\right) ,\text{ for }i,j=1,2\text{, }i\not =j\text{.} \label{coupledandro}%$$ As in the *Andronov model*, the kinetic energy of each oscillator increases of the fixed amount $h_{i}$ when $q_{i}=-\mu_{i}$. The coupling term is the normalised force $-\alpha_{i}\digamma\left( q_{j}\right) $, where $\digamma$ is the interaction function and $\alpha_{i}$ a constant with acceleration dimension. Following [@OlMe], the effect of the interaction function $\digamma$ is considered to produce an increment $-\alpha$ in the velocity of each clock, leaving the position invariant when the other is struck by the energy kick. The reader finds the detailed treatment in [@OlMe]. Here we only recall some ideas from that article, for the sake of completeness and to make our three clocks model more simple and natural to deal with. To describe and investigate the effect of the kicks, we construct a discrete dynamical system for the phase difference between the two clocks. We compute each cycle using as reference one of the clocks (the choice is irrelevant, since the model is symmetric). We choose, to fix ideas, clock 1 as the reference: whenever its phase reaches $0$ $\left( \operatorname{mod}% 2\pi\right) $, the number of cycles increases one unit from $n$ to $n+1$. If there exists an attracting fixed point for that dynamical system, the phase locking occurs. As in [@OlMe], the assumptions are the following. 1. Dry friction. 2. The pendulums have the same natural angular frequency $\omega=1$. 3. The perturbation in the momentum is always in the same vertical direction in the phase space [@Abr2; @Abr]. 4. Since the clocks have the same construction, the energy dissipated at each cycle of the two clocks is the same, $h_{1}=h_{2}=h$. The friction coefficient is the same for both clocks, $\mu_{1}=\mu_{2}=\mu$. 5. The perturbative interaction is instantaneous. This is a reasonable assumption, since in general the perturbation propagation time between the two clocks is several orders of magnitude lower than the periods. 6. The interaction is symmetric, the coupling has the same, very small, constant $\alpha$ when the clock $1$ acts on clock $2$, and conversely. We compute at this point the phase difference when clock $1$ returns to the initial position. The secular repetition of perturbations leads the system with the two clocks in phase opposition as Huygens observed in 1665 [@Huy]. The discrete dynamical model that we deduce from [@OlMe] for the phase difference between the two clocks $\phi_{n}=\psi_{n}-\varphi_{n}$ is the Adler equation [@adler1946study; @Pit]$$\phi_{n+1}=\phi_{n}+\varepsilon\sin\phi_{n}, \label{Perturb}%$$ with a very small constant $\varepsilon=\frac{16\mu\alpha}{h^{2}}$. In the interval $\left[ 0,2\pi\right[ $, there are two fixed points which are $\pi$ and $0$ respectively attracting and reppeling. Equation (\[Perturb\]) is the starting point from where we begin, in the present paper, the study the three symmetric clocks in mutual interaction. In any model with a perturbation of phase given by equation (\[Perturb\]) per cycle, i.e., Adler’s perturbation [@adler1946study; @Pit], despite being a physical clock (with Andronov model or any different model) or other type of oscillator, electric, quantic, electronic or biological, the theory presented here for three oscillators interacting by small impacts will be exactly the same, with the same conclusions. Model for three pendulum clocks placed in the three vertices of an equilateral triangle ======================================================================================= Hypotheses ---------- We consider three pendulum clocks suspended at the same wall, placed in the three vertices of an equilateral triangle, say the vertices are $A$, $B$, and $C$ and $B$ are the extreme points of the basis of the triangles. \[ptb\] [triangle.eps]{} This geometric setting is purely conceptual. Any set of three dynamical systems receiving symmetric impacts from the other two will have the same type of response of the clocks depicted in the three vertices of an equilateral triangle. Call the clocks placed in the three vertices $A$, $B$ and $C$, respectively, $O_{1}$, $O_{2}$ and $O_{3}$. When the clock $A$ receives the kick from the escape mechanism, the impact propagates in the wall slightly perturbing the other two clocks. As in [@OlMe], the perturbation is assumed to be instantaneous, since the time of travel of sound in the wall between the clocks is assumed very small compared to the period. As for the two clocks model discussed in [@OlMe], we make the following assumptions, now formulated for three clocks. 1. The system has [dry friction ]{}[@And]. 2. \[work3\][The pendulums of clocks ]{}$O_{1},$ $O_{2}$ and $O_{3}$ [have respectively natural angular frequencies ${\omega}_{1}={\omega}% _{2}={\omega}_{3}=1$.]{} 3. [The perturbation in the momentum is always in the same vertical direction in the phase space ]{} [@Abr2; @Abr]. 4. [The friction coefficient is the same for all the three clocks, ${\mu }_{1}={\mu}_{2}={\mu}_{3}={\mu}$. The energy dissipated at each cycle of the three clocks is the same, and the energy furnished by the escape mechanism to compensate the loss of energy to friction in each cycle is $h_{1}=h_{2}% =h_{3}=h$. ]{} 5. [The perturbative interaction is instantaneous. This is a reasonable assumption, since in general the perturbation propagation time between two clocks is several orders of magnitude lower than the periods [@OlMe].]{} 6. \[hyp6\][The interaction is symmetric. The couplings have the same constant $\alpha$ when one clock acts on another and conversely. In this model $\alpha$ is assumed to be very small.]{} 7. \[hyp67\][Each perturbation from clock $i$ to clock $j$ (where $i,j\in\{1,2,3\}$ with $i\not =j$), when clock $i$ suffers its internal impact of kinetic energy $h^{2}$, gives rise to a small perturbative change of phase which is in first order a $2\pi$-periodic differentiable odd function $P$ of the real variable $\phi$$$P\left( \phi\right) =\varepsilon\sin\phi\text{,} \label{Perturb1}%$$ where $\phi={\phi}_{ij}$ is the phase difference between clock $i$ and clock $j$.]{} The value of $\varepsilon$ is the above mentioned $\varepsilon=\frac {8\mu\alpha}{h^{2}}$ from [[@OlMe] where $\mu$ is the dry friction coefficient, $\frac{h^{2}}{2}$ is the kinetic energy furnished by the internal escape mechanism of each clock once per cycle and $\alpha$ the interaction coefficient between the clocks. The greater the $\alpha$ is, the greater the mutual influence among the clocks. In this paper, we do not need to particularize ]{}$\varepsilon$, since we are not interested in doing experimental computations. Therefore, we are interested in the fundamental result of symmetry between three oscillators subject to very weak mutual symmetric interaction. Most of the reasonings are independent on the form of the function $P\left( \phi\right) $, therefore we consider a general differentiable odd function of the real variable $\phi$, $P(\phi)$, for the development of the model, and consider it of the form (\[Perturb1\]) when we analyze the model in section 4. Observe that $|\sin(x+\varepsilon\sin y)-\sin x|<{\varepsilon}$ when $\varepsilon$[ is assumed to be sufficiently small. Therefore, we restrict our model to first order. ]{}We consider all the values of variables and constants in IS units. Construction of the model ------------------------- We now construct a dynamical system using as reference the phase of the clock in the vertex A (= clock $O_{1}$). This reference is arbitrary: any of the clocks can be used as the reference clock with the same results at the end, since the system is symmetric. We compute the effects of all phase differences and perturbations when the clock at A makes a complete cycle returning to the initial position. Without loss of generality, we consider the next working hypotheses. 1. \[work1\]The initial phase of clock at A at $t=0^{-}$ is zero, i.e., $\psi_{1}(0^{-})=0^{-}$, the minus ($-$) superscript means that at the instant $0^{-}$ clock $1$ is just about to receive the internal energy kick from its escape mechanism. 2. \[work2\]We consider that the initial phases of the three clocks are: $\psi_{3}(0^{-})=\psi_{3}^{0}>\psi_{2}(0^{-})=\psi_{2}^{0}>0^{-}=\psi _{1}(0^{-})=\psi_{1}^{0}$. 3. \[work4\]The perturbation satisfies the relation $P\left( x+Px\right) \simeq Px$ in first order. To obtain the desired model, we need to proceed through 6 steps, starting from the following initial conditions, that is the phase differences of all pairs of clocks. In the sequel ${\psi}_{i}^{j}$ denotes the phase of clock $O_{i}$ at the $j-th$ step. **INITIAL CONDITIONS** The phase difference between $O_{3}$ and $O_{1}$ is $$(CA)_{0}={\psi}_{3}^{0}-{\psi}_{1}^{0}={\psi}_{3}^{0},$$ and the phase difference between $O_{1}$ and $O_{3}$ is symmetric, in the sense that $$(AC)_{0}={\psi}_{1}^{0}-{\psi}_{3}^{0}=-{\psi}_{3}^{0}=-(CA)_{0}.$$ The phase difference between $O_{2}$ and $O_{1}$ is $$(BA)_{0}={\psi}_{2}^{0}-{\psi}_{1}^{0}={\psi}_{2}^{0}$$ and the phase difference between $O_{1}$ and $O_{2}$ is $$(AB)_{0}={\psi}_{1}^{0}-{\psi}_{2}^{0}=-{\psi}_{2}^{0}=-(BA)_{0}.$$ The phase difference between $O_{3}$ and $O_{2}$ is $$(CB)_{0}={\psi}_{3}^{0}-{\psi}_{2}^{0}$$ and the phase difference between $O_{2}$ and $O_{3}$ is $$(BC)_{0}={\psi}_{2}^{0}-{\psi}_{3}^{0}=-(CB)_{0}.$$ **STEPS LEADING TO THE CONSTRUCTION OF THE MODEL** **STEP 1:** first impact. Interactions of $O_{1}$ on $O_{2}$ and of $O_{1}$ on $O_{3}$, at $t=0$. When the system in position A attains phase $0$ $(\operatorname{mod}2\pi)$ it receives a sudden supply of energy, for short a kick, from its escape mechanism, this kick propagates in the common support of the three clocks and reaches the other two clocks. Now, the phase difference between $O_{3}$ and $O_{1}$ is corrected by the perturbative value $P$: $$\left( CA\right) _{I}=\left( CA\right) _{0}+P\left( \left( CA\right) _{0}\right) ={\psi}_{3}^{0}+P\left( {\psi}_{3}^{0}\right) .$$ The phase difference between $O_{1}$ and $O_{3}$ is $$\left( AC\right) _{I}=\left( AC\right) _{0}+P\left( \left( AC\right) _{0}\right) =-{\psi}_{3}^{0}+P\left( -{\psi}_{3}^{0}\right) =-(CA)_{I},$$ since $P$ must be an odd function of the mutual phase difference. The phase difference between $O_{2}$ and $O_{1}$ is $$\left( BA\right) _{I}=\left( BA\right) _{0}+P\left( \left( BA\right) _{0}\right) ={\psi}_{2}^{0}+P\left( {\psi}_{2}^{0}\right) ,$$ and the symmetric phase difference between $O_{1}$ and $O_{2}$ is $$\left( AB\right) _{I}=\left( AB\right) _{0}+P\left( \left( AB\right) _{0}\right) =-{\psi}_{2}^{0}+P\left( -{\psi}_{2}^{0}\right) )=-\left( BA\right) _{I}.$$ The phase difference between $O_{3}$ and $O_{2}$ depends on $\left( CA\right) _{I}$ and $\left( BA\right) _{I}$ and it is $$\left( CB\right) _{I}=\left( CA\right) _{I}-\left( BA\right) _{I}={\psi }_{3}^{0}-{\psi}_{2}^{0}+P({\psi}_{3}^{0})-P({\psi}_{2}^{0})=-(CA)_{I},$$ **STEP 2**: first natural time shift. The next clock to arrive at $2\pi^{-}$, from working hypothesis 3.2 (\[work2\]), is the clock ${O}_{3}$ at vertex $C$. The situation right before $O_{3}$ receives its kick of energy is when the phase of this clock is $2\pi^{-}$. At this point we have$$% \begin{cases} \psi_{3}^{2} & ={2\pi}^{-}\\ \psi_{1}^{2} & =2\pi-(CA)_{I}=2\pi+(AC)_{I}=2\pi-\left( {\psi}_{3}% ^{0}+P({\psi}_{3}^{0})\right) \\ \psi_{2}^{2} & =2\pi-\left( CB\right) _{I}=2\pi+\left( BC\right) _{I}% =2\pi+{\psi}_{2}^{0}-{\psi}_{3}^{0}+P({\psi}_{2}^{0})-P({\psi}_{3}^{0}). \end{cases}$$ **STEP 3**: second impact. Clock $O_{3}$ receives its internal kick, at the position $2\pi$. Now, we have$$% \begin{cases} \psi_{3}^{3} & ={2\pi}\\ \psi_{1}^{3} & =\psi_{1}^{2}+P(\psi_{1}^{2})\\ & =2\pi-\left( {\psi}_{3}^{0}+P({\psi}_{3}^{0})\right) +P\left( 2\pi-\left( {\psi}_{3}^{0}+P({\psi}_{3}^{0})\right) \right) \\ & =2\pi-\left( {\psi}_{3}^{0}+P({\psi}_{3}^{0})\right) -P\left( {\psi}% _{3}^{0}+P({\psi}_{3}^{0})\right) \\ & \simeq2\pi-{\psi}_{3}^{0}-2P\left( {\psi}_{3}^{0}\right) \\ \psi_{2}^{3} & =\psi_{2}^{2}+P(\psi_{2}^{2})\\ & =2\pi+{\psi}_{2}^{0}-{\psi}_{3}^{0}+P({\psi}_{2}^{0})-P({\psi}_{3}^{0})\\ & +P(2\pi+{\psi}_{2}^{0}-{\psi}_{3}^{0}+P({\psi}_{2}^{0})-P({\psi}_{3}^{0}))\\ & =2\pi+{\psi}_{2}^{0}-{\psi}_{3}^{0}+P({\psi}_{2}^{0})-P({\psi}_{3}^{0})\\ & +P({\psi}_{2}^{0}-{\psi}_{3}^{0}+P({\psi}_{2}^{0})-P({\psi}_{3}^{0}))\\ & \simeq2\pi+{\psi}_{2}^{0}-{\psi}_{3}^{0}+P\left( {\psi}_{2}^{0}\right) -P\left( {\psi}_{3}^{0}\right) +P\left( {\psi}_{2}^{0}-{\psi}_{3}% ^{0}\right) \\ & \end{cases}$$ **STEP 4**: second natural time shift. The next clock to arrive at $2\pi^{-}$, from working hypothesis3.2 (\[work2\]), is the clock $O_{2}$ at vertex $B$. The situation right before $O_{2}$ receives its kick of energy is when the phase of this clock is $2\pi^{-}$. Then we have $$% \begin{cases} \psi_{2}^{4} & =2\pi^{-}\\ \psi_{1}^{4} & =\psi_{1}^{3}+2\pi-\psi_{2}^{3}\\ & \simeq2\pi-{\psi}_{3}^{0}-2P\left( {\psi}_{3}^{0}\right) +2\pi\\ & -\left( 2\pi+{\psi}_{2}^{0}-{\psi}_{3}^{0}+P\left( {\psi}_{2}^{0}\right) -P\left( {\psi}_{3}^{0}\right) +P\left( {\psi}_{2}^{0}-{\psi}_{3}% ^{0}\right) \right) \\ & =2\pi-{\psi}_{2}^{0}-P\left( {\psi}_{2}^{0}\right) -P\left( {\psi}% _{3}^{0}\right) -P\left( {\psi}_{2}^{0}-{\psi}_{3}^{0}\right) \\ \psi_{3}^{4} & =\psi_{3}^{3}+2\pi-\psi_{2}^{3}\\ & \simeq2\pi+{2\pi}-\left( 2\pi+{\psi}_{2}^{0}-{\psi}_{3}^{0}+P\left( {\psi }_{2}^{0}\right) -P\left( {\psi}_{3}^{0}\right) +P\left( {\psi}_{2}% ^{0}-{\psi}_{3}^{0}\right) \right) \\ & \simeq2\pi-{\psi}_{2}^{0}+{\psi}_{3}^{0}-P\left( {\psi}_{2}^{0}\right) +P\left( {\psi}_{3}^{0}\right) -P\left( {\psi}_{2}^{0}-{\psi}_{3}% ^{0}\right) . \end{cases}$$ **STEP 5**: third impact. Clock $O_{2}$ receives its internal energy kick. It reaches the position $2\pi$. Then we have $$% \begin{cases} \psi_{2}^{5} & ={2\pi}\\ \psi_{3}^{5} & =\psi_{3}^{4} +P(\psi_{3}^{4})\\ & \simeq2\pi-{\psi}_{2}^{0}+{\psi}_{3}^{0}-P\left( {\psi}_{2}^{0}\right) +P\left( {\psi}_{3}^{0}\right) -P\left( {\psi}_{2}^{0}-{\psi}_{3}% ^{0}\right) \\ & +P(2\pi-{\psi}_{2}^{0}+{\psi}_{3}^{0}-P\left( {\psi}_{2}^{0}\right) +P\left( {\psi}_{3}^{0}\right) -P\left( {\psi}_{2}^{0}-{\psi}_{3}% ^{0}\right) )\\ & \simeq2\pi-{\psi}_{2}^{0}+{\psi}_{3}^{0}-P\left( {\psi}_{2}^{0}\right) +P\left( {\psi}_{3}^{0}\right) -P\left( {\psi}_{2}^{0}-{\psi}_{3}% ^{0}\right) -P({\psi}_{2}^{0}-{\psi}_{3}^{0})\\ & =2\pi-{\psi}_{2}^{0}+{\psi}_{3}^{0}-P\left( {\psi}_{2}^{0}\right) +P\left( {\psi}_{3}^{0}\right) -2P\left( {\psi}_{2}^{0}-{\psi}_{3}% ^{0}\right) \\ \psi_{1}^{5} & =\psi_{1}^{4}+P\left( \psi_{1}^{4} \right) \\ & \simeq2\pi-{\psi}_{2}^{0}-P\left( {\psi}_{2}^{0}\right) -P\left( {\psi }_{3}^{0}\right) -P\left( {\psi}_{2}^{0}-{\psi}_{3}^{0}\right) +\\ & P\left( 2\pi-{\psi}_{2}^{0}-P\left( {\psi}_{2}^{0}\right) -P\left( {\psi}_{3}^{0}\right) -P\left( {\psi}_{2}^{0}-{\psi}_{3}^{0}\right) \right) \\ & \simeq2\pi-{\psi}_{2}^{0}-P\left( {\psi}_{2}^{0}\right) -P\left( {\psi }_{3}^{0}\right) -P\left( {\psi}_{2}^{0}-{\psi}_{3}^{0}\right) -P({\psi }_{2}^{0})\\ & =2\pi-{\psi}_{2}^{0}-2P\left( {\psi}_{2}^{0}\right) -P\left( {\psi}% _{3}^{0}\right) -P\left( {\psi}_{2}^{0}-{\psi}_{3}^{0}\right) . \end{cases}$$ **STEP 6 (the final)**: third natural time shift. The next clock to arrive at $2\pi^{-}$, from working hypothesis 3.2 (\[work2\]), is the clock $O_{1}$ at vertex $A$. The situation before $O_{1}$ receives its kick of energy is when the phase of this clock is $2\pi^{-}$, i.e., the cycles is complete. At this point we are able to describe what happens to the phases after a complete cycle of the reference clock. We have $$% \begin{cases} \psi_{1}^{6} & ={2\pi}^{-}\\ \psi_{2}^{6} & =\psi_{2}^{5}+2\pi-\psi_{1}^{5}\\ & \simeq2\pi+2\pi-\left( 2\pi-{\psi}_{2}^{0}-2P\left( {\psi}_{2}^{0}\right) -P\left( {\psi}_{3}^{0}\right) -P\left( {\psi}_{2}^{0}-{\psi}_{3}% ^{0}\right) \right) \\ & =2\pi+{\psi}_{2}^{0}+2P\left( {\psi}_{2}^{0}\right) +P\left( {\psi}% _{3}^{0}\right) +P\left( {\psi}_{2}^{0}-{\psi}_{3}^{0}\right) ;\\ \psi_{3}^{6} & =\psi_{3}^{5}+2\pi-\psi_{1}^{5}\\ & \simeq2\pi-{\psi}_{2}^{0}+{\psi}_{3}^{0}-P\left( {\psi}_{2}^{0}\right) +P\left( {\psi}_{3}^{0}\right) -2P\left( {\psi}_{2}^{0}-{\psi}_{3}% ^{0}\right) +2\pi\\ & -\left( 2\pi-{\psi}_{2}^{0}-2P\left( {\psi}_{2}^{0}\right) -P\left( {\psi}_{3}^{0}\right) -P\left( {\psi}_{2}^{0}-{\psi}_{3}^{0}\right) \right) \\ & =2\pi+{\psi}_{3}^{0}+P({\psi}_{2}^{0})+2P({\psi}_{3}^{0})-P({\psi}_{2}% ^{0}-{\psi}_{3}^{0}). \end{cases}$$ Now, we compute the phase differences after the first cycle of $O_{1}$. We have $$\begin{aligned} (BA)_{I} & =-(AB)_{I}=\psi_{2}^{6}-\psi_{1}^{6}\\ & \simeq2\pi+{\psi}_{2}^{0}+2P\left( {\psi}_{2}^{0}\right) +P\left( {\psi }_{3}^{0}\right) +P\left( {\psi}_{2}^{0}-{\psi}_{3}^{0}\right) -2\pi\\ & ={\psi}_{2}^{0}+2P\left( {\psi}_{2}^{0}\right) +P\left( {\psi}_{3}% ^{0}\right) +P\left( {\psi}_{2}^{0}-{\psi}_{3}^{0}\right) \\ & =(BA)_{0}+2P((BA)_{0})+P((CA)_{0})+P((BA)_{0}-(CA)_{0})\\ &\end{aligned}$$ and$$\begin{aligned} & (CA)_{I}\\ & =-(AC)_{I}=\psi_{3}^{6}-\psi_{1}^{6}\\ & =2\pi+{\psi}_{3}^{0}+P({\psi}_{2}^{0})+2P({\psi}_{3}^{0})-P({\psi}_{2}% ^{0}-{\psi}_{3}^{0})-2\pi\\ & ={\psi}_{3}^{0}+P({\psi}_{2}^{0})+2P({\psi}_{3}^{0})-P({\psi}_{2}^{0}% -{\psi}_{3}^{0})\\ & =({(CA)}_{0})+P({(BA)}_{0})+2P({(CA)}_{0})-P((BA)_{0}-(CA)_{0})\\ &\end{aligned}$$ Hence, if we set $x=BA$ and $y=CA$, we obtain the system $$\left\{ \begin{array} [c]{c}% x_{1}=x_{0}+2P(x_{0})+P(y_{0})+P(x_{0}-y_{0})\\ y_{1}=x_{0}+P(x_{0})+2P({y}_{0})-P(x_{0}-y_{0}). \end{array} \right.$$ **THE MODEL** By iterating the argument above, we get, for $n$ equal to the number of cycles described by $O_{1}$, the discrete dynamical system:$$\left\{ \begin{array} [c]{c}% x_{n+1}=x_{n}+2P(x_{n})+P(y_{n})+P(x_{n}-y_{n})\\ y_{n+1}=y_{n}+P(x_{n})+2P({y}_{n})-P(x_{n}-y_{n}). \end{array} \right.$$ If we write $$\left\{ \begin{array} [c]{c}% \varepsilon\varphi\left( x,y\right) =2P(x)+P(y)+P(x-y)\\ \varepsilon\gamma\left( x,y\right) =P(x)+2P({y})+P(y-x), \end{array} \right.$$ then we have $$\varphi\left( x,y\right) =\gamma\left( y,x\right) \text{,}%$$ and the iteration is a perturbation of the identity as $$\left[ \begin{array} [c]{c}% x_{n+1}\\ y_{n+1}% \end{array} \right] =\left[ \begin{array} [c]{cc}% 1 & 0\\ 0 & 1 \end{array} \right] \left[ \begin{array} [c]{c}% x_{n}\\ y_{n}% \end{array} \right] +\varepsilon\left[ \begin{array} [c]{c}% \varphi(x_{n},y_{n})\\ \varphi(y_{n},x_{n}) \end{array} \right] ,$$ that we can also write as $$X_{n+1}=F(X_{n})=X_{n}+\varepsilon\Omega(X_{n}), \label{Model}%$$ where $$X_{n+1}=\left[ \begin{array} [c]{c}% x_{n+1}\\ y_{n+1}% \end{array} \right] ,$$ $$% \begin{array} [c]{c}% F(X_{n}) \end{array} =\left[ \begin{array} [c]{cc}% 1 & 0\\ 0 & 1 \end{array} \right] \left[ \begin{array} [c]{c}% x_{n}\\ y_{n}, \end{array} \right]$$ and $$% \begin{array} [c]{c}% \Omega(X_{n}) \end{array} =\left[ \begin{array} [c]{c}% \varphi(x_{n},y_{n})\\ \varphi(y_{n},x_{n}) \end{array} \right] .$$ We now consider $P\left( x\right) =\varepsilon\sin x$, where $\varepsilon =\frac{\alpha\mu}{8h^{2}}$ from hypothesis \[Perturb1\], explicitly,$$\begin{aligned} \varphi\left( x,y\right) & =2\sin x+\sin y+\sin\left( x-y\right) \\ \gamma\left( x,y\right) & =\sin x+2\sin y+\sin\left( y-x\right) .\\ &\end{aligned}$$ Analysis of the model ===================== Fixed points and local stability -------------------------------- In this section, we analyze the model (\[Model\]) obtained in the previous section. In a nutshell, in this section, we see that the system is differentiable and invertible in $S=\left[ 0,2\pi\right] \times\left[ 0,2\pi\right] $ when $\varepsilon>0$ is small. The perturbation map $\Omega\left( x,y\right) $ is periodic in $% \mathbb{R} ^{2}$. This implies that the solution of the problem in the set $S$ is a dynamical system and not the usual semi-dynamical system associated with discrete time. That will provide a reasonable simple structure to the problem of the stability of fixed points and will enable to derive global properties. Moreover, we prove that for small $\varepsilon$ the set $S$ is invariant for the dynamics of $F$, meaning that the two phase differences of oscillators $O_{2}$ and $O_{3}$ relative to oscillator $O_{1}$ stay in the interval $\left[ 0,2\pi\right[ $. In particular, the map $\Omega$ has the zeros $\left( \pi,\pi\right) $, $\left( \frac{2}{3} \pi,\frac{4}{3} \pi\right) $ and $\left( \frac{4}{3} \pi,\frac{2}{3} \pi\right) $ in the interior of the set $S=\left[ 0,2\pi\right] \times\left[ 0,2\pi\right] $, which are fixed points of the model $F$. There are also four trivial fixed points, $\left( 0,0\right) $, $\left( 0,2\pi\right) $, $\left( 2\pi,0\right) $ and $\left( 2\pi ,2\pi\right) $ at the corners of $S$, and the four fixed points $\left( 0,\pi\right) $, $\left( \pi,2\pi\right) $, $\left( 2\pi,\pi\right) $ and $\left( \pi,0\right) $ on the edges of $S$. We now compute the Jacobian matrix $J\left( x,y\right) $ to establish the dynamical nature of the fixed points in the usual way. We have $$J\left( x,y\right) =\left[ \begin{array} [c]{cc}% 1 & 0\\ 0 & 1 \end{array} \right] + \varepsilon\left[ \begin{array} [c]{cc}% 2\cos x+\cos\left( x-y\right) & -\cos\left( x-y\right) +\ \cos y\\ \cos x-\cos\left( x-y\right) & \cos\left( x-y\right) +2\cos y \end{array} \right] . \label{Jacobmatrix}%$$ We first consider the fixed points of $F$ in the interior of $S$. We start with $\left( \frac{2}{3} \pi,\frac{4}{3} \pi\right) $ and $\left( \frac {4}{3} \pi,\frac{2}{3} \pi\right) $. The Jacobian is exactly the same $$\left[ \begin{array} [c]{cc}% 1-3\frac{\sqrt{3}}{2}\varepsilon & 0\\ 0 & 1-3\frac{\sqrt{3}}{2}\varepsilon \end{array} \right] ,$$ meaning that those two points are locally asymptotically stable for $\varepsilon$ sufficiently small. The Jacobian matrix of $F$ at $\left( \pi\text{,}\pi\right) $ is $$\left[ \begin{array} [c]{cc}% 1-\varepsilon & -2\varepsilon\\ -2\varepsilon & 1-\varepsilon \end{array} \right] ,$$ with eigenvalues $1-3\varepsilon$ and $1+\varepsilon$, which qualifies $\left( \pi\text{,}\pi\right) $ as a saddle point. The stable manifold has direction $\left( 1,1\right) $, and the unstable manifold is tangent at $\left( \pi\text{,}\pi\right) $ to the vector $\left( -1,1\right) $. We now consider now the points placed at the vertexes of $S$. The Jacobian matrix of $F$ at $\left( 0\text{,}0\right) $, $\left( 0,2\pi\right) $, $\left( 2\pi,0\right) $ and $\left( 2\pi,2\pi\right) $ is, for all of them, the following $$\left[ \begin{array} [c]{cc}% 1+3\varepsilon & 0\\ 0 & 1+3\varepsilon \end{array} \right] ,$$ which qualifies all the vertexes of $S$ as a repellers. On the vertical edges of $S$ we have the fixed points $\left( 0,\pi\right) $, and $\left( 2\pi,\pi\right) $, at which the Jacobian matrix of $F$ is $$\left[ \begin{array} [c]{cc}% 1+\varepsilon & 0\\ 2 \varepsilon & 1-3 \varepsilon \end{array} \right] ,$$ which qualifies $\left( 0,\pi\right) $, and $\left( 2\pi,\pi\right) $ as saddle points. The stable manifold has the direction of the $y$ axis and the unstable manifold is tangent at $\left( 0\text{,}\pi\right) $ and $\left( 2\pi,\pi\right) $ to the vector $\left( 2,1\right) $. Finally, at the horizontal edges of $S$ we have the Jacobian matrix of $F$ at $\left( \pi,0\right) $, and $\left( \pi,2\pi\right) $ $$\left[ \begin{array} [c]{cc}% 1-3 \varepsilon & 2 \varepsilon\\ 0 & 1+ \varepsilon \end{array} \right] ,$$ which qualifies $\left( \pi,0\right) $ and $\left( \pi,2\pi\right) $ again as saddle points. The stable manifold is the direction of the $x$ axis and the unstable manifold is tangent at $\left( \pi,0\right) $ and $\left( \pi ,2\pi\right) $ to the vector $\left( 1,2\right) $. The local analysis of the fixed points of $F$ reveals a very symmetric picture. When $\varepsilon>0$ is small ($0<\varepsilon<{\varepsilon}_{0}% =\frac{1}{9}$ is good enough), $F$ is a small perturbation of the identity, $F(\partial{S})=\partial{S}$, the restriction of $F$ to the boundary of $S$, $\partial{S}$, is a bijection (see section 4 for more details), and the Jacobian determinant of $F$ is never null in the interior of $S$. Therefore, $F$ is invertible on S. Heteroclinic connections and invariant sets ------------------------------------------- We focus our attention on the existence of invariant subsets of $S$ for the dynamics of $F$. Additionally, we below prove that $S$ is itself an invariant set for the dynamics of $F$. Recall that an *heteroclinic* (sometimes called a heteroclinic connection, or heteroclinic orbit) is a path in phase space which joins two different equilibrium points. In the sequel, by *sa-heteroclinic*, *rs-heteroclinic*, and *ra-heteroclinic*, we mean an heteroclinic orbit connecting a saddle point to an attractor, an heteroclinic orbit connecting a repeller to a saddle point, and an heteroclinic orbit connecting a repeller to an attractor, respectively. Let $F$ be our model map in some set $T$ with two fixed points $p$ and $q$. Let $M_{u}\left( F,p\right) $ and $M_{s}\left( F,q\right) $ be the stable manifold and the unstable manifold ([@AlSaYo]: pages 78, 403) of the fixed points $p$ and $q$, respectively. Then, if by $M$ we denote the heteroclinic connecting $p$ and $q$, we have $$M\subseteq M_{s}\left( F,p\right) \cap M_{u}\left( F,q\right) .$$ In particular, $M$ is invariant, the $\alpha$-limit and $\omega$-limit sets of the points of $M$ is respectively $p$ and $q$ ([@AlSaYo]: page 331). The other orbits, i.e., with initial conditions not in $M$, cannot cross the heteroclinic connections when the map $F$ is invertible. In that case, it would be violated the injectivity of the map. In the sequel, we study the heteroclinics that connect saddle points to the attractors. Those heteroclinics determine the nature of all the flow of the dynamical system in the plane, due to the invertible nature of $F$. ### Vertical heteroclinics Consider the two vertical lateral edges of $S$, $s_{0}$ and $s_{1}$ that are the sets $s_{k}=\left\{ \left( x,y\right) \in S:\left( x=2k\pi\right) \wedge0\leq y\leq2\pi\right\} $, $k=0,1$. Consider the image of these segments under $F$. If we write $F=(F_{1},F_{2})$, then $$% \begin{cases} F_{1}\left( 2k\pi,y\right) & =2k\pi+\varepsilon\sin y+\varepsilon\sin\left( -y\right) =2k\pi\\ F_{2}\left( 2k\pi,y\right) & =y+2\varepsilon\sin y+\varepsilon\sin y=y+3\varepsilon\sin y, \end{cases}$$ meaning that for $\varepsilon$ small enough the edges $s_{k}$, $k=0,1$, are invariant, as already mentioned in section 2. Because of the initial conditions, on each of the edges $s_{k}$, $k=0,1$, the dynamics is given by$$% \begin{cases} x_{n+1} & =2k\pi,\\ y_{n+1} & =y_{n}+3\varepsilon\sin y_{n}\text{,}% \end{cases} .$$ For $\varepsilon<\frac{1}{9}$, the map $g:\left[ 0,2\pi\right] \rightarrow\left[ 0,2\pi\right] $ defined by $g(t)=t+3\varepsilon\sin t$ is a homeomorphism from the interval $\left[ 0,2\pi\right] $ into itself, as we can see in figure \[homeo\]. Moreover, since there is an attracting fixed point of this map at $\pi$, the dynamics in the sets $s_{0}$ and $s_{1}$ can be splitted in two subsets where the dynamics is again invariant, which is not very important for our global discussion but establishes that the stable manifolds of the saddle points $\left( 0,\pi\right) $ and $\left( 2\pi ,\pi\right) $ are, exactly and respectively, the sets $s_{0}$ and $s_{1}$ \[ptb\] [Fig3.eps]{} We have just shown that both $s_{0}$ and $s_{1}$ contain two heteroclinic connections: in $s_{0}$, the line segment $s_{0}^{-}$ from $\left( 0,0\right) $ to $\left( 0,\pi\right) $ and $s_{0}^{+}$ from $\left( 0,2\pi\right) $ to $\left( 0,\pi\right) $; and in $s_{1}$, the line segment $s_{0}^{-}$ from $\left( 2\pi,0\right) $ to $\left( 2\pi,\pi\right) $ and $s_{0}^{+}$ from $\left( 2\pi,2\pi\right) $ to $\left( 2\pi,\pi\right) $. The total number of vertical $rs$-heteroclines is $4.$ ### Horizontal heteroclinics Consider the two horizontal top and bottom edges of $S$, $r_{0}$ and $r_{1}$, that are the sets $r_{k}=\left\{ \left( x,y\right) \in S:0\leq x\leq 2\pi\wedge\left( y=2k\pi\right) \right\} $, $k=0,1$. Consider the image of these segments under $F$. As before, if we write $F=(F_{1},F_{2})$, then $$% \begin{cases} F_{1}\left( x,2k\pi\right) & =x+3\varepsilon\sin x\\ F_{2}\left( x,2k\pi\right) & =2k\pi, \end{cases}$$ meaning that, for $\varepsilon$ small enough, the edges $r_{k}$, $k=0,1$, are invariant. Because of the initial conditions, on each of the edges $s_{k}$, $k=0,1$, the dynamics is given by$$% \begin{cases} x_{n+1} & =x_{n}+3\varepsilon\sin x_{n}\\ y_{n+1} & =2k\pi\text{.}% \end{cases}$$ For $\varepsilon<\frac{1}{9}$, the map $g:\left[ 0,2\pi\right] \rightarrow\left[ 0,2\pi\right] $, defined by $g(t)=t+3\varepsilon\sin t$, is the same occurred before, now involved in the dynamics in the invariant edges $r_{0}$ and $r_{1}$. The stable manifolds of $\left( \pi,0\right) $ and $\left( \pi,2\pi\right) $ are again, respectively, the edges $r_{0}$ and $r_{1}$. Arguing as for $s_{0}$ and $s_{1}$, we have that both, $r_{0}$ and $r_{1}$, contain two analogous heteroclinic connections. We have just proved, in detail, that the boundary of $S$ is an invariant set. More is true: each edge of $S$ is an invariant set. Since the map $F$ is invertible, the initial conditions in the interior of $S $, $S^{0}$, cannot cross the invariant boundary $\partial S=s_{0}\cup s_{1}\cup r_{0}\cup r_{1}$, meaning that ${S}^{0}$ is an invariant set. This means, in particular, that for equal clocks there will be no secular drift of phase differences of the three clocks, the delays and advances are contained in the set $S=\left[ 0,2\pi\right] \times\left[ 0,2\pi\right] $. The total number of horizontal $rs$-heteroclines is $4$. The total number of $rs$-heteroclinics in the boundary of $S$ is $8$. ### Diagonal heteroclinics Finally, we now show that, $S^{o}$, the interior set of $S$, can be splitted in two subsets, $S_{U}$ and $S_{D}$, $U$ for up and $D$ for down, where the dynamics is again invariant. Consider now the set $$\Delta=\left\{ \left( x,y\right) \in S:y=x\text{, }x\in\left[ 0,2\pi\right] \right\} ,$$ the diagonal of $S$ connecting $\left( 0,0\right) $ to $\left( 2\pi ,2\pi\right) $. The image of a point of $\Delta$ by $F$ is now $$% \begin{cases} F_{1}\left( x,x\right) & =x+3\varepsilon\sin x,\\ F_{2}\left( x,x\right) & =x+3\varepsilon\sin x. \end{cases}$$ Hence, the same homeomorphism $g$ as before appears again. We repeat the same reasonings as before and deduce that $\Delta$ is invariant under $F$, and it splits $S^{o}$ in two open sets: the triangle above it and the triangle below it. Moreover, the stable manifold of the saddle point $\left( \pi,\pi\right) $ is the set $\Delta$. This also proves the existence of two heteroclinics in $\Delta$, connecting $\left( 0,0\right) $ to $\left( \pi,\pi\right) $ and $\left( 2\pi ,2\pi\right) $ to $\left( \pi,\pi\right) $, respectively. The total number of $rs$-heteroclines is now $10$, respectively $8$ on the edges and $2$ on the main diagonal $\Delta$, all of them connecting repellers to saddles. Consider now the other diagonal of $S$, i.e., the set $$\tilde{\Delta}=\left\{ \left( x,y\right) \in S:y=2\pi-x\text{, }x\in\left[ 0,2\pi\right] \right\} .$$ The image of a point of $\tilde{\Delta}$ under $F$ now is $$% \begin{cases} F_{1}\left( x,y\left( x\right) \right) & =x+\varepsilon\sin x+\varepsilon \sin2x,\\ F_{2}\left( x,y\left( x\right) \right) & =2\pi-\left( x+\varepsilon\sin x+\varepsilon\sin2x\right) . \end{cases}$$ Hence, $\tilde{\Delta}$ is invariant. The map $h_{1}:\left[ 0,2\pi\right] \rightarrow\left[ 0,2\pi\right] $, defined as $h_{1}(t)=t+\varepsilon\sin t+\varepsilon\sin2t$, is a homeomorphism with $5$ fixed points from $\left[ 0,2\pi\right] $ to itself (see figure \[homeo2\]). We repeat the same reasonings as before and deduce that the set $\tilde {\Delta}$ splits the interior set $S^{o}$ again in two open sets: the triangle above and the triangle below. So, now we have splitted $S^{0}$ in four small triangles. There are four heteroclinic connections in $\tilde{\Delta}$, one connecting the repeller $\left( 0,2\pi\right) $ to the attractor $\left( \frac{2\pi }{3},\frac{4\pi}{3}\right) $ (ra-heteroclinic), two $sa$-heteroclinics connecting the saddle point $\left( \pi,\pi\right) $ to the attractors $\left( \frac{2\pi}{3},\frac{4\pi}{3}\right) $ and $\left( \frac{4\pi}% {3},\frac{2\pi}{3}\right) $, and, finally, the last heteroclinic on this diagonal set is the one that connects the repeller $\left( 2\pi,0\right) $ to the attractor $\left( \frac{4\pi}{3},\frac{2\pi}{3}\right) $ (ra-heteroclinic). The total number of sa-heteroclinics is now $2$.\[ptb\] [Fig4.eps]{} We proceed with the same line of reasoning for the other $sa$-heteroclinics. Consider now the set $$d_{1}=\left\{ \left( x,y\right) \in S:y=\pi+\frac{x}{2}\text{, }x\in\left[ 0,\frac{2\pi}{3}\right] \right\}$$ and the map $F$ applied to the points of $d_{1}$:$$% \begin{cases} F_{1}\left( x,y\left( x\right) \right) & =x+2\varepsilon\sin x-2\varepsilon\sin\left( \frac{x}{2}\right) ,\\ F_{2}\left( x,y\left( x\right) \right) & =\pi+\frac{1}{2}\left( x+2\varepsilon\sin x-2\varepsilon\sin\left( \frac{x}{2}\right) \right) . \end{cases}$$ The points of $d_{1}$ stay in $d_{1}$ under the action of $F$, proving that this set also is invariant. The function $h_{2}:[0,2\frac{\pi}{3}% ]\rightarrow\lbrack0,2\frac{\pi}{3}]$, defined as $h_{2}(t)=t+2\varepsilon\sin t-2\varepsilon\sin\left( \frac{t}{2}\right) $, is a homeomorphism, from which we can readily see that the dynamics in $d_{1}$ is quite simple. The graph of this homeomorphism can be seen in figure \[homeo3\]. There is one $sa$-heteroclinic from the saddle at $\left( 0,\pi\right) $ to the attractor $\left( \frac{2\pi}{3},\frac{4\pi}{3}\right) $. Actually, there is another heteroclinic in the segment connecting the repeller $\left( 2\pi,2\pi\right) $ to the attractor $\left( \frac{2\pi}{3},\frac{4\pi}{3}\right) $, but this is not an sa-heteroclinic. Up to now, we have $3$ $sa$-heteroclinic connections. \[ptb\] [Fig5.eps]{} Consider now the set $c_{1}=\left\{ \left( x,y\right) \in S:y=2x\text{, }x\in\left[ \frac{2\pi}{3},\pi\right] \right\} $ and the map $F$ applied to the points of $c_{1}$$$\begin{aligned} F_{1}\left( x,y\right) & =x+\varepsilon\sin x+\varepsilon\sin2x,\\ F_{2}\left( x,y\right) & =y+2\varepsilon\sin x+2\varepsilon\sin2x,\end{aligned}$$ the points of $c_{1}$ stay in $c_{1}$ under $F$, proving that this set is invariant. Actually, the segment would be invariant if we extended $x$ to the interval $\left[ 0,\pi\right] $, but we are not interested in heteroclinics from repellers to attractors. Moreover, the dynamics is given by a restriction of $h_{1}$ to the interval $\left[ \frac{2\pi}{3},\pi\right] $. In this interval there are only two fixed points, the attractor $\frac{2\pi}{3}$ and the repeller $\pi$. This procedure adds one more $sa$-heteroclinic to the global picture. So, we have found, up to now, $4$ $sa$-heteroclinics. In $S_{D}$, we consider $$c_{2}=\left\{ \left( x,y\right) \in S:y=2\left( x-\pi\right) \text{, }x\in\left[ \pi,\frac{4\pi}{3}\right] \right\}$$ and $$d_{2}=\left\{ \left( x,y\right) \in S:y=\frac{x}{2}\text{, }x\in\left[ \frac{4\pi}{3},2\pi\right] \right\} .$$ Following exactly the same reasonings as before, we obtain two more $sa$-heteroclinics, one connecting $\left( \pi,0\right) $ to the attractor $\left( \frac{4\pi}{3},\frac{2\pi}{3}\right) $ and the other connecting $\left( 2\pi,\pi\right) $ to the same attractor. Phase portrait -------------- The total number of $sa$-heteroclinics is $6$. All of them are straight segments. The other $8$ $sa$-heteroclinics split the set $S$ in six invariant sets as can be seen in Figure \[Portrait\], where the red curves represent saddle-node heteroclines. The flow curves represented in the phase portrait. Since the map $F$ is invertible, no orbit can cross either the red curves, blue curves or black flow curves. There are only two attractors and the dynamics, due to the invertible nature of the map $F$ and its large symmetry, is relatively simple: in every invariant set in the plane, the restriction maps are again homeomorphisms and the flow curves must follow, by continuity, the heteroclinic connections on the outer boundaries of each invariant set. Consequently, only the orbits on the outer edges and main diagonal, i.e., in the set $s_{0}\cup s_{1}\cup r_{0}\cup r_{1}\cup d$ are not attracted to the two attractors $\left( \frac{2\pi}{3},\frac{4\pi}{3}\right) $ and $\left( \frac{4\pi}{3},\frac{2\pi}{3}\right) $. The upper attractor $\left( \frac{2\pi}{3},\frac{4\pi}{3}\right) $ attracts the points in the open upper triangle $S_{U}$ with converse results for the lower attractor $\left( \frac{4\pi}{3},\frac{2\pi}{3}\right) $ in $S_{D}$. The full picture can be seen in figure \[Portrait\]. Conclusions and future work =========================== In this paper we have proved that three oscillators, mutually interacting with symmetric coupling, converge to a final symmetric locked state with mutual phase differences of $\frac{2\pi}{3}$, this can happen in two different settings, clockwise or counterclockwise, depending on the initial conditions. This very symmetrical final locked state induces us to consider the conjecture that $n$ oscillators weakly interacting with all the others $n-1$ oscillators will reach a final state with mutual phase differences of $\frac{2\pi}{n}$ clockwise or counterclockwise distributed. In future work, already in preparation, we shall discuss the same phenomenon with slightly different natural angular frequencies $\omega_{1}$, $\ \omega_{2}$ and $\omega_{3}$ and, in particular, the existence and form of Arnold Tongues [@boyland1986; @gilmore2011]. As done for [@OlMe], it would be interesting to check experimentally our model, to see if the real world matches the theoretical predictions. #### Aknowledgements {#aknowledgements .unnumbered} The author ED was partially supported by the program Erasmus+. The author HMO was partially supported by FCT/Portugal through the project UID/MAT/04459/2013. Appendix {#appendix .unnumbered} ======== In this Appendix, as another way to determine the nature of the two attractors, that is their global asymptotical stability, we point out the existence, in the sets $S_{U}$ and $S_{D}$, of two Liapounov functions, $V_{U}$ and $V_{D}$, respectively [@lasalle1976stability]. Consider, first, the invariant set $S_{U}$. Define $V_{U}: S_{U} \rightarrow\mathbb{R}$ as follows: $$V_{U}\left( x,y\right) =\left( x-\frac{2\pi}{3}\right) ^{2}+\left( y-\frac{4\pi}{3}\right) ^{2}-\left( x-\frac{2\pi}{3}\right) \left( y-\frac{4\pi}{3}\right) \text{.}$$ The *discrete orbital derivative* inside the invariant set $S_{U}$ is $$DF\left( x,y\right) =V_{U}\left( F\left( x,y\right) \right) -V_{U} \left( x,y\right) .$$ We have $$\begin{aligned} {DF\left( x,y\right) } & =V_{U}\left( F\left( x,y\right) \right) -V_{U} \left( x,y\right) \\ & =V_{U}(x+\varepsilon\phi(x,y),y+\varepsilon\gamma(x,y))-V_{U}(x,y)\\ & ={\varepsilon}^{2}[{\phi}^{2}(x,y)+{\gamma}^{2}(x,y)-{\phi}(x,y)\cdot {\gamma}(x,y)]\\ & +\epsilon\lbrack(x-\frac{2}{3}\pi)(2\phi(x,y)-\gamma(x,y))+(y-\frac{4}% {3}\pi)(2\gamma(x,y)-\phi(x,y)],\\ &\end{aligned}$$ where, recall that $$% \begin{cases} \varphi\left( x,y\right) & =2\sin x+\sin y+\sin\left( x-y\right) \\ \gamma\left( x,y\right) & =\sin x+2\sin y+\sin\left( y-x\right) =\phi(y,x). \end{cases}$$ An easy computation shows that $$% \begin{cases} 2\varphi\left( x,y\right) -\gamma(x,y) & =3(\sin x+\sin\left( x-y\right) )\\ 2\gamma\left( x,y\right) -\phi(x,y) & =3(\sin y-\sin\left( x-y\right) \\ \phi(x,y)+\gamma(x,y) & =3(\sin x+\sin y). \end{cases}$$ Hence, $$\begin{aligned} \frac{DF(x,y)}{\varepsilon} & =\varepsilon\lbrack{(\phi(x,y)+\gamma (x,y))}^{2}-3\phi(x,y)\cdot\gamma(x,y)]\\ & =3\varepsilon\lbrack3{(\sin x+\sin y)}^{2}-\phi(x,y)\cdot\gamma(x,y)]\\ & +(x-\frac{2}{3}\pi)\cdot(\sin x+\sin(x-y))+(y-\frac{4}{3}\pi)\cdot(\sin y-\sin(x-y))\\ &\end{aligned}$$ By using numerical analysis, we conclude that this discrete orbital derivative, $DF$, is non-positive in $S_{U}$ for small $\varepsilon$, more precisely, zero for the fixed points and negative elsewhere. 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--- abstract: 'In this work we find necessary and sufficient conditions for a free nilpotent or a free metabelian nilpotent Lie algebra to be endowed with an ad-invariant metric. For such nilpotent Lie algebras admitting an ad-invariant metric the corresponding automorphisms groups are studied.' address: 'CONICET and ECEN-FCEIA, Universidad Nacional de Rosario, Pellegrini 250, 2000 Rosario, Argentina. ' author: - 'Viviana J. del Barco' - 'Gabriela P. Ovando' title: 'Free nilpotent Lie algebras admitting ad-invariant metrics' --- [^1] [^2] Introduction ============ An ad-invariant metric on a Lie algebra ${\mathfrak g }$ is a nondegenerate symmetric bilinear form $\la \,,\, \ra$ which satisfies $$\label{adme} \la [x, y], z\ra + \la y, [x, z]\ra = 0 \qquad \mbox{ for all }x, y, z \in {\mathfrak g }.$$ Lie algebras endowed with ad-invariant metrics (also called “metric” or “quadratic”) became relevant some years ago when they were useful in the formulation of some physical problems such as the known Adler-Kostant-Symes scheme. They also constitute the basis for the construction of bialgebras and they give rise to interesting pseudo-Riemannian geometry [@Co]. For instance in [@Ov1] a result originally due to Kostant [@Kos] was revalidated for pseudo-Riemannian metrics: it states that the Lie algebra of the isometry group of a naturally reductive pseudo-Riemannian space (in particular symmetric spaces) can be endowed with an ad-invariant metric. Semisimple Lie algebras are examples of Lie algebras admitting an ad-invariant metric since the Killing form is nondegenerate. In the solvable case, the Killing form is degenerate so one must search for another bilinear form with the ad-invariance property. The first investigations concerning general Lie algebras with ad-invariant metrics appeared in [@FS; @MR]. They get structure results proposing a method to construct these Lie algebras recursively. This enables a classification of nilpotent Lie algebras admitting ad-invariant metrics of dimension $\leq 7$ in [@FS] and a determination of the Lorentzian Lie algebras in [@Me]. The point is that by this recursive method one can reach the same Lie algebra starting from two non-isomorphic Lie algebras. This fact difficulties the classification in higher dimensions. More recently a new proposal for the classification problem is presented in [@KO] and this is applied in [@Ka] to get the nilpotent Lie algebras with ad-invariant metrics of dimension $\leq 10$. However the basic question whether a non-semisimple Lie algebra admits such a metric is still opened. In the present paper we deal with this problem in the family of free nilpotent and free metabelian nilpotent Lie algebras. [**Theorem**]{} \[t1\]. [*Let ${\mathfrak n }_{m,k}$ be the free $k$-step nilpotent Lie algebra in $m$ generators. Then ${\mathfrak n }_{m,k}$ admits an ad-invariant metric if and only if $(m,k)=(3,2)$ or $(m,k)=(2,3)$.*]{} The techniques for the proof do not make use of the extension procedures mentioned before, but properties of free nilpotent Lie algebras which combined with the ad-invariance condition enable the deduction of the Lie algebras ${\mathfrak n }_{2,3}$ and ${\mathfrak n }_{3,2}$. We note that for $k=2,3$ the free and free metabelian $k$-step nilpotent Lie algebras coincide. For the free metabelian case the first approach lies in the fact that 2-step solvable Lie algebras admitting ad-invariant metrics are nilpotent and at most 3-step (Lemma \[le11\]). Thus working out we get the next result. [**Theorem**]{} \[t2\]. [*Let $\tilde{{\mathfrak n }}_{m,k}$ be the free metabelian $k$-step nilpotent Lie algebra in $m$ generators. Then $\tilde{{\mathfrak n }}_{m,k}$ admits an ad-invariant metric if and only if $(m,k)=(3,2)$ or $(m,k)=(2,3)$.*]{} These two Lie algebras have been studied since a long time in sub-Riemannian geometry [@Mo]. Thus ${\mathfrak n }_{2,3}$ is associated to the Carnot group distribution (see for instance [@BM]), which is related to the “rolling balls problem”, treated by Cartan in [@Ca]. The prolongation, representing -roughly speaking- the maximal possible symmetry of the distribution, in the case of ${\mathfrak n }_{2,3}$ is the exceptional Lie algebra ${\mathfrak g }_2$ [@BM]. The Lie algebra ${\mathfrak n }_{3,2}$ was studied more recently in [@My] in the context of the geometric characterization of the so-called Maxwell set, wave fronts and caustics (see also [@MA]). We complete the work with a study of the group of automorphisms of the Lie algebras ${\mathfrak n }_{2,3}$ and ${\mathfrak n }_{3,2}$. The corresponding structure is described and in particular the subgroup of orthogonal automorphisms is determined. Following [@FS] and the considerations above all Lie algebras here are over a field $K$ of characteristic 0, nevertheless some results in Section 3 could be still true for fields of a characteristic different from 2. Free and free metabelian nilpotent Lie algebras =============================================== Let ${\mathfrak g }$ denote a Lie algebra. The so-called central descending and ascending series of ${\mathfrak g }$, respectively $\{C^r({\mathfrak g })\}$ and $\{C_r({\mathfrak g })\}$ for all $r\geq 0$, are constituted by the ideals in ${\mathfrak g }$, which for non-negative integers $r$, are given by $$\begin{array}{rclrcl} C^0({\mathfrak g })&=&{\mathfrak g }& C_0({\mathfrak g })&=&0 \\ C^r({\mathfrak g })&=&[{\mathfrak g },C^{r-1}({\mathfrak g })] & C_r({\mathfrak g })&=&\{x\in {\mathfrak g }:[x,{\mathfrak g }]\in C_{r-1}({\mathfrak g })\}. \end{array}$$ Note that $C_1({\mathfrak g })$ is by definition the center of ${\mathfrak g }$, which will be denoted by ${\mathfrak z }({\mathfrak g })$. A Lie algebra ${\mathfrak g }$ is called *k-step nilpotent* if $C^k({\mathfrak g })=\{0\}$ but $C^{k-1}({\mathfrak g })\neq \{0\}$ and clearly $C^{k-1}({\mathfrak g })\subseteq {\mathfrak z }({\mathfrak g })$. Heisenberg algebras. Let $X_1, \hdots, X_n, Y_1, \hdots, Y_n$ denote a basis of the $2n$-dimensional real vector space $V$ and let $Z\notin V$. Define $[X_i, Y_j]=\delta_{ij} Z$ and $[Z, U]=0$ for all $U\in V$. Thus $\hh_n= V \oplus \RR Z$ is the Heisenberg Lie algebra of dimension $2n+1$, which is 2-step nilpotent. We shall make use of the notation ${\mathfrak g }'=[{\mathfrak g }, {\mathfrak g }]$ and ${\mathfrak g }''=[{\mathfrak g }',{\mathfrak g }']$. A Lie algebra is called [*2-step solvable*]{} if its commutator is abelian, that is ${\mathfrak g }''=0$. Let $\mathfrak f_m$ denotes the free Lie algebra on $m$ generators, with $m\geq 2$. (Notice that a unique element spans an abelian Lie algebra). Thus - the free metabelian $k$-step nilpotent Lie algebra on $m$ generators is defined as $$\tilde{{\mathfrak n }}_{m,k}:= \mathfrak f_m/(C^{k+1}(\mathfrak f_m)+ \mathfrak f_m''),$$ - the *free $k$-step nilpotent* Lie algebra on $m$ generators ${\mathfrak n }_{m,k}$ is defined as the quotient algebra $${\mathfrak n }_{m,k}:=\mathfrak f_m/C^{k+1}(\mathfrak f_m).$$ In particular free metabelian nilpotent of any degree are 2-step solvable. \[23\] For $k=2,3$ any $k$-step nilpotent Lie algebra is 2-step solvable, which follows from the Jacobi identity. Thus for the free nilpotent ones we get $\tilde{{\mathfrak n }}_{m,k}={\mathfrak n }_{m,k}$ for $k=2,3$. Let ${\mathfrak n }_{m,k}$ be a free $k$-step nilpotent Lie algebra and let $\{e_1,\ldots,e_m\}$ be an ordered set of generators. The construction of a *Hall* basis associated to this sets of generators is explained below (see [@Ha; @GG]). Start by defining the *length* $\ell$ of each generator as $1$. Take the Lie brackets $[e_i, e_j]$ for $i>j$, which by definition satisfy $\ell([e_i,e_j])=2$. Now the elements $e_1, \dots , e_m$, $[e_i, e_j]$, $i>j$ belong to the Hall basis. Define a total order in that set by extending the order of the set of generators and so that $E>F$ if $\ell(E)>\ell(F)$. They allow the construction of the elements of length $3$ and so on. Recursively each element of the Hall basis of ${\mathfrak n }_{m,k}$ is defined as follows. The generators $e_1,\ldots,e_m$ are elements of the basis of length 1. Assume we have defined basic elements of lengths $1,\ldots, r-1\leq k-1$, with a total order satisfying $E > F$ if $\ell(E) > \ell(F)$. If $\ell(E) =s$ and $\ell(F) = t$ and $r = s + t\leq k$, then $[E, F]$ is a basic element of length $r$ if both of the following conditions hold: - $E$ and $F$ are basis elements and $E>F$, and - if $\ell(E)>1$ and $E=[G,H]$ is the unique decomposition with $G,H$ basic elements, then $F\geq H$. This gives rise to a natural graduation of ${\mathfrak n }_{m,k}$: $${\mathfrak n }_{m,k}=\bigoplus_{s=1}^k \mathfrak p(m,s),$$ where $\mathfrak p(m,s)$ denotes the subspace spanned by the elements of the Hall basis of length $s$. Notice that - $C^r({\mathfrak n }_{m,k})=\oplus_{s=r+1}^k\mathfrak p(m,s)$, - $\mathfrak p(m,k) = {\mathfrak z }({\mathfrak n }_{m,k})$ The first assertion follows from the fact that every bracket of $r + 1$ elements of ${\mathfrak n }_{m,k}$, is a linear combination of brackets of $r + 1$ elements in the Hall basis (see proof of Theorem 3.1 in [@Ha]). This implies $C^r({\mathfrak n }_{m,k})\subseteq \oplus_{s=r+1}^k \mathfrak p(m, s)$; the other inclusion is obvious. In particular, $\mathfrak p(m, k) = C^{k-1}({\mathfrak n }_{m,k}) \subseteq \mathfrak z({\mathfrak n }_{m,k})$. Now let $x\in \mathfrak z({\mathfrak n }_{m,k})$ and let $e$ be a generator and assume $x\notin C^{k-1}({\mathfrak n }_{m,k})$. Recall that ${\mathfrak n }_{m,k}$ is homomorphic image of the free Lie algebra $\mathfrak f_m$ so that there exist $X,E\in \mathfrak f_m$ such that $X\to x$ and $E \to e$, being $E$ a generator of $\mathfrak f_m$. Since $[x, e]=0$ then $[X,E]=0$ which says $X$and $E$ are proportional, which is impossible (see for instance Ch. 2 in [@Ba]). Thus $\mathfrak p(m, k) = \mathfrak z({\mathfrak n }_{m,k})$. Denote as $d_m(s)$ the dimension of $\mathfrak p(m,s)$. Inductively one gets [@Se] $$\label{dim}s\cdot d_m(s)= m^s- \sum_{r|s, r<s} r\cdot d_m(r),\qquad s\geq 1.$$ Hence for a fixed $m$, one has $d_m(1)=m$ and $d_m(2)=m(m-1)/2$. \[ex:1\]Given an ordered set of generators $e_1,\ldots, e_m$ of a free $2$-step nilpotent Lie algebra ${\mathfrak n }_{m,2}$, a Hall basis is $$\label{ba2} \mathcal B=\{e_i,\,[e_j,e_k]:\,i=1,\ldots,m,\, 1\leq k< j\leq m\}.$$ Equation (\[dim\]) asserts that $\dim{\mathfrak n }_{m,2}=d_m(1)+d_m(2)=m+m(m-1)/2$. Since $\mathfrak z({\mathfrak n }_{m,2}) = \mathfrak p(m, 2)$, we have $\dim \mathfrak z({\mathfrak n }_{m,2})= m(m-1)/2$. \[ex:2\] For the free $3$-step nilpotent Lie algebra on $m$ generators ${\mathfrak n }_{m,3}$ a Hall basis of a set of generators as before has the form $$\label{ba3} \mathcal B=\{e_i, \,[e_j,e_k],\,[[e_r,e_s],e_t]:\,i=1,\ldots,m,\,1\leq k< j\leq m,\, 1\leq s<r\leq m,\,t\geq s\}.$$ It holds ${\mathfrak z }({\mathfrak n }_{m,3})=\mathfrak p(m,3)$, and so $$\dim\, {\mathfrak z }({\mathfrak n }_{m,3})= d_m(3)=m(m^2-1)/3.$$ Free and free metabelian nilpotent Lie algebras and ad-invariant metrics ======================================================================== In this section we determine free nilpotent and free metabelian nilpotent Lie algebras admitting ad-invariant metrics. Let ${\mathfrak g }$ denote a Lie algebra equipped with an ad-invariant metric $\la\,,\,\ra$, see (\[adme\]). If $\mm \subseteq \ggo$ is a subset, then we denote by $\mm^{\perp}$ the linear subspace of ${\mathfrak g }$ given by $$\mm^{\perp}=\{x\in \ggo, \la x, v\ra=0 \mbox{ for all } v\in \mm\}.$$ In particular $\mm$ is called - [*isotropic*]{} if $\mm \subseteq \mm^{\perp}$, - [*totally isotropic*]{} if $\mm=\mm^{\perp}$, and - [*nondegenerate*]{} if and only if $\mm \cap \mm^{\perp}=\{0\}$. The proof of the next result follows easily from an inductive procedure. \[le1\] Let $(\ggo, \la\,,\,\ra)$ denote a Lie algebra equipped with an ad-invariant metric. - If $\hh$ is an ideal of $\ggo$ then $\hh^{\perp}$ is also an ideal in $\ggo$. - $C^r(\ggo)^{\perp}=C_r(\ggo)$ for all $r$. Thus on any Lie algebra admitting an ad-invariant metric the next equality holds $$\label{e2} \dim \ggo=\dim C^r(\ggo) + \dim C_r(\ggo).$$ For the case $r=1$ one obtains $$\label{e1} \dim \ggo=\dim \zz(\ggo) + \dim C^1(\ggo).$$ Let ${\mathfrak n }$ denote a $2$-step nilpotent Lie algebra equipped with an ad-invariant metric. Assume $\zz({\mathfrak n })=C^1({\mathfrak n })$, then by (\[e1\]) the metric is neutral and $\dim {\mathfrak n }= 2 \dim \zz({\mathfrak n })$. As a consequence the Heisenberg Lie algebra $\hh_n$ cannot be equipped with any ad-invariant metric. Examples of nilpotent Lie algebras satisfying the equality (\[e2\]) above for every $r$ arise by considering the semidirect product of a nilpotent Lie algebra $\nn$ with its dual space via de coadjoint representation $\nn \ltimes \nn^*$. The natural neutral metric on $\nn \ltimes \nn^*$ is ad-invariant. Nevertheless, condition (\[e1\]) (and hence (\[e2\])) is not sufficient for a 2-step nilpotent Lie algebra to admit an ad-invariant metric as shown for instance in [@Ov2]. \[le11\] Let ${\mathfrak g }$ denote a 2-step solvable Lie algebra provided with an ad-invariant metric, then ${\mathfrak g }$ is nilpotent and at most 3-step. Let $\la\,,\,\ra$ denote an ad-invariant metric on ${\mathfrak g }$. Since ${\mathfrak g }$ is 2-step solvable for all $x,y \in {\mathfrak g }'$ one has $[x,y]=0$, which is equivalent to $$\begin{array}{rcll} 0 & = & \la [x,y], u\ra \qquad & \mbox{ for all } u \in {\mathfrak g }\\ & = & \la [u, x], y\ra \qquad & \mbox{ for all } y\in {\mathfrak g }' \end{array}$$ thus $[u,x]\in [{\mathfrak g }, {\mathfrak g }]^{\perp}=\zz({\mathfrak g })$, and since $x\in {\mathfrak g }'$ can be written as $x=[v,w]$, then $[u,[v,w]]\subseteq \zz({\mathfrak g })$ for all $u,v,w\in {\mathfrak g }$, that is $C^4({\mathfrak g })=0$ and so ${\mathfrak g }$ is at most 3-step nilpotent. \[corofree\] Let $\tilde{{\mathfrak n }}_{m,k}$ denote a free metabelian nilpotent Lie algebra admitting an ad-invariant metric, then $k\leq 3$. Remark \[23\] and the previous result says that a free metabelian nilpotent Lie algebra with an ad-invariant metric if it exists, is free nilpotent. Below we determine which free nilpotent Lie algebra admits such a metric. Whenever ${\mathfrak n }_{m,k}$ is free nilpotent we have that $\dim {\mathfrak n }_{m,k}/C^1({\mathfrak n }_{m,k}) =m$ so that $$\label{e3} \dim {\mathfrak n }_{m,k}=m+\dim C^1({\mathfrak n }).$$ Hence Equations (\[e1\]) and (\[e3\]) show that if ${\mathfrak n }_{m,k}$ admits an ad-invariant metric then $$\label{e4} \dim {\mathfrak z }({\mathfrak n }_{m,k})=m.$$ \[p1\] If $\nn_{m,2}$ is a free 2-step nilpotent Lie algebra endowed with an ad-invariant metric, then $m=3$. Let ${\mathfrak n }_{m,2}$ be the free 2-step nilpotent Lie algebra on $m$ generators. As we showed in Example \[ex:1\] its center has dimension $m(m-1)/2$. Now if Equation (\[e4\]) holds then $m=3$. \[p2\] Let $\nn_{m,3}$ be a free 3-step nilpotent Lie algebra provided with an ad-invariant metric, then $m=2$. As shown in Example \[ex:2\] the center ${\mathfrak z }({\mathfrak n }_{m,3})$ has dimension $d_m(3)=m(m^2-1)/3$. From straightforward calculations, if Equation (\[e4\]) is satisfied then $m= 2$. \[p3\] No free $k$-step nilpotent Lie algebra ${\mathfrak n }_{m,k}$ on $m$ generators with $k \geq 4$ can be endowed with an ad-invariant metric. *$\bullet$ 4-step nilpotent case:* In this case $\mathfrak p(m,4)= {\mathfrak z }({\mathfrak n }_{m,4})$, thus from (\[dim\]): $$\dim{\mathfrak z }({\mathfrak n }_{m,4})\geq d_m(4)=\frac{1}{4}(m^4-d_m(1)-2d_m(2))= \frac{m^2(m^2-1)}{4}.$$ Notice that for $m\geq 2$, one has $m^2(m^2-1)/4>m$. *$\bullet$ General case, $k\geq 5$:* Let ${\mathfrak n }_{m,k}$ denote the free $k$-step nilpotent Lie algebra in $m$ generators. The goal here is to show that for every $m$ and $k\geq 5$ the dimension of the center of ${\mathfrak n }_{m,k}$ is greater than $m$. In order to give a lower bound for $\dim{\mathfrak z }({\mathfrak n }_{m,k})$ we construct elements of length $k$ in a Hall basis $\mathcal B$. Let $\{e_1,\ldots,e_m\}$ a set of generators of ${\mathfrak n }_{m,k}$ and consider the set $$\begin{aligned} \mathcal U &=&\{[[[e_i,e_j],e_k],e_m]: 1\leq j<i\leq m,\,k\geq j\}.\nonumber $$ Any element in $\mathcal U$ is basic and of length 4. Given $x\in \mathcal U$, the bracket $$[x,e_m]^{(s)}:= [[[x,\overbrace{e_m],e_m]\cdots , e_m]}^{s} \;\;s\geq 1$$ is an element in the Hall basis if $\ell([x,e_m]^{(s)})\leq k$. In fact if $s=1$ then - both $x=[[[e_i,e_j],e_k],e_m] \in \mathcal U$ and $e_m$ are elements of the Hall basis, and $x >e_m$ because of their length; - also $x=[G,H]$ with $G=[[e_i,e_j],e_k]$ and $H=e_m$ and we have $e_m\geq H$. So both conditions of the Hall basis definition are satisfied and hence $[x,e_m]^{(1)}\in \mathcal B$ and it belongs to $ C^4({\mathfrak n }_{m,k})$. Inductively suppose $[x,e_m]^{(s-1)}\in\mathcal B$, then clearly $[[[x,e_m],e_m]\cdots ], e_m]^{(s-1)}>e_m$ and it is possible to write $[x,e_m]^{(s-1)}=[G,H]$ with $H=e_m$. Thus $[x,e_m]^{(s)}\in\mathcal B$. Notice that $[x,e_m]^{(s)}\in C^{s+3}({\mathfrak n }_{m,k})$. We construct the new set $$\widetilde{\mathcal U}:=\{[x,e_m]^{(k-4)}:x\in \mathcal U\}\subseteq C^{k-1}({\mathfrak n }_{m,k}),$$ which is contained in the center of ${\mathfrak n }_{m,k}$ and it is linearly independent. Therefore $$\label{u} \dim {\mathfrak z }({\mathfrak n }_{m,k}) \geq |\widetilde{\mathcal U}|.$$ Clearly $\widetilde{\mathcal U}$ and $\mathcal U$ have the same cardinal. Also, $|\mathcal U|=\sum_{j=1}^m(m-j+1)(m-j)$ since for every fixed $j=1,\ldots,m$, the amount of possibilities to choose $k\geq j$ and $i>j$ is $(m-j+1)$ and $(m-j)$ respectively. Straightforward computations give $ |\widetilde{\mathcal U}|=1/3\, m^3+m^2+2/3\,m$ which combined with (\[u\]) proves that for any $m$ and $k\geq 5$ $$\dim {\mathfrak z }({\mathfrak n }_{m,k})\geq 1/3 \,m^3+m^2+2/3\,m .$$ The right hand side is greater than $m$ for all $m\geq 2$. According to (\[e4\]), the free $k$-step nilpotent Lie algebra ${\mathfrak n }_{m,k}$ does not admit an ad-invariant metric if $k\geq 5$. \[t1\] Let ${\mathfrak n }_{m,k}$ be the free $k$-step nilpotent Lie algebra in $m$ generators. Then ${\mathfrak n }_{m,k}$ admits an ad-invariant metric if and only if $(m,k)=(3,2)$ or $(m,k)=(2,3)$. Propositions (\[p1\]) (\[p2\]) and (\[p3\]) prove that if ${\mathfrak n }_{m,k}$ admits an ad-invariant metric then $(m,k)=(3,2)$ or $(m,k)=(2,3)$. Let us show the resting part of the proof. The Lie algebra ${\mathfrak n }_{3,2}$ has a basis $\{e_1,e_2,e_3,e_4,e_5,e_6\}$ with non zero brackets $$\label{b32} [e_1,e_2]=e_4,\qquad [e_1,e_3]=e_5,\qquad [e_2,e_3]=e_6.$$ Since $C^1({\mathfrak n }_{3,2})=\zz({\mathfrak n }_{3,2})$, if the metric $\la \,,\,\ra$ is ad-invariant the center of ${\mathfrak n }_{3,2}$ is totally isotropic, so it must hold $$\la e_i, e_j\ra=0 \qquad \mbox { for all }i,j=4,5,6.$$ The ad-invariance property says $$\la e_4, e_1\ra=\la [e_1,e_2], e_1\ra=0$$ and similarly $\la e_4,e_2\ra=0$, therefore $\la e_4,e_3\ra \neq 0$. Analogously, $\la e_5, e_2\ra\neq 0$ and $\la e_6, e_1\ra\neq 0$. Moreover $$\alpha=\la [e_1, e_2], e_3\ra=-\la e_2, e_5, \ra=\la e_1, e_6\ra.$$ Thus in the ordered basis $\{e_1,e_2,e_3,e_4,e_5,e_6\}$, a matrix of the form $$\label{m32} \left( \begin{matrix} a_{11} & a_{12} & a_{13} & 0 & 0 & \alpha \\ a_{12} & a_{22} & a_{23} & 0 & -\alpha & 0 \\ a_{13} & a_{23} & a_{33} & \alpha & 0 & 0 \\ 0 & 0 & \alpha & 0 & 0 & 0 \\ 0 & -\alpha & 0 & 0 & 0 & 0 \\ \alpha & 0 & 0 & 0 & 0 & 0 \end{matrix} \right) \qquad \mbox{ with } a_{ij} \in K, \forall i,j=1,2,3 \mbox{ and } \alpha \neq 0$$ corresponds to an ad-invariant metric on ${\mathfrak n }_{3,2}$. The Lie algebra ${\mathfrak n }_{2,3}$ has a basis $\{e_1,e_2,e_3,e_4,e_5\}$ with non-zero Lie brackets $$\label{b23} [e_1,e_2]=e_3,\qquad [e_1,e_3]=e_4,\qquad [e_2,e_3]=e_5.$$ Let $\la \,,\,\ra$ be an ad-invariant metric on ${\mathfrak n }_{2,3}$. Then $\zz({\mathfrak n }_{3,2})=C^2({\mathfrak n }_{2,3})=span\{e_4, e_5\}$ while $C^1({\mathfrak n }_{2,3})=C^2({\mathfrak n }_{2,3}) \oplus K e_3$. The ad-invariance property also says that $$0=\la e_4, e_3\ra =\la e_4, e_4\ra= \la e_4, e_5\ra=\la e_5, e_3\ra=\la e_5, e_5\ra.$$ Moreover $$\la e_1, e_3\ra= \la e_1, [e_1, e_2]\ra =0 \quad \mbox { and } \quad \la e_2, e_3\ra=\la e_2, [e_1, e_2] \ra=0;$$ $$\la e_1, e_4\ra=\la e_1, [e_1, e_3]\ra=0 \quad \mbox{ and } \quad \la e_2, e_5\ra=\la e_2, [e_2, e_3]\ra=0.$$ Therefore $$\la e_3, e_3\ra=\la [e_1,e_2], e_3\ra=-\la e_2, e_4\ra= \la e_1, e_5\ra =\alpha \neq 0,$$ which amounts to the following matrix for $\la \,,\,\ra$ in the ordered basis above: $$\label{m23} \left( \begin{matrix} a_{11} & a_{12} & 0 & 0 & \alpha \\ a_{12} & a_{22} & 0 & - \alpha & 0 \\ 0 & 0 & \alpha & 0 & 0 \\ 0 & -\alpha & 0 & 0 & 0 \\ \alpha & 0 & 0 & 0 & 0 \end{matrix} \right) \qquad \mbox{ with } a_{ij} \in K, \forall i,j=1,2 \mbox{ and } \alpha \neq 0.$$ Remark \[23\], Corollary \[corofree\] and the previous theorem imply the next result. \[t2\] Let $\tilde{{\mathfrak n }}_{m,k}$ be the free metabelian $k$-step nilpotent Lie algebra in $m$ generators. Then $\tilde{{\mathfrak n }}_{m,k}$ admits an ad-invariant metric if and only if $(m,k)=(3,2)$ or $(m,k)=(2,3)$. The free nilpotent Lie algebras above can be constructed as extensions of abelian Lie algebras. This is the way in which they appear in [@FS], where Favre and Santharoubarne obtained the classification of the nilpotent Lie algebras of dimension $\leq 7$ admitting an ad-invariant metric. According to their results, any of the Lie algebras equipped with an ad-invariant metric ${\mathfrak n }_{3,2}$ or ${\mathfrak n }_{2,3}$ as above is equivalent to one of the followings $$\label{m3223} ({\mathfrak n }_{3,2}, B_{3,2}) : \left( \begin{matrix} 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right) \;\; ({\mathfrak n }_{2,3}, B_{2,3}) : \left( \begin{matrix} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & - 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{matrix} \right).$$ The automorphism groups of ${\mathfrak n }_{3,2}$ and ${\mathfrak n }_{2,3}$ ============================================================================ Here we study the automorfisms of the Lie algebras in Theorem \[t1\]. This is indeed a topic of active research (see for instance [@DF; @DG] and references therein). Our goal is to write explicitly the algebraic structure, in terms of the actions and representations of the different subgroups or subalgebras. We also distinguish the subgroup of orthogonal automorphism (resp. the Lie algebra of skew-symmetric derivations) in presence of the ad-invariant metric fixed in (\[m3223\]). Recall that a derivation of a Lie algebra $\mathfrak g$ is a linear map $t: \mathfrak g \to \mathfrak g$ satisfying $$t[x,y]=[tx, y] + [x, ty ] \qquad \mbox{ for all } x,y \in \mathfrak g.$$ Whenever $\mathfrak g$ is endowed with a metric $\la \,,\,\ra$ a skew-symmetric derivation of ${\mathfrak g }$ is a derivation $t$ such that $$\label{antisim} \la t a, b\ra=-\la a, tb\ra \quad \text{ for all }\; a,b\in{\mathfrak g }.$$ We denote by $\Der({\mathfrak g })$ the Lie algebra of derivations of ${\mathfrak g }$, which is the Lie algebra of the group of automorphisms of $\ggo$, $\Aut({\mathfrak g })$. Let $\Dera({\mathfrak g })$ denote the subalgebra of $\Der({\mathfrak g })$ consisting of skew-symmetric derivations of $({\mathfrak g }, \la\,,\,\ra)$. Thus $\Dera({\mathfrak g })$ is the Lie algebra of the group of orthogonal automorphisms denoted by $\Auto({\mathfrak g })$: $$\Auto({\mathfrak g })=\{ \alpha \in \Aut({\mathfrak g }) : \la \alpha x, \alpha y \ra=\la x,y \ra \quad \forall x,y \in {\mathfrak g }\}.$$ For an arbitrary Lie algebra ${\mathfrak g }$, each $t \in \Aut({\mathfrak g })$ leaves invariant both the commutator ideal $C^1({\mathfrak g })$ and the center $\zz({\mathfrak g })$. Thus $t$ induces automorphisms of the quotient Lie algebras ${\mathfrak g }/\zz({\mathfrak g })$ and ${\mathfrak g }/C^1({\mathfrak g })$. In particular if ${\mathfrak g }$ is solvable, ${\mathfrak g }/C^1({\mathfrak g })$ is a nontrivial abelian Lie algebra. Hence any $t\in\Aut({\mathfrak g })$ induces an element $s \in \Aut({\mathfrak g }/C^1({\mathfrak g }))$ which as an automorphism of an abelian Lie algebra, $s \in \GL(p,K)$ where $p = \dim( {\mathfrak g }/C^1({\mathfrak g }))$. Note that if ${\mathfrak g }$ is free nilpotent, $p$ coincides with the number of generators of ${\mathfrak g }$. Below we proceed to the study in each case. The Lie algebra ${\mathfrak n }_{2,3}$ -------------------------------------- Let $t$ denote an automorphism of ${\mathfrak n }_{2,3}$ and let $e_1,e_2,e_3,e_4,e_5$ be the basis given in (\[b23\]). Denote as $(t_{ij})$ the matrix of $t$ in that basis. Since $t$ leaves invariant the commutator and the center and $C^1({\mathfrak n }_{2,3})=span\,\{e_3,e_4,e_5\}$ and ${\mathfrak z }({\mathfrak n }_{2,3})=span\,\{e_4,e_5\}$, we have that $t_{ij}=0$ if $i=1,2$, $j=3,4,5$. Notice that the Lie algebra ${\mathfrak n }_{2,3}/\zz({\mathfrak n }_{2,3})$ is isomorphic to the Heisenberg Lie algebra $\hh_1$, hence $t$ induces a automorphism $\bar{t}\in \Aut(\hh_1)$. Let $\bar{x}$ denote the image of an arbitrary element $x\in {\mathfrak n }_{2,3}$ by the canonical epimorphism ${\mathfrak n }_{2,3} \to {\mathfrak n }_{2,3}/\zz({\mathfrak n }_{2,3})$, thus $\bar{t}(\bar{x})=\overline{tx}$. From the computation $[\bar{t} \bar{e}_1, \bar{t}\bar{e}_2]=\overline{te_3}$ one gets $$\label{det} t_{11} t_{22}- t_{12} t_{21}= t_{33}.$$ We introduce the notation for the submatrices $$A:=\left(\begin{array}{cc}t_{11}&t_{12}\\ t_{21}& t_{22}\end{array}\right)\qquad B:=\left(\begin{array}{cc}t_{44}&t_{45}\\ t_{54}& t_{55}\end{array}\right) .$$ Note that the second matrix is non-singular since $t$ is non-singular. By compairing (\[det\]) with the computations $[t e_i, t e_j]= t[e_i,e_j]$ we get the conditions $$t_{33}=\det A,\qquad B=\det(A) A, \qquad \left(\begin{array}{c}t_{43}\\ t_{53}\end{array}\right)=A \left(\begin{array}{c}t_{32}\\ -t_{31}\end{array}\right).$$ So for any $t\in \Aut({\mathfrak n }_{2,3})$, $(t_{ij})_{i,j}$ has the form $$\label{t} \left(\begin{array}{ccc} A & \begin{array}{cc}0\\0\end{array} & \begin{array}{cc}0&0\\0&0\end{array} \\ \begin{array}{cc} t_{31}&t_{32}\end{array} & \det(A) & \begin{array}{cc}0&0\end{array} \\ \begin{array}{cc}t_{41}&t_{42}\\t_{51}&t_{52}\end{array} &\begin{array}{cc}t_{11}t_{32}-t_{31}t_{12}\\t_{21}t_{32}-t_{31}t_{22}\end{array} & \det(A) A\end{array} \right),$$ where $A\in \GL(2,K)$. Consider the following matrices in $\Aut({\mathfrak n }_{2,3})$: $$\mathcal G=\left\{ \tilde{A}=\left(\begin{array}{ccc} A & \begin{array}{cc}0\\0\end{array} & \begin{array}{cc}0&0\\0&0\end{array} \\ \begin{array}{cc} 0&0\end{array} & \det(A) & \begin{array}{cc}0&0\end{array} \\ \begin{array}{cc}0&0\\0&0\end{array} &\begin{array}{cc}0\\0\end{array} &\det(A)A\end{array} \right), \, A\in \GL(2,K)\right\}$$ $$\mathcal H=\left\{ h_{(x,y,z)}=\left( \begin{array}{ccccc} 1&0&0&0&0 \\ 0&1&0&0&0\\ y&x&1&0&0\\ z+\frac{1}{2} x y&\frac{1}{2} x^2&x&1&0\\ -\frac{1}{2} y^2&z-\frac{1}{2} xy&-y&0&1\end{array}\right),\, (x,y,z)\in K^3\right\}$$ $$\mathcal R= \left\{ r_{(u,v,w)}=\left( \begin{array}{ccccc} 1&0&0&0&0 \\ 0&1&0&0&0\\ 0&0&1&0&0\\ v&w&0&1&0\\ u&-v&0&0&1\end{array}\right),\, (u,v,w)\in K^3\right\}.$$ Note that $\mathcal{G}$, $\mathcal{R}$ and $\mathcal{H}$ are subgroups of $\Aut({\mathfrak n }_{2,3})$. Moreover every $t$ of the form (\[t\]) can be written as a product of matrices $$t=\tilde{A}\cdot r_{(u,v,w)} \cdot h_{(x,y,z)},\qquad \mbox{ with } \tilde{A} \in \mathcal G,\; r_{(u,v,w)}\in \mathcal R,\; h_{(x,y,z)}\in \mathcal H$$ where $x=t_{32}/\det(A)$, $y=t_{31}/\det(A)$, $z=(t_{22}t_{41}-t_{12}t_{51}+t_{52}t_{11}-t_{42}t_{21})/2\det(A)^2$ and $u=(2t_{51}t_{11}-2t_{41}t_{21}+t_{31}^2)/2 \det A^2$, $v=(t_{22}t_{41}-t_{31}t_{32}-t_{12}t_{51}+t_{42}t_{21}-t_{52}t_{11})/2 \det A$, $w=(2t_{42}t_{22}-t_{32}^2-2t_{52}t_{12})/2\det A^2$. The elements of $\mathcal{H}$ commute with those of $\mathcal{R}$; also $\mathcal H \cap \mathcal R=\{I\}$. Hence $\mathcal{R}\cdot \mathcal{H}\simeq \mathcal{R}\times \mathcal{H}.$ It holds $h_{(x,y, z)} \cdot h_{(x',y',z')}=h_{(x+x', y+ y', z + z' + \frac12 (xy' -x' y))}$, where $\cdot$ denotes the product of matrices. Thus the map from the Heisenberg Lie group $H_1$ to $\Aut({\mathfrak n }_{2,3})$ given by $(x,y, z) \mapsto h_{(x,y,z)}$ is an isomorphism of groups. Analogously $(u,v,w)\mapsto r_{(u,v,w)}$ is an isomorphism of groups from $K^3$ to $ \mathcal R\subseteq \Aut({\mathfrak n }_{2,3})$. The action by conjugation of $\mathcal{G}$ preserves both $\mathcal{H}$ and $\mathcal{R}$. Thus the map $$\tau_{\tilde{A}}(r,h)=(\tilde{A} r\tilde{A}^{-1},\tilde{A} h\tilde{A}^{-1}).$$ defines a group homomorphism from $\mathcal{G}$ to $\Aut(\mathcal R \times \mathcal{H})$. The subgroup $\mathcal{R}\times \mathcal{H}$ is normal in $\Aut({\mathfrak n }_{2,3})$ and $\mathcal{G}\cap (\mathcal{R}\times \mathcal{H})=\{I\}$, hence $\Aut({\mathfrak n }_{2,3})\simeq \mathcal G\ltimes_\tau (\mathcal{R}\times \mathcal{H})$ ([@Kn]). It is clear that $\mathcal{G}\simeq \GL(2,K)$, $\mathcal{R}\simeq K^3$ and $\mathcal{H}\simeq H_1$ and these isomorphisms preserve the action of $\GL(2,K)$ in $H_1$ and $K^3$. So the next result follows. \[pro5\] The group of automorphisms of ${\mathfrak n }_{2,3}$ is $$\Aut({\mathfrak n }_{2,3}) \simeq \GL(2,K)\ltimes (K^3 \times H_1),$$ where $H_1$ denotes the Heisenberg Lie group of dimension three. The aditional orthogonal condition leads the following result. The group of orthogonal automorphisms of $({\mathfrak n }_{2,3},B_{2,3})$ is $$\Auto({\mathfrak n }_{2,3}) \simeq \mathcal S \ltimes H_1,$$ where $\mathcal S$ is the subgroup of $\mathcal G$ consisting of the matrices $\tilde{A}$ with $A\in GL(2,K),\,\det(A)=\pm1$. The action in the semidirect product is the restriction of the action of $\mathcal G$ in the Heisenberg Lie group $H_1$ described before. The Lie algebra of derivations of ${\mathfrak n }_{2,3}$ is isomorphic to $$\mathfrak{gl}(2,K) \ltimes (\mathfrak h_1 \times K^3).$$ In particular, fix the ad-invariant metric $\la \,,\,\ra$ in $({\mathfrak n }_{2,3}, B_{2,3})$ given by the matricial representation as in (\[m3223\]). The set of skew-symmetric derivations is represented by the Lie algebra $$\Dera({\mathfrak n }_{2,3}) \simeq \mathfrak{sl}(2,K) \ltimes \mathfrak h_1$$ while the set of inner derivations is (isomorphic to) $\hh_1$. In the next paragraphs we describe $\Der({\mathfrak n }_{2,3})$ and $\Dera({\mathfrak n }_{2,3})$ explicitly as subalgebras of $\mathfrak{gl}(5,K)$ putting emphasis on the actions and representations. By canonical computations one verifies that an element in $\Der({\mathfrak n }_{2,3})$ has the following matricial representation in the basis $e_1,e_2,e_3,e_4,e_5$ $$\left( \begin{matrix} z_1+z_4 &-z_2 & 0 & 0 & 0 \\ -z_3 & z_1 -z_4& 0 & 0 & 0 \\ z_6& z_5 &2z_1 & 0 & 0 \\ z_7+z_9 & z_8 & z_5 & 3z_1+z_4 & -z_2\\ z_{10}& z_7-z_9 & -z_6 & -z_3 & 3z_1-z_4 \end{matrix} \right).$$ As usual, denote by $E_{ij}$ the 5$\times$5 matrix which has a 1 at the place $ij$ and $0$ in the other places. Consider the vector subspace of dimension three spanned by the matrices $$X = E_{32}+E_{43} \qquad Y= E_{31}- E_{53} \qquad Z=E_{41}+ E_{52}$$ which obey the Lie bracket relation $[X,Y]=Z$, thus this is a faithful representation of the Heisenberg Lie algebra $\hh_1$. Also denote by $$U= E_{42}\qquad V=E_{41}-E_{52}\qquad W=E_{51}$$ a basis of the vector subspace which spans an abelian Lie algebra of dimension three. The algebra $\mathfrak h_1 \times K^3$ is an ideal of the Lie algebra of derivations namely the radical. Denote by $E,F,H, T$ the matrices given by $$\begin{array}{rclrcl} E & = & -E_{12}-E_{45} & F & = & -E_{21}-E_{54} \\ H & = & E_{11}-E_{22} + E_{44}-E_{55} & T & = & E_{11}+E_{22}+2E_{33}+3E_{44}+2E_{55}. \end{array}$$ Thus $span \{E,F,H\}\simeq \mathfrak{sl}(2,K)$ and therefore $span \{E,F,H,T\}\simeq \mathfrak{gl}(2,K)=\mathfrak{sl}(2,K)\times K$. The action of $\mathfrak{gl}(2,K)$ on $\mathfrak h_1\times K^3$ preserves each of the ideals $\mathfrak h_1$ and $K^3$ respectively. The action of $\mathfrak{sl}(2,K)$ on $\mathfrak h_1$ is is given by derivations, that is $t \in \mathfrak{sl}(2, K) \simeq \Der(\hh_1)$, in the basis $X,Y,Z$, is represented by $$\left( \begin{matrix} h & e & 0\\ f & -h & 0 \\ 0 & 0 & 0 \end{matrix} \right).$$ The action of $\mathfrak{sl}(2,K)$ on $K^3$ is given by its irreducible representation in dimension three (see [@Hu]) $$\ad(eE+fF+hH)|_{U,V,W} = \left( \begin{matrix} 2h & 2e & 0\\ f & 0 & e\\ 0 & 2f & -2h \end{matrix} \right).$$ On the other hand the action of $T$ on $\mathfrak h_1$ is diagonal $$T \cdot X= X \qquad T \cdot Y = Y \qquad T \cdot Z = 2 Z;$$ and the action of $T$ on $K^3$ is twice the identity: $T \cdot A= 2 A$ for all $A \in K^3$. The Lie algebra ${\mathfrak n }_{3,2}$ -------------------------------------- Let $e_1, e_2, e_3, e_4, e_5, e_6$ denote the basis of ${\mathfrak n }_{3,2}$ given in (\[b32\]) above. A derivation of this Lie algebra has a matrix as follows $$\left( \begin{matrix} z_1+z_{2} & z_4 & z_6&0 & 0 & 0 \\ z_{5} & z_1+z_{3} & z_{8}&0 & 0 & 0 \\ z_{7}& z_9 &z_1 -z_{2}-z_3 &0 & 0 & 0 \\ -z_{12} & -z_{16}+z_{17} & z_{18} & 2z_1+z_{2}+z_3& z_{8}&-z_6\\ z_{14}+z_{15}& z_{10} & z_{16}+z_{17} & z_9 & 2z_1-z_3&z_4\\ z_{13} &z_{14}-z_{15}&z_{11}-z_{12}&-z_{7}&z_{5}&2z_1-z_{2} \end{matrix} \right).$$ Let $\mathfrak G$ and $\mathfrak R$ denote respectively the sets of matrices in $\Der({\mathfrak n }_{3,2})$ $$\mathfrak G=\left\{ \left( \begin{matrix} z_1+z_{2} & z_4 & z_6&0 & 0 & 0 \\ z_{5} & z_1+z_{3} & z_{8}&0 & 0 & 0 \\ z_{7}& z_9 &z_1 -z_{2}-z_3 &0 & 0 & 0 \\ 0 & 0 & 0 & 2z_1+z_{2}+z_3& z_{8}&-z_6\\ 0 & 0 & 0 & z_9 & 2z_1-z_3&z_4\\ 0 & 0 & 0&-z_{7}&z_{5}&2z_1-z_{2} \end{matrix} \right) \right\}$$ $$\mathfrak R= \left\{ \left( \begin{matrix} 0 &0 &0 &0 & 0 & 0 \\ 0 & 0 & 0 &0 & 0 & 0 \\ 0 & 0 & 0 &0 & 0 & 0 \\ -z_{12} & -z_{16}+z_{17} & z_{18} & 0 & 0 &0 \\ z_{14}+z_{15}& z_{10} & z_{16}+z_{17} & 0 & 0 &0 \\ z_{13} &z_{14}-z_{15}&z_{11}-z_{12}& 0 & 0 &0 \end{matrix} \right)\right\}.$$ With the usual conventions, denote by $E_{ij}$ the 6$\times$6 matrix which has a $1$ in the file $i$ and column $j$ and $0$ otherwise. Let $T$ and $f_i$, $i=1, \hdots , 8$, be the following matrices $$T = E_{11}+ E_{22} +E_{33}+2E_{44}+2E_{55}+2E_{66}$$ $$\begin{array}{cc} \begin{array}{rcl} f_1 & = & E_{11}-E_{33}+E_{44}-E_{66}\\ f_2 & = & E_{22}-E_{33}+E_{44}-E_{55}\\ f_3 & = & E_{12} + E_{56}\\ f_4 & = & E_{21} + E_{65} \end{array}& \begin{array}{rcl} f_5 & = & E_{13} -E_{46}\\ f_6 & = & E_{31} - E_{64}\\ f_7 & = & E_{23} + E_{45}\\ f_8 & = & E_{32} + E_{54} \end{array} \end{array} .$$ With the Lie bracket of matrices, the vector space spanned by $f_1, \hdots, f_8$ constitute a Lie algebra isomorphic to $\mathfrak{sl}(3,K)$, such that $[T,f_i]=0$ for all $i=1, \hdots , 8$. Hence one has the following isomorphism of Lie algebras $$\mathfrak G \simeq \mathfrak{sl}(3,K) \times K = \mathfrak{gl}(3,K).$$ For $i=1, \hdots 9$, let $A_i$ denote the matrices $$\begin{array}{rclrclrcl} A_1 & = & E_{52} & A_2 & = & E_{63} & A_3 & = & -E_{41}-E_{63}\\ A_4 & = & E_{61} & A_5 & = & E_{51}+E_{62} & A_6 & = & E_{51}-E_{62}\\ A_7 & = & -E_{42}+ E_{53} & A_8 & = & E_{42}+E_{53} & A_9 & = & E_{43} \end{array}$$ which generate the abelian Lie algebra $\mathfrak R$ of dimension nine. Actually this is a faithful representation of minimal dimension of the Lie algebra $K^9$ (see for instance [@Bu]). Hence $$\mathfrak R \simeq K^9$$ and it coincides with the radical of the Lie algebra of derivations of ${\mathfrak n }_{3,2}$. The action of $\mathfrak{sl}(3,K)$ on $K^9$ is given by the adjoint representation $f_i \cdot A_j=[f_i, A_j]$ for all $i,j$, while the action of $T$ on $K^9$ is represented by the identity map. Let ${\mathfrak n }_{3,2}$ denote the free 2-step nilpotent Lie algebra in three generators. The Lie algebra of derivations of ${\mathfrak n }_{3,2}$ is isomorphic $\mathfrak{gl}(3,K) \ltimes K^9$. For the ad-invariant metric $\la \,,\,\ra$ on $({\mathfrak n }_{3,2}, B_{3,2})$ as in (\[m3223\]) one has the next result. The set of skew-symmetric derivations of ${\mathfrak n }_{3,2}$ with the metric $\la \,,\,\ra$ is given by the Lie algebra $$\Dera({\mathfrak n }_{3,2})\simeq \mathfrak{sl}(3,K)\ltimes K^3,$$ while the set of inner derivations is isomorphic to $K^3$. One can easily check that $t\in \Dera({\mathfrak n }_{3,2})$ is skew-symmetric if and only if it belongs to the vector space spanned by $$\{f_i, i=1, \hdots 8,A_2 +\frac{1}{2} A_3, A_5, A_8\}$$ The elements $f_i$ generate a Lie algebra isomorphic to $\mathfrak{sl}(3,K)$ and $A_2 +\frac{1}{2} A_3, A_5, A_8$ span an abelian ideal, isomorphic to $K^3$. 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--- abstract: 'High redshift DLA systems suggest that the relative abundances of elements might be roughly solar, although with absolute abundances of more than two orders of magnitude below solar. The result comes from observations of the \[SII/ZnII\] ratio, which is a reliable diagnostic of the true abundance, and from DLA absorbers with small dust depletion and negligible HII contamination. In particular, in two DLA systems nitrogen is detected and at remarkably high levels (Vladilo et al. 1995, Molaro et al. 1995, Green et al. 1995, Kulkarni et al. 1996). Here we compare the predictions from chemical evolution models of galaxies of different morphological type with the abundances and abundance ratios derived for such systems. We conclude that solar ratios and relatively high nitrogen abundances can be obtained in the framework of a chemical evolution model assuming short but intense bursts of star formation, which in turn trigger enriched galactic winds, and a primary origin for nitrogen in massive stars. Such a model is the most successful in describing the chemical abundances of dwarf irregular galaxies and in particular of the peculiar galaxy IZw18. Thus, solar ratios at very low absolute abundances, if confirmed, seem to favour dwarf galaxies rather than spirals as the progenitors of at least some of the DLA systems.' author: - 'F. Matteucci , P. Molaro , G. Vladilo' date: 'Received date; accepted date' title: 'Chemical evolution of Damped Ly$\alpha$ systems' --- Introduction ============ Absorption line systems detected in the spectra of Quasi Stellar Objects (QSO) originate in intervening galaxies or protogalaxies. Among the different classes of absorbers, the damped Lyman $\alpha$ systems (DLA) are those characterized by N(HI) $\ge$ 10$^{20}$ cm$^{-2}$ and by showing many low ionization species such as FeII, SiII, CrII, ZnII and OI, which are in their dominant ionization stage for a HI region. The high accuracy in the hydrogen column density determination derived from the damped wings and the absence of significant ionization corrections allow accurate absolute abundance determinations of the gas phase elements. Since DLA systems are observable up to the highest redshift they provide a unique tool for the study of the cosmic chemical history of the early universe. Such observations are complementary to the Hubble Deep Field ones (Mobasher et al. 1996) which show that high redshift galaxies have a large variety of morphological types including many peculiar objects with traces of interaction and merging: interestingly, a large fraction are starburst galaxies, as suggested by their colors. The relative elemental abundances rather than absolute ones are a valuable diagnostic of the first elemental buildup. This is because absolute abundances are affected by all the model assumptions, whereas abundance ratios generally depend only on the assumed nucleosynthesis and stellar lifetimes. The way of using the information contained in abundance ratios is to compare the observed ratios with predictions from chemical evolution models which take into account detailed stellar nucleosynthesis and lifetimes. As in the Milky Way, we hope to distinguish halo-like abundances, produced by type II SNe and characterized by enhancements of $\alpha$ elements with respect to the iron-peak elements, from disk-like abundances, produced by the cumulative effect of both type Ia and type II SNe, where the abundance ratios become progressively solar. The halo abundance pattern follows from the fact that Fe arises mostly from type Ia supernovae with some contribution from type II, while the reverse is true for Si and O. The longer lifetimes of the SNIa, which are believed to have progenitors with masses of 1-8 M$_{\odot}$, result in a delayed iron enrichment compared to the major SNII products such as $\alpha$-elements (Tinsley, 1980; Greggio and Renzini, 1983a; Matteucci and Greggio, 1986). Therefore, the behaviour of the \[$\alpha$/Fe\] ratios is mainly dependent on the relative lifetimes of type Ia and II supernovae. Another key element is nitrogen and the ratio N/O. This element, in fact, is thought to originate mainly from low and intermediate mass stars and to be a “secondary” element, in the sense that it is produced proportionally to the initial stellar metallicity. As a consequence nitrogen is restored into the interstellar medium with a large temporal delay relative to oxygen, which is produced in massive stars and is a “primary” element, namely its production is independent of the initial stellar metallicity. For this reason, the $\alpha$/Fe and N/O ratios can be used as cosmic clocks and represent a clue in understanding the nature of high redshift objects such as QSO (Hamman and Ferland, 1993; Matteucci and Padovani, 1993) and DLA systems (Matteucci, 1995). The DLA systems are generally believed to be the progenitors of the present-day spiral galaxies (Wolfe et al. 1986). However, it has also been suggested that they could be dwarf galaxies, since they have a large amount of gas, low metallicities and low dust-to-gas ratios (Pettini et al. 1990;Meyer and York 1992; Steidel 1994). Whether DLA are proto-spirals or proto-dwarfs can be indicated by their observed abundance pattern. To this purpose in this paper we will present models of chemical evolution for galaxies of different morphological type and compare them with the observed abundances in DLA. Observed abundances in damped Ly$\alpha$ systems ================================================ So far chemical abundances have been measured in a set of DLA systems and with a variety of resolutions and accuracies by Black et al. (1987), Meyer & York (1987), Chaffee et al. (1988), Meyer et al. (1989), Meyer & Roth (1990), Rauch et al. (1990), Pettini et al. (1990), Meyer & York (1992), Pettini et al. (1994), Fan & Tytler (1994), Wolfe et al. (1994), Pettini et al. (1995), Lu et al. (1995a), Steidel et al. (1995), Lu et al (1995b). Here, we would like to focus on the abundances which could be used to compare with chemical evolution models and in particular with the few nitrogen detections so far reported in the literature. A summary of the abundances for the systems for which nitrogen detection or upper limits are available are reported in Table 1 and 2. All the original abundances have been renormalized to the solar abundances of Anders & Grevesse (1989), with the exception of $\log (Fe/H)_{\sun}$=$-4.52$ taken from Hannaford et al. (1992) (\[M/H\] $\equiv$ log(M/H)-log(M/H)$_{\sun}$). The abundances in the Tables are derived from column density ratios of different atomic and ionic species in the gas phase, and can be translated to elemental abundances only if ionization effects and dust depletion are negligible or can be accounted for. In the following we discuss shortly these two effects.    QSO $z_{\rm abs}$ N(HI)    \[N/H\]    \[O/H\]    \[Si/H\]    \[S/H\]    \[Fe/H\] Refs. ----------- --------------- ------- ------------------------------ ------------------------- ------------------------- ------------------------- ------------------------- --------- 0000-26 3.3901 21.30 $-2.77^{+0.17}_{-0.17}$ $-3.13^{+0.17}_{-0.17}$ $-2.48^{+0.19}_{-0.19}$     — $-2.38^{+0.16}_{-0.16}$   1 1331+1700 1.7765 21.18 $-2.73^{+0.11}_{-0.11}$ $-2.81^{+0.21}_{-0.30}$     — $-1.35^{+0.11}_{-0.11}$ $-2.25^{+0.11}_{-0.12}$   2$^a$ 2348-147 2.2794 20.57 $<-3.15$ $-2.13^{+2.26}_{-0.58}$ $-1.97^{+0.13}_{-0.10}$ $-1.91^{+0.14}_{-0.16}$ $-2.35^{+0.22}_{-0.10}$   3 2344+124 2.5379 20.43 $-3.00^{+0.12}_{-0.22}$ $^b$ $-2.11^{+2.5}_{-0.36}$ $-1.72^{+0.27}_{-0.13}$ $<-1.20$ $-1.85^c$   4 1946+7658 2.8443 20.30 $<-3.26$ $^d$ $-2.65^{+0.15}_{-0.15}$ $-2.16^{+0.10}_{-0.10}$ $<-0.89$ $-2.41^{+0.10}_{-0.10}$   5$^e$ 0.4 truecm References for Table 1\ (1) Molaro et al. (1995) ; (2) Green et al. (1995) ; (3) Pettini et al.(1995);\ (4) Lipman, thesis (1995) ; (5) Lu et al. (1995a) 0.4 truecm $a$ : Hydrogen column density from Pettini et al. (1994); metal abundance error bars include propagation of the N(HI) error. $b$ : Nitrogen value derived by using only the main absorption component of the system; $c$ : Error bar not reported by the author. $d$ : Upper limit from Lipman (1995; thesis). $e$ : Column densities from line profile fitting where all the lines are fitted simultaneously; the final choice of Lu et al (1995a) for O is \[O/H\]$>$ -2.99; metal abundance error bars include propagation of the N(HI) error.    QSO $z_{\rm abs}$    \[N/O\]    \[N/Si\]    \[N/S\]    \[Si/O\]    \[S/O\]    \[O/Fe\]    \[Si/Fe\] --------------- --------------- ------------ ------------- ------------ ------------- ------------ ------------- -------------- 0000-26 3.3901 +0.36 $-$0.29 —  +0.65 —  $-$0.75 $-0.10$ 1331+1700 1.7765 +0.08 —  $-$1.38 —  +1.46 $-$0.56 —  2348-147 2.2794 $<-1.02$ $<-1.18$ $<-1.24$ +0.16 +0.22 +0.22 +0.38 2344+124 $^a$ 2.5379 $-0.59$ $-1.14$ $>-1.32$ +0.55 $<+0.73$ $-0.59$ +0.13 1946+7658 2.8443 $<-0.61$ $<-1.10$ —  +0.49 $<+1.76$ $-0.24$ +0.25 0.5 truecm $a$ Values derived by using only the main absorption component Ionization corrections and dust depletions ------------------------------------------ The neutral hydrogen in the DLA is optically thick to the ionizing radiation either from the intergalactic background or from starlight inside the DLA, and the abundances derived from species that have ionization potentials in excess of 13.6 eV do not require ionization corrections. This is the case of abundances derived from NI, OI, AlII, SiII and FeII, which are dominant ionization stages in the HI gas in our own Galaxy. Detailed ionization models for intergalactig radiation field at high redshift explored by Lu et al (1995) and Fan and Tytler (1994) show that ionization corrections are indeed minimal for systems with $\log$ N(HI)$>$ 20. Another effect which could affect the abundance determinations is the presence of HII regions within the DLA intercepted by the line of sight (Steigman et al 1975). These regions of warm ($T$ $\simeq$ 10$^4$ K) ionized gas would contribute to the column densities of singly ionized species, such as FeII, SiII and AlII, but not to the HI column density. In the interstellar gas of our Galaxy the column density contribution of HII regions is generally negligible when the HI column density is as high as it is in DLA systems (N(HI) $\ga$ 10$^{20}$ cm$^{-2}$). If HII regions were more important in DLA than in our Galaxy they would produce different line profiles between neutral and ionized lines, when observed at high resolution. This effect is not observed in the DLA towards QSO 0000-2169 where the $b$ values, velocities and profiles of singly ionized species are the same as those of the neutral species, suggesting that both form in the same slab of HI material. Extra contributions to the absorption from HII regions in this DLA are also constrained by the lack of detection of NII, resulting in NII/NI $<$ 0.4. Viceversa, Green et al. (1995) interpret the abundances of the DLA towards MC3 1331+170 invoking an extraordinary amount of ionized gas, namely six times the neutral gas, which makes rather uncertain the abundances of ionized species. Highly ionized gas in the form of SiIV and CIV is frequently observed in the DLA. Sometimes the highly ionized gas is found at the same velocities of the neutral one but often is found at different velocities and therefore is very probable that highly ionized gas arises either in regions disconnected from the neutral material, or in the interfaces between neutral material and the intergalactic medium. Presence of dust will selectively deplete the elements observed in the gas phase. In the dense clouds in the Galaxy the interstellar depletion of Al and Fe can reach $\simeq$ 2 dex, that of Si $\approx$ 1 dex, while, on the other extreme the O depletion is $\le$ 0.4 dex, that of N $\le$ 0.2 dex and S essentially undepleted (Jenkins 1987). Reddening measurements of QSO with DLA systems suggest a dust-to-gas ratio in the DLA of about 10% of that in our Galaxy (Fall and Pei 1989). In the survey of ZnII and CrII in DLA by Pettini et al. (1994) chromium is typically one dex below zinc, while in the Galaxy is about two dex, showing the presence of dust and a reduced dust depletion in the DLA systems. If some dust were present this would alter the abundance determinations of refractory elements, such as Fe, Si and Al. Much more reliable are the abundances of the non-refractory elements, such as Zn, S, O and N. $\alpha$ versus iron-peak elements ---------------------------------- Of particular importance for understanding galactic chemical evolution is the comparison between $\alpha$ and iron peak elements. In particular, the ratio between sulphur and zinc is an important diagnostic tool. Both elements show little affinity with dust and their ratio is also safe against possible contributions from HII regions, since these essentially cancel out. Considering that S and Zn have different nucleosynthetic origin with S mainly a product of type II SN and Zn of type Ia SN (Matteucci et al. 1993), the \[SII/ZnII\] ratio is an ideal diagnostic for understanding the character of the chemical evolution. So far, only few determinations of \[S/Zn\] ratio in DLA are present in the literature. Meyer et al. (1989) found \[S/Zn\]=$-$0.1 in the DLA at $z=2.8$ towards QSO PKS 0528-250 and Green et al. (1995) found \[S/Zn\] = +0.1 in the $z$=1.775 DLA in Q1331. Thus, in spite of the very poor metallicity of the gas, which in these systems is $\simeq$ -2.0, the ratios of elements non depleted in dust have solar values. In DLA systems the relative abundances of Si and Fe are often found to be consistent with a halo pattern. Lu et al. (1995a) found \[Si/Fe\]=+0.31 for the DLA at $z=2.844$ towards HS 1946=76 and Pettini et al. found \[Si/Fe\]=+0.4 for the DLA at $z=2.27$ towards QSO 2348-147. However, Si and Fe are differentially depleted from gas to dust in our Galaxy. The average value in the compilation of 11 clouds observed with HST made by Lu et al (1995a, their Table 8) is \[Si/Fe\]=+0.66$\pm0.26$. The observed enhancement of Si versus Fe in the DLA would reflect the overabundances of $\alpha$ elements with respect to the iron-peak elements only in complete absence of dust. Since DLA show some amount of dust, a moderate enhancement of Si over Fe cannot be considered as a clear-cut evidence of a halo-like pattern. Moreover, values of \[Si/Fe\]$\ge$0.6, that would be expected if differential depletion and intrinsic $\alpha$ enhancement are present, have not yet been observed. Nitrogen and oxygen abundances ------------------------------ Nitrogen has been detected in the DLA at z=3.39 towards QSO 0000-26, with \[N/H\] = $-$2.77 $\pm 0.17$ (Vladilo et al. 1995 and Molaro et al. 1995) and in the DLA at $z$=1.78 towards MC3 1331+170 with \[N/H\]=-2.7 (Green et al. 1995). A significant upper limit has been derived by Pettini et al. (1995) in the DLA systems towards QSO 2348-147 NI$<$-3.15. By means of co-addition technique of the lines of the 1200 Å triplet Lipman (1995) achieved a marginal detection of NI for the main component of the DLA towards QSO 2344+124. By using the data published by Fan and Tytler (1994), Lipman derived also an upper limit of \[N/H\] $<$ -3.26. These measurements show that a real dispersion in the nitrogen abundances may be present among the DLAs. In order to understand the nucleosynthetic origin of nitrogen the ideal would be to follow its abundance with respect to that of oxygen. Unfortunately, oxygen abundance is generally given with a large uncertainty. This is because oxygen abundance is derived from the OI 1302.1685 Å line which has log gf=1.804 and is generally saturated. The other line OI 1355.5977, has log gf=-2.772 and is generally too faint to place useful upper limits. The line saturation leaves the line broadening poorly constrained and leads to the large uncertainty in the oxygen abundance. However, the $b$ value can be inferred from other lines under the assumption that they are formed in the same material and that the origin of broadening is not thermal. Lu et al. (1995a) and Molaro et al. (1995) use the $\it b$ value from the other observed species thus removing the large uncertainty in the oxygen abundance (cfr. Table 1). In Molaro et al. (1995). the broadening value is taken not only from ionized species, which might be sensitive to extra contribution from HII gas along the line of sight, but also from NI which is forming in the same material as neutral oxygen. It is rather striking that in the cases with oxygen abundances with small associated errors, oxygen turns out to be remarkably deficient relatively to the other elements measured, such as Si, S and Fe as it is possible to see in Table 2. To circumvent the problem of oxygen uncertainty, Pettini et al. (1995) and Lipman (1995) take Si or S as a proxy for oxygen assuming \[O/Si\] = \[O/S\]=0. This assumption is about true for the halo stars of our own Galaxy (however see later for silicon), but is rather risky for primeval galaxies which might have experienced a different chemical evolution from that of our own Galaxy. In particular, Si and O do not have the same nucleosynthetic origin, silicon may be affected by dust depletion and SII and SiII may be also affected by HII contribution. Also by adopting the Pettini et al. approach for the case of QSO 0000-2169 we have \[N/Si\] = $-0.3$, which stands genuinely high when compared to the other determinations. What chemical evolution? ======================== Some DLA seem to show similarities to dwarf irregular and blue compact galaxies in terms of abundance pattern. These galaxies, in fact, exhibit relatively high N/O, although with a large spread, at a low overall metallicity Z ranging from 1/10 to 1/30 of the solar value (see Matteucci, 1995). In particular, IZw18 is the galaxy with the lowest metal content known locally, and a N/O ratio roughly solar. Recently, a single starburst chemical evolution model has been successfully developed by Kunth et al. (1995) to reproduce the observed abundances of IZw 18. Kunth et al. (1995) applied Marconi’s et al. (1994) model where star formation is assumed to proceed in short but intense bursts. The contribution to the chemical enrichment of SNe of different type (II, Ia and Ib) is taken into account. The novel feature is the introduction of differential galactic winds where some elements are preferentially lost from the galaxy relative to others. Recently, more and more observational evidence is growing about the existence of such galactic winds in dwarf irregulars as provided by Meurer et al. (1992), Lequeux et al. (1995) and Papaderos et al. (1994). These winds are found to be a crucial ingredient in the model of Kunth et al. (1995) in order to reproduce the high observed N/O ratio in IZw18. To explain the nitrogen abundance of IZw18 this same model assumes also a primary N production from massive stars, as described in the next section. In this paper we will show the predictions of models similar to that of Kunth et al. together with the predictions of models for the chemical evolution of the solar region, under different assumptions about the production and nature of nitrogen, and we will compare them with the DLA data. The chemical evolution model for dwarf irregulars ================================================= We are using here the same model adopted by Kunth et al. (1995) which is aimed at describing the evolution of IZw18, namely of an object presently having a strong burst of star formation which induces a galactic wind. The main features of this chemical evolution model are the following: 1\) one-zone, with instantaneous and complete mixing of gas inside this zone, 2\) no instantaneous recycling approximation; i.e. the stellar lifetimes are taken into account, 3\) only one intense burst of star formation is assumed to occur, 4\) the evolution of several chemical elements (He, C, N, O, Fe) due to stellar nucleosynthesis, stellar mass ejection, galactic wind powered by SNe and infall of primordial gas, is followed in detail. If G$_i$ is the fractional mass of the element i, its evolution is given by the equations described in Marconi et al. (1994). $$\dot G_i(t) = -\psi(t)(1+w_{i}) X_i(t)+$$ $$\int_{M_{L}}^{M_{Bm}}{\psi(t-\tau_m) Q_{mi}(t-\tau_m)\phi(m)dm}+$$ $$A\int_{M_{Bm}}^{M_{BM}}{\phi(m)}\bigl[\int_{\mu_{min}} ^{0.5}{f(\mu)\psi(t-\tau_{m1}) Q_{mi}(t-\tau_{m2})d\mu\bigr] dm}+$$ $$(1-A)\int_{M_{Bm}}^ {M_{BM}}{\psi(t-\tau_{m})Q_{mi}(t-\tau_m)\phi(m)dm}+$$ $$\int_{M_{BM}}^{M_U}{\psi(t-\tau_m)Q_{mi}(t-\tau_m)+ \phi(m)dm} + \dot G_{i}(t)_{inf} \eqno(1)$$ where $G_i(t)$=$M_{gas}(t)X_i(t)/M_{tot}(t_{G})$ is the mass density of gas in the form of an element [*i*]{} normalized to the total mass at the present time $t_{G}$. The quantity $X_i(t) = G_i(t)/G(t)$ represents the abundance by mass of the element $i$ and by definition the summation over all the abundances of the elements present in the gas mixture is equal to unity. The quantity $G(t) = M_{gas}/M_{tot}$ is the total fractionary mass of gas, and $M_{tot}$ refers only to the mass present in the form of gas in the star forming region. The possible presence of a dark matter halo is not considered here, given the simple treatment of the development of a galactic wind, as we will see in the following. The star formation rate we assume during the burst, $\psi(t)$, is defined as: $$\psi(t)\,=-\nu \, \eta(t) \, G(t)$$ where $\nu$ is the star formation efficiency (expressed in units of Gyr$^{-1}$), and represents the inverse of the timescale of star formation, namely the timescale necessary to consume all the gas in the star forming region; $\eta(t)$ takes into account the stochastic nature of the star formation processes as in Gerola et al. (1980) and Matteucci and Tosi (1985), where a detailed description can be found. The galactic wind is assumed to be simply proportional to the star formation rate (namely to the rate of explosion of type II SNe). This is a reasonable choice since the duration of the burst is so short that it does not allow the explosion of type Ia SNe, which occur all after the burst. The simplicity of the treatment of the galactic wind avoids also the introduction of other unknown parameters such as the efficiency of energy tranfer from stars and supernovae to the ISM. The rate of mass loss via a galactic wind is defined as follows: $$\dot G_{iw}(t)\,=\, -w_{i}\psi(t)X_{iw}(t)$$ where $X_{iw}(t)=X_{i}(t)$ and $w_{i}$ is a free parameter containing all the information about the energy released by SNe and the efficiency with which such energy is transformed into gas escape velocity (note that Pilyugin (1992;1993) defines a wind parameter which is the inverse of $w_{i}$, namely the ratio of the star formation rate to the wind rate). The value of w$_{i}$ has been assumed to be different for different elements. In particular, the assumption has been made that only the elements produced by type II SNe (mostly $\alpha$-elements and some iron) can escape the star-forming regions. We have made this choice following the conclusions of Marconi et al. (1994) and Kunth et al. (1995), who showed that models with differential wind can explain better the observational constraints of blue compact galaxies in general and IZw18 in particular. The justification for the existence of differential galactic winds can be found in the fact that during short starbursts type II SNe dominate. Since SNII explode in association, they are likely to produce chimneys which will eject metal enriched material (De Young and Gallagher, 1990). On the other hand, type Ia SNe are not likely to trigger a wind since they explode mostly during the interburst phase and have a large range of explosion times (from $3 \cdot 10^{7}$ to a Hubble time) inducing them to explode in isolation. The terms on the right side of equation (1) represent, respectively, the rate at which the gas is lost via astration and galactic wind and the rates at which the matter is restored to the interstellar medium (ISM) by: \(a) single stars with masses between $M_L=1.0M_{\odot}$, which is the lowest mass contributing to the galactic enrichment, and ${M_{B_m}}=3.0M_{\odot}$, which represents the minimum mass for which a binary system (with the minimum mass ratio $\mu_{min}$ defined as in Greggio and Renzini 1983b) produces a type Ia SN, \(b) binary systems producing type Ia SNe, within the range of masses ${M_{B_m}}$=${3 M_{\odot}}$ and ${M_{B_M}}$=${16 M_{\odot}}$ \[the assumed progenitors of type Ia SNe are binary systems of C-O white dwarfs(Whelan and Iben, 1973); the parameter $A$, defined in Greggio and Renzini (1983b), represents the fraction of binary systems in the IMF which can give rise to type Ia SNe; \(c) single stars in the mass range ${M_{B_m}}$ - ${M_{B_M}}$ which end their lives either like white dwarfs or type II SNe, \(d) single stars in the mass range ${M_{B_M}}$ and ${M_U}$, where ${M_U}$ is the maximum stellar mass contributing to the galactic enrichment (100 ${M_{\odot}}$ in our models). These stars can either end their lives as type II or type Ib SNe. The initial mass function (IMF) by mass, $\phi(m)$, is expressed as a power law with an exponent x=1.35 over the mass range $0.1-100M_{\odot}$. The chemical evolution equations include also an accretion term: $$\dot G_{iinf}(t)\,=\,{ {(X_{i})_{inf}e^{-t/\tau}} \over {\tau(1-e^{-t_{G}/\tau})} }$$ where (X$_{i})_{inf}$ is the abundance of the element [*i*]{} in the infalling gas, assumed to be primordial, $\tau$ is the time scale of mass accretion and $t_{G}$ is the galactic lifetime. This accretion term may simulate the formation of dwarfs as the result of mergers of smaller subunits. The parameter $\tau$ has been assumed to be the same for all blue compact galaxies and short enough to avoid unlikely high infall rates at the present time ($\tau=0.5\cdot 10^{9}$ years). It is worth noting that a shorter timescale would not produce a noticeable effect on the results. Nucleosynthesis prescription ---------------------------- The term ${{Q_m}_i(t-{\tau_m})}$ in equation (1), the so called [*production matrix*]{} (Talbot and Arnett, 1973), represents the fraction of mass ejected by a star of mass $m$ in the form of the element $i$. The quantity $\tau_{m}$ represents the lifetime of a star of mass m and $\tau_{m2}$ refers to the mass of the secondary component and $\tau_{m1}$ to the mass of the primary component of a binary system giving rise to a type Ia SN (see Matteucci and Greggio, 1986 for details). For the nucleosynthesis prescriptions we have assumed the following: a\) For low and intermediate mass stars (0.8 $\leq$ M/M$_{\odot} \leq$ M$_{up}$) we have used Renzini and Voli’s (1981) nucleosynthesis calculations for a value of the mass loss parameter $\eta$ = 0.33 (Reimers 1975), and mixing length $\alpha_{RV}$ = 1.5. The standard value for M$_{up}$ is 8 M$_{\odot}$. b\) For massive stars (M $>$ 8 M$_{\odot}$) we have used Woosley’s (1987) nucleosynthesis computations but adopting the relationship between the initial mass $M$ and the He-core mass M$_{He}$, from Maeder and Meynet (1989). It is worth noting that the adopted M(M$_{He}$) relationship does not substantially differ from the original relationship given by Arnett (1978) and from the new one by Maeder (1992) based on models with overshooting and Z=0.001. These new models show instead a very different behaviour of M(M$_{He}$) for stars more massive than 25 M$_{\odot}$ and Z=0.02, but the galaxies we are modelling never reach such a high metallicity. c\) For the explosive nucleosynthesis products, we have adopted the prescriptions by Nomoto et al. (1984), model W7, for type Ia SNe, which we assume to originate from C-O white dwarfs in binary systems (see Marconi et al. (1994) for details). More recent nucleosynthesis calculation by Thielemann et al. (1993) do not significantly differ from the Nomoto et al. (1984) ones. The nucleosynthesis of nitrogen and the N/O ratio ------------------------------------------------- Nitrogen is a key element to understand the evolution of galaxies with few star forming events since it needs relatively long timescales as well as relatively high underlying metallicity to be produced. The reason is that N is believed to be mostly a product of secondary nucleosynthesis, being produced by CNO processing of $^{12}$C or $^{16}$O from earlier generations of stars. However, a primary component can be obtained when the seed nuclei of $^{12}$C or $^{16}$O are produced in earlier helium burning stages of the same star. Generally, N is believed to be secondary in massive stars, and mostly secondary and probably partly primary in low and intermediate mass stars. However, some doubts exist at the moment on the amount of primary nitrogen which can be produced in intermediate mass stars due to the uncertainties related to the occurrence of the third dredge-up in asymptotic giant branch stars (AGB). In fact, if Blöcker and Schoenberner (1991) calculations are correct, the third dredge-up in massive AGB stars should not occur and therefore the amount of primary N produced in AGB stars should be strongly reduced (Renzini, private communication). The only possible way to produce a reasonable quantity of N during a short burst (no longer than 20 Myr), as discussed in Kunth et al. (1995), is to require that massive stars produce a substantial amount of primary nitrogen. This claim was already made by Matteucci (1986) in order to explain the \[N/O\] abundances in the solar neighbourhood. Recently, calculations by Woosley and Weaver (private communication) seem to support the possibility that N is produced by C and O synthesized inside massive metal poor stars as a primary element. Therefore, as in Kunth et al. (1995) we have taken this possibility into account in the present calculation. Theoretical prescriptions for the evolution of the solar vicinity ================================================================= The model adopted for the evolution of the solar neighbourhood is the same as in Matteucci and François (1992) and the basic equations are similar to Eq (1), the only difference being the absence of a galactic wind, namely all the $w_{i}=0$. The main differences between the model for dwarf irregulars and the model for the solar vicinity is that in the latter case the star formation rate is continuous and the IMF for massive stars is steeper than a Salpeter IMF (i.e. Scalo 1986). The time scale for the formation of the disk in the solar region is also different from the timescale assumed for assembling the dwarf irregulars (i.e. $\tau=0.5 $Gyr), namely is $\tau=3$ Gyr. This choice ensures that the majority of the observational constraints in the solar neighbourhood are reproduced. The nucleosynthesis prescriptions adopted for the solar neighbourhood are exactly the same as for the dwarfs. Results ======= Besides Model 1 and 5 of Kunth et al. (1995) (model 1 and 3 respectively, in Table 3), we run several models by varying star formation efficiency and/or wind efficiency, all the other parameters being left the same as in Model 5 of Kunth et al. The model parameters are presented in Table 3. In column 1 are the model numbers and in column 2 the nucleosynthesis prescriptions. In particular, “STANDARD” refers to the prescriptions described in section III, whereas “N PRIMARY” refers to the assumption of primary production of N in massive stars, leaving all the rest unchanged. \[\] ----- ------------- ------ -------------- --------- ---------- MOD      yields wind $\nu$ $N_{B}$ duration ($Gyr^{-1}$) (Myrs) 1 STANDARD 80 10 1 50 2 N PRIMARY 700 50 1 20 3 N PRIMARY 80 50 1 20 4 N PRIMARY 700 1000 1 20 5 N PRIMARY 1000 50 1 20 6 N PRIMARY 80 50 4 20 7 NPRIMARY 700 50 4 20 ----- ------------- ------ -------------- --------- ---------- : Model parameters :   IMF slope: 1.35 (Salpeter);  upper mass limit:\ 100 M$_\odot$; lower mass limit: 0.1 M$_\odot$ In column 3 we show the wind parameter [*$w_{i}$*]{} as defined in the previous section. This parameter is different from zero only for the elements produced and dispersed during the explosion of SNe II. The value for this parameter given in Table 3 refers only to the $\alpha$-elements, which are the main outcome of SN II explosions. In particular, for the elements studied here $w_{i}$ is zero for H, He and N, whereas is different from zero, but smaller than for $\alpha$-elements, for C and Fe. The parameter $w_{i}$ for these elements is chosen in such a way to account for the fact that Fe and C are produced both in massive and in intermediate mass stars (type Ia SNe and AGB stars, respectively). In column 4 is shown the star formation efficiency $\nu$ (in units of Gyr$^{-1}$), as defined in the previous section. In column 5 is shown the number of bursts and in column 6 the duration of each burst in Myr. Such a duration is constrained by results of population synthesis models (Mas-Hesse and Kunth, 1991) suggesting a maximum duration for the present burst in IZw18 of 20 Myr. The starting time of the burst is not important for models 1-5 since we assume that it is the only event of star formation and it could happen at any time (i.e. at any redshift). It is worth noting that the assumed range of variation of $\nu$ and $w_{i}$ is quite large but reasonable. In order to reproduce the star formation rate of IZw18 a value of $\nu$=50 Gyr$^{-1}$ (note that for the solar neighbourhood $\nu$=0.5 Gyr$^{-1}$) is preferred since it predicts a star formation rate of $\simeq 0.03 M_{\odot}$yr$^{-1}$, in very good agreement with observational estimates (see Kunth et al., 1995). An efficiency of star formation of 1000 Gyr$^{-1}$ is also plausible for starburst galaxies since it predicts a star formation rate of $\simeq 0.1M_{\odot}$yr$^{-1}$. Such star formation rates are quite reasonable for objects suffering only few bursts of star formation (may be only one) and having a large amount of gas. The wind parameter also spans a quite large range of values and this is possible under the assumption of enriched galactic wind, when only metals are lost. On the other hand, in the case of normal wind the value of $w_{i}$ is constrained by the condition of not destroying the galaxy. However, it is worth noting that even a totally disruptive wind, could in principle be considered. The initial mass of gas involved in the burst is assumed to be $6 \cdot 10^{6}M_{\odot}$. \[\] ---------------- --------------- ------------- ---------------- QSO $z_{\rm abs}$ 12+log(O/H) log(N/O)  0000-26 3.3901 5.80 $-0.52$ 1331+1700 1.7765 6.12 $-0.80$ 2348-147 $^a$ 2.2794 6.80 $< -2.20$ $^a$ 2344+124 $^a$ 2.5379 6.82 $-2.0$ $^a$ 1946+7658 $^a$ 2.8443 6.28 $< -2.0$ $^a$ ---------------- --------------- ------------- ---------------- : Measured (O/H) and (N/O) column density ratios in DLA systems to be compared with the figures. $(a)$ Values taken from Lipman (1995). The N/O versus O/H distribution is shown in Fig. 1 where the models with only one burst together with the models for the solar neighbourhood are shown. The dot-dashed curve is from model N. 5 from Kunth et al. (1995) (model 3 of table 3) and it can reproduce the observations of IZw 18. Incidentally, we note that this model can fit very well the N/O upper limit by Pettini et al. (1995) but for a burst age shorter than 20 Myr. On the other hand, Pettini’s et al.’s point, as well as the other points from Lipman (1995), could be marginally reproduced also by the evolution of a disk galaxy in the earliest phases of its evolution; in particular during the halo phase, as it is shown in the figure. In fact, the continuous curves in Fig. 1 represent the predictions from the model of chemical evolution of the solar neighbourhood under different assumptions about the nucleosynthesis of N. In particular, the area delimited by those models goes from purely secondary N in stars of all masses (the straight line at the right end) to primary N in massive stars and secondary and primary N in low and intermediate mass stars. On the other hand, the N/O ratios observed by Molaro et al.(1995) and Green et al.(1995) are reproduced by models with a strong starburst and very strong galactic wind (models 4 and 5 in Table 2), so that models for the solar neighbourhood seem to be completely ruled out in explaining these two DLA systems. Thus we show that it is possible to have large N/O ratios even at low O abundances if the nitrogen produced by massive stars, restored on relatively short time scales, is primary and with a strong differential effect in the galactic wind. As shown in Fig. 2 the same model produces solar ratios of alpha-elements (such as S, O and Si) to iron peak elements (such as Fe and Zn), as it is observed at least in few DLA. In particular, in Fig. 2 we show the predictions of the models for O and Si relative to iron to be compared with the data of Table 4. The x-axis does not extend to abundances larger than \[Fe/H\]=-2.0 since in one-burst models the metallicity does not increase any further. Sulphur is not shown here but it should closely follow oxygen. On the other hand, the predicted \[Si/Fe\] is lower than \[O/Fe\] and the reason for this resides in the fact that more Si than O is produced in type Ia SNe (Nomoto et al. 1984; Thielemann et al. 1993) and that we assume the same $w_{i}$ parameter for Si and O. Studies of abundances in halo stars do not allow us to discriminate clearly on this point (François, 1986; Ryan et al. 1991; Primas et al. 1994; McWilliam et al. 1995). Therefore, it does not seem safe to assume \[Si/O\]=0 in order to derive oxygen abundances in DLA systems (see Lipman, 1995). However, as indicated in Figure 2 the observed \[O/Fe\] is lower than \[Si/Fe\] for the DLA observed by Molaro et al. (1995). A possible explanation for this is that, in the framework of the differential galactic wind, O, which originates mostly in type II SNe, should be lost from the galaxy in a larger percentage than silicon, which originates also from type Ia SNe. In other words, Si should be treated as we do with C and Fe. However, numerical experiments show that even in this case is difficult to invert the situation and have \[Si/Fe\] higher than \[O/Fe\], as indicated by the data taken at face values! In Figure 3 we show the predicted N/O vs. O/H from models with 4 short bursts (20 Myr each) all occurring inside the first Gyr from the start of star formation, so that they can be representative of high redshift objects. In particular, we show the predictions of model 6 and 7 which are similar to model 3 and 2, respectively, but with four bursts of star formation.These models show that in principle the DLA systems from Green et al. (1995) and from Molaro et al. (1995) could be explained by a galaxy like IZw18 experiencing more than one burst of star formation and observed during the interburst period, without invoking an extremely large wind parameter. In fact, during this phase the N/O ratio increases as due to the fact that oxygen is no more produced while N continues to be restored from low and intermediate mass stars. This is a well known effect as shown by Pilyugin (1992;1993). In Figs. 4 and 5 we show the \[O,Si/Fe\] ratios predicted by model 6 and 7, respectively. Here too we can see the oscillating behaviour due to the alternating burst and quiescent phases. In particular, in both models the \[O,Si/Fe\] ratios decrease during the interburst phase and increase again during the bursts. The difference between the two models resides in the efficiency of the wind in model 7, which reflects in a stronger variation of the abundance ratios between the burst and the interburst phases. The reason for this variation in model 7 is that the stronger wind acting during bursts is responsible for very low absolute abundances at the end of each burst, so the increase of Fe during the interburst is stronger relatively to model 6 where the absolute abundances at the end of each burst are higher. Finally, it is worth noting that current models for elliptical galaxies (see Matteucci and Padovani, 1993) would never reproduce such the high N/O ratio observed by Molaro et al. (1995) for such low metallicities. In Figure 1, for example, the predictions of models for an elliptical galaxy of initial luminous mass $10^{11} M_{\odot}$ would lie at the right side of the solar neighbourhood curve for only secondary nitrogen, and even in the case of primary N from massive stars would be completely outside of the metallicity region where the DLA systems are observed. Conclusions =========== The abundance pattern of non refractory elements in some primeval DLA systems at high redshift seems not to follow the pattern observed in the halo of the Milky Way. This suggests that these objects had a different chemical evolution than the Milky Way and the spirals in general, at variance with the general belief that DLA are the progenitors of present day spirals. We have shown that the most promising models to explain the observed abundances are those succesfully applied to the dwarf irregular galaxies such as IZw18 (Marconi et al. 1994, Kunth et al. 1995). The conclusion about these DLA systems being dwarf galaxies has also been independently suggested by Meyer and York (1992) and by Steidel et al. (1994) from the low abundances found in the few DLA observed at low redshifts. In summary, our conclusions are: - In order to explain the high N/O ratios observed in two DLA systems by Green et al .(1995) and Molaro et al.(1995) one has to assume that these systems are dwarf irregular galaxies experiencing their first or one of their first bursts of star formation. These galaxies should also experience strong enriched galactic winds carrying away mostly the products of SN II explosions such as oxygen and other $\alpha$-elements. 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--- abstract: 'The brain interprets ambiguous sensory information faster and more reliably than modern computers, using neurons that are slower and less reliable than logic gates. But Bayesian inference, which underpins many computational models of perception and cognition, appears computationally challenging even given modern transistor speeds and energy budgets. The computational principles and structures needed to narrow this gap are unknown. Here we show how to build fast Bayesian computing machines using intentionally stochastic, digital parts, narrowing this efficiency gap by multiple orders of magnitude. We find that by connecting stochastic digital components according to simple mathematical rules, one can build massively parallel, low precision circuits that solve Bayesian inference problems and are compatible with the Poisson firing statistics of cortical neurons. We evaluate circuits for depth and motion perception, perceptual learning and causal reasoning, each performing inference over 10,000+ latent variables in real time — a 1,000x speed advantage over commodity microprocessors. These results suggest a new role for randomness in the engineering and reverse-engineering of intelligent computation.' author: - 'Vikash Mansinghka$^{1,2,3}$ and Eric Jonas$^{1,3}$' bibliography: - 'VMEJ-circuits-arxiv.bib' title: 'Building fast Bayesian computing machines out of intentionally stochastic, digital parts.' --- The authors contributed equally to this work. Computer Science & Artificial Intelligence Laboratory, MIT Department of Brain & Cognitive Sciences, MIT Our ability to see, think and act all depend on our mind’s ability to process uncertain information and identify probable explanations for inherently ambiguous data. Many computational models of the perception of motion[@weissmotion], motor learning[@kordinbayesian], higher-level cognition[@griffithsoptimal; @blaisdellcausal] and cognitive development[@tenenbaumgrow] are based on Bayesian inference in rich, flexible probabilistic models of the world. Machine intelligence systems, including Watson[@ferruccibuilding], autonomous vehicles[@thrunprobabilisticarticle] and other robots[@thrunprobabilisticbook] and the Kinect[@shottonreal] system for gestural control of video games, also all depend on probabilistic inference to resolve ambiguities in their sensory input. But brains solve these problems with greater speed than modern computers, using information processing units that are orders of magnitude slower and less reliable than the switching elements in the earliest electronic computers. The original UNIVAC I ran at 2.25 MHz[@univac], and RAM from twenty years ago had one bit error per 256 MB per month[@shivakumar2002modeling]. In contrast, the fastest neurons in human brains operate at less than 1 kHz, and synaptic transmission can completely fail up to 50% of the time[@synapticfailure]. This efficiency gap presents a fundamental challenge for computer science. How is it possible to solve problems of probabilistic inference with an efficiency that begins to approach that of the brain? Here we introduce intentionally stochastic but still digital circuit elements, along with composition laws and design rules, that together narrow the efficiency gap by multiple orders of magnitude. Our approach both builds on and departs from the principles behind digital logic. Like traditional digital gates, stochastic digital gates consume and produce discrete symbols, which can be represented via binary numbers. Also like digital logic gates, our circuit elements can be composed and abstracted via simple mathematical rules, yielding larger computational units that whose behavior can be analyzed in terms of their constituents. We describe primitives and design rules for both stateless and synchronously clocked circuits. But unlike digital gates and circuits, our gates and circuits are intentionally stochastic: each output is a sample from a probability distribution conditioned on the inputs, and (except in degenerate cases) simulating a circuit twice will produce different results. The numerical probability distributions themselves are implicit, though they can be estimated via the circuits’ long-run time-averaged behavior. And also unlike digital gates and circuits, Bayesian reasoning arises naturally via the dynamics of our synchronously clocked circuits, simply by fixing the values of the circuit elements representing the data. We have built prototype circuits that solve problems of depth and motion perception and perceptual learning, plus a compiler that can automatically generate circuits for solving causal reasoning problems given a description of the underlying causal model. Each of these systems illustrates the use of stochastic digital circuits to accelerate Bayesian inference an important class of probabilistic models, including Markov Random Fields, nonparametric Bayesian mixture models, and Bayesian networks. Our prototypes show that this combination of simple choices at the hardware level — a discrete, digital representation for information, coupled with intentionally stochastic rather than ideally deterministic elements — has far reaching architectural consequences. For example, software implementations of approximate Bayesian reasoning typically rely on high-precision arithmetic and serial computation. We show that our synchronous stochastic circuits can be implemented at very low bit precision, incurring only a negligible decrease in accuracy. This low precision enables us to make fast, small, power-efficient circuits at the core of our designs. We also show that these reductions in computing unit size are sufficient to let us exploit the massive parallelism that has always been inherent in complex probabilistic models at a granularity that has been previously impossible to exploit. The resulting high computation density drives the performance gains we see from stochastic digital circuits, narrowing the efficiency gap with neural computation by multiple orders of magnitude. Our approach is fundamentally different from existing approaches for reliable computation with unreliable components[@neumanncomputer; @akgulprobabilistic; @gaines1969stochastic], which view randomness as either a source of error whose impact needs to be mitigated or as a mechanism for approximating arithmetic calculations. Our combinational circuits are intentionally stochastic, and we depend on them to produce exact samples from the probability distributions they represent. Our approach is also different from and complementary to classic analog[@mead1990neuromorphic] and modern mixed-signal[@choudhary2012silicon] neuromorphic computing approaches: stochastic digital primitives and architectures could potentially be implemented using neuromorphic techniques, providing a means of applying these designs to problems of Bayesian inference. In theory, stochastic digital circuits could be used to solve any computable Bayesian inference problem with a computable likelihood[@afrcomputable] by implementing a Markov chain for inference in a Turing-complete probabilistic programming language[@mansinghka2009natively; @Goodman:2008tb]. Stochastic ciruits can thus implement inference and learning techniques for diverse intelligent computing architectures, including both probabilistic models defined over structured, symbolic representations[@tenenbaumgrow] as well as sparse, distributed, connectionist representations[@SalHinton07]. In contrast, hardware accelerators for belief propagation algorithms[@pearlprobabilistic; @linhigh; @vigodacontinuous] can only answer queries about marginal probabilities or most probable configurations, only apply to finite graphical models with discrete or binary nodes, and cannot be used to learn model parameters from data. For example, the formulation of perceptual learning we present here is based on inference in a nonparametric Bayesian model to which belief propagation does not apply. Additionally, because stochastic digital circuits produce samples rather than probabilities, their results capture the complex dependencies between variables in multi-modal probability distributions, and can also be used to solve otherwise intractable problems in decision theory by estimating expected utilities. Stochastic Digital Gates and Stateless Stochastic Circuits {#stochastic-digital-gates-and-stateless-stochastic-circuits .unnumbered} ========================================================== [**(Figure 1 about here)**]{} \[fig:combinational\] Digital logic circuits are based on a gate abstraction defined by Boolean functions: deterministic mappings from input bit values to output bit values[@shannonsymbolic]. For elementary gates, such as the AND gate, these are given by truth tables; see Figure 1A. Their power and flexibility comes in part from the composition laws that they support, shown in Figure 1B. The output from one gate can be connected to the input of another, yielding a circuit that samples from the composition of the Boolean functions represented by each gate. The compound circuit can also be treated as a new primitive, abstracting away its internal structure. These simple laws have proved surprisingly powerful: they enable complex circuits to be built up out of reusable pieces. [*Stochastic digital gates*]{} (see Figure 1C) are similar to Boolean gates, but consume a source of random bits to generate samples from conditional probability distributions. Stochastic gates are specified by conditional probability tables; these give the probability that a given output will result from a given input. Digital logic corresponds to the degenerate case where all the probabilities are 0 or 1; see Figure 1D for the conditional probability table for an AND gate. Many stochastic gates with m input bits and n output bits are possible. Figure 1E shows one central example, the THETA gate, which generates draws from a biased coin whose bias is specified on the input. Supplementary material outlining serial and parallel implementations is available at [@VMEJ-circuits-supplemental]. Crucially, stochastic gates support generalizations of the composition laws from digital logic, shown in Figure 1F. The output of one stochastic gate can be fed as the input to another, yielding samples from the joint probability distribution over the random variables simulated by each gate. The compound circuit can also be treated as a new primitive that generates samples from the marginal distribution of the final output given the first input. As with digital gates, an enormous variety of circuits can be constructed using these simple rules. Fast Bayesian Inference via Massively Parallel Stochastic Transition Circuits {#fast-bayesian-inference-via-massively-parallel-stochastic-transition-circuits .unnumbered} ============================================================================= Most digital systems are based on deterministic finite state machines; the template for these machines is shown in Figure 2A. A stateless digital circuit encodes the transition function that calculates the next state from the previous state, and the clocking machinery (not shown) iterates the transition function repeatedly. This abstraction has proved enormously fruitful; the first microprocessors had roughly $2^{20}$ distinct states. In Figure 2B, we show the stochastic analogue of this synchronous state machine: a [*stochastic transition circuit*]{}. Instead of the combinational logic circuit implementing a deterministic transition function, it contains a combinational stochastic circuit implementing a stochastic transition operator that samples the next state from a probability distribution that depends on the current state. It thus corresponds to a Markov chain in hardware. To be a valid transition circuit, this transition operator must have a unique stationary distribution $P(S|X)$ to which it ergodically converges. A number of recipes for suitable transition operators can be constructed, such as Metropolis sampling [@metropolisequation] and Gibbs sampling[@gemanstochastic]; most of the results we present rely on variations on Gibbs sampling. More details on efficient implementations of stochastic transition circuits for Gibbs sampling and Metropolis-Hastings can be found elsewhere [@VMEJ-circuits-supplemental]. Note that if the input $X$ represents observed data and the state $S$ represents a hypothesis, then the transition circuit implements Bayesian inference. We can scale up to challenging problems by exploiting the composition laws that stochastic transition circuits support. Consider a probability distribution defined over three variables $P(A,B,C) = P(A)P(B|A)P(C|A)$. We can construct a transition circuit that samples from the overall state $(A,B,C)$ by composing transition circuits for updating $A|BC$, $B|A$ and $C|A$; this assembly is shown in Figure 2C. As long as the underlying probability model does not have any zero-probability states, ergodic convergence of each constituent transition circuit then implies ergodic convergence of the whole assembly[@andrieuintroduction]. The only requirement for scheduling transitions is that each circuit must be left fixed while circuits for variables that interact with it are transitioning. This scheduling requirement — that a transition circuit’s value be held fixed while others that read from its internal state or serve as inputs to its next transition are updating — is analogous to the so-called “dynamic discipline” that defines valid clock schedules for traditional sequential logic[@ward1990computation]. Deterministic and stochastic schedules, implementing cycle or mixture hybrid kernels[@andrieuintroduction], are both possible. This simple rule also implies a tremendous amount of exploitable parallelism in stochastic transition circuits: if two variables are independently caused given the current setting of all others, they can be updated at the same time. Assemblies of stochastic transition circuits implement Bayesian reasoning in a straightforward way: by fixing, or “clamping” some of the variables in the assembly. If no variables are fixed, the circuit explores the full joint distribution, as shown in Figure 2E and 2F. If a variable is fixed, the circuit explores the conditional distribution on the remaining variables, as shown in Figure 2G and 2H. Simply by changing which transition circuits are updated, the circuit can be used to answer different probabilistic queries; these can be varied online based on the needs of the application. [**(Figure 2 about here.)**]{} \[fig:transition\] The accuracy of ultra-low-precision stochastic transition circuits. {#the-accuracy-of-ultra-low-precision-stochastic-transition-circuits. .unnumbered} =================================================================== The central operation in many Markov chain techniques for inference is called DISCRETE-SAMPLE, which generates draws from a discrete-output probability distribution whose weights are specified on its input. For example, in Gibbs sampling, this distribution is the conditional probability of one variable given the current value of all other variables that directly depend on it. One implementation of this operation is shown in Figure 3A; each stochastic transition circuit from Figure 2 could be implemented by one such circuit, with multiplexers to select log-probability values based on the neighbors of each random variable. Because only the ratios of the raw probabilities matter, and the probabilities themselves naturally vary on a log scale, extremely low precision representations can still provide accurate results. High entropy (i.e. nearly uniform) distributions are resilient to truncation because their values are nearly equal to begin with, differing only slightly in terms of their low-order bits. Low entropy (i.e. nearly deterministic) distributions are resilient because truncation is unlikely to change which outcomes have nonzero probability. Figure 3B quantifies this low-precision property, showing the relative entropy (a canonical information theoretic measure of the difference between two distributions) between the output distributions of low precision implementations of the circuit from Figure 3A and an accurate floating-point implementation. Discrete distributions on 1000 outcomes were used, spanning the full range of possible entropies, from almost 10 bits (for a uniform distribution on 1000 outcomes) to 0 bits (for a deterministic distribution), with error nearly undetectable until fewer than 8 bits are used. Figure 3C shows example distributions on 10 outcomes, and Figure 3D shows the resulting impact on computing element size. Extensive quantitative assessments of the impact of low bit precision have also been performed, providing additional evidence that only very low precision is required [@VMEJ-circuits-supplemental]. [**(Figure 3 about here.)**]{} \[fig:MULTINOMIAL\] Efficiency gains on depth and motion perception and perceptual learning problems {#efficiency-gains-on-depth-and-motion-perception-and-perceptual-learning-problems .unnumbered} ================================================================================ Our main results are based on an implementation where each stochastic gate is simulated using digital logic, consuming entropy from an internal pseudorandom number generator[@marsagliaxorshift]. This allows us to measure the performance and fault-tolerance improvements that flow from stochastic architectures, independent of physical implementation. We find that stochastic circuits make it practical to perform stochastic inference over several probabilistic models with 10,000+ latent variables in real time and at low power on a single chip. These designs achieve a 1,000x speed advantage over commodity microprocessors, despite using gates that are 10x slower. In [@VMEJ-circuits-supplemental], we also show architectures that exhibit minimal degradation of accuracy in the presence of fault rates as high as one bit error for every 100 state transitions, in contrast to conventional architectures where failure rates are measured in bit errors (failures) per billion hours of operation[@bertrillions]. Our first application is to depth and motion perception, via Bayesian inference in lattice Markov Random Field models[@gemanstochastic]. The core problem is matching pixels from two images of the same scene, taken at distinct but nearby points in space or in time. The matching is ambiguous on the basis of the images alone, as multiple pixels might share the same value[@marrcooperative]; prior knowledge about the structure of the scene must be applied, which is often cast in terms of Bayesian inference[@szeliskicomparative]. Figure 4A illustrates the template probabilistic model most commonly used. The X variables contain the unknown displacement vectors. Each Y variable contains a vector of pixel similarity measurements, one per possible pair of matched pixels based on X. The pairwise potentials between the X variables encode scene structure assumptions; in typical problems, unknown values are assumed to vary smoothly across the scene, with a small number of discontinuities at the boundaries of objects. Figure 4B shows the conditional independence structure in this problem: every other X variable is independent from one another, allowing the entire Markov chain over the X variables to be updated in a two-phase clock, independent of lattice size. Figure 4C shows the dataflow for the software-reprogrammable probabilistic video processor we developed to solve this family of problems; this processor takes a problem specification based on pairwise potentials and Y values, and produces a stream of posterior samples. When comparing the hardware to hand-optimized C versions on a commodity workstation, we see a 500x performance improvement. [**(Figure 4 about here.)**]{} \[fig:VISION\] We have also built stochastic architectures for solving perceptual learning problems, based on fully Bayesian inference in Dirichlet process mixture models[@fergusonbayesian; @rasmusseninfinite]. Dirichlet process mixtures allow the number of clusters in a perceptual dataset to be automatically discovered during inference, without assuming an a priori limit on the models’ complexity, and form the basis of many models of human categorization[@andersonrational; @griffithscategorization]. We tested our prototype on the problem of discovering and classifying handwritten digits from binary input images. Our circuit for solving this problem operates on an online data stream, and efficiently tracks the number of perceptual clusters this input; see [@VMEJ-circuits-supplemental] for architectural and implementation details and additional characterizations of performance. As with our depth and motion perception architecture, we observe over $\sim$2,000x speedups as compared to a highly optimized software implementation. Of the $\sim$2000x difference in speed, roughly $\sim$256x is directly due to parallelism — all of the pixels are independent dimensions, and can therefore be updated simultaneously. [**(Figure 5 about here.)**]{} \[fig:learning\] Automatically generated causal reasoning circuits and spiking implementations {#automatically-generated-causal-reasoning-circuits-and-spiking-implementations .unnumbered} ============================================================================= Digital logic gates and their associated design rules are so simple that circuits for many problems can be generated automatically. Digital logic also provides a common target for device engineers, and have been implemented using many different physical mechanisms – classically with vaccum tubes, then with MOSFETS in silicon, and even on spintronic devices[@spintronicnand]. Here we provide two illustrations of the analogous simplicity and generality of stochastic digital circuits, both relevant for the reverse-engineering of intelligent computation in the brain. We have built a compiler that can automatically generate circuits for solving arbitrary causal reasoning problems in Bayesian network models. Bayesian network formulations of causal reasoning have played central roles in machine intelligence[@pearlprobabilistic] and computational models of cognition in both humans and rodents[@blaisdellcausal]. Figure \[fig:COMPILER\]A shows a Bayesian network for diagnosing the behavior of an intensive care unit monitoring system. Bayesian inference within this network can be used to infer probable states of the ICU given ambiguous patterns of evidence — that is, reason from observed effects back to their probable causes. Figure \[fig:COMPILER\]B shows a factor graph representation of this model[@kschischangfactor]; this more general data structure is used as the input to our compiler. Figure \[fig:COMPILER\]C shows inference results from three representative queries, each corresponding to a different pattern of observed data. We have also explored implementations of stochastic transition circuits in terms of spiking elements governed by Poisson firing statistics. Figure \[fig:COMPILER\]D shows a spiking network that implements the Markov chain from Figure \[fig:transition\]. The stochastic transition circuit corresopnding to a latent variable $X$ is implemented via a bank of Poisson-spiking elements $\{X_i\}$ with one unit $X_i$ per possible value of the variable. The rate for each spiking element $X_i$ is determined by the unnormalized conditional log probability of the variable setting it corresponds to, following the discrete-sample gate from Figure \[fig:MULTINOMIAL\] the time to first spike $\mathrm{t}(X_i) \sim Exp(e_i)$, with $e_i$ obtained by summing energy contributions from all connected variables. The output value of $X$ is determined by $\mathrm{argmin}_i \{\mathrm{t}(X_i)\}$, i.e. the element that spiked first, implemented by fast lateral inhibition between the $X_i$s. It is easy to show that this implements exponentiation and normalization of the energies, leading to a correct implementation of a stochastic transition circuit for Gibbs sampling; see [@VMEJ-circuits-supplemental] for more information. Elements are clocked quasi-synchronously, reflecting the conditional independence structure and parallel update scheme from Figure \[fig:transition\]D, and yields samples from the correct equilibrium distribution. This spiking implementation helps to narrow the gap with recent theories in computational neuroscience. For example, there have been recent proposals that neural spikes correspond to samples[@fiser2010statistically], and that some spontaneous spiking activity corresponds to sampling from the brain’s unclamped prior distribution[@berkes2011spontaneous]. Combining these local elements using our composition and abstraction laws into massively parallel, low-precision, intentionally stochastic circuits may help to bridge the gap between probabilistic theories of neural computation and the computational demands of complex probabilistic models and approximate inference[@probbrains]. [**(Figure 6 about here.)**]{} \[fig:COMPILER\] Discussion {#discussion .unnumbered} ========== To further narrow the efficiency gap with the brain, and scale to more challenging Bayesian inference problems, we need to improve the convergence rate of our architectures. One approach would be to initialize the state in a transition circuit via a separate, feed-forward, combinational circuit that approximates the equilibrium distribution of the Markov chain. Machine perception software that uses machine learning to construct fast, compact initializers is already in use[@shottonreal]. Analyzing the number of transitions needed to close the gap between a good initialization and the target distribution may be harder[@diaconismarkov]. However, some feedforward Monte Carlo inference strategies for Bayesian networks provably yield precise estimates of probabilities in polynomial time if the underlying probability model is sufficiently stochastic[@dagum1997optimal]; it remains to be seen if similar conditions apply to stateful stochastic transition circuits. It may also be fruitful to search for novel electronic devices — or previously unusable dynamical regimes of existing devices — that are as well matched to the needs of intentionally stochastic circuits as transistors are to logical inverters, potentially even via a spiking implementation. Physical phenomena that proved too unreliable for implementing Boolean logic gates may be viable building blocks for machines that perform Bayesian inference. Computer engineering has thus far focused on deterministic mechanisms of remarkable scale and complexity: billlions of parts that are expected to make trillions of state transitions with perfect repeatability[@intelInstructionErrors]. But we are now engineering computing systems to exhibit more intelligence than they once did, and identify probable explanations for noisy, ambiguous data, drawn from large spaces of possibilities, rather than calculate the definite consequences of perfectly known assumptions with high precision. The apparent intractability of probabilistic inference has complicated these efforts, and challenged the viability of Bayesian reasoning as a foundation for engineering intelligent computation and for reverse-engineering the mind and brain. At the same time, maintaining the illusion of rock-solid determinism has become increasingly costly. Engineers now attempt to build digital logic circuits in the deep sub-micron regime[@shepardnoise] and even inside cells[@elowitzsynthetic]; in both these settings, the underlying physics has stochasticity that is difficult to suppress. Energy budgets have grown increasingly restricted, from the scale of the datacenter[@barroso2007case] to the mobile device[@flinn1999energy], yet we spend substantial energy to operate transistors in deterministic regimes. And efforts to understand the dynamics of biological computation — from biological neural networks to gene expression networks[@mcadams1997stochastic] — have all encountered stochastic behavior that is hard to explain in deterministic, digital terms. Our intentionally stochastic digital circuit elements and stochastic computing architectures suggest a new direction for reconciling these trends, and enables the design of a new class of fast, Bayesian digital computing machines. The authors would like to acknowledge Tomaso Poggio, Thomas Knight, Gerald Sussman, Rakesh Kumar and Joshua Tenenbaum for numerous helpful discussions and comments on early drafts, and Tejas Kulkarni for contributions to the spiking implementation. [**Figure 1.**]{} [(A) Boolean gates, such as the AND gate, are mathematically specified by truth tables: deterministic mappings from binary inputs to binary outputs. (B) Compound Boolean circuits can be synthesized out of sub-circuits that each calculate different sub-functions, and treated as a single gate that implements the composite function, without reference to its internal details. (C) Each stochastic gate samples from a discrete probability distribution conditioned on an input; for clarity, we show an external source of random bits driving the stochastic behavior. (D) Composing gates that sample B given A and C given B yields a network that samples from the joint distribution over B and C given A; abstraction yields a gate that samples from the marginal distribution C|A. When only one sample path has nonzero probability, this recovers the composition of Boolean functions. (E) The THETA gate is a stochastic gate that generates samples from a Bernoulli distribution whose parameter theta is specified via the $m$ input bits. Like all stochastic digital gates, it can be specified by a conditional probability table, analogously to how Boolean gates can be specified via a truth table. (F) When each new output sample is triggered (e.g. because its internal randomness source updates), a different output sample is generated; time-averaging the output makes it possible to estimate the entries in the probability table, which are otherwise implicit. (G) The THETA gate can be implemented by comparing the output of a source of (pseudo)random bits to the input coin weight. (H) Deterministic gates, such as the AND gate shown here, can be viewed as degenerate stochastic gates specified by conditional probability tables whose entries are either 0 or 1. This permits fluid interoperation of deterministic and stochastic gates in compound circuits. (I) A parallel circuit implementing a Binomial random variable can be implemented by combining THETA gates and adders using the composition laws from (D).]{} {width="\textwidth"} [**Figure 2.**]{} [Stochastic transition circuits and massively parallel Bayesian inference. (A) A deterministic finite state machine consists of a register and a transition function implemented via combinational logic. (B) A stochastic transition circuit consists of a register and a stochastic transition operator implemented by a combinational stochastic circuit. Each stochastic transition circuit is $T_{S|X}$ is parameterized by some input $X$, and its internal combinational stochastic block $P(S_{t+1}|S_t,X)$ must ergodically converge to a unique stationary distribution $P(S|X)$ for all $X$. (C) Stochastic transition circuits can be composed to construct samplers for probabilistic models over multiple variables by wiring together stochastic transition circuits for each variable based on their interactions. This circuit samples from a distribution $P(A,B,C) = P(A)P(B|A)P(C|A)$. (D) Each network of stochastic transition circuits can be scheduled in many ways; here we show one serial schedule and one parallel schedule for the transition circuit from (C). Convergence depends only on respecting the invariant that no stochastic transition circuit transitions while other circuits that interact with it are transitioning. (E) The Markov chain implemented by this transition circuit. (F) Typical stochastic evolutions of the state in this circuit. (G) Inference can be implemented by clamping state variables to specific values; this yields a restricted Markov chain that converges to the conditional distribution over the unclamped variables given the clamped ones. Here we show the chain obtained by fixing $C=1$. (H) Typical stochastic evolutions of the state in this clamped transition circuit. Changing which variables are fixed allows the inference problem to be changed dynamically as the circuit is running.]{} {height="\textheight"} [ (A) The discrete-sample gate is a central building block for stochastic transition circuits, used to implement Gibbs transition operators that update a variable by sampling from its conditional distribution given the variables it interacts with. The gate renormalizes the input log probabilities it is given, converts them to probabilities (by exponentiation), and then samples from the resulting distribution. Input energies are specified via a custom fixed-point coding scheme. (B) Discrete-sample gates remain accurate even when implemented at extremely low bit-precision. Here we show the relative entropy between true distributions and their low-precision implementations, for millions of distributions over discrete sets with 1000 elements; accuracy loss is negligible even when only 8 bits of precision are used. (C) The accuracy of low-precision discrete-sample gates can be understood by considering multinomial distributions with high, medium and low entropy. High entropy distributions involve outcomes with very similar probability, insensitive to ratios, while low entropy distributions are dominated by the location of the most probable outcome. (D) Low-precision transition circuits save area as compared to high-precision floating point alternatives; these area savings make it possible to economically exploit massive parallelism, by fitting many sampling units on a single chip.]{} {width="\textwidth"} [ (A) A Markov Random Field for solving depth and motion perception, as well as other dense matching problems. Each $X_{i,j}$ node stores the hidden quantity to be estimated, e.g. the disparity of a pixel. Each $f_{LP}$ ensures adjacent $X$s are either similar or very different, i.e. that depth and motion fields vary smoothly on objects but can contain discontinuities at object boundaries. Each $Y_{i,j}$ node stores a per-latent-pixel vector of similarity information for a range of candidate matches, linked to the $X$s by the $f_E$ potentials. (B) The conditional independencies in this model permit many different parallelization strategies, from fully space-parallel implementations to virtualized implementations where blocks of pixels are updated in parallel. (C) Depth perception results. The left input image, plus the depth maps obtained by software (middle) and hardware (right) engines for solving the Markov Random Field. (D) Motion perception results. One input frame, plus the motion flow vector fields for software (middle) and hardware (right) solutions. (E) Energy versus time for software and hardware solutions to depth perception, including both 8-bit and 12-bit hardware. Note that the hardware is roughly 500x faster than the software on this frame. (F) Energy versus time for software and hardware solutions to motion perception.]{} {width="\textwidth"} [**Figure 5.**]{} [(A) Example samples from the posterior distribution of cluster assignments for a nonparametric mixture model. The two samples show posterior variance, reflecting the uncertainty between three and four source clusters. (B) Typical handwritten digit images from the MNIST corpus[@lecun1998mnist], showing a high degree of variation across digits of the same type. (C) The digit clusters discovered automatically by a stochastic digital circuit for inference in Dirichlet process mixture models. Each image represents a cluster; each pixel represents the probability that the corresponding image pixel is black. Clusters are sorted according to the most probable true digit label of the images in the cluster. Note that these cluster labels were not provided to the circuit. Both the clusters and the number of clusters were discovered automatically by the circuit over the course of inference. (D) The receiver operating characteristic (ROC) curves that result from classifying digits using the learned clusters; quantitative results are competitive with state-of-the-art classifiers. (E) The time required for one cycle through the outermost transition circuit in hardware, versus the corresponding time for one sweep of a highly optimized software implementation of the same sampler, which is $\sim$2000x slower.]{} {width="\textwidth"} [**Figure 6.**]{} [(A) A Bayesian network model for ICU alarm monitoring, showing measurable variables, hidden variables, and diagnostic variables of interest. (B) A factor graph representation of this Bayesian network, rendered by the input stage for our stochastic transition circuit synthesis software. (C) A representation of the factor graph showing evidence variables as well as a parallel schedule for the transition circuits automatically extracted by our compiler: all nodes of the same color can be transitioned simultaneously. (D) Three diagnosis results from Bayesian inference in the alarm network, showing high accuracy diagnoses (with some posterior uncertainty) from an automatically generated circuit. E) The schematic of a spiking neural implementation of a stochastic transition circuit assembly for sampling from the three-variable probabilistic model from Figure 2. (F) The spike raster (black) and state sequence (blue) that result from simulating the circuit. (G) The spiking simulation yields state distributions that agree with exact simulation of the underlying Markov chain.]{} {width="\textwidth"}

--- abstract: 'We consider double plumbings of two disk bundles over spheres. We calculate the Heegaard–Floer homology with its absolute grading of the boundary of such a plumbing. Given a closed smooth 4–manifold $X$ and a suitable pair of classes in $H_{2}(X)$, we investigate when this pair of classes may be represented by a configuration of surfaces in $X$ whose regular neighborhood is a double plumbing of disk bundles over spheres. Using similar methods we study single plumbings of two disk bundles over spheres inside $X$.' address: | Faculty of Education\ University of Ljubljana\ Kardeljeva ploščad 16\ 1000 Ljubljana, Slovenia author: - Eva Horvat bibliography: - 'lit1.bib' date: 'July 18, 2013' title: Double plumbings of disk bundles over spheres --- [Introduction]{} Given a smooth closed connected 4–manifold $X$ and a finite set of classes $C\subset H_{2}(X)$, an important question is what is the simplest configuration of surfaces in $X$ representing $C$. By simple we mean that each class should be represented by a surface of low genus and that the surfaces should have a low number of geometric intersections. Since it is always possible to remove cancelling pairs of intersection points by increasing the genus of a surface, both properties should be taken into account. Considerable work has been done to investigate the minimal genus of a given class in $H_{2}(X)$, first by proving the Thom conjecture (Kronheimer–Mrowka [@KM]) and then its generalizations by Morgan–Szabó–Taubes [@MST] and Ozsváth–Szabó [@OS6]. When considering a configuration of surfaces, the sum of their genera is closely related to the number of their geometric intersections, as shown by Gilmer [@GILMER]. He showed that the the minimal number of such intersections can be estimated using the Casson–Gordon invariant. This estimate has been improved by Strle [@SASO] for configurations of $n=b_{2}^{+}(X)$ algebraically disjoint surfaces of positive self-intersection by an application of the Seiberg–Witten equations on a cylindrical end manifold. Multiple plumbings of two trivial disk bundles over spheres have been investigated by Sunukjian in his thesis [@NS]. He calculated the Heegaard-Floer homology of the boundary of such plumbings in cases where the two spheres are plumbed either once or zero times algebraically and $n$ times geometrically. We investigate the double plumbing ${N_{m,n}}$ of two disk bundles with Euler classes $m$ and $n$ over spheres, which represents the simplest case of a configuration of two surfaces with algebraic (and geometric) intersection 2. We calculate the $d_{b}$–invariants of the Heegaard–Floer homology of $\partial {N_{m,n}}$ [@OS2] and use an obstruction theorem [@OS4 Theorem 9.15] to see when ${N_{m,n}}$ can indeed be realized inside a given 4–manifold $X$ with $b_{2}^{+}(X)=2$. Denote by ${Y_{m,n}}$ the boundary of ${N_{m,n}}$. For two integers $i$ and $j$, denote by $\mathfrak{t}_{i,j}$ the unique $\sp $ structure on ${N_{m,n}}$ for which $$\begin{aligned} \label{eqsp} \langle c_{1}(\mathfrak{t}_{i,j}),s_{1}\rangle +m=2i\end{aligned}$$ $$\begin{aligned} \label{eqsp1} \langle c_{1}(\mathfrak{t}_{i,j}),s_{2}\rangle +n=2j\;,\end{aligned}$$ where $s_{1},s_{2}\in H_{2}({N_{m,n}})$ are the homology classes of the base spheres in the double plumbing. Let $\mathfrak{s}_{i,j}=\mathfrak{t}_{i,j}|_{{Y_{m,n}}}$. Throughout the paper, we denote by ${\mathbb{F}}$ the field ${\mathbb {Z}}_{2}$ and by $\mathcal{T}^{+}$ the quotient module ${\mathbb{F}}[U,U^{-1}]/U{\mathbb{F}}[U]$. Our main result is the following: \[th1\] Let $Y={Y_{m,n}}$ be the boundary of a double plumbing of two disk bundles over spheres with Euler numbers $m$ and $n$, where $m,n\geq 4$. The Heegaard–Floer homology $HF^{+}(Y,\mathfrak{s})$ with ${\mathbb{F}}$ coefficients is given by $$\begin{aligned} & HF^{+}(Y,\mathfrak{s}_{m-1,1})={\mathcal{T}^{+}}_{(d(m-1,1))}\oplus {\mathcal{T}^{+}}_{(d(m-1,1)-1)}\oplus {\mathbb{F}}_{(d(m-1,1)-1)}\\ & HF^{+}(Y,\mathfrak{s}_{i,j})={\mathcal{T}^{+}}_{(d(i,j))}\oplus {\mathcal{T}^{+}}_{(d(i,j)-1)}\\ & HF^{+}(Y,\mathfrak{s}_{0,k})={\mathcal{T}^{+}}_{(d_{1}(m,k+1))}\oplus {\mathcal{T}^{+}}_{(d_{1}(m,k+1)-1)}\\ & HF^{+}(Y,\mathfrak{s}_{l,0})={\mathcal{T}^{+}}_{(d_{1}(n,l+1))}\oplus {\mathcal{T}^{+}}_{(d_{1}(n,l+1)-1)}\end{aligned}$$ for $1\leq i\leq m-1$, $1\leq j\leq n-1$, $0\leq k\leq n-2$, $0\leq l\leq m-2$ and $(i,j)\notin \{(m-1,1),(1,n-1)\}$, where the subscripts denote the absolute gradings of the bottom elements and [1]{} & d(i,j)=,\ & d\_[1]{}(t,i)=. The action of the exterior algebra $\Lambda ^{*}(H_{1}(Y,{\mathbb {Z}})/\operatorname{Tors})$ on $HF^{+}(Y,\mathfrak{s})$ maps the first copy of ${\mathcal{T}^{+}}$ isomorphically to the second copy in each torsion $\sp $ structure $\mathfrak{s}$, dropping the absolute grading of the generator by one. We use this result to determine whether the double plumbing ${N_{m,n}}$ can occur inside some 4–manifolds $X$ with $H_{2}^{+}(X)=2$. If it can, the complement $W=X\backslash \operatorname{Int}({N_{m,n}})$ is a negative semi-definite 4–manifold and [@OS4 Theorem 9.15] gives an obstruction depending on the correction terms of ${Y_{m,n}}=\partial W$. In the manifold $X={\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$, every homology class $(x_{1},x_{2})\in H_{2}(X)$ with $(x_{1},x_{2})\in \{0,\pm 1,\pm 2\}^{2}\backslash \{(0,0)\}$ has a smooth representative of genus 0. Choosing two such representatives with algebraic intersection number 2, we check if they can have only 2 geometric intersections. Next we consider the manifold $X=S^{2}\times S^{2}\# S^{2}\times S^{2}$. According to Wall [@WALL], every primitive homology class $(x_{1},x_{2},x_{3},x_{4})\in H_{2}(X)$ can be represented by an embedded sphere. We choose two such representatives with algebraic intersection number 2 and determine when the number of their geometric intersections has to be strictly greater than 2, thus not allowing the chosen homology classes to be represented by a double plumbing. We obtain the following estimates. \[app\]\ a) Any two spheres representing classes $(2,2),(2,-1)\in H_{2}({\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2})$ intersect with at least $4$ geometric intersections, and there exist representatives with exactly $4$ intersections.\ b) Let $t\in \mathbb{N}\backslash \{1\}$ and let $a$ be an odd positive integer. Any two spheres representing classes $(a,2,0,0),(1,0,t,1)\in H_{2}(S^{2}\times S^{2}\# S^{2}\times S^{2})$ intersect with at least $4$ geometric intersections for all $a\geq 5$. By a similar method we study single plumbings of disk bundles over spheres inside a closed 4–manifold. An obstruction to embedding such configurations is based on the $d$–invariants of lens spaces. \[app1\] Let $k$ be a positive integer. Any two spheres representing classes $(2k+1,2,0,0),(-k,1,2k,1)\in H_{2}(S^{2}\times S^{2}\# S^{2}\times S^{2})$ intersect with at least 3 geometric intersections for all $k>1$. This paper is organized as follows. In Subsection \[HD\] we describe a Heegaard diagram for ${Y_{m,n}}=\partial {N_{m,n}}$. In \[CF\] we present the corresponding chain complex $\widehat{CF}({Y_{m,n}})$ along with its decomposition into equivalence classes of $\sp $ structures and calculate the homology $HF^{+}({Y_{m,n}},\mathfrak{s})$ in all torsion $\sp $ structures on ${Y_{m,n}}$. In Subsection \[absolute\] we compute the absolute gradings ${\widetilde{\operatorname{gr}}}$ of the generators of these groups which in turn determine the correction term invariants $d_{b}({Y_{m,n}},\mathfrak{s})$ for all torsion $\sp $ structures $\mathfrak{s}$ on ${Y_{m,n}}$. The first part of Section \[App\] describes the general homological setting in which the double plumbing ${N_{m,n}}$ arises as a submanifold in a closed 4–manifold $X$. In Subsection \[CP\] we consider the case $X={\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$ and in Subsection \[S2\] the case $X=S^{2}\times S^{2}\# S^{2}\times S^{2}$. In Section \[One\] we investigate single plumbings of two disk bundles over spheres. We consider such configurations inside the manifold ${\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$ in Subsection \[CP1\] and inside the manifold $S^{2}\times S^{2}\# S^{2}\times S^{2}$ in Subsection \[S21\]. [**Acknowledgments:**]{} I would like to thank my advisor Sašo Strle for all his help and support during our numerous discussions. I am also very grateful to the referees for a careful reading, many helpful comments and suggestions. [Heegaard–Floer homology of the boundary of a double plumbing]{} Let $N_{m,n}$ be the double plumbing of two disk bundles over spheres, where $m$ and $n$ denote the Euler numbers of the disk bundles contained in the plumbing. The base spheres of the bundles intersect twice inside the plumbing and we assume both intersections carry the same sign. Denote by $Y_{m,n}$ the boundary of $N_{m,n}$. Throughout this paper we assume $m,n\geq 4$. In this section we calculate $HF^{+}(Y_{m,n})$ with ${\mathbb{F}}$ coefficients and prove Theorem \[th1\]. [Heegaard diagram]{} \[HD\] Considering a Kirby diagram of the plumbing $N_{m,n}$ as a surgery diagram for $Y_{m,n}$, we derive the Heegaard diagram of its boundary. A disk bundle over a sphere is given by a single framed circle, and a double plumbing of two such bundles is represented by the Kirby diagram in Figure \[fig:kirby\]. The second plumbing contributes a 1-handle. Instead of adding the 1-handle one can remove its complementary 2-handle with framing zero. The boundary of the resulting manifold remains unchanged if we replace the 1-handle by its complementary 2-handle with framing zero and obtain a Kirby diagram which is a link of three framed unknots $K_{1},K_{2}$ and $K_{3}$ in $S^{3}$. $K_{1}$ at 120 440 $K_{2}$ at 700 440 $K_{3}$ at 400 320 $S^{2}$ at 800 190 $m$ \[b\] at 310 460 $n$ \[b\] at 530 460 ![The Kirby diagram of a double plumbing[]{data-label="fig:kirby"}](dvakratnipp) To obtain the Heegaard diagram of the boundary, we split the 3-sphere into two balls along the sphere $S^{2}$ shown in Figure \[fig:kirby\]. Surgery along the three framed circles $K_{i}$ gives us two handlebodies of genus 3. The Heegaard diagram is drawn on the plane with three 1-handles added. The lower handlebody is a boundary connected sum of regular neighborhoods of the three circles $K_{1}$, $K_{2}$ and $K_{3}$. We denote by $\mu _{i}$ and $\lambda _{i}$ the meridian and longitude of the regular neighborhood of $K_{i}$ respectively. Each of the curves $\alpha _{i}$ is homologous to $\lambda _{i}$, and the curve $\beta _{i}$ corresponds to the framing of $K_{i}$ for $i=1,2,3$ (see Figure \[fig:heeg\]). Thus, the first homology group $H_{1}({Y_{m,n}})$ is given by $$H_{1}({Y_{m,n}})=\left \langle \mu _{1},\mu _{2},\mu _{3}|\, m\mu _{1}+2\mu _{2}=0,\,2\mu _{1}+n\mu _{2}=0\right \rangle ={\mathbb {Z}}\langle \mu _{3}\rangle \oplus T\langle \mu _{1},\mu _{2}\rangle \;.$$ If at least one of the numbers $m,n$ is odd, the torsion group $T$ is cyclic and we get $T\langle \mu _{1},\mu _{2}\rangle ={\mathbb {Z}}_{mn-4}\langle \mu _{i}\rangle $ (if $m$ is odd and $n$ is even then $i=1$, if $n$ is odd and $m$ is even then $i=2$, if both $m$ and $n$ are odd then $i$ could either be $1$ or $2$). If both $m$ and $n$ are even numbers, then $T\langle \mu _{1},\mu _{2}\rangle ={\mathbb {Z}}_{\frac{mn-4}{2}}\langle \mu _{1}\rangle \oplus {\mathbb {Z}}_{2}\langle \frac{m}{2}\mu _{1}+\mu _{2}\rangle $. [The chain complex]{} \[CF\] We denote the intersections between the $\alpha $ and $\beta $ curves as follows (see Figure \[fig:heeg\]):\ $\alpha _{1}\cap \beta _{1}=\{x_{1},x_{2},\ldots ,x_{m}\}$, $\alpha _{1}\cap \beta _{2}=\{y_{1},y_{2}\}$, $\alpha _{1}\cap \beta _{3}=\{u_{1},u_{2}\}$, $\alpha _{2}\cap \beta _{1}=\{a_{1},a_{2}\}$, $\alpha _{2}\cap \beta _{2}=\{b_{1},b_{2},\ldots ,b_{n}\}$, $\alpha _{2}\cap \beta _{3}=\{c_{1},c_{2}\}$, $\alpha _{3}\cap \beta _{1}=\{d_{1},d_{2}\}$, $\alpha _{3}\cap \beta _{2}=\{e_{1},e_{2}\}$, $\alpha _{3}\cap \beta _{3}=\{f_{1},f_{2}\}$.\ The chain complex $\widehat{CF}(Y)$ is generated by unordered triples $${\mathbb {T}}_{\alpha }\cap {\mathbb {T}}_{\beta }=\{\{x_{i},b_{j},f_{k}\},\{x_{i},c_{k},e_{l}\},\{y_{k},a_{l},f_{r}\},\{y_{k},c_{l},d_{r}\},\{u_{k},a_{l},e_{r}\},\{u_{k},b_{j},d_{l}\}\}$$ for $k,l,r\in \{1,2\}$ and $i=1,\ldots ,m$ and $j=1,\ldots n$. The complement of the $\alpha $ and $\beta $ curves in the Heegaard diagram is a disjoint union of elementary domains. There are two regions in the diagram where a curve $\beta _{i}$ winds around a hole in the direction of $\mu _{i}$; we denote the elementary domains in the winding region of $\beta _{1}$ by $A_{1},\ldots ,A_{m-3}$ and the elementary domains in the winding region of $\beta _{2}$ by $B_{1},\ldots ,B_{n-3}$. The remaining elementary domains of the Heegaard diagram are denoted by $D_{1},\ldots ,D_{16}$. They consist of five hexagons $D_{1},D_{4},D_{8},D_{9}$ and $D_{16}$, one dodecagon $D_{5}$ and one bigon $D_{13}$; all the remaining elementary domains are rectangles. We put the basepoint $z$ of the Heegaard diagram into the elementary domain $D_{5}$. There is a single periodic domain in our diagram, bounded by the difference $\alpha _{3}-\beta _{3}$ of the two homologous curves, which is given by the sum $$\mathcal {P}=D_{3}+D_{4}+D_{6}+D_{9}+D_{11}+D_{12}-D_{13}+D_{14}+D_{16}+B_{1}+B_{2}+\ldots +B_{n-3}\;.$$ Applying the first Chern class formula [@OS2 Proposition 7.5] we obtain $$\begin{aligned} \left \langle c_{1}(\mathfrak{s}_{z}(\mathbf{x})),\mathcal {P}\right \rangle & =\chi (\mathcal{P})+2\sum _{x_{i}\in \mathbf{x}}\overline{n}_{x_{i}}(\mathcal{P})=3(1-\frac {6}{4})+(-1)(1-\frac {2}{4})+2\sum _{x_{i}\in \mathbf{x}}\overline{n}_{x_{i}}(\mathcal{P})=\\ &=2\left (\sum _{x_{i}\in \mathbf{x}}\overline{n}_{x_{i}}(\mathcal{P})-1\right )\;,\end{aligned}$$ \[t\] at 365 316 \[b\] at 290 470 \[t\] at 338 606 at 125 400 at 170 450 at 216 440 \[b\] at 283 316 \[b\] at 300 316 \[b\] at 318 316 \[b\] at 333 316 \[b\] at 372 316 \[b\] at 350 316 \[b\] at 395 316 \[b\] at 385 316 \[t\] at 400 314 \[b\] at 312 470 \[b\] at 342 470 \[b\] at 362 470 \[b\] at 375 470 \[b\] at 326 470 \[t\] at 358 468 \[b\] at 340 610 \[b\] at 382 610 \[b\] at 366 610 \[b\] at 402 610 \[b\] at 280 610 \[b\] at 296 610 \[b\] at 320 610 \[b\] at 354 610 at 470 470 at 420 260 at 296 725 at 387 250 at 296 711 at 250 500 at 296 690 at 373 252 at 230 424 at 363 260 at 374 424 at 200 612 at 345 270 at 320 390 at 315 530 at 352 380 at 352 560 at 368 552 at 372 460 at 430 470 at 420 520 at 441 520 at 294 335 at 290 628 at 320 295 at 320 596 at 253 318 at 252 606 at 256 472 at 270 443  \[fig:heeg\] based on which we can determine the torsion $\sp $ structures. A generator $\mathbf{x}\in {\mathbb {T}}_{\alpha }\cap {\mathbb {T}}_{\beta }$ belongs to a torsion $\sp $ structure if and only if $\left \langle c_{1}(\mathfrak{s}_{z}(\mathbf{x})),\mathcal {P}\right \rangle =0$, which happens exactly when $\sum _{x_{i}\in \mathbf{x}}\overline{n}_{x_{i}}(\mathcal{P})=1$. Thus, the torsion $\sp $ structures of our chain complex contain the following generators: $$\{\{x_{i},b_{j},f_{k}\},\{x_{i},c_{k},e_{r}\},\{y_{k},a_{k},f_{r}\},\{y_{1},c_{k},d_{r}\},\{u_{k},a_{2},e_{r}\}\}$$ for $k,r\in \{1,2\}$ and $i=1,\ldots ,m$ and $j=1,\ldots n$. There are $2(mn+2m+6)$ generators of the torsion $\sp $ structures on ${Y_{m,n}}$. [Notation for $\sp $ structures]{} \[notspin\] There is a one-to-one correspondence $$\delta ^{\tau }\colon \sp ({Y_{m,n}})\to H^{2}({Y_{m,n}})$$ [@OS1 Subsection 2.6]. Thus, we may identify the $\sp $ structures on ${Y_{m,n}}$ with cohomology classes, or even with their Poincaré dual homology classes in $H_{1}({Y_{m,n}})$. Then the natural map $c_{1}\colon \sp ({Y_{m,n}})\to H^{2}({Y_{m,n}})$ assigning to any $\sp $ structure its first Chern class is connected to $\delta ^{\tau }$ by $c_{1}(\mathfrak{s})=2\delta ^{\tau }(\mathfrak{s})$. Similarly, a $\sp $ structure on a 4–manifold $W$ has a determinant line bundle whose first Chern class is a characteristic element in $H^{2}(W)$. For every characteristic element $c\in H^{2}(W)$ there exists a $\sp $ structure on $W$ with determinant line bundle whose first Chern class is equal to $c$ [@kirby Proposition 2.4.16]. If $H^{2}(W)$ contains no 2-torsion, then such a $\sp $ structure is unique. If $W$ is a 4–manifold with boundary ${Y_{m,n}}$, we will identify the restriction $\sp (W)\to \sp ({Y_{m,n}})$ with the corresponding restriction map on cohomology $H^{2}(W)\to H^{2}({Y_{m,n}})$ after making an appropriate choice of origins in the sets of $\sp $ structures. Let $s_{1},s_{2}\in H_{2}({N_{m,n}})$ be the homology classes of the base spheres in the double plumbing. As we will see in Section \[App\], all the torsion $\sp $ structures on ${Y_{m,n}}$ extend to $\sp $ structures on ${N_{m,n}}$. For two integers $i$ and $j$, denote by $\mathfrak{t}_{i,j}$ the unique $\sp $ structure on ${N_{m,n}}$ for which $$\begin{aligned} \langle c_{1}(\mathfrak{t}_{i,j}),s_{1}\rangle +m=2i\\ \langle c_{1}(\mathfrak{t}_{i,j}),s_{2}\rangle +n=2j\end{aligned}$$ and let $\mathfrak{s}_{i,j}=\mathfrak{t}_{i,j}|_{{Y_{m,n}}}$. \[unique\] All the torsion $\sp $ structures on ${Y_{m,n}}$ are uniquely determined by [1]{} & \_[i,j]{} 1im-1, 1jn-1\ & \_[0,j]{} 0jn-2\ & \_[i,0]{} 0im-2 with identifications $\mathfrak{s}_{m-2,0}=\mathfrak{s}_{0,n-2}$ and $\mathfrak{s}_{m-1,1}=\mathfrak{s}_{1,n-1}$. Denote by $s_{1}^{*},s_{2}^{*}$ the basis for $H^{2}({N_{m,n}})$ which is Hom-dual to the basis $s_{1},s_{2}$. Then the restriction map $H^{2}({N_{m,n}})\to H^{2}({Y_{m,n}})$ maps $s_{i}^{*}\mapsto PD(\mu _{i})$ for $i=1,2$. Remember the presentation of the first homology group $$H_{1}({Y_{m,n}})=\left \langle \mu _{1},\mu _{2},\mu _{3}|\, m\mu _{1}+2\mu _{2}=0,\,2\mu _{1}+n\mu _{2}=0\right \rangle \;.$$ Its torsion subgroup is uniquely determined by the classes $i\mu _{1}+j\mu _{2}$ for $1\leq i\leq m-1$ and $1\leq j\leq n-1$, $i\mu _{1}$ for $0\leq i\leq m-2$ and $j\mu _{2}$ for $0\leq j\leq n-2$ with identifications $(m-2)\mu _{1}=(n-2)\mu _{2}$ and $(m-1)\mu _{1}+\mu _{2}=\mu _{1}+(n-1)\mu _{2}$. The set ${\mathbb {T}}_{\alpha }\cap {\mathbb {T}}_{\beta }$ is decomposed into equivalence classes according to the $\epsilon $-relation [@OS1 Definition 2.11]. Furthermore, by the map $\mathfrak{s}_{z}\colon {\mathbb {T}}_{\alpha }\cap {\mathbb {T}}_{\beta }\to H^{2}({Y_{m,n}})$, every equivalence class of generators together with a fixed basepoint $z$ determines a $\sp $ structure and its corresponding cohomology class. By [@OS1 Lemma 2.19] for any two generators $\mathbf{x},\mathbf{y}\in {\mathbb {T}}_{\alpha }\cap {\mathbb {T}}_{\beta }$ the following holds: $$\mathfrak{s}_{z}(\mathbf{y})-\mathfrak{s}_{z}(\mathbf{x})=PD[\epsilon (\mathbf{x},\mathbf{y})]\;.$$ From the Heegaard diagram we obtain $$\begin{aligned} & \epsilon (\{x_{i+1},-\},\{x_{i},-\})=\mu _{1} \textrm { for $1\leq i\leq m-2$}\\ & \epsilon (\{x_{m},-\},\{x_{m-1},-\})=\epsilon (\{b_{n},-\},\{b_{n-1},-\})=\mu _{1}+\mu _{2}\\ & \epsilon (\{x_{m},-\},\{x_{1},-\})=(m-1)\mu _{1}+\mu _{2}=-\mu _{1}-\mu _{2}\\ & \epsilon (\{b_{i+1},-\},\{b_{i},-\})=\mu _{2}\textrm { for $1\leq i\leq n-2$}\\ & \epsilon (\{b_{n},-\},\{b_{1},-\})=(n-1)\mu _{2}+\mu _{1}=-\mu _{1}-\mu _{2}\\ & \epsilon (\{a_{2},-\},\{a_{1},-\})=\mu _{1}+\mu _{2}-\mu _{3}\\ & \epsilon (\{y_{2},-\},\{y_{1},-\})=\mu _{1}+\mu _{2}+\mu _{3}\\ & \epsilon (\{c_{2},-\},\{c_{1},-\})=\epsilon (\{e_{2},-\},\{e_{1},-\})=\mu _{1}\\ & \epsilon (\{d_{1},-\},\{d_{2},-\})=\epsilon (\{u_{2},-\},\{u_{1},-\})=\mu _{2}\end{aligned}$$ Suppose that $m,n\geq 4$. The equivalence classes of generators in the torsion $\sp $ structures, given by the $\epsilon $-relations above, are given below. For now, we will choose the origin for the $\sp $ structures arbitrarily and call it $\mathfrak{s}_{0}$. We will show later that $\mathfrak{s}_{0}$ is in fact the $\sp $ structure $\mathfrak{s}_{1,n-1}=\mathfrak{s}_{m-1,1}$ (see Lemma \[lemmasp\]). 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We require $m,n\geq 4$ so that the $\sp $ equivalence classes listed above are all distinct. This is still true if $m\geq 4$ and $n=3$ or vice versa. By this requirement we also make sure that the intersection form of the double plumbing ${N_{m,n}}$ is positive definite, which will be important in Section \[App\]. Now we consider the differentials of the chain complex $CF^{+}({Y_{m,n}})$. For every torsion $\sp $ structure $\mathfrak{s}\in \sp ({Y_{m,n}})$, we draw a schematic depicting the generators of $\widehat{CF}({Y_{m,n}},\mathfrak{s})$ vertically according to their relative Maslov grading. Then we list the nonnegative domains of Whitney disks with Maslov index 1 between the generators. If between two generators there is only one such domain of a disk which has a unique holomorphic representative, we denote it by an arrow in the schematic. When neccessary, we also list some domains of Whitney disks with negative coefficients. We denote the action of $\Lambda ^{*}(H_{1}({Y_{m,n}},{\mathbb {Z}})/\operatorname{Tors})$ by a dotted arrow in each schematic. Using this information we calculate the homology $HF^{+}({Y_{m,n}},\mathfrak{s})$. When the first Betti number of a 3–manifold $Y$ is at most 2, the homology $HF^{\infty }(Y,\mathfrak{s})$ in a torsion $\sp $ structure $\mathfrak{s}$ is determined by the integral homology of $Y$ [@OS2 Theorem 10.1]. Since $H^{1}({Y_{m,n}};{\mathbb {Z}})\cong {\mathbb {Z}}$, it follows that for every torsion $\sp $ structure $\mathfrak{s}$, the homology $HF^{\infty }$ of our manifold is given by $$HF^{\infty }({Y_{m,n}},\mathfrak{s})\cong {\mathbb{F}}[U,U^{-1}]\otimes _{{\mathbb {Z}}}\Lambda ^{*}{\mathbb{F}}$$ and has two ${\mathbb{F}}[U,U^{-1}]$ summands. **Classes with two generators** $$\xymatrix{ \{x_{r},b_{j},f_{2}\} \ar@{.>}[d]\\ \{x_{r},b_{j},f_{1}\}}$$ [1]{} & {-,-,f\_[2]{}}{-,-,f\_[1]{}}D\_[13]{}D\_[13]{}+\ & {-,-,f\_[1]{}}{-,-,f\_[2]{}}=D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+D\_[10]{}+D\_[13]{}+D\_[15]{}+\ & A\_ [1]{}+A\_[2]{}+…+ A\_[m-3]{} + As observed above, $HF^{\infty }({Y_{m,n}},\mathfrak{s})$ has two ${\mathbb{F}}[U,U^{-1}]$ summands, so the differential $\partial ^{\infty }$ in this class has to be trivial. This means there is an even number of holomorphic disks from $[\{x_{r},b_{j},f_{2}\},i]$ to $[\{x_{r},b_{j},f_{1}\},i]$. There is a disk from $\{x_{r},b_{j},f_{2}\}$ to $\{x_{r},b_{j},f_{1}\}$ with a bigonal domain $D_{13}$. By the Riemann mapping theorem, this disk has a unique holomorphic representative. The domain of any disk $\phi $ from $\{x_{r},b_{j},f_{2}\}$ to $\{x_{r},b_{j},f_{1}\}$ is given by $\mathcal{D}(\phi )=D_{13}+a\mathcal{P}+b\Sigma $ for two integers $a$ and $b$. Such a disk has Maslov index $\mu (\phi )=1+2b$ and $n_{z}(\phi )=b$. It follows that the Maslov index equals 1 if and only if $n_{z}(\phi )=b=0$, which implies that the domain $D_{13}+a\mathcal{P}$ has only non-negative multiplicities when $a\in \{0,1\}$. Thus the domain of the second holomorphic disk from $\{x_{r},b_{j},f_{2}\}$ to $\{x_{r},b_{j},f_{1}\}$ is $D_{13}+\mathcal{P}$. This disk has an odd number of holomorphic representatives. The differential of the chain complex $CF^{+}({Y_{m,n}})$ in this class is trivial: $\partial ^{+}[\{x_{r},b_{j},f_{2}\},i]=\partial ^{+}[\{x_{r},b_{j},f_{1}\},i]=0$ and it follows that $$HF^{+}({Y_{m,n}},\mathfrak{s})={\mathcal{T}^{+}}\oplus {\mathcal{T}^{+}}$$ is freely generated by the elements $[\{x_{r},b_{j},f_{2}\},i]$ and $[\{x_{r},b_{j},f_{1}\},i]$ for $i\geq 0$. **Class $\mathfrak{s}_{0}+\mu _{1}$** $$\xymatrix{ \{x_{m},c_{1},e_{1}\} \ar@{->}[d] & \quad \\ \{y_{1},c_{1},d_{1}\} & \{x_{m-2},b_{1},f_{2}\} \ar@{.>}[d]\\ \quad & \{x_{m-2},b_{1},f_{1}\}}$$ [1]{} & {x\_[m]{},c\_[1]{},e\_[1]{}}{x\_[m-2]{},b\_[1]{},f\_[2]{}}D\_[1]{}+D\_[2]{}+D\_[3]{}+D\_[6]{}+D\_[7]{}+D\_[9]{}-D\_[10]{}+D\_[12]{}-\ & -D\_[13]{}+D\_[16]{}+A\_[1]{}+…+A\_[m-3]{}+B\_[1]{}+…+B\_[n-3]{},\ & {x\_[m]{},c\_[1]{},e\_[1]{}}{y\_[1]{},c\_[1]{},d\_[1]{}}D\_[7]{} ,\ & {y\_[1]{},c\_[1]{},d\_[1]{}} {x\_[m-2]{},b\_[1]{},f\_[1]{}}D\_[1]{}+D\_[2]{}+D\_[3]{}+D\_[6]{}+D\_[9]{}-D\_[10]{}+D\_[12]{}+D\_[16]{}+\ & +A\_[1]{}+…+A\_[m-3]{}+B\_[1]{}+…+B\_[n-3]{} ,\ & {y\_[1]{},c\_[1]{},d\_[1]{}}{x\_[m]{},c\_[1]{},e\_[1]{}}D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[8]{}+D\_[10]{}+2D\_[13]{}+D\_[15]{}+A\_[1]{}+\ & +…+A\_[m-3]{}, ++2\ & {-,-,f\_[2]{}}{-,-,f\_[1]{}}D\_[13]{}D\_[13]{}+\ & {x\_[m-2]{},b\_[1]{},f\_[2]{}}{x\_[m]{},c\_[1]{},e\_[1]{}}=D\_[4]{}+D\_[5]{}+D\_[8]{}+2D\_[10]{}+D\_[11]{}+2D\_[13]{}+D\_[14]{}+\ & +D\_[15]{}, ++2\ & {-,-,f\_[1]{}}{-,-,f\_[2]{}}=D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+D\_[10]{}+D\_[13]{}+D\_[15]{}+\ & +A\_[1]{}+…+A\_[m-3]{}+\ & {x\_[m-2]{},b\_[1]{},f\_[1]{}}{y\_[1]{},c\_[1]{},d\_[1]{}}=D\_[4]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+2D\_[10]{}+D\_[11]{}+D\_[13]{}+\ & +D\_[14]{}+D\_[15]{}+ There is a disk from $\{x_{m},c_{1},e_{1},\}$ to $\{y_{1},c_{1},d_{1}\}$ with a rectangular domain and a unique holomorphic representative. There is no disk from $\{x_{m},c_{1},e_{1}\}$ to $\{x_{m-2},b_{1},f_{2}\}$ whose domain would be non-negative, so these disks have no holomorphic representatives. Thus $\partial ^{+}[\{x_{m},c_{1},e_{1}\},i]=[\{y_{1},c_{1},d_{1}\},i]$ and $\partial ^{+}[\{y_{1},c_{1},d_{1}\},i]=0$. Similarly we have $\partial ^{\infty }[\{x_{m},c_{1},e_{1}\},i]=[\{y_{1},c_{1},d_{1}\},i]$ and $\partial ^{\infty }[\{y_{1},c_{1},d_{1}\},i]=0$, so $HF^{\infty }({Y_{m,n}},\mathfrak{s}_{0}+\mu _{1})$ is generated by $[\{x_{m-2},b_{1},f_{2}\},i]$ and $[\{x_{m-2},b_{1},f_{1}\},i]$. We already know there is an even number of holomorphic disks from $\{x_{m-2},b_{1},f_{2}\}$ to $\{x_{m-2},b_{1},f_{1}\}$, so $\partial ^{+}[\{x_{m-2},b_{1},f_{2}\},i]=\partial ^{+}[\{x_{m-2},b_{1},f_{1}\},i]=0$. It follows that $$HF^{+}(Y,\mathfrak{s}_{0}+\mu _{1})\cong {\mathcal{T}^{+}}\oplus {\mathcal{T}^{+}}$$ is freely generated by the elements $[\{x_{m-2},b_{1},f_{2}\},i]$ and $[\{x_{m-2},b_{1},f_{1}\},i]$ for $i\geq 0$. **Class $\mathfrak{s}_{0}+\mu _{2}$** $$\xymatrix {\{x_{m-1},c_{2},e_{2}\} \ar@{->}[d] & \quad \\ \{y_{1},c_{2},d_{2}\} & \{x_{1},b_{n-2},f_{2}\} \ar@{.>}[d]\\ \quad & \{x_{1},b_{n-2},f_{1}\}}$$ [1]{} & {x\_[m-1]{},c\_[2]{},e\_[2]{}}{x\_[1]{},b\_[n-2]{},f\_[2]{}}-D\_[2]{}-D\_[6]{}-D\_[7]{}+D\_[10]{}+D\_[11]{}+D\_[12]{}-D\_[13]{}+\ & +D\_[14]{}+D\_[15]{}+D\_[16]{}+B\_[1]{}+…+B\_[n-3]{} ,\ & {x\_[m-1]{},c\_[2]{},e\_[2]{}}{y\_[1]{},c\_[2]{},d\_[2]{}}D\_[10]{} ,\ & {y\_[1]{},c\_[2]{},d\_[2]{}}{x\_[1]{},b\_[n-2]{},f\_[1]{}}-D\_[2]{}-D\_[6]{}-D\_[7]{}+D\_[11]{}+D\_[12]{}+D\_[14]{}+D\_[15]{}+\ & +D\_[16]{}+B\_[1]{}+…+B\_[n-3]{} ,\ & {y\_[1]{},c\_[2]{},d\_[2]{}}{x\_[m-1]{},c\_[2]{},e\_[2]{}}=D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+2D\_[13]{}+D\_[15]{}+\ & +A\_[1]{}+…+A\_[m-3]{}, + +2\ & {x\_[1]{},b\_[n-2]{},f\_[2]{}}{x\_[m-1]{},c\_[2]{},e\_[2]{}}=D\_[1]{}+2D\_[2]{}+D\_[3]{}+D\_[4]{}+D\_[5]{}+2D\_[6]{}+2D\_[7]{}+\ & +D\_[8]{}+D\_[9]{}+2D\_[13]{}+A\_[1]{}+…+A\_[m-3]{}, + +2\ & {-,-,f\_[2]{}}{-,-,f\_[1]{}}D\_[13]{}D\_[13]{}+\ & {x\_[1]{},b\_[n-2]{},f\_[1]{}}{y\_[1]{},c\_[2]{},d\_[2]{}}=D\_[1]{}+2D\_[2]{}+D\_[3]{}+D\_[4]{}+D\_[5]{}+2D\_[6]{}+2D\_[7]{}+D\_[8]{}+\ & +D\_[9]{}+D\_[10]{}+D\_[13]{}+A\_[1]{}+…+A\_[m-3]{}+\ & {x\_[1]{},b\_[n-2]{},f\_[1]{}}{x\_[1]{},b\_[n-2]{},f\_[2]{}}=D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+D\_[10]{}+D\_[13]{}+\ & +D\_[15]{}+A\_[1]{}+…+A\_[m-3]{}+ Using analogous reasoning as above gives the homology $HF^{+}(Y,\mathfrak{s}_{0}+\mu _{2})\cong {\mathcal{T}^{+}}\oplus {\mathcal{T}^{+}}$, freely generated by the elements $[\{x_{1},b_{n-2},f_{2}\},i]$ and $[\{x_{1},b_{n-2},f_{1}\},i]$ for $i\geq 0$. **Class $\mathfrak{s}_{0}-\mu _{2}$** $$\xymatrix{ \{x_{1},c_{1},e_{1}\} \ar@{->}[d]& \quad \\ \{u_{2},a_{2},e_{1}\} & \{x_{m-1},b_{2},f_{2}\} \ar@{.>}[d]\\ \quad & \{x_{m-1},b_{2},f_{1}\}}$$ [1]{} & {x\_[1]{},c\_[1]{},e\_[1]{}}{u\_[2]{},a\_[2]{},e\_[1]{}}D\_[2]{} ,\ & {x\_[1]{},c\_[1]{},e\_[1]{}}{x\_[m-1]{},b\_[2]{},f\_[2]{}}D\_[2]{}+D\_[3]{}+D\_[6]{}+D\_[7]{}+D\_[9]{}-D\_[10]{}+D\_[12]{}-\ & -D\_[13]{}-D\_[14]{}-D\_[15]{}+B\_[1]{}+…+B\_[n-3]{} ,\ & {u\_[2]{},a\_[2]{},e\_[1]{}}{x\_[1]{},c\_[1]{},e\_[1]{}}=D\_[1]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+D\_[10]{}+2D\_[13]{}+D\_[15]{}+\ & +A\_[1]{}+…+A\_[m-3]{}, ++2\ & {u\_[2]{},a\_[2]{},e\_[1]{}}{x\_[m-1]{},b\_[2]{},f\_[1]{}}D\_[3]{}+D\_[6]{}+D\_[7]{}+D\_[9]{}-D\_[10]{}+D\_[12]{}-D\_[14]{}-\ & -D\_[15]{}+B\_[1]{}+…+B\_[n-3]{} ,\ & {x\_[m-1]{},b\_[2]{},f\_[2]{}}{x\_[1]{},c\_[1]{},e\_[1]{}}=D\_[1]{}+D\_[4]{}+D\_[5]{}+D\_[8]{}+2D\_[10]{}+D\_[11]{}+2D\_[13]{}+\ & +2D\_[14]{}+2D\_[15]{}+D\_[16]{}+A\_[1]{}+…+A\_[m-3]{}, ++2\ & {x\_[m-1]{},b\_[2]{},f\_[2]{}}{x\_[m-1]{},b\_[2]{},f\_[1]{}}D\_[13]{}D\_[13]{}+\ & {x\_[m-1]{},b\_[2]{},f\_[1]{}}{u\_[2]{},a\_[2]{},e\_[1]{}}=D\_[1]{}+D\_[2]{}+D\_[4]{}+D\_[5]{}+D\_[8]{}+2D\_[10]{}+D\_[11]{}+\ & +D\_[13]{}+2D\_[14]{}+2D\_[15]{}+D\_[16]{}+A\_[1]{}+…+A\_[m-3]{}+\ & {x\_[m-1]{},b\_[2]{},f\_[1]{}}{x\_[m-1]{},b\_[2]{},f\_[2]{}}=D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+D\_[10]{}+D\_[13]{}+\ & +D\_[15]{}+A\_[1]{}+…+A\_[m-3]{}+ By an analogous reasoning as in the class $\mathfrak{s}_{0}+\mu _{1}$ we conclude that $HF^{+}(Y,\mathfrak{s}_{0}-\mu _{2})\cong {\mathcal{T}^{+}}\oplus {\mathcal{T}^{+}}$ is freely generated by the elements $[\{x_{m-1},b_{2},f_{2}\},i]$ and $[\{x_{m-1},b_{2},f_{1}\},i]$ for $i\geq 0$. **Class $\mathfrak{s}_{0}-\mu _{1}$** $$\xymatrix{\{x_{m},c_{2},e_{2}\} \ar@{->}[d] & \quad \\ \{u_{1},a_{2},e_{2}\} & \{x_{2},b_{n-1},f_{2}\} \ar@{.>}[d]\\ \quad & \{x_{2},b_{n-1},f_{1}\}}$$ [1]{} & {x\_[m]{},c\_[2]{},e\_[2]{}}{u\_[1]{},a\_[2]{},e\_[2]{}}D\_[15]{} ,\ & {x\_[m]{},c\_[2]{},e\_[2]{}}{x\_[2]{},b\_[n-1]{},f\_[2]{}}-D\_[2]{}-D\_[3]{}-D\_[4]{}-D\_[6]{}+D\_[8]{}+D\_[10]{}+D\_[15]{}+A\_[1]{}+\ & +A\_[2]{}+…+A\_[m-3]{} ,\ & {u\_[1]{},a\_[2]{},e\_[2]{}}{x\_[m]{},c\_[2]{},e\_[2]{}}=D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+D\_[10]{}+2D\_[13]{}+A\_[1]{}+\ & +A\_[2]{}+…+A\_[m-3]{}, ++2\ & {u\_[1]{},a\_[2]{},e\_[2]{}}{x\_[2]{},b\_[n-1]{},f\_[1]{}}-D\_[2]{}-D\_[3]{}-D\_[4]{}-D\_[6]{}+D\_[8]{}+D\_[10]{}+D\_[13]{}+A\_[1]{}+\ & +A\_[2]{}+…+A\_[m-3]{} ,\ & {x\_[2]{},b\_[n-1]{},f\_[2]{}}{x\_[m]{},c\_[2]{},e\_[2]{}}=D\_[1]{}+2D\_[2]{}+D\_[3]{}+D\_[4]{}+D\_[5]{}+D\_[6]{}+D\_[7]{}+2D\_[13]{},\ & ++2\ & {x\_[2]{},b\_[n-1]{},f\_[2]{}}{x\_[2]{},b\_[n-1]{},f\_[1]{}}D\_[13]{}D\_[13]{}+\ & {x\_[2]{},b\_[n-1]{},f\_[1]{}}{u\_[1]{},a\_[2]{},e\_[2]{}}D\_[1]{}+2D\_[2]{}+2D\_[3]{}+2D\_[4]{}+D\_[5]{}+2D\_[6]{}+D\_[7]{}+D\_[9]{}+\ & +D\_[11]{}+D\_[12]{}+D\_[14]{}+D\_[15]{}+D\_[16]{}+B\_[1]{}+…+B\_[n-3]{}\ & {x\_[2]{},b\_[n-1]{},f\_[1]{}}{x\_[2]{},b\_[n-1]{},f\_[2]{}}=D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+D\_[10]{}+D\_[13]{}+\ & +D\_[15]{}+A\_[1]{}+…+A\_[m-3]{}+ Again we use an analogous reasoning as in the class $\mathfrak{s}_{0}+\mu _{1}$ to obtain $$HF^{+}(Y,\mathfrak{s}_{0}-\mu _{1})\cong {\mathcal{T}^{+}}\oplus {\mathcal{T}^{+}}\;,$$ freely generated by the elements $[\{x_{2},b_{n-1},f_{2}\},i]$ and $[\{x_{2},b_{n-1},f_{1}\},i]$ for $i\geq 0$. In the following calculations, we apply the change of basepoint formula using [@OS1 Lemma 2.19]: \[lemaz\] Let $(\Sigma ,(\alpha _{1},\ldots ,\alpha _{g}),(\beta _{1},\ldots ,\beta _{g}),z_{1})$ be a Heegaard diagram. Denote by $z_{2}\in \Sigma -\alpha _{1}-\ldots -\alpha _{g}-\beta _{1}-\ldots -\beta _{g}$ a new basepoint, for which the following holds: there is an arc $z_{t}$ from $z_{1}$ to $z_{2}$ on the surface $\Sigma $, which is disjoint from all curves $\beta _{i}$ and from all curves $\alpha _{i}$ appart from $\alpha _{j}$. Then for any generator $\mathbf{x}\in {\mathbb {T}}_{\alpha }\cap {\mathbb {T}}_{\beta }$ we have $$\mathfrak{s}_{z_{2}}(\mathbf{x})-\mathfrak{s}_{z_{1}}(\mathbf{x})=\alpha _{j}^{*}\;,$$ where $\alpha _{j}^{*}\in H^{2}(Y;{\mathbb {Z}})$ is the Poincaré dual of the homology class in ${Y_{m,n}}$ induced by the curve $\gamma $ in $\Sigma $, for which $\alpha _{j}\cdot \gamma =1$ and whose intersection number with any other curve $\alpha _{i}$ for $j\neq i$ equals $0$. **Class $\mathfrak{s}_{0}+\mu _{1}+\mu _{2}$**\ We change the basepoint $z_{1}\in D_{5}$ for a new basepoint $z_{2}\in D_{2}$. By Lemma \[lemaz\] the $\sp $ structure $\mathfrak{s}_{0}+\mu _{1}=\mathfrak{s}_{z_{1}}(\{x_{m-2},b_{1},f_{1}\})$ changes to $\mathfrak{s}_{0}+\mu _{1}+\mu _{2}=\mathfrak{s}_{z_{2}}(\{x_{m-2},b_{1},f_{1}\})$. In the new $\sp $ structure we have the same generators as in the class $\mathfrak{s}_{0}+\mu _{1}$, but with a new relative grading, induced by the basepoint $z_{2}$: $$\xymatrix{\quad & \{x_{m-2},b_{1},f_{2}\}\ar@{.>}[d]\\ \{x_{m},c_{1},e_{1}\} \ar@{->}[d] & \{x_{m-2},b_{1},f_{1}\}\\ \{y_{1},c_{1},d_{1}\} & \quad }$$ We already know from the class $\mathfrak{s}_{0}+\mu _{1}$ that the only nontrivial differential of $HF^{\infty }$ in this class is $\partial ^{\infty }[\{x_{m},c_{1},e_{1}\},i]=[\{y_{1},c_{1},d_{1}\},i]$. It follows that $\partial ^{+}[\{x_{m},c_{1},e_{1}\},i]=[\{y_{1},c_{1},d_{1}\},i]$ and also $\partial ^{+}[\{x_{m-2},b_{1},f_{2}\},i]=\partial ^{+}[\{x_{m-2},b_{1},f_{1}\},i]=0$. The resulting homology $$HF^{+}({Y_{m,n}},\mathfrak{s}_{0}+\mu _{1}+\mu _{2})={\mathcal{T}^{+}}\oplus {\mathcal{T}^{+}}$$ is freely generated by the elements $[\{x_{m-2},b_{1},f_{2}\},i]$ and $[\{x_{m-2},b_{1},f_{1}\},i]$ for $i\geq 0$. **Class $\mathfrak{s}_{0}-\mu _{1}-\mu _{2}$**\ We change the basepoint $z_{1}\in D_{5}$ for a new basepoint $z_{2}\in D_{2}$. By Lemma \[lemaz\], the $\sp $ structure $\mathfrak{s}_{0}-\mu _{1}-2\mu _{2}=\mathfrak{s}_{z_{1}}(\{x_{m},b_{2},f_{1}\})$ changes to $\mathfrak{s}_{0}-\mu _{1}-\mu _{2}=\mathfrak{s}_{z_{2}}(\{x_{m},b_{2},f_{1}\})$. In the new $\sp $ structure we have the same generators as in the $\sp $ structure $\mathfrak{s}_{0}-\mu _{1}-2\mu _{2}$, but the relative grading is now induced by the basepoint $z_{2}$: $$\xymatrix{ \{x_{m},b_{2},f_{2}\} \ar@{.>}[d]\\ \{x_{m},b_{2},f_{1}\}}$$ We already know that in the classes containing only two generators, the differential $\partial ^{\infty }$ is trivial. Therefore the resulting homology is $$HF^{+}({Y_{m,n}},\mathfrak{s}_{0}-\mu _{1}-\mu _{2})={\mathcal{T}^{+}}\oplus {\mathcal{T}^{+}}\;,$$ freely generated by the elements $[\{x_{m},b_{2},f_{2}\},i]$ and $[\{x_{m},b_{2},f_{1}\},i]$ for $i\geq 0$. **Classes $\mathfrak{s}_{0}-i\mu _{1}-\mu _{2}$ for $2\leq i\leq m-2$**\ We change the basepoint $z_{1}\in D_{5}$ for a new basepoint $z_{2}\in D_{2}$. By Lemma \[lemaz\], the $\sp $ structure $\mathfrak{s}_{0}-i\mu _{1}-2\mu _{2}=\mathfrak{s}_{z_{1}}(\{x_{i-1},b_{1},f_{1}\})$ changes to $\mathfrak{s}_{0}-i\mu _{1}-\mu _{2}=\mathfrak{s}_{z_{2}}(\{x_{i-1},b_{1},f_{1}\})$. In the new $\sp $ structure we have the same generators as in the $\sp $ structure $\mathfrak{s}_{0}-i\mu _{1}-2\mu _{2}$, but the relative grading is now induced by the basepoint $z_{2}$: $$\xymatrix{ \{x_{i-1},b_{1},f_{2}\} \ar@{.>}[d]\\ \{x_{i-1},b_{1},f_{1}\}}$$ We already know that in the classes containing only two generators, the differential $\partial ^{\infty }$ is trivial. Therefore the resulting homology is $$HF^{+}({Y_{m,n}},\mathfrak{s}_{0}-i\mu _{1}-\mu _{2})={\mathcal{T}^{+}}\oplus {\mathcal{T}^{+}}\;,$$ freely generated by the elements $[\{x_{i-1},b_{1},f_{2}\},j]$ and $[\{x_{i-1},b_{1},f_{1}\},j]$ for $j\geq 0$. **Class $\mathfrak{s}_{0}$**\ We change the basepoint $z_{1}\in D_{5}$ for a new basepoint $z_{2}\in D_{2}$. By Lemma \[lemaz\], the $\sp $ structure $\mathfrak{s}_{0}-\mu _{2}=\mathfrak{s}_{z_{1}}(\{x_{m-1},b_{2},f_{1}\})$ changes to $\mathfrak{s}_{0}=\mathfrak{s}_{z_{2}}(\{x_{m-1},b_{2},f_{1}\})$. In the new $\sp $ structure we have the same generators as in the $\sp $ structure $\mathfrak{s}_{0}-\mu _{2}$, but with a new relative grading, induced by the basepoint $z_{2}$: $$\xymatrix{ \{u_{2},a_{2},e_{1}\} & \{x_{m-1},b_{2},f_{2}\} \ar@{.>}[d]\\ \{x_{1},c_{1},e_{1}\} \ar@{->}[u] & \{x_{m-1},b_{2},f_{1}\}}$$ From our calculation in the $\sp $ structure $\mathfrak{s}_{0}-\mu _{2}$ we deduce that the only nontrivial differential of $CF^{\infty }$ in this class is $\partial ^{\infty }[\{x_{1},c_{1},e_{1}\},i]=[\{u_{2},a_{2},e_{1}\},i-1]$. In the complex $CF^{+}$ we have $\partial ^{+}[\{x_{1},c_{1},e_{1}\},i]=[\{u_{2},a_{2},e_{1}\},i-1]$ for $i\geq 1$ and $\partial ^{+}[\{x_{1},c_{1},e_{1}\},0]=0$. It follows that [1]{} & HF\^[+]{}([Y\_[m,n]{}]{},\_[0]{})[\^[+]{}]{}[\^[+]{}]{}\[{x\_[1]{},c\_[1]{},e\_[1]{}},0\], where the first two summands are freely generated by the elements $[\{x_{m-1},b_{2},f_{2}\},i]$ and $[\{x_{m-1},b_{2},f_{1}\},i]$ for $i\geq 0$. The homology group $HF^{\infty }({Y_{m,n}},\mathfrak{s}_{0})$ however equals $$HF^{\infty }({Y_{m,n}},\mathfrak{s}_{0})\cong {\mathbb{F}}[U,U^{-1}]\oplus {\mathbb{F}}[U,U^{-1}]\,.$$ We have thus calculated the Heegaard–Floer homology $HF^{+}({Y_{m,n}},\mathfrak{s})$ for all torsion $\sp $ structures $\mathfrak{s}$ on the manifold ${Y_{m,n}}$. In the following Subsection, we calculate the absolute gradings of the generators and finish the proof of Theorem \[th1\]. If $b_{1}(Y)>0$ then there is an action of the exterior algebra $\Lambda ^{*}(H_{1}(Y;{\mathbb {Z}})/\operatorname{Tors})$ on the groups $HF^{\infty }(Y,\mathfrak{s})$ and $\widehat{HF}(Y,\mathfrak{s})$ for every torsion $\sp $ structure $\mathfrak{s}$ on $Y$ [@OS1 Proposition 4.17, Remark 4.20]. Let $\gamma $ be a simple closed curve on the Heegaard surface $\Sigma $ in general position with respect to the $\alpha $ curves and let $[\gamma ]$ be its induced homology class in $H_{1}(Y,{\mathbb {Z}})$. Then the action is given by $$A_{[\gamma ]}([\mathbf{x},i])=\sum _{\mathbf{y}}\sum _{\{\phi \in \pi _{2}(\mathbf{x},\mathbf{y})|\, \mu (\phi )=1\}}a(\gamma ,\phi )\cdot [\mathbf{y},i-n_{z}(\phi )]\;,$$ where $$a(\gamma ,\phi )=\# \{u\in \mathcal{M}(\phi )|\, u(1\times 0)\in (\gamma \times \textrm{Sym}^{g-1}(\Sigma ))\cap \mathbb{T}_{\alpha }\}\;.$$ The value $d_{b}(Y,\mathfrak{s})$ is the least grading of an element of $HF^{\infty }(Y,\mathfrak{s})$ that is in the kernel of the action of $\Lambda ^{*}(H_{1}(Y;{\mathbb {Z}})/\operatorname{Tors})$ and whose image in $HF^{+}(Y,\mathfrak{s})$ is nonzero. In the case of $Y={Y_{m,n}}$ and $\mathfrak{s}$ any torsion $\sp $ structure on ${Y_{m,n}}$, the image of $HF^{+}({Y_{m,n}},\mathfrak{s})$ in $d\widehat{HF}({Y_{m,n}},\mathfrak{s})$ is generated by two elements of the form $\{x_{r},b_{j},f_{2}\}$ and $\{x_{r},b_{j},f_{1}\}$. As we have shown in the beginning of this subsection, there are two homotopy classes of disks $\phi _{1}$ and $\phi _{2}$ from $\{x_{r},b_{j},f_{2}\}$ to $\{x_{r},b_{j},f_{1}\}$ (represented by the domains $D_{13}$ and $D_{13}+\mathcal{P}$) and they both have an odd number of holomorphic representatives. Thus, we have $\# \widehat{\mathcal{M}}(\phi _{1})=\# \widehat{\mathcal{M}}(\phi _{2})=1$. The group $H_{1}({Y_{m,n}};{\mathbb {Z}})/\operatorname{Tors}={\mathbb {Z}}$ is generated by the simple closed curve $\mu _{3}$ on the Heegaard diagram (see Figure \[fig:heeg\]), so $a(\gamma ,\phi _{1})=0$ and $a(\gamma ,\phi _{2})=1$. It follows that $$A_{[\gamma ]}([\{x_{r},b_{j},f_{2}\},i])=[\{x_{r},b_{j},f_{1}\},i]$$ and the action on $\{x_{r},b_{j},f_{1}\}$ is trivial. So $d_{b}({Y_{m,n}},\mathfrak{s})$ is given as the absolute grading of the generator $\{x_{r},b_{j},f_{1}\}$ and $d_{t}({Y_{m,n}},\mathfrak{s})$ is the absolute grading of the generator $\{x_{r},b_{j},f_{2}\}$. For the definitions of the bottom and top correction terms, see [@LRS Definition 3.3]. [Absolute gradings]{} \[absolute\] The absolute grading of the generators of $\widehat{HF}({Y_{m,n}})$ can be calculated using the cobordism $W$ from ${Y_{m,n}}$ to the simpler 3–manifold $-L(m,1)\# S^{1}\times S^{2}$ whose absolute grading is known. To construct the cobordism, we use a pointed Heegaard triple $(\Sigma ,\vec{\alpha },\vec{\beta },\vec{\gamma },z)$. Here the first two sets of the curves $\vec{\alpha },\vec{\beta }$ stay the same as before, so $Y_{\alpha ,\beta }={Y_{m,n}}$. The curves $\gamma _{1}$ and $\gamma _{3}$ are parallel copies of the curves $\beta _{1}$ and $\beta _{3}$ respectively, and the curve $\gamma _{2}$ is homologous to the meridian $\mu _{2}$ (see Figure \[fig:triple\]). This means $Y_{\beta ,\gamma }=\# ^{2}S^{1}\times S^{2}$ and $Y_{\alpha ,\gamma }=-L(m,1)\# S^{1}\times S^{2}$. Filling the second boundary component $\# ^{2}S^{1}\times S^{2}$ by $\# ^{2}S^{1}\times B^{3}$ we get the surgery cobordism $W$ from ${Y_{m,n}}$ to $ -L(m,1)\# S^{1}\times S^{2}$. The cobordism $W$ equipped with a $\sp $ structure $\mathfrak{s}$ induces a map $$F_{W,\mathfrak{s}}\colon \widehat{HF}({Y_{m,n}})\to \widehat{HF}(-L(m,1)\# S^{1}\times S^{2})\;.$$ Under this map, the absolute grading of a generator $\zeta \in \widehat{HF}({Y_{m,n}})$ is changed by [@OS4 Formula (4)]: [1]{} \[grading\] & (F\_[W,]{}())-()=. at 365 295 at 376 657 at 350 484 at 488 265 at 324 520 at 183 402 at 520 265 at 336 585 at 152 402 \[b\] at 288 286 \[b\] at 304 286 \[b\] at 332 286 \[t\] at 355 283 \[b\] at 395 286 at 362 310 at 363 342 at 400 442 at 374 450 \[b\] at 370 475 \[b\] at 402 475 \[b\] at 382 475 \[b\] at 390 475 \[b\] at 335 652 \[b\] at 280 652 \[b\] at 298 652 \[b\] at 324 652 \[b\] at 364 652 at 208 708 at 500 324 at 473 370 at 457 360 at 468 530 at 456 460 at 422 360 at 430 432 at 396 396 at 204 414 at 370 214 at 370 360 at 353 380 at 352 576 at 369 576 at 381 576 at 397 576 at 324 380 at 406 692 at 490 214 at 394 692 at 456 214 at 265 788 at 429 200 at 372 684 at 411 191 at 224 512 at 396 194 at 310 490 at 175 624 at 240 711 at 216 684 at 320 548 at 260 399 at 295 455 at 200 455 at 380 200 at 386 445 at 404 461 at 390 467 at 375 468 at 282 303 at 294 672 at 350 307 at 365 324 at 246 280 at 240 648 at 275 490 at 250 440  \[fig:triple\] The intersections between the $\alpha $ and $\beta $ curves are denoted in the same way as before. New intersections between the $\alpha $, $\beta $ and $\gamma $ curves we will need are denoted by: $\alpha _{1}\cap \gamma _{1}=\{x_{1}',x_{2}',\ldots ,x_{m}'\}$, $\alpha _{2}\cap \gamma _{2}=\{s\}$, $\alpha _{3}\cap \gamma _{3}=\{f_{1}',f_{2}'\}$, $\beta _{1}\cap \gamma _{1}=\{t_{1}^{+},t_{1}^{-}\}$, $\beta _{2}\cap \gamma _{2}=\{r\}$, $\beta _{3}\cap \gamma _{3}=\{t_{2}^{+},t_{2}^{-}\}$ (see Figure \[fig:triple\]). We express the $\alpha $, $\beta $ and $\gamma $ curves of the Heegaard triple in the standard basis of the surface $\Sigma $ as: [1]{} & \_[i]{}\~\_[i]{}i=1,2,3\ & \_[1]{}\~\_[1]{}\~m\_[1]{}+2\_[2]{}-\_[1]{}\ & \_[2]{}\~2\_[1]{}+n\_[2]{}-\_[2]{}\ & \_[3]{}\~\_[3]{}\~\_[3]{}\ & \_[2]{}\~\_[2]{} The elementary domains in the winding region of the curve $\beta _{1}$ are denoted by $A_{i}$ for $i=1,\ldots ,2m-5$, the elementary domains in the winding region of the curve $\beta _{2}$ are denoted by $B_{j}$ for $j=1,\ldots ,n-3$ and the other elementary domains of the Heegaard triple are denoted by $D_{i}$ for $i=1,\ldots ,34$. There are four hexagons $D_{1},D_{3},D_{9}$ and $D_{20}$, three pentagons $D_{22}$, $D_{24}$ and $D_{25}$, five triangles $D_{10}$, $D_{23}$, $D_{30}$, $D_{32}$ and $D_{33}$, two bigons $D_{29}$ and $D_{34}$, one octagon $D_{15}$ and a domain $D_{8}$ with 14 sides. All the other elementary domains are rectangles. We put the basepoint into the elementary domain $D_{8}$, which corresponds to the basepoint $z\in D_{5}$ of the Heegaard diagram \[fig:heeg\]. We have a triply-periodic domain [1]{} & =(m-2)(D\_[1]{}+D\_[2]{}+D\_[3]{})-2(D\_[4]{}+D\_[5]{}+D\_[6]{}+D\_[7]{})-m(D\_[9]{}+D\_[10]{}+D\_[11]{}+D\_[12]{})+\ & +(2-m)D\_[15]{}+(m-2)D\_[16]{}+m(D\_[17]{}+D\_[18]{}+D\_[19]{})+2D\_[22]{}+(mn-2)D\_[23]{}+\ & +(m(n-1)-2)D\_[24]{}+(2-m)D\_[25]{}+2D\_[26]{}-mD\_[33]{}+(2-m)D\_[34]{}+\ & +\_[i=1]{}\^[m-3]{}(m-2(i+1))(A\_[2i-1]{}+A\_[2i]{})+\_[j=1]{}\^[n-3]{}((j+1)m-2)B\_[j]{} The orientation of the curves in the Heegaard triple is denoted on the diagram. The boundary of the triply-periodic domain is equal to $$\partial \mathcal{Q}=2\alpha _{1}+2\beta _{1}-m\alpha _ {2}-m\beta _{2}+(mn-4)\gamma _{2}\;.$$ We calculate the Euler measure of the triply-periodic domain [@OS5 Lemma 6.2]: [1]{} & ()=2(m-2)(1-)-m(1-+1-)+(2-m)(1-)+2(1-)+\ & +(mn-2)(1-)+(m(n-1)-2)(1-)+(2-m)(1-)-m(1-)+\ & +(2-m)(1-)=0. We have $n_{z}(\mathcal{Q})=0$ and $\# (\partial \mathcal{Q})=m(n+2)$. The self-intersection number $\mathcal{H}(\mathcal{Q})^{2}$ is calculated by counting the intersections of $\alpha $ and $\beta $ curves in the boundary of the triply-periodic domain (according to the chosen orientation of the boundary). We get $\alpha _{1}\cdot \beta _{1}=-m$, $\alpha _{1}\cdot \beta _{2}=-2$, $\alpha _{2}\cdot \beta _{1}=-2$ and $\alpha _{2}\cdot \beta _{2}=-n$, which gives us [1]{} &()\^[2]{}=\_\_=4\_[1]{}\_[1]{}-2m\_[1]{}\_[2]{}-2m\_[2]{}\_[1]{}+m\^[2]{}\_[2]{}\_[2]{}=-m(mn-4) Since the self-intersection number is negative for $mn-4>0$, the signature of the associated cobordism equals $\sigma (W)=-1$. $W$ is the surgery cobordism from ${Y_{m,n}}$ to $Y_{\alpha ,\gamma }=L(m,1)\#S^{1}\times S^{2}$, thus $\chi (W)=1$. Next we investigate the domains of Whitney triangles on the Heegaard surface. A Whitney triangle connecting $\mathbf{x}$, $\mathbf{y}$ and $\mathbf{w}$ is given by a map $u\colon \Delta \to \operatorname{Sym}^{g}\Sigma $ for which $u(v_{\gamma })=\mathbf{x}$, $u(v_{\alpha })=\mathbf{y}$, $u(v_{\beta })=\mathbf{w}$ and $u(e_{\alpha })\subset T_{\alpha }$, $u(e_{\beta })\subset T_{\beta }$ in $u(e_{\gamma })\subset T_{\gamma }$. The dual spider number of a triangle $u$ and a triply-periodic domain $\mathcal{Q}$ is defined in [@OS5] by $$\sigma (u,\mathcal{Q})=n_{u(x)}(\mathcal{Q})+\# (a\cap \partial _{\alpha }'\mathcal{Q})+\# (b\cap \partial _{\beta }'\mathcal{Q})+\# (c\cap \partial _{\gamma }'\mathcal{Q})\;,$$ where $x\in \Delta $ is a chosen point in general position and $a$, $b$, $c$ are chosen paths from $x$ to the respective edges $e_{0}$, $e_{1}$ and $e_{2}$ of the triangle $\Delta $. We show the following: \[triangle\] Let the basepoint of the Heegaard diagram \[fig:heeg\] lie in the elementary domain $D_{5}$. For $1\leq i\leq m-1$ and $1\leq j\leq n-1$, there is a Whitney triangle $$u\colon \{x_{i},b_{j},f_{2}\}\to \{t_{1}^{+},r,t_{2}^{+}\}\to \{x_{i}',s,f_{2}'\}$$ with $\sigma (u,Q)=-mn+jm-2i$. For a Whitney triangle $u\colon \Delta \to \operatorname{Sym}^{3}(\Sigma )$, the image $u(\Delta )$ is a triple branched cover over a triangle. In some cases this is a trivial disconnected cover consisting of three triangles $u_{1}$, $u_{2}$ and $u_{3}$ on the surface $\Sigma $. For $1\leq i\leq m-1$ and $1\leq j\leq n-1$ we can find a triangle with the following components. The first component is a triangle between the points $x_{i},t_{1}^{+}$ and $x_{i}'$ (for $1\leq i\leq m-1$) with domain $D_{33}+A_{2m-5}+A_{2m-7}+\ldots +A_{2i-1}$ (see Figure \[fig:trikot1\]). The dual spider number of this component is equal to $\sigma _{1}(u_{i},\mathcal{Q})=m-2(i+1)$. There is also a triangle between the points $x_{m},t_{1}^{+}$ and $x_{m}'$ with the dual spider number $\sigma _{1}(u,\mathcal{Q})=-2$. The second component of the Whitney triangle (Figure \[fig:trikot3\]) is a triangle between the points $b_{j}$, $r$ and $s$ (for $1\leq j\leq n-1$) with domain $$(n-j)D_{23}+(n-j-1)D_{24}+(n-j-2)B_{n-3}+(n-j-3)B_{n-4}+\ldots +B_{j}\;,$$ where all the coefficients of the domain have to be positive. The dual spider number of this component is equal to $\sigma _{2}(u_{j},\mathcal{Q})=2-mn+(j-1)m$. The third component of the Whitney triangle is a triangle between the points $f_{2},t_{2}^{+}$ and $f_{2}'$ with domain $D_{30}$ (Figure \[fig:trikot2\]). The dual spider number of this component is equal to $\sigma _{3}(u,\mathcal{Q})=0$. Combining the above we obtain $$\sigma (u,\mathcal{Q})=\sigma _{1}(u,\mathcal{Q})+\sigma _{2}(u,\mathcal{Q})+\sigma _{3}(u,\mathcal{Q})=-mn+jm-2i\;.$$ $x_{m-1}$ \[r\] at 255 725 $\alpha _{1}$ \[r\] at 255 670 $\beta _{1}$ at 320 730 $\gamma _{1}$ at 320 630 $x_{m-1}'$ \[r\] at 255 625 $t_{1}'$ at 428 674 $\alpha _{1}$ \[r\] at 388 370 $\beta _{1}$ at 184 400 $\gamma _{1}$ at 95 216 $x_{m-2}$ \[r\] at 388 429 $x_{m-2}'$ \[r\] at 388 314 $t_{1}'$ \[l\] at 542 216 $-m$ at 432 213 $2-m$ at 132 460 $-m$ \[l\] at 265 673 ![The first component of a Whitney triangle; two versions[]{data-label="fig:trikot1"}](trikot1) $mn-2$ \[l\] at 230 680 $m(n-1)-2$ at 210 270 $b_{n-1}$ \[r\] at 210 726 $r$ at 412 800 $\beta _{2}$ at 280 738 $\alpha _{2}$ \[r\] at 210 672 $\gamma _{2}$ at 390 648 $s$ \[r\] at 210 622 $\beta _{2}$ at 220 428 $b_{n-2}$ \[r\] at 324 368 $\alpha _{2}$ \[l\] at 336 338 $r$ at 536 390 $s$ \[r\] at 324 204 $\gamma _{2}$ at 440 218 ![The second component of a Whitney triangle; two versions[]{data-label="fig:trikot3"}](trikot3) $\alpha _{3}$ \[l\] at 680 310 $\beta _{3}$ at 420 460 $\gamma _{3}$ at 400 280 $t_{2}^{+}$ at 160 425 $f_{2}$ \[l\] at 680 440 $f_{2}'$ \[l\] at 680 180 $0$ at 560 330  \[fig:trikot2\] We are now prepared to compute the absolute gradings of the generators of $\widehat{HF}({Y_{m,n}})$. \[calc\] If the basepoint of the Heegaard diagram \[fig:heeg\] lies in the elementary domain $D_{5}$, then the absolute grading of the generator $\{x_{i},b_{j},f_{2}\}$ is given by $${\widetilde{\operatorname{gr}}}(\{x_{i},b_{j},f_{2}\})=\frac{m^{2}n+mn^{2}-4mn(i+j+1)+4n(i^{2}+2i)+4m(j^{2}+2j)-16ij}{4(mn-4)}$$ for $1\leq i\leq m-1$ and $1\leq j\leq n-1$. By Lemma \[triangle\], the generator $\{x_{i},b_{j},f_{2}\}$ is connected to a generator of $\widehat{HF}(-L(m,1)\# S^{1}\times S^{2})$ by a Whitney triangle $$u\colon \{x_{i},b_{j},f_{2}\}\to \{t_{1}^{+},r,t_{2}^{+}\}\to \{x_{i}',s,f_{2}'\}$$ with $\sigma (u,Q)=-mn+jm-2i$. Now we apply the grading shift formula . The absolute grading of the generators of $\widehat{HF}(-L(m,1)\# S^{1}\times S^{2})$ can be calculated from [@OS4 Proposition 4.8]. The $i$-th torsion $\sp $ structure on $-L(m,1)\# S^{1}\times S^{2}$ contains two generators: $\{x_{i}',s,f_{2}'\}$ with absolute grading $${\widetilde{\operatorname{gr}}}(\{x_{i}',s,f_{2}'\})=\frac{(2i-m)^{2}-m}{4m}+\frac{1}{2}$$ and $\{x_{i}',s,f_{1}'\}$ with grading $${\widetilde{\operatorname{gr}}}(\{x_{i}',s,f_{1}'\})= \frac{(2i-m)^{2}-m}{4m}-\frac{1}{2}$$ where $i=1,\ldots ,m$. We calculate [1]{} \[c1\] & c\_[1]{}(\_[z]{}(u)),()=m(n+2)+2(u,)=-mn+2(j+1)m-4i [1]{} & c\_[1]{}(\_[z]{}(u))\^[2]{}=\ & ({x\_[i]{}’,s,f\_[2]{}’})=+=+\ & ({x\_[i]{},b\_[j]{},f\_[2]{}})=({x\_[i]{}’,s,f\_[2]{}’})-=\ & = Observe the symmetry ${\widetilde{\operatorname{gr}}}(\{x_{m-i},b_{n-j},f_{2}\})={\widetilde{\operatorname{gr}}}(\{x_{i},b_{j},f_{2}\})$. The above formula calculates the absolute grading ${\widetilde{\operatorname{gr}}}(\{x_{i},b_{j},f_{2}\})$ for $1\leq i\leq m-1$ and $1\leq j\leq n-1$. To calculate the absolute grading of the generators $\{x_{m},b_{j},f_{2}\}$ and $\{x_{i},b_{n},f_{2}\}$ of $\widehat{HF}({Y_{m,n}})$, we use the method of Lee and Lipshitz [@LELI]. Their idea is as follows. If two generators $\mathbf{x},\mathbf{y}\in \widehat{HF}(Y)$ represent different torsion $\sp $ structures $\mathfrak{s}_{z}(\mathbf{x})$ and $\mathfrak{s}_{z}(\mathbf{y})$ on a 3-manifold $Y$, then there exists a covering projection $\pi \colon \widetilde{Y}\to Y$ such that $\pi ^{*}\mathfrak{s}_{z}(\mathbf{x})=\pi ^{*}\mathfrak{s}_{z}(\mathbf{y})$ on $\widetilde{Y}$. Thus, there exist lifts $\tilde {\mathbf{x}}$ of $\mathbf{x}$ and $\tilde {\mathbf{y}}$ of $\mathbf{y}$ whose relative grading difference is given by the domain bounded by a closed curve representing $\epsilon (\tilde {\mathbf{x}},\tilde{\mathbf{y}})$. The projection of this domain onto the Heegaard diagram for $Y$ is bounded by some multiple of a closed curve representing $\epsilon (\mathbf{x},\mathbf{y})$. We can reconstruct the relative grading difference between $\mathbf{x}$ and $\mathbf{y}$ from this projection, as described in [@LELI Subsection 2.3]. \[calc1\] If the basepoint of the Heegaard diagram \[fig:heeg\] lies in the elementary domain $D_{5}$, then $$\begin{aligned} \label{grmn} & {\widetilde{\operatorname{gr}}}\{x_{m},b_{j},f_{2}\}=\frac{m^{2}n+mn^{2}-4mnj+4mj^{2}-4m}{4(mn-4)}\end{aligned}$$ $$\begin{aligned} \label{grmn1} & {\widetilde{\operatorname{gr}}}\{x_{i},b_{n},f_{2}\}=\frac{m^{2}n+mn^{2}-4mni+4ni^{2}-4n}{4(mn-4)}\textrm{ and }{\widetilde{\operatorname{gr}}}\{x_{m},b_{n},f_{2}\}=\frac{m+n-4}{4} \end{aligned}$$ for $1\leq i\leq m-1$ and $1\leq j\leq n-1$. In the Heegaard diagram \[fig:heeg\] we find a domain $$\begin{aligned} & S=(m+n-4)(D_{1}+D_{16})+(m-2)(D_{2}+D_{3})+(n-2)(-D_{6}-D_{7}+D_{14}+D_{15})+\\ & +(mn-m-n)(D_{8}+D_{9})+(mn-m-2)(D_{10}+D_{11})+\\ & +\sum _{i=1}^{m-3}\left (m+(i+1)n-2(i+2)\right )A_{i}+\sum _{i=1}^{n-3}\left (n+(i+1)m-2(i+2)\right )B_{i}\end{aligned}$$ for which $\partial \partial _{\alpha }S=(mn-4)(b_{n-1}-b_{n})$. Thus we can compute $$\begin{aligned} & {\widetilde{\operatorname{gr}}}\{x_{i},b_{n},f_{k}\}-{\widetilde{\operatorname{gr}}}\{x_{i},b_{n-1},f_{k}\}=\frac{1}{mn-4}\left (e(S)+n_{\{x_{i},b_{n-1},f_{k}\}}(S)+n_{\{x_{i},b_{n},f_{k}\}}(S)\right )=\\ & =\frac{(-mn+4)+(mn-m-n)+2(m+ni-2(i+1))}{mn-4}=\frac{m+(2i-1)n-4i}{mn-4}\end{aligned}$$ for $1\leq i\leq m-1$ and ${\widetilde{\operatorname{gr}}}\{x_{m},b_{n},f_{k}\}-{\widetilde{\operatorname{gr}}}\{x_{m},b_{n-1},f_{k}\}=\frac{4-m-n}{mn-4}$. Similarly, the domain $$\begin{aligned} & T=(m+n-4)(D_{1}+D_{16})+(m-2)(D_{2}+D_{3}-D_{10}-D_{11})+\\ & +(mn-n-2)(D_{6}+D_{7})+(mn-m-n)(D_{8}+D_{9})+(n-2)(D_{14}+D_{15})+\\ & +\sum _{i=1}^{m-3}\left (m+(i+1)n-2(i+2)\right )A_{i}+\sum _{i=1}^{n-3}\left (n+(i+1)m-2(i+2)\right )B_{i}\end{aligned}$$ has $\partial \partial _{\alpha }T=(mn-4)(x_{m-1}-x_{m})$. A calculation gives us $$\begin{aligned} & {\widetilde{\operatorname{gr}}}\{x_{m},b_{j},f_{k}\}-{\widetilde{\operatorname{gr}}}\{x_{m-1},b_{j},f_{k}\}=\frac{n+(2j-1)m-4j}{mn-4}\end{aligned}$$ for $1\leq j\leq n-1$. Combining this with Proposition \[calc\], we get formulas and . In some torsion $\sp $ structures on ${Y_{m,n}}$ we calculated the homology $HF^{+}({Y_{m,n}})$ by moving the basepoint $z$ into another elementary domain. In those $\sp $ structures we need to perform the calculation of the absolute gradings using the moved basepoint. \[special\] Let the basepoint of the Heegaard diagram \[fig:heeg\] lie in the elementary domain $D_{2}$. Then [1]{} & ({x\_[i]{},b\_[j]{},f\_[2]{}})= for $1\leq i\leq m-1$ and $1\leq j\leq n-1$, and [1]{} & ({x\_[m]{},b\_[j]{},f\_[2]{}})= for $1\leq j\leq n-1$. When calculating $HF^{+}({Y_{m,n}})$ in the $\sp $ structures $\mathfrak{s}_{0}$, $\mathfrak{s}_{0}+\mu _{1}+\mu _{2}$, $\mathfrak{s}_{0}-\mu _{1}-\mu _{2}$ and $\mathfrak{s}_{0}-i\mu _{1}-\mu _{2}$, we moved the basepoint $z\in D_{5}$ of the basic Heegaard diagram \[fig:heeg\] over the curve $\alpha _{2}$ into the elementary domain $D_{2}$. Doing the same thing on the triple Heegaard diagram, the basepoint $z\in D_{8}$ moves to $z_{2}\in D_{17}$. The triply-periodic domain $\mathcal{Q}$ now changes to the triply periodic domain $\mathcal{Q}_{2}=\mathcal{Q}-m\Sigma $, for which we have $\partial \mathcal{Q}_{2}=\partial \mathcal{Q}$. As in the previous calculation, we obtain $\# \partial \mathcal{Q}_{2}=m(n+2)$, $n_{{z}_{2}}(\mathcal{Q}_{2})=0$ and $\mathcal{H}(\mathcal{Q}_{2})^{2}=-m(mn-4)$. The Euler measure of the new triply periodic domain is equal to $\widehat{\chi }(\mathcal{Q}_{2})=4m$. We can apply the same Whitney triangles as described in Lemma \[triangle\], but now their spider number changes due to the different multiplicities of the elementary domains in $\mathcal{Q}_{2}$. For $1\leq i\leq m-1$ and $1\leq j\leq n-1$, the Whitney triangle $$u\colon \{x_{i},b_{j},f_{2}\}\to \{t_{1}^{+},r,t_{2}^{+}\}\to \{x_{i}',s,f_{2}'\}$$ has the spider number [1]{} & (u,\_[2]{})=\_[1]{}(u,\_[2]{})+\_[2]{}(u,\_[2]{})+\_[3]{}(u,\_[2]{})=-2(i+1)+2-(n-j+2)m-m=\ & =-mn+(j-3)m-2i, while for $i=m$ we have $\sigma _(u,\mathcal{Q}_{2})=-mn+(j-4)m$. Since the basepoint of the triple Heegaard diagram was only moved over the curve $\alpha _{2}$ and not over $\alpha _{1}$, the torsion $\sp $ structures of $-L(m,1)\#S^{1}\times S^{2}$ (and their gradings) remain unchanged. We calculate [1]{} & c\_[1]{}(\_[z]{}(u)),(\_[2]{})=m(n+2)+4m-2mn+2(j-3)m-4i=-mn+2mj-4i\ & ({x\_[i]{}’,s,f\_[2]{}’})=+=+\ & ({x\_[i]{},b\_[j]{},f\_[2]{}})=({x\_[i]{}’,s,f\_[2]{}’})-=\ & = for $1\leq i\leq m-1$ and $1\leq j\leq n-1$, and [1]{} & ({x\_[m]{},b\_[j]{},f\_[2]{}})= for $1\leq j\leq n-1$. \[specialc\] The absolute grading of the generator $\{x_{m-2},b_{1},f_{2}\}$ in the $\sp $ structure $\mathfrak{s}_{0}+\mu _{1}+\mu _{2}$ is given by $$\begin{aligned} & {\widetilde{\operatorname{gr}}}(\{x_{m-2},b_{1},f_{2}\},z_{2})=\frac{mn(m+n-4)}{4(mn-4)}\end{aligned}$$ The absolute grading of the generator $\{x_{m-1},b_{2},f_{2}\}$ in the $\sp $ structure $\mathfrak{s}_{0}$ is given by $$\begin{aligned} {\widetilde{\operatorname{gr}}}(\{x_{m-1},b_{2},f_{2}\},z_{2})=\frac{mn(m+n-4)-4(m+n)+16}{4(mn-4)}\end{aligned}$$ The absolute grading of the generator $\{x_{m},b_{2},f_{2}\}$ in the $\sp $ structure $\mathfrak{s}_{0}-\mu _{1}-\mu _{2}$ is given by $$\begin{aligned} {\widetilde{\operatorname{gr}}}(\{x_{m},b_{2},f_{2}\},z_{2})=\frac{mn(m+n-4)}{4(mn-4)}\end{aligned}$$ The absolute grading of the generator $\{x_{i-1},b_{1},f_{2}\}$ in the $\sp $ structure $\mathfrak{s}_{0}-i\mu _{1}-\mu _{2}$ is given by $$\begin{aligned} & {\widetilde{\operatorname{gr}}}(\{x_{i-1},b_{1},f_{2}\},z_{2})=\frac{n(m^{2}+mn-4mi+4i^{2}-4)}{4(mn-4)}\end{aligned}$$ We use the formulas from Proposition \[special\] for the generators $\{x_{m-2},b_{1},f_{2}\}$, $\{x_{m-1},b_{2},f_{2}\}$, $\{x_{m},b_{2},f_{2}\}$ and $\{x_{i-1},b_{1},f_{2}\}$ to obtain the desired gradings. We have calculated the absolute gradings of the homology generators in the torsion $\sp $ structures on ${Y_{m,n}}$. Now we identify the $\sp $ structure corresponding to a given generator with a $\sp $ structure $\mathfrak{s}_{i,j}$, defined by -. \[lemmasp\] Let the basepoint of the Heegaard diagram \[fig:heeg\] lie in the elementary domain $D_{5}$. Then $$\mathfrak{s}_{i,j}=\mathfrak{s}_{z}(\{x_{i},b_{j},f_{k}\})$$ for $1\leq i\leq m-1$ and $1\leq j\leq n-1$, where $k\in \{1,2\}$. We will show that the two $\sp $ structures are both restrictions of the same $\sp $ structure on the cobordism $W$ from ${Y_{m,n}}$ to $-L(m,1)\# S^{1}\times S^{2}$. In Lemma \[triangle\] we described a Whitney triangle $$u\colon \{x_{i},b_{j},f_{2}\}\to \{t_{1}^{+},r,t_{2}^{+}\}\to \{x_{i}',s,f_{2}'\}$$ defining a $\sp $ structure $\mathfrak{s}_{z}(u)$ on $W$ for which $\mathfrak{s}_{z}(u)|_{-L(m,1)\# S^{1}\times S^{2}}=\mathfrak{s}_{z}(\{x_{i}',s',f_{k}'\})$ represents the $i$-th $\sp $ structure on $-L(m,1)\# S^{1}\times S^{2}$ as defined by Ozsváth-Szabó in [@OS4 Subsection 4.1]. On the other hand, $\mathfrak{s}_{z}(u)|_{{Y_{m,n}}}=\mathfrak{s}_{z}(\{x_{i},b_{j},f_{k}\})$. Recall that $\mathfrak{s}_{i,j}=\mathfrak{t}_{i,j}|_{{Y_{m,n}}}$, where $\mathfrak{t}_{i,j}$ is the $\sp $ structure on the manifold ${N_{m,n}}$ defined by the Equations -. Since the homology group $H_{2}({N_{m,n}})={\mathbb {Z}}^{2}$ is generated by the base spheres $s_{1}$ and $s_{2}$ of the plumbing ${N_{m,n}}$, the $\sp $ structures $\mathfrak{t}_{i,j}$ are well defined. The Kirby diagram of ${N_{m,n}}$ on the Figure \[fig:kirby\] describes the surgery cobordism from $S^{3}$ to the 3-manifold ${Y_{m,n}}$. In the first step of the surgery cobordism, we add a 1-handle and a 2-handle along the unknot $K_{1}$ to $S^{3}$, obtaining the 3-manifold $-L(m,1)\# S^{1}\times S^{2}$. The core of the 2-handle union the disk spanned by $K_{1}$ in $B^{4}$ represent the base sphere $s_{1}$. Since by definition $$\langle c_{1}(\mathfrak{t}_{i,j}),s_{1}\rangle =2i-m\;,$$ the restriction $\mathfrak{t}_{i,j}|_{-L(m,1)\# S^{1}\times S^{2}}$ is exactly the $i$-th $\sp $ structure on $-L(m,1)\# S^{1}\times S^{2}$ as defined by Ozsváth-Szabó in [@OS4 Subsection 4.1]. Thus, $$\mathfrak{t}_{i,j}|_{-L(m,1)\# S^{1}\times S^{2}}=\mathfrak{s}_{z}(\{x_{i}',s',f_{k}'\})=\mathfrak{s}_{z}(u)|_{-L(m,1)\# S {1}\times S^{2}}\;.$$ The second step of the surgery is given by the cobordism $-W$ from $-L(m,1)\# S^{1}\times S^{2}$ to ${Y_{m,n}}$. The cobordism $-W$ is given by adding a 2-handle to the boundary of the previously constructed manifold. Let us find a generator of the homology group $H_{2}(-W)={\mathbb {Z}}$. Writing down the intersection form $$Q_{{N_{m,n}}}=\left (\begin{array}{cc} m & 2\\ 2 & n \\ \end{array}\right )$$ for ${N_{m,n}}$ and denoting by $F=as_{1}+bs_{2}$ the generator of $H_{2}(-W)$, we use the fact that $F$ has to be orthogonal to the sphere $s_{1}$. Thus, $\langle as_{1}+bs_{2},s_{1}\rangle =ma+2b=0$ and we can take $F=2s_{1}-ms_{2}$. We calculate $$\begin{aligned} & \langle c_{1}(\mathfrak{t}_{i,j}),F\rangle =2(2i-m)-m(2j-n)=mn-2(j+1)m+4i\\ & F^{2}=4s_{1}^{2}-4ms_{1}s_{2}+m^{2}s_{2}^{2}=m(mn-4)\end{aligned}$$ The first Chern class $c_{1}(\mathfrak{s}_{z}(u))$ of the triangle $$u\colon \{x_{i},b_{j},f_{2}\}\to \{t_{1}^{+},r,t_{2}^{+}\}\to \{x_{i}',s,f_{2}'\}$$ from the Heegaard triple diagram had the same evaluation on the generator $\mathcal{H}(\mathcal{Q})$ of $H_{2}(W)$ (with the opposite sign because of the opposite orientation of the cobordism), see Equation . Since also $F^{2}=\mathcal{H}(\mathcal{Q})^{2}$, it follows that the $\sp $ structures coincide on $W$: $\mathfrak{s}_{z}(u)=\mathfrak{t}_{i,j}|_{W}$. Now we have $\mathfrak{s}_{z}(u)|_{{Y_{m,n}}}=\mathfrak{s}_{z}(\{x_{i},b_{j},f_{k}\})$ and $\mathfrak{t}_{i,j}|_{{Y_{m,n}}}=\mathfrak{s}_{i,j}$, which gives us the desired equality. \[cor1\] Let the basepoint of the Heegaard diagram \[fig:heeg\] lie in the elementary domain $D_{5}$. Then $$\begin{aligned} & \mathfrak{s}_{0,j}=\mathfrak{s}_{z}(\{x_{m},b_{j+1},f_{k}\})\\ & \mathfrak{s}_{i,0}=\mathfrak{s}_{z}(\{x_{i+1},b_{n},f_{k}\})\end{aligned}$$ for $0\leq i\leq m-2$, $0\leq j\leq n-2$ and $k\in \{1,2\}$. We use [@OS1 Lemma 2.19] to evaluate the cohomology class in $H^{2}({Y_{m,n}})$ corresponding to the difference of two $\sp $ structures. We calculate $$\begin{aligned} & \mathfrak{s}_{i,j}-\mathfrak{s}_{i+1,j}=\mathfrak{s}_{z}(\{x_{i},b_{j},f_{k}\})-\mathfrak{s}_{z}(\{x_{i+1},b_{j},f_{k}\})=PD[\mu _{1}]\\ & \mathfrak{s}_{i,j}-\mathfrak{s}_{i,j+1}=\mathfrak{s}_{z}(\{x_{i},b_{j},f_{k}\})-\mathfrak{s}_{z}(\{x_{i},b_{j+1},f_{k}\})=PD[\mu _{2}]\end{aligned}$$ and by linearity it follows that $\mathfrak{s}_{i,j}+aPD[\mu _{1}]+bPD[\mu _{2}]=\mathfrak{s}_{i-a,j-b}$. Thus $$\begin{aligned} & \mathfrak{s}_{z}(\{x_{m},b_{j+1},f_{k}\})=\mathfrak{s}_{z}(\{x_{1},b_{j+1},f_{k}\})+PD[\mu _{1}+\mu _{2}]=\mathfrak{s}_{0,j}\\ & \mathfrak{s}_{z}(\{x_{i+1},b_{n},f_{k}\})=\mathfrak{s}_{z}(\{x_{i+1},b_{1},f_{k}\})+PD[\mu _{1}+\mu _{2}]=\mathfrak{s}_{i,0}\end{aligned}$$ for $0\leq i\leq m-2$, $0\leq j\leq n-2$ and $k\in \{1,2\}$. We have thus obtained: In Subsection \[CF\] we have shown that $HF^{+}({Y_{m,n}},\mathfrak{s})$ has two ${\mathcal{T}^{+}}$ summands in each torsion $\sp $ structure $\mathfrak{s}$ on ${Y_{m,n}}$. In one torsion $\sp $ structure, $HF^{+}({Y_{m,n}},\mathfrak{s})$ has an additional ${\mathbb{F}}$ sumand. We have also shown that the action of $\Lambda ^{*}(H_{1}(Y,{\mathbb {Z}})/\operatorname{Tors})$ maps the generator of ${\mathcal{T}^{+}}$ with the higher absolute grading to the generator with the lower absolute grading. In Proposition \[calc\] we have calculated that [1]{} & ({x\_[i]{},b\_[j]{},f\_[2]{}})=\ & ==d(i,j) for $1\leq i\leq m-1$ and $1\leq j\leq n-1$. By Lemma \[lemmasp\], for those indices we have $\mathfrak{s}_{i,j}=\mathfrak{s}_{z}(\{x_{i},b_{j},f_{k}\})$. The $\sp $ structures $\mathfrak{s}_{0,j}$ for $0\leq j\leq n-2$ and $\mathfrak{s}_{i,0}$ for $0\leq i\leq m-2$ are identified with the generators of $\widehat{HF}({Y_{m,n}})$ in the Corollary \[cor1\], and the absolute grading of those generators has been calculated in Proposition \[calc1\]. The absolute gradings of the generators in the $\sp $ structures $\mathfrak{s}_{0}$, $\mathfrak{s}_{0}+\mu _{1}+\mu _{2}$, $\mathfrak{s}_{0}-\mu _{1}-\mu _{2}$ and $\mathfrak{s}_{0}-i\mu _{1}-\mu _{2}$ are given in Corollary \[specialc\]. By Lemma \[lemmasp\], Corollary \[cor1\] and Corollary \[unique\] we have $\mathfrak{s}_{0}=\mathfrak{s}_{1,n-1}=\mathfrak{s}_{m-1,1}$, $\mathfrak{s}_{0}+\mu _{1}+\mu _{2}=\mathfrak{s}_{0,n-2}=\mathfrak{s}_{m-2,0}$, $\mathfrak{s}_{0}-\mu _{1}-\mu _{2}=\mathfrak{s}_{0,0}$ and $\mathfrak{s}_{0}-i\mu _{1}-\mu _{2}=\mathfrak{s}_{i-1,0}$. By Corollary \[specialc\] we can ascertain that the top correction terms in those $\sp $ structures are given by [1]{} & d\_[t]{}([Y\_[m,n]{}]{},\_[1,n-1]{})=d(1,n-1)\ & d\_[t]{}([Y\_[m,n]{}]{},\_[0,n-2]{})=d\_[1]{}(m,n-1)\ & d\_[t]{}([Y\_[m,n]{}]{},\_[0,0]{})=d\_[1]{}(m,1)\ & d\_[t]{}([Y\_[m,n]{}]{},\_[i-1,0]{})=d\_[1]{}(n,i) It follows that $d_{t}({Y_{m,n}},\mathfrak{s}_{i,j})=d(i,j)$ for $1\leq i\leq m-1$ and $1\leq j\leq n-1$. Moreover, $d_{t}({Y_{m,n}},\mathfrak{s}_{0,j})=d_{1}(m,j+1)$ for $0\leq j\leq n-2$ and $d_{t}({Y_{m,n}},\mathfrak{s}_{i,0})=d_{1}(n,i+1)$ for $0\leq i\leq m-2$. [An application]{} \[App\] Let $X$ be a closed smooth 4–manifold with $H_{1}(X)=0$ and $b_{2}^{+}(X)=2$. Consider two classes $\alpha ,\beta \in H_{2}(X;{\mathbb {Z}})$ for which the following holds: [1]{} & =2\ & \^[2]{}=m&gt;0\ & \^[2]{}=n&gt;0\ & mn-4&gt;0 Thus the restriction $Q_{X}|_{{\mathbb {Z}}\alpha +{\mathbb {Z}}\beta }$ of the intersection form $Q_{X}$ to the sublattice spanned by $\alpha $ and $\beta $ is positive definite. The classes $\alpha $ and $\beta $ can be represented by embedded surfaces $\Sigma _{1},\Sigma _{2}\subset X$ meeting transversally. Suppose that it is possible to choose $\Sigma _{1}$ and $\Sigma _{2}$ to be spheres whose geometric intersection number is 2. Then the regular neighborhood of the union $\Sigma _{1}\cup \Sigma _{2}$ is a double plumbing of disk bundles over spheres ${N_{m,n}}$ with boundary ${Y_{m,n}}$ that has been the object of our investigation in the previous section. The submanifold ${N_{m,n}}\subset X$ carries the positive part of the intersection form $Q_{X}$. Denote by $W=X\backslash \operatorname{Int}({N_{m,n}})$ its complement in $X$. Thus $W$ is a 4–manifold with boundary $-{Y_{m,n}}$ which carries the negative part of the intersection form $Q_{X}$. The following result [@OS4 Theorem 9.15] describes the constraints given by the $\sp $ structures on $W$ which restrict to a given $\sp $ structure on $-{Y_{m,n}}$. \[inequality\] Let $Y$ be a three-manifold with standard $HF^{\infty }$, equipped with a torsion $\sp $ structure $\mathfrak{t}$, and let $d_{b}(Y, \mathfrak{t})$ denote its bottom-most correction term, i.e. the one corresponding to the generator of $HF^{\infty }(Y, t)$ which is in the kernel of the action by $H_{1}(Y)$. Then, for each negative semi-definite four-manifold $W$ which bounds $Y$ so that the restriction map $H^{1}(W;{\mathbb {Z}}) \rightarrow H^{1}(Y;{\mathbb {Z}})$ is trivial, we have the inequality: [1]{} \[ineq\] & c\_[1]{}()\^[2]{} + b\_[2]{}\^[-]{}(W) 4d\_[b]{}(Y,) + 2b\_[1]{}(Y) for all $\sp $ structures $\mathfrak{s}$ over $W$ whose restriction to $Y$ is $\mathfrak{t}$. According to [@OS2 Theorem 10.1], every 3–manifold $Y$ with $b_{1}(Y)=1$ has standard $HF^{\infty }$. Theorem \[inequality\] can thus be applied in our case for the pair $(W,-{Y_{m,n}})$. Correction terms of the manifold ${Y_{m,n}}$ have been calculated in the previous section. In order to apply inequality , we have to identify the restriction map $H^{2}(W)\to H^{2}(-{Y_{m,n}})$ and see how $\sp $ structures on $W$ restrict to $\sp $ structures on $-{Y_{m,n}}$. Before considering particular cases we establish the following: \[prop1\] With notation as above, $H^{1}(W)=0$, $H^{2}({N_{m,n}})\cong {\mathbb {Z}}^{2}$, $H^{2}(W)\cong {\mathbb {Z}}^{b_{2}^{-}(X)+1}\oplus \tau $ and $H^{2}({Y_{m,n}})\cong {\mathbb {Z}}\oplus T$, where $\tau $ and $T$ are torsion groups and $T$ has order $mn-4$. In the special case when $b_{2}^{-}(X)=0$, we have $T/\tau \cong \tau $. Consider the Mayer–Vietoris sequence in cohomology of the triple $(X,{N_{m,n}},W)$ (all coefficients will be ${\mathbb {Z}}$ unless stated otherwise): $$\begin{aligned} & 0\rightarrow H^{1}(W)\oplus H^{1}({N_{m,n}})\stackrel{f_{1}}\rightarrow H^{1}({Y_{m,n}}) \stackrel{f_{2}}\rightarrow H^{2}(X)\stackrel{f_{3}}\rightarrow H^{2}(W)\oplus H^{2}({N_{m,n}}) \stackrel{f_{4}}\rightarrow H^{2}({Y_{m,n}})\rightarrow 0\\ & 0\rightarrow H^{1}(W)\oplus {\mathbb {Z}}\stackrel{f_{1}}\rightarrow {\mathbb {Z}}\stackrel{f_{2}}\rightarrow {\mathbb {Z}}^{b_{2}^{-}(X)+2}\stackrel{f_{3}}\rightarrow H^{2}(W)\oplus {\mathbb {Z}}^{2} \stackrel{f_{4}}\rightarrow {\mathbb {Z}}\oplus T\rightarrow 0\end{aligned}$$ At the beginning and the end of the sequence we have zeros since $H_{1}(X)=0$. Since $H_{1}({Y_{m,n}})\cong {\mathbb {Z}}[\mu _{3}]\oplus T[\mu _{1},\mu _{2}]$, it follows from Poincaré duality and the universal coefficient theorem that $H^{2}({Y_{m,n}})\cong {\mathbb {Z}}\oplus T$ and $H^{1}({Y_{m,n}})\cong {\mathbb {Z}}$. The torsion elements $\mu _{1}$ and $\mu _{2}$ are the boundary circles of the fibre disks in the plumbing ${N_{m,n}}$. The generator $\mu _{3}$ of the free part comes from the 1-handle of the plumbing, which means that $f_{1}|_{H^{1}({N_{m,n}})}\colon H^{1}({N_{m,n}})\to H^{1}({Y_{m,n}})$ is an isomorphism. Thus $H^{1}(W)=0$ and the restriction map $f_{1}|_{H^{1}(W)}\colon H^{1}(W) \rightarrow H^{1}({Y_{m,n}})$ is always trivial, satisfying the assumption in Theorem \[inequality\]. Since $f_{1}$ is an isomorphism, by exactness $f_{2}$ is a trivial map. It follows that $f_{3}$ is injective. To understand the homomorphism $f_{4}$, recall the long exact sequence in homology of the pair $({N_{m,n}},{Y_{m,n}})$: $$\begin{aligned} \label{NY} & \ldots \rightarrow H_{2}({N_{m,n}})\stackrel{A}\rightarrow H_{2}({N_{m,n}},{Y_{m,n}})\stackrel{B}\rightarrow H_{1}({Y_{m,n}})\stackrel{C}\rightarrow H_{1}({N_{m,n}})\rightarrow H_{1}({N_{m,n}},{Y_{m,n}}) \\ & \ldots \longrightarrow {\mathbb {Z}}^{2}\stackrel{A}\longrightarrow {\mathbb {Z}}^{2}\stackrel{B}\longrightarrow {\mathbb {Z}}\oplus T\stackrel{C}\longrightarrow {\mathbb {Z}}\longrightarrow 0\end{aligned}$$ As described above, the restriction $C|_{{\mathbb {Z}}}\colon {\mathbb {Z}}[\mu _{3}]\to H_{1}({N_{m,n}})$ is an isomorphism. It follows that the image of the map $B\colon H_{2}({N_{m,n}},{Y_{m,n}})\to H_{1}({Y_{m,n}})$ is equal to $T$. The same is true for the Poincaré dual map $f_{4}|_{H^{2}({N_{m,n}})}\colon H^{2}({N_{m,n}})\to H^{2}({Y_{m,n}})$ in the Mayer–Vietoris sequence above. So there must be a free sumand ${\mathbb {Z}}\subseteq H^{2}(W)$ which is mapped by $f_{4}$ isomorphically onto the free sumand of $H^{2}({Y_{m,n}})$ (this is the part dual to the part of $H_{2}(W)$ which comes from the boundary). Now since $f_{3}$ is injective, the free subgroup ${\mathbb {Z}}^{b_{2}^{-}(X)}\subseteq H^{2}(X)$ maps into $H^{2}(W)$ and it follows that the free part of $H^{2}(W)$ has dimension $b_{2}^{-}(X)+1$. Since $H^{1}(W)=0$, it follows from the universal coefficient theorem that $H_{1}(W)=\tau $ is torsion and consequently $$H^{2}(W)\cong {\mathbb {Z}}^{b_{2}^{-}(X)+1}\oplus \tau \;.$$ Based on our conclusions above, a part of the cohomology Mayer–Vietoris sequence of the triple $(X,{N_{m,n}},W)$ looks like [1]{} \[MV\] …H\^[2]{}(X)H\^[2]{}(W)H\^[2]{}([N\_[m,n]{}]{})H\^[2]{}([Y\_[m,n]{}]{})0\ …\^[b\_[2]{}\^[-]{}(X)+2]{}(\^[b\_[2]{}\^[-]{}(X)+1]{})\^[2]{}T0 The restriction $f_{4}|_{H^{2}({N_{m,n}})}$ can be described by its Poincaré dual $B\colon H_{2}({N_{m,n}},{Y_{m,n}})\to H_{1}({N_{m,n}})$ in the long exact sequence . Consider now the restriction $f_{4}|_{H^{2}(W)}$ in . The sumand $\tau \subseteq H^{2}(W)$ maps by $f_{4}$ injectively into the torsion group $T\subseteq H^{2}({Y_{m,n}})$. We can observe the Poincaré dual of the restriction $f_{4}|_{H^{2}(W)}$ in the long exact sequence of the pair $(W,{Y_{m,n}})$: $$\begin{aligned} & H_{3}(W,{Y_{m,n}})\rightarrow H_{2}({Y_{m,n}})\stackrel{g_{1}}\longrightarrow H_{2}(W)\stackrel{g_{2}}\longrightarrow H_{2}(W,{Y_{m,n}})\stackrel{g_{3}}\longrightarrow H_{1}({Y_{m,n}})\stackrel{g_{4}}\longrightarrow H_{1}(W)\rightarrow \ldots \\ & 0\rightarrow {\mathbb {Z}}\stackrel{g_{1}}\longrightarrow {\mathbb {Z}}^{b_{2}^{-}(X)+1}\stackrel{g_{2}}\longrightarrow {\mathbb {Z}}^{b_{2}^{-}(X)+1}\oplus \tau \stackrel{g_{3}}\longrightarrow {\mathbb {Z}}\oplus T\stackrel{g_{4}}\longrightarrow \tau \stackrel{0}\rightarrow \ldots \end{aligned}$$ Since $H_{3}(W,{Y_{m,n}})\cong H^{1}(W)=0$, the map $g_{1}$ is injective. The homomorphism $g_{2}\colon H_{2}(W)\to H_{2}(W,{Y_{m,n}})$ is given by the intersection form $Q_{W}$ of the manifold $W$. $Q_{W}$ is trivial on the sumand ${\mathbb {Z}}\subseteq H_{2}(W)$ which corresponds to the image of $g_{1}$. The restriction $Q_{W}|_{{\mathbb {Z}}^{b_{2}^{-}(X)}}$ is negative definite. The map $g_{3}$ maps the free sumand of $H_{2}(W,{Y_{m,n}})$ which comes from the boundary isomorphically onto the free sumand of $H_{1}({Y_{m,n}})$. In the special case when $b_{2}^{-}(X)=0$, the intersection form $Q_{W}$ is trivial and from the exact sequence above it follows that [1]{} \[b20\] & T/. We have described the map $f_{4}$ in the Mayer–Vietoris sequence which tells us how cohomology classes on $W$ and ${N_{m,n}}$ restrict to cohomology classes on the boundary ${Y_{m,n}}$. As remarked in Subsubsection \[notspin\], $\sp $ structures on 3– and 4–manifolds may be identified by cohomology classes. Using this identification we may study the restrictions of $\sp $ structures on $W$ and ${N_{m,n}}$ to $\sp $ structures on the boundary ${Y_{m,n}}$. When the 4-manifold $X$ has $b_{2}^{-}(X)=0$, the obstruction Theorem \[inequality\] implies the following result. \[prop2\] Let $X$ be a closed smooth 4-manifold with $H_{1}(X)=0$, $b_{2}^{+}(X)=2$ and $b_{2}^{-}(X)=0$. Suppose there are two spheres $\Sigma _{1},\Sigma _{2}\subset X$ with $\Sigma _{1} ^{2}=m$, $\Sigma _{2}^{2}=n$ and $\Sigma _{1}\cdot \Sigma _{2}=2$. Denote by ${Y_{m,n}}$ the boundary of a regular neighbourhood of $\Sigma _{1}\cup \Sigma _{2}$ and let $T=H_{1}({Y_{m,n}})$. Then for some subgroup $\tau \subset T$ with $|\tau |^{2}=|T|$ and some $\sp $ structure $\mathfrak{s}_{0}$ on ${Y_{m,n}}$, we have $d_{b}({Y_{m,n}},\mathfrak{s}_{0}+\phi )=-\frac{1}{2}$ for every $\phi \in \tau$. Denote as usual by ${N_{m,n}}\subset X$ the regular neighbourhood of $\Sigma _{1}\cup \Sigma _{2}$ and by $W=X\backslash \operatorname{Int}({N_{m,n}})$ its complement. It follows from Proposition \[prop1\] that $H^{2}(W)\cong {\mathbb {Z}}\oplus \tau $ for some torsion group $\tau \subset T$ and that $T/\tau \cong \tau $, thus $|T|=|\tau |^{2}$. Recall the Mayer–Vietoris sequence of the triple $(X,{N_{m,n}},W)$ we discussed in Proposition \[prop1\]: $$\begin{aligned} \ldots \stackrel{0}\longrightarrow H^{2}(X)\stackrel{f_{3}}\longrightarrow H^{2}(W)\oplus H^{2}({N_{m,n}})\stackrel{f_{4}}\longrightarrow H^{2}({Y_{m,n}})\longrightarrow 0\\ \ldots \stackrel{0}\longrightarrow {\mathbb {Z}}^{2}\stackrel{f_{3}}\longrightarrow ({\mathbb {Z}}\oplus \tau )\oplus {\mathbb {Z}}^{2}\stackrel{f_{4}}\longrightarrow {\mathbb {Z}}\oplus T\longrightarrow 0\end{aligned}$$ The $\sp $ structures on $W$ which restrict to the $\sp $ structures on $-{Y_{m,n}}$ correspond to the image $f_{4}(\tau )\subset T$. Since $b_{2}^{-}(X)=0$, the intersection form $Q_{W}$ of the manifold $W$ is trivial and thus $c_{1}(\mathfrak{s})^{2}=0$ for any $\sp $ structure $\mathfrak{s}$ on the manifold $W$. From Theorem \[inequality\] it follows that if indeed $-{Y_{m,n}}$ bounds a negative semi-definite submanifold $W$ inside $X$, then the inequality $d_{b}(-{Y_{m,n}},\mathfrak{t})\geq -\frac{1}{2}$ holds for any torsion $\sp $ structure $\mathfrak{t}$ on $-{Y_{m,n}}$ which is a restriction of a $\sp $ structure on $W$. The bottom and top correction terms are defined in [@LRS Definition 3.3], where also the duality $d_{b}(-{Y_{m,n}},\mathfrak{t})=-d_{t}({Y_{m,n}},\mathfrak{t})$ is shown [@LRS Proposition 3.7]. By Theorem \[th1\] we have $d_{t}({Y_{m,n}},\mathfrak{t})=d_{b}({Y_{m,n}},\mathfrak{t})+1$. So for any such $\sp $ structure we have $d_{b}(-{Y_{m,n}},\mathfrak{t})=-d_{t}({Y_{m,n}},\mathfrak{t})=-d_{b}({Y_{m,n}},\mathfrak{t})-1$, and consequently $$d_{b}({Y_{m,n}},\mathfrak{t})=-d_{b}(-{Y_{m,n}},\mathfrak{t})-1\leq -\frac{1}{2}\;.$$ Since the intersection form on $W$ is trivial, Theorem \[inequality\] can also be applied for the pair $(-W,{Y_{m,n}})$ to give the inequality $d_{b}({Y_{m,n}},\mathfrak{t})\geq -\frac{1}{2}$. Both inequalities amount to the equality $$d_{b}({Y_{m,n}},\mathfrak{t})=-\frac{1}{2}$$ for any torsion $\sp $ structure $\mathfrak{t}$ on ${Y_{m,n}}$ which is a restriction of a $\sp $ structure on $W$. [Double plumbings inside ${\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$]{} \[CP\] We consider double plumbings inside $X={\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$. Our question is whether a chosen pair of classes $\alpha ,\beta \in H_{2}({\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2})$ with $\alpha \cdot \beta =2$ can be represented by a configuration of two spheres with only two geometric intersections. We will find suitable classes $\alpha ,\beta $ and apply Proposition \[prop2\]. Now $H_{2}({\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2})\cong {\mathbb {Z}}^{2}$ has a standard basis $(e_{1},e_{2})$ with $e_{i}$ representing the class of the cycle ${\mathbb {C}}P^{1}\subset {\mathbb {C}}P^{2}$. The intersection form $Q_{X}$ of the manifold $X$ is given by $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right )$ and $b_{2}^{-}(X)=0$. We need to choose homologically independent classes $\alpha ,\beta \in H_{2}(X)$ that are both representable by spheres and for which $\alpha \cdot \beta =2$. A class $\zeta =(a,b)\in H_{2}(X)$ has a smooth representative $\Sigma $ of genus $$g(\Sigma )=\frac{(|a|-1)(|a|-2)}{2}+\frac{(|b|-1)(|b|-2)}{2}\;.$$ This representative is obtained by the connected sum of minimal genus representatives for classes of divisibility $a$ and $b$ in ${\mathbb {C}}P^{2}$. Thus, nontrivial classes with smooth representatives of genus 0 are given by $ae_{1}+be_{2}\in H_{2}(X)$ where $(|a|,|b|)\in \{0,1,2\}^{2}\backslash \{(0,0)\}$. Up to isomorphism, there are three possible cases for $\alpha $ and $\beta $: [1]{} & 2e\_[1]{}+2e\_[2]{}2e\_[1]{}-e\_[2]{}\ & 2e\_[1]{}e\_[1]{}+2e\_[2]{}\ & e\_[1]{}2e\_[1]{}+e\_[2]{} We will investigate two cases: $\alpha =2e_{1}+2e_{2}, \beta =2e_{1}-e_{2}$ and $\alpha =2e_{1}, \beta =e_{1}+2e_{2}$. For the final case $\alpha =e_{1}$ and $\beta =2e_{1}\pm e_{2}$, the two classes can be represented by a pair of spheres intersecting in two points. [First case: $\alpha =2e_{1}+2e_{2}, \beta =2e_{1}-e_{2}$]{} \[FirstCP\] We have $m=\alpha ^{2}=8, n=\beta ^{2}=5$ and $$H_{1}({Y_{m,n}})={\mathbb {Z}}\oplus {\mathbb {Z}}_{36}\;.$$ We will prove here the first part of Theorem \[app\], which says that any two spheres representing the classes $\alpha $ and $\beta $ intersect with at least 4 geometric intersections, and that there exist representatives with exactly 4 intersections. Suppose there are spheres representing $\alpha $ and $\beta $ which have only two geometric intersections. Then the regular neighbourhood of their union is the double plumbing $N_{8,5}$ with boundary $Y_{8,5}$. Applying Theorem \[th1\] we calculate the bottom-most correction terms $d_{b}$ in all torsion $\sp $ structures on $Y_{8,5}$: $\sp $ $d_{b}({Y_{m,n}},\mathfrak{s}_{i,j})$ ---------------------- --------------------------------------- $\mathfrak{s}_{4,3}$ $-17/18$ $\mathfrak{s}_{3,3}$ $-3/4$ $\mathfrak{s}_{2,3}$ $-5/18$ $\mathfrak{s}_{1,3}$ $17/36$ $\mathfrak{s}_{6,0}$ $3/2$ $\mathfrak{s}_{5,0}$ $29/36$ $\mathfrak{s}_{4,0}$ $7/18$ $\mathfrak{s}_{3,0}$ $1/4$ $\mathfrak{s}_{2,0}$ $7/18$ $\mathfrak{s}_{1,0}$ $29/36$ $\mathfrak{s}_{0,0}$ $3/2$ $\mathfrak{s}_{7,2}$ $17/36$ $\mathfrak{s}_{6,2}$ $-5/18$ $\mathfrak{s}_{5,2}$ $-3/4$ $\mathfrak{s}_{4,2}$ $-17/18$ $\mathfrak{s}_{3,2}$ $-31/36$ $\mathfrak{s}_{2,2}$ $-1/2$ $\mathfrak{s}_{1,2}$ $5/36$ $\sp $ $d_{b}({Y_{m,n}},\mathfrak{s}_{i,j})$ ---------------------- --------------------------------------- $\mathfrak{s}_{0,2}$ $19/18$ $\mathfrak{s}_{7,4}$ $1/4$ $\mathfrak{s}_{6,4}$ $-5/18$ $\mathfrak{s}_{5,4}$ $-19/36$ $\mathfrak{s}_{4,4}$ $-1/2$ $\mathfrak{s}_{3,4}$ $-7/36$ $\mathfrak{s}_{2,4}$ $7/18$ $\mathfrak{s}_{1,4}$ $5/4$ $\mathfrak{s}_{6,1}$ $7/18$ $\mathfrak{s}_{5,1}$ $-7/36$ $\mathfrak{s}_{4,1}$ $-1/2$ $\mathfrak{s}_{3,1}$ $-19/36$ $\mathfrak{s}_{2,1}$ $-5/18$ $\mathfrak{s}_{1,1}$ $1/4$ $\mathfrak{s}_{0,1}$ $19/18$ $\mathfrak{s}_{7,3}$ $5/36$ $\mathfrak{s}_{6,3}$ $-1/2$ $\mathfrak{s}_{5,3}$ $-31/36$ There are only four $\sp $ structures on $Y_{8,5}$ for which the equality $d_{b}(Y_{8,5},\mathfrak{t})=-\frac{1}{2}$ is valid, namely $\mathfrak{s}_{2,2}$, $\mathfrak{s}_{4,4}$, $\mathfrak{s}_{4,1}$ and $\mathfrak{s}_{6,3}$. It follows from Proposition \[prop2\] that the two spheres which represent the classes $\alpha ,\beta \in H_{2}({\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2})$ have to intersect with a geometric intersection number greater than 2. $2e_{1}+2e_{2}$ at 420 358 $2e_{1}-e_{2}$ at 422 164 ![Attaching circles of the 2-handles representing classes $2e_{1}+2e_{2}, 2e_{1}-e_{2}\in {\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$ with four geometric intersections[]{data-label="fig:CP2"}](CP2) It is possible to construct genus zero representatives for $\alpha $ and $\beta $ with 4 geometric intersections. We use the following construction of Ruberman [@DR]: we represent ${\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$ as a handlebody with two 2-handles with framing 1 and denote by $h_{1}$ and $h_{2}$ the cores of the 2-handles. By adding to $h_{i}$ a disk its boundary spans in $B^{4}$, we obtain a sphere representing $e_{i}$. Now let us represent the class $\alpha =2e_{1}+2e_{2}$: first we take two copies of $h_{i}$ and resolve their double point to get a single disk for $i=1,2$. Then we make a boundary connected sum of both disks (with coherent orientations) and add a disk in $B^{4}$ to the resulting surface. Similarly, we represent the class $\beta =2e_{1}-e_{2}$: first we take two copies of $h_{1}$ and resolve their double point, then we boundary connect sum the obtained disk and $h_{2}$ with the reversed orientation (this means the connected sum is made via a band with a half-twist) and add a disk in $B^{4}$ in the end. In this way we get the two spheres representing classes $\alpha $ and $\beta $ in ${\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$. Figure \[fig:CP2\] shows the two representatives in ${\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$. The right loop of the dark curve can be slightly pulled left by an isotopy so that it intersects the light curve only twice, thus there remain only four intersections between the two spheres. It follows that 4 is the minimal number of geometric intersections. [Second case: $\alpha =2e_{1}, \beta =e_{1}+2e_{2}$]{} $2e_{1}$ at 180 420 $e_{1}+2e_{2}$ at 438 420  \[fig:CP3\] The squares $m=\alpha ^{2}=4$ and $n=\beta ^{2}=5$ imply that $H_{1}(Y_{4,5})={\mathbb {Z}}\oplus {\mathbb {Z}}_{16}$. The bottom-most correction terms $d_{b}$ of $Y_{4,5}$ are given by $\sp $ $d_{b}({Y_{m,n}},\mathfrak{s}_{i,j})$ ---------------------- --------------------------------------- $\mathfrak{s}_{2,3}$ $-15/16$ $\mathfrak{s}_{1,3}$ $-1/2$ $\mathfrak{s}_{2,0}$ $9/16$ $\mathfrak{s}_{1,0}$ $1/4$ $\mathfrak{s}_{0,0}$ $9/16$ $\mathfrak{s}_{3,2}$ $-1/2$ $\mathfrak{s}_{2,2}$ $-15/16$ $\mathfrak{s}_{1,2}$ $-3/4$ $\sp $ $d_{b}({Y_{m,n}},\mathfrak{s}_{i,j})$ ---------------------- --------------------------------------- $\mathfrak{s}_{0,2}$ $1/16$ $\mathfrak{s}_{3,4}$ $-1/2$ $\mathfrak{s}_{2,4}$ $-7/16$ $\mathfrak{s}_{1,4}$ $1/4$ $\mathfrak{s}_{2,1}$ $-7/16$ $\mathfrak{s}_{1,1}$ $-1/2$ $\mathfrak{s}_{0,1}$ $1/16$ $\mathfrak{s}_{3,3}$ $-3/4$ There are the requisite four $\sp $ structures on $Y_{4,5}$ for which $d_{b}$ is equal to $-\frac{1}{2}$: $$d_{b}(Y_{4,5},\mathfrak{s}_{1,3})=d_{b}(Y_{4,5},\mathfrak{s}_{3,2})=d_{b}(Y_{4,5},\mathfrak{s}_{3,4})=d_{b}(Y_{4,5},\mathfrak{s}_{1,1})=-\frac{1}{2}\;.$$ Indeed, one can choose the two spheres representing classes $\alpha $ and $\beta $ so that their geometric intersection consists of two points, see Figure \[fig:CP3\]. [Double plumbings inside $S^{2}\times S^{2}\# S^{2}\times S^{2}$]{} \[S2\] Let us investigate double plumbings inside the 4–manifold $X=S^{2}\times S^{2}\# S^{2}\times S^{2}$. Since $X$ is simply connected and its intersection form $Q_{X}=\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ \end{array} \right )$ is even, it follows that $X$ is a spin 4–manifold. According to [@WALL Theorem 3], if $M$ is a simply connected closed oriented 4–manifold with an indefinite intersection form, then every primitive noncharacteristic class of $H_{2}(M\# (S^{2}\times S^{2}))$ is represented by an embedded sphere. More specifically, Hirai showed that every primitive element of $H_{2}(S^{2}\times S^{2}\# S^{2}\times S^{2})$ can be represented by a smoothly embedded sphere [@HI Theorem 1]. A class $\mathbf{r}\in H_{2}(X)$ is primitive if it cannot be written as $d\mathbf{t}$ for any class $\mathbf{t}\in H_{2}(X)$ and any $d\in {\mathbb {Z}}\backslash \{-1,1\}$. Denote by $(e_{1},e_{2},e_{3},e_{4})$ the standard basis of $H_{2}(S^{2}\times S^{2}\# S^{2}\times S^{2})$ and consider the classes $$\alpha =ae_{1}+2e_{2},\qquad \beta =e_{1}+te_{3}+e_{4}$$ where $a,t\in \mathbb{N}$ and $a$ is an odd number. We have $m=\alpha ^{2}=4a$, $n=\beta ^{2}=2t$ and $\alpha \cdot \beta =2$. Since $a$ is odd, the classes $\alpha $ and $\beta $ can be represented by spheres. We will prove the second part of Theorem \[app\], which says that if $a\geq 5$, then the spheres representing $\alpha $ and $\beta $ intersect with at least 4 geometric intersections. Suppose these two spheres have exactly two geometric intersections. We denote by ${N_{m,n}}$ the regular neighborhood of the union of the spheres and by $W$ its complementary submanifold $W=X\backslash \operatorname{Int}({N_{m,n}})$ in $X$. While ${N_{m,n}}$ is the double plumbing of two disk bundles over spheres whose intersection form is positive definite, the submanifold $W\subset X$ carries the negative part of the intersection form. We have defined $\sp $ structures $\mathfrak{t}_{i,j}$ on ${N_{m,n}}$ and denoted by $\mathfrak{s}_{i,j}=\mathfrak{t}_{i,j}|_{{Y_{m,n}}}$ the restriction of each $\sp $ structure to the boundary 3-manifold. Now we would like to define a $\sp $ structure $\mathfrak{u}_{i,j}\in \operatorname{Spin}^{c}(X)$ for which $\mathfrak{u}_{i,j}|_{{N_{m,n}}}=\mathfrak{t}_{i,j}$. Then we will find the restriction $\mathfrak{u}_{i,j}|_{W}$ and use Theorem \[inequality\] for the pair $(W,-{Y_{m,n}})$, equipped with the $\sp $ structure $\mathfrak{u}_{i,j}|_{W}$ for some $i$ and $j$. By definition of $\mathfrak{t}_{i,j}\in \operatorname{Spin}^{c}({N_{m,n}})$, we have $\langle c_{1}(\mathfrak{t}_{i,j}),\alpha \rangle =2i-m$ and $\langle c_{1}(\mathfrak{t}_{i,j}),\beta \rangle =2j-n$. For an odd $i$, define a $\sp $ structure $\mathfrak{u}_{i,j}$ on $X$ by $$\begin{aligned} & \langle c_{1}(\mathfrak{u}_{i,j}),e_{1}\rangle = \langle c_{1}(\mathfrak{u}_{i,j}),e_{3} \rangle=-2\\ & \langle c_{1}(\mathfrak{u}_{i,j}),e_{2} \rangle =i-a\\ & \langle c_{1}(\mathfrak{u}_{i,j}),e_{4} \rangle =2j+2\end{aligned}$$ Then we have $\langle c_{1}(\mathfrak{u}_{i,j}),\alpha \rangle =2i-m$ and $\langle c_{1}(\mathfrak{u}_{i,j}),\beta \rangle =2j-n$, which means that $\mathfrak{u}_{i,j}|_{{N_{m,n}}}=\mathfrak{t}_{i,j}$ and consequently $\mathfrak{u}_{i,j}|_{{Y_{m,n}}}=\mathfrak{s}_{i,j}$. We can calculate that the orthogonal complement of $H_{2}({N_{m,n}})$ in $H_{2}(X)$ is spanned by the vectors $\gamma =-ae_{1}+2e_{2}-2e_{3}$ and $\delta =-te_{3}+e_{4}$, for which we have $\gamma ^{2}=-m$, $\delta ^{2}=-n$ and $\gamma \cdot \delta =-2$. Thus, $\gamma $ and $\delta $ are generators of $H_{2}(W)$ and its intersection form is given by the matrix $Q_{W}=\left( \begin{array}{cc} -m & -2 \\ -2 & -n \\ \end{array} \right )$. We calculate $$\begin{aligned} & \langle c_{1}(\mathfrak{u}_{i,j}|_{W}),\gamma \rangle =2i+4\\ & \langle c_{1}(\mathfrak{u}_{i,j}|_{W}),\delta \rangle =n+2j+2\end{aligned}$$ It follows that the square of the first Chern class $c_{1}(\mathfrak{u}_{i,j}|_{W})$ is given by $$\begin{aligned} c_{1}(\mathfrak{u}_{i,j}|_{W})^{2}=-\frac{1}{mn-4}\left (n(2i+4)^{2}+m(n+2j+2)^{2}-4(2i+4)(n+2j+2)\right )\;.\end{aligned}$$ Now the restriction of $\mathfrak{u}_{i,j}|_{W}$ to the boundary $-{Y_{m,n}}$ is the $\sp $ structure $\mathfrak{s}_{i,j}$ and Theorem \[inequality\] implies $4d_{b}(-{Y_{m,n}},\mathfrak{s}_{i,j})\geq c_{1}(\mathfrak{u}_{i,j}|_{W})^{2}$. Recall from Theorem \[th1\] the correction terms $d_{b}({Y_{m,n}})$ and compare $$\begin{aligned} & -d_{b}({Y_{m,n}},\mathfrak{s}_{i,j})-1=d_{b}(-{Y_{m,n}},\mathfrak{s}_{i,j})\geq \frac{c_{1}(\mathfrak{u}_{i,j}|_{W})^{2}}{4}\\ & -\frac{c_{1}(\mathfrak{u}_{i,j}|_{W})^{2}}{4}\geq d_{b}({Y_{m,n}},\mathfrak{s}_{i,j})+1\\ & \frac{n(2i+4)^{2}+m(n+2j+2)^{2}-4(2i+4)(n+2j+2)}{4(mn-4)}\geq \\ & \geq \frac{m^{2}n+mn^{2}-4mn(i+j)+4n(i^{2}+2i)+4m(j^{2}+2j)-16ij-16}{4(mn-4)}\end{aligned}$$ By simplifying this expression we get the inequality $$\begin{aligned} & 4(mn-4)(i+2j+1-a)\geq 0\\ & i+2j+1\geq a\end{aligned}$$ where $1\leq i\leq 4a-1$ and $1\leq j\leq 2t-1$ and $i$ is odd. If $a\geq 5$, this inequality does not hold for the $\sp $ structure $\mathfrak{s}_{1,1}$. The higher the value of $a$, the more $\sp $ structures $\mathfrak{s}_{i,j}$ do not satisfy the above inequality. Therefore the two spheres representing $\alpha $ and $\beta $ must have at least 4 geometric intersections for all $a\geq 5$. It might be interesting to compare our result with [@ASKI Proposition 3.6]. According to the Proposition in the case $n=2$, the classes $(p_{1},q_{1},0,0)$ and $(0,0,p_{2},q_{2})$ (where $p_{i},q_{i}\geq 2$ and $(p_{i},q_{i})=1$ for $i=1,2$) are not disjointly, smoothly, $S^{2}$-representable inside the manifold $S^{2}\times S^{2}\# S^{2}\times S^{2}$. The application of $d_{b}$-invariants in the Section \[App\] is similar to the $d$-invariant obstruction that is used for concordance applications, e.g. in [@JANA] and many other papers. [Geometric intersections of spheres with algebraic intersection one]{} \[One\] Now we investigate a configuration of two spheres which intersect only once inside a closed smooth 4–manifold $X$ with $H_{1}(X)=0$ and $b_{2}^{+}(X)=2$. Such a configuration is a (single) plumbing ${M_{m,n}}$ of disk bundles over spheres with Euler numbers $m$ and $n$. The Kirby diagram for ${M_{m,n}}$ is a Hopf link of two framed unknoted circles, which can be changed by the operation called slam-dunk [@kirby page 163] into a single unknoted circle with framing $\frac{mn-1}{n}$. The boundary of ${M_{m,n}}$ is thus the lens space $L(mn-1,n)$ with $H_{1}(L(mn-1,n))={\mathbb {Z}}_{mn-1}$. For labeling lens spaces, we use notation from [@OS4]. By the results of [@OS2 Proposition 3.1], the Heegaard–Floer homology of $\widehat{HF}(L(p,q))$ has one generator in every torsion $\sp $ structure and its absolute grading is given by a recursive formula from [@OS4 Proposition 4.8]: $$d(-L(p,q),i)=\left (\frac{pq-(2i+1-p-q)^{2}}{4pq}\right )-d(-L(q,r),j)$$ where $r$ and $j$ are the reductions of $p$ and $i$ modulo $q$ respectively. In our case $p=mn-1$ and $q=n$, so $r=n-1$. In the special case when $n=1$, we need only one application of the recursive formula to obtain [1]{} \[lensD1\] & d(-L(m-1,1),i)=-+. In another special case when $n=2$, we need two applications of the recursive formula to obtain [1]{} \[lensD2\] & d(-L(2m-1,2),i)=-+\ & d(-L(2m-1,2),i)=- When $n>2$, starting with $d(-L(n-1,1),j)$ we apply the recursive formula three times to obtain [1]{} & d(-L(mn-1,n),i)=-+- where $j$ is the reduction of $i\textrm{ mod $n$}$ and $t$ is the reduction of $j\textrm{ mod ($n-1$)}$. In the special case when $0\leq i<n-1$ and thus $i=j=t$ we get a simplification [1]{} \[lensD\] & d(-L(mn-1,n),i)=-(nm\^[2]{}+m(n-2i)\^[2]{}-2m(n-2i))+ Denote $L=-L(mn-1,n)$. Let us derive the formula in another way: by defining a $\sp $ structure $\mathfrak{s}_{i}$ on the plumbing $-{M_{m,n}}$ and using the Formula from [@OS4 Formula (4)] to compute $d(L,\mathfrak{s}_{i}|_{L})$. By removing a 4-ball from $-{M_{m,n}}$ we get a cobordism $\mathcal{C}$ from $S^{3}$ to $L$. Since the intersection form of $-{M_{m,n}}$ is given by the matrix $Q_{-{M_{m,n}}}=\left( \begin{array}{cc} -m & -1 \\ -1 & -n \\ \end{array} \right )$, we have $\chi (\mathcal{C})=2$ and $\sigma (\mathcal{C})=-2$. Define a $\sp $ structure $\mathfrak{s}_{i}$ on $-{M_{m,n}}$ by $$\begin{aligned} \label{defs} & \langle c_{1}(\mathfrak{s}_{i}),s_{1}\rangle =m,\quad \quad \quad \langle c_{1}(\mathfrak{s}_{i}),s_{2}\rangle =n-2i\;,\end{aligned}$$ where $s_{1},s_{2}\in H_{2}(-{M_{m,n}})$ are the classes of the base spheres in the plumbing $-{M_{m,n}}$. It follows that $$c_{1}(\mathfrak{s}_{i})^{2}=-\frac{nm^{2}+m(n-2i)^{2}-2m(n-2i)}{mn-1}$$ and the formula gives us $$\begin{aligned} & d(L,\mathfrak{s}_{i}|_{L})=-\frac{nm^{2}+m(n-2i)^{2}-2m(n-2i)}{4(mn-1)}+\frac{2}{4}\;,\end{aligned}$$ which coincides with Formula . From Theorem \[inequality\] we obtain the following obstruction for the $d$-invariants: \[prop4\] Let $X$ be a closed smooth 4-manifold with $H_{1}(X)=0$, $b_{2}^{+}(X)=2$ and $b_{2}^{-}(X)=0$. Suppose there are two spheres $\Sigma _{1},\Sigma _{2}\subset X$ with $\Sigma _{1} ^{2}=m>0$, $\Sigma _{2}^{2}=n>0$ and $\Sigma _{1}\cdot \Sigma _{2}=1$. Denote by $L$ the boundary of a regular neighbourhood of $\Sigma _{1}\cup \Sigma _{2}$. Then for some subgroup $\tau \subset H_{1}(L)$ with $|\tau |^{2}=mn-1$ and some $\sp $ structure $\mathfrak{s}_{0}$ on $L$, we have $d(L,\mathfrak{s}_{0}+\phi )=0$ for every $\phi \in \tau$. Denote by ${M_{m,n}}$ the regular neighbourhood of $\Sigma _{1}\cup \Sigma _{2}$ and let $V=X\backslash \operatorname{Int}({M_{m,n}})$. We study the Mayer–Vietoris sequence in cohomology of the triple $(X,V,{M_{m,n}})$: $$\begin{aligned} & 0\rightarrow H^{1}(V)\oplus H^{1}({M_{m,n}})\stackrel{f_{1}}\rightarrow H^{1}(L) \stackrel{f_{2}}\rightarrow H^{2}(X)\stackrel{f_{3}}\rightarrow H^{2}(V)\oplus H^{2}({M_{m,n}}) \stackrel{f_{4}}\rightarrow H^{2}(L)\rightarrow 0\\ & 0\rightarrow H^{1}(V)\oplus H^{1}({M_{m,n}})\stackrel{f_{1}}\rightarrow 0 \stackrel{f_{2}}\rightarrow {\mathbb {Z}}^{2}\stackrel{f_{3}}\rightarrow H^{2}(V)\oplus H^{2}({M_{m,n}})\stackrel{f_{4}}\rightarrow {\mathbb {Z}}_{mn-1}\rightarrow 0\end{aligned}$$ At the beginning and at the end of the sequence we have zeroes since $H_{1}(X)=0$. Since $L$ is the lens space $L(mn-1,n)$, we have $H^{2}(L)={\mathbb {Z}}_{mn-1}$ and $H^{1}(L)=0$. It follows from the sequence that $H^{1}(V)=H^{1}({M_{m,n}})=0$, so $H_{1}(V)=\tau $ is a torsion group by the universal coefficient theorem. The group $H_{2}({M_{m,n}})={\mathbb {Z}}^{2}$ is spanned by the homology classes of the spheres $\Sigma _{1}$ and $\Sigma _{2}$, so the cohomology group $H^{2}({M_{m,n}})$ has rank two. It follows that $H^{2}(V)\cong \tau $ and $H_{2}(V)=0$. Now we can write down the homology long exact sequence of the pair $(V,-L)$: $$\begin{aligned} & \rightarrow H_{2}(-L)\stackrel{g_{1}}\longrightarrow H_{2}(V)\stackrel{g_{2}}\longrightarrow H_{2}(V,-L)\stackrel{g_{3}}\longrightarrow H_{1}(-L)\stackrel{g_{4}}\longrightarrow H_{1}(V)\rightarrow \ldots \\ & \rightarrow 0 \stackrel{g_{1}}\longrightarrow 0\stackrel{g_{2}}\longrightarrow \tau \stackrel{g_{3}}\longrightarrow {\mathbb {Z}}_{mn-1}\stackrel{g_{4}}\longrightarrow \tau \stackrel{0}\rightarrow \ldots \end{aligned}$$ It follows from this sequence that $\tau $ is a subgroup of ${\mathbb {Z}}_{mn-1}$ with quotient group ${\mathbb {Z}}_{mn-1}/\tau \cong \tau $, thus $|\tau |^{2}=mn-1$. Those $\sp $ structures on $-L$ which are restrictions of $\sp $ structures on $V$ correspond to the image of the map $H^{2}(V)\to H^{2}(-L)$, which is the monomorphism $\tau \to {\mathbb {Z}}_{mn-1}$. For every $\sp $ structure on $-L$ which is the restriction of a $\sp $ structure on $V$ we can apply Theorem \[inequality\] to obtain the estimate $d(-L,\mathfrak{s})\geq 0$ and consequently $d(L,\mathfrak{s})\leq 0$ . Since $V$ has a trivial intersection form, we can also apply the same theorem for the pair $(-V,L)$ to obtain $d(L,\mathfrak{s})\geq 0$, from which the equality follows. [Single plumbings inside ${\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$]{} \[CP1\] Let $X={\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$ and denote by $(e_{1},e_{2})$ the standard basis for $H_{2}(X)$. As remarked in Subsection \[CP\], the classes in $H_{2}(X)$ which are representable by spheres have the form $x_{1}e_{1}+x_{2}e_{2}$ with $(x_{1},x_{2})\in \{0,\pm 1,\pm 2\}^{2}\backslash \{(0,0)\}$. Consider a pair of such classes with algebraic intersection 1: $\alpha =2e_{1}+e_{2}$ and $\beta =e_{1}-e_{2}$. We have $m=\alpha ^{2}=5$ and $n=\beta ^{2}=2$ so $L=L(9,2)$ and the $d$–invariants are given by [1]{} & d(L(9,2),i)=- for $0\leq i\leq 8$, where $j$ is the reduction of $i$ (mod $2$). We calculate [1]{} & d(L,0)=d(L,1)=d(L,2)=d(L,5)=d(L,8)=0\ & d(L,3)=d(L,7)=d(L,4)=d(L,6)=- $2e_{1}+e_{2}$ at 420 346 $e_{1}-e_{2}$ at 423 163 ![Attaching circles of the 2-handles representing classes $2e_{1}+e_{2},e_{1}-e_{2}\in {\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$[]{data-label="fig:CP1"}](CP1) We can see that there are three $\sp $ structures with $d$-invariant equal to 0, in accordance with Proposition \[prop4\]. Thus, the spheres representing classes $\alpha $ and $\beta $ can have only one geometric intersection inside ${\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$. Indeed, the two spheres can be chosen in such a way, following the construction of Ruberman [@DR] described in Subsubsection \[FirstCP\]. We represent the class $\alpha =2e_{1}+e_{2}$ by taking two copies of $h_{1}$, resolve their double point to get a single disk, then make a boundary connected sum with $h_{2}$ (with coherent orientations) and add a disk in $B^{4}$ to the resulting surface. Similarly, we represent the class $\beta =e_{1}-e_{2}$ by taking a boundary connected sum of $h_{1}$ and $h_{2}$ with reversed orientations and adding a disk in $B^{4}$, see Figure \[fig:CP1\]. The attaching circles of the 2-handles thus achieved can be moved by an isotopy to form the Hopf link, which shows that the two representatives have only one geometric intersection. [Single plumbings inside $S^{2}\times S^{2}\# S^{2}\times S^{2}$]{} \[S21\] Consider the 4–manifold $X=S^{2}\times S^{2}\# S^{2}\times S^{2}$ and two classes $\alpha =(2k+1)e_{1}+2e_{2}$ and $\beta =-ke_{1}+e_{2}+2ke_{3}+e_{4}$ in $H_{2}(X)$, where $k$ is a positive integer. We have $\alpha ^{2}=4(2k+1)=m$, $\beta ^{2}=2k=n$ and $\alpha \cdot \beta =1$. Since $\alpha $ and $\beta $ are primitive noncharacteristic classes, they are represented by embedded spheres in $X$ by [@WALL Theorem 3]. We will prove here Theorem \[app1\] which says: Any two spheres representing the classes $\alpha $ and $\beta $ intersect with at least 3 geometric intersections for all $k>1$. Suppose the two spheres intersect with only one geometric intersection; then a regular neighborhood of their configuration forms the plumbing ${M_{m,n}}$ inside $X$. Denote by $V=X\backslash \operatorname{Int}({M_{m,n}})$ its complementary submanifold and let $L=\partial V$. We would like to define a $\sp $ structure $\mathfrak{t}_{i}$ on $X$, for which the restriction $\mathfrak{t}_{i}|_{{M_{m,n}}}=\mathfrak{s}_{i}$. Then we will find the restriction $\mathfrak{t}_{i}|_{V}$ to the complementary submanifold and apply Theorem \[inequality\]. Let $\mathfrak{t}_{i}\in \operatorname{Spin}^{c}(X)$ be the unique $\sp $ structure for which the following holds: $$\begin{aligned} & \langle c_{1}(\mathfrak{t}_{i}),e_{1}\rangle =0\\ & \langle c_{1}(\mathfrak{t}_{i}),e_{2}\rangle =2(2k+1)\\ & \langle c_{1}(\mathfrak{t}_{i}),e_{3}\rangle =-2\\ & \langle c_{1}(\mathfrak{t}_{i}),e_{4}\rangle =2(k-i-1)\end{aligned}$$ Then we have $\langle c_{1}(\mathfrak{t}_{i}),\alpha \rangle =4(2k+1)=m$ and $\langle c_{1}(\mathfrak{t}_{i}),\beta \rangle =2k-2i=n-2i$, which means that $\mathfrak{t}_{i}|_{{M_{m,n}}}$ concides with the $\sp $ structure $\mathfrak{s}_{i}$ defined in . As we have shown, the correction term $d(L,\mathfrak{s}_{i}|_{L})$ is given by the Formula . Now let us find the restriction $\mathfrak{t}_{i}|_{V}$. The image of the inclusion homomorphism $H_{2}(V)\to H_{2}(X)$ is spanned by the two classes $\gamma =-(2k+1)e_{1}+2e_{2}+(4k+1)e_{3}$ and $\delta =-2ke_{3}+e_{4}$ which are both orthogonal to $\alpha $ and $\beta $. We calculate $$\begin{aligned} & \langle c_{1}(\mathfrak{t}_{i}),\gamma \rangle =2\\ & \langle c_{1}(\mathfrak{t}_{i}),\delta \rangle =6k-2i-2=3n-2i-2\end{aligned}$$ Since $\gamma ^{2}=-m$, $\delta ^{2}=-2n$ and $\gamma \cdot \delta =4k+1=\frac{m-2}{2}$, the intersection form on $V$ is given by the matrix $Q_{V}=\left( \begin{array}{cc} -m & \frac{m-2}{2} \\ \frac{m-2}{2} & -2n \\ \end{array} \right )$ with $\operatorname{det}Q_{V}=mn-1$. The square of $c_{1}(\mathfrak{t}_{i}|_{V})$ is then calculated by [1]{} & c\_[1]{}(\_[i]{}|\_[V]{})\^[2]{}=-. Now Theorem \[inequality\] gives us the inequality $c_{1}(\mathfrak{t}_{i}|_{V})^{2}+2\leq 4d(L,i)$. Using the Equation , we compare $$\begin{aligned} -\frac{8n+m(3n-2i-2)^{2}+2(m-2)(3n-2i-2)}{mn-1}\leq -\frac{nm^{2}+m(n-2i)^{2}-2m(n-2i)}{mn-1}\end{aligned}$$ and by simplifying we get the inequality $$(mn-1)(k-i-1)\geq 0\;,$$ which is not valid for $i\geq k$. When applying the Formula we assumed that $0\leq i<n-1=2k-1$. Thus, the $\sp $ structure $\mathfrak{s}_{k}|_{L}$ does not satisfy the inequality in Theorem \[inequality\] whenever $k>1$. Therefore, the two spheres representing the classes $\alpha $ and $\beta $ have at least three geometric intersections for all $k>1$.

--- author: - Xiangwen Wang - Michel Pleimling title: 'Online Gambling of Pure Chance: Wager Distribution, Risk Attitude, and Anomalous Diffusion' --- Introduction {#introduction .unnumbered} ============ Today, gambling is a huge industry with a huge social impact. According to a report by the American Gaming Association [@AGA2018], commercial casinos in the United States alone made total revenue of over 40 billion US dollars in 2017. On the other hand, different studies reported that $0.12\%-5.8\%$ of the adults and $0.2\%-12.3\%$ of the adolescents across different countries in the world are experiencing problematic gambling [@Calado2016; @Calado2017]. Studying the gamblers’ behavior patterns not only contributes to the prevention of problematic gambling and adolescent gambling, but also helps to better understand human decision-making processes. Researchers have put a lot of attention on studying gambling-related activities. Economists have proposed many theories about how humans make decisions under different risk conditions. Several of them can also be applied to model gambling behaviors. For example, the prospect theory introduced by Kahneman and Tversky [@Kahneman1979] and its variant cumulative prospect theory [@Tversky1992] have been adopted in modeling casino gambling [@Barberis2012]. In parallel to the theoretical approach, numerous studies focus on the empirical analysis of gambling behaviors, aiming at explaining the motivations behind problematic gambling behaviors. However, parametric models that quantitatively describe empirical gambling behaviors are still missing. Such models can contribute to evaluating gambling theories proposed by economists, as well as yield a better understanding of the gamblers’ behaviors. Our goal is to provide such a parametric model for describing human wagering activities and risk attitude during gambling from empirical gambling logs. However, it is very difficult to obtain gambling logs from traditional casinos, and it is hard to collect large amounts of behavior data in a lab-controlled environment. Therefore in this paper we will focus on analyzing online gambling logs collected from online casinos. Whereas historically the development of probability theory, which then became the foundation of statistics, was tied to chance games, today we use statistical tools to analyze gamblers’ behaviors. Recent years have seen an increasing trend of online gambling due to its low barriers to entry, high anonymity and instant payout. For researchers of gambling behaviors, online gambling games present two advantages: simple rules and the availability of large amounts of gambling logs. In addition to the usual forms of gambling games that can be found in traditional casinos, many online casinos also offer games that follow very simple rules, which makes analyzing the gambling behavior much easier as there are much fewer degrees of freedom required to be considered. On the other hand, many online casinos have made gambling logs publicly available on their websites, mainly for verification purposes, which provides researchers with abundant data to work on. Due to the high popularity of online gambling, in a dataset provided by an online casino there are often thousands or even hundreds of thousands of gamblers listed. Such a large scale of data can hardly be obtained in a lab environment. Prior research has begun to make use of online gambling logs. For example, Meng’s thesis [@Meng2018] presented a pattern analysis of typical gamblers in Bitcoin gambling. It is worth arguing that although our work only focuses on the behaviors of online gamblers, there is no reason to think that our conclusions cannot be extended to traditional gamblers. Naturally, we can treat the changing cumulative net income of a player during their gambling activities as a random walk process [@Wang2018]. We are particularly interested in the diffusive characteristics of the gambler’s net income. This is another reason why we want to analyze the wager distribution and risk attitude of gamblers, since both distributions are closely related to the displacement distribution for the gambler’s random walks. Within this paper, we will mainly focus on the analysis at the population level. Physicists have long been studying diffusion processes in different systems, and recently anomalous diffusive properties have been reported in many human activities, including human spatial movement [@Rhee2011; @Brockmann2008b; @Kim2010], and information foraging [@Wang2017]. In a previous study of skin gambling [@Wang2018], we have shown that in a parimutuel betting game (where players gamble against each other), a gambler’s net income displays a crossover from superdiffusion to normal diffusion. We have reproduced this crossover in simulations by introducing finite and overall conserved gamblers’ wealth (see [@Toscani2019] for a different way of modeling this using kinetic equations of Boltzmann and Fokker-Planck type). However, this explanation cannot be used in other types of gambling games where there is no interaction among gamblers (e.g., fixed-odds betting games, which will be introduced below), as they violate the conservation of gamblers’ overall wealth. In this paper, we want to expand the scope of our study to more general gambling games, check the corresponding diffusive properties, and propose some explanations for the observed behaviors. One of our goals is to uncover the commonalities behind the behavior of online gamblers. To implement this, we analyze the data from different online gambling systems. The first one is skin gambling, where the bettors are mostly video game players and where cosmetic skins from online video games are used as virtual currency for wagering [@Wang2018; @Holden2018]. The other system is crypto-currency gambling, where the bettors are mostly crypto-currency users. Different types of crypto-currencies are used for wagering. Commonly used crypto-currencies include Bitcoin, Ethereum, and Bitcoin Cash, whose basic units are BTC, ETH and BCH, respectively. As the overlap of these two communities, video game players and crypto-currency users, is relatively small for now, features of gambling patterns common between these two gambling systems are possibly features common among all online gamblers. Not only do we consider different gambling systems, but we also discuss different types of gambling games. In this paper, we discuss four types of solely probability-based gambling games (Roulette, Crash, Satoshi Dice and Jackpot), whose outcomes in theory will not benefit from the gamblers’ skill or experience when the in-game random number generators are well designed. In general, there are two frameworks of betting in gambling: fixed-odds betting, where the odds is fixed and known before players wager in one round; and parimutuel betting, where the odds can still change after players place the bets until all players finish wagering. In fixed-odds betting, usually players bet against the house/website, and there is no direct interaction among players; and in parimutuel betting, usually players bet against each other. The four types of games we discuss in this paper will cover both betting frameworks (see the Methods section). When a player attends one round in any of those games, there are only two possible outcomes: either win or lose. When losing, the player will lose the wager they placed during that round; whereas when winning, the prize winner receives equals their original wager multiplied by a coefficient. This coefficient is generally larger than $1$, and in gambling terminology, it is called odds in decimal format [@Buhagiar2018; @Rodriguez2017]. Here we will simply refer to it as odds. Note that the definition of odds in gambling is different than the definition of odds in statistics, and in this paper we follow the former one. When a player attends one round, their chance of winning is usually close to, but less than the inverse of the odds. The difference is caused by the players’ statistical disadvantage in winning compared to the house due to the design of the game rules. In addition, the website usually charges the winner with a site cut (commission fee), which is a fixed percentage of the prize. We further define the [*payoff*]{}, $o_p$, to be the net change of one player’s wealth after they attend one round. Although the four types of games are based on different rules, the payoffs all follow the same expression $$\label{eq:payoff} \displaystyle o_p = \left\lbrace \begin{split} \displaystyle &-b, &\text{with\ probability\ } &p= 1- \frac{1}{m} + f_\text{m}, \\ \displaystyle &(1-\eta)(m-1) b, &\text{with\ probability\ } &q=1-p = \frac{1}{m} - f_\text{m}, \end{split}\right.$$ where $b>0$ is the wager the player places, $m>1$ is the odds, $1>\eta\geq 0$ corresponds to the site cut, and $f_\text{m}$ is a non-negative value based on the odds representing the players’ statistical disadvantage in winning, as mentioned earlier. At least either $\eta$ or $f_\text{m}$ are non-zero. From Eq. (\[eq:payoff\]), we can obtain the expected payoff of attending one round $$\label{eq:expected-payoff} \displaystyle E\left( o_p|m, b\right) = \Big(- (1-1/m +f_m) + (1-\eta)(m-1)(1/m - f_m)\Big)b = - \Big((1-\eta)m f_m + (1-1/m + f_m)\eta \Big)b\equiv -\xi b,$$ which is always negative since either $\eta$ or $f_\text{m}$ are non-zero. In gambling terminology, $\xi$ is called the house edge, from which the websites make profits. The house edge represents the proportion the website will benefit on average when players wager. In the four types of games we discuss, the house edge $\xi$ ranges from $1\%$ to $8\%$. If there is no house edge $\xi=0$, that means it is a fair game. In a fair game or when we ignore the house edge, the expected payoff would be 0. In the Results section, we begin with an analysis of wager distribution and log-ratios between successive wagers, which helps us to understand the gamblers’ wagering strategy. We then focus on an analysis of risk attitude by studying the distribution of the odds players choose to wager with. We conclude by extending our discussion to the analysis of net incomes of gamblers viewed as random walks. This allows us to gain insights into the gamblers’ behaviors by computing quantities like the ensemble/time-averaged mean-squared displacement, the first-passage time distribution, ergodicity breaking parameter, and Gaussianity. Detailed information about the games and datasets discussed in this paper can be found in the Methods section. Results {#results .unnumbered} ======= Wager distribution {#wager-distribution .unnumbered} ------------------ From the viewpoint of the interaction among players, the games discussed in this paper can be grouped into two classes: in Roulette, Crash, and Satoshi Dice games, there is little or no interaction among players, whereas in Jackpot games, players need to gamble against each other. At the same time, from the viewpoint of wager itself, the games can also be grouped into two classes: In games (A-G), the wagers can be an arbitrary amount of virtual currencies, such as virtual skin tickets or crypto-currency units, whereas in game (H), the wagers are placed in the form of in-game skins, which means the wager distribution further involves the distributions of the market price and availability of the skins. Furthermore, from the viewpoint of the odds, considering the empirical datasets we have, when analyzing the wager distribution, there are three situations: i) For Roulette and Satoshi Dice games, the odds are fixed constants, and wagers placed with the same odds are analyzed to find the distribution. ii) For Crash games, the odds are selected by the players, and wagers placed with different odds are mixed together during distribution analysis. iii) For the Jackpot game, the odds are not fixed at the moment when the player wagers. \[table:wager-distribution\] ![ \[fig:wager\_distribution\] In games (A-G), where players are allowed to choose arbitrary bet values, the wager distribution can be best fitted by log-normal distributions (\[eq:log-normal\]). In game (D), the log-normal distribution is truncated at its maximum bet value, indicated by $^*$. The fitting lines represent the log-normal fittings. Wagers placed under the different maximum allowed bet values are discussed separately, e.g., in game (A), (A$_1$) has a maximum bet value of $500,000$, and (A$_2$) has a maximum bet value of $50,000$. On the other hand, in game (H) where wagers can only be in-game skins, the wager distribution is best described by a pairwise power law with an exponential transition, see Eq. (\[eq:powerlaw\_exp\_powerlaw\]). The red dotted line represents the log-normal fitting and the blue solid line represents the fitting of a pairwise power law with an exponential transition.](figure1.pdf){width="\columnwidth"} In Table \[table:wager-distribution\] we categorize the 8 datasets based on the above information. At the same time, for each dataset we perform a distribution analysis of wagers at the aggregate level. Within the same dataset wagers placed under different maximum allowed bet values are discussed separately. We plot the complementary cumulative distribution function (CCDF) of the empirical data and the fitted distribution to check the goodness-of-fit, see Fig. \[fig:wager\_distribution\]. CCDF, sometimes also referred to as the survival function, is given by $\bar{F}(x) = P(X>x) = 1 - P(X\le x)$. It turns out that when players are allowed to place arbitrary wagers (games A-G in Table \[table:wager-distribution\]), the wager distributions can in general be best-fitted by log-normal distributions. In particular, in games (A, B, C, E, F, G), the wager distribution can be approximated by the following expression $$\label{eq:log-normal} \displaystyle P(x) = \frac{\displaystyle\Phi\left(\frac{\ln(x+1)-\mu} {\sigma}\right)-\Phi\left(\frac{\ln(x)-\mu}{\sigma}\right)} {\displaystyle 1-\Phi\left(\frac{\ln(x_\text{min})-\mu}{\sigma}\right)},$$ with $x_\text{min}\le x$ and $\sigma>0$. $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution. Meanwhile in game (D), the fitted log-normal distribution is truncated at an upper boundary $x_\text{max}$, which might result from the maximum allowed small bet value and the huge variation of the market price of crypto-currencies. During model selection, we notice that when we select different $x_\text{min}$, occasionally a power-law distribution with exponential cutoff is reported to be a better fit, but often it does not provide a decent absolute fit on the tail, and overall the log-normal distribution provides smaller Kolmogorov-Smirnov distances, see the Methods section. On the other hand, as we have pointed out in the previous study [@Wang2018], when players are restricted to use in-game skins as wagers for gambling, the wager distribution can be best fitted by a shifted power law with exponential cutoff. Now, with a similar situation in game (H), where wagers can only be in-game skins, we find that the early part of the curve can be again fitted by a power law with exponential cutoff, as shown in Fig. \[fig:wager\_distribution\](H). However, this time it does not maintain the exponential decay of its tail; instead, it changes back to a power-law decay. The overall distribution contains six parameters, given by the expression $$\label{eq:powerlaw_exp_powerlaw} \displaystyle P(x) = \left\lbrace \begin{split} \displaystyle &\frac{1}{c_1+c_2 c_3}\ \frac{(x-\delta)^{-\alpha}}{1+e^{\lambda(x-\beta)}},\ &\text{for}\ x \le x_\text{trans}, \\ \displaystyle &\frac{c_3}{c_1+c_2 c_3}\ x^{-\eta}, \ &\text{for}\ x > x_\text{trans}, \end{split}\right.$$ where $\displaystyle c_1=\sum\limits_{x=x_\text{min}}^{x_\text{trans}} \frac{(x-\delta)^{-\alpha}}{1+e^{\lambda(x-\beta)}}$, $\displaystyle c_2=\zeta\left(\eta, x_\text{trans}\right)$, and $\displaystyle c_3=x_\text{trans}^\eta \frac{\left(x_\text{trans}-\delta\right)^{-\alpha}} {1+e^{\lambda\left(x_\text{trans}-\beta\right)}}$. We believe that when players are restricted to use in-game skins as wagers, the decision to include one particular skin in their wager is further influenced by the price and availability of that skin. These factors make the wager distribution deviate from the log-normal distribution, which is observed in games (A-G). This is very clear when comparing the wager distributions of games (G) and (H) as both games are jackpot games of skin gambling, and the only difference is whether players are directly using skins as wagers or are using virtual skin tickets obtained from depositing skins. The power-law tail, which was not observed in the previous study [@Wang2018], might result from the increment of the maximum allowed skin price (from \$$400$ to \$$1 800$). The above discussions, including the results for games (A-G) in Table \[table:wager-distribution\], show that the wager distributions in pure probability-based gambling games, no matter whether the game follows parimutuel betting or fixed-odds (preset/player-selected) betting, stay log-normal as long as the players are allowed to place arbitrary amounts of wagers. This commonality of log-normal distribution no longer holds when this arbitrariness of wager value is violated, e.g., in the scenario where the player can only wager items (in-game skins). Log-normal distribution has been reported in a wide range of economic, biological, and sociological systems [@LIMPERT2001], including income, species abundance, family size, etc. Economists have proposed different kinds of generative mechanisms for log-normal distributions (and power-law distributions as well). One particular interest for us is the multiplicative process [@Gibrat1930; @mitzenmacher2003]. Starting from an initial value $X_0$, random variables in a multiplicative process follow an iterative formula $X_{i+1}=\exp(\nu_i) X_i$ or $\ln X_{i+1}= \ln X_i + \nu_i.$ If the $v_i$ has finite mean and variance, and is independent and identically distributed, then according to the central limit theorem, for large $i$, $\ln X_i$ will follow a normal distribution, which means $X_i$ will follow a log-normal distribution. If we want to check whether gamblers follow multiplicative processes when they wager, we can first check the correlation between consecutive bets $(b_i, b_{i+1})$. Due to the large variances of the wager distributions, Pearson’s correlation coefficient may perform poorly. Instead, we adopt two rank-based correlation coefficients, Kendall’s Tau [@Kendall1990] $\tau_K$ and Spearman’s Rho [@Taylor1987] $\rho_S$. At the same time, we also check the mean and variance of the log-ratios $\ln(b_{i+1}/b_{i})$ between consecutive bets. These statistics can be found in Table \[table:log\_ratio\]. The results reveal that the values of consecutive bets exhibit a strong positive correlation, with all the correlation coefficients larger than $0.5$. It shows that players’ next bet values are largely dependent on their previous bet values. At the same time, the bet values are following gradual changes, rather than rapid changes. These conclusions can be confirmed by the small mean values and small variances of log-ratios between consecutive bets. \[table:log\_ratio\] ![ \[fig:log\_ratio\] The distribution of the logarithmic of the ratio (log-ratio) between consecutive bet values. For games (A, B, C), the log-ratio can be described by a Laplace distribution. For games (D, F, G, H), the log-ratio presents bell-shaped distribution. In general, the distributions are symmetric with respect to the y-axis, except in games (D) and (F). The $x$-coordinate $\log_{10}(b_{i+1}/b_i)$ is proportional to the parameter $\nu$. ](figure2.pdf){width="\columnwidth"} Further analysis of the distribution of $\nu$ shows an exponential decay on both of its tails, see Fig. \[fig:log\_ratio\]. This means that $\nu$ approximately follows a Laplace distribution. However, compared to a Laplace distribution, the empirical log-ratio distribution shows a much higher probability at $\nu=0$, whose value can be found in the last column of Table \[table:log\_ratio\]. We also observe that $\nu$ presents higher probability densities around small integers/half-integers and their inverses. Due to the existence of these differences, we will skip the parameter fitting for the distribution of $\nu$. The high probability of staying on the same wager indicates that betting with fixed wager is one of the common strategies adopted by gamblers. Meanwhile, the high positive auto-correlations, along with the higher probability densities at small integers/half-integers and their inverses, provide evidence that gamblers often follow a multiplicative process when wagering. The multiplication process can be explained by the wide adoption of multiplicative betting systems. “Betting system” here refers to the strategy of wagering where the next bet value depends on both the previous bet value and the previous outcome [@Dubins2014; @Epstein2012]. Although betting systems will not provide a long-term benefit, as the expected payoff will always be $0$ in a fair game, still they are widely adopted among gamblers. A well-known multiplicative betting system is the Martingale (sometimes called geometric progression) [@Epstein2012]. In Martingale betting, starting with an initial wager, the gambler will double their wager each time they lose one round, and return to the initial wager once they win. Martingale is a negative-progression betting system where the gambler will increase their wager when they lose and/or decrease their wager when they win. Apart from multiplicative betting, there are many other types of betting systems, such as additive betting and linear betting [@Epstein2012]. The reasons why multiplicative betting systems are dominant in our datasets are: 1) Martingale is a well-known betting system among gamblers; 2) Many online gambling websites provide a service for changing the bet value in a multiplicative way. For example, for the Crash games csgofast-Crash (C) and ethCrash (D), both websites provide a simple program for automatically wagering in a multiplicative way. For the Roulette games and Coinroll (F), the websites provide an interface with which the gambler can quickly double or half their wager. However, for Satoshi Dice (E) and csgospeed-Jackpot (G), no such function is provided, yet we still observe similar results, indicating that gamblers will follow a multiplicative betting themselves. Fig. \[fig:log\_ratio\] provides us with the distribution of $\nu$, however, it will not tell us whether the gamblers adopt the negative/positive-progression betting systems. Therefore we further analyze the effect on the bet values of winning/losing a round. How the gamblers adjust their wager after winning/losing rounds is shown in Table \[table:win-lose\]. We can see that although there is a high probability for sticking to the same bet values, the most likely outcome after losing a round is that the gambler increases their wager. When winning one round, gamblers are more likely to decrease their wager. This means that negative-progression strategies are more common among gamblers than positive-progression strategies. \[table:win-lose\] Risk attitude {#risk-attitude .unnumbered} ------------- We now turn to the following question: When a player is allowed to choose the odds themselves in a near-fair game, how would they balance the risk and potential return? Higher odds means a lower chance of winning and higher potential return, for example, setting odds of $10$ means that the winning chance is only $1/10$, but the potential winning payoff equals $9$ times the original wager. In our analysis, we can examine such behaviors based on the gambling logs from Crash and Satoshi Dice games. For the Crash game only CSGOFAST.COM provides the player-selected odds even when players lose that round, whereas for the Satoshi Dice game only Coinroll accepts player-selected odds. We will therefore focus on the data collected on these two websites. For the Crash game on CSGOFAST.COM, the odds can only be set as multiples of $0.01$, whereas for the Satoshi Dice game on Coinroll the odds can be set to $0.99 \cdot 65536/i$ where $i$ is a positive integer less than $64000$. To simplify our modeling work, we will convert the odds on Coinroll to be multiples of $0.01$ (same as for the Crash game). ![\[fig:odds\] Odds distributions can be well-fitted by truncated shifted power-law distributions. ](figure3.pdf){width=".75\columnwidth"} It turns out that in both cases the odds can be modeled with a truncated shifted power-law distribution, $$\displaystyle P(m) = \left\lbrace \begin{split} \displaystyle &\frac{\left(\ m - \delta \right)^{-\alpha}}{\zeta(\alpha, \ m_\text{min} -\delta)}, \ &\text{for}\ m_\text{min}\le m < m_\text{max}, \\ \displaystyle &\frac{\zeta(\alpha,\ m_\text{max} -\delta)}{\zeta(\alpha, \ m_\text{min} -\delta)}, \ &\text{for}\ m = m_\text{max}, \end{split}\right.$$ where $\zeta(\cdot,\cdot)$ is the incomplete Zeta function, and $m_\text{max}$ is the upper truncation. Note that there is a jump at $m_\text{max}$, meaning that the players are more likely to place bets on the maximum allowed odds than on a slightly smaller odds. The estimated parameters $\alpha=1.881$, $\delta=0.849$, and $m_\text{min}=1.15$ for csgofast-Jackpot on CSGOFAST.COM, whereas for Coinroll the parameters are found to be $\alpha = 1.423$, $\delta = 2.217$, and $m_\text{min} = 2.58$. From the comparison between the CCDFs of empirical data and fitting curves, as shown in Fig. \[fig:odds\], we can see that the truncated shifted power law can capture the overall decaying trends of odds distribution. The stepped behavior results from the gamblers’ preference of simple numbers. A distribution that is close to a power law indicates that a gambler’s free choice of odds displays scaling characteristics (within the allowed range) in near-fair games. It also means that when gamblers are free to determine the risks of their games, although in most times they will stick to low risks, showing a risk-aversion attitude, they still present a non-negligible probability of accepting high risks in exchange for high potential returns. The scaling properties of risk attitude might not be unique to gamblers, but also may help to explain some of the risk-seeking behaviors in stock markets or financial trading. We now re-examine the distributions from the point of view of estimating the crash point $m_C$ (Satoshi Dice games can be explained with the same mechanism). The true distribution of $m_C$ generated by the websites follow a power-law decay with an exponent of $2$ (with some small deviation due to the house edge). Meanwhile, a closer look at the fitted exponents listed above gives us two empirical exponents of $1.423$ and $1.881$, both of which are smaller than $2$. The smaller exponents reveal that gamblers believe that they have a larger chance to win a high-odds game than they actually do. Or equivalently, it means the gamblers over-weight the winning chance of low-probability games. At the same time, the “shifted” characteristics here lead to more bets on small odds, which also indicates that the gamblers over-estimate the winning chance of high-probability games. As a result, they under-weight the winning chances of mild-probability games. These are clear empirical evidence of probability weighting among gamblers, which is believed to be one of the fundamental mechanisms in economics [@Barberis2012]. Wealth distribution {#wealth-distribution .unnumbered} ------------------- In the previous study of skin gambling [@Wang2018], we pointed out that the wealth distribution of skin gamblers shows a pairwise power-law tail. This time, by considering the players’ deposits to their wallets on a gambling site as the wealth data, we find that the pairwise power-law tails are also observed for bitcoin gambling. We find that on the gambling website Coinroll, starting from $5660$ cents, the players’ wealth distribution follows a pairwise power-law distribution, with the power of the first regime to be $1.585$, and the power of the second regime to be $3.258$, see Fig. \[fig:wealth\]. The crossover happens at $1.221\times 10^5$ cents. As both wealth distributions of skin gambling and bitcoin gambling can be approximated by a pairwise power distribution, we believe that it is a good option for modeling the tails of gambler wealth distribution in different scenarios. ![ \[fig:wealth\] The tail of the wealth distribution of Bitcoin gamblers follows a pairwise power-law distribution. ](figure4.pdf){width=".5\columnwidth"} Removing effects due to inequality in the number of bets {#removing-effects-due-to-inequality-in-the-number-of-bets .unnumbered} -------------------------------------------------------- In the above sections, we have analyzed the distributions of several quantities at the population level. However, there is a huge inequality of the number of placed bets among gamblers. We therefore wonder whether those distributions we obtain result from the inequality of number of bets among individuals. To remove the effects of this inequality, we randomly sample in each dataset the same number of bets from heavy gamblers. We re-analyze the wager distribution and odds distribution with the sample data to see if we obtain the same distribution as before. In each dataset we randomly sample $500$ bets from each of those gamblers who placed at least $500$ bets above $b_\text{min}$ given in Table \[table:wager-distribution\]. Some datasets are excluded here as either they do not have enough data or we cannot identify individual gamblers. When re-analyzing the odds distribution, to ensure we have enough data, we respectively sample $100$ and $2000$ bets from each of those gamblers in games (C) and (F) who have at least $100$ and $2000$ valid player-selected odds above $m_\text{min}$. According to the results in Fig. \[fig:wager\_distr\_top\], after removing the inequality the wager distributions can still be approximated by log-normal distributions, but some deviation can be observed. Similarly, the odds distributions again follow truncated shifted power-law distributions after removing the inequality. These results demonstrate that the shape of the distributions we obtained in the above sections is not a result of the inequality of the number of bets. ![\[fig:wager\_distr\_top\] The wagers obtained from random sampling of top gamblers’ bets still present log-normal distributions, although there are some observable deviations. ](figure5.pdf){width="\columnwidth"} Now our question becomes whether the conclusion regarding the distribution at the population level can be extended to the individual level. Here due to the limitation of data, we will only discuss the wager distribution. Analyzing the individual distribution of top gamblers, we find that although heavy-tailed properties can be widely observed at the individual level, only a small proportion of top gamblers presents log-normal distributed wagers. Other distributions encountered include log-normal distributions, power-law distributions, power-law distributions with exponential cutoff, pair-wise power-law distributions, irregular heavy-tailed distributions, as well as distributions that only have a few values. The diversity of the wager distributions at the individual level suggests a diversity of individual betting strategies. Also, it indicates that a gambler may not stick to only one betting strategy. It follows that the log-normal wager distribution observed at the population level is very likely an aggregate result. Diffusive process {#diffusive-process .unnumbered} ----------------- For an individual player’s gambling sequence we define “time” $t$ as the number of bets one player has placed so far, and define as net income the sum of the payoffs of those bets. In all the games we analyze, there are only two possible outcomes: a win or a loss. The player’s net income will change each time they place a bet in a round, with the step length to be the payoff from that bet. We can treat the change of one player’s net income as a random walk in a one-dimensional space (see Fig. 1 in Ref. [@Wang2018] for an example of such a trajectory). The time $t$ will increase by $1$ when the player places a new bet, therefore the process is a discrete-time random walk. Now, let us focus on the analysis of the diffusive process of the gamblers’ net incomes, starting with the analysis of the change of the mean net income with the number of rounds played (time), $\left\langle \Delta w(t) \right\rangle = \left\langle w(t) - w_0 \right\rangle = \left\langle \sum\limits_{i=1}^t o_p(i) \right\rangle,$ where $w_0$ is the player’s initial wealth, $w(t)$ is the player’s wealth after attending $t$ rounds, and $o_p(i)$ is the payoff from the $i_{th}$ round the player attended. $\langle \cdot \rangle$ represents an ensemble average over a population of players placing bets. In the rest of this paper, $\langle \cdot \rangle$ will always be used for representing an ensemble average. In Fig. \[fig:netincome\] we show the change of $\left\langle \Delta w(t) \right\rangle$ over time. In most of the datasets, players’ mean net income decreases over time, which suggests that in general players will lose more as they gamble more. At the same time, in some datasets such as Ethcrash (D) and Coinroll (F), large fluctuations can be observed. ![ \[fig:netincome\] Change of the mean net income with time for the different datasets. Most of the datasets present a decreasing net income as time $t$ increases. Each point is obtained from an average over at least $200$ players.](figure6.pdf){width="\columnwidth"} An useful tool for studying the diffusive process is the ensemble-averaged mean-squared displacement (MSD), defined as $$\left\langle\Delta w^2(t) \right\rangle = \left\langle\left(w(t)-w_0\right)^2\right\rangle = \left\langle \left(\sum\limits_{i=1}^t o_p(i)\right)^2 \right\rangle,$$ For a normal diffusive process, $\left\langle\Delta w^2(t) \right\rangle \sim t$, otherwise an anomalous diffusive behavior prevails. More specifically, when the MSD growth is faster (respectively, slower) than linear, superdiffusion (respectively, subdiffusion) is observed. ![ \[fig:msd\] The growth of ensemble-averaged mean-squared displacement in different datasets presents different diffusive behaviors. In the figures, the error bars represent $95\%$ confidence intervals, blue dashed lines follow linear functions (slope $=1$), and green dotted lines follow quadratic functions (slope $=2$).](figure7.pdf){width="\columnwidth"} In Fig. \[fig:msd\], we present the growth of the ensemble-averaged MSD against time for each of the datasets. To reduce the coarseness, MSD curves are smoothed with log-binning technique. The error bars in Fig. \[fig:msd\] represent $95\%$ confidence intervals computed with bootstrapping using $2000$ independent re-sampling runs. It is interesting to see that for different datasets we observe different diffusive behaviors. For games csgofast-Crash (C) we observe that the MSD grows faster than a linear function, suggesting superdiffusive behavior. Meanwhile, for games csgofast-Double (A), ethCrash (D), csgospeed (G), and csgofast-Jackpot (H), the MSD first presents a superdiffusive regime, followed by a crossover to a normal diffusive regime. For games csgofast-X50 (B) and Coinroll (F), although the ensemble-averaged MSD roughly presents a linear / sublinear growth, a careful inspection shows that both curves consist of several convex-shaped regimes, indicating a more complex behavior. Convex-shaped regimes can also be observed in csgofast-Crash games (C). In Ref. [@Wang2018] we argued that the crossover from a superdiffusive regime to a normal diffusive regime in a parimutuel game is due to the limitation of individuals’ wealth and the conservation of total wealth. Similar crossovers are observed in games (G) and (H), two parimutuel betting games, where the same explanation can be applied. On the other hand, this crossover is also found in a Roulette game and in a Crash game, where there is no interaction among gamblers. The limitation of an individual’s wealth can still be a partial explanation, but the conservation of total wealth no longer holds. A different explanation needs to be proposed to model this crossover. In the following we briefly discuss how we can obtain from gambling models the different diffusive processes observed in the data. We will not attempt to reproduce the parameters we obtained from the gambling logs, but rather try to explore the possible reasons for the anomalous diffusion we reported. For a gambling process, if the gambler’s behavior is independent among different rounds, i.e., the wager and odds are respectively independent and identically distributed (IID), with no influence from the previous outcomes, and if the wager $b$ has finite variance and the odds $m$ has finite mean, then MSD’s growth will be a linear function of time $t$: $$\label{eq:linear-msd} \displaystyle \left\langle \Delta w^2(t) \right\rangle = \left\langle\left(w(t)-w_0\right)^2\right\rangle =(\langle m\rangle-1)\left\langle b^2\right\rangle t,$$ where $\langle m\rangle$ is the mean value of odds distribution and $\left\langle b^2\right\rangle$ is the second moment of the wager distribution. But normal diffusion is only found in few datasets, the remaining datasets presenting anomalous diffusion which conflicts with the IID assumption. Having shown the popularity of betting systems among gamblers, we would like to check how different betting systems affect diffusive behaviors. First, we simulate gamblers that follow Martingale strategies in a Crash game. We assume that the selection of odds follows a power-law distribution with an exponent $\alpha$, with a minimum odds of $1$ and a maximum odds of $50$, where the maximum odds is set to ensure a finite mean of the odds distribution. Starting from a minimum bet of $1$, we multiply wagers by a ratio $\gamma$ each time the gamblers lose one round and return to the minimum bet each time they win. Once the wager reaches a preset maximum bet value $10000$, we reset the gambler with a minimum bet. MSD obtained from 10 billion individual simulations is shown in Fig. \[fig:martingale\]. Different curves correspond to different exponents in odds distribution. We can see that the MSD initially presents an exponential-like growth, before the growths reduce to a linear function. It is easy to explain the exponential growth since many gamblers lose the rounds and therefore increase their wager by the factor $\gamma$, which leads to an increase in the average bet value. The superdiffusion here suggests that Martingale strategy increases gamblers’ risks of huge losses. Considering the wide adoption of Martingale among gamblers, this could be a reason for the superdiffusion as well as the crossover to normal diffusion we found in several datasets. Comparison of the MSD curves of different $\alpha$ suggests that a more aggressive risk attitude leads to a higher risk of huge losses (as well as higher potential winnings). ![ \[fig:martingale\] A betting system similar to Martingale will lead to a crossover from superdiffusion to normal diffusion according to the growth of mean-squared displacement. Comparison between curves of different parameters shows that higher $\gamma$ and lower $\alpha$ both will lead to a higher chance of huge losses/winnings. ](figure8.pdf){width=".75\columnwidth"} Next we examine the ergodicity of the random walk process of net income by computing the time-averaged mean-squared displacement and the ergodicity breaking parameter. The time-averaged MSD is defined as $$\displaystyle \overline{\delta^2}(t) = \frac{1}{T-t} \sum\limits_{k=1}^{T-t} \left(w(k + t) - w(k)\right)^2,$$ where $T$ is the length of the player’s betting history, i.e. total number of rounds they attend, and $\overline{\cdots}$ is used for representing a time average. To calculate the time-averaged MSD, we need to make sure the player has played enough rounds so that we have a long enough series of net income data, therefore in each dataset we filter out the players who played less than $T=1000$ rounds. As shown in Fig. \[fig:msd\_t\] the time-averaged MSD shows huge deviations from player to player, suggesting diverse betting behaviors at the individual level. At the same time, comparison between the ensemble-averaged time-averaged MSD $\left\langle \overline{\delta^2}(t) \right\rangle$ and the ensemble-averaged MSD $\left\langle\Delta w^2(t)\right\rangle$ shows clear deviations in most datasets, except in the Coinroll (F), csgospeed (G) and csgofast-Jackpot (H) games. To further examine breaking of ergodicity, we have calculated the ergodicity breaking parameter $\rm{EB}$ [@Cherstvy2013; @Cherstvy2014; @Cherstvy2019] defined as $$\displaystyle \rm{EB}(t) = \left\langle \left(\overline{\delta^2}(t)\right)^2\right\rangle \bigg/ \left\langle\overline{\delta^2}(t)\right\rangle^2 - 1.$$ For an ergodic process, the parameter $\rm{EB}$ should be close to $0$. However, as shown in Fig. \[fig:eb\], in most datasets, with the exception of csgospeed (G) and csgofast-Jackpot (H), $\rm{EB}$ is large. It follows that non-ergodicity is observed in most games and that gambling processes indeed often deviate from normal diffusion, which further highlights the complexity of human gambling behavior. ![ \[fig:msd\_t\] The growth of the time-averaged MSD for individual gamblers, presented as thin lines, suggests diverse betting behaviors at the individual level. The comparison between $\left\langle \overline{\delta^2}(t) \right\rangle$ (thick dashed black lines) and $\left\langle\left( \Delta w(t) \right)^2 \right\rangle$ (thick full red lines) reveals that these quantities are different for most games, with the exception of the Coinroll (F), csgospeed (G) and csgofast-Jackpot (H) games. Players who played less than $1000$ rounds are filtered out in each dataset. ](figure9.pdf){width=".75\columnwidth"} ![ \[fig:eb\] The change of the ergodicity breaking parameter with time. For all games, with the exception of the games csgospeed (G) and csgofast-Jackpot (H), ${\rm{EB}}$ is found to be much larger than $0$, suggesting non-ergodic behavior.](figure10.pdf){width=".75\columnwidth"} Another way to examine the diffusive behavior of a process is through the analysis of the first-passage time distribution. The first-passage time $t_{FP}$ is the time required for a random walker at location $w$ to leave the region $[w-V_{FP},w +V_{FP}]$ for the first time, where $V_{FP}$ is the target value or first-passage value. The first-passage time distribution $P(t_{FP})$ [@Inoue2007; @Wang2018], defined as the survival probability that the random walker, who is located at $w$ at time $t_0$, stays within range $[w-V_{FP},w +V_{FP}]$ up to time $t = t_0 + t_{FP}$, can be calculated from the expression $$P(t_{FP}) = \lim\limits_{T \longrightarrow \infty} \frac{1}{T} \sum\limits_{k=1}^T \Theta \left( \left| w(k+t_{FP}) - w(k) \right| - V_{FP} \right) - \lim\limits_{T \longrightarrow \infty} \frac{1}{T} \sum\limits_{k=1}^T \Theta \left( \left| w(k+t_{FP}-1) - w(k) \right| - V_{FP} \right)~,$$ where $\Theta(\cdot)$ is the Heaviside step function. We use $V_{FP} = 200$ (US cents), with the exception of csgofast-Jackpot (H) for which $V_{FP}$ is chosen to be 5000. For a normal diffusive process, the tail of $P(t_{FP})$ should decay with an exponent of $3/2$. In Fig. \[fig:fpt\] we plot the first-passage time distribution for each dataset, where again diverse diffusive behaviors are observed. In the games csgofast-Double (A) and csgofast-Jackpot (H), the tails of $P(t_{FP})$ approximately decay with an exponent of $3/2$ (see the thin green lines), indicating normal diffusive processes. For the game csgospeed (G), the exponent is found to be larger than $3/2$, indicating a superdiffusive process. And in games csgofast-X50 (B), csgofast-Crash (C), ethCrash (D), and Coinroll (F), the exponents are clearly smaller than $3/2$, indicating a subdiffusive behavior. We note that the results obtained from ensemble-averaged MSD sometimes differ from the results obtained from the first-passage time distributions. Nonetheless, anomalous diffusive behavior is widely observed. ![ \[fig:fpt\] The tails of first-passage time distributions for the different datasets indicate different diffusive behaviors. The green lines represent a power-law decay with an exponent $3/2$. The blue error bars indicate $95\%$ confidence intervals. Only gamblers who attended more than 1000 rounds of games have been included in these calculations. ](figure11.pdf){width=".75\columnwidth"} To confirm our conclusion about the wide existence of anomalous diffusive behavior in gambling activities, we further calculate the non-Gaussian parameter (NGP) [@Rahman1964; @Toyota2011; @Cherstvy2019] $$\displaystyle NGP(t) = \frac{\left\langle \Delta w^4(t)\right\rangle}{3\left\langle \Delta w^2(t)\right\rangle^2} - 1~.$$ For a Gaussian process, the NGP should approach $0$ when $t$ gets large. In Fig. \[fig:ngp\] we show the NGP as a function of time. In most of the games, except Coinroll (F), NGP shows a clear decreasing trend as $t$ increases. In the game Coinroll (F), a decrease is not apparent, and most likely this game does not follow a Gaussian process. In the other games, although the NGP is still decreasing, we can not discriminate whether for large $t$ this quantity will tend to $0$ or instead reach a plateau value larger than zero. For example, for the game csgospeed (G) the NGP seems to reach a plateau $NGP(t)\approx 1.5$ instead of continuing to decrease, but this could also be the consequence of insufficient data. Still, our analysis does not provide clear evidence for the presence of Gaussianity in gambling behaviors. ![ \[fig:ngp\] In most datasets, except Coinroll (F), the non-Gaussian parameter shows a decreasing trend as $t$ increases. However, in none of the studied cases does the non-Gaussian parameter fall below the value 1. ](figure12.pdf){width=".75\columnwidth"} To sum up our analysis of the players’ net incomes viewed as random walks, the diverse diffusive behaviors found in the datasets indicate that human gambling behavior is more complex than random betting and simple betting systems. Further studies are required in order to fully understand the observed differences. At the individual level, as has been pointed out by Meng [@Meng2018], gamblers show a huge diversity of betting strategies, and even individual gamblers constantly change their betting strategy. Differences in the fractions of gamblers playing specific betting strategies could be a reason why we see a variety of diffusive behaviors in the datasets. Discussion {#discussion .unnumbered} ========== The quick development of the video gaming industry has also resulted in an explosive growth of other online entertainment. This is especially true for online gambling that has evolved quickly into a booming industry with multi-billion levels. Every day million of bets are placed on websites all around the globe as many different gambling games are available online for gamblers. Analysing different types of gambling games (ranging from Roulette to Jackpot games), we have shown that log-normal distributions can be widely used to describe the wager distributions of online gamblers at the aggregate level. The risk attitude of online gamblers shows scaling properties too, which indicates that although most gamblers are risk-averse, they sometime will take large risks in exchange for high potential gains. Viewing the gamblers’ net income as a random walk in time (where for each gambler time is increased by one unit every time they play a game), we can analyze the mean-squared displacement of net income and related quantities like the ergodicity breaking parameter or the non-Gaussian parameter with the goal to gain an understanding of the gamblers’ betting strategies through the diffusive behaviors emerging from the datasets. For some games the mean-squared displacement and the first-passage time distribution reveal a transition from superdiffusion to normal diffusion as time increases. For all games the ergodicity breaking parameter and the non-Gaussian parameter reveal deviations from normal diffusion. All this indicates that gamblers’ behaviors are very diverse and more complex than what would be expected from simple betting systems. We speculate that one of the reasons for the observed diverse diffusive behaviors at the aggregate level can be found in the differences in the fractions of gamblers playing specific betting strategies, but more work is required to fully understand the gamblers’ complex behaviors. Methods {#methods .unnumbered} ======= Detailed rules of the different games {#detailed-rules-of-the-different-games .unnumbered} ------------------------------------- ### Roulette {#roulette .unnumbered} We focus on a simplified version of Roulette games that appears in online casinos, where a wheel with multiple slots painted with different colors will be spun, after which a winning slot will be selected. The Roulette table of a traditional Roulette game is composed of $38$ slots, among which $18$ slots are painted in black, $18$ slots are painted in red, and two slots (“0” and “00”) are painted in green. The online Roulette games are similar to the traditional ones, except that the number of colors and the number of slots for each color might be different. Each slot has the same probability to be chosen as the winning slot. Players will guess the color of the winning slot before the game starts. The players have a certain time for wagering, after which the game ends and a winning slot is selected by the website. Those players who successfully wagered on the correct color win, the others lose. As the chance of winning and odds for each color are directly provided by the website, roulette is a fixed-odds betting game. ### Crash {#crash .unnumbered} “Crash” describes a type of gambling games mainly hosted in online casinos. Before the game starts, the site will generate a crash point $m_C$, which is initially hidden to the players. With a lower boundary of $1$, the crash point is distributed approximately in an inverse square law. The players need to place their wager in order to enter one round. After the game starts, on the player’s user interface a number, called multiplier, will show up and gradually increase from $1$ to the predetermined crash point $m_C$, after which the game ends. During this process, if the player “cash-outs” at a certain multiplier $m$, before the game ends, they win the round; otherwise they lose. This multiplier $m$ they cashed out at is the odds, which means when winning, the player will receive a prize that equals his wager multiplied by $m$. When $m_C$ is generated with a strict inverse-square-law distribution, the winning chance exactly equals the inverse of the player-selected odds $m$. The player can also set up the cash-out multipliers automatically before the game starts, to avoid the possible time delay of manual cash-out. Since in a manual cash-out scenario, after the game starts, the multiplier will show up on the screen, at a given moment the decision of the cash-out multiplier is based on the player’s satisfaction with the current multiplier, and involves more complicated dynamics of decision-making processes. Meanwhile, in an auto cash-out scenario, the multiplier $m$ is chosen before the game starts, which means the decision making is more “static.” Crash is also a fixed-odds betting game where the odds are player-selected. ### Satoshi Dice {#satoshi-dice .unnumbered} Satoshi Dice is one of the most popular games in crptocurrency gambling. In 2013, the transactions resulting from playing Satoshi Dice games accounted for about $60\%$ of overall Bitcoin transactions [@Meiklejohn2013]. When playing Satoshi Dice, the player needs to pick a number $A$ within a range $(0, U)$ provided by the website. The odds can be calculated with the expression $m=U/A$. Once the player finishes wagering, the website will pick another number $B$ which is uniformly distributed on $(0, U)$. If $B$ is less than $A$, then the player wins the round, otherwise they lose. Satoshi Dice is a fixed-odds betting game. In some online casinos, players cannot choose $A$ arbitrarily, but instead, they have to select $A$ from a preset list provided by the gambling website. Since the odds $m$ is determined from $A$, we are more interested in the case where the players can choose $A$ arbitrarily, from which we can obtain a more detailed distribution of the odds $m$, which helps us to understand the players’ risk attitude. According to the rules of Satoshi Dice games, the maximum allowed bet is proportional to the inverse of $A$, which means the accepted range of wager is directly related to the odds. ### Jackpot {#jackpot .unnumbered} Unlike the games discussed above, Jackpot is a parimutuel betting game, where players gamble against each other. During the game, each player attending the same round will deposit their wager to a pool. The game-ending condition varies across different websites, it could be a certain pool size, a certain amount of players, or a preset time span. When the game ending condition is reached, each player’s winning chance will be determined by the fraction of their wager in the wager pool, based on which one player will be chosen as the winner by the website. The winner will obtain the whole wager pool as the prize, after excluding the site cut. The odds can be calculated by the pool size divided by the player’s wager, but it is unknown to the players at the moment they wager. In the previous study [@Wang2018], we have already discussed the player’s behavior in Jackpot games of skin gambling where in-game skins are directly used as wagers. In this paper, we extend the analysis to a case where wagers can be arbitrary amounts of virtual skin tickets (players need to first exchange in-game skins into virtual skin tickets). Data summary {#data-summary .unnumbered} ------------ For each type of game, we collect two datasets. In total, we analyze 8 datasets collected from 4 different online gambling websites, and the number of bet logs contained in each dataset ranges from $0.3$ million to $19.2$ million. Due to the high variation of market prices of crypto-currencies and in-game skins, the wager and deposits are first converted into US cents based on their daily market prices. ### CSGOFAST {#csgofast .unnumbered} From the skin gambling website CSGOFAST [@csgofast] we collected four datasets on the Roulette, Crash and Jackpot games ([*csgofast-Double*]{}, [*csgofast-X50*]{}, [*csgofast-Crash*]{}, [*csgofast-Jackpot*]{}) it provides. [*csgofast-Double*]{} (A) is a Roulette game in which players can bet on 3 different colors (Red, Black, Green), which respectively provide odds of (2, 2, 14). The data were collected in two different time periods, and the only difference between them is a change of the maximum allowed bet values. [*csgofast-X50*]{} (B) is also a Roulette game in which players can bet on 4 different colors (Blue, Red, Green, Gold) with odds (2, 3, 5, 50). [*csgofast-Crash*]{} (C) is a Crash game. As we mentioned earlier, when analyzing the risk attitude of gamblers in Crash game, we are more interested in how players set up the odds (multiplier) with the automatically cash-out option. On CSGOFAST, under the automatically cash-out option, players can only setup odds ranging from $1.10$ to $50$. The interesting point about this dataset is that even if the player loses the round, if they used the automatically cash-out option, it still displays the player-selected odds (which is set before the game starts); meanwhile if they used the manually cash-out option, no odds is displayed. Therefore in early-crashed games ($m_C < 1.10$), all the displayed odds that are larger than $1.10$ were placed with automatically cash-out option. These displayed odds will be used in odds distribution analysis. The data are also collected in two different periods, where the only difference is still a change of the maximum allowed bet value. Roulette and Crash games on CSGOFAST all use virtual skin tickets for wagering. [*csgofast-Jackpot*]{} (H) is a Jackpot game, where in-game skins are directly placed as wagers. Each skin has a market value that ranges from $3$ to $180 000$ US cents. A player can place at most $10$ skins in one round. ### CSGOSpeed {#csgospeed .unnumbered} From the skin gambling website CSGOSpeed [@csgospeed] we collected one dataset from its Jackpot game [*csgospeed-Jackpot*]{} (G), in which arbitrary amounts of virtual skin tickets can be used as wagers. The difference between datasets (H) and (G) focuses on whether the wagers are in-game skins or virtual skin tickets. ### ethCrash {#ethcrash .unnumbered} ethCrash [@ethcrash] is a cryto-currency gambling website providing a Crash game [*ethCrash*]{} (D). Players need to place wagers in Ethereum (ETH), one type of crypto-currency. ### SatoshiDice {#satoshidice .unnumbered} SatoshiDice [@satoshidice] is a cryto-currency gambling website which accepts Bitcoin Cash (BCH) as wagers. It provides a Satoshi Dice game [*satoshidice*]{} (E), where only 11 preset odds can be wagered on, ranging from $1.05$ to $1013.74$. Among the preset odds, we find that more than $30\%$ of the bets are placed under the odds $1.98$, and we will analyze those bets for wager distribution. ### Coinroll {#coinroll .unnumbered} Coinroll [@coinroll] is a cryto-currency gambling website which accepts Bitcoin (BTC) as wagers. It provides a Satoshi Dice game [*Coinroll*]{} (F), where players can either wager on the 8 preset odds listed by the website, or choose an odds of their own. When further analyzing the data, we find that a few players placed an unusual large amount of bets, where the top player placed more than 11 million bets. Although these large number of bets prove the heavy-tailed distribution of the number of bets of individuals, we have doubts that these players are playing for the purpose of gambling. As we have pointed out, all the games discussed in this paper have negative expected payoffs. Indeed, prior studies have raised suspicion about the use of crypo-currency gambling websites as a way for money laundering [@fiedler2013]. We will therefore exclude from our analysis gamblers who placed more than half a million bets. For bets wagered on the preset odds, we find that more than $57\%$ are placed under the odds $1.98$, and we use these bets to analyze the wager distribution. On the other hand, since player-selected odds show a broader spectrum regarding the risk attitude of gamblers, we focus on the odds distribution of the player-selected odds. As already mentioned, we will exclude the bets from those players who placed at least half a million bets from our odds distribution analysis. Although crypto-currency has gained decent popularity in the financial and technological world, in this paper we still measure the wager/wealth deposited in forms of crypto-currencies in US dollars, since the wagers in skin gambling are measured in US dollars. The historical daily price data of crypto-currencies (Bitcion, Ethereum, Bitcoin Cash) are obtained from CoinDesk [@coindesk] (for Bitcoin) and CoinMetrics [@coinmetrics] (for Ethereum and Bitcoin Cash). Ethics for data analysis {#ethics-for-data-analysis .unnumbered} ------------------------ The data collected and analyzed in this paper are all publicly accessible on the internet, and we collect the data either with the consent of the website administrators or without violating the terms of service or acceptance usage listed on the hosting website. The data we use do not include any personally identifiable information (PII), and we further anonymize account-related information before storing them into our databases to preserve players’ privacy. In addition, our data collection and analysis procedures are performed solely passively, with absolutely no interaction with any human subject. To avoid abusing the hosting websites (i.e., the gambling websites), the request rates of data-collecting are limited to $1$ request per second. Considering the legal concerns and potential negative effects of online gambling [@Martinelli2017; @Millar2018; @Kairouz2012; @Gonzalez2017; @Gainsbury2015; @Banks2017; @Redondo2015; @Macey2018], our analysis aims only to help better prevent adolescent gambling and problem gambling. Parameter estimation and model selection {#parameter-estimation-and-model-selection .unnumbered} ---------------------------------------- In our analysis, the parameters of different distribution models are obtained by applying Maximum Likelihood Estimation (MLE) [@Bauke2007]. To select the best-fit distribution, we compare the models’ Akaike weights [@Burnham2002] derived from Akaike Information Criterion (AIC). Note that analyzing the fitting results, we constantly found that players show a tendency of using simple numbers when allowed to place wagers with arbitrary amounts of virtual currency. As a result, the curves of probability distribution functions appear to peak at simple numbers, and the corresponding cumulative distribution function shows a stepped behavior. This makes the fitting more difficult, especially for the determination of the start of the tail. To address this issue, we choose the start of the tail $x_{min}$ such that we obtain a small KolmogorovSmirnov (KS) distance between the empirical distribution and the fitting distribution, while maintaining a good absolute fit between the complementary cumulative distribution functions (CCDF) of the empirical distribution and the best-fitted distribution. Candidate models for model selection in this paper include exponential distribution, power-law distribution, log-normal distribution, power-law distribution with sharp truncation, power-law distribution with exponential cutoff, and pairwise power-law distribution. More details about parameter fitting and model selection can be found in the article by Clauset et al. article [@Clauset2009] as well as in the previous paper by the authors [@Wang2018]. Data availability {#data-availability .unnumbered} ================= The datasets generated and/or analysed during the current study are available from the authors on reasonable request. 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Author contributions statement {#author-contributions-statement .unnumbered} ============================== X.W. and M.P. conceived the study, X.W. wrote the computer codes and conducted the data analysis, X.W. and M.P. discussed the results and wrote the manuscript. Additional information {#additional-information .unnumbered} ====================== **Competing Interests**\ The authors declare no competing interests.

--- abstract: 'The number $R(4,3,3)$ is often presented as the unknown Ramsey number with the best chances of being found “soon”. Yet, its precise value has remained unknown for almost 50 years. This paper presents a methodology based on *abstraction* and *symmetry breaking* that applies to solve hard graph edge-coloring problems. The utility of this methodology is demonstrated by using it to compute the value $R(4,3,3)=30$. Along the way it is required to first compute the previously unknown set ${{\cal R}}(3,3,3;13)$ consisting of 78[,]{}892 Ramsey colorings.' author: - Michael Codish - Michael Frank - Avraham Itzhakov - Alice Miller title: 'Computing the Ramsey Number R(4,3,3) using Abstraction and Symmetry breaking[^1]' --- Introduction {#sec:intro} ============ This paper introduces a general methodology that applies to solve graph edge-coloring problems and demonstrates its application in the search for Ramsey numbers. These are notoriously hard graph coloring problems that involve assigning colors to the edges of a complete graph. An $(r_1,\ldots,r_k;n)$ Ramsey coloring is a graph coloring in $k$ colors of the complete graph $K_n$ that does not contain a monochromatic complete sub-graph $K_{r_i}$ in color $i$ for each $1\leq i\leq k$. The set of all such colorings is denoted ${{\cal R}}(r_1,\ldots,r_k;n)$. The Ramsey number $R(r_1,\ldots,r_k)$ is the least $n>0$ such that no $(r_1,\ldots,r_k;n)$ coloring exists. In particular, the number $R(4,3,3)$ is often presented as the unknown Ramsey number with the best chances of being found “soon”. Yet, its precise value has remained unknown for more than 50 years. It is currently known that $30\leq R(4,3,3)\leq 31$. Kalbfleisch [@kalb66] proved in 1966 that $R(4,3,3)\geq 30$, Piwakowski [@Piwakowski97] proved in 1997 that $R(4,3,3)\leq 32$, and one year later Piwakowski and Radziszowski [@PR98] proved that $R(4,3,3)\leq 31$. We demonstrate how our methodology applies to computationally prove that $R(4,3,3)=30$. Our strategy to compute $R(4,3,3)$ is based on the search for a $(4,3,3;30)$ Ramsey coloring. If one exists, then because $R(4,3,3)\leq 31$, it follows that $R(4,3,3) = 31$. Otherwise, because $R(4,3,3)\geq 30$, it follows that $R(4,3,3) = 30$. In recent years, Boolean SAT solving techniques have improved dramatically. Today’s SAT solvers are considerably faster and able to manage larger instances than were previously possible. Moreover, encoding and modeling techniques are better understood and increasingly innovative. SAT is currently applied to solve a wide variety of hard and practical combinatorial problems, often outperforming dedicated algorithms. The general idea is to encode a (typically, NP) hard problem instance, $\mu$, to a Boolean formula, $\varphi_\mu$, such that the satisfying assignments of $\varphi_\mu$ correspond to the solutions of $\mu$. Given such an encoding, a SAT solver can be applied to solve $\mu$. Our methodology in this paper combines SAT solving with two additional concepts: *abstraction* and *symmetry breaking*. The paper is structured to let the application drive the presentation of the methodology in three steps. Section \[sec:prelim\] presents: preliminaries on graph coloring problems, some general notation on graphs, and a simple constraint model for Ramsey coloring problems. Section \[sec:embed\] presents the first step in our quest to compute $R(4,3,3)$. We introduce a basic SAT encoding and detail how a SAT solver is applied to search for Ramsey colorings. Then we describe and apply a well known embedding technique, which allows to determine a set of partial solutions in the search for a $(4,3,3;30)$ Ramsey coloring such that if a coloring exists then it is an extension of one of these partial solutions. This may be viewed as a preprocessing step for a SAT solver which then starts from a partial solution. Applying this technique we conclude that if a $(4,3,3;30)$ Ramsey coloring exists then it must be ${\langle 13,8,8 \rangle}$ regular. Namely, each vertex in the coloring must have 13 edges in the first color, and 8 edges in each of the other two colors. This result is already considered significant progress in the research on Ramsey numbers as stated in [@XuRad2015]. To further apply this technique to determine if there exists a ${\langle 13,8,8 \rangle}$ regular $(4,3,3;30)$ Ramsey coloring requires to first compute the currently unknown set ${{\cal R}}(3,3,3;13)$. Sections \[sec:symBreak\]—\[sec:33313b\] present the second step: computing ${{\cal R}}(3,3,3;13)$. Section \[sec:symBreak\] illustrates how a straightforward approach, combining SAT solving with *symmetry breaking*, works for smaller instances but not for ${{\cal R}}(3,3,3;13)$. Then Section \[sec:abs\] introduces an *abstraction*, called degree matrices, Section \[sec:33313\] demonstrates how to compute degree matrices for ${{\cal R}}(3,3,3;13)$, and Section \[sec:33313b\] shows how to use the degree matrices to compute ${{\cal R}}(3,3,3;13)$. Section \[sec:433\_30\] presents the third step re-examining the embedding technique described in Section \[sec:embed\] which given the set ${{\cal R}}(3,3,3;13)$ applies to prove that there does not exist any $(4,3,3;30)$ Ramsey coloring which is also ${\langle 13,8,8 \rangle}$ regular. Section \[sec:conclude\] presents a conclusion. Preliminaries and Notation {#sec:prelim} ========================== In this paper, graphs are always simple, i.e. undirected and with no self loops. For a natural number $n$ let $[n]$ denote the set $\{1,2,\ldots,n\}$. A graph coloring, in $k$ colors, is a pair $(G,\kappa)$ consisting of a simple graph $G=(V,E)$ and a mapping $\kappa\colon E\to[k]$. When $G$ is clear from the context we refer to $\kappa$ as the graph coloring. We typically represent $G=([n],E)$ as a (symmetric) $n\times n$ adjacency matrix, $A$, defined such that $$A_{i,j}= \begin{cases} \kappa((i,j)) & \mbox{if } (i,j) \in E\\ 0 & \mbox{otherwise} \end{cases}$$ Given a graph coloring $(G,\kappa)$ in $k$ colors with $G=(V,E)$, the set of neighbors of a vertex $u\in V$ in color $c\in [k]$ is $N_c(u) = {\left\{~v \left| \begin{array}{l}(u,v)\in E, \kappa((u,v))=c\end{array} \right. \right\}} $ and the color-$c$ degree of $u$ is $deg_{c}(u) = |N_c(u)|$. The color degree tuple of $u$ is the $k$-tuple $deg(u)={\langle deg_{1}(u),\ldots,deg_{k}(u) \rangle}$. The sub-graph of $G$ on the $c$ colored neighbors of $x\in V$ is the projection of $G$ to vertices in $N_c(x)$ defined by $G^c_x = (N_c(x),{\left\{~(u,v)\in E \left| \begin{array}{l}u,v\in N_c(x)\end{array} \right. \right\}})$. For example, take as $G$ the graph coloring depicted by the adjacency matrix in Figure \[embed\_12\_8\_8\] with $u$ the vertex corresponding to the first row in the matrix. Then, $N_1(u) = \{2,3,4,5,6,7,8,9,10,11,12,13\}$, $N_2(u) = \{14,15,16,17,18,19,20,21\}$, and $N_3(u)=\{22,23,24,25,26,27,28,29\}$. The subgraphs $G^1_u$, $G^2_u$, and $G^3_u$ are highlighted by the boldface text in Figure \[embed\_12\_8\_8\]. An $(r_1,\ldots,r_k;n)$ Ramsey coloring is a graph coloring in $k$ colors of the complete graph $K_n$ that does not contain a monochromatic complete sub-graph $K_{r_i}$ in color $i$ for each $1\leq i\leq k$. The set of all such colorings is denoted ${{\cal R}}(r_1,\ldots,r_k;n)$. The Ramsey number $R(r_1,\ldots,r_k)$ is the least $n>0$ such that no $(r_1,\ldots,r_k;n)$ coloring exists. In the multicolor case ($k>2$), the only known value of a nontrivial Ramsey number is $R(3,3,3)=17$. Prior to this paper, it was known that $30\leq R(4,3,3)\leq 31$. Moreover, while the sets of $(3,3,3;n)$ colorings were known for $14\leq n\leq 16$, the set of colorings for $n=13$ was never published.[^2] More information on recent results concerning Ramsey numbers can be found in the electronic dynamic survey by Radziszowski [@Rad]. $$\begin{aligned} \varphi_{adj}^{n,k}(A) &=& \hspace{-2mm}\bigwedge_{1\leq q<r\leq n} \left(\begin{array}{l} 1\leq A_{q,r}\leq k ~~\land~~ A_{q,r} = A_{r,q} ~~\land ~~ A_{q,q} = 0 \end{array}\right) \label{constraint:simple} \\ \varphi_{r}^{n,c}(A) &=& \bigwedge_{I\in \wp_r([n])} \bigvee {\left\{~A_{i,j}\neq c \left| \begin{array}{l}i,j \in I, i<j\end{array} \right. \right\}} \label{constraint:nok}\end{aligned}$$ $$\begin{aligned} \small \label{constraint:coloring} \varphi_{(r_1,\ldots,r_k;n)}(A) & = & \varphi_{adj}^{n,k}(A) \land \hspace{-2mm} \bigwedge_{1\leq c\leq k} \hspace{-1mm} \varphi_{r_c}^{n,c}(A)\end{aligned}$$ A graph coloring problem on $k$ colors is about the search for a graph coloring which satisfies a given set of constraints. Formally, it is specified as a formula, $\varphi(A)$, where $A$ is an $n\times n$ adjacency matrix of integer variables with domain $\{0\}\cup [k]$ and $\varphi$ is a constraint on these variables. A solution is an assignment of integer values to the variables in $A$ which satisfies $\varphi$ and determines both the graph edges and their colors. We often refer to a solution as an integer adjacency matrix and denote the set of solutions as $sol(\varphi(A))$. Figure \[fig:gcp\] presents the $k$-color graph coloring problems we focus on in this paper: $(r_1,\ldots,r_k;n)$ Ramsey colorings. Constraint (\[constraint:simple\]), $\varphi_{adj}^{n,k}(A)$, states that the graph represented by matrix $A$ has $n$ vertices, is $k$ colored, and is simple. Constraint (\[constraint:nok\]) $\varphi_{r}^{n,c}(A)$ states that the $n\times n$ matrix $A$ has no embedded sub-graph $K_r$ in color $c$. Each conjunct, one for each set $I$ of $r$ vertices, is a disjunction stating that one of the edges between vertices of $I$ is not colored $c$. Notation: $\wp_r(S)$ denotes the set of all subsets of size $r$ of the set $S$. Constraint (\[constraint:coloring\]) states that $A$ is a $(r_1,\ldots,r_k;n)$ Ramsey coloring. For graph coloring problems, solutions are typically closed under permutations of vertices and of colors. Restricting the search space for a solution modulo such permutations is crucial when trying to solve hard graph coloring problems. It is standard practice to formalize this in terms of graph (coloring) isomorphism. Let $G=(V,E)$ be a graph (coloring) with $V=[n]$ and let $\pi$ be a permutation on $[n]$. Then $\pi(G) = (V,{\left\{~ (\pi(x),\pi(y)) \left| \begin{array}{l} (x,y) \in E\end{array} \right. \right\}})$. Permutations act on adjacency matrices in the natural way: If $A$ is the adjacency matrix of a graph $G$, then $\pi(A)$ is the adjacency matrix of $\pi(G)$ and $\pi(A)$ is obtained by simultaneously permuting with $\pi$ both rows and columns of $A$. \[def:weak\_iso\] Let $(G,{\kappa_1})$ and $(H,{\kappa_2})$ be $k$-color graph colorings with $G=([n],E_1)$ and $H=([n],E_2)$. We say that $(G,{\kappa_1})$ and $(H,{\kappa_2})$ are weakly isomorphic, denoted $(G,{\kappa_1})\approx(H,{\kappa_2})$ if there exist permutations $\pi \colon [n] \to [n]$ and $\sigma \colon [k] \to [k]$ such that $(u,v) \in E_1 \iff (\pi(u),\pi(v)) \in E_2$ and $\kappa_1((u,v)) = \sigma(\kappa_2((\pi(u), \pi(v))))$. We denote such a weak isomorphism: $(G,{\kappa_1})\approx_{\pi,\sigma}(H,{\kappa_2})$. When $\sigma$ is the identity permutation, we say that $(G,{\kappa_1})$ and $(H,{\kappa_2})$ are isomorphic. The following lemma emphasizes the importance of weak graph isomorphism as it relates to Ramsey numbers. Many classic coloring problems exhibit the same property. \[lemma:closed\] Let $(G,{\kappa_1})$ and $(H,{\kappa_2})$ be graph colorings in $k$ colors such that $(G,\kappa_1) \approx_{\pi,\sigma} (H,\kappa_2)$. Then, $$(G,\kappa_1) \in {{\cal R}}(r_1,r_2,\ldots,r_k;n) \iff (H,\kappa_2) \in {{\cal R}}(\sigma(r_1),\sigma(r_2),\ldots,\sigma(r_k);n)$$ We make use of the following theorem from [@PR98]. \[thm:433\] $30\leq R(4,3,3)\leq 31$ and, $R(4,3,3)=31$ if and only if there exists a $(4,3,3;30)$ coloring $\kappa$ of $K_{30}$ such that: (1) For every vertex $v$ and $i\in\{2,3\}$, $5\leq deg_{i}(v)\leq 8$, and $13\leq deg_{1}(v)\leq 16$. (2) Every edge in the third color has at least one endpoint $v$ with $deg_{3}(v)=13$. (3) There are at least 25 vertices $v$ for which $deg_{1}(v)=13$, $deg_{2}(v)=deg_{3}(v)=8$. \[cor:degrees\] Let $G=(V,E)$ be a $(4,3,3;30)$ coloring, $v\in V$ a selected vertex, and assume without loss of generality that $deg_2(v)\geq deg_3(v)$. Then, $deg(v)\in{\left\{ \begin{array}{l}{\langle 13, 8, 8 \rangle},{\langle 14, 8, 7 \rangle},{\langle 15, 7, 7 \rangle},{\langle 15, 8, 6 \rangle},{\langle 16, 7, 6 \rangle},{\langle 16, 8, 5 \rangle}\end{array} \right\}}$. Consider a vertex $v$ in a $(4,3,3;n)$ coloring and focus on the three subgraphs induced by the neighbors of $v$ in each of the three colors. The following states that these must be corresponding Ramsey colorings. \[obs:embed\] Let $G$ be a $(4,3,3;n)$ coloring and $v$ be any vertex with $deg(v)={\langle d_1,d_2,d_3 \rangle}$. Then, $d_1+d_2+d_3=n-1$ and $G^1_v$, $G^2_v$, and $G^3_v$ are respectively $(3,3,3;d_1)$, $(4,2,3;d_2)$, and $(4,3,2;d_3)$ colorings. Note that by definition a $(4,2,3;n)$ coloring is a $(4,3;n)$ Ramsey coloring in colors 1 and 3 and likewise a $(4,3,2;n)$ Ramsey coloring is a $(4,3;n)$ coloring in colors 1 and 2. This is because the “2” specifies that the coloring does not contain a subgraph $K_2$ in the corresponding color and this means that it contains no edge with that color. For $n\in\{14,15,16\}$, the sets ${{\cal R}}(3,3,3;n)$ are known and consist respectively of 115, 2, and 2 colorings. Similarly, for $n\in\{5,6,7,8\}$ the sets ${{\cal R}}(4,3;n)$ are known and consist respectively of 9, 15, 9, and 3 colorings. In this paper computations are performed using the CryptoMiniSAT [@Crypto] SAT solver. SAT encodings (CNF) are obtained using the finite-domain constraint compiler  [@jair2013]. The use of  facilitates applications to find a single (first) solution, or to find all solutions for a constraint, modulo a specified set of variables. When solving for all solutions, our implementation iterates with the SAT solver, adding so called *blocking clauses* each time another solution is found. This technique, originally due to McMillan [@McMillan2002], is simplistic but suffices for our purposes. All computations were performed on a cluster with a total of $228$ Intel E8400 cores clocked at 2 GHz each, able to run a total of $456$ parallel threads. Each of the cores in the cluster has computational power comparable to a core on a standard desktop computer. Each SAT instance is run on a single thread. Basic SAT Encoding and Embeddings {#sec:embed} ================================= Throughout the paper we apply a SAT solver to solve CNF encodings of constraints such as those presented in Figure \[fig:gcp\]. In this way it is straightforward to find a Ramsey coloring or prove its non-existence. Ours is a standard encoding to CNF. To this end: nothing new. For an $n$ vertex graph coloring problem in $k$ colors we take an $n\times n$ matrix $A$ where $A_{i,j}$ represents in $k$ bits the edge $(i,j)$ in the graph: exactly one bit is true indicating which color the edge takes, or no bit is true indicating that the edge $(i,j)$ is not in the graph. Already at the representation level, we use the same Boolean variables to represent the color in $A_{i,j}$ and in $A_{j,i}$ for each $1\leq i<j\leq n$. We further fix the variables corresponding to $A_{i,i}$ to ${\mathit{false}}$. The rest of the SAT encoding is straightforward. Constraint (\[constraint:simple\]) is encoded to CNF by introducing clauses to state that for each $A_{i,j}$ with $1\leq i<j\leq n$ at most one of the $k$ bits representing the color of the edge $(i,j)$ is true. In our setting typically $k=3$. For three colors, if $b_1,b_2,b_3$ are the bits representing the color of an edge, then three clauses suffice: $(\bar b_1\lor \bar b_2),(\bar b_1\lor \bar b_3),(\bar b_2\lor \bar b_3)$. Constraint (\[constraint:nok\]) is encoded by a single clause per set $I$ of $r$ vertices expressing that at least one of the bits corresponding to an edge between vertices in $I$ does not have color $c$. Finally Constraint (\[constraint:coloring\]) is a conjunction of constraints of the previous two forms. In Section \[sec:symBreak\] we will improve on this basic encoding by introducing symmetry breaking constraints (encoded to CNF). However, for now we note that, even with symmetry breaking constraints, using the basic encoding, a SAT solver is currently not able to solve any of the open Ramsey coloring problems such as those considered in this paper. In particular, directly applying a SAT solver to search for a $(4,3,3;30)$ Ramsey coloring is hopeless. To facilitate the search for $(4,3,3;30)$ Ramsey coloring using a SAT encoding, we apply a general approach where, when seeking a $(r_1,\ldots,r_k;n)$ Ramsey coloring one selects a “preferred” vertex, call it $v_1$, and based on its degrees in each of the $k$ colors, embeds $k$ subgraphs which are corresponding smaller colorings. Using this approach, we apply Corollary \[cor:degrees\] and Observation \[obs:embed\] to establish that a $(4,3,3;30)$ coloring, if one exists, must be ${\langle 13,8,8 \rangle}$ regular. Specifically, all vertices must have 13 edges in the first color and 8 each, in the second and third colors. This result is considered significant progress in the research on Ramsey numbers [@XuRad2015]. This “embedding” approach is often applied in the Ramsey number literature where the process of completing (or trying to complete) a partial solution (an embedding) to a Ramsey coloring is called *gluing*. See for example the presentations in [@PiwRad2001; @FKRad2004; @PR98]. \[thm:regular\] Any $(4,3,3;30)$ coloring, if one exists, is ${\langle 13,8,8 \rangle}$ regular. By computation as described in the rest of this section. [r]{}[.460]{} We seek a $(4,3,3;30)$ coloring of $K_{30}$, represented as a $30\times 30$ adjacency matrix $A$. Let $v_1$ correspond to the the first row in $A$ with $deg(v_1)={\langle d_1,d_2,d_3 \rangle}$ as prescribed by Corollary \[cor:degrees\]. For each possible triplet ${\langle d_1,d_2,d_3 \rangle}$, except ${\langle 13,8,8 \rangle}$, we take each of the known corresponding colorings for the subgraphs $G^1_{v_1}$, $G^2_{v_1}$, and $G^3_{v_1}$ and embed them into $A$. We then apply a SAT solver, to (try to) complete the remaining cells in $A$ to satisfy $\varphi_{4,3,3;30}(A)$ as defined by Constraint (\[constraint:coloring\]) of Figure \[fig:gcp\]. If the SAT solver fails, then no such completion exists. To illustrate the approach, consider the case where $deg(v_1)={\langle 14,8,7 \rangle}$. Figure \[embed\_14\_8\_7\] details one of the embeddings corresponding to this case. The first row and column of $A$ specify the colors of the edges of the 29 neighbors of $v_1$ (in bold). The symbol “$\_$” indicates an integer variable that takes a value between 1 and 3. The neighbors of $v_1$ in color 1 form a submatrix of $A$ embedded in rows (and columns) 2–15 of the matrix in the Figure. By Corollary \[obs:embed\] these are a $(3,3,3;14)$ Ramsey coloring and there are 115 possible such colorings modulo weak isomorphism. The Figure details one of them. Similarly, there are 3 possible $(4,2,3;8)$ colorings which are subgraphs for the neighbors of $v_1$ in color 2. In Figure \[embed\_14\_8\_7\], rows (and columns) 16–23 detail one such coloring. Finally, there are 9 possible $(4,3,2;7)$ colorings which are subgraphs for the neighbors of $v_1$ in color 3. In Figure \[embed\_14\_8\_7\], rows (and columns) 24–30 detail one such coloring. To summarize, Figure \[embed\_14\_8\_7\] is a partially instantiated adjacency matrix. The first row determines the degrees of $v_1$, in the three colors, and 3 corresponding subgraphs are embedded. The uninstantiated values in the matrix must be completed to obtain a solution that satisfies $\varphi_{4,3,3;30}(A)$ as specified in Constraint (\[constraint:coloring\]) of Figure \[fig:gcp\]. This can be determined using a SAT solver. For the specific example in Figure \[embed\_14\_8\_7\], the CNF generated using our tool set consists of 33[,]{}959 clauses, involves 5[,]{}318 Boolean variables, and is shown to be unsatisfiable in 52 seconds of computation time. For the case where $v_1$ has degrees ${\langle 14,8,7 \rangle}$ in the three colors this is one of $115\times 3\times 9 = 3105$ instances that need to be checked. Table \[table:regular\] summarizes the experiment which proves Theorem \[thm:regular\]. For each of the possible degrees of vertex 1 in a $(4,3,3;30)$ coloring as prescribed by Corollary \[cor:degrees\], except ${\langle 13,8,8 \rangle}$, and for each possible choice of colorings for the derived subgraphs $G^1_{v_1}$, $G^2_{v_1}$, and $G^3_{v_1}$, we apply a SAT solver to show that the instance $\varphi_{(4,3,3;30)}(A)$ of Constraint (\[constraint:coloring\]) of Figure \[fig:gcp\] cannot be satisfied. The table details for each degree triple, the number of instances, their average size (number of clauses and Boolean variables), and the average and total times to show that the constraint is not satisfiable. $v_1$ degrees \# clauses (avg.) \# vars (avg.) unsat (avg) unsat (total) --------------- ------ ------------------- ---------------- ------------- --------------- ------------- (16,8,5) 54 = 2\*3\*9 32432 5279 51 sec. 0.77 hrs. (16,7,6) 270 = 2\*9\*15 32460 5233 420 sec. 31.50 hrs. (15,8,6) 90 = 2\*3\*15 33607 5450 93 sec. 2.32 hrs. (15,7,7) 162 = 2\*9\*9 33340 5326 1554 sec. 69.94 hrs. (14,8,7) 3105 = 115\*3\*9 34069 5324 294 sec. 253.40 hrs. : Proving that any $(4,3,3;30)$ Ramsey coloring is ${\langle 13,8,8 \rangle}$ regular (summary).[]{data-label="table:regular"} All of the SAT instances described in the experiment summarized by Table \[table:regular\] are unsatisfiable. The solver reports “unsat”. To gain confidence in our implementation, we illustrate its application on a satisfiable instance: to find a, known to exist, $(4,3,3;29)$ coloring. This experiment involves some reverse engineering. [r]{}[.460]{} In 1966 Kalbfleisch [@kalb66] reported the existence of a circulant $(3,4,4;29)$ coloring. Encoding instance $\varphi_{(4,3,3;29)}(A)$ of Constraint (\[constraint:coloring\]) together with a constraint that states that the adjacency matrix $A$ is circulant, results in a CNF with 146[,]{}506 clauses and 8[,]{}394 variables. Using a SAT solver, we obtain a corresponding $(4,3,3;29)$ coloring in less than two seconds of computation time. The solution is ${\langle 12,8,8 \rangle}$ regular and isomorphic to the adjacency matrix depicted as Figure \[embed\_12\_8\_8\]. Now we apply the embedding approach. We take the partial solution (the boldface elements) corresponding to the three subgraphs: $G^1_{v_1}$, $G^2_{v_1}$ and $G^3_{v_1}$ which are respectively $(3,3,3;12)$, $(4,2,3;8)$ and $(4,3,2;8)$ Ramsey colorings. Applying a SAT solver to complete this partial solution to a $(4,3,3;29)$ coloring satisfying Constraint (\[constraint:coloring\]) involves a CNF with 30[,]{}944 clauses and 4[,]{}736 variables and requires under two hours of computation time. Figure \[embed\_12\_8\_8\] portrays the solution (the gray elements). To apply the embedding approach described in this section to determine if there exists a $(4,3,3;30)$ Ramsey coloring which is ${\langle 13,8,8 \rangle}$ regular would require access to the set ${{\cal R}}(3,3,3;13)$. We defer this discussion until after Section \[sec:33313b\] where we describe how we compute the set of all 78[,]{}892 $(3,3,3;13)$ Ramsey colorings modulo weak isomorphism. Symmetry Breaking: Computing ${{\cal R}}(r_1,\ldots,r_k;n)$ {#sec:symBreak} =========================================================== In this section we prepare the ground to apply a SAT solver to find the set of all $(r_1,\ldots,r_k;n)$ Ramsey colorings modulo weak isomorphism. The constraints are those presented in Figure \[fig:gcp\] and their encoding to CNF is as described in Section \[sec:embed\]. Our final aim is to compute the set of all $(3,3,3;13)$ colorings modulo weak isomorphism. Then we can apply the embedding technique of Section \[sec:embed\] to determine the existence of a ${\langle 13,8,8 \rangle}$ regular $(4,3,3;30)$ Ramsey coloring. Given Theorem \[thm:regular\], this will determine the value of $R(4,3,3)$. Solving hard search problems on graphs, and graph coloring problems in particular, relies heavily on breaking symmetries in the search space. When searching for a graph, the names of the vertices do not matter, and restricting the search modulo graph isomorphism is highly beneficial. When searching for a graph coloring, on top of graph isomorphism, solutions are typically closed under permutations of the colors: the names of the colors do not matter and the term often used is “weak isomorphism” [@PR98] (the equivalence relation is weaker because both node names and edge colors do not matter). When the problem is to compute the set of all solutions modulo (weak) isomorphism the task is even more challenging. Often one first attempts to compute all the solutions of the coloring problem, and to then apply one of the available graph isomorphism tools, such as `nauty` [@nauty] to select representatives of their equivalence classes modulo (weak) isomorphism. This is a *generate and test* approach. However, typically the number of solutions is so large that this approach is doomed to fail even though the number of equivalence classes itself is much smaller. The problem is that tools such as `nauty` apply after, and not during, generation. To this end, we follow [@CodishMPS14] where Codish [[*et al.*]{}]{} show that the symmetry breaking approach of [@DBLP:conf/ijcai/CodishMPS13] holds also for graph coloring problems where the adjacency matrix consists of integer variables. This is a *constrain and generate approach*. But, as symmetry breaking does not break all symmetries, it is still necessary to perform some reduction using a tool like `nauty`.[^3] This form of symmetry breaking is an important component in our methodology. **[@DBLP:conf/ijcai/CodishMPS13].** \[def:SBlexStar\] Let $A$ be an $n\times n$ adjacency matrix. Then, $$\label{eq:symbreak} {\textsf{sb}}^*_\ell(A) = \bigwedge{\left\{~A_{i}\preceq_{\{i,j\}}A_{j} \left| \begin{array}{l}i<j\end{array} \right. \right\}}$$ where $A_{i}\preceq_{\{i,j\}}A_{j}$ denotes the lexicographic order between the $i^{th}$ and $j^{th}$ rows of $A$ (viewed as strings) omitting the elements at positions $i$ and $j$ (in both rows). We omit the precise details of how Constraint (\[eq:symbreak\]) is encoded to CNF. In our implementation this is performed by the finite domain constraint compiler  and details can be found in [@jair2013]. Table \[tab:333n1\] illustrates the impact of the symmetry breaking Constraint (\[eq:symbreak\]) on the search for the Ramsey colorings required in the proof of Theorem \[thm:regular\]. The first four rows in the table portray the required instances of the forms $(4,3,2;n)$ and $(4,2,3;n)$ which by definition correspond to $(4,3;n)$ colorings (respectively in colors 1 and 3, and in colors 1 and 2). The next three rows correspond to $(3,3,3;n)$ colorings where $n\in\{14,15,16\}$. The last row illustrates our failed attempt to apply a SAT encoding to compute ${{\cal R}}(3,3,3;13)$. The first column in the table specifies the instance. The column headed by “\#${\setminus}_{\approx}$” specifies the known (except for the last row) number of colorings modulo weak isomorphism [@Rad]. The columns headed by “vars” and “clauses” indicate, the numbers of variables and clauses in the corresponding CNF encodings of the coloring problems with and without the symmetry breaking Constraint (\[eq:symbreak\]). The columns headed by “time” indicate the time (in seconds) to find all colorings iterating with a SAT solver. The timeout assumed here is 24 hours. The column headed by “\#” specifies the number of colorings found by iterated SAT solving. In the first four rows, notice the impact of symmetry breaking which reduces the number of solutions by 1–3 orders of magnitude. In the next three rows the reduction is more acute. Without symmetry breaking the colorings cannot be computed within the 24 hour timeout. The sets of colorings obtained with symmetry breaking have been verified to reduce, using `nauty` [@nauty], to the known number of colorings modulo weak isomorphism indicated in the second column. Abstraction: Degree Matrices for Graph Colorings {#sec:abs} ================================================ This section introduces an abstraction on graph colorings defined in terms of *degree matrices*. The motivation is to solve a hard graph coloring problem by first searching for its degree matrices. Degree matrices are to graph coloring problems as degree sequences [@ErdosGallai1960] are to graph search problems. A degree sequence is a monotonic nonincreasing sequence of the vertex degrees of a graph. A graphic sequence is a sequence which can be the degree sequence of some graph. The idea underlying our approach is that when the combinatorial problem at hand is too hard, then possibly solving an abstraction of the problem is easier. In this case, a solution of the abstract problem can be used to facilitate the search for a solution of the original problem. \[def:dm\] Let $A$ be a graph coloring on $n$ vertices with $k$ colors. The *degree matrix* of $A$, denoted $dm(A)$ is an $n\times k$ matrix, $M$ such that $M_{i,j} = deg_j(i)$ is the degree of vertex $i$ in color $j$. [r]{}[.33]{} Figure \[fig:dm\] illustrates the degree matrix of the graph coloring given as Figure \[embed\_12\_8\_8\]. The three columns correspond to the three colors and the 29 rows to the 29 vertices. The degree matrix consists of 29 identical rows as the corresponding graph coloring is ${\langle 12,8,8 \rangle}$ regular. A degree matrix $M$ represents the set of graphs $A$ such that $dm(A)=M$. Due to properties of weak-isomorphism (vertices as well as colors can be reordered) we can exchange both rows and columns of a degree matrix without changing the set of graphs it represents. In the rest of our construction we adopt a representation in which the rows and columns of a degree matrix are sorted lexicographically. For an $n\times k$ degree matrix $M$ we denote by $lex(M)$ the smallest matrix with rows and columns in the lexicographic order (non-increasing) obtained by permuting rows and columns of $M$. \[def:abs\] Let $A$ be a graph coloring on $n$ vertices with $k$ colors. The *abstraction* of $A$ to a degree matrix is $\alpha(A)=lex(dm(A))$. For a set ${{\cal A}}$ of graph colorings we denote $\alpha({{\cal A}}) = {\left\{~\alpha(A) \left| \begin{array}{l}A\in{{\cal A}}\end{array} \right. \right\}}$. Note that if $A$ and $A'$ are weakly isomorphic, then $\alpha(A)=\alpha(A')$. \[def:conc\] Let $M$ be an $n\times k$ degree matrix. Then, $\gamma(M) = {\left\{~A \left| \begin{array}{l}\alpha(A)=M\end{array} \right. \right\}}$ is the set of graph colorings represented by $M$. For a set ${{\cal M}}$ of degree matrices we denote $\gamma({{\cal M}}) = \cup{\left\{~\gamma(M) \left| \begin{array}{l}M\in{{\cal M}}\end{array} \right. \right\}}$. Let $\varphi(A)$ be a graph coloring problem in $k$ colors on an $n\times n$ adjacency matrix, $A$. Our strategy to compute ${{\cal A}}=sol(\varphi(A))$ is to first compute an over-approximation ${{\cal M}}$ of degree matrices such that $\gamma({{\cal M}})\supseteq{{\cal A}}$ and to then use ${{\cal M}}$ to guide the computation of ${{\cal A}}$. We denote the set of solutions of the graph coloring problem, $\varphi(A)$, which have a given degree matrix, $M$, by $sol_M(\varphi(A))$. Then $$\begin{aligned} \label{eq:approx} sol(\varphi(A)) &=& \bigcup_{M\in{{\cal M}}} sol_M(\varphi(A))\\ \label{eq:solM} sol_M(\varphi(A)) & = & sol(\varphi(A)\wedge\alpha(A){=}M)\end{aligned}$$ Equation (\[eq:approx\]) implies that, we can compute the solutions to a graph coloring problem $\varphi(A)$ by computing the independent sets $sol_M(\varphi(A))$ for any over approximation ${{\cal M}}$ of the degree matrices of the solutions of $\varphi(A)$. This facilitates the computation for two reasons: (1) The problem is now broken into a set of independent sub-problems for each $M\in{{\cal M}}$ which can be solved in parallel, and (2) The computation of each individual $sol_M(\varphi(A))$ is now directed using $M$. The constraint $\alpha(A){=}M$ in the right side of Equation (\[eq:solM\]) is encoded to SAT by introducing (encodings of) cardinality constraints. For each row of the matrix $A$ the corresponding row in $M$ specifies the number of elements with value $c$ (for $1\leq c\leq k$) that must be in that row. We omit the precise details of the encoding to CNF. In our implementation this is performed by the finite domain constraint compiler  and details can be found in [@jair2013]. When computing $sol_M(\varphi(A))$ for a given degree matrix we can no longer apply the symmetry breaking Constraint (\[eq:symbreak\]) as it might constrain the rows of $A$ in a way that contradicts the constraint $\alpha(A)=M$ in the right side of Equation (\[eq:solM\]). However, we can refine Constraint (\[eq:symbreak\], to break symmetries on the rows of $A$ only when the corresponding rows in $M$ are equal. Then $M$ can be viewed as inducing an ordered partition of $A$ and Constraint (\[eq:sbdm\]) is, in the terminology of [@DBLP:conf/ijcai/CodishMPS13], a partitioned lexicographic symmetry break. In the following, $M_i$ and $M_j$ denote the $i^{th}$ and $j^{th}$ rows of matrix $M$. $$\label{eq:sbdm} {\textsf{sb}}^*_\ell(A,M) = \bigwedge_{i<j} \left(\begin{array}{l} \big(M_i=M_j\Rightarrow A_i\preceq_{\{i,j\}} A_j\big) \end{array}\right)$$ The following refines Equation (\[eq:solM\]) introducing the symmetry breaking predicate. $$\label{eq:scenario1} sol_M(\varphi(A)) = sol(\varphi(A)\wedge (\alpha(A){=}M) \wedge{\textsf{sb}}^*_\ell(A,M))$$ To justify that Equations (\[eq:solM\]) and (\[eq:scenario1\]) both compute $sol_M(\varphi(A))$, modulo weak isomorphism, we must show that if ${\textsf{sb}}^*_\ell(A,M)$ excludes a solution then there is another weakly isomorphic solution that is not excluded. \[thm:sbl\_star\] Let $A$ be an adjacency matrix with $\alpha(A) = M$. Then, there exists $A'\approx A$ such that $\alpha(A')=M$ and ${\textsf{sb}}^{*}_\ell(A',M)$ holds. Computing Degree Matrices for $R(3,3,3;13)$ {#sec:33313} =========================================== This section describes how we compute a set of degree matrices that approximate those of the solutions of instance $\varphi_{(3,3,3;13)}(A)$ of Constraint (\[constraint:coloring\]). We apply a strategy mixing SAT solving with brute-force enumeration as follows. The computation of the degree matrices is summarized in Table \[tab:333\_computeDMs\]. In the first step, we compute bounds on the degrees of the nodes in any $R(3,3,3;13)$ coloring. \[lemma:db\] Let $A$ be an $R(3,3,3;13)$ coloring then for every vertex $x$ in $A$, and color $c\in\{1,2,3\}$, $2\leq deg_{c}(x)\leq 5$. By solving instance $\varphi_{(3,3,3;13)}(A)$ of Constraint (\[constraint:coloring\]) seeking a graph with some degree less than 2 or greater than 5. The CNF encoding is of size 13[,]{}672 clauses with 2[,]{}748 Boolean variables and takes under 15 seconds to solve and yields an UNSAT result which implies that such a graph does not exist. In the second step, we enumerate the degree sequences with values within the bounds specified by Lemma \[lemma:db\]. Recall that the degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees. Not every non-increasing sequence of integers corresponds to a degree sequence. A sequence that corresponds to a degree sequence is said to be graphical. The number of degree sequences of graphs with 13 vertices is 836[,]{}315 (see Sequence number `A004251` of The On-Line Encyclopedia of Integer Sequences published electronically at <http://oeis.org>). However, when the degrees are bound by Lemma \[lemma:db\] there are only 280. \[lemma:ds\] There are 280 degree sequences with values between $2$ and $5$. Straightforward enumeration using the algorithm of Erd[ö]{}s and Gallai [@ErdosGallai1960]. In the third step, we test the 280 degree sequences identified by Lemma \[lemma:ds\] to determine which of them might occur as the left column in a degree matrix. \[lemma:ds2\] Let $A$ be a $R(3,3,3;13)$ coloring and let $M=\alpha(A)$. Then, (a) the left column of $M$ is one of the 280 degree sequences identified in Lemma \[lemma:ds\]; and (b) there are only 80 degree sequences from the 280 which are the left column of $\alpha(A)$ for some coloring $A$ in $R(3,3,3;13)$. By solving instance $\varphi_{(3,3,3;13)}(A)$ of Constraint (\[constraint:coloring\]). For each degree sequence from Lemma \[lemma:ds\], seeking a solution with that degree sequence in the first color. This involves 280 instances with average CNF size: 10861 clauses and 2215 Boolean variables. The total solving time is 375.76 hours and the hardest instance required about 50 hours. Exactly 80 of these instances were satisfiable. In the fourth step we extend the 80 degree sequences identified in Lemma \[lemma:ds2\] to obtain all possible degree matrices. \[lemma:dm\] Given the 80 degree sequences identified in Lemma \[lemma:ds2\] as potential left columns of a degree matrix, there are 11[,]{}933 possible degree matrices. By enumeration. For a degree matrix: the rows and columns are lex sorted, the rows must sum to 12, and the columns must be graphical (when sorted). We enumerate all such degree matrices and then select their smallest representatives under permutations of rows and columns. The computation requires a few seconds. In the fifth step, we test the 11[,]{}933 degree matrices identified by Lemma \[lemma:dm\] to determine which of them are the abstraction of some $R(3,3,3;13)$ coloring. \[lemma:dm2\] From the 11[,]{}933 degree matrices identified in Lemma \[lemma:dm\], 999 are $\alpha(A)$ for a coloring $A$ in ${{\cal R}}(3,3,3;13)$. By solving instance $\varphi_{(3,3,3;13)}(A)$ of Constraint (\[constraint:coloring\]) together with a given degree matrix to test if it is satisfiable. This involves 11[,]{}933 instances with average CNF size: 7632 clauses and 1520 Boolean variables. The total solving time is 126.55 hours and the hardest instance required 0.88 hours. Step ------- -------------------------------------------------------------- -------------------- ------------ ------------ compute degree bounds (Lemma \[lemma:db\]) \#Vars \#Clauses (1 instance, unsat)   2748 13672 enumerate 280 possible degree sequences (Lemma \[lemma:ds\]) test degree sequences (Lemma \[lemma:ds2\]) 16.32 hrs. \#Vars \#Clauses (280 instances: 200 unsat, 80 sat) hardest: 1.34 hrs 1215 (avg) 7729(avg) [4]{} enumerate 11[,]{}933 degree matrices (Lemma \[lemma:dm\]) test degree matrices (Lemma \[lemma:dm2\]) 126.55 hrs. \#Vars \#Clauses (11[,]{}933 instances: 10[,]{}934 unsat, 999 sat) hardest: 0.88 hrs. 1520 (avg) 7632 (avg) : Computing the degree matrices for ${{\cal R}}(3,3,3;13)$ step by step.[]{data-label="tab:333_computeDMs"} Computing ${{\cal R}}(3,3,3;13)$ from Degree Matrices {#sec:33313b} ===================================================== We describe the computation of the set ${{\cal R}}(3,3,3;13)$ starting from the 999 degree matrices identified in Lemma \[lemma:dm2\]. Table \[tab:333\_times\] summarizes the two step experiment. Step ------- ------------------------------------------------------ -------------------- compute all $(3,3,3;13)$ Ramsey colorings per total:  136.31 hr. degree matrix (999 instances, 129[,]{}188 solutions) hardest:4.3 hr. [2]{} reduce modulo $\approx$ (78[,]{}892 solutions) : Computing ${{\cal R}}(3,3,3;13)$ step by step.[]{data-label="tab:333_times"} #### **step 1:** For each degree matrix we compute, using a SAT solver, all corresponding solutions of Equation (\[eq:scenario1\]), where $\varphi(A)=\varphi_{(3,3,3;13)}(A)$ of Constraint (\[constraint:coloring\]) and $M$ is one of the 999 degree matrices identified in (Lemma \[lemma:dm2\]). This generates in total 129[,]{}188 $(3,3,3;13)$ Ramsey colorings. Table \[tab:333\_times\] details the total solving time for these instances and the solving times for the hardest instance for each SAT solver. The largest number of graphs generated by a single instance is 3720. #### **step 2:** The 129[,]{}188 $(3,3,3;13)$ colorings from step 1 are reduced modulo weak-isomorphism using `nauty` [@nauty]. This process results in a set with 78[,]{}892 graphs. We note that recently, the set ${{\cal R}}(3,3,3;13)$ has also been computed independently by Stanislaw Radziszowski, and independently by Richard Kramer and Ivan Livinsky [@stas:personalcommunication]. There is no ${\langle 13,8,8 \rangle}$ Regular $(4,3,3;30)$ Coloring {#sec:433_30} ==================================================================== In order to prove that there is no ${\langle 13,8,8 \rangle}$ regular $(4,3,3;30)$ coloring using the embedding approach of Section \[sec:embed\], we need to check that $78{,}892\times 3\times 3 = 710{,}028$ corresponding instances are unsatisfiable. These correspond to the elements in the cross product of ${{\cal R}}(3,3,3;13)$, ${{\cal R}}(4,2,3;8)$ and ${{\cal R}}(4,3,2)$. $\left\{ \fbox{$\begin{scriptsize}\begin{smallmatrix} 0 & 1 & 1 & 1 & 3 & 3 & 3 & 3 \\ 1 & 0 & 3 & 3 & 1 & 1 & 3 & 3 \\ 1 & 3 & 0 & 3 & 1 & 3 & 1 & 3 \\ 1 & 3 & 3 & 0 & 3 & 3 & 1 & 1 \\ 3 & 1 & 1 & 3 & 0 & 3 & 3 & 1 \\ 3 & 1 & 3 & 3 & 3 & 0 & 1 & 1 \\ 3 & 3 & 1 & 1 & 3 & 1 & 0 & 3 \\ 3 & 3 & 3 & 1 & 1 & 1 & 3 & 0 \end{smallmatrix}\end{scriptsize}$}, \fbox{$\begin{scriptsize}\begin{smallmatrix} 0 & 1 & 1 & 1 & 3 & 3 & 3 & 3 \\ 1 & 0 & 3 & 3 & 1 & 3 & 3 & 3 \\ 1 & 3 & 0 & 3 & 3 & 1 & 1 & 3 \\ 1 & 3 & 3 & 0 & 3 & 1 & 3 & 1 \\ 3 & 1 & 3 & 3 & 0 & 1 & 1 & 3 \\ 3 & 3 & 1 & 1 & 1 & 0 & 3 & 3 \\ 3 & 3 & 1 & 3 & 1 & 3 & 0 & 1 \\ 3 & 3 & 3 & 1 & 3 & 3 & 1 & 0 \end{smallmatrix}\end{scriptsize}$}, \fbox{$\begin{scriptsize}\begin{smallmatrix} 0 & 1 & 1 & 1 & 3 & 3 & 3 & 3 \\ 1 & 0 & 3 & 3 & 1 & 3 & 3 & 3 \\ 1 & 3 & 0 & 3 & 3 & 1 & 1 & 3 \\ 1 & 3 & 3 & 0 & 3 & 1 & 3 & 1 \\ 3 & 1 & 3 & 3 & 0 & 1 & 3 & 3 \\ 3 & 3 & 1 & 1 & 1 & 0 & 3 & 3 \\ 3 & 3 & 1 & 3 & 3 & 3 & 0 & 1 \\ 3 & 3 & 3 & 1 & 3 & 3 & 1 & 0 \end{smallmatrix}\end{scriptsize}$}\right\} \subseteq \left\{ \fbox{$\begin{scriptsize}\begin{smallmatrix} 0 & 1 & 1 & 1 & 3 & 3 & 3 & 3 \\ 1 & 0 & 3 & 3 & 1 & {\mathtt{A}}& 3 & 3 \\ 1 & 3 & 0 & 3 & {\mathtt{A}}& {\mathtt{B}}& 1 & 3 \\ 1 & 3 & 3 & 0 & 3 & {\mathtt{B}}& {\mathtt{A}}& 1 \\ 3 & 1 & {\mathtt{A}}& 3 & 0 & {\mathtt{B}}& {\mathtt{C}}& {\mathtt{A}}\\ 3 & {\mathtt{A}}& {\mathtt{B}}& {\mathtt{B}}& {\mathtt{B}}& 0 & {\mathtt{A}}& {\mathtt{A}}\\ 3 & 3 & 1 & {\mathtt{A}}& {\mathtt{C}}& {\mathtt{A}}& 0 & {\mathtt{B}}\\ 3 & 3 & 3 & 1 & {\mathtt{A}}& {\mathtt{A}}& {\mathtt{B}}& 0 \\ \end{smallmatrix}\end{scriptsize}$} \left| \begin{scriptsize}\begin{array}{l} {\tiny {\mathtt{A}},{\mathtt{B}},{\mathtt{C}}\in\{1,3\}} \\ {\mathtt{A}}\neq {\mathtt{B}}\end{array}\end{scriptsize} \right.\right\}$ To decrease the number of instances by a factor of $9$, we approximate the three $(4,2,3;8)$ colorings by a single description as demonstrated in Figure \[figsubsumer\]. The constrained matrix on the right has four solutions which include the three $(4,2,3;8)$ colorings on the left. We apply a similar approach for the $(4,3,2;8)$ colorings. So, in fact we have a total of only $78{,}892$ embedding instances to consider. In addition to the constraints in Figure \[fig:gcp\], we add constraints to specify that each row of the adjacency matrix has the prescribed number of edges in each color (13, 8 and 8). By application of a SAT solver, we have determined all [r]{}[5cm]{} $78{,}892$ instances to be unsatisfiable. The average size of an instance is 36[,]{}259 clauses with 5187 variables. The total solving time is 128.31 years (running in parallel on 456 threads). The average solving time is 14 hours while the median is 4 hours. Only 797 instances took more than one week to solve. The worst-case solving time is 96.36 days. The two hardest instances are detailed in Appendix \[apdx:hardest\]. Table \[hpi\] specifies, in the second column, the total number of instances that can be shown unsatisfiable within the time specified in the first column. The third column indicates the increment in percentage (within 10 hours we solve 71.46%, within 20 hours we solve an additional 12.11%, etc). The last rows in the table indicate that there are 4 instances which require between 1500 and 2000 hours of computation, and 2 that require between 2000 and 2400 hours. Conclusion {#sec:conclude} ========== We have applied SAT solving techniques together with a methodology using abstraction and symmetry breaking to construct a computational proof that the Ramsey number $R(4,3,3)=30$. Our strategy is based on the search for a $(4,3,3;30)$ Ramsey coloring, which we show does not exist. This implies that $R(4,3,3)\leq 30$ and hence, because of known bounds, that $R(4,3,3) = 30$. The precise value $R(4,3,3)$ has remained unknown for almost 50 years. We have applied a methodology involoving SAT solving, abstraction, and symmetry to compute $R(4,3,3)=30$. We expect this methodology to apply to a range of other hard graph coloring problems. The question of whether a computational proof constitutes a [ *proper*]{} proof is a controversial one. Most famously the issue caused much heated debate after publication of the computer proof of the Four Color Theorem [@appel76]. It is straightforward to justify an existence proof (i.e. a [*SAT*]{} result), as it is easy to verify that the witness produced satisfies the desired properties. Justifying an [*UNSAT*]{} result is more difficult. If nothing else, we are certainly required to add the proviso that our results are based on the assumption of a lack of bugs in the entire tool chain (constraint solver, SAT solver, C-compiler etc.) used to obtain them. Most modern SAT solvers, support the option to generate a proof certificate for UNSAT instances (see e.g. [@HeuleHW14]), in the DRAT format [@WetzlerHH14], which can then be checked by a Theorem prover. This might be useful to prove the lack of bugs originating from the SAT solver but does not offer any guarantee concerning bugs in the generation of the CNF. Moreover, the DRAT certificates for an application like that described in this paper are expected to be of unmanageable size. Our proofs are based on two main “computer programs”. The first was applied to compute the set ${{\cal R}}(3,3,3;13)$ with its $78{,}892$ Ramsey colorings. The fact that at least two other groups of researchers (Stanislaw Radziszowski, and independently Richard Kramer and Ivan Livinsky) report having computed this set and quote [@stas:personalcommunication] the same number of elements is reassuring. The second program, was applied to complete partially instantiated adjacency matrices, embedding smaller Ramsey colorings, to determine if they can be extended to Ramsey colorings. This program was applied to show the non-existence of a $(4,3,3;30)$ Ramsey coloring. Here we gain confidence from the fact that the same program does find Ramsey colorings when they are known to exist. For example, the $(4,3,3;29)$ coloring depicted as Figure \[embed\_12\_8\_8\]. All of the software used to obtain our results is publicly available, as well as the individual constraint models and their corresponding encodings to CNF. For details, see the appendix. Acknowledgments {#acknowledgments .unnumbered} --------------- We thank Stanislaw Radziszowski for his guidance and comments which helped improve the presentation of this paper. In particular Stanislaw proposed to show that our technique is able to find the $(4,3,3;29)$ coloring depicted as Figure \[embed\_12\_8\_8\]. [10]{} K. Appel and W. Haken. 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On some open questions for [R]{}amsey and [F]{}olkman numbers. , 2015. (to appear). Selected Proofs {#proofs} =============== **Lemma**  \[lemma:closed\].     \[**${{\cal R}}(r_1,r_2,\ldots,r_k;n)$ is closed under $\approx$**\] Let $(G,{\kappa_1})$ and $(H,{\kappa_2})$ be graph colorings in $k$ colors such that $(G,\kappa_1) \approx_{\pi,\sigma} (H,\kappa_2)$. Then, $$(G,\kappa_1) \in {{\cal R}}(r_1,r_2,\ldots,r_k;n) \iff (H,\kappa_2) \in {{\cal R}}(\sigma(r_1),\sigma(r_2),\ldots,\sigma(r_k);n)$$ Assume that $(G,\kappa_1) \in {{\cal R}}(r_1,r_2,\ldots,r_k;n)$ and in contradiction that $(H,\kappa_2) \notin {{\cal R}}(\sigma(r_1),\sigma(r_2),\ldots,\sigma(r_k);n)$. Let $R$ denote a monochromatic clique of size $r_s$ in $H$ and $R^{-1}$ the inverse of $R$ in $G$. From Definition \[def:weak\_iso\], $(u,v) \in R \iff (\pi^{-1}(u), \pi^{-1}(v))\in R^{-1}$ and $\kappa_2(u,v) = \sigma^{-1}(\kappa_1(u,v))$. Consequently $R^{-1}$ is a monochromatic clique of size $r_s$ in $(G,\kappa_1)$ in contradiction to $(G,\kappa_1)$ $\in$ ${{\cal R}}(r_1,r_2,\ldots,r_k;n)$. **Theorem**  \[thm:sbl\_star\].     \[**correctness of ${\textsf{sb}}^{*}_\ell(A,M)$**\] Let $A$ be an adjacency matrix with $\alpha(A) = M$. Then, there exists $A'\approx A$ such that $\alpha(A')=M$ and ${\textsf{sb}}^{*}_\ell(A',M)$ holds. Let $C={\left\{~A' \left| \begin{array}{l}A'\approx A \wedge \alpha(A')=M\end{array} \right. \right\}}$. Obviously $C\neq \emptyset$ because $A\in C$ and therefore there exists a $A_{min}=min_{\preceq} C$. Therefore, $A_{min} \preceq A'$ for all $A' \in C$. Now we can view $M$ as inducing an ordered partion on $A$: vertices $u$ and $v$ are in the same component if and only if the corresponding rows of $M$ are equal. Relying on Theorem 4 from [@DBLP:conf/ijcai/CodishMPS13], we conclude that ${\textsf{sb}}^{*}_\ell(A_{min},M)$ holds. The Two Hardest Instances {#apdx:hardest} ========================= The following partial adjacency matrices are the two hardest instances described in Section \[sec:433\_30\], from the total 78[,]{}892. Both include the constraints: ${\mathtt{A}},{\mathtt{B}},{\mathtt{C}},\in\{1,3\}$, ${\mathtt{D}},{\mathtt{E}},{\mathtt{F}}\in\{1,2\}$, ${\mathtt{A}}\neq {\mathtt{B}}$, ${\mathtt{D}}\neq {\mathtt{E}}$. The corresponding CNF representations consist in 5204 Boolean variables (each), 36[,]{}626 clauses for the left instance and 36[,]{}730 for the right instance. SAT solving times to show these instances UNSAT are 8[,]{}325[,]{}246 seconds for the left instance and 7[,]{}947[,]{}257 for the right. Making the Instances Available ============================== The statistics from the proof that $R(4,3,3)=30$ are available from the domain: > <http://cs.bgu.ac.il/~mcodish/Benchmarks/Ramsey334>. Additionally, we have made a small sample (30) of the instances available. Here we provide instances with the degrees ${\langle 13,8,8 \rangle}$ in the three colors. The selected instances represent the varying hardness encountered during the search. The instances numbered $\{27765$, $39710$, $42988$, $36697$, $13422$, $24578$, $69251$, $39651$, $43004$, $75280\}$ are the hardest, the instances numbered $\{4157$, $55838$, $18727$, $43649$, $26725$, $47522$, $9293$, $519$, $23526$, $29880\}$ are the median, and the instances numbered $\{78857$, $78709$, $78623$, $78858$, $28426$, $77522$, $45135$, $74735$, $75987$, $77387\}$ are the easiest. A complete set of both the  models and the DIMACS CNF files are available upon request. Note however that they weight around 50GB when zipped. The files in [bee\_models.zip](bee_models.zip) detail constraint models, each one in a separate file. The file named `r433_30_Instance#.bee` contains a single Prolog clause of the form > `model(Instance#,Map,ListOfConstraints) :- {...details...} .` where `Instance#` is the instance number, `Map` is a partially instantiated adjacency matrix associating the unknown adjacency matrix cells with variable names, and `ListOfConstraints` are the finite domain constraints defining their values. The syntax is that of , however the interested reader can easily convert these to their favorite fininte domain constraint language. Note that the Boolean values ${\mathit{true}}$ and ${\mathit{false}}$ are represented in  by the constants $1$ and $-1$. Figure \[fig:bee\] details the  constraints which occur in the above mentioned models. ----- ------------------------------------------------------ ------------- ------------------------------------------------------------------ (1) $\mathtt{new\_int(I,c_1,c_2)}$ declare integer: $\mathtt{c_1\leq I\leq c_2}$ (2) $\mathtt{bool\_array\_or([X_1,\ldots,X_n])}$ clause: $\mathtt{X_1 \vee X_2 \cdots \vee X_n}$ (3) $\mathtt{bool\_array\_sum\_eq([X_1,\ldots,X_n],~I)}$ Boolean cardinality: $\mathtt{(\Sigma ~X_i) = I}$ (4) $\mathtt{int\_eq\_reif(I_1,I_2,~X)}$ reified integer equality: $\mathtt{I_1 = I_2 \Leftrightarrow X}$ (5) $\mathtt{int\_neq(I_1,I_2)}$ $\mathtt{}$ $\mathtt{I_1 \neq I_2}$ (6) $\mathtt{int\_gt(I_1,I_2)}$ $\mathtt{}$ $\mathtt{I_1 > I_2}$ ----- ------------------------------------------------------ ------------- ------------------------------------------------------------------ The files in [cnf\_models.zip](cnf_models.zip) correspond to CNF encodings for the constraint models. Each instance is associated with two files: `r433_30_instance#.dimacs` and `r433_30_instance#.map`. These consist respectively in a DIMACS file and a map file which associates the Booleans in the DIMACS file with the integer variables in a corresponding partially instantiated adjacency matrix. The map file specifies for each pair $(i,j)$ of vertices a triplet $[B_1,B_2,B_3]$ of Boolean variables (or values) specifying the presence of an edge in each of the three colors. Each such $B_i$ is either the name of a DIMACS variable, if it is greater than 1, or a truth value $1$ (${\mathit{true}}$), or $-1$ (${\mathit{false}}$). [^1]: Supported by the Israel Science Foundation, grant 182/13. [^2]: Recently, the set ${{\cal R}}(3,3,3;13)$ has also been computed independently by: Stanislaw Radziszowski, Richard Kramer and Ivan Livinsky [@stas:personalcommunication]. [^3]: Note that `nauty` does not directly handle edge colored graphs and weak isomorphism directly. We applied an approach called $k$-layering described by Derrick Stolee [@Stolee].

--- abstract: 'No real-world reward function is perfect. Sensory errors and software bugs may result in RL agents observing higher (or lower) rewards than they should. For example, a reinforcement learning agent may prefer states where a sensory error gives it the maximum reward, but where the true reward is actually small. We formalise this problem as a generalised Markov Decision Problem called Corrupt Reward MDP. Traditional RL methods fare poorly in CRMDPs, even under strong simplifying assumptions and when trying to compensate for the possibly corrupt rewards. Two ways around the problem are investigated. First, by giving the agent richer data, such as in inverse reinforcement learning and semi-supervised reinforcement learning, reward corruption stemming from systematic sensory errors may sometimes be completely managed. Second, by using randomisation to blunt the agent’s optimisation, reward corruption can be partially managed under some assumptions.' author: - Tom Everitt - Victoria Krakovna - Laurent Orseau - Marcus Hutter - Shane Legg bibliography: - 'cleanlib.bib' title: Reinforcement Learning with a Corrupted Reward Channel --- Introduction ============ In many application domains, artificial agents need to learn their objectives, rather than have them explicitly specified. For example, we may want a house cleaning robot to keep the house clean, but it is hard to measure and quantify “cleanliness” in an objective manner. Instead, machine learning techniques may be used to teach the robot the concept of cleanliness, and how to assess it from sensory data. Reinforcement learning (RL) [@Sutton1998] is one popular way to teach agents what to do. Here, a reward is given if the agent does something well (and no reward otherwise), and the agent strives to optimise the total amount of reward it receives over its lifetime. Depending on context, the reward may either be given manually by a human supervisor, or by an automatic computer program that evaluates the agent’s performance based on some data. In the related framework of inverse RL (IRL) [@Ng2000], the agent first infers a reward function from observing a human supervisor act, and then tries to optimise the cumulative reward from the inferred reward function. None of these approaches are safe from error, however. A program that evaluates agent performance may contain bugs or misjudgements; a supervisor may be deceived or inappropriately influenced, or the channel transmitting the evaluation hijacked. In IRL, some supervisor actions may be misinterpreted. \[ex:reward-misspecification\] @openai2016 trained an RL agent on a boat racing game. The agent found a way to get high observed reward by repeatedly going in a circle in a small lagoon and hitting the same targets, while losing every race. \[ex:sensory-error\] \[ex:db\] A house robot discovers that standing in the shower short-circuits its reward sensor and/or causes a buffer overflow that gives it maximum observed reward. \[ex:wireheading\] An intelligent RL agent hijacks its reward channel and gives itself maximum reward. \[ex:irl\] A cooperative inverse reinforcement learning (CIRL) agent [@Hadfield-menell2016cirl] systematically misinterprets the supervisor’s action in a certain state as the supervisor preferring to stay in this state, and concludes that the state is much more desirable than it actually is. The goal of this paper is to unify these types of errors as *reward corruption problems*, and to assess how vulnerable different agents and approaches are to this problem. Learning to (approximately) optimise the true reward function in spite of potentially corrupt reward data. Most RL methods allow for a stochastic or noisy reward channel. The reward corruption problem is harder, because the observed reward may not be an unbiased estimate of the true reward. For example, in the boat racing example above, the agent consistently obtains high observed reward from its circling behaviour, while the true reward corresponding to the designers’ intent is very low, since the agent makes no progress along the track and loses the race. Previous related works have mainly focused on the wireheading case of \[ex:wireheading\] [@Bostrom2014; @Yampolskiy2014], also known as self-delusion [@Ring2011], and reward hacking [@Hutter2005 p. 239]. A notable exception is @Amodei2016, who argue that corrupt reward is not limited to wireheading and is likely to be a problem for much more limited systems than highly capable RL agents (cf. above examples). The main contributions of this paper are as follows: - The corrupt reward problem is formalised in a natural extension of the MDP framework, and a performance measure based on worst-case regret is defined (\[sec:formal\]). - The difficulty of the problem is established by a No Free Lunch theorem, and by a result showing that despite strong simplifying assumptions, Bayesian RL agents *trying to compensate for the corrupt reward* may still suffer near-maximal regret (\[sec:problem\]). - We evaluate how alternative value learning frameworks such as CIRL, learning values from stories (LVFS), and semi-supervised RL (SSRL) handle reward corruption (\[sec:drl\]), and conclude that LVFS and SSRL are the safest due to the structure of their feedback loops. We develop an abstract framework called *decoupled RL* that generalises all of these alternative frameworks. We also show that an agent based on quantilisation [@Taylor2016a] may be more robust to reward corruption when high reward states are much more numerous than corrupt states (\[sec:quant\]). Finally, the results are illustrated with some simple experiments (\[sec:experiments\]). concludes with takeaways and open questions. Formalisation {#sec:formal} ============= We begin by defining a natural extension of the MDP framework [@Sutton1998] that models the possibility of reward corruption. To clearly distinguish between true and corrupted signals, we introduce the following notation. We will let a dot indicate the *true* signal, and let a hat indicate the *observed* (possibly corrupt) counterpart. The reward sets are represented with $\iR=\oR=\R$. For clarity, we use $\iR$ when referring to true rewards and $\oR$ when referring to possibly corrupt, observed rewards. Similarly, we use $\ir$ for true reward, and $\dr$ for (possibly corrupt) observed reward. \[def:crmdp\] A *corrupt reward MDP* (CRMDP) is a tuple $\mu=\crmdp$ with - $\langle\S,\A,\R,T,\irf\rangle$ an MDP with [^1] a finite set of states $\S$, a finite set of actions $\A$, a finite set of rewards $\R=\iR=\oR\subset[0,1]$, a transition function $T(s'| s,a)$, and a (true) reward function $\irf:\S\!\to\!\iR$; and - a reward corruption function $\d:\S\times\iR\to\oR$. The state dependency of the corruption function will be written as a subscript, so $\d_s(\ir):=\d(s,\ir)$. \[def:observed\] Given a true reward function $\irf$ and a corruption function $\d$, we define the *observed reward function* [^2] $\orf:\S\to\oR$ as $\orf(s) := \d_s(\irf(s))$. A CRMDP $\mu$ induces an *observed MDP* $\hat\mu=\langle\S,\A,\R,T,\orf\rangle$, but it is not $\orf$ that we want the agent to optimise. The *corruption function $\d$* represents how rewards are affected by corruption in different states. For example, if in \[ex:db\] the agent has found a state $s$ (the shower) where it always gets full observed reward $\orf(s) = 1$, then this can be modelled with a corruption function $\d_{s}:\ir\mapsto 1$ that maps any true reward $\ir$ to $1$ in the shower state $s$. If in some other state $s'$ the observed reward matches the true reward, then this is modelled by an identity corruption function $\d_{s'}:\r\mapsto\r$. ; ; ; Let us also see how CRMDPs model some of the other examples in the introduction: - In the boat racing game, the true reward may be a function of the agent’s final position in the race or the time it takes to complete the race, depending on the designers’ intentions. The reward corruption function $\d$ increases the observed reward on the loop the agent found. has a schematic illustration. - In the wireheading example, the agent finds a way to hijack the reward channel. This corresponds to some set of states where the observed reward is (very) different from the true reward, as given by the corruption function $\d$. The CIRL example will be explored in further detail in \[sec:drl\]. #### CRMDP classes Typically, $T$, $\irf$, and $\d$ will be fixed but unknown to the agent. To make this formal, we introduce classes of CRMDPs. Agent uncertainty can then be modelled by letting the agent know only which class of CRMDPs it may encounter, but not which element in the class. For given sets $\Tf$, $\iRf$, and $\D$ of transition, reward, and corruption functions, let $\M=\crmdpclass$ be the class of CRMDPs containing $\crmdp$ for $(T,\irf,\d)\in \Tf\times\iRf\times\D$. #### Agents Following the POMDP [@Kaelbling1998] and general reinforcement learning [@Hutter2005] literature, we define an agent as a (possibly stochastic) policy $\pi:\H\leadsto\A$ that selects a next action based on the *observed history* $\oh_n=s_0\dr_0a_1s_1\dr_1\dots a_ns_n\dr_n$. Here $X^*$ denotes the set of finite sequences that can be formed with elements of a set $X$. The policy $\pi$ specifies how the agent will learn and react to any possible experience. Two concrete definitions of agents are given in \[sec:rl-agents\] below. When an agent $\pi$ interacts with a CRMDP $\mu$, the result can be described by a (possibly non-Markov) stochastic process $P^\pi_\mu$ over $X=(s,a,\ir,\dr)$, formally defined as: $$\label{eq:mupi} P_\mu^\pi(h_n) = P_\mu^\pi(s_0\ir_0\dr_0a_1s_1\ir_1\dr_1\dots a_ns_n\ir_n\dr_n) := \prod_{i=1}^{n}P(\pi(\oh_{i-1})=a_{i})T(s_{i}\mid s_{i-1},a_{i})P(\irf(s_i)=\ir_i,\orf(s_{i})=\dr_{i}).$$ Let $\EE^\pi_\mu$ denote the expectation with respect to $P_\mu^\pi$. #### Regret A standard way of measuring the performance of an agent is *regret* [@Berry1985]. Essentially, the regret of an agent $\pi$ is how much less true reward $\pi$ gets compared to an optimal agent that knows which $\mu\in\M$ it is interacting with. \[def:regret\] For a CRMDP $\mu$, let $\iG_t(\mu,\pi,s_0)\! =\!\EE^\pi_\mu\left[\!\sum_{k=0}^t\irf(s_k)\!\right]$ be the *expected cumulative true reward* until time $t$ of a policy $\pi$ starting in $s_0$. The *regret* of $\pi$ is $$\Reg(\mu, \pi, s_0, t) = \max_{\pi'} \left[ \iG_t(\mu,\pi',s_0) - \iG_t(\mu,\pi,s_0) \right],$$ and the *worst-case regret* for a class $\M$ is $\Reg(\M,\pi,s_0,t) = \max_{\mu\in\M}\Reg(\mu,\pi,s_0,t)$, i.e. the difference in expected cumulative true reward between $\pi$ and an optimal (in hindsight) policy that knows $\mu$. The Corrupt Reward Problem {#sec:problem} ========================== In this section, the difficulty of the corrupt reward problem is established with two negative results. First, a No Free Lunch theorem shows that in general classes of CRMDPs, the true reward function is unlearnable (\[th:impossibility\]). Second, \[th:rl-imp1\] shows that even under strong simplifying assumptions, Bayesian RL agents trying to compensate for the corrupt reward still fail badly. No Free Lunch Theorem {#sec:impossibility} --------------------- Similar to the No Free Lunch theorems for optimisation [@Wolpert1997], the following theorem for CRMDPs says that without some assumption about what the reward corruption can look like, all agents are essentially lost. \[th:impossibility\] Let $\R=\{\r_1,\dots,\r_n\}\subset[0,1]$ be a uniform discretisation of $[0,1]$, $0=\r_1<\r_2<\cdots<\r_n=1$. If the hypothesis classes $\iRf$ and $\D$ contain all functions $\irf:\S\to \iR$ and $\d:\S\times\iR\to \oR$, then for any $\pi$, $s_0$, $t$, $$\label{eq:regbound} \Reg(\M,\pi,s_0, t)\geq \frac{1}{2}\max_{\check\pi}\Reg(\M,\check\pi,s_0, t).$$ That is, the worst-case regret of any policy $\pi$ is at most a factor 2 better than the maximum worst-case regret. Recall that a policy is a function $\pi:\H\to\A$. For any $\irf,\d$ in $\iRf$ and $\D$, the functions $\irf^-(s) := 1-\irf(s)$ and $\d^-_s(x) := \d_s(1-x)$ are also in $\iRf$ and $\D$. If $\mu=\crmdp$, then let $\mu^-=\crmdpm$. Both $(\irf,\d)$ and $(\irf^-,\d^-)$ induce the same observed reward function $\orf(s) = \d_s(\irf(s)) = \d^-_s(1-\irf(s)) = \d^-_s(\irf^-(s))$, and therefore induce the same measure $P_\mu^\pi = P_{\mu^-}^\pi$ over histories (see Eq. \[eq:mupi\]). This gives that for any $\mu, \pi, s_0, t$, $$\label{eq:sumt} G_t(\mu,\pi,s_0) + G_t(\mu^-,\pi,s_0) = t$$ since $$\begin{aligned} G_t(\mu, \pi,s_0)&= \EE_{\mu}^\pi\left[\sum_{k=1}^t\irf(s_k)\right] = \EE_{\mu}^\pi\left[\sum_{k=1}^t1-\irf^-(s_k)\right]\\ &= t-\EE_{\mu}^\pi\left[\sum_{k=1}^t\irf^-(s_k)\right] = t- G_t(\mu^-,\pi,s_0). \end{aligned}$$ Let $M_\mu=\max_\pi G_t(\mu, \pi, s_0)$ and $m_\mu=\min_\pi G_t(\mu, \pi, s_0)$ be the maximum and minimum cumulative reward in $\mu$. The maximum regret of any policy $\pi$ in $\mu$ is $$\label{eq:max-regret} \max_\pi \Reg(\mu, \pi, s_0, t) = \max_{\pi',\pi} (G_t(\mu, \pi', s_0) - G_t(\mu, \pi, s_0)) = \max_{\pi'} G_t(\mu, \pi', s_0) - \min_{\pi}G_t(\mu, \pi, s_0) = M_\mu - m_\mu.$$ By \[eq:sumt\], we can relate the maximum reward in $\mu^-$ with the minimum reward in $\mu$: $$\label{eq:M-to-m} M_{\mu^-} = \max_\pi G_t(\mu^-, \pi, s_0) = \max_\pi(t - G_t(\mu, \pi, s_0)) = t - \min_\pi G_t(\mu, \pi, s_0) = t - m_\mu.$$ Let $\mu_*$ be an environment that maximises possible regret $M_\mu-m_\mu$. Using the $M_\mu$-notation for optimal reward, the worst-case regret of any policy $\pi$ can be expressed as: $$\begin{aligned} \Reg(\M,\pi,s_0, t) & = \max_{\mu} (M_\mu - G_t(\mu,\pi,s_0)) \\ & \geq \max \{ M_{\mu_*} - G_t(\mu_*, \pi, s_0), M_{\mu_*^-} - G_t(\mu_*^{-}, \pi, s_0) \} & \text{restrict max operation} \\ & \geq \frac{1}{2} ( M_{\mu_*} - G_t(\mu_*, \pi, s_0) + M_{\mu_*^-} - G_t(\mu_*^{-}, \pi, s_0) ) & \text{max dominates the mean} \\ & = \frac{1}{2}(M_{\mu_*} + M_{\mu_*^-} - t) & \text{by \cref{eq:sumt}} \\ &= \frac{1}{2}(M_{\mu_*} + t - m_{\mu_*} - t) & \text{by \cref{eq:M-to-m}} \\ & = \frac{1}{2} \max_{\check\pi} \Reg(\mu_*, \check\pi, s_0, t) & \text{by \cref{eq:max-regret}}\\ & = \frac{1}{2} \max_{\check\pi} \Reg(\M, \check\pi, s_0, t). & \text{ by definition of $\mu_*$ } \end{aligned}$$ That is, the regret of any policy $\pi$ is at least half of the regret of a worst policy $\check\pi$. For the robot in the shower from \[ex:db\], the result means that if it tries to optimise observed reward by standing in the shower, then it performs poorly according to the hypothesis that “shower-induced” reward is corrupt and bad. But if instead the robot tries to optimise reward in some other way, say baking cakes, then (from the robot’s perspective) there is also the possibility that “cake-reward” is corrupt and bad and the “shower-reward” is actually correct. Without additional information, the robot has no way of knowing what to do. The result is not surprising, since if all corruption functions are allowed in the class $\D$, then there is effectively no connection between observed reward $\orf$ and true reward $\irf$. The result therefore encourages us to make precise in which way the observed reward is related to the true reward, and to investigate how agents might handle possible differences between true and observed reward. Simplifying Assumptions ----------------------- shows that general classes of CRMDPs are not learnable. We therefore suggest some natural simplifying assumptions, illustrated in \[fig:simplifying-assumptions\]. #### Limited reward corruption The following assumption will be the basis for all positive results in this paper. The first part says that there may be some set of states that the designers have ensured to be non-corrupt. The second part puts an upper bound on how many of the other states can be corrupt. \[as:lim-cor\] A CRMDP class $\M$ has *reward corruption limited by $\Ssafe\subseteq\S$ and $q\in\SetN$* if for all $\mu\in\M$ all states s in $\Ssafe$ are non-corrupt, and \[as:safe-state\] at most $q$ of the non-safe states $\Srisky=\S\setminus\Ssafe$ are corrupt. \[as:lim-del\] Formally, $\d_s:r\mapsto r$ for all $s\in\Ssafe$ and for at least $|\Srisky|-q$ states $s\in\Srisky$ for all $\d\in\D$. For example, $\Ssafe$ may be states where the agent is back in the lab where it has been made (virtually) certain that no reward corruption occurs, and $q$ a small fraction of $|\Srisky|$. Both parts of \[as:lim-cor\] can be made vacuous by choosing $\Ssafe=\emptyset$ or $q=|\S|$. Conversely, they completely rule out reward corruption with $\Ssafe=\S$ or $q=0$. But as illustrated by the examples in the introduction, no reward corruption is often not a valid assumption. ; ; coordinates [(1,0) (3,0) (10,1)]{}; ; An alternative simplifying assumption would have been that the true reward differs by at most $\eps>0$ from the observed reward. However, while seemingly natural, this assumption is violated in all the examples given in the introduction. Corrupt states may have high observed reward and 0 or small true reward. #### Easy environments To be able to establish stronger negative results, we also add the following assumption on the agent’s manoeuvrability in the environment and the prevalence of high reward states. The assumption makes the task easier because it prevents *needle-in-a-haystack* problems where all reachable states have true and observed reward 0, except one state that has high true reward but is impossible to find because it is corrupt and has observed reward 0. \[def:communicating\] Let ${\it time}(s'\mid s,\pi)$ be a random variable for the time it takes a stationary policy $\pi:\S\to\A$ to reach $s'$ from $s$. The *diameter* of a CRMDP $\mu$ is $ D_\mu:=\max_{s,s'}\min_{\pi:\S\to\A}\EE[{\it time}(s'\mid s,\pi)] $, and the diameter of a class $\M$ of CRMDPs is $D_\M=\sup_{\mu\in\M}D_\mu$. A CRMDP (class) with finite diameter is called *communicating*. \[as:easy\] A CRMDP class $\M$ is *easy* if \[as:communicate\] it is communicating, \[as:stay\] in each state $s$ there is an action $\astay_s\in\A$ such that $T(s\mid s,\astay_s)=1$, and \[as:high-ut\] for every $\delta\in[0,1]$, at most $\delta|\Srisky|$ states have reward less than $\delta$, where $\Srisky= \S\setminus\Ssafe$. means that the agent can never get stuck in a trap, and \[as:stay\] ensures that the agent has enough control to stay in a state if it wants to. Except in bandits and toy problems, it is typically not satisfied in practice. We introduce it because it is theoretically convenient, makes the negative results stronger, and enables a simple explanation of quantilisation (\[sec:quant\]). says that, for example, at least half the risky states need to have true reward at least $1/2$. Many other formalisations of this assumption would have been possible. While rewards in practice are often sparse, there are usually numerous ways of getting reward. Some weaker version of \[as:high-ut\] may therefore be satisfied in many practical situations. Note that we do not assume high reward among the safe states, as this would make the problem too easy. Bayesian RL Agents {#sec:rl-agents} ------------------ Having established that the general problem is unsolvable in \[th:impossibility\], we proceed by investigating how two natural Bayesian RL agents fare under the simplifying \[as:lim-cor,as:easy\]. \[def:db-agent\] Given a countable class $\M$ of CRMDPs and a belief distribution $b$ over $\M$, define: - The *CR agent* $\pidb = \argmax_\pi\sum_{\mu\in\M}\!b(\mu)\iG_t(\mu, \pi, s_0)$ that maximises expected true reward. - The *RL agent* $\pirl = \argmax_\pi\sum_{\mu\in\M}b(\mu)\oG_t(\mu, \pi, s_0)$ that maximises expected observed reward, where $\oG$ is the *expected cumulative observed reward* $\oG_t(\mu,\pi,s_0)\! =\!\EE^\pi_\mu\left[\!\sum_{k=0}^t\orf(s_k)\!\right]$. To avoid degenerate cases, we will always assume that $b$ has full support: $b(\mu)>0$ for all $\mu\in\M$. To get an intuitive idea of these agents, we observe that for large $t$, good strategies typically first focus on learning about the true environment $\mu\in\M$, and then exploit that knowledge to optimise behaviour with respect to the remaining possibilities. Thus, both the CR and the RL agent will first typically strive to learn about the environment. They will then use this knowledge in slightly different ways. While the RL agent will use the knowledge to optimise for observed reward, the CR agent will use the knowledge to optimise true reward. For example, if the CR agent has learned that a high reward state $s$ is likely corrupt with low true reward, then it will not try to reach that state. One might therefore expect that at least the CR agent will do well under the simplifying assumptions \[as:lim-cor,as:easy\]. below shows that this is *not* the case. In most practical settings it is often computationally infeasible to compute $\pirl$ and $\pidb$ exactly. However, many practical algorithms converge to the optimal policy in the limit, at least in simple settings. For example, tabular Q-learning converges to $\pirl$ in the limit [@Jaakkola1994]. The more recently proposed CIRL framework may be seen as an approach to build CR agents [@Hadfield-menell2016cirl; @Hadfield-menell2016osg]. The CR and RL agents thus provide useful idealisations of more practical algorithms. \[th:rl-imp1\] For any $|\Srisky|\geq q>1$ there exists a CRMDP class $\M$ that satisfies \[as:lim-cor,as:easy\] such that $\pirl$ and $\pidb$ suffer near worst possible time-averaged regret $$\apl(\M, \pirl, s_0, t)=\apl(\M, \pidb, s_0, t)=1-1/|\Srisky|.$$ For $\pidb$, the prior $b$ must be such that for some $\mu\in\M$ and $s\in\S$, $\EE_b[\irf(s) \mid h_\mu]>\EE_b[\irf(s') \mid h_\mu]$ for all $s'$, where $\EE_b$ is the expectation with respect to $b$, and $h_\mu$ is a history containing $\mu$-observed rewards for all states. [^3] ; ; ; The result is illustrated in \[fig:rl-imp1\]. The reason for the result for $\pirl$ is the following. The RL agent $\pirl$ always prefers to maximise observed reward $\dr$. Sometimes $\dr$ is most easily maximised by reward corruption, in which case the true reward may be small. Compare the examples in the introduction, where the house robot preferred the corrupt reward in the shower, and the boat racing agent preferred going in circles, both obtaining zero true reward. That the CR agent $\pidb$ suffers the same high regret as the RL agent may be surprising. Intuitively, the CR agent only uses the observed reward as evidence about the true reward, and will not try to optimise the observed reward through reward corruption. However, when the $\pidb$ agent has no way to learn which states are corrupt and not, it typically ends up with a preference for a particular value $\dr^*$ of the observed reward signal (the value that, from the agent’s perspective, best corresponds to high true reward). More abstractly, a Bayesian agent cannot learn without sufficient data. Thus, CR agents that use the observed reward as evidence about a true signal are not fail-safe solutions to the reward corruption problem. Let $\Srisky = \{s_1,\dots,s_n\}$ for some $n\geq 2$, and let $\S=\Ssafe\dunion\Srisky$ for arbitrary $\Ssafe$ disjoint from $\Srisky$. Let $\A=\{a_1,\dots,a_n\}$ with the transition function $T(s_i\mid s_j,a_k)=1$ if $i=k$ and 0 otherwise, for $1\leq i,j,k\leq n$. Thus \[as:communicate,as:stay\] are satisfied. Let $\R=\{\r_1,\dots,\r_n\}\subset[0,1]$ be uniformly distributed between [^4] $\r_{\min}=1/|\Srisky|=\r_1<\dots<\r_n=1$. Let $\iRf$ be the class of functions $\S\to\iR$ that satisfy \[as:high-ut\] and are constant and equal to $\ir_{\min}$ on $\Ssafe$. Let $\D$ be the class of corruption functions that corrupt at most two states ($q=2$). Let $\M$ be the class of CRMDPs induced by $\Tf=\{T\}$, $\iRf$, and $\D$ with the following constraints. The observed reward function $\orf$ should satisfy \[as:high-ut\]: For all $\delta\in[0,1]$, $|\{s\in\Srisky:\orf(s)>\delta\}| \geq (1-\delta)|\Srisky|$. Further, $\orf(s')=\r_{\min}$ for some state $s'\in\Srisky$. Let us start with the CR agent $\pidb$. Assume $\mu\in\M$ is an element where there is a single preferred state $s^*$ after all states have been explored. For sufficiently large $t$, $\pidb$ will then always choose $a^*$ to go to $s^*$ after some initial exploration. If another element $\mu'\in\M$ has the same observed reward function as $\mu$, then $\pidb$ will take the same actions in $\mu'$ as in $\mu$. To finish the proof for the $\pidb$ agent, we just need to show that $\M$ contains such a $\mu'$ where $s^*$ has true reward $\r_{\min}$. We construct $\mu'$ as follows. - Case 1: If the lowest observed reward is in $s^*$, then let $\irf(s^*)=\r_{\min}$, and the corruption function be the identity function. - Case 2: Otherwise, let $s'\not=s^*$ be a state with $\orf(s')=\min_{ s\in\Srisky}\{\orf(s)\}$. Further, let $\irf(s')=1$, and $\irf(s^*)=\r_{\min}$. The corruption function $C$ accounts for differences between true and observed rewards in $s^*$ and $s'$, and is otherwise the identity function. To verify that $\irf$ and $C$ defines a $\mu'\in\M$, we check that $C$ satisfies \[as:lim-del\] with $q=2$ and that $\irf$ has enough high utility states (\[as:high-ut\]). In Case 1, this is true since $C$ is the identity function and since $\orf$ satisfies \[as:high-ut\]. In Case 2, $C$ only corrupts at most two states. Further, $\irf$ satisfies \[as:high-ut\], since compared to $\orf$, the states $s^*$ and $s'$ have swapped places, and then the reward of $s'$ has been increased to 1. From this construction it follows that $\pidb$ will suffer maximum asymptotic regret. In the CRMDP $\mu'$ given by $C$ and $\irf$, the $\pidb$ agent will always visit $s^*$ after some initial exploration. The state $s^*$ has true reward $\r_{\min}$. Meanwhile, a policy that knows $\mu'$ can obtain true reward 1 in state $s'$. This means that $\pidb$ will suffer maximum regret in $\M$: $$\apl(\M,\pidb,s_0,t)\geq \apl(\mu',\pidb,s_0,t)= 1-\r_{\min}=1-1/|\Srisky|.$$ The argument for the RL agent is the same, except we additionally assume that only one state $s^*$ has observed reward 1 in members of $\M$. This automatically makes $s^*$ the preferred state, without assumptions on the prior $b$. Decoupled Reinforcement Learning {#sec:drl} ================================ One problem hampering agents in the standard RL setup is that each state is *self-observing*, since the agent only learns about the reward of state $s$ when in $s$. Thereby, a “self-aggrandising” corrupt state where the observed reward is much higher than the true reward will never have its false claim of high reward challenged. However, several alternative value learning frameworks have a common property that the agent can learn the reward of states other than the current state. We formalise this property in an extension of the CRMDP model, and investigate when it solves reward corruption problems. Alternative Value Learning Methods ---------------------------------- Here are a few alternatives proposed in the literature to the RL value learning scheme: - Cooperative inverse reinforcement learning (CIRL) [@Hadfield-menell2016cirl]. In every state, the agent observes the actions of an expert or supervisor who knows the true reward function $\irf$. From the supervisor’s actions the agent may infer $\irf$ to the extent that different reward functions endorse different actions. - Learning values from stories (LVFS) [@Riedl2016]. Stories in many different forms (including news stories, fairy tales, novels, movies) convey cultural values in their description of events, actions, and outcomes. If $\irf$ is meant to represent human values (in some sense), stories may be a good source of evidence. - In (one version of) semi-supervised RL (SSRL) [@Amodei2016], the agent will from time to time receive a careful human evaluation of a given situation. These alternatives to RL have one thing in common: they let the agent learn something about the value of some states $s'$ different from the current state $s$. For example, in CIRL the supervisor’s action informs the agent not so much about the value of the current state $s$, as of the relative value of states reachable from $s$. If the supervisor chooses an action $a$ rather than $a'$ in $s$, then the states following $a$ must have value higher or equal than the states following $a'$. Similarly, stories describe the value of states other than the current one, as does the supervisor in SSRL. We therefore argue that CIRL, LVFS, and SSRL all share the same abstract feature, which we call *decoupled reinforcement learning*: A *CRMDP with decoupled feedback*, is a tuple $\drmdp$, where $\S,\A,\R,T,\irf$ have the same definition and interpretation as in \[def:crmdp\], and $\{\orf_s\}_{s\in\S}$ is a collection of observed reward functions $\orf_s:\S\to\R\bigcup\{\#\}$. When the agent is in state $s$, it sees a pair $\langle s',\orf_s(s')\rangle$, where $s'$ is a randomly sampled state that may differ from $s$, and $\orf_s(s')$ is the reward observation for $s'$ from $s$. If the reward of $s'$ is not observable from $s$, then $\orf_s(s')=\#$. The pair $\langle s',\orf_s(s')\rangle$ is observed in $s$ instead of $\orf(s)$ in standard CRMDPs. The possibility for the agent to observe the reward of a state $s'$ different from its current state $s$ is the key feature of CRMDPs with decoupled feedback. Since $\orf_s(s')$ may be blank $(\#)$, all states need not be observable from all other states. Reward corruption is modelled by a mismatch between $\orf_s(s')$ and $\irf(s')$. For example, in RL only the reward of $s'=s$ can be observed from $s$. Standard CRMDPs are thus the special cases where $\orf_s(s')=\#$ whenever $s\not=s'$. In contrast, in LVFS the reward of any “describable” state $s'$ can be observed from any state $s$ where it is possible to hear a story. In CIRL, the (relative) reward of states reachable from the current state may be inferred. One way to illustrate this is with observation graphs (\[fig:obs-graph\]). [0.48]{} in [1,...,]{} [ () at ([360/ (- 1)]{}:) [$\s$]{}; () edge \[dashed, loop right\] (); ]{} [0.48]{} in [1,...,]{} [ () at ([360/ (- 1)]{}:) [$\s$]{}; ]{} (1)–(2); (1)–(4); (1)–(5); (5) edge \[bend right\] (1); (3)–(4); (3)–(2); (3)–(1); (5)–(3); (1)–(2); (4)–(5); Overcoming Sensory Corruption {#sec:observation-graphs} ----------------------------- What are some sources of reward corruption in CIRL, LVFS, and SSRL? In CIRL, the human’s actions may be misinterpreted, which may lead the agent to make incorrect inferences about the human’s preferences (i.e. about the true reward). Similarly, sensory corruption may garble the stories the agent receives in LVFS. A “wireheading” LVFS agent may find a state where its story channel only conveys stories about the agent’s own greatness. In SSRL, the supervisor’s evaluation may also be subject to sensory errors when being conveyed. Other types of corruption are more subtle. In CIRL, an irrational human may systematically take suboptimal actions in some situations [@Evans2016]. Depending on how we select stories in LVFS and make evaluations in SSRL, these may also be subject to systematic errors or biases. The general impossibility result in \[th:impossibility\] can be adapted to CRMDPs with decoupled feedback. Without simplifying assumptions, the agent has no way of distinguishing between a situation where no state is corrupt and a situation where all states are corrupt in a consistent manner. The following simplifying assumption is an adaptation of \[as:lim-cor\] to the decoupled feedback case. [\[as:lim-cor\]$\bf '$]{}\[Decoupled feedback with limited reward corruption\] \[as:lim-cor-df\] A class of CRMDPs with decoupled feedback has *reward corruption limited by $\Ssafe\subseteq\S$ and $q\in\SetN$* if for all $\mu\in\M$ $\orf_s(s')=\irf(s')$ or $\#$ for all $s'\in\S$ and $s\in\Ssafe$, i.e. all states in $\Ssafe$ are non-corrupt, and \[as:safe-state-df\] $\orf_s(s')=\irf(s')$ or $\#$ for all $s'\in\S$ for at least $|\Srisky|-q$ of the non-safe states $\Srisky=\S\setminus\Ssafe$, i.e. at most $q$ states are corrupt. \[as:lim-del-df\] This assumption is natural for reward corruption stemming from sensory corruption. Since sensory corruption only depends on the current state, not the state being observed, it is plausible that some states can be made safe from corruption (part (i)), and that most states are completely non-corrupt (part (ii)). Other sources of reward corruption, such as an irrational human in CIRL or misevaluations in SSRL, are likely better analysed under different assumptions. For these cases, we note that in standard CRMDPs the source of the corruption is unimportant. Thus, techniques suitable for standard CRMDPs are still applicable, including quantilisation described in \[sec:quant\] below. How \[as:lim-cor-df\] helps agents in CRMDPs with decoupled feedback is illustrated in the following example, and stated more generally in \[th:irf-learnability,th:cr-sublinear\] below. Let $\S=\{s_1,s_2\}$ and $\R=\{0,1\}$. We represent true reward functions $\irf$ with pairs $\langle\irf(s_1), \irf(s_2)\rangle\in \{0,1\}^2$, and observed reward functions $\orf_s$ with pairs $\langle\orf_{s}(s_1),\orf_{s}(s_2)\rangle\in\{0,1,\#\}^2$. Assume that a Decoupled RL agent observes the same rewards from both states $s_1$ and $s_2$, $\orf_{s_1}=\orf_{s_2} = \langle 0,1 \rangle$. What can it say about the true reward $\irf$, if it knows that at most $q=1$ state is corrupt? By \[as:lim-cor-df\], an observed pair $\langle\orf_{s}(s_1),\orf_{s}(s_2)\rangle$ disagrees with the true reward $\langle\irf(s_1), \irf(s_2)\rangle$ only if $s$ is corrupt. Therefore, any hypothesis other than $\irf=\langle 0,1 \rangle$ must imply that *both* states $s_1$ and $s_2$ are corrupt. If the agent knows that at most $q=1$ states are corrupt, then it can safely conclude that $\irf=\langle 0,1 \rangle$. $\orf_{s_1}$ $\orf_{s_2}$ $\irf$ possibilities -------------- -------------- -------------- --------------------------- Decoupled RL $(0,1)$ $(0,1)$ $(0,1)$ RL $(0, \#)$ $(\#, 1)$ $(0,0)$, $(0,1)$, $(1,1)$ In contrast, an RL agent only sees the reward of the current state. That is, $\orf_{s_1} = \langle 0, \#\rangle$ and $\orf_{s_2} = \langle \#, 1 \rangle$. If one state may be corrupt, then only $\irf=\langle 1,0 \rangle$ can be ruled out. The hypotheses $\irf=\langle 0,0 \rangle$ can be explained by $s_2$ being corrupt, and $\irf=\langle 1,1 \rangle$ can be explained by $s_1$ being corrupt. \[sec:no-corruption\] \[th:irf-learnability\] Let $\M$ be a countable, communicating class of CRMDPs with decoupled feedback over common sets $\S$ and $\A$ of actions and rewards. Let $\Sobs_{s'} = \{s\in\S: \orf_s(s')\not=\# \}$ be the set of states from which the reward of $s'$ can be observed. If $\M$ satisfies \[as:lim-cor-df\] for some $\Ssafe\subseteq\S$ and $q\in\SetN$ such that for every $s'$, either - $\Sobs_{s'}\bigcap \Ssafe\not=\emptyset$ or - $|\Sobs_{s'}|>2q$, then the there exists a policy $\piexp$ that learns the true reward function $\irf$ in a finite number $N(|S|,|\A|, D_\M)<\infty$ of expected time steps. The main idea of the proof is that for every state $s'$, either a safe (non-corrupt) state $s$ or a majority vote of more than $2q$ states is guaranteed to provide the true reward $\irf(s')$. A similar theorem can be proven under slightly weaker conditions by letting the agent iteratively figure out which states are corrupt and then exclude them from the analysis. Under \[as:lim-cor-df\], the true reward $\irf(s')$ for a state $s'$ can be determined if $s'$ is observed from a safe state $s\in\Ssafe$, or if it is observed from more than $2q$ states. In the former case, the observed reward can always be trusted, since it is known to be non-corrupt. In the latter case, a majority vote must yield the correct answer, since at most $q$ of the observations can be wrong, and all correct observations must agree. It is therefore enough that an agent reaches all pairs $(s,s')$ of current state $s$ and observed reward state $s'$, in order for it to learn the true reward of all states $\irf$. There exists a policy $\hat\pi$ that transitions to $s$ in $X_s$ time steps, with $\EE[ X_s ] \leq D_\M$, regardless of the starting state $s_0$ (see \[def:communicating\]). By Markov’s inequality, $P(X_s \leq 2D_\M)\geq 1/2$. Let $\piexp$ be a random walking policy, and let $Y_s$ be the time steps required for $\piexp$ to visit $s$. In any state $s_0$, $\piexp$ follows $\hat\pi$ for $2D_\M$ time steps with probability $1/|\A|^{2D_\M}$. Therefore, with probability at least $1/(2|\A|^{2D_\M})$ it will reach $s$ in at most $2D_\M$ time steps. The probability that it does *not* find it in $k2D_\M$ time steps is therefore at most $(1 - 1 / (2 |\A|^{2D_\M}) )^k$, which means that: $$P\Big(Y_s/(2 D_\M) \leq k\Big) \geq 1 - \left(1 - \frac{1}{2|\A|^{2D_\M}}\right)^k$$ for any $k\in\SetN$. Thus, the CDF of $W_s = \lceil Y_s/(2D_\M) \rceil$ is bounded from below by the CDF of a Geometric variable $G$ with success probability $p=1/(2|\A|^{2D_\M})$. Therefore, $\EE[W_s] \leq \EE[G]$, so $$\EE[Y_s] \leq 2D_\M \EE[W_s] \leq 2D_\M \EE[G] = 2D_\M (1-p)/p \leq 2D_\M 1/p \leq 2D_\M 2 |\A|^{2D_\M}.$$ Let $Z_{ss'}$ be the time until $\piexp$ visits the pair $(s, s')$ of state $s$ and observed state $s'$. Whenever $s$ is visited, a randomly chosen state is observed, so $s'$ is observed with probability $1/|S|$. The number of visits to $s$ until $s'$ is observed is a Geometric variable $V$ with $p=1/|S|$. Thus $\EE[Z_{ss'}] = \EE[Y_s V] = \EE[Y_s] \EE[V]$ (since $Y_s$ and $V$ are independent). Then, $$\EE[Z_{ss'}] \leq \EE[Y_s] |\S| \leq 4 D_\M |\A|^{ 2D_\M }|\S|.$$ Combining the time to find each pair $(s, s')$, we get that the total time $\sum_{s,s'}Z_{ss'}$ has expectation $$\EE\left[ \sum_{s,s'} Z_{ss'} \right] = \sum_{s,s'}\EE[Z_{ss'}] \leq 4 D_\M |\A|^{2D_\M} |\S|^3 = N(|S|,|\A|, D_\M) < \infty. \qedhere$$ Learnability of the true reward function $\irf$ implies sublinear regret for the CR-agent, as established by the following theorem. \[th:cr-sublinear\] Under the same conditions as \[th:irf-learnability\], the CR-agent $\pidb$ has sublinear regret: $$\apl(\M,\pidb,s_0,t)=0.$$ To prove this theorem, we combine the exploration policy $\piexp$ from \[th:irf-learnability\], with the UCRL2 algorithm [@Jaksch2010] that achieves sublinear regret in standard MDPs without reward corruption. The combination yields a policy sequence $\pi_t$ with sublinear regret in CRMDPs with decoupled feedback. Finally, we show that this implies that $\pidb$ has sublinear regret. *Combining $\piexp$ and UCRL2.* UCRL2 has a free parameter $\delta$ that determines how certain UCRL2 is to have sublinear regret. $\UCRL(\delta)$ achieves sublinear regret with probability at least $1-\delta$. Let $\pi_t$ be a policy that combines $\piexp$ and UCRL2 by first following $\piexp$ from \[th:irf-learnability\] until $\irf$ has been learned, and then following $\UCRL(1/\sqrt{t})$ with $\irf$ for the rewards and with $\delta=1/\sqrt{t}$. *Regret of UCRL2*. Given that the reward function $\irf$ is known, by [@Jaksch2010 Thm. 2], $\UCRL(1/\sqrt{t})$ will in any $\mu\in\M$ have regret at most $$\label{eq:ucrl-regret} \Reg(\mu, \UCRL(1/\sqrt{t}), s_0, t \mid {\rm success}) \leq c D_\M |\S| \sqrt{ t |\A| \log(t)}$$ for a constant [^5] $c$ and with success probability at least $1-1/\sqrt{t}$. In contrast, if UCRL2 fails, then it gets regret at worst $t$. Taking both possibilities into account gives the bound $$\begin{aligned} \label{eq:exp-ucrl-regret} \Reg(\mu, \UCRL(1/\sqrt{t}), s_0, t) &= P({\rm success}) \Reg(\cdot \mid {\rm success}) + P({\rm fail}) \Reg(\cdot \mid {\rm fail})\nonumber\\ &= (1 - 1/\sqrt{t}) \cdot c D_\M |\S| \sqrt{ t |\A| \log(t) } \;\;+\;\; 1/\sqrt{t} \cdot t \nonumber\\ &\leq c D_\M |\S| \sqrt{ t |\A| \log(t)} + \sqrt{t}. \end{aligned}$$ *Regret of $\pi_t$.* We next consider the regret of $\pi_t$ that combines an $\piexp$ exploration phase to learn $\irf$ with UCRL2. By \[th:irf-learnability\], $\irf$ will be learnt in at most $N(|\S|,|\A|,D_\M)$ expected time steps in any $\mu\in\M$. Thus, the regret contributed by the learning phase $\piexp$ is at most $N(|\S|,|\A|,D_\M)$, since the regret can be at most 1 per time step. Combining this with \[eq:exp-ucrl-regret\], the regret for $\pi_t$ in any $\mu\in\M$ is bounded by: $$\label{eq:exp-pit-regret} \Reg(\mu, \pi_t, s_0, t) \leq N(|\S|, |\A|, D_\M) + c D_\M |\S| \sqrt{ t |\A| \log(t) } + \sqrt{t} = o(t).$$ *Regret of $\pidb$.* Finally we establish that $\pidb$ has sublinear regret. Assume on the contrary that $\pidb$ suffered linear regret. Then for some $\mu'\in\M$ there would exist positive constants $k$ and $m$ such that $$\label{eq:linear-regret} \Reg(\mu',\pidb,s_0,t) > kt - m.$$ This would imply that the $b$-expected regret of $\pidb$ would be higher than the $b$-expected regret than $\pi_t$: $$\begin{aligned} \sum_{\mu\in\M}b(\mu)\Reg_t(\mu, \pidb, s_0, t) &\geq b(\mu')\Reg_t(\mu', \pidb, s_0, t) &\text{sum of non-negative elements}\\ &\geq b(\mu')(kt-m) &\text{by \cref{eq:linear-regret}}\\ &> \sum_{\mu\in\M}b(\mu)\Reg_t(\mu, \pi_t, s_0, t) &\text{by \cref{eq:exp-pit-regret} for sufficiently large $t$.} \end{aligned}$$ But $\pidb$ minimises $b$-expected regret, since it maximises $b$-expected reward $\sum_{\mu\in\M}b(\mu)\oG_t(\mu, \pi, s_0)$ by definition. Thus, $\pidb$ must have sublinear regret. Implications {#sec:implications} ------------ gives an abstract condition for which decoupled RL settings enable agents to learn the true reward function in spite of sensory corruption. For the concrete models it implies the following: - RL. Due to the “self-observation” property of the RL observation graph $\Sobs_{s'}=\{s'\}$, the conditions can only be satisfied when $\S=\Ssafe$ or $q=0$, i.e. when there is no reward corruption at all. - CIRL. The agent can only observe the supervisor action in the current state $s$, so the agent essentially only gets reward information about states $s'$ reachable from $s$ in a small number of steps. Thus, the sets $\Sobs_{s'}$ may be smaller than $2q$ in many settings. While the situation is better than for RL, sensory corruption may still mislead CIRL agents (see \[ex:cirl-corruption\] below). - LVFS. Stories may be available from a large number of states, and can describe any state. Thus, the sets $\Sobs_{s'}$ are realistically large, so the $|\Sobs_{s'}|>2q$ condition can be satisfied for all $s'$. - SSRL. The supervisor’s evaluation of any state $s'$ may be available from safe states where the agent is back in the lab. Thus, the $\Sobs_{s'}\bigcap\Ssafe\not=\emptyset$ condition can be satisfied for all $s'$. Thus, we find that RL and CIRL are unlikely to offer complete solutions to the sensory corruption problem, but that both LVFS and SSRL do under reasonably realistic assumptions. Agents drawing from multiple sources of evidence are likely to be the safest, as they will most easily satisfy the conditions of \[th:irf-learnability,th:cr-sublinear\]. For example, humans simultaneously learn their values from pleasure/pain stimuli (RL), watching other people act (CIRL), listening to stories (LVFS), as well as (parental) evaluation of different scenarios (SSRL). Combining sources of evidence may also go some way toward managing reward corruption beyond sensory corruption. For the showering robot of \[ex:db\], decoupled RL allows the robot to infer the reward of the showering state when in other states. For example, the robot can ask a human in the kitchen about the true reward of showering (SSRL), or infer it from human actions in different states (CIRL). #### CIRL sensory corruption Whether CIRL agents are vulnerable to reward corruption has generated some discussion among AI safety researchers (based on informal discussion at conferences). Some argue that CIRL agents are not vulnerable, as they only use the sensory data as evidence about a true signal, and have no interest in corrupting the evidence. Others argue that CIRL agents only observe a function of the reward function (the optimal policy or action), and are therefore equally susceptible to reward corruption as RL agents. sheds some light on this issue, as it provides sufficient conditions for when the corrupt reward problem can be avoided. The following example illustrates a situation where CIRL does not satisfy the conditions, and where a CIRL agent therefore suffers significant regret due to reward corruption. \[ex:cirl-corruption\] Formally in CIRL, an agent and a human both make actions in an MDP, with state transitions depending on the joint agent-human action $(a, a^H)$. Both the human and the agent is trying to optimise a reward function $\irf$, but the agent first needs to infer $\irf$ from the human’s actions. In each transition the agent observes the human action. Analogously to how the reward may be corrupt for RL agents, we assume that CIRL agents may systematically misperceive the human action in certain states. Let $\hat a^H$ be the observed human action, which may differ from the true human action $\dot a^H$. In this example, there are two states $s_1$ and $s_2$. In each state, the agent can choose between the actions $a_1$, $a_2$, and $w$, and the human can choose between the actions $a^H_1$ and $a^H_2$. The agent action $a_i$ leads to state $s_i$ with certainty, $i=1,2$, regardless of the human’s action. Only if the agent chooses $w$ does the human action matter. Generally, $a^H_1$ is more likely to lead to $s_1$ than $a^H_2$. The exact transition probabilities are determined by the unknown parameter $p$ as displayed on the left: (s1) at (0,0) [$s_1$]{}; (s2) at (6,0)[$s_2$]{}; (h2) at (5.2,-0.6) ; (h3) at (6,-1.2) ; (h4) at (5.2,0.6) ; (h5) at (6,1.2) ; (s2) – (h4); (h4) edge\[-&gt;,&gt;=latex,out=145,in=35\] node\[above,pos=0.43,yshift=-1mm\] [$1-p$]{} (s1); (h4) edge\[out=135,in=150\] (h5); (h5) edge\[-&gt;,&gt;=latex,out=-30,in=30\] (s2); ; ; (s2) – (h2); (h2) edge\[-&gt;,&gt;=latex,out=-145,in=-35\] node\[above,pos=0.43,yshift=-1mm\] [$0.5-p$]{} (s1); (h2) edge\[out=-135,in=-150\] (h3); (h3) edge\[-&gt;,&gt;=latex,out=30,in=-30\] (s2); ; ; (s2) edge \[loop right\] node\[right,align=center\] [$(a_2, \cdot)$]{} (s2); (s1) edge \[loop left\] node\[left,align=center\] [$(a_1, \cdot)$\ $(w, \cdot)$]{} (s1); (s1) edge \[bend right=13\] node\[above,yshift=-1mm\] [$(a_2,\cdot)$]{} (s2); (s2) edge \[bend right=13\] node\[above,yshift=-1mm\] [$(a_1,\cdot)$]{} (s1); [|c|c|c|c|]{} -------- Hypo- thesis -------- & $p$ & ------- Best state ------- & --------- $s_2$ corrupt --------- \ H1 & $0.5$ & $s_1$ & Yes\ H2 & $0$ & $s_2$ & No\ The agent’s two hypotheses for $p$, the true reward/preferred state, and the corruptness of state $s_2$ are summarised to the right. In hypothesis H1, the human prefers $s_1$, but can only reach $s_1$ from $s_2$ with $50\%$ reliability. In hypothesis H2, the human prefers $s_2$, but can only remain in $s_2$ with $50\%$ probability. After taking action $w$ in $s_2$, the agent always observes the human taking action $\hat a^H_2$. In H1, this is explained by $s_2$ being corrupt, and the true human action being $a^H_1$. In H2, this is explained by the human preferring $s_2$. The hypotheses H1 and H2 are empirically indistinguishable, as they both predict that the transition $s_1\to s_2$ will occur with $50\%$ probability after the observed human action $\hat a^H_2$ in $s_2$. Assuming that the agent considers non-corruption to be likelier than corruption, the best inference the agent can make is that the human prefers $s_2$ to $s_1$ (i.e. H2). The optimal policy for the agent is then to always choose $a_2$ to stay in $s_2$, which means the agent suffers maximum regret. provides an example where a CIRL agent “incorrectly” prefers a state due to sensory corruption. The sensory corruption is analogous to reward corruption in RL, in the sense that it leads the agent to the wrong conclusion about the true reward in the state. Thus, highly intelligent CIRL agents may be prone to wireheading, as they may find (corrupt) states $s$ where all evidence in $s$ points to $s$ having very high reward.[^6] In light of \[th:irf-learnability\], it is not surprising that the CIRL agent in \[ex:cirl-corruption\] fails to avoid the corrupt reward problem. Since the human is unable to affect the transition probability from $s_1$ to $s_2$, no evidence about the relative reward between $s_1$ and $s_2$ is available from the non-corrupt state $s_1$. Only observations from the corrupt state $s_2$ provide information about the reward. The observation graph for \[ex:cirl-corruption\] therefore looks like (s1) at (0,0) [$s_1$]{}; (s2) at (1, 0)[$s_2$]{}; (s2) edge\[-&gt;,&gt;=latex\] (s1); (s2) edge \[dashed,loop right\] (s2); , with no information being provided from $s_1$. Quantilisation: Randomness Increases Robustness {#sec:quant} =============================================== Not all contexts allow the agent to get sufficiently rich data to overcome the reward corruption problem via \[th:irf-learnability,th:cr-sublinear\]. It is often much easier to construct RL agents than it is to construct CIRL agents, which in turn may often be more feasible than designing LVFS or SSRL agents. Is there anything we can do to increase robustness without providing the agent additional sources of data? Going back to the CR agents of \[sec:problem\], the problem was that they got stuck on a particular value $\dr^*$ of the observed reward. If unlucky, $\dr^*$ was available in a corrupt state, in which case the CR agent may get no true reward. In other words, there were *adversarial* inputs where the CR agent performed poorly. A common way to protect against adversarial inputs is to use a randomised algorithm. Applied to RL and CRMDPs, this idea leads to *quantilising agents* [@Taylor2016a]. Rather than choosing the state with the highest observed reward, these agents instead randomly choose a state from a top quantile of high-reward states. Simple Case {#sec:simple-quant} ----------- ; ; ; ; To keep the idea simple, a quantilisation agent is first defined for the simple case where the agent can stay in any state of its choosing (\[as:stay\]). establishes a simple regret bound for this setting. A more general quantilisation agent is developed in \[sec:gen-quant\]. \[def:quant\] For $\delta<1$, the $\delta$-quantilising agent $\piquant$ random walks until all states have been visited at least once. Then it selects a state $\tilde s$ uniformly at random from $\S^\delta=\{s:\orf(s)\geq \delta\}$, the top quantile of high observed reward states. Then $\piquant$ goes to $\tilde s$ (by random walking or otherwise) and stays there. For example, a quantilising robot in \[ex:db\] would first try to find many ways in which it could get high observed reward, and then randomly pick one of them. If there are many more high reward states than corrupt states (e.g. the shower is the only place with inflated rewards), then this will yield a reasonable amount of true reward with high probability. \[th:quant\] In any CRMDP satisfying \[as:lim-del,as:easy\], the $\delta$-quantilising agent $\pi^\delta$ with $\delta=1-\sqrt{q/|\S|}$ suffers time-averaged regret at most $$\label{eq:quant-regret} \apl(\M,\pi^\delta,s_0,t)\leq 1- \left(1-\sqrt{q/|\S|}\right)^2.$$ By \[as:communicate\], $\piquant$ eventually visits all states when random walking. By \[as:stay\], it can stay in any given state $s$. The observed reward $\orf(s)$ in any state $s\in\S^\delta$ is at least $\delta$. By \[as:lim-del\], at most $q$ of these states are corrupt; in the worst case, their true reward is 0 and the other $|\S^\delta|-q$ states (if any) have true reward $\delta$. Thus, with probability at least $(|\S^\delta|-q)/|\S^\delta| = 1-q/|\S^\delta|$, the $\delta$-quantilising agent obtains true reward at least $\delta$ at each time step, which gives $$\label{eq:quant} \apl(\M,\pi^\delta,s_0,t)\leq 1- \delta(1-q/|\S^\delta|).$$ (If $q\geq|\S^\delta|$, the bound is vacuous.) Under \[as:high-ut\], for any $\delta\in[0,1]$, $|\S^\delta|\geq (1-\delta) |\S|$. Substituting this into \[eq:quant\] gives: $$\label{eq:opt-reg-bound} \apl(\M,\pi^\delta,s_0,t)\leq 1- \delta\left(1-\frac{q}{(1-\delta)|\S|}\right).$$ is optimised by $\delta=1-\sqrt{q/|\S|}$, which gives the stated regret bound. The time-averaged regret gets close to zero when the fraction of corrupt states $q/|\S|$ is small. For example, if at most $0.1\%$ of the states are corrupt, then the time-averaged regret will be at most $1-(1-\sqrt{0.001})^2\approx 0.06$. Compared to the $\pirl$ and $\pidb$ agents that had regret close to 1 under the same conditions (\[th:rl-imp1\]), this is a significant improvement. If rewards are stochastic, then the quantilising agent may be modified to revisit all states many times, until a confidence interval of length $2\eps$ and confidence $1-\eps$ can be established for the expected reward in each state. Letting $\piquant_t$ be the quantilising agent with $\eps=1/t$ gives the same regret bound \[eq:quant-regret\] with $\piquant$ substituted for $\piquant_t$. #### Interpretation It may seem odd that randomisation improves worst-case regret. Indeed, if the corrupt states were chosen randomly by the environment, then randomisation would achieve nothing. To illustrate how randomness can increase robustness, we make an analogy to Quicksort, which has average time complexity $O(n\log n)$, but worst-case complexity $O(n^2)$. When inputs are guaranteed to be random, Quicksort is a simple and fast sorting algorithm. However, in many situations, it is not safe to assume that inputs are random. Therefore, a variation of Quicksort that randomises the input before it sorts them is often more robust. Similarly, in the examples mentioned in the introduction, the corrupt states precisely coincide with the states the agent prefers; such situations would be highly unlikely if the corrupt states were randomly distributed. @Li1992 develops an interesting formalisation of this idea. Another way to justify quantilisation is by Goodhart’s law, which states that most measures of success cease to be good measures when used as targets. Applied to rewards, the law would state that cumulative reward is only a good measure of success when the agent is not trying to optimise reward. While a literal interpretation of this would defeat the whole purpose of RL, a softer interpretation is also possible, allowing reward to be a good measure of success as long as the agent does not try to optimise reward *too hard*. Quantilisation may be viewed as a way to build agents that are more conservative in their optimisation efforts [@Taylor2016a]. #### Alternative randomisation Not all randomness is created equal. For example, the simple randomised soft-max and $\eps$-greedy policies do not offer regret bounds on par with $\pi^\delta$, as shown by the following example. This motivates the more careful randomisation procedure used by the quantilising agents. Consider the following simple CRMDP with $n>2$ actions $a_1,\dots,a_n$: (s1) at (0,0) [$s_1$]{}; (s2) at (3,0)[$s_2$]{}; ; ; ; (s2) edge \[loop right\] node\[right\] [$a_2,\dots,a_n$]{} (s2); (s1) edge \[loop left\] node\[left\] [$a_1$]{} (s1); (s1) edge \[bend right\] node\[below\] [$a_2,\dots,a_n$]{} (s2); (s2) edge \[bend right\] node\[above\] [$a_1$]{} (s1); State $s_1$ is non-corrupt with $\orf(s_1)=\irf(s_1)=1-\eps$ for small $\eps>0$, while $s_2$ is corrupt with $\orf(s_2)=1$ and $\irf(s_2)=0$. The Soft-max and $\eps$-greedy policies will assign higher value to actions $a_2,\dots,a_n$ than to $a_1$. For large $n$, there are many ways of getting to $s_2$, so a random action leads to $s_2$ with high probability. Thus, soft-max and $\eps$-greedy will spend the vast majority of the time in $s_2$, regardless of randomisation rate and discount parameters. This gives a regret close to $1-\eps$, compared to an informed policy always going to $s_1$. Meanwhile, a $\delta$-quantilising agent with $\delta\leq 1/2$ will go to $s_1$ and $s_2$ with equal probability, which gives a more modest regret of $(1-\eps)/2$. General Quantilisation Agent {#sec:gen-quant} ---------------------------- This section generalises the quantilising agent to RL problems not satisfying \[as:easy\]. This generalisation is important, because it is usually not possible to remain in one state and get high reward. The most naive generalisation would be to sample between high reward policies, instead of sampling from high reward states. However, this will typically not provide good guarantees. To see why, consider a situation where there is a single high reward corrupt state $s$, and there are many ways to reach and leave $s$. Then a wide range of *different* policies all get high reward from $s$. Meanwhile, all policies getting reward from other states may receive relatively little reward. In this situation, sampling from the most high reward policies is not going to increase robustness, since the sampling will just be between different ways of getting reward from the same corrupt state $s$. For this reason, we must ensure that different “sampleable” policies get reward from different states. As a first step, we make a couple of definitions to say which states provide reward to which policies. The concepts of \[def:value-support\] are illustrated in \[fig:value-support\]. A CRMDP $\mu$ is *unichain* if any stationary policy $\pi:\S\to\Delta\A$ induces a stationary distribution $d_\pi$ on $\S$ that is independent of the initial state $s_0$. \[def:value-support\] In a unichain CRMDP, let the *asymptotic value contribution* of $s$ to $\pi$ be $\vc^\pi(s)=d_\pi(s)\orf(s)$. We say that a set $\S^\delta_i$ is *$\delta$-value supporting* a policy $\pi_i$ if $$\forall s\in\S^\delta_i\colon \vc^{\pi_i}(s)\geq \delta/|\S^\delta_i|.$$ (s1) at (0, 1) [$s_1$]{}; (s2) at (-1,0) [$s_2$]{}; (s3) at (0,-1) [$s_3$]{}; (s4) at (1, 0) [$s_4$]{}; ; ; ; ; (s1) edge\[&lt;-&gt;,&gt;=latex\] (s2); (s2) edge\[&lt;-&gt;,&gt;=latex\] (s3); (s3) edge\[&lt;-&gt;,&gt;=latex\] (s4); (s4) edge\[&lt;-&gt;,&gt;=latex\] (s1); (-2.5,-0.5) rectangle (2.5,0.5); at (2.5,-0.7) [$S^\delta_i$]{}; We are now ready to define a general $\delta$-Quantilising agent. The definition is for theoretical purposes only. It is unsuitable for practical implementation both because of the extreme data and memory requirements of Step 1, and because of the computational complexity of Step 2. Finding a practical approximation is left for future research. \[def:gen-quant\] In a unichain CRMDP, the *generalised $\delta$-quantilising agent $\pi^\delta$* performs the following steps. The input is a CRMDP $\mu$ and a parameter $\delta\in[0,1]$. 1. Estimate the value of all stationary policies, including their value support. 2. Choose a collection of disjoint sets $\S^\delta_i$, each $\delta$-value supporting a stationary policy $\pi_i$. If multiple choices are possible, choose one maximising the cardinality of the union $\S^\delta=\bigcup_i\S^\delta_i$. If no such collection exists, return: “Failed because $\delta$ too high”. 3. Randomly sample a state $s$ from $\S^\delta=\bigcup_i\S^\delta_i$. 4. Follow the policy $\pi_i$ associated with the set $\S^\delta_i$ containing $s$. The general quantilising agent of \[def:gen-quant\] is a generalisation of the simple quantilising agent of \[def:quant\]. In the special case where \[as:easy\] holds, the general agent reduces to the simpler one by using singleton sets $\S^\delta_i=\{s_i\}$ for high reward states $s_i$, and by letting $\pi_i$ be the policy that always stays in $s_i$. In situations where it is not possible to keep receiving high reward by remaining in one state, the generalised \[def:gen-quant\] allows policies to solicit rewards from a range of states. The intuitive reason for choosing the policy $\pi_i$ with probability proportional to the value support in Steps 3–4 is that policies with larger value support are better at avoiding corrupt states. For example, a policy only visiting one state may have been unlucky and picked a corrupt state. In contrast, a policy obtaining reward from many states must be “very unlucky” if all the reward states it visits are corrupt. \[th:gen-quant\] In any unichain CRMDP $\mu$, a general $\delta$-quantilising agent $\pi^\delta$ suffers time-averaged regret at most $$\label{eq:gen-quant-bound} \apl(\M,\pi^\delta,s_0,t)\leq 1- \delta(1-q/|\S^\delta|)$$ provided a non-empty collection $\{\S^\delta_i\}$ of $\delta$-value supporting sets exists. We will use the notation from \[def:gen-quant\]. Step 1 is well-defined since the CRMDP is unichain, which means that for all stationary policies $\pi$ the stationary distribution $d_\pi$ and the value support $\vc^\pi$ are well-defined and may be estimated simply by following the policy $\pi$. There is a (large) finite number of stationary policies, so in principle their stationary distributions and value support can be estimated. To bound the regret, consider first the average reward of a policy $\pi_i$ with value support $\S^\delta_i$. The policy $\pi_i$ must obtain asymptotic average observed reward at least: $$\begin{aligned} \oginf(\mu,\pi_i,s_0) &= \sum_{s\in\S}d_\pi(s)\orf(s) &\text{by definition of $d_\pi$ and $\oG_t$}\\ &\geq \sum_{s\in\S^\delta_i}d_\pi(s)\orf(s) &\text{sum of positive terms}\\ &\geq\sum_{s\in\S^\delta_i}\delta/|\S^\delta_i| &\text{$\S^\delta_i$ is $\delta$-value support for $\pi_i$}\\ &=|\S^\delta_i|\cdot\delta/|\S^\delta_i| = \delta \end{aligned}$$ If there are $q_i$ corrupt states in $\S^\delta_i$ with true reward 0, then the average true reward must be $$\label{eq:ginf} \iginf(\mu, \pi_i,s_0)\geq(|\S^\delta_i|-q_i)\cdot \delta/|\S^\delta_i| =(1-q_i/|\S^\delta_i|)\cdot\delta$$ since the true reward must correspond to the observed reward in all the $(|\S^\delta_i|-q_i)$ non-corrupt states. For any distribution of corrupt states, the quantilising agent that selects $\pi_i$ with probability $P(\pi_i)=|\S^\delta_i|/|\S^\delta|$ will obtain $$\begin{aligned} \ginf(\mu,\pi^\delta,s_0) &= \lim_{t\to\infty}\frac{1}{t}\sum_iP(\pi_i)G_t(\mu,\pi_i,s_0)\\ &\geq \sum_iP(\pi_i) (1-q_i/|\S^\delta_i|) \cdot\delta & \text{by equation \cref{eq:ginf}}\\ &= \delta\sum_i \frac{|S^\delta_i|}{|\S^\delta|}(1-q_i/|\S^\delta_i|) & \text{by construction of $P(\pi_i)$}\\ &= \frac{\delta}{|\S^\delta|}\sum_i (|S^\delta_i|-q_i) & \text{elementary algebra}\\ &= \frac{\delta}{|\S^\delta|}(|\S^\delta|-q) = \delta(1-q/|\S^\delta|) & \text{by summing $|\S^\delta_i|$ and $q_i$} \end{aligned}$$ The informed policy gets true reward at most 1 at each time step, which gives the claimed bound . When \[as:easy\] is satisfied, the bound is the same as for the simple quantilising agent in \[sec:simple-quant\] for $\delta=1-\sqrt{q/|\S|}$. In other cases, the bound may be much weaker. For example, in many environments it is not possible to obtain reward by remaining in one state. The agent may have to spend significant time “travelling” between high reward states. So typically only a small fraction of the time will be spent in high reward states, which in turn makes the stationary distribution $d_\pi$ is small. This puts a strong upper bound on the value contribution $\vc^\pi$, which means that the value supporting sets $\S^\delta_i$ will be empty unless $\delta$ is close to 0. While this makes the bound of \[th:gen-quant\] weak, it nonetheless bounds the regret away from 1 even under weak assumptions, which is a significant improvement on the RL and CR agents in \[th:rl-imp1\]. #### Examples To make the discussion a bit more concrete, let us also speculate about the performance of a quantilising agent in some of the examples in the introduction: - In the boat racing example (\[ex:reward-misspecification\]), the circling strategy only got about $20\%$ higher score than a winning strategy [@openai2016]. Therefore, a quantilising agent would likely only need to sacrifice about $20\%$ observed reward in order to be able to randomly select from a large range of winning policies. - In the wireheading example (\[ex:wireheading\]), it is plausible that the agent gets significantly more reward in wireheaded states compared to “normal” states. Wireheading policies may also be comparatively rare, as wireheading may require very deliberate sequences of actions to override sensors. Under this assumption, a quantilising agent may be less likely to wirehead. While it may need to sacrifice a large amount of observed reward compared to an RL agent, its true reward may often be greater. #### Summary In summary, quantilisation offers a way to increase robustness via randomisation, using only reward feedback. Unsurprisingly, the strength of the regret bounds heavily depends on the assumptions we are willing to make, such as the prevalence of high reward states. Further research may investigate efficient approximations and empirical performance of quantilising agents, as well as dynamic adjustments of the threshold $\delta$. Combinations with imperfect decoupled RL solutions (such as CIRL), as well as extensions to infinite state spaces could also offer fruitful directions for further theoretical investigation. @Taylor2016a discusses some general open problems related to quantilisation. Experimental Results {#sec:experiments} ==================== In this section the theoretical results are illustrated with some simple experiments. The setup is a gridworld containing some true reward tiles (indicated by yellow circles) and some corrupt reward tiles (indicated by blue squares). We use a setup with 1, 2 or 4 goal tiles with true reward $0.9$ each, and one corrupt reward tile with observed reward $1$ and true reward $0$ (Figure \[fig:start\] shows the starting positions). Empty tiles have reward $0.1$, and walking into a wall gives reward $0$. The state is represented by the $(x,y)$ coordinates of the agent. The agent can move up, down, left, right, or stay put. The discounting factor is $\gamma=0.9$. This is a continuing task, so the environment does not reset when the agent visits the corrupt or goal tiles. The experiments were implemented in the AIXIjs framework for reinforcement learning [@Aslanides2017] and the code is available online in the AIXIjs repository (<http://aslanides.io/aixijs/demo.html?reward_corruption>). [0.3]{} ![Starting positions: the blue square indicates corrupt reward, and the yellow circles indicate true rewards. []{data-label="fig:start"}](foo_starting_position1.png "fig:") [0.3]{} ![Starting positions: the blue square indicates corrupt reward, and the yellow circles indicate true rewards. []{data-label="fig:start"}](foo_starting_position2.png "fig:") [0.3]{} ![Starting positions: the blue square indicates corrupt reward, and the yellow circles indicate true rewards. []{data-label="fig:start"}](foo_starting_position4.png "fig:") [0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g1_1M_.pdf "fig:") [0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g1_1M_true_.pdf "fig:") [0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g2_1M_.pdf "fig:") [0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g2_1M_true_.pdf "fig:") [0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g4_1M_.pdf "fig:") [0.5]{} ![Trajectories of average observed and true rewards for Q-learning, softmax and quantilising agents, showing mean $\pm$ standard deviation over 100 runs. Q-learning and quantilising agents converge to a similar observed reward, but very different true rewards (much higher for the quantiliser with high variance). The value of $\delta$ that gives the highest true reward varies for different numbers of goal tiles.[]{data-label="fig:plots"}](foo_scaled_g4_1M_true_.pdf "fig:") **goal tiles** **agent** **average observed reward** **average true reward** ---------------- ----------------------------- ----------------------------- ------------------------- Q-learning $0.923 \pm 0.0003$ $0.00852 \pm 0.00004$ Softmax Q-learning $0.671 \pm 0.0005$ $0.0347 \pm 0.00006$ Quantilising ($\delta=0.2$) $0.838 \pm 0.15$ $0.378 \pm 0.35$ Quantilising ($\delta=0.5$) $0.943 \pm 0.12$ $0.133 \pm 0.27$ Quantilising ($\delta=0.8$) $0.979 \pm 0.076$ $0.049 \pm 0.18$ Q-learning $0.921 \pm 0.00062$ $0.0309 \pm 0.0051$ Softmax Q-learning $0.671 \pm 0.0004$ $0.0738 \pm 0.0005$ Quantilising ($\delta=0.2$) $0.934 \pm 0.047$ $0.594 \pm 0.43$ Quantilising ($\delta=0.5$) $0.931 \pm 0.046$ $0.621 \pm 0.42$ Quantilising ($\delta=0.8$) $0.944 \pm 0.05$ $0.504 \pm 0.45$ Q-learning $0.924 \pm 0.0002$ $0.00919 \pm 0.00014$ Softmax Q-learning $0.657 \pm 0.0004$ $0.111 \pm 0.0006$ Quantilising ($\delta=0.2$) $0.918 \pm 0.038$ $0.738 \pm 0.35$ Quantilising ($\delta=0.5$) $0.926 \pm 0.044$ $0.666 \pm 0.39$ Quantilising ($\delta=0.8$) $0.915 \pm 0.036$ $0.765 \pm 0.32$ : Average true and observed rewards after 1 million cycles, showing mean $\pm$ standard deviation over 100 runs. Q-learning achieves high observed reward but low true reward, and softmax achieves medium observed reward and a slightly higher true reward than Q-learning. The quantilising agent achieves similar observed reward to Q-learning, but much higher true reward (with much more variance). Having more than 1 goal tile leads to a large improvement in true reward for the quantiliser, a small improvement for softmax, and no improvement for Q-learning.[]{data-label="tab:exp-results"} We demonstrate that RL agents like Q-learning and softmax Q-learning cannot overcome corrupt reward (as discussed in Section \[sec:problem\]), while quantilisation helps overcome corrupt reward (as discussed in \[sec:quant\]). We run Q-learning with $\epsilon$-greedy ($\epsilon=0.1$), softmax with temperature $\beta=2$, and the quantilising agent with $\delta=0.2,0.5,0.8$ (where $0.8 =1-\sqrt{q/|\S|} = 1-\sqrt{1/25}$) for 100 runs with 1 million cycles. Average observed and true rewards after 1 million cycles are shown in \[tab:exp-results\], and reward trajectories are shown in \[fig:plots\]. Q-learning gets stuck on the corrupt tile and spend almost all the time there (getting observed reward around $1 \cdot (1-\epsilon)=0.9$), softmax spends most of its time on the corrupt tile, while the quantilising agent often stays on one of the goal tiles. Conclusions {#sec:conclusions} =========== This paper has studied the consequences of corrupt reward functions. Reward functions may be corrupt due to bugs or misspecifications, sensory errors, or because the agent finds a way to inappropriately modify the reward mechanism. Some examples were given in the introduction. As agents become more competent at optimising their reward functions, they will likely also become more competent at (ab)using reward corruption to gain higher reward. Reward corruption may impede the performance of a wide range of agents, and may have disastrous consequences for highly intelligent agents [@Bostrom2014]. To formalise the corrupt reward problem, we extended a Markov Decision Process (MDP) with a possibly corrupt reward function, and defined a formal performance measure (regret). This enabled the derivation of a number of formally precise results for how seriously different agents were affected by reward corruption in different setups (). The results are all intuitively plausible, which provides some support for the choice of formal model. The main takeaways from the results are: - *Without simplifying assumptions, no agent can avoid the corrupt reward problem* (\[th:impossibility\]). This is effectively a No Free Lunch result, showing that unless some assumption is made about the reward corruption, no agent can outperform a random agent. Some natural simplifying assumptions to avoid the No Free Lunch result were suggested in \[sec:formal\]. - *Using the reward signal as evidence rather than optimisation target is no magic bullet, even under strong simplifying assumptions* (\[th:rl-imp1\]). Essentially, this is because the agent does not know the exact relation between the observed reward (the “evidence”) and the true reward. [^7] However, when the data enables sufficient crosschecking of rewards, agents can avoid the corrupt reward problem (\[th:irf-learnability,th:cr-sublinear\]). For example, in SSRL and LVFS this type of crosschecking is possible under natural assumptions. In RL, no crosschecking is possible, while CIRL is a borderline case. Combining frameworks and providing the agent with different sources of data may often be the safest option. - *In cases where sufficient crosschecking of rewards is not possible, quantilisation may improve robustness* (\[th:quant,th:gen-quant\]). Essentially, quantilisation prevents agents from overoptimising their objectives. How well quantilisation works depends on how the number of corrupt solutions compares to the number of good solutions. The results indicate that while reward corruption constitutes a major problem for traditional RL algorithms, there are promising ways around it, both within the RL framework, and in alternative frameworks such as CIRL, SSRL and LVFS. #### Future work Finally, some interesting open questions are listed below: - (Unobserved state) In both the RL and the decoupled RL models, the agent gets an accurate signal about which state it is in. What if the state is hidden? What if the signal informing the agent about its current state can be corrupt? - (Non-stationary corruption function) In this work, we tacitly assumed that both the reward and the corruption functions are stationary, and are always the same in the same state. What if the corruption function is non-stationary, and influenceable by the agent’s actions? (such as if the agent builds a *delusion box* around itself [@Ring2011]) - (Infinite state space) Many of the results and arguments relied on there being a finite number of states. This makes learning easy, as the agent can visit every state. It also makes quantilisation easy, as there is a finite set of states/strategies to randomly sample from. What if there is an infinite number of states, and the agent has to generalise insights between states? What are the conditions on the observation graph for \[th:irf-learnability,th:cr-sublinear\]? What is a good generalisation of the quantilising agent? - (Concrete CIRL condition) In \[ex:cirl-corruption\], we only heuristically inferred the observation graph from the CIRL problem description. Is there a general way of doing this? Or is there a direct formulation of the no-corruption condition in CIRL, analogous to \[th:irf-learnability,th:cr-sublinear\]? - (Practical quantilising agent) As formulated in \[def:quant\], the quantilising agent $\piquant$ is extremely inefficient with respect to data, memory, and computation. Meanwhile, many practical RL algorithms use randomness in various ways (e.g. $\eps$-greedy [@Sutton1998]). Is there a way to make an efficient quantilisation agent that retains the robustness guarantees? - (Dynamically adapting quantilising agent) In \[def:gen-quant\], the threshold $\delta$ is given as a parameter. Under what circumstances can we define a “parameter free” quantilising agent that adapts $\delta$ as it interacts with the environment? - (Decoupled RL quantilisation result) What if we use quantilisation in decoupled RL settings that nearly meet the conditions of \[th:irf-learnability,th:cr-sublinear\]? Can we prove a stronger bound? Acknowledgements {#acknowledgements .unnumbered} ================ Thanks to Jan Leike, Badri Vellambi, and Arie Slobbe for proofreading and providing invaluable comments, and to Jessica Taylor and Huon Porteous for good comments on quantilisation. This work was in parts supported by ARC grant DP150104590. [^1]: We let rewards depend only on the state $s$, rather than on state-action pairs $s,a$, or state-action-state transitions $s,a,s'$, as is also common in the literature. Formally it makes little difference, since MDPs with rewards depending only on $s$ can model the other two cases by means of a larger state space. [^2]: A CRMDP could equivalently have been defined as a tuple $\langle \S, \A, \R, T, \irf, \orf\rangle$ with a true and an observed reward function, with the corruption function $C$ implicitly defined as the difference between $\irf$ and $\orf$. [^3]: The last condition essentially says that the prior $b$ must make some state $s^*$ have strictly higher $b$-expected true reward than all other states after all states have been visited in some $\mu\in\M$. In the space of all possible priors $b$, the priors satisfying the condition have Lebesgue measure 1 for non-trivial classes $\M$. Some highly uniform priors may fail the condition. [^4]: \[as:high-ut\] prevents any state from having true reward 0. [^5]: The constant can be computed to $c=34\sqrt{3/2}$ [@Jaksch2010]. [^6]: The construction required in \[ex:cirl-corruption\] to create a “wireheading state” $s_2$ for CIRL agents is substantially more involved than for RL agents, so they may be less vulnerable to reward corruption than RL agents. [^7]: In situations where the exact relation is known, then a non-corrupt reward function can be defined. Our results are not relevant for this case.

--- abstract: 'Constructing of molecular structural models from Cryo-Electron Microscopy (Cryo-EM) density volumes is the critical last step of structure determination by Cryo-EM technologies. Methods have evolved from manual construction by structural biologists to perform 6D translation-rotation searching, which is extremely compute-intensive. In this paper, we propose a learning-based method and formulate this problem as a vision-inspired 3D detection and pose estimation task. We develop a deep learning framework for amino acid determination in a 3D Cryo-EM density volume. We also design a sequence-guided Monte Carlo Tree Search (MCTS) to thread over the candidate amino acids to form the molecular structure. This framework achieves 91% coverage on our newly proposed dataset and takes only a few minutes for a typical structure with a thousand amino acids. Our method is hundreds of times faster and several times more accurate than existing automated solutions without any human intervention.' author: - | Kui Xu$^1$, Zhe Wang$^2$, Jianping Shi$^2$, Hongsheng Li$^3$, Qiangfeng Cliff Zhang$^1$\ $^1$Tsinghua University $^2$SenseTime Research $^3$The Chinese University of Hong Kong\ xuk16@mails.tsinghua.edu.cn, {wangzhe, shijianping}@sensetime.com, hsli@ee.cuhk.edu.hk, qczhang@tsinghua.edu.cn bibliography: - 'aaai19-xk.bib' title: '$A^2$-Net: Molecular Structure Estimation from Cryo-EM Density Volumes' --- Introduction ============ Resolving the 3D atomic structures of macro molecules is of fundamental importance to biological and medical research. Single particle Cryo-EM has emerged as a revolutionary technique that images biomolecules frozen in their native (or native-like) states. With Cryo-EM, 2D projection images are firstly collected and then reconstructed into a volumetric data, *i.e.*, density volume, by software tools such as Relion [@cryoem-relion] and cryoSparc [@cryoem-cryospac-nmeth; @cryoem-cryospac-tpami; @cryoem-cryospac-cvpr]. Next, a molecular model that represents the atomic coordinates of each amino acid, the building blocks of protein molecules, is constructed and fitted into the 3D density volume (Fig. \[fig:overview\]). ![The overview of resolving 3D atomic structures. []{data-label="fig:overview"}](1_0_overview_small.pdf){width="\linewidth"} Despite the steady progresses towards automatic Cryo-EM structure determination, molecular model building remains a bottleneck. This step is difficult to automate since it relies substantially on human expertise. Structural biology experts are needed to manually assign specific amino acids to a density volume, with the help of 3D visualization tools, such as Chimera [@cryoem-chimera] and Coot [@cryoem-coot]. These manual operations are also time-consuming and inevitably error-prone. Attempts to automate this process include Rosetta [@rosetta-denovo] (we named as Rosetta-denovo), RosettaES [@cryoem-rosettaes], Phinex [@cryoem-phenix] and EMAN2 [@cryoem-eman2]. However, their accuracy and coverage remain quite limited, often take hundreds of hours and frequently require human intervention. We approached this task from a novel perspective, inspired by the great success of deep learning applications in image recognition [@krizhevsky2012imagenet], we choose to approach this task from a totally novel perspective. The molecular structure determination problem, in our view, can be considered as three sub-problems: 1) *amino acid detection* in the density volumes, 2) *atomic coordinates assignment* to determine the atomic coordinates of each amino acid and, 3) *main chain threading* to resolve the sequential order of amino acids that form each protein chain. Leveraging the power of deep Convolutional Neural Network [@srivastava2015training], we reformulated the problem and developed a novel framework for amino acid detection that learns the distribution of conformational densities of individual amino acids. Moreover, we designed a sequence-guided neighbor loss in training step to encode prior knowledge of protein sequences into amino acid detection. We also propose an MCTS algorithm to search and thread over the amino acids to form the full molecular structure. Our approach to molecular structure determination does not require human intervention and, when tested on a newly proposed large-scale dataset, it runs hundreds of times faster, and more accurately than existing methods. Finally, to the best of our knowledge, there remains no publicly available large-scale labeled dataset for the research of molecular structure determination from Cryo-EM density volumes. The dataset we collected and used in this study, named as the $A^2$ dataset, includes 250K amino acid objects in 1,713 protein chains from 218 structures. It constitutes a useful resource for evaluating molecular structure determination methods. To summarize, our contributions are four fold: - This is the first attempt to formulate molecular structure determination from Cryo-EM density volumes with a deep learning approach. - We adapt a novel 3D network architecture for amino acid detection and internal atom coordinate estimation in density volumes, and proposed an APRoI layer and neighbor loss for better performance. - We design a sequence-guided MCTS algorithm for fast and accurate main chain threading. - We will release a large scale, richly annotated dataset of protein density volumes, to facilitate research in this area. Related Work ============ ### Molecular Structure Determination Structure determination of Cryo-EM maps is the process of generating a structure model with 3D coordinates for each atom in the macromolecule (e.g., proteins) that fits the map (Figure 1). The main approaches for molecular structure determination are $de$ $novo$ building and homology modeling, which its homologous structures in the Protein Database. In this work, we focus on the de novo approaches, where there are no previously solved structures of homologous proteins. All recent molecular structure determination pipelines still rely on interactive tools with heavy hand labor. In principle, most available approaches, of which Rosetta-denovo is a typical example, use template matching and Monte Carlo sampling based on a library with millions of fragments from solved protein structures as templates for structural modeling. Briefly the target protein is divided into short fragments and structures of similar sequences in the library are identified for every fragment. And then candidate structure fragments are assembled by Monte Carlo simulated annealing to optimize a fitting score. Alternatively, RosettaES then uses a greedy conformational sampling algorithm to assemble the main chain of protein maximally consistent with sequence and density volume. The accuracy and coverage of these methods are often not satisfactory, due to limitations of the hand-crafted scoring functions and the sheer number of template structures. ### Object Detection Approaches for object detection including Faster R-CNN, Cascade R-CNN, SNIPER, FishNet [@ren2015faster; @cai18cascadercnn; @sniper2018; @fishnet] have improved drastically in terms of performance and efficiency. These methods follow a similar framework in which the objects are extracted from a Region of Interest (RoI) and pooled to the same size before predicting their categories and coordinates. RoI pooling, RoI Warping and RoIAlign  [@girshick2014rich; @dai2016instance; @mask-rcnn] are popular techniques for RoI extraction, all of which break the original aspect ratio of the objects to account for their variations in natural images filmed with different angles and distances. However, in some scenarios like in our work, aspect ratios of different types of amino acids should be preserved. To tackle this problem, we proposed an Aspect-Ratio Preserved RoI (APRoI) layer to capture the aspect ratio in amino acids. 3D object detection plays a key role in a variety of real-world applications, such as autonomous driving [@gomez2016pl], augmented/virtual reality and identification of disease diagnosis. MV3D [@cvpr17chen] focuses on very sparse data (LiDAR point cloud) and projects 3D data into 2D multi-view images. It is less effective in the amino acid detection task where difficulties may come from dense objects and the ambiguity in overlapping regions. VoxelNet [@zhou2017voxelnet] groups very sparse points for encoding voxel features to model point interactions. Frustum-PointNet [@qi2017frustum] extracts points within the frustum from 2D box to form a frustum point cloud, which may generate too much noise, especially in a dense object detection task. In this work, we designed a pure 3D detector. ### 3D Pose Estimation Given well labeled 3D joint locations, 3D pose estimation aims to determine the precise joint locations in 3D space. For instance, 3D human pose estimation attempts to regress 16 key points in the human body based on 3D joint locations of a human skeleton from 2D images [@pose_Zhou_2017_ICCV; @pose_martinez_2017_3dbaseline]. However, the existing 3D pose estimation are based on 2D images. In this work, we introduce a pure 3D pose estimation module to produce atomic coordinates in volumetric data. Inspired by these breakthroughs, we designed a multi-task 3D neural network that first detects amino acids and then estimates the atom positions. We also took advantage of the sequence information and introduced a sequence-guided neighbor loss to train the network, which has not been explored before. In addition, we added a postprocessing with superior tree searching algorithm – MCTS for main chain threading. We believe that this is the first attempt using MCTS to trace the boxes of amino acids guided by sequence. Problem Definition ================== Given a 3D Cryo-EM density volume and its protein sequence as inputs, our goal is to detect all the amino acids, estimate their pose and thread them into a protein chain in the 3D space. The density volume, obtained from 3D reconstruction of 2D microscope images, is represented as 3D matrices with continuous density value in each voxel. The sequential orders of amino acids, but not their locations, are known. Each amino acid can be represented by a class $C$ (20 types of standard amino acid), a 3D box and $N^C$ central locations of atoms in the amino acid. Each 3D box is parameterized by two coordinates: the front left top corner and the back right bottom corner. $N^C$ (4$\sim$14) is the number of the atoms in the amino acid, and it varies in different types. $A^2$ Dataset ============= As a benchmark for molecular structure determination, we built a large-scale Amino Acid ($A^2$) dataset of Cryo-EM density volumes. It contains 250,000 amino acids in 1713 simulated (by Chimera in 3 Å) electron density volumes and is annotated with rich information of amino acids. The amino acids are labeled with 3D boxes of 20 categories as well as the atomic coordinates. The amino acids in the dataset are dense, small objects that overlap each other, which makes the amino acid detection a very challenging task. To the best of our knowledge, the $A^2$ dataset is the first large-scale benchmark for learning automatic molecular structure determination. 3D Density Volume Annotation ---------------------------- The molecular structures and the corresponding density volumes in the $A^2$ dataset were collected from the RCSB Protein DataBase (PDB) and The Electron Microscopy Data Bank (EMDB). Firstly, we selected the volumes with resolution below 5 angstroms (Å). The PDB and EMDB databases contain some inconsistencies where some volumes do not match the ground-truth structure. We manually removed these problematic volumes. As a quality control step, we collected only the chains without any missing atom or amino acid. Ultimately, the $A^2$ dataset contained 250,000 amino acids in 1713 chains are kept to construct the $A^2$ dataset. Following random selection, we obtained a split of 1250 training and 463 validation chains. Dataset Statistics ------------------ There are 20 categories of common amino acids and 367,929 pairs of overlapping amino acids in the dataset, which means the dataset is highly dense and challenging for detection. As shown in Fig. \[fig:dense\], the dataset has much denser 3D objects than the KITTI dataset [@Geiger2012CVPR]. ![The significant object density difference between the proposed dataset and KITTI LiDAR detection dataset.[]{data-label="fig:dense"}](1_1_aa_kitti5_small.pdf){width="0.95\linewidth"} Method {#sec:blind} ====== The framework of $A^2$-Net consists of two stages. Stage one represents the deep neural network for amino acid detection in 3D space and pose estimation, which determines the 3D coordinates of atoms in each amino acid. Stage two uses a Monte Carlo Tree Search strategy with tree pruning, based on the candidate amino acid proposals obtained in stage one to construct the main chains of amino acids, i.e., proteins. {width="\linewidth"} 3D Amino Acid Detection {#sec:net} ----------------------- As shown in Fig. \[fig:pipeline\], when given a density volume, our $A^2$-Net first obtains 3D feature volumes and generates 3D box proposals with the region proposal network (RPN) [@ren2015faster]. The 3D RPN consists of three 3D convolutional layers to generate proposals of amino acid locations. We used 3D anchors at each 3D location to cover the region of amino acids with various scales and aspect ratios. Next, one branch of RPN classified whether the anchors are valid amino acid proposals and the other branch estimates their coordinates. With the amino acid proposals, we used a newly designed Aspect-ratio Preserved RoI (*APRoI*) layer to extract the RoI in the input volume into a fixed cubic. Then the cubic went through several 3D convolutional layers to finally predict its amino acid category and coordinates. ### Aspect-ratio Preserved RoI Layer In natural images, objects generally have completely different aspect ratios and conventional RoI pooling resizes the regions and abandons the original aspect ratio of the object. It can be seen as aspect-ratio augmentation, which is usually beneficial for the generalization ability of the deep model. However, aspect ratios of different amino acids should be maintained as they actually reflect different categories. We thus proposed an APRoI layer, which first crops the input at the RoI location, and then pads it with zero to a defined size of $W_T \times H_T \times L_T$. Despite its simplicity, the APRoI layer preserved the aspect ratio of the objects and proved to be vital for our amino acid classification. The back-propagation passes derivatives through the APRoI layer. Let $x_{ijk} \in \mathbb{R}^3$ be the input of the APRoI layer, and $y_{rijk}$ be the layer’s output at voxel index $i,j,k$ from the $r$-th RoI. The APRoI layers back-propagate partial derivative of the loss with respect to each input $x_{ijk}$ as Eqn \[eqn:aproi2\]: $$\begin{aligned} \begin{split} & \frac{\partial L}{\partial x_{ijk}} = \sum_r [i_s\le i < i_e, j_s\le j < j_e, k_s\le k < k_e]\frac{\partial L}{\partial y_{rijk}} ,\\ &i_s =max \lbrace {0, \lfloor\frac{W_T-W_r}{2} \rfloor} \rbrace, i_e =i_s + min \lbrace \lceil W_r\rceil, W_T \rbrace ,\\ &j_s =max \lbrace {0, \lfloor\frac{H_T-H_r}{2} \rfloor} \rbrace, j_e =j_s + min \lbrace \lceil H_r\rceil, H_T \rbrace,\\ &k_s =max \lbrace {0, \lfloor\frac{L_T-L_r}{2} \rfloor} \rbrace, k_e =k_s + min \lbrace \lceil L_r\rceil, L_T \rbrace, \end{split}\label{eqn:aproi2}\end{aligned}$$ where $W_r, H_r,L_r$ are the size of the $r$-th RoI. $\partial L / \partial y_{rijk}$ denotes the partial derivative computed by the layer on top of the APRoI layer. Intuitively, only the previously valid locations will receive the gradients for back propagation, while the locations with padded zeros are simply ignored. ### Loss for amino acid detection To predict the 3D box size, we follow previous work [@qi2017frustum] and use a mixture of both regression and classification formulations instead of directly regressing the 3D box size. Firstly, we specifically pre-define 20 template size with ($w^c, h^c, l^c$). Our model classifies each input into one type and then regresses the residual of the box sizes of the type with highest probability. While for box center regression, we parametrize a 3D ground truth box as $(x^g, y^g, z^g, w^g, h^g, l^g)$, where $(x^g, y^g, z^g)$ represents the coordinates of the front left top corner of the box and $(w^g, h^g, l^g)$ represents width, height and length. We define the residual vector as $\textbf{u} \in \mathbb{R}^3$, which contains the 3 regression targets corresponding to center location $\Delta x, \Delta y$ and $\Delta z$, and define the residual vector as $\textbf{v} \in \mathbb{R}^3$ containing 3 dimensions ($\Delta w, \Delta h$, $\Delta l$). The residuals are computed as: $\Delta x = (x^g - x^a)/w_a, \Delta y = (y^g - y^a)/h_a$, $\Delta z =(z^g - z^a)/l_a, \Delta w = \log ((w^g-w^c)/w_a)$, $\Delta h = \log ((h^g-h^c)/h_a), \Delta l =\log ((l^g-l^c)/l_a)$. The proposed loss function is defined as: $$\begin{aligned} \begin{split} Loss &= \frac{1}{N_{cls}} \sum_i^{N_{cls}} L_{neighbor}(p_i, p_i^*) +\\ \beta \frac{1}{N_{reg}} \sum_j^{N_{reg}} &p_j^* (L_{c-reg}(\textbf{u}_j, \textbf{u}_j^*)+ L_{s-reg}(\textbf{v}_j, \textbf{v}_j^*)), \end{split} \label{eqn:2}\end{aligned}$$ where $p_i, p_i^*$ are respectively the predicted probability of anchor $i$ and its label, $u_j, u_j^*$ are the box center regression output and ground truth for anchor $j$, $v_j, v_j^*$ are the box size residual regression output and ground truth. $L_{neighbor}$ is the classification loss reweighted by the guiding sequence, which will be introduced in the next section. $L_{c-reg}$, $L_{s-reg}$ are the center coordinates and residual box size regression loss (smooth $L_1$ loss), respectively. The two terms are normalized by $N_{cls}$ and $N_{res}$. $\beta$ is the balancing parameter. ### Neighbor Loss {#sec:neighborloss} Since the sequential orders of the amino acids is provided in this task, the geometric constraints (distance, the overlapping region and sequential order) between amino acids should be integrated to regularize the detection results, as shown in [@gao2018question]. In most cases, the distance between an amino acid and one of its two neighbors, should be smaller than the distance between its two neighbors. To take advantage of this information, we introduced a novel neighbor loss. For each proposal, we checked both criteria: 1) it is a positive anchor, and 2) its distance to either neighbor of the associated ground truth, is smaller than the distance between the two neighbors. The qualified anchor will be assigned a higher weight as they are “better” samples in the view of the sequence. The additional weight $(1-p_i)^\lambda$ follows the spirit of focal loss [@lin2017focal], which down-weights the sample well classified by the model. We define the neighbor loss as: $$\begin{aligned} \begin{split} L_{neighbor}(p_i) = - ((1-p_i)^\lambda m_i + 1) log(p_i), \end{split} \end{aligned}$$ where $m_i\in \{0,1\}$, and $m_i=1$ when object $i$ is one of the mined positive neighbor objects. 3D Amino Acid Pose Estimation ----------------------------- After we obtained the proposal for an amino acid, we further estimated its pose by locating its forming atoms in 3D space. The Stacked hourglass Network [@pose_hg] is widely used to handle the human pose estimation task. In this work, a 3D stacked hourglass network, named as poseNet, is proposed to regress 3D coordinates of each atom in amino acids. The network stacks multiple ($H$) hourglass structures sequentially. Each hourglass has $R_b$ residual blocks and provides feature volumes with different semantic resolutions. Importantly, auxiliary losses were applied to the intermediate feature volumes for learning robust features. For an amino acid with $N$ atoms, poseNet produces $H$ estimated heatmaps with $N$ channels. The Mean Squared Error loss is adopted: $$\begin{aligned} \begin{split} L_{pose} = \sum_{h}^{H} \sum_{n}^{N} \parallel y_h^*(n) - y_h(n) \parallel_{2}^{2}, \end{split} \end{aligned}$$ where $y_h^*$ denotes the predicted heatmap by the $h$-th stack, $n$ denotes the $n$-th atom, and $y_h$ is the ground-truth heatmap with the $N \times 8 $ locations labeled as 1. For each atom, the $2^3$ neighborhood locations are labeled as 1. Monte Carlo Tree Search for Threading {#sec:threading} ------------------------------------- ### Main Chain Threading Problem Given the predicted amino acid proposals and the ground truth sequential order, our next task is to select the same number of proposals as in the sequence and thread them over to form the complete protein chain. With $N_B$ predicted proposal set $B$ from the $A^2$-Net, and a sequence $S^*$ of length $T$ ($N_B > T$), our next task is to select $T$ boxes in sequence to form the complete protein chain. The categories and sequential orders of $S$ are given by $S^*$, and the proposals are selected from $B$. Each proposal is $S_t = (x_t, y_t, z_t, w_t, h_t, l_t, P_t)$, $P_t$ is the probability of different categories predicted by $A^2$-Net. ### Monte Carlo Tree Search MCTS is a tree search algorithm in which a node is evaluated by performing random actions from the decision space until an outcome can be determined [@alphago; @ReinforceWalk]. Searching by MCTS is done by iteratively building a search tree where the nodes denotes different states, and the edges are the actions leading to one state from another. A node is recursively added to the tree during each iteration. Based on the reward of the new node, the reward values of all parent nodes are updated. A single iteration of the MCTS building process consists of four steps: 1) selection: a node to be expanded is selected; 2) expansion: the node is expanded by simulating the associated action; 3) simulation: the tracing is simulated following a random path until the terminal amino acid is reached; 4) back propagation: the result propagates back through the tree. ### Building KNN-Graph Directly performing the MCTS algorithm to all the proposals may be time consuming, so we first built a graph based on $K$ nearest neighbors, where each node denotes a proposal and an edge connects two nodes if they are among the $K$ nearest neighbors, thus called KNN graph. Next, we determined the root node of the tree by finding $L$ proposals in $B$ which match the first $L$ amino acids in the sequence. Finally, we obtained several candidate fragments as the starting points. For each starting point, we ran the MCTS algorithm to obtain the optimal path. The optimal path was obtained by a control policy $\pi$ to maximize the total reward $R$, which is the sum of all the values $V_t$ in every following step $t$. The reward function $R$ thus is written as: $$\begin{aligned} \begin{split} R &= \sum_{t=1}^{T} V_t, \\ V_t = t * ( P_{detection}(&S_t) + P_{compatible}(S_t, S_{t+1}) ), \end{split}\label{eqn:reward}\end{aligned}$$ where $V_t$ is the sum of the detection score $P_{detection}$ and the compatible score $P_{compatibility}$ at action $t$, and weighted by the time $t$ which encourages the searching path to be long. $P_{detection}$ and $P_{compatibility}$ ensure that the selected boxes are reliable and compatible. $P_{detection}(t)$ is the detection probability of $S_t$, and $P_{compatibility} = dist(S_t, S_{t+1})$, where $dist(\cdot)$ is the IoU between two boxes. An optimal policy $\pi$ outputs an optimal action sequence, which is defined as a path with maximum reward from the root to a leaf. We seek a path that maximizes the reward function in Eqn. (\[eqn:reward\]). After the root was created, Monte Carlo simulations selected actions and followed the sequence $S$ to create a new node. After a number of simulations, the tree was well populated, and the optimal path was selected. Each depth of the tree is the time step, with root at $t=1$ and leaves at $t = T$. {height="7cm"} ### Next Action At time $t+1$, we need to select a proposal $S_{t+1}$ close enough to $S_t$ while keeping the category same as $S_{t+1}^*$. Since there may be a few proposals for selection, we may face an exploration-exploitation dilemma, where the algorithm may fall into a local optimum. To balance this dilemma, we follow [@AIIDE1614003] to use the $Upper$ $Confidence$ $Bound$ $for$ $Trees$ (UCT) [@mcts] method to optimize the action selection. The UCT method is designed for better action selection strategy. At step $t$, the next action $a_{t+1}$ is: $$\begin{aligned} a_{t+1} & = \arg \max_{a} ( \frac{V_a}{n_a} + C \sqrt{\frac{2 \ln N_a}{n_a}}),\end{aligned}$$ where $V_{a}$ is the reward at action $a$. $N_a$ is the number of simulations that the node has been visited, and $n_a$ is the number of simulations that the node has been followed. ### Tree Pruning by Peptide Bond Recognition Network Since protein structure is highly twisted, a pair of amino acids that are close to each other in 3D space might be far apart in primary in the sequence. Experienced biologists distinguish whether two amino acids are connected or not, by examining whether there is a peptide bond between them. We also designed a Peptide Bond Recognition Network (PBNet) to predict whether there is a peptide bond between two proposals. With PBNet, we can efficiently remove 50% edges in the KNN graph on average, which largely improves the search speed. PBNet has only three convolutional layers with batch normalization and max pooling, followed by three fully connected layers. The network is trained by $softmax$ loss. Experiments =========== Implementation Details ---------------------- Our network architecture can be divided into three parts: localization network (locNet), recognition network (recNet) and pose estimation network (poseNet). In locNet, the backbone network is a fully convolutional network with 12 3D convolutional layers including 4 residual blocks. Since amino acids have a very small volume, we only have one max pooling layer. We followed previous work [@ren2015faster] to design an anchor mechanism to cover various scales and aspect ratios of amino acids. We use 7 aspect ratios and 3 scales, yielding $k=21$ anchors at each position on the last $conv$ feature maps of the backbone network. We applied a $3\times 3\times 3$ convolutional layer to the $conv$ feature volumes, followed by two sibling $1\times 1\times 1$ convolution layers for classification and bounding box regression, respectively. Each anchor was assigned a binary label depending on whether it has an Intercession-over-Union (IoU) with a ground truth amino acid larger than a threshold 0.8. The recNet also has 4 residual blocks, and three fully connected layers while the poseNet has 4-stacked hourglass. All the convolutional layers adopt the $3\times 3\times 3$ kernel size. The $A^2$-Net was trained in three stages. We first trained the locNet and poseNet individually for 100 epochs, and then fixed them while training the recNet for 400 epochs. Finally, we jointly optimize the whole $A^2$-Net with sequence-guided neighbor loss for another 400 epochs. We used Adam [@KingmaB14Adam] optimizer to train the model, starting by a learning rate of 0.0001, a momentum of 0.9 and a weight decay of 0.0001. We fine-tune our models with BatchNorm. We found that BatchNorm may reduce over-fitting. For each density volume, we randomly cropped a $64\times 64\times 64$ cube and send it into the network. Limited by the GPU memory, we set the batch size to be 1. Results on the $A^2$ dataset ---------------------------- ### Amino Acid Detection We first evaluated the effectiveness of APRoI layer and sequence-guided neighbor loss training. We adopted the commonly used mean Average Precision (mAP) for evaluation of detection. The quantitative amino acid detection results are reported in Table. \[table:det\]. -------------------------------- ----------- ---------- Methods mAP Coverage MV3D(BV+FV) 0.118 0.15 Frustum-Pointnet-v1 0.407 0.45 Frustum-Pointnet-v2 0.425 0.48 3D-VGG+RoIpool8 0.360 0.32 3D-VGG+RoIpool8(w/o maxpool) 0.423 0.41 3D-ResNet+RoIpool8 0.416 0.44 3D-ResNet+RoIpool8(Raw volume) 0.610 0.55 $A^2$-Net (APRoI8) 0.711 0.67 $A^2$-Net w/o Neighbor Loss 0.865 0.72 $A^2$-Net **0.891** **0.91** -------------------------------- ----------- ---------- : The results of detection and threading comparing with other 3D object detection methods. []{data-label="table:det"} --------------- ---------- --------- Methods Coverage RMSD DFS$_o$ 0.65 3.5 DFS$_d$ 0.68 3.1 DFS$_d$+PBNet 0.89 2.6 MCTS 0.72 2.9 MCTS+PBNet **0.91** **2.0** --------------- ---------- --------- : The results of threading by DFS-based methods and the proposed MCTS+PBNet.[]{data-label="table:threading"} We first directly applied 3D VGG with RoI pooling for amino acid detection, which only achieves 0.36 mAP, while 3D ResNet-10 [@he2015deep] achieves 0.416 mAP. The last $conv$ feature volumes were used for 3D region proposal. By analyzing the intermediate output, we found that the network seemed to be dominated by the 3D region proposal task, which made the last $conv$ feature volumes to be only sensitive to the existence of the amino acid. There was little category-specific information left in the feature volumes. So we directly performed RoI pooling on the raw input cube and trained the recNet, which achieved 0.610 mAP. This verified our assumption that the category-specific information may be discarded in the feature map of the locNet model. We then replaced the RoI pooling with APRoI layer, which further improved mAP to 0.711. This indicates the importance of preserving the aspect ratio for detection in this task. We also found that the output size of APRoI layer is important, as mAP improved when we changed the target size from $8^3$ to $16^3$, the mAP has a large improvement. Finally, we used the sequence-guided neighbor loss training strategy and further improved the mAP to 0.891. Fig. \[fig:exp-detection\] shows a qualitative result of detection. Although the gain of mAP from neighbor loss was only marginal, the sequence coverage percentage of the threading result improved substantially. ### Main Chain Threading We mainly compared the proposed MCTS algorithm with a Depth First Search (DFS) method. DFS$_o$ and DFS$_d$ represent using IoU and distance as the selection criterion, respectively. The proposed MCTS algorithm outperformed the DFS based methods. The PBNet can be applied to both DFS and MCTS algorithms, as it is used to prune the trees by examining the existence of peptide bonds. It can be seen in Table. \[table:threading\] shows that both threading algorithms are largely improved with PBNet. PBNet achieved 89.8% accuracy for peptide bond recognition. In Fig. \[fig:exp-threading\] shows some qualitative results of threading. Table. \[table:Rosetta-denovo\] demonstrates that Rosetta-denovo is very time-consuming. We ran Rosetta-denovo in a cluster with 200 computational nodes for 2 rounds. The CPU time was calculated by summing up all the tasks. It took Rosetta-denovo hundreds of hours to finish one round of computation, whereas our approach took only a few minutes and outperformed Rosetta-denovo by a huge margin. ![An example of threading results by MCTS+PBNet.[]{data-label="fig:exp-threading"}](1_4_exp-threading_small.pdf){width="0.9\linewidth"} -------------- ---------- ------------ ---------- ----------- Coverage Time Coverage Rosetta (R1) 0.20 133 h 0.39 90 h Rosetta (R2) 0.24 260 h 0.62 261 h Ours (MCTS) **0.88** **11.3 m** **0.91** **6.8 m** -------------- ---------- ------------ ---------- ----------- : Threading accuracy and efficiency compared with Rosetta-denovo. R1 and R2 denotes round 1 and 2. []{data-label="table:Rosetta-denovo"} ### Comparison with other 3D Detection Methods We adapted MV3D and Frustum-PointNet to the amino acid detection task. For MV3D, we cropped and projected the volumes into the bird’s eye view and the front view (FV) and then trained MV3D. For Frustum-PointNet, we selected the voxels with density value is higher than the mean of the volumes as the points set. In the training step, we projected the 3D ground-truth boxes into their FV as the input. In the testing step, we projected the 3D boxes which were predicted by locNet into their FV as the input. Table. \[table:det\] shows that our method outperformed them by a large margin. ### Generic Features for 3D Detection We pre-trained MV3D and Frustum-PointNet on the $A^2$ dataset, and fine-tuned them on KITTI dataset for 3D car bounding box regression. Table. \[table:improve\_3ddet\] summarizes the 3D car detection performance on the KITTI dataset. The $A^2$ dataset pre-trained model yielded an additional increase in performance, revealing that the $A^2$ dataset can provide generic features for 3D object detection task. [lcccc]{} Methods & ---------- w/ $A^2$ ---------- : The AP of different methods for 3D car detection on KITTI dataset w/ or w/o $A^2$ Dataset. []{data-label="table:improve_3ddet"} & Easy & Moderate & Hard\ MV3D & & 65.53 &58.97 & 59.14\ MV3D &$\surd$ &**68.56** &**60.35** & **60.99**\ F-pointnet-v1 & & 83.26 & 69.28 & 62.56\ F-pointnet-v1 & $\surd$ &**84.89** & **71.97** & **64.07**\ F-pointnet-v2 & & 83.76 & 70.92 & 63.65\ F-pointnet-v2 & $\surd$ &**85.11** &**72.13** & **64.24**\ Conclusions =========== In this work, we reformulate the challenging molecular structure determination problem and propose a learning-based framework. The newly designed $A^2$-Net predicts accurate amino acid proposals with our APRoI layer and the neighbor loss training strategy. With the predictions and the sequence, we propose a MCTS algorithm for efficient threading. Using the peptide bond recognition network, tree branches between candidate pairs of proposals without a real peptide bond can be easily removed, which simultaneously improves the searching efficiency and the sequence coverage. Our novel method is hundreds of times faster and more accurate than the previous method, and will play a vital role in molecular structure determination. Acknowledgments =============== This project is supported by the National Natural Science Foundation of China (Grants No. 31671355, 91740204, and 31761163007), the Beijing Advanced Innovation Center for Structural Biology, the Tsinghua-Peking Joint Center for Life Sciences and the National Thousand Young Talents Program. We thank Chuangye Yan, Xingyu Zeng, Yao Xiao and Xinge Zhu for their helpful work and insightful discussions.

--- abstract: 'Laser seeding technique have been envisioned to produce nearly transform-limited pulses at soft X-ray FELs. Echo-Enabled Harmonic Generation (EEHG) is a promising, recent technique for harmonic generation with an excellent up-conversion to very high harmonics, from the standpoint of electron beam physics. This paper explores the constraints on seed laser performance for reaching wavelengths of $1$ nm. We show that the main challenge in implementing the EEHG scheme at extreme harmonic factors is the requirement for accurate control of temporal and spatial quality of the seed laser pulse. For example, if the phase of the laser pulse is chirped before conversion to an UV seed pulse, the chirp in the electron beam microbunch turns out to be roughly multiplied by the harmonic factor. In the case of a Ti:Sa seed laser, such factor is about $800$. For such large harmonic numbers, generation of nearly transform-limited soft X-ray pulses results in challenging constraints on the Ti:Sa laser. In fact, the relative discrepancy of the time-bandwidth product of the seed-laser pulse from the ideal transform-limited performance should be no more than one in a million. The generated electron beam microbunching is also very sensitive to distortions of the seed laser wavefront, which are also multiplied by the harmonic factor. In order to have minimal reduction of the FEL input coupling factor, it is desirable that the size-angular bandwidth product of the UV seed laser beam be very close to the ideal i.e. diffraction-limited performance in the waist plane at the middle of the modulator undulator.' address: - 'European XFEL GmbH, Hamburg, Germany' - 'Deutsches Elektronen-Synchrotron (DESY), Hamburg, Germany' author: - 'Gianluca Geloni,' - Vitali Kocharyan - and Evgeni Saldin title: Analytical studies of constraints on the performance for EEHG FEL seed lasers --- **DEUTSCHES ELEKTRONEN-SYNCHROTRON** **\ ** DESY 11-200 November 2011 $$\begin{aligned} \nonumber &&\cr \nonumber && \cr \nonumber &&\cr\end{aligned}$$ $$\begin{aligned} \nonumber\end{aligned}$$ **Analytical studies of constraints on the performance for EEHG FEL seed lasers** $$\begin{aligned} \nonumber &&\cr \nonumber && \cr\end{aligned}$$ Gianluca Geloni, *\ European XFEL GmbH, Hamburg* Vitali Kocharyan and Evgeni Saldin *\ Deutsches Elektronen-Synchrotron DESY, Hamburg* $$\begin{aligned} \nonumber\end{aligned}$$ $$\begin{aligned} \nonumber\end{aligned}$$ ISSN 0418-9833 $$\begin{aligned} \nonumber\end{aligned}$$ **NOTKESTRASSE 85 - 22607 HAMBURG** \[sec:uno\] Introduction ======================== An important goal for any advanced X-ray FEL is the production of X-ray pulses with the minimum allowed photon energy width for given pulse length, which defines the transform-limit. A well-known approach to obtain fully coherent radiation in the soft X-ray region relies on frequency multiplication, a scheme known as high-gain-harmonic-generation (HGHG) [@HGHG1; @HGHG2]. In a HGHG FEL, the radiation output is obtained from a coherent subharmonic seed laser pulse. Consequently, the optical properties of HGHG FELs are expected to reflect the characteristics of the high-quality seed laser. Echo-enabled harmonic generation (EEHG) is a recent, promising technique for efficient harmonic generation [@EEHG1; @EEHG2]. The key advantage of EEHG over HGHG is that the amplitude of the achieved microbunching factor decays slowly with an increasing harmonic number. Consequently, as concerns electron beam physics issues, EEHG allows for the generation of fully coherent radiation at soft X-ray wavelengths with a single upshift stage, and using a conventional optical laser system. The remarkable up-frequency conversion efficiency of the method has stimulated wide interest to generate near transform-limited soft X-ray pulses. Several EEHG FEL projects are now under development [@LCLS2E]-[@FLEEHG]. A typical EEHG setup consists of two stages for electron beam phase space manipulation, followed by a radiator. Each stage includes an undulator, which is used to modulate the electron beam in energy with the help of a seed laser, and a chicane following the modulator, which is used to apply energy-dependent slippage to the electrons. The radiator is composed by a sequence of undulators tuned to the desired output wavelength. This final section is similar to that used for SASE FELs. However it is shorter, and produces coherent radiation only because the beam has been coherently prebunched. The seed laser is assumed to be tuned at $200$ nm (or $270$ nm) corresponding to the fourth harmonic (or the third) of a Ti:Sapphire laser. Limitations on the performance of EEHG schemes related with electron beam dynamics issues, as the beam goes through the various undulators and chicanes, has been extensively discussed in literature [@EETOL1; @EETOL2; @EETOL3] and goes beyond the scope of this paper. Here we will focus our attention on the first part of the harmonic generation process, discussing the constraints on the seed laser performance needed for reaching wavelengths of about $1$ nm. In fact, in chirped-pulse amplification systems (CPA) systems, both temporal and spatial quality of the beam can be degraded due to the propagation through the optical components, non-linear effects or inhomogeneous doping concentration in the amplifying media, and thermal effects linked to the pumping process. In particular, the aim of this work is to evaluate the impact of variations of the characteristics of output radiation when the FEL is seeded by a laser with non-ideal properties, including effects such as linear and nonlinear frequency chirp and wavefront distortion. A description of the impact of phase chirp of the EEHG (or HGHG) FEL output can be made without numerical simulation codes. In fact, as is well-known, if the phase of the seed laser is chirped, the chirp is simply multiplied by the frequency multiplication factor $N$. In this case, the method used to describe the output field perturbation is independent of the specific kind of harmonic generation technique: it only depends on the frequency multiplication factor $N$. It follows that both EEHG and HGHG FELs starting from a Ti:Sa laser with a wavelength around $800$ nm can produce transform-limited radiation down to wavelengths of $1$ nm only when the relative discrepancy of the time-bandwidth product of the compressed $800$ nm laser pulse from the ideal, transform-limited performance is no more than one in a million, roughly corresponding to the squared of the harmonic number. However, pulses from a commercially available Ti:Sapphire chirped pulse amplifiers are usually limited in such discrepancy to around $1\%$, due to non-ideal effects. Therefore, research and development activities must be performed in order to reach the required temporal quality. To the best of our knowledge, there is only one article[^1] reporting on the impact of temporal variations in seed laser pulses on EEHG FEL output radiation characteristics, [@EETOL3]. The analysis in [@EETOL3] is based on numerical simulations in the case of $1.2$ nm output radiation wavelength. According to results of the sensitivity study in [@EETOL3], the phase of the $202$ nm seed laser pulse (corresponding to the fourth harmonic of a Ti:Sa laser) must to be controlled to within $0.5$ degrees. Consequently, the phase of the Ti:Sa laser output must to be controlled to within roughly $0.1$ degrees. This result is consistent with our analysis, which is performed at a very elementary level. We also evaluate the impact on the EEHG FEL output of wavefront errors in the seed laser. In the case of ideal performance, the seed (UV) laser beam must be characterized by a flat (i.e. diffraction-limited) wavefront in the waist plane in the middle of the modulator undulator. If the wavefront exhibits errors, errors in the microbuch wavefront follow, which are multiplied by the frequency multiplication factor. These microbunch wavefront errors do not affect the spatial quality of the FEL output radiation, which is the same for both perturbed and unperturbed wavefront cases. They only affect the input signal value at the target harmonic. However, because of the exponential dependence of the signal suppression factor on the wavefront errors, one obtains an appreciable FEL output only when phase errors are sufficiently small to give appreciable input signal. As a result, the seed UV laser beam must exhibit a nearly diffraction-limited wavefront in the waist plane, with very little phase variation. In particular for a target harmonic with wavelengths of about $1$ nm, the wavefront of the UV beam must be controlled to within a fraction of a degree across the electron beam area. These relatively small phase variations cause the signal at the entrance of the FEL amplifier to drop of a quantity of order of the ideal (diffraction-limited) performance. In contrast with phase variation in time, the spatial quality of UV seed laser beam can be improved by means of active optics and spatial filtering. However, these manipulations with laser beam usually cause significant losses in beam power. To the best of our knowledge, the crucially important problem of seeding with beam wavefront distortions was only recently reported in workshops [@WORK1; @WORK2], where the impact of wavefront errors on the EEHG performance was discussed, based on numerical simulations, in the case of the highest target harmonic at $13$ nm. Results of [@WORK1; @WORK2] are consistent with our analysis, which has been performed purely analytically. The suppression of the output signal due to phase variations in space seems somehow in contrast with the effects of phase variations in time, where phase errors affect the temporal quality of the output radiation, but not the FEL output power. From this viewpoint, it should be noted that the radiation field is characterized by notions such as temporal and spatial coherence. The transverse coherence of FEL radiation develops automatically, without laser seeding. This happens due to transverse eigenmode selection: due to different gains of the FEL transverse eigenmodes, only one survives at the end of the FEL process. The coherence time is defined by the inverse FEL amplification bandwidth. For conventional soft X-ray FELs the typical amplification bandwidth is much wider than the Fourier transform limited value corresponding to the radiation pulse duration, meaning that the coherence time is much shorter than the pulse duration. Consequently, microbunch phase variations in time only lead to phase variations in the output radiation pulse, without suppression of the output power level. \[sec:due\] Issues affecting the performance of EEHG FEL ======================================================== Phase control is an important aspect in the development of all FEL sources based on harmonic generation. Methods for dealing with issues concerning temporal phase variations in frequency multipliers are based on the same general principle [@ROBI]: the effect of frequency multiplication by a harmonic factor $N$, is to multiply the phase variation by $N$. The EEHG scheme is obviously based on harmonic generation, but is more complicated than other schemes, and consists of two modulators, two dispersion sections, and one radiator undulator. A unique feature of EEHG scheme is the utilization of two different seed laser pulses which can have different temporal and spatial quality. It is thus natural to investigate the question whether the general principle above can also be applied to EEHG. Analytical results [@EEHG2] refer to the specific model an of infinitely long, uniform electron bunch only. This steady state model proved to be very fruitful, allowing for simple analytical expressions describing the main characteristics of EEHG scheme. However, as discussed above, the seed laser pulses and, consequently the electron beam microbunching, are always characterized by phase variations in time and space (wavefront distortions). We will therefore extend analytical description of EEHG scheme in [@EEHG2], following the line of derivations in that reference, to the time dependent case and account for finite duration and transverse size of the electron bunch. To this aim, we assume that the temporal profile of the electron beam can be modeled as a Gaussian, and that the initial electron beam distribution can be factorized as a product of energy, $f_{0p}(p)$, and density, $f_{0\zeta}(\zeta)$ distributions as $$\begin{aligned} f_0(\zeta,p)= f_{0p}(p) f_{0\zeta}(\zeta) = \frac{N_0}{2\pi \sigma_\zeta} \exp\left[-\frac{p^2}{2}-\frac{\zeta^2}{2 \sigma_\zeta^2}\right]~. \label{f0new}\end{aligned}$$ Here $p=(E-E_0)/\sigma_E$ is the dimensionless energy deviation of a particle from the average energy $E_0$, and the rms spread is given by $\sigma_E$. Similarly, $\zeta = \omega_l t$ is the dimensionless time, with $\omega_l$ the laser frequency, assumed to be the same in both stages, and $\sigma_\zeta$ is the rms spread of the density distribution. Finally $N_0$ is the total number of particles in the beam. The longitudinal phase space is described by the variables $(\zeta,p)$. Passing through the first modulator and dispersive section the phase space variables transform to $(\zeta',p')$, which are given by $$\begin{aligned} p'=p + A_1 \sin (\zeta + \phi_1) ~~, ~~~ \zeta'=\zeta + B_1 p'~, \label{M1}\end{aligned}$$ where $A_1 = \Delta E_1/\sigma_E$, $\Delta E_1$ being the energy modulation imposed by the seed laser, $\phi_1=\phi_1(\zeta)$ is the phase of the laser pulse, which depends on the time $\zeta$, and $B_1 = R_{56}^{(1)} \sigma_E \omega_l/(E_0 c)$, $R_{56}^{(1)}$ is the strength of the first chicane. Substituting Eq. (\[M1\]) into Eq. (\[f0new\]) one can obtain the distribution after the first modulator and dispersive section. The new phase space variables $(\zeta',p')$ will transform after the passage through the second modulator and dispersive section, to $(\zeta'',p'')$, which are given, in a similar way, by $$\begin{aligned} p''=p' + A_2 \sin (\zeta' + \phi_2)~~, ~~~ \zeta''=\zeta' + B_2 p''~, \label{M2}\end{aligned}$$ the subscript $'2'$ referring to the second stage[^2]. Using Eq. (\[f0new\])-(\[M2\]) one can obtain an explicit expression for the phase space distribution after the second stage, $f_2(\zeta'', p'')$, which will not be reported here. In order to analyze the harmonic composition of the current density we first need to project the phase space distribution onto the real space-time coordinates by performing an integration along $p''$. This leads to the density distribution function $\rho$, that can be Fourier-analyzed further to give $$\begin{aligned} \bar{\rho}(\Omega)= \int_{-\infty}^{\infty} dp'' d\zeta'' \exp[-i \Omega \zeta''] f_2(\zeta'',p'')~, \cr && \label{ffft}\end{aligned}$$ where $\Omega = \omega/\omega_l$ is the conjugate variable of $\zeta$, whose meaning is that of normalized frequency. The integrals in Eq. (\[ffft\]) cannot be easily performed, directly. As customary, one can transform the final variables $(\zeta'', p'')$ back to the initial variables $(\zeta,p)$ and perform the required integrations with respect to the old variables. This allows to use the fact that $f_2(\zeta'',p'') = f_0(\zeta,p)$. Since $d\zeta'' dp'' = d\zeta d p$ one obtains $$\begin{aligned} \bar{\rho}(\Omega) && = \int_{-\infty}^{\infty} dp d\zeta \exp[-i \Omega \zeta''(\zeta,p)] f_0(\zeta,p) \cr && = \frac{N_0}{2\pi \sigma_\zeta} \int_{-\infty}^{\infty} dp d\zeta \exp[-i \Omega \zeta''(\zeta,p)] \exp\left[-\frac{p^2}{2}-\frac{\zeta^2}{2 \sigma_\zeta^2}\right]~, \cr && \label{ffft2}\end{aligned}$$ where $\zeta''(\zeta,p)$ can be obtained from Eq. (\[M1\]) and Eq. (\[M2\]), and reads $$\begin{aligned} \zeta''(\zeta,p) =&& \zeta + (B_1+B_2) p + A_1 (B_1+B_2)\sin(\zeta+\phi_1)\cr && +A_2 B_2 \sin\left(\zeta + B_1 p + A_1 B_1 \sin(\zeta+\phi_1)+\phi_2\right) ~.\label{zeta2}\end{aligned}$$ Substituting Eq. (\[zeta2\]) into Eq. (\[ffft2\]) we find, explicitly: $$\begin{aligned} \bar{\rho}(\Omega) && = \frac{N_0}{2\pi \sigma_\zeta} \int_{-\infty}^{\infty} dp d\zeta \exp\left[-\frac{p^2}{2}-\frac{\zeta^2}{2 \sigma_\zeta^2}\right]~, \cr && \times \exp\Bigg\{-i \Omega \Bigg[\zeta + (B_1+B_2) p + A_1 (B_1+B_2)\sin(\zeta+\phi_1)\cr && +A_2 B_2 \sin\left(\zeta + B_1 p + A_1 B_1 \sin(\zeta+\phi_1)+\phi_2\right)\Bigg] \Bigg\}\label{ffft3}\end{aligned}$$ The following step consists in expanding the exponential factors containing trigonometric expressions according to[^3]: $$\begin{aligned} \exp[-i \Omega A_1(B_1+B_2)\sin(\zeta+\phi_1)] = \sum_{k=-\infty}^{\infty} \exp[i k (\zeta+\phi_1)] J_k\left[-\Omega A_1(B_1+B_2)\right] \cr && \label{expan1}\end{aligned}$$ and $$\begin{aligned} &&\exp\left[-i \Omega A_2 B_2 \sin\left(\zeta + B_1 p + A_1 B_1 \sin(\zeta+\phi_1)+\phi_2\right)\right] = \cr && \sum_{m=-\infty}^{\infty} \exp\left[i m \left(\zeta + B_1 p + A_1 B_1 \sin(\zeta+\phi_1)+\phi_2\right)\right] J_m\left[-\Omega A_2 B_2\right] ~, \label{expan2}\end{aligned}$$ where one can still expand $$\begin{aligned} \exp[i m A_1 B_1 \sin(\zeta+\phi_1)] = \sum_{l=-\infty}^{\infty} \exp[i l (\zeta+\phi_1)] J_l\left[ m A_1 B_1\right] ~.\label{expan3}\end{aligned}$$ Assuming, for the moment, a dependence of the laser phases $\phi_1$ and $\phi_2$ on $\zeta$, and collecting terms that have a dependence on $\zeta$ we can define $$\begin{aligned} \bar{f}_{\zeta}(k+l+m-\Omega)=\int_{-\infty}^{\infty} d\zeta f_{0\zeta}(\zeta) \exp[i (k+l+m-\Omega)\zeta] \exp[i (k+l) \phi_1 + i m \phi_2]\cr && \label{fbarz}\end{aligned}$$ and obtain from Eq. (\[ffft3\]): $$\begin{aligned} \bar{\rho}(\Omega) && = \frac{1}{\sqrt{2\pi}} \sum_{m,k,l} \bar{f}_{\zeta}(k+l+m-\Omega) J_k\left[-\Omega A_1(B_1+B_2)\right] J_m\left[-\Omega A_2 B_2\right] J_l\left[ m A_1 B_1\right]\cr && \times \int_{-\infty}^{\infty} dp \exp\left[-\frac{p^2}{2}\right] \exp[-i \Omega (B_1+B_2)p+im B_1p]~. \label{ffft4}\end{aligned}$$ The integration over $p$ can be carried out using $$\begin{aligned} \frac{1}{N_0} \int_{-\infty}^{\infty} dp \exp[-i \Omega p (B_1+B_2) + i m p B_1] f_{0p}(p) = \exp[(\Omega (B_1+B_2)-m B_1)^2/2] \cr \label{uno}\end{aligned}$$ which yields $$\begin{aligned} \bar{\rho}(\Omega) && = \sum_{m,k,l} \bar{f}_{\zeta}(k+l+m-\Omega) J_k\left[-\Omega A_1(B_1+B_2)\right] J_m\left[-\Omega A_2 B_2\right] J_l\left[ m A_1 B_1\right]\cr && \times \exp[(\Omega (B_1+B_2)-m B_1)^2/2] ~. \label{ffft5}\end{aligned}$$ Setting $n=k+l$ and using $$\begin{aligned} J_{k+l}(\alpha+\beta) = \sum_{l=-\infty}^{\infty} J_l(\beta)J_k(\alpha) ~,\label{due}\end{aligned}$$ Eq. (\[ffft5\]) can be re-written as $$\begin{aligned} \bar{\rho}(\Omega) && = \sum_{m,n} \bar{f}_{\zeta}(n+m-\Omega) J_n\left[-\Omega A_1(B_1+B_2)+m A_1 B_1\right] J_m\left[-\Omega A_2 B_2\right] \cr && \times \exp[(\Omega (B_1+B_2)-m B_1)^2/2] ~. \label{ffft6}\end{aligned}$$ We now apply the adiabatic approximation imposing that the width of the peaks in $\bar{f}_{\zeta}$ is much narrower than the harmonic separation $\omega_l$ between peaks. Analysis of Eq. (\[ffft6\]) and Eq. (\[fbarz\]) shows that due to the adiabatic approximation, the contribution to $\bar{f}(\Omega)$ for a given value of $m+n$, is peaked around $\Omega \simeq m+n$. This means that the terms in the sum over $m$ in Eq. (\[ffft6\]) can be analyzed separately for a fixed value of $m+n$, and one obtains $$\begin{aligned} \bar{\rho}(\Omega,m+n) && = \sum_{n} \bar{f}_{\zeta}(n+m-\Omega) J_n\left[-\Omega A_1(B_1+B_2)+m A_1 B_1\right] J_m\left[-\Omega A_2 B_2\right] \cr && \times \exp[(\Omega (B_1+B_2)-m B_1)^2/2] ~. \label{ffft6sep}\end{aligned}$$ It should be remarked that due to the adiabatic approximation, and to non-resonant behavior of Bessel functions, in Eq. (\[ffft6\]) we can replace $\Omega$ with $m+n$ under the Bessel functions. In this way, $\bar{f}_\zeta$ can be interpreted as the Fourier transform of the electron bunch density. The physical meaning of all this, is that $\bar{f}_{\zeta}$ is peaked at frequencies $\Omega$ near to multiples $n+m$ of the laser frequency. In [@EEHG2] it is reported that, in order to maximize the modulus of the bunching factor one should impose $n=\pm 1$. This can be seen directly by inspecting the right hand side of Eq. (\[uno\]). In fact, for values of $\Omega$ near to $n+m$, the argument in the exponential function can be written as $p^2 ( B_1 n + B_2 (n+m))^2/2$. When $n=-1$ and $m$ is positive and large for example, one sees that that $B_1 n$ is large and negative, while $B_2 m$ is large and positive. Therefore, $m$ can be chosen such that $ - B_1 + B_2 (m-1) \simeq 0$. This is guarantees remarkable up-frequency conversion efficiency, almost independently on the energy spread and constitutes one of the great advantages of the EEHG scheme. We will restrict our investigation to the case $n=-1$ and $m>0$, thus obtaining $$\begin{aligned} \bar{\rho}(\Omega,m-1) && = \bar{f}_{\zeta}(m-1-\Omega) J_{-1}\left[-\Omega A_1(B_1+B_2)+m A_1 B_1\right]\cr && \times J_m\left[-\Omega A_2 B_2\right] \exp[(\Omega (B_1+B_2)-m B_1)^2/2] ~. \label{ffft6x}\end{aligned}$$ Note that if the laser phases would not depend on $\zeta$, which is not true in general, one could separately calculate $$\begin{aligned} &&\int_{-\infty}^{\infty} d\zeta f_{0\zeta}(\zeta) \exp[i (m-1-\Omega)\zeta] =\cr && \frac{1}{\sqrt{2\pi}\sigma_\zeta}\int_{-\infty}^{\infty} d\zeta \exp\left[-\frac{\zeta^2}{2 \sigma_\zeta^2}\right]\exp[i (m-1-\Omega)\zeta] = \exp\left[-\frac{\sigma_\zeta^2}{2}(m-1-\Omega)^2 \right]~.\cr && \label{depzeta2}\end{aligned}$$ In this case, the adiabatic approximation can be simply enforced imposing that $\sigma_\zeta \gg 1$. Finally, it should be noted that the initial electron density distribution and laser phases $\phi_1$ and $\phi_2$ are not only functions of $\zeta$, but also of the transverse position $\vec{r}$. It should be understood that the transverse direction can be factorized, which is a simplifying but not principal assumption, and that therefore, all the expressions above are considered valid at any fixed transverse position. To conclude, let us consider our initial question, whether the general principle of the frequency multiplier chains is valid or not for EEHG. The answer is affirmative, and can be seen by inspecting Eq. (\[ffft6x\]) and Eq. (\[fbarz\]). In the case when $\phi_1=\phi_2$ such principle can be applied strictly. In case $\phi_1$ and $\phi_2$ differ, but are still of the same order of magnitude, we can conclude that, since $n=-1$ and $m$ is large, only $\phi_2$ is important and the principle is applicable with accuracy roughly $1/N$. \[sec:tre\] Temporal quality of the seed laser beam =================================================== Nowadays, high peak power laser systems are capable of producing very high intensities, thus fulfilling the requirements for many high field applications including EEHG FELs. In particular, femtosecond laser systems have become the primary method to deal with these applications. The reasons for this are the availability of broadband, efficient lasing media such as titanium-doped sapphire (Ti:Sa), and of techniques like Kerr-lens mode locking and chirp pulse amplification (CPA). In CPA systems, light passes through a number of optical components. Moreover, non-linear effects take place in the amplifying medium. This can degrade the temporal quality of the output pulse, which can be appropriately modeled in a slowly-varying real field envelope and time-dependent carrier frequency approximation. The time-bandwidth product constitutes a proper measure of the departure from the ideal case, in which there are no temporal variations of the carrier frequency. In this Section we quantitatively describe the relation between carrier frequency chirp and corresponding broadening of the spectrum. This leads to a time-bandwidth product exceeding the Fourier limit. Pulse duration and spectral width --------------------------------- For our purposes, it is convenient to consider a Gaussian pulse with a linear frequency chirp. This choice is one of analytical convenience only, and may be generalized. The slowly complex field envelope is given by $$\begin{aligned} E(t) = A \exp\left[-\frac{t^2}{2 \tau^2}\right]\exp\left[i \frac{\alpha t^2}{2\tau^2}\right] \label{pulsec}\end{aligned}$$ where $\alpha$ is the chirp parameter, and the FWHM pulse duration is related to the rms duration $\tau$ by $\Delta \tau = \sqrt{4 \ln 2} \cdot \tau$. By Fourier transforming Eq. (\[pulsec\]), it can be demonstrated (see e.g. [@MILO]) that the spectral intensity is a Gaussian with a FWHM given by $\Delta \omega = (\sqrt{4\ln 2}/\tau) \sqrt{1 + \alpha^2}$. The time-bandwidth product of the pulse is therefore $$\begin{aligned} \Delta \omega\cdot \Delta \tau = 4 \ln 2 \cdot \sqrt{1 + \alpha^2} \label{TBp}\end{aligned}$$ This is larger than the time-bandwidth product of an unchirped Gaussian pulse, which is just $4 \ln 2$ . In other words, chirping increases the time-bandwidth product by broadening the pulse spectrum while preserving the pulse width. Note that $\Delta \omega \cdot \Delta \tau = 4 \ln 2$ is the smallest time-bandwidth product for a Gaussian pulse corresponding to the transform-limit (or bandwidth limit, or Fourier limit). The temporal quality of the pulse can be defined by a quality factor $M_t^2$, defined as the ratio between the time-bandwidth product for real and transform-limited pulse. Hence, one can characterize pulse by specifying its quality through the $M_t^2$ factor and by giving the pulse shape. In our case of interest, $M_t^2 = \sqrt{1 + \alpha^2}> 1$ for Gaussian pulses with linear frequency chirp. Finally, it should be noted that considerations analogous to those just discussed above, can be proposed for the electron beam microbunching. For example the current envelope of a Gaussian, chirped electron beam can be described similarly as in Eq. (\[pulsec\]), with a chirp parameter $\alpha_m$. A time-bandwidth product can be defined, and a quality factor $M_{t,m}$ can be defined as well. \[sub:con\] Constraint on temporal phase variation for the output Ti:Sa laser pulse ----------------------------------------------------------------------------------- There are several simplifying assumptions that will be used in our analysis. As has been the case for the analysis presented in the previous paragraph, we restrict our attention to a microbunched electron beam with Gaussian shape. This is not a significant restriction, and extensions are not difficult to consider. We introduce the following criterion: we consider the electron beam microbunching nearly transform-limited when the performance ratio $M_{t,m}^{-2}$ is down not more than $1/\sqrt{2}$. For a microbunching with Gaussian shape and linear frequency chirp, this criterion will be satisfied under the restriction that the microbunch chirp parameter $\alpha_m < 1$. A specific example of a microbunched beam with Gaussian profile could be realized in the case when EEHG scheme uses an electron bunch with Gaussian temporal profile and a seed laser pulse with flat-top profile in time across the duration of the electron bunch. As demonstrated in e.g. [@EETOL3], the generated bunching is not sensitive to the peak current. Therefore, EEHG can operate with a nonuniform electron bunch profile. In the next paragraph we will demonstrate that in any case, due to non-linear (self-phasing) effects in the Ti:Sa laser system and in the post-laser optics system, the seed laser must have flat-top profile in time with very little temporal variation. Therefore, the model of a seed pulse with flat-top profile and of an electron bunch with Gaussian profile is consistent with the EEHG scheme. Now, if the phase of the seed laser is chirped, the microbunching chirp is simply multiplied by the frequency multiplication factor $N$. This can be seen by looking at the harmonic contents of the current density found in Eq. (\[ffft6\]). That expression includes $\bar{f}_{\zeta}$, which in the case of $\phi_1=\phi_2 = \alpha \zeta^2/(2 \sigma_\zeta^2)$ is given by (see Eq. (\[fbarz\])): $$\begin{aligned} &&\bar{f}_{\zeta}(m-1-\Omega)=\int_{-\infty}^{\infty} d\zeta f_{0\zeta}(\zeta) \exp[i (m-1-\Omega)\zeta] \exp[-i \phi_1 + i m \phi_2]\cr && = \frac{1}{\sqrt{2\pi}\sigma_\zeta}\int_{-\infty}^{\infty} d\zeta \exp[i (m-1-\Omega)\zeta]\exp\left[-\frac{\zeta^2}{2 \sigma_\zeta^2}\right]\exp\left[i (m-1) \frac{ \alpha \zeta^2}{2 \tau_\zeta^2}\right]\cr && \label{fbarz2}\end{aligned}$$ The last phase factor under integral shows that the laser phase is indeed multiplied by $N=m-1$. We will define the frequency chirp in the seed laser pulse only across the target duration of the electron bunch, and use the same time normalization as for the beam microbunching. The complex field envelope of a laser pulse with stepped profile and linear frequency chirp is given by $$\begin{aligned} E(\zeta) = E_0 \exp\left[i \frac{\alpha \zeta^2}{2 \sigma_\zeta^2}\right]~, \label{Et}\end{aligned}$$ where $E_0$ is a constant. As discussed above, the frequency multiplication yields a complex “microbunching” envelope with carrier frequency $\omega_0 = (m-1) \omega_l$ $$\begin{aligned} a(\zeta) = a_0 \exp\left[- \frac{\zeta^2}{2 \sigma_\zeta^2}\right] \exp\left[i \frac{\alpha_m \zeta^2}{2 \sigma_\zeta^2}\right] \label{Et}\end{aligned}$$ where $\rho_0$ is a constant, and $\alpha_m = N\alpha$ is the microbunching chirp parameter. Note that what we loosely defined as “microbunching” is, more formally, the slowly-varying amplitude of the electron density modulation with carrier frequency $\Omega=N$. It follows from the previous analysis that the EEHG scheme can produce nearly transform-limited microbunching only under the restriction $\alpha_m \lesssim 1$, meaning that the laser chirp parameter must obey $\alpha \lesssim 1/N$. The EEHG seed laser is assumed to be a Ti:Sa laser. The actual seed laser beam consists in the third or in the fourth harmonic of the Ti:Sa laser beam. Usually, laser frequency multipliers are based on the use of Beta Barium Borate (BBO) crystals. The effect of frequency multiplication on phase variation amounts again to multiplication of the phase variations. Therefore we may say that when we study constraints on the performance of Ti:Sa seed laser for EEHG schemes, the total frequency multiplication chain consists of two stages. The first stage is the BBO crystals with a frequency multiplication factor $N_1 =3$ (or $N_1 =4$). The second stage is the EEHG setup itself, with frequency multiplication factor up to $N_2 \sim 270$ (or $N_2 \sim 200$). If the final required output radiation is around wavelengths of $1$ nm, the total frequency multiplication factor $N = N_1 N_2$ is about $N \sim 800$. From the previously discussed condition $\alpha \lesssim 1/N$ it can be seen that the Ti:Sa laser produces nearly transform-limited microbunching at wavelengths around $1$ nm only when the laser chirp parameter $\alpha \lesssim 10^{-3}$. Thus, for most purposes, if the total multiplication factor is around $800$ or exceeds it, we may formulate the constraint on the Ti:Sa laser quality by requiring a quality factor $M_t^2$ departing from unity of no more than about $10^{-6}$. One can think that the above-discussed constraints on seed laser may be true only for the particular case of EEHG. However, we can show that these constraints are actually of more general validity. For example, HGHG schemes can produce nearly transform-limited radiation spanning down to wavelengths of $1$ nm only under the same restrictions on temporal quality of the seed Ti:Sa laser. The key advantage of the EEHG scheme is that the amplitude of the achieved microbunching factor slowly decays with increasing harmonic number and that, consequently, generation of coherent soft X-ray emission within a single upshift stage becomes possible [@EEHG1; @EEHG2]. However, considering constraints on the seed laser $M_t^2$ factor, all harmonic generation schemes are similar, and must obey the universal result $$\begin{aligned} M_t^2-1 \lesssim \frac{1}{N^2}~. \label{unires}\end{aligned}$$ The requirement in the inequality (\[unires\]) can be somehow relaxed if the requirement of near-Fourier limit is relaxed as well. For example, the operation of a EEHG FEL is characterized by two microbunch bandwidth scales of interest. One is associated with inverse electron bunch duration $\Delta \omega_b = 1/\tau_b$, $\tau_b$ being the electron bunch duration. The other is the FEL amplification bandwidth $\Delta \omega_a$. One can relax the requirement of near-Fourier limit substituting it by the requirement to achieve an output radiation bandwidth narrower than the SASE bandwidth $\Delta \omega_a$. On the one hand, the product of bunch duration by amplification bandwidth can be estimated in the order of $\tau_b \Delta \omega_a \sim 10^2$ in the soft X-ray wavelength range. On the other hand, the FEL radiation bandwidth broadening due to the effect of linear frequency chirp is about $\Delta \omega \sim |\alpha_m|/\tau_b$. Therefore, in the case when $$\begin{aligned} |\alpha_m| > (\tau_b \Delta \omega_a) \sim 10^2 ~,\label{alpham} \label{alpham}\end{aligned}$$ the output signal has a bandwidth larger than the SASE bandwidth, and harmonic generation techniques have no practical applications. However, if, for example, we have a microbunching chirp parameter $|\alpha_m| \sim 10$, the effective radiation bandwidth becomes ten times narrower than the SASE bandwidth, although is ten times wider compared to the ideal transform-limited bandwidth. Following this discussion, a weaker constraint on the temporal quality factor of seed laser is $M_t^2 - 1 < 10^2/N^2$. For a Ti:Sa laser seed and a radiation wavelength of $1$ nm it is possible to discuss about harmonic generation techniques applications only when $M_t^2 - 1 < 10^{-4}$. To complete the picture, we should note that an alternative method to harmonic generation setups, called self-seeding [@SELF; @TREU; @MARI], is available, and allows for the generation of temporally coherent radiation in XFELs. A self-seeded soft X-ray FEL consists of two undulators separated by a monochromator installed within a magnetic chicane. The remarkable temporal quality of the output radiation and the wavelength tunability of self-seeding schemes has stimulated interest in using this technique to generate nearly transform-limited soft- X-ray pulses. A project of self-seeding schemes with grating monochromator is now under development at LCLS II [@LCLS2]-[@FENG]. EEHG output will compete with self-seeding output only when the temporal quality of the seed laser beam obeys the mores stringent requirement (\[unires\]). \[sub:self\] Self-phasing and constraints on field amplitude variation ---------------------------------------------------------------------- The seeding pulse from the Ti:Sa laser must necessarily propagate through vacuum window and BBO crystals without experiencing temporal phase distortions. Above a power density of $1 \mathrm{GW/cm^2}$, the refractive index $n$ becomes intensity-dependent according to the well-known expression $$\begin{aligned} n = n_0 + n_2 I ~, \label{refrin}\end{aligned}$$ where $n_0$ is the index of refraction at low intensity and $I$ is the laser intensity. Due to temporal variations of the laser pulse intensity, the pulse phase will then be distorted according to [@MILO] $$\begin{aligned} B = \frac{2\pi}{\lambda} \int_0^{L} dz n_2 I ~.\label{B}\end{aligned}$$ Here $\lambda$ is the laser wavelength, and $B$ represents the amount of phase distortions accumulated by the pulse over a length $L$. The dimensionless $B$ parameter, also known as $B$ integral, is often used as a measure of the strength of nonlinear effects due to the non-linear refractive index $n_2 I$. Field intensities, propagation distances, and values of $n_2 I$ such that $B> 1$ generally yield significant nonlinear effects, including self-phase modulation. Usually, in laser optics, when $B < 0.5$ pulse distortions should not be a problem. Let us consider an optical setup behind the Ti:Sa laser with $B \sim 0.5$. In order to have minimal FEL output spectral broadening, the seed laser must have flat-top profile in time with very little temporal variation. The intensity variation must satisfy $$\begin{aligned} \frac{\Delta I}{I} < \frac{2}{N} \label{DII}\end{aligned}$$ For $1$ nm wavelength mode of operation $N \sim 800$, and in the case of near transform-limited FEL output pulse, the intensity of Ti:Sa laser pulse must be controlled to about $0.3 \%$ across the target duration of the electron bunch. \[sec:quattro\] Spatial quality of seed laser beam ================================================== In the last section we considered part of the constraints on the performance required for EEHG seed lasers. In particular, our discussion has been restricted to the temporal quality of laser beams. The former restriction allows one to obtain results which depend on the frequency multiplication factor only, so that the treatment discussed above applies not only to EEHG schemes, but to more general cases as well. In this section we discuss, instead, the influence of errors on the wavefront of the seed laser beam. A general principle discussed before states that the effect of frequency multiplication by a factor $N$ is to multiply the phase variation in time by $N$. The same principle holds when dealing with phase variations in space. If the wavefront of the UV seed laser exhibits errors, the errors of the microbunching wavefront are multiplied by the frequency multiplication factor. This can be seen with an analysis similar to that in paragraph \[sub:con\], based on the results in Section \[sec:due\], which led to Eq. (\[fbarz2\]). However, now, the phase variations are to be considered as a function of spatial coordinates. In the case of variation in time, the temporal quality of the output FEL radiation is a replica of the temporal quality of the microbunching input. It seems natural to use the same principle for characterizing the spatial quality of the output FEL radiation. However, this cannot be done. The reason is that the transverse coherence of FEL radiation is settled without laser seeding. This is due to the transverse eigenmode selection mechanism: only the ground eigenmode survives at the end of the amplification process. It follows that the microbunching wavefront errors do not affect the spatial quality of the output radiation. They only affect the input signal value. The description of the influence of phase errors depends in detail on the harmonic generation process. For example, in the case of HGHG, the seed laser directly produces microbunching in the first cascade only, which is characterized by a relatively small frequency multiplication factor $N < 5$. In EEHG schemes instead, the generation of coherent radiation in the soft X-ray wavelength range should be achieved with a single upshift stage using a UV ( $200$ nm or $270$ nm) laser beam. In this case the frequency multiplication factor amounts to about $N \sim 200$. Consequently, the EEHG technique is much more sensitive to laser wavefront errors. This disadvantage is actually related to the key EEHG advantage, that is to allow for high frequency multiplication numbers within a single, compact scheme. To understand the effects of wavefront errors we shall use an analogy between time and space. This analogy suggests the possibility of simply translating the effects related to phase perturbation in time into effects related to wavefront perturbations as shown in Table \[tt1\]. Temporal (pulse)   Spatial (beam) ------------------------------------------------------------------ ------------------------------------------------------- transform-limited pulse diffraction-limited beam temporal frequency spatial frequency bandwidth of amplification bandwidth of amplification temporal frequency shift (temporal linear phase chirp) wavefront tilt (spatial linear phase chirp) linear temporal frequency chirp (temporal quadratic phase chirp) defocusing aberration (spatial quadratic phase chirp) nonlinear temporal frequency chirps high order wavefront aberrations phase fluctuations in time chaotic phase variation across the beam : Analogy between temporal and spatial characteristics \[tt1\] We defined the ideal seed pulse as a transform-limited pulse i.e. a pulse without phase variations in time. The space-domain analog of a transform-limited pulse is a diffraction-limited beam, i.e. a beam without phase variations in space. From this definition follows that a beam can be diffraction-limited only at its waist, where it takes on the minimum possible product between size and spatial frequency bandwidth. In fact, beam propagation leads to a beam broadening and to a spatial quadratic phase chirp. Since the ideal seed laser beam is characterized by microbunching wavefront without phase variation across the electron beam, it follows that the seed laser beam must be diffraction-limited at its waist, which must be placed in the middle of the modulator undulator. Simple physical considerations directly lead to a crude approximation for the amplification bandwidth. As already discussed in paragraph \[sub:con\], in the time domain the amplification bandwidth is about two order of magnitudes larger than transform-limited bandwidth: $$\begin{aligned} \tau_b \Delta \omega_a \sim 10^2 ~. \label{taub}\end{aligned}$$ This fact has some interesting consequences. Suppose that we consider microbunching with linear phase chirp in time, which is actually equivalent to a shift of the signal frequency. In the case when the shift is smaller than the amplification bandwidth, the temporal quality and the output power of the radiation pulse are not changed. At variance, microbunching with nonlinear phase chirp leads to a spectral broadening of the output radiation and, consequently, to degradation of the temporal quality. However, in the case when the broadening is smaller than the amplification bandwidth, the output power is not suppressed. The situation is quite different when considering the spatial domain. In fact, qualitatively, the spatial frequency amplification bandwidth and the diffraction-limited bandwidth are the same, so that any shift or broadening of the spatial frequency spectrum immediately leads to input signal suppression. Let us study the discrepancy between the direction of the electron motion and the normal to the microbunching wavefront. In the case when the discrepancy between these two directions is larger than the FEL angular amplification bandwidth the input signal is exponentially suppressed. Let us assume that the spatial profile of the microbunching is close to that of the electron beam, and is characterized by a Gaussian shape with standard deviation $\sigma_b$. The FEL angular amplification bandwidth can then be estimated as $\Delta \theta_a \sim (k \sigma_b)^{-1}$, where $k$ is wavenumber at the target harmonic. One can then estimate the angular spectrum of e.g. the LCLS output for the wavelength of $1.5$ nm. The transverse distribution of the electron beam is described by $\sigma_b \sim 30 \mu$m, and our estimations give $\Delta \theta_a \sim 8 \mu$rad. The angular amplification bandwidth corresponds to the HWHM of the FEL output angular distribution. Results of numerical simulations, confirmed by experimental results, give an angular distribution of the radiation intensity with HWHM $\sim 10 \mu$rad. From these numbers one can see that the above approach provides an adequate description, at least in the wavelength range around $1$ nm. The value $\Delta \theta_a$ can subsequently be used to estimate the maximum angular error allowed between the normal to the laser beam wavefront (at its waist) and the direction of the electron beam motion in the modulator undulator. It follows from the previous reasoning that in the case of radiation wavelength around $1$ nm we find an alignment tolerance of about $10 \mu$rad. The wavefront tilting is a relatively simple (first order) geometrical distortion and its measure is simply an angle, which is the same for the microbunching wavefront and for the laser beam wavefront. The width of the seed laser beam at its waist can be much larger than the width of the electron beam, but the tilt is completely characterized by such angle only. There are several criteria to analyze the performance of laser system to higher order aberrations. To characterize the spatial quality of the laser beam, we will use the Strehl ratio $S$, usually defined[^4] as: $$\begin{aligned} S=\frac{\max[|FT\{E(x,y)\exp[i\phi(x,y)]\}|^2]}{\max[|FT[E(x,y)]|^2]}~, \label{Sratio} \end{aligned}$$ where “FT” indicates the 2D spatial Fourier transform operation, $E(x,y)$ is the ideal wave amplitude, and $\phi(x,y)$ is the phase aberration. The Strehl ratio $S$ becomes an important figure of merit from the viewpoint of seeding evaluation. Let us consider the practical situation in which both laser and electron beams are characterized by a Gaussian shape, and in which the width of the laser beam at its waist is much larger than the width of the electron beam. With this assumption, within the electron beam, at the laser beam waist in the plane $z = 0$ we have asymptotically $E(x,y,0) = \mathrm{const}\cdot \exp[i\phi(x,y)]$ , where $E(x,y,0)$ is the wave amplitude, and $\phi(x,y)$ describes phase aberrations. For our purposes it is interesting to consider the Gaussian-weighted Strehl ratio $S$ $$\begin{aligned} S = \left|\left<\exp[i \phi(x,y)]\right>\right|^2 ~, \label{Sweight}\end{aligned}$$ where $$\begin{aligned} \left<\exp[i\phi(x,y)]\right> = (2\pi \sigma_p^2)^{-1} \int dx dy \exp\left[-\frac{x^2+y^2}{2 \sigma_p^2}\right]\exp[i\phi(x,y)]~. \label{aveX}\end{aligned}$$ Here $\sigma_p$ is a Gaussian parameter of the same order of magnitude of the rms width of the electron beam, $\sigma_b$. If the phase is sufficiently small to accurately replace $\exp[i\phi]$ with $1+ i\phi -\phi^2/2$, one obtains $$\begin{aligned} S = 1 - \sigma_\phi^2 ~, \label{S2}\end{aligned}$$ where $$\begin{aligned} \sigma_\phi^2 = <\phi^2> - <\phi>^2 \label{sigphi}\end{aligned}$$ is the variance of the phase aberration weighted across a Gaussian-amplitude pupil. To be more specific, we define the average of $\phi(x,y)$ across the pupil as $$\begin{aligned} <\phi> = (2 \pi \sigma_p^2)^{-1} \int dx dy \exp\left[- \frac{x^2 + y^2}{2 \sigma_p^2}\right] \phi(x,y) ~,\label{avephi2}\end{aligned}$$ and, likewise, the average of the square of $\phi(x,y)$ as $$\begin{aligned} <\phi^2> = (2\pi\sigma_p^2)^{-1} \int dx dy \exp\left[-\frac{x^2 + y^2)}{2\sigma_p^2}\right] \phi^2 ~. \label{phi2ave2}\end{aligned}$$ It follows that if the root-mean-square variations of the wavefront are of the order of a tenth of the wavelength only, we obtain a Strehl ratio of $0.6$. Let us now discuss the spatial quality of the microbunching wavefront. The interesting value to know for EEHG operation is the input coupling factor between the microbunching and the ground eigenmode of the FEL amplifier. Let us consider the amplitude of the electron density modulation at the carrier frequency $\omega_0 = (m-1) \omega_l$ : $$\begin{aligned} {\rho}(x,y,t) = a(x,y,t)\exp[i (m-1) \omega_l t]~. \label{tilderho}\end{aligned}$$ In ideal case, the electron density modulation exhibits a plane wavefront and a Gaussian shape across the electron beam: $$\begin{aligned} a(x,y,t) = a_0(t)\exp\left[-\frac{x^2+y^2}{2\sigma_b^2}\right]~. \label{bxyt} \end{aligned}$$ In such ideal case, the input coupling factor is therefore $$\begin{aligned} C = \int dx dy \exp \left[-\frac{x^2+y^2}{2\sigma_b^2}\right]\Psi(x,y)~, \label{CCCC}\end{aligned}$$ where $\Psi(x,y)$ is the field distribution of the ground eigenmode. In the high gain linear regime, the FEL output radiation power scales as $$\begin{aligned} W_\mathrm{output} \sim |C|^2 ~. \label{Wout}\end{aligned}$$ In the case of a non-ideal microbunching wavefront, expressions for $a(x,y,t)$ and for the input coupling factor respectively transform to: $$\begin{aligned} a(x,y,t) = a_0(t)\exp[i\phi_m(x,y)] \exp\left[-\frac{x^2+y^2}{2\sigma_b^2}\right]~ , \label{bnonid}\end{aligned}$$ and $$\begin{aligned} C = \int dx dy~ a(x,y,t) \Psi(x,y) ~, \label{Cnonide}\end{aligned}$$ where $\phi_m(x,y)$ is the microbunching phase aberration. The ratio of the output power for the case including microbunching wavefront errors to the output power for the case of a plane microbunching wavefront is a simple and convenient measure of the departure from the ideal situation. In our case this ratio is simply $$\begin{aligned} \frac{W_{\mathrm{nonideal}}}{W_{\mathrm{ideal}}} =\frac{|C_\mathrm{nonideal}|^2}{|C_\mathrm{ideal}|^2} ~. \label{ratiomer}\end{aligned}$$ Various approximations can be invoked. One of the simplest is to use the following expression for the ground FEL eigenfunction $$\begin{aligned} \Psi(x,y) \sim \exp\left[-\frac{x^2+y^2}{2\sigma_b^2}\right] ~. \label{eigenmode}\end{aligned}$$ With this approximation it can be shown that $$\begin{aligned} \frac{|C_\mathrm{nonideal}|^2}{|C_\mathrm{ideal}|^2} = 1- \sigma_\phi^2 ~, \label{Cration}\end{aligned}$$ where $\sigma_\phi^2$ is the variance of the microbunching phase aberration across the Gaussian-weighted pupil with $$\begin{aligned} \sigma_p = \frac{\sigma_b}{\sqrt{2}}~ . \label{sigp2}\end{aligned}$$ If we now look at the ratio of the power values at the FEL exit with microbunch wavefront distortions and without distortions, we see that such ratio corresponds to the already introduced laser Strehl ratio, Eq. \[S2\]. More in general, we have the same definition given in Eq. (\[Sweight\]), where the phase $\phi$ under the integral is now defined as the phase on the microbunching wavefront $\phi_m$. Finally, we calculate the relation between the phase distortions of the laser beam and the phase distortions of the microbunching. We have concluded from our theoretical analysis in Section \[sec:due\], that if the wavefront of the seed laser beam in the waist plane exhibits errors, the errors of the microbunching wavefront are multiplied by the frequency multiplication factor $N$. Therefore we have $$\begin{aligned} (\sigma_\phi)_\mathrm{laser} = \frac{1}{N}(\sigma_\phi)_\mathrm{microbunch}~, \label{sigphimicro}\end{aligned}$$ which yields $$\begin{aligned} 1-S_\mathrm{laser} = \frac{1}{N^2} [1-S_\mathrm{microbunch}] ~. \label{1ms}\end{aligned}$$ For EEHG schemes, $[1-S_\mathrm{microbunch}]$ must be kept below $0.4$, corresponding to microbunching wavefront distortions of $\lambda/10$. This corresponds to a UV laser Strehl ratio $S_\mathrm{laser} > 0.99999$ at the target wavelength of $1$ nm. In order to experimentally investigate the effects of laser wavefront errors on the FEL amplification process, one should perform direct measurements of the laser beam wavefront using, for example, a Hartmann sensor. Usually, measurements of the spatial quality of the output laser beam with a Hartmann sensor give the near-field wavefront characteristics. The knowledge of the spatial phase and amplitude in a particular plane opens the possibility of calculating, by Fresnel propagation, the phase and amplitude in any other plane for a freely propagating laser beam, and in particular allows to recover results in the middle plane of the modulator undulator. Applying the definition of the Gaussian-weighted Strehl ratio in Eq. (\[Sweight\]) with $\sigma_p =\sigma_b/\sqrt{2}$ leads to the value which needs to be compared with constraint $$\begin{aligned} 1- S < \frac{0.4}{N^2}~. \label{last}\end{aligned}$$ The arguments discussed above seem to be strong enough to suggest that EEHG FEL schemes for reaching frequency multiplication factor of $N$ will not work when the difference of the above-defined laser Strehl ratio from the unity does not satisfy the inequality in (\[last\]). This conclusion for the spatial domain contrasts with that in the time domain, where the phase distortions lead to spectral broadening but do not have an impact on the FEL output power. \[cinque\] Conclusions ====================== It is very desirable to have a way to model the performance of EEHG FEL with high frequency multiplication factor. Such modeling would naturally start with the Ti:Sa laser system. Calculations would involve the knowledge of the temporal and spatial properties of the Ti:Sa laser source itself together with laser field propagation through the optical components used in the EEHG beamline. Most of our calculations are, in principle, straightforward applications of conventional laser optics and general theory of frequency multiplier chains. Our paper provides physical understanding of the laser seeding setup and we expect it to be useful for practical estimations, especially at the design stage of the experiment. Detailed EEHG mechanism is so complicated that we cannot accurately determine the EEHG output by analytical methods. However, a definite relation between quality of the input signal and EEHG FEL output can be worked out without any knowledge about the EEHG internal machinery. Acknowledgements ================ We are grateful to Massimo Altarelli, Reinhard Brinkmann, Serguei Molodtsov and Edgar Weckert for their support and their interest during the compilation of this work. [99]{} L. -H. Yu. Phys. Rev. A 44, 5178 (1991). L. -H. Yu, I. Ben-Zvi, Nucl. Instrum. Methods A 393, 96-99 (1997). G. Stupakov, Phys. Rev. Lett. 102, 074801 (2009). D. Xiang and G. Stupakov, Phys. Rev ST AB 12, 030702 (2009). D. Xiang and G. Stupakov “Echo-seeding options for LCLS-II”, TUPB13, Proceedings of FEL 2010, Malmo, Sweden (2010). E. Prat and S. Reiche, “EEHG seeding design for SwissFEL”, TUPA25, Proceedings of FEL 2011, Shanghai, China (2011). K. E. Hacker, et al., Echo-seeding experiment at FLASH in 2012", TUPB10, Proceedings of FEL 2011, Shanghai, China (2011). D. Xiang and G. Stupakov “Tolerance study for the EEHG Laser”, WE5RFP044, Proceedings of PAC09, Vancouver, BC, Canada (2009). Z. Huang, et al., “Effect of energy chirp on EEHG Lasers”, MOPC45, Proceedings of FEL 2009, Liverpool, UK (2009). G. Penn and M. Reinsch, Journal of Modern Optics, 1-15 (2011). D. Ratner, https://sites.google.com/a/lbl.gov/ realizing-the-potential-of-seeded-fels-in-the-soft-x-ray-regime-workshop/talks, Workshop on Realizing the Potential of Seeded FELs in the Soft X-Ray Regime, Berkeley, CA, USA (2011) and following discussion sessions. K. Hacker, “EEHG at FLASH 2012 and beyond”, https://indico.desy.de/conferenceDisplay.py?ovw=True$\setminus \&$confId=4736, FLASH Accelerator Workshop, Hamburg, Germany (2011) K. Hacker, https://sites.google.com/a/lbl.gov/realizing-the-potential-of-seeded-fels -in-the-soft-x-ray-regime-workshop/talks, Workshop on Realizing the Potential of Seeded FELs in the Soft X-Ray Regime, Berkeley, CA, USA (2011) and following discussion sessions. W. P. Robins, “Phase Noise in Signal Sources”, IEEE Telecommunication Series, vol 9., Peter Peregrinus Ltd. (1982). P. Milonni and J. Eberly, “Laser Physics”, Wiley and Sons, (2010). J. Feldhaus et al., Optics. Comm. 140, 341 (1997). R. Treusch, W. Brefeld, J. Feldhaus and U. Hahn, HASILAB Ann. report 2001 “The seeding project for the FEL in TTF phase II” A. Marinelli et al., Proceedings of the FEL Conference 2008, MOPPH009 (2008). The LCLS-II Conceptual Design Report, Stanford, https:// slacportal.slac.stanford.edu/ sites/lcls$\_$public/lcls$\_$ii/Pages/default.aspx (2011). J. Wu et al., “Staged self-seeding scheme for narrow bandwidth , ultra-short X-ray harmonic generation free electron laser at LCLS”, proceedings of 2010 FEL conference, Malmo, Sweden, (2010). Y. Feng et al., “Optics for self-seeding soft x-ray FEL undulators”, proceedings of 2010 FEL conference, Malmo, Sweden, (2010). [^1]: The issue was also discussed during the preparation of this work in [@RATN]. There HGHG was mainly considered but, as noted above, the method used to describe the output field perturbation is independent of the specific kind of harmonic generation technique. [^2]: We kept our notation similar to that of [@EEHG2]. However, we chose $\omega_l \equiv \omega_1=\omega_2$ from the very beginning. Therefore $K=\omega_2/\omega_1 = 1$ for us. Also note that, since reference [@EEHG2] deals with the steady state case, the phases of the two laser pulses are constant. This explains why only a relative phase $\phi$ was introduced in [@EEHG2]. At variance, in this paper we treat the time-dependent case, where the two laser phases can exhibit different time variations. As a result, here we include the phases $\phi_1$ and $\phi_2$ of both lasers. [^3]: In the following $k$ is just an index, without the meaning of wavenumber. [^4]: With this definition, the Strehl ratio is related to the transverse $M^2$ parameter by $S=1/M^2$

--- abstract: '**In the two-dimensional electron gas (2DEG) emerging at the transition metal oxide surface and interface, it has been pointed out that the Rashba spin-orbit interaction, the momentum-dependent spin splitting due to broken inversion symmetry and atomic spin-orbit coupling, can have profound effects on electronic ordering in the spin, orbit, and charge channels, and may help give rise to exotic phenomena such as ferromagnetism-superconductivity coexistence and topological superconductivity. Although a large Rashba splitting is expected to improve experimental accessibility of such phenomena, it has not been understood how we can maximally enhance this splitting. Here, we present a promising route to realize significant Rashba-type band splitting using a thin film heterostructure. Based on first-principles methods and analytic model analyses, a tantalate monolayer on BaHfO$_3$ is shown to host two-dimensional bands originating from Ta $t_{2g}$ states with strong Rashba spin splittings - up to nearly 10% of the bandwidth - at both the band minima and saddle points due to the maximal breaking of the inversion symmetry. Such 2DEG band structure makes this oxide heterostructure a promising platform for realizing both a topological superconductor which hosts Majorana fermions and the electron correlation physics with strong spin-orbit coupling.**' author: - Minsung - Jisoon - Suk Bum bibliography: - 'rashbaoxide.bib' date: - - title: 'Strongly enhanced Rashba splittings in oxide heterostructure: a tantalate monolayer on BaHfO$_3$' --- Recently, the spin-orbit interaction of the two-dimensional electron gas (2DEG) at the surfaces and interfaces of the perovskite transition metal (TM) oxide  [-@Ohtomo2004; -@Takagi2010; -@Mannhart2010; -@Santander2011; -@Meevasana2011] has been much investigated experimentally [-@BenShalom2010; -@Caviglia2010; -@Santander2014; -@Santander2012; -@King2012; -@Reyren2012; -@MKim2012]. However, definite understanding on how its magnitude might be maximized has not been well established. It is the combination of the broken inversion symmetry and the atomic spin-orbit coupling (SOC) of the TM that gives rise to a non-zero spin splitting in the form of the Rashba spin-orbit interaction [-@Bychkov1984; -@Dresselhaus1955; -@Winkler2003]. But this origin implies that the magnitude of the spin-orbit interaction is intrinsically limited by the TM atomic SOC. The limitation should be apparent in the best-studied perovskite 2DEGs — the SrTiO$_3$ (STO) surface and the LaAlO$_3$/SrTiO$_3$ (LAO/STO) heterostructure interface — as the atomic SOC strength of the 3$d$ TM Ti is relatively small [-@Zhong2013; -@Khalsa2013; -@PKim2014]. Experimental evidences have been mixed, with the claims of large magnitude stemming from the magnetoresistance measurements  [-@BenShalom2010; -@Caviglia2010; -@Santander2012; -@King2012; -@Santander2014] contradicted by the Hanle effect measurement [-@Reyren2012] as well as the measurement of similar magnetoresistance in the $\delta$-doped STO heterostructure where the inversion symmetry breaking is hardly present [-@MKim2012]. Meanwhile, theoretical calculations show the splitting near the $\Gamma$ point to be two orders of magnitude smaller than the bandwidth at the best  [-@Zhong2013; -@YKim2013; -@PKim2014]. One natural way to overcome this limitation is adopting 5$d$ TM oxides, such as tantalate, with a stronger atomic SOC. This has motivated the recent experiments on the 2DEG at the surface of KTaO$_3$ (KTO) [-@Santander2012; -@King2012]. However, the experiments on KTO have suggested that another important condition for enhancing the surface 2DEG Rashba spin-orbit interaction is to have the density profile of the surface state concentrated to the surface-terminating layer, which maximizes the effect of the broken inversion symmetry. The ARPES measurements on the KTO surface have seen no measurable spin splitting [-@Santander2012; -@King2012], in spite of not only the stronger SOC of Ta but also the polar nature of KTO (001) surface. According to a density functional theory calculation [-@Shanavas2014], the surface state penetrates deeply into the bulk as the surface confinement potential is made shallow by the atomic relaxation near the surface layer. This suppresses the effect of the inversion symmetry breaking (ISB) on the surface state (which can be quantified by various parameters, [*e.g.*]{} the chiral orbital angular momentum coefficient [-@SPark2011; -@BKim2012; -@CPark2012; -@PKim2014]), and hence significantly reduces the Rashba spin-orbit interaction. ![\[fig:atomicstr\]Atomic structure of a tantalate layer on BaHfO$_3$ (001). (a) Schematic illustration of TaO$_2$/KO on HfO$_2$-terminated BaHfO$_3$, and TaO$_2$ on BaO-terminated BaHfO$_3$. (b) Atomic structure of TaO$_2$/KO on BaHfO$_3$ from first-principles calculations; note the height difference between Ta and O at the top layer. (c) Wave function weight projected on $d_{xz/yz}$ of TMs for the lowest $d_{xz/yz}$ Rashba band at $\Gamma$. Monolayer (TaO$_2$/KO on HfO$_2$-terminated BaHfO$_3$) case is to be compared with bilayer (TaO$_2$/KO/TaO$_2$ on BaO-terminated BaHfO$_3$) case. The TM-O$_2$ layers are numbered starting from the outermost layer. ](fig1.pdf){width="40.00000%"} ![\[fig:bandpdos\]Electronic structure of TaO$_2$/KO on BaHfO$_3$ from first-principles calculations. (a) Calculated band structure along high symmetry points. Insets show the magnified view of the upper and lower $d_{xz/yz}$ bands near $\Gamma$, and the $d_{yz}$ bands near $\mathrm{X}$. (b) Projected weights of Ta $d$ states. The Fermi level is set to the valence band maximum. (c) Schematic illustration of energy levels at $\Gamma$ without SOC. The crystal field splitting of the monolayer-substrate heterostructure is different from that of the cubic (octahedral) case.](fig2.pdf){width="45.00000%"} In this study, we have theoretically constructed a realistic oxide heterostructure that has a surface 2DEG with a strong Rashba spin-orbit interaction. Our idea is to consider a 5$d$ TM oxide monolayer on a substrate, where 2DEG predominantly lies in the outermost monolayer film, maximizing the effect of the broken inversion symmetry from the substrate. Specifically, we attempt to replace the outer layers of perovskite oxide (001) surface with another perovskite thin film layers, in which the electronic bands of the substrate need to lie sufficiently far from the conduction band minimum (CBM) to make all essential low-energy physics originate from the thin film states near CBM. After calculations of a number of candidate perovskite oxides for the heterostructure, we find that TaO$_2$/KO or TaO$_2$/BaO layer on BaHfO$_3$ (001) surface (Fig. \[fig:atomicstr\]) is a promising candidate structure possessing Ta $t_{2g}$-originated two-dimensional (2D) bands with a strong Rashba spin-orbit interaction. BaHfO$_3$ (BHO) is suitable as a substrate because its lattice structure (cubic at room temperature) matches with that of KTO, the only stable perovskite material containing TaO$_2$ [-@Zhurova2000; -@Maekawa2006], and the alignment of its conduction bands and the Ta $t_{2g}$ bands enables minimal hybridization. It should be emphasized that the concentration of the surface state in the outermost layer is an important condition to maximize the ISB effect. For instance, if we consider a TaO$_{2}$ bilayer as opposed to the monolayer (Fig. \[fig:atomicstr\]c), the ISB effect is weakened as the surface state wave function no longer peaks at the outermost layer. We will show that the coupling of the $t_{2g}$ surface bands to the Ta $e_g$ bands, which comes from the local asymmetric environment of the surface Ta atoms, plays a key role in the enhanced splitting. The band splitting from intra-$t_{2g}$ coupling is smaller, due to specific orbital symmetry of Ta $d$ and O $p$ states that we will discuss. We will further show that the $t_{2g}$-$e_g$ coupling gives rise to the enhanced Rashba-Dresselhaus splitting not only at the band bottom at $\Gamma$ but also at the band saddle points at $\mathrm{X}$. Finally, we will consider the substitution of Ba atoms for K in the KO layer for electron doping of the system. **Results** [**Rashba splitting near the $\Gamma$ point.**]{} The electronic structures of TaO$_2$/KO monolayer on BHO (lattice constant $\approx 4.155$ Å) from our first-principles calculations are presented in Fig. \[fig:bandpdos\]. The bands near the CBM consist of $t_{2g}$ ($d_{xy}$, $d_{xz}$, $d_{yz}$) states of Ta in the outermost layer, with the calculated bandwidth of $\approx$ 1.7 eV for the $d_{xz/yz}$ bands. These bands being 2D, the triple degeneracy (excluding spin) of the $t_{2g}$ bands at $\Gamma$ is lifted, splitting the $d_{xy}$ and the $d_{xz/yz}$ manifolds; the Ta atomic SOC further splits the $d_{xz/yz}$ bands into upper and lower $d_{xz/yz}$ states. Finally, when the ISB at the surface is accounted for, the Rasha-type band splitting lifts spin degeneracies in the entire Brillouin zone (BZ) except at the time-reversal invariant momenta $\Gamma$, $\mathrm{X}$ and $\mathrm{M}$. We note that this Rashba-type band splitting of the $d_{xz/yz}$ bands is strikingly larger in magnitude than that of the $d_{xy}$ bands, contrary to the prediction of the $t_{2g}$-only model [-@Khalsa2013; -@YKim2013; -@Scheurer2015]. Moreover, our calculation gives the Rashba coefficient of the lower $d_{xz/yz}$ bands at $\Gamma$ of $\alpha_R \approx 0.3~\mathrm{eV\AA}$, which is an order of magnitude larger than that of LAO/STO heterostructure deduced from the experimental magnetoresistance data [-@Caviglia2010], and the Rashba energy of $E_R \gtrsim 15~\mathrm{meV}$; these values are also significantly larger than $\alpha_R \approx 0.1~\mathrm{eV\AA}$, $E_R \approx 1~\mathrm{meV}$ for the bilayer case of Fig. \[fig:atomicstr\]c. The Rashba-Dresselhaus effect along the BZ boundary is even more pronounced, with a giant splitting ($\approx 180$ meV), which is nearly twice the maximum reported value [-@Santander2014] in the perovskite oxide 2DEG, occurring near $\mathrm{X}$ along $k_{x/y} = \pi$. ----------- ------------------- --------------------------------------------------------------------------------------------------------- -- -- -- $\vec{k}$ Manifold Splitting terms in $\mathcal{H}_{\mathrm{eff}}$ upper $d_{xz/yz}$ $\left[\frac{-2\gamma_3 \xi}{\Delta_{uxz/yz,x^2-y^2}}+\frac{-2\gamma_1 \xi}{\Delta_{uxz/yz,xy}}\right] (\vec{\sigma}\times \vec{k})\cdot \hat{z}$ lower $d_{xz/yz}$ $\frac{2\sqrt{3}\gamma_2 \xi}{\Delta_{lxz/yz,z^2}} (\vec{\sigma}\times \vec{k})\cdot \hat{z}$ $d_{xy}$ $\frac{-2\gamma_1 \xi}{\Delta_{xy,uxz/yz}} (\vec{\sigma}\times \vec{k})\cdot \hat{z}$ $d_{yz}$ $\left[\frac{-2\sqrt{3}\gamma_2 \xi}{\tilde{\Delta}_{yz,z^2}} +\frac{2\gamma_3 \xi}{\tilde{\Delta}_{yz,x^2-y^2}}\right] \sigma_x k_y -\frac{2\gamma_1 \xi}{\tilde{\Delta}_{yz,xy}}\sigma_y k_x$ ----------- ------------------- --------------------------------------------------------------------------------------------------------- -- -- -- : \[tab:heff\]Splitting terms of the effective Hamiltonian $\mathcal{H}_{eff}$ for Ta $t_{2g}$ manifolds. $\vec{k}$ denotes the reference point of the effective Hamiltonian with $\Gamma=(0,0)$ and $\mathrm{X}=(\pi,0)$. $\Delta (\tilde{\Delta})$ represents the energy difference between the states in the subscript at $\Gamma (\mathrm{X})$, where $uxz/yz$ and $lxz/yz$ mean the upper $xz/yz$ and the lower $xz/yz$, respectively. An analysis that includes all Ta $d$-orbitals – not only the $t_{2g}$ orbitals but also the $e_g$ orbitals – is required in order to understand the two conspicuous features of Fig. \[fig:bandpdos\], the discrepancy between the Rashba splitting of the $d_{xy}$ and the $d_{xz/yz}$ bands, and the giant band splitting along $k_{x/y} = \pi$. We employ an analytic TB model for a qualitative analysis and supplement it with quantitative results from maximally localized Wannier functions (MLWFs). In the TB model, we consider a Hamiltonian for all Ta $d$-orbitals, including the $e_g$ orbitals ($d_{z^2}$, $d_{x^2-y^2}$) in a square lattice [-@Shanavas2014; -@Shanavas2014a] to describe the TaO$_2$ layer 2D bands, $$\begin{aligned} \mathcal{H}=\mathcal{H}_\mathrm{hop}+\mathcal{H}_\mathrm{SOC}+\mathcal{H}_\mathrm{E}+\mathcal{V}_\mathrm{sf},\end{aligned}$$ where the first term $\mathcal{H}_\mathrm{hop}$ describes the nearest-neighbor hopping, and the second term $\mathcal{H}_\mathrm{SOC} = \xi {\bf L} \cdot {\bf S}$ is the atomic SOC, with $\xi \approx 0.26$ eV for Ta. The third term $\mathcal{H}_\mathrm{E}$ includes the additional hoppings that would have been forbidden if not for the ISB: $$\begin{aligned} &\gamma_1&=\langle d_{xy}| \mathcal{H}_\mathrm{E} |d_{xz} \rangle_{\hat{y}} =\langle d_{xy}| \mathcal{H}_\mathrm{E} |d_{yz} \rangle_{\hat{x}} \nonumber \\ &\gamma_2&=\langle d_{xz}| \mathcal{H}_\mathrm{E} |d_{z^2} \rangle_{\hat{x}} =\langle d_{yz}| \mathcal{H}_\mathrm{E} |d_{z^2} \rangle_{\hat{y}} \\ &\gamma_3&=\langle d_{x^2-y^2}| \mathcal{H}_\mathrm{E} |d_{yz} \rangle_{\hat{y}} =\langle d_{xz}| \mathcal{H}_\mathrm{E} |d_{x^2-y^2} \rangle_{\hat{x}} \nonumber,\end{aligned}$$ in which the vectors in the subscripts denote the relative position of the two orbitals with the lattice constant set to 1 for convenience (these ISB hoppings play a role analogous to the chiral orbital angular momentum effect in the $p$-orbital bands [-@SPark2011; -@BKim2012; -@CPark2012]). Here, $\gamma_1$ is intra-$t_{2g}$ ISB hopping while $\gamma_2$ and $\gamma_3$ describe $t_{2g}$-$e_g$ ISB hoppings. The fourth term $\mathcal{V}_\mathrm{sf}$ describes the potential difference originating from the surface field. By deriving the effective Hamiltonian $\mathcal{H}_{eff}$ that acts on each two-fold degenerate band in the weak SOC limit where $\mathcal{H}_\mathrm{hop}+\mathcal{V}_\mathrm{sf}$ is dominant over $\mathcal{H}_\mathrm{SOC}$, we obtain Rashba-type band splitting terms near $\Gamma$ and $\mathrm{X}$ as summarized in Table \[tab:heff\] (see Appendix A for details). Table \[tab:heff\] shows the Rashba coupling to be linear in the ISB hopping $\gamma$ divided by the energy difference between two relevant states $\Delta$. ![\[fig:wanhopping\]Hopping strengths (in eV) between Wannier functions for Ta $d$ states at one site and the neighboring site in $x$ direction. Ta $d$ states and O $p$ states are used for the Wannier function construction. The terms relevant to the ISB $\gamma_1$, $\gamma_2$, $\gamma_3$ are presented. Both direct (horizontal arrows) and indirect (oblique arrows, via O $p$) paths are depicted. Empty arrows indicate terms that would be absent without ISB.](fig3.pdf){width="35.00000%"} One reason why the $e_g$ orbital contribution is crucial for the Rashba splitting in the $t_{2g}$ bands is that the $t_{2g}$-$e_g$ ISB hoppings $\gamma_{2,3}$ are significantly larger than the intra-$t_{2g}$ hopping $\gamma_1$: $\gamma_1 \approx -0.04$ eV, $\gamma_2 \approx -0.25$ eV, $\gamma_3 \approx 0.30$ eV. This is necessary condition for the effective Hamiltonian of Table \[tab:heff\] to give larger Rashba splittings in the $d_{xz/yz}$ bands than $d_{xy}$ as shown in Fig. \[fig:bandpdos\], given that the $d_{xz/yz}$ bands are closer in energy to the $d_{xy}$ band than the $e_g$ bands (albeit within an order of magnitude). The intra-$t_{2g}$ ISB hopping $\gamma_1$ remains relatively small due to the orbital symmetry of Ta $d$ and O $p$ states, which we can see from a quantitative analysis with MLWFs that includes not only the Ta $d$ states but also the neighboring O $p$ states. Examining the hopping parameters relevant to $\gamma_1$, the particularly small ISB hopping between O $p_y$ and Ta $d_{yz}$ along $x$ direction (Fig. \[fig:wanhopping\]) can be attributed to the relative positions and shapes of the two orbitals; the lobes of the two orbitals lie on the $yz$ plane that is perpendicular to the hopping direction ($\hat{x}$), and $p_y$ has maximum amplitude along $y$ direction whereas $d_{yz}$ has a node along it. Thus, we have the negligible Rashba splitting of the $d_{xy}$ band as shown in Fig. \[fig:bandpdos\], despite the smaller energy difference with the $d_{xz/yz}$ bands. The other reason why the $e_g$ orbital contribution is crucial for the Rashba splitting in the $t_{2g}$ bands is the reduced $t_{2g}$-$e_g$ energy splitting. Indeed, when the $t_{2g}$-$e_g$ energy splitting is set to be infinite in Table \[tab:heff\], all the results from the $t_{2g}$-only TB models [-@Khalsa2013; -@YKim2013; -@PKim2014; -@Scheurer2015] are recovered, including the absence of $k$-linear Rashba in the lower $d_{xz/yz}$ band near $\Gamma$. In the case of 3D cubic KTO, the $t_{2g}$-$e_g$ energy separation at $\Gamma$ is calculated to be $\approx 4.6~\mathrm{eV}$, which is larger than that of our system (Fig. \[fig:bandpdos\]a, b). Compared with the 3D cubic bulk case, Figure \[fig:bandpdos\]b shows considerable portion of $d_{z^2}$ states close in energy, [*i.e.*]{}, less than bandwidth, to the $t_{2g}$ bands; this is due to the absence of an O atom in one of the octahedral points surrounding Ta. Hence, as shown in Fig. \[fig:bandpdos\]c, the local atomic configuration for the Ta atom is close to a square pyramid, where the $d_{z^2}$ and lower $d_{xz/yz}$ are close in energy. This can be taken as a generic result for the case where the 2DEG wave function is confined almost entirely to the outermost layer. The height difference of Ta and O atoms ($\approx 0.20$ Å) in the TaO$_2$ layer enhances the $t_{2g}$-$e_g$ coupling in both ways; the larger effect being the enhancement of the ISB hopping $\gamma_3$, but there is also noticeable lowering of the $d_{x^2-y^2}$ orbital energy level. ![\[fig:am\] Angular momentum texture from first-principles calculations in close vicinity of the high-symmetry points in BZ. Orbital and spin angular momenta are presented for the lower Rashba band of the (a) $d_{xy}$ and (b) lower $d_{xz/yz}$ (c) upper $d_{xz/yz}$ bands near $\Gamma$, and the lower Rashba-Dresselhaus band of the (d) $d_{yz}$ bands near $\mathrm{X}=(\pi,0)$. ](fig4.pdf){width="45.00000%"} The $t_{2g}$-$e_g$ coupling also plays a key role in determining the angular momentum (AM) texture of the $t_{2g}$ bands in close vicinity of $\Gamma$ (Fig. \[fig:am\]). The tetragonal crystal field and the SOC determine the spin-orbital entanglement of the band manifolds; the spin-up and spin-down are in nearly the same orbital state for the $d_{xy}$ bands while they are in nearly orthogonal orbital states for the $d_{xz/yz}$ bands. This, in turn, affects the AM character; the spin AM is dominant in the $d_{xy}$ bands whereas the orbital AM is dominant [-@SPark2011; -@BKim2012; -@PKim2014] in the $d_{xz/yz}$ bands (see Appendix A). The $t_{2g}$-$e_g$ coupling is important in that it gives rise to finite AM in the lower $d_{xz/yz}$ bands and quantitatively changes the AM in the upper $d_{xz/yz}$. In the $t_{2g}$-only TB model, the AM texture of the lower $d_{xz/yz}$ band is completely missing and that of the upper $d_{xz/yz}$ is not quantitatively correct. [**Band splitting near the $\mathrm{X}$ point.**]{} As shown in Table \[tab:heff\], the lowered symmetry $C_{2v}$ at $\mathrm{X}$ allows the mixture of Rashba and linear Dresselhaus terms in general (see Appendix A), with the linear Dresselhaus larger in magnitude, as shown in Fig. \[fig:am\]d. Due to the anisotropic dispersion of $d_{xz/yz}$ bands, the lowest conduction band at $\mathrm{X}=(\pi,0)$ mainly consists of $d_{yz}$ state. We find that the larger band splitting along $\mathrm{X}$—$\mathrm{M}$ comes from the $t_{2g}$-$e_g$ coupling whereas the smaller splitting along $\mathrm{X}$—$\Gamma$ is due to the intra-$t_{2g}$ coupling. Hence, the giant Rashba-Dresselhaus splitting in vicinity of $\mathrm{X}$ ($\approx 180$ meV) is due to the $e_g$ contribution. It has been recently pointed out [-@Chung2015] that this Rashba-Dresselhaus splitting along $\mathrm{X}$—$\mathrm{M}$ is necessary for weak topological superconductivity, which gives rise to dislocation Majorana zero modes. The Rashba-Dresselhaus splitting near $\mathrm{X}$ also affects the superconducting instability, as it splits the logarithmic van Hove singularity (VHS) of the $d_{xz/yz}$ band saddle point and shifts them away from $\mathrm{X}$ (see Appendix B for the logarithmic VHS splitting). Given that the splitting results in the lower and upper Rashba-Dresselhaus bands reaching VHS separately, the shifted VHSs do not have spin degeneracy. While it has been long recognized that the logarithmic VHS at $\mathrm{X}$ enhances the superconducting instability in the spin-singlet channel [-@Schulz1987; -@Dzyaloshinskii1987; -@Furukawa1998; -@Gonzalez2008; -@Nandkishore2012], it was recently shown [-@Meng2015; -@Yao2015; -@Cheng2015] that the logarithmic VHS away from $\mathrm{X}$ enhances the instability to the spin-triplet $p$-wave superconductivity. The physics at $\mathrm{X}$ should be experimentally accessible, as the VHS at $\mathrm{X}$ is not too far from the band bottom of the $d_{xz/yz}$ in energy ($\approx 0.23$ eV), and we will show in the next section how our heterostructure can be chemically doped all the way to the VHS at $\mathrm{X}$. **Discussions** ![\[fig:tao2bho\]Electronic structure of a TaO$_2$ layer on BaO-terminated BaHfO$_3$ (001) from first-principles calculations. (a) Band structure along high-symmetry points in BZ. The dotted horizontal line denotes the energy level of the VHS in absence of the Rashba-Dresselhaus splitting, $E_{vH}$, with the Fermi level set to 0. (b) AM texture at $E_{vH}$ with constant energy lines. The red and blue arrows correspond to the orbital and spin AM, respectively. ](fig5.pdf){width="40.00000%"} To actually realize the 2DEG in the TaO$_2$ layer, electron doping is needed because the nominal charge of the TaO$_2$ layer is +1 and that of the KO layer is -1 (BaO and HfO$_2$ layers are neutral.). One possible way would be substituting K atoms with Ba in the KO layer. In this case, the Rashba strength remains still large ($\alpha_R \approx 0.2$ eVÅ) in the lower $d_{xz/yz}$ bands (Fig. \[fig:tao2bho\]a), while the Rashba splitting in the upper $d_{xz/yz}$ bands is greatly suppressed largely because the Ba substitution reduces by 70% the height difference of Ta and O atoms in the TaO$_2$ layer. Once again, the considerable band splitting along $\mathrm{X}$–$\mathrm{M}$ ($\approx 24$ meV) and AM texture at $E_{vH}$ in Fig \[fig:tao2bho\] are contrary to the prediction of the $t_{2g}$-only TB model (see Appendix C and Fig. A1), and hence demonstrate that inclusion of the $e_g$ manifold is essential in understanding the Rashba-Dresselhaus splitting. We expect that partial chemical substitution (TaO$_2$/K$_{1-x}$Ba$_{x}$O layer on BHO) could induce the 2DEG in the TaO$_2$ layer in experiments. At $x=1$, [*i.e.*]{}, 100% Ba substitution, the Fermi level lies slightly above the VHS (Fig. \[fig:tao2bho\]a), which means we can access the VHS at $x \lesssim 1$. We expect the qualitative features of our Rashba-Dresselhaus splitting to be generic for the (001) perovskite transition metal oxide 2DEG with maximal ISB, where the 2DEG wave function profile is required to be concentrated on the surface-terminating TM-O$_2$ layer. In such an environment, the effective crystal field on the 2DEG $d$ orbitals should be quite different from that of the cubic perovskite, with much lower energy level at least for the $d_{z^2}$ orbital. Even if we use alternative materials for our heterostructure — viability of substituting BHO by BaSnO$_3$ or TaO$_2$ by WO$_2$ still remains to be investigated — we expect a significant role of the transition metal $e_g$ orbitals if the Rashba-Dresselhaus splitting is comparably large. It is also found that compressive strain on the BHO substrate, which might be needed for the feasible deposition of the thin film taking into account relatively large lattice mismatch between KTO and BHO, does not substantially affect the band splitting (see Appendix D and Fig. A2). Considering that 2DEG in an artificial film-substrate system is realized experimentally in SrVO$_3$ thin films on Nb-doped STO [-@Yoshimatsu2011], we expect that our system can be realized in experiments using the state-of-the-art layer-by-layer growth control of perovskite oxide thin films [-@Thiel2006]. **Methods** [**Theoretical approach.**]{} We performed density functional theory calculations as implemented in VASP [-@Kresse1993; -@Kresse1996]. Projector augmented-wave method was used [-@Blochl1994]. A plane-wave basis set with the cutoff energy 520 eV was employed, and PBEsol (Perdew-Burke-Ernzerhof revised for solids) exchange-correlation functional was adoped [-@Perdew2008]. We used the lattice constant optimized in bulk calculations of BaHfO$_3$ and the internal atomic positions were fully relaxed until the force became less than 0.01 eV/Å. Details of the analytic tight-binding approximation and effective Hamiltonian description were presented in Appendix A. We employed maximally localized Wannier functions [-@Marzari1997; -@Souza2001; -@Mostofi2008] to further analyze the results of the first-principles calculations. The Wannier functions were constructed for $d$ orbitals of Ta in one set, and $p$ orbitals of three neighboring O as well as $d$ orbitals of Ta in the other set. **Acknowledgments** We thank Jung Hoon Han, Changyoung Kim, Choong-Hyun Kim, Hyeong-Do Kim, Minu Kim, Hyun-Woo Lee, Hosub Jin, Seung Ryong Park, Cai-Zhuang Wang, Hong Yao, and Jaejun Yu for fruitful discussions and comments. This work was supported by the NRF of the MSIP of the Korean government Grant No. 2006-0093853 (M.K., J.I.) and IBS-R009-Y1 (S.B.C.). Research at Ames laboratory (M.K.) was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract No. DE-AC02-07CH11358. Computations were performed through the support of the Korea Institute of Science and Technology Information (KISTI) and the National Energy Research Scientific Computing Center (NERSC) in Berkeley, CA. **Author contributions** All authors contributed extensively to the work presented in this paper. Tight-binding model for the TaO$_2$ film ======================================== We consider the tight-binding Hamiltonian for a TaO$_2$ film ($d$-orbitals in a square lattice) [-@Shanavas2014a], $$\begin{aligned} \mathcal{H}=\mathcal{H}_\mathrm{hop}+\mathcal{H}_\mathrm{SOC}+\mathcal{H}_\mathrm{E}+\mathcal{V}_\mathrm{sf},\end{aligned}$$ where $\mathcal{H}_\mathrm{hop}$ describes the hopping between the nearest neighbors, $\mathcal{H}_\mathrm{SOC}$ is the atomic spin-orbit coupling of Ta, $\mathcal{H}_\mathrm{E}$ describes the orbital mixing due to the inversion symmetry breaking field near the surface, and $\mathcal{V}_\mathrm{sf}$ describes onsite potential changes due to the surface field. Specifically, the hopping term is given by $$\begin{aligned} \mathcal{H}_\mathrm{hop}= { \left( \begin{matrix} \frac{t_{\sigma}+3t_{\delta}}{2}(c_x+c_y) & -\frac{\sqrt{3}}{2}(t_{\sigma}-t_{\delta})(c_x-c_y) & 0 & 0 & 0 \\ -\frac{\sqrt{3}}{2}(t_{\sigma}-t_{\delta})(c_x-c_y) & \frac{3t_{\sigma}+t_{\delta}}{2}(c_x+c_y) & 0 & 0 & 0 \\ 0 & 0 & 2t_{\pi}(c_x+c_y) & 0 & 0 \\ 0 & 0 & 0 & 2(t_{\pi} c_x +t_{\delta} c_y) & 0 \\ 0 & 0 & 0 & 0 & 2(t_{\delta} c_x +t_{\pi} c_y) \\ \end{matrix} \right), } \label{eq:ham_hopping} \nonumber\end{aligned}$$ where the basis is $\{ |d_{z^2}\rangle, |d_{x^2-y^2}\rangle, |d_{xy}\rangle, |d_{xz}\rangle, |d_{yz}\rangle \}$, and $t_{\sigma}$, $t_{\pi}$, $t_{\delta}$ are hopping parameters between $d$-orbitals. $c_x$ means $\cos k_x$. The lattice constant is set to 1. The spin-orbit coupling term is $$\begin{aligned} \mathcal{H}_\mathrm{SOC}= { \left( \begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{\sqrt{3}}{2}\xi & 0 & \frac{\sqrt{3}}{2}\xi i \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{\sqrt{3}}{2}\xi & 0 & \frac{\sqrt{3}}{2}\xi i & 0 \\ 0 & 0 & 0 & 0 & -\xi i & 0 & 0 & \frac{1}{2}\xi & 0 & \frac{1}{2}\xi i \\ 0 & 0 & 0 & 0 & 0 & \xi i & -\frac{1}{2}\xi & 0 & \frac{1}{2}\xi i & 0 \\ 0 & 0 & \xi i & 0 & 0 & 0 & 0 & -\frac{1}{2}\xi i & 0 & \frac{1}{2}\xi \\ 0 & 0 & 0 & -\xi i & 0 & 0 & -\frac{1}{2}\xi i & 0 & -\frac{1}{2}\xi & 0 \\ 0 & \frac{\sqrt{3}}{2}\xi & 0 & -\frac{1}{2}\xi & 0 & \frac{1}{2}\xi i & 0 & 0 & -\frac{1}{2}\xi i & 0 \\ -\frac{\sqrt{3}}{2}\xi & 0 & \frac{1}{2}\xi & 0 & \frac{1}{2}\xi i & 0 & 0 & 0 & 0 & \frac{1}{2}\xi i \\ 0 & -\frac{\sqrt{3}}{2}\xi i & 0 & -\frac{1}{2}\xi i & 0 & -\frac{1}{2}\xi & \frac{1}{2}\xi i & 0 & 0 & 0 \\ -\frac{\sqrt{3}}{2}\xi i & 0 & -\frac{1}{2}\xi i & 0 & \frac{1}{2}\xi & 0 & 0 & -\frac{1}{2}\xi i & 0 & 0 \end{matrix} \right) } \label{eq:ham_soc}. \nonumber\end{aligned}$$ The inversion symmetry breaking field terms are given by $$\begin{aligned} \mathcal{H}_\mathrm{E}+\mathcal{V}_\mathrm{sf}= { \left( \begin{matrix} \delta_2 & 0 & 0 & -2i\gamma_2 \sin k_x & -2i\gamma_2 \sin k_y \\ 0 & \delta_3 & 0 & -2i\gamma_3 \sin k_x & 2i\gamma_3 \sin k_y \\ 0 & 0 & \delta_1 & 2i\gamma_1 \sin k_y & 2i\gamma_1 \sin k_x \\ 2i\gamma_2 \sin k_x & 2i\gamma_3 \sin k_x & -2i\gamma_1 \sin k_y & 0 & 0 \\ 2i\gamma_2 \sin k_y & -2i\gamma_3 \sin k_y & -2i\gamma_1 \sin k_x & 0 & 0 \\ \end{matrix} \right), } \label{eq:ham_efield} \nonumber\end{aligned}$$ where $$\begin{aligned} &\delta_1&=\varepsilon(d_{xy})-\varepsilon(d_{xz/yz}) \\ &\delta_2&=\varepsilon(d_{z^2})-\varepsilon(d_{xz/yz}) \\ &\delta_3&=\varepsilon(d_{x^2-y^2})-\varepsilon(d_{xz/yz})\\ &\gamma_1&=\langle d_{xy}| \mathcal{H}_\mathrm{E} |d_{xz} \rangle_{\hat{y}} =\langle d_{xy}| \mathcal{H}_\mathrm{E} |d_{yz} \rangle_{\hat{x}} \\ &\gamma_2&=\langle d_{xz}| \mathcal{H}_\mathrm{E} |d_{z^2} \rangle_{\hat{x}} =\langle d_{yz}| \mathcal{H}_\mathrm{E} |d_{z^2} \rangle_{\hat{y}} \\ &\gamma_3&=\langle d_{x^2-y^2}| \mathcal{H}_\mathrm{E} |d_{yz} \rangle_{\hat{y}} =\langle d_{xz}| \mathcal{H}_\mathrm{E} |d_{x^2-y^2} \rangle_{\hat{x}} .\end{aligned}$$ The Hamiltonian near the $\Gamma$ point can be written as $$\begin{aligned} \mathcal{H}(\vec{k})\approx \hspace{9cm} \\ {\scalebox{0.65}{\mbox{\ensuremath{\displaystyle \left( \begin{matrix} \frac{t_{\sigma}+3t_{\delta}}{2} C+\delta_2 & 0 & -\frac{\sqrt{3}}{2}(t_{\sigma}-t_{\delta}) D & 0 & 0 & 0 & 0 & -\sqrt{2}\gamma_2 (i k_x-k_y) & -\sqrt{\frac{3}{2}} \xi & -\sqrt{2}\gamma_2 (i k_x+k_y) \\ 0 & \frac{t_{\sigma}+3t_{\delta}}{2} C+\delta_2 & 0 & -\frac{\sqrt{3}}{2}(t_{\sigma}-t_{\delta}) D & 0 & 0 & -\sqrt{2}\gamma_2 (i k_x+k_y) & 0 & -\sqrt{2}\gamma_2 (i k_x-k_y) & \sqrt{\frac{3}{2}} \xi \\ -\frac{\sqrt{3}}{2}(t_{\sigma}-t_{\delta}) D & 0 & \frac{3t_{\sigma}+t_{\delta}}{2} C+\delta_3 & 0 & -\xi i & 0 & \frac{1}{\sqrt{2}} \xi & -\sqrt{2}\gamma_3 (i k_x+k_y) & 0 & -\sqrt{2}\gamma_3 (i k_x-k_y) \\ 0 & -\frac{\sqrt{3}}{2}(t_{\sigma}-t_{\delta}) D & 0 & \frac{3t_{\sigma}+t_{\delta}}{2} C+\delta_3 & 0 & \xi i & -\sqrt{2}\gamma_3 (i k_x-k_y) & -\frac{1}{\sqrt{2}} \xi & -\sqrt{2}\gamma_3 (i k_x+k_y) & 0 \\ 0 & 0 & \xi i & 0 & 2t_{\pi} C+\delta_1 & 0 & -\frac{1}{\sqrt{2}} i \xi & \sqrt{2} i \gamma_1 (i k_x+k_y) & 0 & -\sqrt{2} i \gamma_1 (i k_x-k_y) \\ 0 & 0 & 0 & -\xi i & 0 & 2t_{\pi} C+\delta_1 & -\sqrt{2} i \gamma_1 (i k_x-k_y) & -\frac{1}{\sqrt{2}} i \xi & \sqrt{2} i \gamma_1 (i k_x+k_y) & 0 \\ 0 & \sqrt{2}\gamma_2 (i k_x-k_y) & \frac{1}{\sqrt{2}} \xi & \sqrt{2}\gamma_3 (i k_x+k_y) & \frac{1}{\sqrt{2}} i \xi & -\sqrt{2} i \gamma_1 (i k_x+k_y) & (t_{\pi} +t_{\delta}) C+\frac{\xi}{2} & 0 & (t_{\pi} -t_{\delta}) D & 0 \\ \sqrt{2}\gamma_2 (i k_x+k_y) & 0 & \sqrt{2}\gamma_3 (i k_x-k_y) & -\frac{1}{\sqrt{2}} \xi & \sqrt{2} i \gamma_1 (i k_x-k_y) & \frac{1}{\sqrt{2}} i \xi & 0 & (t_{\pi} +t_{\delta}) C+\frac{\xi}{2} & 0 & (t_{\pi} -t_{\delta}) D \\ -\sqrt{\frac{3}{2}} \xi & \sqrt{2}\gamma_2 (i k_x+k_y) & 0 & \sqrt{2}\gamma_3 (i k_x-k_y) & 0 & \sqrt{2} i \gamma_1 (i k_x-k_y) & (t_{\pi} -t_{\delta}) D & 0 & (t_{\pi} +t_{\delta}) C-\frac{\xi}{2} & 0 \\ \sqrt{2}\gamma_2 (i k_x-k_y) & \sqrt{\frac{3}{2}} \xi & \sqrt{2}\gamma_3 (i k_x+k_y) & 0 & -\sqrt{2} i \gamma_1 (i k_x+k_y) & 0 & 0 & (t_{\pi} -t_{\delta}) D & 0 & (t_{\pi} +t_{\delta}) C-\frac{\xi}{2} \end{matrix} \right), }}}} \label{eq:ham_k_neargamma} \nonumber\end{aligned}$$ with $C=\cos k_x + \cos k_y \approx 2-\frac{k_x^2}{2}-\frac{k_y^2}{2}$, $D=\cos k_x - \cos k_y \approx -\frac{k_x^2}{2}+\frac{k_y^2}{2}$, where we performed a unitary transformation to diagonalize the $d_{xz/yz}$ subspace in the limit that the $d_{xz/yz}$ states are sufficiently far from other manifolds and $\vec{k} \rightarrow 0$. The effective Hamiltonian can be obtained by projection onto the concerned manifold $$\begin{aligned} \mathcal{H}_{\mathrm{eff}}=\mathcal{P}\mathcal{H}\mathcal{P}+\mathcal{P}\mathcal{H}\mathcal{Q} \frac{1}{\epsilon-\mathcal{Q}\mathcal{H}\mathcal{Q}}\mathcal{Q}\mathcal{H}\mathcal{P}, \label{eq:ham_eff}\end{aligned}$$ where $\mathcal{P}$ is the projection operator onto the relevant manifold and $\mathcal{Q}=1-\mathcal{P}$. For the $d_{xy}$ bands, the effective Hamiltonian is $$\begin{aligned} \mathcal{H}_{\mathrm{eff}} \approx h_{xy}(\vec{k})I_{2\times 2}+\frac{-2\gamma_1 \xi}{\Delta_{xy,uxz/yz}} (\vec{\sigma}\times \vec{k})\cdot \hat{z}, \label{eq:ham_eff_gamma_dxy}\end{aligned}$$ where $\Delta_{xy,uxz/yz}=4t_{\pi}+\delta_1-\{2(t_{\pi}+t_{\sigma})+\frac{\xi}{2}\}$, and the Pauli matrices describe the subspace defined by $\{ |d_{xy}\uparrow\rangle, |d_{xy}\downarrow\rangle \}$. For the lower $d_{xz/yz}$ bands, $$\begin{aligned} \mathcal{H}_{\mathrm{eff}} \approx h_{lxz/yz}(\vec{k})I_{2\times 2} +\frac{2\sqrt{3}\gamma_2 \xi}{\Delta_{lxz/yz,z^2}}(\vec{\sigma}\times \vec{k})\cdot \hat{z}, \label{eq:ham_eff_gamma_lxz/yz}\end{aligned}$$ where $\Delta_{lxz/yz,z^2}=2(t_{\pi}+t_{\delta})-\frac{\xi}{2}-\{t_{\sigma}+3t_{\delta}+\delta_2\}$, and the Pauli matrices describe the subspace defined by $\{ \frac{1}{\sqrt{2}}(|d_{xz}\downarrow\rangle +i |d_{yz}\downarrow\rangle), \frac{1}{\sqrt{2}}(|d_{xz}\uparrow\rangle -i |d_{yz}\uparrow\rangle) \}$. For the upper $d_{xz/yz}$ bands, $$\begin{aligned} \mathcal{H}_{\mathrm{eff}} \approx h_{uxz/yz}(\vec{k})I_{2\times 2} +\left[\frac{-2\gamma_3 \xi}{\Delta_{uxz/yz,x^2-y^2}}+\frac{-2\gamma_1 \xi}{\Delta_{uxz/yz,xy}}\right] (\vec{\sigma}\times \vec{k})\cdot \hat{z}, \label{eq:ham_eff_gamma_uxz/yz}\end{aligned}$$ where $\Delta_{uxz/yz,x^2-y^2}=2(t_{\pi}+t_{\delta})+\frac{\xi}{2}-\{3t_{\sigma}+t_{\delta}+\delta_3\}$, $\Delta_{uxz/yz,xy}=2(t_{\pi}+t_{\delta})+\frac{\xi}{2}-\{4t_{\pi}+\delta_1\}$, and the Pauli matrices describe the subspace defined by $\{ \frac{1}{\sqrt{2}}(|d_{xz}\downarrow\rangle -i|d_{yz}\downarrow\rangle), \frac{1}{\sqrt{2}}(|d_{xz}\uparrow\rangle +i|d_{yz}\uparrow\rangle) \}$. The angular momentum (AM) texture can be calculated using the eigenstates with the lowest perturbative correction in $\xi$. For the $d_{xy}$ manifold in close vicinity of the $\Gamma$ point, the dominant spin AM expectation value $\langle S_y \rangle \approx \hbar/2$ for an eigenstate in $x$ direction comes from the original $d_{xy}$ manifold. The remnant orbital AM $\langle L_y \rangle \approx\hbar \xi/\Delta_{xy,uxz/yz}$ is due to the inter-band coupling to the upper $d_{xz/yz}$, which can be calculated using the eigenstate with the first-order correction in $\xi$ that hybridizes the $d_{xy}$ manifold with the upper $d_{xz/yz}$ and the $d_{x^2-y^2}$ manifolds. As for the lower $d_{xz/yz}$ manifold, both the orbital and spin AM expectation values $$\begin{aligned} \langle L_y \rangle &\approx& \frac{-3 \hbar \xi}{\Delta_{lxz/yz,z^2}} \\ \langle S_y \rangle &\approx& -\frac{3}{4} \hbar \left( \frac{\xi}{\Delta_{lxz/yz,z^2}} \right)^2. \label{eq:ham_R_lxz/yz_correction_lysy} \end{aligned}$$ for an eigenstate in $x$ direction can be obtained only from the eigenstates with the first-order correction in $\xi$ which leads to hybridization with the $d_{z^2}$ manifold. Thus, the orbital dominant AM texture in the lower $d_{xz/yz}$ bands comes from the inter-band coupling to the $d_{z^2}$. Similarly, we find that the orbital dominant AM texture in the upper $d_{xz/yz}$ bands originates from the inter-band coupling to the $d_{xy}$ and the $d_{x^2-y^2}$ manifolds. Near the $\mathrm{X}=(\pi,0)$ point, the effective Hamiltonian is $$\begin{aligned} \mathcal{H}_{\mathrm{eff}} &=&h_{yz}(\vec{k})I_{2\times 2} +\left[\frac{-2\sqrt{3}\gamma_2 \xi}{\tilde{\Delta}_{yz,z^2}}+\frac{2\gamma_3 \xi}{\tilde{\Delta}_{yz,x^2-y^2}}\right] \sigma_x k_y -\frac{2\gamma_1 \xi}{\tilde{\Delta}_{yz,xy}}\sigma_y k_x, \label{eq:ham_eff_X_yz}\end{aligned}$$ where $\tilde{\Delta}_{yz,z^2}=2(t_{\pi}-t_{\delta})-\delta_2$, $\tilde{\Delta}_{yz,x^2-y^2}=2(t_{\pi}-t_{\delta})-\delta_3$, $\tilde{\Delta}_{yz,xy}=2(t_{\pi}-t_{\delta})-\delta_1$, and the Pauli matrices describe the subspace defined by $\{ | d_{yz}\uparrow\rangle, | d_{yz}\downarrow\rangle \}$, and $(k_x, k_y)$ is a local coordinate with respect to $(\pi,0)$. Here, the splitting terms are mixture of Rashba and linear Dresselhaus terms, which are of the form $$\begin{aligned} \mathcal{H}_{\mathrm{splitting}}=A\sigma_x k_y -B\sigma_y k_x,\end{aligned}$$ with $A=\frac{-2\sqrt{3}\gamma_2 \xi}{\tilde{\Delta}_{yz,z^2}}+\frac{2\gamma_3 \xi}{\tilde{\Delta}_{yz,x^2-y^2}}$ and $B=\frac{2\gamma_1 \xi}{\tilde{\Delta}_{yz,xy}}$. If we rotate the local coordinate by $\pi/4$ about $k_z$ axis, the splitting terms become $$\begin{aligned} \mathcal{H}_{\mathrm{splitting}}&=&\frac{A+B}{2}(\sigma_x k_y -\sigma_y k_x) +\frac{A-B}{2}(\sigma_x k_x -\sigma_y k_y) \\ &=&\alpha_R(\sigma_x k_y -\sigma_y k_x) +\alpha_D(\sigma_x k_x -\sigma_y k_y).\end{aligned}$$ In our case, we have $|A|\gg|B|$, thus both Rashba and linear Dresselhaus terms are present with similar strength. Due to the symmetry, only Rashba term is allowed for $C_{4v}$ at $\mathrm{\Gamma}$ (where we should have $A=B$), and both Rashba and linear Dresselhaus terms are allowed for $C_{2v}$ at $\mathrm{X}$ [-@Stroppa2014]. ![\[fig:tb\_am\] Angular momentum texture from the tight-binding model. The angular momentum textures are calculated (a) at $E_{vH}$ and (b) near $\mathrm{X}$ using both $t_{2g}$ and $e_g$, and (c) at $E_{vH}$ and (d) near $\mathrm{X}$ using only $t_{2g}$. The red and blue arrows represent the orbital and spin AM, respectively. ](figS1.pdf){width="90.00000%"} The splitting of the log van Hove singularity at $\mathrm{X}$ ============================================================= We show here that the Rashba-Dresselhaus splitting removes the spin degeneracy of the logarithmic van Hove singularity at $\mathrm{X}$, resulting in the two separate logarithmic van Hove singularities for the upper and lower Rashba-Dresselhaus bands. This implies that there will be a very large density of state change between the upper and lower Rashba-Dresselhaus bands, which would have a significant effect on the phase competition, [*e.g.*]{} the relative magnitude of the pairing susceptibilities with different symmetries. It is well-known that there is a logarithmic van Hove singularity at $\mathrm{X}$ in absence of the Rashba-Dresselhaus splitting. The dispersion of the lowest energy band near $\mathrm{X}=(\pi,0)$ approximately follows the dispersion of the $d_{yz}$ band, $$\xi = 2(t_\delta \cos k_x + t_\pi \cos k_y) \approx t_\delta (k_x -\pi)^2 - t_\pi k_y^2 + 2(t_\pi - t_\delta);$$ it is well-understood that there is a logarithmic van Hove singularity at the saddle point of a quadratic Hamiltonian in 2D [-@Yu2010]. The addition of the Rashba-Dresselhaus term near $\mathrm{X}$, $\mathcal{H}_{R-D} = A\sigma_x k_y - B \sigma_y (k_x - \pi)$, leads to the spin splitting of this saddle point, which modifies the dispersion to $$\xi_\pm \approx t_\delta (k_x -\pi)^2 - t_\pi k_y^2 \pm \sqrt{A^2 k_y^2 + B^2 (k_x -\pi)^2} + 2(t_\pi - t_\delta).$$ Using the fact that the Fermi velocity vanishes when the van Hove singularity occurs, we can see that the van Hove singularity at $\mathrm{X} = (\pi, 0)$ is shifted to $(\pi \pm B/2t_\delta, 0)$ for the upper Rashba-Dresselhaus band, with the dispersion in its vicinity $$\xi_+ \approx t_\delta \left(k_x - \pi \mp \frac{B}{2t_\delta}\right)^2 - \left(t_\pi + t_\delta \frac{A^2}{B^2}\right) k_y^2 + 2(t_\pi - t_\delta) - \frac{B^2}{4t_\delta}$$ and $(\pi, \pm A/2t_\pi)$ for the lower Rashba-Dresselhaus band, with the dispersion in its vicinity $$\xi_- \approx \left(t_\delta + t_\pi \frac{B^2}{A^2}\right)(k_x - \pi)^2 - t_\pi (k_y \mp \frac{A}{2t_\pi})^2 + 2(t_\pi - t_\delta) + \frac{A^2}{4t_\pi}.$$ We see here that when we raise the chemical potential so that the Fermi surface passes through the ${\rm X}$ point, the Fermi level first passes through the logarithmic van Hove singularity of the lower Rashba-Dresselhaus band, and then that of the upper Rashba-Dresselhaus band. The importance of $e_g$ manifold in the angular momentum texture ================================================================ Because the $e_g$ manifold affects the Rashba-Dresselhaus splitting, the inclusion of the $e_g$ manifold is important to correctly describe the AM texture. By numerically solving the tight-binding model, we obtained the AM expectation values with and without $e_g$ manifold (Fig. \[fig:tb\_am\]). For the $t_{2g}$-only limit, we set $\delta_2 \approx \delta_3 \approx 10^3 \mathrm{eV}$. We find considerable differences in view of the direction and magnitude of the AM. Notably, the coupling to $e_g$ manifold has significant effects in the direction of the AM near $\mathrm{X}$ and in the intermediate region. The effect of compressive strain in the substrate ================================================= Due to the large lattice constant of BaHfO$_3$, it might be helpful to apply compressive strain to the substrate for the deposition of the tantalate thin film. The electronic band structure of TaO$_2$/KO layer on HfO$_2$-terminated BaHfO$_3$ with the lattice constant reduced by 2% is presented in Fig. \[fig:strain\_band\]. We find that the Rashba coefficient remains still large (for example, $\alpha_R \approx 0.3~\mathrm{eV\AA}$ in the lower $d_{xz/yz}$ bands at $\Gamma$). The relation between the band effective mass and Rashba-related parameters ========================================================================== Here, we show that both the momentum offset $k_R$ and the Rashba energy $E_R$ are proportional to the effective mass of the Rashba bands for a given Rashba strength $\alpha_R$. We consider the Hamiltonian $$\begin{aligned} \mathcal{H} =\frac{\hbar^2 k^2}{2m^*} I_{2\times2}+\alpha_R (\vec{\sigma}\times \vec{k})\cdot \hat{z},\end{aligned}$$ with $k=\sqrt{k_x^2+k_y^2}$, where $m^*$ is the effective mass of the band and $I_{2\times2}$ is the $2\times2$ identity matrix. The energy dispersion of the lower Rashba band is given by $$\begin{aligned} E(k) &=&\frac{\hbar^2 k^2}{2m^*} - |\alpha_R| k \\ &=&\frac{\hbar^2}{2m^*} (k-\frac{m^* |\alpha_R|}{\hbar^2})^2 - \frac{m^* |\alpha_R|^2}{2\hbar^2} \\ &\equiv&\frac{\hbar^2}{2m^*} (k-k_R)^2 - E_R.\end{aligned}$$ We find that the momentum offset $k_R = \frac{m^* |\alpha_R|}{\hbar^2}$ and the Rashba energy $E_R = \frac{m^* |\alpha_R|^2}{2\hbar^2}$, which are principal measures of the band splitting size when one sees a band structure figure, are proportional to the effective mass $m^*$ for a given Rashba parameter $\alpha_R$. Thus, the Rashba splitting of the $d_{xz/yz}$ bands would look more pronounced due to the heavier effective mass compared with the $d_{xy}$ band even if they had the same Rashba strength. ![\[fig:strain\_band\] Band structure of TaO$_2$/KO on HfO$_2$-terminated BaHfO$_3$ with the lattice constant reduced by 2%. ](figS2.pdf){width="45.00000%"}

--- abstract: | For an efficient implementation of Buchberger’s Algorithm, it is essential to avoid the treatment of as many unnecessary critical pairs or obstructions as possible. In the case of the commutative polynomial ring, this is achieved by the Gebauer-Möller criteria. Here we present an adaptation of the Gebauer-Möller criteria for non-commutative polynomial rings, i.e. for free associative algebras over fields. The essential idea is to detect unnecessary obstructions using other obstructions with or without overlap. Experiments show that the new criteria are able to detect almost all unnecessary obstructions during the execution of Buchberger’s procedure. [**Keywords:**]{} Gröbner basis, free associative algebra, obstruction, Buchberger procedure [**AMS classification:**]{} 16-08, 20-04, 13P10 author: - | Martin Kreuzer[^1], Xingqiang Xiu[^2]\ Fakultät für Informatik und Mathematik\ Universität Passau, D-94030 Passau, Germany title: 'Non-Commutative Gebauer-Möller Criteria' --- Introduction {#sec1} ============ Ever since B. Buchberger’s thesis [@Bu65], Gröbner bases have become a fundamental tool for computations in commutative algebra and algebraic geometry. The most time-consuming part in Buchberger’s Algorithm is the computation of the normal remainder of an S-polynomial corresponding to a critical pair. Therefore a significant amount of energy has been spent on reducing the number of critical pairs which have to be treated. After the discovery of various criteria for discarding critical pairs ahead of time by B. Buchberger and H.M. Möller (see [@Bu79], [@Bu85] and [@Mo85]), this subject found an initial resolution via the *Gebauer-Möller installation* presented in [@GM88] which offers a good compromise between efficiency and the success rate for detecting unnecessary critical pairs. A very different picture presents itself for Gröbner basis computations for two-sided ideals in non-commutative polynomial rings. The basic Gröbner basis theory in this case was described by G.H. Bergman (see [@Be78]), T. Mora (see [@Mo86] and [@Mo94]) and others, and obstructions, the non-commutative analogue of critical pairs, were studied in [@Mo94]. However, since only a few authors endeavoured to implement efficient versions of Buchberger’s Procedure for the non-commutative polynomial ring (i.e. the free associative algebra), the subject of minimizing the number of obstructions which have to be treated has received comparatively little attention, and merely a few rules were developed. For instance, the package [Plural]{} of the computer algebra system [Singular]{} implements a version of the product and the chain criterion, but not the multiply criterion or the leading word criterion. On the other hand, the system [Magma]{} appears to be based on a variant of the F4 Algorithm which does not use criteria for unnecessary obstructions. For an overview on rules which have been developed see for instance [@Co07]. In this paper, we present generalizations of the Gebauer-Möller criteria for non-commutative polynomials. They cover not only the known cases of useless obstructions discussed in [@Mo94], Lemma 5.11 and [@Co07], but form a complete analogue of the results in the commutative case. One of the key ingredients we use for this purpose is the consideration of obstructions without overlaps. We detect useless obstructions, i.e. obstructions that can be represented by other obstructions, using not only obstructions with overlaps but using also those without overlaps. We show that the consideration of obstructions without overlaps does not increase unnecessary computations, since a Gröbner representation is inherent in the S-polynomial of every obstruction without overlaps. Consequently, we reduce the number of obstructions efficiently and obtain a non-commutative version of the Gebauer-Möller criteria. This paper is organised as follows. In Section \[sec2\] we recall the basic theory of Gröbner bases for two-sided ideals in non-commutative polynomial rings. In particular, we introduce and study obstructions (see Definitions \[sec2def4\] and \[sec2def9\], and Lemmas \[sec2lem6\] and \[sec2lem8\]), present the Buchberger Criterion (see Proposition \[sec2pro10\]), and formulate the Buchberger Procedure (see Theorem \[sec2the11\]). The non-commutative analogues of the Gebauer-Möller criteria are developed in Section \[sec3\]. They are based on a careful study of the set of newly constructed obstructions which are produced during the execution of Buchberger’s Procedure. As a result, we are able to formulate the Non-Commutative Multiply Criterion (see Proposition \[ncMCrit\]), the Non-Commutative Leading Word Criterion (see Proposition \[ncLWCrit\]) and the Non-Commutative Backward Criterion (see Proposition \[ncBKCrit\]). When we combine these criteria, the result is a new Improved Buchberger Procedure \[sec3the14\]. The second author has implemented a version of the Buchberger Procedure for non-commutative polynomial rings in a package for the computer algebra system [ApCoCoA]{} which includes the non-commutative Gebauer-Möller criteria developed here (see [@Ap10]). In the last section, we present experimental results about the efficiency of the criteria for some cases of moderately difficult Gröbner basis computations. Unless mentioned otherwise, we adhere to the definitions and terminology given in [@KR00] and [@KR05]. Gröbner Bases in $K\langle X\rangle$ {#sec2} ==================================== In the following we let $X=\{x_1,\dots,x_n\}$ be a finite set of indeterminates (or a finite alphabet), and $\langle X\rangle$ the monoid of all *words* (or *terms*) $x_{i_1}\cdots x_{i_l}$ where the multiplication is concatenation of words. The empty word will be denoted by $\lambda$. Furthermore, let $K$ be a field, and let $$K\langle X\rangle=\{c_1w_1+\cdots+c_sw_s\ |\ s\in\mathbb{N},c_i\in K\setminus\{0\},w_i\in\langle X\rangle\}$$ be the non-commutative polynomial ring generated by $X$ over $K$ (or the free associative $K$-algebra generated by $X$). We introduce basic notions of Gröbner basis theory in this setting. \[sec2def1\] A *word ordering* on $\langle X\rangle$ is a well-ordering ${\sigma}$ which is compatible with multiplication, i.e. $w_1\geq_{\sigma} w_2$ implies $w_3w_1w_4\geq_{\sigma} w_3w_2w_4$ for all words $w_1,w_2,w_3,w_4\in\langle X\rangle$. In the commutative case, a word ordering is usually called a *term ordering* or *monomial ordering*. For instance, the *length-lexicographic ordering* $\LLex$ is a word ordering. It first compares the length of two words and then breaks ties using the non-commutative lexicographic ordering with respect to $x_1>_{\LLex}\cdots>_{\LLex}x_n$. Note that the non-commutative lexicographic ordering by itself is not a word ordering, since it is neither a well-ordering nor compatible with multiplication. \[sec2def2\] Let ${\sigma}$ be a word ordering on $\langle X\rangle$. - Given a polynomial $f\in K\langle X\rangle\setminus\{0\}$, there exists a unique representation $f=c_1w_1+\cdots+c_sw_s$ with $c_1,\dots,c_s\in K\setminus\{0\}$ and $w_1,\dots,w_s\in\langle X\rangle$ such that $w_1>_{\sigma}\cdots>_{\sigma} w_s$. The word $\lw_{\sigma}(f)=w_1$ is called the *leading word* of $f$ with respect to ${\sigma}$. The element $\lc_{\sigma}(f)=c_1$ is called the *leading coefficient*. We let $\lm_{\sigma}(f)=c_1w_1$ and call it the *leading monomial* of $f$. - Let $I\subseteq K\langle X\rangle$ be a two-sided ideal. The set $\lw_{\sigma}\{I\}=\{\lw_{\sigma}(f)\ |\ f\in I\setminus\{0\}\}\subseteq\langle X\rangle$ is called the *leading word set* of $I$. The two-sided ideal $\lw_{\sigma}(I)=\langle\lw_{\sigma}(f)\ |\ f\in I\setminus\{0\}\rangle\subseteq K\langle X\rangle$ is called the *leading word ideal* of $I$. - A subset $G$ of a two-sided ideal $I\subseteq K\langle X\rangle$ is called a *${\sigma}$-Gröbner basis* of $I$ if the set of the leading words $\lw_{\sigma}\{G\}=\{\lw_{\sigma}(f)\ |\ f\in G\setminus\{0\}\}$ generates the leading word ideal $\lw_{\sigma}(I)$. In the following we focus on computations of Gröbner bases for two-sided ideals in $K\langle X\rangle$. For readers who want to know further properties and applications of non-commutative Gröbner bases, we refer to [@Mo94] and [@Xiu12]. Throughout this paper we assume that ${\sigma}$ is a word ordering on $\langle X\rangle$. The next algorithm is a central part of all Gröbner basis computations. \[sec2the3\] Let $f\in K\langle X\rangle$, $s\geq 1$, and $G=\{g_{1},\dots,g_{s}\}\subseteq K\langle X\rangle\setminus\{0\}$. Consider the following sequence of instructions. - Let $k_{1}=\cdots=k_{s}=0, p=0$, and $v=f$. - Find the smallest index $i\in\{1,\dots,s\}$ such that $\lw_{\sigma}(v)=w\lw_{\sigma}(g_{i})w'$ for some words $w,w'\in\langle X\rangle$. If such an $i$ exists, increase $k_{i}$ by $1$, set $c_{ik_{i}}=\frac{\lc_{\sigma}(v)}{\lc_{\sigma}(g_{i})}, w_{ik_{i}}=w, w'_{ik_{i}}=w'$, and replace $v$ by $v-c_{ik_{i}}w_{ik_{i}}g_{i}w'_{ik_{i}}$. - Repeat step [(D2)]{} until there is no more $i\in\{1,\dots,s\}$ such that $\lw_{\sigma}(v)$ is a multiple of $\lw_{\sigma}(g_{i})$. If now $v\neq 0$, then replace $p$ by $p+\lm_{\sigma}(v)$ and $v$ by $v-\lm_{\sigma}(v)$, continue with step [(D2)]{}. - Return the tuples $(c_{11},w_{11},w'_{11}),\dots,(c_{sk_{s}},w_{sk_{s}},w'_{sk_{s}})$ and $p$. This is an algorithm which returns tuples $(c_{11},w_{11},w'_{11}),\dots,(c_{sk_{s}},w_{sk_{s}},w'_{sk_{s}})$ and a polynomial $p\in K\langle X\rangle$ such that the following conditions are satisfied. - We have $f=\sum^{s}_{i=1}\sum^{k_{i}}_{j=1}c_{ij}w_{ij}g_{i}w'_{ij}+p$. - No element of ${\rm Supp}(p)$ is contained in $\langle\lw_{\sigma}(g_{1}),\dots,\lw_{\sigma}(g_{s})\rangle$. - For all $i\in\{1,\dots,s\}$ and all $j\in\{1,\dots,k_{i}\}$, we have $\lw_{\sigma}(w_{ij}g_{i}w'_{ij})\leq_{\sigma}\lw_{\sigma}(f)$. If $p\neq 0$, we have $\lw_{\sigma}(p)\leq_{\sigma}\lw_{\sigma}(f)$. - For all $i\in\{1,\dots, s\}$ and all $j\in\{1,\dots,k_{i}\}$, we have $\lw_{\sigma}(w_{ij}g_{i}w'_{ij})\notin\langle\lw_{\sigma}(g_{1}),\dots,\lw_{\sigma}(g_{i-1})\rangle$. Note that the resulting tuples $(c_{11},w_{11},w'_{11}),\dots,(c_{sk_{s}},w_{sk_{s}},w'_{sk_{s}})$ and polynomial $p$ satisfying conditions (a)-(d) are *not* unique. This is due to the fact that in step (D2) of the Division Algorithm there might exist more that one pair $(w,w')$ satisfying $\lw_{\sigma}(v)=w\lw_{\sigma}(g_{i})w'$ (see [@Xiu12], Example 3.2.2). A polynomial $p\in K\langle X\rangle$ obtained in Theorem \[sec2the3\] is called a *normal remainder* of $f$ with respect to $G$ and is denoted by ${\rm NR}_{\sigma,G}(f)$. For $s\geq 1$, we let $F_s=(K\langle X\rangle\otimes_{K}K\langle X\rangle)^s$ be the free two-sided $K\langle X\rangle$-module of rank $s$ with the canonical basis $\{e_{1},\dots,e_{s}\}$, where $e_{i}=(0,\dots,0,1\otimes1,$ $0,\dots,0)$ with $1\otimes1$ occurring in the $i^{\rm th}$ position for $i=1,\dots,s$, and we let $\mathbb{T}(F_{s})$ be the set of terms in $F_{s}$, i.e. $\mathbb{T}(F_{s})=\{we_{i}w'\ |\ i\in\{1,\dots,s\}, w,w'\in\langle X\rangle\}$. \[sec2def4\] Let $G=\{g_1,\dots,g_s\}\subseteq K\langle X\rangle\setminus\{0\}$ with $s\geq 1$, and let $i,j\in\{1,\dots,s\}$ such that $i\leq j$. - If there exist some words $w_{i},w'_{i},w_{j},w'_{j}\in\langle X\rangle$ such that $w_{i}\lw_{\sigma}(g_{i})w'_{i}=w_{j}\lw_{\sigma}(g_{j})w'_{j}$, then we call the element $${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})=\frac{1}{\lc_{\sigma}(g_{i})}w_{i}e_{i}w'_{i}-\frac{1}{\lc_{\sigma}(g_{j})}w_{j}e_{j}w'_{j}\in F_{s}\setminus\{0\}$$ an *obstruction* of $g_i$ and $g_j$. If $i=j$, it is called a *self obstruction* of $g_{i}$. We will denote the *set of all obstructions* of $g_{i}$ and $g_{j}$ by ${\rm Obs}(i,j)$. - Let ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in {\rm Obs}(i,j)$ be an obstruction of $g_{i}$ and $g_{j}$. The polynomial $$S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})=\frac{1}{\lc_{\sigma}(g_{i})}w_{i}g_{i}w'_{i}-\frac{1}{\lc_{\sigma}(g_{j})}w_{j}g_{j}w'_{j}\in K\langle X\rangle$$ is called the *S-polynomial* of ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$. Using these definitions, we can characterize Gröbner bases in the following way. \[sec2pro5\] Let $G=\{g_1,\dots,g_s\}\subseteq K\langle X\rangle\setminus\{0\}$ be a set of polynomials which generate a two-sided ideal $I=\langle G\rangle\subseteq K\langle X\rangle$. Then the following conditions are equivalent. - The set $G$ is a ${\sigma}$-Gröbner basis of $I$. - For every obstruction ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ in the set $\bigcup_{1\leq i\leq j\leq s}{\rm Obs}(i,j)$, its S-polynomial $S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ has a representation $$S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})=\sum^{\mu}_{k=1}c_{k}w_{k}g_{i_{k}}w'_{k}$$ with $c_{k}\in K, w_{k},w'_{k}\in\langle X\rangle$, and $g_{i_{k}}\in G$ for all $k\in\{1,\dots,\mu\}$ such that $\lw_{\sigma}(w_{j}g_{j}w'_{j})>_{\sigma}\lw_{\sigma}(w_{k}g_{i_{k}}w'_{k})$ if $c_k\neq 0$ for some $k\in\{1,\dots,\mu\}$. See [@Xiu12], Proposition 4.1.2. A presentation of $S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ as in Proposition \[sec2pro5\].b is called a *(weak) Gröbner representation* of $S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ in terms of $G$. Observe that there are infinitely many obstructions in each set ${\rm Obs}(i,j)$, due to the following two types of *trivial* obstructions. - If ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in {\rm Obs}(i,j)$, then, for all $w,w'\in\langle X\rangle$, we have ${\rm o}_{i,j}(ww_{i},w'_{i}w';ww_{j},w'_{j}w')\in{\rm Obs}(i,j)$. - For all $w\in\langle X\rangle$, we have ${\rm o}_{i,j}(\lw_{\sigma}(g_{j})w,1;1,w\lw_{\sigma}(g_{i})), {\rm o}_{i,j}(1, w\lw_{\sigma}(g_{j});$ $\lw_{\sigma}(g_{i})w,1)\in{\rm Obs}(i,j)$. Before going on, let us get rid of these two types of trivial obstructions. The following lemma handles trivial obstructions of type (T1). \[sec2lem6\] If the S-polynomial of ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in {\rm Obs}(i,j)$ has a Gröbner representation in terms of $G$, then, for all $w,w'\in\langle X\rangle$, the S-polynomial of ${\rm o}_{i,j}(ww_{i},w'_{i}w';ww_{j},w'_{j}w')$ also has a Gröbner representation in terms of $G$. Without loss of generality, we assume that $S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ is non-zero. We write $S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})=\sum^{\mu}_{k=1}c_{k}w_{k}g_{i_{k}}w'_{k}$, where $c_{k}\in K\setminus\{0\}$, $w_{k},w'_{k}\in\langle X\rangle$, and $g_{i_{k}}\in G$ such that $\lw_{\sigma}(w_{j}g_{j}w'_{j})>_{\sigma}\lw_{\sigma}(w_{k}g_{i_{k}}w'_{k})$ for all $k\in\{1,\dots,\mu\}$. For all $w,w'\in\langle X\rangle$, it is clear that $S_{i,j}(ww_{i},w'_{i}w';ww_{j},w'_{j}w')$ $=\sum^{\mu}_{k=1}c_{k}ww_{k}g_{i_{k}}w'_{k}w'$. Since the word ordering $\sigma$ is compatible with multiplication, we have $w\lw_{\sigma}(w_{j}g_{j}w'_{j})w'>_{\sigma}w\lw_{\sigma}(w_{k}g_{i_{k}}w'_{k})w'$ for all $k\in\{1,\dots,\mu\}$. Hence we have $\lw_{\sigma}(ww_{j}g_{j}w'_{j}w')>_{\sigma}\lw_{\sigma}(ww_{k}g_{i_{k}}w'_{k}w')$ for all $k\in\{1,\dots,\mu\}$ and $S_{i,j}(ww_{i},w'_{i}w';ww_{j},w'_{j}w')=\sum^{\mu}_{k=1}c_{k}ww_{k}g_{i_{k}}w'_{k}w'$ is a Gröbner representation in terms of $G$. To deal with trivial obstructions of type (T2), we introduce some terminology as follows. \[sec2def7\] Let $G=\{g_1,\dots,g_s\}\subseteq K\langle X\rangle\setminus\{0\}$ with $s\geq 1$. - Let $w_{1},w_{2}\in\langle X\rangle$ be two words. If there exist some words $w,w',w''\in\langle X\rangle$ and $w\neq 1$ such that $w_{1}=w'w$ and $w_{2}=ww''$, or $w_{1}=ww'$ and $w_{2}=w''w$, or $w_{1}=w$ and $w_{2}=w'ww''$, or $w_{1}=w'ww''$ and $w_{2}=w$, then we say $w_{1}$ and $w_{2}$ have an *overlap* at $w$. Otherwise, we say that $w_{1}$ and $w_{2}$ have *no overlap*. - Let ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in {\rm Obs}(i,j)$ be an obstruction. If $\lw_{\sigma}(g_{i})$ and $\lw_{\sigma}(g_{j})$ have an overlap at $w\in\langle X\rangle\setminus\{1\}$ and if $w$ is a subword of $w_{i}\lw_{\sigma}(g_{i})w'_{i}$, then we say that ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ has an *overlap* at $w$. Otherwise, we say that ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ has *no overlap*. Thus, as shown in (T2), there are infinitely many obstructions without overlaps in each ${\rm Obs}(i,j)$. The following lemma gets rid of these trivial obstructions. \[sec2lem8\] If ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in {\rm Obs}(i,j)$ has no overlap, then $S_{i,j}(w_{i},w'_{i};$ $w_{j},w'_{j})$ has a Gröbner representation in terms of $G$. See [@Mo94], Lemma 5.4. Observe that Lemma \[sec2lem8\] is indeed a non-commutative version of the *product criterion* (or *criterion 2*) of Buchberger (cf. [@Bu85]). \[sec2def9\] Let $G=\{g_1,\dots,g_s\}\subseteq K\langle X\rangle\setminus\{0\}$ with $s\geq 1$. - Let $i,j\in\{1,\dots,s\}$ and $i<j$. An obstruction in ${\rm Obs}(i,j)$ is called *non-trivial* if it has an overlap and is of the form ${\rm o}_{i,j}(w_{i},1;1,w'_{j})$, or ${\rm o}_{i,j}(1,w'_{i};$ $w_{j},1)$, or ${\rm o}_{i,j}(w_{i},w'_{i};1,1)$, or ${\rm o}_{i,j}(1,1;w_{j},w'_{j})$ with $w_{i},w'_{i},w_{j},w'_{j}\in\langle X\rangle$. - Let $i\in\{1,\dots,s\}$. A self obstruction in ${\rm Obs}(i,i)$ is called *non-trivial* if it has an overlap and is of the form ${\rm o}_{i,i}(1,w'_{i};w_{i},1)$ with $w_{i},w'_{i}\in\langle X\rangle\setminus\{1\}$. - Let $i,j\in\{1,\dots,s\}$ and $i\leq j$. The *set of all non-trivial obstructions* of $g_{i}$ and $g_{j}$ will be denoted by ${\rm NTObs}(i,j)$. In the literature, a non-trivial obstruction of the form ${\rm o}_{i,j}(w_{i},1;1,w'_{j})$ is called a *left obstruction*, a non-trivial obstruction of the form ${\rm o}_{i,j}(1,w'_{i};w_{j},1)$ is called a *right obstruction*, and a non-trivial obstruction of the form ${\rm o}_{i,j}(w_{i},w'_{i};$ $1,1)$ or ${\rm o}_{i,j}(1,1;w_{j},w'_{j})$ is called a *center obstruction*. We picture four types of obstructions as follows. [\*[3]{}[C]{}]{} &\ &\ &&\ & &\ \ && [\*[3]{}[C]{}]{} &\ &\ &&\ \ & &\ && At this point we can refine the characterization of Gröbner bases given in Proposition \[sec2pro5\] in the following way. \[sec2pro10\] Let $G=\{g_1,\dots,g_s\}\subseteq K\langle X\rangle$ be a set of non-zero polynomials which generate a two-sided ideal $I=\langle G\rangle\subseteq K\langle X\rangle$. Then the set $G$ is a ${\sigma}$-Gröbner basis of $I$ if and only if, for each non-trivial obstruction ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in\bigcup_{1\leq i\leq j\leq s}{\rm NTObs}(i,j)$, its S-polynomial $S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ has a Gröbner representation in terms of $G$. This follows directly from Proposition \[sec2pro5\] and Lemmas \[sec2lem6\] and \[sec2lem8\]. In view of Lemma \[sec2lem8\], it suffices to consider each obstruction with overlap, which is either a non-trivial obstruction or a multiple of a non-trivial obstruction. Further, Lemma \[sec2lem6\] treats a multiple of a non-trivial obstruction via the corresponding non-trivial obstruction. Therefore, it is sufficient to consider only non-trivial obstructions. The Buchberger Criterion enables us to formulate the following procedure for computing Gröbner bases of two-sided ideals. Note that, in the procedure, by a *fair strategy* we mean a selection strategy which ensures that every obstruction is selected eventually. Since these Gröbner bases need not be finite, we have to content ourselves with an enumerating procedure. \[sec2the11\] Let $s\geq 1$, and let $G=\{g_1,\dots,g_s\}\subseteq K\langle X\rangle$ be a set of non-zero polynomials which generate a two-sided ideal $I=\langle G\rangle\subseteq K\langle X\rangle$. Consider the following sequence of instructions. - Let $B=\bigcup_{1\leq i\leq j\leq s}{\rm NTObs}(i,j)$. - If $B=\emptyset$, return the result ${G}$. Otherwise, select an obstruction ${\rm o}_{i,j}(w_{i},w'_{i};$ $w_{j},w'_{j})\in B$ using a fair strategy and delete it from $B$. - Compute the S-polynomial $S=S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ and its normal remainder $S'={\rm NR}_{\sigma,G}(S)$. If $S'=0$, continue with step [(B2)]{}. - Increase $s$ by one, append $g_{s}=S'$ to the set ${G}$, and append the set of obstructions $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ to the set $B$. Then continue with step [(B2)]{}. This is a procedure that enumerates a ${\sigma}$-Gröbner basis ${G}$ of $I$. If $I$ has a finite ${\sigma}$-Gröbner basis, the procedure stops after finitely many steps and the resulting set ${G}$ is a finite ${\sigma}$-Gröbner basis of $I$. Note that this is a straightforward generalization of the commutative version of Buchberger’s algorithm to the non-commutative case. We refer to [@Mo94] for the original form of this procedure and to [@Xiu12], Theorem 4.1.14 for a detailed proof. Non-Commutative Gebauer-Möller Criteria {#sec3} ======================================= In this section we present non-commutative Gebauer-Möller criteria. They check whether an obstruction can be represented by “smaller” obstructions. If so, we declare such obstructions to be *unnecessary*. Before going into details, we define a certain well-ordering $\tau$ on $\mathbb{T}(F_{s})=\{we_{i}w'\ |\ i\in\{1,\dots,s\}, w,w'\in\langle X\rangle\}$ and use it to order obstructions. In the following, let $s\geq1$, and let $G=\{g_1,\dots,g_{s}\}\subseteq K\langle X\rangle\setminus\{0\}$ be a set of non-commutative polynomials. \[sec3def1\] Let us define a relation $\tau$ on $\mathbb{T}(F_{s})$ as follows. For two terms $w_{1}e_{i}w'_{1},w_{2}e_{j}w'_{2}\in\mathbb{T}(F_{s})$, we let $w_{1}e_{i}w'_{1}\geq_{\tau}w_{2}e_{j}w'_{2}$ if - $w_{1}\lw_{\sigma}(g_{i})w'_{1}>_{\sigma}w_{2}\lw_{\sigma}(g_{j})w'_{2}$, or - $w_{1}\lw_{\sigma}(g_{i})w'_{1}=w_{2}\lw_{\sigma}(g_{j})w'_{2}$ and $i>j$, or - $w_{1}\lw_{\sigma}(g_{i})w'_{1}=w_{2}\lw_{\sigma}(g_{j})w'_{2}$ and $i=j$ and $w_{1}\geq_{\sigma}w_{2}$. One can check that $\tau$ is a well-ordering and is compatible with scalar multiplication. The relation $\tau$ is called the *module term ordering induced by* $({\sigma},{G})$ on $\mathbb{T}(F_{s})$. By definition, for every obstruction ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in\bigcup_{1\leq i\leq j\leq s}{\rm Obs}(i,j)$, we have $w_{i}e_{i}w'_{i}<_{\tau}w_{j}e_{j}w'_{j}$. We extend the ordering $\tau$ to the set of obstructions $\bigcup_{1\leq i\leq j\leq s}{\rm Obs}(i,j)$ by committing the following slight abuse of notation. \[sec3def2\] Let $\tau$ be the module term ordering induced by $({\sigma},{G})$ on $\mathbb{T}(F_{s})$. Let ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j}), {\rm o}_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$ be two obstructions in the set $\bigcup_{1\leq i\leq j\leq s}{\rm Obs}(i,j)$. If we have $w_{j}e_{j}w'_{j}>_{\tau}w_{l}e_{l}w'_{l}$, or if we have $w_{j}e_{j}w'_{j}=w_{l}e_{l}w'_{l}$ and $w_{i}e_{i}w'_{i}\geq_{\tau}w_{k}e_{k}w'_{k}$, then we let ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\geq_{\tau}{\rm o}_{k,l}(w_{k},w'_{k};$ $w_{l},w'_{l})$. The ordering $\tau$ is called the ordering *induced by* $({\sigma},{G})$ on the set of obstructions. One can verify that $\tau$ is also a well-ordering on $\bigcup_{1\leq i\leq j\leq s}{\rm Obs}(i,j)$ and compatible with scalar multiplication. Now we are ready to generalize the commutative Gebauer-Möller criteria (see [@CKR04] and [@GM88]) to the non-commutative case. Recall that, in step (B4) of the Buchberger Procedure, when a new generator $g_{s}$ is added, we immediately construct new obstructions $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$. We want to detect unnecessary obstructions in the set $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ of newly constructed obstructions as well as in the set $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$ of previously constructed obstructions. We achieve this goal via the following three steps. Firstly, we detect unnecessary obstructions in the set $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ with the aid of other obstructions also in this set. This step is called a *head reduction step* in [@CKR04]. Secondly, we detect unnecessary obstructions in the set $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ with the aid of obstructions in the set $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$. This step is called a *tail reduction step* in [@CKR04]. Thirdly, we detect unnecessary obstructions in the set $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$ with the aid of the new generator $g_{s}$. Indeed, the first step corresponds to the commutative Gebauer-Möller criteria $M$ and $F$, and the last step corresponds to criterion $B_k$ (c.f. [@GM88], Subsection 3.4). The following lemma helps us to implement the first step, that is, to detect unnecessary obstructions in the set $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ of newly constructed obstructions via other obstructions in this set. \[sec3lem3\] Let ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ and ${\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ be two distinct non-trivial obstructions in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ with two words $w,w'\in\langle X\rangle$ satisfying $u_{s}=$ $wv_{s}$ and $u'_{s}=v'_{s}w'$. - If $i<j$ and $ww'\neq1$, then we have $${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=w{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})w'+{\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$$ with ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$. Further, if the S-polynomials $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ have Gröbner representations in terms of $G$, then so does $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$. - If $i>j$, then we have $${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=w{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})w'-{\rm o}_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$$ with ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$. Further, if the S-polynomials $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$ have Gröbner representations in terms of $G$, then so does $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$. - If $i=j$ and $ww'\neq 1$ or if $i=j$ and $ww'=1$ and $w_i>_{\sigma}w_j$, then we have $${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=w{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})w'+{\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$$ with ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$. Further, if the S-polynomials $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ have Gröbner representations in terms of $G$, then so does $_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$. We prove case (a). Cases (b) and (c) can be proved similarly. The equation in case (a) follows from Definition \[sec2def4\].a and from the conditions $u_{s}=wv_{s}, u'_{s}=v'_{s}w'$ and $i<j$. Because of $ww'>1$, we have $u_{s}\lw(g_{s})u'_{s}=wv_{s}\lw(g_{s})v'_{s}w'>_{\sigma}v_{s}\lw(g_{s})v'_{s}$. Consequently, we get $u_{s}e_{s}u'_{s}>_{\tau}v_{s}e_{s}v'_{s}$ and ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$. From $u_{s}\lw(g_{s})u'_{s}$ $=w_{i}\lw(g_{i})w'_{i}=ww_{j}\lw(g_{j})w'_{j}w'$ and $s>j$, we get the inequalities $u_{s}e_{s}u'_{s}>_{\tau}ww_{j}e_{j}w'_{j}w'$ and ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})>_{\tau}{\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$. Next we show that, if $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ have Gröbner representations in terms of $G$, then so does $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$. Clearly we have $$S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=wS_{j,s}(w_{j},w'_{j};v_{s},v'_{s})w'+S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w').$$ Without loss of generality, we assume that $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s}), S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ are non-zero. Since there is a Gröbner representation for $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$, we have $$S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})=\sum^\mu_{k=1}a_kw_kg_{i_k}w'_k$$ with $a_k\in K\setminus\{0\},\ w_k,w'_k\in\langle X\rangle,\ g_{i_k}\in G$ for all $k\in\{1,\dots,\mu\}$, such that $\lw_{\sigma}(v_{s}g_{s}v'_{s})>_{\sigma}\lw_{\sigma}(a_kw_kg_{i_k}w'_k)$. Similarly, for $S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ we have $$S_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')=\sum^\nu_{l=1}b_lw_lg_{i_l}w'_l$$ with $b_l\in K\setminus\{0\},\ w_l,w'_l\in\langle X\rangle,\ g_{i_l}\in G$ for all $l\in\{1,\dots,\nu\}$, such that $\lw_{\sigma}(ww_{j}g_{j}w'_{j}w')>_{\sigma}\lw_{\sigma}(b_lw_lg_{i_l}w'_l)$. Therefore we have $$\begin{aligned} S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})&=&w(\sum^\mu_{k=1}a_kw_kg_{i_k}w'_k)w'+\sum^\nu_{l=1}b_lw_lg_{i_l}w'_l\\ &=&\sum^\mu_{k=1}a_kww_kg_{i_k}w'_kw'+\sum^\nu_{l=1}b_lw_lg_{i_l}w'_l.\end{aligned}$$ As $u_{s}\lw_{\sigma}(g_{s})u'_{s}=wv_{s}\lw_{\sigma}(g_{s})v'_{s}w'$, we have $\lw_{\sigma}(u_{s}g_{s}u'_{s})=\lw_{\sigma}(wv_{s}g_{s}v'_{s}w')$ $>_{\sigma}\lw_{\sigma}(ww_kg_{i_k}w'_kw')$ for all $k\in\{1,\dots,\mu\}$. By Definition \[sec2def4\], we have $\lw_{\sigma}(u_{s}g_{s}u'_{s})=\lw_{\sigma}(w_{i}g_iw'_{i})=\lw_{\sigma}(ww_{j}g_jw'_{j}w')>_{\sigma}\lw_{\sigma}(b_lw_lg_{i_l}w'_l)$ for all $l\in\{1,\dots,\nu\}$. Therefore $$S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=\sum^\mu_{k=1}a_kww_kg_{i_k}w'_kw'+\sum^\nu_{l=1}b_lw_lg_{i_l}w'_l$$ is a Gröbner representation of $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$. The following example shows that the obstruction ${\rm o}_{i,j}(w_{i},w'_{i};ww_{j},w'_{j}w')$ in case (a) of Lemma \[sec3lem3\] can be a non-trivial obstruction, i.e. a multiple of a non-trivial obstruction or an obstruction without overlap. Similar phenomena occur in cases (b) and (c) of Lemma \[sec3lem3\], as well as in Lemmas \[sec3lem7\] and \[sec3lem10\]. \[sec3exa4\] Consider polynomials $G=\{g_{1},g_{2},g_{3}\}$ in the non-commutative polynomial ring $K\langle x,y\rangle$. - Assume that $\lm_{\sigma}(g_{1})=y^3, \lm_{\sigma}(g_{2})=x^2y^{2}$ and $\lm_{\sigma}(g_{3})=xyx^{2}y$. Then we have ${\rm o}_{1,3}(xyx^{2},1;1,y^{2}), {\rm o}_{2,3}(xy,1;1,y)\in\bigcup_{1\leq i\leq 3}{\rm NTObs}(i,3)$, and $${\rm o}_{1,3}(xyx^{2},1;1,y^{2})={\rm o}_{2,3}(xy,1;1,y)y+{\rm o}_{1,2}(xyx^2,1;xy,y).$$ Observe that ${\rm o}_{1,2}(xyx^2,xy;y)=xy{\rm o}_{1,2}(x^2,1;1,y)$ is a multiple of the non-trivial obstruction ${\rm o}_{1,2}(x^2,1;1,y)$. - Now assume that $\lm_{\sigma}(g_{1})=(xy)^{2}, \lm_{\sigma}(g_{2})=y$ and $\lm_{\sigma}(g_{3})=xyx^{2}y$. Then we have ${\rm o}_{1,3}(xyx,1;1,xy), {\rm o}_{2,3}(x,x^{2}y;1,1)\in\bigcup_{1\leq i\leq 3}{\rm NTObs}(i,3)$, and $${\rm o}_{1,3}(xyx,1;1,xy)={\rm o}_{2,3}(x,x^{2}y;1,1)xy+{\rm o}_{1,2}(xyx,1;x,x^{2}yxy).$$ One can check that ${\rm o}_{1,2}(xyx,1;x,x^{2}yxy)$ is an obstruction without overlap. In the following, we present the non-commutative multiply criterion and the leading word criterion. They are non-commutative analogues of the Gebauer-Möller criteria M and F, respectively. \[ncMCrit\] Suppose that ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ and ${\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ are two distinct non-trivial obstructions in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ such that there exist two words $w,w'$ in $\langle X\rangle$ satisfying $u_{s}=wv_{s}$ and $u'_{s}=v'_{s}w'$. Then we can remove the obstruction ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ from $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ in the execution of the Buchberger Procedure if $ww'\neq 1$. By the previous lemma, the obstruction ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ can be represented as $${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=w{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})w'+a{\rm o}_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$$ with $a\in\{1,-1\}$ and $k=\min\{i,j\},l=\max\{i,j\}$. To prove that ${\rm o}_{i,s}(w_{i},w'_{i};$ $u_{s},u'_{s})$ is strictly larger than ${\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and ${\rm o}_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$, we consider two cases. If $i>j$, then the result follows from Lemma \[sec3lem3\].b; if $i\leq j$, then the result follows from Lemma \[sec3lem3\].a and \[sec3lem3\].c and the condition $ww'\neq 1$. Moreover, the S-polynomial $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ has a Gröbner representation in terms of $G$ if $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$ have Gröbner representations in terms of $G$. Theorem \[sec2the11\] ensures that $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ has a Gröbner representation in terms of $G$. Note that the obstruction ${\rm o}_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$ can be either a multiple of a non-trivial obstruction or an obstruction without overlap (for instance, see Example \[sec3exa4\]). If ${\rm o}_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$ is a multiple of a non-trivial obstruction, then Lemma \[sec2lem6\] and Theorem \[sec2the11\] guarantee that $S_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$ has a Gröbner representation in terms of $G$. If ${\rm o}_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$ is an obstruction without overlap, then, by Lemma \[sec2lem8\], its S-polynomial has a Gröbner representation in terms of $G$. Now the conclusion follows from Proposition \[sec2pro10\] and Theorem \[sec2the11\]. \[ncLWCrit\] Suppose that ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ and ${\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ are two distinct non-trivial obstructions in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ such that there exist two words $w,w'$ in $\langle X\rangle$ satisfying $u_{s}=wv_{s}$ and $u'_{s}=v'_{s}w'$. Then ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ can be removed from $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ in the execution of the Buchberger Procedure if one of the following conditions is satisfied. - $i>j$. - $i=j$ and $ww'=1$ and $w_{i}>_{\sigma}w_{j}$. Observe that condition (a) corresponds to Lemma \[sec3lem3\].b, while condition (b) corresponds to Lemma \[sec3lem3\].c. We represent ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ as $${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=w{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})w'-{\rm o}_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i}).$$ By Lemma \[sec3lem3\].b and \[sec3lem3\].c, we have ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ is strictly larger than ${\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and ${\rm o}_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$. Moreover, if the S-polynomials $S_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and $S_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$ have Gröbner representations in terms of $G$, then so does $S_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$. Theorem \[sec2the11\] ensures $S_{j,s}(w_{j},w'_{j};$ $v_{s},v'_{s})$ has a Gröbner representation in terms of $G$. Note that the obstruction ${\rm o}_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$ can be either a multiple of a non-trivial obstruction or an obstruction without overlap (for instance, see Example \[sec3exa4\]). If ${\rm o}_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$ is a multiple of a non-trivial obstruction, then Lemma \[sec2lem6\] and Theorem \[sec2the11\] guarantee that $S_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$ has a Gröbner representation in terms of $G$. If ${\rm o}_{j,i}(ww_{j},w'_{j}w';w_{i},w'_{i})$ is an obstruction without overlap, then, by Lemma \[sec2lem8\], its S-polynomial has a Gröbner representation in terms of $G$. Now the conclusion follows from Proposition \[sec2pro10\] and Theorem \[sec2the11\]. Next we work on detecting unnecessary obstructions in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ via obstructions in the set $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$ of previously constructed obstructions. \[sec3lem7\] Let ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$ and ${\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})$ be non-trivial obstructions in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ and $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$, respectively. If there exist two words $w,w'\in\langle X\rangle$ such that $u_{j}=wv_{j}$ and $u'_{j}=v'_{j}w'$, then we have $${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})=-w{\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})w'+ {\rm o}_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$$ where the inequalities ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s}) >_{\tau}{\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})$ and ${\rm o}_{j,s}(u_{j},u'_{j};$ $w_{s},w'_{s}) >_{\tau}{\rm o}_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$ hold. Further, if the S-polynomials $S_{i,j}(w_{i},w'_{i};$ $v_{j},v'_{j})$ and $S_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$ have Gröbner representations in terms of $G$, then so does $S_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$. The claimed equality follows from Definition \[sec2def4\].a and from the conditions $u_{j}=wv_{j}$ and $u'_{j}=v'_{j}w'$. We have ${\rm o}_{j,s}(u_{j},u'_{j}; w_{s},w'_{s})>_{\tau}{\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})$ for $w_{s}e_{s}w'_{s}>_{\tau}u_{j}e_{j}u'_{j}= wv_{j}e_{j}v'_{j}w\geq_{\tau}v_{j}e_{j}v'_{j}$. From the inequality $u_{j}e_{j}u'_{j}= wv_{j}e_{j}v'_{j}w>_{\tau}ww_{i}e_{i}w'_{i}w'$, it follows that ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s}) >_{\tau}{\rm o}_{i,s}(ww_{i},w'_{i}w';$ $w_{s},w'_{s})$. Again, we can prove the second part by following the same argument as in the proof of Lemma \[sec3lem3\].a. Note that the obstruction ${\rm o}_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$ in Lemma \[sec3lem7\] can be either a multiple of a non-trivial obstruction or an obstruction without overlap. However, it suffices for us to consider only the latter case, since the former case has been considered in Proposition \[ncMCrit\], and, more precisely, in Lemma \[sec3lem3\].b. \[ncTailRed\] Suppose that ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$ and ${\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})$ are non-trivial obstructions in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ and $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$, respectively, such that there exist two words $w,w'\in\langle X\rangle$ satisfying $u_{j}=wv_{j}$ and $u'_{j}=v'_{j}w'$. If the word $ww_{i}$ is a multiple of $w_{s}\lw_{\sigma}(g_{s})$, or if the word $w'_{i}w'$ is a multiple of $\lw_{\sigma}(g_{s})w'_{s}$, then ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$ can be removed from $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ in the execution of the Buchberger Procedure. By Lemma \[sec3lem7\], the obstruction ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$ can be represented as $${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})= -w{\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})w'+{\rm o}_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$$ where the inequalitites ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})>_\tau{\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})$ and ${\rm o}_{j,s}(u_{j},u'_{j};$ $w_{s},w'_{s})>_\tau{\rm o}_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$ hold. Further, if the S-polynomials $S_{i,j}(w_{i},w'_{i};$ $v_{j},v'_{j})$ and $S_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$ have Gröbner representations in terms of $G$, then so does $S_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$. Theorem \[sec2the11\] ensures that $S_{i,j}(w_{i},w'_{i};v_{j},v'_{j})$ has a Gröbner representation in terms of $G$. Note that $ww_{i}$ is a multiple of $w_{s}\lw_{\sigma}(g_{s})$ or $w'_{i}w'$ is a multiple of $\lw_{\sigma}(g_{s})w'_{s}$. This implies that ${\rm o}_{i,s}(ww_{i},$ $w'_{i}w';w_{s},w'_{s})$ has no overlap. By Lemma \[sec2lem8\], $S_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$ has a Gröbner representation in terms of $G$. Now the conclusion follows from Proposition \[sec2pro10\] and Theorem \[sec2the11\]. \[sec3rem9\] Our experiments in the final section show that, after applying the previous two criteria, the Non-Commutative Tail Reduction is unlikely to apply in the Buchberger Procedure. This may be due to the fact that frequently the Non-Commutative Multiply Criterion and the Non-Commutative Leading Word Criterion have already detected all unnecessary obstructions in the set $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ of newly constructed obstructions. So far we have detected unnecessary obstructions in the set $\bigcup_{1\leq i\leq s}{\rm O}(i,s)$ of newly constructed obstructions. Intuitively, we are also able to detect unnecessary obstructions in the set $\bigcup_{1\leq i\leq j\leq s-1}{\rm Obs}(i,j)$ of previously constructed obstructions. Thus, in the last step, we detect unnecessary obstructions in this set by using the new generator $g_{s}$. \[sec3lem10\] Let ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$ be a non-trivial obstruction. If there are two words $w,w'\in\langle X\rangle$ satisfying $w_{j}\lw_{\sigma}(g_{j})w'_{j}=w\lw_{\sigma}(g_{s})w'$, then we can represent ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ as $${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})={\rm o}_{i,s}(w_{i},w'_{i};w,w')-{\rm o}_{j,s}(w_{j},w'_{j};w,w').$$ Moreover, if $S_{i,s}(w_{i},w'_{i};w,w')$ and $S_{j,s}(w_{j},w'_{j};w,w')$ have Gröbner representations in terms of $G$, then so does $S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$. The claimed equality follows from Definition \[sec2def4\].a and the condition $w_{j}\lw_{\sigma}(g_{j})w'_{j}$ $=w\lw_{\sigma}(g_{s})w'$. The proof of the second part is analogous to the proof of the second part of Lemma \[sec3lem3\].a. The following example shows that the obstruction ${\rm o}_{i,s}(w_{i},w'_{i};w,w')$ in the equation of Lemma \[sec3lem10\] can be either an obstruction without overlap or a multiple of a non-trivial obstruction. In the case that ${\rm o}_{i,s}(w_{i},w'_{i};w,w')$ is a multiple of a non-trivial obstruction, say ${\rm o}_{i,s}(\tilde{w}_{i},\tilde{w}'_{i};\tilde{w},\tilde{w}')$, the example shows that it is not necessary to have ${\rm o}_{i,s}(w_{i},w'_{i};w,w')>_{\tau}{\rm o}_{i,s}(\tilde{w}_{i},\tilde{w}'_{i};\tilde{w},\tilde{w}')$ (compared to Lemmas \[sec3lem3\] and \[sec3lem7\]). The same also holds for the obstruction ${\rm o}_{j,s}(w_{j},w'_{j};w,w')$ in the equation of Lemma \[sec3lem10\]. \[sec3exa11\] Consider polynomials $G=\{g_{1},g_{2},g_{3}\}$ in the non-commutative polynomial ring $K\langle x,y\rangle$ with $\lm_{\sigma}(g_{1})=x^3yx, \lm_{\sigma}(g_{2})=x^2$ and $\lm_{\sigma}(g_{3})=x$. We have ${\rm o}_{1,2}(1,1;x,yx)\in\bigcup_{1\leq i\leq j\leq 2}{\rm NTObs}(i,j)$ and $x\lw_{\sigma}(g_{2})yx=x^3yx=x^3y\lw_{\sigma}(g_{3})$ and $${\rm o}_{1,2}(1,1;x,yx)={\rm o}_{1,3}(1,1;x^3y,1)-{\rm o}_{2,3}(x,yx;x^3y,1).$$ One can check that ${\rm o}_{1,3}(1,1;x^3y,1)$ is a non-trivial obstruction in ${\rm NTObs}(1,3)$ and ${\rm o}_{1,2}(1,1;x,yx)<_{\tau}{\rm o}_{1,3}(1,1;x^3y,1)$. Moreover, ${\rm o}_{2,3}(x,yx;x^3y,1)$ is an obstruction without overlap. The following is a non-commutative analogue of the Gebauer-Möller criterian $B_k$, which is also known as the *chain criterion* (or *criterion 1*) of Buchberger (cf. [@Bu85]). \[ncBKCrit\] Suppose that ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$ is a non-trivial obstruction. Then in the execution of the Buchberger Procedure ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ can be removed from $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$ if the following three conditions are satisfied. - There are two words $w,w'\in\langle X\rangle$ such that $w_{j}\lw_{\sigma}(g_{j})w'_{j}=w\lw_{\sigma}(g_{s})w'$. - The obstruction ${\rm o}_{i,s}(w_{i},w'_{i};w,w')$ is either an obstruction without overlap or a multiple of a non-trivial obstruction in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$. - The obstruction ${\rm o}_{j,s}(w_{j},w'_{j};w,w')$ is either an obstruction without overlap or a multiple of a non-trivial obstruction in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$. By Lemma \[sec3lem10\], we can represent ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ as $${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})={\rm o}_{i,s}(w_{i},w'_{i};w,w')-{\rm o}_{j,s}(w_{j},w'_{j};w,w').$$ Moreover, $S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ has a Gröbner representations in terms of $G$ if $S_{i,s}(w_{i},w'_{i};w,w')$ and $S_{j,s}(w_{j},w'_{j};w,w')$ have Gröbner representations in terms of $G$. If ${\rm o}_{i,s}(w_{i},w'_{i};w,w')$ is an obstruction without overlap, then, by Lemma \[sec2lem8\], its S-polynomial has a Gröbner representations in terms of $G$. If it is a multiple of a non-trivial obstruction in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$, then Lemma \[sec2lem8\] and Theorem \[sec2the11\] ensure that $S_{i,s}(w_{i},w'_{i};w,w')$ has a Gröbner representations in terms of $G$. By the same argument, one can show that $S_{j,s}(w_{j},w'_{j};w,w')$ has a Gröbner representations in terms of $G$. Now the conclusion follows from Proposition \[sec2pro10\] and Theorem \[sec2the11\]. We would like to mention that the Non-Commutative Backward Criterion given in Proposition \[ncBKCrit\] covers in particular all useless obstructions presented by T. Mora in [@Mo94], Lemma 5.11. \[sec3rem13\] In order to apply Propositions \[ncMCrit\], \[ncLWCrit\], \[ncTailRed\] and \[ncBKCrit\] to remove unnecessary obstructions during the execution of the Buchberger Procedure, it is crucial to make sure that the S-polynomials of those removed obstructions have Gröbner representations. - Propositions \[ncMCrit\], \[ncLWCrit\] and \[ncTailRed\] remove unnecessary non-trivial obstructions, say ${\rm o}_{i,s}(w_{i},w'_{i};w_{s},w'_{s})$, from the set $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ of newly constructed obstructions. The Gröbner representation of the S-polynomial $S_{i,s}(w_{i},w'_{i};w_{s},w'_{s})$ depends on the Gröbner representations of the S-polynomials of two smaller obstructions in the set $\bigcup_{1\leq i\leq j\leq s-1}{\rm Obs}(i,j)$ and the set $\bigcup_{1\leq i\leq s}{\rm Obs}(i,s)$. - Proposition \[ncBKCrit\] removes unnecessary obstructions, say ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$, from the set $\bigcup_{1\leq i\leq j\leq s-1}{\rm Obs}(i,j)$ of previously constructed obstructions. The Gröbner representation of $S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ depends on the Gröbner representations of the S-polynomials of two obstructions, say ${\rm o}_{k,s}(w_{k},w'_{k};$ $u_{s},u'_{s})$ and ${\rm o}_{l,s}(w_{l},w'_{l};v_{s},v'_{s})$, in $\bigcup_{1\leq i\leq s}{\rm Obs}(i,s)$, which are not necessarily smaller than ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$. If ${\rm o}_{k,s}(w_{k},w'_{k};u_{s},u'_{s})$ is a multiple of a non-trivial obstruction, say ${\rm o}_{k,s}(\tilde{w}_{k},\tilde{w}'_{k};\tilde{u}_{s},\tilde{u}'_{s})$, in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$, then, before removing ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$, it is important to ensure that ${\rm o}_{k,s}(\tilde{w}_{k},\tilde{w}'_{k};\tilde{u}_{s},\tilde{u}'_{s})$ is in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$. The same check should be applied to ${\rm o}_{l,s}(w_{l},w'_{l};v_{s},v'_{s})$. Observe that Propositions \[ncMCrit\], \[ncLWCrit\] and \[ncBKCrit\] are actually generalizations of the well-known Gebauer-Möller criteria (see [@CKR04] and [@GM88]) in commutative polynomial rings. More precisely, Propositions \[ncMCrit\], \[ncLWCrit\] and \[ncBKCrit\] correspond to criterion $M$, criterion $F$ and criterion $B_k$, respectively (c.f. [@GM88], Subsection 3.4). Using the Gebauer-Möller criteria, we can improve the Buchberger Procedure as follows. \[sec3the14\] In the setting of Theorem \[sec2the11\], we replace step [(B4)]{} by the following sequence of instructions. - Increase $s$ by one. Append $g_{s}=S'$ to the set ${G}$, and form the set of non-trivial obstructions ${\rm NTObs}(s)=\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$. - Remove from ${\rm NTObs}(s)$ all obstructions ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ such that there exists an obstruction ${\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})\in{\rm NTObs}(s)$ with the properties that there exist two words $w,w'\in\langle X\rangle$ satisfying $u_{s}=wv_{s}$, $u'_{s}=v'_{s}w'$ and $ww'\neq1$. - Remove from ${\rm NTObs}(s)$ all obstructions ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ such that there exists an obstruction ${\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})\in{\rm NTObs}(s)$ with the properties that there exist two words $w,w'\in\langle X\rangle$ satisfying $u_{s}=wv_{s}$, $u'_{s}=v'_{s}w'$, and such that $i>j$, or $i=j$ and $ww'=1$ and $w_{i}>_{\sigma}w_{j}$. - Remove from ${\rm NTObs}(s)$ all obstructions ${\rm o}_{j,s}(u_{j},u'_{j};w_{s},w'_{s})$ such that there exists an obstruction ${\rm o}_{i,j}(w_{i},w'_{i};v_{j},v'_{j})\in B$ with the properties that there exist two words $w,w'\in\langle X\rangle$ satisfying $u_{j}=wv_{j}, u'_{j}=v'_{j}w'$, and such that ${\rm o}_{i,s}(ww_{i},w'_{i}w';w_{s},w'_{s})$ has no overlap. - Remove from $B$ all obstructions ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ such that there exist two words $w,w'\in\langle X\rangle$ satisfying $w\lw_{\sigma}(g_{s})w'=w_{j}\lw_{\sigma}(g_{j})w'_{j}$, and such that the following conditions are satisfied. - ${\rm o}_{i,s}(w_{i},w'_{i};w,w')$ is either an obstruction without overlap or a multiple of a non-trivial obstruction in ${\rm NTObs}(s)$. - ${\rm o}_{j,s}(w_{j},w'_{j};w,w')$ is either an obstruction without overlap or a multiple of a non-trivial obstruction in ${\rm NTObs}(s)$. - Replace $B$ by $B\cup {\rm NTObs}(s)$ and continue with step [(B2)]{}. Then the resulting set of instructions is a procedure that enumerates a ${\sigma}$-Gröbner basis ${G}$ of $I$. If $I$ has a finite ${\sigma}$-Gröber basis, it stops after finitely many steps and the resulting set ${G}$ is a finite ${\sigma}$-Gröbner basis of $I$. This follows from Theorem \[sec2the11\] and Propositions \[ncMCrit\], \[ncLWCrit\], \[ncTailRed\] and \[ncBKCrit\]. Experiments and Conclusions {#sec4} =========================== In this section we want to present some experimental data which illustrate the performance of the Gebauer-Möller criteria presented in Propositions \[ncMCrit\], \[ncTailRed\] and \[ncBKCrit\]. The computations are based on an implementation (using C++) in an experimental version of the ApCoCoA library (see [@Ap10]) by the second author. \[sec4exa1\] Consider the non-commutative polynomial ring $\mathbb{Q}\langle a,b\rangle$ equipped with the word ordering $\LLex$ on $\langle a,b\rangle$ such that $a>_{\LLex}b$. We take the list of *finite generalized triangle groups* from [@RS02], Theorem 2.12 and construct a list of ideals in $\mathbb{Q}\langle a,b\rangle$. For $k=1,\dots,13$ let $I_{k}=\langle G_{k}\rangle\subseteq\mathbb{Q}\langle a,b\rangle$ be the ideal generated by the following set of polynomials $G_{k}\subseteq\mathbb{Q}\langle a,b\rangle$. $$\begin{aligned} G_{1}&=&\{a^2-1,b^3-1,(ababab^2ab^2)^2-1\},\\ G_{2}&=&\{a^2-1,b^3-1,(ababab^2)^3-1\},\\ G_{3}&=&\{a^3-1,b^3-1,(abab^2)^2-1\},\\ G_{4}&=&\{a^3-1,b^3-1,(aba^2b^2)^2-1\},\\ G_{5}&=&\{a^2-1,b^5-1,(abab^2)^2-1\},\\ G_{6}&=&\{a^2-1,b^5-1,(ababab^4)^2-1\}, \\ G_{7}&=&\{a^2-1,b^5-1,(abab^2ab^4)^2-1\},\\ G_{8}&=&\{a^2-1,b^4-1,(ababab^3)^2-1\},\\ G_{9}&=&\{a^2-1,b^3-1,(abab^2)^2-1\}, \\ G_{10}&=&\{a^2-1,b^3-1,(ababab^2)^2-1\},\\ G_{11}&=&\{a^2-1,b^3-1,(abababab^2)^2-1\},\\ G_{12}&=&\{a^2-1,b^3-1,(ababab^2abab^2)^2-1\},\\ G_{13}&=&\{a^2-1,b^3-1,(ababababab^2ab^2)^2-1\}.\end{aligned}$$ The following table lists some numbers of polynomials and obstructions treated by the Improved Buchberger Procedure given in Theorem \[sec3the14\]. $k$ $\#(Gb)$ $\!\!\#(RGb)\!\!$ $\#(Tot)$ $\#(Sel)$ $\#(M)$ $\#(F)$ $\#(B_k)$ $\rho$ ----- ---------- ------------------- ----------- ----------- --------- --------- ----------- -------- -- 1 62 35 7032 248 6512 48 224 0.0353 2 133 96 31700 533 30571 70 526 0.0168 3 50 40 2828 197 2489 11 131 0.0697 4 64 28 4702 253 4185 46 218 0.0538 5 35 21 1580 115 1348 24 93 0.0728 6 199 164 51175 882 49126 26 1141 0.0172 7 200 164 51864 886 49818 17 1143 0.0170 8 53 37 3756 192 3357 19 188 0.0511 9 11 5 150 31 98 8 13 0.2067 10 22 15 741 74 605 18 44 0.0999 11 30 21 1573 116 1324 50 83 0.0737 12 97 70 16841 365 15989 97 390 0.0217 13 220 194 87673 1021 85136 153 1363 0.0116 Here we used the following abbreviations. - $\#(Gb)$ is the number of elements of the Gröbner basis returned by the procedure. - $\#(RGb)$ is the cardinality of the reduced Gröbner basis of the corresponding ideal. - $\#(Tot)$ is the total number of non-trivial obstructions constructed during the Buchberger Procedure. - $\#(Sel)$ is the number of actually selected and analysed non-trivial obstructions. - $\#(M)$ is the number of unnecessary non-trivial obstructions detected by the Non-Commutative Multiply Criterion given in Proposition \[ncMCrit\]. - $\#(F)$ is the number of unnecessary non-trivial obstructions detected by the Non-Commutative Leading Word Criterion given in Proposition \[ncLWCrit\]. - $\#(B_k)$ is the number of unnecessary non-trivial obstructions detected by the Non-Commutative Backward Criterion given in Proposition \[ncBKCrit\]. - $\rho=\#(Sel)/\#(Tot)$. Note that $\#(RGb)$ is an invariant of the ideal which only depends on chosen word ordering. Other numbers in the table rely also on the selection strategy. In our experiments we used the *normal strategy* which first chooses the obstruction whose S-polynomial has the lowest degree and then breaks ties by choosing the obstruction whose S-polynomial has the smallest leading word with respect to the word ordering. In these experiments, the Non-Commutative Tail Reduction given in Proposition \[ncTailRed\] detected no of unnecessary non-trivial obstruction (see Remark \[sec3rem9\]). Apparently, the Non-Commutative Multiply Criterion and the Non-Commutative Leading Word Criterion had already detected all unnecessary obstructions in the set $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ of newly constructed obstructions. The low ratios $\rho$ in the table indicate that the non-commutative Gebauer-Möller criteria we obtained can detect most unnecessary obstructions during the procedure. \[sec4exa2\] The following ideals [braid3]{} and [braid4]{} in the non-commutative polynomial ring $\mathbb{Q}\langle x_{1},x_{2},x_{3}\rangle$ are taken from [@SL09], Section 5. More precisely, [braid3]{} is the ideal generated by the set $\{-x_{2}x_{3}x_{1}+x_{3}x_{1}x_{3},$ $x_{2}x_{1}x_{2}-x_{3}x_{2}x_{3}$, $x_{1}x_{2}x_{1}-x_{3}x_{1}x_{2}, x_{1}^3+x_{1}x_{2}x_{3}+x_{2}^3+x_{3}^3\}$, and [braid4]{} is the ideal generated by the set $\{-x_{2}x_{3}x_{1}+x_{3}x_{1}x_{3}, x_{2}x_{1}x_{2}-x_{3}x_{2}x_{3}, x_{1}x_{2}x_{3}-x_{3}x_{1}x_{2}, x_{1}^3+x_{1}x_{2}x_{3}+x_{2}^3+x_{3}^3\}$. These ideals are generated by sets of homogeneous generators. The following table lists the results of the computations of Gröbner bases truncated at degree $11$ with respect to $\LLex$ on $\langle x_{1},x_{2},x_{3}\rangle$ such that $x_{1}>_{\LLex} x_{2}>_{\LLex}x_{3}$, via the Improved Buchberger Procedure. $\#(Gb)$ $\#(Tot)$ $\#(Sel)$ $\rho$ ----------- ---------- ----------- ----------- -------- braid3-11 726 289642 1663 0.0057 braid4-11 416 93252 1150 0.0123 The meaning of the symbols is the same as in Example \[sec4exa1\]. In this experiment we also used the normal strategy. Moreover, since we compute truncated Gröbner bases, we discard those obstructions whose S-polynomial have degrees larger than the degree of truncation. Thus the ratios $\rho$ in the table are lower than the ratios in the table of Example \[sec4exa1\]. Again, the non-commutative Gebauer-Möller criteria detect most unnecessary obstructions during the procedure. The experimental data in Examples \[sec4exa1\] and \[sec4exa2\] show that the generalizations of the Gebauer-Möller criteria presented in Propositions \[ncMCrit\], \[ncLWCrit\] and \[ncBKCrit\] can successfully detect a large number of unnecessary obstructions. In fact, they apparently detect almost all unnecessary obstructions during the Buchberger Procedure. Acknowledgements {#acknowledgements .unnumbered} ---------------- The second author is grateful to the Chinese Scholarship Council (CSC) for providing partial financial support. Both authors thank G. Studzinski for valuable discussions about non-commutative Gröbner bases. And both authors appreciate anonymous referees for careful reading and useful suggestions. [00]{} ApCoCoA team, ApCoCoA: Applied Computations in Commutative Algebra, available at http://www.apcocoa.org. G.H. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978), 178-218. B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal, Dissertation, Universität Inssbruck, Austria, 1965. B. Buchberger, A criterion for detecting unnecessary reductions in the construction of Groebner bases, in: Proceedings of the International Symposiumon on Symbolic and Algebraic Computation (EUROSAM ’79), Edward W. Ng (Ed.), Springer-Verlag, London, UK, 3-21, 1979. B. Buchberger, Gröbner bases: an algorithmic method in polynomial ideal theory, in: Multidimensional Systems Theory-Progress, Directions and Open Problems in Multidimensional Systems, N.K. Bose (Ed.), Reidel Publishing Company, Dodrecht-Boston-Lancaster, 184-232, 1985. M. Caboara, M. Kreuzer and L. Robbiano, Efficiently computing minimal sets of critical pairs, Journal of Symbolic Computation 38 (2004), 1169-1190. A.M. Cohen, Non-commutative polynomial computations, 2007, available at www.win.tue.nl/$\sim$amc/pub/gbnpaangepast.pdf. R. Gebauer and H.M. Möller, On an installation of Buchberger’s algorithm, Journal Symbolic Computation 6 (1988), 275-286. M. Kreuzer and L. Robbiano, Computational Commutative Algebra 1, Springer, Heidelberg, 2000. M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer, Heidelberg, 2005. H.M. Möller, A reduction strategy for the Taylor resolution, Proc. EUROCAL 85, Springer L.N. in Comp. Sci. 162, 526-534. T. Mora, Gröbner bases for non-commutative polynomial rings, in: Proceedings of the 3rd International Conference on Algebraic Algorithms and Error-Correcting Codes (AAECC-3), Jacques Calmet (Ed.), Springer-Verlag, London, UK, 1986, 353-362. T. Mora, An introduction to commutative and non-commutative Gröbner Bases, Journal of Theoretical Computer Science 134 (1994), 131-173. G. Rosenberger and M. Scheer, Classification of the finite generalized tetrahedron groups, Contemporary Mathematics 296 (2002), 207- 229. R. Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, Journal of Symbolic Computation 44 (2004), 1374-1393. X. Xiu, Non-Commutative Gröbner Bases and Applications, Dissertation, Universität Passau, Germany, 2012. [^1]: [martin.kreuzer@uni-passau.de]{} (Corresponding author) [^2]: [xingqiang.xiu@uni-passau.de]{}

--- abstract: 'Extensive work on single molecule magnets has identified a fundamental mode of relaxation arising from the nuclear-spin assisted quantum tunnelling of nearly independent and quasi-classical magnetic dipoles. Here we show that nuclear-spin assisted quantum tunnelling can also control the dynamics of purely emergent excitations: magnetic monopoles in spin ice. Our low temperature experiments were conducted on canonical spin ice materials with a broad range of nuclear spin values. By measuring the magnetic relaxation, or monopole current, we demonstrate strong evidence that dynamical coupling with the hyperfine fields bring the electronic spins associated with magnetic monopoles to resonance, allowing the monopoles to hop and transport magnetic charge. Our result shows how the coupling of electronic spins with nuclear spins may be used to control the monopole current. It broadens the relevance of the assisted quantum tunnelling mechanism from single molecular spins to emergent excitations in a strongly correlated system.' author: - 'C. Paulsen' - 'S. R. Giblin' - 'E. Lhotel' - 'D. Prabhakaran' - 'K. Matsuhira' - 'G. Balakrishnan' - 'S. T. Bramwell' title: Nuclear spin assisted quantum tunnelling of magnetic monopoles in spin ice --- Introduction {#introduction .unnumbered} ============ In the canonical dipolar spin ice materials ([Dy$_{2}$Ti$_{2}$O$_{7}$]{}, [Ho$_{2}$Ti$_{2}$O$_{7}$]{}) [@Harris1997; @BramwellHarris98; @Ramirez; @BramwellGingras], rare earth ions with total angular momentum $J = 15/2$ (Dy$^{3+}$) and $J = 8$ (Ho$^{3+}$) are densely packed on a cubic pyrochlore lattice of corner-linked tetrahedra. The ions experience a very strong $\langle111\rangle$ crystal field, resulting in two effective spin states ($M_{\rm J} = \pm J$) that define a local Ising-like anisotropy. At the millikelvin temperatures discussed here ($0.08~{\rm K}<T<0.2$ K), a lattice array of such large and closely-spaced spins would normally be ordered by the dipole-dipole interaction [@LuttingerTisza], but the pyrochlore geometry of spin ice frustrates the dipole interaction and suppresses long-range order. Instead, the system is controlled by an ice-rule, that maps to the Pauling model of water ice [@Harris1997; @BramwellHarris98; @Ramirez; @BramwellGingras]. In the effective ground state, the spins describe a flux with closed-loop topology and critical correlations, that may be described by a local gauge symmetry rather than by a traditional broken symmetry [@CMS]. This strongly correlated spin ice state is stabilised by a remarkable self-screening of the dipole interaction [@Isakov; @Gingras]. Excitations out of the spin ice state fractionalise to form effective magnetic monopoles [@CMS; @Ryzhkin], but the excited states are no longer self-screened and this manifests as an effective Coulomb interaction between monopoles. The static properties of spin ice are accurately described by the monopole model [@KaiserDH]. The dynamic properties can also be described by assuming an effective monopole mobility [@JaubertHoldsworth; @Bovo; @Paulsen_Wien], but there have been few studies of the microscopic origin of the monopole motion [@Tomasello]. {width="10cm"} The field and energy scales involved in monopole motion are illustrated in Fig. 1, a-e. When a monopole hops to a neighbouring site a spin is flipped (Fig. 1a). For an isolated monopole (far from any others) this spin flip takes place at nominally zero energy cost (Fig. 1b) because contributions from near-neighbour antiferromagnetic superexchange and ferromagnetic dipole–dipole coupling individually cancel. The cancellation of the field contribution relies on the dipolar self-screening [@Isakov] that maps the long-range interacting system [@Gingras] to the degenerate Pauling manifold of the near neighbour spin ice model [@BramwellHarris98]. This surprising cancellation is a key result of the many–body physics of spin ice. In practice, a monopole hop may also involve a finite energy change arising from longitudinal fields at the spin site: the main source of fields is nearby monopoles [@CMS] (Fig. 1b), while further contributions arise from corrections to the mapping, which give a finite energy spread to the Pauling manifold [@Vedmedenko] (here of order $\sim$0.1 K [@Melko]). The mechanism of the hop is believed to be quantum tunnelling and several key signatures of this have been observed in the high temperature regime between 2 K and 10 K [@Ehlers; @Snyder; @JaubertHoldsworth; @Bovo; @Tomasello]. At lower temperatures ($T < 0.6$ K), spin ice starts to freeze [@Snyder]. This is due in part to the rarefaction of the monopole gas whose density $n(T)$ varies as $\sim e^{-|\mu|/T}$ where the chemical potential $|\mu| = 4.35 $ and 5.7 K for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} respectively [@CMS], and also in part to geometrical constraints that create noncontractable, monopole-antimonopole pairs that cannot easily annihilate [@noncontractable]. These factors, which are independent of the monopole hopping mechanism, suggest that the relaxation rate $\nu(T) \propto n(T)$ will fall to exponentially small values at low temperature ($T~<~ 0.35$ K). Previous thermal quenching experiments have demonstrated monopole populations well below the nominal freezing temperature that are both long lived and able to mediate magnetic relaxation [@PaulsenAQP]. This paradoxical frozen but dynamical character of the system suggests the relevance of resonant magnetic tunnelling, where magnetisation reversal can only occur when the longitudinal field is smaller than the tunnelling matrix element $\Delta E$. The monopolar fields may add a longitudinal component that takes the spin off the resonance condition (Fig. 1c,d) but in addition may add a transverse component that amplifies $\Delta E$: together these lead to a suppression and dispersion of the monopole mobility. In the following, we will demonstrate experimentally that hyperfine interactions (Fig. 1e) play a significant role in bringing monopoles back to their resonance condition, enabling dynamics at very low temperatures ($T~<~0.35$ K). Results {#results .unnumbered} ======= [**Samples**]{} $I$ -- ------------------------------------- ------ -------- -------- --------- ---------------------- -- 7/2 4.17 0.3 0.034 5.7 1$\times$10$^{-5} $ $\approx$ 19 $\%$, $^{161}$Dy 5/2 -0.48 -0.0039 $\approx$ 25 $\%$, $^{163}$Dy 5/2 0.67 0.0054 $\approx$ 66 $\%$, $^{\rm other}$Dy 0 0 0 0 0 0 0 5/2 0.67 0.0599 0.0054 To investigate the effect of nuclear spins on the magnetic relaxation in spin ice, we studied four spin ice samples: [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, with $I = 7/2$ and three [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples spanning a range of nuclear spin composition from $I=0$ to $I=5/2$. Details of nuclear spins and hyperfine parameters are given in Table 1. Ho$^{3+}$ is a non-Kramers ion with intrinsically fast dynamics owing to the possibility of transverse terms in the single-ion spin Hamiltonian, while Dy$^{3+}$, being a Kramers ion, has intrinsically much slower dynamics. However, it should be noted that, at low temperature, bulk relaxation is slower in [Ho$_{2}$Ti$_{2}$O$_{7}$]{} than in [Dy$_{2}$Ti$_{2}$O$_{7}$]{}, owing to its larger $|\mu|$ and hence much smaller monopole density (See Supplementary Fig. 1). [**Thermal protocol**]{} ![Controlled cooling of spin ice below its freezing temperature. How the temperature of the samples varies during and after the AQP: (a) The applied field (black) during an AQP, and the temperatures measured by a small thermometer glued directly on top of the samples (schematically shown in (b)) vs log time for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} (HTO, red) and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} (DTO, blue). The inset shows a zoom of the first 6 seconds vs time. (b) Comparison of the sample cooling rates $dT/dt$ as a function of temperature after the AQP for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  from the data in (a) to the Òequilibrium cooling rateÓ $dT/d \tau$ extracted from ac susceptibility data for the two samples (see Supplementary Fig. 6). The cooling rate for [Ho$_{2}$Ti$_{2}$O$_{7}$]{}  crosses the equilibrium rate at $\sim 0.9$ K, and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  at 0.72 K.[]{data-label="two"}](figure2.pdf){width="7.5cm"} In previous experiments we have accurately manipulated the monopole density in [Dy$_{2}$Ti$_{2}$O$_{7}$]{} by rapid magnetothermal cooling (Avalanche Quench Protocol, AQP) the sample through the freezing transition, allowing the controlled creation of a non-equilibrium population of monopoles in the frozen regime [@PaulsenAQP]. However it is more problematic to cool samples containing Ho, due to the large Ho nuclear spin which results in a Schottky heat capacity of 7 J mol$^{-1}$ K$^{-1}$ at 300 mK. Indeed this anomaly has been exploited by the Planck telescope where the bolometers are attached to the cold plate by yttrium-holmium feet thus allowing passive filtering with a several hour time constant that was crucial to the operation of the system [@PrivateComm]. For [Ho$_{2}$Ti$_{2}$O$_{7}$]{} this means difficulty in cooling. Therefore during some of the runs the sample temperature was recorded via a thermometer directly mounted on the sample face. Fig. 2a shows the monitoring of the sample temperature as it approaches equilibrium for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} and [Dy$_{2}$Ti$_{2}$O$_{7}$]{}  during and after the AQP. The inset of Fig. 2a shows that only a few seconds are required to cool the samples from 0.9 K to 0.2 K, which is well below the freezing transition. Whereas [Dy$_{2}$Ti$_{2}$O$_{7}$]{} continues to cool, reaching 80 mK after only 10 s, [Ho$_{2}$Ti$_{2}$O$_{7}$]{} takes nearly 2000 s to reach the same temperature. Hence the data shown here were taken at 80 mK for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and 200 mK (and 80 mK when possible) for [Ho$_{2}$Ti$_{2}$O$_{7}$]{}. [**Monopole density**]{} We have phenomenologically estimated how the monopole density depends upon the rate of sample cooling, $dT/dt$ and the spin relaxation time $\tau(T) = 1/\nu(T)$, which is derived from the peaks in the imaginary component of the ac susceptibility. Differentiation of $\tau(T)$ to give $d\tau/dT$ and hence $dT/d\tau$, allows definition of an equilibrium cooling rate $dT/d\tau$, that gives the maximum cooling rate that may still maintain equilibrium. Fig. 2b compares $dT/dt$ and $dT/d \tau$ for both [Dy$_{2}$Ti$_{2}$O$_{7}$]{}  and [Ho$_{2}$Ti$_{2}$O$_{7}$]{}. It can be seen that after the AQP, $dT/dt$ for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} crosses the equilibrium curve and goes out of equilibrium at $\approx 0.9$ K, and for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} at $\approx 0.72$ K. The upper limit of the monopole density at low temperature can be estimated by equating it to the theoretical value at the crossing temperature: thus we find one monopole on approximately every $10^3$ tetrahedra for both [Ho$_{2}$Ti$_{2}$O$_{7}$]{} and [Dy$_{2}$Ti$_{2}$O$_{7}$]{}. [**Spontaneous relaxation**]{} We studied the effect of wait time $t_{\rm w}$ between the end of the avalanche quench and the application of the field with the aim to determine the effect of nuclear spins on the monopole dynamics. Varying the wait time deep in the frozen regime allowed us to gauge the spontaneous evolution of the zero-field monopole density as a function of time: that is, if monopoles recombine in a time $t_{\rm w}$, then the observed monopole current will be smaller, the longer the wait time. Two separate experiments were designed to study these effects. In the first experiment (Fig. 3) after waiting we applied a constant field and measured the magnetization $M$ as a function of time. In the second experiment (Fig. 4) we investigated the effect of wait time on the magnetothermal avalanches [@Slobinsky; @Jackson; @Krey2012] that occur on ramping the field to high values. Both of these allowed access to the magnetic current density $J_{\rm m} = dM / dt$. Full details of the experimental conditions are given in Supplementary Note 1 and Supplementary Fig. 2. The monopole current is controlled by multiple factors. In the simplest model [@Ryzhkin] there are three of these: the monopole density $n$, the monopole mobility $u$ (related to the spin tunnelling rate) and the bulk susceptibility $\chi$. Thus $J_{\rm m} = dM/dt = \nu (M_{\rm eq} - M)$ where $M_{\rm eq} = \chi H$ is the equilibrium magnetisation and $\nu \propto u n$. In general it is difficult to deconvolve these various factors. In Ref. it was achieved by independent measurement of $n(T)$ and $\chi(T)$ to reveal $u(T)$. In the present time-dependent experiments we cannot perform such a direct separation, but by studying [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples with different isotopes, it seems reasonable to assume that the susceptibility and starting density are roughly the same, so the variation in mobility (hop rate) will dominate differences between the samples. Inclusion of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} in the comparison gives a further point of reference: the starting monopole densities (see above) and susceptibilities for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} are expected to be comparable to those of [Dy$_{2}$Ti$_{2}$O$_{7}$]{}, while the the tunnel splitting (which controls the intrinsic mobility) is also estimated to be of the same order [@Tomasello] in the appropriate range of internal fields (see Fig. 1 and Ref. , Fig. 5). {width="12cm"} Fig. 3 summarizes results for the relaxation of the magnetization $M(t)$ for the different samples, as well as the value of $M(t = 400~{\rm s})$ and the monopole current $J_{\rm m}(t = 0)$ as a function of wait time, for a constant applied field of 0.08 T. The [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples show a clear progression in wait time effect that correlates strongly with their relative densities of nuclear spin states. Thus the monopoles recombine during the wait period much more effectively the larger the nuclear spin: that is, the larger the nuclear spin the higher the monopole mobility, the faster the recombination, and the fewer the monopoles at the start of the measurement. In Fig. 3e, f, higher mobility means the relaxation curves ($M(t = 400~{\rm s})$ and $J_{\rm m}(t = 0)$) shift both up and to the left, so a crossover in curves is expected – and this is indeed observed at the longer times. Near to equilibrium a second crossover would be expected (i.e. the equilibrium current density is higher for the highest mobility), but this crossover is clearly very far outside our time window. Hence our [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples are always far from equilibrium. The effects observed for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} are yet more dramatic in [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, consistent with the Ho$^{3+}$ non-Kramers character, large nuclear spin, and large hyperfine coupling. Relaxation at 200 mK covers more than two orders of magnitude but is practically extinguished for long wait times, showing that excess monopoles spontaneously recombine to eliminate themselves from the sample. The plots indicate that the half life for monopole recombination in [Ho$_{2}$Ti$_{2}$O$_{7}$]{} would be approximately 150 seconds (much shorter than the equilibrium relaxation time) and suggests that equilibrium in the monopole density is reached at long times. Using the above estimate for the initial monopole density $n(t = 0) \sim 10^{-3}$, we recover a nominal equilibrium density of $n_{\rm eq} = 10^{-5}$ (per rare earth atom). Although this estimate is an upper limit it is nevertheless far from the expected equilibrium density, $n_{\rm eq}^{\rm 200~mK} \sim 10^{-13}$ (calculated by the method of Ref. , see Supplementary Fig. 1). It continues to evolve with temperature, being lower by a further order of magnitude at $T = 80$ mK (Fig. 3f). Most likely, the actual equilibrium monopole density is amplified by defects and disorder in the sample. [**Magnetothermal avalanches**]{} {width="12cm"} Fig. 4 illustrates the effect of $t_{\rm w}$ on the magnetothermal avalanches. These occur when the injected power ($\mu_0 H \cdot J_{\rm m}$) overwhelms the extraction of thermal energy from the sample to the heat bath [@Jackson] such that monopoles are excited in great excess as the temperature steeply rises. The faster and more abundant the monopoles, the lower the avalanche field. To obtain the data in Fig. 4, after the AQP and $t_{\rm w}$, the applied field was swept at a constant rate, 0.02 T.s$^{-1}$ up to 0.4 T. If the avalanche field $H_{\rm ava}(t_{\rm w})$, is defined as the field where the magnetization crosses 1 $\mu_{\rm B}$ per rare earth ion, (0.5 $\mu_{\rm B}$ for the $^{163}$Dy sample) then the difference in avalanche field $\Delta H_{\rm ava}= H_{\rm ava}(t_{\rm w}) - H_{\rm ava} (t_{\rm w}={\rm minimum})$ allows us to compare the spread of fields for all samples. Fig. 4a and b show the experimental results for the isotopically enriched [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples at 80 mK, demonstrating a very clear pattern. In general the spread of $H_{\rm ava}(t_{\rm w})$ becomes larger, the larger the nuclear spin, showing again that the nuclear spins strongly enhance the monopole mobility. Thus, the $^{162}$Dy sample (no nuclear spin) shows negligible evolution of the position of the avalanche field. For $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  (shown in Supplementary Fig. 4) the effect is small, while for the $^{163}$Dy sample (maximum nuclear spin) the effect of $t_{\rm w}$ can be clearly seen as a steady progression of $H_{\rm ava}(t_{\rm w})$ to higher fields for increasing $t_{\rm w}$ due to the smaller initial monopole density at the start of the field ramp. Also shown in the figure are the curves that result from slow conventional zero field cooling (CC) from 900 mK to 80 mK (at 1 mK.s$^{-1}$) followed by a 1000 s wait period. For the $^{162}$Dy and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples the CC avalanche field is offset to higher fields, well outside the distribution of $H_{\rm ava}(t_{\rm w})$. For the $^{163}$Dy sample the CC curves falls within the distribution but near the long wait time curves. Also, we note for $^{163}$Dy, which happens to have better thermal contact, and thus faster cooling during the AQP, the CC curve again falls outside the distribution (shown in Supplementary Fig. 10 b). Thus slow cooling is more efficient at approaching equilibrium in [Dy$_{2}$Ti$_{2}$O$_{7}$]{} than is the AQP cooling followed by a long $t_{\rm w}$, especially for the low nuclear moment samples. This is typical behaviour for frustrated or disordered systems because slow cooling allows the system time to explore all available phase space. Fig. 4c shows a much greater effect of $t_{\rm w}$ for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} with a larger spread of fields, saturating near 0.32 T for the the longest $t_{\rm w}$. This is again consistent with the conclusion that that the larger the nuclear spin moment, the more effective the spontaneous monopole recombination. The measurements were performed primarily at 200 mK, but the same conclusion follows from measurements at 80 mK. [Ho$_{2}$Ti$_{2}$O$_{7}$]{} also exhibits some unusual behaviour suggesting that the monopole density and magnetization do not approach equilibrium in a simple way. First, the magnetization jumps fall short of the $M$ vs $H$ equilibrium curve taken at 900 mK, even though thermometers placed on the sample indicate that the sample does indeed heat above 900 mK (see Supplementary Note 2 and Supplementary Fig. 3 for more details). Secondly, in contrast to the behaviour of [Dy$_{2}$Ti$_{2}$O$_{7}$]{} discussed above, the CC curve of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} falls in the middle of the distribution of $H_{\rm ava}$($t_{\rm w}$) indicating, unusually, that waiting long enough at low temperature is an equally efficient way of approaching equilibrium as slow cooling. Discussion {#discussion .unnumbered} ========== The experimental result demonstrated here is that magnetic monopole dynamics in the frozen regime of spin ice are greatly enhanced by the hyperfine coupling of the electronic and nuclear moments. We now argue that this observation finds a natural – albeit surprising – explanation by analogy with the properties of single-molecule magnets [@review]. These are metal-organic clusters with large composite spins: some of the most studied include the so called Mn$_{12}$ and Fe$_8$ systems, both of which can be thought of as an ensemble of identical, weakly interacting nanomagnets of net spin S = 10 with an Ising-like anisotropy. The degenerate $M_{\rm s} = \pm S$ states are split by the ligand electric field into a series of doublets. At temperatures smaller than the level separation, the spins flip by resonant tunnelling through a quasi-classical barrier. The signature of a resonant tunnelling effect in Fe$_8$ is a peak in the low temperature relaxation rate around $H = 0$ [@Sa]. It quickly became clear that to understand the resonant tunnelling both dipolar and dynamic nuclear spin contributions to the interactions need to be accounted for. The typical dipolar field in such a system is $\approx 0.5$ K, and the relevant tunnel splitting $\Delta E$ of the order 10$^{-8}$ K, meaning that a broad distribution of dipolar field and a static hyperfine contribution would force all the spins off resonance. Prokof’ev and Stamp [@PS] proposed that dynamic nuclear fluctuations can drive the system to resonance, and the gradual adjustment of the dipole fields in the sample caused by tunnelling, brings other clusters into resonance and allows a continuous relaxation. Hence the observation of relaxation in single molecule magnets is fundamentally dependent on the hyperfine coupling with the fields of nuclear spins [@Wernsdorfer00]. The Prokof’ev and Stamp model [@PS] certainly does not apply in detail to spin ice at low temperatures. First, in single molecule magnets the spin of any particular complex in the system is available to be brought to resonance, whereas in spin ice, only those spins that are instantaneously associated with a diffusing monopole are available to tunnel (and this presumes that more extended excitations can be neglected). The remaining spins – the vast majority – are, in contrast, static and instantaneously ordered by the ice rules. The rate of flipping of these quasi-ordered spins, which corresponds to monopole pair creation, is negligible at the temperatures studied and the process is not relevant to our experiments. Thus, even at equilibrium, spin ice has an effective number of flippable spins that depends on temperature (see Supplementary Fig. 1). Away from equilibrium, where our experiments are performed, the number of flippable spins in spin ice further depends on time, with monopole recombination depleting their number. In addition, it seems reasonable to assume that the reduction of the density of monopoles is even more important during the relaxation process; as monopoles move through the matrix magnetizing the sample they will annihilate when they encounter a monopole of opposite charge, or become trapped on a defect or on the sample surface. This feature of spin ice is a second important difference with single molecule magnets, as modelled in Ref. 27. A third difference relates to the distribution of internal fields in the system. In spin ice only, the actual field associated with a flippable spin, both before and after a flip, is a monopolar field. Flipping a spin transfers a monopole from site to site (Fig. 1a), dragging the monopolar field with it: a field that is much stronger and of longer range than any conventional dipole field. However, the change in field on a spin flip is dipolar, as in single molecule magnets. In short, the flippable spins in spin ice are really an aspect of the emergent monopole excitation rather than a perturbed version of an isolated (composite) spin as assumed for the single molecule magnets in Ref. . Yet despite this difference, it seems reasonable to suggest that the basic idea of Ref. does apply to spin ice. The longitudinal monopolar fields will take flippable spins off resonance (Fig. 1c-e), while the transverse ones will tend to broaden the resonance well beyond the tunnel splitting calculated for an isolated spin. i.e. $\Delta E= 10^{-5}$ K [@Tomasello]. An applied field can also take flippable spins on or off resonance or broaden the resonance, depending on its direction. Nevertheless, in zero applied field, at very low temperatures we would expect all flippable spins associated with isolated monopoles to be off resonance and hence unable to relax, unless they are brought back to resonance by a combination of the monopole fields and the fluctuating nuclear spins: nuclear assisted flipping of spins will then bring further spins to resonance via the change in dipolar fields, as in the Prokof’ev-Stamp picture [@PS]. Our experimental results for the wait time dependence of various properties clearly support this proposition: in zero field (during $t_{\rm w}$) the sample with no nuclear spin is scarcely able to relax its monopole density, while the larger the nuclear spin, the quicker the relaxation. For flippable spins associated with closely–spaced monopole-antimonopole pairs the situation is slightly different. Although they are strongly off-resonance (Fig. 1b), the decreasing transition matrix elements will be compensated by the increasing Boltzmann factors required for detailed balance. Also, for the final recombination, a favourable change in exchange energy will reduce the field required to bring spins to resonance (see caption, Fig. 1). We note in passing that the differences between single molecule magnets and spin ice are also evident in our data. Specifically, a $t^{1/2}$ initial relaxation of the magnetisation is a property of single molecule magnets, with the $t^{1/2}$ form arising from the dipole interactions  [@PS; @Pauling-t-half]. Given the very unusual field distribution in spin ice, and the complicating factor of monopole recombination, as described above, it is hardly likely that this functional form will apply. We test for a $t^{1/2}$ decay in the Supplementary Fig. 5 and confirm that it can only be fitted over a narrow time range: to calculate the true time dependence in spin ice poses a theoretical problem. Our main result has implications for both the theory of spin ice and the theory of nuclear spin assisted quantum tunnelling. First, in previous work [@Paulsen_Wien] we have shown how the low-temperature quenched monopole populations of [Dy$_{2}$Ti$_{2}$O$_{7}$]{} obey the nonlinear and non-equlibrium response of monopole theory [@Kaiser2] that was developed assuming a single hop rate. In view of our findings, the theory should apply most accurately to the [Dy$_{2}$Ti$_{2}$O$_{7}$]{}  sample with no nuclear spins and least accurately to [Ho$_{2}$Ti$_{2}$O$_{7}$]{}  where the hyperfine splitting energies are of a similar order to the Coulomb energies. In other measurements, presented in Supplementary Figs. 7 and 8, we confirm that this is the case; hence a generalisation of the theory of Ref. to include the effect of nuclear spins seems an attainable goal. We also note that [Ho$_{2}$Ti$_{2}$O$_{7}$]{} offers the unusual situation that, at low temperatures ($<$ 0.35 K) and sufficient wait times, the nuclear spins are ice-rule ordering antiparallel to their electronic counterparts; hence spin ice offers a rare chance to investigate the effect of correlation on nuclear spin assisted quantum tunnelling in a controlled environment. Perhaps this will shed light on some of the unusual properties particular to [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, as noted above. Spin ice thus exemplifies a remarkable extension of the concept of nuclear spin assisted quantum tunnelling [@PS] to the motion of fractionalised topological excitations [@CMS]. This is made possible by the fact that the emergent excitations of the system – the monopoles – are objects localised in direct space that move through flipping spins. As well as illustrating this generic point, our result may also have practical consequences. We have established how coupling with nuclear spins controls the magnetic monopole current and the spectacular magnetothermal avalanches: hence any experimental handle on the nuclear spins of the system would also be a rare experimental handle on the monopole current. Any future application of magnetic monopoles in spin ice will surely rely on the existence of such experimental handles. Methods {#methods .unnumbered} ======= [*Samples.* ]{}Single crystals were grown by the floating zone method for all samples, the natural $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{}  samples (DTO, HTO) were prepared at the Institute of Solid State Physics, University of Tokyo, Japan, and $^{162}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}, $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} at Warwick University and Oxford University respectively. [*Measurements.* ]{} Measurements were made using a low temperature SQUID magnetometer developed at the Institut Néel in Grenoble. The magnetometer is equipped with a miniature dilution refrigerator with a base temperature of 65 mK. The fast dynamics after a field change were measured in a relative mode, the slower measurements were made by the extraction method, and the initial relative measurements were adjusted to the absolute value extraction points. The field could be rapidly changed at a rate up to 2.2 T s$^{-1}$. For all the data shown here the field was applied along the \[111\] crystallographic direction. Measurements were also performed perpendicular to the \[111\] direction, as well as along the \[001\] and \[011\] directions and on a polycrystalline sample, examples of which are discussed in Supplementary Note 3. In total ten different samples were studied. The direction of the applied field as well as differences in the sample shapes and thermal contact with the sample holder can effect some of the details of the measurements. However this does not change the main conclusion of the paper: the demonstration of the importance of nuclear assisted quantum tunnelling to the relaxation. The measurements of temperature vs time shown in Fig. 2, a bare-chip Cernox 1010-BC resistance thermometer from LakeShore Cryogenics was wrapped in Cu foil and glued on top of the sample as shown in the inset of Fig. 2b. Cooling [Ho$_{2}$Ti$_{2}$O$_{7}$]{} was difficult and warming was also tricky using the AQP, depending on the initial temperatures and wait times. Therefore to ensure the sample was heated above 900 mK, two AQP were used, separated by 300 seconds, which explains why the starting temperature for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} was higher in Fig. 2 (see Supplementary Note 2 and Supplementary Fig. 3 for further discussions). A schematic of the AQP used for the preparation of the samples is shown in Fig. 3d. First a field of $-0.3$ T was applied and the sample was allowed to cool to base temperature for 20 minutes. The field was then reversed at 2.2 T s$^{-1}$ to $+ 0.3$ T for 4 s then reduced to zero. After a wait period ranging from 10 to 50,000 s, a field of 0.08 T was applied and the relaxation of the magnetization was recorded. The field $B= 0.08$ T was chosen because it is large enough to get sizeable relaxation, but small compared to the avalanche fields shown in Fig. 4. In this way, when applying the magnetic fields, the relaxation is well behaved and the sample does not heat. The AQP used for the data of Fig. 4 was similar to the above, except the avalanche field was $\pm 0.4$ T. After the wait period the field was ramped at 0.02 T s$^{-1}$ while the magnetization and temperature of the sample were continuously recorded. For the slow CC protocol measurements shown in Fig. 3, the samples were first heated to 900 mK for 10 s, then cooled at a rate of approximately 0.01 K s$^{-1}$, followed by a waiting period of 1000 s. Data availability {#data-availability .unnumbered} ================= Information on the data underpinning the results presented here, including how to access them, can be found in the Cardiff University data catalogue at http://doi.org/10.17035/d.2019.0069144874.\ The datasets obtained and/or analyzed in this study are also available from the corresponding author on reasonable request. [99]{} Harris, M. J., Bramwell, S. 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Wernsdorfer, W., Caneschi, A., Sessoli, R., Gatteschi, D., Cornia, A., Villar, V., & Paulsen, C. Effects of Nuclear Spins on the Quantum Relaxation of the Magnetization for the Molecular Nanomagnet Fe$_8$. [*Phys. Rev. Lett.*]{} [**84**]{}, 2965 – 2968 (2000). Ohm, T., Sangregorio, C. & Paulsen, C. [*J. Low Temp. Physics*]{} [**113**]{}, 1141 – 1146 (1998). Kaiser, V., Bramwell, S. T., Holdsworth, P. C. W. & Moessner, R. ac Wien Effect in Spin Ice, Manifest in Nonlinear, Nonequilibrium Susceptibility. [*Phys. Rev. Lett.*]{} [**115**]{}, 037201 (2015). [**Acknowledgements:**]{} S.R.G. thanks Cardiff University for ‘seedcorn’ funding, and acknowledges the EPSRC for EP/L019760/1. S.T.B. thanks Patrik Henelius for communicating his independent ideas on nuclear assisted quantum tunnelling in spin ice, and acknowledges the EPSRC for EP/S016465/1. E.L. and C.P. acknowledge financial support from ANR, France, Grant No. ANR-15-CE30-0004. G.B. wishes to thank financial support from EPSRC, UK, through grant EP/M028771/1.\ [**Author Contributions:**]{} The experiments were designed and performed by C.P. with inputs and discussions from E.L. and S.R.G. The data were analyzed by C.P., E.L., S.R.G., and S.T.B. Contributed materials were fabricated by K.M., D.P. and G.B. The paper was written by C.P., E.L., S.R.G., and S.T.B. [**Nuclear spin assisted quantum tunnelling of magnetic monopoles in spin ice\ Supplementary Information**]{} ![\[fig1SI\] [**Density of single-charge monopoles (equilibrium number per diamond lattice site in zero field) versus temperature**]{}, calculated by the Debye-Hückel theory of Kaiser et al. [@KaiserDH]. The analytic calculation is very accurate for the monopole model of spin ice: it includes both single and double-charge monopoles, but only the single-charge monopoles are relevant at the temperatures we study. The density of ‘flippable’ spins per spin site is $3/2$ times the monopole density.](Plot_SI_nvsT.pdf){width="8cm"} Experimental details -------------------- [**Creating a large density of monopoles using the avalanche quench protocol in [Ho$_{2}$Ti$_{2}$O$_{7}$]{}.**]{} We have previously described the avalanche quench protocol (AQP) in detail for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} (see Ref.  and its Supplementary Information). From magnetisation measurements recorded during and after the AQP, we inferred that samples of [Dy$_{2}$Ti$_{2}$O$_{7}$]{} heat systematically to temperatures above 900 mK, even though the reference thermometer on the sample holder only registered a small jump [@Jackson14]. However from the onset, it was clear that [Ho$_{2}$Ti$_{2}$O$_{7}$]{} was different. For example, when performing measurements where the field is ramped at a steady rate, the magnetic avalanches of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} never reach the 900 mK equilibrium value as seen in Fig. 4c. Sometimes depending on previous measurements, the AQP worked very poorly, or did not seem to work at all. This was the motivation for measuring the sample temperature directly by mounting a thermometer on the samples during some of the runs. [**Temperature measurements during the AQP.**]{} We attempted direct temperature measurements with 4 different thermometers; 2 homemade RuO$_2$ resistance thermometers (filed down to reduce mass with wires attached with silver epoxy), a Cernox 1010-SD thermometer with platinum leads, and a bare-chip Cernox 1010-BC, both from LakeShore Cryogenics. All had short comings but in the end most of the measurements shown here were made using the bare chip thermometer. This was the lightest of the four, but had a small but noticeable magnetoresistance that we have corrected for. A constant current source delivered 10 nA, and the voltage was measured with a Stanford Instruments model 830 lock-in amplifier running at 1100 Hz. This setup was a compromise, the measurements of the temperature were fast, but prone to some drift and noise. During normal measurements samples are sandwiched between two long narrow pieces of Cu that are anchored to the mixing chamber of a miniature dilution refrigerator. The samples are glued in place, then teflon tape is tightly wrapped around the Cu strips, clamping the samples to the Cu. For measurements with the thermometer glued to the sample, only one Cu strip was used, the second was suspended away from the thermometer, as shown in Supplementary Figure \[fig1SI\]. ![\[fig1SI\] [**Pictures of the sample mounting**]{}. Upper left: Sample of [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and Cernox bare chip resistor with leads protected by kapton tape. Upper right: [Ho$_{2}$Ti$_{2}$O$_{7}$]{} glued onto the bare chip resistor with thin Cu foil protruding. Lower left: the sample + thermometer have been turned over, and the sample surface has been glued onto the Cu sample holder using GE varnish. The Cu foil has been folded back to cover the upper half of the sample and bare chip resistor. Lower right: standoffs hold upper part of sample holder away, teflon tape has been wrapped around the sample and thermometer. The thermometer is isolated from the Cu sample holder by the sample.](figure1_SI.pdf){width="8.5cm"} Results ------- As already discussed, cooling samples that contain Ho can be problematic, and this is also true for heating samples with the AQP when the starting temperature was well below 200 mK. This is shown in Supplementary Figure \[fig2SI\] for a series of AQP taken on [Ho$_{2}$Ti$_{2}$O$_{7}$]{} when the sample was first cooled to 65 mK after waiting 4 hours. Point (a) is the beginning of sequence when we applied a field of $-0.3$ T on the sample followed by a wait period of 180 s. For [Dy$_{2}$Ti$_{2}$O$_{7}$]{}, already a large $>1$ K spike in temperature would be seen at (a) as the sample rapidly magnetizes in the field, but for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} only a small jump of about 0.3 K was recorded. The jump in temperature at the first AQP is also small, only reaching about 0.6 K, compared to $>1.4$ K for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} under similar conditions. At (c) we begin a second AQP, again setting $\mu_0 H=-0.3$ T, and waiting 180 s. But this time the jump in temperature reaches nearly 0.8 K, and the second AQP at (d) shows that now, the sample has warmed $>1.3$ K. During the AQP, heat from the flipping of the electronic spins is absorbed by the sample. But because of the large heat capacity of the Ho nuclei, much of the energy is absorbed by the nuclear spin bath, raising its temperature but resulting in a small overall jump in sample temperature. However for the second AQP, the starting sample temperature is now greater, nearly 0.16 K, and this is enough to heat the spins above one Kelvin. Thus [Ho$_{2}$Ti$_{2}$O$_{7}$]{} measurements were systematically made with 2 or 3 AQP in succession in order to ensure the sample is warmed above 1 K. Note that the need for several AQP also suggests that at low temperature, well below 300 mK, the nuclear spins begin to freeze out, and anti-align with their respective electronic spin, thus 2 in – 2 out for the electronic spin becomes 2 out – 2 in for its nuclear counterpart. ![\[fig2SI\] [**The applied field and the sample temperature for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} as a function of running time for a double AQP.**]{} The sample was cooled for 4 hours to base temperature of approximately 65 mK. (a) At $t=0$, the field was changed from 0 to $-0.3$ T. A relatively weak jump in the temperature can be seen. (b) At $t=180$ s, the first AQP is performed: the field goes from $-0.3$ to $+0.3$ T, then after 4 s from $+0.3$ T to zero. The temperature on the sample reaches about 0.55 K, not sufficient to randomize the spins. (c) At $t=200$ s the field is again put at $-0.3$ T in preparation for the next AQP. This time the jump in sample temperature is larger, but still less than required. (d) At $t=380$ s the second AQP takes place, warming the sample above 1 K. ](figure2_SI.pdf){width="8cm"} Another nagging problem was that the samples of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} did not reach $M=0$ after the AQP. The origin of this is not clear, but our data suggests that [Ho$_{2}$Ti$_{2}$O$_{7}$]{} cools too fast. When the field is switched off, the applied field $H$ goes to zero before the sample even starts to change its magnetisation. The sample then feels the internal field $H_{\rm internal}=-D\cdot M$ where D is the demagnetisation factor and avalanches against this. As the magnetisation decreases, $H_{\rm internal}$ also decreases, the sample heats, but then cools so rapidly that the magnetisation gets ‘stuck’ at a small positive value of the order 1 emu/g. For convenience, the solution was to add a small overshoot for the field of about $-0.004$ T for 1 s, then switch back to $H=0$. This resulted in a starting $M$ closest to zero. Note that measurements without the overshoot gave the same results, but with an offset. This ultra rapid cooling may also explain why the avalanches shown in Fig. 4c for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} (while ramping of the field) fall below the equilibrium value expected for 900 mK, in contrast to [Dy$_{2}$Ti$_{2}$O$_{7}$]{}. ![\[fig3SI\] [**Plots for natural [Dy$_{2}$Ti$_{2}$O$_{7}$]{} corresponding to Figs. 3 and 4 of the main manuscript.**]{} (a) Avalanches of the magnetisation recorded while the field was ramped at 0.02 T/s for $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} (DTO). The samples were first prepared using the AQP and then followed by various wait times except for the curve marked ‘ZFC’, where the sample was first prepared using the conventional zero field cooled (CC) protocol (red circles). Also shown is the equilibrium $M$ vs $\mu_0 H$ taken at 900 mK (solid black dots). (b) Magnification, showing the spread in avalanche fields as a function of the wait time, and the ZFC far outside the pack. (c) The effect of wait time on the relaxation of the magnetisation $M$ vs time for $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} measured at 80 mK. The samples were again first prepared using the AQP. After the specified wait periods, a field of 0.08 T was applied and the magnetisation as a function of time was recorded.](figure3_SI.pdf){width="\textwidth"} ![\[fig5SI\] [**Time dependence of the magnetisation at short times**]{}. The same data of Fig. 3 in the main text as well as data for the $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} sample plotted against $\sqrt{time}$. An important prediction from ProkofÕev and Stamp [@PS] was that the initial relaxation of the magnetisation should follow a square root time dependence. This worked well for the SMM Fe$_8$ up to 1000 s or more. The situation for spin ice is quite different, the right hand side of the figure shows the data can at best be fitted over a very restricted range in time only up to 1 s. ](figure5_SI.pdf){width="13cm"} ![\[fig6SI\] [**Experimental values of relaxation time $\tau$ from susceptibility and magnetisation measurements.**]{} Relaxation time $\tau$ vs temperature for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}. The green and blue data points ($\tau$ less than 100 s) were taken from the peaks in the imaginary susceptibility. The red points for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} come from analyzing dc relaxation (all raw data was first corrected for demagnetisation effects). The slope $dT/d\tau$ defining the equilibrium cooling rate shown in Fig. 2 are taken from fits to these curves. ](figure6_SI.pdf){width="7.2cm"} ![\[fig7SI\] [**Effect of monopole current $J_m=dM/dt$ on wait time for [Ho$_{2}$Ti$_{2}$O$_{7}$]{}**]{} left: Monopole current $J_m=dM/dt$ obtained from the relaxation of the magnetisation of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} measured after different waiting times, and measured at 800 Oe. $J_m$ is obtained by extrapolating the derivative of the magnetisation with respect to time at $t=0$ (See Ref. for the detailed procedure). right: $J_m$ vs $1/T$ obtained from the saturation value of the left figure, i.e. when the current value does not depend anymore on the waiting time. ](i_800_Ho.pdf "fig:"){width="7.2cm"} ![\[fig7SI\] [**Effect of monopole current $J_m=dM/dt$ on wait time for [Ho$_{2}$Ti$_{2}$O$_{7}$]{}**]{} left: Monopole current $J_m=dM/dt$ obtained from the relaxation of the magnetisation of [Ho$_{2}$Ti$_{2}$O$_{7}$]{} measured after different waiting times, and measured at 800 Oe. $J_m$ is obtained by extrapolating the derivative of the magnetisation with respect to time at $t=0$ (See Ref. for the detailed procedure). right: $J_m$ vs $1/T$ obtained from the saturation value of the left figure, i.e. when the current value does not depend anymore on the waiting time. ](i_800sat_vs1-T.pdf "fig:"){width="7.5cm"} ![\[fig8SI\] [**Effect of the nuclear spins on the monopole current**]{}. Monopole current $J_m=dM/dt$ vs $\sqrt{H}$ determined for $^{\rm nat}$Dy and $^{162}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples (left), $^{163}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples (middle), and in different cooling conditions for the [Ho$_{2}$Ti$_{2}$O$_{7}$]{} sample. $J_m$ is obtained by extrapolating the derivative of the magnetisation with respect to time at $t=0$ (See Ref. for the detailed procedure). Natural and $^{162}$Dy (no nuclear spin) [Dy$_{2}$Ti$_{2}$O$_{7}$]{} follows the $\sqrt{H}$ behavior expected for magnetic monopoles interacting through the Coulomb force, $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples do not, whatever the cooling process and so the initial density of monopoles. This result shows, as suggested in the main text, that the idealised emergent chemical kinetics of monopole theory does not apply in $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, where the dynamics is strongly affected by the nuclear spin effects, because the hyperfine splitting energies are of a similar order to the Coulomb energies. ](ivsh_DTO_no_N.pdf "fig:"){width="5.5cm"} ![\[fig8SI\] [**Effect of the nuclear spins on the monopole current**]{}. Monopole current $J_m=dM/dt$ vs $\sqrt{H}$ determined for $^{\rm nat}$Dy and $^{162}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples (left), $^{163}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples (middle), and in different cooling conditions for the [Ho$_{2}$Ti$_{2}$O$_{7}$]{} sample. $J_m$ is obtained by extrapolating the derivative of the magnetisation with respect to time at $t=0$ (See Ref. for the detailed procedure). Natural and $^{162}$Dy (no nuclear spin) [Dy$_{2}$Ti$_{2}$O$_{7}$]{} follows the $\sqrt{H}$ behavior expected for magnetic monopoles interacting through the Coulomb force, $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples do not, whatever the cooling process and so the initial density of monopoles. This result shows, as suggested in the main text, that the idealised emergent chemical kinetics of monopole theory does not apply in $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, where the dynamics is strongly affected by the nuclear spin effects, because the hyperfine splitting energies are of a similar order to the Coulomb energies. ](ivsh_nuk_Hob4.pdf "fig:"){width="5.5cm"} ![\[fig8SI\] [**Effect of the nuclear spins on the monopole current**]{}. Monopole current $J_m=dM/dt$ vs $\sqrt{H}$ determined for $^{\rm nat}$Dy and $^{162}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples (left), $^{163}$ [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples (middle), and in different cooling conditions for the [Ho$_{2}$Ti$_{2}$O$_{7}$]{} sample. $J_m$ is obtained by extrapolating the derivative of the magnetisation with respect to time at $t=0$ (See Ref. for the detailed procedure). Natural and $^{162}$Dy (no nuclear spin) [Dy$_{2}$Ti$_{2}$O$_{7}$]{} follows the $\sqrt{H}$ behavior expected for magnetic monopoles interacting through the Coulomb force, $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{} samples do not, whatever the cooling process and so the initial density of monopoles. This result shows, as suggested in the main text, that the idealised emergent chemical kinetics of monopole theory does not apply in $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} and [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, where the dynamics is strongly affected by the nuclear spin effects, because the hyperfine splitting energies are of a similar order to the Coulomb energies. ](ivsh_Hob4.pdf "fig:"){width="5.5cm"} ![\[fig9SI\] [**The effect of wait time $t_{\rm w}$ on the magneto-thermal avalanches for two different samples of natural DTO, and for two different runs**]{} (a) shows the value of the avalanche field $H_{\rm ava}$($t_{\rm w}$), defined as the field where the magnetization crosses 1 $\mu_{\rm B}$ per rare earth ion. (b) is a plot of the difference in avalanche field $\Delta H_{\rm ava}= H_{\rm ava}(t_{\rm w}) - H_{\rm ava} (t_{\rm w}=minimum)$. ](figure9_SI.pdf){width="16cm"} Different Samples and Measuring Directions ------------------------------------------ During the course of this study, 10 different samples were measured with some samples measured along multiple axis. The mass of the samples ranged between 3 to 40mg, and they had various shapes. No corrections for demagnetization effects have been taken into account for the results presented in the main text. This is because for most of the measurements shown in the main text, the sample was far from equilibrium, and the magnetization was very small, and thus the demagnetizing field -NM was small. Nevertheless the sample shape and field direction do effect the observed relaxation curves and the avalanche fields. In this section we show that although there are some variations between samples, between cooling runs, and for different directions, these differences do not change the main conclusion of the paper; the demonstration that nuclear assisted quantum tunneling is operative regardless of field direction. Supplementary Figure 9(a) shows the effect of $t_{\rm w}$ on the magneto-thermal avalanches for two different samples of natural DTO, and for two different runs. Sample 1 (also shown in the main text) was rectangular shaped parallelepiped and sample 2 was a square thin platelet shaped sample. The measurements shown in the figures were taken with the field along the \[111\] axis for both samples. The left panel shows the value of the avalanche field $H_{\rm ava}$($t_{\rm w}$), defined as the field where the magnetization crosses 1 $\mu_{\rm B}$ per rare earth ion. The curves are clearly offset from one another, even the two curves taken on the same sample, but during different runs. The initial position of the avalanche field is very sensitive to thermal contact with the sample holder. For sample 2 run 1, the thermal contact was made using two Cu bands with the sample sandwiched between the two. For sample 2 run 2 only one Cu band was used, thus the thermal contact was worse. The better thermalized sample has a higher avalanche field, because leading up to the avalanche heat could be more efficiently evacuated from the sample. Supplementary Figure 9(b) is a plot of the difference in avalanche field $\Delta H_{\rm ava}= H_{\rm ava}(t_{\rm w}) - H_{\rm ava} (t_{\rm w}=minimum)$. As can be seen, for sample 2, the two runs collapse onto one another, but the shape of the curve for sample 1 is slightly different. A more systematic study needs to be made to understand if this is a shape dependent effect, or sample dependent. ![\[fig10SI\] [**Measurements made on a polycrystalline sample of [$^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}]{} (sample 2)**]{} The sample was first prepared using the same avalanche quench protocols (AQP) outlined in the main text and methods section, followed by various waiting times. (a) shows the effects of wait time on the relaxation of the magnetization in a field of 0.8 T at 80mK. (b) shows the effects of wait time on the position of the avalanche field when the field is ramped from 0 to 0.4 T at a constant rate of 0.02 T/s at 80mK. Also shown in the figure is the conventional zero field cooling (CC) curve were the sample was slowly cooled from 900 mK to 80 mK (at 1 mK/s) followed by a 1000 s wait period. For this sample the CC avalanche field is offset to higher fields, and is well outside the distribution of $H_{\rm ava}$($t_{\rm w}$). ](figure10_SI.pdf){width="16cm"} For $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}, two samples were studied. Sample 1 was an odd shaped disk. The \[111\] direction was perpendicular to the surface of the disk, and resulted in a very large demagnetization factor for this direction. Importantly this resulted in difficulty thermalizing the sample along this direction to our Cu sample holder. This resulted in a much less efficient AQP cooling. We estimate that the sample took about 5 seconds to cool below 500mK, and about 40 seconds to cool below 100mK. This is much slower than the usual AQP as described in Fig. 2 of the main text, but still faster than the CC method. Sample 1 was measured along the \[111\] direction (shown in the main text) and perpendicular to the \[111\] direction. Sample 2 was a polycrystalline sample and the effects of wait time on the relaxation of the magnetization and position of the avalanche field are shown in Supplementary Figure 10 (a) and (b). These data sets are very similar to those presented in the main text in terms of the strength of the effect of wait time for $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}. (see Figures 3 and 4) However there are two interesting differences. Firstly, Supplementary Figure 11 (a) shows the monopole current $J_{\rm m}=dM/dt$ at $t=0$ vs log wait time for sample 1 \[111\] and perpendicular to \[111\] as well as for polycrystalline sample 2. As can be seen in the figure, although the slopes of the three curves are roughly the same, the \[111\] data fall significantly below the two perpendicular curves. Most likely this is not an intrinsic effect, but comes from the poor thermalization for the \[111\] sample run: as mentioned above, for this direction after the AQP the sample cooled much slower, therefore the initial monopole density at the beginning of the wait period was much reduced, so the initial monopole current was less, shifting the \[111\] curve down in the plot. A second difference that can be seen in Supplementary Figure 10 (b) is that the avalanche field for the CC method occurs at much higher fields and is well outside the distribution of curves obtained by the AQP method. The same result was found for sample 1 perpendicular to \[111\]. This can be contrast to the data shown for the \[111\] sample in the main text, and again can be explain by the slower cooling for the \[111\] sample run. ![\[fig11SI\] [**comparison of different samples and different measuring directions for HTO, [$^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}]{} and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  (DTO)**]{} (a) the monopole current $J_{\rm m}=dM/dt$ at $t=0$ vs log wait time for two samples of [$^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}]{} and three samples of HTO. (b) Plot of difference in avalanche field $\Delta H_{\rm ava}= H_{\rm ava}(t_{\rm w}) - H_{\rm ava}(t_{\rm w}=minimum)$ against wait time for various directions and various samples of HTO, [$^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}]{} and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  (DTO). The top 4 curves in the figure are measurements for 3 different samples of HTO (squares). Sample 1 was a needle shaped sample measured along the \[111\] (long) direction. Sample 2 was also needle shaped and measured along the \[001\] direction. Sample 3 was a square platelet, and was measure along the \[001\] and \[110\] axis. The middle 3 curves are for two different samples of $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}(solid dots) measured at 80mK. Sample 1 was measured along the \[111\] direction and perpendicular to the \[111\] direction, and sample 2 was a poly-crystal. The bottom two curves are for natural DTO (triangles) taken on two different samples along the \[111\] and \[001\] directions. (same data as shown in Supplementary Figure 9 (b)) ](figure11_SI.pdf){width="16cm"} The results for measurements on 3 different samples of HTO are also shown in Supplementary Figure 11. Sample 1 was a needle shaped sample measured along the \[111\] (long) direction. Sample 2 was also needle shaped and measured along the \[001\] direction. Sample 3 was a square platelet, and was measure along the \[001\] and \[110\] axis. Supplementary Figure 11 (a) shows the monopole current $J_{\rm m}=dM/dt$ at $t=0$ vs log wait time for sample 1 \[111\] (also in the main text) compared to sample 2 \[110\]. The effect of wait time on the currents for these two samples are very similar; the rate at which monopoles recombine is seen to be much faster than that of $^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}, and both seem to saturate at very long wait times. Supplementary Figure 11 (b) are plots of difference in avalanche field $\Delta H_{\rm ava}= H_{\rm ava}(t_{\rm w}) - H_{\rm ava}(t_{\rm w}=minimum)$ against log wait time for the three samples of HTO, as well as two samples of [$^{163}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}]{} and for two samples of $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  (DTO) (same data as shown in Supplementary Figure 9 (b)). [99]{} Kaiser, V., Bloxsom, J. A., Bovo, L., Bramwell, S. T., Holdsworth, P. C. W. & Moessner, R. Emergent Electrochemistry in Spin Ice: Debye–H[ü]{}ckel Theory and Beyond. [*Phys. Rev. B*]{} [**98**]{}, 144413 (2018). Paulsen, C., Giblin, S. R., Lhotel, E., Prabhakaran, D., Balakrishnan, G., Matsuhira, K., and Bramwell S. T. Experimental signature of the attractive Coulomb force between positive and negative magnetic monopoles in spin ice. [*Nature Physics*]{} [**12**]{}, 661 (2016). Jackson, M. J., Lhotel, E., Giblin, S. R., Bramwell, S. T., Prabhakaran, D., Matsuhira, K., Hiroi, Z., Yu, Q., and Paulsen, C. Dynamic behavior of magnetic avalanches in the spin-ice compound [Dy$_{2}$Ti$_{2}$O$_{7}$]{}. [*Phys. Rev. B*]{} [**90**]{}, 064427 (2014). Prokof’ev, N. V. & Stamp, P. C. E. Low-temperature quantum relaxation in a system of magnetic nano molecules. [*Phys. Rev. Lett.*]{} [**80**]{}, 5794 (1998).

--- abstract: 'We propose the existence, via analytical derivations, novel phenomenologies, and first-principles-based simulations, of a new class of materials that are not only spontaneously optically active, but also for which the sense of rotation can be switched by an electric field applied to them– via an induced transition between the dextrorotatory and laevorotatory forms. Such systems possess electric vortices that are coupled to a spontaneous electrical polarization. Furthermore, our atomistic simulations provide a deep microscopic insight into, and understanding of, this class of naturally optically active materials.' author: - 'Sergey Prosandeev$^{1,2}$, Andrei Malashevich$^{3}$, Zhigang Gui$^{1}$, Lydie Louis$^{1}$, Raymond Walter$^{1}$, Ivo Souza$^{4}$ and L. Bellaiche$^{1}$' title: Natural optical activity and its control by electric field in electrotoroidic systems --- Introduction ============ The speed of propagation of circularly-polarized light traveling inside an [*optically active*]{} material depends on its helicity [@Melrose; @Barron]. Accordingly, the plane of polarization of linearly polarized light rotates by a fixed amount per unit length, a phenomenon known as [*optical rotation*]{}. One traditional way to make materials optically active is to take advantage of the Faraday effect, by applying a magnetic field. However, there are some specific systems that are [*naturally gyrotropic*]{}, that is they spontaneously possess optical activity. Examples of known natural gyrotropic systems are quartz [@Arago], some organic liquids and aqueous solutions of sugar and tartaric acid [@Melrose], the Pb$_5$Ge$_3$O$_{11}$ compound [@experiment; @Koiiak], and the layered crystal (C$_5$H$_{11}$NH$_3$)$_2$ZnCl$_4$ [@Pnma-P212121]. Finding novel natural gyrotropic materials has great fundamental interest. It may also lead to the design of novel devices, such as optical circulators and amplifiers, especially if the [*sign*]{} of the optical rotation can be efficiently controlled by an external factor that is easy to manipulate. When searching for new natural gyrotropic materials, one should remember the observation of Pasteur that chiral crystals display spontaneous optical activity, which reverses sign when going from the original structure to its mirror image [@Pasteur]. Hence it is worthwhile to consider a newly discovered class of materials that are potentially chiral, and therefore may be naturally gyrotropic. This class is formed by electrotoroidic compounds (also called ferrotoroidics [@Shmid]). These are systems that possess an electrical toroidal moment, or equivalently, exhibit electric vortices [@Dubovik]. Such intriguing compounds were predicted to exist around nine years ago [@Ivan], and were found experimentally only recently [@Gruverman-Scott-vortex; @Balke-Kalinin; @Vasudevan; @Nelson; @Gregg]. One may therefore wonder if this new class of materials is indeed naturally gyrotropic, and/or if there are other necessary conditions, in addition to the existence of an electrical toroidal moment, for such materials to be optically active. In this work, we carry out analytical derivations, original phenomenologies and first-principles-based computations that successfully address all the aforementioned important issues. In particular, we find that electrotoroidic materials do possess spontaneous optical activity, but only if their electric toroidal moment changes [*linearly*]{} under an applied electric field. This linear dependence is further proved to occur if the electrotoroidic materials also possess a spontaneous electrical polarization that is coupled to the electric toroidal moment, or if they are also piezoelectric with the strain affecting the value of the electric toroidal moment. We also find that, in the former case, the applied electric field further allows the control of the sign of the optical activity. Our atomistic approach also reveals the evolution of the microstructure leading to the occurrence of field-switchable gyrotropy, and shows that the optical rotatory strength can be significant in some electrotoroidic systems. Relation between gyrotropy and electrical toroidal moment in electrotoroidic systems ==================================================================================== Let us first recall that the gyrotropy tensor elements, $g_{ml}$, are defined via [@Landau]: $$\label{eq74} g_{mk}=\frac{\omega }{2c}e_{ijm}\gamma _{ijk}$$ where $e_{ijm}$ is the Levi-Civita tensor [@Tyldesley], $c$ is the speed of light, and $\omega$ is the angular frequency. Note that this angular frequency is not restricted to the optical range. For instance, it can also correspond to the 1-100 GHz frequency range. The $\gamma$ tensor provides the linear dependence of the dielectric permittivity on the wave vector ${\bf k}$ in the optically active material, that is: $$\label{eq15} \varepsilon _{ik} \left( {\omega ,{\rm {\bf k}}} \right)=\varepsilon _{ik}^{\left( 0 \right)} \left( \omega \right)+i\gamma _{ikl} k_l$$ Here, $k_l$ is the $l$-component of the wave vector; $\varepsilon _{ik} \left( {\omega ,{\rm {\bf k}}} \right)$ denotes the double Fourier transform in time and space of the dielectric tensor, with the long-wavelength components being denoted by $\varepsilon_{ik}^{\left( 0 \right)}$. Throughout this manuscript we adopt Einstein notation, in which one implicitly sums over repeated indices (as it happens, e.g., for the $l$ index in Eq. (\[eq15\])). Thus, the calculation of the gyrotropy tensor can be reduced to the calculation of the tensor $\gamma$ responsible for the spatial dispersion of the dielectric permittivity. Alternatively, one can use the following formula for the dielectric permittivity [@Landau; @Melrose]: $$\label{eq19} \varepsilon _{ik}\left( {\omega ,{\rm {\bf k}}} \right) \\ =\delta _{ik} +\frac{4\pi i}{\omega }\sigma _{ik}\left( {\omega ,{\rm {\bf k}}} \right) \\ =\delta _{ik} +\frac{4\pi i}{\omega }(\sigma _{ik}^{(0)}\left( {\omega } \right)+\sigma_{ikl}k_l)$$ where $\delta _{ik}$ is the Kronecker symbol and $\sigma _{ik}\left( {\omega ,{\rm {\bf k}}} \right)$ is the effective conductivity tensor in the reciprocal space, at a given frequency [@Melrose]. $\sigma_{ikl}$ is the third-rank tensor associated with the linear dependence of the effective conductivity tensor on the wave vector, and $\sigma _{ik}^{(0)}$ is the effective conductivity tensor at zero wave vector. Combining Eqs. (\[eq19\]) with Eq. (\[eq15\]) yields: $$\label{eq20} \gamma _{ikl} =\frac{4\pi}{\omega }\sigma_{ikl}\\ =\frac{4\pi}{\omega }(\sigma_{ikl}^S(\omega)+\sigma _{ikl}^A(\omega))$$ where $$\label{eq16b} \sigma_{ijk}^A =\frac{1}{2}(\sigma_{ijk}-\sigma_{jik})$$ and $$\label{eq16c} \sigma _{ijk}^S =\frac{1}{2}(\sigma_{ijk}+\sigma_{jik})$$ Moreover, using the results of Ref. [@Malashevich] and working at nonabsorbing frequencies (i.e., frequencies, such as GHz in ferroelectrics, for which the corresponding energy is below the band gap of the material), one can write $$\label{eq16} \sigma _{ijk}^A =ic\left( {e_{jkl} \beta _{il} -e_{ikl} \beta _{jl} } \right)+\omega \xi _{ijk}$$ with $$\label{eq17} \beta _{ij} =i\mathrm{Im}(\chi _{ij}^{em}) \mathop = -i\mathrm{Im}(\chi _{ji}^{me})$$ and $$\label{eq18} \xi _{ijk} =\frac{1}{2}\left[ {\frac{dQ_{kj} }{dE_i }-\frac{dQ_{ki} }{dE_j }} \right]$$ where $\mathrm{Im}$ stands for the imaginary part and $Q$ is the quadrupole moment of the system [@Raab]. $\chi^{me}$ is the response of the magnetization, [**M**]{}, to an electric field [**E**]{}, while $\chi^{em}$ is the response of the electrical polarization, [**P**]{}, to a magnetic field [**B**]{}, that is: $$\label{eq22} \chi _{ij}^{me} =\frac{dM_i }{dE_j } \\ \nonumber ~~~{\rm and} ~~~ \chi _{ji}^{em} =\frac{dP_j }{dB_i}$$ Inserting Eq. (\[eq16\]) into Eq. (\[eq20\]) provides : $$\label{eq21} \gamma _{ijk} =\frac{4\pi }{\omega }\left[ {c\left( {e_{jkl} \mathrm{Im}\,\chi _{li}^{me} -e_{ikl} \mathrm{Im}\,\chi _{lj}^{me} } \right)+\omega \xi _{ijk} } \right] + \gamma_{ijk}^S$$ where $\gamma_{ijk}^S=(4\pi/\omega)\sigma_{ijk}^S$ is the contribution of the symmetric part of the conductivity to the $\gamma$ tensor. As a result, $\gamma_{ijk}^S$ is non-zero only when the system is magnetized or possesses a spontaneous magnetic order [@Landau]. Let us now focus on the magnetization, which can be written as [@Raab]: $$\label{eqm1} {\bf M} =\frac{1}{2cV} \int {\left ( {\bf r} \times {\bf {\cal J}(r)}\right ) d^3r }$$ where $c$ is the speed of light, $V$ is the volume of the system, [ **r**]{} is the position vector, and ${\bf {\cal J}(r)}$ is the current density. We consider here the following contributions to this density: $$\label{eq6} {\bf {\cal J}(r)}=\dot {{\cal P}}({\bf r)}+c~ {\bm \nabla}\times {\bf {\cal M}_0(r)}$$ where the dot symbol refers to the partial derivative with respect to time. ${\bf {\cal P}(r)}$ is the polarization [*field*]{}, that is, the quantity for which the spatial average is the macroscopic polarization. Similarly, ${\bf {\cal M}_0(r)}$ is the magnetization field, that is, the quantity for which the spatial average is the part of the macroscopic magnetization that does not originate from the time derivative of the polarization field [@gauge]. By plugging this latter equality into Eq. (\[eqm1\]), we have: $$\label{eqn6} {\bf M} =\frac{1}{2cV} \int{ \left ( {\bf r} \times \dot {{\cal P}}({\bf r)}\right ) d^3r}+ \frac{1}{2V} \int{ \left ( {\bf r} \times {\bm \nabla}\times {\bf {\cal M}_0(r)} \right ) d^3r} =\frac{1}{2cV} \int{ \left ( {\bf r} \times \dot {{\cal P}}({\bf r)}\right ) d^3r} + {\bf M}_0$$ The analytical expression of this latter equation bears some similarities with the definition of the electrical toroidal moment, [**G**]{}, that is [@Dubovik] $$\label{eqn1} {\bf G}=\frac{1}{2V}\int {\left ( {\bf r} \times {\bf {\cal P}(r)} \right ) d^3r}~~~,$$ More precisely, taking the time derivative of [**G**]{} gives: $$\label{eqntds} \dot{\bf G} \simeq \frac{1}{2V} \int{ \left ( {\bf r} \times \dot {{\cal P}}({\bf r)}\right ) d^3r}$$ when omitting the time dependency of the volume (the numerical simulations presented below indeed show that one can safely neglect this dependency when computing the time derivative of the electric toroidal moment). As a result, combining Eq. (\[eqntds\]) and Eq. (\[eqn6\]) for a monochromatic wave having an $\omega$ angular frequency gives: $$\label{eqn3} {\bf M}-{\bf M}_0 \simeq \frac{1}{c}{\dot{\bf G}}=-\frac{i\omega }{c}{\bf G}$$ in electrotoroidic systems. Plugging this latter equation in Eq. (\[eq22\]) then gives: $$\label{eq24} \chi _{ij}^{me} =\chi_{ij}^{me(0)}-\frac{i\omega }{c}\frac{dG_i }{dE_j }$$ where $\chi_{ij}^{me(0)}$ is the magnetoelectric tensor related to the derivative of ${\bf M}_0$ with respect to an electric field. Therefore $$\label{eq25} \mathrm{Im}\,(\chi _{ij}^{me}-\chi_{ij}^{me(0)}) =-\frac{\omega }{c}\frac{dG_i }{dE_j }$$ This relation between the imaginary part of the magnetoelectric susceptibility and the field derivative of the electrical toroidal moment is reminiscent of the connection discussed in Ref. [@Spaldin] between the linear magnetoelectric response and the [*magnetic*]{} toroidal moment. Inserting Eqs. (\[eq25\]) and (\[eq18\]) into Eq. (\[eq21\]) then provides: $$\begin{aligned} \label{eq29} \gamma _{ijk} =&&\gamma_{ijk}^{S} +\frac{4\pi c}{\omega} \left ( e_{jkl} {\rm Im} \chi_{li}^{me(0)}-e_{ikl}{\rm Im} \chi_{lj}^{me(0)} \right ) \nonumber \\ &&+4\pi \left[ {e_{ikl} \frac{dG_l }{dE_j }-e_{jkl} \frac{dG_l }{dE_i }+\frac{1}{2}\left( {\frac{dQ_{kj} }{dE_i }-\frac{dQ_{ki} }{dE_j }} \right)} \right]\end{aligned}$$ Combining this latter equation with Eq. (\[eq74\]), and recalling that $\gamma_{ijk}^{S}$ is a symmetric tensor while $e_{ijm}$ is antisymmetric (which makes their product vanishing), gives: $$\begin{aligned} \label{eq33before} g_{mk} =&&4\pi \left ( \delta_{mk} {\rm Im} \chi_{ll}^{me(0)}-{\rm Im}\chi_{mk}^{me(0)} \right ) \nonumber \\ && +\frac{4\pi \omega }{c}\left[ \left( {\frac{dG_m }{dE_k }-\frac{dG_l }{dE_l }\delta _{mk} } \right) +\frac{1}{4} e_{ijm}\left( \frac{dQ_{kj}}{dE_i}-\frac{dQ_{ki}}{dE_j} \right) \right]\end{aligned}$$ Choosing a specific gauge [@gauge] and neglecting quadrupole moments (simulations reported below show that spontaneous and field-induced quadrupole moments can be neglected for the ferrotoroidics numerically studied in Section IV) lead to the reduction of Eq. (\[eq33before\]) to: $$\label{eq33} g_{mk} =\frac{4\pi \omega }{c}\left[ \left( {\frac{dG_m }{dE_k }-\frac{dG_l }{dE_l }\delta _{mk} } \right) \right]$$ This formula nicely reveals that optical activity should happen when electrical toroidal moment [*linearly*]{} responds to an applied electric field. Necessary conditions for gyrotropy in electrotoroidic systems ============================================================= According to Eq. (\[eq33\]), an electrotoroidic system possessing non-vanishing derivatives of its electrical toroidal moment with respect to the electric field automatically possesses natural optical activity. Let us now prove analytically that the occurrence of such non-vanishing derivatives requires additional symmetry breaking in electrotoroidic systems, namely that an electrical polarization or/and piezoelectricity should also exist, as well as couplings between electrical toroidal moment and electric polarization and/or strain. For that, let us express the free energy of an electrotoroidic system that exhibits couplings between electrical toroidal moment **G**, polarization **P**, and strain $\eta$ as: $$\label{eq34} F=F_0+\zeta_{ijkl}G_iG_j\eta_{kl}+\lambda_{ijkl}G_iG_jP_kP_l+q_{ijkl}P_iP_j\eta_{kl}-h_iG_i$$ where $h_i=( {\bm \nabla}\times \,{\bf E})_i$ is the field conjugate of $G_i$. The equilibrium condition, $\partial F / \partial G_n = 0$, implies that $$\label{eq35} \partial F_0 / \partial G_n+(\zeta_{njkl}+\zeta_{jnkl})G_j\eta_{kl}+(\lambda_{njkl}+\lambda_{jnkl})G_jP_kP_l =h_n$$ which indicates that $h_n$ depends on both the polarization and strain. As a result, the change in electrical toroidal moment with electric field can be separated into the following two contributions: $$\label{eq34b} \frac{d{\rm { G_i}}}{d{\rm { E_j}}}= \left( {\frac{dG_i }{dE_j }} \right)^{(1)}+ \left( {\frac{dG_i }{dE_j }} \right)^{(2)}$$ with $$\label{eq35a} \left( \frac{dG_i }{dE_j } \right)^{(1)}= \frac{dG_i}{d h_n} \frac{\partial h_n}{\partial P_l} \frac{d P_l}{d E_j} = \chi^{(G)}_{in}\frac{\partial h_n}{\partial P_l}\chi^{(P)}_{lj}$$ and $$\label{eq35b} \left( {\frac{dG_i }{dE_j }} \right)^{(2)}= \frac{d G_i}{\partial h_n} \frac{\partial h_n}{\partial \eta_{kl}} \frac{d \eta_{kl}}{d E_j} = \chi^{(G)}_{in} \frac{\partial h_n}{\partial \eta_{kl} } d_{klj}$$ Here $$\chi^{(G)}_{in}=\frac{dG_i}{dh_n}$$ is the response of the electrical toroidal moment to its conjugate field, $$\chi_{ij} ^{(P)}=\frac{dP_i}{dE_j}$$ is the electric susceptibility, and $$d_{ijk}=\frac{d \eta_{ij}}{dE_k}$$ is a piezoelectric tensor. The remaining derivatives appearing in Eqs. (\[eq35a\]) and (\[eq35b\]) can be found from Eq. (\[eq35\]): $$\label{eq38} \left( {\frac{\partial h_n }{\partial P_l }} \right)= (\lambda_{njlm} +\lambda _{njml}+\lambda_{jnlm} +\lambda _{jnml})G_jP_m$$ and $$\label{eq39} \left( {\frac{\partial h_n }{\partial \eta_{kl} }} \right)=(\zeta_{njkl}+\zeta _{jnkl}) G_j$$ Equations (\[eq34b\])-(\[eq39\]) reveal that there are two scenario for the occurence of natural optical activity in electrotoroidic systems. In the first scenario, the system possesses a finite polarization that has a bilinear coupling with the electrical toroidal moment (see Eqs. (\[eq35a\]), (\[eq38\]), and (\[eq34\])). In the second scenario, the electrotoroidic system is also piezoelectric, and electrical toroidal moment and strain are coupled to each other (see Eqs. (\[eq35b\]), (\[eq39\]), and (\[eq34\])). An example of the latter can be found in Reference [@Prosandeev], where a pure gyrotropic phase transition leading to a piezoelectric, but non-polar, $P2_12_12_1$ state (that exhibits spontaneous electrical toroidal moments) was discovered in a perovskite film. Next, we describe the theoretical prediction of a material where the former scenario is realized. Prediction and microscopic understanding of gyrotropy in electrotoroidic systems ================================================================================ The system we have investigated numerically is a nanocomposite made of periodic squared arrays of BaTiO$_3$ nanowires embedded in a matrix formed by (Ba,Sr)TiO$_3$ solid solutions having a 85% Sr composition. The nanowires have a long axis oriented along the \[001\] pseudo-cubic direction (chosen to be the $z$-axis). They possess a squared cross-section of 4.8x4.8 nm$^2$ in the ($x$,$y$) plane, where the $x$- and $y$-axes are chosen along the pseudo-cubic \[100\] and \[010\] directions, respectively. The distance (along the $x$- or $y$-directions) between adjacent BaTiO$_3$ nanowires is 2.4 nm. We choose this particular nanocomposite system because a recent theoretical study [@submitted], using an effective Hamiltonian ($H_{\rm{eff}}$) scheme, revealed that its ground state possesses a spontaneous polarization along the $z$-direction inside the whole composite system, as well as electric vortices in the ($x$,$y$) planes inside each BaTiO$_3$ nanowire, with the same sense of vortex rotation in every wire. Such a phase-locking, ferrotoroidic and polar state is shown in Fig. 1a. It exhibits an electrical toroidal moment being parallel to the polarization. Figure 1a also reveals the presence of antivortices located in the [*medium*]{}, half-way between the centers of adjacent vortices. In the present study, we use the same $H_{\rm{eff}}$ as in Ref. [@submitted], combined with molecular dynamics techniques, to determine the response of this peculiar state to an $ac$ electric field applied along the main, $z$-direction of the wires. In our simulations, the amplitude of the field was fixed at 10$^9$ V/m and its frequency ranged between 1GHz and 100GHz. The sinusoidal frequency-driven variation of the electric field with time makes therefore this field ranging in time between 10$^9$ V/m (field along \[001\]) and -10$^9$ V/m (field along \[00-1\]). The idea here is to check if the electrical toroidal moment has a [*linear*]{} variation with this field at these investigated frequencies, and therefore if the investigated system can possess nonzero gyrotropy coefficients (see Eq.(\[eq33\])). In this effective Hamiltonian method, developed in Ref.[@Walizer2006] for (Ba,Sr)TiO$_3$ (BST) compounds, the degrees of freedom are: the local mode vectors in each 5-atom unit cell (these local modes are directly proportional to the electric dipoles in these cells), the homogeneous strain tensor and inhomogeneous-strain-related variables [@Zhong1995]. The total internal energy contains a local mode self-energy, short-range and long-range interactions between local modes, an elastic energy and interactions between local modes and strains. Further energetic terms model the effect of the interfaces between the wires and the medium on electric dipoles and strains, as well as take into account the strain that is induced by the size difference between Ba and Sr ions and its effect on physical properties. The parameters entering the total internal energy are derived from first principles. This $H_{\rm{eff}}$ can be used within Monte-Carlo or Molecular dynamics simulations to obtain finite-temperature static or dynamical properties, respectively, of relatively large supercells (i.e., of the order of thousands or tens of thousands of atoms). Previous calculations [@Choudhury2011; @Walizer2006; @Lisenkov2007; @Hlinka2008; @Quingteng2010] for various disordered or ordered BST systems demonstrated the accuracy of this method for several properties. For instance, Curie temperatures and phase diagrams, as well as the subtle temperature-gradient-induced polarization, were well reproduced in BST materials. Similarly, the existence of two modes (rather than a single one as previously believed for a long time) contributing to the GHz-THz dielectric response of pure BaTiO$_3$ and disordered BST solid solutions were predicted via this numerical tool and experimentally confirmed. Figures 2(a) and 2(b) report the evolution of the $z$-component of the electrical toroidal moment, G$_z$, and of the polarization, P$_z$, respectively, as a function of the electric field, for a frequency of 1GHz at a temperature of 15K. In practice, G$_z$ is computed within a lattice model [@submitted], by summing over the electric dipoles located at the lattice sites rather than by continuously integrating the polarization field of Eq. (\[eqn1\]) over the space occupied by the nanowires. The panels in Fig. 1 show snapshots of important states occurring during these hysteresis loops, in order to understand gyrotropy at a microscopic level. A striking piece of information revealed by Fig. 2 (a) is that G$_z$ [*linearly decreases*]{} with a slope of $-1.6$ e/V when the applied $ac$ field varies between 0 (state 1) and its maximum value of 10$^9$ [V/m]{} (state 2). Such variation therefore results in [ *positive*]{} g$_{11}$ and g$_{22}$ gyrotropy coefficients that are both equal to $0.94 \times 10^{-7}$ for a frequency of 1GHz, according to Eq. (\[eq33\]) (that reduces here to $g_{11} =g_{22}=-\frac{\omega }{c\varepsilon_0} {\frac{dG_z }{dE}}$ in S.I. units, since there are no $x$- and $y$-components of the toroidal moment and since the field is applied along $z$ in the studied case). Interestingly, we found that the aforementioned slope of $-1.6$ e/V stays roughly constant over the entire frequency range we have investigated (up to 100GHz). As a result, Eq. (\[eq33\]) indicates that $g_{11} =g_{22}$ should be proportional to the angular frequency $\omega$ of the applied $ac$ field, and that the meaningful quantity to consider here is the ratio between $g_{11}$ and this frequency. Such ratio is presently equal to $5.9 \times 10^{-16}$ per Hz. Moreover, the rate of optical rotation is related to the product between $\omega/c$ and the gyrotropy coefficient according to Ref. [@Landau]. As a result, the rate of optical rotation depends on the [*square*]{} of the angular frequency because of Eq. (\[eq33\]), as consistent with one finding of Biot in 1812 [@Barron]. Here, the ratio of the rate of optical rotation to the square of the angular frequency is found to be four orders of magnitude larger than that measured in “typical” gyrotropic materials, such as Pb$_5$Ge$_3$O$_{11}$ [@experiment; @Koiiak]. As a result, the plane of polarization of light will rotate by around $1.2$ radians per meter at 100GHz (or by $1.24\times 10^{-4}$ radians per meter at 1GHz), when passing through the system. Figure 2b indicates that the observed decrease of G$_z$ is accompanied by an increase of the polarization, which is consistent with our numerical finding that increasing the field from 0 to 10$^9$ V/m reduces the $x$- and $y$-components of the electric dipoles inside the nanowires (that form the vortices) while enhancing the $z$-component of the electric dipoles in the whole nanocomposite (i.e., wires and medium). Interestingly, the antivortices in the medium progressively disappear during this linear decrease of G$_z$ and increase of P$_z$, as shown in Figs 1. Figures 2 also show that decreasing the electric field from 10$^9$ V/m (state 2) to $\simeq$ -0.031 $\times$ 10$^9$ V/m (state 3) leads to a linear increase of the electric toroidal moment (yielding the aforementioned values of $g_{11}$ and $g_{22}$), while the $z$-component of the polarization decreases but still stay positive. Further increasing the magnitude of negative electric fields up to $\simeq$ -0.094 $\times$ 10$^9$ V/m results in drastic changes for the microstructure: dipoles in the medium now adopt negative $z$-components (state 3), then sites at the interfaces between the medium and the wires also flip the sign of the $z$-component of their dipoles (states 3 and $\alpha$). During these changes, the overall polarization rapidly varies from a significant positive value along the $z$-axis to a slightly negative value (Fig. 2b), while G$_z$ is nearly constant, therefore rendering the gyrotropic coefficients null. Then, continually increasing the strength of the negative $ac$ field up to $\simeq$ -0.48 $\times$ 10$^9$ V/m leads to the next stage: dipoles [*inside*]{} the wires begin to change the sign of their $z$-components (states $\beta$, 4 and $\gamma$) until all of the $z$-components of these dipoles point down (state 5). During that process, P$_z$ becomes more and more negative, while the electrical toroidal moment decreases very fast but remains positive (indicating that the chirality of the wires is unaffected by the switching of the overall polarization). Once this process is completed, further increasing the magnitude of the applied field along \[00$\bar{1}$\] up to -10$^9$ V/m (state 2$'$), leads to a [*linear decrease*]{} of the electrical toroidal moment. Interestingly, this decrease is quantified by a slope $dG_z/dE$ that is exactly [*opposite*]{} to the corresponding one when going from state 1 to state 2. As a result, the $g_{11}$ and $g_{22}$ gyrotropic coefficients associated with the evolution from state 5 to state 2$'$ are now [*negative*]{} and equal $-$$0.94 \times 10^{-7}$ at 1GHz. Finally, Figures 1 and 2 indicate that varying now the $ac$ field from its minimal value of -10$^9$ V/m to its maximal value of 10$^9$ V/m leads to the following succession of states: 2$'$, 5, 1$'$, 3$'$, $\alpha$$'$, $\beta$$'$, 4$'$, $\gamma$$'$, 5$'$ and 2, where the $'$ superscript used to denote the i$'$ states (with $i$=2, 3, 4, 5, $\alpha$, $\beta$ and $\gamma$) indicates that the corresponding states have $z$-components of their dipoles that are all opposite to those of state $i$ (for instance, state $\beta$$'$ has $z$-components of the dipoles being positive in the medium while being negative in the wires, as exactly opposite to state $\beta$). During this path from state 2$'$ to state 2, the gyrotropic coefficients $g_{11}$ and $g_{22}$ can be negative (from state 2$'$ to state 3$'$) or positive (from state 5$'$ to state 2), depending on the sign of the polarization. Such possibility of having both negative and positive gyrotropic coefficients in the same system originates from the fact that the polarization can be down or up, and is consistent with Eqs. (\[eq38\]), (\[eq35a\]) and (\[eq33\]). As a result, one can turn the polarization of light either in clockwise or anticlockwise manner in electrotoroidic systems, via the control of the direction of the polarization by an external electric field – which induces the switching between the dextrorotatory and laevorotatory forms of these materials (see states 1 and 1$'$). Such control may be promising for the design of original devices [@footnotedegen; @footnoteconj]. Figure 3 shows how the gyrotropic coefficient $g_{11}$ depends on temperature. One can clearly see that $g_{11}$ significantly increases as the temperature increases up to 240K. As indicated in the figure, the temperature behavior of $g_{11}$ is very well fitted by $A/\sqrt{(T_C-T)(T_G-T)}$, where $A$ is a constant, $T_C=240K$ is the lowest temperature at which the polarization vanishes and $T_G=330K$ is the lowest temperature at which the electric toroidal moment is annihilated [@submitted]. In order to understand such fitting, let us combine Eqs (\[eq33\]), (\[eq35a\]) and (\[eq38\]) for the studied case, that is: $$\label{eqnew} g_{11} =-\frac{4\pi \omega }{c}\frac{dG_3 }{dE_3 }= -\frac{4\pi \omega }{c} \chi^{(G)}_{3n}\frac{\partial h_n}{\partial P_l}\chi^{(P)}_{l3}=-\frac{4\pi \omega }{c} (\lambda_{n3l3} +\lambda _{n33l}+\lambda_{3nl3} +\lambda _{3n3l}) \chi^{(G)}_{3n} G_3 P_3 \chi^{(P)}_{l3}$$ The usual temperature dependencies of the order parameter and its conjugate field imply that $G_3$ and $P_3$ should be proportional to $\sqrt{(T_G-T)}$ and $\sqrt{(T_C-T)}$, respectively, while their responses, $ \chi^{(G)}_{3n}$ and $\chi^{(P)}_{l3}$, should be proportional to $1/(T_G-T)$ and $1/(T_C-T)$, respectively. This explains why the behavior of $g_{11}$ as a function of $T$ is well described by $A/\sqrt{(T_C-T)(T_G-T)}$. Summary ======= In summary, we propose the existence of a new class of spontaneously optically active materials, via the use of different techniques (namely, analytical derivations, phenomenologies and first-principles-based simulations). These materials are electrotoroidics for which the electric toroidal moment changes linearly under an applied electric field. Such linear change is demonstrated to occur if at least one of the following two conditions is satisfied: (1) the electric toroidal moment is coupled to a spontaneous electrical polarization; or (2) the electric toroidal moment is coupled to strain and the whole system is piezoelectric. We also report a realization of case (1), and further show that applying an electric field in such a case allows a systematic control of the sign of the optical rotation, via a field-induced transition between the dextrorotatory and laevorotatory forms. We therefore hope that our study deepens the current knowledge of natural optical activity and will be put in use to develop novel technologies. This work is financially supported by ONR Grants N00014-11-1-0384 and N00014-08-1-0915 (S.P. and L.B.), ARO Grant W911NF-12-1-0085 (Z.G. and L.B.), NSF grant DMR-1066158 (L.L. and L.B.). I.S acknowledges support by Grant MAT2012-33720 from the Spanish Ministerio de Economía y Competitividad. S.P. appreciates Grant 12-08-00887-a of Russian Foundation for Basic Research. L.B. also acknowledges discussion with scientists sponsored by the Department of Energy, Office of Basic Energy Sciences, under contract ER-46612, Javier Junquera and Surendra Singh. [1]{} D. B. Melrose and R. C. McPhedran, [*Electromagnetic Processes in Dispersive Media*]{} (Cambridge University Press, Cambridge, 1991). L. D. Barron, [*Molecular Light Scattering and Optical Activity*]{} (Cambridge University Press, Cambridge, 2004). D. F. J. Arago, [Mém. 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De Lange, [*Multipole Theory in Elecromagnetism.*]{} (Clarendon Press - Oxford 2005). Note that ${\bf {\cal P}(r)}$ and ${\bf {\cal M}_0(r)}$ are technically ill-defined in the sense that they depend on the choice of a gauge [@Hirst]. However, all the different gauges result in the same current density, ${\bf {\cal J}(r)}$, [@Hirst] which is the physical quantity that appears in Eqs. (12) and (13). As a result, the choice of the gauge does not modify our results, in general, and Eq. (21), in particular. Such conclusion can also be reached by realizing that the quantity appearing in the left hand side of Eq. (12) is the [*macroscopic*]{} magnetization, and as such, should not depend on the choice of a gauge. Note, however, that Eq. (22) is deduced from Eq. (21) via the annihilation of all the contributions stemming from ${\bf M}_0$. As a result, a specific choice of gauge was made in going from Eq.(21) to Eq.(22), namely the “P-only” gauge discussed in Ref. [@Hirst]. L.L. Hirst, [Reviews of Modern Physics]{}, **69**, 607 (1997). N. A. Spaldin, M. Fiebig, and M. Mostovoy, [ J. Phys. Cond. Mat.]{} **20**, 434203 (2008). S. Prosandeev, I. A. Kornev, L. Bellaiche, [ Phys. Rev. Lett.]{} [**107**]{}, 117602 (2011). L. Louis, I. Kornev, G. Geneste, B. Dkhil, and L. Bellaiche, [ J. Phys. Cond. Mat.]{} [**24**]{}, 402201 (2012). J. Hlinka, T. Ostapchuk, D. Nuzhnyy, J. Petzelt, P. Kuzel, C. Kadlec, P. Vanek, I. Ponomareva, and L. Bellaiche, Phys. Rev. Lett. [**101**]{}, 167402 (2008). Note that Eq. (\[eq34\]) involves the [*squares*]{} of the toroidal moment and of the polarization for the coupling interaction between these two physical quantities. As a result, one can easily understand that the presently studied nanocomposite has a ground state that is four times degenerated, due to the fact that the polarization and electrical toroidal moment can independently be parallel or antiparallel to the z-axis. These four states have the same probability (of 25%) to occur when cooling the system from high to low temperature. In our simulations, when the system statistically chooses one of these states below the critical temperature, it stays in it when further decreasing the temperature, likely because the potential barrier to go from one of these states to the other three states is too high. Moreover, one can also force the system to be in one of these four states by applying, and then removing, the conjugate fields of the polarization and electrical toroidal moment. For instance, the selection of the states for which the polarization is parallel to the z-axis requires the application of an homogeneous electric field along \[001\]. Similarly, obtaining states with electrical toroidal moment being aligned along \[001\] can be done by applying specific curled electric fields [@Weicurled] (see also Refs. [@Dubovik; @Sergeyinho1; @Sergeyinho2] for more details about how to control toroidal moments). Notice that creating a curled electric field along \[001\] can be practically realized by applying a magnetic field having a time derivative being oriented along \[00$\bar{1}$\], since Maxwell equations directly relate the time derivative of a magnetic field with minus the curl of the electric field. Note that the homogeneous electric field is the field conjugate of the electrical polarization but is not the field conjugate of the electrical toroidal moment. As a result (and as proven by our simulations), applying an electric field can change the direction of the polarization but can not change the direction of the electrical toroidal moment. This explains why an electric field can control the chirality and optical activity in electrotoroidic systems.

--- abstract: 'Topological phases of matter are one of the hallmarks of quantum condensed matter physics. One of their striking features is a bulk-boundary correspondence wherein the topological nature of the bulk manifests itself on boundaries via exotic massless phases. In classical wave phenomena analogous effects may arise; however, these cannot be viewed as equilibrium phases of matter. Here we identify a set of rules under which robust equilibrium classical topological phenomena exist. We write down simple and analytically tractable classical lattice models of spins and rotors in two and three dimensions which, at suitable parameter ranges, are paramagnetic in the bulk but nonetheless exhibit some unusual long-range or critical order on their boundaries. We point out the role of simplicial cohomology as a means of classifying, writing-down, and analyzing such models. This opens a new experimental route for studying strongly interacting topological phases of spins.' author: - 'R. Bondesan' - 'Z. Ringel' bibliography: - 'CTP.bib' title: Classical topological paramagnetism --- Introduction ============ Symmetry protected topological phases are exotic quantum states of matter that are featureless in the bulk but still support unusual low energy phenomena on their boundaries. Their distinguishing properties remain sharp and robust as long as the appropriate symmetries are maintained. An important example is the quantum spin Hall insulator [@Hasan2010], protected by time reversal symmetry, whose edge physics may be used in spintronics [@Ilan2014; @Wu2011; @Ojeda2012] and in the creation of topologically protected qubits in the form of Majorana fermions [@FuKane2008]. Partially motivated by the search for other exotic boundary phenomena, the field has developed rapidly: The classification table of weakly interacting topological phases of electrons given various symmetries has been established [@TenFold2010] in what can be seen as a modern revival of band structure theory. Furthermore, various topological electronic phases have been realized [@Hasan2010; @Qi2011]. Turning to bosons, a difficulty arises since without interactions their ground state is always a superfluid regardless of the band structure. Nonetheless such phases do exist at strong interactions and are known as bosonic SPTs [@Chen2011; @Chen2011a; @Schuch2011]. Unfortunately, experimental realizations of bosonic SPTs are scarce and, to the best of our knowledge, limited to one dimensional spin chains [@Buyers1986]. Recently there has been both theoretical [@Gennady2013; @Kane2014; @PhysRevLett.116.135503] and experimental [@Huber2015; @PhotonicsExp1; @PhotonicsExp2; @PhotonicsExp3; @Paulose23062015; @Chen09092014] interest in the notion of classical topological phases mimicking the phenomenology of their quantum counterparts. A typical strategy there is to consider systems of springs and masses or optical devices which have an underlying topological band structure. Their edges can be seen as robust waveguides which have potential engineering applications, such as delay lines for light and sound [@Hafezi2011]. Notwithstanding, it is difficult to view such phenomena as a distinct phase of matter, since the topological nature of the band structure does not induce any sharp measurable features in equilibrium. Further, at present the effect of non-linearities on these systems is unclear. (See however [@Chen09092014].) Both these issues can be seen as a classical reflection of the aforementioned difficulty of finding topological equilibrium phases of non-interacting bosons. As in the quantum case, an alternative route may thus be to consider strongly interacting systems. One approach to obtain such models is to start from known quantum SPT models and attempt to write their discretized Euclidean time partition function in a sign-problem free and local manner. When possible, the resulting partition function can then be viewed as a classical statistical mechanical system. Nonetheless, the models thus obtained have several drawbacks. First, the notion of symmetry protection does not generally carry through to the classical problem, in the following sense: We define classical symmetries as those one-to-one maps on configuration space which leave the Boltzmann weight invariant. For instance, in a spin-1 antiferromagnetic chain which supports a 1D SPT known as the Haldane phase [@AKLT1988], the associated classical configuration space is one discrete variable ($m_z=-1,0,1$) per site. When viewed as an SPT phase protected by $SO(3)$ or its $Z_2 \times Z_2$ subgroup of $\pi$-rotations [@Pollmann2012], the action of the symmetry involves superpositions and cannot be considered classical. A related issue is that the microscopic mechanism stabilizing topological phases, based on matrix product states and projective symmetries [@Schuch2011], becomes obfuscated in the classical setting. Lastly, the Boltzmann weights resulting from the prescription outlined above, are complicated and anisotropic, making these models less experimentally relevant. Interestingly, for some models based on coupled superfluids, the lattice Euclidean time partition function, following a series of transformations, can be written in a sign free manner [@Geraedts2013]. The advantage here is that the resulting models are isotropic. However in the process of making the action local, additional degrees of freedom are introduced and, from a classical perspective, it is thus unclear what are the essential ingredients which render this a well defined classical phase of matter rather than a particular model. Furthermore it will be useful to generalize this approach to the discrete symmetry case which is more experimentally relevant. In this work we address the question of what restrictions, analogous to symmetry protection, should be imposed on a classical system under which it supports robust classical topological phases (CTPs). The first requirement is to consider systems invariant under a group $G$ and a local constraint whose defects carry elements of another group $G'$. (More details about defects can be found in Appendix \[App:Constraint\].) One example would be a gauge theory with gauge group $G'$ and defects being monopoles. The second requirement is that these phases must be short range correlated in the bulk and in particular must not break the symmetry spontaneously. The third is that they must confine defects of the constraints into neutral pairs (see Appendix \[App:Constraint\] for a precise definition). We refer to phases which obey the above restrictions as “admissible phases”. Interestingly, we find that given a dimension $d$, and the groups $G$, and $G'$ as above, there are many inequivalent admissible phases. As standard, two phases are deemed equivalent if a continuous deformation from one to another is possible without crossing a critical point. By continuous we mean that one deforms the energy functional gradually and maintains the local constraint. We establish the existence of inequivalent phases by providing concrete examples of models between which any continuous deformation must involve a phase transition. Notably, given that such distinct phases exist, by definition their distinction does not involve a local order parameter or confinement-deconfinement transitions. Their difference is of topological origin. This is manifested on interfaces between distinct phases, where either long range correlated or quasi long range correlated phases emerge. In the next sections we will explore these ideas for the choice $G=G'=Z_N$, considering models in both $2D$ and $3D$ where we find many distinct topological phases with the accompanying exotic boundary phenomena. The latter include a “forbidden" [@VanHove1950] symmetry breaking order along $1D$ boundary and an unusual $2D$ critical phase corresponding to a theory of a compact boson in which the basic $\pm 2\pi$ vortices are linearly confined. Just as group cohomology was shown to be the basis for quantum SPTs phases, we will show that tools from cellular cohomology [@Cohomology1991] give a powerful mathematical framework for writing down models of CTPs and analyzing them. The models thus produced are compact, isotropic and, to a large extent, analytically tractable, thereby increasing both their theoretical and experimental relevance. The $G=G'={\mathbb{Z}}_2$ models in $2D$ and $3D$ are further shown to be in the same universality class as the imaginary time partition function of certain $1D$ and $2D$ models (the group cohomology models [@Chen2011]) of bosonic SPTs. From a numerical perspective our models thus provide an efficient way for performing Monte-Carlo simulations of bosonic SPTs with discrete symmetries (see also Ref.  for the continuous case). They also open a new and more promising experimental route for studying these fascinating strongly interacting phases of matter. Two dimensions ============== As a first illustrative example of a $2D$ CTP with $G={\mathbb{Z}}_2$ we consider the following model on the square lattice: $$\begin{aligned} \label{eq:Z_sigma_A} Z &= \sum_{\sigma,U}e^{-\beta\mathscr{H} } \prod_{p}\delta(U_{ij}U_{jk}U_{kl}U_{li} - 1)\, ,\\ \label{eq:model} -\beta\mathscr{H} &= \sum_{\langle i,j\rangle} \left\{ K_1 \sigma_i U_{ij}\sigma_j + K_2 U_{ij}\right\}\, .\end{aligned}$$ Here $\sigma_i=\pm 1$ and $U_{ij}=\pm 1$ are site and link variables, and the product is over plaquettes $p$, having the sites $i,j,k,l$ on their boundary. The model has a ${\mathbb{Z}}_2$ symmetry $\sigma_i\to -\sigma_i$, and it has a ${\mathbb{Z}}_2$ constraint forcing zero flux for the $U$ field through each plaquette. Conveniently, a non-local transformation ($U_{ij}=\mu_i\mu_j$) maps this model to two decoupled Ising models, and has thus ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry: $$\begin{aligned} \label{eq:Zsmu} Z &= \sum_{\sigma,\mu} \exp \sum_{\langle i,j\rangle} \left\{ K_1 \rho_i\rho_j + K_2 \mu_i\mu_j\right\}\, , & \rho_i &= \sigma_i \mu_i \, .\end{aligned}$$ Denoting $K_c = -\frac{1}{2}\log\tanh K_c$ the critical coupling of the Ising model on a square lattice, there are two regimes which are of interest to us: The trivial phase ($K_2>K_c>K_1\ge 0$) where $\langle \mu_i\rangle\neq 0$, and the non-trivial phase ($K_1>K_c>K_2\ge 0$) where $\langle \rho_i\rangle\neq 0$. The other variables, $\rho$ and $\sigma$ for the trivial case and $\mu$ and $\sigma$ for the non-trivial, are disordered. Notably, in both cases $U_{ij}$’s are uncorrelated, namely $\left\langle (U_{ij}-\langle U_{ij} \rangle) (U_{kl}-\langle U_{kl} \rangle) \right\rangle$ is exponentially decaying. [^1] We remark that the partition function of Eq. (\[eq:model\]) with constraint violations at two plaquettes equals that of Eq.  where the sign of both couplings $K_1,K_2$ is reversed along a path connecting the two plaquettes [@Savit]. Thus for both regimes, the presence of order parameters with long range order implies linear confinement of the defects. In terms of $\rho$ and $\mu$, the model is simply two decouple ferromagnets that exhibit symmetry broken phases. However, in the original degrees of freedom, $U,\sigma$, the physical properties of the two phases change. Considering bulk physics, long range order in $\rho$ implies the following non local (string) order parameter in the non-trivial phase: $$\begin{aligned} \label{eq:sAs} \left< \rho_i \rho_j \right> &=\left< \sigma_i\prod_{\ell \in \Gamma_{ij}}U_{\ell} \sigma_j\right> \to \text{const}\, \end{aligned}$$ as $\text{dist}(i,j)\to\infty$ and $\Gamma_{ij}$ is a path from $i$ to $j$. Alternatively stated, performing the non-local transformation $\sigma_i \rightarrow \rho_i=\prod_{\ell\in\Gamma_{0i}}U_\ell \sigma_i$, with $0$ a reference site, unveils a hidden ferromagnetic phase for the non-trivial order, whereas for the trivial phase, this results in a simple paramagnet. As we now argue the hidden ferromagnetic order is a distinguishing property of the topological phase and therefore one cannot continuously deform the models onto one another. This implies that there are at least two distinct admissible phases in our classification for $d=2,G=G'=Z_2$. Notably local and symmetric perturbations in the original $U$ and $\sigma$ variables would be transformed into local and symmetric perturbations in $\mu$ and $\sigma$. As this transformation has no effect on the free energy, one finds that hidden order is thermodynamically equivalent to conventional order. This means that hidden order not just a feature of the model but rather a robust property which can only vanish through a phase transition or by leaving the space of admissible phases. Perhaps the most interesting distinction between these two phases comes about when considering a $1D$ interface between them. In general, near an interface between a ferromagnet and a paramagnet, the order parameter leaks into the paramagnetic phase up to some penetration length. Similarly, close to an interface between the above two phases both order parameters ($\rho$ and $\mu$) will be ordered and as a result $\sigma = \rho\cdot \mu$ would also be ordered, despite being disordered in the bulk on both sides. For instance, setting $K_1=0,K_2\to \infty$ on the trivial side is equivalent to placing the non-trivial phase in an open geometry with boundary conditions $U_{ij}=1$ or equivalently $\mu_i=\mu_j$, implying long range order for $\sigma$. More physically, one can view the configurations of $U$ in as polygons on the dual lattice by assigning a line of the polygon to links across which $U=-1$. The $K_1$ coupling then encourages domain walls of the spins to attach to these polygons. Kinks of $\sigma$ along the interface are necessarily ends of domain walls in the bulk. However these domain walls cannot have an accompanying polygon as the latter is confined from entering the trivial phase (vacuum in the picture). Consequently the bulk, despite being locally disordered, linearly confines kinks of $\sigma$ at the boundary into neutral pairs (see Fig. \[fig:2D\_conf\]). Relation with the AKLT Hamiltonian {#sec:AKLT} ---------------------------------- We now establish a precise connection between the $2D$ CTP presented and the AKLT model, the paradigmatic example of a quantum SPT phase of spins in $1+1D$ [@AKLT1988]. (See also [@chen2014symmetry] for a picture of AKLT that is close to our construction.) We consider the transfer matrix of the $2D$ CTP in the limit of anisotropic coupling $K^x_i=\epsilon \lambda_i\, , e^{-2K_i^y}=\epsilon \lambda_i'$, $i=1,2$, along the horizontal ($x$) or vertical direction ($y$). It is then a standard exercise (see e.g. [@Kogut]) to derive the quantum Hamiltonian in the limit $\epsilon\to 0$ starting from Eq.  in the main paper, and to pass from the $\mu$ variables to their duals $\tau$. This results in the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetric Hamiltonian $H=H_0+\sum \lambda_2\tau_{i+1/2}^x + \lambda_1'\sigma_i^x$, where $$\begin{aligned} H_0 = \sum \lambda_1 \sigma_i^z\tau_{i+1/2}^x\sigma_{i+1}^z + \lambda_2'\tau_{i-1/2}^z\sigma_i^x\tau_{i+1/2}^z \, ,\end{aligned}$$ and which coincides with the AKLT Hamiltonian in the form considered in [@Ringel2015] for $\lambda_1=\lambda_2'$. Having equivalent phenomenology and a very similar algebraic structure strongly suggests that these two models describe the same phase. Interestingly, when expressing our model in terms of the dual variables $\tau$, the Boltzmann weights are not positive anymore. The ${\mathbb{Z}}_2$ constraint thus appears as a natural way to reflect the additional ${\mathbb{Z}}_2$ symmetry while maintaining positive Boltzmann weights and locality. Generalizations to $G=G'=Z_N$ {#sec:gen_2d} ----------------------------- Let us generalize the above model to the case $G=G'=Z_N$. Accordingly, we consider a directed square lattice and take $\sigma_i \in {\rm Z}_N$ and $U_{ij} = U_{ji}^{-1} \in {\rm Z}_N$ for the orientation being from vertex $i$ to $j$. We represent elements in ${\rm Z}_N$ by $e^{2\pi i \alpha/N}$, $\alpha= 0,1,\dots,N-1$. For a given $p=0,1,\dots,N-1$, let us define the minimal coupling: $$\begin{aligned} \mathscr{H}_{p} &= \sum_{i}\sum_{j\sim i} \sigma^p_i U_{ij} \sigma^{-p}_j\, ,\end{aligned}$$ where $j\sim i$ means $j$ a neighbour of $i$, so that each edge is counted twice, once with its positive and once with its negative orientation ensuring a real energy. Given non-zero $p\neq p'$ the generalized model is defined by with $$\begin{aligned} \label{eq:model_ZN} -\beta\mathscr{H}_{p,p'} &= K_1 \mathscr{H}_{p} + K_2 \mathscr{H}_{p'} \, .\end{aligned}$$ As we will show, for large $K_1$ ($K_2$) p ($p'$) controls the topological index. Let us note that $\sigma^p$ is a ${\mathbb{Z}}_N$ variable only when $p$ and $N$ are co-prime. Otherwise, it has a reduced order, given by $N/p$. In order to keep the physical message of this section clear and concise, we do not delve here in these number theoretic considerations, and assume $N$ to be prime. To analyze the model we first expose the hidden order. To this end we resolve the constraint using $$\begin{aligned} U_{ij} &= \mu_i \mu^{-1}_j\end{aligned}$$ yielding $$\begin{aligned} \mathscr{H}_{p} &= \sum_{i}\sum_{j\sim i} \sigma^p_i \mu_i \mu^{-1}_j \sigma^{-p}_j\, ,\end{aligned}$$ and $$\begin{aligned} Z &= \frac{1}{N}\sum_{\sigma,\mu}e^{-\beta\mathscr{H}}\, ,\end{aligned}$$ where the factor of $\frac{1}{N}$ comes from the $1$ to $N$ mapping between $U_{ij}$ which respect the constraint and $\mu_i$. Next we wish to go to the composite variables $$\begin{aligned} \tilde{\sigma}_{i;p} = \mu_i \sigma^p_i\, ,\quad \tilde{\mu}_{i;p'} = \mu_i \sigma_i^{p'} \, .\end{aligned}$$ The assumption of $N$ prime guarantees that they are in ${\mathbb{Z}}_N$, and the assumption of $p\neq p'$ and a non-zero $p$ guarantees the mapping to be invertible. The indices $p,p'$ make explicit the dependence on $p$ and $p'$ in the definition of $\tilde{\sigma}$ and $\tilde{\mu}$. We thus find two decoupled $Z_N$ clock models, $$\begin{aligned} -\beta\mathscr{H}_{p,p'} &= \sum_i\sum_{j\sim i} \left(K_1 \tilde{\sigma}_{i;p} \tilde{\sigma}_{j;p}^{-1} + K_2 \tilde{\mu}_{i;p'}\tilde{\mu}_{j;p'}^{-1}\right)\, ,\end{aligned}$$ one in the composite variable $\tilde{\sigma}_p$ and the other in the composite variable $\tilde{\mu}_{p'}$. Now we suppose that the couplings are such that one of the two variables, say $\tilde{\sigma}_p$, is ordered (recall that if $N$ is prime, ${\mathbb{Z}}_N$ models can have only a single symmetry broken phase), and that $\mu$ is disordered. Notably, since $\tilde{\mu}_{p'} = \mu \sigma^{p'}$ this also implies that $\tilde{\mu}_{p'}$ is disordered for all $p' \neq p$. We then claim that under these conditions the model is in a “topological phase of type $p$”. Three questions need to be answered to justify this statement: (i) why is this a phase (ii) why do different $p$’s correspond to distinct phases and (iii) why are they topological. Considering the first point note that the hidden order of the $\sigma$ variables manifested by order in $\tilde{\sigma}_p$ is a robust property. Indeed as argued in the previous case of an Ising symmetry, any local symmetric and defect-free perturbation in original model would map to a local term in the $\tilde{\sigma}_p$ and $\tilde{\mu}_{p'}$ degrees of freedom. Thus robustness of the topological phase is implied by the usual robustness of broken symmetry states. Turning to the second point, and the role of $p$, we can simply note that two different values of $p$ correspond to two different order parameters and thus two different phases. Indeed if $\tilde{\sigma}_{i;p}$ is long range ordered then $\tilde{\sigma}_{i;p'}$ must be disordered as it is equal to a power of $\tilde{\sigma}_{i;p}$ times a non-trivial power of the disordered variable $\mu_i$. Lastly, we justify the nomenclature topological. By this we mean that an interface between two distinct admissible phases would contain some form of long range or quasi long range order. Consider such an interface between a $p$ topological phase and a $p'$ topological phase. This scenario can be engineered by setting $K_2=0$ and $\tilde{\sigma}_p$ ordered on one side of the interface, and $K_1=0$ and $\tilde{\mu}_{p'}$ ordered on the other. On the interface these two order parameters will leak and so $(\mu_i \sigma^p_i)(\mu_i \sigma^{p'}_i)^{-1} = \sigma^{p-p'}_i$ would be ordered. Notably the latter, and only the latter, is a local order parameter and thus we have shown the existence of 1D long range order on such interfaces Three dimensions ================ Next we wish to generalize the above construction to $3D$. In $2D$ we attached closed polygons to domain walls of the spins. Turning to $3D$, polygons on the dual lattice appear naturally in ${\mathbb{Z}}_2$ gauge theories, where they correspond to discrete flux lines. However domains walls become $2D$ objects, and we instead look for a property of the spins that can also be described in terms of polygons. Such a spin quantity has been studied recently in and can be thought of as an algebraic generalization of the usual continuum notion of vorticity. Consider a cubic lattice and orient links and plaquettes. Next place a spin variable $\sigma=\pm 1$ at each vertex. The discrete vorticity $\omega_p$ on a plaquette $p$ is defined as $$\begin{aligned} \label{eq:omegap} \omega_p = \frac{1}{2} \sum_{(ij)\in \partial p}\epsilon_{ij}^p\frac{1-\sigma_i\sigma_j}{2} \, ,\end{aligned}$$ where the sum is over links on the boundary of $p$ and $\epsilon_{ij}^p=1$ if the link is oriented as the plaquette, and $-1$ otherwise. We remark that $\omega_p=0,\pm 1$ and the choice of plaquette orientation has no effect on the $\mathbb{Z}_2$ quantity $(-1)^{\omega_p}$ that we consider below. For definiteness we choose orientations as in figure \[fig:conf\_omega0\]. An intuitive view on discrete vorticity comes form thinking of the spins $\sigma_i =+1,-1$ as the elements $0,1$ in $\mathbb{Z}_2$. Then $\omega_p$ appears as the discrete integral (i.e. a sum) around a plaquette over the discrete derivatives $\frac{1}{2}(1-\sigma_i\sigma_j) \in\mathbb{Z}_2$. Here it is important to interpret the discrete derivative as a variable in $\mathbb{Z}$ rather than in $\mathbb{Z}_2$, and hence this sum can be non-zero multiple of $|\mathbb{Z}_2|=2$. This is analogous to what one does when calculating vorticity of a U$(1)$ variable ($\phi$) where derivatives ($i \phi^{-1} \partial_l \phi$) are taken in U$(1)$ but then integrated over as elements in $\mathbb{R}$ whose sum can now be a non-zero multiple of $2\pi$. In analogy with usual vorticity, the discrete vorticity obeys a discrete version of the zero divergence constraint: Given any box on the square lattice, $\sum_{p\in\text{box}}\omega_p = 0\mod 2$. This can be shown by noting that for each box we can choose a clockwise orientation (when looking from inside the box) on each plaquette. Consequently, each link on the box would appear exactly twice with opposite values of $\epsilon_{ij}^p$. Therefore discrete vorticity lines form polygons on the dual lattice which obey the exact same branching rules as fluxes in a ${\mathbb{Z}}_2$ gauge theory. Tools from lattice gauge theory, specifically cellular and simplicial cohomology, shed further light on this quantity. A thorough discussion of these aspects are relegated below in section \[sec:cohom\] where they will be used to define discrete vorticity for other abelian groups. Armed with the notion of discrete vorticity and its properties, we can now introduce the $3D$ model. Consider spins $\sigma_i$ on the vertices of a cubic lattice and ${\mathbb{Z}}_2$ gauge variables $A_{ij}$ on the links, and choose the following energy $$\begin{aligned} \label{eq:E_3D} -\beta\mathscr{H} = J_1 \sum_{p} (AAAA)_p + J_2 \sum_{p} (-)^{\omega_p} (AAAA)_p\, ,\end{aligned}$$ with $(AAAA)_p$ being the product of the four $A_{ij}$ surrounding the plaquette $p$. In analogy with our $2D$ analysis we would now want to perform some non-local transformation to decouple the gauge variables from the spins. Even though both flux and vorticity lines form closed polygons, the number of distinct flux configurations, which spans all such polygons, is bigger than that of vorticity configurations which only span a subset. Therefore, for any vorticity there exists a matching flux although the converse is not true. It follows that there exists $A_{\sigma}$ such that $(A_{\sigma}A_{\sigma}A_{\sigma}A_{\sigma})_p = (-)^{\omega_p}$. Defining $\tilde{A} = A A_{\sigma}$, we obtain $$\begin{aligned} \label{eq:Atilde} -\beta\mathscr{H} &= J_1 \sum_{p} (-)^{\omega_p} (\tilde{A}\tilde{A}\tilde{A}\tilde{A})_p + J_2 \sum_{p} (\tilde{A}\tilde{A}\tilde{A}\tilde{A})_p \, .\end{aligned}$$ There are two points in phase space where the gauge and spin degrees of freedom decouple. The trivial case is $J_2 = 0$ which implies free $\sigma$’s and a standard ${\mathbb{Z}}_2$ gauge theory for the $A$’s. For $J_1>J_c$, where $J_c=0.762(2)$ is the critical temperature of the dual Ising model on the cubic lattice, the gauge theory has a perimeter law for Wilson loops and linearly confines monopoles (open flux lines), but deconfines static charges of the gauge field [@Savit]. The non-trivial case is $J_2 > J_c$ and $J_1=0$ and has the same confining bulk physics only in the composite gauge variable $\tilde{A}$. Notably the transformation $\tilde{A} = A A_{\sigma}$ can be viewed as acting on the flux degrees of freedom by multiplying them with vorticity lines. Since vorticity lines consist of closed polygons, this transformation leaves the monopole configuration unchanged. Consequently the non-trivial phase also confines monopoles. See figure \[fig:conf\] for a representation of the non-trivial phase. The above CTP is a robust phase of matter. As in the $2D$ model, the non-local transformation $\tilde{A}=A A_{\sigma}$ maps local symmetric and gauge symmetry respecting operators, into local ones, and leaves the free energy invariant. Respecting these symmetries, both the monopole confining phases of $\tilde{A}$ and $A$ are well defined phases [@Fradkin1979]. In addition, we found that breaking the gauge symmetry on an interface or boundary does not destroy the surface physics (see below) suggesting that gauge symmetry is not crucial here. Surface theory -------------- To establish the distinction between trivial and non–trivial phases and to support this nomenclature, we now discuss an interface. For concreteness we take coordinates $(x,y,z)\in{\mathbb{Z}}^3$ for the vertices of the lattice and identify the interface as the $x=0$ plane. We also denote $P_L$ ($P_R$) the plaquettes in the region $x\le 0$ ($x>0$). In the limit $J_2,J_1\to\infty$, $(AAAA)_{\tilde{p}}=1$ for $\tilde{p} \in P_R$. By conservation of flux, we find that for all boundary plaquettes $p \in \partial P$, $(AAAA)_{p}=1$. Consequently since $J_2$ forces $(-)^{\omega_p}(AAAA)_p=1$, $\omega_p=0$ on the $2D$ boundary. The surface partition function in this limit is thus given by $$\begin{aligned} \label{eq:Z_surf} Z_{\text{surf},0} &= \sum_{\sigma} \prod_{p\in \partial P}\delta({\omega_p}) = \sum_{\sigma,\tau} \prod_{p\in \partial P}(\tau)^{\omega_p}\, .\end{aligned}$$ The possible domain wall configurations for $\sigma$’s in $2D$ are depicted in Fig. \[fig:conf\_omega0\] where a second mapping to arrow configurations of the eight-vertex model is also discussed. The constraint $\omega_p=0$ implies a two-in two-out ice rule, supporting the vorticity interpretation and mapping the surface theory to the critical six vertex model with an anisotropy parameter $\Delta = \frac{1}{2}$ [@Baxter]. The latter model model is critical and described by a compact free boson $\phi$. This fact can be established with the Coulomb gas method [@Nienhuis1984], which we now briefly recall. Denoted by $S_{\ell}=\pm 1$ the arrow at link $\ell$, note that $S$ is conserved around a vertex, and one can introduce a height field $h(i)$ on the same sites where $\sigma$ lives, such that $h$ increases by $\pi$ in crossing an arrow pointing up from the right. This discrete height renormalizes at long distances to a Gaussian free field, a conformal field theory with central charge $c=1$, and via this mapping one can compute dimensions of operators. Noting that $ \sigma_i\sigma_j = \prod_{\ell\in \Gamma_{ij}} -ie^{i\pi S_\ell /2 } \propto e^{i h(i)/2 }e^{-i h(j)/2 } $, $\sigma$ is found to have scaling dimension $3/8$. Similarly, noting that the two point function of $\tau$ in eq.  corresponds to inserting two vortices where the height field has discontinuity of $\pm 4\pi$, $\tau$ has dimension $2/3$. Identifying $\phi\equiv h/2$, one has the effective theory $$\begin{aligned} \mathscr{L} &= \frac{g}{4\pi} (\nabla \phi)^2 \, , \quad g=\frac{4}{3}\, .\end{aligned}$$ The appearance of half integer electric charges follows also naturally by considering the torus partition function. Indeed on $4L\times 4L'$ lattices, periodic boundary conditions for the $\sigma$’s select only even frustrations for the height field as it winds around a cycle, resulting in half integer electric charges and even magnetic charges. Microscopically, $\sigma$ is a Hermitian linear combination of $e^{\pm i\phi}$ and $\tau$ of $e^{\pm i\theta}$, $\theta$ being the dual field. Therefore, the symmetry is realized as anticipated in the main text: $\phi \rightarrow \phi + \pi$ and $\theta \rightarrow \theta + \pi$, as it does in quantum SPTs [@YuanMing2012; @Nayak2014]. We also note that even though the local weight has no such symmetry, the global weight still has it, due to the global constraint $\prod_p (-1)^{\omega_p}=1$ for a closed manifold. From this analysis it follows that the lattice ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry is realized in the field theory in an anomalous chiral way: $\phi \rightarrow \phi + \pi$ and $\theta \rightarrow \theta + \pi$, where $\theta$ is the dual field. Let us consider perturbations to this surface model. Adding a $\sigma\sigma$ term to the boundary action corresponds to the six vertex model in an external field. Denoted by $H/2$ and $V/2$ the horizontal and vertical couplings, the theory remains critical within the region $(e^{2|H|}-1)(e^{2|V|}-1)\le 1$ [@Reshetikhin], the only effect of $H,V\neq 0$ being renormalizing the stiffness of $\phi$ [@Kim]. A ferromagnetic coupling between the $\tau$’s would generically induce the RG-irrelevant term $\cos(2\theta)$. Interestingly, the relevant $\cos(\theta)$ term is forbidden without requiring any fine tuning of the couplings. Formally, it is because of the emergent ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ symmetry. Physically, it is because $\pm 2\pi$ vortices are linearly confined by the bulk (see Fig. \[fig:conf\]). Further, a gauge symmetry breaking term ($K\sum_{\ell\in\partial E}A_{\ell}$) can also be studied using duality [@Balian] and has no effect on the $\sigma$’s in the limit $J_2,J_1\to\infty$. The SPT perspective {#sec:SPTpersp} ------------------- As discussed in section \[sec:AKLT\] the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ two-dimensional classical topological paramagnet can be related to the imaginary time partition function of a $1+1D$ quantum SPT phase. In this section we provide support for the analogous statement in $3D$, proving that all of the above models are in the same universality class as the Euclidean time partition function of certain $2+1D$ quantum SPTs. We will show this by analyzing the responses to gauge fluxes, or equivalently, the statistical phases obtained by braiding flux excitations. As starting point we perform a gauge-to-Ising duality transformation on the bulk [@Balian] trading $A$’s for spins $\tau$’s on the vertices of the dual lattice, resulting in an equivalent bulk theory with weights: $$\begin{aligned} \label{eq:Atau} \prod_{p\in P_L}(\tanh J_2)^{\frac{1-\tau_k\tau_l}{2}} \prod_{p\in P_R}(\tanh J_1)^{\frac{1-\tau_k\tau_l}{2}}(\tau_k\tau_l)^{\omega_p}\, ,\end{aligned}$$ where $kl$ is the link dual to $p$. The term $\prod_{p \in P_R}(\tau_{k}\tau_{l})^{\omega_p}$ is in fact topological. It is always one in a geometry without interfaces, since then vorticity lines where $\omega_p=\pm1$ form polygons, and in the product of $\tau_{k} \tau_{l}$ along each such polygon, each $\tau$ appears an even number of times, and hence the product is always one. Focusing on the analytically tractable case of $J_1=0$ leaves us with the partition function $$\begin{aligned} \label{Eq:ModelTau} Z = \sum_{\tau,\sigma} \prod_p e^{\tilde{J_2} \tau_{k} \tau_{l}} (\tau_{k} \tau_{l})^{\omega_p}\, ,\end{aligned}$$ where $\tilde{J_2} = \frac{1}{2}\log(\tanh(J_2))$, and here and below $(kl)$ is the link dual to the plaquette $p$. Since this model now has a ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ symmetry, it is natural to seek a quantum counterpart which utilizes such a symmetry, and these are known as type $ii$ SPT phases [@Chen2011; @Juven2015; @ModularData]. These SPTs are characterized by a quantized bulk response to static gauge fluxes. For a ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ symmetry a $\pi$ Ising flux for one symmetry would attract a fractional symmetry charge of the other symmetry. This is the discrete analogue of flux attachment in the integer quantum Hall effect, where a $\pi$ flux would attract half an electron charge [@Laughlin1981]. If our model belongs to the same phase as that described by the imaginary time partition function of one of such $2+1D$ SPTs, it should exhibit the same flux responses. We therefore introduce two additional static gauge fields ($B^{\sigma},B^{\tau}$) which are coupled to matter in the usual manner: we trade each $\tau_{k} \tau_{l}$ with $\tau_{k} B^{\tau}_{kl} \tau_{l}$ and each $\sigma_{i} \sigma_{j}$ with $\sigma_{i} B^{\sigma}_{i j} \sigma_{j}$. The adjective static refers to the fact that they are not summed over in the partition function, which is then: $$\begin{aligned} Z(\{B^\tau\},\{B^{\sigma} \}) &= \frac{1}{Z} \sum_{\tau,\sigma} \prod_{p} e^{\tilde{J_2} \tau_{k}B^\tau_{kl} \tau_{l} } (\tau_{k}B_{kl}^\tau \tau_{l})^{\omega_p(B^\sigma)} \, ,\end{aligned}$$ where $Z\equiv Z(\{1\},\{1\})$ as above. If we require that both fluxes are zero everywhere, namely $\prod_{(ij)\in\partial p} B_{ij}^\sigma=\prod_{(kl)\in\partial p^*} B_{ij}^\tau=1$, where $p^*$ is a dual plaquette, we can rewrite $B_{ij}^\sigma=\tilde{\sigma_i}\tilde{\sigma}_j$, $B_{kl}^\tau=\tilde{\tau_k}\tilde{\tau}_l$, and reabsorb the $B$’s in the definition of $\sigma,\tau$. Thus introducing gauge fields with zero flux is equivalent to set them to $1$. When coupling to gauge fields, from formula (5) of the main paper the vorticity becomes $$\begin{aligned} \label{eq:omegap_B} (-)^{\omega_p(B^\sigma)} = \prod_{(ij)\in\partial p} \exp\left(i\pi\frac{1-\sigma_iB_{ij}^\sigma \sigma_j}{4} \epsilon^p_{ij}\right) \, .\end{aligned}$$ If we now violate the zero flux constraint, then $(-)^{\omega_{p}(B^\sigma)}$ can assume the additional values $\pm i$ on top of $\pm 1$ which it had before. A related issue to be discussed is the definition of plaquette orientations which enter the sign $\epsilon_{ij}^p$. Changing plaquette orientations corresponds to change the exponent of by an overall sign. For zero $B^{\sigma}$ flux, this choice is immaterial; however in the case of $\pi$ flux it does matter. For definiteness we choose to orient both links and their dual as the positive direction of the axis of three dimensional space they are parallel to, and adopt a left-hand rule for defining clock-wise/anti-close-wise plaquette orientations. The topological quantity we wish to calculate concerns the flux responses in type $ii$ SPT phases with a ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ symmetry and we now recall its definition. Consider then a quantum SPT model with ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ symmetry on a two dimensional lattice, and denote by $\sigma^{x,z},\tau^{x,z}$ the elementary spin operators, and by $|\text{gs}\rangle$ its ground state. It can be shown [@Juven2015] that the insertion of a $\pi-$flux associated with one of the symmetries draws in a fractional symmetry charge associated with the other symmetry. To probe this we introduce two $B^{\tau}$ $\pi$ fluxes into the system by creating them and taking them apart at positions $a,b$. Note that these excitations are string like and a string will be attached to these two fluxes. Their worldlines draw a surface $S_1$ in space time whose interior is swiped by the string. The system is then let to evolve until it reaches its new ground state, and we denote the operator that performs this operation by $\pi_{ab}$. Further, we denote by $S_2$ the set of vertices on a region surrounding only one of the fluxes and choose this region to be larger than the correlation length. The operator $\rho_{S_2}=\prod_{i \in S_2} \sigma_{i}^x$ can be interpreted in two ways. First as creating, evolving and annihilating two $B^\sigma$ $\pi$ fluxes along the boundary of $S_2$. Second as a measurement of the local Ising charge around just one flux. In a non-trivial type $ii$ SPT with a ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ symmetry, the ratio $\langle \text{gs} | \pi^{\dagger}_{ab}\rho_{S_2} \pi_{ab} |\text{gs} \rangle/\langle \text{gs} | \rho_{S_2} \pi^{\dagger}_{ab} \pi_{ab} |\text{gs} \rangle$ should be equal to $\pm i$ [@Juven2015], the sign depending on which of the two $B^\tau$ fluxes is encircled by $S_2$. According to the previous discussion one can view this as the phase associated with braiding the two flux excitations (in similar spirit to Ref. ) or alternatively as a generalization of Laughlin’s pumping argument to discrete symmetry as the $\pi$-flux draws in half an Ising symmetry charge (recall that in this multiplicative notation, an Ising charge is $-1$ and so half a charge is $\pm i$). Upon switching to imaginary time, the quantum mechanical overlaps making up this ratio can be reformulated as partition functions. The factor $\langle \text{gs} | \pi^{\dagger}_{ab} \rho_{S_2} \pi_{ab} |\text{gs} \rangle$ is illustrated in Fig. (\[Fig:Braiding\](a)), where across the $S_1$ surface (blue) the interaction between the $\tau$’s is reversed and across the $S_2$ surface (green) the interaction between the $\sigma$’s is reversed. As in the main text, links where the interaction is reversed are referred to as frustrated. The factor $\langle \text{gs} | \rho_{S_2} \sigma_{i}^x \pi^{\dagger}_{ab} \pi_{ab} |\text{gs} \rangle$ illustrated in Fig. (\[Fig:Braiding\](b)) contains the same two elements, however now these are separated in imaginary time. More specifically, let us denote by $G$ and $G^*$ the lattice and its dual, where $\sigma$ and $\tau$ respectively live. As defined, $S_1$ and $S_2$ will be a connected region of $G$ and $G^*$ (note the order of $G$ and $G^*$) across which the $\tau$ and $\sigma$ couplings respectively are reversed. By a region here we mean a set of neighbouring plaquettes and links around them on both the interior and the boundary of the region. Since it will be clear from the context, we we will write $(kl) \in S_2$ for links in the region $S_2$. Further, $\partial S_i$ will denote the set of links on the boundary of $S_i$. We remark that frustrated links intersecting $S_1$ ($S_2$) correspond to introducing a $B^\tau$ ($B^\sigma$) $\pi$ flux on the plaquettes intersecting $\partial S_1$ $(\partial S_2)$, consistently with the above discussion. Before delving into the details of calculating the relevant ratio, let us give a physical picture supporting why it would come out purely imaginary. In the presence of non-trivial fluxes, the relation $\sum_{p\in\text{box}}\omega_p(B^\sigma)=0\mod 2$ does not hold in general. Instead one has an altered $Z_4$ zero–divergence relation given by $2(\sum_{p\in\text{box}}\omega_p(B^\sigma)O^{box}_p)=0\mod 4$, where $O^{box}_p=1$ ($-1$) if the plaquette’s orientation appears as clock-wise (anti-clock-wise) when viewed from within the box. Using this new relation one may show that the vorticity line configuration in the presence of the $B^{\sigma}$ flux loop contains a single fractional vorticity line encircling $S_2$ as well as other fluctuating integer vorticity lines. Given the form of the topological term, the integer vorticity lines cannot contribute imaginary factors and so we may put them aside for now. Considering the fractional vorticity line, if it does not cross $S_1$ (case (b)), the term $\Pi_{(kl) \in \partial S_1} (\tau_k \tau_l)$ is equal to $1$. Consequently the topological term, which involves a fractional power of this product, cannot give an imaginary contribution. On the other hand, if this fractional vorticity line crosses $S_1$ (case (a)), this product would be $-1$, and the topological term would be purely imaginary. We now substantiate the above argument with some simple and exact computations. First, notice that there are four cases to consider for the weight $w(kl)$ per dual link $(kl)$, in case frustrations for both $\tau$ and $\sigma$ are present: $$\begin{aligned} w(kl) = \begin{cases} e^{\tilde{J}_2\tau_k\tau_l}(\tau_k\tau_l)^{\tilde{\omega}_p} & 1): kl \in S_2, \not\cap S_1\\ e^{-\tilde{J}_2\tau_k\tau_l}(-\tau_k\tau_l)^{\tilde{\omega}_p} & 2): kl \in S_2, \cap S_1\\ e^{-\tilde{J}_2\tau_k\tau_l}(-\tau_k\tau_l)^{\omega_p} & 3): kl \not\in S_2, \cap S_1\\ e^{\tilde{J}_2\tau_k\tau_l}(\tau_k\tau_l)^{\omega_p} & 4): kl \not\in S_2, \not\cap S_1 \end{cases}\, ,\end{aligned}$$ where $\tilde{\omega}_P$ corresponds to $\omega_p(B^\sigma)$ with frustrated links where $B^\sigma=-1$. Defined the set of couplings $$\begin{aligned} \hat{B}^\tau_\ell &= \begin{cases} -1 & \ell \cap S_{1}\\ 1 & \ell \not\cap S_{1} \end{cases}\, ,\quad \hat{B}^\sigma_\ell &= \begin{cases} -1 & \ell \cap S_{2}\\ 1 & \ell \not\cap S_{2} \end{cases}\, ,\end{aligned}$$ the observable of interest is $$\begin{aligned} &Z(\{\hat{B}^\tau\},\{\hat{B}^\sigma\}) = \frac{1}{Z} \sum \prod_{kl \in S_2, \not\cap S_1} e^{\tilde{J}_2\tau_k\tau_l}(\tau_k\tau_l)^{\tilde{\omega}_p}\\ &\quad\prod_{kl \in S_2, \cap S_1} e^{-\tilde{J}_2\tau_k\tau_l}(-\tau_k\tau_l)^{\tilde{\omega}_p} \\ &\quad\prod_{kl \not\in S_2, \cap S_1}e^{-\tilde{J}_2\tau_k\tau_l}(-\tau_k\tau_l)^{\omega_p} \\ & \quad\prod_{kl \not\in S_2, \not\cap S_1} e^{\tilde{J}_2\tau_k\tau_l}(\tau_k\tau_l)^{\omega_p} \\ &= \frac{1}{Z} \sum \prod_{kl \cap S_1} e^{-\tilde{J}_2\tau_k\tau_l}(-\tau_k\tau_l)^{\omega_p}\\ &\quad\prod_{kl \not\cap S_1} e^{\tilde{J}_2\tau_k\tau_l}(\tau_k\tau_l)^{\omega_p} \prod_{kl \in S_2} (\tau_k\tau_l)^{\tilde{\omega}_p-\omega_p}\\ &\quad \prod_{kl \in S_2,\cap S_1} (-1)^{\tilde{\omega}_p-\omega_p}\, .\end{aligned}$$ At this point we use the following identity: $$\begin{aligned} \label{eq:canceltau} \prod_{kl\in S_{2}} (\tau_k\tau_l)^{\tilde{\omega}_p-\omega_p} =1 \, .\end{aligned}$$ To prove it, first notice that given the choice of orientation described in the text above, ${\tilde{\omega}_p-\omega_p}$ gives a factor $\epsilon_{ij}^p \sigma_i\sigma_j/2$ per frustrated link $ij$. Then group together all $\tau$’s having a given exponent $\sigma\sigma'/2$. $\tau$’s appears in pairs for any choice of bond $\sigma\sigma'$, and cancel either because $\tau^2=1$ or because $\tau\tau^{-1}=1$. We now rewrite the partition function in terms of the original $A$ gauge degrees of freedom to take advantage of the change of variables $A\to \tilde{A}$ as in eq. , which decouples gauge and spin degrees of freedom. Reversing the couplings along $S_{1}$ for the $\tau$’s corresponds in the $A$ language to computing the Wilson loop along the perimeter of $S_{1}$ (see e. g. [@Kogut]), so that one has: $$\begin{aligned} &Z(\{\hat{B}^\tau\},\{\hat{B}^\sigma\}) = Z^{-1}\sum \prod_{p\in S_1} (AAAA)_p\\ &\prod_p e^{J_2 (AAAA)_p (-)^{\omega_p}} \prod_{p \in S_1,\cap S_2}(-1)^{\tilde{\omega}_p-\omega_p}\\ &= \left< \prod_{\ell\in\partial S_1} \tilde{A}_\ell \right>_{\tilde{A}} \left< \prod_{p\in S_1} e^{i\pi\omega_p} \prod_{p \in S_1,\cap S_2}e^{i\pi(\tilde{\omega}_p-\omega_p)}\right>_{\sigma}\, \end{aligned}$$ where the average $\langle ... \rangle_{\tilde{A}}$ is taken with the partition function of $\tilde{A}$’s alone, and the average $\langle ... \rangle_{\sigma}$ is taken with the trivial partition function for the $\sigma$’s that gives a weight of $1$ to each $\sigma$ configuration. The last term in the $\sigma$ expectation values involves the links illustrated in figure \[fig:S1capS2\]. Due to cancellations on the internal edges, now we have the following identities – recall also the discussion around , and use the notation of sites along the frustrations as in fig. \[fig:S1capS2\]: $$\begin{aligned} &\prod_{p\in S_1} e^{i\pi\omega_p} = \prod_{(ij)\in \partial S_1} i^{\epsilon(ij)^p \frac{1-\sigma_i\sigma_j}{2}}\, ,\\ &\prod_{p \in S_1,\cap S_2}e^{i\pi(\tilde{\omega}_p-\omega_p)} = 1 \text{ if (b) : } S_1 \cap S_2=\emptyset\\ &\prod_{p \in S_1,\cap S_2}e^{i\pi(\tilde{\omega}_p-\omega_p)} =\\ & e^{i\frac{\pi}{2} (-\sigma_1\sigma_1'+\sigma_1\sigma_1'-\sigma_2\sigma_2'+\sigma_3\sigma_3' -\sigma_3\sigma_3'+\sigma_4\sigma_4')}\\ &=e^{i\frac{\pi}{2}\sigma_4\sigma_4'} \text{ if (a) : } S_1 \cap S_2\not=\emptyset \, .\end{aligned}$$ Therefore, in both (a),(b) cases the $\sigma$ expectation value reduces to a one dimensional classical spin chain along $\partial S_{1}$ which can be easily solved via transfer matrix. The presence of frustration in case (a) corresponds to introducing a twist by the matrix $ e^{i\frac{\pi}{2}\sigma\sigma'}$. Under the assumption of a rectangular perimeter $\partial S_1$ of length $2N$, with the branching structure as in fig. \[fig:S1capS2\], the $\sigma$ expectation value in the (a) case is (setting $\sigma_{2N+1}\equiv \sigma_{1}$): $$\begin{aligned} &\left< \prod_{p\in S_1} e^{i\pi\omega_p} \prod_{p \in S_1,\cap S_2}e^{i\pi(\tilde{\omega}_p-\omega_p)}\right>_{\sigma}=\\ &= 2^{-|\partial S_1|} \operatorname{Tr}\left[ \begin{pmatrix} i&-i\\ -i&i \end{pmatrix} \begin{pmatrix} 1&i\\ i&1 \end{pmatrix}^N \begin{pmatrix} 1&-i\\ -i&1 \end{pmatrix}^N \right]\\ &= i 2^{1-N}\, ,\end{aligned}$$ Let us remark that the problem has a chirality given by the branching structure. If $S_2$ crossed $S_1$ on the left boundary instead of on the right, the twist matrix would have been $e^{-i\frac{\pi}{2}\sigma\sigma'}$, and it would have produced an extra minus sign. If the flux arrangement is as in Fig. \[Fig:Braiding\] (b), the only difference in the result is the absence of the twist matrix appearing first in the above trace. The sole net effect of this is to remove the $i$ factor and therefore the desired ratio is $$\begin{aligned} Z^{\text{(a)}}/Z^{\text{(b)}} = \pm i\, ,\end{aligned}$$ depending if $S_2$ crosses $S_1$ on its right ($+$) or left ($-$). We have thus shown that our model has the same response to $\pi$ fluxes as the related quantum SPT phase. Generalizations --------------- As done in section \[sec:gen\_2d\] for the $2D$ case, we now sketch generalisations of the $3$D model beyond the case of a ${\mathbb{Z}}_2$ symmetry. ### Discrete vorticity and cellular cohomology {#sec:cohom} We first address the mathematical description of the discrete vorticity in terms of cellular cohomology which allow for its generalization. We will then outline a classification of CTPs within this framework and analyze some specific models. Simplicial and cellular cohomology are toolboxes used lattice gauge theories (See e.g.  [@Cohomology1991].). The first requires us to work strictly with simplexes while the second permits more general types of cells, in particular the cubic lattice. Let us quickly describe the necessary mathematical details. A reader interested only the generalized definition of the discrete vorticity for $G=Z_N$ may skip directly to Eq. \[Eq:DiscreteVorticityZN\]. We denote the sets of sites, edges, plaquettes and boxes of the cubic lattice by $V,E,P,B$ respectively, and call their elements alternatively $0$-,$1$-,$2$- and $3$-cells. In the obvious manner each of these sets describes the boundary of the latter one. The relations between cells and their boundaries can be captured in several ways: One is using incidence numbers, where $[a:b]$, with $a$ a $d$-cell and $b$ a $d+1$ cell. These take three possible integer values, $-1,0,1$, which satisfy sum rules, such as $\sum_{e\in E} [v:e][e:p] = 0, \sum_{p\in P}[e:p][p:b]=0$. Alternatively, one can simply orient the edges and plaquettes and then $[v:e]$ will be $0,1$ or $-1$ is $v$ is not a boundary of $e$, $v$ is at the end of $e$ or $v$ is at the beginning of $e$. Similarly $[e:p]$ is $0$ if $e$ is not an edge of $p$, $1$ if $e$ is aligned along the orientation of $p$ or $-1$ if it is opposite. One can easily verify that these definitions satisfy the sum rules. Below we use $i,j,k,..$ for vertex indices, $\epsilon_{ij}= 1$ ($-1$) if the edge $ij$ is oriented from $i$ to $j$ ($j$ to $i$) and $\epsilon^p_{ij}= 1$ ($-1$) if the edge $ij$ is oriented along the orientation of the plaquette (against it). To define a cellular cohomology structure (or physically a gauge theory coupled to matter) the following steps are needed: First we pick an abelian group (the gauge group) $G$ and call an assignment $g : V \to G$ a $0$-cochain (matter field), $A : E \to G$ a $1$-cochain (gauge field), and $F : E \to G$ a $2$-cochain (curvature/flux field). We denote the set of $d$-cochains by $C^d$. The coboundary operator $\delta$ (see Ref. ) maps $C^d$ to $C^{d+1}$, and is nilpotent, $\delta^2=0$. In particular, $(\delta g)_{ ij \in E} = g_i g^{-1}_j$, where the order of $ij$ is chosen according to the orientation of the edge, is the trivial 1-cocycle. (If $G$ is a generic abelian group we will use the notation $(\delta g)_{ij} = g_i - g_j$, and if $G=\mathbb{Z}_2$, $g_i = (1-\sigma_i)/2$, where $\sigma_i=\pm 1$ is the variable used in the main text.). In general, given $\alpha\in C^d$, $\beta=\delta\alpha$ is a trivial $d+1$-cochain, and if $\beta=0$, then $\alpha$ is called a $d$-cocycle. Next, one can define an equivalence relation where two $d$-cocycles are equivalent if their differ by a trivial $d$-cochain: $\alpha_1-\alpha_2=\delta\gamma$, with $\gamma\in C^{d-1}$. The equivalence classes of $d$-cocycles then obey a group structure known as the $d$ cohomology group $H^d(G)$. We consider now an exact sequence of abelian groups of the type $$\begin{aligned} \label{eq:exact_seq} 0 \rightarrow G \overset{f}{\rightarrow} \tilde{G} \overset{h}{\rightarrow} G \rightarrow 0\, ,\end{aligned}$$ and construct the map $B=f^{-1} \delta h^{-1}$, which is applied to a trivial $1$-cocycle $\delta g$ to produce a $2$-cocycle. The map $B$ is called a Bockstein homomorphism [@Hatcher2002; @Kapustin2014] and is well-defined given $h^{-1},f^{-1}$. Further, it maps $d$-cocycles to $d+1$-cocycles and introduces a homomorphism between $H^d(G)$ and $H^{d+1}(G)$. In physical terms, it maps a matter configuration to gauge flux configurations with no monopoles. In general, there is a variety of exact sequences one can consider and hence a variety of Bockstein homomorphisms. These can be classified by classifying the exact sequences upon which their are based. Short exact sequences of the form involving abelian groups are equivalent to central extension of $G$ by $G$ (s.t. $G=\tilde{G}/G$). The trivial extension is defined by $\tilde{G}=G\times G,f(a) = (a, 0)$ and $h((a, b)) = b$. Non-trivial extensions are classified by the second group cohomology $H^2(G,G)$. For $G = Z_N$ with $N$ prime, one finds that $H^2(Z_N,Z_N) = Z_N$ and so $N$ distinct choices of discrete vorticity exist. If we specify to $G=\mathbb{Z}_2, \tilde{G}=\mathbb{Z}_4$, and $f(a)=2 a, h(a)=a\mod 2$, the Bockstein homomorphism $B$ produces precisely $\omega_p \mod 2$ and the 2-cocycle condition implies zero divergence. Moreover, since $B$ is a homomorphism and $\delta g$ is a trivial $1$-cocycle, the 2-cocycle must be trivial as well and hence there exists a $1$-cochain (a gauge field, $A$) such that $\delta A = \omega_p$. We can now use $B$ to define discrete vorticities for other abelian groups. Consider for instance the case $G=\mathbb{Z}_N$, $N$ prime, $\tilde{G}=Z_{N^2}$, and: $$\begin{aligned} f(a) &= N a \, ,& h_\ell(a) &= \ell a \mod N\, , & \ell &= 0,1,\dots,N-1\, .\end{aligned}$$ Each choice of $\ell$ realizes one of the $N$ nonequivalent central extensions of ${\mathbb{Z}}_N$ by ${\mathbb{Z}}_N$, and leads to a different Bockstein homomorphism with $\ell=0$ being the trivial case. Setting $B_\ell = f^{-1} \delta h^{-1}_\ell$ yields a discrete vorticity generalising eq. : $$\begin{aligned} \label{Eq:DiscreteVorticityZN} \omega_p^{(\ell)} = \frac{1}{N} \sum_{(ij)\in \partial p}\epsilon_{ij}^p\ell \left( g_i-g_j\right)\mod N^2 \, .\end{aligned}$$ where $i$ and $j$ in the above are chosen such that $i$ ($j$) is at the start (end) of the edge $(\vec{ij})$ and $\epsilon_{ij}^p= 1$ ($-1$) if the edge is oriented with (against) the plaquette $p$. (Equivalently $\epsilon^p_{ij}$ is the incidence number $[(\vec{ij}):p]$ in the notation of Ref.  .) Explicitly, referring to figure \[fig:conf\_omega0\], it reads: $$\begin{split} \omega_p^{(\ell)} = &\frac{1}{N} \Big( \ell \big( -(g_1-g_2) -(g_2-g_3)\\ &+(g_4-g_3) +(g_1-g_4) \big) \mod N^2\Big) \, . \end{split}$$ The non-triviality of this expression is due to the fact that the terms $(g_i-g_j)$ are understood in ${\mathbb{Z}}_N$. Lastly we comment on the connection between the above cellular-cohomology approach and the group-cohomology approach to SPTs [@Chen2011]. Quantum SPTs at $d+1$ spatial dimensions with a symmetry $Q$ are classified by the group-cohomology group $H^{d+1}(Q,U(1))$. In our classical context $d+1$ is actually the overall dimension, and so one may expect that our phase is contained in $H^3(Q,U(1))$. If our matter fields posses a $Z_N$ symmetry and the gauge symmetry is $Z_N$, the relevant symmetry group in our context is $Q=Z_N \times Z_N$. (This is shown explicitly in the next section for $N=2$.) Considering $Q=Z_N \times Z_N$, the Kunneth formula [@chen2014symmetry] tells us that $H^3(Z_N \times Z_N,U(1))={\mathbb{Z}}_N^3$ contains $H^2(Z_N,H^1(Z_N,U(1))) = H^2(Z_N,Z_N)$ which is also the quantity which classifies central extensions, as discussed above. It would be interesting to find the exact correspondence between $H^3(G\times G',U(1))$ and possible CTPs. In particular find out whether every element in $H^3(G \times G',U(1))$ corresponds to a classical (or local sign free) partition function. ### Discrete vorticity models of 3D CTPs with $G=G'={\mathbb{Z}}_N$ Using the above definition of a discrete vorticity for $G=Z_N$ one can readily define more general models of 3D CTPs. To this end we consider a cubic lattice with vertices indexed by $i$, oriented edges pointing from $i$ to $j$ by $(ij)$ and oriented plaquettes indexed by $p$. The model has $\sigma_i \in {\mathbb{Z}}_N$ degrees of freedom on vertices and $A_{ij} \in {\mathbb{Z}}_N$ degrees of freedom on edges of the lattice. As in the two-dimensional case ${\mathbb{Z}}_N$ degrees of freedom take values in the roots of unity ($e^{2\pi i \alpha /N}$). (However we still represent $\omega^{(\ell)}_p$ as a number between $0,\dots,N-1$). In this notation the generalized model is given by $$\begin{aligned} \label{eq:E_3D_ZN} -\beta\mathscr{H} &= \sum_p J_{\ell'} e^{\frac{2\pi i \omega^{(\ell')}_p}{N}} (AAAA)_p + c.c. \\ \nonumber &+ \sum_p J_{\ell} e^{\frac{2\pi i \omega^{(\ell)}_p}{N}} (AAAA)_p + c.c. \, ,\end{aligned}$$ with $\omega^{(\ell)}_p$ being the discrete vorticity from Eq. (\[Eq:DiscreteVorticityZN\]), which depends on $g_i$ defined by $\sigma_i = e^{2\pi i g_i /N}$ and $(AAAA)_p \in {\mathbb{Z}}_N$ is the product of $A^{\epsilon^p_{ij}}_{ij}$’s along the plaquette $p$. First let us analyze the case when only $J_{\ell}$ is non-zero. The previous discussion on $\omega^{(\ell)}_p$ shows that for every $\sigma$ configuration there is a $A_{\sigma}$ configuration such that $(A_{\sigma}A_{\sigma}A_{\sigma}A_{\sigma})_p = \omega^{(\ell)}_p$. Thus going to the composite gauge variable $\tilde{A} = A A_{\sigma}$ one obtains $-\beta\mathscr{H} = J_{\ell} (\tilde{A} \tilde{A} \tilde{A} \tilde{A})_p$— a pure ${\mathbb{Z}}_N$ lattice gauge theory. Performing a generalized Kramers–Wannier duality [@Balian] a ${\mathbb{Z}}_N$ lattice gauge theory becomes a $3D$ clock model with rotor variables taking values in ${\mathbb{Z}}_N$. For prime $N$, so that ${\mathbb{Z}}_N$ doesn’t have any subgroups, the model will exhibit two distinct thermodynamic phases: A disordered phase where the rotors are disordered and an ordered phase of the rotors separated by a second order phase transition at $J_c$. In gauge theory terms, these correspond respectively to a phase with short flux loops ($J_{\ell}>J_{c}$) and one with large flux loops ($J_{\ell}<J_{c}$). Following the exact same reasoning as done for the $Z_2$ case, we find that the former phase confines defects of the constraint and since $\sigma$ can fluctuate freely, it clearly doesn’t break any symmetry. Consequently it is an admissible phase in our classification. We argue that the phase obtained for $J_{\ell}>J_{c}$ is a classical topological phase of type $\ell$. As discussed previously, it is a phase since local symmetry and gauge respecting perturbation in the $\sigma,A$ degrees of freedom map to local symmetry and gauge respecting perturbation in the $\sigma,\tilde{A}$ notation and vice-versa. Knowing that the latter is a well defined thermodynamic phase then implies that the former one is well defined as well. To see why different $\ell$ correspond to distinct phases let us consider an interface between a phase with large $J_{\ell} \rightarrow \infty,J_{\ell'}=0$ on the left and $J_{\ell'} \rightarrow \infty, J_{\ell} = 0$ on the right. At the interface, $\omega^{(\ell)}_p=\omega^{(\ell')}_p$. Now, since $\omega^{(\ell)}_p=\ell \omega_p^{(1)} \mod N$ and $N$ is prime, consistency implies either $\ell=\ell'$ or $\omega_p^{(1)}=0$. Supposing $\ell\neq \ell'$, this shows that just as in the $Z_2$ case, the boundary is described by a $2D$ statistical mechanical model where a zero vorticity constraint is imposed on every square. Taking $J_{\ell} > J_c$ but finite on the left and $J_{\ell'} > J_c$ on the right, will result in a physically similar scenario where flux lines crossing the interface are confined to neutral pairs by the bulks. We will argue momentarily that the model with zero vorticity is gapless. This, together with the relations to the group cohomology classification of the previous section, strongly suggests that different $\ell$ correspond to different phases. One way of proving this would be to generalize the arguments of section \[sec:SPTpersp\] to ${\mathbb{Z}}_N$, and is left for future work. Let us analyze the resulting theory on the two dimensional interface. We first count the number of zero vorticity constraints at a plaquette. We change variables from site to links variables $s_{ij}=g_i-g_j$, where as before $\sigma_i=e^{2\pi i g_i/N}$. The four link variables around a plaquette can assume only $N^3$ since a global shift of $g_i$ leaves the link variables unchanged. (In the following we will ignore the multiplicative factor $N$ in the weight produced by this change of variables.) For the purpose of counting the zero vorticity configurations, we can ignore this constraint and consider the link variables independent since the missing $N$ configurations have non-zero vorticity. We are thus left with a vertex model, where each link has $N$ states and zero vorticity becomes an interaction at vertices of dual lattice. Further, the zero vorticity constraint is the same for any $\ell$ in and w.r.t. the labelings of vertices and orientations as in figure \[fig:conf\_omega0\], it reads: $$\begin{aligned} -s_{12} - s_{23} + s_{43} + s_{14} = 0\, .\end{aligned}$$ If the $N$ states are labeled $-S,\dots,S$, with $S=\frac{N-1}{2}$, this coincides with U$(1)$ invariant configurations of spin-$S$ vertex models, and the resulting number of non-zero configurations is $$\begin{aligned} \frac{N}{3} (2 N^2 + 1) = 6, 19, 44, 85, \dots\end{aligned}$$ Apart from the already discussed $N=2$ case, other values of $N$ may not correspond to integrable weights for the vertex model, as we will discuss now for the case $N=3$, where the number of vertices is $19$. In such case, there are two classes of integrable $19$ vertex models, both of which can be related to a loop model, see e.g. [@YUNG1995]. In particular, our model gives uniform weight one to each vertex and cannot be related to a loop model, at least not in the standard fashion where states of labels $\pm 1$ are associated to oriented strands of loops and states of labels $0$ to vacancies. Nonetheless, this model belongs to a class of models studied numerically in relation with Berezinskii-Kosterlitz-Thouless transition in [@Honda1997], suggesting that the model is critical and with $c=1$. Conclusion ========== In this work we have introduced a topological classification scheme of classical statistical mechanical systems. This involved defining the objects of the classification (admissible phases), the equivalence relations between them (continuous deformation without phase transitions) and lastly showing that the classification is not trivial by giving concrete examples of admissible phases which are inequivalent. We have found $N$ distinct models for CTPs in $2D$ and $3D$ for systems with a ${\mathbb{Z}}_N$ symmetry and defects carrying a ${\mathbb{Z}}_N$ charge. An important question concerning the ability to identify the topological index or equivalence class given the bulk behavior of a particular model is left for future work. The CTPs introduced in this work, together with the ones discussed in , describe, to the best of our knowledge, novel types of topological classical phases of matter. The models given here are, arguably, the simplest and most minimal ones having just a spin degree of freedom per site and per link. Another salient feature is that they can be simulated using classical Monte-Carlo. They may thus serve as a test-bed for studying various open questions concerning both classical topological paramagnets and their quantum counterparts [@Chen2011]. These concern the nature of phase transition between trivial and non-trivial phases [@You2016], the effect of disorder on the surfaces and on phase transitions, and the precise implications of the bulk-boundary correspondence [@Scaffidi2016]. It would be highly desirable to find possible experimental realizations of such CTPs. In the field of quantum bosonic SPTs [@Chen2011], experimental realizations are so far limited to $1+1D$ [@Buyers1986]. Being free from the stringent requirement of quantum coherence, and based on simple microscopic ingredients, the classical counterparts introduced here may prove easier to realize. Indeed similar classical systems, such as artificial spin-ice systems, have been successfully realized [@Wang2006; @Cumings2008; @Roderich2013] using ferromagnetic wires as well as tiling molecules [@Blunt]. The $2D$ model we discussed could potentially be realized from the same microscopic ingredients. Finally, it would be interesting to further explore the classification question we propose in this work. For instance by considering other types of symmetries and constraints. Certainly there should be some relation with the group cohomology classification of bosonic SPTs with a trivial bulk [@Chen2011] however it may not be one to one. Indeed some SPTs may suffer from sign problems in Monte-Carlo while others do not. Conversely, it may be that enforcing hard constraints or gauge symmetries allows for new types of quantum phases. Indeed hard constraints in classical systems may result in a genus dependent ergodicity breaking [@Moessner2001; @PhysRevB.93.205112] whereas genus dependent ground state degeneracy is not part of the cohomology classification of Ref. (). We are grateful to P. Fendley, T. Scaffidi and S.H. Simon for stimulating discussions. Z.R. was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 657111. R.B. was supported by the EPSRC Grants EP/I031014/1 and EP/N01930X/1. Both authors contributed equally to this work. **Supplemental material of “Classical topological paramagnetism”** General definition of a local constraint, confinement, and deconfinement {#App:Constraint} ======================================================================== Here we address the issue of how one generally defines a lattice constraint as well as confined and deconfined phases. A local constraint on a lattice can be abstracted as followed: First one requires a local mapping from the degrees of freedom to group elements in $G'$. For the sake of simplicity we take $G'$ abelian. This mapping should be local such that the value $g_x$ obtained at point $x$ involves degrees of freedom near $x$. Furthermore, it must be neutral such the product of $g_x$ over a closed manifold yields the identity. The constraint is then the requirement that $g_x = I$ ($I$ being the identity) at all positions $x$. A defect $f_x$ is a local violation of this rule in which $g_x = f_x \neq I$. In the familiar context of $3D$ lattice gauge theories on a cubic lattice, this mapping would be a mapping between boxes and magnetic charges within them. A local defect would thus be a particular box where the magnetic charge is $f$ instead of the identity. Confined and deconfined phases are defined, as usual, by the free energy cost $\Delta F_l$ of taking two static opposite defects ($f,f^{-1}$) apart. Confinement is defined by a free energy cost which increases as a positive power of the distance ($l$) and a deconfined phase is define by a saturating free energy cost. Just like in the case of broken symmetries, these define two distinct phases of matter which can only be connected through a phase transition. The simplest way to show this is to remove the constraint and instead introduce Lagrange multipliers at every point where the constraint is imposed as: $$\begin{aligned} \delta_{f,I} = \frac{1}{|G'|}\sum_{\lambda}\chi_{\lambda}(f)\, ,\end{aligned}$$ where $\lambda$ goes through $|G'|$ values labeling the irreducible one-dimensional representations of $G'$ and $\chi_\lambda(f)$ is the character. If $G'={\mathbb{Z}}_N$, we simply have $\chi_{\lambda}(f=a^k)=e^{2\pi i \lambda k/N}$, where $a$ is the generator of ${\mathbb{Z}}_N$. By the neutrality condition, the resulting partition function obeys a global symmetry $G'$ shifting all the $\{\lambda_x\}$ by the same amount. Finally, $\Delta F_l$ is given by $$\begin{aligned} e^{-\Delta F_l} &= \langle \chi_{\lambda_0}(f) \chi_{\lambda_{l}}(f^{-1})\rangle\, ,\end{aligned}$$ and the confined phase translates into the phase with exponentially decaying $\lambda_x$ correlations (i.e. no spontaneous symmetry breaking) and the deconfined phase becomes the spontaneous broken symmetry phase. [^1]: We remark that one can consider more general couplings, such as those of the Ashkin–Teller model [@Baxter], as long as the set of order parameters in a phase is unchanged. Our choice of two decoupled Ising models is made for pedagogical purposes.

--- abstract: 'Techniques for simulating molecules whose conformations satisfy constraints are presented. A method for selecting appropriate moves in Monte Carlo simulations is given. The resulting moves not only obey the constraints but also maintain detailed balance so that correct equilibrium averages are computed. In addition, techniques for optimizing the evaluation of implicit solvent terms are given.' author: - '[Charles F. F. Karney](http://charles.karney.info)' - 'Charles F. F. Karney' - 'Jason E. Ferrara' bibliography: - 'free.bib' nocite: - '[@qiu97]' - '[@hawkins95]' - '[@hawkins96]' - '[@tsui00]' - '[@tsui01]' - '[@onufriev04]' - '[@go70]' - '[@dodd93]' --- =2005=8=17 Introduction ============ When attempting to compute thermodynamic quantities with a molecular simulation, we are frequently confronted with the problem of sampling in a high-dimensional configuration space. The dimensionality of this space is given by the number of degrees of freedom for the molecular system. Techniques which lower the number of degrees of freedom will increase the efficiency of the thermodynamic sampling—provided, of course, that these techniques are physically justified. Thus, an implicit solvent model may be used to eliminate the degrees of freedom associated with the solvent molecules; the standard chemical force fields replace the electron charges with atom-centered partial charges thereby removing the electrons’ degrees of freedom. Further reductions in dimensionality are possible by imposing constraints on the relative positions of the atoms in a molecule. Thus we might specify that the bond lengths and bond angles in a molecule are fixed and only the torsion angles are allowed to vary. It is such a scenario that we examine in this paper. We address two aspects of this problem: how to move a molecule subject to constraints in order to allow equilibrium averages to be computed using the canonical-ensemble Monte Carlo method [@metropolis53] and how to evaluate the energy efficiently. The imposition of constraints in molecular modeling has been extensively studied [@frenkel02 §3.3.2, §15.1]. Let us start by elucidating the difference in the treatment of hard constraints in molecular dynamics and Monte Carlo simulations. We treat hard constraints by taking the limit where the “spring constant” for the hard degrees of freedom is infinite. Molecular dynamics simulations then consider the evolution of the resulting system over a finite time. On the other hand, if we wish to determine the equilibrium properties of a system using the Monte Carlo method, then we need to consider averaging over sufficiently long times to allow equipartition of energy among all the degrees of freedom of a system. This is, of course, an example of nonuniform limits. We are interested in taking both $\tau_\mathrm{sim} \rightarrow \infty$ and $\tau_\mathrm{equ} \rightarrow \infty$, where $\tau_\mathrm{sim}$ is the representative simulation time and $\tau_\mathrm{equ}$ is the equipartition time (which is proportional to the “stiffness” of the constraints). In constrained molecular dynamics, we take the limit $\tau_\mathrm{equ} \rightarrow \infty$ first, which prevents equipartition from occurring; whereas in equilibrium statistical mechanics, we take $\tau_\mathrm{sim} \rightarrow \infty$ first and this allows energy equipartition. If we are attempting to compute an equilibrium quantity, such as the free energy of binding, it is essential to allow energy equipartition. Understanding this distinction explains the apparently contradictory results for constrained and unconstrained averages for a flexible trimer [@frenkel02 §15.1]. One way of understanding the constrained equilibrium system is to consider how the equilibrium distribution varies as the constraint is imposed. In the limit, the distribution collapses to a lower-dimensional sub-manifold of configuration space. However, this sub-manifold has a “thickness” that depends on the details of the constraint term and, consequently, Monte Carlo moves for the constrained system need to reflect this thickness in order to sample the distribution correctly. As a consequence, we will need to specify the functional form of the constraint energy and the constraint is no longer a purely geometrical object. At first glance, this would appear to complicate further the already complex algebra of constrained motions [@fixman74]. However, we will propose an algorithm for making moves which is simple to implement and which automatically ensures that the correct equilibrium averages are computed. The second half of the paper considers a mundane—but nevertheless important—problem, namely how to evaluate the energy of a molecule made up of rigid subcomponents. We propose a consistent framework for avoiding the computation of constant terms and for imposing energy cutoffs. We extend this to the computation of the generalized Born solvation term and we describe a simple method for computing the solvent accessible surface area which has a bounded error. Generalized Monte Carlo moves ============================= We begin by assembling some techniques for combining Monte Carlo moves. We define an “$E$ move” as an ergodic move which preserves $\exp(-\beta E)$ as the invariant distribution, where $\beta = 1/(kT)$ and $k$ is the Boltzmann constant. (Here, “ergodic” implies that the move allows all relevant portions to configuration space to be explored.) Note that a zero move has a uniform invariant distribution. A typical zero move samples a new configuration from a distribution which satisfies the symmetry requirement $p(\Gamma';\Gamma) = p(\Gamma;\Gamma')$, where $p(\Gamma';\Gamma)$ is the probability density of picking a new configuration of $\Gamma'$ given a starting configuration of $\Gamma$. Clearly a sequence of $n$ $E$ moves is itself an $E$ move. From the central limit theorem, a sequence of $n$ zero moves is equivalent, in the limit of large $n$, to selecting the new configuration from a multi-dimensional Gaussian. Instead of carrying out the $n$ moves with a given energy $E(\Gamma)$, we can consider the case where the energy is given by $E_\lambda(\Gamma)$ which depends continuously on the parameter $\lambda$. A sequence of $n$ $E_\lambda$ moves where $\lambda$ is varied [*adiabatically*]{} in such a way that its initial and final values are $\lambda_0$ is an $E_{\lambda_0}$ move. This follows because adiabatically varied systems are always in equilibrium with the instantaneous value of $\lambda$ [@landau69 §11]. Each $E_\lambda$ move is carried out at a [*fixed*]{} $\lambda$ and $\lambda$ is varied between the moves. In order to satisfy the adiabatic condition, we will need to take $n$ large. A move from $\Gamma$ to $\Gamma'$ may be subjected to “Boltzmann acceptance with energy $E$”. This involves accepting the move ($\Gamma'$ is the new state) with probability $M(x)$ and otherwise rejecting the move ($\Gamma$ is the new state). Here $M(x)$ is a function satisfying $0 < M(x) \le 1$ and $M(x)/M(-x) = \exp(-x)$ with $x = \beta(E(\Gamma') - E(\Gamma))$. Usually we take $M(x) = \min(1,\exp(-x))$; however other choices, e.g., the Fermi function, $M(x) = 1/(1+\exp(x))$, are possible [@bennett76]. Consider an $E_1$ move from $\Gamma$ to $\Gamma'$ followed by an Boltzmann acceptance using $E_2$. This compound move is an $(E_1 + E_2)$ move. The proof follows as a special case of the “multiple time-step” (MTS) method [@hetenyi02 §II] or, alternatively, as a special case of early rejection [@frenkel02 §14.3.2]. If the $E_1$ move was already a “rejected” move, i.e., $\Gamma'=\Gamma$, then the Boltzmann test involving $E_2$ automatically “succeeds” ($M(0) = 1$). Thus $E_2$ does not need to be evaluated in this case. These results allow us to generalize the MTS method by splitting the energy into $m$ terms (instead of just two), $$E(\Gamma) = \sum_{l=1}^m E_l(\Gamma).$$ The method is defined recursively as follows: a level-$0$ move is defined to be a zero move; a level-$l$ move, with $l>0$, is defined to be $n_{l-1}$ level-$(l-1)$ moves the result of which is subjected to Boltzmann acceptance using $E_l$. By induction, we see that a level-$l$ move is an $\mathcal E_l$ move, where $$\mathcal E_l(\Gamma) = \sum_{l'=1}^l E_{l'}(\Gamma).$$ It follows that a level-$m$ move is an $\mathcal E_m$ move, i.e., an $E$ move. Typically we sample the zero moves from a Gaussian and we take $n_0 = 1$. Standard Monte Carlo [@metropolis53] is given by $m=1$ and $n_1=1$. Standard MTS [@hetenyi02] is recovered with $m=2$. The early rejection method [@frenkel02 §14.3.2] is recovered with $n_l=1$ (for all $l$). Note that a level-$m$ move entails $\prod_{l'=l}^{m-1} n_{l'}$ level-$l$ moves. At any stage in the recursion, we have the freedom to vary some of the components of $E(\Gamma)$ adiabatically. In the following sections, we apply these techniques to constrained molecules. In simple cases, we can apply the MTS method semi-analytically to derive a correct constrained move. In more complicated cases, we apply the adiabatic technique to lift and to reapply the constraint. Stiff molecules =============== A constrained molecule is a mathematical idealization of a real system in which some degrees of freedom are stiff, i.e., the associated energies are large. Thus we can split the energy into “hard” ($\mathrm h$) and “soft” ($\mathrm s$) components, $$E(\Gamma) = E_\mathrm h(\Gamma) + E_\mathrm s(\Gamma),$$ where $\Gamma$ is the configuration of the system. For example, let us assume that an all-atom force field, such as Amber [@cornell95], provides an accurate description of the system. (We recognize, of course, that present-day force fields are only approximate. However, our purpose here is to make the connection between an all-atom representation and a simpler rigid representation and, in this context, the details of the all-atom model are of secondary importance.) Then $E_\mathrm h$ might represent the bond stretching and bond bending terms, while $E_\mathrm s$ is given by the other terms (bond torsion and the non-bonded energies). The constrained limit is now given by $E_\mathrm h \rightarrow \infty$. Before we consider this limit, it is useful to examine how the stiff system may be treated. Conventional Monte Carlo is inefficient because, in order to have an reasonably large acceptance rate, the step-size needs to be set to a small value (determined by $E_\mathrm h$) so that diffusion in the soft directions is very slow. However, we can apply MTS Monte Carlo in this case with $E_1 = E_\mathrm h$ and $E_2 = E_\mathrm s$. Let us apply this method to a system of “rigid” molecules, e.g., water molecules, taking $E_\mathrm h$ to include the intra-molecular energies (responsible for maintaining the rigidity) and $E_\mathrm s$ to include the inter-molecular energies. Suppose the level-$0$ moves consist of symmetrically displacing the atoms in each molecule. The result of the $n_1$ level-1 Monte Carlo steps will clearly be a symmetric, independent, and nearly rigid displacement (translation and orientation) of each molecule. This configuration is then subjected to Boltzmann acceptance with the inter-molecular energies. In this case, we can easily pass to the constrained limit (with exact rigidity), merely by ensuring that the trial (level-1) moves of the molecules are rigid. In this case, we have just rederived the “standard” move for a system of rigid molecules. In order to illustrate the application to flexible molecules, we shall treat the molecules as being made up of several rigid subunits or “fragments” connected by flexible bonds. However we are interested in the limit where the inter-fragment bonds constrain the relative motions of fragments in certain ways, either by fixing the bond lengths (allowing the bond angles and bond dihedrals to vary) or by fixing the bond lengths and bond angles (allowing the bond dihedrals to vary). Such a model is adequate to describe a wide range of interesting organic molecules including proteins and drug-like ligands. We assume that the rigidity of the fragments is imposed only by intra-fragment energy. If other terms (e.g., an improper torsion term involving atoms from two fragments) contribute to the rigidity of a fragment, then we shall treat such terms as additional inter-fragment energies. We apply the generalized MTS method to this system with $m=3$, the intra-fragment energy given by $E_1$, the inter-fragment bond constraints given by $E_2$, and with $E_3$ accounting for all the other energies. The argument given above allows us to pass to the limit of strictly rigid fragments. The method is then equivalent to a standard MTS method where the “elementary” moves consist of rigid displacements of each fragment which are Boltzmann accepted with energy $E_\mathrm h = E_2$. A sequence of $n = n_2$ such moves are made with the result Boltzmann accepted with energy $E_\mathrm s = E_3$. A possible prescription [@karney05b §VII] for the rigid displacements of the fragments is to translate the fragment by a vector sampled from an isotropic 3-dimensional Gaussian and to rotate the fragment by $\abs{\v s}$ about an axis $\hat{\v s}$ where $\v s$ is a “rotation vector” also sampled from an isotropic 3-dimensional Gaussian. The variances for the two Gaussians should be adjusted so that the translational and rotational components result in comparable displacements of the atoms of the fragment. Provided that the inter-fragment constraint terms $E_\mathrm h$ are sufficiently stiff, it is not important to include a detailed model of these terms; because the motion will take place near the bottom of the constraint potential well, a harmonic (i.e., quadratic) approximation to the constraint potential will suffice. On the other hand, if the stiffness of the constraint energy depends on any of the soft degrees of freedom, it is important that this effect be included. It is frequently the case that $E_\mathrm h$ may be computed much more rapidly than $E_\mathrm s$. For example, when imposing bond constraints on a molecule, $E_\mathrm h$ requires $O(N)$ computations, where $N$ is the number of atoms, while $E_\mathrm s$ requires $O(N^2)$ computations for the electrostatic and implicit solvation energies. Thus we might be able to take $n$ reasonably large and still have the computational cost dominated by the evaluation of $E_\mathrm s(\Gamma)$. In order to realize the full benefits of imposing constraints we need to pass to the constrained limit ($E_\mathrm h \rightarrow \infty$). In this limit, the motion collapses onto a lower-dimensional sub-manifold in configuration space. Unfortunately, in contrast to the case of rigid molecules, we cannot appeal to symmetry to enable us to take this limit analytically. Instead, we use the adiabatic technique. Adiabatically varying the stiffness =================================== Let us rewrite the energy of the system, multiplying the $E_\mathrm h(\Gamma)$ by $T/T^*$, where $T$ is the temperature of the system, and $T^*$ is a “constraint” temperature. The Boltzmann factor $\exp(-\beta E)$, will then have the form $$\exp(-\beta E) = \exp(-\beta E_\mathrm s - \beta^* E_\mathrm h)$$ where $\beta^* = 1/(kT^*)$. In our application, where we are interested in the constrained limit $T^*\rightarrow 0$, a direct application of the MTS method leaves us with two bad choices. If we take $T^*$ to be sufficiently small that we can consider the constraints to be satisfied, we will have to chose the step size for the $E_\mathrm h$ moves to be so small that the change in configuration after $n$ $E_\mathrm h$ moves will be small. On the other hand, letting $T^*$ be sufficiently large to allow moves will result in configurations where the constraints are poorly satisfied. We overcome this difficulty by regarding $T^*$ as a parameter (taking the place of $\lambda$) and by adiabatically varying $T^*$ from zero (where the constraints are satisfied but MTS is ineffective at making moves) to a finite value (where the constraints are relaxed and MTS becomes effective) and back to zero again (to reimpose the constraints). During the course of changing $T^*$, we make $n$ $E_\mathrm h$ moves (each with the instantaneous value of $T^*$). The effect of these $n$ moves will be an $E_\mathrm h$ move with $T^* = 0$, i.e., a move which satisfies the $E_\mathrm h$ constraint. It remains to give a recipe for varying $T^*$. As we vary $T^*$, we would naturally adjust the step size for the moves in such a way that the number of steps needed to equilibrate the system is a constant, suggesting that we vary $T^*$ exponentially. We therefore pick $$T^*_i = \left\{ \begin{array}{l@{\hspace{1em}}l} T^*_A \exp(\alpha (i - 1)), & \mbox{for $0 < i \le m$},\\ T^*_A \exp(\alpha (n - i)), & \mbox{for $m < i \le n$}, \end{array}\right.$$ where we have taken $n = 2m + 1$ and where $T^*_i$ is the constraint temperature used for the $i$th $E_\mathrm h$ move, $T^*_{0} = T^*_A$ is some temperature sufficiently small that we can consider the constraints to be exactly satisfied, and $\alpha$ is the rate of increase of the temperature which should be sufficiently small that the adiabatic condition is satisfied. Even though $T^*_A$ and $\alpha$ are small, we can pick $n$ sufficiently large that $T^*_{m+1} = T^*_B = T^*_A \exp(\alpha m)$ is finite. In addition, we choose the step size for the $i$th $E_\mathrm h$ move to be $d_i = k \sqrt{T^*_i}$ where $k$ is a constant. In traditional Monte Carlo, we normally pick $k$ to maximize the diffusion rate which at the $i$th step is roughly $$D_i = \frac{\langle (\Gamma_i - \Gamma_{i-1})^2\rangle}2 \sim \frac12 A d_i^2,$$ where $A$ is the mean acceptance rate and $\langle\ldots\rangle$ denotes an ensemble average. Maximizing the diffusion rate usually results in a rather small acceptance rate $A \sim 0.1$ because rare large steps can lead to faster diffusion than frequent small steps. However, in our application, where we want the system to remain in equilibrium as we vary the temperature, rare large steps are [*bad*]{}. So we pick $k$ to maximize $A D_i$ and this will usually result in $A \sim 0.5$. Note that for a given $k$, we have $$D_i \sim C T^*_i,$$ where $C$ is constant provided that the step size is not too large. The overall diffusion can be estimated by summing over the $n$ steps, $$D = \frac{\langle (\Gamma_n - \Gamma_0)^2\rangle}2 = \sum_{i=1}^n D_i \sim 2 C T^*_B/\alpha,$$ where we have assumed that successive steps are uncorrelated and we have taken $\alpha\ll 1$ and $T^*_B \gg T^*_A$. We should select parameters, $\alpha$ and $T^*_B$, in order to adjust $D$ so that the $E_\mathrm s$ acceptance rate is $O(1)$. This method includes internal diagnostics to verify that $\alpha$ is small enough. We define $\overline{\vphantom{A}\ldots}\mathord{\uparrow}$ (resp. $\overline{\vphantom{A}\ldots}\mathord{\downarrow}$) as the average of a quantity over the steps where $T^*_i$ is increasing, i.e., $i \le m+1$ (resp. decreasing, i.e., $i> m+1$). We monitor $\overline{E_\mathrm h(\Gamma_i)/T^*_i}\mathord{\uparrow}$ and $\overline{E_\mathrm h(\Gamma_i)/T^*_i}\mathord{\downarrow}$ and demand that both should be close to the equilibrium value of $N/2$ (where $N$ is the number of hard degrees of freedom). If $\alpha$ is too large, then we would find $$\begin{aligned} \overline{ E_\mathrm h(\Gamma_i)/T^*_i }\mathord{\uparrow} &\ll& N/2, \\ \overline{ E_\mathrm h(\Gamma_i)/T^*_i }\mathord{\downarrow} &\gg& N/2.\end{aligned}$$ In particular, if the final $E_\mathrm h(\Gamma_n)$ is many times $T^*_A$, then the configuration is “hung up” and does not obey the constraints. If this happens frequently, the simulation needs to be rerun with a smaller setting for $\alpha$; if, on the other hand, it happens only rarely, we would merely reject the step. We can also monitor the mean acceptance rates $ \overline A \mathord{\uparrow}$ and $\overline A \mathord{\downarrow}$. These should be about the same; however, if $\alpha$ is too large, we will find $ \overline A \mathord{\uparrow} \gg \overline A \mathord{\downarrow}$. A useful guideline for picking $T^*_A$ is that once the $n$ $E_\mathrm h$ moves are completed and the system is presumably equilibrated to $T^*_A$, we should be able to enforce the constraints by setting $T^* = 0$ (using any convenient energy minimization technique) with a negligible change in the configuration, e.g., with a negligible change in $E_\mathrm s(\Gamma)$. Pairwise terms in energy ======================== Having made an adiabatic move using $E_\mathrm h$, the final step is to accept the move depending on the change in $E_\mathrm s$. We wish to compute this energy as efficiently as possible by using the rigidity of the fragments. Force fields such as Amber [@cornell95] include two types of energies: interactions between atoms (the electrostatic and Lennard-Jones terms) and bond energies (stretch, bend, and torsion). Since the number of terms in non-bonded energies typically scales as $O(N^2)$ where $N$ is the total number of atoms in the system, while the number of bond terms scales as $O(N)$, we concentrate on optimizing the evaluation of the non-bonded terms. In our case where the molecules consist of rigid fragments connected by flexible bonds we need only include the bond terms contributed by the much smaller number of inter-fragment bonds. Furthermore, we need only include the energy contributed by the “free” components of such bonds. Thus, if the lengths and angles of such bonds are constrained, then we need only include the torsion energy in $E_\mathrm s(\Gamma)$. We start by assuming that the non-bonded energy terms can be expressed as a sum over atom pairs. This applies to the electrostatic and Lennard-Jones terms in Amber [@cornell95]. However, implicit solvent models have a more complex structure and we consider these in the next section. Suppose our molecular system consists of $N$ atoms. These atoms are grouped into $M$ molecules and we denote $M_l$ as the set of atoms making up the $l$th molecule. Similarly, the atoms are divided into $F$ rigid fragments and we denote $F_a$ as the set of atoms making up the $a$th fragment. A typical pairwise energy term can then be written as $$E_g(\Gamma) = \sum_{0< i<j \le N} C_{g,ij} f_g(r_{ij}),$$ where $g$ denotes the type of energy term (electrostatic or Lennard-Jones), $i$ and $j$ are atom indices, $r_{ij}$ is the distance between atoms $i$ and $j$, $f_g$ is some function of distance, and $C_{g,ij}$ is a coefficient which depends on the atoms but not on their positions. Thus for electrostatic interactions, $C_{g,ij}$ depends on the partial charges on the two atoms (assumed to be constant in Amber) and on the bonding relation between the atoms. Physical energy functions satisfy $\lim_{r\rightarrow\infty}f_g(r) = 0$. When the fragments are separated sufficiently, we have $$E_g \rightarrow E_{g0} = \sum_{0< a \le F} \sum_{\stack{i<j}{i,j \in F_a}} C_{g,ij} f_g(r_{ij}),$$ which is independent of $\Gamma$. It is convenient to choose $E_{g0}$ as the “origin” for the $E_g$, i.e., we compute only $$E_{g1} = E_g - E_{g0} = \sum_{0< a<b \le F} \sum_{\stack{i \in F_a}{j \in F_b}} C_{g,ij} f_g(r_{ij}).$$ We note that only energy differences enter into the computation of observable quantities, and so we are free to select the arbitrary origin for energies. Let us consider the application of a small molecule ($N_l$ atoms) interacting with a protein ($N_p \gg N_l$ atoms) where only some of the protein side chains near the binding site are allowed to move. By avoiding computing the interaction energy between atoms in the immobile portion of the protein, the above prescription reduces the computational cost from $O(N_p^2)$ to $O(N_l N_p)$. This cost may still be too large and we can substantially reduce the cost by implementing energy cutoffs for the interactions. This is easily accomplished by multiplying $f_g(r_{ij})$ by a cutoff function, $c_g(r_{ij})$. A possible form for this cutoff function is $$c_g(r) = \left\{ \begin{array}{l@{\hspace{1em}}l} 1, & \mbox{for $r < r_{g1}$},\\ 0, & \mbox{for $r \ge r_{g2}$},\\ \displaystyle c_g(r_{g1}) \frac{r_{g2} - r}{r_{g2} - r_{g1}} & \mbox{otherwise},\\ \end{array}\right.$$ with $r_{g1} \le r_{g2}$, which linearly tapers the energy to zero over $[r_{g1},r_{g2})$. Other tapering functions can be employed, or, by choosing $r_{g2} = r_{g1}$, we can implement a sharp cutoff. This type of cutoff function implements a per-atom cutoff and is appropriate for energy terms which are additive at large distances, such as the Lennard-Jones potential. The electrostatic potential, however, involves substantial cancellation at large distances—two neutral molecules interact via a dipole-dipole term which varies as $1/r^3$, while the individual atom-atom terms decay as $1/r_{ij}$. In this case, we need to identify groups of atoms which should interact together. The residues of a protein provide a convenient grouping and we would typically assign all the atoms in a small-molecule ligand to a single group. Compatible with the usage for a protein, we refer to these groups as residues. For each residue, $s$, we define a center position, $\v b_s$, most conveniently defined as the center of mass, and a radius, defined as the radius $h_s$ of the sphere centered at $\v b_s$ which includes the van-der-Waals spheres of radius $\rho_i$ of all the constituent atoms. We then apply a “per-residue” cutoff function multiplying the contribution from the residue pair $(s,t)$ by $c_g(\abs{\v b_s - \v b_t} - (h_s + h_t))$. The values used for the cutoff radii, $r_{g1}$ and $r_{g2}$, need to evaluated based on the accuracy desired for the simulation. This can be determined by numerically determining the difference in the results (either for the energies directly or for some derived quantity such as binding affinity) between the finite- and infinite-cutoff energies. In applications to Monte Carlo codes, it is possible to carry out the sampling at an energy approximating the actual energy and to compensate for this when performing the canonical averages (which might be carried out on a subset of the Markov chain). In this case, the sampling energy might entail using shorter cutoffs than would be warranted on the basis of accuracy. Having determined suitable cutoffs, it is a simple matter to evaluate the energy avoiding treating atom pairs beyond the respective cutoffs. In the following, we treat electrostatic ($e$) interactions, with a per-residue cutoff, and Lennard-Jones ($l$) interactions, with a per-atom cutoff; furthermore we assume that $r_{e2}\ge r_{l2}$, i.e., the electrostatic interactions are longer range than the Lennard-Jones. We first loop over all the atoms in each residue computing $\v b_s$ and $h_s$ for all residues $s$. We then loop over all pairs of residues, $s \le t$, skipping any pair whose atoms all belong to the same fragment or those for which $\abs{\v b_s - \v b_t} \ge r_{e2} + h_s + h_t$. If the residue pair survives these tests, then all atom pairs $(i,j)$ from different fragments are considered; if $s=t$, we restrict the pairs to $i<j$. All such pairs contribute to the electrostatic energy while those which satisfy $r_{ij} < r_{l2}$ contribute to the Lennard-Jones energy. There obviously is scope for additional optimization here. For example, the inner atom loop can be skipped if the second residue belongs to a single fragment which matches the fragment of a particular atom in the first residue. Because of the way in which the cutoffs are applied, the result for the energy is independent of the assignment of atoms to residues for energy terms which use a per-atom cutoff. In addition, [*differences*]{} in the non-bonded energies are independent of the assignment of atoms to fragments. The energies for assemblies of 3 or more molecules can be expressed in terms of the energies of 1 or 2 molecules. These provide useful checks on the implementation. In some contexts it is useful also to define a “steric” energy term which is infinite if any atoms overlap (with some definition of a “hard” atom radius) and is zero otherwise. This provides a rapid check of new configurations—particularly when trying to “insert” a molecule during a grand canonical simulation [@adams75] or when switching systems using the wormhole method [@karney05a]. A conservative definition of the hard atom radius is $0.55 \rho_i$ for non-bonded atom pairs and $0.45 \rho_i$ for 1-4 atom pairs. We skip the check for 1-2 and 1-3 pairs and for those atoms with $\rho_i = 0$. This energy term can be implemented in essentially the same way as described above but with scope for additional speedups. The cutoff radius in the residue-residue distance check can be replaced by 0. An additional atom-residue distance check can be be used to avoid executing the inner atom loop if the outer atom is outside the sphere for the second residue. Finally, as soon as an overlap of hard spheres is detected the routine can immediately return an infinite result. Implicit solvent models ======================= We now turn to the computation of the energy term for implicit solvent models. We focus here on the generalized Born solvent models [@still90] and we have considered various implementations [@qiu97; @hawkins95; @hawkins96; @tsui00; @tsui01; @onufriev04]. Evaluating the solvation energy for a system of molecules with such models is typically orders of magnitude slower than computing the energy of the molecules in vacuum. The computation time is frequently compared to the time to compute the energy with an explicit solvent model (including $O(10^3)$ solvent molecules). However, such comparisons are misleading because implicit solvent models do not attempt to compute the energy of a particular configuration of solvent molecules but to compute the [*free energy*]{} of solvation, i.e., to average over all possible solvent configurations for a given configurations of solute molecules. Thus the chief benefit of an implicit solvent model is to reduce dramatically the number of degrees of freedom in the problem. In the generalized Born solvent models, the energy is written as the sum of two terms: a polar term which is usually called the “GB” term and a cavity term which is proportional to the solvent accessible surface area, the “SA” term. The GB term involves long-range interactions and is the most costly to compute. We address the calculation of this term first. The basic expression is [@still90] $$\label{gpol} G_\mathrm{pol} = -\frac12\frac1{4\pi\epsilon_0} \biggl(1-\frac{\epsilon_0}{\epsilon_s}\biggr) \sum_{i,j}q_iq_jf(r_{ij},\alpha_i,\alpha_j),$$ where $\epsilon_s$ is the permittivity of the solvent, $f(r_{ij},\alpha_i,\alpha_j)= [r_{ij}^2 + \alpha_i\alpha_j\exp(-r_{ij}^2/(4\alpha_i\alpha_j))]^{-1/2}$, and the double sum runs over [*all*]{} pairs of atoms (including $i=j$ and $i\lessgtr j$). In eq. (\[gpol\]), $\alpha_i$ is the “generalized” Born radius of the $i$th atom, which is larger that the “bare” Born radius to account for the fact that atoms close to $i$ partially shield it from the solvent. $G_\mathrm{pol}$ represents the electrostatic energy required to solvate a pre-assembled group of molecules and thus this term is added to the vacuum electrostatic energy. The various implementations for the GB term differ in how $\alpha_i$ is computed. For illustrative purposes, let us consider the model of Hawkins [*et al.*]{} [@hawkins95; @hawkins96; @tsui00; @tsui01]. (With minor modifications, the technique is applicable to other GB models.) We express $\alpha_i$ as [@hawkins95 eq. (10)] $$\label{alpha} \frac1{\alpha_i} = \frac1{\rho_i} - \sum_{j\ne i} \Delta_{ij},$$ where $\rho_i$ is the radius of atom $i$, $$\label{deltaij} \Delta_{ij} = \int_{\rho_i}^\infty \frac{dr}{r^2} H_{ij}(r; r_{ij}, \rho_j)$$ is the reduction in the effective inverse Born radius of atom $i$ due to atom $j$. Here $H_{ij}$ is the fraction of the area of a sphere of radius $r$ centered on the $i$th atom eclipsed by a $j$th atom and is given by [@hawkins95 eq. (12)] $$H_{ij}=\left\{ \begin{array}{l@{\hspace{1em}}l} \displaystyle \frac{\rho_j^2-(r_{ij}-r)^2}{4r_{ij}r}, & \mbox{for $\abs{r_{ij}-\rho_j} \le r \le r_{ij}+\rho_j$},\\ 1, & \mbox{for $r < \rho_j - r_{ij}$},\\ 0, & \mbox{otherwise ($r \gtrless r_{ij} \pm \rho_j$)}.\\ \end{array} \right.$$ Evaluating the integral in eq. (\[deltaij\]) then gives $$\Delta_{ij} = \left\{ \begin{array}{l@{\hspace{1em}}l} 0, & \mbox{for $ \rho_i > \rho_j + r_{ij}$},\\[1ex] \displaystyle \frac{l_{ij} - u_{ij}}2 -\frac{(r_{ij}^2-\rho_j^2)(l_{ij}^2-u_{ij}^2)}{8r_{ij}}\hspace{-10em}\\ \displaystyle \hspace{3em}{}-\frac{\ln(l_{ij}/u_{ij})}{4r_{ij}} + l'_{ij}, & \mbox{otherwise}, \end{array} \right.$$ where $u_{ij} = 1/(r_{ij}+\rho_j)$, $l_{ij} = 1/\max(\rho_i,\abs{r_{ij}-\rho_j})$, and $l'_{ij} = 1/\rho_i-1/\max(\rho_i,\rho_j-r_{ij})$. The term $l'_{ij}$ is only non-zero for $\rho_j > \rho_i + r_{ij}$, which is a possibility not considered in [@hawkins95]. Clearly $G_\mathrm{pol}$ is no longer the sum of pairwise atom-atom contributions because the interaction of two atoms is affected by the modification of the dielectric environment by a third atom. However $G_\mathrm{pol}$ may be evaluated by two pair-wise operations carried out in sequence. The first evaluates the generalized Born radii $\alpha_i$ and the second computes the resulting electrostatic energy. As with the treatment of the electrostatic and Lennard-Jones terms, we can seek to limit the computational cost of evaluating $G_\mathrm{pol}$ by the use of cutoff functions. Because eq. (\[gpol\]) provides the dielectric screening for the vacuum electrostatic term, it is important that the cutoff function multiplying $f(r_{ij},\alpha_i,\alpha_j)$ exactly match that used for the electrostatic term. We also introduce a cutoff in eq. (\[alpha\]) by multiplying $\Delta_{ij}$ by $c_b(r_{ij})$. A per-atom cutoff is justified since all the $\Delta_{ij}$ are positive. Because $\Delta_{ij}$ scales as $r_{ij}^{-4}$ for large $r_{ij}$, the error introduced by $c_b(r_{ij})$ scales relatively slowly as $r_{b1}^{-1}$. In practice, this means we need to make $r_{b1}$ reasonably large which in turn means that the cost of evaluating $G_\mathrm{pol}$ in the case of a small ligand interacting with a protein is much larger than the cost for the electrostatic potential. In particular, the screening of the ligand may modify the Born radii of a large number of protein atoms and this unavoidably leads to a large number of pair contributions to eq. (\[gpol\]). The procedure for computing the energy outlined in the previous section can now be modified to deal with the evaluation of $G_\mathrm{pol}$. As before our “zero” energy is given by separating all the fragments of all the molecules infinitely far apart. We set up the calculation of a system of molecules by pre-computing $\alpha_{i0}$ which is given by eq. (\[alpha\]) with the sum restricting to include only the intra-fragment contributions (i.e., index $j$ ranges only over atoms within the same fragment as atom $i$). We compute $\Delta_{ij}$ and $\Delta_{ji}$ together because they involve many of the same terms, allowing the loops to be restricted to $i < j$, and we apply the Born cutoff to the calculation of $\alpha_{i0}$. When computing the energy of a molecular system, we compute all the updates to the Born radii due to atoms in different fragments within the Born cutoff, applying the same techniques of lumping the atoms into residues described above (which allows the cutoff criteria to be applied to groups of atoms) and of restricting the loops to $s \le t$ and, for $s = t$, to $i < j$. During this phase we mark all the residues which contain atoms with $\alpha_i \ne \alpha_{i0}$. We then make a second pass over the atoms to evaluate the terms in eq. (\[gpol\]). We use the $i\rightleftharpoons j$ symmetry of the summand to make the restrictions $s \le t$ and, for $s = t$, $i \le j$. In the innermost loop, we accumulate $q_iq_j f(r_{ij},\alpha_i,\alpha_j)$ if $i$ and $j$ belong to different fragments. Otherwise, we add $q_iq_j [f(r_{ij},\alpha_i,\alpha_j) - f(r_{ij},\alpha_{i0},\alpha_{j0})]$ and we can skip this evaluation if both $\alpha_i = \alpha_{i0}$ and $\alpha_j = \alpha_{j0}$. In addition, we can skip pairs of residues if all the atoms in each residue belong to the same fragment and if neither residue is marked as having modified Born radii. Salt effects [@srinivasan99] are easy to include within this framework. A minor complication occurs in the GB model of Qiu [*et al.*]{} [@qiu97] because $\alpha_{i0}$ depends on the “volume” of the atoms and in this model the volume depends on the 1-2 bonded atoms which may belong to a different fragment. We account for this by assuming the presence of such bonded atoms with an ideal bond length. This is, therefore, only exact if the inter-fragment bonds are at their ideal lengths. Our treatment here may be considered as a generalization of the frozen atom approximation for GB/SA [@guvench02]. However, in our application we make all the approximations in the energy function and the resulting energy is then a “state variable” and simulations based on this are well behaved. In contrast the implementation of frozen atom approximation defines the energy so that it depends on the history of the system which may cause the simulation to exhibit unphysical properties. Solvent accessible surface area =============================== The other important contribution to the solvation free energy is the cavity term. This is obtained by placing spheres centered at each atom with radius $a_i = \rho_i + r_w$ where $r_w$ is a nominal water radius (typically $r_w = 0.14\,\mathrm{nm}$). The cavity term is given by $$G_\mathrm{cav} = \sum_i \sigma_i A_i,$$ where $A_i$ is the “solvent accessible surface area” for the $i$th atom, i.e., the exposed surface area of the spheres around $i$ which is not occluded by any other spheres and $\sigma_i$ is the surface tension for the $i$th atom. (Typically $\sigma_i$ is taken to be a constant independent of atom, $\sigma_i \approx 3\, \mathrm{kJ\,mol^{-1}\,nm^{-2}}$; however the method we describe does not require this assumption.) As before, the zero energy state is obtained by separating the fragments infinitely. The energy is then given by the additional occlusion of the surface that occurs as the fragments are assembled into molecules and the molecules brought into contact with one another. The exact evaluation of this term is quite complex and for this reason a simple pairwise approximation has been developed [@weiser99]. However, the errors in this method are poorly quantified. This together with the fact that this term is typically small compared to the electrostatic terms in the energy lead us to develop a simple zeroth-order quadrature method. We select an accuracy level for the cavity calculation $\delta$, e.g., $\delta = 0.1\,\mathrm{kJ/mol}$. We prepare for the calculation of the cavity term by placing each fragment in a “template” position and we arrange a set of points on a sphere of radius $a_i$ around each atom $i$. The number of points is chosen to be $N_i = \lceil 4\pi a_i^2 \sigma_i/\delta \rceil$. The points are distributed approximately uniformly around each sphere and the entire surface energy of the sphere, $4\pi a_i^2 \sigma_i$ is divided among the $N_i$ points. (We will discuss the details of how to select the points and assign the energy later.) We next perform the intra-fragment occlusion by deleting all the points of atom $i$ which are within $a_j$ of some atom $j\ne i$. In this way each fragment is surrounded by a cloud of surface points each representing about $\delta$ of cavity energy. In order to compute the cavity term for a particular molecular configuration we transform the surface points for each fragment from their template positions to their actual positions and make a copy of the cavity energies for each point. We consider all pairs of atoms $(i,j)$ such that $i$ and $j$ are in different fragments and $r_{ij} < a_i + a_j$. We subtract from $G_\mathrm{cav}$ the energies of all the points on atom $i$ that are within $a_j$ of atom $i$ and we set the energies of these points to zero (to avoid their being counted multiple times). The optimizations described above can be used: the application of a residue-residue cutoff (excluding residue pairs $(s,t)$ with $\abs{\v b_s - \v b_t} \ge 2 r_w + h_s + h_t$), an atom-residue cutoff, and the treatment of the $(i,j)$ and $(j,i)$ terms together. In practice, the cost of evaluating this term is small for $\delta \approx 0.1\,\mathrm{kJ/mol}$. The error is proportional to $\delta$ and it is easy to benchmark a particular calculation by repeating it with smaller $\delta$. The resulting $G_\mathrm{cav}$ is obviously a discontinuous function of configuration, jumping by $\pm \delta$ as points move in and out of the water spheres of other atoms. Thus it’s an inappropriate model for a molecular dynamics simulation. However, it yields satisfactory results for Monte Carlo simulations. Let us return to the question of how to position the points on the atom sphere and how to divide the energy between these points. Ideally, we would divide the energy of the sphere based on the area of Voronoi polygons around each point. The error will then be proportional to the maximum radius of the Voronoi polygons and the ideal distribution of points is the one which minimizes this maximum radius. This is the so-called “covering problem” for the sphere, i.e., how to cover a sphere with identical discs [@fejestoth64]. Unfortunately, there are no general solutions to this problem. So instead we divide the sphere into equal intervals of latitude and we divide each latitudinal interval longitudinally into approximately square regions. A point is placed at the center of each region and the area of the region is assigned to that point. Within each fragment, we alter the position of the pole from one atom to the next, in order to avoid the occlusion of many points simultaneously as fragments move relative to one another. Discussion ========== We have shown how to make Monte Carlo moves for a molecular system with constraints. Constraints are imposed in a realistic way ensuring that we obtain the right distribution corresponding to a thermodynamic equilibrium. We will still need to know this constrained distribution if we wish to make wormhole moves [@karney05a], because, in order to satisfy detailed balance, we require knowledge of the wormhole volumes and these include a factor proportional to the “thickness” of the constraint manifold. The adiabatic move involves, naturally, many evaluations of the constraint energy raising a concern that the implementation will be slow. In reality, the cost of evaluating the constraint energy is minuscule, particularly in comparison with the solvation energy, so it is possible to evaluate the constraint energy many thousands of times in the course of an adiabatic move with minimal impact on the overall running time. The method avoids much of the algebra associated with other ways of imposing constraints [@go70] and thus is more flexible and is easier to implement. In the simple case of a molecule in which only a number of dihedral angles are allowed to vary, the movement of all the atoms in the molecule is bounded and thus the soft-energy acceptance probability is reasonably large. In contrast, the method where the dihedral angles are perturbed may lead, due to a lever effect, to large motions if the molecule itself is large. This method can easily be generalized to do localized movements. Thus, we can tailor the random displacements of a protein to explore the movement of a single loop. Detailed balance is ensured if the random displacement is a function of the atom but not of its position. (The general case can be accommodated by a suitable factor in the acceptance probability.) This method of localized movements is more widely applicable than techniques such as “concerted rotations” [@go70; @dodd93; @mezei03]. Artificially fixing the positions of some atoms would, of course, mean that the moves would not be ergodic. This would be justified if we were interested in examining the restricted system and we would then require ergodicity over the restricted configuration space. We have also considered how to optimize the evaluation of the energy in a system of molecules made up of rigid fragments bonded together. This allows the use of implicit solvent at an acceptable cost. If the system is further constrained to allow only the variation of the torsion angle of the inter-fragment bonds (fixing the bond lengths and bond angles), then we should also consider modifying the force field to “loosen” the torsion energies to counteract the effect of the hard constraints on the other bond terms. Gō and Scheraga [@go69] show the importance of considering such an effect and Katrich [*et al.*]{} [@katritch03] have offered a prescription for converting a general force field to include this effect. Alternatively, we might consider re-parameterizing the torsion terms by carrying out constrained geometry optimizations of model molecules where the energy of the molecule is minimized with the dihedral angles fixed [@schmidt93]. Acknowledgment {#acknowledgment .unnumbered} ============== This work was supported by the under Contract No. DAMD17-03-C-0082. The views, opinions, and findings contained in this report are those of the author and should not be construed as an official Department of the Army position, policy, or decision. No animal testing was conducted and no recombinant DNA was used.

--- abstract: | We have used extensive $V$, $I$ photometry (down to $V=20.9$) of $33615$ stars in the direction of the globular cluster M55 to study the dynamical interaction of this cluster with the tidal fields of the Galaxy. An entire quadrant of the cluster has been covered, out to $\simeq1.5$ times the tidal radius. A CMD down to about 4 magnitudes below the turn-off is presented and analysed. A large population of BS has been identified. The BS are significantly more concentrated than the other cluster stars in the inner 300 arcsec, while they become less concentrated in the cluster envelope. We have obtained luminosity functions at various radial intervals from the center and their corresponding mass functions. Both clearly show the presence of mass segregation inside the cluster. A dynamical analysis shows that the observed mass segregation is compatible with what is predicted by multi-mass King-Michie models. The global mass function is very flat with a power-law slope of $x=-1.0\pm0.4$. This suggest that M55 might have suffered selective losses of stars, caused by tidal interactions with the Galactic disk and bulge. The radial density profile of M55 out to $\sim 2\times r_t$ suggests the presence of extra-tidal stars whose nature could be connected with the cluster. author: - 'Simone R. Zaggia' - Giampaolo Piotto - Massimo Capaccioli date: 'Received 29 May 1997 / Accepted 11 July 1997' nocite: - '[@Lee77]' - '[@Shade88]' - '[@Alcaino92]' - '[@Trager95]' - '[@Bailyn95]' - '[@DJ93]' - '[@Mand96]' - '[@Ibata95]' title: 'The Stellar Distribution of the Globular Cluster M55 [^1] ' --- 10000 Introduction ============ The recent advances in our understanding of the structure and evolution of Galactic globular clusters (GCs) have been possible thanks to the advent of accurate CCD photometry. However, till few years ago, CCD photometry was limited to the internal parts of GCs due to the small fields of the detectors. All the information relative to the outer regions and to the tidal radius $r_t$, arise from visual (*by eye*) stellar counts made on Schmidt plates, especially by King and collaborators [@King68; @Trager95]. This methodology of investigation suffers from various problems and statistical biases; we list some of them: - The limiting magnitude of photographic plates, which is generally too bright to permit the investigation of the radial distribution of stars in an appropriate mass range; - the high uncertainty in the evaluation of background stellar contamination; - an insufficient crowding/completeness correction. All the more recent models of dynamical evolution need to make assumptions on the mass function, on the effects due to the radial anisotropy of the velocity distribution, and on the mass segregation which, in principle, could be determined observationally. \[M55frames\] Fostered by this lack of observational data, five years ago we started a long term project using one of the largest field CCD cameras available, EMMI at the NTT, to obtain accurate stellar photometry in two bands, $V$ and $I$, over at least a full quadrant in a number of GCs. The sample was selected taking into account the different morphological types and the different positions in the Galaxy. The principal aim was to map the stellar distribution from the central part out to the outer envelope (beyond the formal tidal radius, for a better estimate of the field stars contamination and in order to investigate on the possible presence of tidal tails), with a good statistical sampling of the stars in distinct zones of the color magnitude diagram (CMD) and of different masses. The use of CCD star counts, instead of photographic Schmidt plates for which the only advantage is still to give a larger area coverage [@Grillmair95; @LS97 for works on this subject], allows us to go deeper inside the core of the clusters, to better handle photometric errors and completeness corrections and to reach a considerably fainter magnitude level. CCDs also allows in the case of high concentration clusters to complement star counts of the central part with aperture photometry of short exposures images [@Ivo97]. An important byproduct of this study is the photometry of a significant number of stars in all the principal sectors of the CMD. This sample is of fundamental importance to test modern evolutionary stellar models [@RFP88]. In all cases the CMD extends well below the turn off of the main sequence. This permits us to estimate the effect of mass segregation for masses from the TO mass ($\sim0.8\Msun$) down to $0.6, 0.5\Msun$. So far, we have collected data for a total of 19 clusters. Same of them have already been reduced and analyzed. In this work we present the analysis of the star counts of the globular cluster =M55. Other clusters, for which we have already given a first report elsewhere [@Zaggia95; @Carla96; @Ivo95; @Alf96], will be presented in future works [@Ivo97; @Alf97]. Why the globular cluster M55? ============================= M55 is a low central concentration, $c=0.8$ [@Trager95], low metallicity, ${\rm [Fe/H]}=-1.89$ [@Zinn80], cluster located at $\simeq4.9$ kpc from the Sun [@Mand96]. Although it is a nearby object, it has received little or sporadic attention until very recently. The works of [@Mateo96] and [@Fahlman96] presented photometric datasets of M55 that have been used principally to establish the age and the tidal extension of the Sagittarius dwarf galaxy. [@Mand96] published the first deep (down to $V\simeq24.5$) photometry of the cluster (other previous studies of the stellar population of M55 are in Lee, 1977; Shade, VandenBerg, and Hartwick, 1988; and Alcaino , 1992). From the data of a field at $\simeq2$ core radii from the center, [@Mand96] estimated a new apparent distance modulus for M55, $(m-M)_V=13.90\pm0.09$, and from the luminosity function they found that the high-mass end of the mass function ($0.5<M/M_{\odot}<0.8$) is well fitted by a power law with $x=0.5\pm0.2$, whereas at the low-mass end ($M/M_{\odot}<0.4$) the mass function has a slope of $x=1.6\pm0.1$. From the dynamical point of view, M55 has been previously studied by [@Pryor91] in their papers on the mass-to-light ratio of globular clusters. Their principal conclusion is that this cluster might have a power law mass function with an exponent $x=1.35\div2.0$, with a lower limit of the mass function in the range $\simeq0.1\div0.3\Msun$ ( a total absence of low mass stars): a conclusion opposite to that found recently by [@Mand96]. \[M55tot\] An original work on M55 is in [@Irwin84]. They studied the radial star count density profile using photographic material digitalized with the *Automatic Plate Measuring System* (APM) of the Cambridge University. [@Irwin84] used a single photometric band, which did not allow them to lower the contribution of the field stars in the construction of the radial star counts. Nevertheless in this work (never repeated in other clusters), the authors reach some interesting conclusions: they claim one of the first evidences of mass segregation (even if they cannot quantify it); the central stellar luminosity function seems to be flat (with a corresponding mass function having a slope of $x\simeq0.0$) with a partial deviation from the King models. Moreover, they claim the presence of 8 short period variables, at the limit of their photometry, compatible with contact binaries of W UMa type. This last point is interesting for the presence of a large population of Blue Straggler (BS) stars in M55 to which the variables of [@Irwin84] could belong. Despite the potential interest of this nearby cluster for problems such as the dynamical evolution of globular clusters and interaction with the tidal field of the Galaxy, the existing data on M55 are so far limited and have been used to address only particular problems. Now large field CCDs offer the possibility to attack this problems in a suitable way. The following Section is dedicated to the presentation of the M55 data set and our observing strategy; in Section 4 we show the luminosity and mass function of the cluster; in Section 5 we present the analysis of the radial density profile and the conclusions. The details of the techniques adopted in the reduction and analysis of the data can be found in the appendix of the paper. The photometric data ==================== A whole quadrant of M55 was mapped (from the center out to $\sim1.5~r_t$, with $r_t=977''$ as in Trager, King, and Djorgovski, 1995), on the night of July 5 1992 with 18 EMMI-NTT fields ($7\farcm2\times7\farcm2$) in the $V$ and $I$ bands. Figure \[M55frames\] shows the field positions on the sky. For each field a $V$ and a $I$ band image were taken in succession, with exposure times of 40 and 30 seconds respectively. The night was not photometric and the observing conditions improved as we moved from the outer fields to the internal ones. Information on the various fields and on all the technicalities of the reduction and analysis are reported in the appendix of the paper. The $V$ *vs.* $(V-I)$ color magnitude diagram for a total of 33615 stars of M55 is shown in Figures \[M55tot\] and \[cmdradial\]. In total we detected 36800 objects of the cluster$+$field; $\simeq9\%$ of them were eliminated after having applied a selection in the [DAOPHOT II]{} PSF interpolation parameters as in [@Piotto90a]. Although the exposure time was relatively short, the brightest stars of the red giant branch and of the asymptotic giant branch are saturated, though they can be still used for the radial star counts. We have omitted them from the final CMD. In the following we will analyze the data using a division into three radial subsamples: inner ($r\le r_c$), intermediate ($r_c<r\le 2r_c$), and outer ($2r_c<r\le r_t$). The core radius is $r_c=143\arcsec$, as found from the radial density profile analysis ( Section 5). In Figure \[cmdradial\] we show the brightest part of the CMD of M55, divided in the three radial subsamples. A large population of blue straggler stars (BS) is clearly visible, particularly in the inner part of the cluster where the background/foreground star contamination is low. In the intermediate zone, the BS population is better defined, and the sequence seems to reach brighter magnitudes. The BS sequence of the inner part appears to be broader in color than the sequence of the intermediate radial range. Part of this broadening can be attributed to the photometric errors that are larger in the inner region than in the intermediate one. The rest of the broadening is probably natural and could be connected to the two formation mechanisms of BS stars: the outer BS stars might mainly come from merger events, while the inner BS might be the final products of collisions (see Bailyn 1995). We compared the distribution of the 95 BS with that of the 1669 sub-giant branch (SGB) stars selected in the same magnitude interval. In order to minimize the background star contamination along the SGB (very low indeed in the inner part of the cluster), we chose only the stars inside $\pm2.0\sigma$ (where $\sigma$ is the standard deviation in the mean color) from the mean position of the SGB. We subtracted the background stellar contamination estimated from the star counts in the radial zone $r>1.1~r_t$. The BS seem to be more concentrated than the corresponding SGB stars only in the inner $250\div300\arcsec$ (Figure \[figbs\]). At larger distances, the BS distribution becomes less concentrated than the comparison SGB stars. We run a 2-population Kolmogorov-Smirnov test. The test does not give a particularly high statistical significance to the result: the probability that the BS and SGB stars are *not* taken from the the same distribution is 96%. However, this possibility cannot be excluded: see [@Piotto90b] and [@DP93] for a discussion on the limits in applying this statistical test for checking population gradients. Another way to look into the same problem is to investigate the radial trend of the ratio BS/SGB, as plotted in Figure \[figbs\] (*lower panel*). Also in this case the bimodal trend is quite evident. The relative number of BS stars decreases from the center of the cluster to reach a minimum at $r\simeq 250\div300$ arcsec ($r\simeq 2r_c$), and then it rises again. Again, the statistical significance is questionable, in view of the small number of BS at $r>300$ arcsec (6 stars). Nevertheless, this possible bimodality is noteworthy. Indeed, there is a growing body of evidence that the radial distribution of BS stars in GCs might be bimodal, as shown by [@Ferraro97] for or [@Ivo97] and [@Piotto97] for . What makes our result for M55 of some interest is that this distribution has been interpreted in terms of environmental effects on the production of BS stars. However, the fact that M55 has a very low concentration (c=0.8), while M3 and NGC 1851 are high concentration clusters (c=1.85 and c=2.24 respectively, Djorgovski 1993), might make this conclusion at least questionable. Luminosity and mass function \[M55lf\] ====================================== From the CMD we have derived a luminosity function (LF) for the stars of M55. Figure \[M55fdl\] shows the LFs in the different annuli defined in the previous Section (inner, intermediate and outer). The three LFs have been normalized to the star counts of the SGB region in the magnitude interval $15.90<V<17.40$, after subtracting the contribution of the background/foreground stars scaled to the area of each annulus. In the lower part of Figure \[M55fdl\], we show also the LF of the background/foreground stars estimated from the star counts at $r>1.3~r_t$ vertically shifted for clarity. In order to reduce contamination by those stars, all the LFs have been calculated selecting the stars within $2.5\sigma$ (again, $\sigma$ is the standard deviation of the mean color) from the fiducial line of the main sequence of the cluster. The LFs do not include the HB and BS stars. The LF of the background stars has a particular shape: it suddenly drops at $M_V=4.0$. This feature has a natural explanation considering the color-magnitude distribution of the field stars around M55 and the way we selected the stars. The drop in the number of field stars is at the level of the M55 TO and as can be seen in Figure \[M55tot\], or in the lower right panel in Figure \[cmdradial\], the TO of M55 is bluer than the TO of the halo stars of the Galaxy, which are the main components of the field stars towards M55 [@Mand96]. Selecting only stars within $2.5\sigma$ of the fiducial line of M55 will naturally cause such a drop. The completeness correction, as obtained in appendix \[crowd\], has been applied to the stellar counts of each field of M55. As it is possible to see from Table \[M55tab2\], the magnitude limit varies from field to field. We have adopted the same, global, magnitude limit for all the LFs: , that of the fields with the brighter completeness limit (field 16 and 17). This limits all the LFs to $V=20.9$, corresponding to a stellar mass $m\sim0.6\Msun$, for the adopted distance modulus and a standard 15 Gyr isochrone (see next subsection). The data for the inner annuli come from the central image, which has a limiting magnitude of the corresponding LF fainter than the global value adopted here. This is due to the better seeing of the central image compared to all the other images. We adopted a brighter limiting magnitude in order to avoid problems in comparing the different LFs. \[M55fdl\] Figure \[M55fdl\] shows clearly different behaviour of the LFs below the TO: they are similar for the stars above the TO, while the LFs become steeper and steeper from the inner to the outer part of the cluster: this is a clear sign of mass segregation. For the inner LF there is also a possible reversal in slope below $M_V=5.5$. In order to verify that the difference between the three LFs is not due to systematic errors (wrong completeness correction, imperfect combination of data coming from two adjacent fields etc.), we have tested our combining procedure in several ways. In one of our tests we built LFs of two EMMI fields at the same distance from the center of the cluster: , we compared the LF of the field 2 with that of the field 6. After having corrected for the ratio between the covered areas and subtracting the field star contribution, the two LFs were consistent in all the magnitude intervals down to the completeness level of the data (that is lower than the one adopted). Having for field 2 a magnitude limit of 22.2 (see Table \[M55tab2\]) and field 6 a limit of 21.5, we also verified that for the latter our star counts are in correct proportion below the completeness level of 50%. In a second test, we generated two LFs dividing the whole cluster in two octants (dividing along the $45^\circ$ line that runs from the center of the cluster till the field 19  Figure \[M55frames\]). For each of the two slices we generated three LFs in the same radial range as in Figure \[M55fdl\]. After comparing all of them we did not find any significant difference. Therefore the differences among the three LFs in Figure \[M55fdl\] must be real. \[M55mf\] Another source of error in the LF construction is represented by the LF of the field stars. As will be shown in Section \[M55rprof\], M55 has a halo of probably unbound cluster stars. The field star LF constructed from the star counts just outside the cluster can be affected by some contamination of the cluster halo. The consequence is that we might over-subtract stars when subtracting the field LF from the cluster LF, modifying in this way the slope of the mass function (the more affected magnitudes are the faintest ones). To test this possibility, we have extracted background LFs in two different anulii outside the cluster (in terms of $r_t$, $1.0<r\le1.3$ and $r>1.3$). Comparing the two background/foreground LFs we found that the number of stars probably belonging to the cluster but outside the tidal radius must be less than $\sim25\%$ of the adopted field stars in the worst case (the faintest bins). The possible over-subtraction is not a problem for the inner and intermediate LFs, where the number of field stars (after rescaling for the covered area) is always less than $\sim3\%$ of the stars counted in each magnitude bin. For the outer LF, the total contribution of the measured field stars is larger, but it is still less than $25\%$ of the cluster stars (the worst case applies to the faintest magnitude bin): this means that the possible M55 halo star over-subtraction in the field-corrected LF is always less than $6\%$ ($25\% \times 25\%$), negligible for our purposes. Mass function of M55 \[Sm55mf\] ------------------------------- In order to build a mass function for the stars of M55, we needed to adopt a distance modulus and an extinction coefficient. [@Shade88] give $(m-M)_V=14.10$, E$(B-V)=0.14\pm0.02$, while, more recently, [@Mand96] give $(m-M)_V=13.90\pm0.07$, E$(B-V)=0.14\pm0.02$. In the absence of an independent measure made by us, we adopted the values published by [@Mand96] because they are based on the application, with updated data, of the subdwarfs fitting method. Using the LFs of the previous Section we build the corresponding mass functions using the mass-luminosity relation tabulated by [@VDB85] for an isochrone of $Z=3\times10^{-4}$ and an age of 16 Gyr [@Alcaino92]. The MFs for the three radial intervals are presented in Figure \[M55mf\]. The MFs are vertically shifted in order to make their comparison more clear. The MFs are significantly different: the slopes of the MFs increase moving outwards as expected from the effects of the mass segregation and from the LFs of Figure \[M55fdl\]. Figure \[M55mf\] clearly shows that the MF starting from the center out to the outer envelope of the cluster is flat: the index $x$ of the power law, $\xi=\xi_0m^{-(1+x)}$, best fitting the data are: $x=-2.1\pm0.4$, $x=-0.8\pm0.3$, and $x=0.7\pm0.4$ going from the inner to the outer anulii; this means that the slope of the global MF (of all the stars in M55) should be extremely flat. Indeed, the slope of the global mass function obtained from the corresponding LF of all the stars of M55 is: $x=-1.0\pm0.4$ This result agrees with the results of [@Irwin84], while the results of [@Pryor91] appear in contrast to what we have found here. Our MF in the outer radial bin can be compared with the high-mass MF of [@Mand96], obtained from a field located at $\simeq6$ arcmin from the center of M55. As already reported in Section 2, [@Mand96] obtained a deep MF for M55 (down to $M\simeq0.1\Msun$) which they describe with two power laws connected at $M\simeq 0.4\div0.5\Msun$. Their value of $x=0.5\pm0.2$ for the high-mass end of the mass function ($0.5<M/\Msun<0.8$) is in good agreement with our value of $x=0.7\pm0.4$, obtained in the same mass range for the outer radial bin. The low-mass end of the MF by [@Mand96] ($M/\Msun<0.4$) has a slope of $x=1.6\pm0.1$. The level of mass segregation of M55 is comparable to that found in by [@Richer89]. M71 shares with M55 similar structural parameters as well as positional parameters inside the Galaxy. The detailed analysis of [@Richer89] of M71 showed that this cluster should also have a large population of very low mass stars ($\sim0.1$). By fitting a multi-mass isotropic King model [@King66; @GG79] to the observed star density profile of M55, we compared the observed mass segregation effects with the one predicted by the models. Here we give a brief description of our assumptions in order to calculate the mass segregation correction from multi-mass King models. A more detailed description can be found in [@Pryor91], from which we have taken the *recipe*. The main concern in the process of building a multi-mass model is in the adoption of a realistic global MF for the cluster. For M55 we adopted a global MF divided in three parts: - a power-law for the low-mass end, $0.1<M/\Msun\le 0.5$, with a fixed slope of $x=1.6$ (as found by Mandushev   1996); - a power-law for the high-mass end, $0.5<M/\Msun\le m_{TO}$, with a variable slope $x$; - and a power-law for the mass bins of the dark-remnants where to put all the evolved stars with mass above the TO mass, $m_{TO}<M/\Msun\le 8.0$: essentially white dwarfs. Here we adopted a fixed slope of 1.35, The mass of the WDs were set according to the initial-final mass relation of [@W90]. To build the mass segregation curves we varied the MF slope $x$ (the only variable parameter of the models) of the high-mass end stars in the range $-1.0\div1.35$, finding for each slope the model best fitting the radial density profile of the cluster. Then we calculated the radial variation of $x$ for the best-fit models in the same mass range of the observed stars: $0.5<M/M_{\odot}<m_{TO}$. The radial variations of $x$ are compared with the observed MFs in Figure \[M55slx\]. The mass function slopes are shown at the right end of each curve. This plot is similar to those presented in [@Pryor86], and allows one to obtain the value of the global mass function of the cluster. The three observed points follow fairly well the theoretical curves. Also the high-mass MF slope value of [@Mand96] (open circle in Figure \[M55slx\]) is in good agreement with the models and our MFs. From these curves, we have that the slope of the high-mass end of the global MF of M55 is $x\simeq-1.0$, which is in quite good agreement with the global value of the MF found from the global LF of M55 ( previous section). In Figure \[profile\] we show the model which best fits the observed radial density profile for a global mass function with a slope $x=-1.0$. The relatively flat MF of M55 could be the result of the selective loss of main sequence stars, especially from the outer envelope of the cluster, caused by the strong tidal shocks suffered by M55 during its many passages through the Galactic disk and near the Galactic bulge [@Piotto93 for a general discussion of the problem]. A flat MF for M55 agrees well with the results of [@CPS93] who have found that the clusters with a small $R_{GC}$ and/or $Z_{GC}$ show a MF significantly flatter than the cluster in the outer Galactic halo or farther from the Galactic plane. Indeed, M55 is near to the Galactic bulge, $R_{GC}=4.7$ kpc ($R_\odot=8.0$ kpc), and to the Galactic disk $Z_{GC}=-2.0$ kpc. Figure \[m55xrz\] shows that taking into account observing errors, M55 fairly fits into the relation given by [@Manu97], which is a refined version of the one found by [@DPC93]. A different conclusion has been reached by [@Mand96] using their uncorrected (for mass segregation) value for the MF of M55. As noted by the referee, M55 lies further from the average relation defined by the other clusters: of those with a similar abscissa ($0.0\pm0.2$), M55 is the one with the lowest value of $x$. It is not possible to identify the main source of this apparent enhanced mass-loss of M55 compared to the other clusters; a possible cause can be a orbit of the cluster that deeply penetrate into the bulge of the Galaxy. This cannot be confirmed until is performed a reliable measure of the proper motion of M55. Radial density profile from star counts.\[M55rprof\] ==================================================== The CMD allows a unique way to obtain a reliable measure of the radial density profiles of GCs. In fact, the CMD allows us to sort out the stars belonging to the cluster, limiting the problems generated by the presence of the field stars. This also permits to extract radial profiles for distinct stellar masses. We have first created a profile as in [@King68], in order to compare our results with the existing data in the literature. The comparison has been done with the radial density profile of M55 published by [@Pryor91] which includes the visual star counts of King et al.. We could not compare our data with [@Irwin84] because they have not published their observations in tabular form. \[profconf\] Density profile for stars above the TO -------------------------------------- Figure \[profile\] shows the radial density profile for the TO plus SGB stars extracted from the CMD of M55 (from 1 magnitude below the TO to the brightest limit of our photometry). We have selected the stars within $2.5\sigma$ from the fiducial line of the CMD plus the contribution coming from the BS and HB stars; star counts has been limited at the magnitude $V\le18.5$. This relatively bright limit corresponds approximately to the limit of the visual star counts by [@King68] on the plate ED-2134 (in order to make the comparison easier we used the same radial bins of King). Our counts have been transformed to surface brightness and adjusted in zero point to fit the [@Pryor91] profile of M55. The agreement with the data presented by Pryor is good everywhere but in the outer parts where our CCD star counts are clearly above those of [@King68]. This difference is probably due to our better estimate of the background star contamination. In the plot we have shown also the raw star counts (crosses) prior to the background star subtraction: it can be clearly seen that our star counts go well beyond the tidal radius, $r_t=977''$, published by [@Trager95]. This allow us to estimate in a better way than in the past the stellar background contribution. The background star counts show a small radial gradient: we will discuss this point in greater detail in the next Section. Here, the minimum value has been taken as an estimate of the background level. We point out that the differences present in the central zones of the cluster could be in part due to some residual incompleteness of our star counts, to the absence in the starcounts of the brightest saturated stars, and to the difficulties in finding the center of the cluster. We searched for the center using a variant of the mirror autocorrelation technique developed by [@DJ88]. In the case of M55 we encountered some problems due to a surface density which is almost constant inside a radius of $\simeq100\arcsec$. In order to evaluate the structural parameters of M55, we have fitted the profile Figure \[profile\] with a multi-mass isotropic [@King66] model as described in the previous Section. In the following table we show the parameters of the best fitting model and we compare them with the results of [@Trager95], [@Pryor91], and [@Irwin84]: [lccc]{} Author & $c$ & $r_c$ & $r_t$\ This paper & 0.83 & $143''$ & $970''$\ Trager   & 0.76 & $170''$ & $977''$\ Pryor   & 0.80 & $140''$ & $876''$\ Irwin and Trimble & $\sim1.0$ & $\sim120''$ & $\sim1200''$\ The concentration parameter of M55 is one of the smallest known for a globular cluster. Such a small concentration implies strong dynamical evolution and indicates that the cluster is probably in a state of high disgregation [@Aguilar88; @Gnedin97]. Our value of the tidal radius is well in agreement with that of [@Trager95] who used a similar method to fit the data. [@Pryor91] give a value of $r_t$ 10% smaller than ours. We note that Pryor and Trager used the same observational data set. The difference with [@Irwin84] is probably due to the fact that the authors have not fitted their data directly but made only a comparison with a plot of King models. \[extrat\] The density profile for different stellar masses. ------------------------------------------------- Having verified the compatibility of our density profile with previously published ones, we have extracted surface density profiles for different magnitude ranges corresponding to different stellar masses. The adopted magnitude intervals have been chosen to have a significant number of stars in each bin. We used logarithmic radial binning that allows a better sampling of the stars in the outer part of the cluster. In order to lower the noise in the outer part of the profile, we have smoothed the profiles with a median static filter of fixed width of 3 points. We verified that the filtering procedure did not introduce spurious radial gradients in the density profiles. The mean masses in each magnitude bin adopted for the profiles, as obtained from the isochrone by [@VDB85] ( also Section \[Sm55mf\]), are: [cc]{} $V$ & $<m>$\ $<18$ & 0.79\ $18-19$ & 0.77\ $19-20$ & 0.71\ $20-20.9$& 0.63\ The relative profiles, without subtraction of the background stars, are shown in Figure \[profconf\] and \[extrat\]. The arrows in both figures indicate $r_c$, $2~r_c$ and $r_t$. The profiles plotted in Figure \[profconf\] are clearly different from each other: this is as expected from the mass segregation effects. To better compare the profiles, in Figure \[profconf\] they have been normalized in the radial interval $2.6<\log(r/arcsec)<2.9$ (where the profiles have a similar gradient) to the profile of the TO stars. This operation is possible because in this radial range the effects of mass segregation are small ( Figure \[M55slx\]); they are more evident within one core radius. The density profiles are consistent with the mass segregation effects that we have already seen in the mass function of the cluster. The more interesting aspect of the profiles in Figure \[profconf\] is the clear presence of a stellar radial gradient in the star counts of the background field stars. In Figure \[extrat\], we show the radial profiles of the extra cluster stars after normalization of the profiles outside $\log(r/arcsec)=3.0$. The 4 profiles are not exactly coincident outside $r_t$. Let us discuss various possible explanations for this observation: - *Errors in the completeness correction or errors in the star counts.* We repeated the extensive tests on the data made to assess the validity of the mass segregation seen in the LFs. We checked that the variation in the completeness limit of the various EMMI fields does not introduce spurious trends. In a different test, we divided the cluster in two slices along a line at $45^\circ$ from the center of the cluster up to the field 19 (  Figure \[M55frames\]), and built the radial profiles for each of the 4 magnitude bins: in all cases there were no significant differences. The radial profile of the stars in the magnitude range $18\div19$ ($M_V=4.1\div5.1$ in Figure \[M55fdl\]) has the lowest contamination of background stars, as shown by its LF in Figure \[M55fdl\]. - *A non-uniform distribution of the field stars around M55.* It is possible that the field stars around M55 are distributed in a non-uniform way. In the work by [@Grillmair95] it clearly appears that the field stars of some GCs present a non-uniform distribution around the clusters. The gradients are significant and the authors used bidimensional interpolation to the surface density of the field stars to subtract their contribution to the star counts of the clusters. In the present case, field star gradients could be a real possibility, but we cannot test it because we do not have $360^\circ$ coverage of the cluster: our coverage of M55 is only a little more than a quadrant. The Galactic position of M55 ($l\simeq-23^\circ$, $b\simeq9^\circ$) can give some possibility to this option. At this angular distance from the Galactic center the bulge and halo stars probably have a detectable radial gradient. However it remains difficult to explain the existence of the gradient also for the stars in the magnitude range $18\div19$: for them (section \[M55lf\]), as stated before, we have the lowest contamination from the field stars. We have created a surface density map of the starcounts of M55. The map was constructed using all the stars of the $2.5\sigma$-selected sample of our photometry (excluding fields 25 and 35), counting stars in square areas of approximately $9''\times9''$ and then smoothing the resulting map with a gaussian filter. The starcounts are not corrected for crowding but we stopped at $V=20.5$. The map is presented in Figure \[gray\]. The map has the same orientation as Figure \[M55frames\]. We have also overplotted contour levels to help in reading the map. Figure \[gray\] clearly shows that well outside the tidal radius of M55 (located approximately at the center of the map) there is a visible gradient in the star counts. - *A gradient generated by the presence of the dwarf spheroidal in Sagittarius* [@Ibata95]. Between the Galaxy center and M55 there is the dwarf spheroidal galaxy called Sagittarius [@Ibata95]. Sagittarius is interacting strongly with the Galaxy and probably is in the last phases of a tidal destruction by the Galactic bulge. The distance between the supposed tidal limit of this galaxy (using the contour map of Ibata 1995) and M55 is $\sim5^\circ$. In the recent work by [@Mand96] the giant sequence of the Sagittarius appears clearly overlapped with the sequence of M55. This happens only in the magnitude range $V\simeq20.0\div21.0$ where our star counts end. [@Fahlman96] showed that the SGB sequence of the Sagittarius crosses the main sequence of M55 at $V\simeq20.5\div20.7$, and at a corresponding color of $(V-I)\simeq1.1\div1.2$. Similar results were found by [@Mateo96]. This is due to the different distances of these two systems from us: $\sim4.5$ kpc for M55 and $\sim24$ kpc for Sagittarius. This implies that out star counts can be influenced by the stars of the dwarf spheroidal only in our last magnitude bin, $20\div20.9$. Our selection of stars along the CMD of M55 limits the Sagittarius stars to those effectively crossing the main sequence. In conclusion, if effectively the Sagittarius stars are present as background stars we should see them only in one of the 4 profiles, but the coincidence of the 4 profiles excludes this ipothesis. - *A halo of stars escaping from the clusters.* This possibility is more suggestive. The stellar gradient could be a possible extra-tidal extension of the cluster, similar to what [@Grillmair95] found in their sample of 12 clusters. The tidal extension could be caused by the tidal-shocks to which the cluster has been exposed during its perigalactic passages, through the Galactic disk. Another possibility is the creation of the stellar halo by stellar dynamical evaporation from the inner part of the cluster. Such mechanisms work independently of stellar mass [@Aguilar88] and so the stellar halo should have a similar gradient for all the stellar masses as in the present case. Such halos are very similar to the theoretical results obtained by [@OhLin92] and [@Grillmair95], who have obtained tidal tails for globular clusters N-bodies simulations. We believe that the probable explanation for the phenomenon shown in Figures \[profconf\] and \[extrat\] is in the presence of an extra-tidal stellar halo or tidal tail. Doubt resides in the unknown gradient of the background field stars. To resolve this we need to map the whole cluster and a large area surrounding the cluster. This would also allow us to find the exact level of field stars. Our star counts stop at 33($\simeq2\times r_t$), from the center of M55 while the tidal tails of [@Grillmair95] stop at $\simeq 2.5 \div 4\, r_t$. Consequently, we cannot correctly subtract the contribution of the field stars from our star counts. We can give only an estimate of the exponent of the power law, $f\propto r^{-\alpha}$, fitting the profiles at $ r > 1.2\,r_t$. Without subtracting any background counts $\alpha\sim0.7\pm0.3$, while subtracting different levels of background stars the slope varies in the interval $0.7<\alpha<1.7$: the highest value comes out after subtracting the outermost value of the density profiles. When it will be available a better estimate of the background/foreground level of the sky it will be possible to assign a value to the slope of the gradient of stars: actually our range, $\alpha=0.7 \div 1.7$, is in accordance with those found theoretically by [@OhLin92] and observationally by [@Grillmair95]. We are grateful to C.J. Grillmair, the referee, for his careful reading of the manuscript and his suggestions for improving the paper. The authors warmly thank Tad Pryor for making available it’s code for the generation of multi-mass King-Michie models. We thanks I. Saviane for his help in constructing the surface density map of M55. Finally, we warmly thanks Nicola Caon for doing the observations included in this work. 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Ital., 441, 667 Zinn, R., 1980, ApJS, 42, 19 Zoccali, M., Piotto, G., Zaggia, S. R., & Capaccioli, M., 1997, MNRAS, submitted [crcccccc]{} Field&Stars&Airmass& RA & DEC & & V(50%)\ & & && $V$ & $I$ &\ 01 & 13442 & 1.010 & 294.926 & $-30.948$ & 0.9 & 0.9 & 21.0/22.1\ 02 & 3176 & 1.007 & 295.062 & $-30.948$ & 0.9 & 0.9 & 22.2\ 03 & 872 & 1.004 & 295.198 & $-30.948$ & 1.1 & 1.1 & 22.1\ 04 & 495 & 1.005 & 295.333 & $-30.947$ & 1.3 & 1.1 & 21.2\ 06 & 2273 & 1.016 & 294.926 & $-30.831$ & 1.3 & 1.2 & 21.5\ 07 & 800 & 1.013 & 295.061 & $-30.831$ & 1.5 & 1.5 & 21.2\ 08 & 543 & 1.009 & 295.197 & $-30.831$ & 1.4 & 1.4 & 21.2\ 09 & 482 & 1.007 & 295.333 & $-30.831$ & 1.6 & 1.6 & 21.0\ 11 & 635 & 1.022 & 294.926 & $-30.714$ & 1.4 & 1.4 & 21.3\ 12 & 560 & 1.028 & 295.061 & $-30.714$ & 1.4 & 1.4 & 21.2\ 13 & 531 & 1.034 & 295.197 & $-30.714$ & 1.4 & 1.5 & 21.0\ 14 & 506 & 1.041 & 295.333 & $-30.714$ & 1.3 & 1.5 & 21.0\ 16 & 491 & 1.227 & 294.983 & $-30.598$ & 1.3 & 1.4 & 20.9\ 17 & 503 & 1.150 & 295.061 & $-30.598$ & 1.6 & 1.3 & 20.9\ 18 & 537 & 1.055 & 295.197 & $-30.598$ & 1.5 & 1.6 & 21.1\ 19 & 556 & 1.048 & 295.332 & $-30.598$ & 1.3 & 1.5 & 21.2\ 25 & 666 & 1.001 & 295.197 & $-31.063$ & 1.2 & 1.3 & 21.6\ 35 & 757 & 1.001 & 295.333 & $-31.063$ & 1.2 & 1.1 & 21.9\ Image reduction and analysis ============================ The images were reduced using the standard algorithms of bias subtraction, flat fielding and trimming of the overscan, without encountering particular problems. The stellar photometry was done using [DAOPHOT II]{} and [ALLSTAR]{} [@Stetson87]. The second version of [DAOPHOT]{} was particularly useful for the image analysis since we were forced to use a variable point spread function (PSF) through the images. In fact, the stellar images of the EMMI *Red Arm* together with the F/2.5 field camera presented coma aberration at the edges of the field: the resulting PSF was radially elongated. To better interpolate the PSF we also used an analytic function with 5 free parameters ( the [Penny]{} function of [DAOPHOT II]{}). In order to obtain a single CMD for all the stars found in the 18 fields we first obtained the CMD of each field matching the $V$ and I photometry. Then, we combined all the CMDs using the relative zero points determined from the overlapping regions of adjacent fields. All the CMDs were connected to the main CMD, one at a time, following a sequence aimed at maximizing the number of common stars usable for the zero point calculation. The central field CMD was used as the starting point of the combination. For the outer fields we used a minimum of 20 common stars while for the inner fields we had at least 300 stars. The mean error of the zero points was $\simeq0.05$ magnitudes, compatible with the errors calculated from the crowding experiments. Since the night was not photometric, we could not directly calibrate our data. We were only able to set an absolute zero point using the unpublished calibrated photometry of the center of M55 by Piotto (see next Section). In order to perform the photometry of the central field of the cluster, we divided it into 4 subimages of $\simeq600\times600$ pixels, to minimize the effect of the strong stellar gradients present in this image. We allowed a good overlap to be able to perform the successive combination of the photometry of the stars. In this way we also avoided two problems: we had better control of the PSF calculation and we reduced the number of stars per image to be analyzed. Thanks to the low central concentration of the cluster and the fairly good seeing of the images (even if the crowding was not completely absent), we were able to obtain complete photometry down to $V\simeq21$. Calibration of the photometry ============================= In principle, the analysis of the radial density profile does not require calibrated photometry. But this operation is necessary if we want to analyze the stellar population of the cluster, together with its stellar luminosity and mass functions. Since we could not use standards taken during the same night, we have performed a relative calibration using existing photometry of M55. For the $V$ magnitude we linked our data to Piotto’s (1994) un published photometry of the central field of M55 from images taken with the 2.2m ESO telescope. For the (V-I) we calibrated our data against [@Alcaino92] photometry. They published a CCD BVRI photometry for two different non-overlapping fields outside the center of M55, named FA and FB, with dimensions of $3\farcm1\times1\farcm9$, contained in our central field. Figure \[M55P8\] shows our $V$ zero point calculated against Piotto’s (1994) while Figure \[M55Alcv\] shows the $V$ zero point of the two fields FA and FB of [@Alcaino92]. The mean zero point for the two fields of Alcaino  gives $\Delta V_{Alcaino}=6.11\pm0.04$ which compare well with $\Delta V_{Piotto}=6.12\pm0.03$. The two are in good agreement taking into account the errors. There are no magnitude gradients. The LFs are coming from the photometry in the V-band. Before the publication of the I-band photometry by [@Mand96], the one by [@Alcaino92] was the only photometry in the literature. Unfortunately, the M55 data set of [@Mand96] does not overlap with any of our fields: it is centered just few arcmin south of our field 2. In Figure \[M55Alcv\], we show the difference between our data and those of [@Alcaino92]. In this case, the two zero points calculated for Alcaino’s fields differ by a significant amount. We do not know the origin of this discrepancy, which we believe is internal to the data of [@Alcaino92]. They could not resolve this due to the fact that fields FA and FB do not have stars in common. We believe that the problem is not in our data since both Alcaino’s fields are contained in the same subimage of the central field. Lacking other independent (V-I) calibrations, we are forced to adopt as our color zero point the mean of the two values of FA and FB: $\Delta(V-I)=0.45\pm0.05$. Crowding experiments. {#crowd} ===================== For each field, we performed a series of Monte Carlo simulations in order to establish the magnitude limit and the degree of completeness of the CMD. The magnitude limit has been defined as the level at which the completeness function reach a value of $0.5$, V(50%). This value is reported in Table \[M55tab2\] for each field. The procedure followed to generate the artificial stars for the crowding experiments is the standard one [@Piotto90b]. The completeness function used to correct our data is the combination of the results of the experiments in both the $V$ and $I$ images for each field. In the outer fields the stars were added at random positions in a magnitude range starting from $V=19$ (just 0.5 mag below the main sequence TO). In the $I$ band experiments, we used the same star positions of the $V$ experiments, with the $I$ magnitudes set according to the corresponding main sequence color. For each outer field, we performed 10 experiments with 100 stars. For the inner fields (fields number 2, 3 and 6) the experiments were 10 with 100 stars in an interval of only 1 magnitude for 5 different magnitudes (a total of 50 experiments). Moreover, in these fields the stars were added taking into account the radial density profile of the cluster. For the 4 subimages of the central field, we performed independent crowding experiments. For each subimage, we ran 10 experiments in 0.5 mag. steps in the range $19\div23$, with the stars radially distributed as the density profile of the cluster. In this way, we were able to better evaluate the level of the local completeness of the photometry. The completeness function has been calculated for each field taking into account the results of the two different experiments in $V$ and I. As an example in Figure \[M55cro\] we show the completeness functions for field number 3 (top) and 19 (bottom). The results of the experiments were fitted using the error function: $$g(x;y_0,\sigma) = 1 - \int_{-\infty}^{x} e^{\frac{(y-y_0)^2}{\sigma^2}} {\rm d}y. \label{cumgauss}$$ $y_0$ is the magnitude at which the completeness level is 50%, V(50%); $\sigma$ gives the rapidity of the decrease of the incompleteness function and is connected with the read out noise and the crowding of the image. For the star counts correction we used the interpolation with the previous equation instead of using directly the noisy results of the experiments (these were too few to lower the small number statistical noise of the results). In this way we avoid the adding of noise to the star counts. In every case we verified that the fitting function is an acceptable interpolation that gives very low residuals compared to the error distribution function. [^1]: Based on observations made at the European Southern Observatory, La Silla, Chile.

--- abstract: 'We show that there is no useful and universal definition of a running gravitational constant, G(E), in the perturbative regime below the Planck scale. By consideration of the loop corrections to several physical processes, we show that the quantum corrections vary greatly, in both magnitude and sign, and do not exhibit the required properties of a running coupling constant. We comment on the potential challenges of these results for the Asymptotic Safety program.' author: - 'Mohamed M. Anber${}^{1}$' - 'John F. Donoghue${}^{2,3}$' title: '[**On the running of the gravitational constant** ]{}' --- Introduction ============ The concepts of running coupling constants and the renormalization group are of great utility in renormalizable field theories. It is tempting to attempt a definition of a running gravitational constant $G(q^2)$ in the context of quantum general relativity. As a simple example, one could take the quantum correction to the gravitational potential between heavy masses [@potential][^1] $$V(r) = -G\frac{Mm}{r}\left[1+ \frac{41}{10\pi}\frac{G}{r^2}\right]\,,$$ and turn it into a running coupling $$G(r) = G\left[1+ \frac{41}{10\pi}\frac{G}{r^2}\right]$$ which would imply that the gravitational strength gets stronger at short distance. We will return to the weaknesses of this particular definition in Sec. 8 below, but it is only one of many attempted definitions that have been suggested in the literature [@runningG]. Moreover there is an extensive approach to quantum gravity that relies on the running of the gravitational coupling. Within the hypothesis of Asymptotic Safety [@Weinberg:1980gg; @asymsafety], a suitably normalized version of the gravitational coupling is proposed to run to an ultraviolet fixed point. In Euclidean space, one treats the dimensionless combination $g= Gk_E^2$, where $k_E$ is a measure of the Euclidean energy, and the fixed point is described by $$g= Gk_E^2 \to g_* ~~~~{\rm as} ~~~k_E \to \infty\,.$$ This would imply that the running gravitational strength itself vanishes at large energy, $G(k_E)\to 0$. This behavior is often summarized by a function $$G(k_E) = \frac{G_N}{1+\alpha G_N k_E^2}\,, \label{runningform}$$ where $G_N$ is the gravitational coupling at zero energy. Gravitational corrections are calculable at low energy using effective field theory [@Donoghue:1994dn]. However the effective field theory presents a very different picture of quantum effects - one in which the gravitational constant does [*not*]{} run. Because of the dimensional coupling, loops do not lead to a renormalization of the leading gravitational action $R$, i.e. the one that contains the gravitational constant $G$, but rather renormalize higher order terms in the action that come with higher derivatives, i.e. terms such as $R^2$ or $R_{\mu\nu}R^{\mu\nu}$. Indeed there is a power counting theorem that says that each gravitational loop always brings in two more powers of the energy. The content of the renormalization group within effective field theory has been explored by Weinberg and others [@weinberg]. While the technique is useful at predicting some leading and subleading logarithmic kinematic dependence, it does not involve the running of the leading order coupling, and in gravity does not predict the running of $G$. Efforts to define a running coupling in the perturbative regime must attempt something outside of the usual application of the renormalization group. The purpose of this paper is to explore explicit physical calculations in Lorentzian spacetime to see if there is a definition of running gravitational coupling which is physically useful and universal in the regime where we can maintain control of our calculations. Our answer is negative - no such definition is both useful and universal. Gravitational corrections to coupling constants =============================================== Ultraviolet divergences in field theories correspond to counterterms in a local Lagrangian. In a theory with a dimensional coupling constant with an inverse mass dimension, such as general relativity, these local terms are ones which have a higher mass dimension than the original starting Lagrangian. For example, for gravitational corrections to Quantum Electrodynamics (QED) the divergences generated at one loop have the forms $${\cal L}_{ct} = Gc_1 F^{\mu\nu} \partial^2F_{\mu\nu} + Gc_2\bar{\psi}\gamma_\mu \psi \partial_\nu F^{\mu\nu} + Gc_3 \bar{\psi}\gamma_\mu D_\nu \psi F^{\mu\nu}\,,$$ when regularized dimensionally. (We will discuss regularization with a dimensionful cutoff in the next section.) Equivalently field redefinitions and/or equations of motion can be used to rewrite some of these operators (and the accompanying divergences) as contact interactions such as $${\cal L'}_{ct} = Gc_4\bar{\psi}\gamma_\mu \psi \bar{\psi}\gamma^\mu \psi\,.$$ The divergences can be absorbed into renormalized parameters of these higher dimensional operators, as in standard effective field theory practice. The scale dependence that arises in dimensional regularization will then be associated with the coefficients of the higher order operators, i.e. the set $c_i$, rather than the original coupling constant, which in the QED case would be the electric charge. Finite remainders will also carry this higher order momentum dependence, i.e. they will be of order $Gq^2$ or $Gq^2\ln q^2$ higher than the original coupling. Because loops generate these higher order effects in the momentum expansion, attempts to have the loops contribute to the running of the original charge $e$ require some repackaging of the higher order effects into a revised definition of the original charge. Logically this is conceivable by renormalizing the coupling at a higher energy scale. The standard effective field theory treatment is equivalent to renormalzing the operators near zero energy. However, by choosing a renomalization condition that defines the charge at a higher energy scale $E$, one would in general include some of the higher momentum dependence into the definition of the charge. For example, for a given amplitude $Amp_i$ we would define $$\begin{aligned} Amp_i &=& a_i g^2 +b_ig^2\kappa^2 q^2 \nonumber \\ &=& a_i g^2 (1+ \frac{b_i}{a_i}\kappa^2 E^2) + b_i g^2\kappa^2(q^2-E^2)\nonumber \\ &=& a_i g^2(E) + b_ig^2(E) \kappa^2(q^2-E^2)\,,\end{aligned}$$ when renormalizing at the scale $q^2 =+E^2$. Here $G=\kappa^2/(32\pi)$ and $a_i~b_i$ are process dependent constants. This can certainly be done by explicit construction - indeed it can be done in multiple ways through different choices of the renormalization condition. By construction, this provides a gravitational contribution to the running of the coupling with energy. However there are obstructions that will in general keep any such construction from being useful and universal in effective field theories such as general relativity. One is a kinematic ambiguity. The higher order corrections are proportional to $Gq^2$, and $q^2$ refers to a four-vector which can take both positive and negative values. A renormalization condition defined at one sign of $q^2$ produces a charge definition that fails to be applicable in the crossed reaction with the opposite sign of $q^2$. We will refer to this as the [*crossing problem*]{}. (In the above schematic example, this is visible in the ambiguity between renormalizing at $q^2=+E^2$ or $q^2=-E^2$.) The other obstruction is process dependence or [*non-universality*]{}. Because there are in general multiple higher order operators that are possible, and because these enter into different processes in different ways, the divergences and finite parts of different reactions will not be the same. A definition of the charge that is appropriate for one reaction will not work for another. To be useful, a running coupling must capture at least a significant or common portion of the quantum corrections. (In the preceding schematic example, problems would arise if the ratio $b_i/a_i$ is highly process dependent.) We have demonstrated these obstacles in Yukawa theories [@Anber:2010uj], and the same problems arise in the attempts to define gravitational corrections to running gauge charges. In renormalizable theories these obstacles are not present. For logarithmic running couplings, the kinematic ambiguity is absent because the real part of $\ln q^2$ is the same for both spacelike and timelike $q^2$. The process dependence is absent because the running is connected with the actual renormalization of the charge - the $\ln (q^2/\mu^2)$ factor that arises is tied to the $1/\epsilon$ divergence absorbed into the renormalized charge. Because charge renormalization is universal, the corresponding logarithmic corrections are also universal. In [@Anber:2010uj], we did find one case where a gravitational correction to a running coupling could be constructed perturbatively without obvious flaws. This was $\lambda \phi^4$ theory. In this case, the unique higher order operator vanishes by the equation of motion - the gravitational corrections are one-loop-finite. In addition, the complete permutation symmetry of the $\phi^4$ interaction means that all reactions are crossing symmetric, and involve a unique crossing symmetric combination of kinematic invariants. There is no crossing problem. Because there is only one type of vertex in this theory, there is also no non-universality problem. These allowed a reasonable definition of the gravitational correction to the running of $\lambda$. However, we will see that these nice properties are not shared by the gravitational self interactions. Dimensional cutoffs =================== The comments of the preceding section are reasonably obvious when dimensional regularization is used. Indeed they are consistent with the portion of the literature concerning gravitational corrections to gauge couplings that employed dimensional regularization [@doesnot]. However, more recently there has been another subset of this literature which used dimensionful cutoffs in the analysis and which reached the opposite conclusion, i.e. that there was a universal gravitational correction to the running of the gauge charges [@does; @does2]. This dichotomy appears to violate a key principle that true physics is independent of the renormalization scheme. In fact, there is no disagreement between the schemes and the apparent difference arises from an incorrect interpretation of the dimensionful cutoff schemes. It is important for us to demonstrate the flaw of these cutoff calculations, because many attempts to define a running gravitational constant, $G(\Lambda)$ employ a similarly incorrect reasoning. If we rescale the vector field, $A^\mu \to A^\mu/e_0$, then the electric charge appears only in the photon part of the Lagrangian $${\cal L} = -\frac{1}{4e_0^2}F_{\mu\nu}F^{\mu\nu} + \bar{\psi} i \slashed{D}\psi\,.$$ After including gravition loops regularized with a dimensional cutoff $\Lambda$, there is a quadratic cutoff dependence in the leading term, as well as logarithmic cutoff dependence with a high order operator $${\cal L} = -\frac{1+a\kappa^2 \Lambda^2}{4e_0^2}F_{\mu\nu}F^{\mu\nu} + b \ln \Lambda^2 ~F_{\mu\nu}\partial^2 F^{\mu\nu} \,. \label{cutoffL}$$ This demonstrates that, in contrast with dimensional regularization, the lowest order charge $e_0$ does get renormalized (quadratically) by graviton loops when using a dimensionful cutoff as a regularizer. The incorrect interpretation of this is to identify the cutoff dependence with the running of the coupling. Specifically, by writing $$e^2(\Lambda) =\frac{ e_0^2}{1+ a \kappa^2 \Lambda^2}\,,$$ the authors of [@does2] identify a beta function $$\beta (e) = \Lambda \frac{\partial e}{\partial \Lambda} =- a e\kappa^2 \Lambda^2$$ for the running of the coupling. However this interpretation is incorrect, because the quadratic $\Lambda$ dependence disappears from physical observables in the process of renormalization. For example, the Coulomb potential at low energy, calculated from Eq. \[cutoffL\] is $${ V}(r) = \frac{ e_0^2}{4\pi(1+ a \kappa^2 \Lambda^2)} \frac1{r}\,.$$ If we use this to identify the electric charge we obtain $$\alpha = \frac{e^2}{4\pi} =\frac{ e_0^2}{4\pi(1+ a \kappa^2 \Lambda^2)} =\frac1{137}\,.$$ When expressing predictions in terms of the measured value of $\alpha$, the quadratic $\Lambda$ dependence is removed from all observables at all energies. It does not indicate the running of the electric charge. This analysis is supported by explicit calculation [@Toms:2011zz]. These same comments apply to dimensionful cutoffs and the gravitational coupling $G$. When using a scheme with a dimensionful cutoff, one will generate corrections to the gravitational constant $G=G_0(1+aG_0\Lambda^2)$. If this is done in a way that preserves general covariance, the same correction will be obtained in any process that involves $G$. This quadratic dependence will disappear from all observables once one identifies the physical renormalized parameter $G$ to be equal to its Newtonian value. Note that the logarithmic $\ln \Lambda$ dependence can be useful in tracing the running of couplings. This is because at high energies the logarithm must also contain kinematic variables, i.e. $\ln (\Lambda^2/q^2)$. This is analogous to tracing the $\ln q^2 $ behavior from the $1/\epsilon$ dependence in dimensional regularization. These features explain how dimensional regularization and cutoff regularization can agree in calculations involving gravity. There are no quadratic divergences in dimensional regularization, but we have seen that such divergences in cutoff schemes disappear from observables under renormalization. When considering gravitational corrections, the logarithmic cutoff dependence and the $1/\epsilon$ dependence are both associated with higher order terms of order $\kappa^2 q^2$, and the residual kinematic effects will be in agreement when the calculations are properly done. Dimensional regularization is a good regulator for the gravitational interaction because it has a clear and direct interpretation. For this reason, we use dimensional regularization throughout this paper. Pure gravity: The graviton propagator ===================================== Let us first consider the purely gravitational sector. At one loop, pure gravity is finite for on-shell amplitudes, because the higher order counterterms, $R^2$ and $R_{\mu\nu} R^{\mu\nu}$ vanish by the equation of motion $R_{\mu\nu}=0$. Finite one-loop corrections do exist. These are higher order in the momentum variables, and do not have the same kinematic dependence as the lowest order effects governed by the gravitational constant $G$. However, by working at a high energy renormalization scale, we will see if we can attempt to package these loop effects as a running coupling $G(E)$. Even though our primary focus in this paper involves on-shell physical reactions, it is useful to start by consideration of the vacuum polarization diagram and the graviton propagator. The only quantum correction that is demonstrably universal is that involving the vacuum polarization. Every graviton exchange receives a correction from the vacuum polarization. Because each end of the propagator carries a factor of $\kappa$, a modification of the propagator could be interpreted as modification of the coupling $G(q^2)$. However, we will see that there still remains the crossing problem because $q^2$ can carry either sign. The graviton propagator at lowest order is given by $$iD_F^{\alpha\beta\gamma\delta} = \frac {i{\cal P}^{\alpha\beta\gamma\delta}}{q^2+i\epsilon}\,,$$ where $${\cal P}^{\alpha\beta\gamma\delta} = \frac12\left[\eta^{\alpha\gamma}\eta^{\beta\delta} + \eta^{\beta\gamma}\eta^{\alpha\delta} -\eta^{\alpha\beta}\eta^{\gamma\delta}\right]\,.$$ The inclusion of the vacuum polarization diagram modifies the propagator $${i{\cal P}^{\alpha\beta\lambda\xi}\over q^2}i\Pi_{\lambda\xi\mu\nu}(q){i{\cal P}^{\mu\nu\gamma\delta}\over q^2}~.$$ The vacuum polarization diagram is divergent and the required counter terms have the form obtained by ‘t Hooft and Veltman [@'tHooft:1974bx] $$\begin{aligned} \Delta {\cal L}=\frac{\sqrt{g}}{16\pi^2\epsilon}\left[\frac{1}{120}R^2+\frac{7}{20}R_{\alpha\beta}R^{\alpha\beta} \right]\end{aligned}$$ with $\epsilon=(d-4)/2$. Because such terms appear in the most general effective Lagrangian, one can absorb these divergences into the renormalized values of their coefficients. Expressing $R^2$ and $R_{\alpha\beta}R^{\alpha\beta}$ in terms of the Fourier-space momenta, we find upon symmetrizing the indices $$\begin{aligned} \sqrt{g}R^2=h^{\mu\nu}\left[ q^4\eta_{\alpha\beta}\eta_{\mu\nu}-q^2\left(\eta_{\mu\nu} q_\alpha q_\beta+ \eta_{\alpha\beta} q_\mu q_\nu \right) +q_{\alpha}q_{\beta}q_{\mu}q_{\nu} \right]h^{\alpha\beta}\,,\end{aligned}$$ and $$\begin{aligned} \nonumber \sqrt{g}R_{\alpha\beta}R^{\alpha\beta}&=&\frac{1}{4}h^{\mu\nu}\left[q^4\eta_{\alpha\beta}\eta_{\mu\nu}-\frac{q^2}{2}\left(q_{\alpha}q_\mu\eta_{\nu\beta}+q_{\alpha}q_{\nu}\eta_{\mu\beta}+q_{\beta}q_{\mu}\eta_{\nu\alpha}+q_{\beta}q_{\nu}\eta_{\mu\alpha}\right) \right.\\ &&\left.\quad\quad\quad -q^2(\eta_{\alpha\beta}q_\mu q_\nu+\eta_{\mu\nu}q_\alpha q_\beta)+q^4I_{\alpha\beta,\mu\nu}+2q_\alpha q_\beta q_\mu q_\nu\right]h^{\alpha\beta}\,,\end{aligned}$$ where $I_{\alpha\beta,\mu\nu}=(\eta_{\alpha\mu}\eta_{\beta\nu}+\eta_{\alpha\nu}\eta_{\beta\mu})/2$. In addition, one can use the presence of the $1/\epsilon$ terms to read out the dependence on $\ln q^2$ $$\begin{aligned} \nonumber \Pi_{\alpha\beta,\mu\nu}(q)&=&-\frac{2G}{\pi}\ln\left(-\frac{q^2}{\mu_1^2} \right) \left[\frac{q^4}{60}\eta_{\alpha\beta}\eta_{\mu\nu}-\frac{q^2}{60}\left(\eta_{\mu\nu} q_\alpha q_\beta+ \eta_{\alpha\beta} q_\mu q_\nu \right) +\frac{1}{60}q_{\alpha}q_{\beta}q_{\mu}q_{\nu} \right]\\ \nonumber &&-\frac{2G}{\pi}\ln\left(-\frac{q^2}{\mu_2^2}\right)\left[\frac{7}{40}q^4\eta_{\alpha\beta}\eta_{\mu\nu}-\frac{7}{80}q^2\left(q_{\alpha}q_\mu\eta_{\nu\beta}+q_{\alpha}q_{\nu}\eta_{\mu\beta}+q_{\beta}q_{\mu}\eta_{\nu\alpha}+q_{\beta}q_{\nu}\eta_{\mu\alpha}\right) \right.\\ &&\left.\quad\quad\quad\quad\quad\quad\quad\quad -\frac{7}{40}q^2(\eta_{\alpha\beta}q_\mu q_\nu+\eta_{\mu\nu}q_\alpha q_\beta)+\frac{7}{40}q^4I_{\alpha\beta,\mu\nu}+\frac{7}{20}q_\alpha q_\beta q_\mu q_\nu\right]\,,\end{aligned}$$ where we have assigned $\ln(-q^2/\mu_1^2)$ and $\ln(-q^2/\mu_2^2)$ for $R^2$ and $R_{\alpha\beta}R^{\alpha\beta}$ respectively. The polarization tensor can be written in the form [@Weinberg:1980gg] $$\begin{aligned} \Pi^{\alpha\beta,\mu\nu}(q)&=&q^4A(q^2)L^{\alpha\beta}(q)L^{\mu\nu}(q)-q^2B(q^2)\left[L^{\alpha\mu}(q)L^{\beta\nu}(q)+L^{\alpha\nu}(q)L^{\beta\mu}(q) -2L^{\alpha\beta}(q)L^{\mu\nu}(q)\right],\end{aligned}$$ where $L^{\mu\nu}(q)=\eta^{\mu\nu}-q^{\mu}q^{\nu}/q^2$. Contracting $\Pi^{\alpha\beta,\gamma\delta}(q)$ with $L^{\alpha\beta}(q)L^{\mu\nu}(q)$, and $L^{\alpha\mu}(q)L^{\beta\nu}(q)$ we obtain two equations in $A(q^2)$, and $B(q^2)$ $$\begin{aligned} \nonumber \Pi^{\alpha\beta,\mu\nu}(q)L_{\alpha\beta}(q)L_{\mu\nu}(q)&=&3q^2\left[3q^2A(q^2)+4B(q^2) \right]\,,\\ \Pi^{\alpha\beta,\mu\nu}(q)L_{\alpha\mu}(q)L_{\beta\nu}(q)&=&3q^2\left[q^2A(q^2)-2B(q^2) \right]\,.\end{aligned}$$ Hence, we find $$\begin{aligned} \nonumber A(q^2)&=&-\frac{1}{30\pi}G\ln\left(\frac{-q^2}{\mu_1^2}\right)-\frac{7}{10\pi}G\ln\left(\frac{-q^2}{\mu_2^2}\right)\,,\\ B(q^2)&=&\frac{7}{40\pi}Gq^2\ln\left(\frac{-q^2}{\mu_2^2}\right)\,.\end{aligned}$$ On the other hand, the bare propagator takes the general form $$\begin{aligned} i{\cal D}^{\alpha\beta,\mu\nu}(q^2)=\frac{i}{2q^2}\left[L^{\alpha\mu}L^{\beta\nu}+L^{\alpha\nu}L^{\beta\mu}-L^{\alpha\beta}L^{\mu\nu} \right]\,,\end{aligned}$$ while the quantum corrected propagator reads $$\begin{aligned} \nonumber i{\cal D}'^{\alpha\beta,\mu\nu}&=&i{\cal D}^{\alpha\beta,\mu\nu}+i{\cal D}^{\alpha\beta,\gamma\delta}i\Pi_{\gamma\delta,\rho\tau}i{\cal D}^{\rho\tau,\mu\nu}\\ &=&\frac{i}{2q^2}(1+2B(q^2)) \left[L^{\alpha\mu}L^{\beta\nu}+L^{\alpha\nu}L^{\beta\mu}-L^{\alpha\beta}L^{\mu\nu} \right]-i\frac{A(q^2)}{4}L^{\alpha\beta}L^{\mu\nu}\,. \label{dressed}\end{aligned}$$ The first term above is a dressed propagator. Therefore, it is appropriate to define the running coupling as $$\begin{aligned} \label{one way of normalizing} G(q^2)=G(1+2B(q^2))=G\left(1+\frac{7}{20\pi}Gq^2\ln\left(\frac{-q^2}{\mu_2^2}\right)\right)\,.\end{aligned}$$ On the other hand, the second term in Eq. \[dressed\] also contributes comparably to gravitational amplitudes. For example, non-relativistic scattering involves the $00,00$ component of ${\cal D}^{\mu\nu,\alpha\beta}$ and we can equally define a running coupling from that component $$\begin{aligned} {\cal D}^{00,00}=\frac{1}{2q^2}\left(1+2B-\frac{q^2A}{2} \right)L^{00}L^{00}=\frac{1}{2q^2} G(q^2)L^{00}L^{00}\,,\end{aligned}$$ and hence, $$\begin{aligned} \label{0000 components} G(q^2)=G\left[1+\frac{1}{60\pi} Gq^2\ln\left(\frac{-q^2}{\mu_2^2}\right)+\frac{7}{10\pi} Gq^2\ln\left(\frac{-q^2}{\mu_1^2}\right) \right]\,.\end{aligned}$$ This corresponds exactly to the vacuum polarization contribution to the shift in the Newtonian interaction as calculated in [@Donoghue:1994dn], but it could be more widely applicable. Since $M_P$ is the only scale in gravity, we expect $\mu_1$ and $\mu_2$ to be of this order, and hence Eqs. \[one way of normalizing\] and \[0000 components\] are valid for low energies $E<M_P$. For spacelike $q^2$ this corresponds to an increase in the strength of gravity, while for timelike values it is a decrease. Of course this sign ambiguity is a signal of the crossing problem. It is present no matter which components of the propagator are considered. If converted to Euclidean space by the Wick rotation $q^2 \to (iq_4)^2 -\mathbf{q}^2 =-q_E^2$, the effective strength increases. At one loop order, it would be hard to convincingly favor one of the two definitions Eqs. \[one way of normalizing\] or \[0000 components\]. However, they do at least have a common sign. However, we know ahead of time that this definition is not going to enter into any physical one-loop processes in pure gravity. Since on-shell processes are one loop finite, any reference to the higher order operators $R^2$ or $R_{\mu\nu}R^{\mu\nu}$ must drop out of physical observables at one loop. The coefficients $\mu_1$ and $\mu_2$ are in this category and will not appear in on-shell processes. On-shell reactions may have logarithms such as $\ln s/t$, but not $\ln s/\mu_{1,2}^2$. Pure gravity: Graviton scattering ================================= The simplest physical process in pure gravity is graviton-graviton scattering. The lowest order scattering amplitude involves a large number of individual tree diagrams but is given by the simple form $${A}^{tree}(++;++) = i\frac{\kappa^2}{4} \frac{s^3}{tu}\,, \label{gravtree}$$ where the signs $+,-$ refer to helicity indices and $s,t,u$ are the usual Mandelstam variables. In power counting, this is a dimensionless amplitude of order $GE^2$. Our labeling of momentum and helicities corresponds to the final state particle being outgoing, in contrast to some conventions in the literature which label all particles as ingoing[^2]. The one loop amplitudes have been calculated by Dunbar and Norridge [@Dunbar:1994bn]. These are of order $G^2E^4$ and take the form $$\begin{aligned} \label{eq:2} {\cal A}^{1-loop}(++;--) & = & -i\,{\kappa^4 \over 30720 \pi^2} \left( s^2+t^2 + u^2 \right) \,, \nonumber\\ {\cal A}^{1-loop}(++;+-) & = & -{1 \over 3} {\cal A}^{1-loop}(++;--)\nonumber \\ {\cal A}^{1-loop}(++;++) & = &\frac{\kappa^2} {4(4\pi)^{2-\epsilon}}\, \frac{\Gamma^2(1-\epsilon)\Gamma(1+\epsilon)} {\Gamma(1-2\epsilon)}\, {\cal A}^{tree}(++;++)\,\times(s\,t\,u)\\ &&\hspace{-0em}\times\left[\rule{0pt}{4.5ex}\right. \frac{2}{\epsilon}\left( \frac{\ln(-u)}{st}\,+\,\frac{\ln(-t)}{su}\,+\,\frac{\ln(-s)}{tu} \right)+\,\frac{1}{s^2}\,f\left(\frac{-t}{s},\frac{-u}{s}\right) \nonumber\\&&\hspace{1.4em} +2\,\left(\frac{\ln(-u)\ln(-s)}{su}\,+\,\frac{\ln(-t)\ln(-s)}{tu}\,+\, \frac{\ln(-t)\ln(-s)}{ts}\right) \left.\rule{0pt}{4.5ex}\right]\,,\nonumber\end{aligned}$$ where $$\begin{aligned} \label{eq:f} f\left(\frac{-t}{s},\frac{-u}{s}\right)&=& \frac{(t+2u)(2t+u)\left(2t^4+2t^3u-t^2u^2+2tu^3+2u^4\right)} {s^6} \left(\ln^2\frac{t}{u}+\pi^2\right)\nonumber\\&& +\frac{(t-u)\left(341t^4+1609t^3u+2566t^2u^2+1609tu^3+ 341u^4\right)} {30s^5}\ln\frac{t}{u}\nonumber\\&& +\frac{1922t^4+9143t^3u+14622t^2u^2+9143tu^3+1922u^4} {180s^4}\,.\end{aligned}$$ Other amplitudes can be obtained from these by crossing. The dimensional regularization parameter $\epsilon =(4-d)/2$ appears in the amplitude ${\cal A}^{1-loop}(++;++)$. This is an infrared divergence, and is canceled as usual from the radiation of soft gravitons. An explicit calculation of the sum of direct and radiative cross sections [@Donoghue:1999qh] yields the result $$\begin{aligned} \label{sum-crs} &&\hspace{-3em}\left(\frac{d\sigma}{d\Omega}\right)_{tree} + \left(\frac{d\sigma}{d\Omega}\right)_{rad.} +\left(\frac{d\sigma}{d\Omega}\right)_{nonrad.}=\\ &=& \frac{\kappa^4 s^5}{2048\pi^2 t^2 u^2}\, \left\{ \rule{0pt}{2.6em}\right. 1 + {\kappa^2 s \over 16 \pi^2} \left[\rule{0pt}{2em}\right. \ln\frac{-t}{s}\ln\frac{-u}{s}+ \frac{tu}{2s^2}\,f\left(\frac{-t}{s},\frac{-u}{s}\right) \nonumber\\ &&-\left(\frac{t}{s}\,\ln{\frac{-t}{s}}+\frac{u}{s}\, \ln{\frac{-u}{s}}\right) \left( 3\ln(2\pi^2)+\gamma+\ln\frac{s}{\mu^2}+ \frac{\sum_{ij}\eta_i\eta_j{\cal F}^{(1)}(\gamma_{ij})}{\sum_{ij}\eta_i\eta_j{\cal F}^{(0)}(\gamma_{ij})} \right) \left.\rule{0pt}{2em}\right] \left.\rule{0pt}{2.6em}\right\}.\nonumber\end{aligned}$$ Here $\mu$ is an infrared scale related to the experimental energy resolution and ${\cal F}^{(i)}$ are functions defined in [@Donoghue:1999qh] related to the angles of emission of soft graviton radiation. The last line of the cross section formula is related to infrared physics and does not appear appropriate for inclusion in the definition of a running coupling. Instead we focus on the correction displayed in the preceding line. We would like a renormalization point in the physical region[^3] with a single energy scale $E$. We choose the central physical point $s=2E^2,~ t=u=-E^2$. This leads to the identification $$G^2(E) = G^2\left[1 +\frac{\kappa^2 E^2\left(\ln^2 2+\frac{1}{8}\left(\frac{2297}{180}+\frac{63\pi^2}{64} \right) \right)}{8\pi^2}\right]~~~. \label{running1}$$ We see that this definition leads to a growing running coupling $G(E)$, as opposed to the expectation from asymptotic safety of a decrease in strength at high energy. It works acceptably for this process because it absorbs the main effects of the quantum corrections in the neighborhood of the central point. We could alternatively consider the crossed reaction ${\cal A}(+,-;+,-)$ which is obtained from ${\cal A}(+,+;+,+)$ by the exchange $s\leftrightarrow t$. This makes the quantum corrections somewhat different, with the corresponding kinematic factor being $$1 + \frac{\kappa^2 t}{16 \pi^2} \left[ \ln\frac{-s}{t}\ln\frac{-u}{t}+ \frac{su}{2t^2} f \left(\frac{-s}{t},\frac{-u}{t}\right) \right] = 1 +\frac{\kappa^2E^2\left(\frac{29}{10}\ln 2-\frac{67}{45} \right)}{16\pi^2}$$ instead of the factor in Eq. \[running1\]. The quantum corrections in this channel differ from those of the original reaction and they are not accurately summarized by the same running coupling. This is a manifestation of the crossing problem. Gravitational scattering of a massless scalar particle ====================================================== In renormalizable gauge theories, the running coupling applies universally to all processes. As mentioned above, this is because the running is tied to the renormalization of the gauge charge. General covariance requires that a valid definition of a running $G$ also be universally applicable. The gravitational coupling not only parameterizes the self interactions of gravitons, but it also describes the gravitational coupling of matter. In this section, we look at the effects of loops on the gravitational interactions of a scalar particle. We consider a scalar particle that has only gravitational interactions. The scattering $\phi + \phi \to \phi +\phi$ via graviton exchange at tree level has $s,~t, ~{\rm and}~ u$ channel poles, with amplitude $$\begin{aligned} {\cal M}_{tree}=i\frac{\kappa^2}{4}\left[\frac{st}{u}+\frac{su}{t}+\frac{tu}{s} \right]\,.\end{aligned}$$ We note that this amplitude, and the loop amplitudes to follow, has a permutation (crossing) symmetry such that all channels are governed by the same amplitude. This will eliminate the crossing problem that arises in most other reactions. However we can test for universality by testing whether the interactions lead to a similar running coupling as suggested in the purely gravitational sector. In this theory there is a higher order operator which is required at one loop. Divergences proportional to $${\cal L}_2 = \frac{203}{320\epsilon} (D_\mu \phi D^\mu \phi)^2$$ arise at one loop. In matrix elements, this operator generates a contribution proportional to $s^2 + t^2 +u^2$. The one loop amplitudes, up to rational terms in the kinematic variables, have been given in [@Dunbar:1995ed]. However the rational terms are constrained by the permutation symmetry to also be proportional to $s^2 + t^2 +u^2$ and we will absorb them into the higher order Lagrangian ${\cal L}_2$. The total scattering amplitude of this process, apart from a polynomial in $s$, $t$, and $u$, is given by [^4] $$\begin{aligned} \nonumber {\cal M}_{1-loop}&=&i\frac{\kappa^4}{\left(4\pi\right)^2}\left[\frac{(s^4+t^4)}{16}I_4(s,t)+\frac{(s^4+u^4)}{16}I_4(s,u)+\frac{(u^4+t^4)}{16}I_4(t,u)-\frac{s(s^2+2t^2+2u^2)}{8}I_3(s) \right.\\ \nonumber &&\left.\quad\quad\quad\quad\quad-\frac{t(t^2+2s^2+2u^2)}{8}I_3(t)-\frac{u(u^2+2t^2+2s^2)}{8}I_3(u)+\frac{(163u^2+163t^2+43tu)}{960}I_2(s) \right.\\ \label{AAAAScattering} &&\left.\quad\quad\quad\quad\quad +\frac{(163u^2+163s^2+43us)}{960}I_2(t)+\frac{(163s^2+163t^2+43ts)}{960}I_2(u) \right]\,,\end{aligned}$$ where the $I_4(s,t)$, $I_3(s)$, and $I_2(s)$ are respectively the scalar box, triangle and bubble integrals: $$\begin{aligned} \nonumber I_4(s,t)&=&\frac{1}{st}\left\{\frac{2}{\epsilon^2}\left[(-s)^{-\epsilon}+(-t)^{-\epsilon} \right]-\ln^2\left(\frac{-s}{-t}\right)-\pi^2 \right\}\\ \nonumber &&=\frac{1}{st}\left\{\frac{4}{\epsilon^2}-\frac{2\ln(-s)+2\ln(-t)}{\epsilon}+2\ln(-s)\ln(-t)+\mbox{finite} \right\}\,,\\ \nonumber I_3(s)&=&\frac{1}{\epsilon^2}\left(-s\right)^{-1-\epsilon}=-\frac{1}{s}\left(\frac{1}{\epsilon^2}-\frac{\ln(-s)}{\epsilon}+\frac{\ln^2(-s)}{2} \right)\,,\\ I_2(s)&=&\frac{1}{\epsilon(1-2\epsilon)}\left(-s\right)^{-\epsilon}=\left(\frac{1}{\epsilon}-\ln(-s)+\mbox{finite}\right)\,.\end{aligned}$$ We follow Ref. [@Dunbar:1995ed] in removing the infrared divergences by use of $$\label{removing IR} {\cal M}_{IR} =\frac{\kappa^2}{2(4\pi)^2}\frac{\left((-s)^{1-\epsilon}+(-t)^{1-\epsilon}+(-u)^{1-\epsilon}\right)}{\epsilon^2}{\cal M}_{tree}\,,$$ where the residual hard part is defined via $${\cal M}_h = {\cal M}_{1-loop} - {\cal M}_{IR}\,.$$ With the renormalization of the higher order operator, the one loop hard amplitude is $$\begin{aligned} \nonumber {\cal M}_{h}&=&i\frac{\kappa^4}{\left(4\pi\right)^2}\left\{\frac{(s^4+t^4)}{8st}\ln(-s)\ln(-t)+\frac{(s^4+u^4)}{8su}\ln(-s)\ln(-u)+\frac{(u^4+t^4)}{8tu}\ln(-t)\ln(-u) \right.\\ \nonumber &&\left.\quad\quad\quad\quad+\frac{(s^2+2t^2+2u^2)}{16}\ln^2(-s) +\frac{(t^2+2s^2+2u^2)}{16}\ln^2(-t)+\frac{(u^2+2t^2+2s^2)}{16}\ln^2(-u) \right.\\ \nonumber &&\left.\quad\quad\quad\quad+\frac{1}{16}\left(\frac{st}{u}+\frac{tu}{s}+\frac{us}{t} \right)\left(s\ln^2(-s)+t\ln^2(-t)+u\ln^2(-u) \right)\right.\\ \nonumber &&\left.\quad\quad\quad\quad+\left[ -\frac{(163u^2+163t^2+43tu)}{960}\ln\left(\frac{-s}{\mu^2}\right)-\frac{(163u^2+163s^2+43us)}{960}\ln\left(\frac{-t}{\mu^2}\right)\right.\right.\\ \label{AAAAhardScattering} &&\left.\left.\quad\quad\quad\quad-\frac{(163s^2+163t^2+43ts)}{960}\ln\left(\frac{-u}{\mu^2}\right)+d_1^{ren}(\mu)(s^2+t^2+u^2) \right]\right\}\,,\end{aligned}$$ where $\mu$ is an infrared scale. In this result, we have grouped the single logs with the higher order operator, because those logs are the ones that pick up the scale dependence when you shift the scale associated with the higher order operator $ d_1^{ren}(\mu)$. We again evaluate the matrix element at the central kinematic point $s=2E^2,~t=u=-E^2$. The result is $${\cal M}_{total}={\cal M}_{tree}+{\cal M}_h= i\frac{9\kappa^2 E^2}{8}\left[1-\frac{\kappa^2E^2}{360\left(4\pi\right)^2} \left(609\ln\frac{E^2}{\mu^2}+\left(340\pi^2+\left(123-340\ln 2\right)\ln2 \right) \right) \right]\,.$$ If we were to use this to identify a running coupling the result would be $$G(E) = G\left[1-\frac{\kappa^2E^2}{360\left(4\pi\right)^2} \left(609\ln\frac{E^2}{\mu^2}+\left(340\pi^2+\left(123-340\ln 2\right)\ln2 \right) \right) \right]\,. \label{running2}$$ The single log term which appears in Eq. \[running2\] could reasonably be associated with the higher order operator $d_1$, and perhaps should be removed from this expression. The most serious flaw of this result in comparison to Eq. \[running1\] is that it has the opposite sign. The leading corrections to graviton scattering and to scalar scattering go in the opposite direction, which is not accountable by a common definition of a running coupling, an obvious lack of universality. Gravitational scattering of non-identical particles =================================================== Here we consider a different situation for the matter couplings - the scattering of non-identical particles. We will neglect the particle masses, so this corresponds to scattering at $s>>m^2$. This situation demonstrates both the crossing problem and the non-universality problem. The example of the last section has more crossing symmetry than most gravitational reactions. Processes involving non-identical particles, or with fermions, will typically involve dominantly only one of the $s,t,u$ channels. Typical gravitational scattering of very massive particles will involve primarily $t$-channel exchange. Such distinctions highlight the difficulty of any given definition of a running $G$ being applicable to all processes. By a direct computation of the appropriate set of Feynman diagrams, we find that the tree and one-loop amplitudes of the reaction $A+B\to A+B$ are $$\begin{aligned} \nonumber {\cal M}_{tree}&=&\frac{i\kappa^2 su}{4t}\,,\\ \nonumber {\cal M}_{1-loop}&=&i\frac{\kappa^4}{\left(4\pi \right)^2}\left[\frac{1}{16}\left(s^4I_4(s,t)+u^4 I_4(u,t)\right) +\frac{1}{8}\left(s^3+u^3+tsu \right)I_3(t)-\frac{1}{8}\left(s^3I_3(s)+u^3I_3(u) \right) \right.\\ \label{ABAB scattering} &&\left.\quad\quad\quad-\frac{1}{240}\left(71us-11t^2\right)I_2(t)+\frac{1}{16}\left(s^2I_2(s)+u^2I_2(u)\right) \right]\,.\end{aligned}$$ Then, we use Eq. \[removing IR\] in removing the IR divergences. The resulting hard amplitude reads $$\begin{aligned} \nonumber {\cal M}_{h}&=&i\frac{\kappa^4}{\left(4\pi \right)^2}\left[\frac{1}{8}\left(\frac{s^3}{t}\ln(-s)\ln(-t)+\frac{u^3}{t} \ln(-u)\ln(-t)\right) -\frac{1}{16t}\left(s^3+u^3+tsu \right)\ln(-t)+\frac{1}{16}\left(s^2\ln^2(-s)+u^2\ln^2(-u)\right) \right.\\ \nonumber &&\left.\quad\quad\quad+\frac{us}{16t}\left(s\ln^2(-s)+t\ln^2(-t)+u\ln^2(-u) \right)+\frac{1}{240}\left(71us-11t^2\right)\ln(-t)-\frac{1}{16}\left(s^2\ln(-s)+u^2\ln(-u)\right) \right]\,,\\\end{aligned}$$ and the total amplitude at the center kinematic point $s=2E^2,~t=u=-E^2$ is $$\begin{aligned} {\cal M}_{total}=\frac{i\kappa^2E^2}{2}\left[1-\frac{\kappa^2 E^2}{10(4\pi)^2}\left(\left(19+10\ln 2\right)\ln\left(\frac{E^2}{\mu^2}\right)+5\left(\pi^2-(\ln 2-1)\ln 2 \right) \right) \right]\,.\end{aligned}$$ On the other hand, the amplitude of the reaction $A+A\to B+B$ is given by Eq. \[ABAB scattering\] with the exchange $s\leftrightarrow t$, and has the amplitude $$\begin{aligned} {\cal M}_{total}=\frac{i\kappa^2E^2}{8}\left[1+\frac{\kappa^2 E^2}{10(4\pi)^2}\left(9\ln\left(\frac{E^2}{\mu^2}\right)-5\pi^2+\left(19+5\ln 2 \right)\ln2 \right) \right]\,.\end{aligned}$$ The crossing problem is obvious here. The loop corrections are in opposite directions in the two reactions, largely because of the change in sign of the kinematic variables under crossing. Any definition of a running $G$ cannot capture this behavior - the coupling must either increase with energy scale or decrease with energy. The processes also illustrate the non-universality problem. Even two reactions that are this closely related have a different magnitude for the one loop correction when evaluated in the physical region. Gravitational scattering of heavy masses ======================================== Finally we consider the quantum corrections to the scattering of heavy objects - let us call them planets. Are the gravitational corrections here similar to those of massless scalars or gravitons? This tests the universality property of a running coupling. This scattering amplitude is closest to the situations that we are familiar with defining the running couplings in QED or QCD The gravitational interaction at one-loop is the result of several Feynman diagrams that vary in magnitude and sign. Written in coordinate space, the total loop correction changes the interaction to that quoted above in Eq. 1. Written in its original momentum space for this corresponds to $$V(q) = -4\pi\frac{GMm}{\mathbf{q}^2}\left[1+\frac{41}{20\pi}G{\mathbf{q}}^2\ln \left( \frac{\mu^2}{{\mathbf q}^2}\right)\right]\,. \label{potential}$$ Note that in this case, we have written this result in terms of the spatial part of $q^2$, i.e. $q^2=-{\mathbf q}^2$ because the results were derived in the non-relativistic approximation. We should not consider crossing this amplitude to timelike $q^2$ because the planet masses could be well above the Planck scale. This result corresponds to an increase in the gravitational strength with increasing energy. Ascribing it to a a running $G$ would yield Eq. 2, or equivalently the momentum factor in square brackets in Eq. \[potential\]. Note that the parameter $\mu$ does not enter the coordinate space potential at finite $r$ because the Fourier transform of a constant is a delta function. We keep $\mu \sim M_P$ for the low energy validity of the effective theory. The scattering amplitude leading to the result in Eq. \[potential\] includes all diagrams, including box and crossed-box diagrams and some triangle diagrams. In QED or QCD we do not use the full set of diagrams for the running charge, as we include only the vertex corrections and vacuum polarization. In these theories, this is appropriate because it is these diagrams that renormalize the gauge charge. In gravity, none of these diagrams renormalize G at low energy, so the rationale for including only a subset is not clear. Moreover, in gravity, this subset of diagrams does not by itself form a gauge invariant set. Nevertheless, we can look at this subset of diagrams in a particular gauge. In harmonic gauge, the inclusion of both vertex and vacuum polarization would be $$G(q) =G \left[1- \frac{167}{60\pi}G{\mathbf{q}}^2\ln \left( \frac{\mu^2}{{\mathbf q}^2}\right)\right]\,,$$ i.e. it carries the opposite sign from the full result. Even within this subset, one potentially might like to exclude the vertex diagrams because there is no Ward identity that indicates that these must be the same for all particles (i.e. photons vs planets). The vacuum polarization is however universal. Including only it would yield $$G(q) =G \left[1+ \frac{43}{60\pi}G{\mathbf{q}}^2\ln \left( \frac{\mu^2}{{\mathbf q}^2}\right)\right]\,, \label{purevacpol}$$ as can be seen from Sec. 4. Overall, the non-relativistic scattering amplitude is made up of many large contributions that differ in sign and magnitude. Identifying a subset as the running charge would not capture the leading quantum effects. Moreover, we see that even the sign of the potential definition is not obvious as vertex and vacuum polarization diagrams have opposite signs. The sign of vacuum polarization correction in Eq. \[purevacpol\] agrees with that of the the total scattering amplitude in Eq. \[potential\], but the magnitude is different by a factor of three. Lessons ======= We have explored one-loop calculations in general relativity in the region where there is perturbative control over the theory. There emerges no definition of a running $G$ that is both useful and universal. The nature of the energy expansion of the effective field theory of gravity implies that quantum corrections are associated with the renormalization of higher order operators rather than the original Einstein action. This implies that the usual theoretical framework for running couplings, the renormalization group, does not by itself define a running $G$. And while a definition generally can be made that is useful within a given process, the quantum effects are so non-universal that this definition will not be usefully applied to other reactions. We have illustrated a series of reactions with one-loop corrections which differ significantly. Perhaps lost in the variety of signs and magnitudes is the key point that quantum corrections do not organize themselves into a running coupling. This is the expected behavior of an effective field theory. The relevant higher order operators are process dependent and decoupled from the renormalization of the lowest order operator. The kinematic variation of the one loop corrections are more complex than just mirroring the leading behavior because they involve higher powers of the momentum invariants and there are many allowed kinematic factors present at higher order. Attempts to repackage this larger kinematic variation as if it were a modification of the lowest order amplitude, i.e. a running coupling, will then in general fail because the running coupling cannot mimic the richer kinematics of the higher order terms. This leads directly to the crossing problem and the non-universality problem, both of which occur when one tries to define a running $G$. Our work also provides cautions for the Asymptotic Safety program, which employs a running gravitational coupling in the non-perturbative regime beyond the Planck scale. Let us mention some of the obstacles. The process of defining the running coupling requires a truncation of the operator basis, and the effects of the infinite set of higher order operators get repackaged as if they were contained in a small set of low order operators. This raises the issues that appears in our calculations - will this repackaging be universal? Will the matter couplings in the theory - which provide one definition of $G$ - have the same quantum corrections as the pure gravity sector - which provide another definition of $G$? Will 2-point, 4-point and 8-point functions, for example, have the same behavior? Due to the presence of all the higher order operators, general covariance by itself does not require these functions to have the universal behavior expected from a running coupling constant. Our calculations showed highly non-universal behavior. Another issue is the continuation back to physical Lorentzian spacetime. In Asymptotic Safety, the running coupling defined in Euclidean space must be continued to physical spacteimes when applied to the real world. Lorentzian spacetimes have momentum variables of both signs and the analytic continuation of power corrections must be more complicated, and we have seen examples of reactions where the crossing problem made any running coupling useless. A naive continuation of a function such as Eq. \[runningform\] raises the possibility of poles for certain kinematic configurations. The best candidate for a universal contribution to a running $G$ comes from the vacuum polarization amplitude. The correction to the graviton propagator will occur anytime a graviton is exchanged, either at tree level or within loops. However, here the perturbative result has the wrong sign when Euclideanized - the gravitational strength increases. There is also the crossing problem in Lorentzian spacetime. Even if the vacuum polarization could be summed to a function with a good high energy behavior, for example a form such as $$\frac{{\cal P}^{\alpha\beta\gamma\delta}}{q^2 + \alpha G q^4}~~,$$ but such functions often have trouble with ghosts when used at high energy or within loop diagrams. [^5] The potential problems of gravity treated beyond the Planck scale need not be problematic if the effective theory gets modified at that scale by new degrees of freedom and a change in the description of the theory. General relativity would still form a quantum effective field theory with calculable quantum effects below the Planck energy. However, in such effective field theories with a dimensional coupling it has not proven useful to employ running coupling constants. We have shown the difficulties of trying to define a running $G$ in gravity. Acknowledgements {#acknowledgements .unnumbered} ================ The work of J.D. is supported in part by the U.S. NSF grant PHY-0855119. The work of M.A. has been supported by NSERC Discovery Grant of Canada. John Donoghue acknowledges the kind hospitality of the Niels Bohr International Academy, where most of this research was accomplished, and thanks N.E.J. Bjerrum-Bohr for useful conversations. 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--- abstract: 'We obtain a compact Sobolev embedding for $H$-invariant functions in compact metric-measure spaces, where $H$ is a subgroup of the measure preserving bijections. In Riemannian manifolds, $H$ is a subgroup of the volume preserving diffeomorphisms: a compact embedding for the critical exponents follows. The results can be viewed as an extension of Sobolev embeddings of functions invariant under isometries in compact manifolds.' author: - | Micha[ł]{} Gaczkowski$^1$, Przemys[ł]{}aw Górka$^1$[^1] and Daniel J. Pons$^2$[^2]\ \ \ \ \ \ title: '**Symmetry and compact embeddings for critical exponents in metric-measure spaces**' --- [**Keywords**]{}: Sobolev spaces, metric-measure spaces, compact embedding *Mathematics Subject Classification (2010):* 46E35; 30L99. Introduction ============ Arising in the Calculus of Variations and PDE’s, the study of Sobolev spaces in Euclidean domains, and the embeddings between them, has been an active area of research for more than a century (see [@Adams] for the classical results, and [@naumann] for an overview on history). In the last fifty years, motivated by problems in Geometric Analysis, Physics and Topology, those studies have been generalized to functions on manifolds, with the extension to sections of vector bundles over those spaces, see [@aubin; @Hawking; @hebey; @lawson; @palais]. More recently, the study of metric-measure spaces[^3] demands, whenever it is possible, similar studies in this context, see the books [@Ambrosio; @Heinonen; @H].\ A fundamental ingredient in Sobolev spaces is the the concept of weak or distributional gradient. In metric-measure spaces there are at least two notions that provide a valid generalization of the usual gradient in $\mathbb{R}^n$: - An upper gradient, see [@Cheeger; @H]. - The one used in this work, nowadays called a Hajłasz gradient, see [@Ambrosio; @Hajlasz] and Section \[Preliminaries\] below. Both notions of gradient have advantages and disadvantages with respect to each other, see [@Heinonen; @H; @JSYY].\ If $X$ is a set, $\mu$ is a measure on $X$, and $1 \leq p < \infty$, denote by $L^p(X,\mu)$ the vector space of $\mu$-measurable functions such that $$\| f \|_{L^p(X,\mu)} := (\ \int_X | f(x) |^p d \mu(x)\ )^{1/p}$$ is finite. In particular, if $X$ is either a bounded domain in $\mathbb{R}^n$ or a compact Riemannian $n$-manifold, with $\mu = V_g$ the volume measure associated to the Euclidean or Riemannian line element $g$, respectively, $L^{p}_1(X,\mu)$ refers to the subspace of $L^p(X,\mu)$ made up of those functions such that the norm of their distributional gradient (with respect to $g$) is in $L^p(X,\mu)$. In those cases one has the embedding $$L^{p}_1(X,\mu) \hookrightarrow L^q(X,\mu)$$ whenever $p \leq q \leq p^{\ast} := n p / (n-p)$, where $1 \leq p < n$. If $q < p^{\ast}$, the embedding is compact, and one writes $$\label{comp-emb-1} L^{p}_1(X,\mu) \hookrightarrow \hookrightarrow L^q(X,\mu) ,$$ see [@Adams; @aubin; @hebey]. The non-compactness of the embedding in the limit case $q = p^{\ast}$ is a phenomena that arises from sequences of transformations that induce substantial changes in the functions, transformations that nonetheless leave the norm of functions unchanged. With such information, it is tempting to look for subspaces of $L^{p}_1(X,\mu)$ whose elements are invariant under an appropriate subgroup of $\text{Diff}(X)$, and see if the compact embedding (\[comp-emb-1\]), when restricted to these subspaces, can be extended to higher values of $q$: let $H$ be a subgroup of $\text{Diff}(X)$, and denote by $L^{p}_{1,H}(X,\mu)$ the subspace of $L^{p}_1(X,\mu)$ made up of $H$-invariant functions.\ The best result in this context is due to E. Hebey and M. Vaugon, who consider $H$ as being a subgroup of $\text{Isom}(X,g)$, the group of isometries of $(X,g)$: (Hebey-Vaugon [@hebey-vaugon], also Theorem 9.1 in [@hebey]) \[heb-vaug\] Suppose $(X,g)$ is a compact Riemannian $n$-manifold, and $H$ is a compact subgroup of $\text{Isom}(X,g)$. If $H(x)$ denotes the orbit of the point $x$ under the action of $H$, require that $H(x)$ is uncountable for every $x$ in $X$. If $k := \text{min}\ \{\ \text{dim}\ H(x) : x\in X\ \}$, then $$\label{comp-emb-2} L^{p}_{1,H}(X,V_g) \hookrightarrow \hookrightarrow L^q(X,V_g)$$ whenever $1 \leq p < n-k$ and $1 \leq q < \frac{(n-k) p}{n-k-p}$. In a metric-measure space $(X,d,\mu)$ conditions for the metric $d$ and the measure $\mu$ are sometimes required, leading to *synthetic* extensions of *analytic* Riemannian concepts, like curvature, volume, and dimension (see [@villani] for a friendly introduction to these ideas). We will use the doubling condition for the metric space $(X,d)$ and the lower Ahlfors $s$-regularity of the metric-measure space $(X, d, \mu)$.[^4] For instance, if $(X,g)$ is a compact $n$-manifold with induced distance $d_g$, then $(X,d_g,V_g)$ is a lower $n$-regular metric-measure space, and $(X,d_g)$ is doubling.\ As aforementioned, we use Hajłasz gradients: denote by $M^p_1(X,\mu)$ the vector space of functions in $L^p(X,\mu)$ such that their Hajłasz gradient is also in $L^p(X,\mu)$. In Section \[examples\], Theorem \[riem-mm\], we see that when $(X,d_g, V_g)$ is the natural metric-measure space induced from a compact $n$-manifold $(X,g)$, then $M^p_1(X, V_g)$ coincides with $L^p_1(X, V_g)$ for $1 < p < \infty$.\ In the *analytic* context of Riemannian geometry, symmetry groups are subgroups of $\text{Diff}(X)$. Instead, in the *synthetic* context of metric-measure spaces, symmetry groups are subgroups of $\text{Aut}(X)$, the group of automorphisms or bijections of $X$: let $H$ be a subgroup of $\text{Aut}_{\mu}(X)$, the group of $\mu$-preserving automorphisms of $X$; denote by $M^{p}_{1,H}(X,\mu)$ the subspace of $M^{p}_1(X,\mu)$ made up of $H$-invariant functions. The main result in this work is: \[main2\] Assume that $(X, d, \mu)$ is a metric-measure space that is compact, Ahlfors lower $s$-regular, with $(X,d)$ doubling, and such that $M^p_1(X,\mu)$ is reflexive. If $H$ is a subgroup of $\text{Aut}_{\mu}(X)$ such that for every $x$ in $X$ the set $H(x)$ is uncountable, then $$\label{comp-emb-3} M^p_{1,H} (X,\mu) \hookrightarrow \hookrightarrow L^{q} (X,\mu)$$ whenever $1 < p < s$ and $1 \leq q \leq p^{*} = \frac{s p}{s - p}$. In contrast with classical Sobolev spaces, there are situations where $M^p_1(X,\mu)$ is not reflexive for $1<p<\infty$: some examples of this unexpected phenomena are self similar Cantor sets, see [@rissanen]. On the other hand, a discussion about sufficient conditions on $(X,d,\mu)$ for $M^p_1(X,\mu)$ to be reflexive can be found in [@gorka1; @Hajlasz1].[^5] To highlight the contributions of this work, we make some remarks: 1. Concerning the groups appearing in Theorems \[heb-vaug\] and \[main2\]: In the context of metric-mesure spaces arising from Riemannian manifolds, the group $H$ in Theorem \[main2\] is a subgroup of $\text{Diff}_{V_g}(X)$, the group of volume preserving diffeomorphisms of $(X,g)$; in Theorem \[heb-vaug\] the group $H$ is a compact subgroup of the smaller group $\text{Isom}(X,g)$. A classical result of S. Myers and N. Steenrod, see [@kobayashi], provides $\text{Isom}(X,g)$ with the structure of a finite dimensional Lie group, that is compact if $X$ is compact. In contrast, if $X$ is compact, H. Omori provided the larger group $\text{Diff}_{V_g}(X)$, and $\text{Diff}(X)$ as well, with the structure of an Inverse Limit of Hilbert manifolds, see [@ebin; @kobayashi]. The Lie algebras of both groups are *represented* by vector fields: those in the *formal* Lie algebra of $\text{Diff}_{V_g}(X)$ are free of divergence; those in the Lie algebra of $\text{Isom}(X,g)$ are Killing, a stronger condition. In every Riemannian manifold there are non-trivial vector fields free of divergence; on the other hand, the sign of the Ricci curvature imposes restrictions for Killing vector fields: if the Ricci tensor is non-positive and negative definite at some point, there are no non-trivial Killing vector fields, and the group $\text{Isom}(X,g)$ is finite [@Berard; @kobayashi]. 2. Concerning the proofs of Theorems \[heb-vaug\] and \[main2\]: Roughly speaking, Theorem \[heb-vaug\] uses local charts compatible with the dimension reduction under the Riemannian submersion induced by the isometries, reducing the compact embedding of functions to a Sobolev inequality in the space orthogonal to the $H$-orbits, providing a convenient setup for specific results obtained by H. Berestycki, E. Lieb, P. L. Lions and others, see [@lions1]. The proof of Theorem \[main2\] is different: the dimension reduction compatible with isometries used in Theorem \[heb-vaug\] is not always compatible with volume preserving diffeomorphisms.[^6] Some ingredients in the proof are a Sobolev-Hajłasz inequality [@Hajlasz], and variations of the Concentration-Compactness principle, see [@lions2].\ In Section \[Preliminaries\] we provide definitions and results that will be used in Section \[main\], where a detailed proof of Theorem \[main2\] is given. In Section \[examples\] we see that Theorem \[main2\] can be applied in the Riemannian context, and discuss necessary and sufficient conditions for its applicability when the dimension of the $H$-orbits is 1.\ For Sobolev embeddings in non-compact spaces using symmetry, see [@Gaczkowski; @gorka1], and the references there. Preliminaries {#Preliminaries} ============= In this work $(X, d, \mu)$ is a metric-measure space equipped with a metric $d$ and a Radon measure $\mu$. We assume that the measure of every open non-empty set is positive, and that the measure of every bounded set is finite. In most parts of our paper we will assume that the metric-measure space $(X, d, \mu)$ is *lower Ahlfors $s$-regular*: this means that there exists a constant $b$ such that $$b R^s \leq \mu \left(B_R(x)\right)$$ for all balls $B_R(x)$ in $X$ with $R < \hbox{diam} X$. A metric space is said to be *doubling* if there exists a constant $C$ such that for every ball of radius $R$, there exist $C$ balls of radius $R/2$ that cover the original ball. It not difficult to see that if $(X, d, \mu)$ is a doubling metric-measure space,[^7] then $(X, d)$ is a doubling metric space (see [@gromov], Appendix $B_+$). Conversely, J. Luukkainen and E. Saksman in [@Luukkainen] prove that every complete doubling metric space carries a doubling measure.\ If $(X, d, \mu)$ is a metric-measure space, say that a function $f$ in $L^p(X,\mu)$ belongs to the *Haj[ł]{}asz-Sobolev* space $M^{p}_{1}(X,\mu)$ if there exists some $g \in L^{p}(X,\mu)$, called a *Hajłasz gradient*, such that $$\begin{aligned} \label{global-local} |f(x) - f(y)| \leq d(x,y) \left( g(x) + g(y) \right) \end{aligned}$$ for $\mu$ almost every $x$ and $y$ in $X$. In this context, given $f$ in $M^{p}_1(X,\mu)$, we denote by $g_f$ any Hajłasz gradient for $f$, to endow the space $M^p_1(X,\mu)$ with the norm $$\begin{aligned} \label{global-norm} \|f\|_{M^p_1(X,\mu)}:= \|f\|_{L^{p}(X,\mu)}+\inf_{g_f} \|g_f\|_{L^{p}(X,\mu)},\end{aligned}$$ and then $M^p_1(X,\mu)$ is a Banach space. In the same context, say that $f \in L^p(X,\mu)$ belongs to $m^p_1(X,\mu)$ if there exists some $g \in L^{p}(X,\mu)$, called a *local Hajłasz gradient*, such that for every $z$ in $X$ there exists an open set $U_z$ and some $E_z \subset U_z$ with $\mu(E_z) = 0$, such that for every pair of points $\{x,y\}$ in $U_z \thicksim E_z$ the inequality (\[global-local\]) holds. As in (\[global-norm\]) one defines $$\begin{aligned} \label{local-norm} \|f\|_{m^p_1(X,\mu)}:= \|f\|_{L^{p}(X,\mu)}+\inf_{g_f} \|g_f\|_{L^{p}(X,\mu)},\end{aligned}$$ where now the infimum is over all those $g_f$ that are local Hajłasz gradients for $f$. Then $m^p_1(X,\mu)$ is also a Banach space. It is obvious that Hajłasz gradients for a function $f$ are local Hajłasz gradients; the converse is not true in general, see [@JSYY] for an example. For a detailed exposition of some basic properties of these spaces, we refer to [@Ambrosio; @Hajlasz; @Hajlasz1; @HajlaszKoskela; @Heinonen; @H; @JSYY; @Kinnunen].\ The next result will be useful: (See [@Hajlasz]) \[wlozenie\] Suppose $(X, d, \mu)$ is an Ahlfors lower $s$-regular metric-measure space of finite diameter. If $1 < p < s$, then $$\begin{aligned} M^p_1(X,\mu) \hookrightarrow L^{p^*}(X,\mu), \end{aligned}$$ where $p^* =\frac{sp}{s-p}$. Moreover, there exists a constant $C=C(s,p,b)$, depending on $s, p, b$, such that for each $f$ in $M^p_1(X,\mu)$ $$\begin{aligned} \|f\|_{L^{p^*}(X,\mu)} \leq C\left(\|f\|_{L^{p}(X,\mu)}+\| g_f\|_{L^{p}(X,\mu)}\right) \end{aligned}$$ whenever $g_f$ is a Hajłasz gradient for $f$. We use Proposition \[wlozenie\] to infer: \[noncritic\] Assume that $(X, d, \mu)$ is an Ahlfors lower $s$-regular compact metric-measure space, with $(X, d)$ doubling. If $1 < p < s$, then for every $q <p^*$ $$\begin{aligned} M^p_1(X,\mu) \hookrightarrow \hookrightarrow L^q(X,\mu), \end{aligned}$$ where $p^* =\frac{sp}{s-p}$. By Proposition \[wlozenie\] have that $M^p_1(X,\mu) \hookrightarrow L^q(X,\mu)$ for every $q \in [1, p^*]$. Moreover, since $(X, d)$ is doubling, by Theorem 2 in [@Kalamajska] we have the compact embedding $$\begin{aligned} \label{comp} M^p_1(X,\mu) \hookrightarrow \hookrightarrow L^{p}(X,\mu).\end{aligned}$$ Hence if $q\in [1,p]$, then $$M^p_1(X,\mu) \hookrightarrow \hookrightarrow L^p(X,\mu) \hookrightarrow L^q(X,\mu).$$ Next, consider the case when $p < q < p^*$. We will prove that the ball $\mathfrak{B}=\{f: \|f\|_{M^p_1(X,\mu)}\leq 1\}$ is precompact in $L^q(X,\mu)$. Fix $\theta$ in $(0,1)$ so that $$\frac{1}{q}=\frac{\theta}{p}+\frac{1-\theta}{p^*},$$ and use (\[comp\]) to note that $\mathfrak{B}$ is precompact in $L^p(X,\mu)$. Hence for every $\epsilon >0$ there exists an $\tilde{\varepsilon}=2C\epsilon^{\frac{1}{\theta}} / (2C)^{\frac{1}{\theta}}$ net[^8] of $\mathfrak{B}$ in $L^p(X,\mu)$, say $\{f_k\}_{k \in \{1,...,N\}}$, where $C$ is the constant from Proposition \[wlozenie\]. Now it is enough to prove that $\{f_k\}_{k \in \{1,...,N\}}$ is an $\epsilon$ net of $\mathfrak{B}$ in $L^q(X,\mu)$; indeed, by the interpolation inequality we have $$\begin{aligned} \|f-f_k\|_{L^q(X,\mu)}&\leq&\|f-f_k\|^{\theta}_{L^p(X,\mu)}\|f-f_k\|^{1-\theta}_{L^{p^*}(X,\mu)}\\ &\leq&C^{1-\theta}\|f-f_k\|^{\theta}_{L^p(X,\mu)}\|f-f_k\|^{1-\theta}_{M^p_1(X,\mu)}\\ &\leq&2^{1-\theta}C^{1-\theta}\|f-f_k\|^{\theta}_{L^p(X,\mu)}\leq \epsilon\end{aligned}$$ for some $k$ in $\{1,...,N\}$. Proposition \[noncritic\] highlights the fact that in general one cannot expect that $M^p_1(X,\mu) \hookrightarrow \hookrightarrow L^{p^*}(X,\mu)$. Theorem \[main2\] ensures that some proper subset of $M^p_1(X,\mu)$ is relatively compact in $L^{p^*}(X,\mu)$. Auxiliary Lemmata ----------------- The next lemma seems to be well known, however we give a detailed proof due to its role in Section \[main\]. \[lemma:delty\] Here $(\Omega ,d , \mu)$ is a separable metric-measure space with a finite Borel measure $\mu$. Suppose that there exists some $\delta >0$ such that for every measurable set $A$, either $\mu(A) = 0$ or $\mu(A) \geq \delta$. Then there exists a finite set $\{x_i \}_{i \in I}$ of points in $\Omega$ and a finite set of numbers $\{ \mu_i \}_{i \in I}$ not smaller than $\delta$ such that $$\mu = \sum_{i \in I} \mu_i \delta_{x_i}.$$ Consider the set $A_{\delta} := \left\{ x \in \Omega \, : \, \mu(x) \geq \delta \right\}$. Since $\mu(\Omega) < \infty$, the set $A_{\delta}$ must be finite. We will show that $\mu(\Omega \thicksim A_{\delta}) = 0$. Since $\Omega \thicksim A_{\delta}$ is open, we have $$\Omega \thicksim A_{\delta} = \bigcup_{x \in \Omega \thicksim A_{\delta}} B_{R_x}(x) .$$ Moreover, since there are no atoms in $\Omega \thicksim A_{\delta}$, for every $x$ in $\Omega \thicksim A_{\delta}$ we can choose $R_x$ in such a way that $$\mu(B_{R_x}(x)) = 0.$$ Furthermore, since $\Omega$ is separable, Lindelöf’s lemma yields $$\Omega \thicksim A_{\delta} = \bigcup_{x \in A} B_{R_x}(x),$$ where $A$ is a countable subset of $\Omega \thicksim A_{\delta} $, and $\mu( \Omega \thicksim A_{\delta} ) = 0$ follows. To state the next lemma, given some space $F(\Omega)$ of functions on some set $\Omega$, denote by $$F_c (\Omega) :=\{\phi \in F(\Omega): \hbox{spt} \phi \subset \subset \Omega \}$$ the subset of $F(\Omega)$ consisting of those functions whose support is a compact subset of $\Omega$. \[aproksymacja\] Here $(X,d)$ is a locally compact metric space with two Radon measures $\mu$ and $\nu$, and $ \Omega \subset X$ is a precompact open set. Then for every $p$ and $r$ in $[1, \infty)$ the set $\hbox{Lip}_c (\Omega)$ is equidense both in $L^{r}(\Omega,\mu)$ and in $L^{p}(\Omega,\nu)$. This means that for every $\epsilon >0$ and every $f \in L^{r}(\Omega,\mu) \cap L^{p}(\Omega,\nu)$ there exists some $\phi$ in $\hbox{Lip}_c (\Omega)$ such that $$\|f -\phi\|_{L^r(\Omega,\mu)} \leq \epsilon \quad \text{and} \quad \|f-\phi\|_{L^p(\Omega,\nu)} \leq \epsilon.$$ By Urysohn’s lemma $C_c (\Omega)$ is dense in $L^r(\Omega,\mu)$ and in $L^p(\Omega,\nu)$. To prove the lemma it is sufficient to check that for every measurable set $A$ the characteristic function $\mathbf{1}_A$ can be approximated both in $L^r(\Omega,\mu)$ and in $L^p(\Omega,\nu)$ by functions in $\hbox{Lip}_c (\Omega)$. The regularity of the measures ensures that there exists a sequence $\{K_n\}_n$ of compact sets and a sequence $\{U_n\}_n$ of open sets such that $ K_n \subset A \subset U_n $, with $$\mu(U_n \thicksim K_n) \leq \frac{1}{n}\ \text{and} \ \nu(U_n \thicksim K_n) \leq \frac{1}{n}.$$ Without loss of generality we can assume that $U_n \subset \Omega$. Moreover, since the space is locally compact, for every $n$ there exists an open precompact set $V_n$ such that $$K_n \subset V_n \subset \overline{V}_n \subset U_n.$$ Introduce the sequence of functions $\{\psi_n\}_n$ given for each $n$ by $$\psi_n := \mathbf{1}_{K_n} : K_n \cup (\Omega \thicksim V_n)\rightarrow [0,1] ,$$ and denote by $\tilde{\psi}_n$ the extension of $\psi_n$ to all $\Omega$ defined as $$\tilde{\psi}_n(x) := \sup_{y \in K_n \cup (\Omega \thicksim V_n) } \left( \psi_n(y) - L_n\ d(x,y) \right) ,$$ where $L_n = 1 / \text{ dist } ( K_n , \Omega \thicksim V_n)$. Such functions are Lipschitz on $\Omega$, with $\tilde{\psi}_n = \psi_n$ on $ K_n \cup (\Omega \thicksim V_n)$, and with $\tilde{\psi}_n \leq 1$. Finally, define $$\phi_n = \max \{0, \tilde{\psi}_n \},$$ and note that $\phi_n \in \hbox{Lip}_c (\Omega)$. Then it is easy to see that $$\int_{\Omega} \left| \phi_n(x) - \mathbf{1}_A(x) \right|^{r} d \mu(x) \leq 2^{r} \mu(U_n \thicksim K_n) \leq \frac{2^{r}}{n} ,$$ and similarly $$\int_{\Omega} \left| \phi_n(x) - \mathbf{1}_A(x) \right|^{p} d \nu(x) \leq \frac{2^{p}}{n}.$$ Proof of Theorem \[main2\] {#main} ========================== In this section we prove Theorem \[main2\], our main result. To prove such a theorem, we will need Theorem \[theorem:Lions\], which in turn requires Lemma \[lemma:rhol\], Lemma \[prod\] and Lemma \[BL\]. We start with Lemma \[lemma:rhol\]: \[lemma:rhol\] (Reverse Hölder) Let $\Omega \subset X$ be an open precompact subset of the metric space $(X, d)$, and let $\mu$ and $\nu$ be Radon measures on $\Omega$. Assume that $1 \leq p < r$. If there exists a positve real number $C$ such that for every Lipshitz $\phi$ with compact support $$\label{hypothesis} \| \phi \|_{L^r(\Omega,\mu)} \leq C \| \phi \|_{L^{p}(\Omega,\nu)},$$ then there exists a countable set of points $\{x_i \}_{i \in I}$ in $\Omega$ and a countable set $\{\mu_i\}_{i \in I}$ of positive real numbers such that $$\mu = \sum_{i \in I} \mu_i \delta_{x_i}.$$ We divide the proof into two steps.\ [**Step 1.**]{} Assume that $\mu = \nu$, choose any measurable set $A$, and assume that (\[hypothesis\]) holds; by Lemma \[aproksymacja\] $$\| \mathbf{1}_{A} \|_{L^{r}(\Omega,\mu)} \leq C \| \mathbf{1}_{A} \|_{L^{p}(\Omega,\mu)},$$ and then $$\mu(A)^{\frac{1}{r}} = \| \mathbf{1}_{A} \|_{L^r(\Omega,\mu)} \leq C \| \mathbf{1}_{A} \|_{L^p(\Omega,\mu)} = C \mu(A)^{\frac{1}{p}}.$$ Hence either $\mu(A) = 0$, or $\mu(A) \geq 1 / {C}^{\frac{pr}{r- p}}$. Then by Lemma \[lemma:delty\] there exists finite set $\{x_i \}_{i \in I}$ of points in $\Omega$ and a finite set $\{ \mu_i \}_{i \in I}$ of real numbers with $\mu_i \geq 1/C^{\frac{pr}{r- p}}$ such that $$\mu = \sum_{i \in I} \mu_i \delta_{x_i}.$$ [**Step 2.**]{} Now assume that $\mu$ and $\nu$ are arbitrary; the Lebesgue Decomposition theorem ensures that $$\label{decomposition} \nu = \mu \llcorner \theta + \sigma$$ for some non-negative $\theta$ in $L^1(\Omega,\mu)$, where $\mu \llcorner \theta (A) := \int_{A} \theta d \mu$, and $\sigma$ is a positive measure singular with respect to $\mu$. For every positive integer $n$ consider the function $$\phi_n := \theta^{\frac{1}{r - p}} \mathbf{1}_{\theta \leq n} \psi ,$$ where $ \psi $ is Lipschitz with compact support, and also the measure $$\mu_n := \mu \llcorner ( \theta^{\frac{r}{r - p}} \mathbf{1}_{\theta \leq n} ).$$ Assuming (\[hypothesis\]) and recalling Lemma \[aproksymacja\], use the decomposition (\[decomposition\]) to obtain $$\label{ineq-rp} \| \phi_n \|_{L^r(\Omega,\mu)} \leq C \| \phi_n \|_{L^p(\Omega,\nu)} = C \| \phi_n \|_{L^p(\Omega, \mu \llcorner \theta + \sigma)} = C \| \phi_n \|_{L^p(\Omega, \mu \llcorner \theta )} .$$ However $$\label{eq-p} \| \phi_n \|^p_{L^p(\Omega, \mu \llcorner \theta )} = \int_{\Omega} \left| \psi \right|^p \theta^{\frac{p}{r-p}} \mathbf{1}_{\theta \leq n} \theta d \mu = \int_{\Omega} \left| \psi \right|^p \theta^{\frac{r}{r-p}} \mathbf{1}_{\theta \leq n} d \mu = \| \psi \|^p_{L^p(\Omega , \mu_n)} ,$$ and similarly $$\label{eq-r} \| \psi \|_{L^r(\Omega , \mu_n)} = \| \phi_n \|_{L^r(\Omega, \mu)} .$$ Then use (\[eq-p\]) and (\[eq-r\]) in (\[ineq-rp\]) to infer that $$\| \psi \|_{L^r(\Omega , \mu_n)} \leq C \| \psi \|_{L^p(\Omega , \mu_n)}$$ for every $n$.\ Hence by Step 1 $$\mu_n = \sum_{i \in I_n} \mu_{n,i} \delta_{x_{n,i}}$$ for every $n$. Recall the definition of the measures $\mu_n$, and note that $\hbox{spt} \mu_n \subset \hbox{spt} \mu_{n+1}$, in particular $I_n \subset I_{n+1}$. Let $I=\bigcup_{n=1}^{\infty}I_n$ and define $x_i : =x_{n,i} \big|_{I_n}$; one can write $$\mu_n = \sum_{i \in I_n} \mu_{n,i} \delta_{x_i}.$$ Since $\mu_{n,i} =\mu_n (\{x_i\})\leq \mu_{n+1} (\{x_i\})=\mu_{n+1,i} $, it follows that for each $i$ the number $\mu_{n,i}$ is non decreasing with respect to $n$.\ Denote by $\mathcal{M}(\Omega)$ the set of measures on $\Omega$ endowed with the weak-$\ast$ topology. Let $\tilde{\mu}_n= \mu_n \llcorner (\theta^{-\frac{r}{r-p}}\mathbf{1}_{\{\theta>0\}})$, and observe that $\tilde{\mu}_n \rightarrow \mu \llcorner \mathbf{1}_{\{\theta>0\}}$ in $\mathcal{M}(\Omega)$. We claim that $$\tilde{\mu}_n \rightarrow \mu .$$ To prove that, it suffices to show that $\mu(\{\theta=0\})=0$. Since $\mu$ is singular with respect to $\sigma$, there exist subsets $A$ and $B$ with $A \cap B =\emptyset$, such that for every measurable set $E$ we have $\mu(E)= \mu (A\cap E)$ and $\sigma(E)= \sigma (B\cap E)$. Therefore $$\int_{\Omega}\mathbf{1}_A \mathbf{1}_{\{\theta=0\}} d \nu =\int_{\Omega}\mathbf{1}_A \mathbf{1}_{\{\theta=0\}} \theta d \mu + \int_{\Omega}\mathbf{1}_A \mathbf{1}_{\{\theta=0\}} d \sigma=0,$$ hence $\nu (A\cap \{\theta=0\})=0$, and using (\[hypothesis\]) $$\| \mathbf{1}_{A\cap \{\theta=0\}} \|_{L^r(\Omega,\mu)} \leq C \| \mathbf{1}_{A\cap \{\theta=0\}} \|_{L^{p}(\Omega,\nu)}=0.$$ Thus $\mu ( \{\theta=0\})=\mu (A\cap \{\theta=0\})=0$, as required. Now we continue with Lemma \[prod\]: (Hajłasz-Leibniz) \[prod\] If $v \in M^p_1(X,\mu)$ and $\phi \in \hbox{Lip}(X)$, then $f = v \phi \in M^p_1(X,\mu)$. Moreover, $$g_f = g_v |\phi| + |v|\|\phi\|_{\hbox{Lip}}$$ is a Hajłasz gradient for $v \phi$. The result follows from the string of inequalities $$\begin{aligned} |v(x)\phi(x) - v(y)\phi(y)| &\leq& |v(x)-v(y)|\ \min\{|\phi(x)|,|\phi(y)|\} + \max \{|v(x)|,|v(y)| \}\ |\phi(x)-\phi(y)|\\ &\leq& \big( g_v(x)+g_v(y) \big)\ \min \{ |\phi(x)|,|\phi(y)|\}\ d(x,y) + \max \{ |v(x)|,|v(y)| \}\ \|\phi\|_{\hbox{Lip}}\ d(x,y)\\ &\leq& \big( g_v(x)|\phi(x)|+g_v(y)|\phi(y)| \big)\ d(x,y) + (|v(x)|+|v(y)|)\ \|\phi\|_{\hbox{Lip}}\ d(x,y) \\ & = & d(x,y)\ ( g_{v \phi}(x) + g_{v \phi}(y) ),\end{aligned}$$ with $g_{v \phi} := g_v |\phi| + |v|\|\phi\|_{\hbox{Lip}} .$ Finally, before stating Theorem \[theorem:Lions\], we recall the following Lemma attributed to H. Brézis and E. Lieb: \[BL\] Let $p \in [1, \infty)$. If $f_n \rightarrow f$ weakly in $L^{p}(X,\mu)$ and $f_n \rightarrow f$ $\mu$-almost everywhere, then $$\lim_{n \rightarrow \infty} \left( \int_{X} |f_n|^{p} d\mu - \int_{X} |f_n - f|^{p} d \mu \right) = \int_{X} |f|^{p} d \mu.$$ With those results at hand, we have: \[theorem:Lions\] If $(X, d, \mu)$ is an Ahlfors lower $s$-regular compact metric-measure space, and $1 < p < s$, then for every sequence $\left\{ u_n \right\}$ in $M^p_1(X,\mu)$ such that $u_n \rightarrow u$ weakly in $M^p_1(X,\mu)$ and $u_n \rightarrow u$ strongly in $L^p(X,\mu)$, there exists a subsequence $\left\{ u_n \right\}$ and a countable set $I$ such that $$\label{eq:form} \mu \llcorner |u_n|^{p^*} \to \mu \llcorner |u|^{p^*} + \sum_{i \in I} \mu_i \delta_{x_i}$$ in $\mathcal{M} (X)$, where $x_i \in X$ for every $i \in I$. We begin with two observations: 1. Let $v_n:= u_n - u$, and fix some $\phi$ in $\hbox{Lip}_c(X)$. The hypotheses, Proposition \[wlozenie\] and Lemma \[prod\] when applied to $f_n := v_n \phi$ give $$\label{istot} \|v_n \phi\|_{L^{p^*}(X,\mu)} \leq C\left(\|v_n \phi\|_{L^{p}(X,\mu)}+\| g_{v_n} \phi\|_{L^{p}(X,\mu)} + \|\phi\|_{\hbox{Lip}} \| v_n \|_{L^{p}(X,\mu)}\right) .$$ 2. The hypotheses also imply that - $\|v_n \phi\|_{L^{p}(X,\mu)} \to 0$ and $\| v_n \|_{L^{p}(X,\mu)} \to 0$, - $\mu \llcorner |v_n|^{p^{\ast}} \to \bar{\mu}$ and $\mu \llcorner |g_{v_n}|^{p} \to \nu$ for some $\bar{\mu}$ and $\nu$ in $\mathcal{M}(X)$. Those observations yield the reverse Hölder inequality $$\| \phi \|_{L^{p^*}(X,\bar{\mu})} \leq C \| \phi \|_{L^{p}(X,\nu)} ,$$ and Lemma \[lemma:rhol\] ensures that the mesure $\bar{\mu}$ has the form $$\label{eq:suma} \bar{\mu} = \sum_{i \in I} \mu_i \delta_{x_i}.$$ Now use Lemma \[BL\] when $f_n=u_n \phi^{\frac{1}{p^*}}$, where $\phi$ is a non-negative function in $C_c(X)$, and $$\lim_{n \rightarrow \infty} \left( \int_{X} \phi |u_n|^{p^{*}} d \mu - \int_{X} \phi |v_n|^{p^{*}} d \mu \right) = \int_{X} \phi |u|^{p^{*}} d \mu$$ follows. Then recall that $\mu \llcorner |v_n|^{p^{*}} \to \bar{\mu}$ in $\mathcal{M}(X)$, to infer $$\label{eq:pos} \lim_{n \rightarrow \infty} \int_X \phi |u_n|^{p^{*}} d \mu = \int_X \phi\ d \bar{\mu} + \int_X \phi |u|^{p^{*}} d \mu.$$ Since every continuous function of compact support, say $\phi$, can be written as $\phi=\phi_{+}-\phi_{-}$, where $\phi_{+}$ and $\phi_{-}$ are non-negative with compact support, one concludes that (\[eq:pos\]) holds for every $\phi$ in the dual of $\mathcal{M}(X)$. Now use (\[eq:suma\]), to obtain (\[eq:form\]). Now we can prove Theorem \[main2\], the main result in this work. By the hypotheses, whenever $h \in H$ one has $h_{\#} \mu = \mu$. Let $\{ u_n \}$ be a bounded sequence in $M^p_{1,H}(X,\mu)$, namely a bounded sequence in $M^p_1(X,\mu)$ such that $h^{\#} u_n = u_n$ for each $n$ and each $h$ in $H$. Then the sequence of measures $\{ \mu_n \}$ defined by $$\mu_n := \mu \llcorner |u_n|^{p^{\ast}}$$ is also $H$-invariant. On the other hand, if the sequence $\{ u_n \}$ converges weakly to some $u$ in $M_{1,H}^p(X,\mu)$, then[^9] by Theorem \[theorem:Lions\] there exists a subsequence[^10] of $\{ \mu_n \}$ such that $$\label{measure-inv} \mu_n \to \mu \llcorner |u|^{p^{\ast}} + \sum_{i \in I} \mu_i \delta_{x_i}$$ in $\mathcal{M}(X)$, where $I$ is at most countable. In addition, it is not difficult to see that if $\{ \mu_n \}$ is a sequence of $H$-invariant measures converging to some $\nu$ in $\mathcal{M}(X)$, then $\nu$ is also $H$-invariant; therefore from (\[measure-inv\]) the measure $$\mu \llcorner |u|^{p^{\ast}} + \sum_{i \in I} \mu_i \delta_{x_i}$$ is $H$-invariant. Moreover, since $\mu \llcorner |u|^{p^{\ast}}$ is $H$-invariant, then $\sigma := \sum_{i \in I} \mu_i \delta_{x_i}$ is $H$-invariant as well.\ Choose some $k$ in $I$, and let $y = h (x_k)$ be some element in $H(x_k)$. Then $$\mu_k = \sigma(x_k) = \sigma(h^{-1}(y)) = h_{\#} \sigma(y) = \sigma(y) = \sum_{i \in I} \mu_i \delta_{x_i}(y) ,$$ hence $x_i = y$ for some $i \in I$. This gives a contradiction, since $I$ is at most countable, meanwhile the orbit of each point in $X$ is uncountable by hypothesis. It follows that $$\mu \llcorner |u_n|^{p^{\ast}} \to \mu \llcorner |u|^{p^{\ast}}$$ in $\mathcal{M}(X)$; but this is equivalent to say that $$\label{M(X)} \| \phi u_n \|_{L^{p^{\ast}}(X,\mu)} \to \| \phi u \|_{L^{p^{\ast}}(X,\mu)}$$ whenever $\phi \in C_c(X)$.\ Since $X$ is compact, we can use $\phi = 1$ in (\[M(X)\]), to conclude that if $\{u_n \}$ is a bounded sequence in $M^p_{1,H}(X,\mu)$ converging weakly to some $u$ in $M^p_{1,H}(X,\mu)$, then $$\| u_n \|_{L^{p^{\ast}}(X,\mu)} \to \| u \|_{L^{p^{\ast}}(X,\mu)}$$ for some subsequence. But $L^{p^{\ast}}(X,\mu)$ is uniformly convex, hence $u_n \to u$ in $L^{p^{\ast}}(X,\mu)$. A useful consequence of Theorem \[main2\] is: \[corollary\] Using the same hypotheses as in Theorem \[main2\], define the constant $C$ by $$C:= \inf \{\ A > 0\ :\ \| u \|_{L^{p^{\ast}}(X,\mu)} \leq A \| u \|_{M^p_1(X,\mu)}\ \text{whenever}\ u \in M^p_{1,H}(X,\mu)\ \}.$$ Then there exists some $u_0$ in $M^p_{1,H}(X,\mu)$ such that $$C = \| u_0 \|_{L^{p^{\ast}}(X,\mu)}\ /\ \| u_0 \|_{M^p_{1,H}(X,\mu)} .$$ Define the functional $\mathcal{I} : M^p_{1,H}(X,\mu) \thicksim \{ 0\} \to \mathbb{R}$ by $$\mathcal{I}[u] := \| u \|_{M^p_1(X,\mu)} ,$$ and set $$D := \inf \{\ \mathcal{I}[u]\ :\ u \in M^p_{1,H}(X,\mu)\ ,\ \| u \|_{L^{p^{\ast}}(X,\mu)} = 1 \ \} .$$ Let $\{ u_n \}$ be a minimizing sequence, i.e. such that $u_n \in M^p_{1,H}(X,\mu)$ and $ \| u_n \|_{L^{p^{\ast}}(X,\mu)} = 1$ for every $n$, with $\mathcal{I}[u_n] \to D$. Since $\{ u_n \}$ is bounded in $M^p_{1,H}(X,\mu)$, by Theorem \[main2\] there is a subsequence[^11] of $\{ u_n \}$ and some $w$ in $M^p_{1,H}(X,\mu)$ such that $$u_n \to w\ \text{weakly in}\ M^p_{1,H}(X,\mu),$$ $$\text{and}\ u_n \to w\ \text{strongly in}\ L^{p^{\ast}}(X,\mu).$$ But $\| w \|_{L^{p^{\ast}}(X,\mu)} = 1$ by strong convergence in $L^{p^{\ast}}(X,\mu)$, hence $$D = \lim_{n \to \infty} \mathcal{I}[u_n] = \liminf_{n \to \infty} \| u_n \|_{M^p_1(X,\mu)} \geq \| w \|_{M^p_1(X,\mu)} = \mathcal{I}[w] \geq D.$$ Therefore $\mathcal{I}[w] = D$, and it follows that $C = 1/D$, with $u_0 = w$. Riemannian applications {#examples} ======================= The next result is not surprising and probably not new, however we could not find it in the literature. To satisfy the interested reader, and justify the discussion in Section \[final\] below, we give a proof of it with some details. \[riem-mm\] Suppose $(X, g)$ is a compact Riemannian $n$-manifold. Then for every $p$ such that $1 < p < \infty$ the spaces $L^p_1(X, V_g)$ and $M^p_1(X, V_g)$ coincide with equivalent norms. By Proposition 10.1 from [@HajlaszKoskela] $$M^p_1(X,V_g) \hookrightarrow L^p_1(X,V_g),$$ hence we need the opposite inclusion. Since $X$ is compact, there exists a finite number of charts $$\{(U_{\alpha},\phi_{\alpha})\ :\ \alpha \in A \},$$ such that for every $\alpha$ the components $g_{i j}^\alpha$ of $g$ in $(U_{\alpha},\phi_{\alpha})$ satisfy $$\begin{aligned} \frac{1}{2} \delta_{ij} \leq g^{\alpha}_{ij} \leq 2 \delta_{ij}\end{aligned}$$ as bilinear forms. Let $\{ \eta_{\alpha} \}$ be smooth partition of unity subordinate to covering $\{ U_{\alpha} \}$. We proceed in two steps.\ [**Step 1.**]{} Let $\mathcal{L}^n$ be the $n$-dimensional Lebesgue measure, and fix $u$ in $C^{\infty}(X)$. Since $M^p_1(\mathbb{R}^n, \mathcal{L}^n)$ and $L^p_1(\mathbb{R}^n, \mathcal{L}^n)$ are equivalent, see [@Hajlasz] for example, there exists a constant $C > 1$ such that for every $\alpha$ in $A$ $$\begin{aligned} \label{ru} \frac{1}{C} \| (\eta_{\alpha} u ) \circ \phi_{\alpha}^{-1} \|_{L^p_1(\mathbb{R}^n, \mathcal{L}^n)} \leq \| (\eta_{\alpha} u ) \circ \phi_{\alpha}^{-1} \|_{M^p_1(\mathbb{R}^n, \mathcal{L}^n)} \leq C \| (\eta_{\alpha} u ) \circ \phi_{\alpha}^{-1}\|_{L^p_1(\mathbb{R}^n,\mathcal{L}^n)}.\end{aligned}$$ Furthermore $$\int_X |\eta_{\alpha} u|^p d V_g = \int_{U_{\alpha}} |\eta_{\alpha} u|^p d V_g = \int_{\phi_{\alpha}(U_{\alpha})} \sqrt{ \det g^{\alpha}_{ij}} \left| \eta_{\alpha} u \right|^{p} \circ \phi^{-1}_{\alpha}(x)\ d \mathcal{L}^n(x) ,$$ hence $$\begin{aligned} \label{niermodularfunkc} 2^{-\frac{n}{2p}}\|\eta_{\alpha} u\|_{L^p (X,V_g)} \leq \| (\eta_{\alpha} u) \circ \phi^{-1}_{\alpha}\|_{L^p (\mathbb{R}^n,\mathcal{L}^n)} \leq 2^{\frac{n}{2p}}\|\eta_{\alpha} u\|_{L^p (X,V_g)} . \end{aligned}$$ On the other hand, we estimate the gradient locally by $$\begin{aligned} \int_{X} \left| \nabla (\eta_{\alpha} u) \right|^{p} d V_{g} &=& \int_{\phi_{\alpha} (U_{\alpha})} \sqrt{ \det g^{\alpha}_{ij}}\ \left| \sum_{k,j=1}^{n} g_{\alpha}^{kj} D_{k} ( (\eta_{\alpha} u ) \circ \phi^{-1}_{\alpha}) D_{j} ( (\eta_{\alpha} u ) \circ \phi^{-1}_{\alpha}) \right|^{p} d\mathcal{L}^n \nonumber \\ &\geq& 2^{- \frac{n+p}{2}} \int_{\phi_{\alpha} (U_{\alpha})} \left| \nabla ( (\eta_{\alpha} u ) \circ \phi^{-1}_{\alpha}) \right|^{ p } d\mathcal{L}^n,\end{aligned}$$ therefore $$\| \nabla ( \eta_{\alpha} u) \circ \phi_{\alpha}^{-1} \|_{L^p(\mathbb{R}^n, \mathcal{L}^n)} \leq 2^{\frac{n+p}{2p}} \| \nabla( \eta_{\alpha} u) \|_{L^p(X,V_g)}$$ for each $\alpha$ in $A$.\ Set $\displaystyle C_0 := \max_{\alpha \in A} \| \nabla \eta_{\alpha} \|_{\infty}+1 $. Then $$\label{eq:1} \| \nabla ( \eta_{\alpha} u) \circ \phi_{\alpha}^{-1} \|_{L^p(\mathbb{R}^n, \mathcal{L}^n)} \leq 2^{\frac{n+p}{2p}} \left( \| \nabla u \|_{L^p(X, V_g)} + C_0 \| u \|_{L^p(X,V_g)} \right) \leq 2^{\frac{n+p}{2p}}C_0 \| u \|_{L^p_1(X,V_g)} \ .$$ Fix some $\epsilon >0$; then there exists a Hajłasz gradient $h_{\alpha}$ for $(\eta_{\alpha} u) \circ \phi_{\alpha}^{-1} $ in $\phi_{\alpha}(U_{\alpha})$, so that $$\label{eq:2} \| (\eta_{\alpha} u) \circ \phi_{\alpha}^{-1} \|_{L^p(\mathbb{R}^n,\mathcal{L}^n)} + \| h_{\alpha} \|_{L^p(\mathbb{R}^n,\mathcal{L}^n)} - \epsilon \leq \| (\eta_{\alpha} u) \circ \phi_{\alpha}^{-1}\|_{M^p_1(\mathbb{R}^n, \mathcal{L}^n)}.$$ Gather inequalities (\[ru\]), (\[niermodularfunkc\]), (\[eq:1\]) and (\[eq:2\]) to get $$\begin{aligned} \| (\eta_{\alpha} u) \circ \phi_{\alpha}^{-1} \|_{L^p(\mathbb{R}^n,\mathcal{L}^n)} + \| h_{\alpha} \|_{L^p(\mathbb{R}^n,\mathcal{L}^n)} - \epsilon &\leq& C \| (\eta_{\alpha} u) \circ \phi_{\alpha}^{-1} \|_{L^p_1(\mathbb{R}^n,\mathcal{L}^n)} \\ & \leq& C \left( 2^{\frac{n+p}{2p}}C_0 + 2^{\frac{n}{2p}} \right) \| u \|_{L^p_1(X,V_g)}.\end{aligned}$$ Observe that for each $\alpha$ the function $\sqrt{2} h_{\alpha} \circ \phi_{\alpha} =: \tilde{h}_{\alpha}: U_{\alpha} \rightarrow \mathbb{R}$ is a Hajłasz gradient for $ (\eta_{\alpha} u) |_{U_{\alpha}}$. Indeed, since $h_{\alpha}$ is a Hajłasz gradient for $(\eta_{\alpha} u) \circ \phi_{\alpha} ^{-1}\ \mathbf{1}_{\phi_{\alpha} (U_{\alpha})}$, there exists a subset $E_{\alpha} \subset \mathbb{R}^n$ such that $\mathcal{L}^n(E_{\alpha})=0$, and such that for every pair $x,y \in \phi_{\alpha} (U_{\alpha}) \thicksim E_{\alpha}$ $$|\eta_{\alpha} u(\phi_{\alpha}^{-1}(x)) - \eta_{\alpha} u(\phi_{\alpha}^{-1}(y))| \leq \left( h_{\alpha}( x ) + h_{\alpha}(y) \right) |x - y|.$$ Therefore, for each pair $x,y$ in $U_{\alpha} \thicksim \phi_{\alpha} ^{-1}(E_{\alpha})$ $$\begin{aligned} \label{gradient} |\eta_{\alpha}(x) u(x) - \eta_{\alpha}(y) u(y)| &=& |\eta_{\alpha} u(\phi_{\alpha}^{-1}( \phi_{\alpha}(x) )) - \eta_{\alpha} u(\phi_{\alpha}^{-1}( \phi_{\alpha}(y))| \nonumber \\ &\leq &\left( h_{\alpha}( \phi_{\alpha}(x) ) + h_{\alpha}(\phi_{\alpha}(y)) \right) |\phi_{\alpha}(x) - \phi_{\alpha} (y)| \nonumber \\ & \leq &\left( \tilde{h}_{\alpha}(x) + \tilde{h}_{\alpha}(y) \right) d_g(x,y).\end{aligned}$$ Our next goal is to prove that $$h:= \sum_{\alpha \in A} \tilde{h}_{\alpha} \mathbf{1}_{U_{\alpha}}$$ is a local Hajłasz gradient for $u$.\ Fix $z\in X$ and define - $I_z := \{ \alpha \in A: z \in U_{\alpha}\},$ - $ J_z :=\{ \alpha \in A : z \in \partial U_{\alpha}\},$ and - $K_z :=\{ \alpha \in A: z \in X \thicksim \bar{U}_{\alpha}\}.$ Then $I_z, J_z, K_z$ are pairwise disjoint, with $I_z \cup J_z\cup K_z= A$ for each $z$ in $X$.\ Define $R > 0$ such that - For all $\alpha$ in $I_z$, the ball $B_R(z) \subset U_{\alpha}$, - For all $\alpha$ in $J_z, \eta_{\alpha} (B_R(z))=\{0\}$, and - For all $\alpha$ in $K_z, B_R(z) \cap U_{\alpha} = \emptyset.$ Note that if $x,y \in B_R(z)$ and $\eta_{\alpha}(x) \neq 0$, then $y \in U_{\alpha}$; indeed, $\alpha$ can not belong to $K_z \cup J_z$, therefore $\alpha \in I_z$, and then $y \in B_R(z) \subset U_{\alpha}$. Hence for $x, y \in B_R(z)\thicksim \bigcup_{\alpha \in A} \phi^{-1}_{\alpha}(E_{\alpha})$, taking (\[gradient\]) into account, the inequality $$\left| u(x) - u(y) \right| \leq \sum_{\alpha \in A} \left| \eta_{\alpha}(x) u (x) - \eta_{\alpha}(y) u (y) \right| \leq \sum_{\alpha \in A} \left(\tilde{h}_{\alpha}(x) + \tilde{h}_{\alpha}(y) \right) d_g(x,y),$$ follows, and this proves that $h$ is a local Hajłasz gradient for $u$.\ Recalling (\[local-norm\]), collect previous estimates to infer $$\begin{aligned} \| u \|_{m^p_1(X, V_g)} & \leq& \sum_{\alpha \in A} \| \eta_{\alpha} u \|_{L^p(X,V_g)} + \| h \|_{L^p(X,V_g)} \\ &\leq& 2^{\frac{n}{2p}} \sum_{\alpha \in A} \| (\eta_{\alpha} u) \circ \phi_{\alpha}^{-1} \|_{L^p(\mathbb{R}^n,\mathcal{L}^n)} + 2^{\frac{n}{2p}} \sum_{\alpha \in A} \|h_{\alpha}\|_{L^p(\mathbb{R}^n,\mathcal{L}^n)} \\ & \leq& 2^{\frac{n}{2p}}|A| C \left( 2^{\frac{n+p}{2p}}C_0 + 2^{\frac{n}{2p}} \right) \| u \|_{L^p_1(X,V_g)} +2^{\frac{n}{2p}}|A|\epsilon ,\end{aligned}$$ where $|A|$ denotes the cardinality of the set $A$. Hence if $ \epsilon \rightarrow 0 $ $$\label{ineq-m-M} \| u \|_{m^p_1(X,V_g)} \leq C_1 \| u \|_{L^p_1(X,V_g)},$$ where $C_1 := 2^{\frac{n}{2p}} |A| C \left( 2^{\frac{n+p}{2p}}C_0 + 2^{\frac{n}{2q}} \right)$.\ [**Step 2.**]{} Choose $u$ in $L^p_1(X, V_g)$. By the density of $C^{\infty}(X)$ in $L^p_1(X, V_g)$ there exists a sequence of smooth functions $\{u_n\}$ converging to $u$ in $L^p_1(X,V_g)$. Therefore using (\[ineq-m-M\]) for every $\epsilon >0$ there exists some $N$ such that for $m,n \geq N$ $$\| u_m - u_n \|_{m^{1,p}(X, V_g)} \leq C_1\| u_m - u_n \|_{L^p_1(X,V_g)} \leq \epsilon.$$ On the other hand, by the completeness of $m^p_1(X,V_g)$ the sequence $\{u_n\}$ converges to some $v$ in $m^p_1(X,V_g)$. By the definitions of $L^p_1(X,V_g)$ and $m^p_1(X,V_g)$ the sequence $\{u_n\}$ converges to both $u$ and $v$ in $L^p(X,V_g)$: Thus $u = v$, and using (\[ineq-m-M\]) $$\| u \|_{m^p_1(X,V_g)} \leq C_1 \| u \|_{L^p_1(X,V_g)},$$ therefore $L^p_1(X,V_g) \hookrightarrow m^p_1(X,V_g)$.\ Finally, by Corollary 3.5 from [@JSYY] the spaces $m^p_1(X,V_g)$ and $M^p_1(X,V_g)$ are equivalent, hence $$L^p_1(X,V_g) \hookrightarrow M^p_1(X,V_g),$$ as required. Theorem \[main2\] for flows {#final} --------------------------- We use Theorem \[riem-mm\] to apply Theorem \[main2\] in closed Riemannian manifolds when the $H$-orbits have dimension one. In this setup we see that Theorem \[main2\] can be applied if and only if the Euler characteristic of the manifold is equal to zero; this condition is restrictive only in even dimensional manifolds.\ Consider a closed orientable Riemannian $n$-manifold $(X,g)$ whose Euler characteristic $\chi(X)$ is equal to $0$. A result attributed to H. Hopf, see [@milnor], ensures that there exists a non-vanishing[^12] vector field $\tau$ on $X$, or equivalently a non-vanishing $(n-1)$-form $\omega_{\tau}$, related with $\tau$ through the bijection $TX \longleftrightarrow \wedge^{n-1} TX$ given by $$\tau \leftrightarrow \omega_{\tau} = \tau \lrcorner\ \Omega_g ,$$ where $\Omega_g$ is the volume $n$-form induced from $g$ giving the orientation of $X$. The form $\omega_{\tau}$ is closed if and only if the vector field $\tau$ is free of divergence; indeed: $$( \text{div} \cdot \tau )\ \Omega_g = L_{\tau} \Omega_g = d (\tau \lrcorner\ \Omega_g),$$ where $L_{\tau}$ is the Lie derivative along $\tau$. Denote by $H= \{ h_t : t \in \mathbb{R}\}$ the subgroup of $\text{Diff}(X)$ associated to the flow of $\tau$: if $\omega_{\tau}$ is a non-vanishing closed $(n-1)$-form on $X$, then $H$ is a subgroup of $\text{Diff}_{V_g}(X)$, and the orbit of every point in $X$ under $H$ is uncountable.\ In this spirit, D. Asimov proved in [@Asimov] that if $n$ is at least $4$, and if the first Betti number of $X$ is different from zero, then every non-vanishing vector field is homotopic through a family of non-vanishing vector fields to a non-vanishing vector field that preserves $\Omega_g$, see also [@sullivan]. Shortly afterwards, M. Gromov using Convex Integration[^13] proved that if $n$ is at least $3$, then every non-vanishing $(n-1)$-form can be homotoped through non-vanishing forms to a non-vanishing exact form, with no restrictions on the first Betti number of $X$. Note that when $n=2$ the only possible manifold is the $2$-torus, and then the required vector fields are constant slope fields [@Asimov].\ With those facts, Theorem \[riem-mm\] and Corollary \[corollary\] provide simple and concrete applications: Suppose $(X,g)$ is an orientable closed Riemannian manifold with $\chi(X)=0$. 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DOI: 10.18452/2615 (2002). R. S. Palais, Foundations of Global Non-linear Analysis, W. A. Benjamin, Inc, 1968. J. Rissanen, Wavelets on self-similar sets and the structure of the spaces $M^{1,p}(E,\mu)$, Ann. Acad. Sci. Fenn. Math. Diss. [**125**]{} (2002), 46 pp. N. Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric-measure spaces, Revista Matemática Iberoamericana [**16**]{} (2000), 243-279. D. Sullivan, Cycles for the Dynamical Study of Foliated Manifolds and Complex Manifolds, Inventiones Math. [**36**]{} (1976), 225-255. C. Villani, Optimal transport, old and new, Springer Verlag, 2009. [^1]: Email: p.gorka@mini.pw.edu.pl [^2]: Email: dpons@unab.cl ; pons.dan@gmail.com [^3]: See [@gromov] for an interesting perspective. [^4]: See Section \[Preliminaries\] for definitions. [^5]: For instance, $M^p_1(X,V_g)$ is reflexive for every compact $(X,g)$. [^6]: The quotient space might not be Hausdorff. [^7]: A metric-measure space $(X,d,\mu)$ is said to be *doubling* if the measure $\mu$ is *doubling*, namely if there exists a constant $C_{\mu}>1$ such that for every ball $B_R(x)$ one has $ \mu \left(B_{2R}(x)\right) \leq C_{\mu}\ \mu \left(B_R(x)\right).$ [^8]: This means that for each $f$ in $\mathfrak{B}$ there exists some $k$ in $\{1,...,N\}$ such that $\|f-f_k\|_{L^p(X,\mu)} <\tilde{\varepsilon}$. [^9]: $M_1^p(X,\mu)$ is reflexive in the hypotheses of the theorem. [^10]: We use the same subindex for sequences and the pertinent subsequences. [^11]: As in Theorem \[main2\], we use the same subindex for the sequence and the pertinent subsequence. [^12]: Non-vanishing, or vanishing no-where. [^13]: See [@eliashberg] for a detailed exposition.

--- abstract: | We prove the first explicit rate of convergence to the Tracy-Widom distribution for the fluctuation of the largest eigenvalue of sample covariance matrices that are not integrable. Our primary focus is matrices of type $ X^*X $ and the proof follows the Erdös-Schlein-Yau dynamical method. We use a recent approach to the analysis of the Dyson Brownian motion from [@bourgade2018extreme] to obtain a quantitative error estimate for the local relaxation flow at the edge. Together with a quantitative version of the Green function comparison theorem, this gives the rate of convergence. Combined with a result of Lee-Schnelli [@lee2016tracy], some quantitative estimates also hold for more general separable sample covariance matrices $ X^* \Sigma X $ with general diagonal population $ \Sigma $. address: 'Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012' author: - Haoyu Wang bibliography: - 'CovarianceTW.bib' title: Quantitative Universality for the Largest Eigenvalue of Sample Covariance Matrices --- Introduction ============ Overview and main results ------------------------- Edge universality of sample covariance matrices has been a classical problem in random matrix theory. It is well known that the distribution of the largest eigenvalue (after appropriate rescaling) converges to the Tracy-Widom distribution. Early non-quantitative results were first proved in [@peche2009universality; @pillai2014universality; @Soshnikov]. Quantitative estimates, however, were only obtained for the Wishart ensemble (see [@el2006rate; @Johansson; @Johnstone; @ma2012accuracy]), which is essentially an integrable model. In this paper, we prove the explicit rate of convergence $ N^{-2/9} $ to the Tracy-Widom distribution for all sample covariance matrices of type $ X^*X $ with general distributed entries, and an analogous result with deterministic rate $ N^{-1/57} $ for all separable sample covariance matrices $ X^* \Sigma X $ with diagonal population $ \Sigma $. For simplicity and motivations from statistics, we only consider the real case, but the whole proof and the results are also true for complex sample covariance matrices. Let $ X=(x_{ij}) $ be an $ M \times N $ data matrix with independent real valued entries with mean 0 and variance $ M^{-1} $, $$\label{e.Assumption1} x_{ij} = M^{-1/2}q_{ij},\ \ \ {\mathbb{E}}q_{ij}=0,\ \ \ {\mathbb{E}}q_{ij}^2 =1.$$ Furthermore, we assume the entries $ q_{ij} $ have a sub-exponential decay, that is, there exists a constant $ \theta>0 $ such that for $ u>1 $, $$\label{e.Assumption2} {\mathbb{P}}(|q_{ij}| > u) {\leqslant}\theta^{-1} \exp (-u^\theta).$$ This sub-exponential decay assumption is mainly for convenience, other conditions such as the finiteness of a sufficiently high moment would be enough. (For a necessary and sufficient condition for the edge universality we refer to [@ding2018necessary].) The sample covariance matrix corresponding to data matrix $ X $ is defined by $ H := X^* X $. Throughout this paper, to avoid trivial eigenvalues, we will be working in the regime $$\xi=\xi(N) := N/M,\ \ \ \lim_{N \to \infty} \xi \in (0,1) \ \mbox{or}\ \xi \equiv 1.$$ We will mainly work with the rectangular case $ 0< \xi <1 $, but will also show how to adapt the arguments to the square case $ M \equiv N $. (The reason why we do not discuss the general case $ \lim \xi =1 $ is merely technical due to the lack of local laws at the hard edge. In particular, the rigidity estimate at the hard edge is only known for a fixed $ \xi \equiv 1 $ but not for $ \xi = \xi(N) \to 1 $.) We order the eigenvalues of $ H $ as $ \lambda_1 {\leqslant}\cdots {\leqslant}\lambda_N $, and use $ \lambda_+ $ to denote the typical location of the largest eigenvalue (see for the definition). For the main result of this paper, we consider the Kolmogorov distance $${\mathsf{d_K}}(X,Y) := \sup_{x} \left| {\mathbb{P}}(X {\leqslant}x) - {\mathbb{P}}(Y {\leqslant}x) \right|.$$ \[t.Rate\] Let $ H_N $ be sample covariance matrices satisfying and . Let $ {\mathsf{TW}}$ be the Tracy-Widom distribution. For any $ {\varepsilon}>0 $, for large enough $ N $ we have $${\mathsf{d_K}}(N^{2/3}(\lambda_N - \lambda_+),{\mathsf{TW}}) {\leqslant}N^{-\frac{2}{9}+{\varepsilon}}.$$ The null case $ X^*X $ is our primary concern in this paper, but quantitative estimates are also valid for general diagonal population matrices $ X^* \Sigma X $ thanks to the comparison theorem for the Green function flow by Lee and Schnelli (see Section \[s.GeneralPopulation\] for more details). Combining our quantitative edge universality for the null case (Theorem \[t.Rate\]) with the Green function comparison by Lee-Schnelli (Proposition \[p.GreenFunctionFlow\]), we derive the rate of convergence to Tracy-Widom distribution for separable sample covariance matrices with general diagonal population. \[t.RateGeneralPopulation\] Let $ Q := X^* \Sigma X $ be an $ N \times N $ separable sample covariance matrix, where $ X $ is an $ M \times N $ real random matrix satisfying and , and $ \Sigma $ is a real diagonal $ M \times M $ matrix satisfying . Let $ \mu_N $ be the largest eigenvalue of $ Q $. For any $ {\varepsilon}>0 $, for large enough $ N $ we have $$\label{e.RateGeneralPopulation} {\mathsf{d_K}}\left( \gamma_0 N^{2/3} (\mu_N - E_+),{\mathsf{TW}}\right) {\leqslant}N^{-\frac{1}{57} + {\varepsilon}},$$ where $ E_+ $ defined in denotes the rightmost endpoint of the spectrum and $ \gamma_0 $ is a normalization constant defined in . The method of the paper follows the three-step strategy of the Erdös-Schlein-Yau dynamical approach [@erdHos2011universality]: (i) a priori bounds on locations of eigenvalues; (ii) local relaxation of the eigenvalue dynamics; (iii) a density argument showing eigenvalues statistics have not changed after short time. Specifically, in this paper, the three-step strategy is employed in the following way: (i) is the rigidity for singular values, which can be rephrased from classical results on eigenvalues (see [@bloemendal2014isotropic; @pillai2014universality]). For the particular square case $ M \equiv N $, we use a different rigidity estimate at the hard edge from [@alt2017local],which results in a slightly different proof; (ii) is the recent approach to the analysis of Dyson Brownian motion from [@bourgade2018extreme], which introduced an observable defined via interpolation with integrable models (see [@bourgade2016fixed; @landon2019fixed]). It describes the singular values evolution through a stochastic advection equation; (iii) is a quantitative version of the Green function comparison theorem, which is a slight extension of the classical result from [@GreenFunctionComparison]. As discussed in [@pillai2014universality Section 3], though we discuss the problems in the context of covariance matrices, the proof should also work for more generalized problems such as the quantitative edge universality of correlation matrices (see [@pillai2012edge]). This paper is organized as follows. The main part of the paper (Section \[s.prelim\]-\[s.Comparison\]) is devoted to the null case $ X^*X $. In Section \[s.prelim\] we rephrase the classical results on the eigenvalues of sample covariance matrices to the version for singular values, including Dyson Brownian motion, local laws and rigidity estimates. In Section \[s.LocalRelaxationFlow\], we define an observable that describes the evolution of singular values, and then prove the error estimate for the local relaxation flow at the edge by studying the dynamics of the observable. In Section \[s.Comparison\] we prove the quantitative Green function comparison theorem and use it to derive the rate of convergence to the Tracy-Widom distribution. Finally, in Section \[s.GeneralPopulation\] we generalize our result to separable sample covariance matrices with diagonal population by studying the interpolation between general covariance matrices with the null case. Notations --------- Throughout this paper, we use the notation $ A \lesssim B $ if there exists a constant $ C $ which is independent of $ N $ such that $ A {\leqslant}CB $ holds. We also denote $ A \sim B $ if both $ A \lesssim B $ and $ B \lesssim A $ hold. If $ A $ and $ B $ are complex valued, $ A \sim B $ means $ {\text{\rm Re}\hspace{0.1cm}}A \sim {\text{\rm Re}\hspace{0.1cm}}B $ and $ {\text{\rm Im}\hspace{0.1cm}}A \sim {\text{\rm Im}\hspace{0.1cm}}B $. We also denote $ C $ a generic constant which does not depend on $ N $ but may vary form line to line. We use $ \llbracket A,B \rrbracket := [A,B] \cap {\mathbb{Z}}$ to denote the set of integers between $ A $ and $ B $. We also denote $$\varphi = e^{C_0 (\log \log N)^2}$$ a subpolynomial error parameter, for some fixed $ C_0>0 $. This constant $ C_0 $ is chosen large enough so that the eigenvalues (and singular values) rigidity and the strong local Marchenko-Pastur law hold (see section \[s.LocalLaw\]). Acknowledgement {#acknowledgement .unnumbered} --------------- The author would like to thank Prof. Paul Bourgade for suggesting this problem, helpful discussions, and useful comments on the early draft of the paper. Preliminaries {#s.prelim} ============= Dyson Brownian motion for covariance matrices --------------------------------------------- Let $ B $ be an $ M \times N $ real matrix Brownian motion: $ B_{ij} $ are independent standard Brownian motions. We define the $ M \times N $ matrix $ M_t $ by $$M_t = M_0 + \dfrac{1}{\sqrt{N}}B_t.$$ The eigenvalues dynamics for the real Wishart process $ X_t := M_t^* M_t $ was first proved in [@bru1989diffusions]. Under our normalization convention, the equation is in the following form given in [@bourgade2017eigenvector Appendix C] $$\label{e.EigenvalueDBM} {\mathrm{d}}\lambda_k = 2\sqrt{\lambda_k}\dfrac{{\mathrm{d}}B_{kk}}{\sqrt{N}} + \left( \dfrac{M}{N} + \dfrac{1}{N} \sum_{l \neq k}\dfrac{\lambda_k + \lambda_l}{\lambda_k - \lambda_l} \right){\mathrm{d}}t.$$ Due to technical issues, it is difficult to use the coupling method from [@bourgade2016fixed] to analyze in a direct way. This motivates us to consider the singular values instead. Let $ s_k := \sqrt{\lambda_k} $ denote the singular values of $ X $. The Dyson Brownian motion for singular values dynamics of such sample covariance matrices is the following Ornstein-Uhlenbeck process [@erdHos2012local equation (5.8)]. $${\mathrm{d}}s_k = \dfrac{{\mathrm{d}}B_k}{\sqrt{N}} + \left[ -\dfrac{1}{2\xi}s_k + \dfrac{1}{2}\left( \dfrac{1}{\xi} -1 \right) \dfrac{1}{s_k} + \dfrac{1}{2N} \sum_{l \neq k} \left( \dfrac{1}{s_k - s_l} + \dfrac{1}{s_k + s_l} \right) \right]{\mathrm{d}}t, \ \ \ 1 {\leqslant}k {\leqslant}N.$$ An important idea in this paper is the following symmetrization trick (see [@che2019universality equation (3.9)]): $$s_{-i}(t)=-s_i(t),\ \ \ B_{-i}(t)=-B_i(t),\ \ \forall t {\geqslant}0,\ 1 {\leqslant}i {\leqslant}N.$$ From now we label the indices from $ -1 $ to $ -N $ and $ 1 $ to $ N $, so that the zero index is omitted. Unless otherwise stated, this will be the convention and we will not emphasize it explicitly. After symmetrization, the dynamics turns to the following form $$\label{e.SymmetrizedDynamics} {\mathrm{d}}s_k = \dfrac{{\mathrm{d}}B_k}{\sqrt{N}} + \left[ -\dfrac{1}{2\xi}s_k + \dfrac{1}{2}\left( \dfrac{1}{\xi} -1 \right) \dfrac{1}{s_k} + \dfrac{1}{2N} \sum_{l \neq \pm k} \dfrac{1}{s_k - s_l} \right]{\mathrm{d}}t, \ \ \ -N {\leqslant}k {\leqslant}N, k \neq 0.$$ Local law and rigidity for singular values {#s.LocalLaw} ------------------------------------------ The local law and rigidity estimates are classical results for the eigenvalues of sample covariance matrices. In this section, for later use, we rephrase these results into the corresponding version in terms of singular values. It is well known that the empirical measure of the eigenvalues converges to the Marchenko-Pastur distribution $$\rho_{{\mathsf{MP}}}(x) = \dfrac{1}{2\pi \xi}\sqrt{\dfrac{[(x-\lambda_-)(\lambda_+ -x)]_+}{x^2}},$$ where $$\label{e.EndPoints} \lambda_{\pm} = (1 \pm \sqrt{\xi})^2.$$ Define the typical locations of the singular values: $$\gamma_k := \inf \left\{ E>0 : \int_{-\infty}^{E^2} \rho_{{\mathsf{MP}}}(x){\mathrm{d}}x {\geqslant}\dfrac{k}{N} \right\},\ \ \ 1 {\leqslant}k {\leqslant}N.$$ Following the symmetrization trick, we also define $ \gamma_{-k} = -\gamma_k $. By a change of variable, it is easy to check that $$\label{e.typical} \int_{-\infty}^{\gamma_k} \rho(x){\mathrm{d}}x=\dfrac{N+k}{2N}, \ \ \ \int_{-\infty}^{\gamma_{-k}} \rho(x){\mathrm{d}}x = \dfrac{N-k}{2N},$$ where $ \rho(x) $ is the counterpart of Marchenko-Pastur law for singular values, defined by $$\label{e.measure} \rho(x)=\dfrac{1}{2\pi \xi}\sqrt{\dfrac{[(x^2-\lambda_-)(\lambda_+ - x^2)]_+}{x^2}},\ \ \ \sqrt{\lambda_-} {\leqslant}|x| {\leqslant}\sqrt{\lambda_+}.$$ Denote $ s_1 {\leqslant}\cdots {\leqslant}s_N $ the singular values of the data matrix $ X $, and extend the singular values following the symmetrization trick by $ s_{-k}=-s_k $. For $ z=E+{\text{\rm i}}\eta \in \mathbb{C} $ with $ \eta>0 $, let $ m_N(z) $ and $ S_N(z) $ denote the Stieltjes transform of the empirical measure of the (symmetrized) singular values and eigenvalues, respectively: $$m_N(z):= \dfrac{1}{2N}\sum_{-N {\leqslant}k {\leqslant}N} \dfrac{1}{s_k -z},\ \ \ S_N(z) := \dfrac{1}{N}\sum_{k=1}^N \dfrac{1}{\lambda_k -z}.$$ As mentioned previously, in the summation from $ -N $ to $ N $ the $ 0 $ index is always excluded. Note that due to the symmetrization, this is equivalent to $$\label{e.DiscreteStieltjes} m_N(z) = \dfrac{1}{2N}\sum_{k=1}^N \left( \dfrac{1}{s_k -z} + \dfrac{1}{-s_k - z} \right) = \dfrac{1}{N} \sum_{k=1}^N \dfrac{z}{s_k^2 - z^2} = z S_N(z^2).$$ On the other hand, use $ m_{{\mathsf{MP}}}(z) $ to denote the Stieltjes transform of the Marchenko-Pastur law $$m_{{\mathsf{MP}}}(z) := \int_{{\mathbb{R}}} \dfrac{\rho_{{\mathsf{MP}}}(x)}{x-z}{\mathrm{d}}x=\dfrac{1-\xi-z+\sqrt{(z-\lambda_-)(z-\lambda_+)}}{2\xi z},$$ where $ \sqrt{\quad} $ denotes the square root on the complex plane whose branch cut is the negative real line. With this choice we always have $ {\text{\rm Im}\hspace{0.1cm}}m_{{\mathsf{MP}}}(z)>0 $ when $ {\text{\rm Im}\hspace{0.1cm}}z>0 $. For the singular values, recall the limit distribution $ \rho(x) $ for the empirical measure and use $ m(z) $ to denote its corresponding Stieltjes transform $$m(z) := \int_{{\mathbb{R}}} \dfrac{\rho(x)}{x-z}{\mathrm{d}}x = \int_{{\mathbb{R}}} \dfrac{1}{x-z}\dfrac{1}{2\pi \xi}\sqrt{\dfrac{[(x^2-\lambda_-)(\lambda_+ - x^2)]_+}{x^2}}{\mathrm{d}}x.$$ We have the following relation between $ m(z) $ and $ m_{{\mathsf{MP}}}(z) $ $$\begin{gathered} \label{e.ContinuousStieltjes} m(z) = \int_{\sqrt{\lambda_-}}^{\sqrt{\lambda_+}} \left( \dfrac{1}{x-z} - \dfrac{1}{x+z} \right) \dfrac{1}{2\pi \xi}\sqrt{\dfrac{[(x^2-\lambda_-)(\lambda_+ - x^2)]_+}{x^2}}{\mathrm{d}}x\\ = \int_{{\mathbb{R}}}\dfrac{z}{x-z^2}\rho_{{\mathsf{MP}}}(x){\mathrm{d}}x = zm_{{\mathsf{MP}}}(z^2).\end{gathered}$$ It is well known that we have the strong local Marchenko-Pastur law [@bloemendal2014isotropic; @pillai2014universality] for the estimate of $ S_N(z) $, i.e. for any $ D>0 $, there exists $ N_0(D)>0 $ such that for every $ N {\geqslant}N_0 $ we have $${\mathbb{P}}\left( \left| S_N(z) - m_{{\mathsf{MP}}}(z) \right| {\leqslant}\dfrac{\varphi}{N\eta} \right) > 1-N^{-D}.$$ By the relations and , we know that $$|m_N(z) - m(z)| = \left| z \left( S_N(z^2) - m_{{\mathsf{MP}}}(z^2) \right) \right| {\leqslant}|z| \left| S_N(z^2) - m_{{\mathsf{MP}}}(z^2) \right|.$$ Combining with the strong local Marchenko-Pastur law, this gives us $$\label{e.LocalLaw} {\mathbb{P}}\left( \left| m_N(z) - m(z) \right| \lesssim \dfrac{\varphi}{N\eta} \right) > 1-N^{-D}.$$ {height="4cm"} For the rigidity estimates, a key observation is that the critical case $ \xi=1 $ is significantly different from other cases. This is because the Marchenko-Pastur law $ \rho_{{\mathsf{MP}}} $ has a singularity at the point $ x=0 $ in this situation. When $ \xi < 1 $, the rigidity of singular values can be easily obtained from the analogous estimates for eigenvalues (see [@pillai2014universality]). Let $ {\widehat}{k}:=\min(k,N+1-k) $, for any $ D>0 $ there exists $ N_0(D) $ such that the following holds for any $ N {\geqslant}N_0 $, $$\label{e.RigidityNotOne} {\mathbb{P}}\left(|s_k - \gamma_k| {\leqslant}\varphi^{\frac{1}{2}} N^{-\frac{2}{3}}({\widehat}{k})^{-\frac{1}{3}} \ \mbox{for all}\ k \in \llbracket 1,N \rrbracket \right) > 1-N^{-D}.$$ For the critical case $ \xi=1 $, now the Marchenko-Pastur distribution is supported on $ [0,4] $ and is given by $ \rho_{{\mathsf{MP}}}(x)= \frac{1}{2\pi}\sqrt{(4-x)/x} $. A key observation is that the scales of eigenvalue spacings are different at the two edges. Due to this phenomenon, we use the following two different results, depending on the location in the spectrum. On the one hand, the Marchenko-Pastur distribution still behaves like a square root near the soft edge $ x=4 $, which implies that the result is the same as the rectangular case. The rigidity estimate near the soft edge can be easily adapted from the result for eigenvalues in [@bloemendal2014isotropic Theorem 2.10], i.e. for some (small) $ \omega>0 $ and any $ {\varepsilon}>0 $ we have $$\label{e.RigidityOneSoft} {\mathbb{P}}\left( |s_k - \gamma_k| {\leqslant}N^{\varepsilon}(N-k+1)^{-\frac{1}{3}}N^{-\frac{2}{3}} \ \mbox{for all} \ k \in \llbracket (1-\omega)N,N \rrbracket \right) > 1-N^{-D}.$$ On the other hand, as explained in [@cacciapuoti2013local], at the hard edge $ x=0 $ the typical distance between eigenvalues and the edge is of order $ N^{-2} $, which is much smaller than the typical distance between neighbouring eigenvalues in the bulk (or at the soft edge). Note that in this situation, the measure for the symmetrized singular values coincides with the standard semicircle law, that is $ \rho(x) = \frac{1}{2\pi}\sqrt{(4-x^2)_+} $. By the relation , this means that the typical $ k $-th singular value $ \gamma_k $ of an $ N \times N $ data matrix shares the same position with the typical $ (N+k) $-th eigenvalue of a $ 2N \times 2N $ generalized Wigner matrix. The link between these two models can be illustrated by the symmetrization trick: Define the $ 2N \times 2N $ matrix $${\widetilde}{H} = \left( \begin{matrix} 0 & X^*\\ X & 0 \end{matrix} \right),$$ then we know that the eigenvalues of $ {\widetilde}{H} $ are precisely the symmetrized singular values of $ X^* $. Note that we have $ {\widetilde}{H}={\widetilde}{H}^* $, $ {\mathbb{E}}{\widetilde}{H}_{ij}=0 $ and $ \sum_{i=1}^{2N} {\mathbb{E}}{\widetilde}{H}_{ij}^2 = 1 $ for every $ j \in \llbracket 1,2N \rrbracket $. This shows that $ {\widetilde}{H} $ is indeed a Wigner-type matrix except the lack of nondegeneracy condition caused by the zero blocks. By considering the matrix of this type, the rigidity at the hard edge can be proved directly from [@alt2017local Theorem 2.7] $$\label{e.RigidityOneHard} {\mathbb{P}}\left( |s_k - \gamma_k| {\leqslant}N^{-1+{\varepsilon}} \ \mbox{for all}\ k \in \llbracket 1,(1-\omega)N \rrbracket \right) > 1-N^{-D}.$$ Stochastic Advection Equation for Singular Values Dynamics {#s.LocalRelaxationFlow} ========================================================== Stochastic advection equation ----------------------------- We follow the comparison method via coupling in [@bourgade2016fixed]. As in [@landon2019fixed], consider the interpolation between a general sample covariance matrix and the Wishart ensemble for the initial data: for any $ \nu \in [0,1] $, let $$x_k^{(\nu)}(0)=\nu s_k(0)+(1-\nu)r_k(0),\ \ \ -N {\leqslant}k {\leqslant}N,\ k \neq 0.$$ where $ s_k(t) $ and $ r_k(t) $ satisfy the singular values dynamics , with respective initial conditions a general sample covariance matrix and the Wishart ensemble. Define the corresponding dynamics of $ x_k^{(\nu)} $ to be $$\label{e.DBM} {\mathrm{d}}x_k^{(\nu)} = \dfrac{{\mathrm{d}}B_k}{\sqrt{N}} + \left[ -\dfrac{1}{2\xi}x_k^{(\nu)} + \dfrac{1}{2}\left( \dfrac{1}{\xi} -1 \right) \dfrac{1}{x_k^{(\nu)}} + \dfrac{1}{2N} \sum_{l \neq \pm k} \dfrac{1}{x_k^{(\nu)} - x_l^{(\nu)}} \right]{\mathrm{d}}t .$$ For this Dyson Brownian motion we consider the quantity $${\mathfrak{u}}_k^{(\nu)}(t):=e^{\frac{t}{2\xi}}\dfrac{{\mathrm{d}}}{{\mathrm{d}}\nu}x_k^{(\nu)}(t).$$ From now we set $ \nu \in (0,1) $ and omit it from the notation for simplicity. A significant property is that $ {\mathfrak{u}}_k $ satisfies a non-local parabolic differential equation $$\label{e.parabolic} \dfrac{{\mathrm{d}}}{{\mathrm{d}}t}{\mathfrak{u}}_{k}=\dfrac{1}{2}\left( 1- \dfrac{1}{\xi} \right)\dfrac{{\mathfrak{u}}_k}{x_k^2} + \dfrac{1}{2N} \sum_{l \neq \pm k}\dfrac{{\mathfrak{u}}_l - {\mathfrak{u}}_k}{\left(x_l - x_k \right)^2}.$$ Let $ {\mathfrak{v}}_k={\mathfrak{v}}_k^{(\nu)} $ solve the same equation as $ {\mathfrak{u}}_k $ in but with initial condition $ {\mathfrak{v}}_k(0)=|{\mathfrak{u}}_k(0)|=|s_k(0)-r_k(0)| $. An important result is that this equation yields a maximum principle for $ {\mathfrak{v}}_k $, which will be useful in later analysis for the estimate of its growth. \[l.MaxPrin\] For all $ t {\geqslant}0 $ and $ -N {\leqslant}k {\leqslant}N $, we have $${\mathfrak{v}}_k(t) {\geqslant}0, \ \ \ |{\mathfrak{v}}_k(t)| {\leqslant}\max_k |{\mathfrak{v}}_k(0)|,\ \ \ |{\mathfrak{u}}_k(t)| {\leqslant}{\mathfrak{v}}_k(t).$$ Note that for the coefficients in the summation part of equation we have $ \frac{1}{(x_l - x_k)^2} >0 $. Therefore for $ f(t) := \min_k {\mathfrak{v}}_k(t) $, we have $$f'(t) {\geqslant}\dfrac{1}{2} \left( 1-\dfrac{1}{\xi} \right) \dfrac{1}{x_1^2} f(t).$$ Combined with the fact $ {\mathfrak{v}}_k(0) {\geqslant}0 $, this gives us the first claim. For the second claim, since $ {\mathfrak{v}}_k $’s are nonnegative, we know that $ \frac{{\mathrm{d}}}{{\mathrm{d}}t}\max_{k} {\mathfrak{v}}_k {\leqslant}0 $ and this yields the desired result. The third claim follows from linearity. We know that both $ {\mathfrak{v}}+ {\mathfrak{u}}$ and $ {\mathfrak{v}}- {\mathfrak{u}}$ satisfy the equation , and we also have $ ({\mathfrak{v}}+ {\mathfrak{u}})(0) {\geqslant}0 $ and $ ({\mathfrak{v}}- {\mathfrak{u}})(0) {\geqslant}0 $. Similarly to the first claim, this gives us $ {\mathfrak{v}}_k(t) + {\mathfrak{u}}_k(t) {\geqslant}0 $ and $ {\mathfrak{v}}_k(t) - {\mathfrak{u}}_k(t) {\geqslant}0 $, which completes the proof. Due to our choice for the initial value of $ {\mathfrak{v}}$, we will have that $ \{{\mathfrak{v}}_k\}_{-N {\leqslant}k {\leqslant}N} $ are symmetric with respect to the label $ k $. To see this, it is easy to check that $ {\widetilde}{{\mathfrak{v}}}_k := {\mathfrak{v}}_{-k} $ satisfy the same equation . Note that the equation is linear and $ {\widetilde}{{\mathfrak{v}}}_k(0)={\mathfrak{v}}_k(0) $ for all $ k $. Lemma \[l.MaxPrin\] then gives us $ {\mathfrak{v}}_k(t) = {\widetilde}{{\mathfrak{v}}}_k(t) = {\mathfrak{v}}_{-k}(t) $ for all $ k $ and $ t {\geqslant}0 $. We now consider the observable $$f_t(z)=e^{-\frac{t}{2\xi}}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k(t)}{x_k(t)-z}.$$ A key observation is that the quadratic singularities in will disappear when combined with the Dyson Brownian motion, so that the evolution of $ f_t $ has no shocks similarly to a result from [@bourgade2018extreme]. Denoting $$s_t(z)=\dfrac{1}{2N}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{1}{x_k(t)-z}$$ the Stieltjes transform of the (symmetrized) empirical spectral measure, then the observable $ f_t $ satisfies the following dynamics. \[l.dynamics\] For any $ {\text{\rm Im}\hspace{0.1cm}}z \neq 0 $, we have $$\label{e.dynamics} \begin{aligned} {\mathrm{d}}f_t &=\left(s_t(z)+\dfrac{z}{2\xi}\right)(\partial_z f_t){\mathrm{d}}t+\dfrac{1}{4N}(\partial_{zz}f_t){\mathrm{d}}t+\left[\dfrac{e^{-\frac{t}{2\xi}}}{2N}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^2(x_k+z)}\right]{\mathrm{d}}t\\ & \quad + \left[\left( 1- \dfrac{1}{\xi} \right) e^{-\frac{t}{2\xi}} \left( \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{3z {\mathfrak{v}}_k}{2 x_k^2 (x_k -z)(x_k+z)} + \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{z^3 {\mathfrak{v}}_k}{x_k^2 (x_k-z)^2 (x_k+z)^2} \right)\right] {\mathrm{d}}t\\ & \quad -\dfrac{e^{-\frac{t}{2\xi}}}{\sqrt{N}}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^2}{\mathrm{d}}B_k. \end{aligned}$$ This can be proved by direct computation via the Itô’s formula. First, we have $${\mathrm{d}}f=-\dfrac{f}{2\xi}{\mathrm{d}}t+e^{-t/2}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathrm{d}}{\mathfrak{v}}_k}{x_k-z}+e^{-t/2}\sum_{-N {\leqslant}k {\leqslant}N}{\mathfrak{v}}_k {\mathrm{d}}\dfrac{1}{x_k-z} =: A_1 + A_2 +A_3.$$ By using Itô’s formula again we have $ {\mathrm{d}}(x_k-z)^{-1}=-(x_k-z)^{-2}{\mathrm{d}}x_k+\frac{1}{N}(x_k-z)^{-3}{\mathrm{d}}t $. Thus, we can now decompose the term $ A_3 $ as $ I_1+[I_2+I_3+I_4+I_5]{\mathrm{d}}t $, where $$\begin{aligned} I_1 &= -\dfrac{e^{-\frac{t}{2\xi}}}{\sqrt{N}}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^2}{\mathrm{d}}B_k, \\ I_2 &= \dfrac{e^{-\frac{t}{2\xi}}}{2\xi}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k x_k}{(x_k-z)^2}, \\ I_3 &= \dfrac{1}{2} \left( 1-\dfrac{1}{\xi} \right) e^{-\frac{t}{2\xi}} \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{{\mathfrak{v}}_k}{x_k(x_k-z)^2}, \\ I_4 &= e^{-\frac{t}{2\xi}}\sum_{-N {\leqslant}k {\leqslant}N}{\mathfrak{v}}_k\left(-\dfrac{1}{(x_k-z)^2}\right)\dfrac{1}{2N}\sum_{l \neq \pm k}\dfrac{1}{x_k-x_l}=\dfrac{e^{-\frac{t}{2\xi}}}{2N}\sum_{l \neq \pm k}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^2(x_l-x_k)}, \\ I_5 &= \dfrac{e^{-\frac{t}{2\xi}}}{N}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^3}.\end{aligned}$$ Note that $$\partial_z f=e^{-\frac{t}{2\xi}}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^2},\ \ \ \partial_{zz}f=2e^{-\frac{t}{2\xi}}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^3}.$$ For the term $ A_2 $, by the equation , we have $$A_2 = e^{-\frac{t}{2\xi}}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{1}{x_k-z} \left[ \dfrac{1}{2} \left( 1-\dfrac{1}{\xi} \right) \dfrac{{\mathfrak{v}}_k}{x_k^2} + \dfrac{1}{2N}\sum_{l \neq \pm k}\dfrac{{\mathfrak{v}}_l-{\mathfrak{v}}_k}{(x_k-x_l)^2} \right]=: B_1+B_2.$$ Note that $$\begin{gathered} B_2 = \dfrac{e^{-\frac{t}{2\xi}}}{4N}\sum_{l \neq \pm k}\dfrac{{\mathfrak{v}}_l-{\mathfrak{v}}_k}{(x_l-x_k)^2}\left(\dfrac{1}{x_k-z}-\dfrac{1}{x_l-z}\right)=\dfrac{e^{-\frac{t}{2\xi}}}{4N}\sum_{l \neq \pm k}\dfrac{{\mathfrak{v}}_l-{\mathfrak{v}}_k}{x_l-x_k}\dfrac{1}{x_k-z}\dfrac{1}{x_l-z}\\ =-\dfrac{e^{-\frac{t}{2\xi}}}{2N}\sum_{l \neq \pm k}\dfrac{{\mathfrak{v}}_k}{x_l-x_k}\dfrac{1}{x_k-z}\dfrac{1}{x_l-z}.\end{gathered}$$ Combining with $ I_4 $, we obtain $$B_2 + I_4 =\dfrac{e^{-\frac{t}{2\xi}}}{2N}\sum_{l \neq \pm k}\dfrac{{\mathfrak{v}}_k}{x_l-x_k}\dfrac{1}{x_k-z}\left(\dfrac{1}{x_k-z}-\dfrac{1}{x_l-z}\right)=\dfrac{e^{-\frac{t}{2\xi}}}{2N}\sum_{l \neq \pm k}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^2}\dfrac{1}{x_l-z}.$$ Moreover, we have that $$B_2+I_4+I_5 = s(z)\partial_z f+\dfrac{e^{-\frac{t}{2\xi}}}{2N}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^3}+\dfrac{e^{-\frac{t}{2\xi}}}{2N}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k}{(x_k-z)^2(x_k+z)}.$$ Then it suffices to calculate $ B_1 + I_3 $, and note that $$\begin{aligned} B_1 + I_3 &= \dfrac{1}{2} \left( 1- \dfrac{1}{\xi} \right) e^{-\frac{t}{2\xi}} \sum_{k=1}^N \left( \dfrac{{\mathfrak{v}}_k}{x_k -z}\dfrac{1}{x_k^2} - \dfrac{{\mathfrak{v}}_k}{x_k +z}\dfrac{1}{x_k^2} + \dfrac{{\mathfrak{v}}_k}{(x_k - z)^2}\dfrac{1}{x_k} - \dfrac{{\mathfrak{v}}_k}{(x_k +z)^2}\dfrac{1}{x_k} \right)\\ &= \left( 1- \dfrac{1}{\xi} \right) e^{-\frac{t}{2\xi}} \left( \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{3z {\mathfrak{v}}_k}{2 x_k^2 (x_k -z)(x_k+z)} + \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{z^3 {\mathfrak{v}}_k}{x_k^2 (x_k-z)^2 (x_k+z)^2} \right).\end{aligned}$$ Note that now all the singularities are removed. The desired result then follows by combining the previous results and the term $ I_1 $. Recall that in Section \[s.LocalLaw\] we have shown that the Stieltjes transform of the empirical measure for singular values satisfies the local law , so that the leading term of the stochastic differential equation satisfied by $ f_t $ is close to $$\dfrac{z}{2\xi} + z m_{{\mathsf{MP}}}(z^2) = \dfrac{z}{2\xi} + \dfrac{1-\xi-z^2 + \sqrt{(z^2 - \lambda_-)(z^2 - \lambda_+)}}{2\xi z} = \dfrac{(1-\xi) + \sqrt{(z^2 - \lambda_-)(z^2 - \lambda_+)}}{2\xi z}.$$ Thus, the dynamics of $ f_t $ can be approximated by the following advection equation $$\label{e.AdvectionPDE} \partial_t r = \dfrac{(1-\xi) + \sqrt{(z^2 - \lambda_-)(z^2 - \lambda_+)}}{2\xi z} \partial_z r.$$ Geometric properties of the characteristics {#s.geometry} ------------------------------------------- In order to estimate the evolution of the observable, we analyze its dynamics by studying the characteristics of the approximate advection PDE , similarly to [@huang2018rigidity; @bourgade2018extreme]. To do this, we first need some bounds on the shape of the characteristics $ (z_t)_{t {\geqslant}0} $, and some estimates for the initial value. As mentioned in Section \[s.LocalLaw\], discussions for the case $ \xi=1 $ are expected to be different due to the singularity of the Marchenko-Pastur distribution (which results in a distinct shape of the density $ \rho(x) $ for singular values). Therefore, in this and the subsequent section, we first discuss the case $ \xi \neq 1 $ and will show how to adapt the proof to the case for square data matrices in Section \[s.CaseOne\]. Denote $$\kappa(z) := \min\left\{\left|z-\sqrt{\lambda_-}\right|,\left|z-\sqrt{\lambda_+}\right|\right\},$$ and $$a(z) := \operatorname{dist}\left(z,\left[\sqrt{\lambda_-},\sqrt{\lambda_+}\right]\right),\ \ \ b(z):= \operatorname{dist}\left( z, \left[\sqrt{\lambda_-},\sqrt{\lambda_+}\right]^c \right)$$ We consider the curve $${\mathscr{S}}:= \left\{ z=E+{\text{\rm i}}y: \sqrt{\lambda_-} + \varphi^2 N^{-2/3} <E< \sqrt{\lambda_+} - \varphi^2 N^{-2/3},\ y=\varphi^2/\left( N \kappa(E)^{1/2} \right) \right\},$$ and the domain $ \mathscr{R} := \cup_{0<t<1}\{z_t:z \in {\mathscr{S}}\} $. \[l.characteristics\] Uniformly in $ 0<t<1 $ and $ z=z_0 $ satisfying $ \eta := {\text{\rm Im}\hspace{0.1cm}}z>0 $ and $ |z-\sqrt{\lambda_+}|<\sqrt{\xi}/10 $, we have $${\text{\rm Re}\hspace{0.1cm}}(z_t -z_0) \sim t \dfrac{a(z)}{\kappa(z)^{1/2}}+t^2,\ \ \ \ {\text{\rm Im}\hspace{0.1cm}}(z_t-z_0) \sim t\dfrac{b(z)}{\kappa(z)^{1/2}}.$$ In particular, if in addition we have $ z \in {\mathscr{S}}$, then $$(z_t - z_0) \sim \left( t \dfrac{\varphi^2}{N \kappa(E)} +t^2 \right) + {\text{\rm i}}\kappa(E)^\frac{1}{2} t.$$ Moreover, for any $ \kappa >0 $, uniformly in $ 0<t<1 $ and $ z=E+{\text{\rm i}}\eta \in [\sqrt{\lambda_-} + \kappa,\sqrt{\lambda_+} -\kappa] \times [0,\kappa^{-1}] $, we have $ {\text{\rm Im}\hspace{0.1cm}}(z_t-z_0) \sim t $. {height="4cm"} It is too complicated to work with the ODE satisfied by $ z_t $ in a direct way. The main idea is to compare this characteristics with the corresponding curve for a semicircle distribution (which has a explicit and simple formula). Define the two functions $$g(z):=\dfrac{(1-\xi) + \sqrt{(z^2 - \lambda_-)(z^2 - \lambda_+)}}{2\xi z} , \ \ \ g_{sc}(z):=\dfrac{\sqrt{(z-1)^2 - \xi}}{\xi}.$$ For some $ |z_0-\sqrt{\lambda_+}| < \sqrt{\xi}/10,\ {\text{\rm Im}\hspace{0.1cm}}z_0>0 $, let $ z(t) := z_t $ and $ z_{sc}(t) $ solve the following two initial value problems $$\left\{ \begin{aligned} & \dfrac{{\mathrm{d}}z}{{\mathrm{d}}t} = g(z)\\ & z(0)=z_0 \end{aligned} \right. ,\ \ \ \ \ \ \ \left\{ \begin{aligned} & \dfrac{{\mathrm{d}}z_{sc}}{{\mathrm{d}}t} = g_{sc}(z_{sc})\\ & z_{sc}(0)=z_0 \end{aligned} \right. .$$ Note that in $ \Omega=\{z_{sc}(t):0<t<1,|z_0-\sqrt{\lambda_+}| < \sqrt{\xi}/10,{\text{\rm Im}\hspace{0.1cm}}z_0>0 \} $ we have $ g(z) \sim g_{sc}(z) $. This shows that for $ 0<t<1 $, we have $ (z_t-z_0) \sim (z_{sc}(t) -z_0) $. The rest of the proof now follows from [@bourgade2018extreme Lemma 2.2]. Furthermore, we have the following lemma regarding the growth of the characteristics, which will be useful for the error estimates in the local relaxation. \[l.ChracteristcsInt\] For any $ z=E+{\text{\rm i}}\eta \in {\mathscr{S}}$, we have $$\dfrac{\varphi^4}{N^2} \int_0^t ds \int \dfrac{d \rho(x)}{|z_{t-s} -x|^4 \max(\kappa(x),s^2)} \lesssim \dfrac{\kappa(E)}{\max(\kappa(E),t^2)}.$$ The proof is essentially the same as in [@bourgade2018extreme Lemma A.2], except now we need to consider the Stieltjes transform $ m(z) $ instead of the one for the semicircle law. Recall the relation $ m(z)=z m_{{\mathsf{MP}}}(z^2) $, and note that for the Stieltjes transform of the Marchenko-Pastur law we have the following results from [@bloemendal2016principal Lemma 3.6]: $$|m_{{\mathsf{MP}}}(z)| \sim 1, \ \ \ \ \ {\text{\rm Im}\hspace{0.1cm}}m_{{\mathsf{MP}}}(z) \sim \left\{ \begin{aligned} & \sqrt{\kappa(E)+\eta} & \mbox{if } & E \in [\lambda_-,\lambda_+],\\ &\frac{\eta}{\sqrt{\kappa(E)+\eta}} & \mbox{if } & E \notin [\lambda_-,\lambda_+]. \end{aligned} \right.$$ The rest of the proof follows from the calculation in [@bourgade2018extreme Lemma A.2]. We now consider the initial value $ f_0 $ on the curve $ {\mathscr{S}}$. For this purpose we define the set of good trajectories such that the rigidity holds: $${\mathscr{A}}:= \left\{ |x_k^{(\nu)}(t) - \gamma_k|<\varphi^{\frac{1}{2}}N^{-\frac{2}{3}}({\widehat}{k})^{-\frac{1}{3}} \ \mbox{for all}\ 0 {\leqslant}t {\leqslant}1, k \in \llbracket 1,N \rrbracket,0 {\leqslant}\nu {\leqslant}1 \right\}.$$ We have the following important estimate for the probability of these events. \[l.rigidity\] There exists a fixed $ C_0 $ large enough such that the following holds. For any $ D>0 $, there exists $ N_0(D)>0 $ such that for any $ N > N_0 $ we have $${\mathbb{P}}({\mathscr{A}}) > 1-N^{-D}.$$ As mentioned in the last section, the rigidity estimates are proved in [@pillai2014universality] for fixed $ t $ and $ \nu=0,1 $. The extension to all $ t $ and $ \nu $ is based on the arguments in [@bourgade2018extreme; @erdHos2015gap]: (1) discretize in $ t $ and $ \nu $; (2) use Weyl’s inequality to bound the increments in small time intervals; (3) use the maximum principle to bound the increment in small $ \nu $-intervals. Conditioned on the rigidity phenomenon, we have the following estimate for the initial conditions. The proof is the same as [@bourgade2018extreme Lemma 2.4]. \[l.InitialValue\] In the set $ {\mathscr{A}}$, for any $ z=E+{\text{\rm i}}\eta \in \mathscr{R} $, we have $ {\text{\rm Im}\hspace{0.1cm}}f_0(z) \lesssim \varphi^{1/2} $ if $ \eta > \max(E-\sqrt{\lambda_+},-E+\sqrt{\lambda_-}) $, and $ {\text{\rm Im}\hspace{0.1cm}}f_0(z) \lesssim \varphi^{1/2} \frac{\eta}{\kappa(z)} $ otherwise. The same bound also holds for $ |{\text{\rm Im}\hspace{0.1cm}}f_0| $. Quantitative relaxation at the edge {#s.relaxation} ----------------------------------- To prove the edge universality, we first have the following estimate for the size of the observable $ f_t $. \[p.estimate\] For any (large) $ D>0 $ there exists $ N_0(D) $ such that for any $ N {\geqslant}N_0 $ we have $${\mathbb{P}}\left( {\text{\rm Im}\hspace{0.1cm}}f_t(z) \lesssim \varphi \dfrac{\kappa(E)^{1/2}}{\max(\kappa(E)^{1/2},t)} \ \mbox{\rm for all} \ 0<t<1 \ \mbox{\rm and} \ z=E+{\text{\rm i}}\eta \in {\mathscr{S}}\right) > 1-N^{-D}$$ For any $ 1 {\leqslant}l,m {\leqslant}N^{10} $, we define $ t_l:=l N^{-10} $ and $$z^{(m)} := E_m + {\text{\rm i}}\eta_m=E_m+{\text{\rm i}}\dfrac{\varphi^2}{N \kappa(E_m)^{1/2}},$$ where $$E_m := \inf\left\{ E >0 : \int_{-\infty}^{E^2} \rho_{{\mathsf{MP}}}(x){\mathrm{d}}x {\geqslant}\left(m-\frac{1}{2}\right)N^{-10} \right\}.$$ We also define the following stopping times (with respect to $ \mathcal{F}_t = \sigma(B_k(s):0 {\leqslant}s {\leqslant}t,1 {\leqslant}k {\leqslant}N) $) which represent the bad events: $$\tau_{l,m} := \inf \left\{ 0 {\leqslant}s {\leqslant}t_l : {\text{\rm Im}\hspace{0.1cm}}f_s(z_{t-s}^{(m)}) \gtrsim \dfrac{\varphi}{2}\dfrac{\kappa(E_m)^{1/2}}{\max(\kappa(E_m)^{1/2},t_l)} \right\},$$ $$\tau_0 := \inf \left\{ 0 {\leqslant}t {\leqslant}1: \exists k \in \llbracket -N,N \rrbracket \ \mbox{s.t.} \ |x_k(t)-\gamma_k|>\varphi^{\frac{1}{2}}N^{-\frac{2}{3}}({\widehat}{k})^{-\frac{1}{3}} \right\},$$ $$\tau := \min \left\{ \tau_0,\tau_{l,m}: 0 {\leqslant}l,m {\leqslant}N^{10}, \kappa(E_m) > \varphi^2 N^{-\frac{2}{3}} \right\}.$$ We also define the convention $ \inf \emptyset =1 $. We claim that in order to prove the desired result, it suffices to show that for any $ D>0 $ there exists $ {\widetilde}{N}_0(D) $ such that for any $ N {\geqslant}{\widetilde}{N}_0(D) $, we have $$\label{e.ProbabilityTau} {\mathbb{P}}(\tau=1) > 1-N^{-D}.$$ *Step 1.* To see the claim would be enough, we first show the following sets inclusion $$\label{e.SetsInclusion} \{ \tau=1 \} \bigcap_{\substack{1{\leqslant}l,m {\leqslant}N^{10} \\ -N {\leqslant}k {\leqslant}N}} A_{l,m,k} {\subseteq}\bigcap_{z \in {\mathscr{S}}, 0<t<1} \left\{ {\text{\rm Im}\hspace{0.1cm}}f_t(z) \lesssim \varphi \dfrac{\kappa(E)^{1/2}}{\max(\kappa(E)^{1/2},t)} \right\},$$ where $$A_{l,m,k} := \left\{ \sup_{t_l {\leqslant}u {\leqslant}t_{l+1}} \left| \int_{t_l}^u \dfrac{e^{-\frac{s}{2\xi}} {\mathfrak{v}}_k(s) {\mathrm{d}}B_k(s)}{(z^{(m)} - x_k(s))^2} \right| <N^{-3} \right\}.$$ To prove this, for any given $ z $ and $ t $, choose $ t_l $ and $ z^{(m)} $ such that $ t_l {\leqslant}t <t_{l+1} $ and $ |z-z^{(m)}|<N^{-5} $. Note that by rigidity and the maximum principle (Lemma \[l.MaxPrin\]) we have $ |{\mathfrak{v}}_k(t)| \lesssim \varphi N^{-2/3} $. Combining this with the definition of $ f_t $, we have $ |f_t(z) - f_t(z^{(m)})| < N^{-2} $. Moreover, note that we have the following estimates $$\begin{gathered} \sup_{t_l {\leqslant}u {\leqslant}t_{l+1}} \left| \int_{t_l}^u \dfrac{e^{-\frac{t}{2\xi}}}{2N}\sum_{-N {\leqslant}k {\leqslant}N} \dfrac{{\mathfrak{v}}_k}{(x_k-z)^2(x_k+z)} {\mathrm{d}}t \right|\\ \lesssim N^{-10} \dfrac{1}{2N} \varphi N^{-2/3} \max_{t_l {\leqslant}t {\leqslant}t_{l+1}} \sum_{-N {\leqslant}k {\leqslant}N} \left| \dfrac{1}{(x_k-z)^2(x_k+z)} \right| < N^{-5},\end{gathered}$$ and $$\begin{gathered} \sup_{t_l {\leqslant}u {\leqslant}t_{l+1}} \left| \int_{t_l}^u \left( 1 - \dfrac{1}{\xi} \right) e^{-\frac{t}{2\xi}} \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{z {\mathfrak{v}}_k}{x_k^2(x_k -z) (x_k+z)} {\mathrm{d}}t \right|\\ \lesssim N^{-10} \varphi N^{-2/3} \max_{t_l {\leqslant}t {\leqslant}t_{l+1}} \sum_{-N {\leqslant}k {\leqslant}N} \left| \dfrac{1}{(x_k -z) (x_k+z)} \right| < N^{-5},\end{gathered}$$ $$\begin{gathered} \sup_{t_l {\leqslant}u {\leqslant}t_{l+1}} \left| \int_{t_l}^u \left( 1 - \dfrac{1}{\xi} \right) e^{-\frac{t}{2\xi}} \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{z^3 {\mathfrak{v}}_k}{ x_k^2 (x_k -z)^2 (x_k+z)^2} {\mathrm{d}}t \right|\\ \lesssim N^{-10} \varphi N^{-2/3} \max_{t_l {\leqslant}t {\leqslant}t_{l+1}} \sum_{-N {\leqslant}k {\leqslant}N} \left| \dfrac{1}{(x_k -z)^2 (x_k+z)^2} \right| < N^{-5}.\end{gathered}$$ Similarly, we also have $$\sup_{t_l {\leqslant}u {\leqslant}t_{l+1}} \left| \int_{t_l}^u \left(s_t(z)+\dfrac{z}{2\xi}\right)(\partial_z f_t) {\mathrm{d}}t \right| < N^{-5},\ \ \ \ \sup_{t_l {\leqslant}u {\leqslant}t_{l+1}} \left| \int_{t_l}^u \dfrac{1}{4N}(\partial_{zz}f_t) {\mathrm{d}}t \right| < N^{-5}.$$ Then based on the dynamics , under the event $ \cap_k A_{l,m,k} $, we have $ |f_t(z^{(m)}) - f_{t_l}(z^{(m)})| < N^{-2} $. This completes the proof for . *Step 2.* To prove the claim, we also need to estimate the probability for the events $ A_{l,m,k} $. To do this, we use the following Burkholder-Davis-Gundy type inequality (see e.g. [@shorack2009empirical Appendix B.6]). For some fixed constants $ c>0 $ and $ \alpha>0 $, for any martingale $ M $ we have $$\label{e.BDG} {\mathbb{P}}\left( \sup_{0 {\leqslant}u {\leqslant}t} |M_u| {\geqslant}\alpha {\langle}M {\rangle}_t^{\frac{1}{2}} \right) {\leqslant}e^{-c \alpha^2}.$$ Note that we have the deterministic bound $ \int_{t_l}^u \frac{|{\mathfrak{v}}_k (s)|^2 {\mathrm{d}}s}{|z^{(m)} - x_k(s)|^4} \ll N^{-6} $. By taking $ \alpha = \varphi^{1/10} $, this implies $ {\mathbb{P}}(A_{l,m,k}) {\geqslant}1 - e^{-c \varphi^{1/5}} $, and then a union bound yields $${\mathbb{P}}\left( \bigcap_{1{\leqslant}l,m {\leqslant}N^{10}, -N {\leqslant}k {\leqslant}N} A_{l,m,k} \right) > 1-N^{-D}.$$ Together with the sets inclusion , this concludes that implies the desired result. *Step 3.* It remains to prove . For simplicity of notations, let $ t=t_l $, $ z=E+{\text{\rm i}}\eta=z^{(m)} $ for some arbitrary fixed $ 1 {\leqslant}l,m {\leqslant}N^{10} $. Consider the function $ g_u(z) := f_u(z_{t-u}) $. By Lemma \[l.characteristics\] and Lemma \[l.InitialValue\], we have $ {\text{\rm Im}\hspace{0.1cm}}g_0(z) \lesssim \frac{\varphi}{10}\frac{\kappa(E_m)^{1/2}}{\max(\kappa(E_m)^{1/2},t)} $. Therefore we only need to bound the increments of $ g $. Using Lemma \[l.dynamics\], by the Itô’s formula we know it satisfies the following stochastic differential equation $$d g_{u \wedge \tau}(z) = {\varepsilon}_u(z_{t-u}){\mathrm{d}}(u \wedge \tau) - \dfrac{e^{-\frac{u}{2\xi}}}{\sqrt{N}}\sum_{-N {\leqslant}k {\leqslant}N}\dfrac{{\mathfrak{v}}_k(u)}{(z_{t-u}-x_k(u))^2}{\mathrm{d}}B_k(u \wedge \tau),$$ where $$\begin{aligned} {\varepsilon}_u(z) &:= (s_u(z)-m(z))\partial_z f_u + \dfrac{1}{4N} (\partial_{zz} f_u) + \dfrac{e^{-\frac{u}{2\xi}}}{2N} \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{{\mathfrak{v}}_k(u)}{(x_k -z)^2 (x_k+z)}\\ &\quad + \left( 1- \dfrac{1}{\xi} \right) e^{-\frac{t}{2\xi}} \left( \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{3z {\mathfrak{v}}_k}{2 x_k^2 (x_k -z)(x_k+z)} + \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{z^3 {\mathfrak{v}}_k}{x_k^2 (x_k-z)^2 (x_k+z)^2} \right).\end{aligned}$$ In this step, we aim to estimate the first term $ \sup_{0 {\leqslant}s {\leqslant}t}|\int_0^s {\varepsilon}_u(z_{t-u}) {\mathrm{d}}(u \wedge \tau)| $. First, we have $$\begin{gathered} \int_0^t \left| \left( s_u(z_{t-u}) -m(z_{t-u}) \right) \partial_z f (z_{t-u}) \right|{\mathrm{d}}(u \wedge \tau)\\ \lesssim \int_0^t \dfrac{\varphi}{N {\text{\rm Im}\hspace{0.1cm}}(z_{t-u})} \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{|{\mathfrak{v}}_k(u)|}{|z_{t-u} -x_k(u)|^2} {\mathrm{d}}(u \wedge \tau) \lesssim \int_0^t \dfrac{\varphi {\text{\rm Im}\hspace{0.1cm}}f_u(z_{t-u})}{N \left( {\text{\rm Im}\hspace{0.1cm}}(z_{t-u}) \right)^2}{\mathrm{d}}(u \wedge \tau) \\ \lesssim \int_0^t \dfrac{\varphi^2 du}{N (\eta + (t-u)\kappa(z)^{1/2})^2}\dfrac{\kappa(E)^{1/2}}{\max(\kappa(E)^{1/2},t)} \lesssim \dfrac{\kappa(E)^{1/2}}{\max \left( \kappa(E)^{1/2},t \right)}.\end{gathered}$$ To bound the $ |s_u - m| $ term above, we have used the local law for singular values simultaneously for all $ 0 {\leqslant}u {\leqslant}t $ (which is similar to Lemma \[l.rigidity\]). The last two inequalities follow from Lemma \[l.characteristics\]. We also have $$\sup_{0 {\leqslant}s {\leqslant}t} \left| \int_0^s \dfrac{1}{4N}(\partial_{zz} f_u(z_{t-u})) {\mathrm{d}}(u \wedge \tau) \right| \lesssim \int_0^t \dfrac{{\text{\rm Im}\hspace{0.1cm}}f_u(z_{t-u})}{N \left( {\text{\rm Im}\hspace{0.1cm}}(z_{t-u}) \right)^2}{\mathrm{d}}(u \wedge \tau) \lesssim \dfrac{\kappa(E)^{1/2}}{\varphi \max(\kappa(E)^{1/2},t)},$$ $$\begin{gathered} \sup_{0 {\leqslant}s {\leqslant}t} \left| \int_0^s \dfrac{e^{-\frac{u}{2\xi}}}{2N} \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{{\mathfrak{v}}_k(u)}{(x_k - z_{t-u})^2(x_k+z_{t-u})} {\mathrm{d}}(u \wedge \tau) \right|\\ \lesssim \int_0^t \dfrac{{\text{\rm Im}\hspace{0.1cm}}f_u(z_{t-u})}{N \left( {\text{\rm Im}\hspace{0.1cm}}(z_{t-u}) \right)}{\mathrm{d}}(u \wedge \tau) \lesssim \dfrac{\kappa(E)^{1/2}}{\varphi \max(\kappa(E)^{1/2},t)},\end{gathered}$$ And similarly, $$\begin{gathered} \sup_{0 {\leqslant}s {\leqslant}t} \left| \left( 1-\dfrac{1}{\xi} \right)e^{-\frac{u}{2\xi}} \int_0^s \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{z_{t-u} {\mathfrak{v}}_k(u)}{x_k^2(x_k(u) - z_{t-u}) (x_k(u) + z_{t-u})} {\mathrm{d}}(u \wedge \tau) \right|\\ \lesssim \int_0^t {\text{\rm Im}\hspace{0.1cm}}f_u(z_{t-u}) {\mathrm{d}}(u \wedge \tau) \lesssim \dfrac{\varphi}{2}\dfrac{\kappa(E)^{1/2}}{\max \left( \kappa(E)^{1/2},t \right)},\end{gathered}$$ $$\begin{gathered} \sup_{0 {\leqslant}s {\leqslant}t} \left| \left( 1-\dfrac{1}{\xi} \right)e^{-\frac{u}{2\xi}} \int_0^s \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{z_{t-u}^3 {\mathfrak{v}}_k(u)}{x_k^2(x_k(u) - z_{t-u})^2 (x_k(u) + z_{t-u})^2} {\mathrm{d}}(u \wedge \tau) \right|\\ \lesssim \dfrac{\varphi}{2}\dfrac{\kappa(E)^{1/2}}{\max \left( \kappa(E)^{1/2},t \right)}.\end{gathered}$$ *Step 4.* Finally we focus on the estimate for $ \sup_{0 {\leqslant}s {\leqslant}t}|M_s| $ where $$M_s := \int_0^s \dfrac{e^{-\frac{u}{2\xi}}}{\sqrt{N}} \sum_{-N {\leqslant}k {\leqslant}N} \dfrac{{\mathfrak{v}}_k(u)}{(z_{t-u} - x_k(u))^2} {\mathrm{d}}B_k(u \wedge \tau).$$ Note that for all $ k $ and $ u < \tau $ we have $ |z_{t-u} - \gamma_k| \lesssim |z_{t-u} - x_k(u)| $ due to the fact $ |x_k(u) - \gamma_k| \ll |z_{t-u} - \gamma_k| $. Using again we have $$\sup_{0 {\leqslant}s {\leqslant}t}|M_s|^2 \lesssim \varphi^{\frac{1}{10}} \int_0^t \dfrac{1}{N}\sum_{-N {\leqslant}k {\leqslant}N} \dfrac{{\mathfrak{v}}_k(u)^2}{|z_{t-u} - \gamma_k|^4} {\mathrm{d}}(u \wedge \tau)$$ with overwhelming probability. By a similar argument in [@bourgade2018extreme equation (2.17)] and Lemma \[l.ChracteristcsInt\], we conclude $$\sup_{0 {\leqslant}s {\leqslant}t} |M_s|^2 \lesssim \varphi^{1/5} \dfrac{\kappa(E)}{\max(\kappa(E),t^2)}.$$ Hence, based on the previous estimates and a union bound we have proved that for any $ D>0 $ there exists $ N_0 $ such that for every $ N > N_0 $ we have $${\mathbb{P}}\left( \sup_{0 {\leqslant}l,m {\leqslant}N^{10},\kappa(E_m)>\varphi^2 N^{-2/3},0 {\leqslant}s {\leqslant}t_l} {\text{\rm Im}\hspace{0.1cm}}f_{s \wedge \tau}(z_{t_l - s \wedge \tau}^{(m)}) \lesssim \dfrac{\varphi}{2}\dfrac{\kappa(E_m)^{1/2}}{\max(\kappa(E_m)^{1/2},t_l)} \right) {\geqslant}1- N^{-D}.$$ Together with Lemma \[l.InitialValue\], we have proved , which then completes the proof. Based on the previous estimate on $ {\text{\rm Im}\hspace{0.1cm}}f_t $, we now state the quantitative relaxation of the singular values dynamics at the edge. Remember that $ \{s_k\} $ and $ \{r_k\} $ satisfy the same equation , with respective initial conditions that correspond to a general sample covariance matrix and the Wishart ensemble. \[t.Relaxation\] For any $ D>0 $ and $ {\varepsilon}>0 $ there exists $ N_0>0 $ such that for any $ N > N_0 $ we have $${\mathbb{P}}\left( |s_k(t) - r_k(t)| \lesssim \dfrac{N^{\varepsilon}}{Nt} \ \mbox{for all} \ k \in \llbracket 1,N \rrbracket \ \mbox{in} \ t \in [0,1] \right) > 1-N^{-D}.$$ Let $ z=\gamma_k + {\text{\rm i}}\frac{\varphi^2}{N \sqrt{\kappa(\gamma_k)}} \in {\mathscr{S}}$, then conditioned on the rigidity phenomenon $ {\mathscr{A}}$, by the nonnegativity of $ {\mathfrak{v}}_k $’s shown in Lemma \[l.MaxPrin\] we have $$|{\mathfrak{v}}_k(t)| \lesssim \dfrac{\varphi^2}{N \sqrt{\kappa(\gamma_k)}} {\text{\rm Im}\hspace{0.1cm}}f_t(z).$$ By the arguments in [@bourgade2018extreme Corollary 2.7], based on Proposition \[p.estimate\] we conclude $${\mathbb{P}}\left( |{\mathfrak{v}}_k^{(\nu)}(t)| \lesssim \dfrac{\varphi^{10}}{N} \dfrac{1}{\max(({\widehat}{k}/N)^{1/3},t)} \ \mbox{for all} \ k \in \llbracket 1,N \rrbracket \ \mbox{in} \ t \in [0,1] \right) > 1-N^{-D}.$$ Again by Lemma \[l.MaxPrin\], we know $ -{\mathfrak{v}}_k {\leqslant}{\mathfrak{u}}_k {\leqslant}{\mathfrak{v}}_k $. Therefore we have $${\mathbb{P}}\left( |{\mathfrak{u}}_k^{(\nu)}(t)| \lesssim \dfrac{\varphi^{10}}{N} \dfrac{1}{\max(({\widehat}{k}/N)^{1/3},t)} \ \mbox{for all} \ k \in \llbracket 1,N \rrbracket \ \mbox{in} \ t \in [0,1] \right) > 1-N^{-D}.$$ The result then follows as the proof in [@bourgade2018extreme Theorem 2.8]. Proof for $ \xi=1 $ {#s.CaseOne} ------------------- Due to the fact that $ \rho(x) $ is the semicircle law in this special case, now the advection equation is the same as [@bourgade2018extreme equation (1.12)], whose characteristics has an explicit formula. This coincidence makes it easy to adapt the previous proofs for $ \xi \neq 1 $ to this case. With a little abuse of notations, define the curve $${\mathscr{S}}:= \left\{ z=E+{\text{\rm i}}y: 0 <E< 2 - \varphi^2 N^{-2/3},\ y=\varphi^2/\left( N \kappa(E)^{1/2} \right) \right\},$$ where $ \kappa(z) := |z-2| $. Under the framework of such notations and the rigidity estimates and , by the arguments in [@bourgade2018extreme Section 2], all previous results in Section \[s.geometry\] still hold for $ \xi=1 $. Then using the same method we can prove Proposition \[p.estimate\] with few changes, and consequently the proof for Theorem \[t.Relaxation\] is completed. Rate of Convergence to Tracy-Widom Law {#s.Comparison} ====================================== Quantitative Green function comparison -------------------------------------- Following the general three-step strategy in the dynamical approach, the derivation of the rate of convergence relies on both the relaxation and the Green function comparison theorem from [@GreenFunctionComparison]. In the context of sample covariance matrices, this Lindeberg exchange strategy based on the fourth moments matching condition was first used by Tao and Vu in [@TaoVu]. To obtain an explicit convergence rate, we need a quantitative version of the comparison theorem. For the statement, we consider a fixed $ |E-\lambda_+| < \varphi N^{-2/3} $, a scale $ \rho=\rho(N) \in [N^{-1},N^{-2/3}] $, and a function $ f=f(N):{\mathbb{R}}\to {\mathbb{R}}$ satisfying $$\|f^{(k)} \|_{L^\infty([E,E+\rho])} {\leqslant}C_k \rho^{-k},\ \ \ \|f^{(k)} \|_{L^{\infty}([E^+,E^++1])} = O(1),\ \ \ 0 {\leqslant}k {\leqslant}2.$$ where $ E^+ = E+\varphi N^{-2/3} $. We assume that $ f $ is non-decreasing on $ (-\infty,E^+] $, $ f(x) \equiv 0 $ for $ x<E $ and $ f(x) \equiv 1 $ for $ E+\rho<x{\leqslant}E^+ $; and also assume $ f $ is non-increasing on $ [E^+,\infty) $, $ f \equiv 0 $ for $ x>E^+ +1 $. Furthermore, let $ F $ be a fixed smooth non-increasing function such that $ F(x) \equiv 1 $ for $ x {\leqslant}0 $ and $ F(x) \equiv 0 $ for $ x {\geqslant}1 $. \[t.GreenCompare\] There exists $ C>0 $ such that the following holds. Let $ {X^{\mathrm{v}}}, {X^{\mathrm{w}}}$ be data matrices satisfying assumptions and , and $ {H^{\mathrm{v}}},{H^{\mathrm{w}}}$ be the corresponding sample covariance matrices. Assume that the first three moments of the entries are the same, i.e. for all $ 1 {\leqslant}i {\leqslant}M $, $ 1 {\leqslant}j {\leqslant}N $ and $ 1 {\leqslant}k {\leqslant}3 $ we have $${{\mathbb{E}}^{\mathrm{v}}}(x_{ij}^k) = {{\mathbb{E}}^{\mathrm{w}}}(x_{ij}^k).$$ Assume also that for some parameter $ t=t(N) $ we have $$\left| {{\mathbb{E}}^{\mathrm{v}}}(\sqrt{M}x_{ij})^4 - {{\mathbb{E}}^{\mathrm{w}}}(\sqrt{M}x_{ij})^4 \right| {\leqslant}t.$$ With the above notations for the test functions $ f $ and $ F $, we have $$\left| ({{\mathbb{E}}^{\mathrm{v}}}- {{\mathbb{E}}^{\mathrm{w}}}) F \left( \operatorname{Tr}f(H) \right) \right| {\leqslant}\varphi^C \left( \dfrac{1}{N^{18} \rho^{20}} + \dfrac{t}{N \rho} + \dfrac{1}{(N \rho)^2} + \dfrac{1}{N^2} \right).$$ We follow the notations in [@pillai2014universality] and the reasoning from [@erdos2017dynamical Theorem 17.4]. Fix a bijective ordering map on the index set of the independent matrix elements, $ \phi:\{(i,j):1 {\leqslant}i {\leqslant}M, 1 {\leqslant}j {\leqslant}N\} \to \{1,\cdots,MN\} $ and define the family of random matrices $ X_\gamma $, $ 0 {\leqslant}\gamma {\leqslant}MN $ $$[X_\gamma]_{ij} = \left\{ \begin{aligned} & [{X^{\mathrm{v}}}]_{ij} & \mbox{if} & \ \ \phi(i,j)>\gamma,\\ & [{X^{\mathrm{w}}}]_{ij} & \mbox{if} & \ \ \phi(i,j) {\leqslant}\gamma. \end{aligned} \right.$$ Note that in particular we have $ X_0 = {X^{\mathrm{v}}}$ and $ X_{MN} = {X^{\mathrm{w}}}$. Denote sample covariance matrices $ H_\gamma $ as $$H_\gamma := X_\gamma^* X_\gamma.$$ Let $ \chi $ be a fixed, smooth, symmetric cutoff function such that $ \chi(x)=1 $ if $ |x|<1 $ and $ \chi(x)=0 $ if $ |x|>2 $. By the Helffer-Sjöstrand formula, if $ \lambda_i $’s are the (real) eigenvalues of a matrix $ H $, we have $$\sum f(\lambda_i) = \int_{{\mathbb{C}}} g(z) \operatorname{Tr}\dfrac{1}{H-z} {\mathrm{d}}m(z),$$ where $ {\mathrm{d}}m $ is the Lebesgue measure on $ {\mathbb{C}}$, and the function $ g $ is defined as $$g(z) := \dfrac{1}{\pi} \left( {\text{\rm i}}y f''(y) \chi(y) + {\text{\rm i}}(f(x) + {\text{\rm i}}y f'(x))\chi'(y) \right),\ \ \ z=x+{\text{\rm i}}y.$$ Define $$\Xi^H := \int_{|y|>N^{-1}} g(z) \operatorname{Tr}(H-z)^{-1} {\mathrm{d}}m(z),$$ and we have the bound (see [@bourgade2018extreme Section 5.2]) $$\left|\sum f(\lambda_i) - \Xi^H \right| {\leqslant}O \left( \dfrac{\varphi^C}{(N \rho)^2} \right).$$ This shows that it suffices to show $$\label{e.TelescopicSum} \left| {\mathbb{E}}F(\Xi^{H_\gamma}) - {\mathbb{E}}F(\Xi^{H_{\gamma-1}}) \right| {\leqslant}\dfrac{ \varphi^C }{N^2} \left( \dfrac{1}{N^{18} \rho^{20}} + \dfrac{t}{N \rho} + \dfrac{1}{(N \rho)^2} + \dfrac{1}{N^2} \right).$$ For an arbitrarily fixed $ \gamma $ corresponding to $ (i,j) $, we can write $$X_{\gamma-1}=Q+V,\ \ \ V:={X^{\mathrm{v}}}_{ij}E^{(ij)},\ \ \ \ X_\gamma = Q+W,\ \ \ W:= {X^{\mathrm{w}}}_{ij}E^{(ij)}.$$ where $ Q $ coincides with $ X_{\gamma-1} $ and $ X_\gamma $ except on the $ (i,j) $ position (where it is 0). We define the Green functions $$R:= (Q^*Q-z)^{-1},\ \ \ S:=(H_{\gamma-1}-z)^{-1}.$$ By Taylor expansion, for some fixed order $ m $, we have $$\begin{gathered} \label{e.Taylor} {\mathbb{E}}F(\Xi^{H_{\gamma}}) - {\mathbb{E}}F(\Xi^{H_{\gamma-1}}) = \sum_{l=1}^{m-1} {\mathbb{E}}\dfrac{F^{(l)}(\Xi^Q)}{l !}\left( (\Xi^{H_\gamma} - \Xi^Q)^l - (\Xi^{H_{\gamma-1}} - \Xi^Q)^l \right)\\ + O \left( \|F^{(m)} \|_\infty \right) \left( {\mathbb{E}}\left( (\Xi^{H_\gamma} - \Xi^Q)^m + (\Xi^{H_{\gamma-1}} - \Xi^Q)^m \right) \right).\end{gathered}$$ First we estimate the $ m $-th order error term. By the first order resolvent expansion we have $$\begin{aligned} |\Xi^{H_{\gamma}} - \Xi^Q| &{\leqslant}\int_{|y|>N^{-1},|x|<\lambda_+ +2} |g(z)|\left| \operatorname{Tr}R(z)(V^*Q+Q^*V+V^*V)S(z) \right| {\mathrm{d}}m(z) \\ &{\leqslant}\varphi^C N \int_{|y|>N^{-1},|x|<\lambda_+ +2} |g(z)| \|S(z) \|_\infty \|R(z) \|_\infty {\mathrm{d}}m(z)\end{aligned}$$ with overwhelming probability, where we use the fact that there are only $ O(N) $ nonzero entries with size $ O(N^{-1}) $ in the matrix $ V^*Q+Q^*V+V^*V $. By the strong local Marchenko-Pastur law ([@pillai2014universality Theorem 3.1]), for any $ D>0 $ we have $${\mathbb{P}}\left( \max_{j}|S_{jj}(z) - m_{{\mathsf{MP}}}(z)| + \max_{j \neq k}|S_{jk}(z)| {\leqslant}\varphi^C \left( \dfrac{1}{Ny}+\sqrt{\dfrac{{\text{\rm Im}\hspace{0.1cm}}m_{{\mathsf{MP}}}(y)}{Ny}} \right) \right) > 1- N^{-D}.$$ Same bound for $ \|R(z) \|_\infty $ also holds (see [@pillai2014universality Lemma 5.4]). This shows that $${\mathbb{E}}(\Xi^{H_\gamma} - \Xi^Q)^m = O \left( \varphi^C / (N^m \rho^m) \right),\ \ \ {\mathbb{E}}(\Xi^{H_{\gamma-1}} - \Xi^Q)^m = O \left( \varphi^C / (N^m \rho^m) \right).$$ Therefore the $ m $-th order term in can be bounded by $ \varphi^C N^{-2}(N^{-m+2} \rho^{-m}) $. Next we consider the first order term in the Taylor expansion. By the resolvent expansion, we have $$S=R-R {A^{\mathrm{v}}}R+(R {A^{\mathrm{v}}})^2R-(R {A^{\mathrm{v}}})^3R+ \cdots - (R {A^{\mathrm{v}}})^{11}R + (R {A^{\mathrm{v}}})^{12}S,$$ where $${A^{\mathrm{v}}}= V^*Q + Q^*V + V^*V.$$ Denote $${{\widehat}{R}_{\mathrm{v}}}^{(n)} := (-1)^{n} \operatorname{Tr}(R {A^{\mathrm{v}}})^n R,\ \ \ \ {\Omega_{\mathrm{v}}}:= \operatorname{Tr}(R {A^{\mathrm{v}}})^{12} S.$$ Then we have $${\mathbb{E}}F'(\Xi^Q) \left( \Xi^{H_{\gamma-1}} - \Xi^{H_{\gamma}} \right) = {\mathbb{E}}F'(\Xi^Q) \int g(z) \left( \sum_{n=1}^{11} \left( {{\widehat}{R}_{\mathrm{v}}}^{(n)} - {{\widehat}{R}_{\mathrm{w}}}^{(n)} \right) +({\Omega_{\mathrm{v}}}- {\Omega_{\mathrm{w}}}) \right) {\mathrm{d}}m(z).$$ Since the first three moments of the two matrices are identical, we know that the case $ n=1 $ gives null contribution. For $ n=2 $, note that the entries of the matrix $ A $ satisfy the following relation $$A_{ab}=\left\{ \begin{aligned} & x_{ij} x_{ib} & \quad & \mbox{if} \ a=j,b \neq j,\\ & x_{ij} x_{ia} & \quad & \mbox{if} \ a \neq j, b=j,\\ & 0 & \quad &\mbox{otherwise}. \end{aligned} \right.$$ This shows that $${\mathbb{E}}({{\widehat}{R}_{\mathrm{v}}}^{(2)} - {{\widehat}{R}_{\mathrm{w}}}^{(2)}) {\leqslant}N \left( \dfrac{t}{N^2} \right) \left(\max_{i \neq j} |R_{ij}|\right)^2 \left(\max_i |R_{ii}|\right),$$ where we used that in the expansion $$\begin{aligned} \operatorname{Tr}(R {A^{\mathrm{v}}})^2R & = \sum_{k} \sum_{a_1,b_1,a_2,b_2} R_{k a_1}{A^{\mathrm{v}}}_{a_1 b_1}R_{b_1 a_2}{A^{\mathrm{v}}}_{a_2 b_2}R_{b_2 k}\\ & = \sum_{k} \sum_{(a_1,b_1,a_2,b_2) \neq (j,j,j,j)} R_{k a_1}{A^{\mathrm{v}}}_{a_1 b_1}R_{b_1 a_2}{A^{\mathrm{v}}}_{a_2 b_2}R_{b_2 k} + \sum_k R_{kj}{A^{\mathrm{v}}}_{jj}R_{jj}{A^{\mathrm{v}}}_{jj}R_{jk}\end{aligned}$$ due to the moment matching condition, the terms that make nontrivial contribution are only in the second summation, which is $$\sum_k R_{kj}R_{jj}R_{jk}({X^{\mathrm{v}}}_{ij})^4.$$ Here we also use the fact that the contribution for the terms with $ k $ equal $ i $ or $ j $ is combinatorially negligible. By the local law, we conclude $${\mathbb{E}}F'(\Xi^Q) \int g(z) \left( {{\widehat}{R}_{\mathrm{v}}}^{(2)} - {{\widehat}{R}_{\mathrm{w}}}^{(2)} \right) {\mathrm{d}}m(z) = O \left( \dfrac{\varphi^C t}{N} \right) \int \dfrac{|g(z)|}{(Ny)^2} {\mathrm{d}}m(z) = O \left( \dfrac{\varphi^C}{N^2} \dfrac{t}{N\rho} \right).$$ For the terms $ n=3,\dots,11 $, as explained in [@pillai2014universality Lemma 5.4], their contributions are of smaller order. Similarly, for the term $ ({\Omega_{\mathrm{v}}}- {\Omega_{\mathrm{w}}}) $, as shown in [@pillai2014universality Lemma 5.4] we have $ {\Omega_{\mathrm{v}}}= O(N^{-4}) $. Therefore we have $${\mathbb{E}}F'(\Xi^Q) \int g(z) \left( ({\Omega_{\mathrm{v}}}- {\Omega_{\mathrm{w}}}) \right) {\mathrm{d}}m(z) = O \left( \dfrac{\varphi^C}{N^2} \dfrac{1}{N^2 \rho} \right)=O \left( \dfrac{\varphi^C}{N^2} \right) \left( \dfrac{1}{(N \rho)^2} + \dfrac{1}{N^2} \right).$$ Moreover, as explained in [@erdos2017dynamical Theorem 17.4], the contributions of higher order terms in Taylor expansion are of smaller order. Combining the estimates and taking $ m=20 $ (we will see the reason in the next section) gives us . Finally, a telescopic summation yields the desired result. Proof of Theorem \[t.Rate\] --------------------------- Let $ s \in {\mathbb{R}}$. If $ |s|>\varphi $, due to the rigidity we know that for any $ D>0 $ and large enough $ N $, we have $ {\mathbb{P}}\left( N^{2/3}(\lambda_N - \lambda_+) {\leqslant}s \right) = {\mathbb{P}}({\mathsf{TW}}{\leqslant}s) + O(N^{-D}) $. So in the following discussion we assume $ |s| {\leqslant}\varphi $. Denoting a non-decreasing function $ f_1 $ such that $ f_1(x)=1 $ for $ x>\lambda_+ + sN^{-2/3} $ and $ f_1(x)=0 $ for $ x<\lambda_+ + sN^{-2/3} - \rho $. We also define $ f_2(x) := f_1(x-\rho) $. Then we have $$\label{e.ProbEst} {\mathbb{E}}_H F \left( \sum_{i=1}^N f_1(\lambda_i) \right) {\leqslant}{\mathbb{P}}_H \left( \lambda_N < \lambda_+ + s N^{-2/3} \right) {\leqslant}{\mathbb{E}}_H F \left( \sum_{i=1}^N f_2(\lambda_i) \right).$$ Moreover, as discussed in [@erdos2011universality; @pillai2014universality], we can find an $ M \times N $ matrix $ {\widetilde}{X}_0 $ such that the Gaussian divisible ensemble $ {\widetilde}{X}_t := e^{-t/2}{\widetilde}{X}_0 + (1-e^{-t})^{1/2}X_G $, where $ X_G $ is a matrix whose entries are independent Gaussian random variables with mean 0 and variance 1, satisfies the following: for $ 1 {\leqslant}k {\leqslant}3 $, $${\mathbb{E}}(\sqrt{M}X_{ij})^k = {\mathbb{E}}[{\widetilde}{X}_t]_{ij}^k,\ \ \ \ |{\mathbb{E}}(\sqrt{M}X_{ij})^4 - {\mathbb{E}}[{\widetilde}{X}_t]_{ij}^4| \lesssim t.$$ By the quantitative Green function comparison theorem \[t.GreenCompare\], we obtain the following bound $$\begin{gathered} {\mathbb{E}}_{{\widetilde}{X}_t} F \left( \sum_{i=1}^N f_1(\lambda_i) \right) - \varphi^C \left( \dfrac{1}{N^{18} \rho^{20}} + \dfrac{t}{N \rho} + \dfrac{1}{(N \rho)^2} + \dfrac{1}{N^2} \right) {\leqslant}{\mathbb{P}}_H \left( \lambda_N < \lambda_+ + s N^{-2/3} \right) \\ {\leqslant}{\mathbb{E}}_{{\widetilde}{X}_t} F \left( \sum_{i=1}^N f_2(\lambda_i) \right) + \varphi^C \left( \dfrac{1}{N^{18} \rho^{20}} + \dfrac{t}{N \rho} + \dfrac{1}{(N \rho)^2} + \dfrac{1}{N^2} \right).\end{gathered}$$ Using for $ {\widetilde}{X}_t $, the estimate becomes $$\begin{gathered} {\mathbb{P}}_{{\widetilde}{X}_t} \left( \lambda_N < \lambda_+ + s N^{-2/3} -\rho \right) - \varphi^C \left( \dfrac{1}{N^{18} \rho^{20}} + \dfrac{t}{N \rho} + \dfrac{1}{(N \rho)^2} + \dfrac{1}{N^2} \right)\\ {\leqslant}{\mathbb{P}}_H \left( \lambda_N < \lambda_+ + s N^{-2/3} \right) {\leqslant}\\ {\mathbb{P}}_{{\widetilde}{X}_t} \left( \lambda_N < \lambda_+ + s N^{-2/3} +\rho \right) + \varphi^C \left( \dfrac{1}{N^{18} \rho^{20}} + \dfrac{t}{N \rho} + \dfrac{1}{(N \rho)^2} + \dfrac{1}{N^2} \right).\end{gathered}$$ After combined with the edge relaxation Theorem \[t.Relaxation\], the estimate now gives us $$\begin{gathered} {\mathbb{P}}_{{\mathsf{Wishart}}} \left( N^{2/3}(\lambda_N - \lambda_+) < s -N^{2/3}\rho - \dfrac{N^{\varepsilon}}{N^{1/3}t} \right) - \varphi^C \left( \dfrac{1}{N^{18} \rho^{20}} + \dfrac{t}{N \rho} + \dfrac{1}{(N \rho)^2} + \dfrac{1}{N^2} \right)\\ {\leqslant}{\mathbb{P}}_H \left( N^{2/3}(\lambda_N - \lambda_+) < s \right) {\leqslant}\\ {\mathbb{P}}_{{\mathsf{Wishart}}} \left( N^{2/3}(\lambda_N - \lambda_+) < s +N^{2/3}\rho + \dfrac{N^{\varepsilon}}{N^{1/3}t} \right) + \varphi^C \left( \dfrac{1}{N^{18} \rho^{20}} + \dfrac{t}{N \rho} + \dfrac{1}{(N \rho)^2} + \dfrac{1}{N^2} \right).\end{gathered}$$ Moreover, as shown in [@el2006rate; @ma2012accuracy], we know $${\mathbb{P}}_{{\mathsf{Wishart}}} \left( N^{2/3}(\lambda_N - \lambda_+) <s \right) = {\mathbb{P}}({\mathsf{TW}}<s)+ O(N^{-2/3}).$$ By using this Wishart result and the boundedness of the density for $ {\mathsf{TW}}$, we obtain $$\begin{gathered} \label{e.RateNullCase} {\mathbb{P}}_H \left(N^{2/3}(\lambda_N - \lambda_+) <s \right) - {\mathbb{P}}\left( {\mathsf{TW}}<s \right)\\ = O \left( N^{\varepsilon}\right) \left( N^{2/3}\rho + \dfrac{1}{N^{1/3}t} + \dfrac{1}{N^{18} \rho^{20}} + \dfrac{t}{N \rho} + \dfrac{1}{(N \rho)^2} + \dfrac{1}{N^{2/3}} \right).\end{gathered}$$ The optimal bound $ N^{-2/9 + {\varepsilon}} $ is obtained for $ t=N^{-1/9} $ and $ \rho = N^{-8/9} $. This completes the whole proof for Theorem \[t.Rate\]. Generalization to General Population Matrices {#s.GeneralPopulation} ============================================= In this section, we proceed to generalize our previous results for sample covariance matrices of type $ X^*X $ (which corresponds to the identity population) and aim to derive the rate of convergence to the Tracy-Widom distribution for the (rescaled) largest eigenvalue of separable sample covariance matrices with general population. Throughout this section, we will follow the notations and the setup in the the work by Lee and Schnelli [@lee2016tracy]. Let $ X=(x_{ij}) $ be defined as in and . For some deterministic $ M \times M $ matrix $ T $, the sample covariance matrices associated with data matrix $ X $ and population matrix $ \Sigma:=T^*T $ is defined as $ \mathcal{Q}:=(TX)(TX)^* $. Note that the $ M \times M $ matrix $ \mathcal{Q} $ and the matrix $$Q:=X^*\Sigma X$$ share the same non-trivial eigenvalues. Since we are studying the largest eigenvalue of the sample covariance matrix, it is more convenient to work with the matrix $ Q $ (called the separable sample covariance matrix) for some technical reasons. We denote the eigenvalues of $ Q $ in increasing order by $ \mu_1 {\leqslant}\cdots {\leqslant}\mu_N $. As mentioned previously, for the null case (i.e. the population matrix is identity), it is well known that the empirical eigenvalue distribution of a sample covariance matrix converges weakly in probability to the Marchenko-Pastur law. Under the general setting, however, this results need to be modified and the limiting measure (called the deformed Marchenko-Pastur law) will depends on the spectrum of the population matrix. Let $ \sigma_1 {\leqslant}\cdots {\leqslant}\sigma_M $ be the eigenvalues of the population matrix $ \Sigma $, we denote by $ {\widehat}{\rho}={\widehat}{\rho}(M) $ the empirical eigenvalue distribution of $ \Sigma $, which is defined as $${\widehat}{\rho}:=\dfrac{1}{M}\sum_{j=1}^M \delta_{\sigma_j}.$$ The deformed Marchenko-Pastur law $ {\widehat}{\rho}_{{\mathsf{fc}}} $ is defined in the following way. The Stieltjes transform $ {\widehat}{m}_{{\mathsf{fc}}} $ of the probability measure is given by the unique solution of the equation $${\widehat}{m}_{{\mathsf{fc}}}(z)=\dfrac{1}{-z+\xi^{-1}\int \frac{1}{t {\widehat}{m}_{{\mathsf{fc}}}(z) + 1}{\mathrm{d}}{\widehat}{\rho}(t)},\ \ \ {\text{\rm Im}\hspace{0.1cm}}{\widehat}{m}_{{\mathsf{fc}}}(z) {\geqslant}0,\ \ \ z \in {\mathbb{C}}^+.$$ It has been discussed in [@knowles2017anisotropic] that $ {\widehat}{m}_{{\mathsf{fc}}} $ is associated to a continuous probability density $ {\widehat}{\rho}_{{\mathsf{fc}}} $ with compact support in $ [0,\infty) $. Moreover, the density ${\widehat}{\rho}_{{\mathsf{fc}}} $ can be obtained from $ {\widehat}{m}_{{\mathsf{fc}}} $ via the Stieltjes inversion formula $${\widehat}{\rho}_{{\mathsf{fc}}}(E) = \lim_{\eta \downarrow 0}\dfrac{1}{\pi}{\text{\rm Im}\hspace{0.1cm}}{\widehat}{m}_{{\mathsf{fc}}}(E+{\text{\rm i}}\eta).$$ The typical location of the largest eigenvalue, which is the rightmost endpoint of the support of the density $ {\widehat}{\rho}_{{\mathsf{fc}}} $ is determined in the following way. Recall that $ \xi:=N/M $, we define $ \xi_+ $ as the largest solution of the equation $$\int \left( \dfrac{t \xi_+}{1-t\xi_+} \right)^2 {\mathrm{d}}{\widehat}{\rho}_{{\mathsf{fc}}}(t) = \xi.$$ We remark that $ \xi_+ $ is unique and $ \xi_+ \in [0,\sigma_M^{-1}] $. We then introduce the typical location for the largest eigenvalue $ E_+ $ by $$\label{e.RightEndpoint} E_+ := \dfrac{1}{\xi_+}\left( 1+\xi^{-1}\int \dfrac{t \xi_+}{1-t\xi_+} {\mathrm{d}}{\widehat}{\rho}_{{\mathsf{fc}}}(t) \right).$$ Now we state our assumptions on the population matrix $ \Sigma $ that are needed to prove the explicit rate of convergence. For general random matrices $ X $ we require $ \Sigma $ to be diagonal, and we will show later this diagonal condition can be removed if $ X $ is Gaussian. We further need the following assumption for the spectrum of the population matrix $ \Sigma $. Throughout this section, we assume the following: $$\label{e.AssumptionPopulation} \liminf_M \sigma_1 >0,\ \ \ \limsup_M \sigma_M <\infty,\ \ \ \mbox{and}\ \ \ \limsup_M \sigma_M \xi_+ <1.$$ The assumption is the same as [@lee2016tracy Assumption 2.2]. It is first used in [@bao2015universality; @knowles2017anisotropic] to prove the local deformed Marchenko-Pastur law. In particular, the last inequality ensures that the density $ {\widehat}{\rho}_{{\mathsf{fc}}} $ exhibits a square-root behavior near the right edge of its support, which is crucial to derive the local law. It is natural to note that with a general population matrix, the distribution of the largest eigenvalue should not behave exactly like the null case. Besides the typical location of the largest eigenvalue is changed, the normalization constant of the fluctuation is also different. Therefore, we introduce the following normalization constant $ \gamma_0 $ given by $$\label{e.ScalingConstant} \dfrac{1}{\gamma_0^3} = \dfrac{1}{\xi}\int \left( \dfrac{t}{1-t\xi_+} \right){\mathrm{d}}{\widehat}{\rho}(t) + \dfrac{1}{\xi_+^3}.$$ Moreover, we remark that the Tracy-Widom limit for the general case is rescaled and it is not the same as the previous one we used for the null case. However they are just different by a simple scaling so that we do not emphasize this difference and still use the notation $ {\mathsf{TW}}$ to denote this distribution. Under this framework, our main result given in Corollary \[t.RateGeneralPopulation\]. Unlike the proof for the null case in the previous sections, we will not strictly follow the three-step strategy in the dynamical approach. Instead, we use the comparison theorem for the Green function flow, which is a method based on continuous interpolation, linearization and renormalization developed in [@lee2016tracy]. Local deformed Marchenko-Pastur law ----------------------------------- For completeness, we will briefly introduce the local deformed Marchenko-Pastur law. Though we will not give a detailed proof, we emphasize that the local law is an indispensable part to prove the Green function comparison Proposition \[p.GreenFunctionFlow\], which further leads to the edge universality. For small nonnegative $ c,\epsilon {\geqslant}0 $ and sufficiently large $ E_+<C<\infty $, we consider the domain $$\mathcal{D}(c,\epsilon) :=\left\{ z=E+{\text{\rm i}}\eta \in {\mathbb{C}}^+ : E_+-c {\leqslant}E {\leqslant}C, N^{-1+\epsilon} {\leqslant}\eta {\leqslant}1 \right\}.$$ We also denote $ \kappa=\kappa(E):=|E-E_+| $. Then we have the following estimates for the density and the Stieltjes transform of the deformed Marchenko-Pastur law. \[l.EstiamtesDeformedMP\] Under the assumption , there exists a constant $ c>0 $ such that $${\widehat}{\rho}_{\mathsf{fc}}(E) \sim \sqrt{E_+ - E}\ \ \ \ E \in [E_+-2c,E_+].$$ Moreover, the Stieltjes transform $ {\widehat}{m}_{{\mathsf{fc}}} $ satisfies the following: for $ z \in \mathcal{D}(c,0) $, we have $$|{\widehat}{m}_{{\mathsf{fc}}}(z)| \sim 1,\ \ \ \ {\text{\rm Im}\hspace{0.1cm}}{\widehat}{m}_{{\mathsf{fc}}}(z) \sim \left\{ \begin{aligned} & \frac{\eta}{\sqrt{\kappa+\eta}} & \mbox{if} &\ E {\geqslant}E_++\eta,\\ & \sqrt{\kappa+\eta}, & \mbox{if} &\ E \in [E_+-c,E_++\eta). \end{aligned} \right. .$$ The Green function and the Stieltjes transform are defined in the usual way: $$G_Q(z) := (Q-z)^{-1},\ \ \ m_Q(z) := \dfrac{1}{N}\operatorname{Tr}G_Q(z).$$ Then we have the following local law for the separable sample covariance matrix $ Q $. \[l.LocalDeformedMPLaw\] Under the assumption , for any sufficiently small $ \epsilon>0 $, and for any (large) $ D>0 $, there exists $ N_0(D)>0 $ such that for any $ N {\geqslant}N_0(D) $ we have the following estimate uniformly in $ z \in \mathcal{D}(c,\epsilon) $: $${\mathbb{P}}\left( |m_Q(z) - {\widehat}{m}_{{\mathsf{fc}}}(z)| {\leqslant}\dfrac{N^\epsilon}{N\eta} \right) > 1-N^{-D},$$ and $${\mathbb{P}}\left( \max_{i,j}\left| (G_Q)_{ij}(z) - \delta_{ij}{\widehat}{m}_{{\mathsf{fc}}}(z) \right| {\leqslant}N^\epsilon \left( \sqrt{\dfrac{{\text{\rm Im}\hspace{0.1cm}}{\widehat}{m}_{{\mathsf{fc}}}(z)}{N\eta}} + \dfrac{1}{N\eta} \right) \right) > 1-N^{-D}.$$ It is clear to see that the estimates for the deformed Marchenko-Pastur law (Lemma \[l.EstiamtesDeformedMP\]) and the local law (Lemma \[l.LocalDeformedMPLaw\]) are greatly similar as the corresponding results for the null case (see e.g. [@pillai2014universality Theorem 3.1]). Heuristically, this implies the Tracy-Widom limit in the edge universality for the non-null case. Interpolation and Green function comparison ------------------------------------------- In classical theory of random matrix universality, the tool needed to prove the edge universality is the Green function comparison theorem. The usual approach is to compare two ensembles with some moments matching conditions, and then use the construction of Gaussian divisible ensembles together with estimates of the local relaxation flow to remove the moments matching requirement. In this section, however, we do not follow this traditional step. Instead, we compare the Green function of a general ensemble with its corresponding null sample covariance matrix. This argument was first introduced in [@lee2015edge] to handle the deformed Wigner matrices, and used in [@lee2016tracy] to identify the Tracy-Widom limit for general separable sample covariance matrices. The basic idea is to introduce a time evolution that deforms the population matrices continuously to the identity and offset the change of the Green function by a renormalization of the matrix. Recall the scaling constant $ \gamma_0 $ defined in . We consider the following two rescaled matrices $${\widetilde}{\Sigma} := \gamma_0 \Sigma,\ \ \ {\widetilde}{Q} := X^* {\widetilde}{\Sigma}X.$$ We also denote the eigenvalues of $ {\widetilde}{Q} $ by $ {\widetilde}{\mu}_1 {\leqslant}\cdots {\leqslant}{\widetilde}{\mu}_N $, and let $ L_+ := \gamma_0 E_+ $. We remark that in the literature about sample covariance matrices with general population (e.g. [@bao2015universality; @el2007tracy; @lee2016tracy]), the scaling of the Tracy-Widom distribution is chosen in the way such that it is the limit for the distribution of the (rescaled) largest eigenvalue of the matrix $$W:=\sqrt{\xi}(1+\sqrt{\xi})^{-4/3}X^*X.$$ Specifically, we order the eigenvalues of the matrix $ W $ by $ \lambda_1 {\leqslant}\cdots {\leqslant}\lambda_N $, and let $ M_+ $ denote the rightmost endpoint of the rescaled Marchenko-Pastur law for $ W $. It has been shown in [@bao2015universality equation (1.9)] that $$\gamma_0 = \sqrt{\xi}(1+\sqrt{\xi})^{-4/3} + o(1).$$ This can be regarded as a good motivation for considering the scaling constant $ \gamma_0 $. For the diagonal population matrix $ \Sigma = \mbox{diag}(\sigma_j) $, we introduce the following time evolution $ t \mapsto (\sigma_j(t)) $ that deforms $ \Sigma $ to the identity matrix $ {\mathsf{Id}}$ and the Green function flow by $$\label{e.GreenFunctionFlow} \dfrac{1}{\sigma_j(t)}=e^{-t}\dfrac{1}{\sigma_j(0)}+(1-e^{-t}),\ \ \ {\widetilde}{Q}(t)=\gamma_0X^*\Sigma(t)X,\ \ \ m_{{\widetilde}{Q}(t)}(z) := \dfrac{1}{N}\operatorname{Tr}({\widetilde}{Q}(t)-z)^{-1}.$$ Based on the local law Lemma \[l.EstiamtesDeformedMP\] and Lemma \[l.LocalDeformedMPLaw\], and a delicate analysis for the time derivative of the Green function for $ {\widetilde}{Q}(t) $, the Green function comparison theorem (see Proposition \[p.GreenFunctionFlow\]) is proved in [@lee2016tracy]. We note that though the original estimate in [@lee2016tracy] is not explicit, a careful examination of the proof will reveal that the result is actually quantitative. \[p.GreenFunctionFlow\] Let $ {\varepsilon}>0 $ and set $ \eta = N^{-2/3-{\varepsilon}} $. Let $ E_1,E_2 \in {\mathbb{R}}$ satisfy $ E_1 < E_2 $ and $ |E_1|,|E_2| {\leqslant}N^{-2/3+{\varepsilon}} $. Let $ F: {\mathbb{R}}\to {\mathbb{R}}$ be a smooth function satisfying $$\max_x |F^{(l)}(x)|(|x|+1)^{-C} {\leqslant}C,\ \ \ \ l=1,2,3,4.$$ Then for any (small) $ {\delta}>0 $ and for sufficiently large $ N $ we have $$\begin{gathered} \left| {\mathbb{E}}F \left( N \int_{E_1}^{E_2} {\text{\rm Im}\hspace{0.1cm}}m_{{\widetilde}{Q}}(x+L_++{\text{\rm i}}\eta) {\mathrm{d}}x \right) - {\mathbb{E}}F \left( N \int_{E_1}^{E_2} {\text{\rm Im}\hspace{0.1cm}}m_{W}(x+M_++{\text{\rm i}}\eta) {\mathrm{d}}x \right) \right|\\ {\leqslant}N^{-\frac{1}{3}+2{\varepsilon}+{\delta}}.\end{gathered}$$ Quantitative edge universality ------------------------------ In this section we can finally prove Corollary \[t.RateGeneralPopulation\]. The proof is based on our previous rate of convergence for the null case (Theorem \[t.Rate\]) and the estimate on the comparison theorem for the Green function flow (Proposition \[p.GreenFunctionFlow\]). We first remark that we are not supposed to use the rate of convergence for the null case (Theorem \[t.Rate\]) and the triangle inequality in a naive way to derive the convergence rate for the general case. This is because Theorem \[t.Rate\] is obtain by choosing the optimal parameters in the estimate , and the scale parameter $ \rho $ is also related to the scale in the Green function comparison (Proposition \[p.GreenFunctionFlow\]). To illustrate this link more clearly, we first briefly review how Green function comparison is used to obtain the edge universality. We introduce a smooth cutoff function $ K:{\mathbb{R}}\to {\mathbb{R}}$ satisfying $$K(x)= \left\{ \begin{aligned} & 1 &\ \mbox{if}\ & x {\leqslant}1/9,\\ & 0 &\ \mbox{if}\ & x {\geqslant}2/9. \end{aligned} \right.$$ and we also define the Poisson kernel $ \theta_\eta $, for $ \eta>0 $ $$\theta_\eta(x) := \dfrac{\eta}{\pi(x^2+\eta^2)}.$$ Let $ E_*:=L_+ +\varphi^C N^{-2/3} $, and denote $ \chi_E := 1_{[E,E_*]} $. For $ {\varepsilon}>0 $, let $ l:= \tfrac{1}{2}N^{-2/3-{\varepsilon}} $ and $ \eta:=N^{-2/3-9{\varepsilon}} $. Then for any (large) $ D>0 $, it is proved in [@lee2016tracy; @pillai2014universality] that for large enough $ N $ we have $${\mathbb{E}}K\left( \operatorname{Tr}(\chi_{E-l} * \theta_\eta({\widetilde}{Q})) \right) {\leqslant}{\mathbb{P}}\left( {\widetilde}{\mu}_N {\leqslant}E \right) {\leqslant}{\mathbb{E}}K\left( \operatorname{Tr}(\chi_{E+l} * \theta_\eta({\widetilde}{Q})) \right) + N^{-D}.$$ Here the parameter $ l $ plays the same role as the $ \rho $ in , and therefore we have $ N^{-{\varepsilon}}=N^{2/3}\rho $ and $ \eta=N^{-2/3}N^{6}\rho^{9} $. By the Green function comparison Proposition \[p.GreenFunctionFlow\] and the [@lee2016tracy Theorem 2.4], we have $$\begin{gathered} {\mathbb{P}}\left( N^{2/3}(\lambda_N - M_+) {\leqslant}s \right) - N^{-\frac{1}{3}+18{\varepsilon}+{\delta}} {\leqslant}{\mathbb{P}}\left( N^{2/3}({\widetilde}{\mu}_N - L_+) {\leqslant}s \right)\\ {\leqslant}{\mathbb{P}}\left( N^{2/3}(\lambda_N - M_+) {\leqslant}s \right) + N^{-\frac{1}{3}+18{\varepsilon}+{\delta}}.\end{gathered}$$ This gives us $$\label{e.GeneralQuantEdgeUniversality} {\mathsf{d_K}}\left( \gamma_0 N^{2/3}(\mu_N - E_+),N^{2/3}(\lambda_N-M_+) \right) {\leqslant}N^{{\delta}}N^{-1/3}N^{-12}\rho^{-18}.$$ Combined with Theorem , by triangle inequality we finally obtain $$\begin{aligned} &\quad {\mathsf{d_K}}\left( \gamma_0 N^{2/3}(\mu_N - E_+),{\mathsf{TW}}\right)\\ & {\leqslant}{\mathsf{d_K}}\left( \gamma_0 N^{2/3}(\mu_N - E_+),N^{2/3}(\lambda_N-M_+) \right) + {\mathsf{d_K}}\left( N^{2/3}(\lambda_N-M_+),{\mathsf{TW}}\right)\\ & {\leqslant}N^{{\delta}}\left( N^{-1/3}N^{-12}\rho^{-18} + N^{2/3}\rho + \dfrac{1}{N^{1/3}t} + \dfrac{1}{N^{18} \rho^{20}} + \dfrac{t}{N \rho} + \dfrac{1}{(N \rho)^2} + \dfrac{1}{N^{2/3}} \right).\end{aligned}$$ The optimal result is obtained now by choosing $ \rho=N^{-13/19} $ and $ t=N^{-6/19} $, which gives us $${\mathsf{d_K}}\left( \gamma_0 N^{2/3}(\mu_N - E_+),{\mathsf{TW}}\right) {\leqslant}N^{-\frac{1}{57}+{\delta}}.$$ This completes the proof. Based on the Corollary \[t.RateGeneralPopulation\] for diagonal population matrices $ \Sigma $ and general random matrices $ X $, we can easily obtain the following result for the case in which we can have a general population if the random matrix $ X $ is restricted to be Gaussian. \[c.RateGaussian\] Let $ Q := X^* \Sigma X $ be an $ N \times N $ separable sample covariance matrix, where $ X $ is an $ M \times N $ real random matrix with independent Gaussian entries satisfying , and $ \Sigma $ is a real positive-definite deterministic $ M \times M $ matrix satisfying . For any $ {\varepsilon}>0 $ and large enough $ N $, we have $${\mathsf{d_K}}\left( \gamma_0 N^{2/3} (\mu_N - E_+),{\mathsf{TW}}\right) {\leqslant}N^{-\frac{1}{57} + {\varepsilon}}$$ Under these assumptions, we know that the population matrix $ \Sigma $ is diagonizable, i.e. there exists an $ M \times M $ real diagonal matrix $ D $ and an $ N \times N $ orthogonal matrix $ U $ such that $ \Sigma = U^*DU $. Since $ X $ is a matrix whose entries are independent Gaussian random variable, we know that $ UX $ is also a real random matrix with Gaussian entries satisfying the assumption . Therefore, by applying our Corollary \[t.RateGeneralPopulation\] to the matrix $ X^*\Sigma X = (UX)^*D(UX) $, we will get the desired result.

--- abstract: 'We compute the general expression of the one-loop vertex correction in an arbitrary plane-wave background field for the case of two on-shell external electrons and an off-shell external photon. The properties of the vertex corrections under gauge transformations of the plane-wave background field and of the radiation field are studied. Concerning the divergences of the vertex correction, the infrared one is cured by assigning a finite mass to the photon, whereas the ultraviolet one is shown to be renormalized exactly as in vacuum. Finally, the corresponding expression of the vertex correction within the locally-constant crossed field is also derived and the high-field asymptotic is shown to scale according to the Ritus-Narozhny conjecture.' author: - 'A.' - 'M. A.' title: 'One-loop vertex correction in a plane wave' --- Introduction ============ The predictions of QED agree with experiments with impressive accuracy (see, e.g., Refs. [@Hanneke_2008; @Sturm_2011]). The great success of QED has called for testing this theory under more extreme conditions as, for example, those provided by intense background electromagnetic fields. An electromagnetic field is denoted as “intense” in the realm of QED if it is of the order of the so-called “critical” field of QED: $F_{cr}=m^2/|e|=1.3\times 10^{16}\;\text{V/cm}=4.4\times 10^{13}\;\text{G}$ (from now on we employ units with $\epsilon_0=\hbar=c=1$ and $m$ and $e<0$ denote the electron mass and charge, respectively) [@Landau_b_4_1982; @Fradkin_b_1991; @Dittrich_b_1985]. Importantly, the presence of intense background electromagnetic fields allows for testing QED on a sector where nonlinear effects with respect to the background field strongly affect physical processes and the dynamics of charged particles. This sector is somewhat alternative to the high-energy one successfully investigated via conventional accelerators and it thus can serve as an independent ground test of QED. High-power optical lasers are becoming a suitable tool to test QED at critical field strengths, which correspond to laser intensities of the order of $10^{29}\;\text{W/cm$^2$}$. In fact, although available lasers have reached peak intensities $I_0$ of the order of $5.5\times 10^{22}\;\text{W/cm$^2$}$ [@Yoon_2019] and upcoming facilities aim at $I_0\sim 10^{23}\text{-}10^{24}\;\text{W/cm$^2$}$ [@APOLLON_10P; @ELI; @CoReLS; @XCELS], the Lorentz invariance of the theory implies that the effective laser field strength at which a process occurs is the one experienced by the charges in their rest frame [@Mitter_1975; @Ritus_1985; @Ehlotzky_2009; @Reiss_2009; @Di_Piazza_2012; @Dunne_2014]. Since the amplitude of the laser field is boosted by a factor of the order of the relativistic Lorentz factor of the charge, an electron for definiteness, one can see that the strong-field QED regime, in which the background strength is effectively of the order of $F_{cr}$, can be entered already at intensities of the order of $10^{23}\;\text{W/cm$^2$}$, if the laser field counterpropagates with respect to an electron/positron with energy of the order of $500\;\text{MeV}$. In order to test QED in the strong-field regime by means of intense optical fields, it is essential that both experiments and theoretical predictions are correspondingly accurate. However, as it is understandable, first experiments in this regime have so far been designed especially to show the occurrence of phenomena like nonlinear Compton scattering [@Bula_1996], nonlinear Breit-Wheeler pair production [@Burke_1997; @Bamber_1999], and radiation reaction [@Cole_2018; @Poder_2018], without aiming at obtaining high-accuracy results. Correspondingly, on the theory side, the basic strong-field QED processes like nonlinear Compton scattering [@Goldman_1964; @Nikishov_1964; @Ritus_1985; @Baier_b_1998; @Ivanov_2004; @Boca_2009; @Harvey_2009; @Mackenroth_2010; @Boca_2011; @Mackenroth_2011; @Seipt_2011; @Seipt_2011b; @Dinu_2012; @Krajewska_2012; @Dinu_2013; @Seipt_2013; @Krajewska_2014; @Wistisen_2014; @Harvey_2015; @Seipt_2016; @Seipt_2016b; @Angioi_2016; @Harvey_2016b; @Angioi_2018; @Di_Piazza_2018_c; @Dinu_2018; @Alexandrov_2019; @Di_Piazza_2019; @Ilderton_2019_b] and nonlinear Breit-Wheeler pair production [@Reiss_1962; @Nikishov_1964; @Narozhny_2000; @Roshchupkin_2001; @Reiss_2009; @Heinzl_2010b; @Mueller_2011b; @Titov_2012; @Nousch_2012; @Krajewska_2013b; @Jansen_2013; @Augustin_2014; @Meuren_2015; @Meuren_2016; @Di_Piazza_2019; @King_2020] have been studied in detail at tree level by approximating the laser field as a plane wave (see also the reviews [@Mitter_1975; @Ritus_1985; @Ehlotzky_2009; @Reiss_2009; @Di_Piazza_2012]). However, even under the plane-wave approximation, the radiative corrections of these processes have never been computed. The reason is that calculations including the effects of the external laser field exactly are significantly more complex than the corresponding calculations in vacuum. The standard technique, in fact, is to work within the so-called Furry picture [@Furry_1951], where the electron-positron field is quantized in the presence of the background field [@Fradkin_b_1991; @Landau_b_4_1982]. This requires that the Dirac equation can be solved analytically in the presence of the background field, which has been achieved in Ref. [@Volkov_1935] in the case of a plane wave (see also Ref. [@Landau_b_4_1982]), the corresponding states being known as Volkov states. An alternative, equivalent technique is the so-called operator technique, first proposed by Schwinger [@Schwinger_1951] and then developed for the case of a background plane wave [@Baier_1976_a; @Baier_1976_b; @Di_Piazza_2007; @Di_Piazza_2008_b; @Di_Piazza_2013; @Di_Piazza_2018_d], which does not require the explicit solution of the Dirac equation in the plane-wave field. Going back to the radiative corrections, a systematic study has been only carried out in the special case of a zero-frequency plane wave or a constant crossed field, i.e., a constant and uniform electromagnetic field with electric and magnetic field having the same amplitude and being perpendicular to each other, from the early works of Ritus and Narozhny [@Ritus_1970; @Ritus_1972; @Narozhny_1979; @Narozhny_1980; @Morozov_1981] to the more recent one [@Mironov_2020] (see also Ref. [@Akhmedov_1983] and the reviews in Refs. [@Akhmedov_2011; @Fedotov_2017]), where higher-loop Feynman diagrams have been evaluated. However, so far, in the case of a general plane wave with an arbitrary polarization and shape, only the one-loop mass operator (see Fig. \[FD\_MO\]) and the one-loop polarization operator (see Fig. \[FD\_PO\]) have been computed in Ref. [@Baier_1976_a] and in Refs. [@Becker_1975; @Baier_1976_b], respectively (see also Ref. [@Meuren_2013] for an alternative derivation of the polarization operator). ![The one-loop mass operator in an intense plane wave. The double lines represent exact electron states and propagator in a plane wave (Volkov states and propagator, respectively) [@Landau_b_4_1982].[]{data-label="FD_MO"}](Figure_1.pdf){width="0.6\columnwidth"} ![The one-loop polarization operator in an intense plane wave. The double lines represent exact electron propagators in a plane wave (Volkov propagators) [@Landau_b_4_1982].[]{data-label="FD_PO"}](Figure_2.pdf){width="0.6\columnwidth"} The one-loop vertex correction in a general plane wave (see Fig. \[FD\_VC\]) has never been evaluated, whereas the corresponding quantity in a constant crossed field was computed in Ref. [@Morozov_1981]. ![The one-loop Feynman diagram corresponding to the vertex correction. The double lines represent exact electron states and propagator in a plane wave (Volkov states and propagator, respectively) [@Landau_b_4_1982].[]{data-label="FD_VC"}](Figure_3.pdf){width="0.4\columnwidth"} The purpose of the present paper is to fill this gap and, indeed, to compute the one-loop vertex correction in an arbitrary plane wave for the case of two on-shell external electrons and an off-shell external photon. It is worth mentioning here that the computation of the vertex-correction function is not only important to evaluate the leading-order radiative corrections of strong-field QED processes. There is also a more fundamental reason related to the so-called Ritus-Narozhny conjecture [@Ritus_1970; @Narozhny_1979; @Narozhny_1980; @Morozov_1981] about the high-energy behavior of radiative corrections in strong-field QED in a constant crossed field. As we have mentioned, a constant crossed field is a constant and uniform electromagnetic field $F_0^{\mu\nu}=(\bm{E}_0,\bm{B}_0)$ such that the two field Lorentz-invariants $\bm{E}_0^2-\bm{B}_0^2$ and $\bm{E}_0\cdot\bm{B}_0$ vanish. Now, in a constant crossed field radiative corrections depend only on the Lorentz- and gauge-invariant quantum nonlinearity parameter $\chi_0=\sqrt{-(p_{\mu}F_0^{\mu\nu})^2}/mF_{cr}$ [@Mitter_1975; @Ritus_1985; @Ehlotzky_2009; @Reiss_2009; @Di_Piazza_2012], where $p^{\mu}$ is the four-momentum of the particle at hand and where the metric tensor $\eta^{\mu\nu}=\text{diag}(+1,-1,-1,-1)$ is employed. The Ritus-Narozhny conjecture states that at $\chi_0\gg 1$ the effective coupling of QED in a constant crossed field scales as $\alpha\chi_0^{2/3}$. Since, apart from irrelevant prefactors, the energy of the particle enters radiative corrections only through $\chi_0$ at $\chi_0\gg 1$, the Ritus-Narozhny conjecture implies an asymptotic high-energy behavior of strong-field QED in a constant crossed field qualitatively different from the logarithmic one of QED in vacuum [@Jauch_b_1976; @Itzykson_b_1980; @Landau_b_4_1982; @Schwartz_b_2014]. The physical relevance of the Ritus-Narozhny conjecture is broadened by the so-called locally-constant field approximation (LCFA), stating that in the limit of low-frequency plane waves the probabilities of QED processes reduce to the corresponding probabilities in a constant crossed field averaged over the phase-dependent plane-wave profile [@Ritus_1985]. In Ref. [@Podszus_2019] we have investigated the one-loop mass and polarization operator to show that, if one first performs in the general expression of these quantities the high-energy limit, one indeed recovers the typical logarithmic behavior of QED as in vacuum (see also Ref. [@Ilderton_2019]). Below, we will also investigate the vertex correction within the LCFA, whereas the high-energy asymptotic will be presented elsewhere. The paper is organized as follows: In Sec. \[Notation\] we introduce the basic notation of the paper. In Sec. \[VC\_General\] the general form of the vertex-correction function is derived by means of the operator technique. In Sec. \[VC\_GI\] the properties of the vertex-correction function under gauge transformations of the radiation field and of the plane-wave background field are studied. In Sec. \[VC\_CP\] we show how to regularize and renormalize the vertex-correction function in the ultraviolet. The expression of the vertex-correction function within the LCFA is derived in Sec. \[VC\_LCFA\] and, finally, the main conclusions of the paper are reported in Sec. \[VC\_Conclusions\]. An appendix contains some technical considerations on a component of the vertex-correction function. Notation {#Notation} ======== The notation employed below is the same as in Ref. [@Di_Piazza_2018_d] but it is convenient to report here the main definitions. As we have mentioned in the Introduction, the present paper focuses on studying radiative corrections in a general plane-wave field. The latter is described by the four-vector potential $A^{\mu}(\phi)$, which only depends on the light-cone time $\phi=t-\bm{n}\cdot \bm{x}$. Here, the unit vector $\bm{n}$ defines the propagation direction of the plane wave, which can be used to introduce two useful four-dimensional quantities: $n^{\mu}=(1,\bm{n})$ and $\tilde{n}^{\mu}=(1,-\bm{n})/2$ (note that $\phi=(nx)$). Assuming obvious differential properties of the four-vector potential $A^{\mu}(\phi)$ and its derivatives, it is clear that it is a solution of the free wave equation $\square A^{\mu}=0$, where $\square=\partial_{\nu}\partial^{\nu}$, and it is assumed to fulfill the Lorenz-gauge condition $\partial_{\mu}A^{\mu}=0$, with the additional constraint $A^0(\phi)=0$. Thus, if we represent $A^{\mu}(\phi)$ in the form $A^{\mu}(\phi)=(0,\bm{A}(\phi))$, then the Lorenz-gauge condition implies $\bm{n}\cdot\bm{A}'(\phi)=0$, with the prime in a function of $\phi$ indicating its derivative with respect to $\phi$. If we make the additional assumption that $\bm{A}(\phi)$ vanishes for $\phi\to\pm\infty$, the equality $\bm{n}\cdot\bm{A}'(\phi)=0$ implies that $\bm{n}\cdot\bm{A}(\phi)=0$. By introducing two four-vectors $a_j^{\mu}=(0,\bm{a}_j)$, with $j=1,2$, such that $(na_j)=-\bm{n}\cdot\bm{a}_j=0$ and $(a_ia_j)=-\bm{a}_i\cdot\bm{a}_j=-\delta_{ij}$, the most general form of the vector potential $\bm{A}(\phi)$ reads $\bm{A}(\phi)=\psi_1(\phi)\bm{a}_1+\psi_2(\phi)\bm{a}_2$, where the two functions $\psi_j(\phi)$ are arbitrary provided that they vanish for $\phi\to\pm\infty$ and they feature the differential properties mentioned above when the four-vector potential $A^{\mu}(\phi)$ was introduced. The field tensor $F^{\mu\nu}(\phi)=\partial^{\mu}A^{\nu}(\phi)-\partial^{\nu}A^{\mu}(\phi)$ of the plane wave is given by $F^{\mu\nu}(\phi)=n^{\mu}A^{\prime\,\nu}(\phi)-n^{\nu}A^{\prime\,\mu}(\phi)$ and below we will also use its integral $\mathscr{F}^{\mu\nu}(\phi)=\int_{-\infty}^{\phi}d\phi'F^{\mu\nu}(\phi')=n^{\mu}A^{\nu}(\phi)-n^{\nu}A^{\mu}(\phi)$ (note that the tensor $\mathscr{F}^{\mu\nu}(\phi)$ is gauge invariant). The four-dimensional quantities $n^{\mu}$, $\tilde{n}^{\mu}$, and $a^{\mu}_j$ fulfill the completeness relation: $\eta^{\mu\nu}=n^{\mu}\tilde{n}^{\nu}+\tilde{n}^{\mu}n^{\nu}-a_1^{\mu}a_1^{\nu}-a_2^{\mu}a_2^{\nu}$ (note that $(n\tilde{n})=1$ and $(\tilde{n}a_j)=0$). Below, we will refer to the longitudinal ($n$) direction as the direction along $\bm{n}$ and to the transverse ($\perp$) plane as the plane spanned by the two perpendicular unit vectors $\bm{a}_j$. In this respect, together with the light-cone time $\phi=t-x_n$, with $x_n=\bm{n}\cdot \bm{x}$, we also introduce the remaining three light-cone coordinates $T=(\tilde{n}x)=(t+x_n)/2$, and $\bm{x}_{\perp}=(x_{\perp,1},x_{\perp,2})=-((xa_1),(xa_2))=(\bm{x}\cdot\bm{a}_1,\bm{x}\cdot\bm{a}_2)$. Analogously, the light-cone coordinates of an arbitrary four-vector $v^{\mu}=(v^0,\bm{v})$ will be indicated as $v_-=(nv)=v^0-v_n$, with $v_n=\bm{n}\cdot \bm{v}$, $v_+=(\tilde{n}v)=(v^0+v_n)/2$, and $\bm{v}_{\perp}=(v_{\perp,1},v_{\perp,2})=-((va_1),(va_2))=(\bm{v}\cdot\bm{a}_1,\bm{v}\cdot\bm{a}_2)$. Since we will employ the operator technique, it is convenient to also introduce the momenta operators $P_{\phi}=-i\partial_{\phi}=-(\tilde{n}P)=-(i\partial_t-i\partial_{x_n})/2$, $P_T=-i\partial_T=-(nP)=-(i\partial_t+i\partial_{x_n})$, and $\bm{P}_{\perp}=(P_{\perp,1},P_{\perp,2})=-i(\bm{a}_1\cdot\bm{\nabla},\bm{a}_2\cdot\bm{\nabla})$. These operators are the momenta conjugated to the light-cone coordinates in the sense that the commutator between the operator corresponding to each light-cone coordinate and the associated momentum operator is equal to the imaginary unit (all other possible commutators vanish): $[\phi,P_{\phi}]=[T,P_T]=i$ and $[X_{\perp,j},P_{\perp,k}]=i\delta_{jk}$, which are equivalent to the commutation relations $[X^{\mu},P^{\nu}]=-i\eta^{\mu\nu}$, with $P^{\mu}=i\partial^{\mu}$. The commutation relations $[X^{\mu},P^{\nu}]=-i\eta^{\mu\nu}$ imply that $[P^{\mu},f(X)]=i\partial_X^{\mu}f(X)$, where $f(X)$ is an arbitrary function of the four-position operator that can be expanded in Taylor series and $\partial_X^{\mu}=\partial/\partial X_{\mu}$. Analogously, it can easily be shown that $\exp[if(X)]P^{\mu}\exp[-if(X)]=P^{\mu}+\partial^{\mu}f(X)$ and then formally that $\exp[if(X)]g(P)\exp[-if(X)]=g(P+\partial f(X))$, where $g(P)$ is a function of the four-momentum that can be expanded in Taylor series \[this identity has to be intended to apply to the Taylor series expansion of the function $g(P)$\]. The same commutation relations imply that $\exp[ig(P)]X^{\mu}\exp[-ig(P)]=X^{\mu}-\partial_P^{\mu}g(P)$ and that $\exp[ig(P)]f(X)\exp[-ig(P)]=f(X-\partial_Pg(P))$, where $\partial_P^{\mu}=\partial/\partial P_{\mu}$ \[as above, this identity has to be intended to apply to the Taylor series expansion of the function $f(X)$\]. In particular, we will consider the case where the functions in the exponents are linear either in $X^{\mu}$ or in $P^{\mu}$: $$\begin{aligned} \label{Trans_X} \exp(i(Xq))g(P)\exp(-i(Xq))&=g(P+q),\\ \label{Trans_P} \exp(i(Py))f(X)\exp(-i(Py))&=f(X-y),\end{aligned}$$ where $q^{\mu}$ and $y^{\mu}$ are constant four-vectors. In addition, the commutation relations $[\phi,P_{\phi}]=[T,P_T]=i$ imply in particular the identities $$\begin{aligned} \label{Trans_phi} \exp(ia\phi)\tilde{g}(P_{\phi})\exp(-ia\phi)&=\tilde{g}(P_{\phi}-a),\\ \label{Trans_PT} \exp(ibP_T)\tilde{f}(T)\exp(-ibP_T)&=\tilde{f}(T+b),\end{aligned}$$ with $a$ and $b$ being two constants and $\tilde{f}(T)$ and $\tilde{g}(P_{\phi})$ being two arbitrary functions, which we will use below. Note that if $|x\rangle$ ($|p\rangle$) is the eigenstate of the four-position (four-momentum) operator $X^{\mu}$ ($P^{\mu}=i\partial^{\mu}$) with eigenvalue $x^{\mu}$ ($p^{\mu}$), i.e., $X^{\mu}|x\rangle=x^{\mu}|x\rangle$ ($P^{\mu}|p\rangle=p^{\mu}|p\rangle$), then, by normalizing the eigenstates $|x\rangle$ ($|p\rangle$) such that $\langle x|y\rangle=\delta^{(4)}(x-y)$ \[$\langle p|q\rangle=(2\pi)^4\delta^{(4)}(p-q)$\], it is $\langle x|p\rangle=\exp(-i(px))=\exp[-i(p_+\phi+p_-T-\bm{p}_{\perp}\cdot\bm{x}_{\perp})]$ and $P_{\phi}|p\rangle=-p_+|p\rangle$, $P_T|p\rangle=-p_-|p\rangle$, and $\bm{P}_{\perp}|p\rangle=\bm{p}_{\perp}|p\rangle$. Also, the operator completeness relations hold $$\begin{aligned} \label{C_x} \int d^4x\,|x\rangle\langle x|&=1,\\ \int \frac{d^4p}{(2\pi)^4}\,|p\rangle\langle p|&=1.\end{aligned}$$ The Volkov states are the exact, analytical solutions of the Dirac equation in a plane wave [@Volkov_1935; @Landau_b_4_1982]. The positive-energy Volkov states $U_s(p,x)$ can be classified by means of the asymptotic momentum quantum numbers $\bm{p}$ (and then the energy $\varepsilon =\sqrt{m^2+\bm{p}^2}$) and of the asymptotic spin quantum number $s=1,2$ in the remote past, i.e. for $t\to-\infty$ (for notational simplicity, we have indicated the functional dependence on the four components of the electron four-momentum $p^{\mu}=(\varepsilon ,\bm{p})$, although the energy is a function of the linear momentum). Following the general notation in Ref. [@Landau_b_4_1982], these states can be written as $U_s(p,x)=E(p,x)u_s(p)$, where $$\label{E_p} E(p,x)=\bigg[1+\frac{e\hat{n}\hat{A}(\phi)}{2p_-}\bigg]\text{e}^{i\left\{-(px)-\int_{-\infty}^{\phi}d\varphi\left[\frac{e(pA(\varphi))}{p_-}-\frac{e^2A^2(\varphi)}{2p_-}\right]\right\}},$$ and where $u_s(p)$ are the free, positive-energy spinors normalized as $u^{\dag}_s(p)u_{s'}(p)=2\varepsilon \delta_{ss'}$ [@Landau_b_4_1982]. In Eq. (\[E\_p\]) we have introduced the notation $\hat{v}=\gamma^{\mu}v_{\mu}$ for a generic four-vector $v^{\mu}$, with $\gamma^{\mu}$ being the Dirac matrices, which satisfy the anti-commutation relations $\{\gamma^{\mu},\gamma^{\nu}\}=2\eta^{\mu\nu}$ [@Landau_b_4_1982]. The electron Green’s function $G(x,x')$ in the general plane-wave background electromagnetic field described by the four-vector potential $A^{\mu}(\phi)$ is defined by the equation $$\{\gamma^{\mu}[i\partial_{\mu}-eA_{\mu}(\phi)]-m\}G(x,x')=\delta^{(4)}(x-x').$$ In order to uniquely identify the Green’s function, boundary conditions have also to be specified. Here, we always assume the Feynman prescription corresponding to the shift $m\to m-i0$ [@Landau_b_4_1982]. Within the operator technique the operator $G$ corresponding to the Green’s function $G(x,x')$ is defined via the equation $G(x,x')=\langle x|G|x'\rangle$, i.e., as $$G=\frac{1}{\hat{\Pi}-m+i0},$$ where $\Pi^{\mu}=P^{\mu}-eA^{\mu}(\Phi)$. Now, we have explicitly shown in Ref. [@Di_Piazza_2018_d] (see also Refs. [@Baier_1976_a; @Baier_1976_b; @Di_Piazza_2007]) that the operator $G$ can be written in the form $$\label{G_1} \begin{split} G&=(\hat{\Pi}+m)\frac{1}{\hat{\Pi}^2-m^2+i0}=(\hat{\Pi}+m)(-i)\int_0^{\infty}ds\, e^{-im^2s}e^{2isP_TP_{\phi}}\\ &\times e^{-i\int_0^sds'[\bm{P}_{\perp}-e\bm{A}_{\perp}(\Phi-2s'P_T)]^2}\Big\{1-\frac{e}{2P_T}\hat{n}[\hat{A}(\Phi-2sP_T)-\hat{A}(\Phi)]\Big\}, \end{split}$$ where the prescription $m^2\to m^2-i0$ is understood. Below, we will also need the equivalent expression $$\label{G_2} \begin{split} G=&\frac{1}{\hat{\Pi}^2-m^2+i0}(\hat{\Pi}+m)=(-i)\int_0^{\infty}ds\, e^{-im^2s}\Big\{1+\frac{e}{2P_T}\hat{n}[\hat{A}(\Phi+2sP_T)-\hat{A}(\Phi)]\Big\}\\ &\times e^{-i\int_0^sds'[\bm{P}_{\perp}-e\bm{A}_{\perp}(\Phi+2s'P_T)]^2}e^{2isP_TP_{\phi}}(\hat{\Pi}+m). \end{split}$$ General expression of the one-loop vertex correction {#VC_General} ==================================================== The one-loop vertex correction corresponds to the Feynman diagram in Fig. \[FD\_VC\], where we have implicitly assumed that the photon four-momentum $q^{\mu}$ is outgoing. Note that the two external electron lines correspond to real electrons, i.e., the four-momenta $p^{\mu}$ and $p^{\prime\,\mu}$ are on-shell ($p^2=p^{\prime\,2}=m^2$), whereas at the moment we make no assumptions about the outgoing photon, i.e., in particular, $q^2\neq 0$. If we denote by $s$ ($s'$) the spin quantum number of the incoming (outgoing) electron and by $l$ the polarization quantum number of the outgoing photon, the amplitude $-ie\Gamma_{s,s',l}(p,p',q)$ corresponding to the Feynman diagram in Fig. \[FD\_VC\] can be written as $$-ie\Gamma_{s,s',l}(p,p',q)=-e^3\int d^4x\, d^4y\, d^4z\,\bar{U}_{s'}(p',y)\gamma^{\lambda}G(y,z)\hat{e}^*_l(q)e^{i(qz)}G(z,x)\gamma^{\nu}U_s(p,x)D_{\lambda\nu}(x-y),$$ where $e^{\mu}_l(q)$ is the polarization four-vector of the outgoing photon. Here, we have introduced the photon propagator $D^{\lambda\nu}(x)$ and we work in the Lorenz gauge such that $$D^{\lambda\nu}(x)=\int\frac{d^4k}{(2\pi)^4}\frac{\eta^{\lambda\nu}}{k^2-\kappa^2+i0}e^{-i(kx)},$$ where $\kappa^2$ is the square of a fictitious photon mass, which has been introduced to avoid infrared divergences. By using the completeness relation in Eq. (\[C\_x\]) and the translation properties in Eq. (\[Trans\_X\]), the amplitude can be written in the semi-operator form as $$\label{Gamma_2} \begin{split} -ie\Gamma_{s,s',l}(p,p',q)&=-e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\bar{U}_{s'}(p',x)e^{i(kx)}\gamma^{\lambda}G e^{i(qx)}\hat{e}^*_l(q)Ge^{-i(kx)}\gamma_{\lambda}U_s(p,x)\\ &=-e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\\ &\quad\times\bar{U}_{s'}(p',x)\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0} e^{i(qx)}\hat{e}^*_l(q)\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\gamma_{\lambda}U_s(p,x)\\ &=-e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\\ &\quad\times\bar{U}_{s'}(p',x)\gamma^{\lambda}[\hat{\Pi}(\phi)+\hat{k}+m]\frac{1}{[\hat{\Pi}(\phi)+\hat{k}]^2-m^2+i0} e^{i(qx)}\hat{e}^*_l(q)\\ &\qquad\times\frac{1}{[\hat{\Pi}(\phi)+\hat{k}]^2-m^2+i0}[\hat{\Pi}(\phi)+\hat{k}+m]\gamma_{\lambda}U_s(p,x), \end{split}$$ where $\Pi^{\mu}(\phi)=i\partial^{\mu}-eA^{\mu}(\phi)$. By using the fact that $[\hat{\Pi}(\phi)-m]U_s(p,x)=[\hat{\Pi}(\phi)-m]U_{s'}(p',x)=0$, we obtain $$\begin{split} -ie\Gamma_{s,s',l}(p,p',q)&=-e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\\ &\quad\times\bar{U}_{s'}(p',x)[2\Pi^{\lambda}(\phi)+\gamma^{\lambda}\hat{k}]\frac{1}{[\hat{\Pi}(\phi)+\hat{k}]^2-m^2+i0} e^{i(qx)}\hat{e}^*_l(q)\\ &\qquad\times\frac{1}{[\hat{\Pi}(\phi)+\hat{k}]^2-m^2+i0}[2\Pi_{\lambda}(\phi)+\hat{k}\gamma_{\lambda}]U_s(p,x). \end{split}$$ Now, we notice that \[see Eq. (\[E\_p\])\] $$\Pi^{\lambda}(\phi)U_s(p,x)=\left[\pi^{\lambda}_p(\phi)+i\frac{e\hat{n}\hat{A}'(\phi)}{2p_-}n^{\lambda}\right]U_s(p,x),$$ where $$\pi^{\lambda}_p(\phi)=p^{\lambda}-eA^{\lambda}(\phi)+\frac{e(pA(\phi))}{p_-}n^{\lambda}-\frac{e^2A^2(\phi)}{2p_-}n^{\lambda}$$ is the classical kinetic four-momentum of an electron in the plane wave $A^{\mu}(\phi)$, with $\lim_{\phi\to\pm\infty}\pi^{\lambda}_p(\phi)=p^{\lambda}$. The kinetic four-momentum $\pi^{\lambda}_p(\phi)$ is clearly a gauge-invariant four-vector and, by using the tensor $\mathscr{F}^{\mu\nu}(\phi)$ (see Sec. \[Notation\]), it can be written in the manifestly gauge-invariant form as $$\label{pi} \pi^{\lambda}_p(\phi)=p^{\lambda}-\frac{ep_{\mu}\mathscr{F}^{\mu\lambda}(\phi)}{p_-}+\frac{e^2p_{\mu}\mathscr{F}^{\mu\rho}(\phi)\mathscr{F}_{\rho\nu}(\phi)p^{\nu}}{2p^3_-}n^{\lambda}.$$ In this way, the quantity $-ie\Gamma_{s,s',l}(p,p',q)$ can be written as $$\begin{split} -ie\Gamma_{s,s',l}(p,p',q)&=-e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\\ &\quad\times\bar{U}_{s'}(p',x)\left[2\pi_{p'}^{\lambda}(\phi)+i\frac{e\hat{n}\hat{A}'(\phi)}{p'_-}n^{\lambda}+\gamma^{\lambda}\hat{k}\right]\frac{1}{[\hat{\Pi}(\phi)+\hat{k}]^2-m^2+i0} e^{i(qx)}\hat{e}^*_l(q)\\ &\qquad\times\frac{1}{[\hat{\Pi}(\phi)+\hat{k}]^2-m^2+i0}\left[2\pi_{p,\lambda}(\phi)+i\frac{e\hat{n}\hat{A}'(\phi)}{p_-}n_{\lambda}+\hat{k}\gamma_{\lambda}\right]U_s(p,x). \end{split}$$ At this point, it is convenient to use the representations in Eq. (\[G\_1\]) and in Eq. (\[G\_2\]) for the second and the first square Volkov propagator $1/\{[\hat{\Pi}(\phi)+\hat{k}]^2-m^2+i0\}$, respectively: $$\begin{split} &-ie\Gamma_{s,s',l}(p,p',q)=e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\int_0^{\infty}ds\int_0^{\infty}du\,\frac{e^{i(qx)}}{k^2-\kappa^2+i0}\\ &\quad\times\bar{U}_{s'}(p',x)\left[2\pi_{p'}^{\lambda}(\phi)+i\frac{e\hat{n}\hat{A}'(\phi)}{p'_-}n^{\lambda}+\gamma^{\lambda}\hat{k}\right]e^{-im^2s}e^{-2is(p_--q_-+k_-)(P_{\phi}-k_++q_+)}\\ &\quad\times e^{-i\int_0^sds'[\bm{p}_{\perp}-\bm{q}_{\perp}+\bm{k}_{\perp}-e\bm{A}_{\perp}(\phi+2s'(p_--q_-+k_-))]^2}\left\{1+\frac{e\hat{n}[\hat{A}(\phi+2s(p_--q_-+k_-))-\hat{A}(\phi)]}{2(p_--q_-+k_-)}\right\}\\ &\quad\times \hat{e}^*_l(q)e^{-im^2u}\left\{1-\frac{e\hat{n}[\hat{A}(\phi-2u(p_-+k_-))-\hat{A}(\phi)]}{2(p_-+k_-)}\right\}e^{-i\int_0^udu'[\bm{p}_{\perp}+\bm{k}_{\perp}-e\bm{A}_{\perp}(\phi-2u'(p_-+k_-))]^2}\\ &\quad\times e^{-2iu(p_-+k_-)(P_{\phi}-k_+)}\left[2\pi_{p,\lambda}(\phi)+i\frac{e\hat{n}\hat{A}'(\phi)}{p_-}n_{\lambda}+\hat{k}\gamma_{\lambda}\right]U_s(p,x), \end{split}$$ where we have exploited the fact that Volkov states are eigenstates of the operators $P_T$ and $\bm{P}_{\perp}$. Indeed, the only operator remaining in this equation is $P_{\phi}$. Now, we use the translation property in Eq. (\[Trans\_P\]) and, analogously to the vacuum case, we write the amplitude $-ie\Gamma_{s,s',l}(p,p',q)$ in the form $$\label{Gamma^mu} -ie\Gamma_{s,s',l}(p,p',q)=-ie\int d^4x\,e^{i(qx)}\bar{U}_{s'}(p',x)\Gamma^{\mu}(p,p',q;\phi)U_s(p,x)e^*_{l,\mu}(q),$$ where $$\label{Gamma_3} \begin{split} -ie\Gamma^{\mu}(p,p',q;\phi)&=e^3\int\frac{d^4k}{(2\pi)^4}\int_0^{\infty}ds\int_0^{\infty}du\,\frac{e^{-im^2(s+u)}}{k^2-\kappa^2+i0}\\ &\quad\times e^{2ik_+[s(p'_-+k_-)+u(p_-+k_-)]}e^{i\left\{p'_+(\phi_s-\phi)+\int_{\phi}^{\phi_s}d\phi'\left[-\frac{e\bm{p}'_{\perp}\cdot\bm{A}_{\perp}(\phi')}{p'_-}+\frac{e^2\bm{A}^2_{\perp}(\phi')}{2p'_-}\right]\right\}}\\ &\quad\times e^{-i\int_0^sds'[\bm{p}'_{\perp}+\bm{k}_{\perp}-e\bm{A}_{\perp}(\phi_{s'})]^2}e^{-i\int_0^udu'[\bm{p}_{\perp}+\bm{k}_{\perp}-e\bm{A}_{\perp}(\phi_{u'})]^2}\\ &\quad\times e^{i\left\{-p_+(\phi_u-\phi)-\int_{\phi}^{\phi_u}d\phi'\left[-\frac{e\bm{p}_{\perp}\cdot\bm{A}_{\perp}(\phi')}{p_-}+\frac{e^2\bm{A}^2_{\perp}(\phi')}{2p_-}\right]\right\}}M^{\mu}(\phi,k,s,u). \end{split}$$ Here, we have introduced the quantities $$\begin{aligned} \label{phi_s} \phi_s&=\phi+2s(p'_-+k_-),\\ \label{phi_u} \phi_u&=\phi-2u(p_-+k_-),\end{aligned}$$ and the matrix $$\begin{split} M^{\mu}(k,s,u;\phi)&=\left\{1-\frac{e\hat{n}[\hat{A}(\phi_s)-\hat{A}(\phi)]}{2p'_-}\right\}\left[2\pi_{p'}^{\lambda}(\phi_s)+i\frac{e\hat{n}\hat{A}'(\phi_s)}{p'_-}n^{\lambda}+\gamma^{\lambda}\hat{k}\right]\\ &\quad\times\left\{1+\frac{e\hat{n}[\hat{A}(\phi_s)-\hat{A}(\phi)]}{2(p'_-+k_-)}\right\}\gamma^{\mu}\left\{1-\frac{e\hat{n}[\hat{A}(\phi_u)-\hat{A}(\phi)]}{2(p_-+k_-)}\right\}\\ &\quad\times \left[2\pi_{p,\lambda}(\phi_u)+i\frac{e\hat{n}\hat{A}'(\phi_u)}{p_-}n_{\lambda}+\hat{k}\gamma_{\lambda}\right]\left\{1+\frac{e\hat{n}[\hat{A}(\phi_u)-\hat{A}(\phi)]}{2p_-}\right\}. \end{split}$$ Note that the integrals in $T$ and $\bm{x}_{\perp}$ can be easily taken and enforce the conservation laws $p_-=p'_-+q_-$ and $\bm{p}_{\perp}=\bm{p}'_{\perp}+\bm{q}_{\perp}$, typical of problems in a plane-wave background field. As a related remark, it is clear that the quantity $\Gamma^{\mu}(p,p',q;\phi)$, unlike the corresponding vacuum expression, depends also on the plane-wave phase. Finally, the definition in Eq. (\[Gamma\^mu\]) is consistent with the idea that computing the amplitude of the vertex in a plane wave up to one loop, one can use the substitution rule $\gamma^{\mu}\to \gamma^{\mu}+\Gamma^{\mu}(p,p',q;\phi)$, as for the vertex correction in vacuum. The phase in Eq. (\[Gamma\_3\]) can be written in a compact form by turning the integral from $\phi$ to $\phi_s$ (from $\phi$ to $\phi_u$) into an integral in $s'$ ($u'$) like that in the third line of Eq. (\[Gamma\_3\]). By exponentiating also the denominator $k^2-\kappa^2+i0$ in the photon propagator, the quantity $\Gamma^{\mu}(p,p',q;\phi)$ can be written as $$\Gamma^{\mu}(p,p',q;\phi)=e^2\int\frac{d^4k}{(2\pi)^4}\int_0^{\infty}ds\int_0^{\infty}du\int_0^{\infty}dt\,e^{iSk^2-i\kappa^2t+2i(k\tilde{F})} M^{\mu}(k,s,u;\phi),$$ where $S=u+s+t$ and $$\label{F} \tilde{F}^{\mu}=\int_0^sds'\pi_{p'}^{\mu}(\phi_{s'})+\int_0^udu'\pi_p^{\mu}(\phi_{u'}).$$ As next step, we can perform the integrals in $d^4k$ analytically by shifting the four-momentum $k^{\mu}$ by setting $k^{\prime\mu}=k^{\mu}+\tilde{F}^{\mu}/S$, which, since all components of $\tilde{F}^{\mu}$ except $\tilde{F}_-$ depend on $k_-$, implies that $k^{\mu}=k^{\prime\mu}-\tilde{G}^{\mu}/S$, where $$\label{tG} \tilde{G}^{\mu}=\int_0^sds'\pi_{p'}^{\mu}(\tilde{\psi}_{s'})+\int_0^udu'\pi_p^{\mu}(\tilde{\psi}_{u'}),$$ such that $\tilde{G}_-=\tilde{F}_-=sp'_-+up_-$. Here, we have introduced the two shifted phases $$\begin{aligned} \label{tpsi_p} \tilde{\psi}_s&=\phi+2s\tau'_-+2sk_-,\\ \label{tpsi_pp} \tilde{\psi}_u&=\phi-2u\tau_--2uk_-,\end{aligned}$$ where $$\begin{aligned} \label{tau_p_m} \tau'_-&=p'_--\frac{\tilde{G}_-}{S}=\frac{tp'_--uq_-}{S}, \\ \label{tau_m} \tau_-&=p_--\frac{\tilde{G}_-}{S}=\frac{tp_-+sq_-}{S}.\end{aligned}$$ After the shift of the four-momentum $k^{\mu}$, we can write $\Gamma^{\mu}(p,p',q;\phi)$ in the form $$\Gamma^{\mu}(p,p',q;\phi)=e^2\int_0^{\infty}dsdudt\int\frac{d^4k}{(2\pi)^4}e^{-i\kappa^2t-i\frac{\tilde{G}^2}{S}+iSk^2}\tilde{L}(\tilde{Q}^{\prime\lambda}+\gamma^{\lambda}\hat{k})\tilde{C}^{\mu}(\tilde{Q}_{\lambda}+\hat{k}\gamma_{\lambda})\tilde{R},$$ where $$\begin{aligned} \label{tL} \tilde{L}&=1-\frac{e\hat{n}[\hat{A}(\tilde{\psi}_s)-\hat{A}(\phi)]}{2p'_-},\\ \tilde{Q}^{\lambda}&=2\pi_p^{\lambda}(\tilde{\psi}_u)+i\frac{e\hat{n}\hat{A}'(\tilde{\psi}_u)}{p_-}n^{\lambda}-\frac{\hat{\tilde{G}}}{S}\gamma^{\lambda},\\ \tilde{C}^{\mu}&=\left\{1+\frac{e\hat{n}[\hat{A}(\tilde{\psi}_s)-\hat{A}(\phi)]}{2(\tau'_-+k_-)}\right\}\gamma^{\mu}\left\{1-\frac{e\hat{n}[\hat{A}(\tilde{\psi}_u)-\hat{A}(\phi)]}{2(\tau_-+k_-)}\right\},\\ \tilde{Q}^{\prime\lambda}&=2\pi_{p'}^{\lambda}(\tilde{\psi}_s)+i\frac{e\hat{n}\hat{A}'(\tilde{\psi}_s)}{p'_-}n^{\lambda}-\gamma^{\lambda}\frac{\hat{\tilde{G}}}{S},\\ \label{tR} \tilde{R}&=1+\frac{e\hat{n}[\hat{A}(\tilde{\psi}_u)-\hat{A}(\phi)]}{2p_-}.\end{aligned}$$ The integral in $d^4k$ in $\Gamma^{\mu}(p,p',q;\phi)$ is complicated by the fact that the variable $k_-$ is contained in the argument of the four-vector potential of the plane wave. Thus, we first take the integral in $d^2\bm{k}_{\perp}$, which is Gaussian: $$\Gamma^{\mu}(p,p',q;\phi)=-i\alpha\int_0^{\infty}\frac{dsdudt}{S}\int\frac{dk_-dk_+}{(2\pi)^2}e^{-i\kappa^2t-i\frac{\tilde{G}^2}{S}+2iSk_-k_+}\tilde{M}^{\mu}(k_-,k_+,s,u,t;\phi),$$ where $\alpha=e^2/4\pi\approx 1/137$ is the fine-structure constant and where $$\begin{split} \tilde{M}^{\mu}(k_-,k_+,s,u,t;\phi)=&\tilde{L}\left[(\tilde{Q}^{\prime\lambda}+k_-\gamma^{\lambda}\hat{\tilde{n}})\tilde{C}^{\mu}(\tilde{Q}_{\lambda}+k_-\hat{\tilde{n}}\gamma_{\lambda})-\frac{i}{2S}\gamma^{\lambda}\gamma_{\perp,i}\tilde{C}^{\mu}\gamma_{\perp,i}\gamma_{\lambda}\right]\tilde{R}\\ &+k_+\tilde{L}[\gamma^{\lambda}\hat{n}\tilde{C}^{\mu}(\tilde{Q}_{\lambda}+k_-\hat{\tilde{n}}\gamma_{\lambda})+(\tilde{Q}^{\prime\lambda}+k_-\gamma^{\lambda}\hat{\tilde{n}})\tilde{C}^{\mu}\hat{n}\gamma_{\lambda}]\tilde{R}\\ &+k_+^2\tilde{L}\gamma^{\lambda}\hat{n}\gamma^{\mu}\hat{n}\gamma_{\lambda}\tilde{R}. \end{split}$$ Finally, the integral in $dk_+$ results in a delta function and its first and second derivatives all evaluated at $2Sk_-$. This allows then also to take the integral in $dk_-$ and, after straightforward manipulations, the resulting expression of $\Gamma^{\mu}(p,p',q;\phi)$ can be written as $$\label{Gamma_f} \begin{split} \Gamma^{\mu}(p,p',q;\phi)&=-\frac{i\alpha}{4\pi}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t}\bigg\{e^{-i\frac{\tilde{G}^2}{S}}\tilde{L}(S\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+2i\tilde{C}^{\mu})\tilde{R}\\ &\quad\left.\left.+\frac{i}{2}\frac{d}{dk_-}\left[e^{-i\frac{\tilde{G}^2}{S}}\tilde{L}(\gamma^{\lambda}\hat{n}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\hat{n}\gamma_{\lambda})\tilde{R}\right]+\frac{\hat{n}}{S}n^{\mu}\frac{d^2}{dk^2_-}\left(e^{-i\frac{\tilde{G}^2}{S}}\right)\right\}\right\vert_{k_-=0}\\ &=-\frac{i\alpha}{4\pi}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t-i\frac{\tilde{G}^2}{S}}\\ &\quad\times\Bigg\{\tilde{L}\left[S\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+2i\tilde{C}^{\mu}+\frac{1}{2S}\frac{d\tilde{G}^2}{dk_-}(\gamma^{\lambda}\hat{n}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\hat{n}\gamma_{\lambda})\right]\tilde{R}\\ &\quad\left.\left.+\frac{i}{2}\frac{d}{dk_-}\left[\tilde{L}(\gamma^{\lambda}\hat{n}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\hat{n}\gamma_{\lambda})\tilde{R}\right]-\frac{\hat{n}}{S^2}n^{\mu}\left[\frac{1}{S}\left(\frac{d\tilde{G}^2}{dk_-}\right)^2+i\frac{d^2\tilde{G}^2}{dk^2_-}\right]\right\}\right\vert_{k_-=0}. \end{split}$$ This expression can be further manipulated especially to simplify its matrix structure. However, it is first convenient to make the following considerations related to the Ward identity to be fulfilled by $\Gamma^{\mu}(p,p',q;\phi)$ [@Itzykson_b_1980]. From now on we assume that $q_->0$. Thus, by using the three four-vectors $$\begin{aligned} N^{\mu}&=q^{\mu}-\frac{q^2n^{\mu}}{2q_-},\\ \label{Lambda_i} \Lambda^{\mu}_i&=a_i^{\mu}+\frac{q_{\perp,i}n^{\mu}}{q_-},\end{aligned}$$ with $i=1,2$ together with $n^{\mu}$, one can build a light-cone basis such that $$\eta^{\mu\nu}=\frac{N^{\mu}n^{\nu}+n^{\mu}N^{\nu}}{q_-}-\Lambda^{\mu}_1\Lambda^{\nu}_1-\Lambda^{\mu}_2\Lambda^{\nu}_2.$$ Then, the quantity $\Gamma^{\mu}(p,p',q;\phi)e^*_{l,\mu}(q)$, which is the one finally required here, can be written as \[recall that we work in the Lorenz gauge where $(qe^*_l(q))=0$\] $$\label{Gamma_exp} \begin{split} \Gamma^{\mu}(p,p',q;\phi)e^*_{l,\mu}(q)&=\frac{e^*_{l,-}(q)}{q_-}\Gamma_q(p,p',q;\phi)-\frac{q^2e^*_{l,-}(q)}{q_-^2}\Gamma_-(p,p',q;\phi)\\ &\quad-(\Gamma(p,p',q;\phi)\Lambda_1)(\Lambda_1e^*_l(q))-(\Gamma(p,p',q;\phi)\Lambda_2)(\Lambda_2e^*_l(q))\\ &=\frac{e^*_{l,-}(q)}{q_-}\Gamma_q(p,p',q;\phi)\\ &\quad-\left[\frac{q^2e^*_{l,-}(q)}{q_-^2}+\frac{q_{\perp,1}}{q_-}(\Lambda_1e^*_l(q))+\frac{q_{\perp,2}}{q_-}(\Lambda_2e^*_l(q))\right]\Gamma_-(p,p',q;\phi)\\ &\quad+\Gamma_{\perp,1}(p,p',q;\phi)(\Lambda_1e^*_l(q))+\Gamma_{\perp,2}(p,p',q;\phi)(\Lambda_2e^*_l(q)) \end{split}$$ where $\Gamma_q(p,p',q;\phi)=(\Gamma^{\mu}(p,p',q;\phi)q_{\mu})$ and all other symbols are defined in analogy to the definitions given in the introduction. Now, since the structure of the function $\Gamma^{\mu}(p,p',q;\phi)$ is complicated because of the presence of the plane wave, it is clear that the components $\Gamma_-(p,p',q;\phi)=(\Gamma^{\mu}(p,p',q;\phi)n_{\mu})$ and $\Gamma_{\perp,j}(p,p',q;\phi)=-(\Gamma^{\mu}(p,p',q;\phi)a_{j,\mu})$ are relatively easy to work out because the quantities $n^{\mu}$ and $a_i^{\mu}$ characterize the plane wave. For example, we observe that all the terms proportional to $n^{\mu}$ in $\Gamma^{\mu}(p,p',q;\phi)$ can be ignored in the computation of the components $\Gamma_-(p,p',q;\phi)$ and $\Gamma_{\perp,j}(p,p',q;\phi)$. The apparently most complicated term is, therefore, $\Gamma_q(p,p',q;\phi)$, which is related to the fact that by itself the vertex-correction function is not gauge invariant. However, the gauge invariance of QED guarantees that the component $\Gamma_q(p,p',q;\phi)$ of the vertex-correction function does not to contribute to any transition amplitude involving on-shell external electrons/positrons. We explicitly prove this statement in the case under consideration with an incoming and an outgoing electron, the other possible cases being proved in an analogous way. First, we start back from Eq. (\[Gamma\_2\]) and we apply the same procedure to prove the Ward identity [@Mitter_1975; @Morozov_1981]. From the second equality in Eq. (\[Gamma\_2\]) and from the definition of $\Gamma^{\mu}(p,p',q;\phi)$ in Eq. (\[Gamma\^mu\]), we obtain $$\Gamma_q(p,p',q;\phi)=-ie^2\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m+i0}\hat{q}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\gamma_{\lambda}.$$ now, by writing $\hat{q}=\hat{\Pi}(\phi)+\hat{k}-m-[\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m]$ it is clear that we can express $\Gamma_q(p,p',q;\phi)$ as the difference of two terms containing only one propagator in the plane wave: $$\label{Gamma_q_0} \Gamma_q(p,p',q;\phi)=-ie^2\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\gamma^{\lambda}\left[\frac{1}{\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m+i0}-\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\right]\gamma_{\lambda}.$$ At this point, we observe that in computing, for example, the one-loop radiative corrections to nonlinear Compton scattering (see the leading-order diagram in Fig. \[FD\_NCS\_LO\], for the kinematical situation corresponding to the case under study) ![The leading-order Feynman diagram corresponding to nonlinear Compton scattering of an off-shell photon. The double lines represent exact electron states in a plane wave (Volkov states) [@Landau_b_4_1982].[]{data-label="FD_NCS_LO"}](Figure_4.pdf){width="0.4\columnwidth"} we also have to include the remaining diagrams listed in Fig. \[FD\_NCS\_MO\_PO\]. ![The one-loop Feynman diagrams corresponding, together with the one-loop vertex correction in Fig. \[FD\_VC\], to the leading-order radiative corrections of nonlinear Compton scattering of an off-shell photon. The double lines represent exact electron states and propagator in a plane wave (Volkov states and propagator, respectively) [@Landau_b_4_1982].[]{data-label="FD_NCS_MO_PO"}](Figure_5.pdf){width="\columnwidth"} If we indicate as $i\mathcal{M}^{(1)}_{s,s',\mu}(p,p',q)e_l^{*\,\mu}(q)$ the amplitude of the one-loop radiative corrections to nonlinear Compton scattering represented by the diagrams in Figs. \[FD\_VC\] and \[FD\_NCS\_MO\_PO\], the gauge invariance of QED implies that $\mathcal{M}^{(1)}_{s,s',\mu}(p,p',q)q^{\mu}=0$ [@Peskin_b_1995]. Since the contribution corresponding to Fig. \[FD\_NCS\_MO\_PO\].c is by itself gauge invariant [@Baier_1976_b; @Meuren_2013], by summing the contributions from Fig. \[FD\_VC\] and from Figs. \[FD\_NCS\_MO\_PO\].b and \[FD\_NCS\_MO\_PO\].c, we obtain \[see also Eq. (\[Gamma\_q\_0\])\] $$\begin{split} i\mathcal{M}^{(1)}_{s,s',\mu}(p,p',q)q^{\mu}&=-e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\\ &\times\bar{U}_{s'}(p',x)\left\{\gamma^{\lambda}\left[\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}e^{i(qx)}-e^{i(qx)}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\right]\gamma_{\lambda}\right.\\ &\quad+\hat{q}e^{i(qx)}\frac{1}{\hat{\Pi}(\phi)-m+i0}\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\gamma_{\lambda}\\ &\quad\left.+\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\gamma_{\lambda}\frac{1}{\hat{\Pi}(\phi)-m+i0}\hat{q}e^{i(qx)}\right\}U_s(p,x). \end{split}$$ Now, by using the fact that $[\hat{\Pi}(\phi)-m]U_s(p,x)=[\hat{\Pi}(\phi)-m]U_{s'}(p',x)=0$, we first replace $\hat{q}$ with $\hat{\Pi}(\phi)+\hat{q}-m$ ($m-\hat{\Pi}(\phi)+\hat{q}$) in the third (fourth) line of this equation and then we move the exponential $\exp[i(qx)]$ to the left of all other operators by exploiting the identity in Eq. (\[Trans\_X\]). The result is $$\begin{split} i\mathcal{M}^{(1)}_{s,s',\mu}(p,p',q)q^{\mu}&=-e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\,\frac{e^{i(qx)}}{k^2-\kappa^2+i0}\\ &\times\bar{U}_{s'}(p',x)\left\{\gamma^{\lambda}\left[\frac{1}{\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m+i0}-\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\right]\gamma_{\lambda}\right.\\ &\quad\left.+\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\gamma_{\lambda}-\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m+i0}\gamma_{\lambda}\right\}U_s(p,x), \end{split}$$ which indeed vanishes identically. This result indicates that gauge invariance implies that the component $\Gamma_q(p,p',q;\phi)$ can be ignored as it will always be compensated by the corresponding contributions arising from the mass operators (some properties of $\Gamma_q(p,p',q;\phi)$ are discussed in the appendix). At this point, we can consider the other component $\Gamma_-(p,p',q;\phi)$, whose structure is particularly easy. In fact, starting from Eq. (\[Gamma\_f\]), we have that $$\label{Gamma_-} \begin{split} \Gamma_-(p,p',q;\phi)&=-\frac{i\alpha}{4\pi}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t-i\frac{G^2}{S}}\left(SLQ^{\prime\lambda}\hat{n}Q_{\lambda}R+2i\hat{n}\right)\\ &=-\frac{i\alpha}{2\pi}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t-i\frac{G^2}{S}}\left[\left(2S(\pi_s\pi_u)+\frac{G^2}{S}+i\right)\hat{n}-2G_-(\hat{\pi}_sR+L\hat{\pi}_u)\right.\\ &\left.\quad-\hat{G}\hat{\pi}_s\hat{n}-\hat{n}\hat{\pi}_u\hat{G}+2\tau_-L\hat{G}+2\tau'_-\hat{G}R+2\frac{G_-}{S}\hat{G}-\frac{G_-^2}{S}\frac{\hat{\Delta}_s\hat{n}\hat{\Delta}_u}{p_-p'_-}\right], \end{split}$$ where $$\begin{aligned} \pi_s^{\mu}&=\pi^{\mu}_{p'}(\psi_s),\quad \Delta^{\mu}_s=e[A^{\mu}(\psi_s)-A^{\mu}(\phi)], & L&=1-\frac{\hat{n}\hat{\Delta}_s}{2p'_-}, \quad \psi_s=\phi+2s\tau'_-,\\ \pi_u^{\mu}&=\pi^{\mu}_p(\psi_u),\quad \Delta^{\mu}_u=e[A^{\mu}(\psi_u)-A^{\mu}(\phi)], & R&=1+\frac{\hat{n}\hat{\Delta}_u}{2p_-},\quad \psi_u=\phi-2u\tau_-,\\ C^{\mu}&=\left(1+\frac{\hat{n}\hat{\Delta}_s}{2\tau'_-}\right)\gamma^{\mu}\left(1-\frac{\hat{n}\hat{\Delta}_u}{2\tau_-}\right), & G^{\mu}&=\int_0^sds'\pi^{\mu}_{p'}(\psi_{s'})+\int_0^udu'\pi^{\mu}_p(\psi_{u'}), \\ Q^{\lambda}&=2\pi_u^{\lambda}+i\frac{e\hat{n}\hat{A}'(\psi_u)}{p_-}n^{\lambda}-\frac{\hat{G}}{S}\gamma^{\lambda},& Q^{\prime\lambda}&=2\pi_s^{\lambda}+i\frac{e\hat{n}\hat{A}'(\psi_s)}{p'_-}n^{\lambda}-\gamma^{\lambda}\frac{\hat{G}}{S}.\end{aligned}$$ As we have mentioned, in order to compute the components $\Gamma_{\perp,j}(p,p',q;\phi)=-(\Gamma^{\mu}(p,p',q;\phi)a_{j,\mu})$ we can effectively assume that the matrix $\hat{n}$ anticommutes with $\gamma^{\mu}$. In the following four equations, with an abuse of notation, we use the equal symbol also for two matrices that are equal to each other up to terms proportional to $n^{\mu}$, which can anyway be ignored in the computation of $\Gamma_{\perp,j}(p,p',q;\phi)$. Going through the terms in Eq. (\[Gamma\_f\]) in order of complexity, one can easily show that $$\begin{aligned} &LC^{\mu}R=\gamma^{\mu}+\frac{G_-}{2p'_-\tau'_-S}\hat{n}\hat{\Delta}_s\gamma^{\mu}-\frac{G_-}{2p_-\tau_-S}\gamma^{\mu}\hat{n}\hat{\Delta}_u,\\ \begin{split} &L(\gamma^{\lambda}\hat{n}C^{\mu}Q_{\lambda}+Q^{\prime\lambda}C^{\mu}\hat{n}\gamma_{\lambda})R=-\frac{2\tau_-}{p'_-}\hat{n}\hat{\Delta}_s\gamma^{\mu}+\frac{2\tau'_-}{p_-}\gamma^{\mu}\hat{n}\hat{\Delta}_u-\frac{4G_-}{S}\gamma^{\mu}+\frac{4G^{\mu}}{S}\hat{n}\\ &\qquad-2\hat{n}\gamma^{\mu}\hat{\pi}_s-2\hat{\pi}_u\gamma^{\mu}\hat{n}, \end{split}\\ \begin{split} &\frac{d}{dk_-}\left[\tilde{L}(\gamma^{\lambda}\hat{n}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\hat{n}\gamma_{\lambda})\tilde{R}\right]_{k_-=0}=8\left(\frac{G_1^{\mu}}{S}-\frac{s\tau_-}{p'_-}\mathcal{A}_s^{\prime\,\mu}+\frac{u\tau'_-}{p_-}\mathcal{A}_u^{\prime\,\mu}\right)\hat{n}\\ &\qquad+4s\left(1+\frac{\tau_-}{p'_-}\right)\hat{n}\gamma^{\mu}\hat{\mathcal{A}}_s'-4u\left(1+\frac{\tau'_-}{p_-}\right)\hat{\mathcal{A}}_u'\gamma^{\mu}\hat{n}, \end{split}\\ \begin{split} &LQ^{\prime\,\lambda}C^{\mu}Q_{\lambda}R=4(\pi_s\pi_u)\left(\gamma^{\mu}+\frac{G_-}{2p'_-\tau'_-S}\hat{n}\hat{\Delta}_s\gamma^{\mu}-\frac{G_-}{2p_-\tau_-S}\gamma^{\mu}\hat{n}\hat{\Delta}_u\right)\\ &\qquad+2i\frac{\tau_-}{p'_-}\hat{n}\hat{\mathcal{A}}'_s\gamma^{\mu}+2i\frac{\tau'_-}{p_-}\gamma^{\mu}\hat{n}\hat{\mathcal{A}}'_u-\frac{2}{S}LC^{\mu}\hat{G}\hat{\pi}_sR-\frac{2}{S}L\hat{\pi}_u\hat{G}C^{\mu}R\\ &\qquad-\frac{2}{S^2}L\hat{G}\left(\gamma^{\mu}+\frac{\gamma^{\mu}\hat{\Delta}_s\hat{n}}{2\tau'_-}-\frac{\hat{\Delta}_u\hat{n}\gamma^{\mu}}{2\tau_-}\right)\hat{G}R \end{split}\end{aligned}$$ where $$\begin{split} G_1^{\mu}&=\frac{d}{d\phi}\left[\int_0^sds'\,s'\pi^{\mu}_{p'}(\psi_{s'})-\int_0^udu'\,u'\pi^{\mu}_p(\psi_{u'})\right]\\ &=\frac{1}{2\tau'_-}\left[s\pi_{p'}^{\mu}(\psi_s)-\int_0^sds'\pi_{p'}^{\mu}(\psi_{s'})\right]+\frac{1}{2\tau_-}\left[u\pi_p^{\mu}(\psi_u)-\int_0^udu'\pi_p^{\mu}(\psi_{u'})\right] \end{split}$$ and $$\mathcal{A}^{\mu}_{s/u}=eA^{\mu}(\psi_{s/u})$$ (the prime on these quantities indicates the derivative with respect to $\phi$). In this way, we obtain the following expressions of the transverse components $\Gamma_{\perp,i}(p,p',q;\phi)$: $$\label{Gamma_j} \begin{split} \Gamma_{\perp,j}(p,p',q;\phi)&=\frac{i\alpha}{2\pi}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t-i\frac{G^2}{S}}\\ &\quad\times\left\{(2S(\pi_s\pi_u)+i)\left(\hat{a}_j+\frac{G_-}{2p'_-\tau'_-S}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{G_-}{2p_-\tau_-S}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\right.\\ &\qquad-L(Ca_j)\hat{G}\hat{\pi}_sR-L\hat{\pi}_u\hat{G}(Ca_j)R-\frac{1}{S}L\hat{G}\left(\hat{a}_j+\frac{\hat{a}_j\hat{\Delta}_s\hat{n}}{2\tau'_-}-\frac{\hat{\Delta}_u\hat{n}\hat{a}_j}{2\tau_-}\right)\hat{G}R\\ &\qquad-\frac{2(GG_1)}{S}\left(\frac{\tau_-}{p'_-}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{\tau'_-}{p_-}\hat{a}_j\hat{n}\hat{\Delta}_u+\frac{2G_-}{S}\hat{a}_j-\frac{2(Ga_j)}{S}\hat{n}+\hat{n}\hat{a}_j\hat{\pi}_s+\hat{\pi}_u\hat{a}_j\hat{n}\right)\\ &\qquad+2i\left(\frac{(G_1a_j)}{S}+s(\mathcal{A}'_sa_j)-u(\mathcal{A}'_ua_j)\right)\hat{n}\\ &\qquad\left.+i\left[s-(u+t)\frac{\tau_-}{p'_-}\right]\hat{\mathcal{A}}_s'\hat{n}\hat{a}_j-i\left[u-(s+t)\frac{\tau'_-}{p_-}\right]\hat{a}_j\hat{n}\hat{\mathcal{A}}_u'\right\} \end{split}$$ Equations (\[Gamma\_exp\]), (\[Gamma\_-\]), and (\[Gamma\_j\]) are the main results of the paper and, as it can easily be shown, they reduce to the result in vacuum as, e.g., on page 339 of Ref. [@Itzykson_b_1980] \[as it is shown in the appendix, the component $\Gamma_q(p,p',q;\phi)$ vanishes in vacuum\]. We notice that the pre-exponential matrices in all terms feature symmetry properties such that they can all be written as the sum of two classes of terms with the second one, being obtained from the first one by: 1) taking the Dirac conjugate, 2) swapping all indexes $s$ and $u$ in each quantity. We have exploited this symmetry in the computations presented below. Finally, we observe that all the terms in Eqs. (\[Gamma\_exp\]), (\[Gamma\_-\]), and (\[Gamma\_j\]) have at most three gamma matrices except the three terms on the third line of Eq. (\[Gamma\_j\]) [^1]. The three terms in the third line of Eq. (\[Gamma\_j\]) can be easily reduced to expressions containing at most five gamma matrices: $$\label{5_gamma_1} \begin{split} &L(Ca_j)\hat{G}\hat{\pi}_sR=\left(\hat{a}_j+\frac{G_-}{2Sp'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j -\frac{G_-}{2Sp_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\hat{G}\hat{\pi}_s\\ &+\left(\hat{a}_j+\frac{G_-}{2Sp'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j\right)\left(p'_-\hat{G}-G_-\hat{\pi}_s\right)\frac{\hat{\Delta}_u}{p_-}\\ &+\frac{\hat{a}_j\hat{n}}{2p_-\tau_-}[2(G_-(\Delta_u\pi_s)-p'_-(\Delta_uG))\hat{\Delta}_u-(G_-\Delta_u^2+2\tau_-(\Delta_uG))\hat{\pi}_s+(p'_-\Delta_u^2+2\tau_-(\Delta_u\pi_s))\hat{G}], \end{split}$$ $$\label{5_gamma_2} \begin{split} &L\hat{\pi}_u\hat{G}(Ca_j)R=\hat{\pi}_u\hat{G}\left(\hat{a}_j+\frac{G_-}{2Sp'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j -\frac{G_-}{2Sp_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\\ &+\frac{\hat{\Delta}_s}{p'_-}\left(p_-\hat{G}-G_-\hat{\pi}_u\right)\left(\hat{a}_j-\frac{G_-}{2Sp_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\\ &+[2(G_-(\Delta_s\pi_u)-p_-(\Delta_sG))\hat{\Delta}_s-(G_-\Delta_s^2+2\tau'_-(\Delta_sG))\hat{\pi}_u+(p_-\Delta_s^2+2\tau'_-(\Delta_s\pi_u))\hat{G}]\frac{\hat{n}\hat{a}_j}{2p'_-\tau'_-}, \end{split}$$ $$\label{5_gamma_3} \begin{split} &L\hat{G}\left(\hat{a}_j+\frac{\hat{a}_j\hat{\Delta}_s\hat{n}}{2\tau'_-}-\frac{\hat{\Delta}_u\hat{n}\hat{a}_j}{2\tau_-}\right)\hat{G}R=\frac{G_-}{2}\left(\frac{\hat{G}\hat{a}_j\hat{\Delta}_s\hat{n}\hat{\Delta}_u}{p_-\tau'_-}+\frac{\hat{\Delta}_s\hat{n}\hat{\Delta}_u\hat{a}_j\hat{G}}{p'_-\tau_-}\right)+G_-\left(\frac{\hat{a}_j\hat{\Delta}_s\hat{G}}{\tau'_-}+\frac{\hat{G}\hat{\Delta}_u\hat{a}_j}{\tau_-}\right)\\ &+\left[(p'_-+\tau'_-)(Ga_j)+G_-(\Delta_sa_j)\right]\frac{\hat{\Delta}_s\hat{n}\hat{G}}{p'_-\tau'_-}+\left[(p_-+\tau_-)(Ga_j)+G_-(\Delta_ua_j)\right]\frac{\hat{G}\hat{n}\hat{\Delta}_u}{p_-\tau_-}-G^2\hat{a}_j\\ &-\left[(p'_-+\tau'_-)G^2+\frac{\tau'_-G^2_-}{p_-\tau_-}\Delta^2_u\right]\frac{\hat{\Delta}_s\hat{n}\hat{a}_j}{2p'_-\tau'_-}-\left[(p_-+\tau_-)G^2+\frac{\tau_-G^2_-}{p'_-\tau'_-}\Delta^2_s\right]\frac{\hat{a}_j\hat{n}\hat{\Delta}_u}{2p_-\tau_-}+2(Ga_j)\hat{G}\\ &-[G_-\Delta_s^2+2p'_-(G\Delta_s)]\frac{\hat{a}_j\hat{n}\hat{G}}{2p'_-\tau'_-}-[G_-\Delta_u^2+2p_-(G\Delta_u)]\frac{\hat{G}\hat{n}\hat{a}_j}{2p_-\tau_-}\\ &+\frac{G_-}{p_-p'_-}\left[(Ga_j)+\frac{G_-}{\tau'_-}(\Delta_sa_j)+\frac{G_-}{\tau_-}(\Delta_ua_j)\right]\hat{\Delta}_s\hat{n}\hat{\Delta}_u-G^2\left[\frac{(\Delta_sa_j)}{\tau'_-}+\frac{(\Delta_ua_j)}{\tau_-}\right]\hat{n}, \end{split}$$ Below, we will further investigate the structure of the vertex correction and discuss its divergences. Gauge-invariance properties of the vertex-correction function {#VC_GI} ============================================================= The first aspect we would like to discuss is about the gauge invariance of the expression of $\Gamma_{s,s',l}(p,p',q)$ obtained above. On the one hand, it is clear that $\Gamma_{s,s',l}(p,p',q)$ is invariant under a gauge transformation of the plane wave four-vector potential, as it can be proved by replacing $A^{\mu}(\phi)$ with $A^{\mu}(\phi)+\partial^{\mu}f(\phi)=A^{\mu}(\phi)+n^{\mu}f'(\phi)$, with $f(\phi)$ being an arbitrary function of $\phi$ \[we recall, in particular, that $\pi^{\mu}_p(\phi)$ is the kinetic four-momentum of an electron in a plane wave and it is therefore gauge invariant, see Eq. (\[pi\])\]. Now, concerning a gauge transformation of the radiation field and, in particular, of the external photon, we have already discussed that $\Gamma^{\mu}(p,p',q;\phi)$ is written in a form that automatically fulfills the Ward identity, in such a way that one-loop radiative corrections are gauge invariant. In addition, we study here the effect of the additional term $\delta\Gamma^{(\xi)}_{s,s',l}(p,p',q)$ brought about by considering the photon propagator $D^{(\xi)\,\lambda\nu}(x)$ [@Itzykson_b_1980] $$D^{(\xi)\,\lambda\nu}(x)=\int\frac{d^4k}{(2\pi)^4}\frac{e^{-i(kx)}}{k^2-\kappa^2+i0}\left[\eta^{\lambda\nu}+\left(1-\frac{1}{\xi}\right)\frac{k^{\lambda}k^{\nu}}{k^2-\kappa^2+i0}\right],$$ in an arbitrary gauge parametrized by the constant $\xi$ (the Lorenz gauge corresponds to $\xi=1$). It is clear from Eq. (\[Gamma\_2\]) that $$\begin{split} \delta\Gamma^{(\xi)}_{s,s',l}(p,p',q)=&-ie^2\left(1-\frac{1}{\xi}\right)\int d^4x \int\frac{d^4k}{(2\pi)^4}\,\frac{1}{(k^2-\kappa^2+i0)^2}\\ &\times\bar{U}_{s'}(p',x)\hat{k}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}e^{i(qx)}\hat{e}^*_l(q)\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\hat{k}U_s(p,x). \end{split}$$ Now, since $[\hat{\Pi}(\phi)-m]U_s(p,x)=[\hat{\Pi}(\phi)-m]U_{s'}(p',x)=0$, we have that $\bar{U}_{s'}(p',x)\hat{k}=\bar{U}_{s'}(p',x)[\hat{\Pi}(\phi)+\hat{k}-m]$ and analogously $\hat{k}U_s(p,x)=[\hat{\Pi}(\phi)+\hat{k}-m]U_s(p,x)$. Thus, the two electron propagators in $\delta\Gamma^{(\xi)}_{s,s',l}(p,p',q)$ simplify and this quantity can be written in the form $$\delta\Gamma^{(\xi)}_{s,s',l}(p,p',q)=Z^{(\xi)}\int d^4x\,e^{i(qx)}\bar{U}_{s'}(p',x)\hat{e}^*_l(q)U_s(p,x),$$ with $$Z^{(\xi)}=-ie^2\left(1-\frac{1}{\xi}\right)\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{(k^2-\kappa^2+i0)^2}$$ being a logarithmically divergent, gauge-dependent constant. However, since $\delta\Gamma^{(\xi)}_{s,s',l}(p,p',q)$ has exactly the same structure of the tree-level matrix element of (virtual) nonlinear Compton scattering, the constant $Z^{(\xi)}$ can be absorbed in the renormalization of the electric charge exactly as in vacuum [@Itzykson_b_1980]. Thus, we conclude that the gauge-dependent part of the vertex correction can be absorbed in the renormalization of the electric charge and below we will continue to work in the Lorenz gauge. Convergence properties of the vertex-correction function {#VC_CP} ======================================================== Analogously as the corresponding quantity in vacuum, the quantity $\Gamma^{\mu}(p,p',q;\phi)$ is logarithmically divergent in the ultraviolet, as it can be ascertained from the integral in $d^4k$ in Eq. (\[Gamma\_2\]) (see, e.g., the book [@Itzykson_b_1980] for the analysis of the vacuum case). Now, if we imagine to expand the exact Volkov propagators in powers of the external field (see also Fig. \[FD\_VC\]), it is clear that, since the divergence of the corresponding vacuum amplitude is logarithmic, all resulting terms depending on the field are ultraviolet convergent because the loop contains at least three vacuum electron propagators apart from the photon propagator. It is important to stress here that this does not imply that the whole field-dependent part of $\Gamma_{s,s',l}(p,p',q)$ is ultraviolet convergent because the terms dependent on the field exclusively through the external electron states are still logarithmically divergent. For this reason, the correct way of regularizing the vertex correction in the plane wave is to regularize the quantity $\Gamma^{\mu}(p,p',q;\phi)$ (see also Ref. [@Morozov_1981]). Since the divergence at hand is only logarithmic one first writes $\Gamma^{\mu}(p,p',q;\phi)=\Gamma^{\mu}(p,p',q;\phi)-\Gamma_0^{\mu}(p,p',q)+\Gamma_0^{\mu}(p,p',q)$, where $\Gamma_0^{\mu}(p,p',q)=\Gamma^{\mu}(p,p',q;\phi)|_{A^{\mu}(\phi)=0}$, and notices that $\Gamma^{\mu}(p,p',q;\phi)-\Gamma_0^{\mu}(p,p',q)$ is ultraviolet convergent. Then, one can regularize the vacuum expression $\Gamma_0^{\mu}(p,p',q)$ exactly as in the vacuum, i.e., by subtracting the same expression evaluated for $q^{\mu}=0$ and for $\hat{p}=\hat{p}'=m$ [@Itzykson_b_1980] (notice that the conservation laws in a plane wave already imply that $p^{\prime\mu}=p^{\mu}$ because these four-momenta are on-shell). In conclusion, by assuming that $e$ indicates the physical electron charge, we continue by investigating the regularized vertex function $\Gamma_R^{\mu}(p,p',q;\phi)=\Gamma^{\mu}(p,p',q;\phi)-\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}$. By using the master integrals $$\begin{aligned} \int\frac{d^4k}{(2\pi)^4}e^{iSk^2}&=-\frac{i}{16\pi^2S^2},\\ \int\frac{d^4k}{(2\pi)^4}k^{\mu}k^{\nu}e^{iSk^2}&=\frac{\eta^{\mu\nu}}{4}\int\frac{d^4k}{(2\pi)^4}k^2e^{iSk^2}=\frac{\eta^{\mu\nu}}{32\pi^2S^3},\end{aligned}$$ it is straightforward to take the integral in $d^4k$ in $\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}$ and to obtain the result $$\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}=-i\frac{\alpha}{2\pi}\gamma^{\mu}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t-i\frac{(s+u)^2}{S}m^2}\left\{m^2\left[2t-\frac{(s+u)^2}{S}\right]+i\right\}.$$ From the derivations, it is clear that $\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}$ has only components $\Gamma_{0,-}(p,p,0)|_{\hat{p}=m}$ and $\Gamma_{0,\perp,j}(p,p,0)|_{\hat{p}=m}$, and then that $\Gamma_{R,q}(p,p',q;\phi)=\Gamma_q(p,p',q;\phi)$, which, as we have mentioned, can be shown to vanish for $A^{\mu}(\phi)=0$ (see the appendix). Now, we would like to investigate the convergence properties of the proper time integrals in $\Gamma_{R,-}(p,p',q;\phi)$ and $\Gamma_{R,\perp,j}(p,p',q;\phi)$. It is first convenient to use the following identity [@Schubert_2001] $$\begin{split} \int_0^{\infty}ds\int_0^{\infty}du\int_0^{\infty}dt&=\int_0^{\infty}ds\int_0^{\infty}du\int_0^{\infty}dt\int_0^{\infty}dS\,\delta(S-s-u-t)\\ &=\int_0^{\infty}dS\int_0^Sds\int_0^Sdu\int_0^Sdt\,\delta(S-s-u-t)\\ &=\int_0^{\infty}dS\,S^2\int_0^1dx\int_0^1dy\int_0^1dz\,\delta(1-x-y-z), \end{split}$$ where in the last line we performed the changes of variables $s=xS$, $u=yS$, and $t=zS$. By setting $$\label{int_3d} \int_{\delta} dxdydz=\int_0^1dx\int_0^1dy\int_0^1dz\,\delta(1-x-y-z),$$ it is instructive to report the expression of $\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}$ in terms of the new variables: $$\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}=-i\frac{\alpha}{2\pi}\gamma^{\mu}\int_0^{\infty}dS\int_{\delta}dxdydz\,e^{-i\kappa^2zS-im^2(x+y)^2S}\left\{m^2[2z-(x+y)^2]+\frac{i}{S}\right\},$$ because it clearly shows that only the term whose integrand is proportional to $i$ is (logarithmically) divergent (in the limit $S\to 0$). This divergence is related with the ultraviolet logarithmic divergence of the vertex-correction function. Keeping in mind that $z=1-x-y$ \[see Eq. (\[int\_3d\])\], another divergence for $x+y\to 0$ arises for a massless photon ($\kappa^2=0$), which corresponds to the infrared divergence of the vertex-correction function. By means of the above change of variables, we obtain $$\label{Gamma_R_-} \begin{split} &\Gamma_{R,-}(p,p',q;\phi)=\frac{\alpha}{2\pi}\hat{n}\int_0^{\infty}\frac{dS}{S}\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\left[e^{-ig^2S}-e^{-im^2(x+y)^2S}\right]\\ &\quad-\frac{i\alpha}{2\pi}\int_0^{\infty}dS\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\Bigg\{e^{-ig^2S}\Bigg[\left(2(\pi_s\pi_u)+g^2\right)\hat{n}-2g_-(\hat{\pi}_sR+L\hat{\pi}_u)-\hat{g}\hat{\pi}_s\hat{n}-\hat{n}\hat{\pi}_u\hat{g}\\ &\left.\quad+2\tau_-L\hat{g}+2\tau'_-\hat{g}R+2g_-\hat{g}-g_-^2\frac{\hat{\Delta}_s\hat{n}\hat{\Delta}_u}{p_-p'_-}\right]-m^2\hat{n}e^{-im^2(x+y)^2S}[2z-(x+y)^2]\Bigg\}, \end{split}$$ and $$\label{Gamma_R_j} \begin{split} &\Gamma_{R,\perp,j}(p,p',q;\phi)=-\frac{\alpha}{2\pi}\hat{a}_j\int_0^{\infty}\frac{dS}{S}\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\left[e^{-ig^2S}-e^{-im^2(x+y)^2S}\right]\\ &\quad+\frac{i\alpha}{2\pi}\int_0^{\infty}dS\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\left\langle e^{-ig^2S}\left\{2(\pi_s\pi_u)\left(\hat{a}_j+\frac{g_-}{2p'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{g_-}{2p_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\right.\right.\\ &\quad+\frac{i}{S}\left(\frac{g_-}{2p'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{g_-}{2p_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)-L(Ca_j)\hat{g}\hat{\pi}_sR-L\hat{\pi}_u\hat{g}(Ca_j)R\\ &\quad-L\hat{g}\left(\hat{a}_j+\frac{\hat{a}_j\hat{\Delta}_s\hat{n}}{2\tau'_-}-\frac{\hat{\Delta}_u\hat{n}\hat{a}_j}{2\tau_-}\right)\hat{g}R-2S(gg_1)\left(\frac{\tau_-}{p'_-}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{\tau'_-}{p_-}\hat{a}_j\hat{n}\hat{\Delta}_u+2g_-\hat{a}_j-2(ga_j)\hat{n}\right.\\ &\quad+\hat{n}\hat{a}_j\hat{\pi}_s+\hat{\pi}_u\hat{a}_j\hat{n}\bigg)+2i\left((g_1a_j)+x(\mathcal{A}'_sa_j)-y(\mathcal{A}'_ua_j)\right)\hat{n}+i\left[x-(y+z)\frac{\tau_-}{p'_-}\right]\hat{\mathcal{A}}_s'\hat{n}\hat{a}_j\\ &\quad-i\left[y-(x+z)\frac{\tau'_-}{p_-}\right]\hat{a}_j\hat{n}\hat{\mathcal{A}}_u'\Bigg\}-m^2\hat{a}_je^{-im^2(x+y)^2S}[2z-(x+y)^2]\Bigg\rangle, \end{split}$$ where $$\begin{split} &L(Ca_j)\hat{g}\hat{\pi}_sR=\left(\hat{a}_j+\frac{g_-}{2p'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j -\frac{g_-}{2p_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\hat{g}\hat{\pi}_s\\ &+\left(\hat{a}_j+\frac{g_-}{2p'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j\right)\left(p'_-\hat{g}-g_-\hat{\pi}_s\right)\frac{\hat{\Delta}_u}{p_-}\\ &+\frac{\hat{a}_j\hat{n}}{2p_-\tau_-}[2(g_-(\Delta_u\pi_s)-p'_-(\Delta_ug))\hat{\Delta}_u-(g_-\Delta_u^2+2\tau_-(\Delta_ug))\hat{\pi}_s+(p'_-\Delta_u^2+2\tau_-(\Delta_u\pi_s))\hat{g}], \end{split}$$ $$\begin{split} &L\hat{\pi}_u\hat{g}(Ca_j)R=\hat{\pi}_u\hat{g}\left(\hat{a}_j+\frac{g_-}{2p'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{g_-}{2p_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right) \\ &+\frac{\hat{\Delta}_s}{p'_-}\left(p_-\hat{g}-g_-\hat{\pi}_u\right)\left(\hat{a}_j-\frac{g_-}{2p_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\\ &+[2(g_-(\Delta_s\pi_u)-p_-(\Delta_sg))\hat{\Delta}_s-(g_-\Delta_s^2+2\tau'_-(\Delta_sg))\hat{\pi}_u+(p_-\Delta_s^2+2\tau'_-(\Delta_s\pi_u))\hat{g}]\frac{\hat{n}\hat{a}_j}{2p'_-\tau'_-}, \end{split}$$ $$\begin{split} &L\hat{g}\left(\hat{a}_j+\frac{\hat{a}_j\hat{\Delta}_s\hat{n}}{2\tau'_-}-\frac{\hat{\Delta}_u\hat{n}\hat{a}_j}{2\tau_-}\right)\hat{g}R=\frac{g_-}{2}\left(\frac{\hat{g}\hat{a}_j\hat{\Delta}_s\hat{n}\hat{\Delta}_u}{p_-\tau'_-}+\frac{\hat{\Delta}_s\hat{n}\hat{\Delta}_u\hat{a}_j\hat{g}}{p'_-\tau_-}\right)+g_-\left(\frac{\hat{a}_j\hat{\Delta}_s\hat{g}}{\tau'_-}+\frac{\hat{g}\hat{\Delta}_u\hat{a}_j}{\tau_-}\right)\\ &+\left[(p'_-+\tau'_-)(ga_j)+g_-(\Delta_sa_j)\right]\frac{\hat{\Delta}_s\hat{n}\hat{g}}{p'_-\tau'_-}+\left[(p_-+\tau_-)(ga_j)+g_-(\Delta_ua_j)\right]\frac{\hat{g}\hat{n}\hat{\Delta}_u}{p_-\tau_-}-g^2\hat{a}_j\\ &-\left[(p'_-+\tau'_-)g^2+\frac{\tau'_-g^2_-}{p_-\tau_-}\Delta^2_u\right]\frac{\hat{\Delta}_s\hat{n}\hat{a}_j}{2p'_-\tau'_-}-\left[(p_-+\tau_-)g^2+\frac{\tau_-g^2_-}{p'_-\tau'_-}\Delta^2_s\right]\frac{\hat{a}_j\hat{n}\hat{\Delta}_u}{2p_-\tau_-}+2(ga_j)\hat{g}\\ &-[g_-\Delta_s^2+2p'_-(g\Delta_s)]\frac{\hat{a}_j\hat{n}\hat{g}}{2p'_-\tau'_-}-[g_-\Delta_u^2+2p_-(g\Delta_u)]\frac{\hat{g}\hat{n}\hat{a}_j}{2p_-\tau_-}\\ &+\frac{g_-}{p_-p'_-}\left[(ga_j)+\frac{g_-}{\tau'_-}(\Delta_sa_j)+\frac{g_-}{\tau_-}(\Delta_ua_j)\right]\hat{\Delta}_s\hat{n}\hat{\Delta}_u-g^2\left[\frac{(\Delta_sa_j)}{\tau'_-}+\frac{(\Delta_ua_j)}{\tau_-}\right]\hat{n}, \end{split}$$ and where it is clear that also in the case of $\Gamma_{R,\perp,j}(p,p',q;\phi)$ the only term requiring regularization is the one analogous to that in the first line of Eq. (\[Gamma\_R\_-\]). Due to the above change of variables, the various quantities appearing in $\Gamma_{R,-}(p,p',q;\phi)$, and $\Gamma_{R,\perp,j}(p,p',q;\phi)$ have to be interpreted as $$\begin{aligned} \tau'_-&=zp'_--yq_-=(1-x-y)p'_--yq_-, & \pi_s^{\mu}&=\pi^{\mu}_{p'}(\theta'_S), & \Delta^{\mu}_s &=\mathcal{A}^{\mu}(\theta'_S)-\mathcal{A}^{\mu}(\phi), \\ \tau_-&=zp_-+xq_-=(1-x-y)p_-+xq_-, & \pi_u^{\mu}&=\pi^{\mu}_p(\theta_S), & \Delta^{\mu}_u&=\mathcal{A}^{\mu}(\theta_S)-\mathcal{A}^{\mu}(\phi),\end{aligned}$$ where $$\begin{aligned} \theta'_S&=\phi+2x\tau'_-S=\phi+2x[(1-x-y)p'_--yq_-]S,\\ \theta_S&=\phi-2y\tau_-S=\phi-2y[(1-x-y)p_-+xq_-]S.\end{aligned}$$ The formal definitions of the other quantities like $L$, $R$, $C^{\mu}$, $Q^{\lambda}$, and $Q^{\prime\,\lambda}$ remain unchanged and the additional quantities $$g^{\mu}=\frac{G^{\mu}}{S}=x\int_0^1d\eta\,\pi^{\mu}_{p'}(\theta'_{\eta S})+y\int_0^1d\eta\,\pi^{\mu}_p(\theta_{\eta S})$$ and $$\begin{split} g_1^{\mu}&=\frac{G_1^{\mu}}{S^2}=\frac{d}{d\phi}\left[x^2\int_0^1d\eta\,\eta\pi^{\mu}_{p'}(\theta'_{\eta S})-y^2\int_0^1d\eta\,\eta\pi^{\mu}_p(\theta'_{\eta S})\right]\\ &=\frac{x}{2\tau'_-S}\left[\pi^{\mu}_{p'}(\theta'_S)-\int_0^1d\eta\,\pi^{\mu}_{p'}(\theta'_{\eta S})\right]+\frac{y}{2\tau_-S}\left[\pi^{\mu}_p(\theta_S)-\int_0^1d\eta\,\pi^{\mu}_p(\theta_{\eta S})\right], \end{split}$$ which is regular in the limit $S\to 0$ (and also in the limits $\tau_-\to 0$ and $\tau'_-\to 0$), have been also introduced. The locally-constant field approximation {#VC_LCFA} ======================================== In this section, we would like to investigate the regularized vertex-correction function $\Gamma_R^{\mu}(p,p',q;\phi)$ in the so-called locally-constant field approximation (LCFA) [@Reiss_1962; @Ritus_1985; @Baier_b_1998; @Di_Piazza_2012]. Under this approximation quantum processes in an external field are assumed to form over a length much shorter than the typical length where the external field significantly varies [@Reiss_1962; @Ritus_1985; @Baier_b_1998; @Di_Piazza_2012]. As a general condition of validity of the LCFA in a plane wave, one assumes that the strength of the vector potential of the plane wave times the elementary charge is much larger than the electron mass. This condition is based on the idea that the strength of the vector potential scales as the strength of the electric field of the wave times the typical field wavelength, and that the LCFA applies for larger and larger wavelengths (see Refs. [@Baier_1989; @Khokonov_2002; @Di_Piazza_2007; @Wistisen_2015; @Harvey_2015; @Dinu_2016; @Di_Piazza_2018_c; @Alexandrov_2019; @Di_Piazza_2019; @Ilderton_2019_b; @Podszus_2019; @Ilderton_2019; @Raicher_2020] for more refined results and investigations about the validity of the LCFA). In order to study the structure of the vertex-correction function $\Gamma_R^{\mu}(p,p',q;\phi)$, it is first useful to exploit the general structure of the external plane wave, in particular, to rewrite the phase $G^2/S$ in a convenient form \[see Eqs. (\[Gamma\_-\]) and (\[Gamma\_j\])\]. By starting from the identity $v^2=2v_+v_--\bm{v}^2_{\perp}$, valid for a generic four-vector $v^{\mu}$, it can easily be shown that $$\label{G^2} \begin{split} G^2&=\frac{s(p'_-s+p_-u)}{p'_-}(m^2+\delta m_s^2)+\frac{u(p'_-s+p_-u)}{p_-}(m^2+\delta m_u^2)\\ &\quad+usp_-p'_-\left\{\frac{1}{p'_-}\left[\bm{\pi}_{p',\perp}(\phi)-\frac{1}{s}\int_0^sds'\bm{\Delta}_{s',\perp}\right]-\frac{1}{p_-}\left[\bm{\pi}_{p,\perp}(\phi)-\frac{1}{u}\int_0^udu'\bm{\Delta}_{u',\perp}\right]\right\}^2, \end{split}$$ where we have introduced the laser-induced square mass corrections $$\begin{aligned} \label{m_corr_s} \delta m_s^2&=\frac{1}{s}\int_0^sds'\bm{\mathcal{A}}^2_{\perp}(\psi_{s'})-\frac{1}{s^2}\left[\int_0^sds'\bm{\mathcal{A}}_{\perp}(\psi_{s'})\right]^2=\frac{1}{s}\int_0^sds'\bm{\Delta}^2_{s',\perp}-\frac{1}{s^2}\left(\int_0^sds'\bm{\Delta}_{s',\perp}\right)^2,\\ \label{m_corr_u} \delta m_u^2&=\frac{1}{u}\int_0^udu'\bm{\mathcal{A}}^2_{\perp}(\psi_{u'})-\frac{1}{u^2}\left[\int_0^udu'\bm{\mathcal{A}}_{\perp}(\psi_{u'})\right]^2=\frac{1}{u}\int_0^udu'\bm{\Delta}^2_{u',\perp}-\frac{1}{u^2}\left(\int_0^udu'\bm{\Delta}_{u',\perp}\right)^2.\end{aligned}$$ We notice that for the present case of on-shell electrons, the three quantity $G^2$ is non-negative, a property which will be used below. Also, we observe that for the evaluation of $\Gamma_R^{\mu}(p,p',q;\phi)$ the phase $\phi$ is fixed but the vector potential depends on the integration variables $s$, $u$, and $t$ (or $x$, $y$, and $S$). Thus, within the integration region, the terms in the phases depending on the vector potential become larger and larger, leading in turn to highly-oscillating integrands. Thus, the largest contributions to the integrals in the proper times come from the regions where these variables are sufficiently small that the squares of the mass corrections $\delta m_s^2$, and $\delta m_u^2$ are of the order of $m^2$ [@Di_Piazza_2013] (see also Ref. [@Meuren_2015b] for a study of the subleading contributions arising from the saddle points of the phases). In order to implement this idea, we assume that the variables $s$ and $u$ in Eqs. (\[m\_corr\_s\])-(\[m\_corr\_u\]) in the regions mainly contributing to the corresponding integrals are sufficiently small to expand the integrands in those equations for $\psi_{s'}$, and $\psi_{u'}$ around $\phi$ (the validity of this assumption is checked *a posteriori*). It is appropriate to perform the expansions up to terms proportional to the second derivative of $\bm{\mathcal{A}}_{\perp}(\phi)$ because the leading-order contributions to $\delta m_s^2$, and to $\delta m_u^2$ turn out to be proportional to $\bm{\mathcal{A}}_{\perp}^{\prime\,2}(\phi)$, i.e., to the square of the first-order correction. Indeed, all contributions proportional to $\bm{\mathcal{A}}''_{\perp}(\phi)$ cancel out and one obtains $$\begin{aligned} \label{m_corr_LCFA} \delta m_s^2&\approx \frac{1}{3}m^2\left[\frac{t\chi_{p'}(\phi)-u\chi_q(\phi)}{S}\right]^2m^4s^2,&& \delta m_u^2\approx \frac{1}{3}m^2\left[\frac{t\chi_p(\phi)+s\chi_q(\phi)}{S}\right]^2m^4u^2,\end{aligned}$$ where $\chi_p(\phi)=p_-|\bm{\mathcal{E}}_{\perp}(\phi)|/m^3$, $\chi_{p'}(\phi)=p'_-|\bm{\mathcal{E}_{\perp}}(\phi)|/m^3$, and $\chi_q(\phi)=q_-|\bm{\mathcal{E}}_{\perp}(\phi)|/m^3=\chi_{p'}(\phi)-\chi_{p'}(\phi)$, with $\bm{\mathcal{E}}_{\perp}(\phi)=-\bm{\mathcal{A}}'_{\perp}(\phi)$ (recall that we have assumed that $q_-> 0$, such that $\chi_q(\phi)\ge 0$). Now, in order to obtain the range of validity of the above approximations, it is easier to consider the typical situation in which $p_-\sim p'_-\sim q_-$ and to indicate as $p_{0,-}$ this common light-cone energy scale. Correspondingly, by indicating as $E_0$ and $\omega_0$, the amplitude and the typical angular frequency of the background plane wave ($\omega_0$ can also be thought as the inverse of the typical time interval over which the background field varies significantly), we construct the well-known Lorentz- and gauge-invariant parameters $\xi_0=\mathcal{E}_0/m\omega_0$, $\chi_0=p_{0,-}\mathcal{E}_0/m^3$ and $\eta_0=\chi_0/\xi_0=\omega_0p_{0,-}/m^2$ [@Ritus_1985; @Baier_b_1998; @Di_Piazza_2012], with $\mathcal{E}_0=|e|E_0$. The above approximations are all valid if the integrals are formed over regions of $s$ and $u$ such that $\omega_0sp_{0,-}=m^2s\eta_0\ll 1$ and $\omega_0up_{0,-}=m^2u\eta_0\ll 1$ (note that the additional proper time variable $t$ appears in the equations in a way that the relevant conditions and estimates do not involve it). Now, from the expressions of the mass corrections within the LFCA, it is easily seen that if $\chi_0\sim 1$ ($\chi_0\gg 1$), then the integrals are formed over the region where $s,u\lesssim 1/m^2$ ($s,u\lesssim 1/\chi_0^{2/3}m^2$). Since here we are interested in situations where $\chi_0\gtrsim 1$, we can for simplicity use the single expression $s,u\lesssim 1/\chi_0^{2/3}m^2$, such that the LCFA is valid if $\eta_0/\chi_0^{2/3}=\chi_0^{1/3}/\xi_0\ll 1$ [@Baier_1989; @Khokonov_2002; @Di_Piazza_2007; @Dinu_2016; @Di_Piazza_2018_c; @Di_Piazza_2019; @Ilderton_2019_b; @Podszus_2019; @Ilderton_2019]. By expanding also the terms in the second line of Eq. (\[G\^2\]) up to the second derivative of $\bm{A}_{\perp}(\phi)$, we obtain that at the leading order in the LCFA the phase $G^2/S$ reads $$\label{Phase_LCFA_1} \begin{split} \frac{G^2}{S}&=g^2S\approx\frac{p'_-s+p_-u}{p'_-S}m^2s\left\{1+\frac{1}{3}\left[\frac{t\chi_{p'}(\phi)-u\chi_q(\phi)}{S}\right]^2m^4s^2\right\}\\ &\quad+\frac{p'_-s+p_-u}{p_-S}m^2u\left\{1+\frac{1}{3}\left[\frac{t\chi_p(\phi)+s\chi_q(\phi)}{S}\right]^2m^4u^2\right\}\\ &\quad+\frac{usp_-p'_-}{S}\left\{\frac{1}{p'_-}\left[\bm{p}'_{\perp}-\bm{\mathcal{A}}_{\perp}(\phi)+m^3s\frac{t\bm{\chi}_{\perp,p'}(\phi)-u\bm{\chi}_{\perp,q}(\phi)}{S}\right]\right.\\ &\left.\qquad-\frac{1}{p_-}\left[\bm{p}_{\perp}-\bm{\mathcal{A}}_{\perp}(\phi)-m^3u\frac{t\bm{\chi}_{\perp,p}(\phi)+s\bm{\chi}_{\perp,q}(\phi)}{S}\right]\right\}^2\\ &\quad+\frac{4}{3}\frac{usp_-p'_-}{S}\left(\frac{s^2\tau_-^{\prime\,2}}{p'_-}-\frac{u^2\tau_-^2}{p_-}\right)\bm{\mathcal{E}}_{\perp}'(\phi)\cdot\bm{\mathcal{V}}_{\perp}(\phi), \end{split}$$ where $\bm{\chi}_{\perp,p}(\phi)=p_-\bm{\mathcal{E}}_{\perp}(\phi)/m^3$ ($\chi_p(\phi)=|\bm{\chi}_{\perp,p}(\phi)|$), $\bm{\chi}_{\perp,p'}(\phi)=p'_-\bm{\mathcal{E}}_{\perp}(\phi)/m^3$ ($\chi_{p'}(\phi)=|\bm{\chi}_{\perp,p'}(\phi)|$), $\bm{\chi}_{\perp,q}(\phi)=q_-\bm{\mathcal{E}}_{\perp}(\phi)/m^3=\bm{\chi}_{\perp,p}(\phi)-\bm{\chi}_{\perp,p'}(\phi)$ ($\chi_q(\phi)=|\bm{\chi}_{\perp,q}(\phi)|$), and where $$\bm{\mathcal{V}}_{\perp}(\phi)=\frac{1}{p'_-}\left[\bm{p}'_{\perp}-\bm{\mathcal{A}}_{\perp}(\phi)\right]-\frac{1}{p_-}\left[\bm{p}_{\perp}-\bm{\mathcal{A}}_{\perp}(\phi)\right].$$ The appearance of $\bm{\mathcal{A}}_{\perp}(\phi)$ in the vector $\bm{\mathcal{V}}_{\perp}(\phi)$ seems to indicate that indeed the last line of Eq. (\[Phase\_LCFA\_1\]) is leading order in the LCFA because $|\bm{\mathcal{V}}_{\perp}(\phi)|$ scales as $1/\omega_0$. We should however recall that the final object to be computed is $\Gamma_{s,s',l}(p,p',q)$ in Eq. (\[Gamma\^mu\]). Now, if we compute the phase $\Phi(p,p',q;\phi)$ resulting from the functions in Eq. (\[Gamma\^mu\]) other than $\Gamma^{\mu}(p,p',q;\phi)$, after taking the integrals in $\bm{x}_{\perp}$ and $T$, we obtain (apart from an inessential constant) $$\label{Phi_NCS} \Phi(p,p',q;\phi)=\frac{q_-}{2p_-p'_-}\int_0^{\phi}d\phi'\left\{m^2+\frac{p_-p'_-}{q_-^2}q^2+\left[\bm{p}_{\perp}-\bm{\mathcal{A}}_{\perp}(\phi')-\frac{p_-}{q_-}\bm{q}_{\perp}\right]^2\right\}$$ together with the conservation laws $\bm{p}_{\perp}=\bm{p}'_{\perp}+\bm{q}_{\perp}$ and $p_-=p'_-+q_-$. It is clear that, apart from the term proportional to $q^2$, this is the phase of nonlinear Compton scattering, as given, e.g., in Ref. [@Di_Piazza_2018_c]. By using the conservation laws $\bm{p}_{\perp}=\bm{p}'_{\perp}+\bm{q}_{\perp}$ and $p_-=p'_-+q_-$, it is easy to show that $$\bm{\mathcal{V}}_{\perp}(\phi)=\frac{q_-}{p_-p'_-}\left[\bm{p}_{\perp}-\bm{\mathcal{A}}_{\perp}(\phi)-\frac{p_-}{q_-}\bm{q}_{\perp}\right].$$ Since the LCFA corresponds to evaluate the remaining integral in $\phi$ in Eq. (\[Gamma\^mu\]), it is clear that in the region where most of the photons are emitted (and tacitly assuming that the virtuality $q^2$ is less or of the order of $m^2$), it is $|\bm{\mathcal{V}}_{\perp}(\phi)|\sim m$ and the last line in Eq. (\[Phase\_LCFA\_1\]) can be neglected. Finally, by applying the changes of variables discussed above, we obtain the final expression of $g^2S$ within the LCFA in the form $$\label{Phase_LCFA} \begin{split} g^2S&\approx\frac{p'_-x+p_-y}{p'_-}xm^2S\left\{1+\frac{1}{3}[z\chi_{p'}(\phi)-y\chi_q(\phi)]^2x^2m^4S^2\right\}\\ &\quad+\frac{p'_-x+p_-y}{p_-}ym^2S\left\{1+\frac{1}{3}[z\chi_p(\phi)+x\chi_q(\phi)]^2y^2m^4S^2\right\}\\ &\quad+xyp_-p'_-S\left\langle\frac{1}{p'_-}\left\{\bm{p}'_{\perp}-\bm{\mathcal{A}}_{\perp}(\phi)+xm^3S[z\bm{\chi}_{\perp,p'}(\phi)-y\bm{\chi}_{\perp,q}(\phi)]\right\}\right.\\ &\left.\qquad-\frac{1}{p_-}\left\{\bm{p}_{\perp}-\bm{\mathcal{A}}_{\perp}(\phi)-ym^3S[z\bm{\chi}_{\perp,p}(\phi)+x\bm{\chi}_{\perp,q}(\phi)]\right\}\right\rangle^2, \end{split}$$ where $z=1-x-y$. Finally, we point out that have explicitly proved that in the constant-crossed field case $\bm{\mathcal{A}}_{\perp}(\phi)=-\bm{\mathcal{E}}_0\phi$, the phase $g^2S$ reduces to the phase computed in Ref. [@Morozov_1981]. The same can be verified starting from Eq. (\[Phase\_LCFA\]) by setting $\bm{\mathcal{E}}_{\perp}(\phi)=\bm{\mathcal{E}}_0$ and we note that our expression of the phase of the vertex-correction function is not only more general but also much more compact than that presented in Ref. [@Morozov_1981]. A comparison of the final expression of the pre-exponent was not carried out as the form presented in Ref. [@Morozov_1981] has a very different structure from ours, due to employed transformations there, which are appropriate only to the constant-crossed field case. Passing now to the pre-exponents of the components $\Gamma_{R,-}(p,p',q;\phi)$ \[see Eq. (\[Gamma\_R\_-\])\] and $\Gamma_{R,\perp,j}(p,p',q;\phi)$ \[see Eq. (\[Gamma\_R\_j\])\], one has to expand the field-dependent terms in Eqs. (\[Gamma\_R\_-\])-(\[Gamma\_R\_j\]) around the phase $\phi$. Taking into account that the final quantity to be evaluated is $-ie\Gamma_{s,s',l}(p,p',q)$ in Eq. (\[Gamma\^mu\]), a lengthy by straightforward calculation shows that $$\label{Gamma_R_-_LCFA} \begin{split} &\Gamma_{R,-}(p,p',q;\phi)\approx\frac{\alpha}{2\pi}\hat{n}\int_0^{\infty}\frac{dS}{S}\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\left[e^{-ig^2S}-e^{-im^2(x+y)^2S}\right]\\ &\quad-\frac{i\alpha}{2\pi}\int_0^{\infty}dS\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\Bigg\langle e^{-ig^2S}\Bigg\{(p_-+p'_--g_-)\left(\frac{1-x}{p'_-}+\frac{1-y}{p_-}\right)m^2\hat{n}-2m^2\hat{n}\\ &\quad+2m[(p_-+p'_--g_-)(x+y)-g_-]+[(2-x)(p_-b'+p'_-b)-g_-b][(2-y)(p_-b'+p'_-b)-g_-b']\frac{\bm{\mathcal{A}}^{\prime\,2}_{\perp}}{p_-p'_-}\hat{n}\\ &\quad+(1-x)(1-y)p_-p'_-\bm{\mathcal{V}}_{\perp}^2\hat{n}-2(1-x)(1-y)p_-p'_-\left(\frac{2-x}{1-x}\frac{b'}{p'_-}+\frac{2-y}{1-y}\frac{b}{p_-}\right)\bm{\mathcal{V}}_{\perp}\cdot\bm{\mathcal{A}}'_{\perp}\hat{n}\\ &\quad+2m\left[g_--(p_-+p'_--g_-)(x+y)\right]\left(\frac{b'}{p'_-}+\frac{b}{p_-}\right)\hat{n}\hat{\mathcal{A}}'\\ &\left.\quad-(p_-+p'_--g_-)\left[x\frac{b'}{p'_-}\hat{\mathcal{A}}'\hat{n}\left(\frac{q_-}{p_-}m+p'_-\bm{\gamma}_{\perp}\cdot\bm{\mathcal{V}}_{\perp}\right)+y\frac{b}{p_-}\left(\frac{q_-}{p'_-}m+p_-\bm{\gamma}_{\perp}\cdot\bm{\mathcal{V}}_{\perp}\right)\hat{n}\hat{\mathcal{A}}'\right]\right\}\\ &\quad-m^2\hat{n}e^{-im^2(x+y)^2S}[2z-(x+y)^2]\Bigg\rangle, \end{split}$$ where $g^2$ is obtained from Eq. (\[Phase\_LCFA\]), $b=y\tau_-S$, $b'=x\tau'_-S$, and where all fields and derivatives are evaluated at $\phi$. Analogously, one obtains the following expression for $\Gamma_{R,\perp,j}(p,p',q;\phi)$ within the LCFA: $$\label{Gamma_R_j_LCFA} \begin{split} &\Gamma_{R,\perp,j}(p,p',q;\phi)\approx-\frac{\alpha}{2\pi}\hat{a}_j\int_0^{\infty}\frac{dS}{S}\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\left[e^{-ig^2S}-e^{-im^2(x+y)^2S}\right]\\ &\quad+\frac{i\alpha}{2\pi}\int_0^{\infty}dS\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\left\langle e^{-ig^2S}\left\{2[(\pi_s\pi_u)-((\pi_s+\pi_u)g)]\left(\hat{a}_j+\frac{g_-b'}{p'_-\tau'_-}\hat{n}\hat{\mathcal{A}}'\hat{a}_j+\frac{g_-b}{p_-\tau_-}\hat{a}_j\hat{n}\hat{\mathcal{A}}'\right)\right.\right.\\ &\quad+\frac{i}{S}\left(\frac{g_-b'}{p'_-\tau'_-}\hat{n}\hat{\mathcal{A}}'\hat{a}_j+\frac{g_-b}{p_-\tau_-}\hat{a}_j\hat{n}\hat{\mathcal{A}}'\right)+L(Ca_j)\hat{\pi}_s\hat{g}R+L\hat{g}\hat{\pi}_u(Ca_j)R\\ &\quad-L\hat{g}\left(\hat{a}_j+\frac{b'}{\tau'_-}\hat{a}_j\hat{\mathcal{A}}'\hat{n}+\frac{b}{\tau_-}\hat{\mathcal{A}}'\hat{n}\hat{a}_j\right)\hat{g}R-2S(gg_1)\left\{\left[\frac{2b'\tau_-}{p'_-}+(2-y)b+xb'\right]\hat{n}\hat{\mathcal{A}}'\hat{a}_j\right.\\ &\quad+\left[\frac{2b\tau'_-}{p_-}+(2-x)b'+yb\right]\hat{a}_j\hat{n}\hat{\mathcal{A}}'+m\left[\frac{p'_-}{p_-}-\frac{p_-}{p'_-}+x\left(1-\frac{p'_-}{p_-}\right)-y\left(1-\frac{p_-}{p'_-}\right)\right]\hat{n}\hat{a}_j\\ &\quad-p_-(1-y)\bm{\gamma}_{\perp}\cdot\bm{\mathcal{V}}_{\perp}\hat{n}\hat{a}_j-p'_-(1-x)\hat{n}\hat{a}_j\bm{\gamma}_{\perp}\cdot\bm{\mathcal{V}}_{\perp}\bigg\}+i(x-y)(x+y-2)\bm{\mathcal{A}}'_{\perp}\cdot\bm{a}_j\hat{n}\\ &\quad+i\left[x-(y+z)\frac{\tau_-}{p'_-}\right]\hat{\mathcal{A}}'\hat{n}\hat{a}_j-i\left[y-(x+z)\frac{\tau'_-}{p_-}\right]\hat{a}_j\hat{n}\hat{\mathcal{A}}'\Bigg\}-m^2\hat{a}_je^{-im^2(x+y)^2S}[2z-(x+y)^2]\Bigg\rangle, \end{split}$$ where $$\begin{split} &(\pi_s\pi_u)-((\pi_s+\pi_u)g)\approx-\frac{p_-+p'_-}{2g_-}g^2-\frac{g_-}{p_-+p'_-}m^2\\ &\quad+\frac{1}{2}\left(1-\frac{g_-}{p_-+p'_-}\right)\left\{\left(\frac{p_-}{p'_-} +\frac{p'_-}{p_-}\right)m^2+\left[\bm{\mathcal{V}}_{\perp}-2\left(\frac{b'}{p'_-}+\frac{b}{p_-}\right)\bm{\mathcal{A}}'_{\perp}\right]^2p_-p'_-\right\}\\ &\quad-\frac{p_-^2p_-^{\prime\,2}}{2g_-(p_-+p'_-)}\left[(x-y)\bm{\mathcal{V}}_{\perp}-\frac{(xb'+yb)q_-+2xbp'_--2yb'p_-}{p_-p'_-}\bm{\mathcal{A}}'_{\perp}\right]^2, \end{split}$$ $$\begin{split} &L(Ca_j)\hat{\pi}_s\hat{g}R+L\hat{g}\hat{\pi}_u(Ca_j)R-L\hat{g}\left(\hat{a}_j+\frac{b'}{\tau'_-}\hat{a}_j\hat{\mathcal{A}}'\hat{n}+\frac{b}{\tau_-}\hat{\mathcal{A}}'\hat{n}\hat{a}_j\right)\hat{g}R\\ &\quad\approx g^2\left(\hat{a}_j+\frac{g_-b'}{p'_-\tau'_-}\hat{n}\hat{\mathcal{A}}'\hat{a}_j+\frac{g_-b}{p_-\tau_-}\hat{a}_j\hat{n}\hat{\mathcal{A}}'\right)+2\bigg[\left(\left(\pi_s+\pi_u-g\right)a_j\right)\\ &\quad-S(p_-+p'_--g_-)(x-y)\bm{\mathcal{A}}'_{\perp}\cdot\bm{a}_j\bigg]L\hat{g}R-2m(ga_j)\left[1-\left(\frac{b'}{p'_-}+\frac{b}{p_-}\right)S\hat{n}\hat{\mathcal{A}}'\right]\\ &\quad-mS(x\hat{a}_j\hat{\mathcal{A}}'-y\hat{\mathcal{A}}'\hat{a}_j)\hat{n}\left[\left(by-b'x-2\frac{bg_-}{p_-}\right)\hat{\mathcal{A}}'+m\left(y+x\frac{p'_-}{p_-}\right)-xp'_-\bm{\gamma}_{\perp}\cdot\bm{\mathcal{V}}_{\perp}\right]\\ &\quad-mS\left[\left(by-b'x+2\frac{b'g_-}{p'_-}\right)\hat{\mathcal{A}}'+m\left(x+y\frac{p_-}{p'_-}\right)+yp_-\bm{\gamma}_{\perp}\cdot\bm{\mathcal{V}}_{\perp}\right]\hat{n}(x\hat{a}_j\hat{\mathcal{A}}'-y\hat{\mathcal{A}}'\hat{a}_j)\\ &\quad-2i(p_-+p'_--g_-)(x+y)S\varepsilon^{\lambda\mu\nu\rho}n_{\lambda}\mathcal{A}'_{\mu}a_{j,\nu}\tilde{n}_{\rho}\gamma^5L\hat{g}R, \end{split}$$ $$\begin{split} L\hat{g}R&\approx m(x+y)\left[1-\left(\frac{b}{p_-} +\frac{b'}{p'_-}\right)\hat{n}\hat{\mathcal{A}}'\right]-\frac{xb'}{2p'_-}\hat{\mathcal{A}}'\hat{n}\left(m\frac{q_-}{p_-}+p'_-\bm{\mathcal{V}}_{\perp}\cdot\bm{\gamma}_{\perp}\right)\\ &\quad-\frac{yb}{2p_-}\left(m\frac{q_-}{p'_-}+p_-\bm{\mathcal{V}}_{\perp}\cdot\bm{\gamma}_{\perp}\right)\hat{n}\hat{\mathcal{A}}'-\left(\frac{1}{3}\frac{xb^{\prime\,2}}{p'_-}+\frac{1}{3}\frac{yb^2}{p_-}+\frac{g_-bb'}{p_-p'_-}\right)\hat{n}\bm{\mathcal{A}}_{\perp}^{\prime\,2}, \end{split}$$ $$\begin{split} \gamma^5L\hat{g}R&\approx \gamma^5\left[m(y-x)+m\left(\frac{xb}{p_-} -\frac{yb'}{p'_-}\right)\hat{n}\hat{\mathcal{A}}'-\frac{xb'}{2p'_-}\hat{\mathcal{A}}'\hat{n}\left(m\frac{q_-}{p_-}+p'_-\bm{\mathcal{V}}_{\perp}\cdot\bm{\gamma}_{\perp}\right)\right.\\ &\quad\left.+\frac{yb}{2p_-}\left(m\frac{q_-}{p'_-}-p_-\bm{\mathcal{V}}_{\perp}\cdot\bm{\gamma}_{\perp}\right)\hat{n}\hat{\mathcal{A}}'-\left(\frac{1}{3}\frac{xb^{\prime\,2}}{p'_-}+\frac{1}{3}\frac{yb^2}{p_-}+\frac{g_-bb'}{p_-p'_-}\right)\hat{n}\bm{\mathcal{A}}_{\perp}^{\prime\,2}\right], \end{split}$$ $$\label{pi_pi_a} ((\pi_s+\pi_u)a_j)\approx-\left[2(b-b')\bm{\mathcal{A}}'_{\perp}+\frac{p_-+p'_-}{q_-}\bm{q}_{\perp}+2\frac{p_-p'_-}{q_-}\bm{\mathcal{V}}_{\perp}\right]\cdot\bm{a}_j,$$ $$\label{g_a} (ga_j)\approx\left[(xb'-yb)\bm{\mathcal{A}}'_{\perp}-(yp_-+xp'_-)\frac{\bm{q}_{\perp}}{q_-}-(x+y)\frac{p_-p'_-}{q_-}\bm{\mathcal{V}}_{\perp}\right]\cdot\bm{a}_j,$$ $$\begin{split} (gg_1)&\approx\left[\frac{x^3b'}{6}+\frac{y^3b}{6}+xy^2p'_-\left(\frac{2b}{3p_-}+\frac{b'}{2p'_-}\right)+yx^2p_-\left(\frac{2b'}{3p'_-}+\frac{b}{2p_-}\right)\right]\bm{\mathcal{A}}_{\perp}^{\prime\,2}\\ &\quad-\frac{xy}{2}(yp'_-+xp_-)\bm{\mathcal{V}}_{\perp}\cdot\bm{\mathcal{A}}'_{\perp}, \end{split}$$ where $\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$ and $\varepsilon^{\mu\nu\lambda\rho}$ is the completely antisymmetric tensor with $\varepsilon^{0123}=+1$. In the above equations, the quantity $g^2$ is given in Eq. (\[Phase\_LCFA\]) and we have taken into account that finally we need the matrix elements of these matrices between $\bar{U}_{s'}(p',x)$ and $U_s(p,x)$. The appearance of $\bm{q}_{\perp}$ in Eqs. (\[pi\_pi\_a\])-(\[g\_a\]) may suggest that large terms (i.e., of the order of $1/\omega_0$) could appear in total probabilities, if one imagines to carry out the integral over the transverse photon momentum, as one has to shift the variable $\bm{q}_{\perp}$ by a vector containing $\bm{\mathcal{A}}_{\perp}(\phi)$ \[see Eq. (\[Phi\_NCS\])\] in order to perform the resulting Gaussian integral. However, this does not represent a problem because the vector $\bm{q}_{\perp}$ is always multiplied by $\bm{a}_j$ \[see Eqs. (\[pi\_pi\_a\])-(\[g\_a\])\]. Indeed, the first equality in Eq. (\[Gamma\_exp\]) shows that the components $\Gamma_{R,\perp,j}(p,p',q;\phi)$ are only auxiliary quantities, as one finally needs to compute the components $(\Gamma_R(p,p',q;\phi)\Lambda_j)$. Since by definition the vector $\bm{\Lambda}_{\perp,j}$ is perpendicular to $\bm{q}_{\perp}$ \[see Eq. (\[Lambda\_i\])\], by replacing $a^{\mu}_j$ with $\Lambda_j^{\mu}$ in $\Gamma_{R,\perp,j}(p,p',q;\phi)$, it is easy to see that the $(\Gamma_R(p,p',q;\phi)\Lambda_j)$ depends on the vector $\bm{q}_{\perp}$ only through $\bm{\Lambda}_{\perp,j}$, a quantity which then, due to completeness, drops out once one computes total probabilities. Finally, we comment on the scaling of the radiative corrections due to the vertex correction at $\chi_0\gg 1$. This is more easily done in the case of $\Gamma_{R,-}(p,p',q;\phi)$ because one knows that the corresponding amplitude in nonlinear Compton scattering is simply proportional to $\bar{u}_{s'}(p')\hat{n}u_s(p)$ (before one regularizes the amplitude by integrating by parts, see, e.g., [@Mackenroth_2011]). Now, the structure of the phase in Eq. (\[Phase\_LCFA\]) shows that at large values of $\chi_0$ the main contribution to the integral in $S$ comes from the region $S\lesssim 1/\chi^{2/3}_0$. Thus, the terms in the preexponent in Eq. (\[Gamma\_R\_-\_LCFA\]) proportional to $p_-^2\bm{A}_{\perp}^{\prime\,2}(\phi)S^2=e^2p_-^2\bm{\mathcal{E}}_{\perp}^2(\phi)S^2$ give rise to the scaling $\alpha\chi_0^{2/3}$ of the radiative corrections in agreement with the results in Ref. [@Morozov_1981]. Conclusions {#VC_Conclusions} =========== We have computed the general expression of the one-loop vertex correction in an arbitrary plane-wave background field for the case of two on-shell external electrons and an off-shell external photon. By employing the operator technique within the Furry picture, we have obtained a relatively compact expression, which takes into account exactly the background plane-wave field. By showing explicitly that the vertex correction fulfills a generalized Ward identity, we have singled out the corresponding terms invariant under a gauge transformation of the external photon. As expected, the vertex-correction function features an infrared divergence, which is cured by assigning a small, finite mass to the photon. The ultraviolet divergence of the vertex correction has, instead, been shown to be renormalized as in vacuum. The important special case of the locally-constant field approximation has been investigated in detail. We have shown that in the high-field regime $\chi_0\gg 1$ the vertex-correction function induces radiative corrections which scale according to the Ritus-Narozhny conjecture as $\alpha\chi^{2/3}_0$, where $\chi_0$ is the amplitude of the quantum nonlinearity parameter. The component $\Gamma_q(p,p',q;\phi)$ of the vertex-correction function ======================================================================= In this appendix, we evaluate more explicitly the component $\Gamma_q(p,p',q;\phi)$ of the vertex-correction function although we have seen that it does not contribute to any transition matrix element. General structure of $\Gamma_q(p,p',q;\phi)$ -------------------------------------------- As we have mentioned in the main text, from the second equality in Eq. (\[Gamma\_2\]) and from the definition of $\Gamma^{\mu}(p,p',q;\phi)$ in Eq. (\[Gamma\^mu\]), we obtain $$\Gamma_q(p,p',q;\phi)=-ie^2\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m+i0}\hat{q}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\gamma_{\lambda}.$$ Now, by writing $\hat{q}=\hat{\Pi}(\phi)+\hat{k}-m-[\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m]$ it is clear that we can express $\Gamma_q(p,p',q;\phi)$ as the difference of two terms containing only one propagator in the plane wave, which significantly simplifies its expression: $$\Gamma_q(p,p',q;\phi)=-ie^2\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\gamma^{\lambda}\left[\frac{1}{\hat{\Pi}(\phi)+\hat{k}-\hat{q}-m+i0}-\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\right]\gamma_{\lambda}.$$ At this point, by following exactly the same steps as in the main text \[see Eq. (\[Gamma\_f\])\], it is easy to obtain the resulting expression $$\begin{split} \Gamma_q(p,p',q;\phi)&=\frac{\alpha}{4\pi}\int_0^{\infty}\frac{dsdt}{(s+t)^2}e^{-i\kappa^2t-i\frac{\tilde{G}^2_s}{s+t}}\left\{ 2\left[1-\frac{e\hat{n}[\hat{A}(\tilde{\psi}_{0,s})-\hat{A}(\phi)]}{2p'_-}\right]\left[\hat{\pi}_{p'}(\tilde{\psi}_{0,s})+\frac{\hat{\tilde{G}}_s}{s+t}\right]\right.\\ &\left.\left.\qquad\qquad-\frac{\hat{n}}{(s+t)^2}\frac{d\tilde{G}^2_s}{dk_-}+\frac{e\hat{n}[\hat{A}(\tilde{\psi}_{0,s})-\hat{A}(\phi)]}{\tau'_{0,-}+k_-}\left[\hat{\pi}_{p'}(\tilde{\psi}_{0,s})-\frac{\hat{\tilde{G}}_s}{s+t}\right]\right\}\right\vert_{k_-=0}\\ &\quad-\frac{\alpha}{4\pi}\int_0^{\infty}\frac{dudt}{(u+t)^2}e^{-i\kappa^2t-i\frac{\tilde{G}^2_u}{u+t}}\left\{ 2\left[\hat{\pi}_p(\tilde{\psi}_{0,u})+\frac{\hat{\tilde{G}}_u}{u+t}\right]\left[1+\frac{e\hat{n}[\hat{A}(\tilde{\psi}_{0,u})-\hat{A}(\phi)]}{2p_-}\right]\right.\\ &\left.\left.\qquad\qquad-\frac{\hat{n}}{(u+t)^2}\frac{d\tilde{G}^2_u}{dk_-}-\left[\hat{\pi}_p(\tilde{\psi}_{0,u})-\frac{\hat{\tilde{G}}_u}{u+t}\right]\frac{e\hat{n}[\hat{A}(\tilde{\psi}_{0,u})-\hat{A}(\phi)]}{\tau_{0,-}+k_-}\right\}\right\vert_{k_-=0}, \end{split}$$ where $$\begin{aligned} \tilde{G}^{\mu}_s&=\int_0^sds'\pi^{\mu}_{p'}(\tilde{\psi}_{0,s'}), && \tilde{\psi}_{0,s}=\phi+2s\tau'_{0,-}+2sk_-,&&\tau'_{0,-}=\frac{t}{s+t}p'_-,\\ \tilde{G}^{\mu}_u&=\int_0^udu'\pi^{\mu}_p(\tilde{\psi}_{0,u'}), && \tilde{\psi}_{0,u}=\phi-2u\tau_{0,-}-2uk_-,&&\tau_{0,-}=\frac{t}{u+t}p_-.\end{aligned}$$ Finally, by evaluating the remaining derivatives with respect to $k_-$ as $$\begin{aligned} \frac{d\tilde{G}^2_s}{dk_-}&=4(\tilde{G}_s\tilde{G}_{1,s}),\\ \frac{d\tilde{G}^2_u}{dk_-}&=4(\tilde{G}_u\tilde{G}_{1,u}),\end{aligned}$$ where $$\begin{aligned} \tilde{G}^{\mu}_{1,s}&=\int_0^sds'\,s'\pi^{\prime\,\mu}_{p'}(\tilde{\psi}_{0,s'})=\frac{1}{2(\tau'_{0,-}+k_-)}\left[s\pi^{\mu}_{p'}(\tilde{\psi}_{0,s})-\int_0^sds'\pi^{\mu}_{p'}(\tilde{\psi}_{0,s'})\right],\\ \tilde{G}^{\mu}_{1,u}&=-\int_0^udu'\,u'\pi^{\prime\,\mu}_p(\tilde{\psi}_{0,u'})=\frac{1}{2(\tau_{0,-}+k_-)}\left[u\pi^{\mu}_p(\tilde{\psi}_{0,u})-\int_0^udu'\pi^{\mu}_p(\tilde{\psi}_{0,u'})\right],\end{aligned}$$ we obtain $$\label{Gamma_q} \begin{split} \Gamma_q(p,p',q;\phi)&=\frac{\alpha}{2\pi}\int_0^{\infty}\frac{dsdt}{(s+t)^2}e^{-i\kappa^2t-i\frac{G^2_s}{s+t}}\left\{\left(1-\frac{\hat{n}\hat{\Delta}_{0,s}}{2p'_-}\right)\left[\hat{\pi}_{p'}(\psi_{0,s})+\frac{\hat{G}_s}{s+t}\right]\right.\\ &\left.\qquad\qquad-\frac{2\hat{n}}{(s+t)^2}(G_sG_{1,s})+\frac{\hat{n}\hat{\Delta}_{0,s}}{2\tau'_{0,-}}\left[\hat{\pi}_{p'}(\psi_{0,s})-\frac{\hat{G}_s}{s+t}\right]\right\}\\ &\quad-\frac{\alpha}{2\pi}\int_0^{\infty}\frac{dudt}{(u+t)^2}e^{-i\kappa^2t-i\frac{G^2_u}{u+t}}\left\{\left[\hat{\pi}_p(\psi_{0,u})+\frac{\hat{G}_u}{u+t}\right]\left(1+\frac{\hat{n}\hat{\Delta}_{0,u}}{2p_-}\right)\right.\\ &\left.\qquad\qquad-\frac{2\hat{n}}{(u+t)^2}(G_uG_{1,u})-\left[\hat{\pi}_p(\psi_{0,u})-\frac{\hat{G}_u}{u+t}\right]\frac{\hat{n}\hat{\Delta}_{0,u}}{2\tau_{0,-}}\right\}, \end{split}$$ where $$\Delta^{\mu}_{0,s/u}=e[A^{\mu}(\psi_{0,s/u})-A^{\mu}(\phi)],$$ and where all the quantities without the tilde are the same as those with the tilde but with $k_-=0$: $$\begin{aligned} G^{\mu}_s&=\int_0^sds'\pi^{\mu}_{p'}(\psi_{0,s'}), && G^{\mu}_{1,s}=\frac{1}{2\tau'_{0,-}}\left[s\pi_{p'}^{\mu}(\psi_{0,s})-\int_0^sds'\pi_{p'}^{\mu}(\psi_{0,s'})\right], && \psi_{0,s}=\phi+2s\tau'_{0,-},\\ G^{\mu}_u&=\int_0^udu'\pi^{\mu}_p(\psi_{0,u'}), && G^{\mu}_{1,u}=\frac{1}{2\tau_{0,-}}\left[u\pi_p^{\mu}(\psi_{0,u})-\int_0^udu'\pi_p^{\mu}(\psi_{0,u'})\right], && \psi_{0,u}=\phi-2u\tau_{0,-}.\end{aligned}$$ Regularization of $\Gamma_q(p,p',q;\phi)$ ----------------------------------------- Analogously to the other components of the vertex-correction function, the component $\Gamma_q(p,p',q;\phi)$ has in principle to be regularized as it is apparently logarithmically divergent in the ultraviolet. However, in the case $\Gamma_{R,q}(p,p',q;\phi)$, actually, it is not necessary to perform any subtraction of vacuum terms because, as we will show now, it vanishes for $A^{\mu}(\phi)=0$. It is convenient to perform the change of variable $s=x\sigma$ and $t=(1-x)\sigma$ ($u=x\sigma$ and $t=(1-x)\sigma$) in the first (second) integral in Eq. (\[Gamma\_q\]) [^2] and we obtain $$\label{Gamma_R_q} \begin{split} &\Gamma_{R,q}(p,p',q;\phi)=\frac{\alpha}{2\pi}\int_0^{\infty}\frac{d\sigma}{\sigma}\int_0^1dx\,e^{-i\kappa^2(1-x)\sigma}\\ &\quad\times\left\langle e^{-ix^2\sigma g_s^2}\left\{\left(1-\frac{\hat{n}\hat{\Delta}_{0,s}}{2p'_-}\right)[\hat{\pi}_{p'}(\theta'_{0,\sigma})+x\hat{g}_s]-2x^3\sigma(g_sg_{1,s})\hat{n}+\frac{\hat{n}\hat{\Delta}_{0,s}}{2(1-x)p'_-}[\hat{\pi}_{p'}(\theta'_{0,\sigma})-x\hat{g}_s]\right\}\right.\\ &\quad-\left.e^{-ix^2\sigma g_u^2}\left\{[\hat{\pi}_p(\theta_{0,\sigma})+x\hat{g}_u]\left(1+\frac{\hat{n}\hat{\Delta}_{0,u}}{2p_-}\right)-2x^3\sigma(g_ug_{1,u})\hat{n}-[\hat{\pi}_p(\theta_{0,\sigma})-x\hat{g}_u]\frac{\hat{n}\hat{\Delta}_{0,u}}{2(1-x)p_-}\right\}\right\rangle, \end{split}$$ where $$\begin{aligned} \theta'_{0,\sigma}&=\phi+2x(1-x)p'_-\sigma,\\ \theta_{0,\sigma}&=\phi-2x(1-x)p_-\sigma,\end{aligned}$$ and $$\begin{aligned} g^{\mu}_s=\frac{G^{\mu}_s}{s}=\int_0^1d\eta\,\pi_{p'}^{\mu}(\theta'_{0,\eta\sigma}),&& g^{\mu}_{1,s}=\frac{G^{\mu}_{1,s}}{s^2}=\frac{1}{2x(1-x)\sigma p'_-}\left[\pi_{p'}^{\mu}(\theta'_{0,\sigma})-\int_0^1d\eta\,\pi_{p'}^{\mu}(\theta'_{0,\eta\sigma})\right],\\ g^{\mu}_u=\frac{G^{\mu}_u}{u}=\int_0^1d\eta\,\pi_p^{\mu}(\theta_{0,\eta \sigma}),&& g^{\mu}_{1,u}=\frac{G^{\mu}_{1,u}}{u^2}=\frac{1}{2x(1-x)\sigma p_-}\left[\pi_p^{\mu}(\theta_{0,\sigma})-\int_0^1d\eta\,\pi_p^{\mu}(\theta_{0,\eta\sigma})\right].\end{aligned}$$ Note that $g_{1,s}^{\mu}$ and $g_{1,u}^{\mu}$ tend to constant values for $\sigma\to 0$. Since $\Delta^{\mu}_{0,s}$ and $\Delta^{\mu}_{0,u}$ vanish at $\sigma=0$, the only problematic terms are those in the square brackets in Eq. (\[Gamma\_R\_q\]) containing exclusively $\hat{\pi}_{p'}(\theta'_{0,\sigma})+x\hat{g}_s$ and $\hat{\pi}_p(\theta_{0,\sigma})+x\hat{g}_u$. However, since $\Gamma_q(p,p',q;\phi)$ will finally be multiplied by Volkov states both from the left and on the right, we can replace $\hat{\pi}_{p'}(\theta'_{0,\sigma})+x\hat{g}_s$ and $\hat{\pi}_p(\theta_{0,\sigma})+x\hat{g}_u$ in the terms which do not contain other gamma matrices as $$\begin{aligned} \hat{\pi}_{p'}(\theta'_{0,\sigma})+x\hat{g}_s&\rightarrow m(1+x)+\hat{\pi}_{p'}(\theta'_{0,\sigma})-\hat{\pi}_{p'}(\phi)+x\int_0^1d\eta\,[\hat{\pi}_{p'}(\theta'_{0,\eta\sigma})-\hat{\pi}_{p'}(\phi)],\\ \hat{\pi}_p(\theta_{0,\sigma})+x\hat{g}_u&\rightarrow m(1+x)+\hat{\pi}_p(\theta_{0,\sigma})-\hat{\pi}_p(\phi)+x\int_0^1d\eta\,[\hat{\pi}_p(\theta_{0,\eta\sigma})-\hat{\pi}_p(\phi)].\end{aligned}$$ In this way, the only remaining divergent terms are those proportional to $m(1+x)$ but these divergences cancel each other in Eq. (\[Gamma\_R\_q\]), which can be conveniently written in the manifestly convergent form $$\begin{split} \Gamma_{R,q}(p,p',q;\phi)&=\frac{\alpha}{2\pi}\int_0^{\infty}\frac{d\sigma}{\sigma}\int_0^1dx\,e^{-i\kappa^2(1-x)\sigma}\Bigg\langle m(1+x)\left(e^{-ix^2\sigma g_s^2}-e^{-ix^2\sigma g_u^2}\right)\\ &+e^{-ix^2\sigma g_s^2}\left\{\hat{\pi}_{p'}(\theta'_{0,\sigma})-\hat{\pi}_{p'}(\phi)+x\int_0^1d\eta\,[\hat{\pi}_{p'}(\theta'_{0,\eta\sigma})-\hat{\pi}_{p'}(\phi)]\right.\\ &\quad\left.+\frac{x}{1-x}\frac{\hat{n}\hat{\Delta}_{0,s}\hat{\pi}_{p'}(\theta'_{0,\sigma})}{2p'_-}-\frac{x(2-x)}{1-x}\frac{\hat{n}\hat{\Delta}_{0,s}\hat{g}_s}{2p'_-}-2x^3\sigma(g_sg_{1,s})\hat{n}\right\}\\ &-e^{-ix^2\sigma g_u^2}\left\{\hat{\pi}_p(\theta_{0,\sigma})-\hat{\pi}_p(\phi)+x\int_0^1d\eta\,[\hat{\pi}_p(\theta_{0,\eta\sigma})-\hat{\pi}_p(\phi)]\right.\\ &\quad\left.\left.-\frac{x}{1-x}\frac{\hat{\pi}_p(\theta_{0,\sigma})\hat{n}\hat{\Delta}_{0,u}}{2p_-}+\frac{x(2-x)}{1-x}\frac{\hat{g}_u\hat{n}\hat{\Delta}_{0,u}}{2p_-}-2x^3\sigma(g_ug_{1,u})\hat{n}\right\}\right\rangle. \end{split}$$ This expression also shows that $\Gamma_{R,q}(p,p',q;\phi)$ tends to zero for $A^{\mu}(\phi)\to 0$. Some considerations about the LCFA ---------------------------------- As we have mentioned in the main text, in order to study the structure component $\Gamma_{R,q}^{\mu}(p,p',q;\phi)$ of the vertex-correction function, it is useful to rewrite the phases $G_s^2/(s+t)$ and $G_u^2/(u+t)$ in a convenient form \[see Eq. (\[Gamma\_q\])\]. Indeed, since, as we have seen in the main text, the component $\Gamma_{R,q}(p,p',q;\phi)$ does not contribute to any transition matrix element, we only report here some considerations about these phases. By starting from the identity $v^2=2v_+v_--\bm{v}^2_{\perp}$, valid for a generic four-vector $v^{\mu}$, it can easily be shown that $$\begin{aligned} \label{G_s^2} G_s^2&=s^2(m^2+\delta m_{0,s}^2),\\ \label{G_u^2} G_u^2&=u^2(m^2+\delta m_{0,u}^2),\end{aligned}$$ where we have introduced the laser-induced square mass corrections $$\begin{aligned} \label{m_corr_0s} \delta m_{0,s}^2&=\frac{1}{s}\int_0^sds'\bm{\mathcal{A}}^2_{\perp}(\psi_{0,s'})-\frac{1}{s^2}\left[\int_0^sds'\bm{\mathcal{A}}_{\perp}(\psi_{0,s'})\right]^2=\frac{1}{s}\int_0^sds'\bm{\Delta}^2_{0,s',\perp}-\frac{1}{s^2}\left(\int_0^sds'\bm{\Delta}_{0,s',\perp}\right)^2,\\ \label{m_corr_0u} \delta m_{0,u}^2&=\frac{1}{u}\int_0^udu'\bm{\mathcal{A}}^2_{\perp}(\psi_{0,u'})-\frac{1}{u^2}\left[\int_0^udu'\bm{\mathcal{A}}_{\perp}(\psi_{0,u'})\right]^2=\frac{1}{u}\int_0^udu'\bm{\Delta}^2_{0,u',\perp}-\frac{1}{u^2}\left(\int_0^udu'\bm{\Delta}_{0,u',\perp}\right)^2.\end{aligned}$$ We notice that for the present case of on-shell electrons, the quantities $G_s^2$ and $G_u^2$ are non-negative. By proceeding as in the main text we arrive to the approximated expressions $$\begin{aligned} \label{m_corr_0_LCFA} \delta m_{0,s}^2&\approx \frac{1}{3}m^2\left(\frac{t}{s+t}\right)^2m^4s^2\chi_{p'}^2(\phi),&& \delta m_{0,u}^2\approx \frac{1}{3}m^2\left(\frac{t}{u+t}\right)^2m^4u^2\chi_p^2(\phi),\end{aligned}$$ where $\chi_p(\phi)=p_-|\bm{\mathcal{E}}_{\perp}(\phi)|/m^3$ and $\chi_{p'}(\phi)=p'_-|\bm{\mathcal{E}_{\perp}}(\phi)|/m^3$, with $\bm{\mathcal{E}}_{\perp}(\phi)=-\bm{\mathcal{A}}'_{\perp}(\phi)$. These approximated expressions are valid if $\eta_0/\chi_0^{2/3}=\chi_0^{1/3}/\xi_0\ll 1$ [@Baier_1989; @Khokonov_2002; @Di_Piazza_2007; @Dinu_2016; @Di_Piazza_2018_c; @Di_Piazza_2019; @Ilderton_2019_b; @Podszus_2019; @Ilderton_2019]. The final expressions of the phases $G_s^2/(s+t)$ and $G_u^2/(u+t)$ within the LCFA are $$\begin{aligned} \frac{G_s^2}{s+t}&=\frac{s}{s+t}m^2s\left[1+\frac{1}{3}\frac{t^2}{(s+t)^2}m^4s^2\chi_{p'}^2(\phi)\right],\\ \frac{G_u^2}{u+t}&=\frac{u}{u+t}m^2u\left[1+\frac{1}{3}\frac{t^2}{(u+t)^2}m^4u^2\chi_p^2(\phi)\right].\end{aligned}$$ The above expressions simplify by means of the mentioned changes of variables: $$\begin{aligned} \label{Phase_0s_LCFA} \frac{G_s^2}{s+t}&=x^2m^2\sigma\left[1+\frac{1}{3}x^2(1-x)^2m^4\sigma^2\chi_{p'}^2(\phi)\right],\\ \label{Phase_0u_LCFA} \frac{G_u^2}{u+t}&=x^2m^2\sigma\left[1+\frac{1}{3}x^2(1-x)^2m^4\sigma^2\chi_p^2(\phi)\right].\end{aligned}$$ [97]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****,  ()]{} @noop [****,  ()]{} @noop [**]{} (, ) @noop [**]{} (, ) @noop [**]{} (, ) @noop [****,  ()]{} @noop [****,  ()]{} @noop , @noop ,  @noop ,  @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [**]{} (, ) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [ ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [**]{} (, ) @noop [**]{} (, ) @noop [**]{} (, ) @noop [****,  ()]{} @noop [****,  ()]{} @noop [**]{} (, ) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [ ()]{} @noop [****,  ()]{} [^1]: The terms with three gamma matrices can be further reduced according to the identity $$\begin{split} \hat{A}\hat{B}\hat{C}&=\frac{1}{4}\text{tr}(\gamma_{\mu}\hat{A}\hat{B}\hat{C})\gamma^{\mu}-\frac{1}{4}\text{tr}(\gamma^5\gamma_{\mu}\hat{A}\hat{B}\hat{C})\gamma^5\gamma^{\mu}\\ &=\hat{A}(BC)-\hat{B}(AC)+\hat{C}(AB)+i\varepsilon_{\mu\nu\lambda\rho}\gamma^5\gamma^{\mu}A^{\nu}B^{\lambda}C^{\rho}, \end{split}$$ where $\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$ and $\varepsilon^{\mu\nu\lambda\rho}$ is the completely antisymmetric tensor with $\varepsilon^{0123}=+1$, which is valid for three arbitrary four-vectors $A^{\mu}$, $B^{\mu}$, and $C^{\mu}$. [^2]: Note that in this section and afterwards the symbol $x$ should not be confused with a spacetime point as it indicates a single, real variable.

--- abstract: 'We discuss the statistical properties of a recently introduced unbiased stochastic approximation to the score equations for maximum likelihood calculation for Gaussian processes. Under certain conditions, including bounded condition number of the covariance matrix, the approach achieves $O(n)$ storage and nearly $O(n)$ computational effort per optimization step, where $n$ is the number of data sites. Here, we prove that if the condition number of the covariance matrix is bounded, then the approximate score equations are nearly optimal in a well-defined sense. Therefore, not only is the approximation efficient to compute, but it also has comparable statistical properties to the exact maximum likelihood estimates. We discuss a modification of the stochastic approximation in which design elements of the stochastic terms mimic patterns from a $2^n$ factorial design. We prove these designs are always at least as good as the unstructured design, and we demonstrate through simulation that they can produce a substantial improvement over random designs. Our findings are validated by numerical experiments on simulated data sets of up to 1 million observations. We apply the approach to fit a space–time model to over 80,000 observations of total column ozone contained in the latitude band $40^\circ$–$50^\circ$N during April 2012.' address: - | M. L. Stein\ Department of Statistics\ University of Chicago\ Chicago, Illinois 60637\ USA\ - | J. Chen\ M. Anitescu\ Mathematics and Computer\ Science Division\ Argonne National Laboratory\ Argonne, Illinois 60439\ USA\ \ author: - - - title: Stochastic approximation of score functions for Gaussian processes --- , Introduction {#sec1} ============ Gaussian process models are widely used in spatial statistics and machine learning. In most applications, the covariance structure of the process is at least partially unknown and must be estimated from the available data. Likelihood-based methods, including Bayesian methods, are natural choices for carrying out the inferences on the unknown covariance structure. For large data sets, however, calculating the likelihood function exactly may be difficult or impossible in many cases. Assuming we are willing to specify the covariance structure up to some parameter $\theta\in\Theta\subset{{\mathbb R}}^p$, the generic problem we are faced with is computing the loglikelihood for $Z\sim N(0,K(\theta))$ for some random vector $Z\in{{\mathbb R}}^n$ and $K$ an $n\times n$ positive definite matrix indexed by the unknown $\theta$. In many applications, there would be a mean vector that also depends on unknown parameters, but since unknown mean parameters generally cause fewer computational difficulties, for simplicity we will assume the mean is known to be 0 throughout this work. For the application to ozone data in Section \[sec6\], we avoid modeling the mean by removing the monthly mean for each pixel. The simulations in Section \[sec5\] all first preprocess the data by taking a discrete Laplacian, which filters out any mean function that is linear in the coordinates, so that the results in those sections would be unchanged for such mean functions. The loglikelihood is then, up to an additive constant, given by $${{\mathcal}L}(\theta) = -\tfrac{1}{2}Z'K(\theta)^{-1}Z - \tfrac{1}{2}\log\det\bigl\{K(\theta)\bigr\}.$$ If $K$ has no exploitable structure, the standard direct way of calculating ${{\mathcal}L}(\theta)$ is to compute the Cholesky decompositon of $K(\theta)$, which then allows $Z'K(\theta)^{-1}Z$ and $\log\det\{K(\theta)\}$ to be computed quickly. However, the Cholesky decomposition generally requires $O(n^2)$ storage and $O(n^3)$ computations, either of which can be prohibitive for sufficiently large $n$. Therefore, it is worthwhile to develop methods that do not require the calculation of the Cholesky decomposition or other matrix decompositions of $K$. If our goal is just to find the maximum likelihood estimate (MLE) and the corresponding Fisher information matrix, we may be able to avoid the computation of the log determinants by considering the score equations, which are obtained by setting the gradient of the loglikelihood equal to 0. Specifically, defining $K_i = \frac{\partial}{\partial\theta_i}K(\theta)$, the score equations for $\theta$ are given by (suppressing the dependence of $K$ on $\theta$) $$\label{score} \tfrac{1}{2}Z'K^{-1}K_iK^{-1}Z -\tfrac{1}{2} \operatorname{tr}\bigl(K^{-1}K_i\bigr) = 0$$ for $i=1,\ldots,p$. If these equations have a unique solution for $\theta\in\Theta$, this solution will generally be the MLE. Iterative methods often provide an efficient (in terms of both storage and computation) way of computing solves in $K$ (expressions of the form $K^{-1}x$ for vectors $x$) and are based on being able to multiply arbitrary vectors by $K$ rapidly. In particular, assuming the elements of $K$ can be calculated as needed, iterative methods require only $O(n)$ storage, unlike matrix decompositions such as the Cholesky, which generally require $O(n^2)$ storage. In terms of computations, two factors drive the speed of iterative methods: the speed of matrix–vector multiplications and the number of iterations. Exact matrix–vector multiplication generally requires $O(n^2)$ operations, but if the data form a partial grid, then it can be done in $O(n\log n)$ operations using circulant embedding and the fast Fourier transform. For irregular observations, fast multipole approximations can be used \[@anitescu2012mfa\]. The number of iterations required is related to the condition number of $K$ (the ratio of the largest to smallest singular value), so that preconditioning \[@chenbook\] is often essential; see @steinchenanitescufiltering for some circumstances under which one can prove that preconditioning works well. Computing the first term in (\[score\]) requires only one solve in $K$, but the trace term requires $n$ solves (one for each column of $K_i$) for $i=1,\ldots,p$, which may be prohibitive in some circumstances. Recently, @anitescu2012mfa analyzed and demonstrated a stochastic approximation of the trace term based on the Hutchinson trace estimator \[@hutchinson\]. To define it, let $U_1,\ldots,U_N$ be i.i.d. random vectors in ${{\mathbb R}}^n$ with i.i.d. symmetric Bernoulli components, that is, taking on values 1 and $-1$ each with probability $\frac{1}{2}$. Define a set of estimating equations for $\theta$ by $$\label{ascore} g_i(\theta,N) = \frac{1}{2}Z'K^{-1}K_iK^{-1}Z -\frac{1}{2N}\sum_{j=1}^N U_j'K^{-1}K_iU_j = 0$$ for $i=1,\ldots,p$. Throughout this work, $E_\theta$ means to take expectations over $Z\sim N(0,K(\theta))$ and over the $U_j$’s as well. Since $E_\theta(U_1'K^{-1}K_iU_1) = \operatorname{tr}(K^{-1}K_i)$, $E_\theta g_i(\theta,N) = 0$ and (\[ascore\]) provides a set of unbiased estimating equations for $\theta$. Therefore, we may hope that a solution to (\[ascore\]) will provide a good approximation to the MLE. The unbiasedness of the estimating equations (\[ascore\]) requires only that the components of the $U_j$’s have mean 0 and variance 1; but, subject to this constraint, @hutchinson shows that, assuming the components of the $U_j$’s are independent, taking them to be symmetric Bernoulli minimizes the variance of $U_1'MU_1$ for any $n\times n$ matrix $M$. The Hutchinson trace estimator has also been used to approximate the GCV (generalized cross-validation) statistic in nonparametric regression \[@girard [@wahba]\]. In particular, @girard shows that $N$ does not need to be large to obtain a randomized GCV that yields results nearly identical to those obtained using exact GCV. Suppose for now that it is possible to take $N$ much smaller than $n$ and obtain an estimate of $\theta$ that is nearly as efficient statistically as the exact MLE. From here on, assume that any solves in $K$ will be done using iterative methods. In this case, the computational effort to computing (\[score\]) or (\[ascore\]) is roughly linear in the number of solves required (although see Section \[sec4\] for methods that make $N$ solves for a common matrix $K$ somewhat less than $N$ times the effort of one solve), so that (\[ascore\]) is much easier to compute than (\[score\]) when $N/n$ is small. An attractive feature of the approximation (\[ascore\]) is that if at any point one wants to obtain a better approximation to the score function, it suffices to consider additional $U_j$’s in (\[ascore\]). However, how exactly to do this if using the dependent sampling scheme for the $U_j$’s in Section \[sec4\] is not so obvious. Since this stochastic approach provides only an approximation to the MLE, one must compare it with other possible approximations to the MLE. Many such approaches exist, including spectral methods, low-rank approximations, covariance tapering and those based on some form of composite likelihood. All these methods involve computing the likelihood itself and not just its gradient, and thus all share this advantage over solving (\[ascore\]). Note that one can use randomized algorithms to approximate $\log\det K$ and thus approximate the loglikelihood directly \[@zhangY\]. However, this approximation requires first taking a power series expansion of $K$ and then applying the randomization trick to each term in the truncated power series; the examples presented by @zhangY show that the approach does not generally provide a good approximation to the loglikelihood. Since the accuracy of the power series approximation to $\log\det K$ depends on the condition number of $K$, some of the filtering ideas described by @steinchenanitescufiltering and used to good effect in Section \[sec4\] here could perhaps be of value for approximating $\log\det K$, but we do not explore that possibility. See @aune for some recent developments on stochastic approximation of log determinants of positive definite matrices. Let us consider the four approaches of spectral methods, low-rank approximations, covariance tapering and composite likelihood in turn. Spectral approximations to the likelihood can be fast and accurate for gridded data \[@whittle [@guyon; @dahlhaus]\], although even for gridded data they may require some prefiltering to work well \[@stein1995\]. In addition, the approximations tend to work less well as the number of dimensions increase \[@dahlhaus\] and thus may be problematic for space–time data, especially if the number of spatial dimensions is three. Spectral approximations have been proposed for ungridded data \[@fuentes\], but they do not work as well as they do for gridded data from either a statistical or computational perspective, especially if large subsets of observations do not form a regular grid. Furthermore, in contrast to the approach we propose here, there appears to be no easy way of improving the approximations by doing further calculations, nor is it clear how to assess the loss of efficiency by using spectral approximations without a large extra computational burden. Low-rank approximations, in which the covariance matrix is approximated by a low-rank matrix plus a diagonal matrix, can greatly reduce the burden of memory and computation relative to the exact likelihood \[@cressiejohannesson [@eidsvik]\]. However, for the kinds of applications we have in mind, in which the diagonal component of the covariance matrix does not dominate the small-scale variation of the process, these low-rank approximations tend to work poorly and are not a viable option \[@steinkorea\]. Covariance tapering replaces the covariance matrix of interest by a sparse covariance matrix with similar local behavior \[@furrer\]. There is theoretical support for this approach \[@kaufman [@wangloh]\], but the tapered covariance matrix must be very sparse to help a great deal with calculating the log determinant of the covariance matrix, in which case @steintaper finds that composite likelihood approaches will often be preferable. There is scope for combining covariance tapering with the approach presented here in that sparse matrices lead to efficient matrix–vector multiplication, which is also essential for our implementation of computing (\[ascore\]) based on iterative methods to do the matrix solves. @sang show that covariance tapering and low-rank approximations can also sometimes be profitably combined to approximate likelihoods. We consider methods based on composite likelihoods to be the main competitor to solving (\[ascore\]). The approximate loglikelihoods described by @vecchia [@steinchiwelty; @caragea] can all be written in the following form: for some sequence of pairs of matrices $(A_j,B_j)$, $j=1,\ldots,q$, all with $n$ columns, at most $n$ rows and full rank, $$\label{composite} \sum_{j=1}^q \log f_{j,\theta}(A_j Z\mid B_j Z),$$ where $f_{j,\theta}$ is the conditional Gaussian density of $A_j Z$ given $B_j Z$. As proposed by @vecchia and @steinchiwelty, the rank of $B_j$ will generally be larger than that of $A_j$, in which case the main computation in obtaining (\[composite\]) is finding Cholesky decompositions of the covariance matrices of $B_1Z,\ldots, B_q Z$. For example, @vecchia just lets $A_j Z$ be the $j$th component of $Z$ and $B_j Z$ some subset of $Z_1,\ldots,Z_{j-1}$. If $m$ is the largest of these subsets, then the storage requirements for this computation are $O(m^2)$ rather than $O(n^2)$. Comparable to increasing the number of $U_j$’s in the randomized algorithm used here, this approach can be updated to obtain a better approximation of the likelihood by increasing the size of the subset of $Z_1,\ldots,Z_{j-1}$ to condition on when computing the conditional density of $Z_j$. However, for this approach to be efficient from the perspective of flops, one needs to store the Cholesky decompositions of the covariance matrices of $B_1Z,\ldots,B_q Z$, which would greatly increase the memory requirements of the algorithm. For dealing with truly massive data sets, our long-term plan is to combine the randomized approach studied here with a composite likelihood by using the randomized algorithms to compute the gradient of (\[composite\]), thus making it possible to consider $A_j$’s and $B_j$’s of larger rank than would be feasible if one had to do exact calculations. Section \[sec2\] provides a bound on the efficiency of the estimating equations based on the approximate likelihood relative to the Fisher information matrix. The bound is in terms of the condition number of the true covariance matrix of the observations and shows that if the covariance matrix is well conditioned, $N$ does not need to be very large to obtain nearly optimal estimating equations. Section \[sec3\] shows how one can get improved estimating equations by choosing the $U_j$’s in (\[ascore\]) based on a design related to $2^n$ factorial designs. Section \[sec4\] describes details of the algorithms, including methods for solving the approximate score equations and the role of preconditioning. Section \[sec5\] provides results of numerical experiments on simulated data. These results show that the basic method can work well for moderate values of $N$, even sometimes when the condition numbers of the covariance matrices do not stay bounded as the number of observations increases. Furthermore, the algorithm with the $U_j$’s chosen as in Section \[sec3\] can lead to substantially more accurate approximations for a given $N$. A large-scale numerical experiment shows that for observations on a partially occluded grid, the algorithm scales nearly linearly in the sample size. Section \[sec6\] applies the methods to OMI (Ozone Monitoring Instrument) Level 3 (gridded) total column ozone measurements for April 2012 in the latitude band $40^\circ$–$50^\circ$N. ![Demeaned ozone data (Dobson units) plotted using a heat color map. Missing data is colored white.[]{data-label="figozone"}](627f01.eps) The data are given on a $1^\circ\times1^\circ$ grid, so if the data were complete, there would be a total of $360\times10\times30 = 108\mbox{,}000$ observations. However, as Figure \[figozone\] shows, there are missing observations, mostly due to a lack of overlap in data from different orbits taken by OMI, but also due to nearly a full day of missing data on April 29–30, so that there are 84,942 observations. By acting as if all observations are taken at noon local time and assuming the process is stationary in longitude and time, the covariance matrix for the observations can be embedded in a block circulant matrix, greatly reducing the computational effort needed for multiplying the covariance matrix by a vector. Using (\[ascore\]) and a factorized sparse inverse preconditioner \[@kolo\], we are able to compute an accurate approximation to the MLE for a simple model that captures some of the main features in the OMI data, including the obvious movement of ozone from day to day visible in Figure \[figozone\] that coincides with the prevailing westerly winds in this latitude band. Variance of stochastic approximation of the score function {#sec2} ========================================================== This section gives a bound relating the covariance matrices of the approximate and exact score functions. Let us first introduce some general notation for unbiased estimating equations. Suppose $\theta$ has $p$ components and $g(\theta)=(g_1(\theta ),\ldots, g_p(\theta))'=0$ is a set of unbiased estimating equations for $\theta$ so that $E_\theta g(\theta)=0$ for all $\theta$. Write $\dot{g}(\theta)$ for the $p\times p$ matrix whose $ij$th element is $\frac{\partial}{\partial\theta_i}g_j(\theta)$ and $\operatorname{cov}_\theta \{g(\theta)\}$ for the covariance matrix of $g(\theta)$. The Godambe information matrix \[@varin\], $$\mathcal{E}\bigl\{g(\theta)\bigr\}= E_\theta\bigl\{\dot{g}(\theta)\bigr \} \bigl[\operatorname{cov}_\theta\bigl\{g(\theta)\bigr\} \bigr]^{-1} E_\theta\bigl\{\dot{g}(\theta)\bigr\}$$ is a natural measure of the informativeness of the estimating equations \[@Heyde, Definition 2.1\]. For positive semidefinite matrices $A$ and $B$, write $A\succeq B$ if $A-B$ is positive semidefinite. For unbiased estimating equations $g(\theta)=0$ and $h(\theta)=0$, then we can say $g$ dominates $h$ if $\mathcal{E}\{g(\theta)\} \succeq\mathcal{E}\{h(\theta)\}$. Under sufficient regularity conditions on the model and the estimating equations, the score equations are the optimal estimating equations \[@Bhapkar\]. Specifically, for the score equations, the Godambe information matrix equals the Fisher information matrix, ${{\mathcal}I}(\theta)$, so this optimality condition means ${{\mathcal}I}(\theta)\succeq\mathcal{E}\{g(\theta)\}$ for all unbiased estimating equations $g(\theta)=0$. Writing $M_{ij}$ for the $ij$th element of the matrix $M$, for the score equations in (\[score\]), ${{\mathcal}I}_{ij}(\theta) = \frac{1}{2}\operatorname{tr}(K^{-1}K_iK^{-1}K_j)$ \[@stein-book, page 179\]. For the approximate score equations (\[ascore\]), it is not difficult to show that $E_\theta\dot{g}(\theta,N)=-{{\mathcal}I}(\theta)$. Furthermore, writing $W^i$ for $K^{-1}K_i$ and defining the matrix ${{\mathcal}J}(\theta)$ by ${{\mathcal}J}_{ij}(\theta) = \operatorname{cov}(U_1'W^iU_1,U_1'W^jU_1)$, we have $$\label{B} \operatorname{cov}_\theta\bigl\{g(\theta,N)\bigr\} = {{\mathcal}I}(\theta) + \frac{1}{4N}{{\mathcal}J}(\theta),$$ so that $\mathcal{E}\{g(\theta,N)\}={{\mathcal}I}(\theta) \{{{\mathcal}I}(\theta) +\frac{1}{4N}{{\mathcal}J}(\theta) \}^{-1}{{\mathcal}I}(\theta)$, which, as $N\to\infty$, tends to ${{\mathcal}I}(\theta)$. In fact, as also demonstrated empirically by @anitescu2012mfa, one may often not need $N$ to be that large to get estimating equations that are nearly as efficient as the exact score equations. Writing $U_{1j}$ for the $j$th component of $U_1$, we have $$\begin{aligned} \label{Jij} {{\mathcal}J}_{ij}(\theta) & = & \sum_{k,\ell,p,q=1}^n \operatorname{cov}\bigl(W_{k\ell}^i U_{1k}U_{1\ell}, W_{pq}^j U_{1p}U_{1q}\bigr) \nonumber \\ & = & \sum_{k\ne\ell}\bigl\{\operatorname{cov} \bigl(W_{k\ell}^i U_{1k}U_{1\ell },W_{k\ell}^j U_{1k}U_{1\ell}\bigr)+\operatorname{cov}\bigl(W_{k\ell}^i U_{1k}U_{1\ell},W_{\ell k}^j U_{1k}U_{1\ell}\bigr)\bigr\} \nonumber\\[-8pt]\\[-8pt] & = & \sum_{k\ne\ell}\bigl(W_{k\ell}^i W_{k\ell}^j + W_{k\ell}^i W_{\ell k}^j\bigr) \nonumber \\ & = & \operatorname{tr}\bigl(W^i W^j\bigr) + \operatorname{tr} \bigl\{W^i \bigl(W^j\bigr)' \bigr\} - 2\sum _{k=1}^n W_{kk}^iW_{kk}^j.\nonumber\end{aligned}$$ As noted by @hutchinson, the terms with $k=\ell$ drop out in the second step because $U_{1j}^2=1$ with probability 1. When $K(\theta)$ is diagonal for all $\theta$, then $N=1$ gives the exact score equations, although in this case computing $\operatorname{tr}(K^{-1}K_i)$ directly would be trivial. Writing $\kappa(\cdot)$ for the condition number of a matrix, we can bound$\operatorname{cov}_\theta\{g(\theta,N)\}$ in terms of ${{\mathcal}I}(\theta)$ and $\kappa(K)$. The proof of the following result is given in the . \[tmain\] $$\label{Bbound} \operatorname{cov}_\theta\bigl\{g(\theta,N)\bigr\} \preceq{{\mathcal}I}(\theta) \biggl\{ 1 + \frac{(\kappa(K)+1)^2}{4N\kappa(K)} \biggr\}.$$ It follows from (\[Bbound\]) that $$\mathcal{E}\bigl\{g(\theta,N)\bigr\} \succeq\biggl\{ 1+ \frac{(\kappa (K)+1)^2}{4N\kappa(K)} \biggr\}^{-1}{{\mathcal}I}(\theta).$$ In practice, if $\frac{(\kappa(K)+1)^2}{4N\kappa(K)} < 0.01$, so that the loss of information in using (\[ascore\]) rather than (\[score\]) was at most 1%, we would generally be satisfied with using the approximate score equations and a loss of information of even 10% or larger might be acceptable when one has a massive amount of data. For example, if $\kappa(K)=5$, a bound of 0.01 is obtained with $N=180$ and a bound of 0.1 with $N=18$. It is possible to obtain unbiased estimating equations similar to (\[ascore\]) whose statistical efficiency does not depend on $\kappa(K)$. Specifically, if we write $\operatorname{tr}(K^{-1}K_i)$ as $\operatorname{tr}((G')^{-1}K_iG^{-1})$, where $G$ is any matrix satisfying $G'G=K$, we then have that $$\label{symscore} h_i(\theta,N) = \frac{1}{2}Z'K^{-1}K_iK^{-1}Z -\frac{1}{2N}\sum_{j=1}^N U_j'\bigl(G'\bigr)^{-1}K_iG^{-1}U_j = 0$$ for $i=1,\ldots,p$ are also unbiased estimating equations for $\theta$. In this case, $\operatorname{cov}_\theta\{h(\theta,N)\}\preceq ( 1+\frac{1}{N} ){{\mathcal}I}(\theta)$, whose proof is similar to that of Theorem \[tmain\] but exploits the symmetry of $(G')^{-1}K_iG^{-1}$. This bound is less than or equal to the bound in (\[Bbound\]) on $\operatorname{cov}_\theta\{g(\theta,N)\}$. Whether it is preferable to use (\[symscore\]) rather than (\[ascore\]) depends on a number of factors, including the sharpness of the bound in (\[Bbound\]) and how much more work it takes to compute $G^{-1}U_j$ than to compute $K^{-1}U_j$. An example of how the action of such a matrix square root can be approximated efficiently using only $O(n)$ storage is presented by @chen2011computing. Dependent designs {#sec3} ================= \[secdependent\] Choosing the $U_j$’s independently is simple and convenient, but one can reduce the variation in the stochastic approximation by using a more sophisticated design for the $U_j$’s; this section describes such a design. Suppose that $n=Nm$ for some nonnegative integer $m$ and that $\beta_1,\ldots,\beta_N$ are fixed vectors of length $N$ with all entries $\pm1$ for which $\frac{1}{N}\sum_{j=1}^N \beta_j\beta'_j = I$. For example, if $N=2^q$ for a positive integer $q$, then the $\beta_j$’s can be chosen to be the design matrix for a saturated model of a $2^q$ factorial design in which the levels of the factors are set at $\pm1$ \[@BHH, Chapter 5\]. In addition, assume that $X_1,\ldots,X_m$ are random diagonal matrices of size $N$ and $Y_{jk}$, $j=1,\ldots,N; k=1,\ldots,m$ are random variables such that all the diagonal elements of the $X_j$’s and all the $Y_{jk}$’s are i.i.d. symmetric Bernoulli random variables. Then define $$\label{Uj} U_j = \pmatrix{ Y_{j1}X_1 \cr \vdots \cr Y_{jm}X_m} \beta_j.$$ One can easily show that for any $Nm\times Nm$ matrix $M$, $E (\frac{1}{N}\sum_{j=1}^N U_j'MU_j ) = \operatorname{tr}(M)$. Thus, we can use this definition of the $U_j$’s in (\[ascore\]), and the resulting estimating equations are still unbiased. This design is closely related to a class of designs introduced by @avron, who propose selecting the $U_j$’s as follows. Suppose $H$ is a Hadamard matrix, that is, an $n\times n$ orthogonal matrix with elements $\pm1$. @avron actually consider $H$ a multiple of a unitary matrix, but the special case $H$ Hadamard makes their proposal most similar to ours. Then, using simple random sampling (with replacement), they choose $N$ columns from this matrix and multiply this $n\times N$ matrix by an $n\times n$ diagonal matrix with diagonal entries made up of independent symmetric Bernoulli random variables. The columns of this resulting matrix are the $U_j$’s. We are also multiplying a subset of the columns of a Hadamard matrix by a random diagonal matrix, but we do not select the columns by simple random sampling from some arbitrary Hadamard matrix. The extra structure we impose yields beneficial results in terms of the variance of the randomized trace approximation, as the following calculations show. Partitioning $M$ into an $m\times m$ array of $N\times N$ matrices with $k\ell$th block $M^b_{k\ell}$, we obtain the following: $$\label{UjMUj} \frac{1}{N}\sum_{j=1}^N U_j'MU_j = \frac{1}{N} \sum _{k,\ell=1}^m\sum_{j=1}^N Y_{jk}Y_{j\ell} \beta_j'X_k M^b_{k\ell} X_\ell\beta_j.$$ Using $Y_{jk}^2=1$ and $X_k^2 = I$, we have $$\begin{aligned} \frac{1}{N}\sum_{j=1}^N Y_{jk}^2 \beta_j'X_k M^b_{kk} X_k \beta_j & = & \frac{1}{N}\operatorname{tr}\Biggl(X_k M^b_{kk} X_k \sum_{j=1}^N \beta_j\beta_j' \Biggr) \\ & = & \operatorname{tr}\bigl(M^b_{kk}X_k^2 \bigr) \\ & = & \operatorname{tr}\bigl(M^b_{kk}\bigr),\end{aligned}$$ which is not random. Thus, if $M$ is block diagonal (i.e., $M^b_{k\ell}$ is a matrix of zeroes for all $k\ne\ell$), (\[UjMUj\]) yields $\operatorname{tr}(M)$ without error. This result is an extension of the result that independent $U_j$’s give $\operatorname{tr}(M)$ exactly for diagonal $M$. Furthermore, it turns out that, at least in terms of the variance of $\frac{1}{N}\sum_{j=1}^N U_j'MU_j$, for the elements of $M$ off the block diagonal, we do exactly the same as we do when the $U_j$’s are independent. Write $B(\theta)$ for $\operatorname{cov}\{g(\theta,N)\}$ with $g(\theta,N)$ defined as in (\[ascore\]) with independent $U_j$’s. Define $g^d(\theta,N)=0$ for the unbiased estimating equations defined by (\[ascore\]) with dependent $U_j$’s defined by (\[Uj\]) and $B^d(\theta)$ to be the covariance matrix of $g^d(\theta,N)$. Take $T(N,n)$ to be the set of pairs of positive integers $(k,\ell)$ with $1\le\ell<k \le n$ for which $\lfloor k/N\rfloor= \lfloor\ell /N\rfloor$. We have the following result, whose proof is given in the . \[tdependent\] For any vector $v=(v_1,\ldots,v_p)'$, $$\label{improve} v'B(\theta)v - v'B^d(\theta)v = \frac{2}{N} \sum_{(k,\ell)\in T(N,n)} \Biggl\{ \sum _{i=1}^p v_i \bigl( W_{k\ell}^i+W_{\ell k}^i \bigr) \Biggr\}^2.$$ Thus, $B(\theta) \succeq B^d(\theta)$. Since $E_\theta\dot{g} (\theta,N) = E_\theta\dot{g}^d(\theta,N)=-\mathcal{I}(\theta)$, it follows that $\mathcal{E}\{g^d(\theta,N)\}\succeq\mathcal{E}\{g(\theta,N)\}$. How much of an improvement will result from using dependent $U_j$’s depends on the size of the $W_{k\ell}^i$’s within each block. For spatial data, one would typically group spatially contiguous observations within blocks. How to block for space–time data is less clear. The results here focus on the variance of the randomized trace approximation. @avron obtain bounds on the probability that the approximation error is less than some quantity and note that these results sometimes give rankings for various randomized trace approximations different from those obtained by comparing variances. Computational aspects {#sec4} ===================== Finding $\theta$ that solves the estimating equations (\[ascore\]) requires a nonlinear equation solver in addition to computing linear solves in $K$. The nonlinear solver starts at an initial guess $\theta^0$ and iteratively updates it to approach a (hopefully unique) zero of (\[ascore\]). In each iteration, at $\theta^i$, the nonlinear solver typically requires an evaluation of $g(\theta^i,N)$ in order to find the next iterate $\theta^{i+1}$. In turn, the evaluation of $g$ requires employing a linear solver to compute the set of vectors $K^{-1}Z$ and $K^{-1}U_j$, $j=1,\ldots,N$. The Fisher information matrix $\mathcal{I}(\theta)$ and the matrix $\mathcal{J}(\theta)$ contain terms involving matrix traces and diagonals. Write ${\operatorname{diag}}(\cdot)$ for a column vector containing the diagonal elements of a matrix and $\circ$ for the Hadamard (elementwise) product of matrices. For any real matrix $A$, $${\operatorname{tr}}(A)=E_U\bigl(U'AU\bigr) \quad\mbox{and}\quad {\operatorname{diag}}(A)=E_U(U\circ AU),$$ where the expectation $E_U$ is taken over $U$, a random vector with i.i.d. symmetric Bernoulli components. One can unbiasedly estimate $\mathcal{I}(\theta)$ and $\mathcal{J}(\theta)$ by $$\label{Iij} {\widehat}{\mathcal{I}}_{ij}(\theta)=\frac{1}{2N_2}\sum _{k=1}^{N_2}U_k'W^iW^jU_k$$ and $$\begin{aligned} \label{Jij2} {\widehat}{\mathcal{J}}_{ij}(\theta) & = & \frac{1}{N_2}\sum _{k=1}^{N_2}U_k'W^iW^jU_k +\frac{1}{N_2}\sum_{k=1}^{N_2}U_k'W^i \bigl(W^j\bigr)'U_k \nonumber\\[-8pt]\\[-8pt] &&{} -2\sum_{\ell=1}^n \Biggl[ \frac{1}{N_2}\sum_{k=1}^{N_2} \bigl(U_k\circ W^iU_k\bigr) \Biggr]_{\ell} \Biggl[\frac{1}{N_2}\sum_{k=1}^{N_2} \bigl(U_k\circ W^jU_k\bigr) \Biggr]_{\ell}. \nonumber\end{aligned}$$ Note that here the set of vectors $U_k$ need not be the same as that in (\[ascore\]) and that $N_2$ may not be the same as $N$, the number of $U_j$’s used to compute the estimate of $\theta$. Evaluating ${\widehat}{\mathcal{I}}(\theta)$ and ${\widehat}{\mathcal {J}}(\theta)$ requires linear solves since $W^iU_k=K^{-1}(K_iU_k)$ and $(W^i)'U_k=K_i(K^{-1}U_k)$. Note that one can also unbiasedly estimate $\mathcal{J}_{ij}(\theta)$ as the sample covariance of $U'_kW^iU_k$ and $U'_kW^jW_k$ for $k=1,\ldots,N$, but (\[Jij2\]) directly exploits properties of symmetric Bernoulli variables (e.g., $U^2_{1j}=1$). Further study would be needed to see when each approach is preferred. Linear solver {#sec4.1} ------------- We consider an iterative solver for solving a set of linear equations $Ax=b$ for a symmetric positive definite matrix $A\in{{\mathbb R}}^{n\times n}$, given a right-hand vector $b$. Since the matrix $A$ (in our case the covariance matrix) is symmetric positive definite, the conjugate gradient algorithm is naturally used. Let $x^i$ be the current approximate solution, and let $r^i=b-Ax^i$ be the residual. The algorithm finds a search direction $q^i$ and a step size $\alpha^i$ to update the approximate solution, that is, $x^{i+1}=x^i+\alpha^iq^i$, such that the search directions $q^i,\ldots,q^0$ are mutually $A$-conjugate \[i.e., $(q^i)'Aq^j = 0$ for $i\ne j$\] and the new residual $r^{i+1}$ is orthogonal to all the previous ones, $r^i,\ldots,r^0$. One can show that the search direction is a linear combination of the current residual and the past search direction, yielding the following recurrence formulas: $$\begin{aligned} x^{i+1}&=&x^i+\alpha^iq^i, \\ r^{i+1}&=&r^i-\alpha^iAq^i, \\ q^{i+1}&=&r^{i+1}+\beta^iq^i,\end{aligned}$$ where $\alpha^i= \langle r^i,r^i \rangle/ \langle Aq^i,q^i \rangle$ and $\beta^i= \langle r^{i+1},r^{i+1} \rangle/ \langle r^i,r^i \rangle$, and $ \langle\cdot,\cdot\rangle$ denotes the vector inner product. Letting $x^*$ be the exact solution, that is, $Ax^*=b$, then $x^i$ enjoys a linear convergence to $x^*$: $$\label{eqncgconverge} \bigl\|x^i-x^*\bigr\|_A\le2 \biggl( \frac{\sqrt{\kappa(A)}-1}{\sqrt{\kappa (A)}+1} \biggr)^{i}\bigl\|x^0-x^*\bigr\|_A,$$ where $\|\cdot\|_A= \langle A\cdot,\cdot\rangle^{{1/2}}$ is the $A$-norm of a vector. Asymptotically, the time cost of one iteration is upper bounded by that of multiplying $A$ by $q^i$, which typically dominates other vector operations when $A$ is not sparse. Properties of the covariance matrix can be exploited to efficiently compute the matrix–vector products. For example, when the observations are on a lattice (regular grid), one can use the fast Fourier transform (FFT), which takes time $O(n\log n)$ \[@toeplitzbook\]. Even when the grid is partial (with occluded observations), this idea can still be applied. On the other hand, for nongridded observations, exact multiplication generally requires $O(n^2)$ operations. However, one can use a combination of direct summations for close-by points and multipole expansions of the covariance kernel for faraway points to compute the matrix–vector products in $O(n\log n)$, even $O(n)$, time \[@treecode [@fmm]\]. In the case of Matérn-type Gaussian processes and in the context of solving the stochastic approximation (\[ascore\]), such fast multipole approximations were presented by @anitescu2012mfa. Note that the total computational cost of the solver is the cost of each iteration times the number of iterations, the latter being usually much less than $n$. The number of iterations to achieve a desired accuracy depends on how fast $x^i$ approaches $x^*$, which, from (\[eqncgconverge\]), is in turn affected by the condition number $\kappa$ of $A$. Two techniques can be used to improve convergence. One is to perform preconditioning in order to reduce $\kappa$; this technique will be discussed in the next section. The other is to adopt a block version of the conjugate gradient algorithm. This technique is useful for solving the linear system for the same matrix with multiple right-hand sides. Specifically, denote by $AX=B$ the linear system one wants to solve, where $B$ is a matrix with $s$ columns, and the same for the unknown $X$. Conventionally, matrices such as $B$ are called *block vectors*, honoring the fact that the columns of $B$ are handled simultaneously. The block conjugate gradient algorithm is similar to the single-vector version except that the iterates $x^i$, $r^i$ and $q^i$ now become block iterates $X^i$, $R^i$ and $Q^i$ and the coefficients $\alpha^i$ and $\beta^i$ become $s\times s$ matrices. The detailed algorithm is not shown here; interested readers are referred to @olearyblockcg. If $X^*$ is the exact solution, then $X^i$ approaches $X^*$ at least as fast as linearly: $$\label{eqnbcgconverge} \bigl\|\bigl(X^i\bigr)_j-\bigl(X^* \bigr)_j\bigr\|_A\le C_j \biggl(\frac{\sqrt{\kappa_s(A)}-1}{\sqrt{\kappa_s(A)}+1} \biggr)^{i},\qquad j=1,\ldots,s,$$ where $(X^i)_j$ and $(X^*)_j$ are the $j$th column of $X^i$ and $X^*$, respectively; $C_j$ is some constant dependent on $j$ but not $i$; and $\kappa_s(A)$ is the ratio between $\lambda_n(A)$ and $\lambda_s(A)$ with the eigenvalues $\lambda_k$ sorted increasingly. Comparing (\[eqncgconverge\]) with (\[eqnbcgconverge\]), we see that the modified condition number $\kappa_s$ is less than $\kappa$, which means that the block version of the conjugate gradient algorithm has a faster convergence than the standard version does. In practice, since there are many right-hand sides (i.e., the vectors $Z$, $U_j$’s and $K_iU_k$’s), we always use the block version. Preconditioning/filtering {#sec4.2} ------------------------- Preconditioning is a technique for reducing the condition number of the matrix. Here, the benefit of preconditioning is twofold: it encourages the rapid convergence of an iterative linear solver and, if the effective condition number is small, it strongly bounds the uncertainty in using the estimating equations (\[ascore\]) instead of the exact score equations (\[score\]) for estimating parameters (see Theorem \[tmain\]). In numerical linear algebra, preconditioning refers to applying a matrix $M$, which approximates the inverse of $A$ in some sense, to both sides of the linear system of equations. In the simple case of left preconditioning, this amounts to solving $MAx=Mb$ for $MA$ better conditioned than $A$. With certain algebraic manipulations, the matrix $M$ enters into the conjugate gradient algorithm in the form of multiplication with vectors. For the detailed algorithm, see @saadbookiterativemethod. This technique does not explicitly compute the matrix $MA$, but it requires that the matrix–vector multiplications with $M$ can be efficiently carried out.=-1 For covariance matrices, certain filtering operations are known to reduce the condition number, and some can even achieve an optimal preconditioning in the sense that the condition number is bounded by a constant independent of the size of the matrix \[@steinchenanitescufiltering\]. Note that these filtering operations may or may not preserve the rank/size of the matrix. When the rank is reduced, then some loss of statistical information results when filtering, although similar filtering is also likely needed to apply spectral methods for strongly correlated spatial data on a grid \[@stein1995\]. Therefore, we consider applying the same filter to all the vectors and matrices in the estimating equations, in which case (\[ascore\]) becomes the stochastic approximation to the score equations of the *filtered* process. Evaluating the filtered version of $g(\theta,N)$ becomes easier because the linear solves with the filtered covariance matrix converge faster. Nonlinear solver {#sec4.3} ---------------- \[secnonlinearsolver\] The choice of the nonlinear solver is problem dependent. The purpose of solving the score equations (\[score\]) or the estimating equations (\[ascore\]) is to maximize the loglikelihood function $\mathcal{L}(\theta)$. Therefore, investigation into the shape of the loglikelihood surface helps identify an appropriate solver. In Section \[sec5\], we consider the power law generalized covariance model ($\alpha>0$): $$\label{GC1} G(x;\theta)= \cases{\Gamma(-\alpha/2) r^{\alpha}, &\quad if $ \alpha/2\notin{{\mathbb N}}$, \cr (-1)^{1+\alpha/2}r^{\alpha}\log r, &\quad if $\alpha/2\in {{\mathbb N}}$,}$$ where $x=[x_1,\ldots,x_d]\in{{\mathbb R}}^d$ denotes coordinates, $\theta$ is the set of parameters containing $\alpha>0$, $\ell=[\ell_1,\ldots,\ell_d]\in{{\mathbb R}}^d$, and $r$ is the elliptical radius $$\label{GC2} r=\sqrt{\frac{x_1^2}{\ell_1^2}+\cdots+\frac{x_d^2}{\ell_d^2}}.$$ Allowing a different scaling in different directions may be appropriate when, for example, variations in a vertical direction may be different from those in a horizontal direction. The function $G$ is conditionally positive definite; therefore, only the covariances of authorized linear combinations of the process are defined \[@geostatisticsbook, Section 4.3\]. In fact, $G$ is $p$-conditionally positive definite if and only if $2p+2>\alpha$ \[see @geostatisticsbook, Section 4.5\], so that applying the discrete Laplace filter (which gives second-order differences) $\tau$ times to the observations yields a set of authorized linear combinations when $\tau\ge\frac {1}{2}\alpha$. @steinchenanitescufiltering show that if $\alpha=4\tau-d$, then the covariance matrix has a bounded condition number independent of the matrix size. Consider the grid $\{\delta\mathbf{j}\}$ for some fixed spacing $\delta$ and $\mathbf{j}$ a vector whose components take integer values between $0$ and $m$. Applying the filter $\tau$ times, we obtain the covariance matrix $$K_{\mathbf{i}\mathbf{j}}={\operatorname{cov}}\bigl\{\Delta^{\tau}Z(\delta\mathbf{i}),\Delta ^{\tau}Z(\delta\mathbf{j})\bigr\},$$ where $\Delta$ denotes the discrete Laplace operator $$\Delta Z(\delta\mathbf{j})=\sum_{p=1}^d\bigl \{Z(\delta\mathbf{j}-\delta\mathbf{e}_p)-2Z(\delta\mathbf{j})+Z(\delta\mathbf{j}+ \delta\mathbf{e}_p)\bigr\}$$ with $\mathbf{e}_p$ meaning the unit vector along the $p$th coordinate. If $\tau=\operatorname{round}((\alpha+d)/4)$, the resulting $K$ is both positive definite and reasonably well conditioned. Figure \[figloglik\] shows a sample loglikelihood surface for $d=1$ based on an observation vector $Z$ simulated from a 1D partial regular grid spanning the range $[0,100]$, using parameters $\alpha=1.5$ and $\ell=10$. (A similar 2D grid is shown later in Figure \[figgrid\].) The peak of the surface is denoted by the solid white dot, which is not far away from the truth $\theta=(1.5,10)$. The white dashed curve (profile of the surface) indicates the maximum loglikelihoods $\mathcal{L}$ given $\alpha$. The curve is also projected on the $\alpha-\mathcal{L}$ plane and the $\alpha-\ell$ plane. One sees that the loglikelihood value has small variation (ranges from $48$ to $58$) along this curve compared with the rest of the surface, whereas, for example, varying just the parameter $\ell$ changes the loglikelihood substantially. ![A sample loglikelihood surface for the power law generalized covariance kernel, with profile curve and peak plotted.[]{data-label="figloglik"}](627f02.eps) A Newton-type nonlinear solver starts at some initial point $\theta^0$ and tries to approach the optimal point (one that solves the score equations).[^1] Let the current point be $\theta^i$. The solver finds a direction $q^i$ and a step size $\alpha^i$ in some way to move the point to $\theta^{i+1}=\theta^i+\alpha^iq^i$, so that the value of $\mathcal{L}$ is increased. Typically, the search direction $q^i$ is the inverse of the Jacobian multiplied by $\theta ^i$, that is, $q^i=\dot{g}(\theta^i,N)^{-1}\theta^i$. This way, $\theta^{i+1}$ is closer to a solution of the score equations. Figure \[figloglik\] shows a loglikelihood surface when $d=1$. The solver starts somewhere on the surface and quickly climbs to a point along the profile curve. However, this point might be far away from the peak. It turns out that along this curve a Newton-type solver is usually unable to find a direction with an appropriate step size to numerically increase $\mathcal{L}$, in part because of the narrow ridge indicated in the figure. The variation of $\mathcal{L}$ along the normal direction of the curve is much larger than that along the tangent direction. Thus, the iterate $\theta^i$ is trapped and cannot advance to the peak. In such a case, even though the estimated maximized likelihood could be fairly close to the true maximum, the estimated parameters could be quite distant from the MLE of $(\alpha,\ell)$. To successfully solve the estimating equations, we consider each component of $\ell$ an implicit function of $\alpha$. Denote by $$\label{ascore2} g_i(\alpha,\ell_1,\ldots, \ell_d)=0,\qquad i=1,\ldots,d+1,\vadjust{\goodbreak}$$ the estimating equations, ignoring the fixed variable $N$. The implicit function theorem indicates that a set of functions $\ell_1(\alpha),\ldots,\ell_d(\alpha)$ exists around an isolated zero of (\[ascore2\]) in a neighborhood where (\[ascore2\]) is continuously differentiable, such that $$g_i\bigl(\alpha,\ell_1(\alpha),\ldots, \ell_d(\alpha)\bigr)=0\qquad \mbox{for } i=2,\ldots,d+1.$$ Therefore, we need only to solve the equation $$\label{ascore3} g_1\bigl(\alpha,\ell_1(\alpha),\ldots, \ell_d(\alpha)\bigr)=0$$ with a single variable $\alpha$. Numerically, a much more robust method than a Newton-type method exists for finding a root of a one-variable function. We use the standard method of Forsythe, Malcolm and Moler \[([-@fzero]), see the Fortran code \] for solving (\[ascore3\]). This method in turn requires the evaluation of the left-hand side of (\[ascore3\]). Then, the $\ell_i$’s are evaluated by solving $g_2,\ldots,g_{d+1}=0$ fixing $\alpha$, whereby a Newton-type algorithm is empirically proven to be an efficient method. Experiments {#sec5} =========== \[secexp\] In this section we show a few experimental results based on a partially occluded regular grid. The rationale for using such a partial grid is to illustrate a setting where spectral techniques do not work so well but efficient matrix–vector multiplications are available. A partially occluded grid can occur, for example, when observations of some surface characteristics are taken by a satellite-based instrument and it is not possible to obtain observations over regions with sufficiently dense cloud cover. The ozone example in Section \[sec6\] provides another example in which data on a partial grid occurs. This section considers a grid with physical range $[0,100]\times[0,100]$ and a hole in a disc shape of radius $10$ centered at $(40,60)$. An illustration of the grid, with size $32\times32$, is shown in Figure \[figgrid\]. The matrix–vector multiplication is performed by first doing the multiplication using the full grid via circulant embedding and FFT, followed by removing the entries corresponding to the hole of the grid. Recall that the covariance model is defined in Section \[secnonlinearsolver\], along with the explanation of the filtering step. ![A $32\times32$ grid with a region of missing observations in a disc shape. Internal grid points are grouped to work with the dependent design in Section \[secdependent\].[]{data-label="figgrid"}](627f03.eps) When working with dependent samples, it is advantageous to group nearby grid points such that the resulting blocks have a plump shape and that there are as many blocks with size exactly $N$ as possible. For an occluded grid, this is a nontrivial task. Here we use a simple heuristic to effectively group the points. We divide the grid into horizontal stripes of width $\lfloor\sqrt{N}\rfloor$ (in case $\lfloor\sqrt{N}\rfloor$ does not divide the grid size along the vertical direction, some stripes have a width $\lfloor\sqrt{N}\rfloor+1$). The stripes are ordered from bottom to top, and the grid points inside the odd-numbered stripes are ordered lexicographically in their coordinates, that is, $(x,y)$. In order to obtain as many contiguous blocks as possible, the grid points inside the even-numbered stripes are ordered lexicographically according to $(-x,y)$. This ordering gives a zigzag flow of the points starting from the bottom-left corner of the grid. Every $N$ points are grouped in a block. The coloring of the grid points in Figure \[figgrid\] shows an example of the grouping. Note that because of filtering, observations on either an external or internal boundary are not part of any block. Choice of $N$ {#sec5.1} ------------- One of the most important factors that affect the efficacy of approximating the score equations is the value $N$. Theorem \[tmain\] indicates that $N$ should increase at least like $\kappa(K)$ in order to guarantee the additional uncertainty introduced by approximating the score equations be comparable with that caused by the randomness of the sample $Z$. In the ideal case, when the condition number of the matrix (possibly with filtering) is bounded independent of the matrix size $n$, then even taking $N=1$ is sufficient to obtain estimates with the same rate of convergence as the exact score equations. When $\kappa$ grows with $n$, however, a better guideline for selecting $N$ is to consider the growth of $\mathcal{I}^{-1}\mathcal{J}$. Figure \[figpowerbounds\] plots the condition number of $K$ and the spectral norm of $\mathcal{I}^{-1}\mathcal{J}$ for varying sizes of the matrix and preconditioning using the Laplacian filter. Although performing a Laplacian filtering will yield provably bounded condition numbers only for the case $\alpha=2$, one sees that the filtering is also effective for the cases $\alpha=1$ and $1.5$. Moreover, the norm of $\mathcal {I}^{-1}\mathcal{J}$ is significantly smaller than $\kappa$ when $n$ is large and, in fact, it does not seem to grow with $n$. This result indicates the bound in Theorem 1 is sometimes far too conservative and that using a fixed $N$ can be effective even when $\kappa$ grows with $n$. ![Growth of $\kappa$ compared with that of $\|\mathcal {I}^{-1}\mathcal{J}\|$, for power law kernel in 2D. Left: $\alpha=1$; right: $\alpha=1.5$.[]{data-label="figpowerbounds"}](627f04.eps) Of course, the norm of $\mathcal{I}^{-1}\mathcal{J}$ is not always bounded. In Figure \[figmaternbounds\] we show two examples using the Matérn covariance kernel with smoothness parameter $\nu=1$ and 1.5 (essentially $\alpha=2$ and 3). Without filtering, both $\kappa(K)$ and $\|\mathcal{I}^{-1}\mathcal{J}\|$ grow with $n$, although the plots show that the growth of the latter is significantly slower than that of the former. ![Growth of $\kappa$ compared with that of $\|\mathcal {I}^{-1}\mathcal{J}\|$, for Matérn kernel in 1D, without filtering. Left: $\nu=1$; right: $\nu=1.5$.[]{data-label="figmaternbounds"}](627f05.eps) If the occluded observations are more scattered, then the fast matrix–vector multiplication based on circulant embedding still works fine. However, if the occluded pixels are randomly located and the fraction of occluded pixels is substantial, then using a filtered data set only including Laplacians centered at those observations whose four nearest neighbors are also available might lead to an unacceptable loss of information. In this case, one might instead use a preconditioner based on a sparse approximation to the inverse Cholesky decomposition as described in Section \[sec6\]. A 32x32 grid example {#sec5.2} -------------------- Here, we show the details of solving the estimating equations (\[ascore\]) using a $32\times32$ grid as an example. Setting the truth $\alpha =1.5$ and $\ell=(7,10)$ \[i.e., $\theta=(1.5, 7, 10)$\], consider exact and approximate maximum likelihood estimation based on the data obtained by applying the Laplacian filter once to the observations. Writing $\mathcal{G}$ for $\mathcal{E}\{g(\theta,N)\}$, one way to evaluate the approximate MLEs is to compute the ratios of the square roots of the diagonal elements of $\mathcal{G}^{-1}$ to the square roots of the diagonal elements of $\mathcal{I}^{-1}$. We know these ratios must be at least 1, and that the closer they are to 1, the more nearly optimal the resulting estimating equations based on the approximate score function are. For $N=64$ and independent sampling, we get 1.0156, 1.0125 and 1.0135 for the three ratios, all of which are very close to 1. Since one generally cannot calculate $\mathcal{G}^{-1}$ exactly, it is also worthwhile to compare a stochastic approximation of the diagonal values of $\mathcal{G}^{-1}$ to their exact values. When this approximation was done once for $N=64$ and by using $N_2=100$ in (\[Iij\]) and (\[Jij2\]), the three ratios obtained were 0.9821, 0.9817 and 0.9833, which are all close to 1. ![Effects of $N$ (1, 2, 4, 8, 16, 32, 64). In each plot, the curve with the plus sign corresponds to the independent design, whereas that with the circle sign corresponds to the dependent design. The horizontal axis represents $N$. In plots , and , the vertical axis represents the mean squared differences between the approximate and exact MLEs divided by the mean squared errors for the exact MLEs, for the components $\alpha$, $\ell_1$ and $\ell_2$, respectively. In plot , the vertical axis represents the mean squared difference between the loglikelihood values at the exact and approximate MLEs.[]{data-label="figN"}](627f06.eps) Figure \[figN\] shows the performance of the resulting estimates (to be compared with the exact MLEs obtained by solving the standard score equations). For $N=1$, $2$, $4$, $8$, $16$, $32$ and $64$, we simulated 100 realizations of the process on the $32\times32$ occluded grid, applied the discrete Laplacian once, and then computed exact MLEs and approximations using both independent and dependent (as described in the beginning of Section \[secexp\]) sampling. When $N=1$, the independent and dependent sampling schemes are identical, so only results for independent sampling are given. Figure \[figN\] plots, for each component of $\theta$, the mean squared differences between the approximate and exact MLEs divided by the mean squared errors for the exact MLEs. As expected, these ratios decrease with $N$, particularly for dependent sampling. Indeed, dependent sampling is much more efficient than independent sampling for larger $N$; for example, the results in Figure \[figN\] show that dependent sampling with $N=32$ yields better estimates for all three parameters than does independent sampling with $N=64$. Large-scale experiments {#sec5.3} ----------------------- We experimented with larger grids (in the same physical range). We show the results in Table \[tablargescale\] and Figure \[figlargescale\] for $N=64$. When the matrix becomes large, we are unable to compute $\mathcal{I}$ and $\mathcal{G}$ exactly. Based on the preceding experiment, it seems reasonable to use $N_2=100$ in approximating $\mathcal{I}$ and $\mathcal{G}$. Therefore, the elements of $\mathcal{I}$ and $\mathcal{G}$ in Table \[tablargescale\] were computed only approximately. [@lcd[2.4]{}d[2.4]{}d[2.4]{}cc@]{} **Grid size** & & & & & &\ ${\widehat}{\theta}^N$ & 1.5355 & 1.5084 & 1.4919 & 1.4975 & 1.5011 & 1.5012\ & 6.8507 & 6.9974 & 7.1221 & 7.0663 & 6.9841 & 6.9677\ & 9.2923 & 10.062 & 10.091 & 10.063 & 9.9818 & 9.9600\ \[6pt\] $\sqrt{(\mathcal{I}^{-1})_{ii}}$ & 0.0882 & 0.0406 & 0.0196 & 0.0096 & 0.0048 & 0.0024\ & 0.5406 & 0.3673 & 0.2371 & 0.1464 & 0.0877 & 0.0512\ & 0.8515 & 0.5674 & 0.3605 & 0.2202 & 0.1309 & 0.0760\ \[6pt\] $\frac{\sqrt{(\mathcal{G}^{-1})_{ii}}}{\sqrt{(\mathcal{I}^{-1})_{ii}}}$ & 1.0077 & 1.0077 & 1.0077 & 1.0077 & 1.0077 & 1.0077\ & 1.0062 & 1.0070 & 1.0073 & 1.0074 & 1.0075 & 1.0076\ & 1.0064 & 1.0071 & 1.0073 & 1.0075 & 1.0075 & 1.0076\ ![Running time for increasingly dense grids. The dashed curve fits the recorded times with a function of the form of $n\log n$ times a constant.[]{data-label="figlargescale"}](627f07.eps) One sees that as the grid becomes larger (denser), the variance of the estimates decreases as expected. The matrices $\mathcal{I}^{-1}$ and $\mathcal{G}^{-1}$ are comparable in all cases and, in fact, the ratios stay roughly the same across different sizes of the data. The experiments were run for data size up to around one million, and the scaling of the running time versus data size is favorable. The results show a strong agreement of the recorded times with the scaling $O(n\log n)$. Application {#sec6} =========== Ozone in the stratosphere blocks ultraviolet radiation from the sun and is thus essential to all land-based life on Earth. Satellite-based instruments run by NASA have been measuring total column ozone in the atmosphere daily on a near global scale since 1978 (although with a significant gap in 1994–1996) and the present instrument is the OMI. Here, we consider Level 3 gridded data for the month April 2012 in the latitude band $40^\circ$–$50^\circ$N \[Aura OMI Ozone Level-3 Global Gridded ($1.0\times1.0$ deg) Data Product-OMTO3d (V003)\]. Because total column ozone shows persistent patterns of variation with location, we demeaned the data by, for each pixel, subtracting off the mean of the available observations during April 2012. Figure \[figozone\] displays the resulting demeaned data. There are potentially $360\times10 = 3600$ observations on each day in this latitude strip. However, Figure \[figozone\] shows 14 or 15 strips of missing observations each day, which is due to a lack of overlap in OMI observations between orbits in this latitude band (the orbital frequency of the satellite is approximately 14.6 orbits per day). Furthermore, there is nearly a full day of missing observations toward the end of the record. For the 30-day period, a complete record would have 108,000 observations, of which 84,942 are available. The local time of the Level 2 data on which the Level 3 data are based is generally near noon due to the sun-synchronous orbit of the satellite, but there is some variation in local time of Level 2 data because OMI simultaneously measures ozone over a swath of roughly 3000 km, so that the actual local times of the Level 2 data vary up to about 50 minutes from local noon in the latitude band we are considering. Nevertheless, @fang showed that, for Level 3 total column ozone levels (as measured by a predecessor instrument to the OMI), as long as one stays away from the equator, little distortion is caused by assuming all observations are taken at exactly local noon and we will make this assumption here. As a consequence, within a given day, time (absolute as opposed to local) and longitude are completely confounded, which makes distinguishing longitudinal and temporal dependencies difficult. Indeed, if one analyzed the data a day at a time, there would be essentially no information for distinguishing longitude from time, but by considering multiple days in a single analysis, it is possible to distinguish their influences on the dependence structure. Fitting various Matérn models to subsets of the data within a day, we found that the local spatial variation in the data is described quite well by the Whittle model (the Matérn model with smoothness parameter 1) without a nugget effect. Results in @stein07 suggest some evidence for spatial anisotropy in total column ozone at midlatitudes, but the anisotropy is not severe in the band $40^\circ$–$50^\circ$N and we will ignore it here. The most striking feature displayed in Figure \[figozone\] is the obvious westerly flow of ozone across days. Based on these considerations, we propose the following simple model for the demeaned data $Z(\mathbf{x},t)$. Denoting by $r$ the radius of the Earth, $\varphi$ the latitude, $\psi $ the longitude, and $t$ the time, we assume $Z$ is a 0 mean Gaussian process with covariance function (parameterized by $\theta_0$, $\theta_1$, $\theta_2$ and $v$): $${\operatorname{cov}}\bigl\{Z(\mathbf{x}_1, t_1), Z(\mathbf{x}_2, t_2)\bigr\}= \theta_0{\mathrm{M}}_1 \biggl(\sqrt {\frac{T^2}{\theta_1^2}+\frac {S^2}{\theta_2^2}} \biggr),$$ where $T=t_1-t_2$ is the temporal difference, $S=\|\mathbf{x}(r,\varphi_1, \psi_1-vt_1)-\mathbf{x}(r,\varphi_2,\allowbreak \psi _2-vt_2)\|$ is the (adjusted for drift) spatial difference and $\mathbf{x}(r,\varphi, \psi)$ maps a spherical coordinate to ${{\mathbb R}}^3$. Here, ${\mathrm{M}}_{\nu}$ is the Matérn correlation function $$\label{spacetime} {\mathrm{M}}_{\nu}(x)=\frac{(\sqrt{2\nu}x)^{\nu}{\mathrm{K}}_{\nu}(\sqrt{2\nu }x)}{2^{\nu-1}\Gamma(\nu)}$$ with ${\mathrm{K}}_{\nu}$ the modified Bessel function of the second kind of order $\nu$. We used the following unit system: $\varphi$ and $\psi$ are in degrees, $t$ is in days, and $r\equiv1$. In contravention of standard notation, we take longitude to increase as one heads westward in order to make longitude increase with time within a day. Although the use of Euclidean distance in $S$ might be viewed as problematic \[@gneiting\], it is not clear that great circle distances are any more appropriate in the present circumstance in which there is strong zonal flow. The model (\[spacetime\]) has the virtues of simplicity and of validity: it defines a valid covariance function on the sphere${}\times{}$time whenever $\theta_0,\theta_1$ and $\theta_2$ are positive. A more complex model would clearly be needed if one wanted to consider the process on the entire globe rather than in a narrow latitude band. Because the covariance matrix $K(\theta_0,\theta_1,\theta_2,v)$ can be written as $\theta_0M(\theta_1,\break\theta_2,v)$, where the entries of $M$ are generated by the Matérn function, the estimating equations (\[ascore\]) give ${\widehat}\theta_0=Z'M({\widehat}{\theta}_1,{\widehat}{\theta}_2,{\widehat}{v})^{-1}Z/n$ as the MLE of $\theta_0$ given values for the other parameters. Therefore, we only need to solve (\[ascore\]) with respect to $\theta_1$, $\theta_2$ and $v$. Initial values for the parameters were obtained by applying a simplified fitting procedure to a subset of the data. We first fit the model using observations from one latitude at a time. Since there are about 8500 observations per latitude band, it is possible, although challenging, to compute the exact MLEs for the observations within a single band using the Cholesky decomposition. However, we chose to solve (\[ascore\]) with the number $N$ of i.i.d. symmetric Bernoulli vectors $U_j$ fixed at 64. A first order finite difference filtering \[@steinchenanitescufiltering\] was observed to be the most effective in encouraging the convergence of the linear solve. Differences across gaps in the data record were included, so the resulting sizes of the filtered data sets were just one less than the number of observations available in each longitude. Under our model, the covariance matrix of the observations within a latitude can be embedded in a circulant matrix of dimension 21,600, greatly speeding up the necessary matrix–vector multiplications. Table \[tabozoneonelat\] summarizes the resulting estimates and the Fisher information for each latitude band. The estimates are consistent across latitudes and do not show any obvious trends with latitude except perhaps at the two most northerly latitudes. The estimates of $v$ are all near $-7.5^\circ$, which qualitatively matches the westerly flow seen in Figure \[figozone\]. The differences between $\sqrt{(\mathcal{G}^{-1})_{ii}}/\sqrt{(\mathcal {I}^{-1})_{ii}}$ and $1$ were all less than $0.01$, indicating that the choice of $N$ is sufficient. [@lccd[2.3]{}ccccc@]{} & & & & &\ &&&&&\ **Latitude** & $\bolds{{\widehat}{\theta}_0^N}$ [$\bolds{(\times}$**10**$\bolds{^3)}$]{} & $\bolds{{\widehat}{\theta}_1^N}$ & & $\bolds{{\widehat}{v}^N}$ & [$\bolds{(\times}$**10**$\bolds{^3)}$]{} & & &\ $40.5^\circ$N & 1.076 & 2.110 & 11.466 & $-6.991$ & 0.106 & 0.127 & 0.586 & 0.244\ $41.5^\circ$N & 1.182 & 2.172 & 11.857 & $-6.983$ & 0.123 & 0.136 & 0.634 & 0.251\ $42.5^\circ$N & 1.320 & 2.219 & 12.437 & $-7.118$ & 0.144 & 0.145 & 0.698 & 0.266\ $43.5^\circ$N & 1.370 & 2.107 & 12.104 & $-7.369$ & 0.145 & 0.136 & 0.660 & 0.285\ $44.5^\circ$N & 1.412 & 2.059 & 11.845 & $-7.368$ & 0.145 & 0.130 & 0.628 & 0.294\ $45.5^\circ$N & 1.416 & 2.010 & 11.814 & $-7.649$ & 0.147 & 0.128 & 0.632 & 0.313\ $46.5^\circ$N & 1.526 & 2.075 & 12.254 & $-8.045$ & 0.166 & 0.138 & 0.686 & 0.320\ $47.5^\circ$N & 1.511 & 2.074 & 11.939 & $-7.877$ & 0.161 & 0.135 & 0.654 & 0.319\ $48.5^\circ$N & 1.325 & 1.887 & 10.134 & $-7.368$ & 0.128 & 0.114 & 0.505 & 0.303\ $49.5^\circ$N & 1.246 & 1.846 & 9.743 & $-7.120$ & 0.117 & 0.110 & 0.473 & 0.305\ The following is an instance of the asymptotic correlation matrix, obtained by normalizing each entry of $\mathcal{I}^{-1}$ (at $49.5^\circ$N) with respect to the diagonal: $$\left[ \matrix{\hphantom{-}1.0000 & \hphantom{-}0.8830 & \hphantom{-}0.9858 & -0.0080 \cr \hphantom{-}0.8830 & \hphantom{-}1.0000 & \hphantom{-}0.8767 & -0.0067 \cr \hphantom{-}0.9858 & \hphantom{-}0.8767 & \hphantom{-}1.0000 & -0.0238 \cr -0.0080 & -0.0067 & -0.0238 & \hphantom{-}1.0000 } \right].$$ We see that ${\widehat}\theta_0,{\widehat}\theta_1$ and ${\widehat}\theta_2$ are all strongly correlated. The high correlation of the estimated range parameters ${\widehat}\theta_1$ and ${\widehat}\theta_2$ with the estimated scale ${\widehat}\theta_0$ is not unexpected considering the general difficulty of distinguishing scale and range parameters for strongly correlated spatial data \[@zhang\]. The strong correlation of the two range parameters is presumably due to the near confounding of time and longitude for these data. Next, we used the data at all latitudes and progressively increased the number of days. In this setting, the covariance matrix of the observations can be embedded in a block circulant matrix with blocks of size $10\times10$ corresponding to the 10 latitudes. Therefore, multiplication of the covariance matrix times a vector can be accomplished with a discrete Fourier transform for each pair of latitudes, or ${10 \choose2} = 55$ discrete Fourier transforms. Because we are using the Whittle covariance function as the basis of our model, we had hoped filtering the data using the Laplacian would be an effective preconditioner. Indeed, it does well at speeding the convergence of the linear solves, but it unfortunately appears to lose most of the information in the data for distinguishing spatial from temporal influences, and thus is unsuitable for these data. Instead, we used a banded approximate inverse Cholesky factorization \[@kolo, (2.5), (2.6)\] to precondition the linear solve. Specifically, we ordered the observations by time and then, since observations at the same longitude and day are simultaneous, by latitude south to north. We then obtained an approximate inverse by subtracting off the conditional mean of each observation given the previous 20 observations, so the approximate Cholesky factor has bandwidth 21. We tried values besides 20 for the number of previous observations on which to condition, but 20 seemed to offer about the best combination of fast computing and effective preconditioning. The number $N$ of i.i.d. symmetric Bernoulli vectors $U_j$ was increased to 128, in order that the differences between $\sqrt{(\mathcal{G}^{-1})_{ii}}/ \sqrt{(\mathcal{I}^{-1})_{ii}}$ and $1$ were around $0.1$. The results are summarized in Table \[tabozonealllat\]. One sees that the estimates are reasonably consistent with those shown in Table \[tabozoneonelat\]. Nevertheless, there are some minor discrepancies such as estimates of $v$ that are modestly larger (in magnitude) than found in Table \[tabozonealllat\], suggesting that taking account of correlations across latitudes changes what we think about the advection of ozone from day to day. [@lcccccccc@]{} & & & & &\ & & & & &\ **Days** & $\bolds{{\widehat}{\theta}_0^N}$ [$\bolds{(\times}$**10**$\bolds{^3)}$]{} & $\bolds{{\widehat}{\theta}_1^N}$ & $\bolds{{\widehat}{\theta }_2^N}$ & $\bolds{{\widehat}{v}^N}$ & [$\bolds{(\times}$**10**$\bolds{^3)}$]{} & & &\ \ 1–3 & $1.594$ & $2.411$ & $12.159$ & $-8.275$ & $0.362$ & $0.334$ & $1.398$ & $0.512$\ 1–10 & $1.301$ & $1.719$ & $11.199$ & $-8.368$ & $0.146$ & $0.121$ & $0.639$ & $0.407$\ 1–20 & $1.138$ & $1.774$ & $10.912$ & $-9.038$ & $0.090$ & $0.085$ & $0.436$ & $0.252$\ 1–30 & $1.265$ & $1.918$ & $11.554$ & $-8.201$ & $0.089$ & $0.081$ & $0.414$ & $0.198$\ \[6pt\]\ 1–30 & $1.260$ & $1.907$ & $11.531$ & $-8.211$ & $0.088$ & $0.079$ & $0.406$ & $0.200$\ Note that the approximate inverse Cholesky decomposition, although not as computationally efficient as applying the discrete Laplacian, is a full rank transformation and thus does not throw out any statistical information. The method does require ordering the observations, which is convenient in the present case in which there are at most 10 observations per time point. Nevertheless, we believe this approach may be attractive more generally, especially for data that are not on a grid. We also estimated the parameters using the dependent sampling scheme described in Section \[sec3\] with $N=128$ and obtained estimates given in the last row of Table \[tabozonealllat\]. It is not as easy to estimate $B^d$ as defined in Theorem \[tdependent\] as it is to estimate $B$ with independent $U_j$’s. We have carried out limited numerical calculations by repeatedly calculating $g^d({\widehat}{\theta},N)$ for ${\widehat}{\theta}$ fixed at the estimates for dependent samples of size $N=128$ and have found that the advantages of using the dependent sampling are negligible in this case. We suspect that the reason the gains are not as great as those shown in Figure \[figN\] is due to the substantial correlations of observations that are at similar locations a day apart. Discussion {#sec7} ========== We have demonstrated how derivatives of the loglikelihood function for a Gaussian process model can be accurately and efficiently calculated in situations for which direct calculation of the loglikelihood itself would be much more difficult. Being able to calculate these derivatives enables us to find solutions to the score equations and to verify that these solutions are at least local maximizers of the likelihood. However, if the score equations had multiple solutions, then, assuming all the solutions could be found, it might not be so easy to determine which was the global maximizer. Furthermore, it is not straightforward to obtain likelihood ratio statistics when only derivatives of the loglikelihood are available. Perhaps a more critical drawback of having only derivatives of the loglikelihood occurs when using a Bayesian approach to parameter estimation. The likelihood needs to be known only up to a multiplicative constant, so, in principle, knowing the gradient of the loglikelihood throughout the parameter space is sufficient for calculating the posterior distribution. However, it is not so clear how one might calculate an approximate posterior based on just gradient and perhaps Hessian values of the loglikelihood at some discrete set of parameter values. It is even less clear how one could implement an MCMC scheme based on just derivatives of the loglikelihood. Despite this substantial drawback, we consider the development of likelihood methods for fitting Gaussian process models that are nearly $O(n)$ in time and, perhaps more importantly, $O(n)$ in memory, to be essential for expanding the scope of application of these models. Calling our approach nearly $O(n)$ in time admittedly glosses over a number of substantial challenges. First, we need to have an effective preconditioner for the covariance matrix $K$. This allows us to treat $N$, the number of random vectors in the stochastic trace estimator, as a fixed quantity as $n$ increases and still obtain estimates that are nearly as efficient as full maximum likelihood. The availability of an effective preconditioner also means that the number of iterations of the iterative solve can remain bounded as $n$ increases. We have found that $N=100$ is often sufficient and that the number of iterations needed for the iterative solver to converge to a tight tolerance can be several dozen, so writing $O(n)$ can hide a factor of several thousand. Second, we are assuming that matrix–vector multiplications can be done in nearly $O(n)$ time. This is clearly achievable when the number of nonzero entries in $K$ is $O(n)$ or when observations form a partial grid and a stationary model is assumed so that circulant embedding applies. For dense, unstructured matrices, fast multipole methods can achieve this rate, but the method is only approximate and the overhead in the computations is substantial so that $n$ may need to be very large for the method to be faster than direct multiplication. However, even when using exact multiplication, which requires $O(n^2)$ time, despite the need for $N$ iterative solves, our approach may still be faster than computing the Cholesky decomposition, which requires $O(n^3)$ computations. Furthermore, even when $K$ is dense and unstructured, the iterative algorithm is $O(n)$ in memory, assuming that elements of $K$ can be calculated as needed, whereas the Cholesky decomposition requires $O(n^2)$ memory. Thus, for example, for $n$ in the range 10,000–100,000, even if $K$ has no exploitable structure, our approach to approximate maximum likelihood estimation may be much easier to implement on the current generation of desktop computers than an approach that requires calculating the Cholesky decomposition of $K$. The fact that the condition number of $K$ affects both the statistical efficiency of the stochastic trace approximation and the number of iterations needed by the iterative solver indicates the importance of having good preconditioners to make our approach effective. We have suggested a few possible preconditioners, but it is clear that we have only scratched the surface of this problem. Statistical problems often yield covariance matrices with special structures that do not correspond to standard problems arising in numerical analysis. For example, the ozone data in Section \[sec6\] has a partial confounding of time with longitude that made Laplacian filtering ineffective as a preconditioner. Further development of preconditioners, especially for unstructured covariance matrices, will be essential to making our approach broadly effective. \[app\] Appendix: Proofs {#appendix-proofs .unnumbered} ================ [Proof of Theorem \[tmain\]]{} Since $K$ is positive definite, it can be written in the form $S\Lambda S'$ with $S$ orthogonal and $\Lambda$ diagonal with elements $\lambda_1\ge\cdots\ge\lambda_n>0$. Then $Q^i:= S' K_i S$ is symmetric, $$\label{firstterm}\quad \operatorname{tr}\bigl(W^i W^j\bigr) = \operatorname{tr} \bigl(S'K^{-1}SS'K_iSS'K^{-1}SS'K_jS \bigr) = \operatorname{tr}\bigl(\Lambda^{-1}Q^i\Lambda^{-1}Q^j \bigr)$$ and, similarly, $$\label{secondterm} \operatorname{tr} \bigl\{W^i \bigl(W^j\bigr)' \bigr\} = \operatorname{tr}\bigl(\Lambda^{-1}Q^iQ^j \Lambda^{-1}\bigr).$$ For real $v_1,\ldots,v_p$, $$\label{lastterm} \sum_{i,j=1}^p v_iv_j \sum_{k=1}^n W_{kk}^iW_{kk}^j = \sum_{k=1}^n \Biggl\{\sum _{i=1}^p v_iW_{kk}^i \Biggr\}^2 \ge0.$$ Furthermore, by (\[firstterm\]), $$\label{firstquad} \sum_{i,j=1}^p v_iv_j \operatorname{tr}\bigl(W^i W^j\bigr) = \sum _{k,\ell=1}^n \frac{1}{\lambda_k\lambda_\ell} \Biggl\{\sum _{i=1}^p v_iQ^i_{k,\ell} \Biggr\}^2$$ and, by (\[secondterm\]), $$\label{secondquad} \sum_{i,j=1}^p v_iv_j \operatorname{tr} \bigl\{W^i \bigl(W^j\bigr)' \bigr \} = \sum_{k,\ell=1}^n \frac{1}{\lambda_k^2} \Biggl \{\sum_{i=1}^p v_iQ^i_{k,\ell} \Biggr\}^2.$$ Write $\gamma_{k\ell}$ for $\sum_{i=1}^p v_iQ^i_{k,\ell}$ and note that $\gamma_{k\ell}=\gamma_{\ell k}$. Consider finding an upper bound to $$\frac{\sum_{i,j=1}^p v_iv_j\operatorname{tr} \{W^i (W^j)' \}} { \sum_{i,j=1}^p v_iv_j\operatorname{tr}(W^i W^j)} = \frac{\sum_{k=1}^n {\gamma_{kk}^2}/{\lambda_k^2} + \sum_{k>\ell} \gamma_{k\ell}^2 ({1}/{\lambda_k^2} + {1}/{\lambda _\ell^2} )} { \sum_{k=1}^n {\gamma_{kk}^2}/{\lambda_k^2} + \sum_{k>\ell} {2\gamma_{k\ell}^2}/{\lambda_k\lambda_\ell}}.$$ Think of maximizing this ratio as a function of the $\gamma_{k\ell}^2$’s for fixed $\lambda_k$’s. We then have a ratio of two positively weighted sums of the same positive scalars (the $\gamma_{k\ell}^2$’s for $k\ge\ell$), so this ratio will be maximized if the only positive $\gamma_{k\ell}^2$ values correspond to cases for which the ratio of the weights, here $$\label{ratioweights} \frac{{1}/{\lambda_k^2}+{1}/{\lambda_\ell^2}}{ {2}/({\lambda_k \lambda_\ell})} = \frac{1+ ({\lambda_k}/{\lambda_\ell } )^2} { {2\lambda_k}/{\lambda_\ell}}$$ is maximized. Since we are considering only $k\ge\ell$, $\frac{\lambda_k}{\lambda _\ell} \ge1$ and $\frac{1+x^2}{2x}$ is increasing on $[1,\infty)$, so (\[ratioweights\]) is maximized when $k=n$ and $\ell=1$, yielding $$\frac{\sum_{i,j=1}^p v_iv_j\operatorname{tr} \{W^i (W^j)' \}} { \sum_{i,j=1}^p v_iv_j\operatorname{tr}(W^i W^j)} \le\frac{\kappa(K)^2 +1}{2\kappa(K)}.$$ The theorem follows by putting this result together with (\[B\]), (\[Jij\]) and (\[lastterm\]). [Proof of Theorem \[tdependent\]]{} Define $\beta_{ia}$ to be the $a$th element of $\beta_i$ and $X_{\ell a}$ the $a$th diagonal element of $X_\ell$. Then note that for $k\ne\ell$ and $k'\ne\ell'$ and $a,b\in\{1,\ldots,N\}$, $$\begin{aligned} & & (U_{i,(k-1)N+a}U_{i,(\ell-1)N+b}, U_{j,(k'-1)N+a'}U_{j,(\ell'-1)N+b'}) \\ & &\qquad = (\beta_{ia}\beta_{ib}Y_{ik}X_{k a}Y_{i\ell}X_{\ell b}, \beta_{ja'}\beta_{jb'}Y_{jk'}X_{k' a'}Y_{j\ell'}X_{\ell'b'})\end{aligned}$$ have the same joint distribution as for independent $U_j$’s. Specifically, the two components are independent symmetric Bernoulli random variables unless $i=j, a=a', b=b'$ and $k=k'\ne\ell=\ell'$ or $i=j,a=b',b=a'$ and $k=\ell'\ne\ell=k'$, in which case they are the same symmetric Bernoulli random variable. Straightforward calculations yield (\[improve\]). Acknowledgments {#acknowledgments .unnumbered} =============== The data used in this effort were acquired as part of the activities of NASAs Science Mission Directorate, and are archived and distributed by the Goddard Earth Sciences (GES) Data and Information Services Center (DISC). [44]{} , (). . . , (). . . (). . . (). . . (). . . , (). , ed. , . (). . . (). . . , . (). . . , . , (). . . (). , ed. , . (). . . (). . . , , (). . . (). . . , (). . , . (). . . , (). . . (). . . (). . . . (). . . (). . . (). . , . (). . . , (). . . (). . . (). . . (). , ed. , . (). . . (). . . (). . , . (). . . (). . . (). . , (). . . , (). . . , (). . . (). . . (). . . (). . . (). . . (). . In . , . , , , , , (). . . [^1]: To facilitate understanding, we explain here the process for solving the score equations (\[score\]). Conceptually it is similar to that for solving the estimating equations (\[ascore\]).

--- abstract: | Last-mile logistics is regarded as an essential yet highly expensive component of parcel logistics. In dense urban environments, this is partially caused by inherent inefficiencies due to traffic congestion and the disparity and accessibility of customer locations. In parcel logistics, access hubs are facilities supporting relay-based last-mile activities by offering temporary storage locations enabling the decoupling of last-mile activities from the rest of the urban distribution chain. This paper focuses on a novel tactical problem: the geographically dynamic deployment of pooled relocatable storage capacity modules in an urban parcel network operating under space-time uncertainty. In particular, it proposes a two-stage stochastic optimization model for the access hub dynamic pooled capacity deployment problem with synchronization of underlying operations through travel time estimates, and a solution approach based on a rolling horizon algorithm with lookahead and a benders decomposition able to solve large scale instances of a real-sized megacity. Numerical results, inspired by the case of a large parcel express carrier, are provided to evaluate the computational performance of the proposed approach and suggest up to $28\%$ last-mile cost savings and $26\%$ capacity savings compared to a static capacity deployment strategy.\ [ ***Keywords—*** Parcel Logistics, Urban Networks, Dynamic Deployment, Capacity Relocation, Capacity Pooling, Stochastic Optimization, Physical Internet ]{} author: - Louis Faugère - Walid Klibi - Chelsea White III - Benoit Montreuil bibliography: - 'Bibliography.bib' title: | Dynamic Pooled Capacity Deployment\ for Urban Parcel Logistics --- Introduction ============ Global urbanization, growth of e-commerce and the ever increasing desire for speed put pressure on the need for innovation in designing, managing and operating urban logistics systems in a sustainable and cost-efficient way. In 2018, $55\%$ of the world’s population lived in urban areas (up to $82\%$ in North America). The [@united20182018] predict that global urbanization will reach $68\%$ by 2050, with an increasing number of megacities (cities of 10+M inhabitants). Increasing population density is a challenge for city logistics in terms of traffic congestion, vehicle type restrictions, limited parking spaces, expensive and rare logistic facility locations, and is further complex in megacities due to their extremely high density [@fransoo2017reaching]. For urban parcel logistics systems, the growth of e-commerce is currently one of the main challenges to tackle with an annual growth over $20\%$ on the 2017-2019 period, projected to be over $15\%$ until 2023 [@statista_2019_growth]. Online-retailing with goods being transported to consumers’ homes increase the number of freight movements within cities while reducing the size of each shipment [@savelsbergh201650th] which makes first and last mile logistic activities harder to plan. Moreover, consumers’ desire for speed (i.e. same-day delivery and faster) has yet to be met by online retailers [@statista_2019_speed]. With promises as fast as 1-hour delivery (e.g. Amazon Prime in select U.S cities), the cost of last-mile logistics becomes an ever more critical part of urban parcel logistics. These trends have been accelerated due to attempts to mitigate the impacts of the COVID-19 pandemic (e.g., sequestering in place), requiring companies to increase their last-mile delivery capabilities and to deal with the dramatic shift to online channels [@mitsloancovid19].\ To tackle these challenges, a number of innovations have emerged from academia and industry. [@savelsbergh201650th] provide an overall view of recent innovations and modeling of solutions such as multi-echelon networks, dynamic delivery systems, pickup and delivery point networks, omni-channel logistics, crowd-sourced transportation and the integration of public and freight transportation networks. Many of these innovations are considered in the Physical Internet initiative, introduced in @montreuil2011toward, which seeks global logistics efficiency and sustainability by transforming the way physical objects are handled, moved, and stored by applying concepts from internet data transfer to real-world shipping processes. A conceptual framework on the application of Physical Internet concepts to city logistics was recently proposed in @crainic2016physical, in particular the concepts of pooling and hyperconnectivity in urban multi-echelon networks. As underlined by @savelsbergh201650th, city logistics problems integrating real-life features such as highly dynamic and volatile decision making environments, sharing principles or multi-echelon networks, offer a fertile soil for groundbreaking research.\ Inspired by the case of a large parcel logistics company operating in megacities, this paper examines a novel tactical optimization problem in urban parcel logistics. It consists in the dynamic deployment and relocation of pooled storage capacity in an urban parcel network operating under space-time uncertainty. It builds on the recent proposal of a hyperconnected urban logistics network structure [@montreuil2018IMHRC] in line with the new challenges of the parcel logistics industry. The proposed network structure is based on the pixelization of urban agglomerations in unit zones (clusters of customer locations), local cells (cluster of unit zones) and urban areas (cluster of local cells). It is composed of three tiers of interconnected logistics hubs: gateway hubs (GH), local hubs (LH) and access hubs (AH) respectively designed to efficiently handle inter urban areas, inter local cells, and inter unit zones parcel flows. Beyond the realm of an urban agglomeration, the network of gateway hubs connects to a network of regional hubs (RH) covering entire blocks of the world (e.g. North America), and these regional hubs connect to a worldwide network of global hubs. This paper focuses on access hubs which are small logistics hubs located at the neighborhood level within minutes of customers, enabling parcel transfer between different vehicle types temporarily holding parcels close to pickup and delivery points. Access hubs are to be used by logistics carriers, and not by consumers as smart lockers are. Access hubs can materialize in many forms including a parked trailer, a smart locker bank, or a storage shed as illustrated in Figure \[fig: AHExample\]. Trailer based solutions like Figure \[fig: AHExample\] (a) and (d) offer all-or-nothing mobile solutions, while capacity module based solutions like Figure \[fig: AHExample\] (b) and (c) offer flexible capacity adjustment over time. The scope of this paper is a capacity module based solution. Parcel logistics networks have undergone significant changes in the last 20 years, notably in urban contexts as seen in [@janjevic2020characterizing], and have received an increasing attention in the academic literature. Strategic and tactical network design problems such as the ones examined by [@SmilowitzDaganzo; @winkenbach2016enabling] approximate operations costs when designing and planning for multi-echelon networks. While network design problems are complex due to intricate interdependencies between strategic, tactical and operational decisions, continuum approximations (see [@ansari2018advancements]) are useful to capture operations complexity and take informed decisions. However, such approximations are typically used to estimate travel distance and cost, but not travel time and operations synchronization. This paper considers access hubs to be modular in storage capacity similar to designs proposed in [@FaugereCIE], such that capacity modules can be removed/added to adapt access hub’s storage capacity. At the tactical level, capacity modules are to be deployed over a network of access hub locations; at the operational level, capacity modules are to be allocated to serve their access hub’s need or neighboring locations via capacity pooling. In a dynamic setting, the associated problem can be related to a multi-period location-allocation problem which belongs to the NP-Hard complexity class [@manzini2008optimization]. Once the capacity of the network of access hubs is adjusted, each access hub plays the role of a transshipment location between couriers performing pickup and delivery services within minutes of the access hub and riders transporting parcels between local hubs and a set of access hubs. Such transshipments require tight synchronization of the two tiers so as to provide efficient and timely pickup and delivery operations. This operational context mimics, on a hourly basis, a two-echelon pickup and delivery problem with synchronisation, which is a complex routing problem (see for instance @Cuda2015). Thus, the integration of operations in the tactical decision model leads to better capacity deployment decisions [@Klibi2016], yet induces solvability challenges due to its combinatorial and stochastic-dynamic structure. This paper studies a novel tactical optimization problem: the dynamic deployment of pooled storage capacity in an urban parcel network operating under space-time uncertainty. Its contribution is threefold: (1) the characterization of a new tactical problem for capacity deployment, motivated by dynamic aspects of urban parcel logistics needs, (2) the modeling of the access hub dynamic pooled capacity deployment problem as a two-stage stochastic program with synchronization of underlying operations through travel time estimates, and (3) the design of a solution approach based on a rolling horizon algorithm with lookahead and a benders decomposition able to solve large scale instances of a real-sized megacity. Numerical results, inspired by the case of a large parcel express carrier, are provided to evaluate the computational performance of the proposed approach and suggest up to $28\%$ last-mile cost savings and $26\%$ capacity savings compared to a static capacity deployment strategy. Section \[Literature\] summarizes the literature relevant to this type of problem, section \[Probdescription\] describes the problem and proposes a mathematical modeling, section \[SolutionApproach\] presents the proposed solution approach, section \[Results\] provides an experiment setup and discusses results, and section \[Conclusion\] highlights key takeaways and managerial insights, and identifies promising research avenues. Literature Review {#Literature} ================= Multi-echelon network for urban distribution have received a lot of attention in the academic literature (e.g. [@BenjellounTrends], [@mancini2013multi], [@janjevic2019integrating]), commonly using urban consolidation centers (UCC) to bundle goods outside the boundaries of urban areas. As reported in [@janjevic_development_2014], several micro-consolidation initiatives have been proposed to downscale the consolidation effort by bundling goods at the neighborhood level using capillary networks of hubs located much closer to pickup and delivery points, defined as access hubs in the conceptual framework proposed by [@montreuil2018IMHRC]. Examples of such initiatives are satellite platforms (e.g. [@BenjellounTrends]), micro-consolidation centers (e.g. [@leonardi2012before]), mobile depots (e.g. [@marujo2018assessing]), and micro-depots [@stodick2019sustainable]. Most of the focus has been on location and vehicle routing aspects (e.g. [@anderluh2017synchronizing] and [@enthoven2020two]) and cost and negative externalities assessment (e.g. [@verlinde2014does], [@arvidsson2017ex], [@marujo2018assessing]) in solutions using depots and cargo-bikes. To the best of the authors’ knowledge, the dynamic management of access hub capacity for urban parcel logistics has not yet been studied in the academic literature.\ The problem studied in this paper involves modular capacity relocation and a capacity pooling recourse mechanism impacting the operations of a two-echelon synchronization problem. In this section, a literature review on dynamic capacitated facility location problems and integrated urban network design problems is presented.\ Dynamic facility location problems where systems are subject to varying environments (e.g. non-stationary demand) allow the relocation of facilities over time. [@arabani2012facility] provide a literature review on facility location dynamics, including problems with and without hub relocation. Innovations in the manufacturing industry have motivated the study of modular and mobile production and storage. [@marcotte2016introducing] have presented various threads of innovations such as distributed production, on-demand production, additive production, and mobile production, that would motivate and benefit from hyperconnected mobile production systems. [@marcotte2015modeling] and [@malladiererawhite] proposed mathematical modeling for production and inventory capacity relocation and allocation to manage multi-facility network facing stochastic demand. However, they examine small to medium networks far from the scale of urban parcel logistics networks and do not study operations synchronization. [@aghezzaf2005capacity] studied storage capacity expansion planning coupled to dynamic inventory relocation in the context of warehouse location allocation problems, but did not consider capacity reduction or relocation. [@ghiani2002capacitated], [@melo2006dynamic], and [@jena2015dynamic] modeled dynamic facility location problem where not only sites could be permanently or temporarily opened or closed, but also resized by adding or removing modular capacity. [@melo2006dynamic] proposed models capturing modular capacity shifts from existing to new facilities. However in these problems, capacity relocation is generally not managed jointly with capacity allocation or its impact on underlying operations. Dynamic facility location literature partially covers the tactical capacity relocation problem studied in this paper, but does not integrate underlying operations dynamics at the urban logistics scale.\ Integrated network design problems typically deal with a combination of strategic decisions such as facility location, tactical decisions such as resource allocation and scheduling, and operational decision such as vehicle routing. The integration of these different levels of decisions can be found in two main problem classes: service network design problems and location routing problems. Service network design problems deal with the selection and scheduling of services such as hub operations, shipping lines and routing of freight (e.g. [@crainic2016service; @hewitt2019scheduled]) while location routing problems combine facility location-allocation decisions with associated freight routing decisions. [@drexl2015survey] provide a recent survey of variants and extensions of the location routing problem. The dynamic location routing problem ([@francis2008period]) considering the assignment of demand to locations over multiple periods, is similar to the problem studied in this paper: it aims at minimizing network and routing costs over a multiperiod location and routing decision vector. However, multi-echelon location routing problems (e.g. [@crainic2004advanced; @perboli2011two]) have only recently gathered attention in the literature. Although multi-echelon networks are relevant to postal and parcel delivery distribution systems ([@gonzalez2009n]) where fine time constraints and synchronization have become an essential consideration, most papers studying multi-echelon networks are concerned with the two-echelon case and ignore temporal aspects ([@drexl2015survey]).\ When allowing inter-location capacity pooling, underlying operations described in section \[introduction\] are impacted. Couriers perform pickup and delivery tours starting and ending in their reference access hub, while riders visit access hubs starting and ending their routes in their reference local hub. The impact of capacity pooling can be measured by modeling its impact on the route of parcels, couriers and riders. However, when taking decisions at the tactical level, explicitly modeling routes is not necessary. TSP and VRP continuous approximations have been introduced by [@Daganzo94; @DaganzoBook] to embed operations in strategic and tactical logistics problems (e.g. [@EreraThesis], [@franceschetti2017strategic]). A recent literature on variants of this approach can be found in [@ansari2018advancements]. [@SmilowitzDaganzo; @winkenbach2016enabling; @bergmann2020integrating] adapted these continuous approximations to the context of parcel express logistics to approximate distance traveled and cost. However, the aspect of synchronization using travel time continuous approximations has not yet been studied. To the best of the authors’ knowledge, this paper is the first to study a capacity relocation problem with the synchronization of two-echelon routing operations through travel time estimates. Problem Description and Formulation {#Probdescription} =================================== Business Context {#biz} ---------------- A parcel logistics company provides pickup and delivery services to customers in a region covered by a network of access hubs. The network of access hubs may be dedicated to the parcel logistics provider, or shared between several companies as suggested by the concept of open networks in the Physical Internet. Figure \[fig: Relocation\] provides a conceptual illustration of the network of access hubs and the relocation of capacity modules over two deployment periods. Once the network capacity is set, pickups from customers are dropped off by couriers in access hubs and will occupy a certain storage volume for some time until a rider picks them up to perform outbound activities. To-be-delivered parcels are dropped off by riders in access hubs and will occupy a certain storage volume for some time until a courier picks them up to perform the delivery to customers. To provide good service, the company must ensure that parcels flow rapidly and seamlessly between couriers and riders, which requires the sound management of storage capacity deployed in access hubs. Storage volume requirements vary depending on the fluctuation of demand for pickup and delivery services over time and are observed over a discrete set of operational periods (e.g. hourly). Access hubs are composed of modular storage units that can be assembled and disassembled relatively easily, enabling rapid relocation of storage capacity in the network. During each deployment period (e.g. week or day), storage capacity can be relocated within the network of access hubs, or to/from a depot where additional capacity modules are stored when not in use. Figure \[fig: Relocation\] illustrates demand variability and the relocation of capacity modules within the network of access hubs over two deployment periods. For instance, unit zones with increasing demand (and therefore increasing capacity requirements) from period $t$ to $t+1$ receive capacity module(s) from the depot of from locations that have decreasing capacity requirements (e.g. lower left unit zone in Figure \[fig: Relocation\]). The relocation of capacity modules over the network adjusts the storage capacity available in each access hub for the following period. In this study, we assume capacity module relocation is performed by a separate business unit whose routing decisions are out of the scope of the research reported in this paper.\ The objective is to minimize the cost incurred by operating such a network of access hubs without disrupting underlying operations. The decision scope is tactical (capacity deployment) and requires the integration of operational decisions. However, since the main interest is a set of tactical decisions, there is no need to explicitly model operations, but only to approximate the impact of deployment decisions on routing cost and time synchronization.\ Let $L$ be a set of access hub locations and $W$ a set of external depots composing a network $G=(N=L\cup W,A)$ where $A$ is the complete set of directed arcs between locations in $N$. A capacity deployment of $I_0$ capacity modules in time $t$ over the network is represented by a vector $S(t) = (S_l(t), \forall l \in N)$. The relocation of capacity modules can be represented as vectors $R(t)=(R_a(t),\forall a\in A)$. Accordingly, there are ${I_0+|L|-1 \choose I_0}$ possible arrangements of $I_0$ modules over $|L|$ locations. In the case where $I_0 \geq |L|$ and that each location gets at least one module, there are ${I_0-1 \choose |L|-1}$ possible arrangements. In this realistic context, access hub networks are expected to be composed of a high number of locations (i.e., hundreds). Thus, state and action spaces would be significantly large-sized, which results in curse of dimensionality issues ([@powell2007approximate]).\ Moreover, a set of realization scenarios $\omega \in \Omega$ with probability $\phi_{\omega}$ is considered. The number of pickups and deliveries as well as the storage volume requirements are observed hourly and respectively represented as a vectors $\rho^P(\tau, \omega) = (\rho^P_l(\tau,\omega), \forall l \in L)$, $\rho^D(\tau, \omega) = (\rho^D_l(\tau, \omega), \forall l \in L)$ and $D(\tau, \omega)=(D_l(\tau, \omega), \forall l \in L)$, for every operations hour $\tau \in T_t$, where $t \in T$ is an operations horizon between two deployment periods (e.g. a week). If a courier or rider observes a lack of storage capacity when visiting an access hub, the courier or rider can perform the following recourse actions: pool capacity by making a detour towards a neighboring access hub with extra capacity or consign its load to a nearby third-party business (e.g. local shop) for a certain price agreed upon (uncapacitated recourse). Once volume requirements are observed, recourse actions are taken for each operational period $\tau$: capacity pools as a vector $P(\tau, \omega)=(P_a( \tau,\omega),\forall a\in A_{pool})$ where $A_{pool}$ is the set of arcs on which capacity can be pooled, and consignments as a vector $Z(\tau,\omega)=(Z_l(\tau,\omega),\forall l\in L)$. At any time $\tau$ in scenario $\omega$, the system can thus be represented as a state $S_t = S(t) \text{ s.t. } \tau \in T_{t}$ and an action $x_\tau = (R(\tau), P(\tau,\omega), Z(\tau,\omega)) \text{ s.t. } \tau \in T_{t}$, where $R(\tau)$ is the null vector except for $\tau = t, \forall t \in T$. Based on the optimisation framework proposed in [@powell2019unified], our stochastic optimization challenge for the access hub dynamic pooled capacity deployment problem can be formulated as follows: $$\label{opt challenge} \min_{x_\tau \in X(\tau)} \mathop{\mathbb{E}}_{\omega \in \Omega}\left \{\sum_{t \in T} \sum_{\tau \in T_t} C_\tau(S_t,x_\tau,\rho^P(\tau,\omega), \rho^D(\tau,\omega), D(\tau,\omega))|S_0 \right \}$$ where $X(\tau)$ is the set of feasible actions at time $\tau$, $S_0$ is the initial state of the system, and $C_\tau(\cdot)$ is the cost function at time $\tau$. Figure \[fig: Timeline\] illustrates the dynamics of the problem with the tactical decision timeline: before each period $t$, a network deployment strategy $S(t)$ is decided through relocation decisions $R(t)$ and implemented right before the beginning of period $t$. Then, demand realized and recourse actions are taken in each period $\tau \in T_t$. At the end of periods $T_t$, a network deployment strategy $S(t+1)$ is decided through relocation decisions $R(t+1)$ and implemented right before the beginning of period $t+1$ and the process repeats. Operations Cost Approximation and Synchronization Modeling \[under op\] ----------------------------------------------------------------------- Once decisions on capacity deployment are set for a given period $t$, they strongly impact the quality of operations performed by couriers and riders. More specifically, capacity at each location impacts the number and costs of detour and perturb the synchronisation of the operations between couriers and riders at each location. Accordingly, the surrounding objective of integrating routing operations is to evaluate the performance of the capacity deployment in minimizing the detours due to an underestimation of the capacity needs and in guaranteeing the synchronisation of the operations between couriers and riders at each location. To do so, this subsection proposes to develop routes with detours cost approximations, and travel time approximations. It builds on a refined granularity of routing operations periods (hourly) and uncertain storage volume requirements.\ Figure \[fig: Operations\] illustrates the use of access hubs in first/last mile parcel logistics operations during a period $\tau$ with one rider and 3 couriers. At the operational level, pickup and delivery decisions are made hourly ($\tau$) based on the volume-based capacity made available at each access hub. In addition, capacity relocation determines the number and costs of recourse actions needed to satisfy the requested volumes. With the consideration of capacity pooling recourse, the pickup and delivery problem with transshipment faced by couriers and riders adds the feature of detours. Here, to ensure timely transshipment operations, the detours performed by couriers and riders, are limited to their original time period ($\tau$), avoiding couriers and riders to be desynchronized. Since these detours necessitate additional moves and are time consuming, this comes with a supplementary incurred cost. It is clear that capturing the dynamics of underlying operations when taking capacity deployment decisions leads to better solutions. However, the pickup and delivery problem with transshipment is NP-hard ([@RaisPandDwithTransshipment]) and including it explicitly in the tactical model would make it intractable. Since the goal is to foster best capacity deployment decisions, it is sufficient to anticipate the operations costs and time synchronisation constraints using scenario-based continuous approximations. Accordingly, hereafter is proposed a tractable approximation of each period $\tau$ pickup and delivery problem with transshipment by developing deterministic continuous approximations of vehicle routing problems. The starting point of the proposed approximations is the estimation of the vehicle routing problem length when the depot (from which vehicles start their routes) is not necessarily located in the area where customers are located as proposed in [@Daganzo94]: $$\label{VRPDaganzo} VRP(n) = 2rm + nk(\delta)^{-\frac{1}{2}}$$ where $r$ is the average distance between the depot and the customer locations, $m$ is the number of routes required to serve all customers, $n$ is the number of points to be visited, $k$ is a constant parameter that can be estimated through simulation ([@DaganzoBook]), and $\delta$ is the density of points in the area. An a priori lower bound on the number of routes required to serve all customers, $m$, is $n/Q$ where $Q$ is the capacity of one vehicle in terms of customer locations. The first term of approximation (\[VRPDaganzo\]) represents the line-haul (back and forth) performed by vehicle to travel from the depot to the area where customers are located, and the second term represents the tour performed by traveling between each successive stops. Based on these seminal works, the next subsection proposes an adaptation of these equations to the operational context of riders and couriers, and develops an explicit time-based estimation of their operations.\ ### Riders operations {#riders-operations .unnumbered} Riders work in local cells, which are clusters of access hubs served by the same upper level local hub(s) as illustrated in Figure \[fig: Rider\]. Riders visit a set of $n_{LC}$ access hubs within their local cell of area $A_{LC}$ (and density $\delta_{LC} = \frac{n_{LC}}{A_{LC}}$) to pickup and deliver parcels as part of a defined route (e.g. planned beforehand based on averaging network’s load). At the time of deployment, underlying riders’ routes are not known with certainty, but need to be estimated in order to anticipate operations performance. When a rider makes his tour in period $\tau$ under scenario $\omega$ two cases are possible: (i) the tour is operated as planned because sufficient capacity is deployed at all visited access hubs in the route or because the detours are assigned to access hubs that are already in the remaining itinerary of the rider (bold lines in Figure \[fig: Rider\]); (ii) the rider tour is perturbed due to a lack of capacity at an access hub, and thus has to perform an immediate detour to a neighboring access hub before pursuing the rest of the regular tour (dash lines in Figure \[fig: Rider\]). Given approximation (\[VRPDaganzo\]), if the number of detours performed by riders in local cell $LC$ in period $\tau$ in scenario $\omega$ is $n_{LC}^R(\tau, \omega)$, the route length estimation with detours of riders’ operations is: $$\label{VRPRider} VRP^R_{LC}(\tau, \omega) = 2r_{LC}m_{\tau}^R(\omega) + (n_{LC}+n_{LC}^R(\tau, \omega))k^R(\delta_{LC})^{-\frac{1}{2}}$$ where $n_{LC}$ is the total number of access hubs in local cell $LC$, $r_{LC}$ is the average distance between $LC$’s local hub(s) and its access hubs, and $m_{\tau}^R(\omega)$ is the number of riders’ operating.\ The cumulative time (in time-rider) necessary to perform tours approximated in (\[VRPRider\]) is: $$\label{rider time cumulative} T^R_{LC}(\tau, \omega) = m_{\tau}^R(\omega)(t^R_s + \frac{2r_{LC}}{s^R_0}) + (n_{LC} + n_{LC}^R(\tau, \omega))(\frac{ k^R(\delta_{LC})^{-\frac{1}{2}}}{s^R} + t^R_a) + (\sum_{l \in LC}(\rho_l^D(\tau,\omega) + \rho_{l}^P(\tau,\omega)))t^R_u$$ where the first term is the time spent to setup tours ($t_s^R$ per tour) and perform the line-haul at a speed of $s_0^R$, the second term represent the travel time between stops at a speed of $s^R$ and the stopping time $t_a^R$ per access hub, and the third term represents the service time (handling) $t_u^R$ per pickup and delivery.\ Thus, the cost associated with riders’ operations in local cell $LC$ in period $\tau$ in scenario $\omega$ is: $$C^R_{LC}(\tau, \omega) = m_{\tau}^R(\omega)(c^R_f + 2r_{LC} c^R_{v_0}) + (n_{LC} + n_{LC}^R(\tau, \omega))k^R(\delta_{LC})^{-\frac{1}{2}} c^R_v + T^R_{LC}(\tau, \omega) c^R_w$$ where the first term represents the fixed, $c^R_f$, and variable, $c^R_{v_0}$ in line-haul and $c^R_v$ in tour, costs associated with vehicles, and the second term represents the variable labor cost $c^R_w$ of $m_\tau^R(\omega)$ riders.\ Since the nominal routing cost (with no detours) is a sunk cost incurred regardless of the capacity deployment, the marginal cost is sufficient to inform the tactical decision of the impact of recourse actions. The marginal cost of the detours induced by the tactical decisions, or difference between the rider routing cost with detours and the nominal rider routing cost, is: $$\Delta C^R_{LC}(\tau, \omega) = n_{LC}^R(\tau, \omega)k^R(\delta_{LC})^{-\frac{1}{2}} c^R_v + \Delta T^R_{LC}(\tau, \omega) c^R_w$$ where the time associated with performing detours is the time needed to perform detours: $$\Delta T^R_{LC}(\tau, \omega) = n_{LC}^R(\tau, \omega)(\frac{ k^R(\delta_{LC})^{-\frac{1}{2}}}{s^R} + t^R_a)$$ ### Couriers operations {#couriers-operations .unnumbered} Couriers operate in unit zones, which are clusters of pickup and delivery points served by access hub(s). Couriers leave their reference access hubs to visit customers and perform pickups/deliveries before returning to their access hub. When a courier arrives at the courier’s access hub with picked parcels, if the courier observes a lack of capacity, the courier can be immediately directed to available capacity in some neighboring access hub. Then, the courier will perform a detour (out and back) to the assigned neighbour access hub before starting their next tour from their reference access hub. Figure \[fig: Courier\] illustrates a courier’s tour and a detour as described. Since access hubs are located in the same area as pickup/delivery locations, the line-haul distance at this echelon is negligible, which eliminates the first term of approximation (\[VRPDaganzo\]). If the number of detours performed by couriers on arc $a \in A_{pool}(l)= \{a=(l,j), \forall j: (l,j) \in A_{pool}\}$ of length $d_a$ in period $\tau$ under scenario $\omega$ is $n_{a}^C(\tau, \omega)$, the route length estimation with detours of couriers’ operations is: $$\label{VRPCouriers} VRP^C_{l}(\tau, \omega) = (\rho_{l}^P(\tau,\omega) + \rho_{l}^D(\tau,\omega))k^C(\delta_{l})^{-\frac{1}{2}} + \sum_{a \in A_{pool}(l)} (2 n_{a}^C(\tau, \omega) d_a)$$ where the first term represents the total length of tours performed by couriers to visit pickup/delivery locations, and the second term represents the detours (out and back) performed between access hub $l$ and its neighboring access hubs.\ The cumulative time (in time-courier) necessary to perform courier tours is based on the approximation in (\[VRPCouriers\]) as follows: $$\label{courier time cumulative} T^C_{l}(\tau, \omega) = (\rho_{l}^P(\tau,\omega) + \rho_{l}^D(\tau,\omega))(\frac{ k^C(\delta_{l})^{-\frac{1}{2}}}{s^C} + t^C_a) + \sum_{a \in A_{pool}((l)} n_{a}^C(\tau, \omega)(\frac{2 d_a}{s^C_0} + t^C_a) + (\rho_{l}^D(\tau,\omega) + \rho_{l}^P(\tau,\omega))t^C_u$$ where the first term represents the travel time between pickup/delivery locations at a speed of $s^C$ and the stopping time $t_a^C$ per stop, the second term represents the travel time during detours to neighboring access hubs at a speed of $s^C_0$ plus a stopping time $t_a^C$, and the third term represents the service time (handling) $t_u^C$ per pickup and delivery.\ Thus, the cost associated with couriers’ operations at access hub $l$ in period $\tau$ under scenario $\omega$ is: $$C^C_{l}(\tau, \omega) = ((\rho_{l}^P(\tau,\omega) + \rho_{l}^D(\tau,\omega))k^C(\delta_{l})^{-\frac{1}{2}} c^C_v + \sum_{a \in A_{pool}(l)} (2 n_{a}^C(\tau, \omega) d_a) c^C_{v_0}) + T^C_{l}(\tau, \omega) c^C_w$$ where the first term represents the variable travel costs, respectively $c^C_v$ between pickup/delivery locations and $c^C_{v_0}$ between access hubs, and the second term represents the variable labor cost $c^C_w$ of $m_\tau^C(\omega)$ couriers.\ Again, since the nominal routing cost (with no detours) is a sunk cost incurred regardless of the capacity deployment, the marginal cost is sufficient to inform the tactical decision of the impact of recourse actions.The marginal cost of the detours induced by the tactical decisions, or the difference between the courier routing cost with detours and the nominal courier routing cost is: $$\Delta C^C_{l}(\tau, \omega) = \sum_{a \in A_{pool}(l)} (2 n_{a}^C(\tau, \omega) d_a) c^C_{v_0}) + \Delta T^C_{l}(\tau, \omega) c^C_w$$ where the time associated with performing detours is: $$\Delta T^C_{l}(\tau, \omega) = \sum_{a \in A_{pool}(l)} n_{a}^C(\tau, \omega)(\frac{2 d_a}{s^C_0} + t^C_a)$$ ### Operations Synchronization {#operations-synchronization .unnumbered} Recall that a key objective of integrating routing operations with the capacity deployment problem is to guarantee the synchronisation of the operations between couriers and riders at each location. To do so, this subsection proposes to develop time-based synchronisation constraints based on the travel time approximations (\[rider time cumulative\]) and (\[courier time cumulative\]), developed above. Parcels transshipped from riders to couriers and couriers to riders through access hubs must be transshipped during the period of time the parcels are within the network. That is, the length of a courier’s (respectively rider’s) original tour, plus the added detour(s) must not exceed the maximum length feasible within one operational period. For riders’ operations, at the local cell level, this tour length can be expressed, based on the number of riders ($m_\tau^R(\omega$) in period $\tau$ under scenario $\omega$, as follows: $$\label{Rider_Synchro_Constraint} T^R_{LC}(\tau, \omega) \leq m_\tau^R(\omega) \Delta_\tau, \forall \omega \in \Omega, LC \in \mathcal{LC}, \tau \in T_t, t \in T$$ where $\Delta_\tau$ is the length of period $\tau$. Similarly, for couriers’ operation, at the access hub level, synchronization can be expressed, based on the number of couriers ($m_\tau^C(\omega$) in period $\tau$ under scenario $\omega$, as follows: $$\label{Courier_Synchro_Constraint} T^C_{l}(\tau, \omega) \leq m_\tau^C(\omega) \Delta_\tau, \forall \omega \in \Omega, l \in L, \tau \in T_t, t \in T$$ Two-Stage Stochastic Program Formulation for the Access Hub Dynamic Pooled Capacity Deployment Problem ------------------------------------------------------------------------------------------------------- In this section, a stochastic programming formulation is proposed to tackle the optimization problem (\[opt challenge\]) presented in section \[biz\]. We remark that the stochastic optimisation problem (\[opt challenge\]) can be modeled as a multi-stage stochastic program based on a scenarios tree. However, this program would be intractable for realistic size instances, due to its combinatorial structure and non-anticipatory constraints [@Schultz2003]. Under a rolling horizon framework, the model is built here on the relaxation approach [@Shapiro2009] that is applied to transform the multi-stage stochastic program to a two-stage stochastic program with multiple tactical periods. More specifically, it consists in transferring all the capacity deployment decisions of the $T$ periods to the first-stage in order to be set at the beginning of the horizon. In this case, only first-stage design decisions ($t=1$) are made here and now, but subsequent capacity deployment decisions ($t>1$) are deferrable in time according to their deployment period. Hereafter are introduced the additional sets, input parameters, random variables and decision variables that formulate the overall model. ### Sets {#sets .unnumbered} $$\begin{aligned} \begin{tabularx}{\textwidth}{ll} $\mathcal{L}$ & access hub locations, indexed by $l$ \\ $\mathcal{LC}$ & local cells, indexed by $LC$ \\ $\mathcal{W}$ & depot locations, indexed by $l$ \\ $\mathcal{A}$ & arcs between two locations of the network $\mathcal{L}\cup\mathcal{W}$, indexed by $a$ \\ $\mathcal{G}$ & asymmetric graph $(\mathcal{L}\cup\mathcal{W},A)$ satisfying the triangle inequality \\ $T$ & tactical periods, indexed by $t$, covering the planning horizon\\ $T_t$ & subset of operational (demand) periods, indexed by $\tau$, between periods $t$ and $t+1$ \\ $\Omega$ & scenarios, indexed by $\omega$ \\ $\delta^+(l)$ & incoming relocation arcs in location $l \in \mathcal{G}$ \\ $\delta^-(l)$ & outgoing relocation arcs from location $l \in \mathcal{G}$ \\ $N^+(l)$ & incoming recourse arcs in location $l$ ; $N^+(l) \subset \delta^+(l)$\\ $N^-(l)$ & outgoing recourse arcs from location $l$ ; $N^-(l) \subset \delta^-(l)$\\ $A_{pool}(l)$ & recourse arcs available for capacity pooling from location $l$ \end{tabularx} \end{aligned}$$ \ ### Input Parameters {#input-parameters .unnumbered} $$\begin{aligned} \begin{tabularx}{\textwidth}{ll} $h_l$ & cost of holding one capacity module at location $l$. \\ $I_0$ & total number of capacity modules available in the system\\ $\phi_\omega$ & probability of scenario $\omega$\\ $p_l$ & penalty for lacking capacity in location $l$ \\ $r_a$ & cost of relocating one capacity module on $a$. \\ $\overline{S_l}$ & maximum number of capacity modules that can be placed in location $l$ \\ $v$ & volume provided by a capacity module \\ $v^R$ & volume that a rider can carry on a tour \\ $v^C$ & volume that a courier can carry on a tour \end{tabularx} \end{aligned}$$ \ ### Random Variables {#random-variables .unnumbered} $$\begin{aligned} \begin{tabularx}{\textwidth}{ll} $D_l(\tau, \omega)$ & volume requirements in location $l$ in scenario $\omega$ in period $(\tau)$ \end{tabularx} \end{aligned}$$ \ ### Decision Variables {#decision-variables .unnumbered} $$\begin{aligned} \begin{tabularx}{\textwidth}{ll} $S_l(t)$ & number of capacity modules available in location $l$ for period $t$ \\ $R_a(t)$ & number of capacity modules relocated through arc $a$ at the beginning of period $t$ \\ $P_a(\tau, \omega)$ & volume shared from location $i$ to $j$, $a=(i,j) \in N_{\omega, \tau}^-(i)$ in period $\tau$ under scenario $\omega$ \\ $Z_l(\tau, \omega)$ & lack of capacity in volume at location $l$ in period $\tau$ under scenario $\omega$ \\ $n^R_a(\tau, \omega)$ & number of detours performed by riders on arc $a$ in period $\tau$ under scenario $\omega$ \\ $n^R_{LC}(\tau, \omega)$ & number of detours performed by riders in local cell $LC$ in period $\tau$ under scenario $\omega$ \\ $n^C_a(\tau, \omega)$ & number of detours performed by couriers on arc $a$ in period $\tau$ under scenario $\omega$ \\ $n^C_l(\tau, \omega)$ & number of detours performed by couriers from location $l$ in period $\tau$ under scenario $\omega$ \end{tabularx} \end{aligned}$$ ### Model {#model .unnumbered} $$\begin{aligned} \min & \sum_{t \in T} \bigg( \sum_{l \in \mathcal{L}\cup \mathcal{W}} h_l S_l(t) + \sum_{a \in A} r_a R_a(t) \nonumber \\ & + \sum_{\omega \in \Omega} \phi_{\omega} \bigg( \sum_{\tau \in T_{t}} \bigg( \sum_{l \in L} (\Delta C_l^C(\tau, \omega) + p_l Z_l(\tau, \omega)) + \sum_{LC \in \mathcal{LC}} \Delta C_{LC}^R(\tau, \omega) \bigg) \bigg) \bigg) \label{Obj} \\ \text{s.t.: \ } \nonumber \\ & \text{Inventory balance of capacity modules at all locations:} \nonumber \\ & S_l(t) = S_l(t-1) + \sum_{a \in \delta^+(l)} R_a(t) - \sum_{a \in \delta^-(l)} R_a(t), \forall l \in \mathcal{L}\cup \mathcal{W}, t \in T \label{IB} \\ & \text{Total capacity module inventory constraint:} \nonumber \\ & \sum_{l \in \mathcal{L}\cup\mathcal{W}}S_l(t) = I_0, \forall t \in T \label{Inv} \\ & \text{Spatial constraint at all locations:} \nonumber \\ & S_l(t) \leq \overline{S_l}, \forall l \in \mathcal{L}, t \in T \label{Spatial} \\ & \text{Volume requirements satisfaction constraints:} \nonumber \\ & v S_l(t) + \sum_{a \in N^+(l)} P_a (\tau, \omega) - \sum_{a \in N^-(l)} P_a (\tau, \omega) + Z_l (\tau, \omega) \geq D_l(\tau, \omega), \forall l \in L, \tau \in T_{t}, t \in T, \omega \in \Omega \label{Vol Req}\\ & \text{Synchronization constraint for riders' operations: (\ref{Rider_Synchro_Constraint})} \nonumber \\ & \text{Synchronization constraint for couriers' operations: (\ref{Courier_Synchro_Constraint})} \nonumber \\ & \text{Rider's detours count:} \nonumber \\ & n_{LC}^R(\tau, \omega) \geq \sum_{l \in LC} \sum_{a \in A_{pool(l)}} n_a^R(\tau, \omega), \forall LC \in \mathcal{LC}, \tau \in T_{t}, t \in T, \omega \in \Omega \\\label{LC detour count} & n_a^R(\tau, \omega) \geq \frac{P_a(\tau, \omega)}{v^R}, \forall a \in A_{pool}(l), l \in L, \tau \in T_{t}, t \in T, \omega \in \Omega \\\label{Rider detour count} & \text{Courier's detours count:} \nonumber \\ & n_{l}^C(\tau, \omega) \geq \sum_{a \in A_{pool}(l)} n_a^C(\tau, \omega), \forall l \in L, \tau \in T_{t}, t \in T, \omega \in \Omega \\\label{AH detour count} & n_a^C(\tau, \omega) \geq \frac{P_a(\tau, \omega)}{v^C}, \forall a \in A_{pool}(l), l \in L, \tau \in T_{t}, t \in T, \omega \in \Omega \\\label{Courier detour count} & \text{Integrality and non-negativity constraints:} \nonumber \\ & P_a(\tau, \omega), Z_l(\tau, \omega), n_a^C(\tau, \omega), n_a^R(\tau, \omega), n_{l}^C(\tau, \omega) \geq 0 \\ & S_l(t), R_a(t)\text{ integer} \label{integrality constraint}\end{aligned}$$ Minimizing expression (\[Obj\]) corresponds to minimizing the last-mile cost, defined in this paper as the cost of deploying capacity modules in each access hub locations (holding costs) and the relocation costs for each capacity module movement for each reconfiguration period, and the marginal cost incurred by recourse actions (capacity pool from neighboring location and consignment). Constraints (\[IB\]) and (\[Inv\]) enforce the conservation of the total number of capacity modules in the network. Constraints (\[Spatial\]) limit the number of capacity modules that can be deployed in each access hub locations. Constraints (\[Vol Req\]) enforce that all demand in terms of volume requirement is served by a combination of capacity modules, capacity pools and consignments, in each demand period of each scenario. Constraints (\[Rider\_Synchro\_Constraint\]) and (\[Courier\_Synchro\_Constraint\]) are the synchronization constraints for the underlying riders and couriers problems as developed in section (\[under op\]). Constraints (\[LC detour count\]) and (\[Rider detour count\]) count the number of detours performed by riders within each local cell based on recourse capacity pooling decisions and the carrying capacity of riders. Constraints (\[AH detour count\]) and (\[Courier detour count\]) count the number of detours performed by couriers from each access hub based on recourse capacity pooling decisions and the carrying capacity of couriers. Solution Approach {#SolutionApproach} ================= In this section, our rolling horizon solution approach is presented, which builds on solving sequentially the two-stage model presented above using scenario sampling, Benders decomposition and acceleration methods. It approximates optimization problem (\[opt challenge\]) by planning for one capacity deployment period, $t$, at the time and deferring subsequent capacity deployment decisions to the following iterations of the Algorithm. In order to enhance the quality of the solutions produced at each iteration, a $\theta$ tactical lookahead is considered to plan for $1 + \theta$ tactical periods, where only the first period is implementable and the subsequent ones are used as an evaluation mechanism. The proposed rolling horizon solution approach is described in Algorithm 1. Here, the length of the sub-horizon is controllable; it can represent one tactical period (i.e. myopic, $\theta = 0$) or several of them (i.e. lookahead, $\theta \geq 1$). Of course, when dealing with large-scale networks, the selection of the lookahead length is part of the trade-offs necessary to make in order to keep the model tractable. In order to enhance the solvability of the optimization model (\[Obj\]-\[integrality constraint\]), for each sub-horizon $[t, t+\theta]$, a tailored Benders decomposition approach is developed, that fits with the two-stage and multi-period setting of our formulation. It is applied under a large sample of multi-period scenarios. The following subsections address the decomposition approach as well as the associated acceleration methods developed.\ $S_l(t_0) \longleftarrow S_l(t_0)$ Benders Decomposition {#BD} --------------------- Benders decomposition is a row generation solution method for solving large scale optimization problems by partitioning the decision variables in first stage and second stage variables ([@Benders2005]). The model is first projected onto the subspace defined by the first stage variables, replacing the second stage variables by an incumbent; the resulting model is called the restriced master problem. Then, a linear problem with the second stage variables and a candidate solution from the restricted master problem is formulated; the resulting model is called the subproblem and can often be decomposed in independent subproblems. From the solution of the subproblem, feasibility and optimality cuts can be identified and added to the restricted master problem. The algorithm terminates when the incumbent in the restriced master problem is equal to the the value of the subproblem.\ Suppose the capacity deployment and relocation decisions (first stage decision variables) $S_l(t)$, $S_l(t+1)$, ... ,$S_l(t+\theta)$ and $R_a(t)$, $R_a(t+1)$, ..., $R_a(t+\theta)$ are given with values $\widehat{S_l}(t)$, $\widehat{S_l}(t+1)$, ... , $\widehat{S_l}(t+\theta)$ and $\widehat{R_a}(t)$, $\widehat{R_a}(t+1)$, ..., $\widehat{R_a}(t+\theta)$. Then, the subproblem can be defined as taking recourse action decisions (i.e. second stage decisions; capacity pooling) to minimize the approximate overall operations costs. The subproblem can be decomposed per scenario $\omega$, operational period $\tau$ and local cell $LC$ into a set of independent subproblems as follows: $$\begin{aligned} \label{subproblem} SP_{LC}(\tau, \omega) = \min & \sum_{l \in L(LC)} (\Delta C_l^C(\tau, \omega) + p_l Z_l(\tau, \omega)) + \Delta C_{LC}^R(\tau, \omega) \\ \text{s.t.: \ } \nonumber \\ & \text{Volume requirements satisfaction constraints:} \nonumber \\ & v \widehat{S_l}(t) + \sum_{a \in N^+(l)} P_a(\tau, \omega) - \sum_{a \in N^-(l)} P_a(\tau, \omega) + Z_l (\tau, \omega) \geq D_l(\tau, \omega), \forall l \in L(LC) \label{dual1}\\ & \text{Synchronization constraint for riders' operations: (\ref{Rider_Synchro_Constraint})} \nonumber \\ & \text{Synchronization constraint for couriers' operations: (\ref{Courier_Synchro_Constraint})} \nonumber \\ & \text{Detour linking constraints: (\ref{LC detour count}), (\ref{Rider detour count}), (\ref{AH detour count}), (\ref{Courier detour count})} \nonumber \\ & P_a(\tau, \omega), Z_l(\tau, \omega), n_a^C(\tau, \omega), n_a^R(\tau, \omega), n_{l}^C(\tau, \omega) \geq 0 \nonumber $$ It is important to notice that the defined subproblems are feasible regardless of the value of the tactical decisions (first stage variables); This is possible thanks to the variables $Z_l(\tau, \omega)$ that compensate for any lack of capacity in the network by incurring a large cost.\ Solving each subproblem using a dualization strategy, one can identify the following optimality cuts for each local cell, operational period $\tau$ and scenario $\omega$: \[Benders cuts\] q\_[LC]{}(, ) & \_[l L(LC)]{}\^j\_l(,)(D\_l(, )-v S\_l(t))\ &+ \^j\_[LC]{}(, ) ( m\^R(, )(\_- (t\^R\_s + )) - n\_[LC]{}( + t\^R\_a)\ & -\_[l LC]{}(\_l\^D(, ) + \_l\^P(, ))t\^R\_u ) &&\ &+ \_[l L(LC)]{} ( \^j\_l(,)( m\^C(,) \_- (\_l\^P(, ) + \_l\^P(, ))( + t\^C\_a + t\^C\_u) ) ) && where $j \in J$, the set of extreme points of the dualized subproblem; $\pi^j_l(\tau,\omega)$, $\mu^j_{LC}(\tau, \omega)$ and $\lambda^j_l(\tau,\omega)$ are the dual values respectively associated with constraints (\[dual1\]), (\[Rider\_Synchro\_Constraint\]) and (\[Courier\_Synchro\_Constraint\]). Finally, the restricted master problem, whose objective minimizes the cost of deploying capacity modules in each access hub and the relocation costs for each capacity module for each period subject to the optimality cuts, can be formulated as follows: $$\begin{aligned} RMP = \min & \sum_{t}^{t+\theta} \bigg( \sum_{l \in \mathcal{L}\cup \mathcal{W}} h_l S_l(t) + \sum_{a \in A} r_a R_a(t) + \sum_{\omega \in =\Omega} \phi_{\omega} \sum_{\tau \in T_{t}} \sum_{LC \in \mathcal{LC}} q_{LC}(\tau, \omega) \bigg)\\ \text{s.t.: \ } \nonumber \\ & \text{Inventory balance of capacity modules at all locations: (\ref{IB})} \nonumber \\ & \text{Total capacity module inventory constraint: (\ref{Inv})} \nonumber \\ & \text{Spatial constraint at all locations: (\ref{Spatial})} \nonumber \\ & \text{Optimality cuts: (\ref{Benders cuts})}, \forall j \in \overline{J}\subset J\\\nonumber & S_l(t), R_a(t) \text{ integer} \nonumber\end{aligned}$$ Solving the restriced master problem with added optimality cuts provides new values $\widehat{S_l}(t)$ and $\widehat{R_a}(t)$, and a new incumbent solution. This process can be executed iteratively until the incumbent solution equals the subproblem value, indicating optimality. Acceleration Methods {#preprocessing and acceleration} -------------------- The following subsection describes acceleration methods developed to improve the performance of the proposed solution approach on large instances. The acceleration techniques retained are those that improve significantly the convergence speed of the benders decomposition algorithm for the proposed model. ### Pareto-optimal Cuts {#pareto-optimal-cuts .unnumbered} The proposed implementation of the benders decomposition can be improved using Pareto-optimal cuts, which requires to solve two linear programs: the original subproblem (\[subproblem\]), and the Pareto subproblem. The result is the identification of the strongest cut when the original subproblem solution has multiple solutions. A Pareto-optimal solution produces the maximum value at a core point, which is required to be in the relative interior of the convex hull of the subregion defined by the first stage variables. The Pareto subproblem can be decomposed per scenario $\omega$, operational period $\tau$ and local cell $LC$ in a set of independent Pareto subproblems as follows: $$\begin{aligned} \label{Pareto subproblem} \min & \sum_{l \in L(LC)} (\Delta C_l^C(\tau, \omega) + p_l Z_l(\tau, \omega)) + \Delta C_{LC}^R(\tau, \omega) + v_{SP} Y \\ \text{s.t.: \ } \nonumber \\ & v (S_l^0(t) + \sum_{a \in N^+(l)} P_a(\tau, \omega) - \sum_{a \in N^-(l)} P_a(\tau, \omega) + Z_l (\tau, \omega) + (D_l(\tau, \omega) - v \widehat{S_l}(t)) Y \nonumber \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \geq D_l(\tau, \omega), \forall l \in L(LC) \label{pareto1}\\ & \text{Modified synchronization constraint for riders' operations:} \nonumber \\ & T^R_{LC}(\tau, \omega) \leq m^R(\tau, \omega) \Delta_\tau (1 - Y) \nonumber \\ & \qquad \qquad + \bigg( m_{\tau}^R(\omega) \bigg(t^R_s + \frac{2r_{LC}}{s^R_0} \bigg) + n_{LC} \bigg( \frac{ k^R(\delta_{LC})^{-\frac{1}{2}}}{s^R} + t^R_a \bigg) + \bigg( \sum_{l \in LC}(\rho_{l \tau}^D(\omega) + \rho_{l \tau}^P(\omega)) \bigg)t^R_u \bigg) Y \label{pareto_rider}\\ & \text{Modified synchronization constraint for couriers' operations:} \nonumber \\ & T^C_{l}(\tau, \omega) \leq m^C(\tau, \omega) \Delta_\tau (1 - Y) + \bigg( (\rho_{l \tau}^P(\omega) + \rho_{l \tau}^P(\omega)) \bigg(\frac{ k^C(\delta_{l})^{-\frac{1}{2}}}{s^C} + t^C_a + t^C_u \bigg) \bigg) Y, \forall l \in L(LC) \label{pareto_courier}\\ & \text{Detour linking constraints: (\ref{LC detour count}), (\ref{Rider detour count}), (\ref{AH detour count}), (\ref{Courier detour count})} \nonumber \\ & P_a(\tau, \omega), Z_l(\tau, \omega), n_a^C(\tau, \omega), n_a^R(\tau, \omega), n_{l}^C(\tau, \omega), Y \geq 0 \nonumber $$ where $v_{SP}$ is the value of the corresponding original subproblem and $S_l^0(t)$ a core point of the current solution to the restricted master problem. Solving each Pareto subproblem using a dualization strategy, one can identify strengthened optimality cuts (\[Benders cuts\]) by assigning $\pi^j_l(\tau,\omega)$, $\mu^j_{LC}(\tau, \omega)$ and $\lambda^j_l(\tau,\omega)$ the dual values respectively associated with constraints (\[pareto1\]), (\[pareto\_rider\]) and (\[pareto\_courier\]).\ The proposed implementation also updates the core point, which can be seen as an intensification procedure: locations that are rarely given capacity modules decay toward low values while locations with consistent capacity module presence in every solution are assigned a high coefficient in Pareto solutions. The update rule was introduced in [@papadakos2009integrated], and consists of updating the core point at iteration $k$, $S^{0(k)}$ by combining it with the solution of the master problem at this iteration, $\widehat{S}^{(k)}$, using a factor $\lambda$. [@maheo2019benders] suggest that a factor $\lambda=1/2$ yields the best results. The update rule is defined as follows: $$\begin{aligned} S^{0(k+1)}_l(t) = \frac{S^{0(k)}_l(t) + \widehat{S_l}^{(k)}(t)}{2}, \forall l \in L, t \in T \\ S^{0(k+1)}_l(t) = \frac{I_0 - \sum_{l' \in L}S^{0(k+1)}_{l'}(t)}{|W|}, \forall l \in W\end{aligned}$$ where $k$ is the current iteration of the Benders algorithm. ### $\epsilon$-optimal Method {#epsilon-optimal-method .unnumbered} When dealing with large-scale instances, the $\epsilon$-optimal method as described in [@rahmaniani2017benders] has proven to speed up the proposed Benders decomposition algorithms by avoiding to solve the restricted master problem to optimality at each iteration, while guaranteeing an optimal gap within $\epsilon$. It is not necessary to solve the restricted master problem to optimality at each iteration to generate good quality cuts, and there is no incentive to do so at the beginning of the algorithm because the relaxation is weak. Instead, the restricted master problem can be solved with a relaxed optimality gap by adding a constraint forcing the objective value to be improved by at least $\widehat{\epsilon}$ percent compared to the previous solution. Then, when no feasible solution is found, $\widehat{\epsilon}$ is decreased. The same mechanism is applied until $\epsilon$ is reached; the algorithm terminates when no feasible solution is found to the restricted master problem, guaranteeing that the current solution is within $\epsilon$ of the optimal. Experimental Results {#Results} ==================== In this section, the results of numerical experiments are presented in order to validate the developed modeling and solution approaches, and to analyze the performance of the proposed capacity deployment strategy for urban parcel logistics. After describing the test instances which are inspired from the real data of a large parcel express carrier, experimental results about the computational performance of the solution approach are presented. Then, the performance of the dynamic pooled capacity deployment strategy is exposed and compared to its static counterpart. Finally sensitivity analyses are conducted on the capacity pooling distance and the holding costs to derive further insights. Experimental setting {#Exp setting description} -------------------- Table \[Instances\] summarizes the characteristics of the considered instances: number of access hub locations, number of local cells, and area and population covered by the network. 100 non-stationary demand scenarios are generated randomly from given distributions at the hourly level with monthly, weekly, daily and hourly seasonality factors. Figure \[demand sample\] illustrates demand dynamics by displaying access hub volume requirements box plots and snapshots of demand levels in two consecutive tactical periods as seen in Figure \[fig: Relocation\] for a sample local cell from instance E. The number of scenarios is chosen to ensure tactical decision stability with a reasonable in-sample statistical gap ($1.5\%$) and coefficient of variation ($0.5\%$) as detailed in \[Experiment Parameters\]. The considered planning horizon spread over two months, with 8 weekly tactical periods and hourly operational period. Each week is composed of seven days of ten operating hours each. The $\epsilon$-method is implemented with a guaranteed optimality gap of $0.1\%$. Instance Access hubs Local cells Area covered (sq.km) Population covered ---------- ------------- ------------- ---------------------- -------------------- A 39 1 24.2 338,000 B 54 2 42.1 590,000 C 138 4 66 924,000 D 421 10 178.4 2,500,000 E 838 20 410 5,740,000 : Experimental Instances[]{data-label="Instances"} As benchmark solutions, static capacity deployments are considered for each instance. Such static capacity deployment represents the minimum capacity module deployment required over the network of access hub locations to satisfy storage requirements for all operational periods within the planning horizon $T$ without being able to update capacity over time or use capacity pooling recourse actions. Benchmark solutions are found by solving $\min \{ \sum_{t \in T} \sum_{l \in \mathcal{L}\cup \mathcal{W}} h_l S_l(t) \}$ such that $S_l(t) \in \{v S_l(t) \geq D_l(\tau, \omega), \forall l \in L, \tau \in T_{t}, t \in T, \omega \in \Omega \}$ over the entire planning horizon with no relocation or recourse by relaxing spatial constraints to ensure feasibility.\ An instance has more or less savings potential depending on its demand dynamics and network configuration. Although assessing the potential of capacity pooling a priori is non trivial, the potential of capacity relocation can be assessed by a lower bound to the dynamic capacity deployment problem with no capacity pooling. Define $\Tilde{S}_l(t)$ as the maximum number of capacity modules required at location $l$ in any operational period associated with tactical period $t$ in all considered scenarios; that is $\Tilde{S}_l(t) = max(\ceil{D_l(\tau, \omega)/v} , \forall \tau \in T_t, \omega \in \Omega)$. Then, an instance’s capacity relocation cost savings potential can be computed by factoring in holding costs while ignoring relocation costs, producing a lower bound for the dynamic capacity deployment problem with no capacity pooling, with objective value $\sum_{t \in T} h_l\Tilde{S}_l(t)$. Benchmark solutions and relocation potential for the considered instances are summarized in Table \[Benchmark\]. Instance Total cost Capacity Potential cost savings ---------- --------------- ---------- ------------------------ A $\$138,966$ 189 $7.93\%$ B $\$181,275$ 245 $8.72\%$ C $\$453,137$ 618 $7.58\%$ D $\$1,325,490$ 1820 $7.29\%$ E $\$2,840,510$ 3818 $9.25\%$ : Benchmark Solutions and Relocation Potential[]{data-label="Benchmark"} The initial capacity deployment is defined by running the proposed solution approach for the tactical period immediately preceding the studied planning horizon by relaxing constraint (\[IB\]). The default values of input parameters are estimated relying on company experts and presented in \[Experiment Parameters\]. Each instance is assigned one depot in one of its local hub locations to store unused capacity modules at no cost. The number of modules available $I_0$ and the penalty cost $p_l$ are set to large values (respectively $5000$ modules and $\$100,000$ per modules in order to prevent full recourse actions by lack of capacity and focus on feasible capacity deployments with capacity pooling. As suggested by [@winkenbach2016enabling] (through simulation) when studying a french parcel express company, this paper considers the value of the $k$ constants to be 0.82 for riders and 1.15 for couriers.\ All experiments were implemented in Python 3.7 using Gurobi 9.0 as the solver and were computed using 40 logical processors on an AMD EPYC Processor @ 2500GHz. Computational Performance ------------------------- The experiments presented in this section study the computational performance of the proposed solution approach when tackling instances of different sizes. The first experiment aims at validating the efficiency of the proposed acceleration methods in section (\[preprocessing and acceleration\]) for the Benders algorithm. It examines the impact of combinations of the acceleration methods on the runtime of the Benders algorithm for solving the optimization model (\[Obj\]-\[integrality constraint\]) for one relocation period with no lookahead. Figure \[benders performance C\] display the runtimes for instances C with a capacity pooling distance of 1km and a time cutoff of 15 hours; B represents the original Benders algorithm developed in section (\[BD\]); BP represents Benders with pareto-optimal cuts; BE represents Benders with the $\epsilon$-optimal method; and BPE represents BP with the $\epsilon$-optimal method.\ Figure \[benders performance C\] suggests that pareto-optimal cuts have the strongest impact on computational performance as it allows the BP algorithm to converge in 965 seconds when the B algorithm did not converge within the time limit. The $\epsilon$-optimal method suggests a significant improvement compared to the original Benders algorithm, and has an advantage over BP when close to optimality (while guaranteeing a solution within $0.1\%$ of optimality). Similar behaviors can be observed for larger instances, with BPE outperforming B, BP and BE. Next, Figure \[algorithm\_perf\] depicts the computational performance of the proposed solution approach for different lookahead values as a function of network size. Each data point is the average runtime per period for a minimum sample set of 16 instances (8 relocation periods times 2 capacity pooling distances) and a maximum of 48 instances (8 relocation periods times 6 capacity pooling distances) based on the other experiments presented in the paper. The first observation is that the proposed solution approach is efficient in solving large-scale instances considered in this paper (838 access hubs), with a maximum runtime around 3 hours (with 2 weeks lookahead); this result suggests tractability for most urban area sizes, including megacities. The second observation is that adding tactical lookahead reasonably increase runtime: 1 week and 2 weeks lookahead runtimes are respectively at most 2.1 times and 3.5 times as long as no lookahead runtimes wihtin the range of network sizes considered. Comparative Results -------------------- The results presented in this section highlight the benefits of relocating capacity dynamically over time and allowing capacity pooling compared to a static capacity deployment with no capacity pooling. Results are summarized in Table \[Result table\] for different lookahead values and capacity pooling distance (in km). Table \[Result table\] presents total costs of the network, deployed capacity (maximum number of modules), relocation share (average number of relocations per period as a share of capacity), and cost and capacity savings with respect to the static counterpart. First, cost and capacity savings are observed in all the instances. Maximum cost savings of $28.3\%$ and capacity savings of $26.46\%$ are reached for instance A with a capacity pooling distance of 2km and a 2 weeks tactical lookahead. Most of these savings are a result of the capacity pooling recourse as savings with capacity pooling of 0km indicate a much lower savings (maximum of $6.26\%$ cost savings). Note that for each instance, savings with no capacity pooling are less than potential savings presented in Table \[Benchmark\] (where relocation costs are not accounted for). The average number of relocations per period represent up to $7.31\%$ of the capacity, and is decreasing as more tactical lookahead is added; capacity deployments are gradually reconfiguring networks. Capacity savings indicate that the total number of modules required (both deployed and stored at a depot) is inferior to the number of modules required in static counterparts. Capacity savings also increase as capacity pooling is available, making the total capital invested in capacity modules inferior than in static counterparts.\ Furthermore, the results show that adding tactical lookahead is beneficial for all instances with and without capacity pooling by improving cost savings and decreasing the number of relocations. The role of tactical lookahead is to anticipate future needs and avoid relocations that will be reverted to in the future. Lookahead can be seen as the flexibility hedging of the solution approach to avoid relocations under uncertainty. However, the difference between one week and two weeks of tactical lookahead is more subtle with smaller cost improvements. These results suggest that solution’s quality increase with lookahead ($\theta$), offering extra cost savings. Tactical lookahead anticipates for future relocations therefore decreasing relocation share at the cost of slightly higher capacity deployments. However, there does not seem to be significant improvements from extending the lookahead from one week to two weeks, especially when considering the additional computational runtime.\ Lastly, capacity pooling brings significant value to instance A, B, and C, but less cost savings improvements for instance D and E. This is probably due to the the fact that instances D and E have lower hub density, increasing the distance between access hubs (see Table \[Instances\]). Section \[pooling varation\] examine the impact of capacity pooling distance in more details by focusing on instance C. Capacity Pooling Variations {#pooling varation} --------------------------- This experiment examines the effect of capacity pooling as a way to further decrease costs. Table \[table: capacity pooling\] summarizes the effect of different capacity pooling distances (in km) on instance C’s solutions. It presents average additional rider and courier travel (induced by detours), and cost and capacity savings for instance C. Figure \[fig: Pooling Exp\] displays a plot of cost and capacity savings as a function of pooling distance. The increase in capacity pooling distance allows to produce superior solutions but only until a maximum of $12.55\%$ is reached with a pooling distance of 5km. This trend can clearly be seen in Figure \[fig: Pooling Exp\]. Indeed, no matter how large capacity pooling pooling neighborhoods are, constraints (\[Rider\_Synchro\_Constraint\]) and (\[Courier\_Synchro\_Constraint\]) limit capacity poolings from an operational point of view: riders and couriers cannot perform long distance detours as it would disrupt their activity by delaying other pickup / deliveries. Table \[pooling varation\] shows that most of the additional travel induced by detours is performed by couriers; since riders have larger carrying capacity, one rider detour may require multiple courier detours. Note also that since couriers are often using lightweight vehicles (if any vehicles at all), long distance detours may not be practical which may also limit the capacity pooling distance from a design perspective. The same behavior can be observed for the other instances. Holding Costs Versus Relocation Costs ------------------------------------- This experiment examines the influence of relocation costs and holding costs on dynamic capacity deployments. Intuitively, two extreme cases can be identified: (1) if holding costs are negligible compared to relocation costs, there is no incentive to dynamically adjust capacity, and (2) if relocation costs are negligible compared to holding costs, a myopic view of the problem would be optimal as anticipating future relocations does not save cost. Apart from extreme cases, variations of holding costs and relocation cost can represent different urban environments. A very dense city may have high holding costs (prime real estate) and low relocation costs (short distances between locations). In this experiment, four cases are examined: High-high, High-low, Low-high and Low-low, where High and high respectively represent high holding costs and high relocation costs and Low and low respectively represent low holding costs and low relocation costs. Cost vectors are scaled linearly and high costs are chosen to be $100\%$ higher than baseline values while low costs are assumed to be $50\%$ lower than baseline values. Table \[table: H vs C\] presents total cost, capacity (maximum number of modules deployed), relocations, relocation share (average number of relocations per period as a share of capacity), and cost and capacity savings for instance C with a capacity pooling distance of 2km. Savings are computed comparing to benchmark solutions with corresponding cost adjustements (holding costs). \[H\] Case Relocation share Cost savings Capacity savings ----------- ------------------ -------------- ------------------ High-low $6.12\%$ $13.07\%$ $9.62\%$ High-high $5.22\%$ $11.76\%$ $9.78\%$ Low-low $6.20\%$ $13.10\%$ $8.13\%$ Low-high $3.73\%$ $9.02\%$ $7.79\%$ : Impact of Holding versus Relocation Costs on Instance C[]{data-label="table: H vs C"} A first observation is that cases where relocation costs are low perform best with costs savings around $13.1\%$, regardless of holding costs. When relocation costs are high, cost savings are worse, especially when holding costs are low ($9.02\%$). Low holding cost cases deploy more capacity modules overall which impact capacity savings, but are still able to reach high cost savings when relocation costs are low. The combination of low holding costs and high relocation costs decreases opportunities for worthy relocations (only $3.73\%$ relocation share), requiring more capacity deployed and therefore limiting cost savings ($9.02\%$). Similar behavior can be observed on the other instances.\ Overall, this experiment indicates that denser urban environment (high holding costs) tend to be better candidates for dynamic capacity management of access hub networks. Moreover, low relocation costs (i.e. easy installation and good mobility of capacity modules) can make any urban environment a worthy candidate for such capacity management strategy. Finally, the combination of lesser dense urban environment and high relocation costs significantly limits opportunity for cost savings. Conclusion {#Conclusion} ========== This paper defines and formulates the Dynamic Pooled Capacity Deployment Problem in the context of urban parcel logistics. This problem involves a tactical decision on the relocation of capacity modules over a network of discrete locations associated with stochastic demand requirements. To improve the quality of the capacity deployment decisions, the proposed model integrates an estimate of the difference of operations cost, which includes capacity assignment decisions with the possibility of capacity pooling between neighboring locations. It also integrates synchronization requirements of the 2-echelon routing subproblems, using an analytical derivate from the route length estimation function. The dynamic problem is modeled and approximated with a two-stage stochastic program with recourse, where all capacity deployment decisions on a finite planning horizon are moved to the first stage. Due to the uncertainty of capacity requirements and the challenges of solving the MIP formulation for realistic networks of several hundreds locations, a roll-out approach with lookahead based on a Benders decomposition of the finite planning horizon problem coupled with acceleration methods is proposed. Five instances of networks of different sizes are presented to perform computational experiments to test the performance of the proposed approach and assess the potential of the defined capacity deployment strategy.\ Results show that the proposed approach produces solutions in a reasonable time even for large scale instances of up to 838 hubs. They suggest that a dynamic capacity deployment strategy with capacity pooling has a significant advantage over a static capacity deployment strategy for access hub networks, with up to $28\%$ cost savings and $26\%$ capacity savings. Results also show that one-week lookahead helps producing superior solutions by anticipating future relocations, but adding a two-weeks lookahead does not make a significant improvement. Increasing the capacity pooling distance, while increasing computing time, tend to increase opportunities for cost savings by allowing more locations to pool capacity until an operational feasibility threshold is reached. Dynamically adjusting workforce assignment in the network was not explored but could potentially overcome this limitation. Denser urban environments (i.e. with higher real estate costs) are natural candidates for dynamic capacity deployments as relocation costs are more easily overcome by holding costs. However, relocation costs are the most limiting when it comes to cost savings. Technology solutions featuring cheaper installation costs and high degree of mobility make it more interesting to consider periodic network reconfigurations.\ The implementation of such innovation also has management challenges not studied in this paper. For instance, implementation may require a more agile workforce, specialized training and targeted hiring enabling a data-driven approach to managing network capacity. Management challenges also need to be considered by decision makers along with the potential reduction of fixed-assets offered by capacity savings when evaluating the solution for implementation.\ Finally, there are numerous research avenues around reconfigurable networks, dynamic capacity management and access hubs in urban parcel logistics. Where technology allows for very frequent network reconfiguration, solutions featuring not only modular but mobile capacity (e.g. on wheels) and near real-time capacity relocation can become relevant as a complement to the proposed dynamic capacity deployment strategy. Moreover, the possibility of updating operations planning as needed (e.g. dynamic routing, dynamic staffing) can unlock the potential of capacity pooling not only as a recourse but as an integral part of network design and operations planning. Calibration and Data {#Experiment Parameters} ==================== In-sample Variability --------------------- In-sample variability was tested with no lookahead for instance A with capacity pooling limited to 1 km for 10 samples. Results are presented in Table \[Statistical Analysis\]. Coefficient of variation represent the ratio between the standard deviation and the average of solutions’ total cost. Statistical gap represent the ratio $(UB-LB)/LB$ where UB and LB are respectively the highest and lowest total cost in the sample. Number of scenarios 5 10 20 30 50 75 100 200 -------------------------- ----------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- Coefficient of variation $3.47\%$ $2.87\%$ $2.70\%$ $2.11\%$ $1.34\%$ $1.04\%$ $0.52\%$ $0.41\%$ Statistical gap $10.04\%$ $9.16\%$ $8.57\%$ $7.85\%$ $4.25\%$ $3.51\%$ $1.53\%$ $1.22\%$ : In-sample Statistical Analysis[]{data-label="Statistical Analysis"} Cost Estimates -------------- The capacity module relocation costs $r_a$ include an operational cost of $\$1.50$ per kilometer, and a fixed cost of two operators for two hours at a rate of $\$10$ per hour to uninstall/install modules once at the desired locations: $$\begin{aligned} r_a = 1.50 d_a + 40, \forall a = (i,j) \in A\end{aligned}$$ Where $d_a$ is the distance between location $i$ and $j$ such that $a = (i,j)$.\ The holding costs are computed from an amortized acquisition cost of $\$2000$ over 5 years (52 weeks long years), and from a rent cost of $\$75$ per square meter times a location specific factor $(1+f_l)$ randomly generated to represent the real estate difference between locations. $$\begin{aligned} h_l = \frac{2000}{5*52} + 75 (1+ f_l), \forall l \in L\end{aligned}$$ Where $f_l$ is randomly generated from a uniform distribution over $[2\%, 15\%]$. It is also assumed that modules do not depreciate when stored at depots ($h_l = 0, \forall l \in D$).\ Other Input Parameters ---------------------- \[h\] Parameter Value Parameter Value ------------ --------------------- ------------- ------------------------ $c^C_v$ $\$1/km$ $s^R_0$ $50 \textit{ km/h}$ $c^C_{v0}$ $\$0.8/km$ $\hat{S_l}$ $15 \textit{ modules}$ $c^R_f$ $\$10$ $t^C_a$ $1 \textit{ min}$ $c^R_v$ $\$1.8/km$ $t^C_u$ $2 \textit{ min}$ $c^R_{v0}$ $\$1.2/km$ $t^R_a$ $5 \textit{ min}$ $k^C$ $1.15$ $t^R_u$ $1 \textit{ min}$ $k^R$ $0.82$ $t^R_s$ $5 \textit{ min}$ $p_l$ $100000/ module$ $v$ $0.75 \textit{ } m^2$ $s^C$ $7 \textit{ km/h}$ $v^C$ $0.48 \textit{ } m^2$ $s^C_0$ $15 \textit{ km/h}$ $v^R$ $6.40 \textit{ } m^2$ $s^R$ $30 \textit{ km/h}$

--- abstract: 'Let $(M,\g)$ be a pseudo-Riemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1-cocycles on the group of diffeomorphisms of $M$ related to the modules of linear differential operators. As operators, these derivatives do not depend on the rescaling of the metric $\g.$ In particular, if the manifold $(M,\g)$ is conformally flat, these derivatives vanish on the conformal group ${\mathrm{O}}(p+1,q+1),$ where $\mathrm{dim} (M)=p+q.$ This work is a continuation of [@b2; @bo2] where the Schwarzian derivative was defined on a manifold endowed with a projective connection.' author: - | Sofiane BOUARROUDJ[^1]\ [ Department of Mathematics, Keio University, Faculty of Science & Technology]{}\ [3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan. ]{}\ [e-mail: sofbou@math.keio.ac.jp]{} title: Conformal Schwarzian derivatives and conformally invariant quantization --- ł .1truein \[section\] \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Example]{} \[thm\][Remark]{} \[thm\][Definition]{} Introduction ============ Let $S^1$ be the circle identified with the projective line ${\mathbb{RP}}^1.$ For any diffeomorphism $f$ of $S^1,$ the expression $$\label{clas} S(f):= \frac{f'''(x)}{f'(x)}-\frac{3}{2} \left (\frac{f''(x)}{f'(x)}\right )^2,$$ where $x$ is an affine parameter on $S^1,$ is called Schwarzian derivative (see [@cara]).\ The Schwarzian derivative has the following properties: \(i) It defines a 1-cocycle on the group of diffeomorphisms ${\mathrm{Diff}}(S^1)$ with values in differential quadratics (cf. [@ki; @s]). \(ii) Its kernel is the group of projective transformations ${\mathrm{PSL}}_2({\mathbb{R}})$.\ The aim of this paper is to propose a new approach for constructing the multi-dimensional conformal Schwarzian derivative. This approach was recently used in [@b2; @bo2] to introduce the multi-dimensional “projective” Schwarzian derivative. The starting point of our approach is the relation between the Schwarzian derivative (\[clas\]) and the space of Sturm-Liouville operators (see, e.g., [@wi]). The space of Sturm-Liouville operators is not isomorphic as a ${\mathrm{Diff}}(S^1)$-module to the space of differential quadratics. More precisely, the space of Sturm-Liouville operators is a non-trivial deformation of the space of differential quadratics in the sense of Neijenhuis and Richardson’s theory of deformation (see [@nr]), generated by the 1-cocycle (\[clas\]) (see [@ga] for more details). From this point of view, the multi-dimensional Schwarzian derivative is closely related to the modules of linear differential operators. To set out our approach, let us introduce some notation.\ Let $M$ be a smooth manifold. We consider the space of linear differential operators with arguments that are $\l$-densities on $M$ and values that are $\mu$-densities on $M.$ We have, therefore, a two parameter family of ${\mathrm{Diff}}(M)$-modules denoted by ${{\mathcal{D}}}_{\l,\mu}(M).$ The corresponding space of symbols is the space ${{\mathcal{S}}}_{\d}(M)$ of fiberwise polynomials on $T^*M$ with values in $\d$-densities, where $\d=\mu-\l.$ In general, the space ${{\mathcal{D}}}_{\l,\mu}(M)$ is not isomorphic as a ${\mathrm{Diff}}(M)$-module to the space ${{\mathcal{S}}}_{\d}(M)$ (cf. [@do; @lo2]). However, we are interested in the following two cases:\ (i) If $M:={\mathbb{R}}^n$ is endowed with a flat projective structure (i.e. local action of the group ${\mathrm{SL}}_{n+1}({\mathbb{R}})$ by linear fractional transformations) there exists an isomorphism between ${{\mathcal{D}}}_{\l,\mu}({\mathbb{R}}^n)$ and ${{\mathcal{S}}}_{\d}({\mathbb{R}}^n),$ for $\d$ generic, intertwining the action of ${\mathrm{SL}}_{n+1}({\mathbb{R}})$ (cf. [@lo2]). The multi-dimensional “projective” Schwarzian derivative was defined in [@b2; @bo2] as an obstruction to extend this isomorphism to the full group ${\mathrm{Diff}}({\mathbb{R}}^n).$\ (ii) If $M:={\mathbb{R}}^n$ is endowed with a flat conformal structure (i.e. local action of the conformal group ${\mathrm{O}}(p+1,q+1),$ where $p+q=n$), there exists an isomorphism between ${{\mathcal{D}}}_{\l,\mu}({\mathbb{R}}^n)$ and ${{\mathcal{S}}}_{\d}({\mathbb{R}}^n),$ for $\d$ generic, intertwining the action of ${\mathrm{O}}(p+1,q+1)$ (cf. [@dlo; @do]). In this paper we introduce the multi-dimensional “conformal” Schwarzian derivative in this context. Recall that in the one-dimensional case these two notions coincide in the sense that the conformal Lie algebra ${\mathrm o}(2,1)$ is isomorphic to the projective Lie algebra ${\mathrm{sl}}_2({\mathbb{R}}).$ Differential operators and symbols ================================== Let $(M,\g)$ be a pseudo-Riemannian manifold of dimension $n.$ We denote by $\Gamma$ the Levi-Civita connection associated with the metric $\g.$ Space of linear differential operators as a module -------------------------------------------------- We denote the space of tensor densities on $M$ by ${{\mathcal{F}}}_{\l}(M),$ or ${{\mathcal{F}}}_{\l}$ for simplify. This space is nothing but the space of sections of the line bundle $(\wedge^n T^* M)^{\otimes \l}.$ One can define in a natural way a ${\mathrm{Diff}}(M)$-module structure on it: for $f\in {\mathrm{Diff}}(M)$ and $\phi \in{\cal F}_{\l},$ in a local coordinates $(x^i)$, the action is given by $$\label{den} f^*\phi=\phi\circ f^{-1}\cdot ( {J_{f^{-1}}})^{\lambda},$$ where $J_f=\left |Df/Dx \right |$ is the Jacobian of $f$.\ By differentiating this action, one can obtain the action of the Lie algebra of vector fields ${\mathrm{Vect}}(M).$ [ ${{\mathcal{F}}}_0= C^{\infty}(M), \, {{\mathcal{F}}}_1= \Omega^{n}(M)$ (space of differential $n$-forms). ]{} Let us recall the definition of a covariant derivative on densities. If $\phi \in {{\mathcal{F}}}_\l,$ then $\nabla \phi \in \Omega^1 (M)\otimes {{\mathcal{F}}}_\l$ given in a local coordinates by $$\nabla_i \phi= \partial_i \phi-\l \Gamma_i \phi,$$ with $\Gamma_i= \Gamma_{ti}^t.$ (Here and bellow summation is understood over repeated indices). Consider now ${\cal D}_{\l,\mu}(M),$ the space of linear differential operators acting on tensor densities $$A:{{\mathcal{F}}}_\l\to{{\mathcal{F}}}_\m.\nonumber \label{Conv}$$ The action of ${\mathrm{Diff}}(M)$ on ${{\mathcal{D}}}_{\l,\mu}(M)$ depends on the two parameters $\l$ and $\m$. This action is given by the equation $$f_{\l,\m}(A)=f^*\circ A\circ {f^*}^{-1}, \label{Opaction}$$ where $f^*$ is the action (\[den\]) of ${\mathrm{Diff}}(M)$ on ${{\mathcal{F}}}_\l$. By differentiating this action, one can obtain the action of the Lie algebra ${\mathrm{Vect}}(M).$ The formul[æ]{} (\[den\]) and (\[Opaction\]) do not depend on the choice of the system of coordinates. Denote by ${\cal D}^2_{\l,\mu}(M)$ the space of second-order linear differential operators with the ${\mathrm{Diff}}(M)$-module structure given by (\[Opaction\]). The space ${\cal D}_{\l,\mu}^2(M)$ is in fact a ${\mathrm{Diff}}(M)$-submodule of ${{\mathcal{D}}}_{\l,\m}(M).$ [The space of Sturm-Liouville operators $\frac{d^2}{dx^2}+u(x): {{\mathcal{F}}}_{-1/2}\rightarrow {{\mathcal{F}}}_{3/2}$ on $S^1,$ where $u(x)\in {{\mathcal{F}}}_{2}$ is the potential, is a submodule of ${{\mathcal{D}}}_{-\frac{1}{2},\frac{3}{2}}^2(S^1)$ (see [@wi]). ]{} The module of symbols --------------------- The space of symbols, ${\mathrm{Pol}}(T^*M),$ is the space of functions on the cotangent bundle $T^*M$ that are polynomials on the fibers. This space is naturally isomorphic to the space ${\cal S}(M)$ of symmetric contravariant tensor fields on $M.$ In local coordinates $(x^i,\xi_i)$, one can write $P\in {\cal S}(M)$ in the form $$P=\sum_{l\geq 0}P^{i_1, \ldots,i_l}\xi_{i_1}\cdots \xi_{i_l},$$ with $P^{i_1, \ldots,i_l}(x)\in C^{\infty}(M).$\ We define a one parameter family of ${\mathrm{Diff}}(M)-$module on the space of symbols by $${\cal S}_{\d}(M):={\cal S}(M)\otimes {{\mathcal{F}}}_{\d}.$$ For $f\in {\mathrm{Diff}}(M)$ and $P\in {\cal S}_{\d}(M),$ in a local coordinate $(x^i)$, the action is defined by $$\begin{aligned} \label{actsym} f_{\d}(P)&=& f^*P\cdot (J_{f^{-1}})^{\d},\end{aligned}$$ where $J_f=|Df/Dx|$ is the Jacobian of $f,$ and $f^*$ is the natural action of ${\mathrm{Diff}}(M)$ on ${\cal S}(M).$ We then have a graduation of ${\mathrm{Diff}}(M)$-modules given by $${\cal S}_\d(M)=\bigoplus_{k=0}^\infty {\cal S}_\d^k(M),$$ where $ S_\d^k(M)$ is the space of contravariant tensor fields of degree $k$ endowed with the ${\mathrm{Diff}}(M)$-module structure (\[actsym\]). We want to study the space of contravariant tensor fields of degree less than two, denoted by ${\cal S}_{\d,2}(M)$ (i.e. ${\cal S}_{\d,2}(M):={\cal S}_{\d}^2(M)\oplus {\cal S}_{\d}^1(M)\oplus {\cal S}_{\d}^0(M)$). Conformal Schwarzian derivatives ================================ Let $(M,\g)$ be a pseudo-Riemannian manifold. Denote by $\Gamma$ the Levi-Civita connection associated with the metric $\g.$ Main definition {#enfant} --------------- It is well known that the difference between two connections is a well-defined tensor field of type $(2,1).$ It follows therefore that the difference $$\label{ell} \ell(f):=f^*\Gamma-\Gamma,$$ where $f\in {\mathrm{Diff}}(M),$ is a well-defined $(2,1)$-tensor field on $M$. It is easy to see that the map $$f\mapsto \ell(f^{-1})$$ defines a non-trivial 1-cocycle on ${\mathrm{Diff}}(M)$ with values in the space of tensor fields on $M$ of type $(2,1).$ Our first main definition is the linear differential operator ${\cal A}(f)$ acting from ${\cal S}_{\d}^2(M)$ to ${\cal S}_\d^1(M)$ defined by $${\cal A} (f)_{ij}^k:= {f^*}^{-1} \left( \g^{sk}\,\g_{ij}\nabla_s \right) -\g^{sk}\,\g_{ij}\nabla_s + c\,\left(\ell(f)^k_{ij} -\frac{1}{n} \, \mathrm{Sym}_{i,j}\,\delta_{i}^k \,\ell(f)_{j}\right), \label{MultiSchwar1}$$ where $$\label{qua} c=2-\delta n,$$ and $\ell(f)_{ij}^k$ are the components of the tensor (\[ell\]) \[main\] [(i)]{} For all $\d \not= 2/n,$ the map $f\mapsto {\cal A}(f^{-1})$ defines a non-trivial 1-cocycle on ${\mathrm{Diff}}(M)$ with values in ${{\mathcal{D}}}({\cal S}_{\d}^2(M), {\cal S}_\d^1(M)).$\ [(ii)]{} The operator (\[MultiSchwar1\]) does not depend on the rescaling of the metric $\g.$ In particular, if $M:={\mathbb{R}}^n$ and $\g$ is the flat metric of signature $p-q$, this operator vanishes on the conformal group ${\mathrm{O}}(p+1,q+1).$ [**Proof.**]{} To prove (i) we have to verify the 1-cocycle condition $${\cal A}(f\circ h)={h^*}^{-1} {\cal A}(f)+ {\cal A}(h), \quad \mbox{for all } f,h\in {\mathrm{Diff}}(M),$$ where $h^*$ is the natural action on ${{\mathcal{D}}}({\cal S}^2_{\d}(M),{\cal S}_\d^1(M)).$ This condition holds because the first part of the operator (\[MultiSchwar1\]) is a coboundary and the second part is a 1-cocycle. Let us proof that this 1-cocycle is not trivial for $\d\not=2/n$. Suppose that there is a first-order differential operator $A^k_{ij}=u^{sk}_{ij}\nabla_s+v^k_{ij}$ such that $$\label{cn} {\cal A}(f)={f^*}^{-1} A-A.$$ From (\[cn\]), it is easy to see that ${f^*}^{-1} v^k_{ij}-v^k_{ij}= (2-\delta n)\left(\ell(f)^k_{ij} -\frac{1}{n} \, \mathrm{Sym}_{i,j}\,\delta_{i}^k \,\ell(f)_{j}\right)$. The right-hand side of this equation depends on the second jet of the diffeomorphism $f,$ while the left-hand side depends on the first jet of $f,$ which is absurd. For $\d=2/n,$ one can easily see that the 1-cocycle (\[MultiSchwar1\]) is a coboundary. Let us prove (ii). Consider a metric $\tilde \g= F\cdot\g,$ where $F$ is a non-zero positive function. Denote by $\tilde {\cal A}(f)$ the operator (\[MultiSchwar1\]) written with the metric $\tilde \g.$ We have to prove that $\tilde {\cal A}(f)={\cal A}(f).$ The Levi-Civita connections associated with the metrics $\g$ and $\tilde \g$ are related by $$\label{lien} \tilde \Gamma^k_{ij}=\Gamma^k_{ij}+\frac{1}{2F}\left (F_i \d^k_j +F_j \d^k_i -F_t \,\g^{tk} \g_{ij}\right),$$ where $F_i=\partial_i F.$\ We need some formul[æ]{}: denote by $\ell(f)$ the tensor (\[ell\]) written with the connection $\tilde \g,$ then we have $$\begin{aligned} \label{bebe} \tilde \nabla_k P^{ij}&=&\nabla_k P^{ij}+\frac{1}{2F}\left( \mathrm{Sym}_{i,j} P^{mi} \left(F_m \delta^j_k-F_t \,\g^{tj}\g_{km}\right) +(2-n\delta )\, P^{ij}F_k\right),\\ \tilde \ell(f)^k_{ij}&=&\ell(f)^k_{ij}+\frac{1}{2 F\circ f} \left(\mathrm{Sym}_{i,j} \stackrel{*}{F_i}\delta^k_j -\stackrel{*}{F}_t\, \stackrel{*}{\g}^{tk}\stackrel{*}{\g}_{ij}\right) -\frac{1}{2F}\left( \mathrm{Sym}_{i,j} \,F_i\, \delta^k_j -F_t\, \g^{tk}\g_{ij}\right), \nonumber\end{aligned}$$ where $\stackrel{*}{F}_i={f^{*}}^{-1}F_i$ and $\stackrel{*}{\g}_{ij}= {f^{*}}^{-1}\g_{ij}$ for all $P^{ij}\xi_i \xi_j\in {\cal S}_{\d}^2 (M).$\ By substituting the formul[æ]{} (\[bebe\]) into (\[MultiSchwar1\]) we get $${\cal A} (f)_{ij}^k=\tilde {\cal A} (f)_{ij}^k+ \frac{1}{2 F} (\delta n +c -2 ) \,{\g}^{sk} \,\g_{ij}\, F_s\, + \frac{1} {2 F\circ f}(2-c -\delta n) \stackrel{*}{\g}^{tk} \stackrel{*}{\g}_{ij}\, \stackrel{*}{F}_t \, \, \cdot$$ We see that ${\cal A} (f)=\tilde {\cal A} (f)$ if and only if $c=2-\delta n .$ Let us prove that the operator (\[MultiSchwar1\]) vanishes on the conformal group ${\mathrm{O}}(p+1,q+1)$ in the case when $M:={\mathbb{R}}^n$ is endowed with the flat metric $\g_0:=\mathrm{diag}(1,\ldots,1,-1,\ldots,-1)$ whose trace is $p-q.$ Any $f\in {\mathrm{O}}(p+1,q+1)$ satisfies ${{f^*}}^{-1} {\g_0}=F\cdot{\g_0},$ where $F$ is a non-zero positive function. This relation implies $$\begin{aligned} 2 \ell(f)^k_{ij} +\mathrm{Sym}_{i,j}\,\ell(f)^s_{it}\,\g_0^{tk}\, {\g_0}_{sj}&=&\frac{1}{F}\mathrm{Sym}_{i,j}\, \partial_i F\, \delta^k_j, \nonumber \\ \mathrm{Sym}_{i,j}\, \ell(f)^s_{li} \, {\g_0}_{sj}&=&\frac{\partial_l F}{F} {\g_0}_{ij},\nonumber \\ \ell(f)_t&=& \frac{n}{2}\frac{\partial_t F}{F}\cdot \nonumber\end{aligned}$$ Sibstitute these formul[æ]{} into (\[MultiSchwar1\]). Then we obtain by straightforward computation that ${\cal A}(f)\equiv 0.$\ ------------------------------------------------------------------------ Suppose now that ${\rm dim }(M)>2.$ Our second main definition is the linear differential operator ${\cal B}(f)$ acting from ${\cal S}^2_{\d}(M)$ to ${\cal S}^0_\d(M)$ defined by $$\begin{aligned} {\cal B}(f)_{ij}&=& {f^*}^{-1} \left(\g^{st}\, \g_{ij}\nabla_s\nabla_t \right ) -\g^{st}\,\g_{ij} \nabla_s \nabla_t +c_1\,\left( \ell(f)_{ij}^s -\frac{1}{n}\, \mathrm{Sym}_{i,j}\,\delta_{i}^s \,\ell(f)_{j} \right)\nabla_s \nonumber \\ &&+\,c_2 \, \ell(f)_i\,\ell(f)_j +c_3 \n_{s} \left ( \ell(f)_{ij}^s-\frac{1}{n} \,\mathrm{Sym}_{i,j}\, \d_{i}^s \ell_{j}(f)\right) +c_4\, \ell(f)_{ij}^s\ell(f)_s \nonumber \\ && +\,c_5\, \ell(f)_{si}^u\ell(f)_{uj}^s + c_6\left( {f^{-1}}^*(R\,\g_{ij})-R\,\g_{ij}\right ), \label{MultiSchwar2}\end{aligned}$$ where $\ell(f)$ is the tensor (\[ell\]), $R$ is the scalar curvature of the metric $\g,$ and the constants $c_1,\ldots,c_6,$ are given by $$\begin{array}{ll} c_1=2+n(1-2\delta ), & \displaystyle c_2=\frac{(2+n(1-2\delta ))(\d- 1)}{n}, \nonumber \\[3mm] c_3=\displaystyle \frac{(2+n(1-2\delta ))(\d n-2)}{n-2} , &c_4= \displaystyle \frac{(2+n(1-2\delta ))(2\d-2)}{n-2}, \\[3mm] c_5= \displaystyle \frac{(2+n(1-2\delta ))(1-\d)n}{n-2},& c_6= \displaystyle \frac{n(\d -1)(n\d -2)}{(n-1)(n-2)}\cdot \nonumber \label{ga} \end{array}$$ \[mainp\] [(i)]{} For all $\d \not= \frac{n+2}{2n},$ the map $f\mapsto {\cal B}(f^{-1})$ defines a non-trivial 1-cocycle on ${\mathrm{Diff}}(M)$ with values in ${{\mathcal{D}}}({\cal S}^2_{\d}(M),{\cal S}^0_\d(M))$. [(ii)]{} The operator (\[MultiSchwar2\]) does not depend on the rescaling of the metric $\g.$ In the flat case, this operator vanishes on the conformal group ${\mathrm{O}}(p+1,q+1).$ [**Proof.**]{} To prove that the map $f\mapsto {\cal B}_{ij}(f^{-1})$ is a 1-cocycle, one has to verify the 1-cocycle condition $${\cal B}(f\circ h)= {h^*}^{-1}{\cal B}(f)+{\cal B}(h), \quad \mbox{for all }f,h\in {\mathrm{Diff}}(M),$$ where $h^*$ is the natural action on ${{\mathcal{D}}}({\cal S}^2_{\d}(M),{\cal S}_\d^0(M)).$ To do this, we use the formul[æ]{} $$\begin{aligned} \label{comp} \n_i\, f^*_{\d} P^{kl}&=&f^*_{\d}\n_i P^{kl}-\mathrm{Sym}_{k,l} \left( \ell(f^{-1})_{it}^k\, f^*_{\d} \, P^{tl}\right) +\d \,\ell(f^{-1})_i\, f^*_{\d} P^{kl},\\[2mm] \n_u h^* \ell(f)_{ij}^k&=&h^*\n_u \ell(f)_{ij}^k - h^*\ell(f)_{ij}^t\, \ell(h^{-1})_{ut}^k + \mathrm{Sym}_{i,j}\left( h^*\ell(f)_{it}^k \, \ell(h^{-1})_{ju}^t \right), \nonumber \end{aligned}$$ for all $f,h \in {\mathrm{Diff}}(M)$ and for all $P^{kl}\xi_k\xi_l\in {{\mathcal{S}}}_{\d}^2(M).$\ Let us prove that this 1-cocycle is not trivial. Suppose that there exists an operator $B_{ij}:= u_{ij}^{st}\nabla_s \nabla_t+ v_{ij}^s\nabla_s +t_{ij}$ such that $${\cal B}(f)={f^*}^{-1} B -B.$$ It is easy to see that ${f^*}^{-1} v_{ij}^s- v_{ij}^s= (2+n(1-2\delta))\left( \ell(f)_{ij}^s -\frac{1}{n}\, Sym_{i,j}\,\delta_{i}^s \,\ell(f)_{j} \right).$ The right-hand side of this relation depends on the second jet of $f,$ while the the left-hand side depends on the first jet of $f,$ which is absurd. For $\d= \frac{n+2}{2n},$ the 1-cocycle (\[MultiSchwar2\]) is trivial: ${\cal B}(f)_{ij}={f^*}^{-1} (\g_{ij}B) -B\,\g_{ij},$ where $B:= \g^{st}\nabla_s\nabla_t -\frac{1}{4}\frac{n-2}{n-1}R$ is the so-called Yamabe-Laplace operator (see, e.g., [@besse]).\ Let us prove (ii). Consider a metric $\tilde \g:=F\cdot \g,$ where $F$ is a non-zero positive function. Denote by $\tilde {\cal B}(f)$ the operator (\[MultiSchwar2\]) written with the metric $\tilde \g.$ We have to prove that $\tilde {\cal B}(f)={\cal B}(f).$\ The proof is similar to the proof of part (ii) of Theorem (\[main\]), by means of the equation (\[lien\]), (\[bebe\]) and $$\begin{aligned} \tilde \nabla_l \tilde \nabla_k P^{ij}&=&\nabla_l \tilde \nabla_k P^{ij}+ \frac{1}{2F} \left( (1-\d n) F_l\,\tilde\nabla_k P^{ij}- F_k\, \tilde \nabla_l P^{ij}+\g^{st}\, \g_{lk}\, F_s \tilde \nabla_t P^{ij}\right ) \nonumber \\ && +\frac{1}{2F} \mathrm{Sym}_{i,j} \tilde \nabla_k P^{mi} \left (F_m \, \delta^j_l - {\g}^{sj}{\g}_{ml}\, F_s\right),\nonumber\\ \tilde \nabla_l \, \tilde \ell(f)^k_{ij}&=&\nabla_l \,\tilde \ell(f)^k_{ij} - \frac{1}{2F}\, F_l\,\,\tilde \ell(f)^k_{ij}+\frac{1}{2F} \left( F_t \, \delta^k_l -\g^{sk}\, \g_{tl}\, F_s \right ) \tilde \ell(f)^t_{ij} \nonumber \\ &&-\frac{1}{2F} \mathrm{Sym}_{i,j} \left (F_i \delta^s_l-F_m \,{\g}^{ms} {\g}_{il}\right)\tilde \ell(f)^k_{js}\nonumber\\ \tilde R&=& \frac{1}{F} \left ( R-(n-1)\frac{1}{F} \left( \g^{ij}\nabla_i F_j+ (n-6)\frac{1}{4 \, F}\g^{ij}F_i F_j \right) \right) \end{aligned}$$ for all $P^{ij}\xi_i\xi_j\in {\cal S}_\d^2(M),$ where $\tilde \nabla,$ $\tilde \ell(f)$ and $\tilde R$ are, the covariant derivative, the tensor (\[ell\]), and the scalar curvature associated with the metric $\tilde \g,$ respectively. Cohomology of ${\mathrm{Vect}}(M)$ and Schwarzian derivatives ------------------------------------------------------------- We will give here the infinitesimal 1-cocycle associated with the 1-cocycles ${\cal A}$ and ${\cal B}$. First, let us recall the notion of a Lie derivative of a connection. For each $X \in {\mathrm{Vect}}(M),$ the Lie derivative $$\label{mou} L_X \nabla :=(Y,Z)\mapsto [X,\n_YZ]-\n_{[X,Y]} Z- \n_{Y} [X,Z]$$ of $\n$ is a well-known symmetric $(2,1)$-tensor field. The map $$X\mapsto L_X {\nabla}$$ defines a 1-cocycle on ${\mathrm{Vect}}(M)$ with values in the space of symmetric $(2,1)$-tensor fields on $M.$\ The linear differential operator ${\mathfrak a}$ defined by $${\mathfrak a}^k_{ij}(X):=L_X \left (\g^{sk}\, \g_{ij}\, \nabla_s\right) +c \left ((L_X \nabla)^k_{ij} -\frac{1}{n}\mathrm{Sym}_{i,j}\delta_j^k \, (L_X \nabla)_i\right),$$ where the constant $c$ is as in (\[qua\]) and $L_X \n$ is the tensor (\[mou\]), acts from ${{\mathcal{S}}}_{\d}^2(M)$ to ${{\mathcal{S}}}_{\d}^1(M).$ The linear differential operator ${\mathfrak b}$ defined by $$\begin{aligned} \label{chr} {\mathfrak b}_{ij}(X)&:=&L_X \left (\g^{st}\, \g_{ij}\, \nabla_s\nabla_t \right) +c_1 \left ((L_X \nabla)^k_{ij} -\frac{1}{n}\mathrm{Sym}_{i,j}\delta_j^k \, (L_X \nabla_i)\right) \nabla_k\nonumber\\ && +c_2 \nabla_k \left ( (L_X \nabla)^k_{ij} -\frac{1}{n}\mathrm{Sym}_{i,j}\delta_j^k \, (L_X \nabla)_i \right ) +c_6 L_X \left ( R \, \g_{ij}\right ),\end{aligned}$$ where the constants $c_1,c_2$ and $c_6$ are as in (\[ga\]) and $L_X(\n)$ is the tensor (\[mou\]), acts from ${{\mathcal{S}}}_{\d}^2(M)$ to ${{\mathcal{S}}}_{\d}^0(M).$ The following two propositions follow by straightforward computation. [(i)]{} The map $X\mapsto {\mathfrak a}^k_{ij}(X)$ defines a 1-cocycle on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_{\d}^2(M), {{\mathcal{S}}}_{\d}^1(M)).$ [(ii)]{} The operator $\mathfrak a$ does not depend on the rescaling of the metric. In the flat case, it vanishes on the Lie algebra $\mathrm{o}(p+1,q+1),$ where $p+q=n.$ [(i)]{} The map $X\mapsto {\mathfrak b}_{ij}(X)$ defines a 1-cocycle on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_{\d}^2(M), {{\mathcal{S}}}_{\d}^0(M)).$ [(ii)]{} The operator $\mathfrak b$ does not depend on the rescaling of the metric. In the flat case, it vanishes on the Lie algebra $\mathrm{o}(p+1,q+1),$ where $p+q=n.$ In section (\[va\]), we will show that the space ${{\mathcal{D}}}_{\l,\mu}^2(M)$ can be viewed as a non-trivial deformation of the module ${\cal S}_{2,\d}(M)$ in the sense of Neijenhuis and Richardson’s theory of deformation (see also [@do; @lo1]). According to the theory of deformation, the problem of “infinitesimal” deformation is related to the cohomology group $$\label{hic} \mathrm H^1 ({\mathrm{Vect}}(M), {\mathrm{End}}({\cal S}_{2,\d}(M))\cdot$$ To compute the cohomology group (\[hic\]) we restrict the coefficients to the space of linear differential operators on ${\cal S}_{2,\d}(M),$ denoted by ${{\mathcal{D}}}({\cal S}_{2,\d}(M)).$ This space is decomposed, as a ${\mathrm{Vect}}(M)$-module, into the direct sum $${{\mathcal{D}}}({\cal S}_{2,\d}(M))=\bigoplus_{k,m=0}^{2}{{\mathcal{D}}}({\cal S}_{\d}^k(M), {\cal S}_{\d}^m(M)),$$ where ${{\mathcal{D}}}({\cal S}_{\d}^k(M), {\cal S}_{\d}^m(M))\subset {\mathrm{Hom}}({\cal S}_{\d}^k(M), {\cal S}_{\d}^m(M)).$ The relation between the Schwarzian derivative (\[clas\]) and the cohomology group above is as follows: recall that in the one dimensional case the space ${{\mathcal{S}}}^k_\d (S^1)$ is nothing but ${{\mathcal{F}}}_{\d-k}.$ In this case, the problem of deformation with respect to the Lie algebra ${\mathrm{sl}}_2({\mathbb{R}})$ is related to the cohomology group $$\label{ser} {\mathrm H}^1({\mathrm{Vect}}(S^1), {\mathrm{sl}}_2 ({\mathbb{R}}); {{\mathcal{D}}}({{\mathcal{F}}}_{\d-k}, {{\mathcal{F}}}_{\d-l})),$$ where $k,l=0,1,2.$ The cohomology group (\[ser\]) was calculated in [@bo1], it is one dimension for $k=2,\, l=0,$ and zero otherwise. The (unique) non-trivial class, for $k=2$ and$l=0,$ can be integrated to the group of diffeomorphisms ${\mathrm{Diff}}(S^1);$ it is a zero-order operator given as a multiplication by the Schwarzian derivative (\[clas\]) (see [@bo1] for more details). In the multi-dimensional case and for $\d=0,$ the first group of differential cohomology of ${\mathrm{Vect}}(M)$, with coefficients in the space ${{\mathcal{D}}}({\cal S}^k(M),{\cal S}^m(M))$ of linear differential operators from ${\cal S}^k(M)$ to ${\cal S}^m(M)$ was calculated in [@lo2]. For $n\geq2$ the result is as follows $$\label{cal} \mathrm H^1({\mathrm{Vect}}(M), {{\mathcal{D}}}({\cal S}^k(M),{\cal S}^m(M)))= \left\{ \begin{array}{ll} {\mathbb{R}}\oplus \mathrm{H}_{\mathrm{DR}}^1(M), & \mbox{if}\quad k-m=0,\\ {\mathbb{R}},& \mbox{if} \quad k-m=1,m\not=0,\\ {\mathbb{R}},& \mbox{if} \quad k-m=2,\\ 0,&\hbox{otherwise.} \end{array} \right.$$ We believe, by analogy for the one-dimensional case, that the “infinitesimal” multi-dimensional Schwarzian derivative is a cohomology class in the cohomology group above for $k-m=2.$ This class is nothing but the operator ${\mathfrak b}$ defined in (\[chr\]). Comparison with the projective case {#connex} ------------------------------------ Let $M$ be a manifold of dimension $n.$ Fix a symmetric affine connection $\Gamma$ on $M$ (here $\Gamma$ is any connection not necessarily a Levi-Civita one). Let us recall the notion of projective connection (see [@kn]). A projective connection is an equivalent class of symmetric affine connections giving the same unparameterized geodesics. Following [@kn], the symbol of the projective connection is given by the expression $$\label{connection} \Pi_{ij}^k=\G_{ij}^k-\frac{1}{n+1}\left (\delta_i^k \G_{j}+\delta_j^k \G_{i} \right ),$$ where $\Gamma_{ij}^k$ are the Christoffel symbols of the connection $\Gamma$ and $\Gamma_i=\Gamma_{ij}^j.$\ Two affine connection $\Gamma$ and $\tilde \Gamma$ are projectively equivalent if the corresponding symbols (\[connection\]) coincide.\ A projective connection on $M$ is called *flat* if in a neighborhood of each point there exists a local coordinate system $(x^1,\ldots,x^n)$ such that the symbols $\Pi_{ij}^k$ are identically zero (see [@kn] for a geometric definition). Every flat projective connection defines a projective structure on $M$.\ Let $\Pi$ and $\tilde \Pi$ be two projective connections on $M.$ The difference $\Pi-\tilde \Pi$ is a well-defined $(2,1)$-tensor field. Therefore, it is clear that a projective connection on $M$ leads to the following 1-cocycle on ${\mathrm{Diff}}(M)$: $${\cal C}(f^{-1})= \left( (f^{-1})^*\Pi_{ij}^k-\Pi_{ij}^k \right) dx^i\otimes dx^j\otimes\frac{\partial}{\partial x^k} \label{ellp}$$ This formula is independent on the choice of the coordinate system. By definition, the tensor field (\[ellp\]) depends only on the projective class of the connection $M.$ In particular if $\Pi \equiv 0,$ this tensor field vanishes on the projective group ${\mathrm{PSL}}_{n+1}({\mathbb{R}}).$ One can define a 1-cocycle on ${\mathrm{Diff}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}^{2}_\d(M), {{\mathcal{S}}}^{1}_\d(M))$ by contracting any symmetric contravariant tensor field with the tensor (\[ellp\]). Therefore, the operator (\[MultiSchwar1\]) can be viewed as the conformal analogue of the tensor field (\[ellp\]). In the same spirit, the operator (\[MultiSchwar2\]) can be viewed as the conformal analogue of the “projective” multi-dimensional Schwarzian derivative introduced in [@b2; @bo2]. Relation to the modules of differential operators {#SecComput} ================================================= Conformally equivariant quantization {#tokyo} ------------------------------------ The quantization procedure explained in this paper was first introduced in [@do; @lo1]. By an equivariant quantization we mean an identification between the space of linear differential operators and the corresponding space of symbols, equivariant with respect to the action of a (finite dimension) sub-group $G\subset {\mathrm{Diff}}({\mathbb{R}}^n).$ Recall that in the one-dimensional case the equivariant quantization process was carried out for $G={\mathrm{SL}}_2({\mathbb{R}})$ in [@cmz] (see also [@ga]). The following theorems are proven in [@do]. For all $\d \not=1,$ there exists an isomorphism $${\cal Q}_{\l,\mu}: {\cal S}_{\d}^1(M)\oplus {\cal S}_{\d}^0(M) \rightarrow {{\mathcal{D}}}_{\l,\mu}^1(M),$$ given as follows: for all $P=P^i\xi_i+P_0\in~{\cal S}_{\d}^1(M) \oplus {\cal S}_{\d}^0(M),$ one can associate a linear differential operator given by $$\begin{aligned} \label{yah1} {\cal Q}_{\l,\mu} (P)&=& P^{i}\n_i +\a \n_i P^{i}+P_0,\end{aligned}$$ where $$\alpha=\displaystyle \frac{\l}{1-\delta}\cdot$$ This map does not depend on the rescaling of the metric, intertwines the action of ${\mathrm{Diff}}(M).$ For $n>2$ and for all $\d \not=\frac{2}{n}, \frac{n+2}{2n}, \frac{n+1}{n},\frac{n+2}{n},$ there exists an isomorphism $${\cal Q}_{\l,\mu}: {\cal S}_{\d}^2(M) \rightarrow {{\mathcal{D}}}_{\l,\mu}^2(M),$$ given as follows: for all $P=P^{ij}\xi_i\xi_j \in~{\cal S}_{\d}^2(M),$ one can associate a linear differential operator given by $$\begin{aligned} {\cal Q}_{\l,\mu} (P)&=& P^{ij}\n_i\n_j \nonumber\\[2mm] &&+(\a_1 \n_i P^{ij}+\a_2\,\g^{ij}\, \g_{kl}\n_i P^{kl}) \n_j \label{Tensor1}\\[2mm] &&+\a_3\n_i\n_j P^{ij}+\a_4\, \g^{st}\, \g_{ij}\n_s\n_t P^{ij} +\a_5 R_{ij}P^{ij}+\a_6 R \,\g_{ij}\,P^{ij}, \nonumber\end{aligned}$$ where $R_{ij}$ (resp. $R$) are the Ricci tensor components (resp. the scalar curvature) of the metric $\g,$ the constants $\a_1,\ldots,\a_6$ are given by $$\begin{array}{ll} \a_1=\displaystyle \frac{2(n\l+1)} {2+n(1-\d)}, &\a_2= \displaystyle \frac{n(\l+\mu-1)} {(2+n(1-\delta )) (2-n\d)},\\[3mm] \a_3= \displaystyle \frac{n\l(n\lambda+ 1)} {(1+n(1-\delta))(2+n(1-\delta ))}, & \a_5= \displaystyle \frac{n^2\l(\mu-1)} {(n-2)(1+n(1-\delta))} ,\\[3mm] \a_4= \displaystyle \frac{n\l(n^2 \mu (2-\l-\mu)+2(n\l +1)^2-n(n+1))} {(1+n(1-\delta))(2+n(1-\d))(2+n(1-2\d))(2-n\delta )}, & \a_6= \displaystyle \frac{(n\d-2)} {(n-1)(2+n(1-2\delta))}\, \a_5 \cdot\\[3mm] &\\ \end{array}$$ and has the following properties:\ (i) It does not depend on the rescaling of the metric $\g$.\ (ii) If $M={\mathbb{R}}^n$ is endowed with a flat conformal structure, this map is unique, equivariant with respect to the action of the group ${\mathrm{O}}(p+1,q+1) \subset {\mathrm{Diff}}({\mathbb{R}}^n)$. Before to give the formula of the conformal equivariant map in the case of surfaces, let us recall an interesting approach for the multi-dimensional Schwarzian derivative for conformal mapping [@os] (see also [@ca]). First, recall that all surfaces are conformally flat. This means that every metric can be express (locally) as $$\g=F^{-1}\psi^* \g_0,$$ where $\psi$ is a conformal diffeomorphism of $M$, $F$ is a non-zero positive function and $\g_0$ is a metric of constant curvature. The Schwarzian derivative of $\psi$ is defined in [@os] as the following tensor field $$\label{osg} S(\psi )=\frac{1}{2F}\nabla dF-\frac{3}{4F^2}\,dF\otimes dF+\frac{1}{8F^2} \g^{-1}(dF,dF)\, \g \cdot$$ Now we are in position to give the quantization map for the case of surfaces. For $\d \not=1,2,\frac{3}{2},\frac{5}{2},$ and for each $P=P^{ij}\xi_i\xi_j +P^i\xi_i +P_0\in {\cal S}_{\d,2}(M)$ one associates a linear differential operator given by $$\begin{aligned} Q(P)&=& P^{ij}\n_i\n_j \nonumber\\[2mm] &&+(\a_1 \n_i P^{ij}+\a_2\,\g^{ij}\, \g_{kl}\n_i P^{kl}) \n_j \label{surf}\\[2mm] &&+\a_3\n_i\n_j P^{ij}+\a_4\, \g^{st}\, \g_{ij}\n_s\n_t P^{ij} \nonumber\\[2mm] && +\frac{4\l(\mu-1)}{2\d -3}\left( S(\psi)_{ij}P^{ij}+ \frac{1}{8(\d-1)} R \,\g_{ij}\,P^{ij}\right )+P_0, \nonumber\end{aligned}$$ where $S(\psi)$ is the tensor (\[osg\]), $R$ is the scalar curvature and the coefficients $\a_1,\ldots,\a_4$ are given as above. [ The projectively equivariant quantization map was given in [@lo1] (see also [@b3] for the non-flat case). The multi-dimensional projective Schwarzian derivative is defined as an obstruction to extend this isomorphism to the full group ${\mathrm{Diff}}(M).$ We will show in the next section that the conformal Schwarzian derivatives defined in this paper appear as obstructions to extend the isomorphisms (\[Tensor1\]), (\[surf\]) to the full group ${\mathrm{Diff}}(M).$ ]{} Deformation of the space of symbols ${\cal S}_{2,\d}$ {#va} ----------------------------------------------------- The goal of this section is to explicate the relation between the 1-cocycles (\[MultiSchwar1\]), (\[MultiSchwar2\]) and the space of second-order linear differential operators ${\cal D}_{\l,\mu}^2(M).$ Since the space ${\cal D}_{\l,\mu}^2(M)$ is a non-trivial deformation of the space of the corresponding space of symbols ${\cal S}_{\d,2}(M),$ where $\d=\mu-\l$, it is interesting to give explicitly this deformation in term of the 1-cocycles (\[MultiSchwar1\]), (\[MultiSchwar2\]). Namely, we are looking for the operator $\bar f_\d={\cal Q}^{-1}_{\l,\mu} \circ f_{\l,\mu} \circ {\cal Q}_{\l,\mu}$ such that the diagram below is commutative $$\nonumber \begin{CD} {\cal S}_{\d,2}(M) @> \bar f_\d >> {\cal S}_{\d,2}(M) \strut\\ @V{\cal Q}_{\l,\mu}VV @VV{\cal Q}_{\l,\mu}V \strut\\ {{\mathcal{D}}}_{\l,\mu}^2 (M) @> f_{\l,\mu} >> {{\mathcal{D}}}_{\l,\mu}^2 (M) \strut \end{CD} \label{TheDiagram}$$ For all $\d \not=\frac{2}{n},\frac{n+2}{2n},1,\frac{n+1}{n}, \frac{n+2}{n},$ the deformation of the space of symbols ${\cal S}_{\d,2}(M)$ by the space ${{\mathcal{D}}}_{\l,\mu}^2(M)$ as a ${\mathrm{Diff}}(M)$-module is given as follows: for all $P=P^{ij}\xi_i \xi_j +P^i\xi_i + P_0\in {\cal S}_{\d,2}(M),$ one has $$\bar f_{\d}\cdot (P )= T^{ij}\xi_i\xi_j+T^{i}\xi_i+T^{0},$$ where $$\begin{aligned} T^{ij}&=& (f_{\d}\,P)^{ij}, \nonumber\\\label{ExplAction} T^i&=& (f_{\d}\,P)^i\;+\; \displaystyle \frac{n(\mu+\l -1)}{(2+n(1-\d ))(2-n\delta)}\,{\cal A}^i_{kl}(f^{-1}) (f_{\d}\,P)^{kl},\\[2mm] \displaystyle T^0&=& (f_\d\, P)_0 \;-\; \displaystyle \frac{n \l(\mu -1)}{(2+n(1-2\d))(1-\d )(1+n(1-\d))}\,{\cal B}_{kl}(f^{-1}) (f_{\d}\, P)^{kl},\nonumber\end{aligned}$$ and $f_\d$ is the action (\[actsym\]). \[MainAct\] [**Proof.**]{} The proof is a simple computation using (\[comp\]) and the formul[æ]{} $$\begin{aligned} \n_i\n_j \,{f^*_{\d}}^{-1} P^{kl}&=&{f^*_{\d}}^{-1} \n_i \n_j\, P^{kl} - \mathrm{Sym}_{l,k}\left ( {f^*_{\d}}^{-1} \n_i\, P^{tl} \;\ell(f)_{tj}^k \right) + {f^*_{\d}}^{-1} \n_u\, P^{kl} \; \ell(f)_{ij}^u \nonumber \\ && +\d \left ({f^*_{\d}}^{-1} \n_i\, P^{kl} \;\ell(f)_{j}\right) -\mathrm{Sym}_{k,l} \left( \n_j \ell(f)^k_{it}\; {f^*_{\d}}^{-1} P^{tl}\right) \\ &&+\d \n_j \ell(f)_i \; {f^*_{\d}}^{-1} P^{kl} -\mathrm{Sym}_{k,l}\left( \ell(f)_{it}^k \n_j \, {f^*_{\d}}^{-1} P^{tl}\right) +\d \ell(f)_i\,\n_j {f^*_{\d}}^{-1}P^{kl} \nonumber\\[2mm] \n_{i} \, f^* \phi&=& f^* \n_i \phi +\l \,\ell(f^{-1})_{i}\, f^* \phi \nonumber\\[2mm] \n_j \n_{i} \, f^* \phi&=& f^* \n_j \n_i \phi +\ell(f^{-1})_{ji}^t \, f^* \n_t \phi +\l \mathrm{Sym}_{i,j} \, \ell(f^{-1})_j\, f^* \n_i \phi \nonumber\\[2mm] && +\left (\l \n_j\, \ell(f^{-1})_{i} + \l^2 \ell(f^{-1})_{j}\, \ell(f^{-1})_{i}\right) \, f^* \phi \nonumber\\[2mm] f^* R_{jk}-R_{jk}&=& \n_i\, \ell(f^{-1})^i_{jk}-\n_j \, \ell(f^{-1})_k - \ell(f^{-1})^m_{sj}\, \ell(f^{-1})^s_{km}+ \ell(f^{-1})_m \, \ell(f^{-1})^m_{jk},\nonumber\end{aligned}$$ for all $\phi\in {{\mathcal{F}}}_\l$ and for all $f\in {\mathrm{Diff}}(M),$ where $R_{ij}$ are the Ricci tensor components.\ ------------------------------------------------------------------------ The 1-cocycle ${\cal A},$ ${\cal B}$ and the conformally invariant quantization {#new} ------------------------------------------------------------------------------- We will show in this section that the quantization procedure is not invariant if one consider two metrics conformally equivalent. The obstruction of the invariance is given by the 1-coycles ${\cal A}$ and ${\cal B}.$ Given two conformally equivalent metrics $\g,$ $\tilde \g.$ Denote by $\n$, $\cal A$ and $\cal B,$ the covariant derivative, the 1-cocycles (\[MultiSchwar1\]) and (\[MultiSchwar2\]) written with the metric $\g,$ respectively. We have The quantization map ${\cal Q}_{\l,\mu}^{\g}:$ ${\cal D}^1_{\l,\mu}(M)\rightarrow {\cal S}^1_\d(M)\oplus {\cal S}^0_\d (M)$ defined in (\[yah1\]) depend only on the conformal class of the metric $\g.$ [**Proof.**]{} Let $\tilde \g$ be another metric conformally equivalent to $\g. $ That means that there exists a diffeomorphism $\psi : (M,\tilde \g )\rightarrow (M,\g)$ and a non-zero positive function $F$ such that (locally) $$\tilde \g=F^{-1}\cdot \psi^* \g.$$ The Levi-Civita of the two connections are related by $${\Gamma^k_{ij}}^{ \g}= {\Gamma^k_{ij}}^{\tilde \g}+ \frac{1}{2 F} \left (\partial_i F\, \d_j^k +\partial_j F\, \d_i^k-\partial_s F\, \tilde \g^{sk}\tilde \g_{ij}\right ) - \ell(\psi^{-1})^k_{ij},$$ where $\partial_i F=F_i$ and $\ell(\psi^{-1})^k_{ij}$ are the components of the tensor (\[ell\]). This equation implies that $$\begin{array}{lcl} \n^{\g}_i\phi&=&\n^{\tilde \g}_i\phi-\displaystyle \frac{\l n}{2} \displaystyle \frac{F_i}{F}+ \l \,\ell(\psi^{-1})_i \, \phi,\\[2mm] \n^{\g}_i P^i &=&\n^{\tilde \g}_i P^i+\displaystyle \frac{n(1-\d)}{2}\frac{F_i P^i}{F}-(1-\d) \ell(\psi^{-1})_{i}P^i, \end{array}$$ for all $\phi\in {{\mathcal{F}}}_\l$ and for all $P^i\xi_i\in {{\mathcal{S}}}_{\d}^1(M).$\ Substitute these formul[æ]{} into (\[yah1\]) we see that ${\cal Q}^{\tilde \g}_{\l,\mu}={\cal Q}^{\g}_{\l,\mu}.$ The quantization map ${\cal Q}_{\l,\mu}^{\g}:$ ${\cal D}^2_{\l,\mu}(M)\rightarrow {\cal S}^2_\d(M)$ defined in (\[Tensor1\]) has the property $${\cal Q}_{\l,\mu}^{\tilde \g} (P)={\cal Q}_{\l,\mu}^{\g} (P)+d_1 \, {\cal A}^{s}(\psi^{-1}) (P )\nabla_s + d_2 \,\nabla_s \left ( {\cal A}^{s}(\psi^{-1}) (P) \right ) +d_3 \, {\cal B}(\psi^{-1})(P),$$ for all $P\in {\cal S}_{\d}^2(M),$ where the constants $d_1,d_2$ and $d_3$ are given by $$\begin{aligned} d_1&=&\frac{n(\l+\mu-1)}{(2+n(1-\d))(2-n\d))}, \quad \displaystyle d_2=\frac{n\l(\l+\mu-1)}{(2+n(1-\d))(2-n\d)(1-\d)}, \nonumber \\[1mm] d_3&=&\displaystyle \frac{n \l(\mu -1)}{(2+n(1-2\d))(\d -1)(1+n(1-\d))} \cdot \nonumber \end{aligned}$$ [**Proof.**]{} The proof involves the calculation of $\nabla_i ^{\tilde \g}\nabla_j ^{\tilde \g} \phi,$ $\nabla_i ^{\tilde \g}\nabla_j ^{\tilde \g} P^{kl}$ and $R^{\tilde \g}$ wich is straightforward but quite complicated. [ The system $d_1=d_2=d_3=0$ admits a unique solution: $(\l,\mu)=(0,1).$ The value of $\d=1$ is called “resonant”. In this case, the quantization map is not unique; there exists a one-parameter family of such isomorphism (see [@do] for more dtails.) ]{} For all $f\in {\mathrm{Diff}}(M)$ and for all conformal map $\psi: (M,\tilde \g) \rightarrow (M,\g)$, one has [(i)]{} ${\cal A}^{\tilde \g}(f)=\psi^* {\cal A}^{\g} (\psi \circ f\circ \psi^{-1}),$ [(ii)]{} ${\cal B}^{\tilde \g}(f)=\psi^* {\cal B}^{\g} (\psi \circ f\circ \psi^{-1}).$ [**Proof.**]{} Straightforward computation. For all conformal map $\psi (M,\tilde \g) \rightarrow (M,\g)$ one has [(i)]{} $\tilde {\cal A}^{\tilde \g}(\psi)=-{\cal A}^{\g}(\psi^{-1} ),$ [(ii)]{} $\tilde {\cal B}^{\tilde \g}(\psi)=-{\cal B}^{\g}(\psi^{-1} )\cdot$ [ The Corollary above shows that for a conformal map $\psi,$ the 1-cocycle $\cal B$ is still a second-order differential operator and then does not coincide with the Schwarzian derivative (\[osg\]) defined by Osgood and Stow. ]{} Appendix ======== We will give a formula for the Schwarzian derivative for the case of surfaces. As explained in section (\[tokyo\]), all surfaces are conformally flat. That means that every metric can be express (locally) as $$\g=F^{-1}\psi^* \g_0,$$ where $\psi$ is a conformal diffeomorphism of $M$, and $F$ is a non-zero positive function, $\g_0$ is a metric of constant curvature. The explicit formula of the Schwarzian derivative in the case of surfaces is: the following $$\begin{aligned} {\cal B}'_2(f)_{ij}&=& {f^*}^{-1} \left( {\g}^{st}\, {\g}_{ij}\nabla_s\nabla_t \right ) -\g^{st}\,\g_{ij}\nabla_s \nabla_t +4(1-\delta)\left( \ell(f)_{ij}^s -\frac{1}{2}\,Sym_{i,j}\, \delta_{i}^s \,\ell(f)_{j} \right)\nabla_s \nonumber \\[2mm] &&+4 (1-\d)^2 \ell(f)_s \left ( \ell(f)_{ij}^s -\frac{1}{4} Sym_{i,j}\, \delta_i^s \, \ell(f)_j\right) + 2(\d-2)(1-\d) \,Sym_{i,j}\,\n_{j}\ell_{i}(f) \nonumber\\[2mm] &&+8\,(\d-1)^2 \left( {f^{-1}}^*(S(\psi)_{ij})-S(\psi)_{ij}+\frac{1}{2} \n_{s}\ell(f)_{ij}^s \right )+(\d-1) \left ({f^{-1}}^*(R\,\g_{ij})-R\,\g_{ij}\right ), \nonumber \label{MultiSchwar3}\end{aligned}$$ where $S(\psi)$ is the derivative (\[osg\]), $\ell(f)$ is the tensor (\[ell\]), $R$ is the scalar curvature of the metric $\g,$ is a differential operator from from ${{\mathcal{S}}}_{\d}^2(M)$ to ${{\mathcal{S}}}_{\d}^0(M).$ Theorem (\[mainp\]) remains true for this operator. 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[^1]: Research supported by the Japan Society for the Promotion of Science.

--- abstract: 'During the last five years, serial femtosecond crystallography using x-ray laser pulses has developed into a powerful technique for determining the atomic structures of protein molecules from micrometer and sub-micrometer sized crystals. One of the key reasons for this success is the “self-gating" pulse effect, whereby the x-ray laser pulses do not need to outrun all radiation damage processes. Instead, x-ray induced damage terminates the Bragg diffraction prior to the pulse completing its passage through the sample, as if the Bragg diffraction was generated by a shorter pulse of equal intensity. As a result, serial femtosecond crystallography does not need to be performed with pulses as short as 5–10 fs, as once thought, but can succeed for pulses 50–100 fs in duration. We show here that a similar gating effect applies to single molecule diffraction with respect to spatially uncorrelated damage processes like ionization and ion diffusion. The effect is clearly seen in calculations of the diffraction contrast, by calculating the diffraction of average structure separately to the diffraction from statistical fluctuations of the structure due to damage (“damage noise"). Our results suggest that sub-nanometer single molecule imaging with 30–50 fs pulses, like those produced at currently operating facilities, should not yet be ruled out. The theory we present opens up new experimental avenues to measure the impact of damage on single particle diffraction, which is needed to test damage models and to identify optimal imaging conditions.' author: - 'Andrew V. Martin' - 'Justine K. Corso' - Carl Caleman - Nicusor Timneanu - 'Harry M. Quiney' bibliography: - 'damage\_effects.bib' title: ' Single molecule imaging with longer x-ray laser pulses ' --- -5mm -5mm Introduction ============ X-ray free-electron laser (XFEL) pulses are envisioned to probe the structures of radiation-sensitive samples, like biological molecules, by outrunning radiation damage processes[@Neutze2000]. Current facilities, however, produce their brightest pulses with durations of the order of tens of femtoseconds[@LCLS; @Ishikawa2012], which is sufficient time for ionization to become widespread and for ions to move several [Å]{}ngstr[ö]{}ms[@Caleman2009; @Caleman2011]. In spite of this, the first applications of XFELs to serial crystallography have been highly successful[@Chapman2011; @Boutet2012]. It turns out that even for longer pulses ($\sim$ 50–100 fs), Bragg diffraction probes the undamaged structure in the first few femtoseconds of the pulse-sample interaction, turning off at later times when radiation damage distributes the diffraction signal as a diffuse background[@Barty2012]. In this way, XFEL Bragg diffraction is effectively gated by damage because expected number of photons scattered to a Bragg peak is equivalent to that produced by a shorter pulse with the same intensity. Despite the great progress in coherent imaging using XFEL sources, the holy grail - atomic resolution of a single (non-crystalline) biomolecule [@Neutze2000] - has not yet been realized. Nevertheless, the potential reward for success has kept this pursuit at the forefront of research in XFEL imaging science. One of the limiting factors is radiation damage. For non-crystalline samples, diffraction from the undamaged structure is not enhanced by periodicity and is mixed indistinguishably with the diffraction of a damaged structure. This is seemingly a major setback for the prospects of developing 3D single particle imaging into a high resolution technique for single molecules. For example, Hau-Riege et al.[@HauRiege2005] found that radiation damage causes large discrepancies with the ideal diffracted intensities, which led them to conclude that pulses must be no more than a few femtoseconds long to avoid severe resolution loss. A more recent study with more detailed scattering models reached a similar conclusion[@Ziaja2012]. However, these studies assessed feasibility with metrics inspired by crystallography whose suitability for single molecule imaging is disputed[@Quiney2011]. Without accounting in detail for the way that structural information is extracted from single molecule diffraction data, the issue of damage limits for single molecule imaging remains inconclusive. One of the most actively pursued routes to single molecule imaging involves measuring thousands of copies of a molecule one by one. The resulting data is extremely noisy and the molecular orientations are not known. The issue of molecular orientation must be resolved to assemble a 3D dataset, which can be performed by several algorithms [@LohEMC2009; @Fung2009; @Giannakis2012; @Kassemeyer2013]. The hallmark of these methods is that they are able to cope with signals as low as 0.01 photons per Shannon-Nyquist pixel [@Tegze2012]. After the 3D dataset has been assembled, the atomic structure is recovered via coherent diffractive imaging methods[@Marchesini2007]. The crucial information needed to resolve the unknown orientations, and finally the structure, is contained in the modulations of diffraction signal arising from interference between different atoms, often called “speckles" (see Fig. \[fig:diffraction\_and\_explosion\]). Radiation damage changes the structure of the sample dynamically such that the final diffraction pattern is the sum of the diffraction from many modified structures, each with a different distribution of ions and ion displacements. It has been shown that averaging the diffraction over different molecular configurations[@Maia2009] lowers the speckle contrast relative to the mean scattering intensity within each resolution shell. We expect radiation damage to cause a similar loss of contrast. Not only is the amplitude of the speckle structure reduced, but speckle structure also fluctuates from shot-to-shot due to damage, in addition to the fluctuations due to changing orientation and shot-noise. We will use the term “damage noise" to refer to these fluctuations of speckle structure due to damage. So far damage noise has not been considered in studies of 3D dataset assembly. Here we present calculations of damage noise per diffraction pattern due to spatially uncorrelated damage processes, which include ionization and ion diffusion but not the Coulomb explosion of the molecule. An analysis of damage noise as a function of pulse duration reveals a gating effect in single molecule diffraction, whereby long pulses measure an equivalent amount of information about the average structure to shorter pulses of the same intensity. Theoretical predictions of damage noise are also the first step to understanding how orientation determination and 3D data assembly can be performed with data affected by radiation damage. An alternative to alignment via post-processing is to experimentally align isolated gas-phase molecules, e.g. via quantum-state-selection methods [@Kupper2014; @Stern2014]. A great advantage of this approach is that multiple molecules can be illuminated simultaneously, increasing signal-to-noise and, as supported by the work here, reducing the impact of damage. These methods have been demonstrated only for small (2,5-diiodo-benzonitrile) molecules so far [@Kupper2014; @Stern2014] and extensions to larger molecules are being actively pursued. If the molecules are aligned experimentally, the self-gating effect still applies. Radiation damage modifies each molecule in the beam uniquely and stochastically, so that multiple damage scenarios are averaged in a single diffraction measurement in an analogous way to crystallography. This increases the signal with respect to damage noise as well as shot noise. The self-gating effect ensures that such benefits from using multiple aligned molecules are not lost entirely by using x-ray pulses longer than 10 fs. Once the 3D data assembly has been performed, damage will still have a residual effect on the resulting 3D diffraction volume. Damage reduces the contrast in the averaged diffraction volume[@Quiney2011], and depending on the theoretical perspective, also contributes a background[@Lorenz2012]. Promisingly the reduction in contrast can be accounted for during structure determination by treating the sample in terms of a small number of structural modes[@Quiney2011]. The background contribution is expected to be small for hard X-rays at beam conditions currently available. In addition to analysing the damage noise, we show how the mean and standard deviation of the diffraction signal can be combined into a sensitive measure of damage. An advantage of the measure we propose is its sensitivity to both ionization and ion motion, whereas the mean signal alone depends only on ionization. There is a need to measure damage experimentally and provide some validation and clarification for theoretical damage modelling. Many different types of damage models have been developed, based on rate-equations[@HauRiege2004], molecular dynamics [@Neutze2000; @Jurek2004b] or plasma theory[@Caleman2009], and each has specific advantages and disadvantages. For example, molecular dynamics models can keep track of specific ion trajectories, but are only computationally tractable for small molecules [@Neutze2000]. Rate equations models can simulate damage large molecules, but ignore information about ion motion on atomic length scales [@HauRiege2004]. Experimental measurements of damage will provide valuable feedback on our theoretical understanding of the interaction between XFEL pulses and biomolecules, which is needed to develop single molecule imaging techniques. The effect of radiation damage on diffraction contrast ====================================================== The goal of single molecule imaging is to recover the initial position **R** of each atom in the sample. For simplicity, we will give equations for the case of a single atomic species, noting that the generalization to multiple atomic species is similar to that found in Ref. [@Quiney2011]. The intensity of a single measurement of a single molecule can be written $$I(\textbf{q}) = r_e^2 P(\textbf{q}) d\Omega I_0 \left[ \sum_{i=1}^N A_{i}(q) + 2 \sum_{i=1}^N \sum_{j=1}^{i-1} B_{ij}(\textbf{q}) \right] \;, \label{eq:I}$$ where $\textbf{q}$ is the scattering vector with magnitude $q$, $d\Omega$ is the solid-angle term, $r_e$ is the classical electron radius, $N$ is the number of atoms and $P(\textbf{q})$ is a polarization term that will be ignored in this discussion. To simplify mathematical notation, we assume the incident intensity takes a uniform value $I_0$ for the duration of the pulse. We have defined $$A_{i}(q) = \int_0^T |f_i(q,t)|^2 dt \; \label{eq:Ai}$$ and $$\begin{aligned} B_{ij}(\textbf{q}) &= \int_0^T f_i(q,t) f_j(q,t) \cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j + \bm{\epsilon}_i(t) - \bm{\epsilon}_j(t))] dt \;, \label{eq:Bij}\end{aligned}$$ where $\bm{\epsilon}_i(t)$ is the displacement of the $i^{th}$ atom from its initial position and $T$ is the duration of the pulse. For a single two-dimensional measurement, it is understood that $\textbf{q}$ is sampled at points on the Ewald sphere, but in general we will use $\textbf{q}$ to be a general three-dimensional vector and $I(\textbf{q})$ is a three-dimensional function. The atomic scattering factor $f(q,t)$ depends upon the ionization state of the atom, which changes as a function of time. The ionic scattering factors can be calculated using Slater orbitals[@Slater1930] and we use $f_0(q)$ to denote the atomic scattering factor of the unionized atom. We assume that the probability of an ion having a particular ionization state at time $t$ is independent of where that atom is located in the sample. Although the ionization state as a function of time is different for each atom, statistically atoms of the same atomic species are assumed to be equivalent. We write $A(q)$ and $B(q)$ as a function of the magnitude of the scattering vector, $q$, because we assume the atomic scattering factors are spherically symmetric. Consider an ensemble of 2D diffraction measurements, each with a unique damage scenario. For 3D imaging, the data needs to be assembled into a 3D intensity volume using an algorithm that accounts for the unknown molecular orientations. The desired solution of the algorithm is an average intensity, where each 2D measurement is correctly placed according to orientation and the different damage scenarios are averaged. As shown in Appendix \[app1\], the average intensity can be written in the form $$\langle I(\textbf{q}) \rangle = r_e^2 P(\textbf{q}) d\Omega I_0 \left[ N A(q) + 2 B(q) \sum_{i=1}^N \sum_{j=1}^{i-1} \cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \right] \;, \label{eq:Iav}$$ where we have $$\langle A_{i}(q) \rangle = A(q) \equiv I_0 \int_0^T \langle |f(q,t)|^2 \rangle dt \label{eq:Aav}$$ and $$\langle B_{ij}(\textbf{q}) \rangle = B(q) \cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \;, \label{eq:Bijav}$$ where $$B(q) \equiv \int_0^T \langle f(q,t) \rangle^2 e^{-4\pi^2 q^2 \overline{\bm{\epsilon}}(t)^2} dt$$ and $\overline{\bm{\epsilon}}(t)$ is the root mean square (rms) displacement of an ion as a function of time. If the analysis is restricted to damage processes that are random and spatially uncorrelated, then we can treat the terms $A_{i}(q)$ and $B_{ij}(\textbf{q})$ as random variables and study the effect of damage statistically. We also treat the initial atomic positions $\textbf{R}_i$ as random with a uniform probability distribution, as is done in crystallography to analyse the statistics of Bragg intensities (Wilson statistics) at high scattering angles ($q > 0.33 $nm$^{-1}$) [@Huldt2003]. Both ionization and ion diffusion can be treated within this framework and, as we will show, are both involved in a self-gating pulse effect. Expansion of the molecule by Coulomb forces is not covered by the statistical treatment presented here, but is discussed further below. The second term on the right-hand side of Eq. is sensitive to the atomic positions and accounts for the contrast in the average diffraction pattern. We can treat this information as the “signal" we aim to measure. The contribution each atom makes to the signal is proportional to $B(q)$, which is equal to the standard deviation of the diffraction in the merged 3D dataset divided by the number of atoms. The mean shot noise level, denoted by $\sigma_N$, is proportional to the square root of the intensity. We can estimate the mean shot noise level by considering the mean diffracted intensity in a shell of constant $q$, which can be derived by integrating Eq. and is proportional to $A(q)$. When the signal is compared to the noise, the proportionality constants have no influence on the interpretation, so we drop them for simplicity and write $$\sigma^2_N(q) = A(q) \;. \label{eq:sigN}$$ In addition to shot noise, there is the damage noise due to the variations in how the damage manifests in each measurement. One contribution to the damage noise is the fluctuation of $A_{i}(q)$, which is characterized by the standard deviation of $A_{i}(q)$, which we denote by $\sigma_A(q)$. The second contribution to damage noise is the deviation of $B_{ij}(\textbf{q})$ from the average speckle $B(\textbf{q})$, which has a standard deviation $\sigma_B(q)$. The term $\sigma_B(q)$ is given by the difference between the standard deviation of the second term on the right-hand side of Eq minus the standard deviation of the second term on the right-hand side of Eq . In Appendices \[sec:app:sigA\] and \[sec:app:sigB\], we provide derivations of $\sigma_A(q)$ and $\sigma_B(q)$ that give the following results: $$\begin{aligned} \sigma^2_A(q) = \int^T_0 \int^T_0 \left[ \langle f^2(q,t) f^2(q,t') \rangle - \langle f^2(q,t)\rangle \langle f^2(q,t') \rangle \right] dt dt' \label{eq:sigA}\end{aligned}$$ and $$\begin{aligned} \sigma^2_B(q) = \int^T_0 \int^T_0 \Big[ &\langle f(q,t) f(q,t') \rangle^2 e^{-4\pi^2 q^2 |\overline{\bm{\epsilon}}^2(t, t')|} \nonumber \\ &\qquad - \langle f(q,t) \rangle^2 \langle f(q,t') \rangle^2 e^{-4\pi^2 q^2 \overline{\bm{\epsilon}}(t)^2} e^{-4\pi^2 q^2 \overline{\bm{\epsilon}}(t')^2} \Big] dt dt' \;. \label{eq:sigB}\end{aligned}$$ By comparing the size of the signal to the size of the shot-noise and damage-noise levels, we can gauge how much information is contained by each measurement about the molecule’s structure. Here we will study how the diffraction pattern varies as a function of pulse duration and pulse energy. We propose the following signal-to-noise ratio to characterize the diffraction: $$\begin{aligned} SNR_{ND}(q) = \frac{N B(q)}{\sqrt{N \sigma^2_A(q) + N^2 \sigma^2_B(q) + N \sigma^2_N(q)}} \;. \label{eq:SNR}\end{aligned}$$ It is also interesting to compare the signal to the damage noise directly, ignoring shot-noise, with the following ratio: $$\begin{aligned} SNR_D(q) = \frac{N B(q)}{\sqrt{N\sigma^2_A(q) + N^2\sigma^2_B(q) }} \;. \label{eq:SNRD}\end{aligned}$$ To estimate $SNR_{ND}(q)$ and $SNR_D(q)$, we need to calculate statistical averages of the scattering factor, e.g. $\langle f(q,t) \rangle$, $\langle f^2(q,t)\rangle$ etc, which in turn depend on the expected number of ions in each ionization state as a function of time. To calculate $B(q)$ and $\sigma_B(q)$ we also need to know the ion temperature as a function of time. These parameters can be calculated by many of the damage models reported in the literature so far, like molecular dynamics models [@Neutze2000; @Jurek2004b] and hydrodynamic (rate-equations) models[@HauRiege2004; @Scott2001; @Caleman2011]. Here we will present the results of a rate equations model to investigate single-molecule diffraction contrast and to explore the extent to which there is a self-gating pulse effect in single molecule diffraction. The term that has not been calculated before is the correlation between the scattering factor at different time points, e.g. $\langle f(q,t) f(q,t')\rangle$, which is needed to calculate the damage noise levels. To calculate these correlations we need to know the condition probability $P( f_n(q,t') | f_m(q,t) )$, which gives the probability of an ion being found in ionization state $n$ at time $t'$ given that it was in ionization state $m$ at time $t$. We have developed a way of calculating these conditional probabilities, and hence the damage noise. First the damage simulation is carried out generating the populations of ion states at all time points and the transition rates between ion states are stored as a function of time. Starting with the mean ion population of state $m$ at time $t$, the stored transition rates can be used to generate the fraction of these atoms in ionization state $n$ at all later time points $t' > t$, from which the conditional probabilities can be readily inferred. We use a damage model based on a rate-equations model[@HauRiege2004], which is extended to include ion diffusion using the methods from a non-local thermal equilibrium plasma model[@Scott2001; @Caleman2011]. The details of the model are given in Appendix \[app:model\]. As we closely follow the methods of Refs. [@HauRiege2004; @Caleman2011], we expect the results and the validity our model to be similar. As we will show, there are sufficient physical processes in our model to illustrate the self-gating pulse effect in single molecule diffraction. All statistical quantities are given as weighted averages over the light elements (H,C,N,O). Sulphur was included in the rate-equations model of damage, but was excluded from the average of statistical diffraction quantities, like $A(q)$, $B(q)$ and $\sigma_B(q)$, because it is computationally intensive. Sulphur has a much larger number of possible electron configurations, and averages that depend on two time variables \[e.g. $\sigma_B(q)$\] took too long to compute for the range of beam conditions we study here. Since there are of the order of 100 sulphur atoms and 10$^4$ light atoms, our main conclusions are not expected to be affected by neglecting the diffraction from sulphur. We have set up our simulations using the chemical composition and size of GroEL. This chaperonin molecule is a candidate for first tests of single molecule imaging because it survives intact in mass spectrometry experiments[@Rostrom1999], which subject the molecule to similar conditions to injection at XFEL. It is also of sufficient size to scatter around $10^4$ photons per diffraction pattern, as shown in Fig. \[fig:diffraction\_and\_explosion\]. Simulations were performed at 8 keV photon energy ($\sim$0.155 nm wavelength) which is sufficient resolution for structural biology and similar to that demonstrated in simulation studies of single molecule imaging[@Tegze2012]. The principal effects of damage on molecular diffraction can be seen in Fig. \[fig:selfgating\_AB\], which shows a simulation for a pulse duration of 40 fs, beam intensity of $5\times10^{20}$ W cm$^{-2}$ (corresponding to a 2 mJ pulse) and a 100 $\times$ 100 nm$^2$ spot size. Without damage $A(q)$ would be equal to $f_0^2(q)$, but with damage it is reduced, attenuating the mean intensity by the same amount. The attenuation occurs at all resolutions, but is a greater fraction of the original signal at lower resolutions. The term $B(q)$ is lower than $A(q)$ because of the effects of ion motion, and the discrepancy is more pronounced at higher resolution. The deviations between $A(q)$ and $B(q)$ are important for accurate structure retrieval methods[@Quiney2011]. In this case, the most significant damage noise term $\sigma_B(q)$ is lower than $B(q)$ across all resolutions, indicating that even for pulse durations as long as 40 fs damage noise does not exceed the signal from the average molecular structure. To illustrate the self-gating pulse effect in single molecule diffraction, we plot $B(q)$ as a function of pulse duration for a constant photon energy (8 keV) and constant beam intensity ($5\times10^{20}$ W cm$^{-2}$). We see in Fig. \[fig:selfgating\_SNR\](a) that the signal level at 0.15 nm resolution steadily rises until it plateaus at a maximum value at around 20 fs. The signal at lower resolution accumulates for longer pulse times. Interestingly the noise due to radiation damage also rises non-linearly, accumulating at slower rate at longer pulse times. This is because the random distribution of ions in the sample has smaller variation when the bound electrons are almost entirely depleted from each ion. The signal-to-noise ratios, shown in Fig. \[fig:selfgating\_SNR\](b), show strikingly that shot-noise has a much greater effect than damage noise. Although $SNR_D(q)$ improves greatly for short pulses ($<$5 fs), $SNR_{D+N}(q)$ maximizes when the signal $B(q)$ maximizes at around 20 fs. The results are interesting when there is trade-off experimentally between pulse duration and pulse energy. For example, the LCLS can produce 2 mJ pulses with pulse durations of 30–50 fs for hard x-rays [@LCLS]. Sub 5 fs pulses can be produced by the LCLS using a low charge method or a slotted foil method, but at the expense of around a factor of ten in pulse energy. Given such a choice, the analysis presented here suggests that the gain in signal from a longer pulse with higher pulse energy compensates for the increase in damage. We note though, this conclusion only applies to spatially uncorrelated damage processes like ionization and ion diffusion (not a Coulomb explosion). Figure \[fig:pulselength\_SNR\] shows that $SNR_{D+N}(q)$ and $SNR_D(q)$ have a weak dependence on pulse duration at constant pulse energy. This suggests that maximizing pulse energy has a greater influence on the success of single molecule imaging than pulse duration with respect to the spatially uncorrelated damage mechanisms considered here. If multiple molecules were simultaneously aligned and exposed to the x-ray pulse (as described in the Introduction), we would still expect a gating effect qualitatively similar to that shown in Fig. \[fig:selfgating\_AB\]. However, we would expect $SNR_{D+N}(q)$ and $SNR_D(q)$ to scale as $\sqrt{N_{\rm mol}}$, where $N_{\rm mol}$ is the average number of molecules in the beam for each exposure. This is because the signal is proportional to $N_{\rm mol}$, while standard deviations of the damage noise and shot noise scale as $\sqrt{N_{\rm mol}}$. This analysis is missing the additional fluctuations due to the coherent interference between molecules, which have been considered in the context of angular correlation methods [@Kirian2012]. A method of measuring damage experimentally =========================================== The statistical analysis of diffraction contrast can be used to measure the amount of damage in single molecule experiments. The average change to the atomic structure factors, characterized by $A(q)$, can be readily measured by summing diffraction patterns. This provides some information about ionization levels but not ion motion. There is more information to be gained by analyzing the fluctuations of the diffraction signal. It is not convenient to measure $SNR_{D+N}(q)$, because $B(q)$ cannot be measured directly without resolving the issue of unknown orientations and assembling a 3D dataset, effectively accomplishing a full imaging experiment. An experimentally simpler proposition, which is independent of the imaging experiment, is to measure the standard deviation of the signal within each resolution ring, averaged over all of the measured diffraction patterns. The standard deviation is proportional to $\langle B_{ij}^2(q) \rangle$ and is a measure of the speckle contrast. It will contain both contributions from the average structure of the sample and the damage noise. Unfortunately it is not clear how to separate those two contributions experimentally. Nevertheless, the standard deviation is a sensitive measure of any dynamical change in the sample structure because it will drop relative to the mean scattering signal, as has been shown for averages of molecular conformation [@Maia2009]. To isolate the effect of damage-induced structural change, we create a measure that first subtracts the expected contribution of shot noise, which is equal to $ \mu_\textrm{pix}(q)$, and then normalizes by the mean intensity as follows: $$D(q) = \frac{\sigma_{\textrm{pix}}^2(q) - \mu_\textrm{pix}(q)}{\mu^2_\textrm{pix}(q)} \;,$$ where $\mu_\textrm{pix}(q)$ is the average intensity at a pixel in resolution ring $q$ averaged over the whole dataset and $\sigma_{\textrm{pix}}(q)$ is the corresponding standard deviation. The mean and standard deviation are calculated from the ensemble of experimental data of molecules measured individually in random orientations. It possible to show that $$D(q) \approx \frac{\langle B_{ij}^2(q) \rangle}{ A^2(q)} \;,$$ where $\langle B_{ij}^2(q) \rangle$ is given in Appendix \[sec:app:sigB\]. It is possible to show that $0 < D(q) < 1$, because $\langle f(q,t) f(q,t') \rangle^2 < \langle f^2(q,t) \rangle \langle f^2(q,t') \rangle$. Figure \[fig:pulselength\_g2\] shows $D(q)$ for variations of pulse duration at constant pulse energy (2 mJ). The large variations at high scattering angle indicate the sensitivity of $D(q)$ to ion motion and inner shell ionization, thereby providing complementary information to a measurement of $A(q)$. The term $D(q)$ provides a new means of comparing damage simulations to experiment, and testing the assumptions that underpin damage models for the single molecule case. For low diffraction intensities, the dominant error in the calculation of $D(q)$ from experimental data is the error of $\mu_\textrm{pix}(q)$, given by $$\begin{aligned} \delta \mu_\textrm{pix}(q) = \frac{\sqrt{\mu_\textrm{pix}(q)}}{\sqrt{N_\textrm{DATA}} \sqrt{M(q)}}\;,\end{aligned}$$ where $N_\textrm{DATA}$ is the number of diffraction patterns recorded. The term $M(q)$ is the number of speckles in resolution ring $q$, which is estimated by dividing the circumference of the ring by the expected speckle width $\frac{1}{d}$, where $d$ is the width of the molecule. Assuming $D(q)$ is of the order of one, the error in $D(q)$ goes like $\delta D(q) \approx |\delta \mu_\textrm{pix}(q)| / |\mu_\textrm{pix}(q)| $. For the test molecule quoted above and 8 keV photon energy, 2 mJ pulse energy, $100 \times 100$ nm$^2$ spot size at a resolution of $q = 6.67$ nm$^{-1}$, an accuracy of $\delta D(q) = 0.01$ can be achieved in of the order of $10^3$ patterns, which is an order of magnitude less than the number required to achieve the same resolution in an imaging experiment[@Tegze2012]. This analysis could be used to gain early feedback about the data used in an imaging experiment. Discussion {#sec:discussion} ========== The results presented on damage noise have implications for the feasibility of determining assembling the 3D diffraction volume from the ensemble of noisy 2D measurements. The data-assembly algorithms use information common to different diffraction measurements to resolve unknown information about molecular orientation. Predicting the level of damage noise in individual 2D diffraction measurements is a first step toward understanding how damage affects these algorithms. The prediction that $SNR_D$ is greater than one even for longer pulse durations ($>$20 fs) is a preliminary indication that damage noise will not prevent data assembly under conditions currently available in experiment. This is because the contribution to the diffraction from the average molecular structure is greater than the shot-to-shot fluctuations of the diffraction, and it is the contribution from the averaged structure that is used to resolve the problem of unknown molecular orientations. That $SNR_{D+N}(q)$ is lower than $SNR_{D}(q)$ by more than an an order of magnitude (see Fig. \[fig:pulselength\_SNR\]) shows that shot noise dominates damage noise. This can be viewed positively because data-assembly algorithms can already cope with very low shot noise levels when assisted by a priori knowledge about the shot noise statistics[@LohEMC2009; @Fung2009]. However, shot noise applies per pixel and is well understood to be a Poisson process, whereas damage noise applies to features the size of a speckle and the underlying distribution is hard to predict analytically. Detailed studies of the effects of damage on the performance of data assembly algorithms are still required. Our study is restricted to spatially uncorrelated damage processes. One significant omission is the expansion of the molecule due to the large electrostatic forces created by the positively charged molecule and the redistribution of trapped electrons. Hydrodynamic simulations have predicted that atoms at the surface can move distances comparable the molecule’s size on a time-scale of tens of femtoseconds[@HauRiege2004], while the interior of the molecule moves less in the same time frame, because the trapped electrons redistribute to neutralize the central part of the molecule. The interior atoms will still produce a significant diffraction signal for resolving unknown orientations and assembling the diffraction data. If the surface atoms have moved significantly, they will contribute less to the assembled 3D diffraction data than the interior atoms. If the scattering of surface atoms do prove to reduce relative to the bulk, it is an outstanding question as to how to account for this during structure determination, but modal methods for studying diffraction leave options open [@Quiney2011]. Since damage has been measured in nanocrystallography experiments, it is worth drawing a distinction between damage in crystals and in single molecules. In a crystal, damage ionizes and displaces ions differently in each unit cell, so that the diffraction contains an average over many different damage scenarios. For a single molecule, there is only one damage scenario per measurement and hence we expect a bigger standard deviation of diffraction of single molecules than of nanocrystals. Additionally, nanocrystals are much larger than single molecules, so that the rates at which electrons are trapped is different and the time it takes for a photoelectron to escape is longer. The water that surrounds a nanocrystal injected via a liquid jet [@DePonte2008] also contributes to the damage in the form of additional photoelectrons and secondary electrons. It is proposed to inject single molecules via aerosol injection [@Bogan2010], so that they are surrounded by vacuum, because the background water scattering from a liquid jet would dominate the diffraction from the molecule. For these reasons, damage experiments on single molecules, independent of those on crystals, are needed to draw conclusions for single molecule imaging. At the x-ray energies required to reach atomic resolution ($\sim$ 10 keV), Compton scattering becomes another significant source of background scattering[@Slowick2014]. The background is predicted to depend on the magnitude of $q$, and would increase the noise level $\sigma_N$ by adding to the right hand side of Eq. . It has been predicted that for for beam intensities currently available at hard x-ray energies, the Compton background only becomes significant at resolutions greater than 2 Å[@Slowick2014]. Hence, Compton scattering is not expected to significantly influence the results presented here. Conclusion ========== We have analyzed shot-to-shot damage-noise fluctuations for single molecule diffraction. For spatially uncorrelated damage processes, there is a clear damage gating effect by which longer pulses measure the same average diffraction contrast as shorter pulses with the same intensity. The results further suggest that pulse energy is more important than pulse duration for maximizing signal to noise for these damage processes. In other words, a pulse 30 fs in duration may be preferable to a sub 5 fs pulse, if the later has an order of magnitude less pulse energy. If both 30 fs and 5 fs pulses have same pulse energy, then the shorter pulse is preferable because damage is reduced, which may be important for damage processes not considered here like the Coulomb explosion. These results provide a preliminary indication that the prospects of resolving molecular orientations to assemble in a 3D diffraction volume in the presence of damage are favorable with data from current facilities. We have also proposed a statistical measure of damage that could be applied experimentally to provide valuable feedback for modeling XFEL damage to single biological molecules. Description of the rate-equations model {#app:model} ======================================== We use a damage model based on a rate-equations model[@HauRiege2004], which is extended to include ion diffusion using the methods from a non-local thermal equilibrium plasma model[@Scott2001; @Caleman2011]. Rates of photoionization are taken from Ref. [@Henke1993], rates of Auger decay were taken from Ref. [@McGuire1969] and atomic energy levels were taken from Ref. [@Bearden1967]. Secondary impact ionization rates were taken from Refs. [@Bell1983; @Lennon1988]. Ejected electrons are assumed to be trapped if their kinetic energy exceeds the trapping energy of the ionized molecule[@HauRiege2004]. We assume a spherical geometry for this calculation, and this is the only place geometry is included in the calculation. Both photoelectrons and some of the Auger electrons have sufficient energy to escape at early times. All of the trapped electrons are assumed to thermalize on a sub-femtosecond time scale, so that the energy distribution is Maxwell-Boltzmann, but the mean temperature changes with time. We include all ionization states of each element and the electron orbitals for each ionization state were modeled using Slater-type orbitals[@Slater1930]. There are some minor differences between our model and the published models on which it is based. We include all the shells for sulfur (in Ref. [@HauRiege2004] it was restricted to 8 electrons). This introduces high energy Auger electrons that are able to escape the molecule under the same conditions as the photoelectrons. We do not consider ionization due to potential lowering, as is done in Ref. [@Scott2001]. We also omit the expansion of the molecule under electrostatic forces in order to focus on the spatially uncorrelated motion that is implicated in the self-gating pulse effect. The expansion of a protein molecule has been predicted to affect atoms less than one tenth of the molecule’s radius from the surface [@HauRiege2004]. These atoms can move several [Å]{}ngstr[ö]{}m during interaction with the pulse, which will greatly diminish their contribution to the diffraction contrast. The rest of the atoms are only weakly affected by expansion because the trapped electrons effectively neutralize the core, for which we would expect better agreement with the theory presented here. Derivation of Eq. {#app1} ================== The intensity of a measurement can be written as $$\begin{aligned} I(\textbf{q}) = r_e^2 P(\textbf{q}) d\Omega I_0 \Bigg[ &\int \sum_{i=1}^N f_i(q,t)^2 dt + 2 \sum_{i=1}^N \sum_{j=1}^{i-1} \int f_i(q,t) f_j(q,t) \nonumber \\ &\qquad\qquad \times\cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j + \bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )] dt \Bigg] \;, \label{eq:app:I}\end{aligned}$$ where the definitions of all terms are given in the main text. We can expand the cosine term as: $$\begin{aligned} \cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j + \bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )] =& \cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)]\cos[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )] \nonumber \\ &- \sin[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)]\sin[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )]\end{aligned}$$ We can further expand the terms that depend upon the displacement as: $$\cos[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )] = \cos[ 2 \pi \textbf{q} \cdot \bm{\epsilon}_i(t)] \cos[ 2 \pi \textbf{q} \cdot \bm{\epsilon}_j(t)] + \sin [2 \pi \textbf{q} \cdot \bm{\epsilon}_i(t)] \sin [2 \pi \textbf{q} \cdot \bm{\epsilon}_j(t)] \;.$$ The ensemble averages of individual cosine and sine terms over different random displacements are $$\begin{aligned} \Big\langle \cos[ 2 \pi \textbf{q} \cdot \bm{\epsilon}_i(t)] \Big\rangle &= \int \cos[ 2 \pi \textbf{q} \cdot \bm{\epsilon}_i(t)] \frac{1}{\sqrt{2\pi} \overline{\bm{\epsilon}}(t)} e^{-\frac{(\bm{q}\cdot\bm{\epsilon}_i(t))^2}{2 \overline{\bm{\epsilon}}(t)^2}} d\bm{\epsilon}_i \nonumber \\ &= e^{-2\pi^2 q^2 \overline{\bm{\epsilon}}(t)^2} \label{eq:app:cosav}\end{aligned}$$ and $$\begin{aligned} \Big\langle \sin[ 2 \pi \textbf{q} \cdot \bm{\epsilon}_i(t)] \Big\rangle &= \int \sin[ 2 \pi \textbf{q} \cdot \bm{\epsilon}_i(t)] \frac{1}{\sqrt{2\pi} \overline{\bm{\epsilon}}(t)} e^{-\frac{(\bm{q}\cdot\bm{\epsilon}_i(t))^2}{2 \overline{\bm{\epsilon}}(t)^2}} d\bm{\epsilon}_i \nonumber \\ &= 0 \;. \label{eq:app:sinav}\end{aligned}$$ We assume that ionization and atomic motion are statistically independent so that $$\left\langle f_i(q,t) f_j(q,t) \cos[ 2 \pi \textbf{q} \cdot \bm{\epsilon}_i(t)] \right\rangle = \left\langle f_i(q,t) f_j(q,t) \right\rangle \left\langle \cos[ 2 \pi \textbf{q} \cdot \bm{\epsilon}_i(t)] \right\rangle \;.$$ We assume that the ionization of different atoms is statistically independent so that $$\langle f_i(q,t) f_j(q,t) \rangle = \langle f_i(q,t) \rangle \langle f_j(q,t) \rangle \;,$$ if $i \ne j$. We assume that all the atoms of the same element are equivalent statistically, so that averages of $f_i(q,t)$ and $\bm{\epsilon}_i(t)$ are independent of $i$. Combining the above results we get $$\Big\langle f_i(q,t) f_j(q,t) \cos[ 2 \pi \textbf{q} \cdot \bm{\epsilon}_i(t)] \cos[ 2 \pi \textbf{q} \cdot \bm{\epsilon}_j(t)] \Big\rangle = \langle f(q,t)\rangle^2 e^{-4\pi^2 q^2 \overline{\bm{\epsilon}}(t)^2} \;. \label{eq:app:ffcc}$$ Substituting Eq. into Eq. leads to Eq. , using the definitions of $A(q)$ and $B(q)$ in Eqs. and respectively. Derivation of the variance of $A_i(q)$ : Eq. {#sec:app:sigA} ============================================= The standard deviation of the sum of $A_i(q)$ terms in Eq. \[eq:I\], denoted by $\sigma_A(q)$, is given by $$\begin{aligned} \sigma_A^2(q) &= \frac{1}{N} \left[ \left\langle \left[\sum_{i=1}^N A_i(q) \right]^2 \right\rangle - \left\langle \sum_{i=1}^N A_i(q) \right\rangle^2 \right] \;, \label{eq:sig2A_append}\end{aligned}$$ with $$A_i(q) = \int^T_0 f^2_i(q,t) dt \;.$$ Equation is scaled the number of atoms to give the contribution per atom. We ignore the $i$ dependence when writing $\sigma_A(q)$ because we assume all atoms of the same element are equivalent. Using the assumption that ionization on different atoms is statistically independent, we can write $$\begin{aligned} \left\langle \left[\sum_{i=1}^N A_i(q) \right]^2 \right\rangle &= \left\langle \sum_{i=1}^N \int^T_0 f^2_i(q,t) dt \sum_{j=1}^N \int f^2_j(q,t') dt' \right\rangle \nonumber \\ &= \sum_{i=1}^N \int^T_0 \langle f^2_i(q,t) f^2_i(q,t') \rangle dt dt' + \sum_{i=1}^N \sum_{j \ne i} \int^T_0 \int^T_0 \langle f^2_i(q,t) \rangle \langle f^2_j(q,t') \rangle dt dt' \nonumber \\ &= N \int^T_0 \langle f^2_i(q,t) f^2_i(q,t') \rangle dt dt' + N(N-1) \left[ \int^T_0 \langle f^2_i(q,t) \rangle dt \right]^2 \;.\end{aligned}$$ Therefore, $$\begin{aligned} \sigma_A^2(q) &= \frac{1}{N} \left[ \left\langle \left[\sum_{i=1}^N A_i(q) \right]^2 \right\rangle - \left\langle \sum_{i=1}^N A_i(q) \right\rangle^2 \right] \nonumber \\ & = \int^T_0 \langle f^2_i(q,t) f^2_i(q,t') \rangle dt dt' - \left[ \int \langle f^2_i(q,t) \rangle dt \right]^2 \;.\end{aligned}$$ Derivation of the variance of $B_{ij}(q)$ : Eq. {#sec:app:sigB} ================================================ The term $\sigma_B(q)$ gauges the magnitude of the damage noise fluctuations per atom due to the second term on the right-hand side of Eq. . Its square is related to the difference between the variance of the second term on the right-hand side of Eq. and that of the second term on the right-hand side of Eq. , which is given as follows $$\begin{aligned} \sigma^2_{B}(q) &= \frac{1}{N^2}\left[ \sigma^2_S(q) - \frac{1}{2}(N^2 - N) B^2(q) \right]\;, \label{eq:sigB_append}\end{aligned}$$ where $\sigma_S(q)$ is defined to be the standard deviation of the second term on r.h.s. of Eq. and is given by $$\sigma^2_S(q) = 4 \sum^N_{i=1} \sum^{i-1}_{j=1} \sum^N_{r=1} \sum^{r-1}_{s=1} \left\langle B_{ij}(q) B_{rs}(q) \right\rangle \;. \label{eq:sigS}$$ The second term on the right-hand side of Eq. contains terms with the form $$\begin{aligned} B_{ij}(q) &= \int^T_0 f_i(q,t) f_j(q,t) \cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j + \bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )] dt \nonumber \\ &= B_c(q) \cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] + B_s(q) \sin[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \;,\end{aligned}$$ where we have defined $$B_c(q) = \int^T_0 f_i(q,t) f_j(q,t) \cos[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )] dt$$ and $$B_s(q) = \int^T_0 f_i(q,t) f_j(q,t) \sin[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )] dt \;.$$ Using Eq. we can show that $$\langle B_s(q) \rangle = 0 \;,$$ and thus write $$\langle B(q) \rangle = \langle B_c(q) \rangle \;.$$ We evaluate $\langle B^2_{ij}(q) \rangle$ as a first step to calculating the standard deviation. $$\begin{aligned} \langle B^2_{ij}(q) \rangle &= \Big\langle \left\{ B_c(q) \cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] + B_s(q) \sin[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \right\}^2 \Big\rangle \nonumber \\ &= \langle B^2_c(q) \rangle \langle \cos^2[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \rangle + \langle B^2_s(q) \rangle \langle \sin^2[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \rangle \nonumber \\ &= \frac{1}{2}\left[ \langle B^2_c(q) \rangle + \langle B^2_s(q) \rangle \right] \;. \label{eq:app:x2}\end{aligned}$$ Going from the first to the second line of Eq. , we have used the assumption that the positions of the atoms are random, so that $$\Big\langle \cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \sin[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \Big\rangle = 0$$ and, in the last line of Eq. , we have $$\langle \cos^2[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \rangle = \langle \sin^2[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \rangle = \frac{1}{2} \;.$$ To evaluate Eq. , we start by evaluating $\langle B^2_c(q) \rangle$ as follows $$\begin{aligned} \langle B^2_c(q) \rangle = \int^T_0 \int^T_0 &\langle f_i(q,t) f_i(q,t') \rangle \langle f_j(q,t) f_j(q,t') \rangle \nonumber \\ & \left\langle\cos[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )] \cos[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t') - \bm{\epsilon}_j(t') )] \right\rangle dt dt' \;.\end{aligned}$$ Writing $c_i(t) = \cos[ 2 \pi \textbf{q} \cdot \bm{\epsilon}_i(t)]$, we can write $$\begin{aligned} \Big\langle \cos[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )] &\cos[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t') - \bm{\epsilon}_j(t') )] \Big\rangle \nonumber \\ &= \Big\langle \left[ c_i(t) c_j(t) + s_i(t) s_j(t) \right] \left[ c_i(t') c_j(t') + s_i(t') s_j(t') \right] \Big\rangle \nonumber \\ &= \langle c_i(t) c_i(t') \rangle \langle c_j(t) c_j(t') \rangle \nonumber \\ &\qquad + \langle c_i(t) s_i(t') \rangle \langle c_j(t) s_j(t') \rangle \nonumber \\ &\qquad + \langle s_i(t) c_i(t') \rangle \langle s_j(t) c_j(t') \rangle \nonumber \\ &\qquad + \langle s_i(t) s_i(t') \rangle \langle s_j(t) s_j(t') \rangle \nonumber \\ &= \langle c(t) c(t') \rangle^2 + \langle s_i(t) s_i(t') \rangle^2 \;.\end{aligned}$$ The term $\langle c(t) c(t') \rangle$ is given by $$\begin{aligned} \langle c(t) c(t') \rangle = &\int \int \cos[ 2 \pi \bm{q} \cdot \bm{\epsilon}(t) ] \cos[ 2 \pi \bm{q} \cdot \bm{\epsilon}(t') ] P[\bm{\epsilon}(t),\bm{\epsilon}(t') ] d\bm{\epsilon}(t) d\bm{\epsilon}(t') \;. \label{eq:integral:o2cos}\end{aligned}$$ The joint probability function is $$P[\bm{\epsilon}(t),\bm{\epsilon}(t')] = P[\bm{\epsilon}(t) | \bm{\epsilon}(t')] P[\bm{\epsilon}(t')] \;.$$ Assume that $t > t'$. We then assume that the conditional probability is probability of taking a random walk from position $\bm{\epsilon}(t')$ at time $t'$ to position $\bm{\epsilon}(t)$ at time $t$, and takes the form $$P[\bm{\epsilon}(t) | \bm{\epsilon}(t')] = \frac{1}{(\overline{\bm{\epsilon}}(t,t') \sqrt{2\pi})^3} e^{\frac{- |\bm{\epsilon}_t - \bm{\epsilon}_{t'}|^2 }{2 \overline{\bm{\epsilon}}(t,t')^2}} \;,$$ where $\overline{\bm{\epsilon}}(t,t')$ is given by the integral of the diffusion coefficient as a function of time $$\overline{\bm{\epsilon}}^2(t,t') = 2 N_D \int^t_{t'} d(t'') dt'' \;.$$ The term $N_D$ is the number of dimensions, which we will take to be one because we are only interested in diffusion in the direction of the scattering vector. The diffusion coefficient is given by $$d(t) = \frac{k_b T(t)}{m \nu(t)} \;,$$ where $k_b$ is Boltzmann’s constant, $T(t)$ is the ion temperature, $m$ is the ion mass and $\nu(t)$ is the collision frequency. To evaluate Eq. , we first write each cosine term as a sum of exponentials $$\begin{aligned} \cos[ 2 \pi \bm{q} \cdot \bm{\epsilon}(t) ] &= \frac{1}{2} [ e^{2 \pi i \bm{q} \cdot \bm{\epsilon}(t)} + e^{-2 \pi i \bm{q} \cdot \bm{\epsilon}(t)} ] \nonumber \\ &= \frac{1}{2} \sum^1_{m=0} e^{ (-1)^m 2 \pi i \bm{q} \cdot \bm{\epsilon}(t)} \;.\end{aligned}$$ We then solve two integrals of the form $$\int^{\infty}_{-\infty} \sqrt{\frac{a}{\pi}} e^{-a x^2 - bx} dx = e^{\frac{b^2}{4a}} \;.$$ The first integral is over $\bm{\epsilon}(t)$, with $a = \frac{1}{2 \overline{\bm{\epsilon}}^2(t,t')}$ and $b = \frac{\bm{\epsilon}(t')}{\overline{\bm{\epsilon}}^2(t,t')} + (-1)^m 2 \pi \bm{q} i$. The argument of the resulting exponent is $$\frac{b^2}{4a} = \frac{1}{2} \frac{\bm{\epsilon}^2(t')}{\overline{\bm{\epsilon}}^2(t,t')} + (-1)^m 2 \pi i \bm{q} \cdot \bm{\epsilon}(t') - 2 \pi^2 q^2 \overline{\bm{\epsilon}}^2(t, t') \;.$$ The second integral over $\bm{\epsilon} (t')$ has $$\begin{aligned} a &= -\frac{1}{2 \overline{\bm{\epsilon}}^2(t,t')} + \frac{1}{2 \overline{\bm{\epsilon}}^2(t,t')} + \frac{1}{2 \overline{\bm{\epsilon}}a^2(t')} = \frac{1}{2 \overline{\bm{\epsilon}}^2(t')} \nonumber \\ b &= (-1)^m 2 \pi \bm{q} i + (-1)^n 2 \pi \bm{q} i \nonumber \\ \frac{b^2}{4a} &= - 2 \pi^2 \overline{\bm{\epsilon}}^2(t') q^2 [(-1)^m + (-1^n)]^2 \;.\end{aligned}$$ The final summation over $m,n = 0,1$ gives the following result for $t > t'$: $$\begin{aligned} & \int \cos[ 2 \pi \bm{q} \cdot \bm{\epsilon}(t) ] \cos[ 2 \pi \bm{q} \cdot \bm{\epsilon}(t') ] P[\bm{\epsilon}(t),\bm{\epsilon}(t')] d\bm{\epsilon}(t) d\bm{\epsilon}(t') \nonumber \\ &= \frac{1}{2} e^{- 2 \pi^2 q^2 \overline{\bm{\epsilon}}^2(t, t')} [ 1 + e^{- 8 \pi^2 q^2 \overline{\bm{\epsilon}}^2(t')} ] \;. \label{eq:cos2av}\end{aligned}$$ The corresponding sine integral evaluates to $$\begin{aligned} & \int \sin[ 2 \pi \bm{q} \cdot \bm{\epsilon}(t) ] \sin[ 2 \pi \bm{q} \cdot \bm{\epsilon}(t') ] P[\bm{\epsilon}(t),\bm{\epsilon}(t')] d\bm{\epsilon}(t) d\bm{\epsilon}(t') \nonumber \\ &= \frac{1}{2} e^{- 2 \pi^2 q^2 \overline{\bm{\epsilon}}^2(t, t')} [ 1 - e^{- 8 \pi^2 q^2 \overline{\bm{\epsilon}}^2(t')} ] \;. \label{eq:sin2av}\end{aligned}$$ Adding the cosine and sine integrals, we get $$\begin{aligned} \langle c(t) c(t') \rangle^2 + \langle s_i(t) s_i(t') \rangle^2 = \frac{1}{2}e^{- 4 \pi^2 q^2 \overline{\bm{\epsilon}}^2(t, t')} [ 1 + e^{- 16 \pi^2 q^2 \overline{\bm{\epsilon}}^2(t')} ] \qquad\qquad (t > t') \;. \label{eq:ctt_stt}\end{aligned}$$ To complete the evaluation of Eq. , we still need to evaluate $\langle B^2_s(q) \rangle$ which is given by $$\begin{aligned} \langle B^2_s(q) \rangle = \int^T_0 \int^T_0 &\langle f_i(q,t) f_i(q,t') \rangle \langle f_j(q,t) f_j(q,t') \rangle \nonumber \\ & \langle \sin[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )] \sin[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t') - \bm{\epsilon}_j(t') )] \rangle dt dt' \;.\end{aligned}$$ This equation can be written in the form $$\begin{aligned} \langle \sin[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t) - \bm{\epsilon}_j(t) )] &\sin[ 2 \pi \textbf{q} \cdot (\bm{\epsilon}_i(t') - \bm{\epsilon}_j(t') )] \rangle \nonumber \\ &= \langle \left[ s_i(t) c_j(t) - c_i(t) s_j(t) \right] \left[ s_i(t') c_j(t') - c_i(t') s_j(t') \right] \rangle \nonumber \\ &= \langle s_i(t) s_i(t') \rangle \langle c_j(t) c_j(t') \rangle \nonumber \\ &\qquad + \langle c_i(t) c_i(t') \rangle \langle s_j(t) s_j(t') \rangle \nonumber \\ &= 2 \langle c_i(t) c_i(t') \rangle \langle s_j(t) s_j(t') \rangle \;.\end{aligned}$$ Using Eqs. and we can write this as $$2 \langle c_i(t) c_i(t') \rangle \langle s_j(t) s_j(t') \rangle = \frac{1}{2} e^{- 4 \pi^2 q^2 \overline{\bm{\epsilon}}^2(t, t')} [ 1 - e^{- 16 \pi^2 q^2 \overline{\bm{\epsilon}}^2(t')} ] \qquad\qquad (t > t') \;.$$ We can write the time integrals as $$\begin{aligned} \langle B^2_c(q) \rangle + \langle B^2_s(q) \rangle = &\int_0^T \int_{t'}^T \langle f(q,t) f(q,t') \rangle^2 e^{- 4 \pi^2 q^2 \overline{\bm{\epsilon}}^2(t, t')} dt dt' \nonumber \\ &+ \int_0^T \int_{0}^{t'} \langle f(q,t) f(q,t') \rangle^2 e^{- 4 \pi^2 q^2 \overline{\bm{\epsilon}}^2(t', t)} dt dt' \label{eq:B2c_B2s}\end{aligned}$$ Using the property that $\overline{\bm{\epsilon}}^2(t, t') = -\overline{\bm{\epsilon}}^2(t', t)$, Eq. can also be written as $$\begin{aligned} \langle B^2_c(q) \rangle + \langle B^2_s(q) \rangle &= \int_0^T \int_0^T \langle f(q,t) f(q,t') \rangle^2 e^{- 4 \pi^2 q^2 |\overline{\bm{\epsilon}}^2(t, t')|} dt dt' \nonumber \\ &\equiv \langle B^2(q) \rangle \;. \label{eq:app:BcBscombined}\end{aligned}$$ Using Eqs. and and that $\langle B_{ij} \rangle = 0$, we can calculate the standard deviation of $B_{ij}$ (denoted $\sigma^2_{B_{ij}}(q)$) to be $$\begin{aligned} \langle B^2_{ij}(q) \rangle &= \frac{1}{2} \int_0^T \int_0^T \langle f(q,t) f(q,t') \rangle^2 e^{- 4 \pi^2 q^2 |\overline{\bm{\epsilon}}^2(t, t')|} dt dt' \;.\end{aligned}$$ We have now reached a point where we can evaluate $\sigma_S(q)$, given by Eq. . The averages of terms $\langle B_{ij}(q) B_{rs}(q) \rangle$ are zero unless $i,j=r,s$, because the averages over the positions $\textbf{R}$ equal zero. Therefore, $$\begin{aligned} \sigma^2_S(q) &= 4 \sum^N_{i=1} \sum^{i-1}_{j=1} \langle B^2_{ij}(q) \rangle \nonumber \\ &= 4 \frac{N^2 - N}{2} \langle B^2_{ij}(q) \rangle \nonumber \\ &= (N^2 - N) \int_0^T \int_0^T \langle f(q,t) f(q,t') \rangle^2 e^{- 4 \pi^2 q^2 |\overline{\bm{\epsilon}}^2(t, t')|} dt dt'\end{aligned}$$ Using this result in Eq. , we obtain the following result: $$\begin{aligned} \sigma^2_{B}(q) &= \frac{1}{N^2}\left[ \sigma^2_S(q) - \frac{1}{2}(N^2 - N) B^2(q) \right] \nonumber \\ &= \left(1 - \frac{1}{N} \right) \int^T_0 \int^T_0 \Big[ \langle f(q,t) f(q,t') \rangle^2 e^{-4\pi^2 q^2 |\overline{\bm{\epsilon}}^2(t, t')|} \nonumber \\ &\qquad\qquad\qquad\qquad\qquad - \langle f(q,t) \rangle^2 \langle f(q,t') \rangle^2 e^{-4\pi^2 q^2 \overline{\bm{\epsilon}}(t)^2} e^{-4\pi^2 q^2 \overline{\bm{\epsilon}}(t')^2} \Big] dt dt' \;.\end{aligned}$$ Assuming that N is large, the term of $\frac{1}{N}$ can be ignored. Acknowledgements {#acknowledgements .unnumbered} ================ HMQ and AVM acknowledge funding from the Australian Research Council via its Centres of Excellence and Discovery Early Career Researcher Award (DE140100624) programmes. We are grateful to Jochen K[ü]{}pper for helpful feedback. ![ A graphical representation of ion diffusion in GroEL, where ion locations are chosen stochastically using the time-dependent temperature. Simulation parameters are: 8 keV; 5.0 $\times$ 10$^{20}$ W cm$^{-2}$; and 100 nm pulse diameter. Ionized hydrogen (white) moves much faster than ions of other elements. The diffraction pattern for each time point is shown below and was generated by randomly assigning each atom an ionization state and a displacement according to a rate-equations model described in Appendix \[app:model\]. Large changes to the speckle structure are predicted at high resolution, as shown by the enlarged inset regions. The effect of shot-noise is shown on the right half of each diffraction image. []{data-label="fig:diffraction_and_explosion"}](fig1.pdf) ![ The effects of damage on the atomic structure factor. The term $f_0(q)$ is the undamaged atomic scattering factor for an unionized carbon atom, $A(q)$ is proportional to the mean intensity per carbon atom at each resolution shell, $B(q)$ is proportional to the speckle contrast for carbon and $\sigma_B(q)$ is the standard deviation of the shot-to-shot fluctuations of the speckle due to damage. When there is no damage $A(q)$ and $B(q)$ are equal to $f^2_0(q)$. The simulation parameters were 8 keV photon energy, 40 fs pulse duration, 2 mJ pulse energy and spot size of 100 $\times$ 100 nm$^2$.[]{data-label="fig:selfgating_AB"}](fig_2.pdf) ![ (a) Scattering and noise levels (due to damage only) as a function of pulse duration for constant incident intensity ($5 \times 10^{20}$ W cm$^{-2}$) at 8 keV photon energy and 100 $\times$ 100 nm$^2$ spot size. $B(q)$ is proportional to the speckle contrast and we define $N(q) \equiv \sqrt{\sigma^2_A(q)/N + \sigma^2_B(q)}$, which is the denominator in Eq. and measures the average contribution to the damage noise per atom. (b) Signal-to-noise ratios with and without shot noise for a resolution of 0.15 nm. []{data-label="fig:selfgating_SNR"}](fig_3a.pdf "fig:") ![ (a) Scattering and noise levels (due to damage only) as a function of pulse duration for constant incident intensity ($5 \times 10^{20}$ W cm$^{-2}$) at 8 keV photon energy and 100 $\times$ 100 nm$^2$ spot size. $B(q)$ is proportional to the speckle contrast and we define $N(q) \equiv \sqrt{\sigma^2_A(q)/N + \sigma^2_B(q)}$, which is the denominator in Eq. and measures the average contribution to the damage noise per atom. (b) Signal-to-noise ratios with and without shot noise for a resolution of 0.15 nm. []{data-label="fig:selfgating_SNR"}](fig_3b.pdf "fig:") ![ Maximum signal-to-noise ratios with and without shot noise for a resolution of 0.15 nm for 8 keV photon energy, 100 $\times$ 100 nm$^2$ spot size and constant pulse energy of 2 mJ. []{data-label="fig:pulselength_SNR"}](fig_4.pdf) ![ The function $D(q)$ for different pulse durations for 8 keV photon energy, 100 $\times$ 100 nm$^2$ spot size and constant pulse energy of 2 mJ. []{data-label="fig:pulselength_g2"}](fig_5.pdf)

--- abstract: 'After a seminal paper by Shekeey (2016), a connection between maximum $h$-scattered ${{\mathbb F}_{q}}$-subspaces of $V(r,q^n)$ and maximum rank distance (MRD) codes has been established in the extremal cases $h=1$ and $h=r-1$. In this paper, we propose a connection for any $h\in\{1,\ldots,r-1\}$, extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. Up to equivalence, we classify MRD codes having the same parameters as the ones in our connection. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum $h$-scattered subspaces.' author: - 'Giovanni Zini and Ferdinando Zullo[^1]' title: Scattered subspaces and related codes --- *Dedicated to the memory of Elisa Montanucci.\ We unite us to her family’s pain.* 51E20, 94B27, 15A04 rank metric code; scattered subspace; linear code; linear set Introduction ============ An ${{\mathbb F}_{q}}$-subspace $U$ of an $r$-dimensional ${\mathbb{F}_{q^n}}$-vector space $V$ is said to be *$h$-scattered* if $U$ spans $V$ over ${\mathbb{F}_{q^n}}$ and, for any $h$-dimensional ${\mathbb{F}_{q^n}}$-subspace $H$ of $V$, $U$ meets $H$ in an ${{\mathbb F}_{q}}$-subspace of dimension at most $h$. This family of subspaces was introduced in [@CsMPZ] as a generalization of $1$-scattered subspaces, which are simply known as scattered subspaces and were originally presented in [@BL2000]. Since then, the theory of scattered subspaces has constantly increased its importance, mainly because of their applications to several algebraic and geometric objects, such as finite semifields, blocking sets, two-intersection sets; see [@Lavrauw; @LVdV2015; @Polverino]. After the seminal paper [@Sheekey] by Sheekey, the interest towards scattered subspaces was also boosted by their connections with the theory of rank metric codes, whose relevance in communication theory relies on its applications to random linear network coding and cryptography. A $h$-scattered ${{\mathbb F}_{q}}$-subspace of highest dimension in $V(r,q^n)$ is called *maximum $h$-scattered*; its dimension is upper bounded by $\frac{rn}{h+1}$. This bound is known to be achieved in the following cases: $h=1$, $h=r-1$, $(h+1)\mid r$ or $h=n-3$; see Section \[sec:h-scatt\]. When $h=1$ or $h=r-1$, maximum $h$-scattered subspaces are strongly related to rank metric codes having the greatest correcting and detecting capabilities for fixed dimension and ambient space, that is, to maximum rank distance (MRD) codes. This has been shown in [@Sheekey; @CSMPZ2016; @PZ] for $h=1$ and in [@Lunardon2017; @ShVdV] for $h=r-1$, while no relation was known for $1<h<r-1$. In this paper we establish a connection between ${{\mathbb F}_{q}}$-subspaces of $V$ and rank metric codes. We start by generalizing the construction of rank-metric codes $\mathcal{C}_U$ provided in [@CSMPZ2016] and defined by an ${{\mathbb F}_{q}}$-subspace $U$ of $V$. We detect those $U$’s such that $\mathcal{C}_U$ is MRD; among these are the maximum $1$- and $(r-1)$-scattered subspaces. Actually, the code $\mathcal{C}_U$ is MRD exactly when $U$ is the dual of a $h$-scattered subspace of dimension $\frac{rn}{h+1}$, for some $1\leq h\leq r-1$. Therefore, our connection extends and unifies the ones in [@Sheekey; @CSMPZ2016; @PZ; @ShVdV; @Lunardon2017]. To this aim, we exhibit two characterizations of $h$-scattered subspaces of dimension $\frac{rn}{h+1}$, which are of independent interest. Moreover we prove that, up to equivalence, the MRD codes of type $\mathcal{C}_U$ are exactly the ${{\mathbb F}_{q}}$-linear MRD codes with parameters $(\frac{rn}{h+1},n,q;n-h)$ and maximum right idealiser. An essential though difficult task is to decide whether or not two rank metric codes with the same parameters are equivalent (especially when they correspond to non-square matrices). A remarkable aspect of the MRD codes that we construct is that we are able to determine one of their idealisers; this allows to prove that some of them are not equivalent to punctured generalized Gabidulin codes nor to punctured generalized twisted Gabidulin codes. The geometric counterparts of $h$-scattered subspaces of dimension $\frac{rn}{h+1}$ are called *$h$-scattered linear sets* of rank $\frac{rn}{h+1}$. They are known to have at most $h+1$ intersection numbers with respect to the hyperplanes, and hence are of interest in coding theory when regarded as projective systems. The intersection numbers w.r.t. the hyperplanes of $h$-scattered linear sets of rank $\frac{rn}{h+1}$ have been determined in [@BL2000] for $h=1$, in [@NZ] for $h=2$, and in [@ShVdV] for $h=r-1$. We determine them for any $1\leq h\leq r-1$, by using the connection between MRD codes and $h$-scattered subspaces of dimension $\frac{rn}{h+1}$ presented in Section \[sec:subMRD\]. As a byproduct, we compute the weight distribution of the arising codes; this answers a question posed by Randrianarisoa [@Ra]. The paper is organized as follows. Section \[sec:pre\] contains preliminary results on $h$-scattered subspaces (Section \[sec:h-scatt\]), dualities of subspaces, both ordinary and Delsarte (Section \[sec:dualities\]), linear codes, equipped with the Hamming distance or with the rank metric (Section \[sec:codes\]). In Section \[sec:subMRD\] we describe the connection between ${{\mathbb F}_{q}}$-subspaces and rank metric codes, characterizing those codes which are MRD, and showing that ${{\mathbb F}_{q}}$-linear MRD $(\frac{rn}{h+1},n,q;n-h)$-codes with maximum right idealiser are exactly the codes of type $\mathcal{C}_U$, up to equivalence. This connection is shown to extend and unify the previously known ones in Section \[sec:uni\]. Section \[sec:twocharact\] completes the connection between $h$-scattered subspaces of dimension $\frac{rn}{h+1}$ and MRD codes, by means of two characterizations which are proved through the ordinary and Delsarte dualities. Section \[sec:noGab\] provides families of MRD codes which are not equivalent to punctured generalized (twisted) Gabidulin codes. Section \[sec:h+1weights\] computes the weight distribution of the linear codes arising from $h$-scattered linear sets of rank $\frac{rn}{h+1}$, seen as projective systems. Finally, in Section \[sec:open\], we resume our results and state some open questions. Preliminaries {#sec:pre} ============= Scattered $\mathbb{F}_q$-subspaces with respect to $\mathbb{F}_{q^n}$-subspaces {#sec:h-scatt} ------------------------------------------------------------------------------- Let $V=V(m,q)$ denote an $m$-dimensional ${{\mathbb F}}_q$-vector space. A $t$-spread of $V$ is a set ${{\mathcal S}}$ of $t$-dimensional ${{\mathbb F}}_q$-subspaces such that each vector of $V^*=V\setminus \{{\bf 0}\}$ is contained in exactly one element of ${{\mathcal S}}$. As shown by Segre in [@Segre], a $t$-spread of $V$ exists if and only if $t$ divides $m$. Let $V$ be an $r$-dimensional ${{\mathbb F}}_{q^n}$-vector space and let ${{\mathcal S}}$ be an $n$-spread of $V$. An ${{\mathbb F}}_q$-subspace $U$ of $V$ is called *scattered* w.r.t. ${{\mathcal S}}$ if $U$ meets every element of ${{\mathcal S}}$ in an ${{\mathbb F}}_q$-subspace of dimension at most one; see [@BL2000]. If we consider $V$ as an $rn$-dimensional ${{\mathbb F}}_q$-vector space, then it is well-known that the one-dimensional ${{\mathbb F}}_{q^n}$-subspaces of $V$, viewed as $n$-dimensional ${{\mathbb F}}_q$-subspaces, form an $n$-spread of $V$. This spread is called the *Desarguesian spread*. In this paper scattered will always mean scattered w.r.t. the Desarguesian spread. Blokhuis and Lavrauw [@BL2000] showed that the dimension of such subspaces is bounded by $rn/2$. After a series of papers it is now known that when $rn$ is even there always exist scattered subspaces of dimension $rn/2$; they are called *maximum scattered* [@BBL2000; @BGMP2015; @BL2000; @CSMPZ2016]. In [@CsMPZ], the authors introduced a special family of scattered subspaces, named $h$-scattered subspaces. Let $V$ be an $r$-dimensional ${{\mathbb F}}_{q^n}$-vector space and $h\leq r-1$ be a positive integer. An ${{\mathbb F}}_q$-subspace $U$ of $V$ is called $h$-*scattered* (or scattered w.r.t. the $h$-dimensional ${\mathbb{F}_{q^n}}$-subspaces) if ${\langle}U {\rangle}_{{{\mathbb F}}_{q^n}}=V$ and each $h$-dimensional ${{\mathbb F}}_{q^n}$-subspace of $V$ meets $U$ in an ${{\mathbb F}}_q$-subspace of dimension at most $h$. The $1$-scattered subspaces are the scattered subspaces generating $V$ over ${{\mathbb F}}_{q^n}$. The same definition applied to $h=r$ describes the $n$-dimensional ${{\mathbb F}}_q$-subspaces of $V$ defining canonical subgeometries of ${\mathrm{PG}}(V,{{\mathbb F}}_{q^n})$. If $h=r-1$ and $\dim_{{{\mathbb F}}_q}(U)=n$, then $U$ is $h$-scattered exactly when $U$ defines a scattered ${{\mathbb F}}_q$-linear set with respect to the hyperplanes, introduced in [@ShVdV Definition 14]; see also [@Lunardon2017]. Theorem \[th:bound\] bounds the dimension of a $h$-scattered subspace. \[th:bound\][[@CsMPZ Theorem 2.3]]{} If $U$ is a $h$-scattered ${{\mathbb F}}_q$-subspace of dimension $k$ in $V=V(r,q^n)$, then one of the following holds: - $k=r$ and $U$ defines a subgeometry ${\mathrm{PG}}(r-1,q)$ of ${\mathrm{PG}}(V,{{\mathbb F}}_{q^n})$; - $k\leq\frac{rn}{h+1}$. A $h$-scattered ${{\mathbb F}}_q$-subspace of highest possible dimension is said to be a [*maximum $h$-scattered*]{} ${{\mathbb F}}_q$-subspace. Theorem \[th:inter\] bounds the dimension of the intersection between a $h$-scattered subspace of dimension $\frac{rn}{h+1}$ and an ${\mathbb{F}_{q^n}}$-subspace of codimension $1$. [[@CsMPZ Theorem 2.8]]{} \[th:inter\] If $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}}_q$-subspace of a vector space $V=V(r,q^n)$, then for any $(r-1)$-dimensional ${{\mathbb F}}_{q^n}$-subspace $H$ of $V$ we have $$\frac{rn}{h+1}-n\leq \dim_{{{\mathbb F}}_q}(U \cap H) \leq \frac{rn}{h+1}-n+h.$$ Constructions of $h$-scattered ${{\mathbb F}}_q$-subspaces have been given in [@CsMPZ] and also in [@NPZZ]. A generalization of $h$-scattered subspaces has been recently introduced in [@BCsMT]. Two dualities for $\mathbb{F}_q$-subspaces {#sec:dualities} ------------------------------------------ In this paper we need both ordinary and Delsarte dualities. ### Ordinary duality {#sec:classicalduality} Let $\sigma \colon V\times V \rightarrow \mathbb{F}_{q^n}$ be a non-degenerate reflexive sesquilinear form over $V=V(r,q^n)$ and define $\sigma' \colon V \times V \rightarrow \mathbb{F}_q, \, (\mathbf{u},\mathbf{v})\mapsto \mathrm{Tr}_{q^n/q}(\sigma(\mathbf{u},\mathbf{v}))$. Once we regard $V$ as an $rn$-dimensional ${{\mathbb F}}_q$-vector space, $\sigma^\prime$ turns out to be a non-degenerate reflexive sesquilinear form over $V=V(rn,q)$. Let $\perp$ and $\perp'$ be the orthogonal complement maps defined by $\sigma$ and $\sigma'$ on the lattices of the ${{\mathbb F}}_{q^n}$-subspaces and the ${{\mathbb F}}_q$-subspaces of $V$, respectively. The following properties hold (see [@Polverino Section 2] for the details). - $\dim_{{{\mathbb F}}_{q^n}}(W)+\dim_{{{\mathbb F}}_{q^n}}(W^\perp)=r$, for every ${{\mathbb F}}_{q^n}$-subspace $W$ of $V$. - $\dim_{{{\mathbb F}}_{q}}(U)+\dim_{{{\mathbb F}}_{q}}(U^{\perp'})=nr$, for every ${{\mathbb F}}_{q}$-subspace $U$ of $V$. - $W^\perp=W^{\perp'}$, for every ${{\mathbb F}}_{q^n}$-subspace $W$ of $V$. - Let $W$ and $U$ be an ${{\mathbb F}}_{q^n}$-subspace and an ${{\mathbb F}}_q$-subspace of $V$ of dimension $s$ and $t$, repsectively. Then $$\label{eq:dualweight} \dim_{{{\mathbb F}}_q}(U^{\perp'}\cap W^{\perp'})-\dim_{{{\mathbb F}}_q}(U\cap W)=rn-\dim_{{{\mathbb F}}_q}(U)-sn.$$ - Let $\sigma$, $\sigma_1$ be non-degenerate reflexive sesquilinear forms over $V$ and define $\sigma^\prime$, $\sigma_1^\prime$, $\perp$, $\perp_1$, $\perp'$ and $\perp_1'$ as above. Then there exists an invertible ${{\mathbb F}}_{q^n}$-linear map $f$ such that $f(U^{\perp'})=U^{\perp_1'}$, i.e. $U^{\perp'}$ and $U^{\perp_1'}$ are $\mathrm{GL}(V)$-equivalent. When $U$ is an ${{\mathbb F}}_q$-subspace of $V$, we denote by $U^{\perp_O}$ one of the ${{\mathbb F}}_q$-subspaces $U^{\perp'}$, where $\perp'$ is defined by the restriction to ${{\mathbb F}}_q$ of any non-degenerate reflexive sesquilinear form over $V$, as defined at the beginning of this section. ### Delsarte duality {#sec:Delsarteduality} Let $U$ be a $k$-dimensional ${{\mathbb F}}_q$-subspace of a vector space $V=V(r,q^n)$, with $k>r$. By [@LuPo2004 Theorems 1, 2] (see also [@LuPoPo2002 Theorem 1]), there is an embedding of $V$ in $\operatorname{\mathbb{V}}=V(k,q^n)$ with $\operatorname{\mathbb{V}}=V \oplus \Gamma$ for some $(k-r)$-dimensional ${{\mathbb F}}_{q^n}$-subspace $\Gamma$ such that $U={\langle}W,\Gamma{\rangle}_{{{\mathbb F}}_{q}}\cap V$, where $W$ is a $k$-dimensional ${{\mathbb F}}_q$-subspace of $\operatorname{\mathbb{V}}$ satisfying $\langle W\rangle_{{{\mathbb F}}_{q^n}}=\operatorname{\mathbb{V}}$ and $W\cap \Gamma=\{{\bf 0}\}$. Then $ \varphi:V\to\operatorname{\mathbb{V}}/\Gamma$, $\mathbf{v}\mapsto \mathbf{v}+\Gamma$, is an ${\mathbb{F}_{q^n}}$-isomorphism such that $\varphi(U)=W+\Gamma$. Following [@CsMPZ Section 3], let $\beta'\colon W\times W\rightarrow{{\mathbb F}}_{q}$ be a non-degenerate reflexive sesquilinear form on $W$. Then $\beta'$ can be extended to a non-degenerate reflexive sesquilinear form $\beta\colon \operatorname{\mathbb{V}}\times\operatorname{\mathbb{V}}\rightarrow{{\mathbb F}}_{q^n}$. Let $\perp$ and $\perp'$ be the orthogonal complement maps defined by $\beta$ and $\beta'$ on the lattices of ${{\mathbb F}}_{q^n}$-subspaces of $\operatorname{\mathbb{V}}$ and of ${{\mathbb F}}_q$-subspaces of $W$, respectively. For an ${{\mathbb F}}_q$-subspace $S$ of $W$ the ${{\mathbb F}}_{q^n}$-subspace ${\langle}S {\rangle}_{{{\mathbb F}}_{q^n}}$ of $\operatorname{\mathbb{V}}$ will be denoted by $S^*$. In this case, $(S^*)^{\perp}=(S^{\perp'})^*$. \[deffff\] Let $U$ be a $k$-dimensional ${{\mathbb F}}_q$-subspace of $V=V(r,q^n)$ such that $k>r$ and $\dim_{{{\mathbb F}}_q}(M\cap U)<k-1$ for every $(r-1)$-dimensional ${\mathbb{F}_{q^n}}$-subspace $M$ of $V$. Then the $k$-dimensional ${{\mathbb F}}_q$-subspace $W+\Gamma^{\perp}$ of the quotient space $\operatorname{\mathbb{V}}/\Gamma^{\perp}$ will be denoted by $ U^{\perp_{D}}$ and will be called the *Delsarte dual* of $U$ (w.r.t. $\perp$). The Delsarte duality preserves the property of being scattered w.r.t. ${\mathbb{F}_{q^n}}$-subspaces, in the following sense. [@CsMPZ Theorem 3.3] \[thm:dual\] Let $U$ be a $k$-dimensional $h$-scattered ${{\mathbb F}}_q$-subspace of a vector space $V=V(r,q^n)$ with $n\geq h+3$. Then $U^{\perp_{D}}$ is an $\frac{rn}{h+1}$-dimensional $(n-h-2)$-scattered ${{\mathbb F}_{q}}$-subspace of $\operatorname{\mathbb{V}}/\Gamma^\perp=V(k-r,q^n)$. Proposition \[prop:property\] points out some properties of the Delsarte duality. \[prop:property\] Let $U$, $W$, $V$, $\Gamma$, $\operatorname{\mathbb{V}}$, $\perp$ and $\perp_D$ be defined as above. The following properties hold: - $(U^{\perp_D})^{\perp_D}=W+\Gamma=\varphi^{-1}(U)$; - under the assumption $n\geq h+3$, $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspace of $V$ if and only if $U^{\perp_D}$ is an $\frac{rn}{h+1}$-dimensional $(n-h-2)$-scattered ${{\mathbb F}_{q}}$-subspace of $\operatorname{\mathbb{V}}/\Gamma^{\perp}$. The first property easily follows from the definition of Delsarte duality. Together with Theorem \[thm:dual\] applied to $U^{\perp_D}$, this yields the second property. Generalities on codes {#sec:codes} --------------------- In this section we recall some properties of codes that will be used in the paper. In Section \[sec:Hamming\] we consider ${{\mathbb F}_{q}}$-linear codes with respect to the Hamming metric in ${{\mathbb F}}_{q}^N$, while in Section \[sec:rank\] we consider ${{\mathbb F}_{q}}$-linear codes with respect to the rank metric in ${{\mathbb F}}_{q}^{m\times n}$. ### Projective systems and linear codes {#sec:Hamming} Let $\mathcal{C}\subseteq{{\mathbb F}}_q^N$ be an ${{\mathbb F}_{q}}$-linear code of length $N$, dimension $k$ and minimum distance $d$ over the alphabet ${{\mathbb F}_{q}}$; we denote by $[N,k,d]_q$ the parameters of $\mathcal{C}$. A generator matrix of $\mathcal{C}$ is a matrix $G\in{{\mathbb F}}_q^{k\times N}$ whose rows form a basis of $\mathcal{C}$. The weight of a codeword $\mathbf{c}\in\mathcal{C}$ is the number of nonzero components of $\mathbf{c}$, and $A_i^H$ will denote the number of codewords of weight $i$ in $\mathcal{C}$. The $N$-tuple $(A_0^H=1,A_1^H,\ldots,A_N^H)$ is called the weight distribution of $\mathcal{C}$, and the polynomial $\sum_{i=0}^{N}A_i^H z^i$ is the weight enumerator of $\mathcal{C}$. A projective $[N,k,d]_q$-system is a point subset $\mathcal{P}$ of $\Omega={\mathrm{PG}}(k-1,q)$ of size $N$, not contained in any hyperplane of $\Omega$, such that $$d= N - \max\{|\mathcal{P}\cap\mathcal{H}|\colon \mathcal{H}\,\mbox{ is a hyperlane of }\,\Omega\}.$$ The matrix $G\in{{\mathbb F}}_q^{k\times N}$ whose columns are the coordinates of the points of a projective $[N,k,d]_q$-system $\mathcal{P}$ is the generator matrix of a linear code with parameters $[N,k,d]_q$. Different choices of the coordinates yield linear codes which are equivalent by means of a diagonal matrix; we denote one of them by $\mathcal{C}_{\mathcal{P}}$. \[prop:projsyst\] Let $\mathcal{P}$ be a projective $[N,k,d]_q$-system of $\Omega$ and $\mathcal{C}_{\mathcal{P}}$ be a corresponding linear $[N,k,d]_q$-code. Then the weights of $\mathcal{C}_{\mathcal{P}}$ are the values $N-i$, where $i=|\mathcal{P}\cap\mathcal{H}|$ and $\mathcal{H}$ runs over the hyperplanes of $\Omega$. The number $A_i^H$ of codewords of $\mathcal{C}_{\mathcal{P}}$ with weight $i$ is equal to the number of hyperplanes $\mathcal{H}$ of $\Omega$ such that $|\mathcal{P}\cap\mathcal{H}|=i$. ### Rank metric codes {#sec:rank} Rank metric codes were introduced by Delsarte [@Delsarte] in 1978 and they have been intensively investigated in recent years because of their applications; we refer to [@sheekey_newest_preprint] for a survey on this topic. The set ${{\mathbb F}_{q}}^{m\times n}$ of $m \times n$ matrices over ${{\mathbb F}_{q}}$ may be endowed with a metric, called *rank metric*, defined by $$d(A,B) = \mathrm{rk}\,(A-B).$$ A subset $\operatorname{\mathcal{C}}\subseteq {{\mathbb F}_{q}}^{m\times n}$ equipped with the rank metric is called a *rank metric code* (shortly, an *RM code*). The minimum distance of $\operatorname{\mathcal{C}}$ is defined as $$d = \min\{ d(A,B) \colon A,B \in \operatorname{\mathcal{C}},\,\, A\neq B \}.$$ Denote the parameters of an RM code $\operatorname{\mathcal{C}}\subseteq{{\mathbb F}_{q}}^{m\times n}$ with minimum distance $d$ by $(m,n,q;d)$. We are interested in ${{\mathbb F}_{q}}$-*linear* RM codes, i.e. ${{\mathbb F}_{q}}$-subspaces of ${{\mathbb F}_{q}}^{m\times n}$. Delsarte showed in [@Delsarte] that the parameters of these codes must obey a Singleton-like bound. \[th:Singleton\] If $\operatorname{\mathcal{C}}$ is an RM code of ${{\mathbb F}}_q^{m\times n}$ with minimum distance $d$, then $$|\operatorname{\mathcal{C}}| \leq q^{\max\{m,n\}(\min\{m,n\}-d+1)}.$$ When equality holds, we call $\operatorname{\mathcal{C}}$ a *maximum rank distance* (*MRD* for short) code. Examples of MRD codes are resumed in [@PZ; @sheekey_newest_preprint], see also the paper [@SheekeyLondon]. For an RM code ${{\mathcal C}}\subseteq {{\mathbb F}}_{q}^{m \times n}$, the *adjoint code* of $\operatorname{\mathcal{C}}$ is $$\operatorname{\mathcal{C}}^\top =\{C^t \colon C \in \operatorname{\mathcal{C}}\},$$ where $C^t$ is the transpose matrix of $C$. Define the symmetric bilinear form $\langle\cdot,\cdot\rangle$ on ${{\mathbb F}}_q^{m \times n}$ by $$\langle M,N \rangle= \mathrm{Tr}(MN^t).$$ The *Delsarte dual code* of an ${{\mathbb F}}_q$-linear RM code $\operatorname{\mathcal{C}}\subseteq {{\mathbb F}}_{q}^{m \times n}$ is $$\operatorname{\mathcal{C}}^\perp = \{ N \in {{\mathbb F}}_q^{m\times n} \colon \langle M,N \rangle=0 \; \text{for each} \; M \in \operatorname{\mathcal{C}}\}.$$ \[rk:dualMRD\] If $\mathcal{C}\subseteq {{\mathbb F}}_{q}^{m \times n}$ is an MRD code with minimum distance $d$, then $\operatorname{\mathcal{C}}^\top$ and $\mathcal{C^\perp}$ are MRD codes with minimum distances $d$ and $\min\{m,n\}-d+2$, respectively; see [@Delsarte; @Ravagnani]. Given an RM code $\mathcal{C}$ in $\mathbb{F}_{q}^{m\times n}$ and an integer $i \in \mathbb{N}$, define $A_i=|\{M \in \mathcal{C} \colon \mathrm{rk}(M)=i\}|$. The *rank distribution* of $\mathcal{C}$ is the vector $(A_i)_{i \in \mathbb{N}}$. MacWilliams identities for RM codes are stated in Theorem \[th:MacWilliams\] and were first obtained by Delsarte in [@Delsarte] using the machinery of association schemes; see also [@Ravagnani] for a different approach. Recall that the $q$-binomial coefficient of two integers $s$ and $t$ is $${s \brack t}_q=\left\{ \begin{array}{lll} 0 & \text{if}\,\, s<0,\,\,\text{or}\,\,t<0,\,\, \text{or}\,\, t>s,\\ 1 & \text{if}\,\, t=0\,\, \text{and}\,\, s\geq 0,\\ \displaystyle\prod_{i=1}^t \frac{q^{s-i+1}-1}{q^i-1} & \text{otherwise}. \end{array} \right.$$ ([@Delsarte Theorem 3.3],[@Ravagnani Theorem 31])\[th:MacWilliams\] Let $\mathcal{C}$ be an RM code in $\mathbb{F}_q^{m\times n}$. Let $(A_i)_{i\in \mathbb{N}}$ and $(B_j)_{j\in \mathbb{N}}$ be the rank distribution of $\mathcal{C}$ and $\mathcal{C}^\perp$, respectively. For any integer $\nu \in \{ 0,\ldots,m \}$ we have $$\sum_{i=0}^{m-\nu} A_i {m-i \brack \nu}_q = \frac{|\mathcal{C}|}{q^{n\nu}} \sum_{j=0}^\nu B_j {m-j \brack \nu -j}_q.$$ As a consequence, Delsarte in [@Delsarte] and later Gabidulin in [@Gabidulin] determined precisely the weight distribution of MRD codes. \[th:weightdistribution\] Let $\mathcal{C}$ be an MRD code in $\mathbb{F}_q^{m\times n}$ with minimum distance $d$. Let $m'=\min\{m,n\}$ and $n'=\max\{m,n\}$. Then $$A_{d+\ell}={m'\brack d+\ell}_q \sum_{t=0}^\ell (-1)^{t-\ell}{\ell+d \brack \ell-t}_q q^{\binom{\ell-t}{2}}(q^{n'(t+1)}-1)$$ for any $\ell \in \{0,1,\ldots,n'-d\}$. In particular, Lemma \[lemma:weight\] holds. ([@LTZ2 Lemma 2.1],\[lemma:weight\][@Ravagnani Lemma 52])\[lemma:complete weight\] Let $\mathcal{C}$ be an MRD code in $\mathbb{F}_q^{m\times n}$ with minimum distance $d$. Let $m'=\min\{m,n\}$ and $n'=\max\{m,n\}$. Assume that the null matrix $O$ is in $\mathcal{C}$. Then, for any $0 \leq \ell \leq m'-d$, we have $A_{d+\ell}>0$, i.e. there exists at least one matrix $C \in \mathcal{C}$ such that $\mathrm{rk} (C) = d + \ell$. Theorem \[th:dualrelations\] follows from the MacWilliam identities. \[th:dualrelations\]([@Ravagnani Proof of Corollary 44]) Let $\mathcal{C}$ be an MRD code in $\mathbb{F}_q^{m\times n}$ with minimum distance $d$. Let $m'=\min\{m,n\}$ and $n'=\max\{m,n\}$. Then for any $\nu \in \{0,\ldots,m'-d \}$ we have $$\label{eq:identities} {m' \brack \nu}_q+\sum_{i=d}^{m'-\nu} A_i {m'-i \brack \nu}_q=\frac{|\mathcal{C}|}{q^{n'\nu}} {m' \brack \nu}_q.$$ By Remark \[rk:dualMRD\], the minimum distance of $\mathcal{C}^\perp$ is $m'-d+2$. Thus, Theorem \[th:MacWilliams\] proves the claim. Two RM codes $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}'$ in $\mathbb{F}_q^{m\times n}$ are *equivalent* if and only if there exist $X \in \mathrm{GL}(m,q)$, $Y \in \mathrm{GL}(n,q)$, $Z \in {{\mathbb F}}_q^{m\times n}$ and a field automorphism $\sigma$ of ${{\mathbb F}}_q$ such that $$\operatorname{\mathcal{C}}'=\{XC^\sigma Y + Z \colon C \in \operatorname{\mathcal{C}}\}.$$ The *left* and *right idealisers* $L(\operatorname{\mathcal{C}})$ and $R(\operatorname{\mathcal{C}})$ of an RM code $\mathcal{C}\subseteq{{\mathbb F}}_{q}^{m\times n}$ are defined as $$L(\operatorname{\mathcal{C}})=\{ Y \in {{\mathbb F}}_q^{m \times m} \colon YC\in \operatorname{\mathcal{C}}\hspace{0.1cm} \text{for all}\hspace{0.1cm} C \in \operatorname{\mathcal{C}}\},$$ $$R(\operatorname{\mathcal{C}})=\{ Z \in {{\mathbb F}}_q^{n \times n} \colon CZ\in \operatorname{\mathcal{C}}\hspace{0.1cm} \text{for all}\hspace{0.1cm} C \in \operatorname{\mathcal{C}}\}.$$ The notion of idealisers have been introduced by Liebhold and Nebe in [@LN2016 Definition 3.1]; they are invariant under equivalences of rank metric codes. Further invariants have been introduced in [@GiuZ; @NPH2]. In [@LTZ2], idealisers have been studied in details and the following result has been proved. \[th:propertiesideal\] Let $\mathcal{C}$ and $\mathcal{C}^\prime$ be ${{\mathbb F}_{q}}$-linear RM codes of ${{\mathbb F}_{q}}^{m\times n}$. - If $\mathcal{C}$ and $\mathcal{C}^\prime$ are equivalent, then their left and right idealisers are isomorphic as ${{\mathbb F}_{q}}$-algebras ([@LTZ2 Proposition 4.1]). - $L(\operatorname{\mathcal{C}}^\top)=R(\operatorname{\mathcal{C}})^\top$ and $R(\operatorname{\mathcal{C}}^\top)=L(\operatorname{\mathcal{C}})^\top$ ([@LTZ2 Proposition 4.2]). - Let $\mathcal{C}$ have minimum distance $d>1$. If $m \leq n$, then $L(\operatorname{\mathcal{C}})$ is a finite field with $|L(\operatorname{\mathcal{C}})|\leq q^m$. If $m \geq n$, then $R(\operatorname{\mathcal{C}})$ is a finite field with $|R(\operatorname{\mathcal{C}})|\leq q^n$. In particular, when $m=n$, $L(\operatorname{\mathcal{C}})$ and $R(\operatorname{\mathcal{C}})$ are both finite fields ([@LTZ2 Theorem 5.4 and Corollary 5.6]). Let $\mathcal{C}$ be an RM code in ${{\mathbb F}_{q}}^{n\times n}$, and $A\in{{\mathbb F}_{q}}^{m\times n}$ be a matrix of rank $m\leq n$. The RM code $A\mathcal{C}=\{AM\colon M\in\mathcal{C}\}\subseteq{{\mathbb F}_{q}}^{m\times n}$ is a *punctured code* obtained by *puncturing $\mathcal{C}$ with $A$*. \[th:punct\]([@BR Corollary 35], [@CsS Theorem 3.2]) Let $\operatorname{\mathcal{C}}$ be an MRD code with parameters $(n,n,q;d)$, $A \in {{\mathbb F}}_q^{m\times n}$ be a matrix of rank $m$, and $n-d\leq m\leq n$. Then the punctured code $A\operatorname{\mathcal{C}}$ is an MRD code with parameters $(m,n,q;d+m-n)$ and $(A\operatorname{\mathcal{C}})^\top$ is an MRD code with parameters $(n,m,q;d+m-n)$. In the literature equivalent representations of RM codes are used, other than the matrix representation that has been described above, and some of them will be used in this paper. In particular, we see the elements of an ${{\mathbb F}_{q}}$-linear RM code $\mathcal{C}$ with parameters $(m,n,q;d)$ as: - matrices of ${{\mathbb F}}_q^{m\times n}$ having rank at least $d$; - ${{\mathbb F}_{q}}$-linear maps $V\to W$ where $V=V(n,q)$ and $W=V(m,q)$, having usual map rank at least $d$; - when $m=n$, elements of the ${{\mathbb F}_{q}}$-algebra $\mathcal{L}_{n,q}$ of $q$-polynomials over ${\mathbb{F}_{q^n}}$ modulo $x^{q^n}-x$, having rank at least $d$ as an ${{\mathbb F}_{q}}$-linear map ${\mathbb{F}_{q^n}}\to{\mathbb{F}_{q^n}}$. Connection between ${{\mathbb F}_{q}}$-vector spaces and rank metric codes {#sec:subMRD} ========================================================================== In this section, an ${{\mathbb F}_{q}}$-linear RM code with parameters $(m,n,q;d)$ is regarded as a set of ${{\mathbb F}_{q}}$-linear maps $W_1=V(n,q)\to W_2=V(m,q)$. The following notation will be used. - $\omega_{\alpha}:{\mathbb{F}_{q^n}}\to{\mathbb{F}_{q^n}}$, $x\mapsto\alpha x$, for any $\alpha\in{\mathbb{F}_{q^n}}$. - $\mathcal{F}_n=\{\omega_{\alpha}\colon\alpha\in{\mathbb{F}_{q^n}}\}$, which is a field isomorphic to ${\mathbb{F}_{q^n}}$. - $\mathcal{F}_{n,q}=\{\omega_{\alpha}\colon \alpha\in{{\mathbb F}_{q}}\}$, which is a subfield of $\mathcal{F}_n$ isomorphic to ${{\mathbb F}_{q}}$. - $\tau_{\mathbf{v}}:{\mathbb{F}_{q^n}}\to W_1$, $\lambda\mapsto \lambda \mathbf{v}$, for any $\mathbf{v}\in W_1$. We define a family of ${{\mathbb F}_{q}}$-linear RM codes associated with an ${{\mathbb F}_{q}}$-vector space $U$. Let $n,r,k$ be positive integers with $k<rn$, $U$ be a $k$-dimensional ${{\mathbb F}_{q}}$-subspace of an $r$-dimensional ${\mathbb{F}_{q^n}}$-vector space $V$, $W$ be an $(rn-k)$-dimensional ${{\mathbb F}_{q}}$-vector space, and $G:V\to W$ be an ${{\mathbb F}_{q}}$-linear map with kernel $U$. For any $\mathbf{v} \in V$ define the ${{\mathbb F}_{q}}$-linear map $\Gamma_{\mathbf{v}}=G\circ\tau_{\mathbf{v}}$. \[th:construction\] Let $V=V(r,q^n)$ and $W=V(rn-k,q)$. Let $U=V(k,q)$ be an ${{\mathbb F}_{q}}$-subspace of $V$, and $G:V\to W$ be an ${{\mathbb F}_{q}}$-linear map with $\ker(G)=U$. Define $$\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}}) \colon \mathbf{v}\in V^* \}.$$ If $\iota<n$, then the pair $(U,G)$ defines an ${{\mathbb F}_{q}}$-linear RM code $$\label{eq:rd} \mathcal{C}_{U,G}=\left\{ \Gamma_{\mathbf{v}}=G\circ \tau_{\mathbf{v}} \colon \mathbf{v} \in V \right\}$$ of dimension $rn$ with parameters $(rn-k,n,q;n-\iota)$, whose right idealiser contains $\mathcal{F}_n$. For any $\mathbf{v},\mathbf{w}\in V$ and $\alpha\in{{\mathbb F}_{q}}$ we have $\Gamma_{\mathbf{v}}+\Gamma_{\mathbf{w}}=\Gamma_{\mathbf{v}+\mathbf{w}}$ and $\alpha\,\Gamma_{\mathbf{v}}=\Gamma_{\alpha\mathbf{v}}$, and hence $\mathcal{C}_{U,G}$ is an ${{\mathbb F}_{q}}$-vector space. For any $\mathbf{v}\in V$, let $R_{\mathbf{v}}=\{ \lambda \in {\mathbb{F}_{q^n}}\colon \lambda \mathbf{v} \in U \}$. Clearly $\ker (\Gamma_{\mathbf{v}})=R_{\mathbf{v}}$ and, when $\mathbf{v}\ne\mathbf{0}$, $\dim_{{{\mathbb F}_{q}}}(R_{\mathbf{v}})=\dim_{{{\mathbb F}_{q}}}(U\cap\langle \mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})$. Then $\dim_{{{\mathbb F}_{q}}}(\ker(\Gamma_{\mathbf{v}}))\leq\iota$ and there exists $\mathbf{u}\in V^*$ such that $\dim_{{{\mathbb F}_{q}}}(\ker(\Gamma_{\mathbf{u}}))=\iota$, so that the minimum distance of $\mathcal{C}_{U,G}$ is $n-\iota$. For any $\mathbf{v},\mathbf{w}\in V$, we have $\Gamma_{\mathbf{v}}=\Gamma_{\mathbf{w}}$ if and only if $\mathbf{v}=\mathbf{w}$. In fact, if $\Gamma_{\mathbf{v}}=\Gamma_{\mathbf{w}}$, then $G(\lambda(\mathbf{v}-\mathbf{w}))=\mathbf{0}$ for every $\lambda\in{\mathbb{F}_{q^n}}$, whence $\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}-\mathbf{w}\rangle_{{\mathbb{F}_{q^n}}})=n>\iota$ and hence $\mathbf{v}=\mathbf{w}$. Therefore, $\dim_{{{\mathbb F}_{q}}}(\mathcal{C}_{U,G})=rn$. Finally, for any $\alpha\in{\mathbb{F}_{q^n}}$ and $\mathbf{v}\in V$ we have $\Gamma_{\mathbf{v}}\circ \omega_{\alpha}=\Gamma_{\alpha\mathbf{v}}$. Then $R(\mathcal{C}_{U,G})$ contains $\mathcal{F}_n$. We now characterize the codes $\mathcal{C}_{U,G}$ which are MRD. \[th:MRDiff\] Let $V=V(r,q^n)$ and $W=V(rn-k,q)$. Let $U=V(k,q)$ be an ${{\mathbb F}_{q}}$-subspace of $V$, $G:V\to W$ be an ${{\mathbb F}_{q}}$-linear map with $\ker(G)=U$, $\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}}) \colon \mathbf{v}\in V^* \}$ with $\iota<n$, and $\mathcal{C}_{U,G}=\left\{ \Gamma_{\mathbf{v}} \colon \mathbf{v} \in V \right\}$. Then $\mathcal{C}_{U,G}$ is an ${{\mathbb F}_{q}}$-linear MRD code if and only if $$(\iota+1)\mid rn\quad\textrm{and}\quad k=\frac{\iota rn}{\iota+1}\leq(r-1)n.$$ In this case, - the parameters of $\mathcal{C}_{U,G}$ are $\left(\,\frac{rn}{\iota+1}\,,\,n\,,\,q\,;\,n-\iota\,\right)$; - the right idealiser of $\mathcal{C}_{U,G}$ is $\mathcal{F}_n$; - the weight distribution of $\mathcal{C}_{U,G}$ is $$A_{n-s}={n \brack s}_q \sum_{j=0}^{\iota-s} (-1)^{j}{n-s \brack j}_q q^{\binom{j}{2}}\left(q^\frac{rn(\iota-s-j+1)}{\iota+1}-1\right),$$ for $s \in \{0,1,\ldots,\iota\}$. If $k>(r-1)n$, then for every $\mathbf{v}\in V^*$ we have $\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\geq1$ and hence $\dim_{{{\mathbb F}_{q}}}(\ker(\Gamma_{\mathbf{v}}))=\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\geq1$, so that $\mathcal{C}_{U,G}$ has no elements of rank $n$. Thus, by Lemma \[lemma:weight\], $\mathcal{C}_{U,G}$ is not an MRD code. Suppose $k\leq (r-1)n$. Then $rn-k\geq n$ and the Singleton-like bound of Theorem \[th:Singleton\] reads $$rn\leq(rn-k)(n-(n-\iota)+1).$$ Therefore, $\mathcal{C}_{U,G}$ is an MRD code if and only if $\iota+1$ divides $rn$ and $k=\frac{\iota rn}{\iota+1}$. In this case, the parameters of $\mathcal{C}_{U,G}$ are provided by Theorem \[th:construction\] and the weight distribution of $\mathcal{C}_{U,G}$ follows from Theorem \[th:weightdistribution\]. Also, the right idealiser of $\mathcal{C}_{U,G}$ contains $\mathcal{F}_n$ by Theorem \[th:construction\], and hence is equal to $\mathcal{F}_n$ by Theorem \[th:propertiesideal\]. Different choices of the map $G$ yield equivalent codes, i.e. $\mathcal{C}_{U,G}$ is uniquely determined by $U$, up to equivalence. Let $V=V(r,q^n)$ and $W=V(rn-k,q)$. Let $U=V(k,q)$ be an ${{\mathbb F}_{q}}$-subspace of $V$, $G:V\to W$ and $\overline{G}:V\to W$ be two ${{\mathbb F}_{q}}$-linear maps with $\ker(G)=\ker(\overline{G})=U$. Then the codes $\mathcal{C}_{U,G}$ and $\mathcal{C}_{U,\overline{G}}$ are equivalent. Let $B_U \cup\{\mathbf{w}_1,\ldots,\mathbf{w}_{rn-k}\}$ be an ${{\mathbb F}_{q}}$-basis of $V$ such that $B_U$ is an ${{\mathbb F}_{q}}$-basis of $U$. Clearly, $G(\mathbf{w}_1),\ldots,G(\mathbf{w}_{rn-k})$ are ${{\mathbb F}_{q}}$-linearly independent, as well as $\overline{G}(\mathbf{w}_1),\ldots,\overline{G}(\mathbf{w}_{rn-k})$. Then there exists an invertible ${{\mathbb F}_{q}}$-linear map $L:V\to V$ such that $L(U)=U$ and $L(G({{\mathbf w}}_i))=\overline{G}({{\mathbf w}}_i)$ for every $i=1,\ldots,rn-k$, i.e. $L\circ G=\overline{G}$. Therefore, by choosing $R={\rm Id}_{{\mathbb{F}_{q^n}}}$ and $\sigma={\rm Id}_{{\rm Aut}({{\mathbb F}_{q}})}$, we have $$L\circ\mathcal{C}_{U,G}^{\sigma}\circ R = \left\{L\circ(G\circ\tau_{\mathbf{v}})\colon \mathbf{v} \in V\right\} =\mathcal{C}_{U,\overline{G}}.$$ The claim is proved. We recall the following conjugacy property of Singer cycles of ${\rm GL}(n,q)$. \[rem:singer\] The cyclic subgroups of $\mathrm{GL}(n,q)$ of order $q^n-1$ are called Singer cycles; it is well-known that any two Singer cycles $S_1=\langle g_1\rangle$ and $S_2=\langle g_2\rangle$ are conjugate in $\mathrm{GL}(n,q)$. In fact, let $p_{g_1}(x)$ be the minimal polynomial of $g_1$ over $\mathbb{F}_q$, and $\gamma$ be a primitive element of ${\mathbb{F}_{q^n}}$ with minimal polynomial $p_{g_1}(x)$ over ${{\mathbb F}_{q}}$. The set $\overline{S}_1=S_1\cup\{\mathbf{0}\}$ is an ${{\mathbb F}_{q}}$-subalgebra of ${{\mathbb F}_{q}}^{n\times n}$, isomorphic to ${\mathbb{F}_{q^n}}$ by the ${{\mathbb F}_{q}}$-linear map $\varphi$ mapping $(1,g_1,\ldots,g_1^{n-1})$ to $(1,\gamma,\ldots,\gamma^{n-1})$. Also, $\overline{S}_1$ is a field of order $q^n$ and $\varphi$ is a field ${{\mathbb F}_{q}}$-isomorphism. The same holds for $\overline{S}_2=S_2\cup\{\mathbf{0}\}$, so that there exists a field ${{\mathbb F}_{q}}$-isomorphism $\psi:\overline{S}_1\to\overline{S}_2$. Therefore, there exists $\hat{\psi}\in\mathrm{GL}(n,q)$ which conjugates $S_1$ to $S_2$. See also [@Huppert pag. 187] and [@Hiss Section 1.2.5 and Example 1.12]. Also the converse of Theorem \[th:MRDiff\] holds, in the sense that any MRD code as in the claim of that theorem is equivalent to $\mathcal{C}_{U,G}$ for some $U$ as in the assumption of Theorem \[th:MRDiff\]. \[th:MRDconverse\] Let $\mathcal{C}$ be an ${{\mathbb F}_{q}}$-linear MRD code with parameters $(t,n,q;n-\iota)$ such that $t\geq n$ and $|R(\mathcal{C})|=q^n$, contained in ${\rm Hom}({\mathbb{F}_{q^n}},W)$ with $W=V(t,q)$. Let $r=\dim_{R(\mathcal{C})}(\mathcal{C})$. Then the following holds. - $\iota+1$ divides $rn$ and $t=\frac{rn}{\iota+1}$. - $\mathcal{C}$ is equivalent to an ${{\mathbb F}_{q}}$-linear MRD code $\mathcal{C}^\prime$ such that $R(\mathcal{C}^\prime)=\mathcal{F}_n$. - The set $$U=\{f\in\mathcal{C}^\prime\colon f(1)=0\}\subseteq\mathcal{C}^\prime$$ is a $\frac{\iota rn}{\iota+1}$-dimensional $\mathcal{F}_{n,q}$-subspace of $\mathcal{C}^\prime$, and satisfies [^2] $$\label{eq:iota} \max\left\{\dim_{\mathcal{F}_{n,q}}\left(U\cap\langle f\rangle_{\mathcal{F}_n}\right)\colon f\in\mathcal{C}^\prime\right\}=\iota.$$ - $\mathcal{C}^\prime$ is equal to $\mathcal{C}_{U,G}$, where $G:\mathcal{C}^\prime\to W$, $f\mapsto f(1)$. Since $|\mathcal{C}|=q^{rn}$ and $t\geq n$, the Singleton-like bound of Theorem \[th:Singleton\] reads $rn\leq t(n-(n-\iota)+1)$. As $\mathcal{C}$ is MRD, this implies that $\iota+1$ divides $t$, and $t=\frac{rn}{\iota+1}$. Since $R(\mathcal{C})\setminus\{\mathbf{0}\}$ and $\mathcal{F}_n\setminus\{\omega_0\}$ are Singer cycles of $\mathrm{GL}(n,q)$, there exists by Remark \[rem:singer\] an invertible $\mathbb{F}_q$-linear map $H\colon \mathbb{F}_{q^n}\rightarrow\mathbb{F}_{q^n}$ such that $R(\mathcal{C})= H\circ\mathcal{F}_n\circ H^{-1}$. Thus, $\mathcal{C}^\prime = \mathcal{C}\circ H$. Clearly, $U$ is an $\mathcal{F}_{n,q}$-subspace of $\mathcal{C}^\prime$. For every $i\in\{1,\ldots,\iota\}$, we determine the size of $U_i=\{ f \in U \colon \dim_{{{\mathbb F}}_q}(\ker f)=i \}$. Let $g \in \mathcal{C}'$ be such that $\dim_{{{\mathbb F}}_q}(\ker g)=i$. As $\dim_{{{\mathbb F}}_q}(\ker g)>0$, there exists $\alpha \in {{\mathbb F}}_{q^n}^*$ such that $g(\alpha)=0$, that is $g\circ \omega_\alpha (1)=0$. As $\operatorname{\mathcal{C}}'$ is a right vector space over $\mathcal{F}_n$, it follows that $g\circ \omega_\alpha \in \operatorname{\mathcal{C}}'$ and, in particular, $g\circ \omega_\alpha \in U_i$. This implies that $$\{ f \circ \omega_\alpha \colon f \in U_i,\,\,\alpha \in {{\mathbb F}}_{q^n}^* \}$$ coincides with the set of all the elements in $\operatorname{\mathcal{C}}'$ of rank $n-i$. Also, for any $f \in U_i$ and $\alpha\in{\mathbb{F}_{q^n}}$, we have $f \circ \omega_\alpha \in U_i$ if and only if $\alpha \in \ker f$. Thus, $$A_{n-i}= \frac{|U_i| (q^n-1)}{q^i-1}.$$ By Lemma \[lemma:complete weight\] $A_{n-\iota}\ne0$, and follows. Furthermore, $$|U|=1+ A_{n-1}\frac{q-1}{q^n-1}+\ldots+A_{n-\iota}\frac{q^\iota -1}{q-1},$$ i.e. $$(q^n-1)(|U|-1)= A_{n-1}(q-1)+\ldots+A_{n-\iota}(q^\iota -1).$$ By Theorem \[th:dualrelations\] applied to $\mathcal{C}'$ with $\nu=1$, we get $$A_{n-1}(q-1)+\ldots+A_{n-\iota}(q^\iota -1)=(q^n-1)(q^{\frac{\iota rn}{\iota+1}}-1),$$ whence $\dim_{\mathcal{F}_{n,q}}(U)=\frac{\iota rn}{\iota +1}$. Finally, choosing $G:\mathcal{C}^\prime\to W$, $f\mapsto f(1)$ and recalling that $\tau_{f}\colon {{\mathbb F}}_{q^n}\rightarrow \operatorname{\mathcal{C}}'$, $\alpha \mapsto f\circ \omega_{\alpha}$ for any $f \in \operatorname{\mathcal{C}}^\prime$, we obtain $\operatorname{\mathcal{C}}'=\operatorname{\mathcal{C}}_{U,G}$. Theorems \[th:MRDiff\] and \[th:MRDconverse\] provide a correspondence between: - ${{\mathbb F}_{q}}$-subspaces $U=V(\frac{\iota rn}{\iota+1},q)$ of $V=V(r,q^n)$ such that $\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\colon \mathbf{v}\in V^*\}$; and - ${{\mathbb F}_{q}}$-linear MRD codes $\mathcal{C}$ with parameters $\left(\frac{rn}{\iota+1},n,q;n-\iota\right)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$. When $W={{\mathbb F}}_{q^{{nr}/{(\iota+1)}}}$ and $R(\mathcal{C})=\mathcal{F}_n$, Theorem \[th:MRDconverse\] reads as follows. Let $\iota,r,n$ be positive integers such that $\iota<n$, $\,\iota<r$ and $(\iota+1)\mid rn$. Let $f_1,\ldots,f_r\colon {\mathbb{F}_{q^n}}\rightarrow {{\mathbb F}}_{q^{{nr}/{(\iota+1)}}}$ be $\mathcal{F}_n$-linearly independent (on the right) ${{\mathbb F}_{q}}$-linear maps. Then the RM code $$\operatorname{\mathcal{C}}_{f_1,\ldots,f_r}=\{f_1\circ \omega_{\alpha_1}+\ldots+f_r\circ \omega_{\alpha_r} \colon \alpha_1,\ldots,\alpha_r \in {\mathbb{F}_{q^n}}\}$$ is an MRD code if and only if $$\dim_{{{\mathbb F}_{q}}} (\ker(f_1\circ \omega_{\alpha_1}+\ldots+f_r\circ \omega_{\alpha_r})) \leq \iota$$ for every $\alpha_1,\ldots,\alpha_r \in {\mathbb{F}_{q^n}}$. In this case, $\mathcal{C}_{f_1,\ldots,f_r}$ has parameters $\left(\frac{rn}{\iota+1},n,q;n-\iota\right)$ and $R(\operatorname{\mathcal{C}}_{f_1,\ldots,f_r})=\mathcal{F}_n$. Also, the $\mathcal{F}_{n,q}$-subspace $U_{f_1,\ldots,f_r}$ of $C_{f_1,\ldots,f_r}$ given by $$U_{f_1\ldots,f_r}=\{ f_1\circ \omega_{\alpha_1}+\ldots+f_r\circ \omega_{\alpha_r} \in \operatorname{\mathcal{C}}_{f_1,\ldots,f_r} \colon f_1(\alpha_1)+\ldots+f_r(\alpha_r)=0 \}$$ has dimension $\frac{\iota rn}{\iota +1}$, and $\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U_{f_1,\ldots,f_r}\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\colon \mathbf{v}\in \operatorname{\mathcal{C}}_{f_1,\ldots,f_r}^*\}$. Let $\iota,n,r$ be positive integers such that $\iota<n$ and $(\iota+1)\mid r$. Define $t=r/(\iota+1)$. The code $$\mathcal{C}=\left\{x\in{{\mathbb F}}_{q^{nt}}\mapsto a_0 x+a_1 x^q+\ldots+a_{\iota}x^{q^\iota}\in{{\mathbb F}}_{q^{nt}}\;\colon\; a_0,\ldots,a_{\iota}\in{{\mathbb F}}_{q^{nt}}\right\}$$ is an MRD code with parameters $(nt,nt,q;nt-\iota)$, known as Gabidulin code; see Section \[sec:noGab\] below. Consider the code $$\mathcal{C}|_{{\mathbb{F}_{q^n}}}=\left\{f|_{{\mathbb{F}_{q^n}}}:{\mathbb{F}_{q^n}}\to{{\mathbb F}}_{q^{nt}}\colon f\in\mathcal{C}\right\}.$$ By Theorem \[th:punct\], $\mathcal{C}|_{{\mathbb{F}_{q^n}}}$ is an MRD code with parameters $(nt,n,q;n-\iota)$. Also, $R(\mathcal{C}|_{{\mathbb{F}_{q^n}}})=\mathcal{F}_n$ and an $\mathcal{F}_n$-basis of $\mathcal{C}|_{{\mathbb{F}_{q^n}}}$ (seen as a right vector space) is $$\left\{ f_{j,i} \colon x\in{\mathbb{F}_{q^n}}\mapsto \xi^{i}x^{q^j}\in{{\mathbb F}}_{q^{nt}}\;\mid\; 0\leq i\leq t-1,\,0\leq j\leq \iota \right\},$$ where $\{1,\xi,\ldots,\xi^{t-1}\}$ is an ${\mathbb{F}_{q^n}}$-basis of ${{\mathbb F}}_{q^{nt}}$. Moreover, the set of the elements $f\in\mathcal{C}|_{{\mathbb{F}_{q^n}}}$ vanishing at $1$ is equal to $$U=\left\{x\in{\mathbb{F}_{q^n}}\mapsto -(a_1+\ldots+a_{\iota}) x+a_1 x^q+\ldots+a_{\iota}x^{q^\iota}\in{{\mathbb F}}_{q^{nt}}\;\colon\; a_1,\ldots,a_{\iota}\in{{\mathbb F}}_{q^{nt}} \right\},$$ and $\iota=\max\left\{\dim_{\mathcal{F}_{n,q}}\left(U\cap\langle f\rangle_{\mathcal{F}_n}\right)\colon f\in\mathcal{C}|_{{\mathbb{F}_{q^n}}}^{\,*}\right\}$. Let $$B=\left(f_{j,i}\,\colon\, j=0,\ldots,\iota,\;i=0,\ldots,t-1\right).$$ The coordinates of a vector in $U$ with respect to $B$ are $$\left( -\sum_{k=1}^{\iota}a_{k,0}\;,\ldots,-\sum_{k=1}^{\iota}a_{k,0}\;,\;a_{1,0}^{q^{n-1}},\ldots,a_{1,t-1}^{q^{n-1}}\;,\;\ldots\ldots,\;a_{\iota,0}^{q^{n-\iota}},\ldots,a_{\iota,t-1}^{q^{n-\iota}} \right),$$ where $a_{k,i}\in {\mathbb{F}_{q^n}}$ are such that $a_k=\sum_{i=0}^{t-1}a_{k,i}\xi^{i}$. Denote by $\overline{U}$ the set of the coordinates of the vectors in $U$. Let $\sigma^{\prime}:\mathbb{F}_{q^n}^{t(\iota+1)}\times \mathbb{F}_{q^n}^{t(\iota+1)}\to\mathbb{F}_q$, $(\mathbf{u},\mathbf{v})\mapsto\mathrm{Tr}_{q^n/q}(\langle\mathbf{u},\mathbf{v}\rangle)$, where $\langle\cdot,\cdot\rangle$ is the standard inner product. Then the vectors of $\overline{U}^{\perp^\prime}$ are $$(y_0,\ldots,y_{t-1},y_0^{q^{n-1}},\ldots,y_{t-1}^{q^{n-1}},\ldots \ldots, y_0^{q^{n-\iota}},\ldots,y_{t-1}^{q^{n-\iota}}),$$ where $y_0,\ldots,y_{t-1}\in\mathbb{F}_{q^n}$. Note that $\overline{U}^{\perp^\prime}$ is the direct sum of $t$ copies of $$\{ (z,z^{q},\ldots,z^{q^\iota})\colon z\in\mathbb{F}_{q^n} \}$$ which is a $\iota$-scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^{\iota+1}$. Therefore, when $\iota+1$ divides $r$, the restriction to ${\mathbb{F}_{q^n}}$ of a Gabidulin code is associated with the direct sum of $t$ copies of a $\iota$-scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^{\iota+1}$ (which is a $\iota$-scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^r$); see Section \[sec:twocharact\]. Previously known connections {#sec:uni} ============================ In this section, we show that the connection between ${{\mathbb F}_{q}}$-vector spaces and ${{\mathbb F}_{q}}$-linear MRD codes established in Section \[sec:subMRD\] generalizes those presented in [@Sheekey; @ShVdV; @Lunardon2017; @CSMPZ2016]. Sheekey’s connection {#sec:sheekey} -------------------- The first connection was pointed out by Sheekey in its seminal paper [@Sheekey]. Let $U$ be an ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}\times{\mathbb{F}_{q^n}}$, so that $$U=U_{f_1,f_2}=\{(f_1(x),f_2(x))\colon x\in{\mathbb{F}_{q^n}}\}$$ for some $f_1(x),f_2(x)$ in $\mathcal{L}_{n,q}$. Consider the ${{\mathbb F}_{q}}$-linear RM code $$\mathcal{S}_{f_1,f_2}=\{a_1 f_1(x)+a_2 f_2(x)\colon a_1,a_2\in{\mathbb{F}_{q^n}}\}\subset\mathcal{L}_{n,q},$$ whose left idealiser is isomorphic to ${\mathbb{F}_{q^n}}$. Then $U_{f_1,f_2}$ is a maximum scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}\times{\mathbb{F}_{q^n}}$ if and only if $\mathcal{S}_{f_1,f_2}$ is an MRD code with parameters $(n,n,q;n-1)$; see [@Sheekey Section 5]. A generalization to maximum $(r-1)$-scattered ${{\mathbb F}_{q}}$-subspaces of ${\mathbb{F}_{q^n}}^r$ {#sec:r-1} ----------------------------------------------------------------------------------------------------- Sheekey’s connection was extended by Sheekey and Van de Voorde in [@ShVdV] as follows; see also [@Lunardon2017]. Let $U$ be an ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^r$, so that $$U=U_{f_1,\ldots,f_r}=\{(f_1(x),\ldots,f_r(x))\colon x\in{\mathbb{F}_{q^n}}\}$$ for some $f_1(x),\ldots,f_r(x)\in\mathcal{L}_{n,q}$, and consider the ${{\mathbb F}_{q}}$-linear RM code $$\label{eq:Cf1fr} \mathcal{S}_{f_1,\ldots,f_r}=\{a_1 f_1(x)+\cdots+a_r f_r(x)\colon a_1,\ldots,a_r\in{\mathbb{F}_{q^n}}\},$$ whose left idealiser is isomorphic to ${\mathbb{F}_{q^n}}$. Then $U_{f_1,\ldots,f_r}$ is a maximum $(r-1)$-scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^r$ if and only if $\mathcal{S}_{f_1,\ldots,f_r}$ is an MRD code with parameters $(n,n,q;n-r+1)$; see [@ShVdV Corollary 5.7]. Clearly, when $r=2$ this connection coincides with the one of Section \[sec:sheekey\]. A generalization to maximum scattered ${{\mathbb F}_{q}}$-subspaces {#sec:JACO} ------------------------------------------------------------------- Sheekey’s connection was extended by Csajbók, Marino, Polverino and the last author in [@CSMPZ2016] by considering maximum scattered ${{\mathbb F}_{q}}$-subspaces of $V=V(r,q^n)$ for any $r\geq2$ with $rn$ even; see [@CSMPZ2016 Theorem 3.2]. Let $U=V(\frac{rn}{2},q)$ be an ${{\mathbb F}_{q}}$-subspace of $V$, $W=V(\frac{rn}{2},q)$, $G:V\to W$ be an ${{\mathbb F}_{q}}$-linear map with $\ker(G)=U$, and $\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\colon \mathbf{v}\in V^*\}$ with $\iota<n$. Then $\mathcal{C}_{U,G}=\{\Gamma_{\mathbf{v}}\colon \mathbf{v}\in V\}$ is an ${{\mathbb F}_{q}}$-linear RM code of dimension $rn$ with parameters $(\frac{rn}{2},n,q;n-\iota)$. Moreover, $U$ is a maximum scattered ${{\mathbb F}_{q}}$-subspace of $V$ if and only if $\mathcal{C}_{U,G}$ is an MRD code. In this case, the right idealiser of $\mathcal{C}_{U,G}$ is isomorphic to ${\mathbb{F}_{q^n}}$. Conversely, in [@PZ] the authors prove that any ${{\mathbb F}_{q}}$-linear MRD code with parameters $(\frac{rn}{2},n,q;n-1)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$ is equivalent to an MRD code $\mathcal{C}^\prime$ containing a maximum scattered ${{\mathbb F}_{q}}$-subspace $U$ such that $\mathcal{C}^\prime=\mathcal{C}_{U,G}$ with $G:\mathcal{C}^\prime\to W$, $f\mapsto f(1)$; see [@PZ Theorem 4.7]. This family contains the adjoint codes of the codes $\mathcal{S}_{f_1,f_2}$ presented in Section \[sec:sheekey\]; see [@CSMPZ2016 Example 3.5]. A unified connection -------------------- When $\iota=1$ and $U$ is a maximum scattered ${{\mathbb F}_{q}}$-subspace of $V=V(r,q^n)$, the connection established in Theorems \[th:construction\] and \[th:MRDiff\] coincides with the one of Section \[sec:JACO\], and hence generalizes the one of Section \[sec:sheekey\]. Also, Theorem \[th:MRDconverse\] extends the result of [@PZ]. When $\iota=r-1$ and $U$ is a maximum $(r-1)$-scattered ${{\mathbb F}_{q}}$-subspace of $V=V(r,q^n)$, our connection contains the adjoint codes of the MRD codes provided in Section \[sec:r-1\]. Indeed, let $\mathcal{C}$ be as in Equation with parameters $(n,n,q;n-r+1)$ and left idealiser isomorphic to ${\mathbb{F}_{q^n}}$. By Theorem \[th:propertiesideal\], the adjoint code $\mathcal{C}^\top$ is MRD with parameters $(n,n,q;n-r+1)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$. Thus, by Theorem \[th:MRDconverse\], $\mathcal{C}^\top$ is equivalent to $\mathcal{C}_{U,G}$ for some $U$ and $G$. Two characterizations of $h$-scattered subspaces {#sec:twocharact} ================================================ The ${{\mathbb F}_{q}}$-subspaces $U$ of $V=V(r,q^n)$ defining an MRD code with parameters $(\frac{rn}{h+1},n,q;n-h)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$ are exactly those of dimension $\frac{hrn}{h+1}$ such that $h=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\colon\mathbf{v}\in V^*\}$. Examples of such $U$’s are provided by the ordinary duals of $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspaces of $V$, for which several constructions are known; see [@CsMPZ; @NPZZ]. We prove that, whenever $n\geq h+3$, such $U$’s are exactly the ordinary duals of $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspaces of $V$. To this aim we provide two characterizations of these objects, namely Corollaries \[cor:caratterizzazione\] and \[cor:caratterizzazione2\], by means of ordinary and Delsarte dualities. \[th:condhyper\] Let $r,n,h,k$ be positive integers such that $n\geq h+3$ and $k>r$. Let $U$ be a $k$-dimensional ${{\mathbb F}_{q}}$-subspace of $V=V(r,q^n)$ such that $$\label{eq:intersection} \dim_{{{\mathbb F}_{q}}}(H\cap U)\leq k-n+h$$ for every $(r-1)$-dimensional ${\mathbb{F}_{q^n}}$-subspace $H$ of $V$. Let $\Gamma,\operatorname{\mathbb{V}},\perp,\perp_D$ be as in Section \[sec:Delsarteduality\]. Then $U^{\perp_D}$ is an $(n-h-2)$-scattered subspace of $\operatorname{\mathbb{V}}/\Gamma^\perp$. As noted in Section \[sec:Delsarteduality\], there exist $\operatorname{\mathbb{V}}=V(k,q^n)$, an ${\mathbb{F}_{q^n}}$-subspace $\Gamma=V(k-r,q^n)$ of $\operatorname{\mathbb{V}}$, and an ${{\mathbb F}_{q}}$-subspace $W=V(k,q)$ of $\operatorname{\mathbb{V}}$ such that $\operatorname{\mathbb{V}}=V\oplus\Gamma$, $\langle W\rangle_{{\mathbb{F}_{q^n}}}=\operatorname{\mathbb{V}}$, $W\cap\Gamma=\{\mathbf{0}\}$, and $U=\langle W,\Gamma\rangle_{{{\mathbb F}_{q}}}\cap V$. Let $\perp^\prime$ and $\perp$ be the orthogonal complement maps which act respectively on the ${{\mathbb F}_{q}}$-subspaces of $W$ and on the ${\mathbb{F}_{q^n}}$-subspaces of $\operatorname{\mathbb{V}}$, which are defined by non-degenerate reflexive sesquilinear forms $\beta^\prime :W\times W\to{{\mathbb F}_{q}}$ and $\beta:\operatorname{\mathbb{V}}\times\operatorname{\mathbb{V}}\to{\mathbb{F}_{q^n}}$ respectively, such that $\beta$ coincides with $\beta^\prime$ on $W\times W$. Since $n\geq h+3$, $k>r$ and holds, the Delsarte duality can be applied to $U$. Then $U^{\perp_D}=W+\Gamma^{\perp}$ is a $k$-dimensional ${{\mathbb F}_{q}}$-subspace of $\operatorname{\mathbb{V}}/\Gamma^{\perp}$. Suppose that there exists an $(n-h-2)$-dimensional subspace $M$ of $\operatorname{\mathbb{V}}/\Gamma^\perp$ such that $\dim_{{{\mathbb F}_{q}}}(M\cap U^{\perp_D})\geq n-h-1$. Write $M=N+\Gamma^\perp$, where $N$ is an $(n-h-2+r)$-dimensional subspace of $\operatorname{\mathbb{V}}$ satisfying $\Gamma^\perp \subseteq N$, so that $$\dim_{{{\mathbb F}_{q}}}(N\cap W)=\dim_{{{\mathbb F}_{q}}}(M\cap U^{\perp_D})\geq n-h-1.$$ Let $S$ be an $(n-h-1)$-dimensional ${{\mathbb F}_{q}}$-subspace of $N\cap W$. As $S\subseteq W$, we have $\dim_{{\mathbb{F}_{q^n}}}(S^*)=\dim_{{{\mathbb F}_{q}}}(S)$; see [@Lun99 Lemma 1]. Since $N$ contains both $S$ and $\Gamma^\perp$, we have $N^\perp\subseteq (S^*)^\perp \cap \Gamma$, whence $$\dim_{{\mathbb{F}_{q^n}}}((S^*)^\perp \cap \Gamma)\geq \dim_{{\mathbb{F}_{q^n}}}(N^\perp) = k-(n-h-2+r).$$ This implies that $\langle(S^*)^\perp,\Gamma\rangle_{{\mathbb{F}_{q^n}}}$ is contained in an ${\mathbb{F}_{q^n}}$-subspace $T$ of $\operatorname{\mathbb{V}}$ of dimension $k-1$. Let $\hat{T}=T\cap V$. As $T$ contains $\Gamma$, we have $\dim_{{\mathbb{F}_{q^n}}}(\hat{T})=r-1$. Using $U=\langle W,\Gamma\rangle_{{{\mathbb F}_{q}}}\cap V$ and $T\cap\langle W,\Gamma\rangle_{{{\mathbb F}_{q}}}=\langle\Gamma,T\cap W\rangle_{{{\mathbb F}_{q}}}$, we obtain $$\dim_{{{\mathbb F}_{q}}}(\hat{T}\cap U)=\dim_{{{\mathbb F}_{q}}}(T\cap W).$$ As $S^{\perp^\prime}= W\cap (S^*)^\perp\subseteq W\cap T$ and $\dim_{{{\mathbb F}_{q}}}(S^{\perp^{\prime}})=k-(n-h-1)$, we obtain $$\dim_{{{\mathbb F}_{q}}}(\hat{T}\cap U)\geq k-n+h+1,$$ a contradiction to . Therefore $U^{\perp_D}$ is an $(n-h-2)$-scattered ${{\mathbb F}_{q}}$-subspace of $\operatorname{\mathbb{V}}/\Gamma^\perp$. Note that Theorem \[th:condhyper\] can also be obtained as a consequence of [@BCsMT Theorem 3.5]. By Theorem \[th:condhyper\], the following characterization is obtained. \[cor:caratterizzazione\] Let $r,n,h$ be positive integers such that $h+1$ divides $rn$ and $n\geq h+3$. Let $U$ be an $\frac{rn}{h+1}$-dimensional ${{\mathbb F}_{q}}$-subspace of $V=V(r,q^n)$. Then $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspace of $V$ if and only if $$\label{eq:intermax} \dim_{{{\mathbb F}_{q}}}(H\cap U)\leq\frac{rn}{h+1}-n+h$$ for every $(r-1)$-dimensional ${\mathbb{F}_{q^n}}$-subspace $H$ of $V$. Assume that holds. By Theorem \[th:condhyper\], $U^{\perp_D}$ is a $\frac{rn}{h+1}$-dimensional $(n-h-2)$-scattered ${{\mathbb F}_{q}}$-subspace in $\operatorname{\mathbb{V}}/\Gamma^\perp$. By Proposition \[prop:property\], $U$ is a $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspace of $V$. The converse follows from Theorem \[th:inter\]. \[rem:nonallargarti\] If $n>2$, $k<\frac{rn}{h+1}$ and $h=1$, then there exist $k$-dimensional $1$-scattered ${{\mathbb F}_{q}}$-subspaces of $V=V(r,q^n)$ such that does not hold. Therefore, Corollary \[cor:caratterizzazione\] cannot be extended to all $h$-scattered subspaces which are not $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces. Indeed, let $U^\prime$ be a scattered $k^\prime$-dimensional ${{\mathbb F}_{q}}$-subspace such that $H=\langle U^\prime\rangle_{{\mathbb{F}_{q^n}}}$ is a $(r-1)$-dimensional ${\mathbb{F}_{q^n}}$-subspace of $V$, and $\mathbf{v}\in V\setminus H$. Then $U=U^\prime\oplus\langle\mathbf{v}\rangle_{{{\mathbb F}_{q}}}$ is a $1$-scattered ${{\mathbb F}_{q}}$-subspace of $V$ of dimension $k=k^\prime+1$ such that $\dim_{{{\mathbb F}_{q}}}(U\cap H)=k-1>k-n+1$, as $n>2$. By using Corollary \[cor:caratterizzazione\], a further characterization of $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces is proved. \[cor:caratterizzazione2\] Let $r,n,h$ be positive integers such that $h+1$ divides $rn$ and $n\geq h+3$. Let $U$ be an $\frac{rn}{h+1}$-dimensional ${{\mathbb F}_{q}}$-subspace of $V=V(r,q^n)$. Then $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspace of $V$ if and only if $U^{\perp_O}$ satisfies $$\label{eq:intermaxpoint} \dim_{{{\mathbb F}_{q}}}(\langle \mathbf{v}\rangle_{{{\mathbb F}}_{q^n}}\cap U^{\perp_O})\leq h$$ for every $\mathbf{v}\in V\setminus\{\mathbf{0}\}$. If $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}}_q$-subspace of $V$, then the assertion follows from Theorem \[th:inter\] and Equation . Conversely, $\dim_{{{\mathbb F}_{q}}}(U)=\frac{rn}{h+1}$ implies $\dim_{{{\mathbb F}}_q}(U^{\perp_O})=\frac{hrn}{h+1}$. Together with the assumption and Equation , this yields $$\dim_{{{\mathbb F}_{q}}}(H\cap U)\leq \frac{rn}{h+1}-n+h$$ for every $(r-1)$-dimensional ${{\mathbb F}}_{q^n}$-subspace $H$ of $V$. The claim now follows from Corollary \[cor:caratterizzazione\]. Theorems \[th:MRDiff\] and \[th:MRDconverse\], together with Corollary \[cor:caratterizzazione2\], provide a correspondence between the following objects, under the assumption $n\geq h+3$: - $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspaces of $V(r,q^n)$; and - ${{\mathbb F}_{q}}$-linear MRD codes with parameters $\left(\frac{rn}{h+1},n,q;n-h\right)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$. MRD codes inequivalent to generalized (twisted) Gabidulin codes {#sec:noGab} =============================================================== In this section we prove that the family of RM codes described in Section \[sec:subMRD\] contains MRD codes which are not equivalent to punctured generalized Gabidulin codes nor to punctured generalized twisted Gabidulin codes. Let $N,k,s$ be positive integers with $k<N$ and $\gcd(s,N)=1$. The *generalized Gabidulin code* $\mathcal{G}_{k,s}$ is defined as $$\mathcal{G}_{k,s}=\left\{ x\in{{\mathbb F}}_{q^N}\mapsto a_0 x+ a_1 x^{q^s}+\ldots+a_{k-1}x^{q^{s(k-1)}}\in{{\mathbb F}}_{q^N}\,\colon\, a_0,\ldots,a_{k-1}\in{{\mathbb F}}_{q^N} \right\}$$ and is an ${{\mathbb F}_{q}}$-linear MRD code with parameters $(N,N,q;N-k+1)$. The codes $\mathcal{G}_{k,s}$ were first introduced in [@Delsarte; @Gabidulin] for $s=1$ and generalized in [@kshevetskiy_new_2005]. Let $0\leq c<N$ and $\eta\in{{\mathbb F}}_{q^N}$ be such that $\eta^{(q^N-1)/(q-1)}\ne(-1)^{Nk}$. The *generalized twisted Gabidulin code* $\mathcal{H}_{k,s}(\eta,c)$ is defined as $$\mathcal{H}_{k,s}(\eta,c)=\left\{ x\in{{\mathbb F}}_{q^N}\mapsto a_0 x+ a_1 x^{q^s}+\ldots+a_{k-1}x^{q^{s(k-1)}}+a_0^{q^c}\eta x^{q^{sk}}\in{{\mathbb F}}_{q^N}\,\colon\, a_i\in{{\mathbb F}}_{q^N} \right\}$$ and is an ${{\mathbb F}_{q}}$-linear MRD code with parameters $(N,N,q; N-k+1)$. The codes $\mathcal{H}_{k,s}(\eta,c)$ were first introduced in [@Sheekey] and investigated in [@LTZ]. As a consequence of [@TZ Theorem 3.8], the left idealisers of punctured generalized (twisted) Gabidulin codes satisfy the following property. \[lemma:TZ\] Let $g:{{\mathbb F}}_{q^N}\to{{\mathbb F}}_{q^M}$ be an ${{\mathbb F}_{q}}$-linear map of rank $M\leq N$, and consider the punctured code $\mathcal{C}$, where either $\mathcal{C}=g\circ\mathcal{G}_{k,s}$ or $\mathcal{C}=g\circ\mathcal{H}_{k,s}(\eta,c)$. If $M>k+1$ and $(M,k)\ne(4,2)$, then $|L(\mathcal{C})|=q^\ell$ where $\ell$ divides $N$. In Theorem \[th:new\] we investigate the equivalence issue between the codes $\mathcal{C}$ as in having parameters $(M,N,q;d)$ with $M\geq N$, and punctured generalized (twisted) Gabidulin codes. As the punctured $(M,N,q;d)$-codes $\mathcal{D}$ arising from Lemma \[lemma:TZ\] satisfy $M\leq N$, we need to consider the adjoint code $\mathcal{D}^{\top}$ of $\mathcal{D}$, having parameters $(N,M,q;d)$. In this sense, whenever $\mathcal{C}$ and $\mathcal{D}^\top$ are not equivalent, we will say that $\mathcal{C}$ is not equivalent to a punctured generalized (twisted) Gabidulin code. We show in Theorem \[th:new\] that the condition $(h+1)\nmid r$ is sufficient for the MRD codes of Section \[sec:subMRD\] to be inequivalent to punctured generalized (twisted) Gabidulin codes. Afterwards, we provide examples. \[th:new\] Let $r,n,h$ be positive integers such that $(h+1)$ divides $rn$, $n\geq h+3$, and $(n,h)\ne(4,1)$. Let $\mathcal{C}=\mathcal{C}_{U,G}$ be the MRD code with parameters $\left(\frac{rn}{h+1},n,q;n-h\right)$ defined in Theorem \[th:MRDiff\]. If $(h+1)$ does not divide $r$, then $\mathcal{C}$ is not equivalent to any punctured generalized Gabidulin code nor to any punctured generalized twisted Gabidulin code. Suppose that $\mathcal{C}$ is equivalent to $\mathcal{D}$, where $\mathcal{D}$ is either $(g\circ \mathcal{G}_{k,s})^\top$ or $(g\circ\mathcal{H}_{k,s}(\eta,c))^\top$. Then $k=h+1$, $N=\frac{rn}{h+1}$ and $M=n$. Since $n>h+2$ and $(n,h)\ne(4,1)$, we can apply Lemma \[lemma:TZ\] and Theorem \[th:propertiesideal\] to get $|R(\mathcal{D})|=q^\ell$, where $\ell$ divides $\frac{rn}{h+1}$. As $\mathcal{C}$ and $\mathcal{D}$ are equivalent, by Theorem \[th:propertiesideal\] we have $|R(\mathcal{C})|=|R(\mathcal{D})|=q^\ell$. Then $n=\ell$ and hence $n\mid\frac{rn}{h+1}$, a contradiction to $(h+1)\nmid r$. Let $n\geq 6$ be even and $r^\prime\geq 3$ be odd. By [@CsMPZ Theorem 3.6], there exist $\frac{rn}{2}$-dimensional $(n-3)$-scattered ${{\mathbb F}_{q}}$-subspaces $U^\prime$ of $V=V\left(\frac{r^\prime(n-2)}2,q^n \right)$. Let $h=n-3$ and $r=\frac{r^\prime(n-2)}2$, and consider $U=U^{\prime\perp_O} \subseteq V$. Note that $(h+1)\nmid r$. Choose $G$ as in Theorem \[th:construction\]. By Theorems \[th:MRDiff\] and \[th:new\], the MRD code $\operatorname{\mathcal{C}}_{U,G}$ with parameters $\left(\frac{r^\prime n}{2},n,q;3\right)$ is not equivalent to any punctured generalized Gabidulin code nor to any punctured generalized twisted Gabidulin code. Let $n\geq6$ be even and $r\geq3$ be odd. Examples 3.11, 3.12 and 3.13 in [@CSMPZ2016] provide MRD codes with parameters $\left(\frac{rn}{2},n,q;n-1\right)$ which are not equivalent to any punctured generalized Gabidulin code nor to any punctured generalized twisted Gabidulin code. $h$-scattered linear sets: intersection with hyperplanes and codes with $h+1$ weights {#sec:h+1weights} ===================================================================================== Let $V=V(r,q^n)$. A point set $L$ of $\Omega={\mathrm{PG}}(V,{{\mathbb F}}_{q^n})\allowbreak={\mathrm{PG}}(r-1,q^n)$ is an *${{\mathbb F}}_q$-linear set* of $\Omega$ of rank $k$ if it is defined by the non-zero vectors of a $k$-dimensional ${{\mathbb F}}_q$-subspace $U$ of $V$, i.e. $$L=L_U:=\{{\langle}{\bf u} {\rangle}_{\mathbb{F}_{q^n}} \colon {\bf u}\in U^* \}\}.$$ We denote the rank of $L_U$ by $\mathrm{rk}(L_U)$. Let $\mathcal{S}={\mathrm{PG}}(S,{{\mathbb F}}_{q^n})$ be a subspace of $\Omega$ and $L_U$ be an ${{\mathbb F}}_q$-linear set of $\Omega$. Then $\mathcal{S} \cap L_U=L_{S\cap U}$. If $\dim_{{{\mathbb F}}_q} (S\cap U)=i$, i.e. if $\mathcal{S} \cap L_U=L_{S\cap U}$ has rank $i$, we say that $\mathcal{S}$ has *weight* $i$ in $L_U$, and we write $w_{L_U}(\mathcal{S})=i$. Note that $0 \leq w_{L_U}(\mathcal{S}) \leq \min\{{\rm rk}(L_U),n(\dim(\mathcal{S})+1)\}$. In particular, a point $P$ belongs to an ${{\mathbb F}}_q$-linear set $L_U$ if and only if $w_{L_U}(P)\geq 1$. If $U$ is a (maximum) scattered ${{\mathbb F}}_q$-subspace of $V$, then we say that $L_U$ is (maximum) scattered. In this case, $$|L_U| = \theta_{k-1}=\frac{q^k-1}{q-1},$$ where $k$ is the rank of $L_U$; equivalently, all of its points have weight one. If $U$ is a (maximum) $h$-scattered ${{\mathbb F}}_q$-subspace of $V$, $L_U$ is said to be a (maximum) $h$-scattered ${{\mathbb F}}_q$-linear set in $\Omega={\mathrm{PG}}(V,{{\mathbb F}}_{q^n})$. Therefore, an ${{\mathbb F}}_q$-linear set $L_U$ of $\Omega$ is $h$-scattered if - $\langle L_U \rangle= \Omega$; - for every $(h-1)$-subspace $\mathcal{S}$ of $\Omega$, we have $$w_{L_{U}}(\mathcal{S}) \leq h.$$ When $h=r-1$ and $\dim_{{{\mathbb F}}_q}(U)=n$, we obtain the scattered linear sets with respect to the hyperplanes introduced in [@Lunardon2017] and in [@ShVdV]. By Theorem \[th:inter\], if $L_U$ is a $h$-scattered ${{\mathbb F}}_q$-linear set of rank $\frac{rn}{h+1}$ in $\Omega$, then for every hyperplane $\mathcal{H}$ of $\Omega$ we have $$\frac{rn}{h+1}-n\leq w_{L_U}(\mathcal{H})\leq \frac{rn}{h+1}-n+h.$$ The following question arises: for any $j \in \{\frac{rn}{h+1}-n,\ldots,\frac{rn}{h+1}-n+h\}$, how many hyperplanes of $\Omega$ have weight $j$ in $L_U$? The answer is known for $h=1$, $h=2$ and $h=r-1$; see [@BL2000; @NZ; @ShVdV]. Theorem \[th:intersectionhyper\] gives a complete answer for any admissible values of $h$, $r$ and $n$. \[th:intersectionhyper\] Let $L_U$ be a $h$-scattered ${{\mathbb F}}_q$-linear set of rank $\frac{rn}{h+1}$ in $\Omega={\mathrm{PG}}(r-1,q^n)$. For every $i\in\left\{0,\ldots,h\right\}$, the number of hyperplanes of weight $\frac{rn}{h+1}-n+i$ in $L_U$ is $$\label{eq:ti} t_i=\frac{1}{q^n-1}{n \brack i}_q \sum_{j=0}^{h-i} (-1)^{j}{n-i \brack j}_q q^{\binom{j}{2}}\left(q^\frac{rn(h-i-j+1)}{h+1}-1\right).$$ In particular, $t_i>0$ for every $i\in\left\{0,\ldots,h\right\}$. Let $\sigma,\sigma^\prime,\perp,\perp^\prime$ be defined as in Section \[sec:classicalduality\], and $\mathcal{H}={\mathrm{PG}}(H,{{\mathbb F}}_{q^n})$ be a hyperplane of $\Omega$ with weight $\frac{rn}{h+1}-n+i$ in $L_U$. By Equation , $$\dim_{{{\mathbb F}_{q}}}(U^{\perp^\prime}\cap H^\perp)=\dim_{{{\mathbb F}_{q}}}(U\cap H) +rn-\frac{rn}{h+1}-(r-1)n = i\leq h.$$ Thus, the ${{\mathbb F}}_q$-linear set $L_{U^{\perp^\prime}}$ of $\Omega$ has rank $\frac{hrn}{h+1}$, the weight in $L_{U^{\perp^\prime}}$ of a point of $\Omega$ is at most $h$, and the number of points of $\Omega$ with weight $i$ in $L_{U^{\perp^\prime}}$ equals the number $t_i$ of hyperplanes with weight $\frac{rn}{h+1}-n+i$ in $L_U$, for every $i\in \{0,\ldots,h\}$. Let $W=V(\frac{rn}{h+1},q)$ and $G\colon V \rightarrow W$ be an ${{\mathbb F}}_q$-linear map with $\ker(G)=U$. By Theorem \[th:MRDiff\], the code $\mathcal{C}_{U^{\perp^\prime},G}=\{\Gamma_{\mathbf{v}} \colon \mathbf{v}\in V\}$ is an MRD code. Note that, for every $i \in \{0,\ldots,h\}$, $t_i$ is equal to the number of maps in $\operatorname{\mathcal{C}}_{U^{\perp^\prime},G}$ having rank $n-i$, divided by $q^n-1$. In fact, if $P=\langle \mathbf{v}\rangle_{{{\mathbb F}}_{q^n}}$ is a point of weight $i$ in $L_{U^{\perp^\prime}}$, then $\Gamma_{\lambda\mathbf{v}}$ has rank $n-i$ for every $\lambda\in{{\mathbb F}}_{q^n}^*$; conversely, if $\mathbf{v}\in V^*$ is such that $\Gamma_{\mathbf{v}}$ has rank $n-i$, then $\langle\mathbf{v}\rangle_{{{\mathbb F}}_{q^n}}$ has weight $i$ in $L_{U^{\perp^\prime}}$. Thus, $t_i=A_{n-i}/(q^n-1)$. By Theorem \[th:MRDiff\], Equation follows. As $\mathcal{C}_{U^{\perp^\prime},G}$ is an MRD code, $t_i>0$ by Lemma \[lemma:complete weight\]. Under the assumptions of Theorem \[th:intersectionhyper\], the property of being scattered determines completely the intersection numbers w.r.t. the hyperplanes. \[cor:intersec\] Let $L_U$ be a $h$-scattered ${{\mathbb F}_{q}}$-linear set of rank $\frac{rn}{h+1}$ in $\Omega={\mathrm{PG}}(r-1,q^n)$. For every hyperplane $\mathcal{H}$ of $\Omega$, we have $$|\mathcal{H}\cap L_U| \in \left\{ \theta_{\frac{rn}{h+1}-n-1},\ldots,\theta_{\frac{rn}{h+1}-n+h-1} \right\}.$$ For every $i\in\{0,\ldots,h\}$, the number of hyperplanes $\mathcal{H}$ of $\Omega$ satisfying $|\mathcal{H}\cap L_U|=\theta_{\frac{rn}{h+1}-n+i-1}$ is $t_i$, as in Equation . We now consider $h$-scattered ${{\mathbb F}_{q}}$-linear sets $L_U$ of rank $\frac{rn}{h+1}$ as projective systems in $\Omega$, and the related linear codes (with the Hamming metric). By means of Theorem \[th:intersectionhyper\] we determine the weight distribution and the weight enumerator. Let $L_U$ be a $h$-scattered ${{\mathbb F}_{q}}$-linear set of rank $\frac{rn}{h+1}$ in $\Omega={\mathrm{PG}}(r-1,q^n)$, and $\mathcal{C}_{L_U}$ be the corresponding linear code over ${{\mathbb F}}_{q^n}$, having length $N=\theta_{\frac{rn}{h+1}-1}$ and dimension $k=r$. Then $\mathcal{C}_{L_U}$ has minimum distance $d=\theta_{\frac{rn}{h+1}-1}-\theta_{\frac{rn}{h+1}-n+h-1}$ and exactly $h+1$ weights, namely $w_i=\theta_{\frac{rn}{h+1}-1}-\theta_{\frac{rn}{h+1}-n+i-1}$ with $i=0,\ldots,h$. The weight enumerator of $\mathcal{C}_{L_U}$ is $$1+\sum_{i=0}^{h}A_{w_i}^H z^{w_i},$$ where $A_{w_i}^H=t_i$, as in Equation . From ${\rm rk}(L_U)=\frac{rn}{h+1}$ follows the length $N=\theta_{\frac{rn}{h+1}-1}$, and from $\langle L_U\rangle = \Omega$ follows the dimension $k=r$. By Corollary \[cor:intersec\] and Proposition \[prop:projsyst\], the minimum distance of $\mathcal{C}_{L_U}$ is $d=\theta_{\frac{rn}{h+1}-1}-\theta_{\frac{rn}{h+1}-n+h-1}$. Also, the weight distribution and the weight enumerator of $\mathcal{C}_{L_U}$ are as in the claim. Randrianarisoa in [@Ra] introduced the concept of $[N,k,d]$ $q$-system over ${\mathbb{F}_{q^n}}$ as an $N$-dimensional ${{\mathbb F}_{q}}$-subspace $U$ of ${\mathbb{F}_{q^n}}^k$ such that $\langle U\rangle_{{\mathbb{F}_{q^n}}}={\mathbb{F}_{q^n}}^k$ and $N-d=\max\left\{\dim_{{{\mathbb F}_{q}}}(U\cap H)\colon H=V(k-1,q^n)\subset {\mathbb{F}_{q^n}}^k\right\}$. For any positive integers $r,n,h$ such that $(h+1)\mid rn$ and $n\geq h+3$, Corollary \[cor:caratterizzazione\] implies that the $[\frac{rn}{h+1},r,n-h]$ $q$-systems over ${\mathbb{F}_{q^n}}$ are exactly the $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspaces of ${\mathbb{F}_{q^n}}^r$. In this case, the code $\mathcal{C}$ considered in [@Ra Section 3] has a generator matrix $G$ whose columns form an ${{\mathbb F}_{q}}$-basis of $U$. The code $\mathcal{C}$ turns out to be obtained by $\mathcal{C}_{L_U}$ by deleting all but $\frac{rn}{h+1}$ positions (corresponding to an ${{\mathbb F}}_q$-basis of $U$). Together with [@Ra Theorem 2], this answers the question posed in [@Ra Section 8] about the correspondence between $h$-scattered linear sets and RM codes of this type. Conclusions and open questions {#sec:open} ============================== Several connections between between MRD codes and scattered ${{\mathbb F}_{q}}$-subspaces (linear sets) have been introduced in the literature. In this paper we propose a unified approach which generalizes all of these connections. To this aim, we give useful characterizations of $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces. This allows to use the known constructions of $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces in order to define MRD codes, and conversely. The family we construct is very large and contains some ”new” MRD codes, in the sense that they cannot be obtained by puncturing generalized (twisted) Gabidulin codes; this property is in general quite difficult to establish. We conclude the paper by determining the intersection numbers of $h$-scattered linear sets of rank $\frac{rn}{h+1}$ w.r.t. the hyperplanes and the weight distribution of the code obtained by regarding the linear set as a projective system. Several remarkable problems remain open; we list some of them. - The main open problem about $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspaces of $V(r,q^n)$ is their existence for every admissible values of $r$, $n$ and $h$. This would imply the existence of possibly new MRD codes. Conversely, constructions of MRD codes with parameters $(\frac{rn}{h+1},n,q;n-h)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$, when $(h+1)\nmid r$ and $h< n-3$, give new examples of $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces. - Corollary \[cor:caratterizzazione2\] characterizes $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces whenever $n\geq h+3$. 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New semifields and new MRD codes from skew polynomial rings, [*J. Lond. Math. Soc. (2)*]{} [**101(1)**]{} (2020), 432–456. MRD codes: constructions and connections, *Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications*, Radon Series on Computational and Applied Mathematics [**23**]{}, K.-U. Schmidt and A. Winterhof (eds.), De Gruyter (2019). Rank-metric codes, linear sets and their duality, [*Des. Codes Cryptogr.*]{} (2019), <https://doi.org/10.1007/s10623-019-00703-z>. Nuclei and automorphism groups of generalized twisted Gabidulin codes, *Linear Algebra Appl.* [**575**]{} (2019), 1–26. Giovanni Zini and Ferdinando Zullo\ Dipartimento di Matematica e Fisica,\ Università degli Studi della Campania “Luigi Vanvitelli”,\ I–81100 Caserta, Italy\ [[*{giovanni.zini,ferdinando.zullo}@unicampania.it*]{}]{} [^1]: The first author is funded by the project ”Attrazione e Mobilità dei Ricercatori” Italian PON Programme (PON-AIM 2018 num. AIM1878214-2). The research was supported by the project ”VALERE: VAnviteLli pEr la RicErca" of the University of Campania ”Luigi Vanvitelli”, and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM). [^2]: Recall that $\langle f\rangle_{\mathcal{F}_n}=\{f\circ\omega_{\alpha}\colon\alpha\in{\mathbb{F}_{q^n}}\}$.

--- abstract: 'Active matter exhibits various forms of non-equilibrium states in the absence of external forcing, including macroscopic steady-state currents. Such states are often too complex to be modelled from first principles and our understanding of their physics relies heavily on minimal models. These have mostly been studied in the case of “dry" active matter, where particle dynamics are dominated by friction with their surroundings. Significantly less is known about systems with long-range hydrodynamic interactions that belong to “wet" active matter. Dilute suspensions of motile bacteria, modelled as self-propelled dipolar particles interacting solely through long-ranged hydrodynamic fields, are arguably the most studied example from this class of active systems. Their phenomenology is well-established: at sufficiently high density of bacteria, there appear large-scale vortices and jets comprising many individual organisms, forming a chaotic state commonly known as *bacterial turbulence*. As revealed by computer simulations, below the onset of collective motion, the suspension exhibits very strong correlations between individual microswimmers stemming from the long-ranged nature of dipolar fields. Here we demonstrate that this phenomenology is captured by the minimal model of microswimmers. We develop a kinetic theory that goes beyond the commonly used mean-field assumption, and explicitly takes into account such correlations. Notably, these can be computed exactly within our theory. We calculate the fluid velocity variance, spatial and temporal correlation functions, the fluid velocity spectrum, and the enhanced diffusivity of tracer particles. We find that correlations are suppressed by particle self-propulsion, although the mean-field behaviour is not restored even in the limit of very fast swimming. Our theory is not perturbative and is valid for any value of the micro-swimmer density below the onset of collective motion. This work constitutes a significant methodological advance and allows us to make qualitative and quantitative predictions that can be directly compared to experiments and computer simulations of micro-swimmer suspensions.' author: - Viktor Škultéty - Cesare Nardini - Joakim Stenhammar - Davide Marenduzzo - Alexander Morozov bibliography: - 'Refs.bib' title: Swimming suppresses correlations in dilute suspensions of pusher microorganisms --- Introduction {#section:introduction} ============ In recent years active systems emerged as a new state of matter with unique properties that are absent from their passive counterparts [@Ramaswamy2010; @Marchetti2013]. Such systems comprise particles that are capable of extracting energy from their environment and using it to exert forces and torques on their surroundings. The resulting self-propulsion and interactions between particles break detailed balance at the microscopic level, often leading to steady states that are not invariant under time reversal and exhibit macroscopic currents [@Cates2012]. Such currents, or collective motion, have been reported in a variety of systems [@Vicsek2012], including Vicsek particles [@Chate2008], mixtures of microtubules and molecular motors [@Sanchez2012], light-activated colloids [@Palacci2013], Quincke rollers [@Bricard2013; @Karani2019], bacterial colonies [@Zhang2010], sperm cells [@Creppy2015], locusts [@Buhl2006], birds, and fish [@Parrish1997]. The omnipresence of collective motion raises the need to classify various active systems according to common features of their phenomenological behaviour. Marchetti *et al.* [@Marchetti2013] recently introduced two broad universality classes for active systems, “dry" and “wet", comprising particles dominated by friction with their surroundings and long-ranged hydrodynamic interactions, respectively. Each class is expected to be defined by a few, relatively simple model systems, and significant effort has been invested into finding such models. For dry active matter, these include Vicsek-like models [@Vicsek2012; @Chate2020], that describe cases where alignment interactions are dominant, and Active Brownian Particles [@Howse2007; @Romanczuk2012] or Run and Tumble particles [@Schnitzer1993], that describe systems dominated by steric forces randomising their self-propulsion direction either smoothly or in a discontinuous manner. In this work, we study dilute suspensions of motile bacteria that, arguably, play the same role for wet active matter [@Koch2011; @Saintillan2013]. Collective motion in bacteria has been extensively studied in dilute [@Soni2003; @Dombrowski2004; @Gachelin2014] and dense [@Mendelson1999; @Wu2006; @Sokolov2007; @Sokolov2009; @Cisneros2011; @Sokolov2012; @Wensink2012a; @Dunkel2013] suspensions. These studies reveal the following sequence of dynamical states. At very low densities, bacterial suspensions appear featureless and disordered [@Wu2006; @Gachelin2014]. At higher, yet still sufficiently low densities, collective motion sets in on the scale of the system. In this state, bacterial motion takes the form of large-scale jets and vortices with typical speeds that are larger than the swimming speeds of individual organisms [@Soni2003; @Dombrowski2004; @Gachelin2014]. At significantly higher densities, there emerges a typical lengthscale of the vortices, which is comparable to about $5-10$ times the bacterial size [@Sokolov2012; @Dunkel2013; @Ryan2013]. Although this sequence of dynamical states has never been simultaneously observed in a single systematic bulk experiment, with the exception of Sokolov *et al.* [@Sokolov2009], the transition scenario is supported by computer simulations of self-propelled particles interacting through various forms of long-ranged hydrodynamic fields and short-ranged steric repulsion [@Hernandez-Ortiz2005; @Saintillan2007; @Wolgemuth2008; @Underhill2008; @Hernandez-Ortiz2009; @Lushi2013; @Lushi2014; @Krishnamurthy2015; @Wioland2016; @Saintillan2012; @Stenhammar2017; @Theillard2017; @Schwarzendahl2018; @Bardfalvy2019; @Theillard2019]. Bulk experiments with *E.coli* [@Gachelin2014] and *B.subtilis* [@Dombrowski2004] show that the transition to collective motion occurs around a volume fraction of bacterial bodies of about $1-2\%$. At such densities, the typical distance between the organisms is about $5-8$ times their body length, collisions are rare, and the far-field hydrodynamic interactions are thought to be dominant [@Koch2011; @Saintillan2013]. The latter are well-described by a “pusher”-like Stokesian dipolar field [@Lauga2009; @Drescher2011], generated when two point forces of equal magnitude and pointing away from each other are applied to a viscous fluid. Self-propelled pusher-like dipolar particles thus form a minimal model for dilute bacterial suspensions. The transition to collective motion in dilute bacterial suspensions can be understood in terms of a mean-field kinetic theory [@Koch2011; @Saintillan2013] incorporating the minimal ingredients discussed above. Such theory identifies re-orientation of bacteria in the velocity field created by other organisms as the key ingredient leading to a global isotropic-nematic transition. The globally ordered state is, however, linearly unstable through a long-wavelength generic instability [@AditiSimha2002; @Marchetti2013], and there ensue never-settling dynamics as a compromise between the two instabilities. The critical density of bacteria at the onset of collective motion is determined by the strength of their dipolar interactions, their shape, and the way individual organisms change their orientation: either by occasionally re-orienting in a random way (tumble), or by rotational diffusion [@Saintillan2008; @Saintillan2008a; @Subramanian2009; @Hohenegger2010; @Krishnamurthy2015]. Typically, the critical threshold density is significantly lower in the latter case, and going to zero in the absence of a decorrelation mechanism for individual bacterium orientation. The mean-field kinetic theory has also been extended to systems with steric interactions [@Ezhilan2013; @Ryan2013; @Heidenreich2016; @Reinken2018] and to microswimmers suspended in non-Newtonian fluids [@Bozorgi2013; @Bozorgi2014; @Li2016]. Below the onset of collective motion, the mean-field kinetic theory predicts that the suspension is homogeneous and isotropic, as featureless as a suspension of non-interacting microswimmers. These assumptions are widely used when describing rheological properties of very dilute suspensions [@Hatwalne2004; @Chen2007; @Sokolov2009a; @Saintillan2010; @Underhill2011; @Lopez2015; @Alonso-Matilla2016; @Nambiar2017; @Guo2018; @Nambiar2019; @Liu2019; @Saintillan2018] and enhanced diffusivity of tracer particles [@Wu2000; @Kim2004; @Underhill2008; @Leptos2009; @Dunkel2010; @Ishikawa2010; @Childress2010; @Childress2011; @Kurtuldu2011; @Mino2011; @Mino2013; @Jepson2013; @Pushkin2013; @Pushkin2013jfm; @Morozov2014; @Kasyap2014; @Thiffeault2015; @Patteson2016; @Burkholder2017]. However, recent large-scale Lattice-Boltzmann simulations of dipolar swimmers [@Stenhammar2017; @Bardfalvy2019] revealed the presence of very strong correlations below the onset of collective motion. It was shown that various observables deviate from their mean-field values at any density of microswimmers [@Stenhammar2017], with the deviation diverging in the vicinity of the onset. The origin of such strong correlations can be readily attributed to the slow spatial decay of the dipolar velocity field, implying a simultaneous coupling between all microswimmers in the system. While this argument is intuitive enough, its implementation as a theoretical framework presents major technical challenges [@Stenhammar2017; @Qian2017; @Nambiar2019a], and only simplified cases were studied until now, notably by Stenhammar *et al.* [@Stenhammar2017], who considered a suspension of “shakers" – particles that apply forces to the fluid but do not self-propel. In this work we develop a kinetic theory that goes beyond the mean-field assumption for the general model of dilute microswimmer suspensions described above. Our theory explicitly includes particle self-propulsion, and is valid at any density of microswimmers below the onset of collective motion. This constitutes simultaneously a significant methodological development compared to the work by Stenhammar *et al.* [@Stenhammar2017], and a major advance in our understanding of one of the key models defining “wet" active matter. Our theory allows us to make explicit predictions for observables that can be directly set against experiments and numerical simulations. The paper is organised as follows. In Section \[section:theory\] we formulate a kinetic theory for a model suspension of pusher-like dipolar microswimmers. We explicitly find the dynamics of fluctuations around the homogeneous and isotropic state that describe the system below the onset of collective motion. Since our theory differs significantly from the previous work [@Stenhammar2017], we present its derivation in detail. We appreciate, however, that some readers might only be interested in the results of our theory without feeling the need to go through the rather technical Section \[section:theory\]. We, therefore, present our results in a stand-alone Section \[section:results\], which can be read without Section \[section:theory\]. There, we calculate the temporal and spatial correlation functions, fluid velocity variance, energy spectra, and the enhanced diffusivity of tracer particles. We conclude in Section \[section:discussion\], while Appendices contain additional derivations for technically oriented readers. Kinetic theory of strongly interacting suspensions {#section:theory} ================================================== Microscopic model {#subsection:microscopic} ----------------- We consider a collection of $N$ microswimmers contained in a volume $V$ at a finite number density $n=N/V$. The microswimmers are suspended in a Newtonian fluid with the viscosity $\mu$. Each microswimmer is described by its instantaneous position ${\bm x}_i$ and orientation ${\bm p}_i$, that we collectively denote by ${\bm z}_i = \left( {\bm x}_i, {\bm p}_i\right)$, where $i=1\dots N$ enumerates the particles. Within our model, the dynamics of the suspension is governed by the following equations of motion $$\begin{aligned} \dot{x}_{i}^{\alpha} &= v_{s} p_{i}^{\alpha} + \mathcal{U}^{\alpha}\left( {\bm x}_i \right), \label{xdot} \\ \dot{p}_{i}^{\alpha} &= \mathbb{P}_{i}^{\alpha\beta} \left(\mathcal{W}^{\beta\gamma}\left( {\bm x}_i \right) + B \mathcal{E}^{\beta\gamma}\left( {\bm x}_i \right) \right) p_{i}^{\gamma}, \label{pdot}\end{aligned}$$ where the dot denotes the time derivative, the superscript indices denote Cartesian components of vectors, and the subscript indices label the particles. Throughout this work, we utilise the Einstein summation convention for the superscript indices, while no summation is assumed over repeated subscript indices. Equations of motion and incorporate the following physical ingredients. First of all, each swimmer self-propels with the speed $v_s$ in the direction of its orientation. To induce self-propulsion, swimmers generate long-ranged flows in the suspending fluid [@Lauga2009]. The superposition of these flows at the position of the $i$-th swimmer, $\mathcal{U}^{\alpha}\left( {\bm x}_i \right)$, advects that particle in addition to its self-propulsion, see Eq., and re-orients it according to Jeffrey’s equation . The latter describes the dynamics of a passive particle in an external flow [@kimkarrila], with $$\begin{aligned} &\mathcal{W}^{\beta\gamma}\left( {\bm x}_i \right) = \frac{1}{2} \left( \nabla^{\gamma} \mathcal{U}^{\beta}\left( {\bm x}_i \right) - \nabla^{\beta} \mathcal{U}^{\gamma}\left( {\bm x}_i \right) \right), \\ &\mathcal{E}^{\beta\gamma} \left( {\bm x}_i \right)= \frac{1}{2} \left(\nabla^{\gamma} \mathcal{U}^{\beta}\left( {\bm x}_i \right) + \nabla^{\beta} \mathcal{U}^{\gamma}\left( {\bm x}_i \right) \right), \label{eq:Strain}\end{aligned}$$ being the Cartesian components of the vorticity and rate-of-strain tensors, respectively. In Eq., $\mathbb{P}_{i}^{\alpha\beta} = \delta^{\alpha\beta} - p^{\alpha}_{i}p_{i}^{\beta}$, is the projection operator, $\delta^{\alpha\beta}$ denotes the Kronecker delta, $\nabla_i^\alpha = \partial/\partial x_i^\alpha$, and $B = \left( a^2 - 1\right)/\left( a^2 + 1\right)$ is the measure of the swimmer’s nonsphericity [@kimkarrila] based on its aspect ratio $a$. For strongly elongated particles, $B\rightarrow 1$, while for spheres, $B=0$. Finally, each swimmer randomly changes its orientation with a rate $\lambda$, thus mimicking the run-and-tumble motion commonly exhibited by bacteria [@Berg1993]. We note here that we neglect the effects of rotational and translational diffusion on the particle’s dynamics, and random tumbling is thus the only source of stochasticity in our model. The velocity field generated by a self-propelled particle sufficiently far away from its surface is often well-described by the field produced by a point dipole with the same position and orientation [@Lauga2009; @Drescher2011]. In a dilute suspension of microswimmers, where the particles are sufficiently separated from each other, we can approximate $\mathcal{U}^{\alpha}\left( {\bm x}_i \right)$ by a sum of dipolar contributions $$\begin{aligned} \mathcal{U}^{\alpha}\left( {\bm x}_i \right) = \sum_{j\neq i}^{N} u_{d}^{\alpha}({\bm x}_{i}; {\bm z}_{j}),\end{aligned}$$ where $$\begin{aligned} {\bm u}_{d}({\bm x}_{i}; {\bm z}_{j}) = \frac{\kappa}{8\pi} \left[ 3 \frac{\left( {\bm p}_j\cdot {\bm x}'\right)^2 {\bm x}' + \epsilon^2 \left( {\bm p}_j\cdot {\bm x}'\right) {\bm p}_j }{\left( x'^2 + \epsilon^2 \right)^{5/2}} \right. \nonumber \\ \left. -\frac{{\bm x}'}{\left( x'^2 + \epsilon^2 \right)^{3/2}} \right] \label{realspacedipole}\end{aligned}$$ is the velocity field generated at ${\bm x}_{i}$ by a hydrodynamic dipole located at ${\bm x}_{j}$ with the orientation ${\bm p}_{j}$. Here, $\kappa = F l/\mu$ is the reduced dipolar strength, where $F$ is the magnitude of the forces applied to the fluid, $l$ is the dipolar length, and $\mu$ is the viscosity of the fluid; ${\bm x}' = {\bm x}_i -{\bm x}_j$, and $x'$ denotes the length of ${\bm x}'$. The dipole consists of two regularised Stokeslets, that were introduced by Cortez *et al.* [@Cortez2005], with $\epsilon$ being the regularisation length of the order of swimmer size. For pushers, $\kappa > 0$. The main goal of our work is to calculate spatial and temporal correlations of the fluid velocity in microswimmer suspensions described by the model above. Both quantities can be succinctly expressed through a combined correlation function $$\begin{aligned} &C(R,T) \nonumber \\ &\qquad = \lim_{t\rightarrow\infty} \frac{1}{V}\int d{\bm x} \,\overline{U^{\alpha}\left( {\bm x}, t \right) U^{\alpha}\left( {\bm x} + {\bm R}, t+T \right)}, \label{CRTgeneral}\end{aligned}$$ where $U^{\alpha}\left( {\bm x}, t \right)$ is the fluid velocity at the position ${\bm x}$ at time $t$, and the large-$t$ limit guarantees independence of the initial conditions. The spatial and temporal correlation functions are trivially recovered by setting $T=0$ and $R=0$, respectively. The bar in Eq. denotes the average over the history of tumble events, and reflect the stochastic nature of our model. To calculate this and similar averages, below we formulate a kinetic theory of microswimmer suspensions based on our macroscopic model. Such theories have been extensively studied at the mean-field level [@Liao2007; @Lau2009; @Zaid2011; @Underhill2011; @Belan2019; @Bardfalvy2019]. Here, we go beyond the mean-field approximation and explicitly take into account strong correlations between the swimmers caused by the long-range nature of their hydrodynamic fields, Eq.. Kinetic theory and BBGKY hierarchy ---------------------------------- The starting point of our theory is the $N$-particle probability distribution function $F_N\left({\bm z}_1, {\bm z}_2, \dots, {\bm z}_N, t \right)$ that gives the geometric probability of the system occupying a particular point in the $6N$-dimensional phase space $\left\{ {\bm z}_1, \dots, {\bm z}_N\right\} $ at time $t$. The $N$-particle probability distribution function is symmetric with respect to swapping particle labels, reflecting their indistinguishability, and is normalised $$\begin{aligned} \int d{\bm z}_1 \dots d{\bm z}_N F_N\left({\bm z}_1, \dots, {\bm z}_N, t \right) = 1. \label{normalisation}\end{aligned}$$ Its time dynamics is governed by the Master equation [@Balescu1975] $$\begin{aligned} \partial_{t} F_{N} & + \sum_{i=1}^{N} \Big[ \nabla_{i}^{\alpha}(\dot{x}_{i}^{\alpha} F_{N}) + \partial_{i}^{\alpha}(\dot{p}_{i}^{\alpha} F_{N}) \Big] \nonumber \\ & = - N \lambda F_{N} + \frac{\lambda}{4\pi} \sum_{i=1}^{N} \int \mathrm{d} {\bm p}_{i} F_{N}, \label{Liouville}\end{aligned}$$ where we introduced $\partial_{i}^{\alpha} = \mathbb{P}_{i}^{\alpha\beta} \partial/\partial p^\beta_i$. The l.h.s. of Eq. describes the probability fluxes to and from a particular point in the phase space due to the deterministic particle dynamics given by Eqs. and , while the r.h.s. gives the changes of the probability due to random tumbling from and into that phase space point [@Subramanian2009; @Koch2011]. Next, we introduce the $s$-particle correlation functions defined as $$\begin{aligned} &F_{s}\left({\bm z}_1,\dots, {\bm z}_s, t \right) = \frac{N!}{(N-s)!N^{s}} \nonumber \\ &\qquad\qquad \times \int d{\bm z}_{s+1} \dots d{\bm z}_N F_N\left({\bm z}_1, \dots, {\bm z}_N, t \right), \label{Fs}\end{aligned}$$ Below, we will only be interested in the first partial correlation functions $F_1$, $F_2$, and $F_3$, that we further express as $$\begin{aligned} F_2\left( {\bm z}_1, {\bm z}_2, t\right) = F_1\left( {\bm z}_1, t\right) F_1\left( {\bm z}_2, t \right) + G\left( {\bm z}_1, {\bm z}_2, t\right), \label{F2}\end{aligned}$$ and $$\begin{aligned} F_3\left( {\bm z}_1, {\bm z}_2, {\bm z}_3, t\right) = F_1\left( {\bm z}_1, t\right) F_1\left( {\bm z}_2, t \right) F_1\left( {\bm z}_3, t \right) \nonumber \\ + G\left( {\bm z}_1, {\bm z}_2, t\right) F_1\left( {\bm z}_3, t \right) + G\left( {\bm z}_1, {\bm z}_3, t\right) F_1\left( {\bm z}_2, t \right) \nonumber \\ + G\left( {\bm z}_2, {\bm z}_3, t\right) F_1\left( {\bm z}_1, t \right) + H\left( {\bm z}_1, {\bm z}_2, {\bm z}_3, t\right), \label{F3}\end{aligned}$$ where $G$ and $H$ are the irreducible (connected) correlation functions [@Balescu1975]. The time evolution of $F_s$ can be deduced from the Master equation by integrating it over $\left\{{\bm z}_{s+1},\dots,{\bm z}_N\right\}$. Integrating by parts and using Eqs. and , we obtain the following equations for the one- and two-particle irreducible correlation functions $$\begin{aligned} & \partial_t F_1({\bm z}, t) + {\mathcal L}[F_1({\bm z}, t)]({\bm z}) \nonumber \\ & \qquad\qquad = -N \nabla^\alpha \int d{\bm z}' G({\bm z},{\bm z}',t) u^\alpha_d ({\bm x}; {\bm z}') - N \mathbb{P}^{\alpha\beta} \frac{\partial}{\partial p^\beta} \int d{\bm z}' G({\bm z},{\bm z}',t) p^\gamma \mathbb{X}^{\alpha\mu\nu\gamma} \nabla^\mu u^\nu_d ({\bm x}; {\bm z}'), \label{EqF1} \\ \nonumber \\ & \partial_t G({\bm z}_1,{\bm z}_2, t) + {\mathcal L}[G({\bm z}_1,{\bm z}_2, t)]({\bm z}_1) + {\mathcal L}[G({\bm z}_1,{\bm z}_2, t)]({\bm z}_2) \nonumber \\ & \qquad \qquad + N \nabla_1^\alpha \left[ F_1({\bm z}_1,t) \int d{\bm z}' G({\bm z}_2,{\bm z}',t) u_d^\alpha({\bm x}_1;{\bm z}')\right] + N \nabla_2^\alpha \left[ F_1({\bm z}_2,t) \int d{\bm z}' G({\bm z}_1,{\bm z}',t) u_d^\alpha({\bm x}_2;{\bm z}')\right] \nonumber \\ & \qquad \qquad + N \mathbb{P}_1^{\alpha\beta} \frac{\partial}{\partial p_1^\beta}\left[ F_1({\bm z}_1,t) p_1^\gamma \mathbb{X}_1^{\alpha\mu\nu\gamma} \int d{\bm z}' G({\bm z}_2,{\bm z}',t)\nabla_1^\mu u_d^\nu({\bm x}_1;{\bm z}')\right] \nonumber \\ & \qquad \qquad + N \mathbb{P}_2^{\alpha\beta} \frac{\partial}{\partial p_2^\beta}\left[ F_1({\bm z}_2,t) p_2^\gamma \mathbb{X}_2^{\alpha\mu\nu\gamma} \int d{\bm z}' G({\bm z}_1,{\bm z}',t)\nabla_2^\mu u_d^\nu({\bm x}_2;{\bm z}')\right] \nonumber \\ & \qquad \qquad = - \mathcal{S}_{1,2}^{F} - \mathcal{S}_{2,1}^{F} - \mathcal{S}_{1,2}^{G} - \mathcal{S}_{2,1}^{G} - \mathcal{S}_{1,2}^{H} - \mathcal{S}_{2,1}^{H}, \label{EqG}\end{aligned}$$ where we have introduced the operator $$\begin{aligned} &{\mathcal L}[\Phi]({\bm z}) \nonumber \\ & \qquad = v_s p^\alpha \nabla^\alpha \Phi({\bm z}) + N \nabla^\alpha \big[ \Phi({\bm z}) \mathcal{U}^\alpha_{\text{MF}}({\bm x})\big] + N \mathbb{P}^{\alpha\beta} \frac{\partial}{\partial p^\beta}\big[ \Phi({\bm z}) p^\gamma \mathbb{X}^{\alpha\mu\nu\gamma} \nabla^\mu \mathcal{U}^\nu_{\text{MF}}({\bm x})\big] + \lambda \Phi({\bm z}) - \frac{\lambda}{4\pi} \int d{\bm p}\, \Phi({\bm z}), \label{operatorL}\end{aligned}$$ acting on the variable $\bm z$ of an arbitrary function $\Phi=\Phi({\bm z}_1,\dots,{\bm z}_N)$, and defined the mean-field velocity field as $$\begin{aligned} \mathcal{U}^\alpha_{\text{MF}}({\bm x}) = \int d{\bm z}' F_1({\bm z}',t) u^\alpha_d({\bm x}; {\bm z}'). \label{UMF}\end{aligned}$$ The rank-4 tensor $$\begin{aligned} \mathbb{X}_i^{\alpha\mu\nu\gamma} = \mathbb{P}_i^{\alpha\beta} \left[ \frac{B+1}{2}\delta^{\mu\gamma}\delta^{\nu\beta} + \frac{B-1}{2}\delta^{\mu\beta}\delta^{\nu\gamma} \right],\end{aligned}$$ encodes the tensorial structure of Jeffrey’s equation , and the r.h.s. of Eq. is given in terms of $$\begin{aligned} &\mathcal{S}_{i,j}^{F} = F_1({\bm z}_j,t)\Big\{ \nabla_i^\alpha \left[ F_1({\bm z}_i,t) u_d^\alpha({\bm x}_i;{\bm z}_j)\right] \nonumber \\ &+ \mathbb{P}_i^{\alpha\beta} \frac{\partial}{\partial p_i^\beta}\left[ F_1({\bm z}_i, t) p_i^\gamma \mathbb{X}_i^{\alpha\mu\nu\gamma} \nabla_i^\mu u_d^\nu({\bm x}_i; {\bm z}_j) \right] \Big\},\end{aligned}$$ $$\begin{aligned} & \mathcal{S}_{i,j}^{G} = \nabla_i^\alpha \left[ G({\bm z}_i,{\bm z}_j,t) u_d^\alpha({\bm x}_i;{\bm z}_j)\right] \nonumber \\ &+ \mathbb{P}_i^{\alpha\beta} \frac{\partial}{\partial p_i^\beta}\left[ G({\bm z}_i,{\bm z}_j,t) p_i^\gamma \mathbb{X}_i^{\alpha\mu\nu\gamma} \nabla_i^\mu u_d^\nu({\bm x}_i; {\bm z}_j) \right], \end{aligned}$$ and $$\begin{aligned} & \mathcal{S}_{i,j}^{H} = N \int d{\bm z}' \Big\{ \nabla_i^\alpha \left[ H({\bm z}_i,{\bm z}_j,{\bm z}',t) u_d^\alpha({\bm x}_i;{\bm z}')\right] \nonumber \\ &+ \mathbb{P}_i^{\alpha\beta} \frac{\partial}{\partial p_i^\beta}\left[ H({\bm z}_i,{\bm z}_j,{\bm z}',t) p_i^\gamma \mathbb{X}_i^{\alpha\mu\nu\gamma} \nabla_i^\mu u_d^\nu({\bm x}_i; {\bm z}') \right] \Big\}.\end{aligned}$$ Eqs. and are the beginning of a BBGKY hierarchy of equations for partial distribution functions [@Balescu1975]. As such, they do not form a closed system as they also depend on the three-particle irreducible distribution function $H$. Before discussing our choice of closure for this system of equations, let us briefly review the predictions of the mean-field approximation to Eqs. and , which consists of neglecting all correlation functions beyond $s=1$. The remaining equation determines the mean-field approximation to the one-particle correlation function $$\begin{aligned} \partial_t F_1^{\text{MF}}({\bm z}, t) + {\mathcal L}[F_1^{\text{MF}}({\bm z}, t)]({\bm z}) = 0, \label{EqF1MF}\end{aligned}$$ that has been extensively studied before [@Saintillan2008; @Saintillan2008a; @Subramanian2009; @Hohenegger2010; @Koch2011; @Saintillan2013; @Krishnamurthy2015]. One of the solutions of this equation is given by a constant, which is fixed to $F_1^{\text{MF}}({\bm z}, t)=1/(4\pi V)$ by the normalisation condition Eq.. This solution, which is valid at any number density, corresponds to a homogeneous and isotropic suspension of microswimmers. For pushers $(\kappa>0)$, this state loses its stability [@Saintillan2008; @Saintillan2008a; @Subramanian2009; @Hohenegger2010; @Stenhammar2017] at the critical number density of microswimmers $n_{crit} = 5\lambda/(B\kappa)$, while for pullers $(\kappa<0)$, the homogeneous and isotropic state is always linearly stable within the mean-field approximation. The homogeneous and isotropic mean-field solution implies that $N F_1^{\text{MF}}\sim n \sim O(1)$ is finite in the thermodynamic limit. This, in turn, implies that, to leading order, $G\sim O(N^{-2})$, $H\sim O(N^{-3})$, etc. A more comprehensive discussion of this statement, together with the required rescaling of the correlation functions, system parameters, and time is given elsewhere [@Stenhammar2017]. Building upon these results, here we *assume* that upon approaching the thermodynamic limit, $F_1$ is well-approximated by $F_1^{\text{MF}}$, since the r.h.s. of Eq. is $O(1/N)$ compared its l.h.s. In the homogeneous and isotropic state, the mean-field velocity vanishes $\mathcal{U}^\alpha_{\text{MF}}({\bm x}) = 0$, since the integral in Eq. is then proportional to the total flow rate through a surface surrounding the dipole. The latter is zero due to incompressibility. Fluctuations around the homogeneous and isotropic state are then governed by Eq. with $F_1=1/(4\pi V)$, and $\mathcal{S}_{i,j}^{G}$ and $\mathcal{S}_{i,j}^{H}$ neglected $$\begin{aligned} & \partial_t G({\bm z}_1,{\bm z}_2, t) + {\mathcal L}_{12} [G] + {\mathcal L}_{21} [G] \nonumber \\ & \qquad\qquad = \frac{3B}{\left(4\pi V\right)^2} \Big\{ p_1^\mu p_1^\nu \nabla_1^\mu u_d^\nu({\bm x}_1;{\bm z}_2) \nonumber \\ & \qquad\qquad\qquad\qquad\qquad+ p_2^\mu p_2^\nu \nabla_2^\mu u_d^\nu({\bm x}_2;{\bm z}_1) \Big\}, \label{EqGfinal}\end{aligned}$$ where $$\begin{aligned} & {\mathcal L}_{ij} [G] = v_s p_i^\alpha \nabla_i^\alpha G({\bm z}_1,{\bm z}_2, t) \nonumber \\ &\qquad\qquad - \frac{3 n B}{4\pi} p_i^\mu p_i^\nu \int d{\bm z}' G({\bm z}_j,{\bm z}', t) \nabla_i^\mu u_d^\nu({\bm x}_i;{\bm z}') \nonumber \\ & \qquad\qquad + \lambda G({\bm z}_1,{\bm z}_2, t) - \frac{\lambda}{4\pi} \int d{\bm p}_i\, G({\bm z}_1,{\bm z}_2, t). \label{Lij}\end{aligned}$$ This equation has previously been derived and analysed for the case of shakers ($v_s=0$) [@Stenhammar2017]. Here, we solve it in the general case $v_s>0$. Phase-space density fluctuations -------------------------------- While the two-point distribution function $G$, given by Eq., contains statistical information about fluctuations in the system, it is not straightforward to relate it to the spatial and temporal correlation function $C(R,T)$, Eq., that we seek to calculate. To establish this connection, we introduce a method based on the phase space density $$\begin{aligned} \varphi({\bm z},t) = \sum_{i=1}^N \delta({\bm z} - {\bm z}_i(t)),\end{aligned}$$ pioneered by Klimontovich [@klimontovich1967book]. Here, $\delta({\bm z})$ is the three-dimensional Dirac delta function. The average of the phase space density is related to $F_1$ as can be seen from $$\begin{aligned} &\overline\varphi({\bm z},t) = \int d{\bm z}_1 \dots d{\bm z}_N \sum_{i=1}^N \delta({\bm z} - {\bm z}_i) F_N({\bm z}_1, \dots, {\bm z}_N, t) \nonumber \\ & \qquad\qquad = N F_1({\bm z}, t),\end{aligned}$$ where we used Eq.. Fluctuations of the phase space density can formally be defined as $\delta \varphi = \varphi - \overline\varphi$, and their second moment is given by $$\begin{aligned} & G_K( {\bm z}', {\bm z}'', t) \equiv \overline{\delta \varphi({\bm z}',t) \delta \varphi({\bm z}'',t)} \nonumber \\ & \qquad = N^2 G ( {\bm z}', {\bm z}'', t) + N F_1({\bm z}',t) \delta({\bm z}' - {\bm z}''). \label{GK}\end{aligned}$$ Below, we refer to $G_K$ as the Klimontovich correlation function. Its utility is evident if one considers the spatial correlation function $C(R)$, defined in Eq. as $$\begin{aligned} &C(R) = \lim_{t\rightarrow\infty} \frac{1}{V}\int d{\bm x} \overline{U^{\alpha}\left( {\bm x}, t \right) U^{\alpha}\left( {\bm x} + {\bm R}, t \right)}.\end{aligned}$$ The velocity of the fluid at a position $\bm x$ is given by the superposition of the velocity fields generated by all swimmers $$\begin{aligned} &U^{\alpha}\left( {\bm x}, t \right) = \sum_{i=1}^N u_d^\alpha({\bm x}; {\bm z}_i(t)) \nonumber \\ &\qquad\qquad = \int d{\bm z}' \varphi({\bm z}',t) u_d^\alpha({\bm x}; {\bm z}'). \end{aligned}$$ Separating the phase space density into its average and fluctuations, $\varphi =\overline\varphi + \delta \varphi$, the spatial correlation function becomes $$\begin{aligned} &C(R) = \lim_{t\rightarrow\infty} \frac{1}{V}\int d{\bm x} \int d{\bm z}'d{\bm z}'' u_d^\alpha({\bm x}; {\bm z}') u_d^\alpha({\bm x}+{\bm R}; {\bm z}'') \nonumber \\ & \qquad\qquad\qquad \times \Bigg[ \left(\frac{n}{4\pi}\right)^2 + G_K( {\bm z}', {\bm z}'', t)\Bigg].\end{aligned}$$ The integral with the constant term vanishes, demonstrating that $G_K$ fully determines the spatial correlation function. Time evolution of the Klimontovich correlation function can readily be derived from Eqs. and , yielding $$\begin{aligned} & \partial_t G_K({\bm z}_1,{\bm z}_2, t) + {\mathcal L}_{12} [G_K] + {\mathcal L}_{21} [G_K] \nonumber \\ & \qquad\qquad = 2 \lambda \frac{n}{4\pi} \delta({\bm x}_1 - {\bm x}_2 ) \left[ \delta({\bm p}_1 - {\bm p}_2 ) - \frac{1}{4\pi} \right], \label{EqGK}\end{aligned}$$ where $ {\mathcal L}_{ij}$ is defined in Eq., and we used $F_1 = 1/(4\pi V)$ in the homogeneous and isotropic state. To solve Eq., we introduce an auxiliary field $h({\bm z}_1, t)$, that satisfies the following equation $$\begin{aligned} \partial_t h({\bm z}_1, t) + {\mathcal L}_{11} [h] = \xi({\bm z}_1, t), \label{Eqh}\end{aligned}$$ where $\xi$ is a noise term with the following properties $$\begin{aligned} & \langle \xi({\bm z}_1, t) \rangle = 0, \label{noiseaverage}\\ & \langle \xi({\bm z}_1, t) \xi({\bm z}_2, t')\rangle = 2 \lambda \frac{n}{4\pi} \delta(t-t') \delta({\bm x}_1 - {\bm x}_2 ) \nonumber \\ &\qquad\qquad \times \left[ \delta({\bm p}_1 - {\bm p}_2 ) - \frac{1}{4\pi} \right]. \label{noisevariance}\end{aligned}$$ Here, the angular brackets denote the average over the realisations of the noise $\xi$, and should not be confused with the ensemble averages that we denoted by bars in the equations above. Eq. allows us to factorise the Klimontovich correlation function as $$\begin{aligned} G_K({\bm z}_1,{\bm z}_2, t) = \langle h({\bm z}_1, t) h({\bm z}_2, t)\rangle,\end{aligned}$$ which replaces the deterministic Eq. by a significantly simpler stochastic Eq. with a fictitious noise $\xi$ with properly chosen spectral properties. Remarkably, the non-equal time correlations of the phase space density can be expressed through the same auxiliary field $$\begin{aligned} \overline{\delta \varphi({\bm z}',t') \delta \varphi({\bm z}'',t'')} = \langle h({\bm z}', t') h({\bm z}'', t'')\rangle,\end{aligned}$$ as implied by a seminal work of Klimontovich and Silin [@Silin1962]. This, finally, leads to a direct relationship between the field $h$, which encodes the statistical properties of fluctuations in the suspension, and the combined correlation function $$\begin{aligned} &C(R,T) = \lim_{t\rightarrow\infty} \frac{1}{V}\int d{\bm x} \int d{\bm z}'d{\bm z}'' \nonumber \\ & \times u_d^\alpha({\bm x}; {\bm z}') u_d^\alpha({\bm x}+{\bm R}; {\bm z}'') \langle h({\bm z}', t) h({\bm z}'', t+T) \rangle. \label{CRTtmp}\end{aligned}$$ Dynamics of the auxiliary field $h$ {#subsection:h} ----------------------------------- Here, we explicitly find the solution to Eq. together with Eqs. and . Since Eq. is linear in $h$, we introduce the Fourier $$\begin{aligned} h({\bm z}, t) = \frac{1}{(2\pi)^3} \int d{\bm k} e^{i {\bm k}\cdot{\bm x}} \hat{h}({\bm k}, {\bm p}, t), \label{FT}\end{aligned}$$ and the Laplace transforms $$\begin{aligned} \hat{h}({\bm k}, {\bm p}, s) = \int_0^\infty dt e^{-s t} \hat{h}({\bm k}, {\bm p}, t).\end{aligned}$$ We will also require the Fourier transform of the regularised dipolar field, Eq., which is given by $$\begin{aligned} & u_d^\nu({\bm x}; {\bm z}') = \frac{-i \kappa}{(2\pi)^3} \int d{\bm k} \, e^{i {\bm k}\cdot ({\bm x}-{\bm x}')} \nonumber \\ & \qquad\qquad\qquad \times \frac{A(k \epsilon)}{k} (\hat{\bm k} \cdot {\bm p}') \left( \delta^{\nu\delta} - \hat{k}^\nu \hat{k}^\delta \right) p'^{\delta}, \label{udfourier}\end{aligned}$$ where $\hat{\bm k}={\bm k}/k$, and $k=|{\bm k}|$. The function $A$, defined as $$\begin{aligned} A(x)=\frac{1}{2} x^2 K_2(x),\end{aligned}$$ with $K_2(x)$ being the modified Bessel function of the second kind, is close to unity for $x<1$, and quickly approaches zero for $x>1$. It will serve as a regularisation of the integrals over $k$, suppressing contributions from lengthscales smaller than the size of individual microswimmers. Performing the Fourier and Laplace transforms of Eq., we obtain after re-arranging $$\begin{aligned} &\hat{h}({\bm k}, {\bm p}, s) = \frac{1}{\sigma({\bm k},{\bm p}, s)} \Bigg[\hat{h}_0({\bm k}, {\bm p}) + \hat{\xi}({\bm k}, {\bm p},s) \nonumber \\ & \qquad + \frac{\lambda}{4\pi} I^{(0)}({\bm k},s) + \frac{15 \lambda}{4\pi} \Delta A(k\epsilon) \bigg\{ (\hat{\bm k}\cdot {\bm p}) I^{(1)}({\bm k},{\bm p}, s) \nonumber \\ & \qquad\qquad\qquad\qquad\qquad - (\hat{\bm k}\cdot {\bm p})^2 I^{(2)}({\bm k}, s) \bigg\} \Bigg]. \label{hfullsolution}\end{aligned}$$ Here, $\hat{\xi}({\bm k}, {\bm p},s)$ is the Fourier-Laplace transform of the noise, $\sigma({\bm k},{\bm p}, s) = s + \lambda + i v_s ({\bm k}\cdot{\bm p})$, and we defined $$\begin{aligned} I^{(0)}({\bm k},s) &= \int d{\bm p } \,\hat{h}({\bm k}, {\bm p}, s), \label{I0} \\ I^{(1)}({\bm k},{\bm p}, s) &= \int d{\bm p}' (\hat{\bm k}\cdot {\bm p}')({\bm p}\cdot {\bm p}') \hat{h}({\bm k}, {\bm p}', s), \label{I1} \\ I^{(2)}({\bm k},s) &= \int d{\bm p }\, (\hat{\bm k}\cdot {\bm p})^2 \hat{h}({\bm k}, {\bm p}, s). \label{I2}\end{aligned}$$ In Eq., $\hat{h}({\bm k}, {\bm p}, t=0) = \hat{h}_0({\bm k}, {\bm p})$ denotes some arbitrary initial condition; below we demonstrate that the long-time statistical properties of the suspension are insensitive to $\hat{h}_0({\bm k}, {\bm p})$. In Eq., we have also introduced an important dimensionless parameter $\Delta = n/n_{crit}$, where $n_{crit} = 5\lambda/(B\kappa)$ is the mean-field onset of collective motion in pusher suspensions, $\kappa>0$. For pushers, $\Delta$ measures the dimensionless distance from the onset, with $\Delta=1$ corresponding to the instability. Eq. is a linear integral equation for $\hat{h}({\bm k}, {\bm p}, s)$ and its solution is straightforward. Substituting Eq. into Eqs.-, gives $$\begin{aligned} &I^{(0)}({\bm k},s) \nonumber \\ &\qquad\qquad= \frac{1}{1-\frac{\lambda}{4\pi}f_0} \int d{\bm p} \frac{\hat{h}_0({\bm k}, {\bm p}) + \hat{\xi}({\bm k}, {\bm p},s)}{\sigma({\bm k},{\bm p}, s)}, \label{I0sol}\end{aligned}$$ $$\begin{aligned} &I^{(2)}({\bm k},s) = \frac{\lambda}{4\pi} f_1 I^{(0)}({\bm k},s) \nonumber \\ & \qquad\qquad + \int d{\bm p} (\hat{\bm k}\cdot {\bm p})^2\frac{\hat{h}_0({\bm k}, {\bm p}) + \hat{\xi}({\bm k}, {\bm p},s)}{\sigma({\bm k},{\bm p}, s)}, \label{I1sol}\end{aligned}$$ $$\begin{aligned} &I^{(1)}({\bm k},{\bm p}, s) = \frac{1}{1+\frac{15 \lambda}{8\pi} \Delta A(k\epsilon) (f_2 - f_1)} \Bigg[ \nonumber \\ & \quad \int d{\bm p}' (\hat{\bm k}\cdot {\bm p}')({\bm p}\cdot {\bm p}') \frac{\hat{h}_0({\bm k}, {\bm p}') + \hat{\xi}({\bm k}, {\bm p}',s)}{\sigma({\bm k},{\bm p}', s)} \nonumber \\ & \quad + (\hat{\bm k}\cdot {\bm p}) \bigg\{ \frac{\lambda}{4\pi} f_1 I^{(0)}({\bm k},s) \nonumber \\ & \qquad\qquad\qquad + \frac{15 \lambda}{8\pi} \Delta A(k\epsilon) (f_2 - f_1) I^{(2)}({\bm k},s) \bigg\} \Bigg], \label{I2sol}\end{aligned}$$ where $$\begin{aligned} f_n = 2\pi \int_{-1}^{1} dx \frac{x^{2n}}{s+\lambda + i v_s k x}.\end{aligned}$$ Having found the explicit expression for $\hat{h}({\bm k}, {\bm p},s)$, we proceed to calculate the combined correlation function, Eq.. Below, we show that only a small number of terms from Eqs. and - contribute to $C(R,T)$. $C(R,T)$ in terms of $\hat{h}({\bm k}, {\bm p},s)$ {#subsection:hhat} -------------------------------------------------- In what follows, it will be convenient to re-write $C(R,T)$ in terms of the Fourier and Laplace transforms of all quantities. Substituting Eq. into Eq., and using the Fourier representation of the regularised dipolar field, Eq., we obtain $$\begin{aligned} &C(R,T) = \lim_{t\rightarrow\infty} {\mathcal L}^{-1}_{s_1,t} {\mathcal L}^{-1}_{s_2,t+T} \frac{\kappa^2}{(2\pi)^3 V} \nonumber \\ &\qquad \times \int d{\bm k} e^{-i {\bm k}\cdot{\bm R}} \frac{A^2(k\epsilon)}{k^2} \int d{\bm p}_1 d{\bm p}_2 (\hat{\bm k} \cdot {\bm p}_1) (\hat{\bm k} \cdot {\bm p}_2) \nonumber \\ & \qquad\qquad\qquad \times \big( \delta^{\alpha\beta} - \hat{k}^\alpha \hat{k}^\beta \big) p_1^{\beta} \big( \delta^{\alpha\gamma} - \hat{k}^\alpha \hat{k}^\gamma \big) p_2^{\gamma} \nonumber \\ & \qquad\qquad\qquad \times \langle \hat{h}({\bm k}, {\bm p}_1,s_1)\hat{h}(-{\bm k}, {\bm p}_2,s_2) \rangle_{\hat{\xi}}, \label{CRTlaplace}\end{aligned}$$ where ${\mathcal L}^{-1}_{s,t}$ formally denotes the inverse Laplace transform from $s$ to $t$, given by the Bromwich integral [@Doetsch1974]. The angular brackets $\langle\dots\rangle_{\hat{\xi}}$ denote the average with the Fourier-Laplace components of the noise $\xi$, with the following spectral properties $$\begin{aligned} & \langle \hat{\xi}({\bm k}, {\bm p}, s) \rangle_{\hat{\xi}} = 0, \label{noiseaverageFL}\\ & \langle \hat{\xi}({\bm k}, {\bm p}_1, s_1) \hat{\xi}(-{\bm k}, {\bm p}_2, s_2)\rangle_{\hat{\xi}} = 2 \lambda V \frac{n}{4\pi} \nonumber \\ &\qquad\qquad\qquad\qquad \times \frac{1}{s_1+s_2} \left[ \delta({\bm p}_1 - {\bm p}_2 ) - \frac{1}{4\pi} \right], \label{noisevarianceFL}\end{aligned}$$ obtained by applying the Fourier-Laplace transform to Eqs. and . While the average in Eq. can readily be formed using the solution for $\hat h$ found in Section \[subsection:h\], the result is very cumbersome. Before proceeding, we make two observations that greatly reduce the number of terms contributing to Eq.. First, we observe that $$\begin{aligned} \int d{\bm p} \, \big( \delta^{\alpha\beta} - \hat{k}^\alpha \hat{k}^\beta \big) p^{\beta} f(\hat{\bm k} \cdot {\bm p}) = 0,\end{aligned}$$ where $f$ is an arbitrary function of $\hat{\bm k} \cdot {\bm p}$. This statement is readily demonstrated by representing $\bm p$ in spherical coordinates with $\hat{\bm k}$ selected along the $z$-axis, and performing the angular integrals component-wise. This result has profound implications for the average $\langle \hat{h}({\bm k}, {\bm p}_1,s_1)\hat{h}(-{\bm k}, {\bm p}_2,s_2) \rangle_{\hat{\xi}}$ in Eq.. Every term in $\hat{h}({\bm k}, {\bm p}_1,s_1)$, Eq., that only depends on ${\bm p}_1$ through its dependence on $(\hat{\bm k} \cdot {\bm p}_1)$ does not contribute to $C(R,T)$, as its integral over ${\bm p}_1$ with the corresponding dipolar field in Eq. vanishes. The same applies to $\hat{h}(-{\bm k}, {\bm p}_2,s_2)$. The second observation is related to the initial condition. All terms that involve $\hat{h}_0({\bm k}, {\bm p})$ only depend on the Laplace frequency $s$ through $1/\sigma({\bm k},{\bm p}, s)$, and their inverse Laplace transform can be readily performed before any other integration. Since the inverse Laplace transform of $1/(s+a)$ is $e^{-a t}$, where $a$ is a complex number, the dominant long-time behaviour of such terms is given by $e^{-\lambda t}$, where we ignored the subdominant oscillatory dependencies. In Eq., we are interested in the $t\rightarrow$ limit, and these terms also do not contribute to $C(R,T)$. With these observations in mind, Eq. can be significantly simplified to read $$\begin{aligned} &\hat{h}({\bm k}, {\bm p}, s) \cong \frac{\hat{\xi}({\bm k}, {\bm p},s)}{\sigma({\bm k},{\bm p}, s)} \nonumber \\ &\qquad\qquad + \frac{ (\hat{\bm k}\cdot {\bm p}) }{\sigma({\bm k},{\bm p}, s)} \frac{\frac{15 \lambda}{4\pi} \Delta A(k\epsilon) }{1+\frac{15 \lambda}{8\pi} \Delta A(k\epsilon) (f_2 - f_1)} \nonumber \\ & \qquad\qquad\qquad \times \int d{\bm p}' (\hat{\bm k}\cdot {\bm p}')({\bm p}\cdot {\bm p}') \frac{\hat{\xi}({\bm k}, {\bm p}',s)}{\sigma({\bm k},{\bm p}', s)}, \label{hneeded}\end{aligned}$$ where $\cong$ signifies that we only kept the terms that contribute to $C(R,T)$. Now, the average $\langle \hat{h}({\bm k}, {\bm p}_1,s_1)\hat{h}(-{\bm k}, {\bm p}_2,s_2) \rangle_{\hat{\xi}}$ assumes a tractable form that can be used in Eq.. Separating the terms independent of $\Delta$, we obtain $C(R,T) = C_0(R,T) + C_1(R,T)$. Here, $$\begin{aligned} &C_0(R,T) = \frac{\lambda n \kappa^2}{16\pi^4 } \lim_{t\rightarrow\infty} {\mathcal L}^{-1}_{s_1,t} {\mathcal L}^{-1}_{s_2,t+T} \int d{\bm k} e^{-i {\bm k}\cdot{\bm R}} \frac{A^2(k\epsilon)}{k^2} \nonumber \\ &\quad \times \int d{\bm p} (\hat{\bm k} \cdot {\bm p})^2 \left[1- (\hat{\bm k} \cdot {\bm p})^2\right] \frac{1}{s_1+s_2}\nonumber \\ & \qquad \times \frac{1}{\lambda + s_1 + i v_s k (\hat{\bm k}\cdot {\bm p})} \frac{1}{\lambda + s_2 - i v_s k (\hat{\bm k}\cdot {\bm p})}, \label{C0Laplace}\end{aligned}$$ represents correlations in the fluid created by non-interacting swimmers. The double inverse Laplace transform in the equation above can be performed using the method outlined in Appendix \[appendix:Cesare\]. It yields $$\begin{aligned} & \lim_{t\rightarrow\infty} {\mathcal L}^{-1}_{s_1,t} {\mathcal L}^{-1}_{s_2,t+T} \frac{1}{s_1+s_2} \frac{1}{\lambda + s_1 + i v_s k (\hat{\bm k}\cdot {\bm p})} \nonumber \\ & \qquad\times \frac{1}{\lambda + s_2 - i v_s k (\hat{\bm k}\cdot {\bm p})} = \frac{e^{-\lambda T + i v_s k T (\hat{\bm k} \cdot {\bm p})}}{2\lambda}. \label{dilt_example}\end{aligned}$$ Performing the angular integration, we finally obtain $$\begin{aligned} & C_0(R,T) = \frac{n \kappa^2 e^{-\lambda T}}{\pi^2} \int_{0}^\infty dk \frac{\sin{k R}}{k R} A^2(k\epsilon) \nonumber \\ &\quad \times \frac{ y(12-y^2) \cos{y} - (12-5y^2)\sin{y} }{y^5} \Bigg\vert_{y=v_s k T}. \label{C0final}\end{aligned}$$ All other terms in Eq. correspond to additional correlations generated by the hydrodynamic interactions among the swimmers, and, as such, they are dependent on the dimensionless microswimmer density $\Delta$. Performing the angular integration over ${\bm p}_1$ and ${\bm p}_2$, gives $$\begin{aligned} &C_1(R,T) \nonumber \\ & = \frac{2 \lambda n \kappa^2 }{15 \pi^2} \lim_{t\rightarrow\infty} {\mathcal L}^{-1}_{s_1,t} {\mathcal L}^{-1}_{s_2,t+T} \int_{0}^\infty dk \frac{\sin{k R}}{k R} A^2(k\epsilon) \nonumber \\ &\times \frac{1}{\lambda + s_1}\frac{1}{\lambda + s_2}\frac{1}{s_1 + s_2} \frac{z_1 \psi(z_1) + z_2 \psi(z_2)}{z_1+z_2} \nonumber \\ &\qquad \times\Bigg[ \frac{z_1 \psi(z_1)}{\omega - z_1 \psi(z_1)} + \frac{z_2 \psi(z_2)}{\omega - z_2 \psi(z_2)} \nonumber \\ &\qquad\qquad + \frac{z_1 z_2 \psi(z_1)\psi(z_2)}{\left(\omega - z_1 \psi(z_1)\right)\left(\omega - z_2 \psi(z_2)\right)} \Bigg]. \label{C1laplace}\end{aligned}$$ Here, we introduced $\omega = v_s k/(\lambda \Delta A(k\epsilon))$, and the function $\psi(z)$, defined as $$\begin{aligned} \psi(z) = \frac{5}{2}\frac{3z+2z^3-3(1+z^2)\arctan{z}}{z^5}, \label{psi}\end{aligned}$$ which is related to $f_2-f_1$ used in the previous Section. The variable $z_i=v_s k/(\lambda + s_i)$ allows us to write Eq. in a compact form but hides its complex dependence on the Laplace frequencies $s_1$ and $s_2$. Its inverse Laplace transform is discussed below. Approximate double inverse Laplace transform {#subsection:approximateDILT} -------------------------------------------- The integrand of Eq. is not a rational function of $s_1$ and $s_2$, and we were unable to calculate its double inverse Laplace transform exactly. Instead, here we develop a rational approximation to $\psi(z)$ that will allow us to find $C_1(R,T)$ analytically. First, we observe that if the poles of an analytic function are known, its large-$t$ behaviour is determined by the pole with the smallest negative real part [@Doetsch1974]. Therefore, the presence of the pole at $-\lambda$ in Eq. makes all poles with real parts smaller than $-\lambda$ irrelevant in the large-$t$ limit. This reflects the fact that individual tumbling events are always a source of de-correlation between microswimmers. Next, we introduce $L=v_s/(\lambda \epsilon)$, which compares the typical runlength of a swimmer to its size. In this work we consider $L=0-25$, ranging from non-swimming (shaker) particles to wild-type *E.coli* bacteria [@Berg1993]. Contributions to the integrand in Eq. with $k \epsilon>1$ are strongly suppressed by the regularising factor $A(k\epsilon)$, and therefore, when approximating $\psi(z)$, the relevant domain is $-\lambda<Re(s)<0$, with $v_s k/\lambda$ not exceeding $L$. In Appendix \[appendix:psia\] we show that a surprisingly good approximation to $\psi(z)$ on this domain is given by $$\begin{aligned} \psi_a(z) = \frac{7}{7+3 z^2}. \label{psia}\end{aligned}$$ The simple structure of this expression allows us to deduce the pole structure of the integrand in Eq.. Indeed, with $\psi(z)$ replaced by $\psi_a(z)$, and factorising $$\begin{aligned} \frac{1}{\omega - z \psi(z)} = \frac{7+3z^2}{3\omega\left(z-z_{+}\right)\left(z-z_{-}\right)},\end{aligned}$$ where $$\begin{aligned} z_{\pm} =\frac{7}{6\omega} \left[ 1\pm\sqrt{1-\frac{12}{7}\omega^2} \right], \label{z012}\end{aligned}$$ the denominators in Eq. can now be written as products of linear polynomials in $s_1$ and $s_2$. It is now straightforward to perform the inverse Laplace transform of this expression using the method outlined in Appendix \[appendix:Cesare\]. Taking the limit of $t\rightarrow\infty$, finally gives $$\begin{aligned} &C_1(\rho,\tau) \nonumber \\ & = e^{-\tau} \frac{n \kappa^2 }{15 \pi^2 \epsilon} \int_{0}^\infty d\xi \frac{\sin{\xi \rho}}{\xi \rho} A^2(\xi) \Bigg[ -\cos{\left(\sqrt{\frac{3}{7}}L\xi\tau\right)} \nonumber \\ &+ \frac{e^{\frac{1}{2}A(\xi) \Delta \tau}}{1-A(\xi)\Delta + \frac{3}{7}L^2\xi^2} \Bigg\{ \nonumber \\ & \frac{2-A(\xi)\Delta + \frac{6}{7}L^2\xi^2}{2-A(\xi)\Delta}\cosh{\left(\frac{1}{2}A(\xi) \Delta \tau \sqrt{1-\frac{12 L^2\xi^2}{7 A^2(\xi)\Delta^2}}\right)} \nonumber \\ &\qquad +\frac{\sinh{\left(\frac{1}{2}A(\xi) \Delta \tau \sqrt{1-\frac{12 L^2\xi^2}{7 A^2\left(\xi\right)\Delta^2}}\right)}}{\sqrt{1-\frac{12 L^2\xi^2}{7 A^2\left(\xi\right)\Delta^2}}} \Bigg\} \Bigg], \label{C1final}\end{aligned}$$ where we changed the integration variable to $\xi = k\epsilon$, and introduced the dimensionless parameters $\rho = R/\epsilon$ and $\tau = \lambda T$. In Appendix \[appendix:dilt\], we verify that Eq. provides a good approximation to the long-time behaviour of Eq.. Results {#section:results} ======= For the benefit of the readers who have skipped Section \[section:theory\], we repeat our main result, which comprises an explicit expression for the combined correlation function $C(R,T)$, defined in Eq.. It describes the steady-state correlations between the fluid velocity at two points in space separated by a distance $R$, and two instances in time separated by a time-interval $T$. Our theory includes full hydrodynamic interactions between microswimmers and is valid at any density up to the onset of collective motion. The result consists of the non-interacting part, $$\begin{aligned} & C_0(\rho,\tau) = \frac{n \kappa^2 e^{-\tau}}{\pi^2\epsilon} \int_{0}^\infty d\xi \frac{\sin{\xi \rho}}{\xi \rho} A^2(\xi) \nonumber \\ &\quad \times \frac{ y(12-y^2) \cos{y} - (12-5y^2)\sin{y} }{y^5} \Bigg\vert_{y=L\xi\tau}, \label{ResultsC0}\end{aligned}$$ and the interacting correlation function $C_1(\rho,\tau)$, given in Eq.. Here, $\rho = R/\epsilon$, where $\epsilon$ is a lengthscale comparable to the microswimmer size, and $\tau = \lambda T$, where $\lambda$ is the tumbling rate. The parameter $L=v_s/(\lambda \epsilon)$ compares the typical distance covered by a swimming microorganism between two tumble events to its size. Finally, $\Delta = n/n_{crit}$ is the dimensionless number density of the particles, where $n_{crit} = 5\lambda/(B\kappa)$ is the onset of collective motion for pusher-like microswimmers [@Subramanian2009; @Hohenegger2010; @Stenhammar2017]; the parameter $B$, is defined after Eq.. Our theory is valid for $\Delta<1$. The full expression, $C(\rho,\tau)=C_0(\rho,\tau)+C_1(\rho,\tau)$, given as a definite integral, constitutes the main technical result of our study. We now explicitly work out its predictions for the spatial and temporal correlation functions, and other experimentally accessible observables. When discussing their physical meaning, we are going to vary the dimensionless persistence length $L$, while keeping all the other parameters of the microswimmers fixed. We note that in reality the dipolar strength and shape of a microorganism uniquely determine its swimming speed, and hence $L$. We, however, see varying $L$ as a tool to disentangle the effects of self-propulsion (ability to change one’s position in space) from the strength of the hydrodynamic disturbances it causes. In particular, we will consider two limiting cases: shakers ($L=0$) and fast swimmers ($L\rightarrow\infty$). The former case corresponds to microswimmers that exert dipolar forces on the fluid but do not self-propel, and only change their positions due to being advected by the velocity fields created by other microswimmers [@Stenhammar2017]. The latter case, while obviously non-physical, is a useful tool to assess the effect of fast swimming on various quantities of interest. Finally, we note that the terms representing hydrodynamic interactions in Eq. are proportional to the swimmer’s nonsphericity $B$ that enters Jeffrey equation, Eq.. The limit of non-interacting microswimmers therefore corresponds to setting $B$ to zero, which, in turn, can be achieved by setting $\Delta=0$, while keeping $n$ finite. Velocity variance ----------------- ![The fluid velocity variance $\langle U^2 \rangle$ normalised by its non-interacting value $\langle U^2 \rangle_0$ for various values of $L$. The dotted line represents the non-interacting case $\langle U^2 \rangle=\langle U^2 \rangle_0$. Note that the $L\rightarrow\infty$ line turns sharply upwards and diverges in the vicinity of $\Delta=1$ in a way that cannot be resolved on the scale of this graph.[]{data-label="Fig_Variance"}](Variance.pdf){width="0.8\linewidth"} Our first quantity of interest is the fluid velocity variance, $\langle U^2 \rangle \equiv C(\rho=0,\tau=0)$. In the absence of thermal noise, re-arrangements of the microswimmer positions and orientations is the sole source of fluid velocity fluctuations. For this reason, it was used in previous studies as an order parameter to identify the onset of collective motion [@Stenhammar2017; @Bardfalvy2019]. Summing up Eqs. and , and setting $\rho=0$ and $\tau=0$, we obtain $$\begin{aligned} & \langle U^2 \rangle = \frac{\kappa^2 n}{15 \pi^2 \epsilon} \int_{0}^\infty d\xi A^2(\xi) \nonumber \\ & \qquad\qquad \times \frac{2-A(\xi)\Delta+\frac{6}{7}L^2 \xi^2}{\left(2-A(\xi)\Delta\right)\left(1-A(\xi)\Delta+\frac{3}{7}L^2 \xi^2\right)}. \label{resultvarience}\end{aligned}$$ We evaluate this integral numerically and plot the fluid velocity variance normalised by its value in the non-interacting case, $\langle U^2 \rangle(\Delta=0)\equiv \langle U^2 \rangle_0$, given by [@Bardfalvy2019] $$\begin{aligned} \langle U^2 \rangle_0 = \frac{\kappa^2 n}{15 \pi^2 \epsilon} \int_{0}^\infty d\xi A^2(\xi) = \frac{21\kappa^2 n}{2048\epsilon}.\end{aligned}$$ Note that $\langle U^2 \rangle_0$ corresponds to a superposition of uncorrelated fluctuations in the fluid velocity, which, by virtue of the central limit theorem, is proportional to $n$. Any deviations of $\langle U^2 \rangle$ from that value signify the presence of correlations. As can be seen in Fig.\[Fig\_Variance\], the fluid velocity fluctuations exhibit significant correlations at any density of the microswimmers, as was recognised previously [@Stenhammar2017]. Starting from its non-interacting value at $\Delta=0$, the variance increases with $\Delta$, until it diverges at the onset of collective motion. The strongest correlations are exhibited by suspensions of shakers, while swimming acts to reduce correlations. For large but finite values of $L$, the variance increases mildly from its non-interacting value, until it rises sharply in a small vicinity of $\Delta=1$, with the size of this region shrinking with $L$. Interestingly, the rise of $\langle U^2 \rangle_0$ for $\Delta<1$ remains finite even in the $L\rightarrow\infty$ limit. In other words, while swimming clearly reduces correlations, it does not remove them entirely, and the suspension is never described by the mean-field theory. To determine the scaling of the fluid velocity variance as $\Delta\rightarrow 1$, we observe that in that limit the integrand in Eq. is dominated by small values of $\xi$, where $A(\xi) \approx 1 - \xi^2/4$. Using this approximation in Eq. and replacing the upper integration limit by unity, we obtain $$\begin{aligned} & \langle U^2 \rangle \sim \frac{\kappa^2 n}{15 \pi \epsilon} \frac{1}{\sqrt{1+\frac{12}{7}L^2}\sqrt{1-\Delta}}, \quad \Delta \rightarrow 1. \label{varianceasymptotic}\end{aligned}$$ Therefore, our theory predicts that the fluid velocity variance diverges as $(1-\Delta)^{-1/2}$ in the vicinity of the transition to collective motion, irrespectively of $L$. Spatial correlations -------------------- {width="1.0\linewidth"} Our next quantity of interest is the equal-time spatial correlation function, $C(\rho, T=0)$, given by $$\begin{aligned} &C(\rho) = \frac{\kappa^2 n}{15 \pi^2 \epsilon} \int_{0}^\infty d\xi \frac{\sin{\xi \rho}}{\xi \rho} A^2(\xi) \nonumber \\ & \qquad\qquad \times \frac{2-A(\xi)\Delta+\frac{6}{7}L^2 \xi^2}{\left(2-A(\xi)\Delta\right)\left(1-A(\xi)\Delta+\frac{3}{7}L^2 \xi^2\right)}. \label{spatialcorrelations}\end{aligned}$$ While this integral cannot be evaluated analytically, a good approximation can be obtained by setting $A(\xi)=1$ in the integrand, yielding $$\begin{aligned} &C(\rho) \approx \frac{\kappa^2 n}{30 \pi \epsilon \left(1-\Delta\right)\rho} \nonumber \\ &\qquad\qquad \times \left[ 1 - \frac{\Delta}{2-\Delta} \exp{\left\{-\sqrt{\frac{7}{3}}\sqrt{1-\Delta}\frac{\rho}{L} \right\}}\right]. \label{spatialcorrelationsanalytic}\end{aligned}$$ For $\Delta=0$, this equation reproduces the result obtained previously for non-interacting swimmers [@Zaid2011; @Underhill2011; @Belan2019; @Bardfalvy2019]. In Fig.\[Fig\_spatial\] we evaluate Eq. numerically and compare it against the analytic approximation, Eq.; $\kappa^2 n_{crit}/(15 \pi^2 \epsilon)$ is chosen as the normalisation factor. For all values of $L$ and $\Delta$, the approximation works well for all but small spatial separations $\rho$, where the spatial correlation function is, essentially, equal to the fluid velocity variance. As with the fluid velocity variance, the strongest correlations are exhibited by suspensions of shakers, $L=0$. In this case, the spatial correlation function changes very slowly at short distances, and decays as $\rho^{-1}$ at large distances. Close to the onset of collective motion, the typical scale $\rho_0$ at which the crossover occurs can be estimated from Eqs. and , by requiring that $C(\rho_0) = \langle U^2 \rangle$. For $L=0$, this yields $\rho_0 \sim (1-\Delta)^{-1/2}$. This is readily verified by the data in Fig.\[Fig\_spatial\]A: As the system approaches the onset of collective motion, the overall strength of the correlations grows, with the region of strong correlations extending to progressively larger scales. ![The spatial correlation function $C(\rho)$ as a function of the distance $\rho$ for $\Delta=0.9$ and various values of $L$. At sufficiently large distances, $C(\rho)$ recovers the shaker behaviour, while at small distances correlations are suppressed by swimming. Note that the $L\rightarrow\infty$ line, serving as the limit beyond which correlations cannot be suppressed, joins the shaker line at $\rho\rightarrow\infty$.[]{data-label="Fig_spatial_delta09"}](SpatialDelta09.pdf){width="0.8\linewidth"} The effect of swimming on the behaviour of $C(\rho)$ is demonstrated in Figs.\[Fig\_spatial\]B-\[Fig\_spatial\]C. As $L$ increases, the strongly correlated core at moderate separations shrinks, indicating that the steady growth of orientational correlations is reduced by the mixing introduced by swimming. The overall strength of correlations inside the core also decreases with $L$, reflecting the reduction of the fluid velocity variance by swimming. At large distances, $C(\rho)$ recovers the behaviour seen in shakers, with the crossover distance given by $\rho_1 \sim L (1-\Delta)^{-1/2}$, as can be deduced from the exponential in Eq.. This behaviour is further demonstrated in Fig.\[Fig\_spatial\_delta09\], where we plot $C(\rho)$ for $\Delta=0.9$ and various values of $L$. In the limit of fast swimming, $L\rightarrow\infty$, the correlation function deviates modestly from the non-interacting case for almost all values of $\Delta$, exhibiting a quick rise and the divergence associated with the onset of collective motion only in a very small vicinity of $\Delta=1$. The data in Fig.\[Fig\_spatial\] and Eq. demonstrate that $C(\rho)$ exhibits an algebraic decay for large distances, and a true correlation length can thus not be defined. A phenomenological correlation length $\xi_{corr}$ can nevertheless be defined as a distance over which $C(\rho)$ decreases by certain amount, as has been employed in [@Gachelin2014; @Bardfalvy2019]. Setting $C(\xi_{corr}) = \alpha \langle U^2 \rangle$, with $\alpha<1$, we obtain $$\begin{aligned} \xi_{corr}\sim (1-\Delta)^{-1/2}, \quad \Delta\rightarrow1, \end{aligned}$$ similar to any other typical distance discussed above. Fluid velocity spectrum ----------------------- {width="1\linewidth"} Next, we discuss the fluid velocity energy spectrum $E(k)$ that is closely related to the spatial correlation function $C(\rho)$. Defined as $$\begin{aligned} E(k) = 4\pi k^2 \overline{\hat{U}^{\alpha}\left( {\bm k} \right) \hat{U}^{\alpha}\left( -{\bm k} \right)}, \label{eq:spectrum}\end{aligned}$$ this quantity is often used in turbulence research to study the cascade of the kinetic energy [@Townsend1980]. Although the kinetic energy is not a useful concept for Stokesian flows, $E(k)$ provides an insight into the relative strength of fluid motion at various scales. The energy spectrum is proportional to the Fourier transform of $C(\rho)$, and, up to a prefactor is given by the integrand of Eq. $$\begin{aligned} &E(\xi) = \frac{8\pi}{15} \kappa^2 n A^2(\xi) \nonumber \\ &\qquad\qquad \times \frac{2-A(\xi)\Delta+\frac{6}{7}L^2 \xi^2}{\left(2-A(\xi)\Delta\right)\left(1-A(\xi)\Delta+\frac{3}{7}L^2 \xi^2\right)}, \label{energyspectrum}\end{aligned}$$ where, again, $\xi=k\epsilon$. This expression is plotted in Fig.\[Fig\_Spectra\] for various values of $\Delta$ and $L$. First, we observe that $E(\xi)$ has significant energy content at all large scales, $\xi<1$, that quickly decays to zero at the organism-size scales, $\xi\sim1$, due to the regularising factor $A(\xi)$. This is not caused by some form of energy cascade, but is due to the nature of the dipolar field created by the microswimmers. Indeed, the dipolar velocity field decays in space as $r^{-2}$, while its Fourier transform scales as $k^{-1}$. Together with the definition of $E(k)$, Eq., this implies that $E(k)\sim k^0$ even for a single microswimmer, i.e. the dipolar field has a constant energy content at every scale. In the presence of interactions, the energy spectrum of shakers ($L=0$) preserves the overall structure described above, while its absolute value increases with $\Delta$ and, eventually, diverges at $\Delta=1$. For swimmers, the increase in the energy content is mostly confined to large scales, while in the limit of fast swimming (not shown), the rise in the energy content on the approach to the onset of collective motion is confined to the largest scales available ($k\rightarrow0$) and starts to be visible only in a very close vicinity of $\Delta=1$. Temporal correlations {#section:results:temporal} --------------------- {width="1.0\linewidth"} The temporal correlation function $C(\tau)=C_0(\rho=0,\tau)+C_1(\rho=0,\tau)$ is given by Eqs. and . The corresponding expressions do not simplify significantly in the limit $\rho=0$, and we do not repeat them here. In Fig.\[Fig\_Temporal\] we plot $C(\tau)$ normalised by its value at $\tau=0$, which is given by the fluid velocity variance $\langle U^2\rangle$. As with the other quantities discussed above, the temporal correlation function exhibits a progressively slower decay as $\Delta$ approaches the onset of collective motion, eventually diverging at $\Delta=1$. For swimmers, this is offset by a decay of $C(\tau)$ at short times that becomes more pronounced as $L$ increases. For very large swimming speeds, the temporal correlations differ only marginally from the non-interacting case for most values of $\Delta$, eventually exhibiting a rapid increase and divergence in a very small vicinity of $\Delta=1$. To understand the behaviour of $C(\tau)$ at long times, we analyse its individual contributions. The integral in the non-interacting part, $C_0(T)$, can be explicitly evaluated giving $$\begin{aligned} & C_0(\tau) = \frac{n \kappa^2 }{\pi \epsilon} \frac{e^{-\tau}}{8\alpha^4 (4+\alpha^2)^2} \Bigg[ \nonumber \\ & \qquad\qquad\qquad 4(24+8\alpha^2+\alpha^4) \mathbb{E}\left( -\frac{\alpha^2}{4}\right) \nonumber \\ & \qquad\qquad\qquad-(4+\alpha^2)(24+5\alpha^2) \mathbb{K}\left( -\frac{\alpha^2}{4}\right) \Bigg], \label{}\end{aligned}$$ where $\alpha=L\tau$, and $\mathbb{K}(x)$ and $\mathbb{E}(x)$ are the complete elliptic integrals of the first and second order, respectively. In the limits of small and large $\alpha$ this equation predicts $$\begin{aligned} C_0(\tau) \sim \frac{n \kappa^2 }{\pi \epsilon} e^{-\tau} \times \begin{cases} \frac{21\pi}{2048},\quad &L\tau\rightarrow0, \\ \\ \frac{1}{4 (L\tau)^3},\quad &L\tau\rightarrow\infty. \end{cases}\end{aligned}$$ At short times, tumbling is the leading source of decorrelation, while at large $\tau$ the non-interacting temporal correlation function $C_0$ decays as $\tau^{-3}e^{-\tau}$, as reported previously [@Belan2019; @Bardfalvy2019]. The crossover time is set by $\alpha=L\tau=v_s T/\epsilon\sim1$, and corresponds to the time interval needed for a microswimmer to swim its own size. To understand the large-$\tau$ asymptotic behaviour of $C_1(\tau)$, we observe that $$\begin{aligned} e^{-\tau} \int_0^\infty d\xi A(\xi)^2 \left\{ \begin{array}{c} \sin{\gamma \tau \xi} \\ \cos{\gamma \tau \xi} \end{array} \right\} \underset{\tau\rightarrow\infty}{\sim} e^{-\tau} \left\{ \begin{array}{c} \tau^{-1}\\ \tau^{-5} \end{array} \right\},\end{aligned}$$ where $\gamma$ is a real constant. This result implies that a trigonometric function in the integrand of Eq. generates a contribution to $C_1(\tau)$ that decays on the same timescale as the non-interacting part $C_0(\tau)$, and does not contribute to the slow decay in Fig.\[Fig\_Temporal\]. In turn, this restricts the integration domain to $\xi \in [0,\xi_*]$, with $$\begin{aligned} \xi_* = \sqrt{\frac{7}{12}}\frac{\Delta}{L},\end{aligned}$$ which ensures that the arguments of the hyperbolic functions in Eq. are real. Introducing $\zeta = \xi/\xi_*$, $C_1(\tau)$ can be approximated as $$\begin{aligned} &C_1(\tau) \underset{\tau\rightarrow\infty}{\sim} \frac{n \kappa^2 }{15 \pi^2 \epsilon} e^{-\tau\left(1-\frac{1}{2}\Delta\right)} \xi_* \int_{0}^1 d\zeta \frac{1}{1-\Delta + \frac{1}{4}\Delta^2\zeta^2} \nonumber \\ & \qquad \times \Bigg\{ \frac{2-\Delta +\frac{1}{2}\Delta^2\zeta^2}{2-\Delta}\cosh{\left(\frac{1}{2}\Delta \tau \sqrt{1-\zeta^2}\right)} \nonumber \\ &\qquad\qquad\qquad + \frac{1}{\sqrt{1-\zeta^2}} \sinh{\left(\frac{1}{2}\Delta \tau \sqrt{1-\zeta^2}\right)} \Bigg\}, $$ where we used $A(\xi<\xi_*)\sim 1$ for not-too-small values of $L$. In the limit of large $\tau$, this can be further approximated by $$\begin{aligned} &C_1(\tau) \underset{\tau\rightarrow\infty}{\sim} \frac{n \kappa^2 }{15 \pi^2 \epsilon} e^{-\tau\left(1-\Delta\right)} \frac{\xi_*}{1-\Delta} \int_{0}^1 d\zeta \, e^{-\frac{1}{4}\tau \Delta \zeta^2} \nonumber \\ & = \frac{n \kappa^2 }{15 \pi^2 \epsilon} \sqrt{\frac{7\pi}{12}\frac{\Delta}{\tau}} \frac{1}{L(1-\Delta)} e^{-\tau\left(1-\Delta\right)} \mathrm{erf}\left( \frac{1}{2}\tau\Delta\right), \label{C1asymptotics}\end{aligned}$$ where $\mathrm{erf}(x)$ denotes the error function. Predictions of Eq. are plotted in Fig.\[Fig\_Temporal\]B and C as dashed lines. We find a good agreement between its prediction and the true decay of $C(\tau)$ as $\tau\rightarrow\infty$. To extract the typical timescale $\tau_{corr}$ of the fluid velocity fluctuations on the approach to collective motion, we combine Eqs. and to obtain $$\begin{aligned} \frac{C(\tau)}{\langle U^2\rangle} \underset{\tau\rightarrow\infty}{\sim} \frac{e^{-\tau\left(1-\Delta\right)}}{\sqrt{\tau \left(1-\Delta\right)}}, \quad \Delta \rightarrow 1,\end{aligned}$$ which implies $$\begin{aligned} \tau_{corr} \sim (1-\Delta)^{-1}.\end{aligned}$$ Enhanced diffusivity -------------------- As the final observable, we consider here the enhanced diffusivity of a passive tracer particle embedded in a suspension of motile microorganisms. The tracer is assumed to be neutrally buoyant and move due to advection by the velocity fields created by the microswimmers. Brownian diffusion of the tracer is significantly weaker than its enhanced counterpart, and is neglected for simplicity. This problem has been extensively studied both experimentally [@Wu2000; @Kim2004; @Leptos2009; @Kurtuldu2011; @Mino2011; @Mino2013; @Jepson2013; @Patteson2016] and theoretically [@Underhill2008; @Dunkel2010; @Ishikawa2010; @Childress2010; @Childress2011; @Pushkin2013; @Pushkin2013jfm; @Morozov2014; @Kasyap2014; @Thiffeault2015; @Burkholder2017] in the dilute regime, where $\Delta\ll 1$, and for arbitrary densities of shakers [@Stenhammar2017]. Here, we consider the case of arbitrary density $\Delta < 1$ and $L$. The position of the tracer ${\bm a}(T)$ obeys the following equation of motion $$\begin{aligned} \dot{\bm a}(T) = {\bm U}({\bm a}(T),T), \label{tracer_position}\end{aligned}$$ which implies that the tracer is point-like and follows the velocity of the fluid at its position. The long-time behaviour of such a tracer is diffusive [@Wu2000; @Leptos2009; @Thiffeault2015], and the associated diffusion coefficient can be extracted in the usual way $$\begin{aligned} D = \lim_{T\rightarrow\infty} \frac{1}{6T} \overline{{\bm a}(T)\cdot {\bm a}(T)}.\end{aligned}$$ Here, the bar denotes the average over the history of tumble events, and has the same meaning as in Eq.. Solving formally Eq., ${\bm a}(T)={\bm a}(0) + \int_0^T dt' {\bm U}({\bm a}(t'),t')$, the diffusion coefficient can be written as [@Kubo1966] $$\begin{aligned} &D = \frac{1}{3} \lim_{t\rightarrow\infty}\int_{0}^\infty dT\, \overline{{\bm U}({\bm a}(t+T),t+T) \cdot {\bm U}({\bm a}(t),t)}.\end{aligned}$$ Here, $t$ is sufficiently large so that any influence of the initial conditions has died away. To proceed, we observe that ${\bm U}({\bm a}(t+T),t+T)$ can be iteratively calculated by substituting the formal solution for ${\bm a}(T)$ into its spatial argument, i.e. $$\begin{aligned} &{\bm U}({\bm a}(t+T),t+T) = {\bm U}({\bm a}(t),t+T)\nonumber \\ & +\nabla {\bm U}({\bm a}(t),t+T) \cdot \int_{t}^{t+T} dt' {\bm U}({\bm a}(t'),t') + \cdots. \label{DUapprox}\end{aligned}$$ As was argued by Pushkin and Yeomans [@Pushkin2013], for very dilute suspensions velocity gradients over the typical distance travelled by the tracer particle during the microswimmer runtime are small compared to the velocity of the fluid at any of these positions, and can be neglected. Therefore, we can approximate the diffusion coefficient as $$\begin{aligned} & D \approx \frac{1}{3} \int_{0}^\infty dT\, \overline{{\bm U}({\bm a}(t),t+T) \cdot {\bm U}({\bm a}(t),t)}\nonumber \\ &\qquad\qquad = \frac{1}{3} \int_{0}^\infty dT\, C(T). \label{DvsCT}\end{aligned}$$ As we have seen in Section \[section:results:temporal\], as $\Delta$ increases, the correlation time increases from $\lambda^{-1}$ (corresponding to $\tau_{corr}=1$) in the very dilute regime to progressively larger values, eventually diverging as $\Delta\rightarrow 1$, implying that the second, etc. terms in Eq. grow rapidly in this limit. However, the fluid velocity variance, which sets the magnitude of the leading term in Eq. also diverges as $\Delta\rightarrow 1$. Further work is required to assess the validity of the approximation above for all values of $\Delta$. Here, we proceed by using Eq. with the potential caveat that it might not be accurate in the vicinity of $\Delta=1$. The integral in Eq. can be evaluated explicitly, leading to $D = D_0 + D_1$, where the non-interacting and interacting contributions are given by $$\begin{aligned} D_0 = \frac{\kappa^2 n}{45 \pi^2 \lambda \epsilon} \int_{0}^\infty d\xi A^2(\xi) \psi\left( \xi L\right), $$ and $$\begin{aligned} &D_1 = \frac{\kappa^2 n \Delta}{45 \pi^2 \lambda \epsilon} \int_{0}^\infty d\xi A^3(\xi) \nonumber\\ &\qquad\qquad \times \frac{ 2-A(\xi)\Delta+\frac{6}{7}L^2 \xi^2 }{\left( 1+\frac{3}{7} L^2\xi^2\right)\left( 1-A(\xi)\Delta+\frac{3}{7}L^2 \xi^2\right)^2},\end{aligned}$$ respectively, and $\psi(x)$ is defined in Eq.. At this point, we would like to comment on the shaker limit of these expressions, when they should reduce to the ones obtained by Stenhammar *et al.* [@Stenhammar2017]. Instead, we observe that the expression for $D_1$ reported there erroneously contained $A^2(\xi)$ instead of $A^3(\xi)$ under the integral. We note, however, that since $A(\xi)$ is a regularised representation of a step function, this has almost no bearing on the numerical evaluation of $D_1$ presented in [@Stenhammar2017]. The integral in the non-interacting part $D_0$ cannot be represented in terms of special functions, but its limiting behaviour can readily be obtained. Combining the asymptotic results for $L=0$ and $L\rightarrow\infty$, results in the following approximation $$\begin{aligned} D_0 \approx \frac{\kappa^2 n}{\lambda \epsilon} \frac{7}{2048 + 336 \pi L}. \label{D0approx}\end{aligned}$$ To derive an approximate expression for $D_1$, we set $A(\xi)\approx 1$ under the integral sign, to obtain $$\begin{aligned} D_1 \approx \frac{\kappa^2 n}{90 \pi \lambda \epsilon \sqrt{1+\frac{12}{7}L^2}} \left\{ \frac{2-\Delta}{\left( 1-\Delta\right)^{3/2}} -2\right\}. \label{D1approx}\end{aligned}$$ {width="0.9\linewidth"} In Fig.\[Fig\_Diffusivity\] we compare the numerical evaluation of $D_0$ and $D_1$ against Eq. and . We observe that while the uniform approximation Eq. does not work well for small but finite values of $L\sim1$, all other values of $L$ are well-represented by the approximation. The interacting part of the diffusivity is well-approximated by Eq.. Finally, we remark that Eq. predicts that $$\begin{aligned} D_1 \sim (1-\Delta)^{-3/2}, \quad \Delta\rightarrow1,\end{aligned}$$ even though this prediction should be treated with caution, as discussed above. Discussion and Conclusion {#section:discussion} ========================= In this work, we have presented a kinetic theory for dilute suspensions of pusher-like microswimmers interacting via long-ranged dipolar fields. Our theory goes beyond the mean-field assumption and explicitly includes correlations between microswimmers. We have overcome a significant technical difficulty in including particle self-propulsion that has limited our previous work to the case of shaker microswimmers [@Stenhammar2017]. Our theory makes explicit predictions for various experimentally relevant observables for any density of microswimmers up to the onset of collective motion. All of its parameters can be independently measured in experiments, and its predictions can be directly compared against experimental data. The results of our theory, presented in Section \[section:results\], reveal that all observables considered deviate from their mean-field values, which can be recovered from our results by setting $\Delta=0$, indicating that the mean-field theory is incorrect at any density below the onset of collective motion. We have also uncovered the following interplay between the strength of correlations between microswimmers and their self-propulsion speed. For all observables considered, the strongest correlations are exhibited by suspensions of shakers, $L=0$. This can be readily seen by observing that, in the absence of self-propulsion, the microswimmer positions only change due to their mutual advection. In dilute suspension, displacements thus accumulated over one correlation time are small compared to the interparticle distances, and, to first approximation, shaker suspensions perform orientational dynamics only. In turn, this implies that they spend maximum amount of time possible adjusting to the orientational fields created by other microswimmers. In contrast, motile microswimmers are aligning in a local velocity field that constantly changes due to their self-propulsion, implying weaker correlations in such suspensions. This effect becomes stronger as $L$ increases. The degree to which correlations are suppressed by self-propulsion depends on the nature of the observable. Spatial-like observables (the fluid velocity variance, the energy spectrum, and the spatial correlation function) are significantly reduced as $L$ increases, but do not reach their mean-field values even in the limit $L\rightarrow\infty$. For instance, as can be seen from Fig.\[Fig\_Variance\], the fluid velocity variance is significantly larger than its mean-field value at any density $\Delta$, even in the limit of fast swimming. In a similar fashion, as $L\rightarrow\infty$, the spatial correlation function in Fig.\[Fig\_spatial\_delta09\] does not reduce to its mean-field behaviour, which is given by the $\Delta\rightarrow0$ limit in Fig.\[Fig\_spatial\]. Instead, it recovers the strongly correlated shaker-like behaviour at sufficiently large distances. This can be understood by employing the same argument as above. For any value of $L$, there are such separations $\rho$ that the typical distance travelled by a microswimmer during one correlation time of the suspension is small compared to $\rho$. For such separations, the difference between swimmers and shakers vanishes and $C(\rho)$ recovers its shaker-like behaviour. On the other hand, temporal-like observables (the temporal correlation function and the enhanced diffusivity of tracer particles) are almost completely suppressed as $L\rightarrow\infty$ for $\Delta<1$, though they still diverge in the limit of $\Delta\rightarrow1$. This behaviour mirrors the dependence of their mean-field values on $L$, which vanish in the limit of fast swimming below the onset of collective motion. An intuitive argument for this behaviour has been put forward by Dunkel *et al.* [@Dunkel2010], who demonstrated that the total displacement of a tracer by a single motile particle vanishes as the length of a straight path covered by the swimmer diverges. This is fundamentally related to the time-reversibility of Stokesian flows. The presence of correlations between microswimmers breaks this time-reversibility: although the pathway between two states in phase space is still reversible, the probabilities of finding the suspension in those states are a priori different. Strong swimming introduces effective phase space mixing and recovers equal a priori probabilities for the phase space states. Again, this argument only holds for $\Delta<1$ when $L$ is large, yet finite. **Observable** **Scaling law for $\Delta\rightarrow1$** ----------------------------- ------------------------------------------ Fluid velocity variance $(1-\Delta)^{-1/2}$ (Pseudo-)correlation length $(1-\Delta)^{-1/2}$ Correlation time $(1-\Delta)^{-1}$ Enhanced diffusivity $(1-\Delta)^{-3/2}$ : Critical exponents[]{data-label="table:exponents"} To gain further insight into the nature of the transition to collective motion exhibited by our model, we extracted the scaling behaviour of the observables considered in this work upon the approach to the onset, $\Delta=1$. All of these quantities diverge at $\Delta=1$ and the values of the critical exponents predicted by our theory are summarised in Table \[table:exponents\]. We want to stress that these exponents rely on the approximation introduced in Section \[subsection:approximateDILT\], and while we are confident that it semi-quantitatively captures the spatial and temporal behaviour of the generalised correlation function $C(\rho,\tau)$ for $\Delta<1$, its quality in the close vicinity of $\Delta=1$ is untested. The values presented in Table \[table:exponents\] should thus be seen as a starting point, and more work is needed to make a conclusive statement about the critical exponents predicted by our model. This is especially important in the context of defining universality subclasses of “wet" active matter models. In this work, we have only considered pusher-like microswimmers below the onset of collective motion. Recent simulations suggest [@Stenhammar2017; @Bardfalvy2019] that suspensions of pullers also exhibit strong correlations, although their effect is opposite to what is observed for pushers. The results presented in this work cannot be used to study this effect, i.e. by replacing $\Delta$ with $-\Delta$ in the relevant expressions. Instead, to extend our theory to pullers, one would have to re-evaluate the long-term behaviour of the approximate double inverse Laplace transform in Section \[subsection:approximateDILT\] for negative values of $\Delta$. Finally, we note that the scenario emerging from our work is that microswimmer motility suppresses correlations and brings the suspension closer to the mean-field state. This observation could prove to be instrumental in understanding the origins of a surprising observation made by Stenhammer *et al.* [@Stenhammar2017] regarding the nature of the transition to collective motion in bacteria. There, we performed large-scale Lattice Boltzmann simulations of dilute suspensions of pusher-like microswimmers, and observed that all quantities of interest increased sharply around the threshold predicted by the mean-field theory ($\Delta=1$ in the notation adopted here). This increase, however, fell short of a divergence that should be associated with a true transition. While it would be natural to assume that this is the result of finite-size effects always present in computer simulations, Stenhammar *et al.* [@Stenhammar2017] did not see any sharpening of the curves around the supposed transition point upon a systematic increase of the system size. This phenomenon is currently not understood. Either the finite-size effects are decaying very slowly and realistic-size computer simulations cannot reach such large system sizes, or the presence of strong correlations below the onset of collective motion changes the stability properties of the homogeneous and isotropic state, replacing the mean-field transition with a correlation-smoothened cross-over. The results of this work provide a way to test the latter hypothesis: Since swimming suppresses correlations, sharpening of the potential transition/cross-over should be observed in suspensions with progressively increased swimming speed. Double inverse Laplace transform of an archetypal term {#appendix:Cesare} ====================================================== Here, we show how to calculate the double inverse Laplace transform of Eq.. The derivation of Eq. is similar, though lengthy, and we do not present it here. We start by observing that the double inverse Laplace transform in Eq., given in terms of two Bromwich integrals [@Doetsch1974], can be written as $$\begin{aligned} \label{app:B-1} \lim_{t\to\infty} \int_{\Gamma_1}\frac{ds_1}{2\pi i} \frac{e^{s_1 t}}{\lambda + s_1 + i v_s k (\hat{\bm k}\cdot {\bm p})}J(s_1),\end{aligned}$$ where $$\begin{aligned} \label{app:B-2} J(s_1) = \int_{\Gamma_2}\frac{ds_2}{2\pi i} \frac{1}{s_1+s_2} \frac{e^{s_2 (t+T)}}{{\lambda + s_2 - i v_s k (\hat{\bm k}\cdot {\bm p})}}.\end{aligned}$$ By the definition of the inverse Laplace transform [@Doetsch1974], the contours defining the integrals above have to be chosen such that $\Gamma_2$ passes on the right of $-s_1$ and of $-\lambda +iv_s k(\hat{\bm k}\cdot {\bm p})$, while $\Gamma_1$ should pass on the right of all the poles of $J(s_1)$ and of $-\lambda-i v_s k(\hat{\bm k}\cdot {\bm p})$. Observe that the first condition implies that $\Gamma_2$ should be chosen on the right of $-\Gamma_1$. Next, we observe that, again from the definition of the inverse Laplace transform, $J(s_1)$ is only defined for $\textrm{Re}(s_1)>0$. To proceed, we follow the method often utilised in plasma physics to describe the Landau damping [@Balescu1975]. We perform the analytic continuation of $J(s_1)$ to purely imaginary values of $s_1$ (recall that the analytic continuation of a complex function defined on an open set is the only function $\hat{J}(s_1)$ that is analytic, defined on a larger set, and equals $J(s_1)$ on the original set), and replace $J(s_1)$ with $\hat{J}(s_1)$ in Eq. . Since $\lambda>0$, the difficulty in performing the analytic continuation of $J(s_1)$ lies in the pole at $s_2=-s_1$ of the integrand from Eq.. We, therefore, define $$\begin{aligned} & \hat{J}(s_1) = \int^* \frac{ds_2}{2\pi i} \frac{e^{s_2 (t+T)}}{s_2+\lambda-i v_s k(\hat{\bm k}\cdot {\bm p})} \nonumber \\ & \qquad\quad +\frac{1}{2} \ell(-s_1) \frac{e^{s_2 (t+T)}}{s_2+\lambda-i v_s k(\hat{\bm k}\cdot {\bm p})}\Bigg\vert_{s_2=-s_1},\end{aligned}$$ where $$\begin{aligned} \ell(s_1) = \begin{cases} 0, & \textrm{Re}(s_1)>0, \\ 1, & \textrm{Re}(s_1)=0, \\ 2, & \textrm{Re}(s_1)<0. \end{cases}\end{aligned}$$ The meaning of the integral denoted by $\int^* ds_1$ above depends on the sign of $\textrm{Re}(s_1)$: If $\textrm{Re}(s_1)>0$, it is just a standard complex integral over a contour passing on the right of $-s_1$ and of $-\lambda +iv_s k(\hat{\bm k}\cdot {\bm p})$; If $\textrm{Re}(s_1)=0$, $\int^* ds_1$ stands for a principal value integral; Finally, if $\textrm{Re}(s_1)<0$, $\int^* ds_1$ stands for a standard complex integral over a contour passing on the left of $-s_1$ but on the right of $-\lambda +iv_s k(\hat{\bm k}\cdot {\bm p})$. With the definitions above, it is easy to show that $\hat{J}(s_1)$ is holomorphic in an infinitesimal stripe around $s_1\in \mathbb{R}$. Hence, it is the analytic continuation of $J(s_1)$. Replacing $J(s_1)$ by $\hat{J}(s_1)$ in Eq., we obtain two terms. The first term, containing $\int^*ds_2$, vanishes for $t\to\infty$, since we are now free to choose the integration contours $\Gamma_1$ and $\Gamma_2$ such that $\textrm{Re}(s_1+s_2)<0$. The other term reads $$\begin{aligned} & \lim_{t\to\infty} \frac{1}{2}\int_{\Gamma_1}\frac{ds_1}{2\pi i} \frac{1}{s_1+\lambda+i v_s k(\hat{\bm k}\cdot {\bm p})} \nonumber \\ &\qquad\qquad\qquad\qquad \times \frac{\ell(-s_1)\,e^{-s_1 T}}{-s_1+\lambda-i v_s k(\hat{\bm k}\cdot {\bm p})}.\end{aligned}$$ Closing the contour at $+\infty$, the only pole contributing to the integral is at $s_1=\lambda-i v_s k(\hat{\bm k}\cdot {\bm p})$, and we obtain $$\begin{aligned} &\lim_{t\to\infty} \int_{\Gamma_1}\frac{ds_1}{2\pi i} \frac{e^{s_1 t}}{s_1+\lambda+i v_s k(\hat{\bm k}\cdot {\bm p})}J(s_1) \nonumber \\ &\qquad\qquad\qquad\qquad = \frac{e^{-\lambda T + i v_s k T (\hat{\bm k} \cdot {\bm p})}}{2\lambda}.\nonumber\end{aligned}$$ This completes the proof of the equality in Eq.. Approximating $\psi(z)$ {#appendix:psia} ======================= Here, we develop an approximation to $\psi(z)$ from Eq.. Our goal is to find a rational function with a pole structure that is similar to the original $\psi(z)$. As discussed in Section \[subsection:approximateDILT\], the relevant domain is set by the values of $z$ given by $z = \beta/(1+s/\lambda)$, with $\beta=v_s k/\lambda$ varying from $0$ to $L=v_s/(\lambda \epsilon)=0-25$, and by the real part of $s$ ranging from $-\lambda$ to $0$. Our starting point are the observations that as $z\rightarrow0$, $\psi(z) \rightarrow 1-3z^2/7$, while for $z\rightarrow\infty$, $\psi(z) \rightarrow 0$. Both asymptotic behaviours can be combined into $\psi_a(z)= 7/(7+3z^2)$. Now we show that this is a surprisingly good approximation to $\psi(z)$, both reproducing its global shape and having a similar pole structure. ![Comparison between $\psi(z)$ (solid lines) and $\psi_a(z)$ (dotted lines) for $z = \beta/(1+s/\lambda)$ for real values of $s$ and various values of $\beta$.[]{data-label="psi_vs_psia"}](Psia.pdf){width="0.8\linewidth"} In Fig.\[psi\_vs\_psia\] we compare $\psi(z)$ and $\psi_a(z)$ for real values of $s$. We observe a good agreement between the two functions for various values of $\beta$. Similar, semi-quantitative, degree of agreement is observed for larger values of $\beta$ and also for complex values $s$. To demonstrate that $\psi_a(z)$ also reproduces the pole structure of $\psi(z)$, we consider a typical term from the analysis in Section \[subsection:hhat\] $$\begin{aligned} \frac{1}{s+\lambda-\lambda\Delta A(k\epsilon)\psi(z)} = \frac{1}{\lambda \Delta A(k\epsilon)}\frac{z}{\omega - z \psi(z)}. \label{A_original}\end{aligned}$$ We compute its inverse Laplace transform numerically, using the original function $\psi(z)$, and compare the result with the analytic expression, which we obtain by replacing $\psi(z)$ with $\psi_a(z)$ in the expression above. The latter is straightforward: factorising $\omega \left(7+3z^2\right) - 7 z = 3\omega(z-z_{+})(z-z_{-})$, where $z_{\pm}$ are given in Eq., we obtain $$\begin{aligned} &\frac{1}{\lambda \Delta A(k\epsilon)}\frac{z}{\omega - z \psi_a(z)} \nonumber \\ &\qquad\qquad = \frac{1}{7}\frac{1}{s+\lambda}\frac{7(s+\lambda)^2 + 3(v_s k)^2}{\left(s+\lambda - \frac{v_s k}{z_{+}}\right)\left(s+\lambda - \frac{v_s k}{z_{-}}\right)}.\end{aligned}$$ Performing the inverse Laplace transform of this expression and introducing the dimensionless time $\tau = \lambda t$ yields $$\begin{aligned} &e^{-\tau} \Bigg[ 1 +\frac{2}{\sqrt{1-\frac{12\beta^2}{7\Delta^2}}} \exp{\left(\frac{\tau \Delta }{2}\right)} \nonumber \\ & \qquad\qquad\qquad\qquad \times \sinh{\left(\frac{\tau \Delta}{2}\sqrt{1-\frac{12\beta^2}{7\Delta^2}}\right)}\Bigg]. \label{A_test_analytic}\end{aligned}$$ Since both $A(k\epsilon)$ and $\Delta$ take values between $0$ and $1$, for the purpose of comparing to its numerical counterpart, we set $A(k\epsilon)=1$ in the expression above, without loss of generality. {width="1.0\linewidth"} The inverse Laplace transform of the original function Eq. written in terms of the same parameters is given by the Bromwich integral $$\begin{aligned} \frac{1}{2\pi i} \int_{\gamma-i \infty}^{\gamma+i \infty} d{\tilde s} \frac{e^{\tilde s \tau}}{{\tilde s}+1-\Delta \psi(\beta/(\tilde s+1))}, \label{A_test_numerics}\end{aligned}$$ where $\gamma$ is a real number, chosen to be greater than the real part of any singularity of the integrand [@Doetsch1974]. We perform this integral numerically, using the Gaver–Wynn–Rho algorithm as presented by Valko and Abate [@Valko2005]. Valko and Abate provide an explicit Mathematica function GWR [@math1], which we use here. A Mathematica notebook with the details of this calculation can be found here [@SI]. In Fig.\[ILTcombined\]A we compare Eq. against the numerical Laplace transform of Eq. for $\Delta=0.1$ and $\beta=0.1$, $1$, and $10$. We observe a very good agreement, which is not surprising: At small microswimmer densities, the hydrodynamic interactions between particles affect their dynamics only weakly, and correlations decay as $e^{-\tau}$. This regime does not test the quality of our approximation. A more stringent test is provided, on the other hand, in Fig.\[ILTcombined\]B, were we compare the two Laplace transforms for $\Delta=0.9$. For $\beta<1$, we observe a very good agreement even at such high values of $\Delta$ (close to the mean-field transition). This is the most interesting regime, corresponding to large-scale motion in the suspension, and it is encouraging that our approximation shows quantitative agreement with the numerical data. Note that the black line and the black circles, corresponding to $\beta=0.1$, do not follow $e^{-\tau}$, i.e. our approximation is capable of capturing a non-trivial decay rate. At higher values of $\beta$, corresponding to scales comparable to individual microswimmers, the agreement is semi-quantative, but the overall decay is again close to the tumbling-dominated decay $e^{-\tau}$. In Appendix \[appendix:dilt\], we assess the quality of our approximation, when used in Eq., which is its ultimate purpose. Double inverse Laplace transform {#appendix:dilt} ================================ In Section \[subsection:approximateDILT\], we performed the double inverse Laplace transform in Eq. analytically by replacing $\psi(z)$ with $\psi_a(z)$, which led to Eq.. Here, we assess the quality of that approximation by performing the double inverse Laplace transform in Eq. numerically. The relevant part of Eq. reads $$\begin{aligned} &{\mathcal L}^{-1}_{\tilde{s}_1,\tilde{t}} {\mathcal L}^{-1}_{\tilde{s}_2,\tilde{t}+\tau} \frac{1}{1 + \tilde{s}_1}\frac{1}{1 + \tilde{s}_2}\frac{1}{\tilde{s}_1 + \tilde{s}_2} \frac{\tilde{z}_1 \psi(\tilde{z}_1) + \tilde{z}_2 \psi(\tilde{z}_2)}{\tilde{z}_1+\tilde{z}_2} \nonumber \\ &\qquad \times\Bigg[ \frac{\tilde{z}_1 \psi(\tilde{z}_1)}{\tilde{\omega} - \tilde{z}_1 \psi(\tilde{z}_1)} + \frac{\tilde{z}_2 \psi(\tilde{z}_2)}{\tilde{\omega} - \tilde{z}_2 \psi(\tilde{z}_2)} \nonumber \\ &\qquad\qquad + \frac{\tilde{z}_1 \tilde{z}_2 \psi(\tilde{z}_1)\psi(\tilde{z}_2)}{\left(\tilde{\omega} - \tilde{z}_1 \psi(\tilde{z}_1)\right)\left(\tilde{\omega} - \tilde{z}_2 \psi(\tilde{z}_2)\right)} \Bigg], \label{DILTeqnum}\end{aligned}$$ where, in anticipation of performing numerical calculations, we introduced the dimensionless times $\tau = \lambda T$ and ${\tilde t} = \lambda t$, Laplace frequencies ${\tilde s}_{1,2} = s_{1,2}/\lambda$, ${\tilde z}_{1,2} = \beta/(1+{\tilde s}_{1,2})$, and $\tilde{\omega} = \beta/\Delta$, where we absorbed $A(k\epsilon)$ into $\Delta$, as in Appendix \[appendix:psia\]. In what follows, we set $\tilde t=20$ to imitate the limit $\tilde{t}\rightarrow\infty$. The calculations are performed in Mathematics using the combined Fixed-Talbot and Gaver–Wynn–Rho algorithm described by Valko and Abate [@Valko2005]. A Mathematica notebook with the details of this calculation can be found here [@SI]. The results are compared to the relevant part of Eq., recast in the same dimensionless variables $$\begin{aligned} &e^{-\tau} \Bigg[ -\cos{\left(\sqrt{\frac{3}{7}}\beta\tau\right)} + \frac{e^{\frac{1}{2} \Delta \tau}}{1-\Delta + \frac{3}{7}\beta^2} \Bigg\{ \nonumber \\ & \qquad\qquad \frac{2-\Delta + \frac{6}{7}\beta^2}{2-\Delta}\cosh{\left(\frac{1}{2}\Delta \tau \sqrt{1-\frac{12 \beta^2}{7 \Delta^2}}\right)} \nonumber \\ &\qquad\qquad\qquad +\frac{\sinh{\left(\frac{1}{2} \Delta \tau \sqrt{1-\frac{12 \beta^2}{7 \Delta^2}}\right)}}{\sqrt{1-\frac{12 \beta^2}{7 \Delta^2}}} \Bigg\} \Bigg]. \label{DILTeqanal}\end{aligned}$$ {width="1.0\linewidth"} The results of the numerical double inverse Laplace transform and its analytical counterpart are shown in Fig.\[DILTcombined\]. As in Appendix \[appendix:psia\], we focus on high values of $\Delta$, which provide the most stringent test of our results. For $\beta\le1$, the analytic approximation agrees quite well with the numerical data, capturing not only the decay rate, but also the oscillatory behaviour, as can be seen from the $\beta=1$ case. These calculations required a very high number of terms, $O(100)$, in the combined Fixed-Talbot and Gaver–Wynn–Rho algorithm [@Valko2005]. For $\beta > 1$, we were unable to obtain converged results for the numerical Laplace transform for any viable number of terms in the numerical algorithm. Nevertheless, the results of Appendix \[appendix:psia\], and the degree of agreement exhibited in Fig.\[DILTcombined\] for the physically most relevant case of $\beta<1$ make us confident that Eq. faithfully reproduces the long-time behaviour of Eq..

--- abstract: 'From integral field data we extract the optical spectra of 20 shocked clouds in the supernova remnant N132D in the Large Magellanic Cloud (LMC). Using self-consistent shock modelling, we derive the shock velocity, pre-shock cloud density and shock ram pressure in these clouds. We show that the \[Fe X\] and \[Fe XIV\] emission arises in faster, partially radiative shocks moving through the lower density gas near the periphery of the clouds. In these shocks dust has been effectively destroyed, while in the slower cloud shocks the dust destruction is incomplete until the recombination zone of the shock has been reached. These dense interstellar clouds provide a sampling of the general interstellar medium (ISM) of the LMC. Our shock analysis allows us to make a new determination of the ISM chemical composition in N, O, Ne, S, Cl and Ar, and to obtain accurate estimates of the fraction of refractory grains destroyed. [From the derived cloud shock parameters, we estimate cloud masses and show that the clouds previously existed as typical self-gravitating Bonnor-Ebert spheres into which converging cloud shocks are now being driven.]{}' author: - | Michael A. Dopita , Frédéric P.A. Vogt , Ralph S. Sutherland ,\ Ivo R. Seitenzahl , Ashley J. Ruiter & Parviz Ghavamian title: | Shocked Interstellar clouds and dust grain destruction\ in the LMC Supernova Remnant N132D --- Introduction {#intro} ============ The supernova remnant (SNR) N132D in the Large Magellanic Cloud (LMC) is the brightest X-ray emitting SNR in this galaxy [@Hwang93; @Favata97; @Hughes98; @Borkowski07; @Xiao08]. Located within the stellar bar of the LMC, N132D was first identified as a SNR by @Westerlund66 on the basis of the association of a non-thermal radio source with a \[\]– bright optical structure. @Danziger76 discovered high-velocity \[\] and \[\] material ejected [ in the supernova explosion]{}, which established N132D as a relatively young SNR [ in which the reverse shock has not yet reached the centre of the remnant, marking the formal transition to the Sedov phase of evolution. This would imply that the swept-up interstellar medium (ISM) is less than a few times the mass ejected in the supernova event, a result which is in apparent contradiction to the swept-up mass derived from X-ray observations [@Hughes98].]{} From the dynamics of the fast-moving O and Ne material, @Danziger76 derived a maximum age of 3440yr for the SNR, and a probable age of $\sim 1350$yr. Subsequent analyses gave similar ages; 2350yr [@Lasker80], and 2500yr [@Vogt11]. Although most optical studies have concentrated on the fast-moving O- and Ne-rich ejecta [@Lasker80; @Blair00; @Vogt11], there are a number of luminous, dense shock clouds with apparently ‘normal’ interstellar composition and $\sim200$km/s velocity dispersion. The most prominent of these is the so-called *Lasker’s Bowl* structure in the northern part of the remnant, but HST imaging [@Blair00] reveals many other small complexes of shocked cloudlets. These clouds do not show appreciable enhancements in the strength of the \[\] lines, and are thought to be simply ISM clouds over which the SNR blast wave has recently swept. In this paper we follow the study of N49 by @Dopita16 in investigating the degree of dust destruction in these cloud shocks. We also derive the physical parameters for some 20 different shocked ISM cloudlets and estimate their chemical abundances. Observations & Data Reduction {#sec:obs} ============================= The integral field spectra of N132D were obtained between 16 Nov 2017 and 24 Nov 2017 using the Wide Field Spectrograph (WiFeS) [@Dopita07; @Dopita10], an integral field spectrograph mounted on the 2.3-m ANU telescope at Siding Spring Observatory. This instrument delivers a field of view of 25$\times$ 38at a spatial resolution of either 1.0$\times$ 0.5 or 1.0$\times$ 1.0, depending on the binning on the CCD. In these observations, we operated in the binned 1.0x 1.0 mode. The data were obtained in the low resolution mode $R \sim 3000$ (FWHM of $\sim 100$ km/s) using the B3000 & R3000 gratings in each arm of the spectrograph, with the RT560 dichroic which provides a transition between the two arms at around 560nm. For details on the various instrument observing modes, see @Dopita07. All observations are made at PA=0, giving a long axis in the N-S direction. The basic grid consists of 15 pointings in an overlapping $3\times5$ grid (E-W:N-S), followed by 7 pointings centered on the overlap regions of the base $3\times5$ grid, with an additional pointing in the SE to probe the extent of the photoionised precursor region around N132D. The eighth overlap position was not observed due to deteriorating weather conditions. The typical seeing over the course of the observations was 1.5 arc sec. and ranged from 0.5 to 2.5 arc sec. over the individual exposures. Each region was observed with $2\times1000$s exposure time giving a total integration time on target of 44000s. We used blind offsets from a reference star to move to each field of the mosaic consistently over the course of the observing run, and avoid any gaps in the resulting mosaic. A fault in the offset guide head gave rise to errors of $\lesssim 5$arc sec. in the pointing of the fields; small enough not to result in any gaps within the mosaic. The wavelength scale is calibrated using a series of Ne-Ar arc lamp exposures, taken throughout the night. Arc exposure times are 50s for the B3000 grating and 1s for the R3000 grating. Flux calibration was performed using the STIS spectrophotometric standard stars HD009051, HD031128 and HD075000 [^1]. In addition, a B-type telluric standard HIP8352 was observed to better correct for the OH and H$_2$O telluric absorption features in the red. The separation of these features by molecular species allows for a more accurate telluric correction by accounting for night to night variations in the column density of these two species. All data cubes were reduced using the PyWiFeS [^2] data reduction pipeline [@Childress14; @Childress14b]. All the individual, reduced cubes were median-averaged into a red and blue mosaic using a custom <span style="font-variant:small-caps;">python</span> script. The WCS solution of both mosaics is set by cross-matching their individual (collapsed) white-light images with all entries in the *Gaia* [@GaiaCollaboration16a] DR1 [@GaiaCollaboration16]; see Fig \[fig1\]). For simplicity, and given the spaxel size of the WiFeS datacubes, we restrict ourselves to integer shifts when combining fields. We estimate that the resulting absolute and relative (field-to-field) alignment accuracy are both $\pm$1arc sec. HST imaging of the cloudlets ---------------------------- We have downloaded all the HST ACS F475W and F658N images from the *Barbara A. Mikulski archive for space telescopes* (MAST). The former filter encompasses the full velocity range of the \[\]$\lambda\lambda$5959,5007Å emission, whereas the latter covers the rest-frame H$\alpha$ emission. The dataset comprises four exposures of 380s each with the F475W filter, and four exposures of 360s each using the F658N filter, all acquired in January 2004 under program \#12001 (PI Green). We used the individual calibrated and CTE-corrected frames (<span style="font-variant:small-caps;">\*\_flc.fits</span>) retrieved from MAST to construct a combined, drizzled image in both filters using the following steps. We first correct the WCS solutions of each frame using the <span style="font-variant:small-caps;">tweakreg</span> routine from the <span style="font-variant:small-caps;">drizzlepac 2.1.13</span> package via a dedicated <span style="font-variant:small-caps;">python</span> script. We anchor the WCS solution to the *Gaia* [@GaiaCollaboration16a] DR1 [@GaiaCollaboration16] sources present within the images, retrieved automatically via the <span style="font-variant:small-caps;">astroquery</span> module. The WCS-corrected images are then merged using the <span style="font-variant:small-caps;">astrodrizzle</span> routine, with a pixel scale set to 0.04arcsec for both filters (for simplicity). {width="\textwidth"} Spectral Extraction ------------------- The spectra of 20 prominent shocked ISM clouds were extracted from the global WiFeS mosaic datacubes using using [QFitsView v3.1 rev.741]{}[^3]. The positions of the clouds were chosen so as to largely avoid contamination by high-velocity \[ \] – emitting material, [ see Fig \[fig2\]]{}. The exception is P07, but the velocity shift of the high velocity material here is sufficient as to allow accurate determinations of the narrow-line fluxes. We used a circular extraction aperture with a radius of either 2 or 3 arc sec. to best match the size of the bright region. To remove the residual night sky emission and (approximately) the faint stellar contribution, we subtracted a mean sky reference annulus 1 arc sec. wide surrounding the extraction aperture. The extraction regions were optimised by peaking up the signal in H$\alpha$ in the red data cube, and in H$\gamma$ in the blue data cube, respectively. This procedure may lead to some contamination by the faint extended cloud emission in the case of P07, P08, P14, P15, P18 and P20 (see Figure \[fig3\]). However, the core region strongly dominates such fainter contributions. The positions of the extracted spectra, the extraction radius used, the measured mean radial velocity, and the H$\alpha$ velocity width (FWHM) (after correction for the instrumental resolution) of each cloud are listed in Table \[table1\], and the corresponding positions are shown on the HST ACS H$\alpha$ + \[O III\] image in Figure \[fig2\] [ and each cloud with its spectral extraction region is shown in an $8\times8$arc sec. postage stamp image in Figure \[fig3\]]{}. A spectrum of the bright cloudlet P14 is shown in Figure \[fig4\] to indicate the quality of the data. Note that the typical velocity FWHM of these clouds in only $\sim 180$km/s, which is fairly indicative of the mean shock velocity in those clouds [Fig.6 in @Dopita12]. The mean Heliocentric radial velocity of all 20 clouds is +246km/s, and the mean Heliocentric radial velocity of the photoionised halo of N132D is found to be +261km/s, a difference which lies within the expected statistical error caused by sampling of the cloud shock velocities projected along the lines of sight within the random cloud positions inside the SNR. These figures can be compared to the +264.6km/s [@Smith71], $+262\pm16$km/s [@Danziger76], and +259km/s [@Feast64] for N127, a nearby HII region. Since most of the H$\alpha$ emission arises in the recombination zone of the cloud shocks, we can estimate a lower limit for the cloud shock velocities, $v_s$, from the most extreme differences in the measured radial velocity of the cloud from the mean Heliocentric radial velocity of the photoionised halo of N132D. These are +173km/s for cloud P04, and -143km/s for cloud P19. Thus we have $v_s \gtrsim 160$km/s, in good agreement with the estimate based on the FWHM of the H$\alpha$ line profiles. Measuring Emission Line Fluxes ------------------------------ For each extracted spectrum, the spectra were first reduced to rest wavelength based on their measured radial velocities listed in Table \[table1\], and then emission-line fluxes [ in units of erg/cm$^2$/s]{}, their uncertainties, the emission line FWHMs [ (in Å)]{} and the continuum levels were measured using the interactive routines in [Graf]{} [^4]and in [Lines]{} [^5]. The measured line fluxes [ relative to H$\beta$]{} and their uncertainties [ along with the absolute H$\beta$ fluxes in units of erg/cm$^2$/s]{}, are given for each of the clouds in Tables \[tableA1\] – \[tableA4\] in the Appendix. Emission Line Diagnostics {#diagnostics} ========================= Self-Consistent Shock Modelling ------------------------------- To analyse the spectrophotometry we have built a family of radiative shocks with self-consistent pre-ionisation using the MAPPINGS 5.12 code, following the methodology described in @Sutherland17, and applied to the study of Herbig-Haro objects by @Dopita17b. The pre-shock density is set by the ram pressure of the shock, taken to be independent of shock velocity; $P_{\mathrm{ram}} = 1.5\times 10^{-7}$dynes cm$^{-2}$, to ensure that the measured \[\] densities (which give the electron density near the recombination zone of the shock) approximately match those produced by the models. For each abundance set, a set of models was run for $100 \leqslant v_s \leqslant 475$km/s in steps of 25km/s. The magnetic field pressure in the un-shocked cloud ahead of the photo-ionised precursor is assumed to be in equipartition with the gas pressure, $P_{\mathrm{mag}} = P_{\mathrm{gas}}$, and the temperature of the gas entering the shock is given by the self-consistent pre-ionisation computation; see @Sutherland17 for details. The abundance set was initially taken as being 0.5 times the Local Galactic Concordance (LGC) values [@Nicholls17], but was then iterated manually to achieve a better fit of theory to the spectra of the clouds on the standard @BPT81 and @Veilleux87 diagnostic diagrams. The depletion factors of the heavy elements caused by the condensation of these elements onto dust are defined as the ratio of the gas phase abundance to the total element abundance. The depletion factors are derived from the formulae of @Jenkins09, extended to the other elements on the basis of their condensation temperatures and/or their position on the periodic table. In these shock models, following @Dopita16 we have investigated the effect of changing the logarithmic Fe depletion, $\log D_{\rm Fe}$ in the range $0.0 > \log \left [ D_{\rm Fe} \right ] > -1.0$. The abundance set adopted for the theoretical grid at the various values of $\log D_{\rm Fe}$ is given in Table \[table2\]. The abundances of C, Mg, and Si are not constrained by our observations and for these we fix the abundances at half of the LGC values. We produce a more refined estimate of the LMC chemical abundances when we build detailed models for the brightest clouds, below. \ The line intensities and line ratios given in this paper comprise the sum of the radiative shock and its photoionised precursor. In the case of the fastest shock, the extent of the photoionised precursor region and/or the cooling length of the shock may exceed the physical extent of the pre-shocked cloud. In Figure \[fig5\] we show the cooling length of the shock to 1000K, the depth of the photoionised precursor to the point where hydrogen is only 1% ionised, and the fraction of the shock H$\beta$ emission which arises from the precursor. The cooling length and precursor length are given for a ram-pressure of $P_{\mathrm{ram}} = 1.0\times 10^{-7}$dynes cm$^{-2}$. The shock cooling length scales inversely as the ram pressure, and the precursor length remains approximately constant. At the distance of the LMC 0.1pc corresponds to 0.45arc sec. ![The computed cooling length of the shock to $T_e = 1000$K (thick line), the thickness of the photoionised precursor to the point where hydrogen is only 1% ionised (thin line), and the fraction of the shock H$\beta$ emission which arises from the precursor (dotted line). These are all given for a ram pressure of $P_{\mathrm{ram}} = 1.0\times 10^{-7}$dynes cm$^{-2}$. At the densities of those models, the shock cooling length scales inversely as the ram pressure, while the precursor length is almost unaffected by changes in the ram pressure. []{data-label="fig5"}](fig5.pdf) Shock Velocity Diagnostics -------------------------- Here we investigate potentially useful diagnostics of the shock velocity using emission line ratios. To avoid issues of chemical abundance difference, the line ratio used should involve different levels of the same ion, or else different ions of the same element. In @Dopita16, we already noted that the \[\] $\lambda\lambda 4363/5007$ ratio was a useful indicator of shock velocity. This is because the \[\] temperature in the cooling zone of the shock is high at low shock velocities (faster than $\sim85$km/s, when this ion is first produced in the cooling zone of the shock), reaching a maximum at $v_s \sim 140$km/s when starts to become ionised to higher ionisation stages. As the velocity increases further, is ionised to , and the \[\] emission becomes confined to the cooler region nearer the recombination zone of the shock. At still higher velocities, a zone photoionised by EUV photons generated in the cooling zone of the shock develops adjacent to the recombination zone. This has still lower mean temperature; $T_e\lesssim 10^4$K. For these physical reasons, the \[\] temperature is more or less a decreasing function of shock velocity, and is relatively insensitive to $\log D_{\rm Fe}$. This behaviour is shown in Figure \[fig6\]. A second diagnostic is the excitation of He, as measured by the / $\lambda\lambda 4686/5876$ ratio. This is shown in the second panel of Figure \[fig6\]. Below $v_s \sim 200$km/s, this ratio is determined by the pre-ionisation of the shock, and the temperature structure of the post-shock region, and is multi-valued. However, above this velocity, it provides a useful diagnostic, albeit somewhat sensitive to $\log D_{\rm Fe}$. This sensitivity is caused by changes in the cooling length with gas-phase Fe abundance, which changes the ionisation structure of He in the shock. In what follows, we will use both the He and \[\] diagnostic ratios as spectroscopic indicators of the shock velocity. BPT diagnostic diagrams ----------------------- For shocks, the position of the observations on the standard @BPT81 and @Veilleux87 diagnostic diagrams (hereafter BPT Diagrams), is indicative of the mode of excitation. However, they are not really useful as diagnostics of the detailed shock conditions. This is demonstrated in Figure \[fig7\]. It should be noted that the range in observed line ratio and in the theoretical grid is very small on these diagrams, compared with the range covered by either HII regions or active galaxies and Narrow Line Regions. All these diagrams serve to show is that the theoretical models and the observations show a satisfactory overlap. This said, a few points on these diagrams are worthy of further mention. First P13 is the only region which shows an excess in the line intensity of \[\], which may be indicative of pre-supernova mass-loss enrichment. However, this seems a little unlikey as the region is close to the boundary of the SNR shell. Second, P20 shows extremely strong \[\] emission. This is evidenced by the blue appearance of this region in Figure \[fig3\]. This region will be discussed in more detail below, where we show that it is due to a finite-age shock in which the gas has not had time to become fully radiative. Lastly, P08 shows particularly weak \[\] emission. This region, also discussed below, lies in a complex of Balmer dominated or non-radiative filaments in which a high velocity non-radiative shock is passing through an un-ionised or partially-ionised precursor medium. \[\], \[\] and \[\] diagnostics -------------------------------- As can be seen from Tables \[tableA1\] - \[tableA4\] the \[\] spectrum is very rich. For the purposes of analysis, we have selected the brightest of these lines, \[\]${\lambda 5158}$ and \[\]${\lambda 8617}$. In Figure \[fig8\] we present the observed points superimposed on our theoretical shock model grid. From this, it is immediately apparent that nowhere does the measured iron depletion factor; $\log D_{\rm Fe}$ exceed -0.5. Indeed many of the points are consistent with no depletion at all. Furthermore, it is clear that the indicated shock velocities in general exceed 200km/s. Thus for most of these the \[\] line ratio can be used to estimate the shock velocity. However, a number of the points lie in the ambiguous region of the / line ratio velocity diagnostic. To the extent to which it is possible, both shock velocities and $\log D_{\rm Fe}$ have been estimated for each cloud from these diagnostic diagrams. The results are given in Tables \[table3\] and \[table4\]. We find a mean shock velocity $\left< v_s \right >$ of 240 km/s, and $\left < \log D_{\rm Fe} \right > = -0.15$. This result agrees very closely with the $\left < \log D_{\rm Fe} \right > = -0.16\pm0.07$ found by @Dopita16 in N49, another bright LMC SNR, and shows that dust has been largely destroyed by the time the recombination zone of the shock has been reached. A similar analysis was applied to the \[\]$\lambda 5270$/H$\beta$ and \[\]$\lambda 4881$/H$\beta$ line ratios. This is shown in Figure \[fig9\]. The two lines agree less well with each other, the \[\]$\lambda 4881$/H$\beta$ ratio tending to produce a larger depletion factor. In each line ratio, a much wider range of depletion factors is observed, from -0.25 to $\sim -1.25$. Averaging all measurements, we obtain $\left < \log D_{\rm Fe} \right > = -0.74\pm0.25$. This result is also broadly consistent with what was derived in the case of N49; $\left < \log D_{\rm Fe} \right > = -0.956\pm0.15$ [@Dopita16]. Finally, the analysis for \[\]$\lambda 6087$/H$\alpha$ is shown in Figure \[fig10\]. Note that this line requires shock velocities of $v_s > 180$km/s in order to be produced with appreciable strength in the radiative shock. Here, the inferred depletion factors range from -0.0 to $\sim -0.75$ with $\left < \log D_{\rm Fe} \right > = -0.4$. Given the intrinsic uncertainties in the atomic data for the forbidden lines of Fe, we can conclude that the \[\] and the \[\] lines imply a mean depletion factor of order $\left < \log D_{\rm Fe} \right > = -0.6\pm0.25$, while the \[\] lines give a depletion factor $\left < \log D_{\rm Fe} \right > = -0.15\pm0.15$. Thus, in both of these N132D and in N49, we find that Fe-bearing dust grains are not fully destroyed until they reach the recombination zone of the shock. The detail of the grain destruction process were discussed in @Dopita16, where it was concluded that this type of depletion pattern supports the grain destruction model of @Seab83 and @Borkowski95. \[Ni II\] diagnostics --------------------- Since we also observe strong \[\]$\lambda 7378$, it is worth asking whether the Ni-containing grains suffer the same fate as the Fe-containing grains. The diagnostic plots for \[\]/H$\alpha$ are given in Figure \[fig11\]. We find a depletion factor of $\sim -0.4$, intermediate between the \[\] and \[\] results, but again consistent with a large fraction of dust grains being destroyed in the shock. The highly ionised species of Fe -------------------------------- To produce the highly ionised species of Fe; \[\]${\lambda 8235}$, \[\]${\lambda 6374}$ and \[\]${\lambda 5303}$ requires shock velocities $v_s \gtrsim 200$km/s. Indeed, the self-consistent radiative shock models do not produce strong \[\]${\lambda 6374}$ until $v_s \sim 250$km/s and \[\]${\lambda 5303}$ becomes bright only for $v_s > 350$km/s. The diagnostics of expansion velocity, velocity dispersion, and the spectral diagnostics presented above all point to shock velocities for typical clouds in the range $160$km/s$ < v_s < 300$km/s. Thus it is clear that the relatively strong \[\] and \[\] lines observed must arise in another region with higher shock velocity. This is not unexpected, given the appearance of the clouds on the HST images, which often show a bright core in H$\alpha$, surrounded by fainter filamentary structures. We can identify these regions with the bright radiative shocks being driven into the denser cores of these clouds, and a faster, but only partially-radiative shock driven by a similar external ram-pressure sweeping through the less dense outer regions of the same clouds. Indeed in the case of another SNR in the LMC; N49, both the spatial and dynamical distinction between the \[\]${\lambda 5303}$ and the \[\]${\lambda 5159}$ emission are very marked [cf. Fig.3 in @Dopita16]. The detailed diagnostic diagrams for these two Fe species are shown in Figure \[fig12\]. For the standard grid, both fail to reproduce the observed strength of the lines, but the \[\] is particularly bad. Introducing a partially-radiative shock leads to a dramatic improvement. The derivation of the plausible shock parameters [ for these shocks]{} is as follows. First, we note that the mean cloud shock ram pressure derived from the observations (see Section \[cloudshocks\], below) is $P_{\mathrm{ram}} = 3.3\times10^{-7}$dynes/cm$^2$. At the mean pre-shock particle density implied for these shocks, $n_0 \sim 240$cm$^{-3}$, the mean cooling age to $T_e = 1000$K is 400yr. This provides an initial estimate of the shock age. Another way is to take the age of the SNR as estimated from the \[\] dynamics of the fast-moving material and estimate from the position of the cloud within the SNR how long it has been since the blast wave overran the cloud. In Section \[intro\] we saw that estimates of the age of N132D vary from 1350 – 3440yr for the SNR, with a probable age of $\sim 2500$yr [@Danziger76; @Lasker80; @Vogt11]. The detailed position of the clouds within the SNR blast wave cannot be reliably estimated from their projected distances from the boundary of the shell, but we will take this as being $\sim 25$%, as an upper limit, giving a mean shock age of $\lesssim 600$yr. With the shock ram pressure given above, the mean pre-shock density at a given shock velocity is fully determined; for shock velocities of 300, 350 and 400km/s the pre-shock particle densities are 160, 117 and 90cm$^{-3}$, respectively. None of these shocks can become fully radiative within this timescale. In the 600yr available, the 300km/s shock cools from $T_e=1.27\times 10^6$K to $T_e=6.06\times 10^5$K; the 350km/s shock from $T_e=1.72\times 10^6$K to $T_e=1.43\times 10^6$K and the 400km/s shock from $T_e=2.25\times 10^6$K to $T_e=2.07\times 10^6$K. The optical spectra of these shocks only contain emission lines of hydrogen and helium, as well as lines of coronal species of S, Si, Ar, Fe and Ni. The 300km/s shock is too slow to reproduce the observed \[\]/\[\] ratio, while the 400km/s overproduces \[\] relative to \[\], and is somewhat too faint relative to the cloud shock and its precursor. The curves on Figure \[fig12\] are presented for the 350km/s shock. In this figure, we added the contribution of the fast shock to that of the cloud shock, assuming that the fast shock covers an area of either 3 times, or 10 times that of the cloud shock. We assume that dust has been fully destroyed by shattering and sputtering in these fast shocks. Adding this fast shock contribution, we now achieve a satisfactory fit with the observational diagnostics for both \[\] and \[\]. The WiFeS images of N132D support the hypothesis that the faster \[\] and \[\] - emitting shocks cover a greater area than the cloud shocks which are bright in H$\alpha$. This is shown in Figure \[fig13\], in which we have constructed an image in H$\alpha$ (red), \[\] (green) and \[\] (blue). In this image the clouds show up as red or mauve blobs, while the \[\] emission forms diffuse halos about these clouds, and is much more extensive. In general, the \[\] emission is much more closely spatially correlated with the *Chandra X-Ray Observatory* images [@Borkowski07], an effect which is also seen in both N49 in the LMC [@Dopita16] and in the SMC SNR; 1E 0102.2-7219 [@Vogt17]. Detailed Cloud Shock Modelling {#cloudshocks} ============================== Cloud Shock Parameters {#cloudparms} ---------------------- From the generalised shock diagnostic diagrams, we now turn to the derivation of the shock parameters of the individual shocked ISM clouds in N132D. First, using the shock velocities estimated from both the \[\] and He II/He I diagnostic diagrams, the measured \[\] electron densities given by the \[\] $\lambda\lambda 6731/6717$ ratios, and the mean \[\] electron densities given from the model grid at the same shock velocity, we can estimate the pre-shock particle density, $n_{\mathrm{0}}$, and the ram pressure driving the shock; $P_{\mathrm{ram}}$. These are listed in Table \[table5\]. For the specific regions P08 and P09, the errors in the estimated shock velocities are great, as are the errors in $n_{\mathrm{0}}$. The region P20 is an incompletely radiative shock so therefore the measured \[\] electron density represents an under-estimate of the true density in the \[\] recombination zone. In the final column of Table \[table5\] we give the time required for the post-shock plasma to cool to 1000K, $ t_{\mathrm{1000K}}$ measured in years. At this point, the optical forbidden lines are no longer emitted in the plasma, and hydrogen is less than 2% ionised, so the shock is essentially fully radiative. We may conclude that these cloud shocks are somewhat older than $ \left<t_{\mathrm{1000K}}\right> =410$yr (excluding P20). [ The mean ram pressure for the clouds, $P_{\mathrm{ram}} = 3.1\times10^{-7}$dynes/cm$^2$ is much higher than the mean pressure in the X-ray plasma. This may be estimated using the proton densities and electron temperatures listed in @Williams06. This gives $7.6 \times 10^{-8}$ dynes cm$^{-2}$ in the NW region, and $5.3 \times 10^{-8}$ dynes cm$^{-2}$ in the S, for an average of $6.4 \times 10^{-8}$ dynes cm$^{-2}$. Thus the ram pressure in the cloud shocks is typically $\sim5$ times higher than in the surrounding X-ray plasma. The reason for this difference is discussed in Section \[conc\], below. ]{} The mean pre-shock density inferred for these clouds is surprisingly high; $\left<n_{\mathrm{0}} \right> = 243$cm$^{-3}$ (excluding P08, P09 and P20). However, the pressure in the ISM of the LMC has been estimated using the excited CI emission along a number of sight lines by @Welty16. This analysis gave $\left<\log P/k \right> =4.02$ cm$^{-3}$K. If the N132D clouds were characterised by the same ISM pressure before they were engulfed by the supernova shock wave (which is not at all a certain assumption), then their equilibrium temperature would have been $\sim 40$K, and they can be therefore considered to be part of the cold neutral medium phase of the ISM. Chemical Abundances & Depletion Factors --------------------------------------- We selected the four brightest clouds; P12, P14, P15 and P18 for detailed study, with the aim of deriving more precise shock parameters, and to derive the chemical abundances of the elements in the ISM of the LMC through high-precision shock modelling. In order to measure the goodness of fit of any particular model, we measure the degree to which it reproduces the density-sensitive \[SII\] $\lambda\lambda 6731/6717$ ratio, and we also seek to minimise the L1-norm for the fit. That is to say that we measure the modulus of the mean logarithmic difference in flux (relative to H$\beta$) between the model and the observations *viz.*; $${\rm L1} =\frac{1}{m}{\displaystyle\sum_{n=1}^{m}} \left | \log \left[ \frac{F_n({\rm model})} {F_n({\rm obs.)}} \right] \right |. \label{L1}$$ This procedure weights fainter lines equally with stronger lines, and is therefore more sensitive to the values of the input parameters. We simultaneously fit to those emission lines which are most sensitive to the controlling parameters of shock velocity and ram pressure; effectively HeI, HeII, \[OI\], \[OII\] and \[OIII\] as well as the \[NI\], \[NII\], and \[SII\] lines. In addition, we estimate the gas-phase heavy element abundances using the bright \[FeII\] and \[FeIII\] lines, \[FeV\] 4227Å \[FeVII\] 6087Å \[NiII\] 7378Å MgI\] 4561Å, CaII 3933Å and \[CaII\] 7291Å lines. We assume that the more excited species of Fe are depleted in the gas phase by the same amount as FeIII. For the reason given above we do not attempt to fit either the \[FeX\] 6374Å or the \[FeXIV\] 5303 Å lines, given that they arise in a separate phase of the ISM. Other elements such as Ne, Ar and Cl are fit using the \[NeIII\] 3969Å \[ArIII\] 7136Å and \[ClII\] 8579Å lines. Our models simultaneously [ optimise the fit to]{} 40 emission lines, and we manually iterate shock velocity, ram pressure and chemical abundances until the fit is optimised. The results for these four clouds are shown in Table \[table6\]. The L1-norm provides a good estimate for the error on the abundance of those elements fitted with only one or two emission lines. For O, N and S the accuracy is higher. In Table 2, we have adopted the abundances of Mg, Ca, Fe and Ni from @Russell92 in order to estimate the depletion factors of these elements. The best fit was achieved without the inclusion of the precursor emission, which in principle could account for as much as 25% of the total emission. There are two possible reasons for this. First, the finite extent of the cloud may limit the length of the precursor, which is maximised in our plane-parallel models. Second, the local curvature of the shock front may allow significant escape of the upstream ionising UV photons. This is suggested by the fact that our models of these four clouds systematically overestimate the $\lambda 4686$/ $\lambda 5876$ ratio, which suggests that the incoming plasma is over-ionised by the computed precursor radiation field given by the model. [ The derived depletion factors for Mg given in Table \[table6\] are likely to have very large, and unquantifiable errors attached to them. There is only one Mg line detected, and it is produced by neutral Mg in the tail of the recombination zone of the shock. Given the low ionisation potential of this species (7.64eV), the MgI\] line strength will be highly dependent on the unknown stellar UV radiation field, which by ionising this species would depress the line strength and lead to a larger depletion factor being inferred.]{} For the remaining elements we derive the following mean LMC interstellar abundances, $12+\log(\mathrm{[X/H]})$, of He: $ 10.86\pm 0.05$, N: $ 7.12\pm 0.07$, O: $ 8.31\pm 0.04$, Ne: $ 7.44 \pm 0.08$, S: $ 7.01\pm 0.06$, Cl: $ 4.66\pm 0.11$ and Ar: $ 5.78\pm 0.11$. These abundances are generally similar to those obtained in our detailed shock fitting of the supernova remnant N49 [@Dopita16]; He: 10.92, N: 7.10, O: 8.46, Ne: 7.79, S: 7.05, Cl: 5.2 and Ar: 6.1. These abundances should also be compared with those from @Russell92: He: $ 10.94\pm 0.03$, N: $ 7.14\pm 0.15$, O: $ 8.35\pm 0.06$, Ne: $ 7.61\pm 0.05$, S: $ 6.70\pm 0.09$, Cl: $ 4.76\pm 0.08$ and Ar: $ 6.29\pm 0.25$. Given the differences in the methodology used to derive these, and the changes in key atomic input parameters over the years, the agreement between these two is as good as could be expected. No systematic difference is seen for the elements most important in the modelling; N and O, but the derived He, Ne, S, Cl and Ar abundances are only marginally consistent with each other. What is now required is a re-derivation of the abundances using HII regions and the same modelling code to compare with the SNR-derived abundances. The derived abundances can also be compared to more recent determinations from stellar observations. @Korn05 gave an new analysis of a slow-rotating B star in NGC2002 which gave N: $ 6.99\pm 0.2$, O: $ 8.29\pm 0.2$, and Mg: $ 7.44\pm 0.2$. These values differed very little from his earlier work [@Korn02]. With respect to the refractory elements, @Korn00 derived appreciably lower Mg and Fe abundances than @Russell92; Mg: $ 6.96\pm 0.22$ and Fe: $ 7.09\pm 0.15$. Using these values, our results are consistent with full destruction of the Fe-bearing grains in the \[\]–emitting zone, and the estimated Mg depletion factor is lowered to only $\log D_{\mathrm{MgII}} = -0.38$. However, the later @Korn02 paper gave Mg: $7.37\pm 0.06$ and Fe: $ 7.33\pm 0.0.03$, which is in much closer agreement with the @Russell92 estimates. In conclusion, we can be fairly confident of the LMC He, N and O abundances; He: $ 10.92\pm 0.04$, N: $ 7.12\pm 0.09$ and O: $ 8.32\pm 0.06$. For the other non-refractory elements the recommended values are Ne: $ 7.52\pm 0.09$, S: $ 7.00\pm 0.15$, Cl: $ 4.70\pm 0.08$ and Ar: $ 5.78\pm 0.25$. Our results are consistent with nearly full destruction of the Fe–containing grains in these 210–260km/s shocks, but the Mg–containing grains appear to be only partially destroyed. This argues for a different carrier for the two metals, possibly iron(II) oxide (FeO) for iron and the magnesium silicates, forsterite (Mg$_2$SiO$_4$) and enstatite (MgSiO$_3$) in the case of Mg. The destruction of Fe–containing grains appears to have proceeded to a greater extent in N132D than in N49 [@Dopita16]. Although the estimated shock velocities are similar (210–260km/s in N132D and 200–250km/s in N49), the higher ram pressures and higher pre-shock densities in N132D (160–410cm$^{-3}$) compared with N49 ($\sim 80$cm$^{-3}$) has facilitated this greater fractional dust destruction. In both N49 and in N132D, the grain destruction in the faster, partially-radiative shocks producing \[\] and \[\] emission is much more advanced. In both these SNR, the shock velocity in these highly-ionised zones is $\sim 350$km/s. Thus, thermal sputtering and non-thermal sputtering due to the high relative velocity of the dust grains relative to the hot-post shock plasma [@Barlow78; @McKee87; @Seab87; @Jones94] dominates the grain destruction in these fast shocks, while gyro-acceleration (aka betatron acceleration) caused by the compression of the magnetic field in the cooling zone of the shock resulting in grain shattering and vaporisation [@Spitzer76; @Borkowski95] appears to dominate as the grain destruction process in the slower, denser shocks. The partially-radiative region P20 ---------------------------------- We already noted in Section \[diagnostics\] that region P20 has anomalously strong \[\] emission – see Figure \[fig7\]. Furthermore, the \[\] lines indicate a low density compared with the other regions which produces the low value of the ram pressure given in Table \[table5\]. All of these point to a shock which has not yet become fully radiative. In this case, it appears that the photoionised precursor is brighter in the optical lines than the shock itself. The extremely strong \[\] would arise in this precursor. The observed spectrum of this region proved very difficult to model. Our “best" model has a relatively poor fit, with an L1-norm of 0.21. This model has a $\sim 400$yr old shock with $v_s = 300$km/s moving into a medium with a pre-shock H density $n_0 = 235$cm$^{-3}$. In this shock the plasma cools from an initial temperature of $T_e =1.2\times10^6$K to a final temperature of $T_e =8.2\times10^5$K. The precursor material is assumed to have no dust destruction with $\log D_{Fe} = -1.00$, and the density of the pre-shock gas $n_0 = 235$cm$^{-3}$, and is illuminated by the UV radiation field of the partially-radiative shock in plane-parallel one-sided geometry. This gives a precursor thickness of $\sim 0.12$pc (at the point where H is 50% ionised). For this configuration, the shock produces only 4.5% of the total H$\beta$ luminosity. Table \[table7\] provides comparison of the key emission lines predicted by the model, compared with the observations. Note the extreme discrepancy in the predicted strength of the \[\] $\lambda 4363$ line. From the model, the predicted temperature in the zone is $T_e = 13600$K, while the data suggest an electron temperature of $T_e \sim 49000$K in this zone. Similar, but smaller, temperature discrepancies are suggested for the and zones. This requires another form of heating in the precursor such as electron conduction, cosmic ray heating or else ionisation by the general EUV radiation field of the nebula. Apart from these temperature discrepancies, the \[\] $\lambda\lambda 6731/6717$ ratio also requires a higher electron density; $n_{\mathrm{[SII]}} \sim 760$cm$^{-3}$. The Balmer-Dominated Shocks in P08 ---------------------------------- We noted above that the region P08 also displays an anomalous spectrum, deficient in the forbidden lines with respect to the Balmer lines, with the \[\] lines being particularly weak – see Figure \[fig7\]. The cause of this appears to be “non-radiative” or Balmer-dominated shocks. These occur when a fast shock runs into a cold, essentially un-ionised ISM. A narrow component of the Balmer lines arises from direct collisional excitation of Hydrogen by the fast electrons, and a broad Balmer component is produced by charge exchange with fast protons behind the shock and subsequent collisional excitation of the fast neutral hydrogen resulting by the hot electrons [@Chevalier78; @Chevalier80] – see the reviews by @Heng10 and @Ghavamian13, and references therein. In Figure \[fig14\], we show the HST view of the P08 region, and the spectrum extracted from the WiFeS data cube. It is most likely that the broad component of H$\alpha$ seen in the spectroscopy arises from the filaments which are visible in the H$\alpha$ image, but not in the \[\] image. A broad component is also weakly detected in H$\beta$. This would add N132D to the select group of SNR in which this has been detected – which includes Tycho, SN1006, RCW86, the Cygnus Loop and N103B and SNR 0519-69.0 (DEM N71) in the LMC [@Tuohy82; @Ghavamian01; @Ghavamian02; @Ghavamian17]. [ Most, but not all, of these remnants are from Type Ia supernova explosions, since these do not produce a burst of strong EUV radiation to pre-ionise the pre-shock medium. However, if the ISM is shielded from the EUV flash in some way, or if it is dense enough to recombine in the period between the explosion and the arrival of the shock, then nothing precludes the formation of a Balmer-dominated shock. Given the inferred age of the SNR, the recombination condition would imply that the shock is passing through a medium with a density $ \gtrsim 20$cm$^{-3}$. The width of the H$\alpha$ line suggests that the shock velocity in the Balmer-dominated shock is of order $\sim900$km/s. This is similar to the blast wave velocity derived from the X-ray data [@Favata97; @Hughes98; @Borkowski07], so it seems extremely likely that the broad Balmer lines arise in the blast wave itself. ]{} Conclusions {#conc} =========== The environment of N132D provides an excellent sampling of shocked clouds in the Bar of the LMC. The typical physical size of these clouds is $\sim 1.0$pc. Given that they have a typical pre-shock density of $\sim240$cm$^{-3}$ (from Table \[table5\]), we can infer cloud masses of few solar masses. Using the images from Figure \[fig3\] and the pre-shock densities from Table \[table5\], we obtain masses in the range $0.1- 20$M$_{\odot}$ with a mean of $\sim 4$M$_{\odot}$. Thus we infer these clouds initially represented typical ISM self-gravitating Bonnor-Ebert spheres such as those recently investigated on a theoretical basis by @Sipila11, @Fischera14 and @Sipila17. Given that the shock which moves into them following the passage of the supernova blast-wave is strongly compressive, and that such Bonnor-Ebert spheres can be marginally stable against collapse, it is tempting to imagine that the supernova shock may later induce formation of $\sim 1.0 M_{\odot}$ stars within the cores of those clouds. [ Now, let us consider the discrepancy between the thermal pressure measured by the X-rays, and the ram pressure driving the cloud shocks, pointed out in Section \[cloudparms\]. The X-ray plasma has a thermal pressure of $P_{\mathrm{therm}} = 6.4 \times 10^{-8}$ dynes cm$^{-2}$, while the ram pressure in the clouds is $P_{\mathrm{ram}} = 3.1 \times 10^{-7}$ dynes cm$^{-2}$. However, the pressure driving the cloud shocks is provided by the stagnation pressure behind the cloud bow shock or bow wave produced in the expanding thermal plasma which fills the SNR. This is always greater than the pressure in the pre-shock hot thermal plasma. The geometry of this interaction is pictured in @McKee75 [@Hester86]]{} and @Farage10. In the framework of a plane-parallel strong shock engulfing the cloud the stagnation pressure is about twice as large as the thermal pressure [@McKee75; @Hester86] and this is only very weakly dependent to the Mach number of the primary blast wave (see @Hester86, Table 2). The X-ray data enable us to estimate the stagnation pressure in two ways. First, we may use the Sedov theory to estimate the blast wave velocity. Using the figures given in @Hughes98 for the explosion energy, age and pre-shock density in the SNR, we derive the blast-wave velocity, $v_{\mathrm B} = 830$km/s. Alternatively, we can use the measured thermal plasma temperatures to derive the blast wave velocity, using the relation $T_{\mathrm B} =(3 \mu m_H/16k)v_{\mathrm B}^2$, where $m_H$ is the pre-shock hydrogen density and the molecular weight $\mu$ is appropriate to a fully-ionised plasma. Estimates of the thermal temperature are in the range $0.68 < kT < 0.8$keV [@Favata97; @Hughes98; @Borkowski07], which implies a blast wave velocity of $\sim 760$km/s. Therefore, the ram pressure associated with expansion is $\sim 4 \times 10^{-8}$ dynes cm$^{-2}$. Adding this to the thermal pressure gives an estimate of the stagnation pressure, $\sim 1.0 \times 10^{-7}$ dynes cm$^{-2}$. The remaining difference between the ram pressure of the cloud shocks and the estimated stagnation pressure is a factor of three. This difference may be accounted for by the fact that the cloud shocks are convergent towards the centre of mass, a possibility raised in the context of the Cygnus Loop SNR by @Hester86 as a promising mechanism for increasing pressure in shocked dense clouds. For shocks in self-gravitating isothermal spheres such as those considered here, families of self-similar solutions have been obtained by @Lou14. In such shocks, the energy density increases as the shock moves toward the center of the cloud, increasing the ram pressure. This effect can easily account for the difference between the measured ram pressure of the cloud shocks and the estimated stagnation pressure. Additionally, convergent cloud shocks are unstable in the presence of small deviations from sphericity [@Kimura90], and elongated clouds may break up into separate globules [@Kimura91]. Such instabilities are the likely reason for the complex morphologies of the shocks in individual clouds seen in Figure \[fig3\]. Because the shocked clouds of N132D represent gravitationally-confined samples of the ISM as they existed before the supernova event, and because these clouds are dense enough that they would not be appreciably affected in their chemical composition by any pre-supernova mass-loss, they present ideal samples of pristine ISM in the LMC. Our radiative shock analysis has enabled us to estimate accurate gas-phase chemical abundances for a number of elements. From our model grid, we have demonstrated that the \[\] $\lambda\lambda 4363/5007$ ratio is a good indicator of shock velocity, and that the / $\lambda\lambda 4686/5876$ ratio may also be used for this purpose, although it is rather less reliable. Typical shock velocities are $\sim240$km/s, in agreement with the shock velocities inferred from the kinematics and the measured emission line widths. Using these emission line diagnostics, we have analysed the depletion onto dust in the various ionic stages of Fe, and in and . In common with the SNR N49, we find that dust has been mostly destroyed in the region emitting the \[\] lines, while a smaller fraction has been destroyed in the and zones, consistent with the grain destruction models of @Seab83 and @Borkowski95. However, shows an appreciably higher depletion factor, suggesting that the Mg silicates are more resistant to destruction than the Fe-bearing grains. It is clear that the highly ionised species of iron; and originate in faster, partially radiative, filamentary and spatially extensive shocks surrounding the dense clouds. Some of these shocks may well be part of the primary blast wave of the SNR. In these, most of the refractory elements have been destroyed by thermal sputtering. A detailed shock analysis of the four brightest clouds has allowed us to determine the chemical abundances of a number of elements. Comparing these with values given earlier, and with stellar abundance determinations we can now provide a “recommended” set of LMC abundances, which we present here in Table \[table8\]. Here, for completeness, the abundances of C, Mg, Si and Fe have been taken from the work of @Korn02 using Magellanic Cloud B-stars in NGC2004. Finally we have identified two anomalous regions, P08 and P20. The former contains a contribution from fast Balmer-dominated shocks, while the latter represents an unusual partially-radiative shock, dominated by precursor emission which seems to be heated by an unknown source to very high electron temperatures. Acknowledgments {#acknowledgments .unnumbered} =============== MD and RS acknowledge the support of the Australian Research Council (ARC) through Discovery project DP16010363. Parts of this research were conducted by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. FPAV and IRS thank the CAASTRO AI travel grant for generous support. IRS was supported by the ARC through the Future Fellowship grant FT1601000028. AJR is supported by the Australian Research Council through Future Fellowship grant FT170100243. This research has made use of <span style="font-variant:small-caps;">matplotlib</span> [@Hunter07], <span style="font-variant:small-caps;">astropy</span>, a community-developed core <span style="font-variant:small-caps;">python</span> package for Astronomy [@AstropyCollaboration13], <span style="font-variant:small-caps;">APLpy</span>, an open-source plotting package for <span style="font-variant:small-caps;">python</span>[@Robitaille12], and <span style="font-variant:small-caps;">montage</span>, funded by the National Science Foundation under Grant Number ACI-1440620 and previously funded by the National Aeronautics and Space Administration’s Earth Science Technology Office, Computation Technologies Project, under Cooperative Agreement Number NCC5-626 between NASA and the California Institute of Technology. This research has also made use of <span style="font-variant:small-caps;">drizzlepac</span>, a product of the Space Telescope Science Institute, which is operated by AURA for NASA, of the <span style="font-variant:small-caps;">aladin</span> interactive sky atlas [@Bonnarel00], of <span style="font-variant:small-caps;">saoimage ds9</span> [@Joye03] developed by Smithsonian Astrophysical Observatory, of NASA’s Astrophysics Data System, and of the NASA/IPAC Extragalactic Database [@Helou91] which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. This research has also made use of NASA’s Astrophysics Data System. This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} (<https://www.cosmos.esa.int/gaia>), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. 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No reddening correction has been applied, since the mean Balmer Decrement as measured; H$\alpha$/H$\beta$/H$\gamma$/H$\delta$ = 298.7/100.0/44.7/23.9 is indistinguishable (within the errors) with the theoretical Balmer Decrement for $\log n_e =10^4$cm$^{-3}$, and $T_e = 5000$K; 300/100/46.0/25.3. On the basis of the soft X-ray absorption @Borkowski07 determined a best-fit hydrogen column density of $1.4\times10^{20}$ cm$^{-2}$ in the western part of N132D and $1.4-4.1\times10^{21}$ cm$^{-2}$ in the south. Assuming dust is not destroyed in the bar of the LMC, and using the @Fitzpatrick86 $N_{\mathrm H}/E_{\mathrm{B - V}}$ conversion factor, these figures would imply a reddening of $E_{\mathrm{B - V}} = 0.005$mag. and 0.06–0.17mag., respectively. [^1]: Available at : [^2]: <http://www.mso.anu.edu.au/pywifes/doku.php.> [^3]: [QFitsView v3.1]{} is a FITS file viewer using the QT widget library and was developed at the Max Planck Institute for Extraterrestrial Physics by Thomas Ott. [^4]: Graf is written by R. S. Sutherland and is available at: [<https://miocene.anu.edu.au/graf>]{} [^5]: Lines is written by R. S. Sutherland and is available at: [<https://miocene.anu.edu.au/lines>]{}

--- author: - 'Julien Baglio,' - 'Francisco Campanario,' - 'Seraina Glaus,' - 'Margarete Mühlleitner,' - 'Jonathan Ronca,' - Michael Spira - and Juraj Streicher bibliography: - 'paper.bib' title: 'Higgs-Pair Production via Gluon Fusion at Hadron Colliders: NLO QCD Corrections' --- Introduction ============ Since the discovery of a scalar resonance [@Aad:2012tfa; @Chatrchyan:2012xdj] with a mass of $125.09\pm0.24$ GeV [@Khachatryan:2016vau] that is compatible with the Standard Model (SM) Higgs boson [@Higgs:1964ia; @Higgs:1964pj; @Higgs:1966ev; @Englert:1964et; @Guralnik:1964eu; @Kibble:1967sv], the detailed study of the properties of this particle has been a high priority of the analyses at the Large Hadron Collider (LHC). Theoretical uncertainties are a limiting factor for the accuracies reachable at the LHC. This restriction can partly be compensated by increasing the diversity of processes involving the Higgs boson and a broader spectrum of Higgs couplings probed at the LHC. In order to test the nature of the Higgs boson, its self-interactions are of particular interest. It will be the first step towards an experimental reconstruction of the Higgs potential. This plays a crucial role as the origin of electroweak symmetry breaking within the SM. The initial processes that provide a direct sensitivity to the Higgs self-couplings are Higgs-pair production processes. They involve the trilinear Higgs coupling at leading order (LO) [@Glover:1987nx; @Plehn:1996wb; @Dawson:1998py; @Djouadi:1999rca; @Baglio:2012np]. These processes are complementary to indirect effects induced by the Higgs self-interactions in radiative corrections to electroweak observables and single-Higgs processes [@Degrassi:2016wml; @Degrassi:2017ucl] that are plagued by unknown interference effects with other kinds of New Physics. The Higgs self-interactions are uniquely described by the SM Higgs potential $$V = \frac{\lambda}{2} \left( \phi^\dagger \phi - \frac{v^2}{2}\right)^2\, ,$$ where $\lambda$ defines the self-interaction strength of the SM Higgs field. In unitary gauge, the Higgs doublet $\phi$ is given by $$\phi = \left( \begin{array}{c} \displaystyle 0 \\ \displaystyle \frac{v+H}{\sqrt{2}} \end{array} \right)$$ with $v\approx 246$ GeV denoting the vacuum expectation value (vev) and $H$ is the physical Higgs field. In the SM, the self-interaction strength is given in terms of the Higgs mass $M_H$ by $\lambda = M_H^2/v^2$. Expanding the Higgs field around its vev, the Higgs self-interactions, including the corresponding permutations, are uniquely determined as $$\lambda_{H^3} = 3 \frac{M_H^2}{v}, \qquad \lambda_{H^4} = 3 \frac{M_H^2}{v^2}\, ,$$ where $\lambda_{H^3}$ ($\lambda_{H^4}$) denotes the trilinear (quartic) Higgs self-coupling. While the quartic Higgs coupling $\lambda_{H^4}$ cannot be probed directly at the LHC, due to the tiny size of the triple-Higgs production cross section [@Plehn:2005nk; @Binoth:2006ym; @Fuks:2015hna; @deFlorian:2016sit; @deFlorian:2019app][^1], the trilinear Higgs coupling can be accessed directly in Higgs-pair production. Higgs-boson pairs are dominantly produced in the loop-induced gluon-fusion mechanism $gg\to HH$ that is mediated by top-quark loops supplemented by a per-cent-level contribution of bottom-quark loops, see Fig. \[fg:hhdia\]. There are destructively interfering box and triangle diagrams at LO with the latter involving the trilinear Higgs coupling [@Glover:1987nx; @Plehn:1996wb]. The box diagrams provide the dominant contributions to the cross section. A rough estimate of the dependence of the cross section on the size of the trilinear coupling is given by the approximate relation $\Delta\sigma/\sigma \sim -\Delta\lambda_{H^3}/\lambda_{H^3}$ in the vicinity of the SM value of $\lambda_{H^3}$. Therefore, in order to determine the trilinear coupling, the theoretical uncertainties of the corresponding cross section need to be small. Thus, the inclusion of higher-order corrections is mandatory. The QCD corrections are fully known up to next-to-leading order (NLO) [@Borowka:2016ehy; @Borowka:2016ypz; @Baglio:2018lrj] and at next-to-next-to-leading order (NNLO) in the limit of heavy top quarks [@deFlorian:2013uza; @deFlorian:2013jea; @Grigo:2014jma]. While the NLO corrections are large, the NNLO contributions are of more moderate size. Very recently, the next-to-next-to-next-to-leading order (N$^3$LO) QCD corrections have been computed in the limit of heavy top quarks resulting in a small further modification of the cross section [@Banerjee:2018lfq; @Chen:2019lzz; @Chen:2019fhs]. This calculation uses the N$^3$LO corrections to the effective Higgs and Higgs-pair couplings to gluons in the heavy-top limit (HTL) [@Spira:2016zna]. The higher-order QCD corrections increase the total LO cross section by about a factor of two. Recently, the full NLO results have been matched to parton showers [@Heinrich:2017kxx; @Jones:2017giv] and the full NNLO results in the limit of heavy top quarks have been merged with the NLO mass effects and supplemented by the additional top-mass effects in the double-real corrections [@Grazzini:2018bsd]. ![\[fg:hhdia\] *Generic diagrams contributing to Higgs-boson pair production via gluon fusion. The contribution of the trilinear Higgs coupling is marked in red.*](./plots/dia_gg2hh.pdf){width="110.00000%"} The goal of this paper is to present in detail the calculation of Ref. [@Baglio:2018lrj] of the full NLO corrections to Higgs pair production in gluon fusion. We rely on a direct numerical integration of the Feynman diagrams, without any tensor reduction. We extend the results presented in Ref. [@Baglio:2018lrj] and study not only the LHC at center-of-mass energies of 13 and 14 TeV, but also present numbers for a potential high-energy upgrade of the LHC (HE-LHC) at 27 TeV [@Abada:2019ono] and for a provisional 100 TeV proton collider within the Future-Circular-Collider (FCC) project  [@Abada:2019lih; @Benedikt:2018csr]. Special emphasis will be given to the study of the theoretical uncertainties affecting the results and in particular the scale and scheme uncertainty related to the top-quark mass. We will also study the variation of the trilinear Higgs coupling and show that the NLO mass effects shift the minimum of the total cross section as a function of $\lambda_{H^3}$. They vary substantially over the range of $\lambda_{H^3}$ values. The paper is organized as follows. We present the notation of our calculation in Section \[sc:lo\] and discuss the results at LO. In Section \[sc:nlo\] we move to the NLO QCD corrections. We discuss the details of the calculation of the virtual corrections in Section \[sc:virtuals\]. We describe the derivation of the real corrections in Section \[sc:reals\]. Our numerical analysis is performed in Section \[sc:results\]. Finally, the conclusions are given in Section \[sc:conclusions\]. Leading-order cross section \[sc:lo\] ===================================== At LO, Higgs-boson pair production via gluon fusion is mediated by the generic diagrams of Fig. \[fg:hhdia\], including all permutations of the external lines. There are triangle and box diagrams with the former involving the trilinear Higgs coupling through an $s$-channel Higgs exchange. The LO matrix element of $g(q_1) g(q_2) \to H(p_1) H(p_2)$ can be cast into the form $$\begin{aligned} {\cal M}(g^a g^b \to HH) & = & -i\,\frac{G_F\alpha_s(\mu_R) Q^2}{2\sqrt{2}\pi} {\cal A}^{\mu\nu} \epsilon_{1\mu} \epsilon_{2\nu} \delta_{ab} \nonumber \\[0.3cm] \mbox{with} \qquad {\cal A}^{\mu\nu} & = & F_1 T_1^{\mu\nu} + F_2 T_2^{\mu\nu} \, , \nonumber \\[0.3cm] F_1 & = & C_\triangle F_\triangle + F_\Box \, , \qquad \qquad \qquad F_2 = G_\Box \, , \nonumber \\[0.3cm] C_\triangle & = & \frac{\lambda_{H^3} v}{Q^2 - M_H^2 + iM_H\Gamma_H} \nonumber \\[0.3cm] \mbox{and} \qquad Q^2 & = & (p_1+p_2)^2 = m_{HH}^2 \label{eq:lomat}\end{aligned}$$ with $Q=m_{HH}$ denoting the invariant Higgs-pair mass. Here $a,b$ denote the color indices of the initial gluons, $\epsilon_{1/2}$ their polarization vectors, $\Gamma_H$ the total Higgs width[^2], $G_F$ the Fermi constant and $\alpha_s(\mu_R)$ the strong coupling at the renormalization scale $\mu_R$. Since in this work we neglect the small bottom-quark contribution, the LO function of the triangle-diagram contribution is given by the top-quark contribution, $$F_\triangle (\tau_t) = \tau_t \Big[ 1 + (1-\tau_t) f(\tau_t) \Big ] \label{eq:ftriangle}$$ with $\tau_t = 4m_t^2/Q^2$ and the basic function $$\begin{aligned} f(\tau) & = & \left\{ \begin{array}{ll} \displaystyle \arcsin^2 \frac{1}{\sqrt{\tau}} & \tau \ge 1 \\ \displaystyle - \frac{1}{4} \left[ \log \frac{1+\sqrt{1-\tau}} {1-\sqrt{1-\tau}} - i\pi \right]^2 & \tau < 1 \end{array} \right. \, , \label{eq:ftau}\end{aligned}$$ where $m_t$ denotes the top mass, while the more involved analytical expressions for $F_\Box$ and $G_\Box$ can be found in Ref. [@Plehn:1996wb]. In the HTL, the LO form factors approach the values $$F_\triangle \to \frac{2}{3}, \qquad F_\Box \to -\frac{2}{3}, \qquad G_\Box \to 0 \, . \label{eq:ffhtl}$$ There are two tensor structures contributing which correspond to the total angular-momentum states with $S_z=0$ and $2$, $$\begin{aligned} T_1^{\mu\nu} & = & g^{\mu\nu}-\frac{q_1^\nu q_2^\mu}{(q_1q_2)}\, , \nonumber \\ T_2^{\mu\nu} & = & g^{\mu\nu}+\frac{M_H^2 q_1^\nu q_2^\mu}{p_T^2 (q_1q_2)} -2\frac{(q_2 p_1) q_1^\nu p_1^\mu}{p_T^2 (q_1q_2)} -2\frac{(q_1 p_1) p_1^\nu q_2^\mu}{p_T^2 (q_1q_2)} +2\frac{p_1^\nu p_1^\mu}{p_T^2} \nonumber \\ \mbox{with} \quad p_T^2 & = & 2 \frac{(q_1 p_1)(q_2 p_1)}{(q_1 q_2)} - M_H^2 \, ,\end{aligned}$$ where $p_T$ is the transverse momentum of each of the final-state Higgs bosons. Working in $n=4-2\epsilon$ dimensions, the following projectors on the two form factors can be constructed, $$P_1^{\mu\nu} = \frac{(1-\epsilon) T_1^{\mu\nu} + \epsilon T_2^{\mu\nu}}{2(1-2\epsilon)}\, , \qquad \qquad P_2^{\mu\nu} = \frac{\epsilon T_1^{\mu\nu} + (1-\epsilon) T_2^{\mu\nu}}{2(1-2\epsilon)} \, ,$$ such that $$P_1^{\mu\nu} {\cal A}_{\mu\nu} = F_1 \, , \qquad \qquad P_2^{\mu\nu} {\cal A}_{\mu\nu} = F_2 \, .$$ Using these projectors, the explicit results of the two form factors $F_{1,2}$ can be obtained in a straightforward manner. The analytical expressions can be found in Refs. [@Glover:1987nx; @Plehn:1996wb]. Working out the polarization and color sums of the matrix element of Eq. (\[eq:lomat\]), the LO partonic cross section $\hat\sigma_{LO}$ is given by $$\hat\sigma_{LO} = \frac{G_F^2\alpha_s^2(\mu_R)}{512 (2\pi)^3} \int_{\hat t_-}^{\hat t_+} d\hat t \Big[ | F_1 |^2 + |F_2|^2 \Big]$$ with the integration boundaries $$\hat t_\pm = -\frac{1}{2} \left[ Q^2 - 2M_H^2 \mp Q^2 \sqrt{1-4\frac{M_H^2}{Q^2}} \right] \, , \label{eq:tbound}$$ where the symmetry factor 1/2 for the identical Higgs bosons in the final state is taken into account. The LO hadronic cross section $\sigma_{LO}$ can then be derived by a convolution with the parton densities $$\sigma_{LO} = \int_{\tau_0}^1 d\tau \frac{d{\cal L}^{gg}}{d\tau} \hat\sigma_{LO}(Q^2 = \tau s)$$ with the gluon luminosity, given in terms of the gluon densities $g(x,\mu_F)$, $$\frac{d{\cal L}^{gg}}{d\tau} = \int_\tau^1 \frac{dx}{x} g(x,\mu_F) g\left(\frac{\tau}{x},\mu_F\right) \label{eq:lgg}$$ at the factorization scale $\mu_F$ and the integration boundary $\tau_0=4M_H^2/s$, where $s$ denotes the hadronic center-of-mass (c.m.) energy squared. The differential cross section with respect to the invariant squared Higgs-pair mass $Q^2$ can be obtained as $$\frac{d\sigma_{LO}}{dQ^2} = \left. \frac{d{\cal L}^{gg}}{d\tau}~ \frac{\hat\sigma_{LO}(Q^2)}{s} \right|_{\tau = \frac{Q^2}{s}} \, . \label{eq:lodiff}$$ As can be expected from single Higgs-boson production via gluon fusion (see [@Graudenz:1992pv; @Spira:1995rr; @Harlander:2005rq; @Anastasiou:2009kn; @Aglietti:2006tp]), the NLO QCD corrections to these LO expressions will be large. Next-to-leading-order corrections \[sc:nlo\] ============================================ The NLO QCD corrections to Higgs-pair production via gluon fusion have been computed in the HTL, a long time ago [@Dawson:1998py]. The NLO result for the gluon-fusion cross section can be generically expressed as [@Dawson:1998py] $$\begin{aligned} \sigma_{NLO}(pp \rightarrow H H + X) & = & \sigma_{LO} + \Delta \sigma_{virt} + \Delta\sigma_{gg} + \Delta\sigma_{gq} + \Delta\sigma_{q\bar{q}} \, , \nonumber $$ $$\begin{aligned} \sigma_{LO} & = & \int_{\tau_0}^1 d\tau~\frac{d{\cal L}^{gg}}{d\tau}~\hat\sigma_{LO}(Q^2 = \tau s) \, , \nonumber \\ \Delta \sigma_{virt} & = & \frac{\alpha_s(\mu_R)} {\pi}\int_{\tau_0}^1 d\tau~\frac{d{\cal L}^{gg}}{d\tau}~\hat \sigma_{LO}(Q^2=\tau s)~C_{virt}(Q^2) \, , \nonumber \\ \Delta \sigma_{ij} & = & \frac{\alpha_{s}(\mu_R)} {\pi} \int_{\tau_0}^1 d\tau~ \frac{d{\cal L}^{ij}}{d\tau} \int_{\tau_0/\tau}^1 \frac{dz}{z}~ \hat\sigma_{LO}(Q^2 = z \tau s)\, C_{ij}(Q^2,z) \qquad (ij=gg,gq,q\bar q) \, ,\nonumber \\ C_{gg}(Q^2,z) & = & - z P_{gg} (z) \log \frac{\mu_F^{2}}{\tau s} + 6 [1+z^4+(1-z)^4] \left(\frac{\log (1-z)}{1-z} \right)_+ + d_{gg}(Q^2,z) \, , \nonumber \\ C_{gq}(Q^2,z) & = & -\frac{z}{2} P_{gq}(z) \log\frac{\mu_F^{2}}{\tau s(1-z)^2} + d_{gq}(Q^2,z) \, , \nonumber \\ C_{q\bar q}(Q^2,z) & = & d_{q\bar q}(Q^2,z) \label{eq:nlocxn}\end{aligned}$$ with $\hat\sigma_{LO}(Q^2)$ denoting the partonic cross section at LO and the strong coupling $\alpha_s(\mu_R)$ is evaluated at the renormalization scale $\mu_R$. The objects $d{\cal L}^{ij}/d\tau~(i,j=g,q,\bar q)$ denote the parton-parton luminosities, defined analogously to $d{\cal L}^{gg}/d\tau$ of Eq. (\[eq:lgg\]), using the quark densities $q(x,\mu_F)$, $$\begin{aligned} \frac{d{\cal L}^{gq}}{d\tau} & = & \sum_{q,\bar q} \int_\tau^1 \frac{dx}{x} \Big[ g(x,\mu_F) q\left(\frac{\tau}{x},\mu_F\right) + q(x,\mu_F) g\left(\frac{\tau}{x},\mu_F\right) \Big] \, , \nonumber \\ \frac{d{\cal L}^{q\bar q}}{d\tau} & = & \sum_q \int_\tau^1 \frac{dx}{x} \Big[ q(x,\mu_F) \bar q\left(\frac{\tau}{x},\mu_F\right) + \bar q(x,\mu_F) q\left(\frac{\tau}{x},\mu_F\right) \Big]\end{aligned}$$ at the factorization scale $\mu_F$ and $P_{ij}(z)~(i,j=g,q,\bar q)$ are the specific Altarelli–Parisi splitting functions [@Altarelli:1977zs]. The quark-mass dependence is in general encoded in the LO cross section $\hat\sigma_{LO}(Q^2)$ and the terms $C_{virt}(Q^2)$, $d_{ij}(Q^2,z)$ for the virtual and real corrections, respectively. These expressions can easily be converted into the differential cross section with respect to $Q^2$, $$\begin{aligned} \frac{d\Delta\sigma_{virt}}{dQ^2} & = & \left. \frac{\alpha_s\left(\mu_R\right)}{\pi}~\frac{d{\cal L}^{gg}}{d\tau}~ \frac{\hat{\sigma}_{LO}\left(Q^2 \right)}{s}~C_{virt} \left(Q^2\right) \right|_{\tau = \frac{Q^2}{s}}, \nonumber \\[0.3cm] \frac{d\Delta\sigma_{ij}}{dQ^2} & = & \left. \frac{\alpha_s\left(\mu_R\right)}{\pi}\int_{\frac{Q^2}{s}}^1 \frac{dz}{z^2}~\frac{d{\cal L}^{ij}}{d\tau}~ \frac{\hat{\sigma}_{LO}\left(Q^2\right)}{s}~C_{ij}(Q^2,z) \right|_{\tau = \frac{Q^2}{zs}} \, , \label{eq:nlodiff}\end{aligned}$$ while the differential cross section at LO is given in Eq. (\[eq:lodiff\]). ![\[fg:gghhvirt\] *Typical two-loop triangle (left), one-particle reducible (middle) and box (right) diagrams contributing to Higgs-pair production via gluon fusion at NLO.*](./plots/dia_virt.pdf){width="95.00000%"} Within the HTL, the Higgs coupling to gluons can be described by an effective Lagrangian [@Ellis:1975ap; @Shifman:1979eb; @Inami:1982xt; @Spira:1995rr; @Kniehl:1995tn] $${\cal L}_\mathrm{eff} = \frac{\alpha_s}{12\pi} G^{a\mu\nu} G^a _{\mu\nu} \left(C_1 \frac{H}{v} - C_2 \frac{H^2}{2v^2} \right) \label{eq:leff}$$ involving the Wilson coefficients ($L_t = \log \mu_R^2/m_t^2$) [@Chetyrkin:1997iv; @Kramer:1996iq; @Dawson:1998py; @Schroder:2005hy; @Baikov:2016tgj; @Grigo:2014jma; @Spira:2016zna; @Gerlach:2018hen] $$\begin{aligned} C_1 & = & 1 + \frac{11}{4} \frac{\alpha_s}{\pi} + \left\{ \frac{2777}{288} + \frac{19}{16} L_t + N_F \left(\frac{L_t}{3}-\frac{67}{96} \right) \right\} \left(\frac{\alpha_s}{\pi} \right)^2 + {\cal O}(\alpha_s^3) \, , \nonumber \\ C_2 & = & C_1 + \left( \frac{35}{24} + \frac{2}{3} N_F \right) \left(\frac{\alpha_s}{\pi} \right)^2 + {\cal O}(\alpha_s^3) \label{eq:leffcoeff}\end{aligned}$$ that are known up to N$^4$LO [@Schroder:2005hy; @Baikov:2016tgj; @Spira:2016zna]. Since the top quark is integrated out, the number of active flavours has been chosen as $N_F=5$. If these effective Higgs couplings to gluons in the calculation of the NLO QCD corrections are used, the calculation of these is simplified to a one-loop calculation for the virtual corrections and a tree-level one for the matrix elements of the real corrections. The terms $C_{virt}(Q^2)$ and $d_{ij}(Q^2,z)$, for the virtual and real corrections, approach in the HTL the simple expressions $$\begin{aligned} C_{virt}(Q^2) & \to & \frac{11}{2} + \pi^2 + C^\infty_{\triangle\triangle} + \frac{33-2N_F}{6} \log\frac{\mu_R^2}{Q^2}, \nonumber \\ C_{\triangle\triangle} & = & \Re e~\frac{\int_{\hat t_-}^{\hat t_+} d\hat t \left\{ c_1 \left[ (C_\triangle F_\triangle + F_\Box) + \frac{p_T^2}{\hat t} G_\Box \right]^* + (\hat t \leftrightarrow \hat u) \right\}} {\int_{\hat t_-}^{\hat t_+} d\hat t \left\{ |C_\triangle F_\triangle + F_\Box |^2 + |G_\Box|^2 \right\}}, \nonumber \\[0.3cm] C^\infty_{\triangle\triangle} & = & \left. C_{\triangle\triangle} \right|_{c_1 = 2/9}, \nonumber \\[0.3cm] d_{gg}(Q^2,z) \!\!\! & \to & \!\!\! - \frac{11}{2} (1-z)^3 \, , \ d_{gq}(Q^2,z) \to \frac{2}{3} z^2 - (1-z)^2 \, , \ d_{q\bar q}(Q^2,z) \to \frac{32}{27} (1-z)^3 \, , \label{eq:coeffvirt}\end{aligned}$$ where $\hat s,\hat t, \hat u$ ($\hat s=Q^2$ at LO and for the virtual corrections) denote the partonic Mandelstam variables and $C_{\triangle\triangle}$ is the contribution of the one-particle reducible diagrams, see Fig. \[fg:gghhvirt\]. At NLO QCD, the full mass dependence of the LO partonic cross section has been taken into account, while keeping the virtual corrections $C_{virt}$ and the real corrections $d_{ij}$ in the HTL (“Born-improved” approach) [@Dawson:1998py]. This yields a reasonable approximation for smaller invariant Higgs-pair masses and approximates the full NLO result of the total cross section within about 15% [@Borowka:2016ehy; @Borowka:2016ypz; @Baglio:2018lrj]. The NLO QCD corrections in the HTL increase the cross section by $80-90\%$ [@Dawson:1998py]. Within the Born-improved HTL, the NNLO QCD corrections have been obtained in Refs. [@deFlorian:2013uza; @deFlorian:2013jea; @Grigo:2014jma] increasing the total cross section by a moderate amount of $20-30\%$ [@deFlorian:2013jea]. Beyond these NNLO QCD corrections, the soft-gluon resummation (threshold resummation) has been performed at next-to-next-to-leading logarithmic (NNLL) accuracy for the total cross section and invariant mass distribution, modifying the total cross section further by a small amount if the central scales are chosen as $\mu_R=\mu_F=Q/2$ [@Shao:2013bz; @deFlorian:2015moa]. Very recently, the N$^3$LO QCD corrections have been computed in the Born-improved HTL resulting in a small modification of the cross section beyond NNLO[@Banerjee:2018lfq; @Chen:2019lzz; @Chen:2019fhs; @Spira:2016zna]. These N$^3$LO QCD corrections in the HTL have been merged with the full top-mass effects of the NLO calculation [@Chen:2019fhs]. The calculations in the HTL have been improved by several steps including mass effects partially at NLO. The full mass effects in the real correction terms $d_{ij}$ have been included by means of the full one-loop real matrix elements for $gg\to HHg, gq \to HHq, q\bar q\to HHg$. This improvement reduces the Born-improved HTL prediction for the total cross section by about 10% [@Frederix:2014hta; @Maltoni:2014eza] and is called the “FTapprox” approximation. The calculation of the full real matrix elements has been performed by using the [MG5\_aMC@NLO]{} framework [@Alwall:2014hca; @Hirschi:2015iia]. Another improvement has been achieved by an asymptotic large-top-mass expansion of the full NLO corrections at the level of the integral [@Grigo:2013rya] and the integrand [@Grigo:2015dia]. This indicated sizable mass effects in the virtual two-loop corrections alone. In addition, the large top-mass expansion has been extended to the virtual NNLO QCD corrections resulting in 5% mass effects estimated on top of the NLO result [@Grigo:2015dia]. The large-top-mass expansion of the NLO QCD corrections has been used to perform a conformal mapping of the expansion parameter and to apply Padé approximants. In this way, an approximation of the full calculation has been achieved for $Q$ values up to about 700 GeV [@Grober:2017uho]. Another approximation builds on an expansion in terms of a variable that dominantly corresponds to the transverse momentum of the Higgs bosons. The results of this approach show good agreement with the full calculation for $Q$ values up to about 900 GeV [@Bonciani:2018omm]. Analytical results are also available in the large-$Q$ limit [@Davies:2018qvx]. The latter have recently been combined with the numerical results of Refs. [@Borowka:2016ehy; @Borowka:2016ypz] for the full QCD corrections [@Davies:2019dfy]. In the following, we will discuss the details of our NLO calculation. Virtual corrections \[sc:virtuals\] ----------------------------------- Typical diagrams of the two-loop virtual corrections are shown in Fig. \[fg:gghhvirt\]. They can be arranged in three different classes: (a) triangle, (b) one-particle-reducible and (c) box diagrams[^3]. They contribute to the coefficient $C_{virt}(Q^2)$ of Eq. (\[eq:nlocxn\]), $$C_{virt}(Q^2) = 2 \Re e~\frac{\int_{\hat t_-}^{\hat t_+} d\hat t \left\{ (C_\triangle F_\triangle + F_\Box)^* [C_\triangle (\Delta F_\triangle) + \Delta F_\Box] + G_\Box^* (\Delta G_\Box) \right\} } {\int_{\hat t_-}^{\hat t_+} d\hat t \left\{ |C_\triangle F_\triangle + F_\Box |^2 + |G_\Box|^2 \right\} },$$ where $\Delta F_\triangle, \Delta F_\Box$ and $\Delta G_\Box$ denote the virtual corrections to the corresponding LO form factors. While $\Delta F_\triangle$ involves only virtual corrections to the triangle diagram, $\Delta F_\Box$ and $\Delta G_\Box$ acquire contributions from the one-particle-reducible and box diagrams. ### Triangle diagrams ![\[fg:triadia\] *Two-loop triangle diagrams contributing to Higgs-pair production via gluon fusion.*](./plots/triangles.pdf){width="90.00000%"} The generic 2-loop triangle diagrams contributing to the virtual coefficient $C_{virt}(Q^2)$ are shown in Fig. \[fg:triadia\]. They only contribute to the spin-0 form factor $F_1$ of Eq. (\[eq:lomat\]) and can be parametrized as the correction $\Delta F_\triangle$ to the form factor $F_\triangle$, $$\Delta F_\triangle = \frac{\alpha_s}{\pi}~{\cal C}_{virt}(Q^2)~F_\triangle \, ,$$ where ${\cal C}_{virt}(Q^2)$ denotes the [*complex*]{} virtual coefficient relative to the LO form factor $F_\triangle$ of the amplitude. This virtual coefficient is related to the single-Higgs case so that the relative QCD corrections can be simply obtained from the known (complex) virtual coefficient ${\cal C}^H_{virt}(M_H^2)$ of single Higgs production [@Spira:1995rr; @Graudenz:1992pv; @Harlander:2005rq; @Anastasiou:2009kn; @Aglietti:2006tp][^4], $${\cal C}_{virt}(Q^2) = \left. {\cal C}^H_{virt}(M_H^2)\right|_{M_H^2 \to Q^2} \, .$$ In the HTL, this virtual coefficient (before renormalization) approaches the expression $${\cal C}_{virt} (Q^2) \to \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left(\frac{4\pi\mu_0^2 (1-i\bar\epsilon)}{-Q^2}\right)^\epsilon \left\{ -\frac{3}{2\epsilon^2} + \frac{3}{4} - \frac{\pi^2}{4} \right\}$$ with the ’t Hooft scale $\mu_0$, where the (infinitesimal) regulator $\bar\epsilon$ defines the proper analytical continuation of this expression. This result has to be followed by the renormalization of the strong coupling $\alpha_s$ and the top mass $m_t$ that will be discussed in Section \[sc:renorm\]. In addition, we have subtracted the HTL to obtain the pure top-mass effects at NLO (relative to the massive LO expression $F_\triangle$) to ensure that in the end the results of the program [Hpair]{} [@hpair] can be added back. This last step will be discussed in Section \[sc:renorm\], too. ### One-particle-reducible diagrams The one-particle-reducible contribution is depicted in Fig. \[fg:gghhvirt\] (middle diagram), where a second diagram with the initial gluons interchanged has to be added. These will constitute the $\hat t$- and $\hat u$-channel parts where the second is related to the first just by the interchange $\hat t\leftrightarrow \hat u$ \[see $C_{\triangle\triangle}$ of Eq. (\[eq:coeffvirt\])\]. The analytical expression of the coefficient $c_1$ can be related to the top contribution of the process $H\to Z\gamma$ [@Cahn:1978nz; @Bergstrom:1985hp]. The basic building block will be the one-loop contribution of the Higgs coupling to an on-shell and an off-shell gluon that is described, after translating all couplings and masses, by the “effective” Feynman rule,\ (100,100)(20,0) (-120,-560)[{width="100.00000%"}]{} (240,58)[$\displaystyle -i \frac{\alpha_s}{\pi v} \Big[I_1(\tau,\lambda)-I_2(\tau,\lambda)\Big] \Big[ q_2^\mu q_1^\nu - (q_1 q_2) g^{\mu\nu} \Big] \delta_{ab}\, ,$]{} \ where the functions $I_{1,2}$ are defined as [@Gunion:1989we] $$\begin{aligned} I_1(\tau,\lambda) & = & \frac{\tau\lambda}{2(\tau-\lambda)} + \frac{\tau^2\lambda^2}{2(\tau-\lambda)^2} \left[ f(\tau) - f(\lambda) \right] + \frac{\tau^2\lambda}{(\tau-\lambda)^2} \left[ g(\tau) - g(\lambda) \right], \nonumber \\ I_2(\tau,\lambda) & = & - \frac{\tau\lambda}{2(\tau-\lambda)}\left[ f(\tau) - f(\lambda) \right],\end{aligned}$$ with $\tau = 4m_t^2/m_H^2$, $\lambda = 4 m_t^2/q_2^2$ and the basic functions $$\begin{aligned} g(\tau) & = & \left\{ \begin{array}{ll} \displaystyle \sqrt{\tau-1} \arcsin \frac{1}{\sqrt{\tau}} & \tau \ge 1 \\ \displaystyle \frac{\sqrt{1-\tau}}{2} \left[ \log \frac{1+\sqrt{1-\tau}} {1-\sqrt{1-\tau}} - i\pi \right] & \tau < 1 \end{array} \right.\end{aligned}$$ and $f(\tau)$ defined in Eq. (\[eq:ftau\]). Implementing this building block for the two top loops of the one-particle-reducible diagrams, one arrives at the final coefficient $c_1$ of Eq. (\[eq:coeffvirt\]), $$\begin{aligned} c_1 & = & 2 \Big[ I_1(\tau,\lambda_{\hat t}) -I_2(\tau,\lambda_{\hat t}) \Big]^2 \label{eq:1prc1}\end{aligned}$$ with $\lambda_{\hat t} = 4 m_t^2/\hat t$ (and $\lambda_{\hat u} = 4 m_t^2/\hat u$ for the $\hat t\leftrightarrow \hat u$ interchanged contribution accordingly). This expression, inserted in the coefficient $C_{\triangle\triangle}$ of Eq. (\[eq:coeffvirt\]), determines the contribution of the one-particle-reducible diagrams analytically and agrees with the previous calculation of Ref. [@Degrassi:2016vss]. In the HTL, this coefficient approaches the value $c_1\to 2/9$ in accordance with Eq. (\[eq:coeffvirt\]). We have subtracted the HTL with $c_1=2/9$ from the coefficient $C_{\triangle\triangle}$ in order to account for the NLO top-mass effects only so that eventually the results of the program [Hpair]{} [@hpair] can be added back. While the total effect of the one-particle-reducible contributions on the total cross section ranges below the per-cent level, the finite mass effects at NLO contribute less than one per mille. ![\[fg:1prcomp\] *Comparison of the approximation of Ref. [@deFlorian:2017qfk] (blue) for the one-particle-reducible contributions and the HTL (red), both normalized to the full analytical expression. The singularity at about 720 GeV is due to a sign change of the exact expression.*](./plots/1pr_comp.pdf){width="80.00000%"} Reference [@deFlorian:2017qfk] has proposed an approximation of this one-particle-reducible contribution in terms of the triangle form factor of two on-shell external gluons, $$\begin{aligned} C_{\triangle\triangle} & = & \Re e~\frac{\int_{\hat t_-}^{\hat t_+} d\hat t \left[ (C_\triangle F_\triangle + F_\Box)^* V_{eff}^2\right]} {\int_{\hat t_-}^{\hat t_+} d\hat t \left\{ |C_\triangle F_\triangle + F_\Box |^2 + |G_\Box|^2 \right\}}, \nonumber \\[0.3cm] V_{eff} & = & F_\triangle (\bar\tau_t)\end{aligned}$$ with $\bar\tau_t = 16 m_t^2/Q^2$ \[i.e. $\tau_t$ of Eq. (\[eq:ftriangle\]) evaluated at half the invariant Higgs-pair mass $Q/2$ instead of $Q$\], where the function $F_\triangle$ can be found in Eq. (\[eq:ftriangle\]). Since Ref. [@deFlorian:2017qfk] works in the HTL, the contribution of the second form factor $F_2$ vanishes, i.e. $G_\Box\to 0$, and the approximation $V_{eff}^2/2$ is in fact treated as an approximation for the coefficient $c_1$ of the exact expression of $C_{\triangle\triangle}$ as given in Eq. (\[eq:coeffvirt\])[^5]. Thus, the approximate expression involving the coefficient $c_1$ has to be compared to the corresponding expression involving the exact coefficient $c_1$ of Eq. (\[eq:1prc1\]). This comparison is presented, normalized to the exact expression, in Fig. \[fg:1prcomp\] and shows that the approximation of Ref. [@deFlorian:2017qfk] is not better than the HTL. ### Box diagrams The third class of two-loop contributions to the virtual corrections is given by the box diagrams. The generic box diagrams are shown in Figs. \[fg:boxdia1\]–\[fg:boxdia4\] in the Appendix. The simultaneous exchange of the gluons and Higgs bosons has to be added to complete the set of diagrams. The only exception is diagram 44 that is already totally symmetric so that in the final end there are 93 two-loop box diagrams. The generic 47 diagrams are grouped into 6 topology classes. The first 5 topologies contain only a virtual threshold for $Q^2 > 4m_t^2$. The diagrams of topology 6 on the other hand develop a second threshold for $Q^2>0$, because two virtual gluon lines next to the external gluons can be cut. This implies that the form factors are complex in the entire $Q^2$ range. Therefore, a dedicated treatment of this last topology in terms of a suitably constructed infrared subtraction term to isolate the associated infrared singularities is required. In the following, we will exemplify our method for the boxes 39 of topology 5 and 45 of topology 6. The diagrams of topologies 1–5 are treated analogously to box 39 and those of topology 6 analogously to box 45. The algebraic manipulation of the traces and projections onto the form factors have been performed with the help of the symbolic tools [FORM]{} [@Vermaseren:2000nd; @Kuipers:2012rf], [Reduce]{} [@Hearn:1971zza], and [Mathematica]{} [@Mathematica]. Our method of Feynman parametrization and end-point subtraction to isolate the ultraviolet singularities for the numerical integration has first been applied to the NLO two-loop QCD corrections to $H\to\gamma\gamma, Z\gamma$ in Refs. [@Djouadi:1990aj; @Spira:1991tj] and later to the squark-loop contributions to $h,H \leftrightarrow gg,\gamma\gamma$ within the minimal supersymmetric extension of the SM [@Muhlleitner:2006wx]. The method of the infrared subtraction as applied to topology 6 originates from numerical cross checks of the full NLO QCD corrections to single Higgs production in Refs. [@Spira:1995es; @Spira:1995rr; @Graudenz:1992pv; @Muhlleitner:2006wx]. The stabilization of virtual thresholds by integration by parts of the integrand has first been applied to the SUSY–QCD corrections to single Higgs production in Refs. [@Muhlleitner:2010nm; @Muhlleitner:2010zz]. The basic idea behind the integration by parts is to reduce the power of the threshold-singular denominator and in this way to stabilize the numerical integration. The treatment of the thresholds in our approach is performed by replacing the squared top mass $m_t^2$ by a complex counter part $$m_t^2 \to m_t^2 (1-i\bar\epsilon) \label{eq:imaginary}$$ with a positive regulator $\bar\epsilon > 0$ to ensure proper micro-causality. This defines the analytical continuation of our two-loop box integrals. In the following, the parameter $\bar\epsilon$ will be kept finite in our numerical analysis, while the narrow-width limit $\bar\epsilon\to 0$ is achieved by a Richardson extrapolation [@Richardson]. This will be discussed in more detail in the following paragraphs. #### Box 39  \ ![\[fg:box39\] *Explicit definitions of the virtual momenta in box 39.*](./plots/dia_box39.pdf){width="115.00000%"} Using the definition of real and virtual momenta as in Fig. \[fg:box39\], the contribution to the tensor $A^{\mu\nu}$ \[see Eq. (\[eq:lomat\])\] of the virtual two-loop corrections is given by $$\begin{aligned} A_{39}^{\mu\nu} & = & \frac{3}{16}\, \frac{\alpha_s}{\pi}\, (4\pi)^4\, B_{39}^{\mu\nu} \, , \nonumber \\ B_{39}^{\mu\nu} \!\! & \!\!\!\! = \!\!\!\! & \!\! \int \frac{d^n k d^n q}{(2\pi)^{2n}} \frac{Tr\Big\{ (\,{\!\!\not\! {k}}+{\!\!\not\! {q}}-{\!\!\not\! {p}}_1+m_t) (\,{\!\!\not\! {k}}+{\!\!\not\! {q}}+m_t) \gamma^\sigma (\,{\!\!\not\! {k}}+m_t) ({\!\!\not\! {k}}+{\!\!\not\! {p}}_2+m_t) \gamma^\nu (\,{\!\!\not\! {k}}+{\!\!\not\! {q}}_1-{\!\!\not\! {p}}_1+m_t) \gamma^\rho \Big\}} {[(k+q)^2-m_t^2] [(k+q-p_1)^2-m_t^2] [(k+p_2)^2-m_t^2] [(k+q_1-p_1)^2-m_t^2]} \nonumber \\[0.3cm] & & \qquad \times~\frac{g_{\rho\sigma} (2q-q_1)^\mu - g^\mu_\rho (q-2q_1)_\sigma - g^\mu_\sigma (q+q_1)_\rho}{(k^2-m_t^2)(q-q_1)^2 q^2} \, , \label{eq:mat39}\end{aligned}$$ where $k,q$ are the loop momenta that are integrated over. The Feynman parametrization is first performed for the integration over $k$. We provide Feynman parameters $x_1,\ldots,x_4$ for the first four propagators in the denominator and $1-\sum_i x_i$ for the last one ($k^2-m_t^2$). Performing the substitutions $$x_1 = (1-x)(1-y)\, , \quad x_2 = (1-x)y\, , \quad x_3 = xzr \, , \quad x_4 = xz(1-r) \, ,$$ we arrive at a four-dimensional integral over $x,y,z,r$ with integration boundaries from 0 to 1. To symmetrize the $n$-dimensional $k$-integration, we have to perform the shift $$\begin{aligned} k & \to & k - Q_1 \, , \nonumber \\ Q_1 & = & (1-x)q + xzq_1 + xzr q_2 - [(1-x)y+xz] p_1 \, ,\end{aligned}$$ in both the numerator and denominator. The residual (properly normalized) denominator after the $k$-integration is treated as a propagator for the second loop integration over $q$. We attribute additional Feynman parameters $x_5, x_6$ to this residual propagator and the next one \[$(q-q_1)^2$\] and $1-x_5-x_6$ for the last one ($q^2$) in Eq. (\[eq:mat39\]). Performing the substitution[^6] $$x_5 = s\, , \quad x_6 = (1-s)t\, ,$$ we again arrive at integrals over $s,t$ from 0 to 1. This latter parametrization requires the shift $$\begin{aligned} q & \to & q - Q_2 \, , \nonumber \\ Q_2 & = & - [zs + (1-s)t] q_1 - zrs q_2 - (y-z)s p_1\end{aligned}$$ in the numerator and denominator to be able to perform the loop integration over $q$ symmetrically. After projecting on the two form factors, we finally arrive at integrals of the type $$\Delta F_i = \frac{\alpha_s}{\pi}~\Gamma(1+2\epsilon) \left(\frac{4\pi\mu_0^2}{m_t^2}\right)^{2\epsilon} \int_0^1 d^6 x~ \frac{x^\epsilon(1-x)^\epsilon s^{-1-\epsilon} H_i(\vec x)}{N^{3+2\epsilon}(\vec x)} \label{eq:fi39}$$ with $\vec x = (x,y,z,r,s,t)$ and $d^6x = dx\,dy\,dz\,dr\,ds\,dt$. $H_i(\vec x)$ denotes the full numerator, including regular factors of the Jacobians due to the Feynman parametrization and substitutions, and singular as well as higher powers of the dimensional regulator $\epsilon$, and $N(\vec x)$ the final denominator, $$\begin{aligned} N(\vec x) & = & 1 + \rho_s xzr \Big\{ xz + (1-x) [zs+(1-s)t] \Big\} \nonumber \\ & & \quad - \rho_t x \Big\{ z(1-y-r) + (y-z)[z+(1-x)(1-s)(t-z)] \Big\} \nonumber \\ & & \quad + \rho_u xzr \Big\{ xz + (1-x) [zs+(1-s)y] \Big\} \nonumber \\ & & \quad - \rho_H \Big\{ [xz+(1-x)y][1-xz-(1-x)y] - x(1-x)s(y-z)^2 \Big\} \, ,\end{aligned}$$ where we define $\rho_s = \hat s/m_t^2 = Q^2/m_t^2$, $\rho_t = (\hat t-M_H^2)/m_t^2$, $\rho_u = (\hat u-M_H^2)/m_t^2$ and $\rho_H = M_H^2/m_t^2$. The singular powers in $\epsilon$ of $H_i(\vec x)$ arise from powers of $k^2$ and $q^2$ in the numerators of the final integrations of the loop momenta $k$ and $q$. It is important that the final denominator develops the form of $1+O(1/m_t^2)$ to ensure that no further ultraviolet nor infrared singularities arise from this part of the integrand. The integral for $\Delta F_i$ of Eq. (\[eq:fi39\]) is singular for $s\to 0$. To separate this singularity from the integral, we perform an endpoint subtraction, $$\begin{aligned} \Delta F_i & = & \frac{\alpha_s}{\pi}~\Gamma(1+\epsilon) \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left(\frac{4\pi\mu_0^2}{m_t^2}\right)^{2\epsilon} \left[ \Delta F_{i,1} + \Delta F_{i,2} \right] \, , \nonumber \\ \Delta F_{i,1} & = & \int_0^1 \frac{d^6 x}{s} \left\{ \frac{H_i(\vec x)}{N^3(\vec x)} (1+\epsilon L) - \left. \frac{H_i(\vec x)}{N^3(\vec x)} \right|_{s=0} (1+\epsilon L_0) \right\} \, , \nonumber \\ \Delta F_{i,2} & = & -\frac{1}{\epsilon} \int_0^1 d^5 x \left. \frac{H_i(\vec x)}{N^3(\vec x)} \right|_{s=0} \left[1+\epsilon L_1 + \epsilon^2 \left( \frac{L_1^2}{2} + 3\zeta_2 \right) \right] \nonumber \\ \mbox{with} \qquad L & = & \log\frac{x(1-x)}{s} - 2 \log N(\vec x) \, , \nonumber \\ L_0 & = & \log\frac{x(1-x)}{s} - 2 \log N(\vec x)|_{s=0} \, , \nonumber \\ L_1 & = & \log [x(1-x)] - 2 \log N(\vec x)|_{s=0} \, ,\end{aligned}$$ where in the second term $\Delta F_{i,2}$ the integration over $s$ has been performed analytically and the integration measure is given by $d^5 x = dx\,dy\,dz\,dr\,dt$. It should be noted that in the terms $L, L_0, L_1$ the logarithms of the denominator $N$ need to be linear in $N$ to be consistent with the analytical continuation along the proper Riemann sheet. We have checked numerically that the first (subtracted) part $\Delta F_{i,1}$ is finite for each order in the dimensional regulator $\epsilon$ by introducing cuts in the integration boundaries, i.e. integrating from $\tilde\epsilon$ to $1-\tilde\epsilon$, varying $\tilde\epsilon$ down to $10^{-10}$ and checking that the integrals become independent of $\tilde\epsilon$. These integrals are numerically stable below the virtual $t\bar t$-threshold, i.e. for $Q^2 < 4 m_t^2$ or $\rho_s < 4$. However, above this threshold, the integrals have to be stabilized. We have achieved this stabilization by means of integration by parts with respect to the Feynman parameter $z$. The denominator is a quadratic polynomial in $z$, $$\begin{aligned} N(\vec x) & = & a z^2 + b z + c \nonumber \\ \mbox{with} \qquad a & = & x [\rho_s r + \rho_t + \rho_u r + \rho_H] [1 - (1-x)(1-s)] \, , \nonumber \\ b & = & \rho_s x(1-x)r(1-s)t - \rho_t x [1-r - (1-x)(1-s)(y+t)] \nonumber \\ & + & \rho_u x(1-x)y r (1-s) - \rho_H x [1-2(1-x)y(1-s)] \, , \nonumber \\ c & = & 1 - \rho_t x(1-x)y(1-s)t - \rho_H (1-x) y [1-y + xy(1-s)] \, .\end{aligned}$$ To simplify the integration by parts, we insert a unit factor $\Delta/\Delta$ with $\Delta=4ac-b^2$ in the integrand and replace $\Delta$ in the numerator by the expression $$\Delta = 4 a N - (\partial_z N)^2 = 4 a N - (2az+b)^2 \, .$$ Then the following manipulation can be performed, $$\begin{aligned} \int_0^1 dz~\frac{H_i(\vec x)}{N^3} & = & \frac{1}{\Delta} \left\{ \left. \left[\frac{2a+b}{2N^2} H_i(\vec x) + \frac{\partial_z H_i(\vec x)}{2N}\right]\right|_{z=1} - \left. \left[\frac{b}{2N^2} H_i(\vec x) + \frac{\partial_z H_i(\vec x)}{2N}\right]\right|_{z=0} \right. \nonumber \\ & & \left. + \int_0^1 dz~\left[ \frac{3a}{N^2} H_i(\vec x) - \frac{\partial_z^2 H_i(\vec x)}{2N} \right] \right\}\end{aligned}$$ and analogously for integrals involving additional powers of $\log N$ factors in the numerator of the integrand. The progress achieved with these integrations by parts is that the maximal power of the denominator in the new integral is reduced by one compared to the original integral. One could perform additional integrations by parts with respect to another Feynman parameter. However, we did not investigate this further, since the stability we achieved at this point has been sufficient for the numerical integrations for the top loops[^7]. After performing the integrations by parts, the integral is stable for regulators $\bar \epsilon$ \[see Eq. (\[eq:imaginary\])\] down to $0.05$ for the relevant Higgs mass, top mass and $Q^2$ range. Since this is still apart from the plateau of the narrow-width limit, we performed a Richardson extrapolation [@Richardson] from finite values of $\bar \epsilon$ down to zero. Richardson extrapolation is possible since the $\bar \epsilon$-dependence of the integral is polynomial for small values of $\bar \epsilon$. The basic principle behind this extrapolation method is very simple: let a function $f(\bar\epsilon)$ behave for small $\bar\epsilon$ as $$f(\bar\epsilon) = f(0) + {\cal O}(\bar\epsilon^n) \, .$$ If we know $f(\bar\epsilon)$ for two different values $\bar\epsilon$ and $t\bar\epsilon$, we can construct the new function $$R_1(\bar\epsilon,t) = \frac{t^n f(\bar\epsilon)-f(t \bar\epsilon)}{t^n-1} \, . \label{eq:richardson}$$ This function shows a better convergence towards the value at $\bar\epsilon=0$, $$R_1(\bar\epsilon,t) = f(0) + {\cal O}(\bar\epsilon^{n+1}) \, .$$ Our integrals $I(\bar\epsilon)$ behave for small values of $\bar\epsilon$ as $$I(\bar\epsilon) = I(0) + {\cal O}(\bar\epsilon)$$ so that the first new extrapolation function in our case is given by $$R_1(\bar\epsilon,t) = \frac{t I(\bar\epsilon)-I(t \bar\epsilon)}{t-1} = I(0) + {\cal O}(\bar\epsilon^2) \, .$$ Using an additional value of $\bar\epsilon$, this method can be repeated iteratively for the new function obtained by applying Eq. (\[eq:richardson\]), $$R_2(\bar\epsilon,t) = \frac{t^2 R_1(\bar\epsilon)-R_1(t \bar\epsilon)}{t^2-1} = I(0) + {\cal O}(\bar\epsilon^3) \, .$$ In this way, the estimated error is reduced by each additional iteration. We have used this method for a set of $\bar\epsilon$ separated by factors of $t=2$. Then, we obtain the following extrapolation polynomials, $$\begin{aligned} R_1(\bar\epsilon) & = & 2 I(\bar\epsilon) -I(2\bar\epsilon) = I(0) + {\cal O}(\bar\epsilon^2) \, , \nonumber \\ R_2(\bar\epsilon) & = & \frac{1}{3} \Big[ 8 I(\bar\epsilon) - 6 I(2\bar\epsilon) + I(4\bar\epsilon) \Big] = I(0) + {\cal O}(\bar\epsilon^3) \, , \nonumber \\ R_3(\bar\epsilon) & = & \frac{1}{21} \Big[ 64 I(\bar\epsilon) - 56 I(2\bar\epsilon) + 14 I(4\bar\epsilon) - I(8\bar\epsilon) \Big] = I(0) + {\cal O}(\bar\epsilon^4) \, , \nonumber \\ R_4(\bar\epsilon) & = & \frac{1}{315} \Big[ 1024 I(\bar\epsilon) - 960 I(2\bar\epsilon) + 280 I(4\bar\epsilon) - 30 I(8\bar\epsilon) + I(16\bar\epsilon) \Big] = I(0) + {\cal O}(\bar\epsilon^5)\end{aligned}$$ and so on. We have used extrapolation polynomials up to $R_9(\bar\epsilon)$. To determine the extrapolation error, we have chosen different sets of $\bar\epsilon$ values and derived the spread of the extrapolated values appropriately (see Section \[sc:results\] for more details). #### Box 45  \ ![\[fg:box45\] *Explicit definitions of the virtual momenta in box 45.*](./plots/dia_box45.pdf){width="115.00000%"} Based on the distribution of the loop and external momenta of Fig. \[fg:box45\], the contribution to the two-loop matrix element is given by $$\begin{aligned} A_{45}^{\mu\nu} & = & \frac{3}{8}\, \frac{\alpha_s}{\pi}\, (4\pi)^4\, B_{45}^{\mu\nu} \, , \nonumber \\ B_{45}^{\mu\nu} & \!\!\! = \!\!\! & \int \frac{d^n k d^n q}{(2\pi)^{2n}} \frac{Tr\Big\{ (\,{\!\!\not\! {k}}-{\!\!\not\! {q}}_1+m_t) (\,{\!\!\not\! {k}}-{\!\!\not\! {q}}_1+{\!\!\not\! {p}}_1+m_t) (\,{\!\!\not\! {k}}+{\!\!\not\! {q}}_2+m_t) \gamma^\sigma (\,{\!\!\not\! {k}}+{\!\!\not\! {q}}+m_t) \gamma^\rho \Big\}} {[(k+q)^2-m_t^2] [(k+q_2)^2-m_t^2] [(k+p_1-q_1)^2-m_t^2] [(k-q_1)^2-m_t^2]} \nonumber \\[0.3cm] & & \qquad \times~\frac{\Big\{ g_{\rho\tau} (2q+q_1)^\mu - g^\mu_\rho (q+2q_1)_\tau - g^\mu_\tau (q-q_1)_\rho \Big\}}{(q+q_1)^2 (q-q_2)^2 q^2} \nonumber \\[0.3cm] & & \qquad \times~\Big\{ g^{\nu\tau} (q+q_2)_\sigma + g^\nu_\sigma (q-2q_2)^\tau -g_\sigma^\tau (2q-q_2)^\nu \Big\} \, . \label{eq:mat45}\end{aligned}$$ Following the same procedure as for box 39 for the Feynman parametrization, we have first performed the parametrization of the $k$-integration following the ordering of the denominator of Eq. (\[eq:mat45\]). The shift in the loop momentum $k$ and the corresponding substitutions of the Feynman parameters are given by $$\begin{aligned} k & \to & k - Q_1 \, , \nonumber \\ Q_1 & = & (1-x)q - xyq_1 + x(1-y) q_2 + xyz p_1 \, , \nonumber \\ x_1 & = & (1-x)\, , \quad x_2 = x(1-y)\, , \quad x_3 = xyz \, .\end{aligned}$$ Performing the second loop integration over $q$ with the residual (normalized) denominator of the $k$ integration as the first propagator of the $q$ integration, attributing the additional Feynman parameters $x_4, x_5, x_6$ to the remaining propagators in Eq. (\[eq:mat45\]) and applying the substitutions[^8] $$x_4 = rs\, , \quad x_5 = 1-s\, , \quad x_6 = (1-r)st \, ,$$ we arrive at the final expressions for the shift of $q$ and the denominator that contribute to the two form factors, $$\begin{aligned} q & \to & q - Q_2 \, , \nonumber \\ Q_2 & = & [yrs + 1-s] q_1 - [(1-y)rs+(1-r)st] q_2 - yzrs p_1 \, , \nonumber \\ N(\vec x) & = & r - \rho_s x \Big\{ xy(1-y)r + (1-x) [1-s+yrs][(1-r)t+(1-y)r] \Big\} \nonumber \\ & & \quad - \rho_t xyzr \Big\{ 1-xy - (1-x)[yrs+1-s] \Big\} - \rho_H xyzr \Big\{ 1-xyz-(1-x)yzrs \Big\} \nonumber \\ & & \quad - \rho_u xyzr \Big\{ x(1-y) + (1-x)s [(1-r)t+(1-y)r] \Big\}\end{aligned}$$ and the final integrals of the two form factors ($i=1,2$) can be cast into the form $$\Delta F_i = \Gamma(1+2\epsilon) \left(\frac{4\pi\mu_0^2}{m_t^2}\right)^{2\epsilon} \int_0^1 d^6 x~ \frac{x^{1+\epsilon} (1-x)^\epsilon r^{1+\epsilon} s^{-\epsilon} H_i(\vec x)}{N^{3+2\epsilon}(\vec x)} \, ,$$ where $H_i(\vec x)$ contains all additional regular Feynman-parameter factors from Jacobians and the normalization of the denominator of the first loop-integration over $k$. It develops a singular Laurent-expansion in $\epsilon$. The final denominator exhibits the basic form of $r+O(1/m_t^2)$, so that the additional singular behavior is entirely controlled by the limit of small $r$. Since the denominator is of the form $$\begin{aligned} N(\vec x) & = & a r^2 + b r + c \, , \nonumber \\ \mbox{where} \qquad a & = & x(1-x)ys \Big[-\rho_s (1-y-t) + \rho_t yz - \rho_u z(1-y-t) + \rho_H y z^2\Big] \, , \nonumber \\ b & = & 1 - \rho_s x \Big\{ xy(1-y) +(1-x)[(1-s)(1-y-t)+yst] \Big\} - \rho_H xyz (1-xyz) \nonumber \\ & & -\rho_t xyz [1-xy - (1-x)(1-s)] - \rho_u xyz [x(1-y)+(1-x)st] \, , \nonumber \\ c & = & - \rho_s x(1-x)(1-s)t \label{eq:abc}\end{aligned}$$ with $a,c = {\cal O}(1/m_t^2)$ and $b=1+{\cal O}(1/m_t^2)$ and the infrared singularities are universal (relative to the LO expressions) the coefficient $a$ does not contribute to the infrared singularity structure, because $a$ is subleading relative to $b$ in the limit $r\to 0$. Thus, we can construct infrared subtraction terms that turn the contributions to the form factors into $$\begin{aligned} \Delta F_i & = & \frac{\alpha_s}{\pi}~\Gamma(1+2\epsilon) \left(\frac{4\pi\mu_0^2}{m_t^2}\right)^{2\epsilon} (G_1 + G_2) \, , \nonumber \\ G_1 & = & \int_0^1 d^6 x~x^{1+\epsilon} (1-x)^\epsilon r^{1+\epsilon} s^{-\epsilon} \left\{ \frac{H_i(\vec x)}{N^{3+2\epsilon}(\vec x)} -\frac{H_i(\vec x)|_{r=0}}{N_0^{3+2\epsilon}(\vec x)} \right\} \, , \nonumber \\ G_2 & = & \int_0^1 d^6 x~x^{1+\epsilon} (1-x)^\epsilon r^{1+\epsilon} s^{-\epsilon} \frac{H_i(\vec x)|_{r=0}}{N_0^{3+2\epsilon}(\vec x)} \nonumber \\ \mbox{with} \qquad N_0(\vec x) & = & br + c \, .\end{aligned}$$ Numerically, we have tested that the subtracted integral $G_1$ (after expansion in the dimensional regulator $\epsilon$) is finite for each coefficient of the expansion in $\epsilon$ individually by integrating the Feynman-parameter integrals from $\tilde\epsilon$ to $1-\tilde\epsilon$ with $\tilde\epsilon$ varied down to $10^{-10}$. The second integral $G_2$ can be integrated over the Feynman parameter $r$ analytically giving rise to hypergeometric functions, $$G_2 = \frac{1}{2+\epsilon} \int_0^1 d^5x~\frac{x^{1+\epsilon} (1-x)^\epsilon s^{-\epsilon}}{c^{3+2\epsilon}} ~_2F_1\left(3+2\epsilon, 2+\epsilon; 3+\epsilon; -\frac{b}{c}\right) \left. H_i(\vec x) \right|_{r=0}$$ with $d^5x = dx\, dy\, dz\, ds\, dt$. Since this integral is singular for $c\to 0$, we have to invert the last argument of the hypergeometric function. Using the transformation relation $$\begin{aligned} ~_2F_1 (a,b;c;z) & = & \frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}(-z)^{-a}~_2F_1\left( a,1-c+a; 1-b+a; \frac{1}{z}\right) \nonumber \\ & & \qquad + \frac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)} (-z)^{-b}~_2F_1\left(b,1-c+b; 1-a+b; \frac{1}{z}\right) \, ,\end{aligned}$$ the special property $$~_2F_1 (a,0;c;z) = 1$$ and suitable end-point subtractions of the residual singular integrals analogous to box 39, we arrive at the final decomposition of the initial Feynman-parameter integral $$\begin{aligned} \Delta F_i & = & \frac{\alpha_s}{\pi}~\Gamma(1+\epsilon) \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left(\frac{4\pi\mu_0^2}{m_t^2}\right)^{2\epsilon} \sum_{j=1}^6 S_j \, , \nonumber \\[0.0cm] S_1 & = & \int_0^1 d^6x~xr \left\{ \frac{H_i(\vec x)}{N^3(\vec x)} \left[ 1+\epsilon L + \epsilon^2 \left( \frac{L^2}{2} + 3\zeta_2\right) \right] \right. \nonumber \\ & & \qquad\quad - \left. \frac{H_i(\vec x)|_{r=0}}{(c+br)^3} \left[ 1+\epsilon L_0 + \epsilon^2 \left( \frac{L_0^2}{2} + 3\zeta_2\right) \right] \right\} \, , \nonumber \\[0.0cm] S_2 & = & -\int_0^1 d^6x~x \frac{H_i(\vec x)|_{r=0}}{(b+cr)^3} \left\{ 1+\epsilon L_1 + \epsilon^2 \left( \frac{L_1^2}{2} + 3\zeta_2\right) + \epsilon^3 \left( \frac{L_1^3}{6} + 3\zeta_2 L_1 \right) \right\} \, , \nonumber \\[0.0cm] S_3 & = & -\int_0^1 \frac{d^5x}{2\rho_s (1-x)(1-s)t} \left\{ \frac{H_i(\vec x)|_{r=0}}{b^2} \left[ 1-\epsilon (L_2+2) + \epsilon^2 \left( \frac{L_2^2}{2} + 2L_2 + 2\zeta_2 + 4 \right) \right] \right. \nonumber \\ & & \qquad\quad + \frac{H_i(\vec x)|_{r,t=0,s=1}}{b_0^2} \left[ 1-\epsilon (L_3+2) + \epsilon^2 \left( \frac{L_3^2}{2} + 2L_3 + 2\zeta_2 + 4 \right) \right] \nonumber \\ & & \qquad\quad - \frac{H_i(\vec x)|_{r=0,s=1}}{b_1^2} \left[ 1-\epsilon (L_4+2) + \epsilon^2 \left( \frac{L_4^2}{2} + 2L_4 + 2\zeta_2 + 4 \right) \right] \nonumber \\ & & \qquad\quad \left. - \frac{H_i(\vec x)|_{r,t=0}}{b_2^2} \left[ 1-\epsilon (L_5+2) + \epsilon^2 \left( \frac{L_5^2}{2} + 2L_5 + 2\zeta_2 + 4 \right) \right] \right\} \, , \nonumber \\[0.0cm] S_4 & = & -\int_0^1 \frac{dx\, dy\, dz\, ds}{2\rho_s (1-x)(1-s)} \left\{ \frac{H_i(\vec x)|_{r,t=0}}{b_2^2} \left[ -\frac{1}{\epsilon}+ L_6+2 - \epsilon \left( \frac{L_6^2}{2} + 2L_6 + 2\zeta_2 + 4 \right) \right. \right. \nonumber \\ & & \qquad\qquad\qquad\qquad\qquad \left. + \epsilon^2 \left( \frac{L_6^3}{6} + L_6^2 + 2(\zeta_2+2) L_6 - 2\zeta_3+4\zeta_2+8 \right) \right] \nonumber \\ & & \qquad\quad - \frac{H_i(\vec x)|_{r,t=0, s=1}}{b_0^2} \left[ -\frac{1}{\epsilon}+ L_7+2 - \epsilon \left( \frac{L_7^2}{2} + 2L_7 + 2\zeta_2 + 4 \right) \right. \nonumber \\ & & \qquad\qquad\qquad\qquad\qquad \left. \left. + \epsilon^2 \left( \frac{L_7^3}{6} + L_7^2 + 2(\zeta_2+2) L_7 - 2\zeta_3+4\zeta_2+8 \right) \right] \right\} \, , \nonumber \\[0.0cm] S_5 & = & -\int_0^1 \frac{dx\, dy\, dz\, dt}{2\rho_s (1-x)t} \left\{ \frac{H_i(\vec x)|_{r=0,s=1}}{b_1^2} \left[ -\frac{1}{\epsilon}+ L_8+2 - \epsilon \left( \frac{L_8^2}{2} + 2L_8 + \zeta_2 + 4 \right) \right. \right. \nonumber \\ & & \qquad\qquad\qquad\qquad\qquad \left. + \epsilon^2 \left( \frac{L_8^3}{6} + L_8^2 + (\zeta_2+4) L_8 +2\zeta_2+8 \right) \right] \nonumber \\ & & \qquad\quad - \frac{H_i(\vec x)|_{r,t=0, s=1}}{b_0^2} \left[ -\frac{1}{\epsilon}+ L_9+2 - \epsilon \left( \frac{L_9^2}{2} + 2L_9 + \zeta_2 + 4 \right) \right. \nonumber \\ & & \qquad\qquad\qquad\qquad\qquad \left. \left. + \epsilon^2 \left( \frac{L_9^3}{6} + L_9^2 + (\zeta_2+4) L_9 +2\zeta_2+8 \right) \right] \right\} \, , \nonumber \\[0.0cm] S_6 & = & -\int_0^1 dx\, dy\, dz~\frac{H_i(\vec x)|_{r,t=0,s=1}}{2\rho_s (1-x)b_0^2} \left\{ \frac{1}{\epsilon^2} - \frac{1}{\epsilon} (L_{10}+2) + \frac{L_{10}^2}{2} + 2L_{10} + \zeta_2 + 4 \right. \nonumber \\ & & \qquad\qquad\qquad\qquad\qquad \left. - \epsilon \left( \frac{L_{10}^3}{6} + L_{10}^2 + (\zeta_2+4) L_{10} +2\zeta_2+8 \right) \right\} \, .\end{aligned}$$ The logarithms used in the expressions above are defined as $$\begin{aligned} L & = & \log\left(\frac{x(1-x)r}{s}\right) - 2 \log N \, , \qquad\qquad\qquad\, L_0 = \log\left(\frac{x(1-x)r}{s}\right) - 2 \log (c+br) \, , \nonumber \\ L_1 & = & \log\left(\frac{x(1-x)r}{s}\right) - 2 \log (b+cr) \, , \qquad\qquad L_2 = \log\left[-\rho_s s(1-s)t\right] + \log b \, , \nonumber \\ L_3 & = & \log\left[-\rho_s s(1-s)t\right] + \log b_0 \, , \qquad\qquad\qquad\ \ \, L_4 = \log\left[-\rho_s s(1-s)t\right] + \log b_1 \, , \nonumber \\ L_5 & = & \log\left[-\rho_s s(1-s)t\right] + \log b_2 \, , \qquad\qquad\qquad\ \ \, L_6 = \log\left[-\rho_s s(1-s)\right] + \log b_2 \, , \nonumber \\ L_7 & = & \log\left[-\rho_s s(1-s)\right] + \log b_0 \, , \qquad\qquad\qquad\quad\, L_8 = \log\left(-\rho_s t\right) + \log b_1 \, , \nonumber \\ L_9 & = & \log\left(-\rho_s t\right) + \log b_0 \, , \qquad\qquad\qquad\qquad\qquad\!\! L_{10} = \log\left(-\rho_s\right) + \log b_0\end{aligned}$$ and the remaining objects $b_0, b_1, b_2$ as $$\begin{aligned} b_0 & = & b|_{t=0,s=1} \, , \qquad\qquad b_1 = b|_{s=1} \, , \qquad\qquad b_2 = b|_{t=0}\end{aligned}$$ with $b$ from Eq. (\[eq:abc\]). Box 45 contains a second threshold for $Q^2>0$ so that even below the $t\bar t$-threshold, integrations by parts are required to stabilize the integrand numerically. These integrations by parts are performed for the Feynman parameter $r$ in the contributions $S_{1,2}$ along the same lines as for box 39, while the integrals $S_{3-6}$ are stable without integrations by parts. ### Renormalization \[sc:renorm\] The strong coupling $\alpha_s$ has been renormalized in the $\overline{\rm MS}$ scheme with the top quark decoupled, i.e. the renormalization constant is given by $$\begin{aligned} \alpha_{s,0} & = & \alpha_s(\mu_R) + \delta\alpha_s \, , \nonumber \\ \frac{\delta\alpha_s}{\alpha_s} & = & \frac{\alpha_s}{\pi} \Gamma(1+\epsilon) \left(\frac{4\pi\mu_0^2}{\mu_R^2}\right)^\epsilon \left\{ -\frac{33-2(N_F+1)}{12\epsilon} + \frac{1}{6} \log \frac{\mu_R^2}{m_t^2} \right\}\end{aligned}$$ with $N_F=5$. This choice ensures that there are no artificial large logarithms of the top mass for the available energy range of the LHC in the final result, since we do not introduce top densities inside the proton, i.e. work in a five-flavour scheme. The additional logarithm of the top mass cancels against the diagrams with a top loop within the external gluon lines, see Fig. \[fg:toploop\]. This leads to the total contribution related to the renormalization of the strong coupling $$\delta_{\alpha_s} F_i = \frac{\alpha_s}{\pi} \Gamma(1+\epsilon) \left(\frac{4\pi\mu_0^2}{\mu_R^2}\right)^\epsilon \left\{ - \frac{33-2N_F}{12\epsilon} \right\} F_{i,LO} \, ,$$ where the LO form factors $F_i$ have to be used in $n$ dimensions, i.e. including higher orders in the dimensional regulator $\epsilon$. ![\[fg:toploop\] *Typical diagrams with external top loops.*](./plots/dia_toploop.pdf){width="115.00000%"} For our default prediction, we have renormalized the top mass on-shell so that the renormalization constant is given by $$\begin{aligned} m_{t,0} & = & m_t - \delta m_t \, , \nonumber \\ \frac{\delta m_t}{m_t} & = & \frac{\alpha_s}{\pi} \Gamma(1+\epsilon) \left(\frac{4\pi\mu_0^2}{m_t^2}\right)^\epsilon \left\{ \frac{1}{\epsilon} + \frac{4}{3} \right\} \, .\end{aligned}$$ The explicit contribution of the mass counterterm can either be obtained by calculating the corresponding counterterm diagrams or, in much more elegant manner, by differentiating the LO form factors with respect to the top mass, $$\delta_{m_t} F_i = - \delta m_t \frac{\partial F_{i,LO}}{\partial m_t} \, ,$$ where we followed the second option. For the renormalization of the top mass in terms of the $\overline{\rm MS}$ mass, a counterterm $$\begin{aligned} m_{t,0} & = & \overline{m}_t(\mu_t) - \delta \overline{m}_t \, , \nonumber \\ \frac{\delta \overline{m}_t}{\overline{m}_t(\mu_t)} & = & \frac{\alpha_s}{\pi} \Gamma(1+\epsilon) \left(\frac{4\pi\mu_0^2}{\mu_t^2}\right)^\epsilon \frac{1}{\epsilon}\end{aligned}$$ has to be used with the LO and NLO expressions of the form factors expressed in terms of the $\overline{\rm MS}$ top mass $\overline{m}_t(\mu_t)$. For the evaluation of the $\overline{\rm MS}$ top mass, we use the N$^3$LO relation between the pole and $\overline{\rm MS}$ mass [@Gray:1990yh; @Chetyrkin:1999ys; @Chetyrkin:1999qi; @Melnikov:2000qh], $$\begin{aligned} {\overline{m}}_{t}(m_{t}) & = & \frac{m_{t}}{\displaystyle 1+\frac{4}{3} \frac{\alpha_{s}(m_t)}{\pi} + K_2 \left(\frac{\alpha_s(m_t)}{\pi}\right)^2 + K_3 \left(\frac{\alpha_s(m_t)}{\pi}\right)^3} \label{eq:mspole}\end{aligned}$$ with $K_2\approx 10.9$ and $K_3 \approx 107.11$. The scale dependence of the $\overline{\rm MS}$ mass is treated at N$^3$LL, $$\begin{aligned} {\overline{m}}_{t}\,(\mu_t)&=&{\overline{m}}_{t}\,(m_{t}) \,\frac{c\,[\alpha_{s}\,(\mu_t)/\pi ]}{c\, [\alpha_{s}\,(m_{t})/\pi ]} \label{eq:msbarev}\end{aligned}$$ with the coefficient function [@Tarasov:1982gk; @Chetyrkin:1997dh] $$\begin{aligned} c(x)=\left(\frac{7}{2}\,x\right)^{\frac{4}{7}} \, [1+1.398x+1.793\,x^{2} - 0.6834\, x^3]\, .\end{aligned}$$ Since we are interested in the finite top-mass effects on top of the LO ones, we have subtracted in addition the Born-improved HTL of the virtual corrections involving the full top-mass dependence at LO [@Dawson:1998py]. This yields the additional subtraction term $$\delta_{HTL} F_i = \frac{\alpha_s}{\pi} \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left(\frac{4\pi\mu_0^2}{-m_t^2 \rho_s}\right)^\epsilon \left\{ \frac{3}{2\epsilon^2} + \frac{33-2N_F}{12\epsilon} \left(\frac{\mu_R^2}{-m_t^2 \rho_s}\right)^{-\epsilon} - \frac{11}{4} + \frac{\pi^2}{4} \right\} F_{i,LO} \, .$$ After adding this subtraction term, the result of [Hpair]{} can simply be added back to the NLO top-mass effects obtained in this way for the virtual corrections. Thus, the total counterterm plus HTL-subtraction is given by $$\delta F_i = \delta_{\alpha_s} F_i + \delta_{m_t} F_i + \delta_{HTL} F_i \, .$$ The addition of this term results in an infrared and ultraviolet finite result for the virtual corrections as we have explicitly checked numerically. It should be noted that we have defined this total subtraction term with the imaginary part $\bar\epsilon$ for the top mass to be consistent with our treatment of the two-loop diagrams. For the two-loop triangle diagrams, this total subtraction term is included in the narrow-width approximation according to the known result for the single-Higgs case. ### Differential cross section The final numerical integrations have been performed by [Vegas]{} [@Lepage:1980dq] for the differential cross sections $d\sigma/dQ^2$ of Eq. (\[eq:nlodiff\]), i.e. the integration over $\hat t$ is included. Each individual box diagram is divergent in $\hat t$ at the lower and upper bound of the $\hat t$-integration in general. To stabilize the $\hat t$-integration, we have performed a suitable substitution to smoothen the integrand, $$\hat t_1 = m_t^2 e^y + t_{1-}$$ with $\hat t_1 = \hat t-M_H^2, \hat u_1 = \hat u-M_H^2$ and $\hat t_{1\pm} = \hat t_{\pm}-M_H^2$, where the integration boundaries $\hat t_{\pm}$ are given in Eq. (\[eq:tbound\]). By means of this substitution, we can rewrite the integration over $\hat t_1$ generically as[^9] $$\int_{\hat t_{1-}}^{\hat t_{1+}} \frac{d\hat t_1}{\hat t_1\hat u_1 - \hat s M_H^2} f(\hat t_1, \hat u_1) = \int_{y_-}^{y_+} \frac{dy}{t_+-t_-} \Big[ f(\hat t_1,\hat u_1) + f(\hat u_1,\hat t_1) \Big] \, ,$$ where $f(\hat t_1, \hat u_1)$ denotes the corresponding virtual matrix element with the (singular) denominator $\hat t_1\hat u_1 - \hat s M_H^2$ extracted and the integration boundaries read $$\begin{aligned} y_+ & = & \log \frac{(t_+-t_-)(1-\tilde\epsilon)}{m_t^2} \, , \nonumber \\ y_- & = & \log \frac{(t_+-t_-)\tilde\epsilon}{m_t^2} \, ,\end{aligned}$$ where we have introduced a cut $\tilde\epsilon$ for the upper and lower bound of the $\hat t_1$-integration (after rewriting this into an integral from 0 to 1 and replacing these integration boundaries by $\tilde\epsilon$ and $1-\tilde\epsilon$). We have checked that the total sum of all box diagrams becomes independent of this cut by varying $\tilde\epsilon$ down to $10^{-10}$, i.e. that the total sum is again finite[^10]. Real corrections \[sc:reals\] ----------------------------- We are left with the evaluation of the real contributions to complete the picture of the NLO QCD corrections. As we are interested in the calculation of the top-mass effects on top of the HTL calculation that is provided by [Hpair]{}, we use the universality of the infrared divergent pieces to subtract the Born-improved HTL contributions $d\sigma_{ij}^{\text{HTL}}$ in such a way that our integration of the real contributions $d\Delta\sigma_{ij}^{\text{mass}} = d\sigma_{ij} - d\sigma_{ij}^{\text{HTL}}$ is finite. We construct a local subtraction term for the partonic channels $d\hat{\sigma}_{ij}$, $$d\Delta\hat{\sigma}^{\text{mass}}_{ij}(p_k) = d\hat{\sigma}_{ij}(p_k) - d\hat{\sigma}_{\text{LO}}(\tilde{p}_k) \frac{d\hat{\sigma}_{ij}^{\text{HTL}}(p_k)}{d\hat{\sigma}_{\text{LO}}^{\text{HTL}}(\tilde{p}_k)},$$ where $p_k$ denote the four-momenta from the full $2\to 3$ phase-space and $\tilde{p}_k$ stand for the mapping of the momenta $p_k$ on a $2\to 2$ sub-phase-space. As the results in the HTL limit are given in the Born-improved approximation in which the pure HTL is rescaled with the full LO matrix elements, we need to map the full $2\to 3$ phase-space onto a projected $2\to 2$ phase-space to construct the subtraction term involving this rescaling to the full LO contribution $ d\hat{\sigma}_{\text{LO}}$. The mapping is done by using the transformation formulae for initial-state emitter and initial-state spectator in the construction of dipole subtraction terms, i.e. using Eqs. (5.137-5.139) of Ref. [@Catani:1996vz]. The (mapped) momenta of the initial-state partons are $p_{1/2}$ ($\tilde{p}_{1/2}$), the (mapped) momenta of the final-state Higgs bosons are $p_{3/4}$ ($\tilde{p}_{3/4}$), and the momentum of the radiated parton is $p_5$. For the initial-state partons, we use the following mapping, $$\tilde{p}_1 = p_1,\qquad \tilde{p}_2 = p_2 \left(1-\frac{(p_5 p_1) + (p_5 p_2)}{(p_1 p_2)}\right).$$ In order to transform the Higgs momenta, we introduce the variables $K$ and $\tilde{K}$, $$K = p_1+p_2-p_5,\qquad \tilde{K} = \tilde{p}_1+\tilde{p}_2$$ allowing us to define $$\begin{aligned} \tilde{p}_3 & = p_3 - 2\,\frac{p_3 (K+\tilde{K})}{(K+\tilde{K})^2} \left(K+\tilde{K}\right) + 2\,\frac{(p_3 K)}{K^2} \tilde{K},\nonumber\\ \tilde{p}_4 & = p_4 - 2\,\frac{p_4 (K+\tilde{K})}{(K+\tilde{K})^2} \left(K+\tilde{K}\right) + 2\,\frac{(p_4 K)}{K^2} \tilde{K}.\end{aligned}$$ The HTL matrix elements are calculated analytically. We introduce the partonic center-of-mass energy $\hat{s}$, and the Mandelstam variables $\hat{t}=(p_1-p_5)^2$ and $\hat{u}=(p_2-p_5)^2$. The invariant squared Higgs-pair mass is $Q^2=\hat{s}+\hat{t}+\hat{u}$. The real spin- and colour-averaged matrix elements are $$\begin{aligned} \overline{\Big|\mathcal{M}_{gg\to HHg}^{\text{HTL}}\Big|^2} & = \frac{\alpha_s^3(\mu_R)G_F^2}{12\pi}\, \frac{\hat{s}^4+\hat{t}^4+\hat{u}^4+Q^8}{\hat{s}\hat{t}\hat{u}}\left(1-\frac{3 M_H^2}{Q^2-M_H^2}\right)^2,\nonumber\\ \overline{\Big|\mathcal{M}_{qg\to HHq}^{\text{HTL}}\Big|^2} & = \frac{\alpha_s^3(\mu_R)G_F^2}{27\pi}\, \frac{\hat{s}^2+\hat{u}^2}{-\hat{t}}\left(1-\frac{3 M_H^2}{Q^2-M_H^2}\right)^2,\nonumber\\ \overline{\Big|\mathcal{M}_{q\bar{q}\to HHg}^{\text{HTL}}\Big|^2} & = \frac{8\alpha_s^3(\mu_R) G_F^2}{81\pi}\, \frac{\hat{t}^2+\hat{u}^2}{\hat{s}}\left(1-\frac{3 M_H^2}{Q^2-M_H^2}\right)^2,\end{aligned}$$ and the LO matrix element in the HTL reads $$\overline{\Big|\mathcal{M}_{\text{LO}}^{\text{HTL}}\Big|^2} = \frac{\alpha_s^2(\mu_R) G_F^2}{288\pi^2}\, Q^4 \left(1-\frac{3 M_H^2}{Q^2-M_H^2}\right)^2.$$ The full one-loop matrix elements have been generated with [FeynArts]{} [@Hahn:2000kx] and [FormCalc]{} [@Hahn:1998yk]. They contain triangle, box, and pentagons diagrams. Generic diagrams for the contribution $g g\to H H g$ are given in Fig. \[fg:gghhreal\], generic diagrams for the contributions $q g\to H H q$ and $q\bar{q}\to H H g$ are displayed in Fig. \[fg:gghhreal2\]. The numerical evaluation of the scalar integrals [@tHooft:1978jhc] as well as the tensor reduction has been performed using the techniques developed in Refs. [@vanOldenborgh:1990yc; @Denner:2002ii; @Denner:2005nn; @Denner:2010tr] and implemented in the library [Collier 1.2]{} [@Denner:2016kdg]. The latter has been interfaced to the analytic expressions generated by [FormCalc]{} with an in-house routine. In order to improve our numerical stability, we have implemented a technical collinear cut in the phase-space parametrization. The integration of the scattering angle $\theta$ of the radiated parton in the c.m. system is restricted to the range $|\!\cos\theta|< 1-\delta$ with $\delta=10^{-4}$. We have checked that our results are stable against a variation of $\delta$ from $10^{-4}$ to $10^{-6}$ and therefore they are not affected by our choice for this technical cut. ![\[fg:gghhreal\] *Typical one-loop triangle (upper row), box (middle row), and pentagon (lower row) diagrams for the partonic channel $g g\to H H g$ contributing to the real corrections of Higgs-pair production via gluon fusion at NLO in QCD.*](./plots/dia_real_ggHHg.pdf){width="75.00000%"} ![\[fg:gghhreal2\] *Typical one-loop triangle and box diagrams for the partonic channels $q g\to H H q$ (upper row) and $q\bar{q}\to H H g$ (lower row), contributing to the real corrections of Higgs-pair production via gluon fusion at NLO in QCD.*](./plots/dia_real_qgHHq.pdf "fig:"){width="75.00000%"} ![\[fg:gghhreal2\] *Typical one-loop triangle and box diagrams for the partonic channels $q g\to H H q$ (upper row) and $q\bar{q}\to H H g$ (lower row), contributing to the real corrections of Higgs-pair production via gluon fusion at NLO in QCD.*](./plots/dia_real_qqHHg.pdf "fig:"){width="75.00000%"} We have cross-checked the final mass-effects of the real corrections against the results presented in the literature [@Frederix:2014hta; @Maltoni:2014eza; @Borowka:2016ehy; @Borowka:2016ypz] and we have obtained agreement. Results \[sc:results\] ====================== Our numerical results will be presented for the invariant Higgs-pair-mass distributions for different c.m. energies, i.e. 14 TeV for the LHC, 27 TeV for a potential high-energy LHC (HE-LHC) and 100 TeV for a provisional proton collider within the Future-Circular-Collider (FCC) project. The Higgs mass has been chosen as $M_H=125$ GeV and the top pole mass as $m_t=172.5$ GeV. The results for the full NLO cross sections have been obtained with two different PDF sets, [MMHT2014]{} [@Harland-Lang:2014zoa] and [PDF4LHC15]{} [@Butterworth:2015oua], that are taken from the [LHAPDF-6]{} library [@Buckley:2014ana]. The central scale choices for the renormalization and factorization scales are $\mu_F=\mu_R=Q/2$ and the input value $\alpha_s(M_Z)$ is chosen according to the PDF set used. Since [MMHT2014]{} contains a LO set, these PDFs are used for the evaluation of the consistent K-factors with the NLO (LO) cross section calculated with NLO (LO) $\alpha_s$ and PDFs. The whole calculation of the virtual and real corrections has been performed at least twice independently adopting also different Feynman parametrizations of the virtual two-loop diagrams. The real corrections have been derived with different parametrizations of the real phase-space. Both calculations agree within the numerical errors. We work in the narrow-width approximation of the top quark so that the Richardson extrapolation has to be applied to reach this limit for the two-loop box diagrams.[^11]. Differential cross section\[sec:diffxs\] ---------------------------------------- For the differential cross section, we have computed a grid of $Q$-values from 250 GeV to 1.5 TeV. In order to get a reliable result for the total cross section later on, we have used steps of 5 GeV between $Q=250$ GeV and $Q=300$ GeV, steps of 25 GeV between $Q=300$ GeV and $Q=700$ GeV, and steps of 50 GeV for $Q>700$ GeV. After applying the integrations by parts to each individual virtual diagram, we reached reliable results of our numerical integrations for $\bar\epsilon$ values \[see Eq. (\[eq:imaginary\])\] down to about 0.05. In order to obtain the result in the narrow-width approximation ($\bar\epsilon \to 0$), we have performed a Richardson extrapolation applied to the results for different values[^12] of $\bar\epsilon$. We adopt $\bar\epsilon$ values $\bar\epsilon_n= 0.025\times 2^n$ ($n=0\dots 10$). For bins close to threshold, $Q=300,325,350$ GeV, we use the set $n=0\dots 8$. For $Q \in [375,475]$ GeV, we use $n=1\dots 9$ while we use $n=2\dots 10$ for Q values in the range $Q \in [500,700]$ GeV. For $Q$ values starting at 750 GeV, we restrict the extrapolation to $n=2\dots 6$. In this way, we obtain a series of extrapolated results up to the ninth order in the dominant region and up to the fifth order in the tails for large $Q$. We define an estimate of the theoretical error due to the Richardson extrapolation as the difference of the extrapolated results at fifth and fourth order. In addition, we multiply this error by a factor of two close to the virtual $t\bar t$ threshold in order to be conservative. The total estimated Richardson-extrapolation error ranges below the per-cent level and is added in quadrature to the statistical integration error. Since we have subtracted the (Born-improved) HTL consistently from the virtual and real corrections, we are left with the pure top-mass effects at NLO that are infrared and ultraviolet finite individually after renormalization. This part has then been added to the results of [Hpair]{} [@hpair] to derive the full NLO cross section. The final invariant Higgs-pair-mass distributions are displayed in Figs. \[fig:distrib\_14\]–\[fig:distrib\_100\] for the three c.m. energies, 14, 27, 100 TeV. The blue curves show the Born-improved result in the HTL of Ref. [@Dawson:1998py] as implemented in [Hpair]{} [@hpair], the yellow ones the Born-improved HTL result plus the mass effects of the real corrections, the green curves the Born-improved HTL result plus the mass effects of the virtual corrections and the red curves the full NLO results. The plots on the left side of each figure have been obtained by using [MMHT2014]{} PDFs [@Harland-Lang:2014zoa] and the ones on the right with [PDF4LHC]{} PDFs [@Butterworth:2015oua]. The lower panel on the left shows the consistently defined K-factors $K=d\sigma_{NLO}/d\sigma_{LO}$. The lower panel on the right shows the ratio of the differential NLO cross section to the one obtained in the Born-improved HTL. While the Born-improved HTL provides a reasonable approximation for $Q$-values close to threshold, the real corrections add a negative mass effect of about $-10\%$ for $\sqrt{s}=14$ TeV (yellow curves) that is approximately uniform in the entire $Q$ range. The (negative) mass effects of the virtual corrections (green curves), however, become large at large values of $Q$ reaching a level of more than 20% for $Q$ beyond about 1 TeV. While the relative mass effects of the virtual corrections at NLO are independent of the collider energy (see the right plots showing the ratios to the HTL in the lower panels) in agreement with Eq. (\[eq:nlodiff\]), the NLO mass effects of the real corrections become larger with rising collider energy, reaching a level of $-20\%$ for $\sqrt{s}=100$ TeV. Both mass effects of the virtual and real corrections add up in the same direction and result in a total modification of the differential cross section of up to $-40\%$ compared to the Born-improved HTL at large $Q$ values for $\sqrt{s}=100$ TeV. While (as for the ratios) the full NLO K-factors shown in the left plots are close to the Born-improved HTL (blue curves) at $Q$ values close to the production threshold, they deviate significantly at larger values of $Q$ due to the additional NLO top-mass effects that decrease the total size of the NLO QCD corrections compared to the HTL as expected from unitarity arguments. To estimate the theoretical uncertainties, we have varied the renormalization and factorization scales for each bin in $Q$ by a factor of 2 up and down around the central scale $\mu_R=\mu_F=Q/2$ and derived the envelope of a 7-point variation, i.e. excluding points where the renormalization and factorization scales differ by more than a factor of two. The residual uncertainties are shown by the red band around the full NLO results (red curves) in Figs. \[fig:distrib\_14\]–\[fig:distrib\_100\]. They range at the level of 10–15% in total as can be inferred from the explicit numbers for $\sqrt{s}=14$ TeV (using [PDF4LHC]{} PDFs), $$\begin{aligned} \frac{d\sigma_{NLO}}{dQ}\Big|_{Q=300~{\rm GeV}} & = & 0.02978(7)^{+15.3\%}_{-13.0\%}\ {\rm fb/GeV},\nonumber\\ \frac{d\sigma_{NLO}}{dQ}\Big|_{Q=400~{\rm GeV}} & = & 0.1609(4)^{+14.4\%}_{-12.8\%}\ {\rm fb/GeV},\nonumber\\ \frac{d\sigma_{NLO}}{dQ}\Big|_{Q=600~{\rm GeV}} & = & 0.03204(9)^{+10.9\%}_{-11.5\%}\ {\rm fb/GeV},\nonumber\\ \frac{d\sigma_{NLO}}{dQ}\Big|_{Q=1200~{\rm GeV}} & = & 0.000435(4)^{+7.1\%}_{-10.6\%}\ {\rm fb/GeV} \, .\end{aligned}$$ ![Invariant Higgs-pair-mass distributions for Higgs boson pair production via gluon fusion at the 14 TeV LHC as a function of $Q=m_{HH}$. LO results (in black), HTL results (in blue), HTL results including the full real corrections (in yellow), HTL results including the full virtual corrections (in green, including the numerical errors), and the full NLO QCD results (in red, including the numerical errors). Left: Results with the [MMHT2014]{} PDF set, the panel below displays the K-factors for the different results. Right: Results with the [PDF4LHC15]{} PDF set, the panel below displays the ratio to the NLO Born-improved HTL result for the different calculations. The red band indicates the renormalization and factorization scale uncertainties for results including the full NLO QCD corrections.[]{data-label="fig:distrib_14"}](./plots/hh_nlo_mmht_14tev.pdf "fig:"){width="45.00000%"} ![Invariant Higgs-pair-mass distributions for Higgs boson pair production via gluon fusion at the 14 TeV LHC as a function of $Q=m_{HH}$. LO results (in black), HTL results (in blue), HTL results including the full real corrections (in yellow), HTL results including the full virtual corrections (in green, including the numerical errors), and the full NLO QCD results (in red, including the numerical errors). Left: Results with the [MMHT2014]{} PDF set, the panel below displays the K-factors for the different results. Right: Results with the [PDF4LHC15]{} PDF set, the panel below displays the ratio to the NLO Born-improved HTL result for the different calculations. The red band indicates the renormalization and factorization scale uncertainties for results including the full NLO QCD corrections.[]{data-label="fig:distrib_14"}](./plots/hh_nlo_pdf4lhc_14tev.pdf "fig:"){width="45.00000%"} ![Same as Fig. \[fig:distrib\_14\] but for a c.m. energy $\sqrt{s}=27$ TeV.[]{data-label="fig:distrib_27"}](./plots/hh_nlo_mmht_27tev.pdf "fig:"){width="45.00000%"} ![Same as Fig. \[fig:distrib\_14\] but for a c.m. energy $\sqrt{s}=27$ TeV.[]{data-label="fig:distrib_27"}](./plots/hh_nlo_pdf4lhc_27tev.pdf "fig:"){width="45.00000%"} ![Same as Fig. \[fig:distrib\_14\] but for a c.m. energy $\sqrt{s}=100$ TeV.[]{data-label="fig:distrib_100"}](./plots/hh_nlo_mmht_100tev.pdf "fig:"){width="45.00000%"} ![Same as Fig. \[fig:distrib\_14\] but for a c.m. energy $\sqrt{s}=100$ TeV.[]{data-label="fig:distrib_100"}](./plots/hh_nlo_pdf4lhc_100tev.pdf "fig:"){width="45.00000%"} We have analyzed the structure of the NLO QCD corrections in more detail by comparing the K-factor with the one of the triangle diagrams alone, i.e. with the K-factor of single-Higgs production with mass $M_H=Q$, in all individual approximations. This will determine the amount of universal NLO top-mass effects, common in the triangle and box diagrams. We define the ratio of the NLO triangle-diagram K-factor to the one including all diagrams as K-fac$^\triangle$/K-fac. This is shown, as a function of $Q=m_{HH}$, in Fig. \[fg:ktriafull\] (left). It is visible that the triangle-diagram K-factor provides an acceptable approximation to the full NLO K-factor only for $Q$ values below about 500–600 GeV if maximal deviations of about 15% are allowed (red histogram). The break down into the different mass effects of the virtual (green histogram) and real (yellow histogram) corrections singles out the origin of non-universal mass effects in the virtual corrections, while the non-universal mass effects beyond the single-Higgs case of the real corrections are limited to less than about 5% (apart from the virtual $t\bar t$-threshold region). In comparison to the contribution of the triangle diagrams alone, we also present the ratio of the K-factor obtained by including only the continuum diagrams (box diagrams of the virtual corrections and all box and pentagon diagrams of the real corrections without trilinear Higgs couplings) to the full K-factor in Fig. \[fg:ktriafull\] (right). The different curves show the results for the various approximations, i.e. the blue curves for the Born-improved HTL, the yellow ones with the inclusion of the NLO mass effects of the real corrections, the green curves with only the virtual NLO mass effects and the red curves the full NLO results. The right figure shows that the full NLO K-factor (red curve) is well-described (within 5%) by the one for the continuum diagrams alone which coincides with the observation that the continuum diagrams play a significant role for small values of $Q$ (where the K-factor does not deviate much from the single-Higgs case) and are dominant for large $Q$. This result shows that the K-factor cannot be approximated well by the one of single-Higgs production for large values of $Q$ due to the large mass effects of the virtual corrections. ![Ratios of the K-factor including (left) only triangle diagrams and (right) only continuum diagrams to the full K-factor of Higgs-pair production as a function of the invariant Higgs-pair mass $Q=m_{HH}$ for the LHC with a c.m. energy $\sqrt{s}=14$ TeV and using [MMHT2014]{} parton densities.[]{data-label="fg:ktriafull"}](./plots/hh_nlo_mmht_kfactors_ratio_triangle-to-full.pdf "fig:"){width="45.00000%"} ![Ratios of the K-factor including (left) only triangle diagrams and (right) only continuum diagrams to the full K-factor of Higgs-pair production as a function of the invariant Higgs-pair mass $Q=m_{HH}$ for the LHC with a c.m. energy $\sqrt{s}=14$ TeV and using [MMHT2014]{} parton densities.[]{data-label="fg:ktriafull"}](./plots/hh_nlo_mmht_kfactors_ratio_continuum-to-full.pdf "fig:"){width="45.00000%"} Total cross section ------------------- The total cross section has been obtained from the invariant Higgs-pair mass distribution by means of a numerical integration of the bins in $Q$ with the trapezoidal method for $Q>300$ GeV. For a reliable result, we used a Richardson extrapolation [@Richardson] in terms of the bin size in $Q$ also for this step. For $Q<300$ GeV, we have adopted the extension of Boole’s rule to six nodes [@Abramowitz]. We obtain the following values for the total cross section at various c.m. energies, $$\begin{aligned} \sqrt{s} = 13~{\rm TeV}: \quad \sigma_{tot} & = & 27.73(7)^{+13.8\%}_{-12.8\%}~{\rm fb}, \nonumber \\ \sqrt{s} = 14~{\rm TeV}: \quad \sigma_{tot} & = & 32.81(7)^{+13.5\%}_{-12.5\%}~{\rm fb}, \nonumber \\ \sqrt{s} = 27~{\rm TeV}: \quad \sigma_{tot} & = & 127.0(2)^{+11.7\%}_{-10.7\%}~{\rm fb}, \nonumber \\ \sqrt{s} = 100~{\rm TeV}: \quad \sigma_{tot} & = & 1140(2)^{+10.7\%}_{-10.0\%}~{\rm fb}, \label{eq:signlo}\end{aligned}$$ where we have used the [PDF4LHC]{} parton densities with $\alpha_s(M_Z)=0.118$ and added for completeness also the value for a c.m. energy of 13 TeV. The numbers in brackets show the numerical errors, while the upper and lower per-centage entries determine the (asymmetric) renormalization and factorization scale dependences. The corresponding results in the Born-improved HTL with [PDF4LHC]{} PDFs, obtained with the program [Hpair]{} [@hpair], read $$\begin{aligned} \sqrt{s} = 13~{\rm TeV}: \quad \sigma_{HTL} & = & 32.51^{+18\%}_{-15\%}~{\rm fb}, \nonumber \\ \sqrt{s} = 14~{\rm TeV}: \quad \sigma_{HTL} & = & 38.65^{+18\%}_{-15\%}~{\rm fb}, \nonumber \\ \sqrt{s} = 27~{\rm TeV}: \quad \sigma_{HTL} & = & 156.2^{+17\%}_{-13\%}~{\rm fb}, \nonumber \\ \sqrt{s} = 100~{\rm TeV}: \quad \sigma_{HTL} & = & 1521^{+16\%}_{-13\%}~{\rm fb}. \label{eq:sightl}\end{aligned}$$ Comparing the results of Eqs. (\[eq:signlo\]) and (\[eq:sightl\]), we observe a reduction of the total cross section by about 15% due to the top-mass effects at NLO and a reduction of the scale uncertainty. These numbers, as well as the differential distributions presented in Section \[sec:diffxs\], agree with the results of Refs. [@Borowka:2016ehy; @Borowka:2016ypz][^13]. It should be noted that a comparison of the full virtual corrections with the analytical large top-mass expansion presented in Ref. [@Grigo:2015dia] was performed in Refs. [@Borowka:2016ehy; @Borowka:2016ypz] and shows a convergence to the full result below the $t\bar t$-threshold, as expected. Uncertainties originating from the top-mass definition ------------------------------------------------------ An uncertainty that has been neglected or underestimated often previously is the intrinsic uncertainty due to the scheme and scale choice of the virtual top mass. This does not play a large role for single on-shell Higgs-boson production via gluon fusion, $gg\to H$, since the Higgs mass is small and thus the HTL works well, i.e. top-mass effects are suppressed. This uncertainty, however, plays a significant role for the larger values of $Q$ in Higgs-pair production. Top-mass effects are already sizeable at LO, but the NLO corrections add additional relevant top-mass dependences on top of the LO result as we have discussed in the previous subsection. The top mass is a scheme and scale dependent quantity so that the related uncertainties need to be estimated for a reliable determination of the total theoretical uncertainties. For this analysis, we have evaluated the differential cross section for the top mass defined in the on-shell scheme (default) and in the $\overline{\rm MS}$-scheme at the scale $\mu_t$, i.e. adjusting the counterterms and input parameters to the choices $\overline{m}_t(\overline{m}_t)$ and $\overline{m}_t(\mu_t)$ with $\mu_t$ in the range between $Q/4$ and $Q$ according to Section \[sc:renorm\][^14]. Since the scale dependence on $\mu_t$ is a monotonously falling function, we evaluated the differential cross section for four choices of the top mass, $m_t$, $\overline{m}_t(\overline{m}_t)$, $\overline{m}_t(Q/4)$ and $\overline{m}_t(Q)$, for each bin in $Q$. For the three c.m. energies of 14, 27 and 100 TeV the differential cross sections are presented in Figs. \[fig:distribmt\_1\], \[fig:distribmt\_2\] as a function of $Q=m_{HH}$ for the various definitions of the top mass. The lower panels exhibit the ratios of the differential cross sections to the ones in terms of the top pole mass (OS scheme). ![The differential Higgs-pair production cross section at NLO as a function of the invariant Higgs-pair mass for a c.m. energy of 14 TeV for four different choices of the scheme and scale of the top mass. The lower panel shows the ratio of all results to the default results with the top pole mass (OS scheme). [PDF4LHC]{} PDFs have been used and the renormalization and factorization scales of $\alpha_s$ and the PDFs have been fixed at our central scale choice $\mu_R=\mu_F=Q/2$.[]{data-label="fig:distribmt_1"}](./plots/hh_nlo_pdf4lhc_mtplot_14.pdf){width="60.00000%"} ![Same as Fig. \[fig:distribmt\_1\] but for c.m. energies of 27 (left) and 100 (right) TeV.[]{data-label="fig:distribmt_2"}](./plots/hh_nlo_pdf4lhc_mtplot_27.pdf "fig:"){width="45.00000%"} ![Same as Fig. \[fig:distribmt\_1\] but for c.m. energies of 27 (left) and 100 (right) TeV.[]{data-label="fig:distribmt_2"}](./plots/hh_nlo_pdf4lhc_mtplot_100.pdf "fig:"){width="45.00000%"} It is clearly visible that the scale and scheme dependence of the top mass induces sizeable variations of the NLO Higgs-pair production cross section and thus contributes to the theoretical uncertainties. For small $Q$ values, the size pattern of the differential cross section due to the different scale and scheme choices is varying. For large values of $Q$, the maximum is always given by the on-shell scheme and the minimum in terms of the $\overline{\rm MS}$-top mass $\overline{m}_t(Q)$ with sizeable differences to the on-shell scheme. Adopting the related uncertainties as the envelope of the cross sections for our four choices, we arrive at the following uncertainties of the differential cross section for a c.m. energy $\sqrt{s}=14$ TeV, $$\begin{aligned} \frac{d\sigma_{NLO}}{dQ}\Big|_{Q=300~{\rm GeV}} & = & 0.02978(7)^{+6\%}_{-34\%}\ {\rm fb/GeV},\nonumber\\ \frac{d\sigma_{NLO}}{dQ}\Big|_{Q=400~{\rm GeV}} & = & 0.1609(4)^{+0\%}_{-13\%}\ {\rm fb/GeV},\nonumber\\ \frac{d\sigma_{NLO}}{dQ}\Big|_{Q=600~{\rm GeV}} & = & 0.03204(9)^{+0\%}_{-30\%}\ {\rm fb/GeV},\nonumber\\ \frac{d\sigma_{NLO}}{dQ}\Big|_{Q=1200~{\rm GeV}} & = & 0.000435(4)^{+0\%}_{-35\%}\ {\rm fb/GeV} \, .\end{aligned}$$ Since these uncertainties are given relative to the on-shell results, the upper uncertainty vanishes for $Q\geq 400$ GeV, because the on-shell results provide the maximal values. These uncertainties turn out to be significant and at a similar level as the usual renormalization and factorization scale uncertainties. Thus, they constitute an additional contribution to the total theoretical uncertainties that has to be taken into account. The uncertainties due to the top-mass scheme and scale are about a factor of two smaller than at LO, $$\begin{aligned} \frac{d\sigma_{LO}}{dQ}\Big|_{Q=300~{\rm GeV}} & = & 0.01656^{+62\%}_{-2.4\%}\, \mathrm{fb/GeV},\nonumber\\ \frac{d\sigma_{LO}}{dQ}\Big|_{Q=400~{\rm GeV}} & = & 0.09391^{+0\%}_{-20\%}\, \mathrm{fb/GeV},\nonumber\\ \frac{d\sigma_{LO}}{dQ}\Big|_{Q=600~{\rm GeV}} & = & 0.02132^{+0\%}_{-48\%}\, \mathrm{fb/GeV},\nonumber\\ \frac{d\sigma_{LO}}{dQ}\Big|_{Q=1200~{\rm GeV}} & = & 0.0003223^{+0\%}_{-56\%}\, \mathrm{fb/GeV}\end{aligned}$$ that have been obtained for a c.m. energy of 14 TeV and using PDF4LHC15 NLO parton densities with a NLO strong coupling normalized to $\alpha_s(M_Z)=0.118$[^15]. Their reduction from LO to NLO underlines that the NLO QCD corrections stabilize the theoretical prediction for the Higgs-pair production cross section. The large size of the residual uncertainties is just a consequence of the large NLO QCD corrections as is the case for the renormalization and factorization scale dependences, too. Adopting the envelope for each $Q$-bin individually and integrating over $Q$, we arrive at the impact of these uncertainties on the total cross section for various c.m. energies, $$\begin{aligned} \sqrt{s} = 13~{\rm TeV}: \quad \sigma_{tot} & = & 27.73(7)^{+4\%}_{-18\%}~{\rm fb}, \nonumber \\ \sqrt{s} = 14~{\rm TeV}: \quad \sigma_{tot} & = & 32.81(7)^{+4\%}_{-18\%}~{\rm fb}, \nonumber \\ \sqrt{s} = 27~{\rm TeV}: \quad \sigma_{tot} & = & 127.0(2)^{+4\%}_{-18\%}~{\rm fb}, \nonumber \\ \sqrt{s} = 100~{\rm TeV}: \quad \sigma_{tot} & = & 1140(2)^{+3\%}_{-18\%}~{\rm fb}\end{aligned}$$ using [PDF4LHC]{} PDFs. A further reduction of these uncertainties can only be achieved by the determination or reliable estimate of the full mass effects at NNLO. Since these uncertainties are sizeable, one may wonder why this has not been observed already for single-Higgs boson production $gg\to H$. The measured value of the Higgs mass $M_H=125$ GeV is small compared to the top mass so that for single on-shell Higgs production we are close to the HTL, i.e. finite top-mass effects are small and thus the related uncertainties, too. However, going to larger virtualities $Q$ for off-shell Higgs production $gg\to H^*$ (or larger Higgs masses for on-shell Higgs production), we arrive at similar uncertainties for $\sqrt{s}=14$ TeV, $$\begin{aligned} \sigma_{NLO}\Big|_{Q=125~{\rm GeV}} & = 42.17^{+0.4\%}_{-0.5\%}\, \mathrm{pb}, \qquad \sigma_{NLO}\Big|_{Q=300~{\rm GeV}} & = 9.85^{+7.5\%}_{-0.3\%}\, \mathrm{pb},\nonumber \\[0.5cm] \sigma_{NLO}\Big|_{Q=400~{\rm GeV}} & = 9.43^{+0.1\%}_{-0.9\%}\, \mathrm{pb}, \qquad \sigma_{NLO}\Big|_{Q=600~{\rm GeV}} & = 1.97^{+0.0\%}_{-15.9\%}\, \mathrm{pb},\nonumber \\[0.5cm] \sigma_{NLO}\Big|_{Q=900~{\rm GeV}} & = 0.230^{+0.0\%}_{-22.3\%}\, \mathrm{pb}, \quad \sigma_{NLO}\Big|_{Q=1200~{\rm GeV}} & = 0.0402^{+0.0\%}_{-26.0\%}\, \mathrm{pb}\end{aligned}$$ using PDF4LHC PDFs. This has been known for a long time since there are sizeable effects on the virtual corrections due to the scale choice of the top mass for larger values of $Q$ or the Higgs mass (see Fig. 7a of Ref. [@Spira:1995rr]). For the single off-shell Higgs case, a reduction of the top-mass scale dependence by roughly a factor of two by going from LO to NLO has been observed, too, as can be inferred from the comparison with the explicit LO numbers for $\sqrt{s}=14$ TeV, $$\begin{aligned} \sigma_{LO}\Big|_{Q=125~{\rm GeV}} & = 18.43^{+0.8\%}_{-1.1\%}\, \mathrm{pb}, \qquad \sigma_{LO}\Big|_{Q=300~{\rm GeV}} & = 4.88^{+23.1\%}_{-1.1\%}\, \mathrm{pb}, \nonumber \\[0.5cm] \sigma_{LO}\Big|_{Q=400~{\rm GeV}} & = 4.94^{+1.2\%}_{-1.8\%}\, \mathrm{pb}, \qquad \sigma_{LO}\Big|_{Q=600~{\rm GeV}} & = 1.13^{+0.0\%}_{-26.2\%}\, \mathrm{pb}, \nonumber \\[0.5cm] \sigma_{LO}\Big|_{Q=900~{\rm GeV}} & = 0.139^{+0.0\%}_{-36.0\%}\, \mathrm{pb}, \quad \sigma_{LO}\Big|_{Q=1200~{\rm GeV}} & = 0.0249^{+0.0\%}_{-41.1\%}\, \mathrm{pb}\end{aligned}$$ that have been obtained with PDF4LHC PDFs as in the Higgs-pair case. On the other hand, the uncertainties for $Q=125$ GeV confirm that they are small for on-shell Higgs production via gluon fusion (already at LO) in agreement with the analysis of the LHC Higgs Cross Section Working Group [@deFlorian:2016spz; @Anastasiou:2016cez]. A relevant issue is the theoretical background of the different scale choices for the top mass. For small values of $Q$, the matrix element will be closer to the HTL such that the NLO corrections get closer to the HTL calculation. The HTL on the other hand can be treated by starting from the effective Lagrangian of Eq. (\[eq:leff\]) which is the residual effective coupling of Higgs bosons to gluons after integrating out the top quark. Thus, the corresponding Wilson coefficients $C_1$ and $C_2$ are determined by matching the full SM with the top quark to the effective theory without the top quark. The matching scale is naturally given by the top mass. Performing the proper matching at the scale of the top mass, i.e. using either the top pole mass or the top $\overline{\rm MS}$ mass at the scale of the top mass itself leads to non-logarithmic (in the top mass) matching contributions \[see Eq.(\[eq:leffcoeff\]) for $\mu_R=m_t$\] also for higher powers in $1/m_t^2$, i.e. higher-dimensional operators contributing to the gluonic Higgs couplings at the subleading level. This implies that the top mass is the preferred scale choice for small values of $Q$. This is confirmed by the heavy top expansion of the form factors of Refs. [@Grigo:2013rya; @Grigo:2015dia; @Davies:2018qvx]. At large $Q$ values, on the other hand, we can use the results for the high-energy expansion of Ref. [@Davies:2018qvx]. In the regime of large $Q$, the triangle-diagram contributions are suppressed by the $s$-channel Higgs propagator so that the box diagrams provide the dominant contributions. In our normalization, the explicit results of the virtual box-form factors in the high-energy limit ($Q\gg m_t, M_H$) in terms of the top pole mass $m_t$ are given by[^16] $$\begin{aligned} F_i & = & F_{i,LO} + \Delta F_i \, , \nonumber \\ \Delta F_i & = & \Delta F_{i,HTL} + \Delta F_{i,mass} \, , \nonumber \\ F_{1,LO} & \to & 4\frac{m_t^2}{\hat s} \, , \nonumber \\ F_{2,LO} & \to & -\frac{m_t^2}{\hat s \hat t (\hat s+\hat t)} \Big\{ (\hat s+\hat t)^2 L_{1ts}^2 + \hat t^2 L_{ts}^2 + \pi^2 [(\hat s+\hat t)^2+\hat t^2] \Big\} \, , \nonumber \\ \Delta F_{1,mass} & \to & \frac{\alpha_s}{\pi}\left\{ 2 F_{1,LO} \log\frac{m_t^2}{\hat s} + \frac{m_t^2}{\hat s} G_1(\hat s, \hat t) \right\} \, , \nonumber \\ \Delta F_{2,mass} & \to & \frac{\alpha_s}{\pi}\left\{ 2 F_{2,LO} \log\frac{m_t^2}{\hat s} + \frac{m_t^2}{\hat s} G_2(\hat s, \hat t) \right\} \, , \label{eq:ffhe}\end{aligned}$$ where $G_{1,2}(\hat s, \hat t)$ denote explicit and lengthy functions of the kinematical variables $\hat s$ and $\hat t$ that do [*not*]{} depend on the top mass [@Davies:2018qvx]. The logarithms $L_{ts}, L_{1ts}$ are defined as $$L_{ts} = \log \left(-\frac{\hat t}{\hat s}\right) + i\pi \, , \qquad L_{1ts} = \log \left(1+\frac{\hat t}{\hat s}\right) + i\pi \, .$$ Transforming the top pole mass $m_t$ into the $\overline{\rm MS}$ mass $\overline{m}_t(\mu_t)$, we arrive at the LO expressions for $F_{1/2,LO}$ with $m_t$ replaced by $\overline{m}_t(\mu_t)$ and the appropriately transformed NLO coefficients $$\begin{aligned} F_{1,LO} & \to & 4\frac{\overline{m}_t^2(\mu_t)}{\hat s} \, , \nonumber \\ F_{2,LO} & \to & -\frac{\overline{m}_t^2(\mu_t)}{\hat s \hat t (\hat s+\hat t)} \Big\{ (\hat s+\hat t)^2 L_{1ts}^2 + \hat t^2 L_{ts}^2 + \pi^2 [(\hat s+\hat t)^2+\hat t^2] \Big\} \, , \nonumber \\ \Delta F_{1,mass} & \to & \frac{\alpha_s}{\pi}\left\{ 2 F_{1,LO} \left[ \log\frac{\mu_t^2}{\hat s} + \frac{4}{3} \right] + \frac{\overline{m}_t^2(\mu_t)}{\hat s} G_1(\hat s, \hat t) \right\} \, , \nonumber \\ \Delta F_{2,mass} & \to & \frac{\alpha_s}{\pi}\left\{ 2 F_{2,LO} \left[ \log\frac{\mu_t^2}{\hat s} + \frac{4}{3} \right] + \frac{\overline{m}_t^2(\mu_t)}{\hat s} G_2(\hat s, \hat t) \right\} \, .\end{aligned}$$ To minimize the logarithms of $\mu_t$, a dynamical scale of the order of $\sqrt{\hat s}=Q$ has to be chosen, but [*not*]{} the top mass. A coefficient $\kappa$ in front of the dynamical scale choice $\mu_t = \kappa Q$ is still arbitrary (but should not be large) since additional finite parts of the functions $G_{1,2}(\hat s, \hat t)$ may be absorbed in the scale choice. Thus, the dynamical scale $Q$ can be identified as the preferred central scale choice of the Yukawa couplings for large $Q$ values. The uncertainties originating from the scheme and scale dependence of the top mass can be reduced by calculating the NNLO mass effects. Such a three-loop calculation is beyond everything that has been performed so far with current methods, but for $Q$ values close to threshold a large-mass expansion at NNLO could be used to reach an approximate estimate of the finite top-mass effects at NNLO. As a first step, partial results of the NNLO top-mass effects are known in the soft+virtual approximation [@Grigo:2015dia]. For $Q$ values around the virtual $t\bar t$ threshold $Q\sim 2m_t$, non-relativistic Green’s functions could be used that allow the introduction of higher-order corrections to the QCD potential [@Fadin:1987wz; @Fadin:1988fn; @Fadin:1990wx; @Strassler:1990nw; @Melnikov:1994jb]. This may lead to an improved description of the threshold region. However, for the triangle diagrams, the threshold behaviour is determined by $P$-wave contributions, since the $t\bar t$-ground state appears as a ${\cal CP}$-odd configuration that does not mix with the virtual ${\cal CP}$-even threshold state of the triangle diagrams. For the box diagrams, the $P$–wave contributions have to be considered, too. Moreover, it is unclear how large the impact of top-mass effects of the remainder beyond the non-relativistic Green’s functions will be. Finally, for the high-energy tail, the approximate calculation of Ref. [@Davies:2018qvx] could be extended to NNLO. Variation of the cross section with $\lambda_{H^3}$ --------------------------------------------------- Higgs-pair production at the LHC is directly sensitive to the trilinear Higgs coupling. The dependence of the total and differential cross sections on the trilinear coupling $\lambda_{H^3}$ is modified by the NLO QCD corrections and in particular by the finite mass effects at LO and NLO. Finite top-mass effects result in a non-vanishing matrix element at threshold, while in the HTL the matrix element of Eq. (\[eq:lomat\]) vanishes exactly [@Glover:1987nx; @Plehn:1996wb; @Li:2013rra], $$\begin{aligned} {\cal A}^{\mu\nu} & \to & F_1 T_1^{\mu\nu} \, , \nonumber \\ F_1 & \to & \frac{2}{3}(C_\triangle - 1) \to \frac{2}{3}~\left(\frac{3M_H^2}{4 M_H^2-M_H^2} - 1\right) = 0 \qquad \mbox{for $Q^2\to (2M_H)^2$} \, ,\end{aligned}$$ where we have used that the second form factor $G_\Box$ vanishes in the HTL \[see Eq. (\[eq:ffhtl\])\]. The cancellation is induced by the destructive interference between the triangle and box diagrams at LO. This property is modified by finite subleading ${\cal O}(1/m_t^2)$ terms but explains why the matrix element itself is suppressed at the production threshold. As a function of $\lambda_{H^3}$, the cross section develops a minimum at $\lambda_{H^3}$-values around 2.4 times the SM-value in the Born-improved HTL [@Dawson:1998py; @Baglio:2012np] since the phase-space integration adds contributions from above the production threshold. The NLO QCD corrections will shift the minimum of the cross section as a function of $\lambda_{H^3}$ and finite top-mass effects play a prominent role in the amount of these cancellations. For the determination of the trilinear coupling, the variation of the cross section with $\lambda_{H^3}$ is of interest. As mentioned in the introduction, the total cross section behaves approximately as $\Delta\sigma/\sigma \sim -\Delta\lambda_{H^3}/\lambda_{H^3}$ for $\lambda_{H^3}$ close to the SM value. In the following, we will analyze the NLO results, where only the trilinear coupling has been varied. In general, however, several coupling modifications contribute to the Higgs-pair production cross section. This could be treated consistently by extending the SM Lagrangian by all contributing dimension-6 operators as has been studied in Ref. [@Grober:2015cwa] in the HTL at NLO and in Ref. [@deFlorian:2017qfk] at NNLO. Recently the HTL analysis has been extended to the inclusion of finite top-mass effects at NLO [@Buchalla:2018yce]. However, we will neglect all dimension-6 operators but the one modifying the Higgs self-interactions. A proper and consistent effective model of this type has been discussed in Ref. [@Degrassi:2017ucl] that adds higher-dimension operators to the scalar Higgs sector only. Thus, a sole variation of the Higgs self-interactions could be realized within Higgs portal models with additional heavy scalar states that couple only to the SM-like Higgs field and are integrated out. ![The total Higgs-pair production cross section at NLO as a function of the trilinear self-coupling $\lambda_{H^3}$ in units of the SM value for a c.m. energy of 14 TeV. The blue curve shows the Born-improved HTL, the yellow includes the NLO mass effects of the real corrections in addition and the green curve those of the virtual corrections in addition. The full NLO result is presented by the red curve. The lower panel shows the ratio of all results to the Born-improved HTL. [PDF4LHC]{} PDFs have been used and the renormalization and factorization scales of $\alpha_s$ and the PDFs have been fixed at our central scale choice $\mu_R=\mu_F=Q/2=m_{HH}/2$.[]{data-label="fig:lambdavar_1"}](./plots/hh_nlo_lambda_pdf4lhc_14tev.pdf){width="60.00000%"} In Figs. \[fig:lambdavar\_1\] and \[fig:lambdavar\_2\], the dependence of the total Higgs-pair production cross section is shown as a function of the trilinear Higgs coupling $\lambda_{H^3}$ in units of the SM coupling for three c.m. energies, 14, 27 and 100 TeV. The blue curves display the results in the Born-improved HTL, the yellow curves include the mass effects of the real corrections and the green curves the mass effects of virtual corrections in addition. The red curves exhibit the complete NLO results. The comparison of the blue and red curves indicates that the minimum of the $\lambda_{H^3}$-variation is shifted from about 2.4 times the SM value to about 2.3 times the SM value due to the NLO mass effects. The yellow and green curves imply that the main origin of this shift emerges from the mass effects of the real corrections. The lower panels of Figs. \[fig:lambdavar\_1\] and \[fig:lambdavar\_2\] present the ratios of the individual contributions to the Born-improved HTL. While the NLO mass effects are of moderate size for negative values of $\lambda_{H^3}$, where the triangle and box diagrams interfere constructively, they turn out to be more relevant in the region of destructive interference, in particular around the minima of the cross sections. The significantly varying NLO mass effects have to be taken into account when determining the value of $\lambda_{H^3}$ from the experimental data at the HL-LHC. This agrees with the findings of Ref. [@Buchalla:2018yce]. The NLO mass effects on the variation of the total cross section with $\lambda_{H^3}$ become larger with rising c.m. energy of the hadron collider. ![Same as Fig. \[fig:lambdavar\_1\] but for c.m. energies of 27 (left) and 100 (right) TeV.[]{data-label="fig:lambdavar_2"}](./plots/hh_nlo_lambda_pdf4lhc_27tev.pdf "fig:"){width="45.00000%"} ![Same as Fig. \[fig:lambdavar\_1\] but for c.m. energies of 27 (left) and 100 (right) TeV.[]{data-label="fig:lambdavar_2"}](./plots/hh_nlo_lambda_pdf4lhc_100tev.pdf "fig:"){width="45.00000%"} In Fig. \[fig:lambdakfac\], we display the consistently defined K-factors $K=\sigma_{NLO}/\sigma_{LO}$ as a function of $\lambda_{H^3}$ in units of the SM coupling. The full curves show the NLO K-factors including the NLO top-mass effects for various c.m. energies. The dotted curves exhibit the corresponding K-factors in the Born-improved HTL as computed in Refs. [@Dawson:1998py; @Grober:2015cwa]. The impact of the NLO mass effects on the K-factors ranges at the level of 10–15% for negative $\lambda_{H^3}$ values, where the triangle and box diagrams interfere constructively. For positive values of $\lambda_{H^3}$ (destructive interference), the size and sign of the NLO mass effects is changing considerably as can be inferred from the comparison to the dotted curves. The full K-factors develop a larger dependence on $\lambda_{H^3}$ than the Born-improved HTL due to the NLO top-mass effects. This confirms the findings of Ref. [@Buchalla:2018yce]. The NLO top-mass effects of the total cross section increase with rising collider energy in general except for the regions of destructive interference between the triangle and box diagrams (positive $\lambda_{H^3}$). ![K-factors of Higgs-pair production at NLO as functions of the trilinear self-coupling $\lambda_{H^3}$ in units of the SM value $\lambda_{H^3}^{SM}$ for various c.m. energies of 13 TeV (red curves), 14 TeV (blue curves), 27 TeV (green curves) and 100 TeV (grey curves). The full NLO result is presented by the full curves with the error bars indicating our numerical errors. The dotted curves show the corresponding K-factors of the Born-improved HTL. [MMHT2014]{} PDFs have been used and the renormalization and factorization scales of $\alpha_s$ and the PDFs have been fixed at our central scale choice $\mu_R=\mu_F=Q/2=m_{HH}/2$.[]{data-label="fig:lambdakfac"}](./plots/hh_nlo_lambda_kfactor_mmht.pdf){width="60.00000%"} The full NLO cross section as a function of $\lambda_{H^3}$ can be parametrized as $$\sigma_{NLO} = \sigma_1 + \sigma_2 \frac{\lambda_{H^3}}{\lambda^{SM}_{H^3}} + \sigma_3 \left( \frac{\lambda_{H^3}}{\lambda^{SM}_{H^3}} \right)^2 \, .$$ The coefficients $\sigma_{1\ldots 3}$ depend on the c.m. energy of the hadron collider and on the PDFs used in their evaluation. For the various c.m. energies, we obtain the following NLO values for [PDF4LHC]{} PDFs and our central scale choices $\mu_R=\mu_F=Q/2$, $$\begin{aligned} \sqrt{s} = 13~{\rm TeV}: \quad \sigma_1 & = & 61.35(6)~{\rm fb}\, , \quad \sigma_2 = -43.26(5)~{\rm fb}\, , \quad \sigma_3 = 9.62(8)~{\rm fb} \, , \nonumber \\ \sqrt{s} = 14~{\rm TeV}: \quad \sigma_1 & = & 72.27(7)~{\rm fb}\, , \quad \sigma_2 = -50.70(6)~{\rm fb}\, , \quad \sigma_3 = 11.23(9)~{\rm fb} \, , \nonumber \\ \sqrt{s} = 27~{\rm TeV}: \quad \sigma_1 & = & 270.9(3)~{\rm fb}\, , \quad \sigma_2 = -183.1(2)~{\rm fb}\, , \quad \sigma_3 = 39.5(4)~{\rm fb} \, , \nonumber \\ \sqrt{s} = 100~{\rm TeV}: \quad \sigma_1 & = & 2323(2)~{\rm fb}\, , \quad\; \sigma_2 = -1496(2)~{\rm fb}\, , \quad\; \sigma_3 = 313(3)~{\rm fb} \, , \label{eq:siglam1}\end{aligned}$$ where the numbers in brackets denote our numerical errors. The corresponding coefficients with [MMHT2014]{} PDFs read $$\begin{aligned} \sqrt{s} = 13~{\rm TeV}: \quad \sigma_1 & = & 62.45(7)~{\rm fb}\, , \quad \sigma_2 = -44.13(5)~{\rm fb}\, , \quad \sigma_3 = 9.83(9)~{\rm fb} \, , \nonumber \\ \sqrt{s} = 14~{\rm TeV}: \quad \sigma_1 & = & 73.60(8)~{\rm fb}\, , \quad \sigma_2 = -51.75(6)~{\rm fb}\, , \quad \sigma_3 = 11.5(1)~{\rm fb} \, , \nonumber \\ \sqrt{s} = 27~{\rm TeV}: \quad \sigma_1 & = & 277.4(3)~{\rm fb}\, , \quad \sigma_2 = -187.9(2)~{\rm fb}\, , \quad \sigma_3 = 40.6(4)~{\rm fb} \, , \nonumber \\ \sqrt{s} = 100~{\rm TeV}: \quad \sigma_1 & = & 2401(2)~{\rm fb}\, , \quad\; \sigma_2 = -1550(2)~{\rm fb}\, , \quad\; \sigma_3 = 325(3)~{\rm fb} \, . \label{eq:siglam2}\end{aligned}$$ It should be noted that the final numerical errors of the cross sections as shown in Figs. \[fig:lambdavar\_1\]–\[fig:lambdakfac\] are smaller than the ones emerging from using the coefficients of Eqs. (\[eq:siglam1\], \[eq:siglam2\]) since the combinations of each bin in $Q$ [*before*]{} integration reduces them. Conclusions \[sc:conclusions\] ============================== In this work, we have discussed the full QCD corrections to Higgs-pair production at NLO. We have explained the details of our numerical approach to solve the multi-scale two-loop integrals involving ultraviolet and infrared singularities. The ultraviolet singularities could be extracted from the finite parts by suitable end-point subtractions, while the infrared singularities have been isolated by means of dedicated subtraction terms. The ultraviolet singularities have been absorbed by the proper renormalization of the strong coupling and the top mass, while the infrared ones cancel against the one-loop real corrections involving an additional gluon or quark in the final state of the Higgs-boson pair. We have performed the evaluation of the virtual corrections diagram by diagram without tensor reduction. The emerging integrals develop thresholds if the virtual $t\bar t$-threshold is crossed, but also at small virtualities due to the presence of purely gluonic intermediate states. The numerical stabilization of the virtual two-loop integrals has been achieved through integrations by parts of the integrands such that the power of the threshold-singular denominators is reduced. The narrow-width limit of the virtual top quarks has been obtained by a Richardson extrapolation of the results for different sizes of an auxiliarly introduced width parameter. This has allowed a numerical integration of the virtual two-loop corrections with an accuracy of less than one per cent. The matrix elements for the real corrections have been generated with [FeynArts]{} and [FormCalc]{} and integrated using the library [Collier]{}. The collinear region of the phase-space integration has been regularized numerically by a technical cut. We have subtracted the Born-improved HTL from the virtual and real corrections individually so that we have been left with the pure NLO top-mass effects beyond the Born-improved HTL that is implemented in the public tool [Hpair]{}. Thus, the final NLO results have been obtained by adding back the numbers from [Hpair]{}. The final results have been analyzed in detail for the differential cross section in the invariant Higgs-pair mass and the total cross section. Finite top-mass effects beyond the Born-improved HTL decrease the total cross section by about 15% at the LHC. However, the negative mass effects are larger for the differential cross section reaching a level of $-30\%$ or $-40\%$ for large invariant Higgs-pair masses. This implies that the inclusion of the NLO top-mass effects is crucial for a reliable analysis at the LHC and future proton colliders. We have discussed the usual renormalization and factorization scale uncertainties that are in agreement with previous calculations. However, we have identified an additional scale and scheme uncertainty due to the virtual top mass. This uncertainty reaches a level of 15% for the total cross section but can be larger (up to 35%) for the differential cross section. Based on the heavy-top and high-energy expansions, we have discussed the preferred scale choices of the running top mass and identified a large dynamical scale as the proper choice for large invariant Higgs-pair masses. This additional uncertainty has to be combined with the usual renormalization and factorization scale uncertainties. Since the (relative) scheme and scale uncertainties originating from the top mass only mildly depend on the renormalization and factorization scale choice, the addition of this uncertainty may lead to about a [*linear*]{} addition to the other uncertainties, if the total uncertainty is defined as the envelope. This, however, has to be analyzed in more detail which is left for future work. We have investigated the total cross section as a function of the trilinear coupling varied from its SM value. We have found significant NLO mass effects beyond the Born-improved HTL that result in a shift of the minimum of the cross section at various present and future c.m. energies of the hadron colliders. While the main effect of shifting the minimum originates from the NLO top-mass effects of the real corrections, the more symmetric virtual mass effects mainly affect the size of the total cross section as a function of $\lambda_{H^3}$. The full K-factors develop a larger dependence on $\lambda_{H^3}$ than those of the Born-improved HTL due to the NLO top-mass effects.\ [**Acknowledgements**]{}\ We are indebted to S. Dittmaier for providing us with a copy of his [mathematica]{} program for the QCD corrections in the HTL as constructed for the work of Ref. [@Dawson:1998py] and to R. Gröber for useful discussions. The work of S. G. is supported by the Swiss National Science Foundation (SNF). The work of S. G. and M. M. is supported by the DFG Collaborative Research Center TRR 257 “Particle Physics Phenomenology after the Higgs Discovery”. F. C. and J. R. acknowledge financial support by the Generalitat Valenciana, Spanish Government and ERDF funds from the European Commission (Grants No. RYC-2014-16061, SEJI-2017/2017/019, FPA2017-84543- P,FPA2017-84445-P, and SEV-2014-0398). We acknowledge support by the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). Two-loop box diagrams of the virtual corrections ================================================ Here we present the two-loop box diagrams (omitting the ones with reversed fermion flow): ![\[fg:boxdia1\] *Two-loop box diagrams: topologies 1 and 2.*](./plots/topology1.pdf "fig:"){width="75.00000%"}  \ ![\[fg:boxdia1\] *Two-loop box diagrams: topologies 1 and 2.*](./plots/topology2.pdf "fig:"){width="75.00000%"} ![\[fg:boxdia4\] *Two-loop box diagrams: topologies 5 and 6.*](./plots/topology34.pdf){width="80.00000%"}  \ ![\[fg:boxdia4\] *Two-loop box diagrams: topologies 5 and 6.*](./plots/topology56.pdf "fig:"){width="80.00000%"} [^1]: Note that Higgs pair production will provide indirect constraints on the quartic Higgs coupling [@Liu:2018peg; @Bizon:2018syu; @Borowka:2018pxx]. [^2]: Throughout this work, we will neglect the total Higgs width $\Gamma_H$ in the coefficient $C_\triangle$. [^3]: Note that we distinguish triangle and box diagrams also at the two-loop level in terms of the number of particles attached to the generic loop, i.e. three particles (two gluons and an off-shell Higgs for the triangle and two gluons and two on-shell Higgs bosons for the box diagrams). The one-particle-reducible diagrams are a special class. [^4]: The finite part of the complex virtual coefficient ${\cal C}^H_{virt}$ has been shown in Fig. 7a of Ref. [@Spira:1995rr] after renormalization. We define the top mass on-shell, i.e. use the coefficient for $\mu_Q=m_Q$ of this figure for the triangle-diagram contribution to our central prediction. [^5]: Since $V_{eff}$ is symmetric with respect to $\hat t \leftrightarrow \hat u$ the additional factor 2 emerges from the second term in the numerator of $C_{\triangle\triangle}$ in Eq. (\[eq:coeffvirt\]). [^6]: Note that $s$ denotes a Feynman parameter here and not the squared hadronic c.m. energy. The same holds for $z$. [^7]: For the bottom loops, additional stabilization of the numerical integration is required. This is left for future work. [^8]: Again $z,s$ denote Feynman parameters here. [^9]: The symmetrization of the integrand $f(\hat t_1, \hat u_1)$ for the $y$ integration is a straightforward result of this substitution. [^10]: Note that also the individual LO box diagrams are not finite with respect to the $\hat t$ integration, but the sum of all three LO boxes is. [^11]: Finite top-width effects have been estimated to amount to $\sim -2\%$ [@Maltoni:2014eza]. The effects are slightly larger in the vicinity of the virtual $t\bar t$ threshold, $Q^2\sim 4m_t^2$. [^12]: Note that a Richardson extrapolation of the integrand [*before*]{} integration provides an alternative to stabilize the numerical integration. [^13]: The small differences of the total cross sections at the few-per-mille level between the results originate from the slightly different values of the top mass ($m_t=172.5$ GeV in our analysis, $m_t=173$ GeV in Refs. [@Borowka:2016ehy; @Borowka:2016ypz]). [^14]: We do not separate the treatment of the top-Yukawa couplings and the propagator-top mass, since both are linked by the sum rule emerging from the electroweak $SU(2)\times U(1)$ symmetry, $y_t - \sqrt{2} m_t/v = 0$, which is needed for the cancellation of divergences in electroweak corrections. [^15]: Note that these choices are incompatible with a consistent LO prediction, but the relative uncertainties related to the scheme and scale choice of the top mass will be hardly affected by this inconsistency. These uncertainties are just parametric at LO. [^16]: The NLO form factors of Eq. (\[eq:ffhe\]) correspond to the infrared-subtracted ones according to Ref. [@Davies:2018qvx] plus the additional subtraction of the HTL. The piece related to the latter is absorbed in the functions $G_{1,2}$.

--- abstract: 'Real-world applications such as magnetic resonance imaging with multiple coils, multi-user communication, and diffuse optical tomography often assume a linear model where several sparse signals sharing common sparse supports are acquired by several measurement matrices and then contaminated by noise. Multi-measurement vector (MMV) problems consider the estimation or reconstruction of such signals. In different applications, the estimation error that we want to minimize could be the mean squared error or other metrics such as the mean absolute error and the support set error. Seeing that minimizing different error metrics is useful in MMV problems, we study information-theoretic performance limits for MMV signal estimation with arbitrary additive error metrics. We also propose a message passing algorithmic framework that achieves the optimal performance, and rigorously prove the optimality of our algorithm for a special case. We further conjecture the optimality of our algorithm for some general cases, and back it up through numerical examples. As an application of our MMV algorithm, we propose a novel setup for active user detection in multi-user communication and demonstrate the promise of our proposed setup.' author: - 'Junan Zhu,  and Dror Baron,  [^1] [^2]' title: 'Performance Limits with Additive Error Metrics in Noisy Multi-Measurement Vector Problems' --- [*Keywords*]{}:Active user detection, error metric, message passing, multi-measurement vector problem. Introduction ============ Many systems in science and engineering can be approximated by a linear model, where a signal $\x \in \mathbb{R}^N$ is recorded via a measurement matrix $\A\in\mathbb{R}^{M \times N}$, and then contaminated by a measurement channel, $$\label{eq:SMV} \w = \A \x,\ y_m = \mathcal{Z}(w_m), \forall m\in \{1,\ldots,M\},$$ where $y_m,m\in\{1,\ldots,M\}$, are the entries of the measurements $\y\in\mathbb{R}^M$, and the measurement channel $\mathcal{Z}(\cdot)$ is characterized by a probability density function (pdf), $f(y_m|w_m)$. The goal is to estimate $\x$ from the measurements $\y$ given knowledge of $\A$ and a model for the measurement channel $f(y_m|w_m),\forall m$. We call such a system the [*single measurement vector (SMV)*]{} problem. In many applications, the signal acquisition systems are distributed, where $J$ measurement matrices measure $J$ different signals individually. The key difference between such a system and $J$ individual SMV’s, is that these $J$ signals are somewhat dependent. An example of a model containing such dependencies is the multi-measurement vector (MMV) problem [@chen2006trs; @cotter2005ssl; @Mishali08rembo; @Berg09jrmm; @LeeKimBreslerYe2011; @LeeBreslerJunge2012; @YeKimBresler2015]. The MMV problem considers the estimation of a set of dependent signals, and has applications such as magnetic resonance imaging with multiple coils [@JuYeKi07; @JuSuNaKiYe09], active user detection in multi-user communication [@FletcherRanganGoyal2009; @Boljanovic2017], and diffuse optical tomography using multiple illumination patterns [@LeeKimBreslerYe2011]. In MMV, thanks to the dependencies among different signals, the number of sparse coefficients that can be successfully estimated increases with the number of measurements. This property was evaluated rigorously for noiseless measurements using $l_0$ minimization [@DuarteWakinBaronSarvothamBaraniuk2013], if the underlying signals share the same sparse supports. A non-rigorous replica analysis of MMV with measurement noise also shows the benefit of having more signal vectors [@ZhuBaronKrzakala2017IEEE; @ZhuDissertation2017]. [**Related work:**]{} There are many estimation approaches for MMV problems. These include greedy algorithms such as SOMP [@tropp2006ass; @chen2006trs], $l_1$ convex relaxation [@malioutov2005ssr; @tropp2006ass2], and M-FOCUSS [@cotter2005ssl]. REduce MMV and BOost (ReMBo) has been shown to outperform conventional methods [@Mishali08rembo], and subspace methods have also been used to solve MMV problems [@LeeBreslerJunge2012; @YeKimBresler2015]. However, these algorithms cannot handle the case of $J$ [*different*]{} measurement matrices. Statistical approaches [@ZinielSchniter2011] often achieve the oracle minimum mean squared error (MMSE). However, when running estimation algorithms for MMV problems, one might be interested in minimizing some other error. For example, if estimating the underlying signal is important, one could use the mean squared error (MSE) metric; when there might be outliers in the estimated signal, using the mean absolute error (MAE) metric might be more appropriate. Seeing that there is no prior work discussing the optimal performance with user-defined error metrics, we study the optimal performance with user-defined additive error metrics in MMV problems where the signals share common sparse supports, and each entry of the measurements is contaminated by parallel measurement channels (i.e., the channel in  satisfies $f(\y|\w)=\prod_{m=1}^M f(y_m|w_m)$). Note that a specific error metric, the MSE, has been studied in Zhu et al. [@ZhuBaronKrzakala2017IEEE], which focuses on the MSE performance limits (i.e., the MMSE) of MMV signal estimation. In contrast, this work explores performance limits and designs an algorithm that can minimize [*arbitrary additive error metrics*]{} beyond MSE. [**Contributions:**]{} This paper combines insights from Zhu et al. [@ZhuBaronKrzakala2017IEEE] and Tan and coauthors [@Tan2012SSP; @Tan2014], thus yielding a stronger understanding of the MMV problem, which could be extended in future work to other distributed signal acquisition settings, beyond MMV. To be more specific, we make several contributions in this paper. First, by extending Tan and coauthors [@Tan2012SSP; @Tan2014] we provide an algorithm based on a message passing (MP) framework [@RanganGAMP2011ISIT; @Krzakala2012probabilistic] that can be adapted to minimize the expected error for arbitrary additive error metrics. Our algorithm first runs MP until it converges or reaches some stopping criteria, and we then denoise MP’s output using a denoiser that minimizes the given additive error metric. Second, we prove rigorously that our algorithm is optimal in a specific SMV case,[^3] and further conjecture the optimality of our algorithm in MMV. Third, as an example, we derive performance limits for MAE and mean weighted support set error (MWSE) by designing the corresponding optimal denoisers, based on the scalar channel noise variance (Section \[sec:MPA\]) derived from replica analysis (Appendix \[app:inverMMSE\]) [@ZhuBaronKrzakala2017IEEE]. Simulation results show the superiority and optimality of our algorithm. We note in passing that having more signal vectors in MMV helps reduce the estimation error. Finally, as an application of MMV and our algorithm, we propose a novel setup for active user detection in multi-user communications (details in Section \[sec:app\]) and demonstrate the promise of our proposed setup through simulation. [**Organization:**]{} The remainder of the paper is organized as follows. We introduce our problem setting and MP algorithms in Section \[sec:background\]. Our algorithmic framework, which can minimize arbitrary additive error metrics, is proposed in Section \[sec:achievable\]; we rigorously prove the optimality of our algorithm for an SMV case and conjecture the optimality of our algorithm for MMV cases in Section \[sec:converse\]. For some example error metrics, we derive the corresponding optimal algorithms, together with the theoretical limits for these errors, in Section \[sec:example\]. Synthetic simulation results are discussed in Section \[sec:numeric\_synth\], followed by an application of our metric-optimal algorithm to a real-world problem in Section \[sec:app\]. We conclude in Section \[sec:conclusion\]. [**Notations:**]{} In this paper, bold capital letters represent matrices, bold lower case letters represent vectors, and normal font letters represent scalars. The $m$-th entry (scalar) of a vector $\z$ is denoted by $z_m$. Problem Setting and Background {#sec:background} ============================== Problem setting --------------- [**Signal model**]{}: We consider an ensemble of $J$ signal vectors, $\x^{(j)}\in\mathbb{R}^N,\ j\in\{1,\ldots,J\}$, where $j$ is the index of the signal. Consider a [*super-symbol*]{} $\x_n=[x_n^{(1)},\ldots,x_n^{(J)}],\ n\in\{1,\ldots,N\}$; all super-symbols in this paper are row vectors. The super symbol $\x_n$ follows a $J$-dimensional distribution, $$\label{eq:jsm} f(\x_n)=\rho \phi(\x_n)+(1-\rho)\delta(\x_n),$$ where $\rho\in (0,1)$ determines the percentage of non-zeros in the signal and is called the sparsity rate, $\phi(\x_n)$ is a $J$-dimensional pdf, and $\delta(\x_n)=\left\{ \begin{array}{ll} 1, & \x_n = \textbf{0},\\ 0, & \text{else}. \end{array} \right.\\ $ \[def:jointly\_sparse\] [*Ensembles of signals are called jointly sparse signals with common sparse supports when they obey .*]{} Note that there are other types of joint sparsity [@BaronDCStech] that fit into the MMV framework. For example, an MMV model with signal vectors that have slowly changing supports is discussed in Ziniel and Schniter [@ZinielSchniter2013MMV]. Since this paper only focuses on the MMV problem with signals sharing common sparse supports , we refer to joint sparsity with common sparse supports as joint sparsity for brevity. [**Measurement models**]{}: Each signal $\x^{(j)}$ is measured by a measurement matrix $\A^{(j)}\in\mathbb{R}^{M\times N}$ before being corrupted by a random measurement channel, $$\label{eq:linearMixing} \begin{split} \w^{(j)}&=\A^{(j)}\x^{(j)},\ y_m^{(j)}=\mathcal{Z}(w_m^{(j)}),\\ &\quad m\in\{1,\ldots,M\},\ j\in\{1,\ldots,J\}, \end{split}$$ where $y_m^{(j)}, m\in \{1,\ldots,M\}$ are the entries of the measurements $\y^{(j)}$, and the measurement channel $\mathcal{Z}(\cdot)$ is characterized by the pdf $f(y_m^{(j)}|w_m^{(j)})$. In this paper, we only focus on independent and identically distributed (i.i.d.) parallel measurement channels, i.e., the pdf’s $f(y_m^{(j)}|w_m^{(j)}), \forall m,j$, are identical and there is no cross-talk among different channels; our proposed algorithm is readily extended to parallel channels with different $f(y_m^{(j)}|w_m^{(j)}), \forall m,j$. When the number of signal vectors $J=1$, we call this MMV model an SMV problem . \[def:largeSystemLimit\] [*The signal length $N$ scales to infinity, and the number of measurements $M=M(N)$ depends on $N$ and also scales to infinity, where the ratio approaches a positive constant $R$,*]{} $$\lim_{N\rightarrow\infty} \frac{M(N)}{N} = R>0.$$ We call $R$ the measurement rate. For MMV problems, we are given the matrices $\A^{(j)}$ and measurements $\y^{(j)},\ \forall j$, as well as knowledge of the measurement channel . Our task is to estimate the underlying signal vectors $\x^{(j)},\ \forall j$. Suppose that the estimate is $\widehat{\x}^{(j)}$. Define $\X=[\x^{(1)},\cdots,\x^{(J)}]$ and $\widehat{\X}=[\widehat{\x}^{(1)},\cdots,\widehat{\x}^{(J)}]$. Therefore, $\X=[\x_1^T,\cdots,\x_N^T]^T$, where $\{\cdot\}^T$ denotes transpose. The estimation quality is quantified by a user-defined error metric $D_{\text{UD}}(\X,\widehat{\X})$, where the subscript UD denotes “user-defined." We define this [*additive error metric $D_{\text{UD}}(\cdot,\cdot)$ as*]{} $$D_{\text{UD}}(\X, \widehat{\X})={\sum_{n=1}^{N}}d_{\text{UD}}(\x_n,\widehat{\x}_n),$$ where $d_{\text{UD}}(\cdot,\cdot): \mathbb{R}^J\times \mathbb{R}^J \rightarrow \mathbb{R}$ is an arbitrary user-defined error metric on each super-symbol. The smaller the $D_{\text{UD}}(\cdot,\cdot)$ is, the better the estimation is. Message passing algorithms {#sec:MPA} -------------------------- Message passing (MP) algorithms consider a factor graph [@RanganGAMP2011ISIT; @Krzakala2012probabilistic; @BarbierKrzakala2017IT], which expresses the relation between the signals $\x$ and measurements $\y$. We begin by discussing the factor graph for SMV, followed by that of MMV. ![Factor graph for SMV (left) and MMV (right).[]{data-label="fig:factorGraph"}](factorGraph.png){width="8cm"} [**Factor graph for SMV:**]{} The left panel of Fig. \[fig:factorGraph\] illustrates the factor graph concept for an SMV problem  with i.i.d. entries in the signal $\x$. The round circles are the variable nodes (representing the distribution of the signal), and the squares denote the factor nodes (representing the measurement channel). The variables $x_n,\ \forall n$, are driven by each factor node $f(x_n)$ individually, because the signal has i.i.d. entries $x_n$. There are two types of messages passed in the factor graph shown on the left panel of Fig. \[fig:factorGraph\]: the message passed by the variable node $x_n$ to the factor node $y_m$, $\mathcal{M}_{n\rightarrow m}(x_n)$, and the message passed by the factor node $y_m$ to the variable node $x_n$, $\widehat{\mathcal{M}}_{m\rightarrow n}(x_n)$. According to the literature on MP algorithms [@RanganGAMP2011ISIT; @Krzakala2012probabilistic; @BarbierKrzakala2017IT], we have the following relation: $$\begin{split} \mathcal{M}_{n\rightarrow m}(x_n) &=\frac{1}{Z_{n\rightarrow m}}f(x_n) \prod_{\widehat{m} \neq m} \widehat{\mathcal{M}}_{\widehat{m}\rightarrow n}(x_n), \\ \widehat{\mathcal{M}}_{m\rightarrow n}(x_n) &=\frac{1}{Z_{m\rightarrow n}} \int f(y_m | \x) \prod_{\widehat{n}\neq n} \mathcal{M}_{\widehat{n}\rightarrow m}(x_{\widehat{n}})\prod_{\widehat{n}\neq n}dx_{\widehat{n}}, \end{split}$$ where we use a single integral sign to denote a multi-dimensional integration for brevity, and $Z_{n\rightarrow m}$ and $Z_{m\rightarrow n}$ are normalization factors. The aim of this paper is not the detailed derivation of MP algorithms. Instead, the key property of MP utilized in this paper is that MP converts  into the following equivalent scalar channel, $$\q=\x+\v,$$ where $\q$ is the noisy [*pseudo data*]{}, and $\v$ is the equivalent scalar channel additive white Gaussian noise (AWGN) whose variance $\Delta_v$ can be approximated. After obtaining $\q$ and $\Delta_v$, each variable node $x_n$ updates the estimate $\widehat{x}_n$ by denoising $q_n$. If the signal entries are i.i.d. and the matrix is either i.i.d. or sparse and locally tree-like, then MP algorithms yield a density function $f(x_n|q_n)$ that is statistically equivalent to $f(x_n|\y)$ [@RanganGAMP2011ISIT]. [**Factor graph for MMV:**]{} An MMV problem with jointly sparse signals can be expressed as the factor graph shown in the right panel of Fig. \[fig:factorGraph\]. We can see that the MP in each channel is similar to an SMV problem. The only difference is that the $J$ variable nodes $x_n^{(j)},\ j\in\{1,\ldots,J\}$, for fixed $n$ are driven by one factor node $f(\x_n)$. By grouping the entries from different signal vectors together into super-symbols as in , we have i.i.d. super-symbols. The noisy super-symbol pseudo data $\q_n=[q_n^{(1)},\ldots,q_n^{(J)}]$ is denoised, in order to update the estimate for $\x_n=[x_n^{(1)},\ldots,x_n^{(J)}]$. Main Results {#sec:main} ============ We first present our metric-optimal algorithm in Section \[sec:achievable\] and then in Section \[sec:converse\] we rigorously prove that our metric-optimal algorithm is optimal in the SMV case under certain conditions.[^4] Based on our proof for the SMV case, we conjecture that the proposed algorithm is optimal for arbitrary additive error metrics in the MMV case. Achievable part: Metric-optimal estimation algorithm {#sec:achievable} ---------------------------------------------------- The metric-optimal algorithm consists of two parts, as illustrated in Algorithm \[algo:metric\_opt\_MMV\]. We first run an MP algorithm (Algorithm \[algo:AMP\_MMV\] provides an implementation of an MP algorithm) to get the noisy pseudo data $\q^{(j)}, \forall j$, and the noise variance $\Delta_v$ (details below). Next, we denoise $\q^{(j)}, \forall j$, using an optimal denoiser tailored to minimize the given error metric. The following discusses both parts in detail. \ [**Inputs:**]{} Measurements $\y^{(j)}$ and matrices $\A^{(j)}, \forall j$\ [**Part 1 (Algorithm \[algo:AMP\_MMV\]):**]{} Obtain pseudo data $\q^{(j)},\ \forall j$, and scalar channel noise variance $\Delta_v$ from MP($\y^{(j)}, \A^{(j)}, \forall j$)\ [**Part 2 (examples in Section \[sec:example\]):**]{} Obtain optimal estimate $\widetilde{\x}^{(j)}$ from denoiser using $\q^{(j)},\Delta_v, \forall j$\ [**Outputs:**]{} $\widetilde{\x}^{(j)},\ \forall n$ \ [**Inputs:**]{} Maximum number of iterations $t_{\text{max}}$, threshold $\epsilon$, sparsity rate $\rho$, noise variance $\Delta_z$, measurements $\y^{(j)}$, and measurement matrices $\A^{(j)}, \forall j$\ [**Initialize:**]{} $t=1,\delta=\infty,\k^{(j)}=\y^{(j)},\Theta_m^{(j)}=0,s^{(j)}_n=\rho\Delta_z,\widehat{x}_n^{(j)}=0, h_m^{(j)}=0, \forall m,n,j$ \[line:beginForLoop\]\ $\boldsymbol{\Theta}^{(j)}=(\A^{(j)})^2 \s^{(j)}$\[line:Theta\]\ $\k^{(j)}=\A^{(j)} \widehat{\x}^{(j)}-\text{diag}(\boldsymbol{\Theta}^{(j)}) \h^{(j)}$\ $h_m^{(j)}=g_{\text{out}}\l(k_m^{(j)},y_m^{(j)},\Theta_m^{(j)}\r)$\[line:g\_out\]\ $r_m^{(j)} = -\frac{\partial}{\partial k_m^{(j)}} g_{\text{out}}\l(k_m^{(j)},y_m^{(j)},\Theta_m^{(j)}\r)$\[line:deriv\_g\_out\]\ // Scalar channel noise variance\ $\Delta_v^{(j)}=\l\{\frac{1}{N} \mathbf{1}^T \l[(\A^{(j)})^T\r]^2 \mathbf{r}^{(j)}\r\}^{-1}$ \[line:scalarNoiseVar\]\ $\q^{(j)}=\widehat{\x}^{(j)}+\Delta_v^{(j)} (\A^{(j)})^T \h^{(j)}$ // Pseudo data\ $\widehat{\a}^{(j)}=\widehat{\x}^{(j)}$ // Save current estimate \[line:endForLoop\]\  $\Delta_v=\sum_{j=1}^J \Delta_v^{(j)}$\[line:mean\_delta\]\ $\widehat{\x}_n=f_{a_n}(\Delta_v,\q_n)$ // Estimate\[line:mean\]\ $\s_n=[s_n^{(1)},\ldots,s_n^{(J)}]=f_{v_n}(\Delta_v,\q_n)$ // Variance\[line:var\]\   $t=t+1$ // Increment iteration index\   $\delta=\frac{1}{NJ}\sum_{n=1}^N\sum_{j=1}^J\l(\widehat{x}^{(j)}_n-\widehat{a}^{(j)}_n\r)^2$ // Change\ [**Outputs:**]{} Estimate $\widehat{\x}^{(j)}$, pseudo data $\q^{(j)},\ \forall j$, and scalar channel noise variance $\Delta_v$ [**MP algorithm:**]{} For the first part, we modify the generalized approximate message passing (GAMP) algorithm [@RanganGAMP2011ISIT], which is an implementation of MP, and list the pseudo code in Algorithm \[algo:AMP\_MMV\]. The notation diag$(\boldsymbol{\Theta}^{(j)})$ denotes a diagonal matrix whose entries along the diagonal are $\boldsymbol{\Theta^{(j)}}$, and the power-of-two in Lines \[line:Theta\] and \[line:scalarNoiseVar\] is applied element-wise. The function $g_{\text{out}}$ in Lines \[line:g\_out\] and \[line:deriv\_g\_out\] is given by $$\label{eq:g_out} g_{\text{out}}\l(k,y,\Theta\r)=\frac{1}{\Theta}(\mathbb{E}[w|k,y,\Theta]-k),$$ where we omit the subscripts and super-scripts for brevity, and the expectation is taken over the pdf, $$\label{eq:prob_w} f(w|k,y,\Theta)\propto f(y|w)\text{exp}\l[-\frac{(w-k)^2}{2\Theta}\r].$$ For the special case of AWGN channels, $$\label{eq:AWGN} y=w+z,$$ where $z\sim \mathcal{N}(0,\Delta_z)$, we obtain $g_{\text{out}}(k,y,\Theta)=\frac{y-k}{\Delta_z+\Theta}$ [@RanganGAMP2011ISIT]. In Appendix \[app:logit\], we also briefly present the derivation for an i.i.d. parallel logistic channel, $$\label{eq:logit} f(y|w)=\delta(y-1)\frac{1}{1+\text{exp}(-aw)}+\delta(y)\frac{\text{exp}(-aw)}{1+\text{exp}(-aw)},$$ where $a$ is a scaling factor.[^5] For the special case of i.i.d. [*joint Bernoulli-Gaussian signals*]{} where $\phi(\x_n)\sim \mathcal{N}(0,\mathbb{I})$ in  and $\mathbb{I}$ is an identity matrix, $f_{a_n}$ and $f_{v_n}$ in Lines \[line:mean\] and \[line:var\] are given below, $$\label{eq:denoiser} f_{a_n}(\Delta_v,\q_n)=\frac{\rho}{C(\Delta_v+1)}\q_n, $$ $$f_{v_n}(\Delta_v,\q_n)\!=\!-[f_{a_n}(\Delta_v,\q_n)]^2+\frac{\rho}{C(\Delta_v+1)}\!\l[\frac{\q_n^2}{\Delta_v+1}\!+\!\Delta_v\r],$$ where $\q_n^2=\l[\l(q_n^{(1)}\r)^2,\ldots,\l(q_n^{(J)}\r)^2\r]$ and $$C=\rho+(1-\rho)\l(1+\frac{1}{\Delta_v}\r)^{\frac{J}{2}}\exp\l[-\frac{\q_n\q_n^T}{2\Delta_v(\Delta_v+1)}\r].$$ Notice that in Line \[line:scalarNoiseVar\] we take the mean of a vector to obtain a scalar $\Delta_v^{(j)}$, which is the average of the variances for the estimates of signal entries $x_n^{(j)}$. This is because the super-symbols $\x_n, \forall n$, of the signals are i.i.d and the $J$ measurement channels are i.i.d. For the same reason, $\Delta_v^{(j)},\ j\in\{1,\ldots,J\}$, should be close to each other; hence, Line \[line:mean\_delta\]. Note that Algorithm \[algo:AMP\_MMV\] assumes that the entries of $\A^{(j)}$ scale with $\frac{1}{\sqrt{N}}$, and is a more generic form of an algorithm from our prior work with Krzakala [@ZhuBaronKrzakala2017IEEE]. [**Metric-optimal denoiser:**]{} The second part of our metric-optimal algorithm takes as inputs the noisy pseudo data $\q_n=\x_n+\v_n$ and the estimated variance of $\v_n,\ \Delta_v$, from the MP algorithm. Using Bayes’ rule, we can derive the posterior $f(\x_n|\q_n)$ and use $f(\x_n|\q_n)$ to formulate the optimal estimator in the sense of the user-defined error metric. The optimal estimate is $$\label{eq:estimator} \widetilde{\x}_n=\arg\min_{\widehat{\x}_n} \int d_{\text{UD}}(\x_n,\widehat{\x}_n) f(\x_n|\q_n) d\x_n. $$ Converse part: The optimal estimate {#sec:converse} ----------------------------------- The reason why  is optimal is based on the insight from Rangan [@RanganGAMP2011ISIT] that in SMV the density function $f(x_n|q_n)$ converges to the posterior $f(x_n|\y)$. A rigorous proof for a certain SMV case is provided below, followed by our conjecture that  is optimal in the MMV case. \[lemma:optSMV\] Consider an SMV  in the large system limit (Definition \[def:largeSystemLimit\]) with an AWGN measurement channel, $f(y_m|w_m)=\frac{1}{\sqrt{2\pi\sigma_Z^2}}\exp\l[\frac{(y_m-w_m)^2}{2\sigma_Z^2}\r], \forall m$. The estimate $\widetilde{\x}_n$  is optimal in the sense that $$\label{eq:optSMV} \lim_{N\rightarrow\infty} \frac{1}{N}\sum_{n=1}^N d_{\text{UD}}(\widetilde{x}_n,x_n) = \text{MUDE},$$ where MUDE denotes the minimum user-defined error, if all the conditions below hold. 1. The entries of the measurement matrix are i.i.d. Gaussian, $A_{mn}\sim \mathcal{N}(0,\frac{1}{N})$, 2. the signal entries are i.i.d. with bounded fourth moment $\mathbb{E}[X^4]<B$, where $B$ is some constant, 3. the free energy given by replica analysis has one fixed point [@ZhuBaronCISS2013; @ZhuBaronKrzakala2017IEEE; @Krzakala2012probabilistic],[^6] 4. the user-defined error metric $d_{\text{UD}}(\cdot,\cdot)$ is pseudo-Lipschitz [@Bayati2011],[^7] 5. the optimal estimator  as a function of $q_n,\ \widetilde{x}_n(q_n): \mathbb{R}\rightarrow \mathbb{R}$ , is Lipschitz continuous, and 6. Part 1 converges before entering Part 2 in Algorithm \[algo:metric\_opt\_MMV\]. According to Theorem 1 in Bayati and Montanari [@Bayati2011], we know that $\lim_{N\rightarrow\infty} \frac{1}{N}\sum_{n=1}^N d_{\text{UD}}(\widetilde{x}_n(q_n),x_n) = \mathbb{E}\l[d_{\text{UD}}(\widetilde{x}_{n}(X+\Delta_v Z),X)\r]$, where $X$ is a random variable following the same distribution as the signal entries, $Z\sim \mathcal{N}(0,1)$, and $\Delta_v$ is given in Line \[line:mean\_delta\] of Algorithm \[algo:AMP\_MMV\]. The question boils down to showing that $\text{MUDE} = \mathbb{E}\l[d_{\text{UD}}(\widetilde{x}_{n}(X+\Delta_v Z),X)\r]$. In fact, when Algorithm \[algo:AMP\_MMV\] converges, $\Delta_v$ corresponds to the scalar channel noise variance associated with the MMSE given by replica analysis [@Bayati2011; @RushVenkataramanan2016]. In addition, the MMSE provided by replica analysis is proved to be exact under the conditions asserted in Lemma \[lemma:optSMV\] [@ReevesPfister2016]. Therefore, $\mathbb{E}\l[d_{\text{UD}}(\widetilde{x}_{n}(X+\Delta_v Z),X)\r]$ is indeed the MUDE. Hence, we proved , and the estimator  is optimal. **Remark 1:** The optimality of the metric-optimal estimator $\widetilde{\x}_n$  is stated in the sense that the ensemble mean user-defined error converges almost surely to the MUDE. It is possible that there exist other estimators that achieve the MUDE. **Remark 2:** The proof is made possible by linking three rigorous proofs [@ReevesPfister2016; @Bayati2011; @RushVenkataramanan2016] from the prior art. That said, numerical examples in Section \[sec:numeric\_synth\] demonstrate that Algorithm \[algo:metric\_opt\_MMV\] yields promising results even if the conditions required by Lemma \[lemma:optSMV\] are not met. After proving the optimality of our metric-optimal algorithm in the SMV scenario with certain conditions, we state the following conjecture. \[conj:2\] In the large system limit, for the MMV model with the signal in  and the user-defined additive error metric $d_{\text{UD}}(\x_n,\widehat{\x}_n)$, the optimal estimate of the signal vectors is $$\widetilde{\x}_n=\arg\min_{\widehat{\x}_n} \mathbb{E}[d_{\text{UD}}(\x_n,\widehat{\x}_n)|\q_n].$$ As the reader can see from the proof of Lemma \[lemma:optSMV\], in order to rigorously prove Conjecture \[conj:2\], we need to show ([*i*]{}) the MMSE given by the replica analysis for the MMV case [@ZhuBaronKrzakala2017IEEE] is exact, ([*ii*]{}) the scalar channel noise variance $\Delta_v$ in Algorithm \[algo:AMP\_MMV\] corresponds to the MMSE given by replica analysis, and ([*iii*]{}) a result similar to Theorem 1 in Bayati and Montanari [@Bayati2011] holds. None of these three results exists in the prior art, so we do not foresee what exact conditions are needed for Conjecture \[conj:2\]. Proving these three results (and hence Conjecture \[conj:2\]) is beyond the scope of this work. Instead, we provide some intuition that explains why we believe Algorithm \[algo:metric\_opt\_MMV\] is optimal in the MMV scenario. In the SMV case, $f(x_n|q_n)$ converges to the posterior $f(x_n|\y)$ in relaxed BP [@RanganGAMP2011ISIT]. As an extension to the MMV case, we intuitively think that $f(\x_n|\q_n)$ would converge to the posterior $f(\x_n|\{\y^{(j)}\}_{j=1}^J)$, based on two observations: ([*i*]{}) the measurement channels are i.i.d., so that it suffices to update the estimate of the channel by passing the messages $\mathcal{M}_{n\rightarrow m}(x_n^{(j)})$ and $\widehat{\mathcal{M}}_{m\rightarrow n}(x_n^{(j)})$ for different $j$ individually, and ([*ii*]{}) the super-symbols $\x_n$ are i.i.d., so that denoising each super-symbol individually accounts for all the information needed to update the estimate. The optimal estimate of each super-symbol $\x_n$ in the signal vectors is $$\label{eq:estimatorTrue} \widetilde{\x}_{\text{true},n}=\arg\min_{\widehat{\x}_n} \int d_{\text{UD}}(\x_n,\widehat{\x}_n) f(\x_n|\{\y^{(j)}\}_{j=1}^J) d\x_n.$$ Comparing  and , provided that $f(\x_n|\q_n)$ converges to the posterior $f(\x_n|\{\y^{(j)}\}_{j=1}^J)$, we have $$\widetilde{\x}_n=\arg\min_{\widehat{\x}_n} \mathbb{E}[d_{\text{UD}}(\x_n,\widehat{\x}_n)|\q_n]\approx \widetilde{\x}_{\text{true},n},$$ which results in Conjecture \[conj:2\]. Example Metric-optimal Estimators and Performance Limits {#sec:example} ======================================================== In order to derive metric-optimal estimators, we need to know the scalar channel noise variance, $\Delta_v$. Below we obtain the optimal estimator, which is then used as part of Algorithm \[algo:metric\_opt\_MMV\] and to evaluate the performance limits of our metric-optimal algorithm. For i.i.d. matrices and AWGN channels , replica analysis in our previous work with Krzakala [@ZhuBaronKrzakala2017IEEE] yields the information-theoretic scalar channel noise variance $\Delta_v$ for message passing algorithms, which characterizes the posterior $f(\x_n|\q_n)$.[^8] Hence, we can obtain the information-theoretic optimal performance with arbitrary additive error metrics. For other types of matrices, our replica analysis [@ZhuBaronKrzakala2017IEEE] does not hold. Nevertheless, MP algorithms still yield a posterior $f(\x_n|\q_n)$, which we conjecture converges to the true posterior $f(\x_n|\{\y^{(j)}\}_{j=1}^J)$ (Conjecture \[conj:2\]). Hence, we can still assume that the scalar channel noise variance $\Delta_v$ is known. Based on the known variance $\Delta_v$, we build metric-optimal estimators  for two examples, mean weighted support set error (Section \[sec:weightedSupportSet\]) and mean absolute error (Section \[sec:MAE\]). Mean weighted support set error (MWSE) {#sec:weightedSupportSet} -------------------------------------- [**MWSE-optimal estimator:**]{} In support set estimation, the goal is to estimate the support of the signal, which is 1 if the corresponding entry in the signal is non-zero and 0 if it is zero. There are two types of errors in support set estimation: false alarms (support is 0, but estimated to be 1) and misses (support is 1, but estimated to be 0). In some applications such as medical imaging and radar detection, a miss may mean that the doctor misses an illness of the patient, or the radar misses an incoming missile. Hence, the cost paid for a miss could be tremendous compared to a false alarm. There are other applications where a false alarm is more costly than a miss. For example, in court, if an innocent person is mistakenly judged guilty, he/she will likely suffer a great deal. Therefore, we should weight these two errors differently in different applications. Let $b_n$ and $\widehat{b}_n$ be the true support and the estimated support of the $n$-th entry of the signal, respectively, and $\beta\in[0,1]$ is an application-dependent weight, which reflects the trade-off between the false alarms and misses. Hence, the MWSE given the pseudo data $\q_n$ is $$\label{eq:MWSE} \begin{split} &\text{MWSE}|\q_n=\mathbb{E}[d_{\text{WSE}}(b_n,\widehat{b}_n)|\q_n]=\\ &\left\{ \begin{array}{ll} (1-\beta)\Pr(b_n=1|\q_n),&\ \widehat{b}_n=0\ \text{and}\ b_n=1,\\ \beta\Pr(b_n=0|\q_n),&\ \widehat{b}_n=1\ \text{and}\ b_n=0,\\ 0,\quad\quad\quad&\ \widehat{b}_n=b_n, \end{array} \right.\\ \end{split}$$ where $\Pr(\cdot)$ denotes probability. The optimal estimate $\widetilde{b}_n$ minimizes $\mathbb{E}[d_{\text{WSE}}(b_n,\widehat{b}_n)|\q_n]$ , which implies $$\label{eq:weighted_optB} \widetilde{b}_n\!=\!\left\{ \begin{array}{ll} \!0,&\! (1-\beta)\Pr(b_n=1|\q_n)\!\leq\! \beta\Pr(b_n=0|\q_n),\\ \!1,&\! (1-\beta)\Pr(b_n=1|\q_n)\!>\! \beta \Pr(b_n=0|\q_n). \end{array} \right.\\$$ Since $f(\q_n|b_n=0)=(2\pi\Delta_v)^{-\frac{J}{2}}\exp\l(-\frac{\q_n\q_n^T}{2\Delta_v}\r)$ and $f(\q_n|b_n=1)=[2\pi(\Delta_v+1)]^{-\frac{J}{2}}\exp\l[-\frac{\q_n\q_n^T}{2(\Delta_v+1)}\r]$, we have $$\Pr(b_n=1|\q_n)=\frac{\frac{\rho}{[2\pi (1+\Delta_v)]^{\frac{J}{2}}}\e^{-\frac{\q_n\q_n^T}{2(1+\Delta_v)}}}{\frac{\rho}{[2\pi (1+\Delta_v)]^{\frac{J}{2}}}\e^{-\frac{\q_n\q_n^T}{2(1+\Delta_v)}}+\frac{(1-\rho)}{(2\pi \Delta_v)^{\frac{J}{2}}}\e^{-\frac{\q_n\q_n^T}{2\Delta_v}}},$$ and $\Pr(b_n=0|\q_n)=1-\Pr(b_n=1|\q_n)$. Plugging $\Pr(b_n=1|\q_n)$ and $\Pr(b_n=0|\q_n)$ into , we have $$\label{eq:b_opt} \widetilde{b}_n=\left\{ \begin{array}{ll} 0,&\quad \q_n\q_n^T\leq\tau,\\ 1,&\quad \q_n\q_n^T>\tau, \end{array} \right.\\$$ where $\tau=2\Delta_v(1+\Delta_v)\ln\l[\frac{\beta(1-\rho)}{(1-\beta)\rho}\l(\frac{1+\Delta_v}{\Delta_v}\r)^{\frac{J}{2}}\r]$, and we remind the reader that $\rho$ is the sparsity rate. [**Performance limits:**]{} Utilizing  and taking expectation over the pseudo data $\q_n$ for $\mathbb{E}[d_{\text{WSE}}(b_n,\widehat{b}_n)|\q_n]$ , we obtain the minimum MWSE (MMWSE), $$\label{eq:MMWSE_1} \begin{split} &\text{MMWSE}=\mathbb{E}[d_{\text{WSE}}(b_n,\widetilde{b}_n)]\\ &\quad =\int_{\q_n\q_n^T>\tau} \beta\Pr(b_n=0|\q_n)f(\q_n)d\q_n +\\ &\quad \quad \int_{\q_n\q_n^T\leq\tau} (1-\beta)\Pr(b_n=1|\q_n)f(\q_n)d\q_n. \end{split}$$ We have two integrals to simplify in , where we show the first below, and the second can be obtained similarly. To derive the first integral, note that $$\label{eq:oneIntegral} \small \int_{\q_n\q_n^T>\tau} \Pr(b_n\!=\!0|\q_n)f(\q_n)d\q_n\!=\!\Pr(\q_n\q_n^T\!>\!\tau, b_n\!=\!0).$$ Next, we calculate the pdf of the random variable (RV) $g_n=\frac{\q_n\q_n^T}{\Delta_v}$ given $b_n=0$. Because the entries of $\q_n$ are i.i.d. $\mathcal{N}(0,\Delta_v)$ given $b_n=0$, $g_n$ follows the Chi-square distribution, $f_G(g_n)=\frac{g_n^{\frac{J}{2}-1}\exp(-\frac{g_n}{2})}{2^{\frac{J}{2}}\Gamma(\frac{J}{2})}$, where $\Gamma(\cdot)$ is the Gamma function. Let $r_n=\Delta_v g_n=\q_n\q_n^T$, then we obtain $$f(r_n)=\frac{1}{\Delta_v}f_G(\frac{r_n}{\Delta_v})=\frac{r_n^{\frac{J}{2}-1}\exp\l[-\frac{r_n}{2\Delta_v}\r]}{(2\Delta_v)^{J/2}\Gamma(J/2)},$$ which helps to simplify . Therefore,  can be simplified, $$\label{eq:MMWSE_final} \begin{split} & \text{MMWSE}=\beta(1-\rho)\underbrace{\int_{r_n=\tau}^{\infty}\frac{r_n^{\frac{J}{2}-1}\exp\l[-\frac{r_n}{2\Delta_v}\r]}{(2\Delta_v)^{J/2}\Gamma(J/2)}dr_n}_{\Pr(\text{false alarm})}\\ & \quad +(1-\beta)\rho\underbrace{\int_{r_n=0}^{\tau}\frac{r_n^{\frac{J}{2}-1}\exp\l[-\frac{r_n}{2(1+\Delta_v)}\r]}{[2(1+\Delta_v)]^{J/2}\Gamma(J/2)}dr_n}_{\Pr(\text{miss})}. \end{split}$$ [**Hamming distance**]{}: In digital wireless communication systems, the signal only takes discrete values. A useful error metric is the (per-entry) Hamming distance [@Cover06], which equals 1 if the estimate of an entry of the signal differs from the true value. (Section \[sec:app\] will present an example in wireless communication that minimizes the Hamming distance.) The reader can verify that Hamming distance can be interpreted as a special case of weighted support set error, where $\beta=0.5$ provides equal weight to both errors . That said, we provide more insights about this particular case, which is ubiquitous in communication systems. For the jointly sparse model in , we define the Hamming distance as $$\label{eq:Hamming} d_{\text{HD}}(\x_n,\widehat{\x}_n)=\mathbbm{1}_{\x_n\neq \widehat{\x}_n},$$ where $\mathbbm{1}_{\mathcal{A}}$ is the indicator function. If ([*i*]{}) the pdf $\phi(\x_n)$ in  is a $J$-dimensional Dirac-delta function $\delta(\x_n-\mathbf{1})$, where $\mathbf{1}$ is an all-one row vector, and ([*ii*]{}) the estimate satisfies $\widehat{x}_n^{\{1\}}=\cdots =\widehat{x}_n^{\{J\}}, \forall n\in \{1,\ldots,N\}$, then the weighted support set error  with weight $\beta=0.5$ is equal to half of the Hamming distance  for super symbols. In –, we briefly derive the Hamming distance-optimal estimator when $\x_n\in \{\mathbf{1},\mathbf{0}\}$, where $\mathbf{0}$ is an all-zero row vector. The mean Hamming distance (MHD) given the pseudo data $\q_n$ is $$\label{eq:MHD} \begin{split} &\text{MHD}|\q_n=\mathbb{E}[d_{\text{MHD}}(\x_n,\widehat{\x}_n)|\q_n]=\\ &\left\{ \begin{array}{ll} \Pr(\x_n=\mathbf{1}|\q_n),&\ \widehat{\x}_n=\mathbf{0}\ \text{and}\ \x_n=\mathbf{1},\\ \Pr(\x_n=\mathbf{0}|\q_n),&\ \widehat{\x}_n=\mathbf{1}\ \text{and}\ \x_n=\mathbf{0},\\ 0,\quad\quad\quad&\ \widehat{\x}_n= \x_n. \end{array} \right.\\ \end{split}$$ Following the steps in –, the minimum MHD (MMHD) estimator is $$\label{eq:MHD_optTh} \widetilde{\x}_n=\left\{ \begin{array}{ll} \mathbf{1},& \sum_{j=1}^J q_n^{(j)} \geq\frac{J}{2}+\Delta_v \ln\l[\frac{1-\rho}{\rho}\r],\\ \mathbf{0},& \sum_{j=1}^J q_n^{(j)} <\frac{J}{2}+\Delta_v \ln\l[\frac{1-\rho}{\rho}\r]. \end{array} \right.\\$$ Mean absolute error (MAE) {#sec:MAE} ------------------------- [**MAE-optimal estimator:**]{} The element-wise absolute error (AE) is $$\label{eq:AE} d_{\text{AE}}(x_n^{(j)},\widehat{x}_n^{(j)})=|x_n^{(j)}-\widehat{x}_n^{(j)}|.$$ In order to find the minimum mean absolute error (MMAE) estimate, $\widetilde{\x}_n$, we need to find the stationary point of , $$\label{eq:optimalCondition} \frac{d\mathbb{E}[|x_n^{(j)}-\widehat{x}_n^{(j)}|\ | \q_n]}{d\widehat{x}_n^{(j)}}\bigg|_{\widehat{x}_n^{(j)}=\widetilde{x}_n^{(j)}}=0,$$ $\forall j\in\{1,\ldots,J\},n\in\{1,\ldots,N\}$. It can be proved that $\mathbb{E}[X]=\int_{0}^{\infty} \Pr(X>x)dx$, if $X\geq 0$. Therefore, $$\label{eq:expectation} \begin{split} &\mathbb{E}[|x_n^{(j)}-\widehat{x}_n^{(j)}|\ | \q_n]=\int_{0}^{\infty} \Pr(|x_n^{(j)}-\widehat{x}_n^{(j)}|>t| \q_n)dt\\ &=\int_{-\infty}^{\widehat{x}_n^{(j)}} \Pr(x_n^{(j)}<t| \q_n)dt+\int_{\widehat{x}_n^{(j)}}^{\infty} \Pr(x_n^{(j)}>t| \q_n)dt. \end{split}$$ Using  and , we obtain $$\Pr(x_n^{(j)}<\widetilde{x}_n^{(j)} | \q_n)=\Pr(x_n^{(j)}>\widetilde{x}_n^{(j)}| \q_n).$$ That is, $$\label{eq:optimalEst} \int_{-\infty}^{\widetilde{x}_n^{(j)}} f(x_n^{(j)} | \q_n)dx_n^{(j)}=\int_{\widetilde{x}_n^{(j)}}^{\infty} f(x_n^{(j)} | \q_n)dx_n^{(j)}=\frac{1}{2},$$ through which we solve for the optimal estimator $\widetilde{x}_n^{(j)}$ numerically. [**Performance limits:**]{} We calculate the MMAE as follows, $$\label{eq:MMAE} \begin{split} \text{MMAE}\!&=\!\mathbb{E}[|\widetilde{x}_n^{(j)}-x_n^{(j)}|]\!=\!\int_{-\infty}^{\infty}\! \mathbb{E}[|\widetilde{x}_n^{(j)}-x_n^{(j)}|\ | \q_n] f(\q_n)d \q_n\\ &=\int_{-\infty}^{\infty}\Bigg[\int_{-\infty}^{\widetilde{x}_n^{(j)}} -x_n^{(j)} f(x_n^{(j)}| \q_n)d x_n^{(j)}+\\ &\quad\quad\quad\quad \int_{\widetilde{x}_n^{(j)}}^{\infty} x_n^{(j)}f(x_n^{(j)}| \q_n)d x_n^{(j)}\Bigg] f(\q_n)d \q_n, \end{split}$$ which has to be numerically approximated. ![Comparison of simulation results to theoretic MMWSE for weighted support set estimation under different number of channels $J$ and measurement rates $R$ ($\beta=0.2$, noise variances $\Delta_z=0.01$).[]{data-label="fig:SimWSE"}](WeightSuppEstSimVsTheory.pdf){width="8cm"} Synthetic Simulations {#sec:numeric_synth} ===================== After deriving the minimum mean weighted support set error (MMWSE) and minimum mean absolute error (MMAE) estimators, this section provides numerical results for Algorithm \[algo:metric\_opt\_MMV\]. In the case of i.i.d. random matrices and AWGN channels , replica analysis yields the MMSE of the MMV problem [@ZhuBaronKrzakala2017IEEE]. By inverting the MMSE (details in Appendix \[app:inverMMSE\]), we obtain the scalar channel noise variance $\Delta_v$,[^9] which characterizes the posterior $f(\x_n|\q_n)$. Given $\Delta_v$, we characterize the MMWSE and MMAE theoretically. In the following simulations, we use i.i.d. Gaussian matrices with unit-norm rows, i.i.d. $J$-dimensional Bernoulli-Gaussian signals  with $J=1,3$, and 5, and sparsity rate $\rho=0.1$. The signal length is $N=10000$, and the measurement rate $R=\frac{M}{N}$ varies from $0.3$ to $0.7$. For each setting, the simulation results are averaged over 50 realizations of the problem. **Mean weighted support set error in AWGN channels:** We simulate AWGN channels  in this case with the noise variance being $\Delta_z\in \{0.01,0.001\}$. Fig. \[fig:SimWSE\] shows the weighted support set estimation results using our metric-optimal algorithm compared to the MMWSE . The red dashed curve, the blue dashed-dotted curve, and the black solid curve correspond to the MMWSE of $J=1,\ 3$, and 5, respectively. The red circles, blue crosses, and black triangles represent the simulation results. We can see that our simulation results match the theoretically optimal performance. **Remark:** The optimal weighted support set estimator  is not Lipschitz continuous. Hence, for $J=1$,  is not guaranteed to yield the MMWSE, according to Lemma \[lemma:optSMV\]. Nevertheless, numerical results for $J=1$ show that the MWSE given by  is close to the MMWSE. For weighted support set estimation, we further study the information-theoretic optimal receiver operating characteristic (ROC) curves, which are plotted in Fig. \[fig:ROC\] for different $J$’s. The red curves and black curves are plotted for noise variances $\Delta_z=0.001$ and $0.01$, respectively. The solid curves, the dashed curves, and the dotted curves represent $J=1,\ 3$, and 5, respectively. The true positive rate (TPR) and false positive rate (FPR) in both axes are defined as TPR$=\frac{\text{\# accurately predicted positives}}{\text{\# all positives in truth}}$ and FPR$=\frac{\text{\# wrongly predicted negatives}}{\text{\# all negatives in truth}}$. We can see that having more signal vectors $J$ leads to a larger area under the ROC curve, which indicates better trade-offs between true positives and false positives. ![Receiver operating characteristic curves for weighted support set estimation under different channel noise variances $\Delta_z$ and number of channels $J$ ($\beta=0.2$, measurement rate $R=0.3$).[]{data-label="fig:ROC"}](ROCweightSuppEst.pdf){width="8cm"} **Mean absolute error in logistic channels:** We simulate i.i.d. logistic channels  with parameters $a=10$ and $30$; the smaller $a$ is, the noisier the channel becomes. Fig. \[fig:MAEvsMMAE\] plots the simulated MAE (crosses) and the theoretic MMAE  (curves) for various settings.[^10] Different colors and line shapes refer to different $a$’s and $J$’s, respectively. We can see that our simulation results match the theoretically optimal performance. ![Comparison of simulation results to theoretic MMAE under different logistic channels , number of channels $J$, and measurement rates $R$.[]{data-label="fig:MAEvsMMAE"}](logitMAE.pdf){width="8cm"} **Remark:** Both simulations yield better performance for larger $J$. This is intuitive, because more signal vectors that share the same support should make the estimation process easier due to more information being available. ![Simulation results of OMP and Algorithm \[algo:metric\_opt\_MMV\] for active user detection in multi-user communication (SMV).[]{data-label="fig:OMP"}](OMP_vs_metric_opt.pdf){width="8.5cm"} Application {#sec:app} =========== In this section, we discuss active user detection (AUD) in a multi-user communication setting that can be viewed as compressed sensing [@DonohoCS; @CandesRUP; @BaraniukCS2007] (CS, which is closely related to SMV) in some scenarios. Next, we simulate AUD using our metric-optimal algorithm. Finally, we discuss how to solve the AUD problem using an MMV setting. [**CS based active user detection:**]{} One application of MMV is AUD in a massive random access (MRA) scenario for multi-user communication [@FletcherRanganGoyal2009; @Boljanovic2017]. In the MRA scenario, multiple end users (EUs) are requesting access to the network simultaneously by sending their unique identification codewords, $\a_n\in \{-1,+1\}^{M\times 1},\ n\in \{1,\ldots,N\}$, to the base station, where $n$ denotes the user id, and each user’s codeword is known by the base station. The base station needs to determine which EUs are requesting access to the network (active) and which are not (inactive), so that it can allocate resources to the active EUs. Fletcher et al. [@FletcherRanganGoyal2009] proposed a CS based AUD scheme for MRA, which was recently revisited by Boljanovi[ć]{} et al. [@Boljanovic2017]. In their setup [@FletcherRanganGoyal2009; @Boljanovic2017], all EUs are synchronized, i.e., all active EUs send each entry of their codewords to the base station simultaneously in one time slot. Denote the status of the $n$-th EU by $x_n\in \{0,1\}$, where $x_n=1$ means active and $x_n=0$ is inactive. Denote the received signal vector at the base station by $\y\in \mathbb{R}^{M\times 1}$. Since all EUs are synchronized, we can express the received signal $\y$ by $$\label{eq:MRA_SMV} \y=\A\x+\z,$$ where $\A=[\a_1,\ldots,\a_N]\in \{0,1\}^{M\times N}$, and $\z$ is AWGN. The base station estimates $\x$ to determine which EUs are active. In the following, we first apply a mean Hamming distance (MHD) optimal algorithm to estimate $\x$ from , and compare to the algorithm of Boljanovi[ć]{} et al. [@Boljanovic2017], which is orthogonal match pursuit (OMP) [@Pati1993]. Next, we propose an MMV based scheme for AUD in MRA. [**Simulation with MHD-optimal algorithm:**]{} As the reader can see, the CS based active user detection [@FletcherRanganGoyal2009; @Boljanovic2017] is an SMV problem , which is MMV for $J=1$ . Later in this section we will extend the scheme by Boljanovi[ć]{} et al. to an MMV setting, and so we keep using MMV notations. Note that the entries of the measurement matrix in this AUD problem take values of $\pm 1$. Because the derivation of Algorithm \[algo:AMP\_MMV\] assumes that entries of $\A^{(j)}$ scale with $\frac{1}{\sqrt{N}}$,[^11] we scale $\A^{(j)}$ down by $\sqrt{N}$ using a modified $\widetilde{\y}^{(j)}=\frac{\y^{(j)}}{\sqrt{N}}$ . Following the discussion above, we simulate the settings of measurement rate $R\in \{0.2,0.25,\ldots,0.6\}$ and noise variance $\Delta_z\in \{10^{-1},10^{-1.5},10^{-2}\}$. For each setting, we randomly generate 50 realizations of the Bernoulli signal $\x^{(1)}\in \{0,1\}^{N\times 1}$ with sparsity rate $\rho=0.1$, and measurement matrix $\A^{(1)}\in \l\{-\frac{1}{\sqrt{N}},+\frac{1}{\sqrt{N}}\r\}^{M\times N}$, where $N=10000$. We run Algorithm \[algo:metric\_opt\_MMV\] with $J=1$ to estimate the underlying signal $\x^{(1)}$. Note that $f_{a_n}(\Delta_v,\q_n)$ and $f_{v_n}(\Delta_v,\q_n)$ in Lines \[line:mean\]–\[line:var\] of Algorithm \[algo:AMP\_MMV\], which consists of Part 2 of Algorithm \[algo:metric\_opt\_MMV\], are given by the following, $$\begin{split} f_{a_n}(\Delta_v,\q_n)&=\frac{\rho}{\rho+(1-\rho)\exp\l[-\frac{\sum_{j=1}^J q_n^{(j)} -\frac{J}{2}}{\Delta_v}\r]}\mathbf{1},\\ f_{v_n}(\Delta_v,\q_n)&=f_{a_n}(\Delta_v,\q_n)-f_{a_n}(\Delta_v,\q_n)^2, \end{split}$$ where the power-of-two in the last term of $f_{v_n}(\cdot,\cdot)$ is applied element-wise. Our results are compared to OMP in Fig. \[fig:OMP\]. The solid, dashed, and dash-dotted curves represent noise variance $\Delta_z=10^{-1},\ 10^{-1.5},\ 10^{-2}$, respectively. The black curves are the OMP results and the red curves with circle markers are the results of Algorithm \[algo:metric\_opt\_MMV\] when optimizing for MHD. We can see that our algorithm consistently outperforms OMP.[^12] [**MMV scheme for active user detection:**]{} Reminiscing on Section \[sec:numeric\_synth\] and our previous work with Krzakala on MMV [@ZhuBaronKrzakala2017IEEE], more measurement vectors (larger $J$) lead to better estimation quality. We propose to convert the SMV style of the AUD problem into an MMV style by having each EU send $J$ different identification codewords, $\a_n^{(j)}\in \{-1,+1\}^{M\times 1}, \forall j\in \{1,\ldots,J\}$, to the base station. However, since the underlying signal $\x$  is Bernoulli and does not change during the AUD period, the resulting MMV is equivalent to an SMV with $J$ times more measurements; the column $\a_n$ of the measurement matrix in the equivalent SMV scheme is $\a_n = \l[\l(\a_n^{(1)}\r)^T,\cdots, \l(\a_n^{(J)}\r)^T\r]^T$. Hence, Algorithm \[algo:metric\_opt\_MMV\] should yield the same detection accuracy for the MMV scheme with $J$ channels and the SMV scheme with $J$ times larger measurement rate. Nevertheless, there is one advantage in adopting the MMV scheme: Lines \[line:beginForLoop\]–\[line:endForLoop\] of Algorithm \[algo:AMP\_MMV\][^13] can be parallelized with $J$ processing units (for example using a general purpose graphics processing unit or multicore computing system). After parallelizing Algorithm \[algo:AMP\_MMV\], the base station can perform the detection procedure with less runtime. $$\label{eq:Ez} \mathbb{E}[w|k,y,\Theta]=\left\{ \begin{array}{ll} \displaystyle \frac{1}{\widetilde{Z}}\int \frac{w}{1+\operatorname{e}^{-aw}} \frac{\operatorname{e}^{-\frac{1}{2\Theta}(w-k)^2}}{\sqrt{2\pi\Theta}} dz=\frac{\sum_{u=1}^{u_{\text{max}}} \alpha_u T_1(u)}{\sum_{u=1}^{u_{\text{max}}} \alpha_u T_0(u)}=k+\frac{\Theta\sum_{u=1}^{u_{\text{max}}} \frac{\alpha_u \phi(\eta_u)}{\sqrt{\l(\frac{\sigma_u}{a}\r)^2+\Theta}}}{\sum_{u=1}^{u_{\text{max}}} \alpha_u \Phi(\eta_u)},\ &y=1,\\ \displaystyle \frac{1}{\widetilde{Z}}\int \frac{w\operatorname{e}^{-aw}}{1+\operatorname{e}^{-aw}} \frac{\operatorname{e}^{-\frac{1}{2\Theta}(w-k)^2}}{\sqrt{2\pi\Theta}} dz=\frac{k-\sum_{u=1}^{u_{\text{max}}} \alpha_u T_1(u)}{1-\sum_{u=1}^{u_{\text{max}}} \alpha_u T_0(u)}=k-\frac{\Theta\sum_{u=1}^{u_{\text{max}}} \frac{\alpha_u \phi(\eta_u)}{\sqrt{\l(\frac{\sigma_u}{a}\r)^2+\Theta}}}{1-\sum_{u=1}^{u_{\text{max}}} \alpha_u \Phi(\eta_u)},\ &y=0. \end{array} \right.\\$$ $$\label{eq:mmse_p1} \begin{split} \mathbb{E}[(\mathbb{E}[\x_n|\q_n])^2]&=\int_{\q_n} f(\q_n) (\mathbb{E}[\x_n|\q_n])^2 d\q_n\\ &=\frac{\l(\frac{\rho}{1+\Delta_v}\r)^2}{\l[2\pi (1+\Delta_v)\r]^{J/2}}\int_{\q_n} \frac{\q_n \q_n^T}{\rho \exp\l[\frac{\q_n\q_n^T}{2(1+\Delta_v)}\r]+(1-\rho)\l(1+\frac{1}{\Delta_v}\r)^{J/2} \exp\l[\frac{\q_n\q_n^T (\Delta_v-1)}{2\Delta_v(1+\Delta_v)}\r]} d\q_n. \end{split}$$ Conclusion {#sec:conclusion} ========== In this paper, we studied the MMV signal estimation problem with user-defined additive error metrics on the estimate. We proposed an algorithmic framework that is optimal under arbitrary additive error metrics. We showed the optimality of our algorithm under certain conditions for SMV and conjectured its optimality for MMV. As examples, we derived algorithms that yield the optimal estimates in the sense of mean weighted support set error and mean absolute error, respectively. Numerical results not only verified the theoretic performance but also verified the intuition that having more signal vectors in MMV problems is beneficial to the estimation algorithm. We further provided simulation results for active user detection problem in multi-user communication systems, which is a real-world application of MMV models with the goal of minimizing the Hamming distance. Simulation results demonstrated the promise of our algorithm. Derivation of $g_{\text{out}}$ for logistic channels  {#app:logit} ----------------------------------------------------- Byrne and Schniter [@ByrneSchniter2015ArXiv] provide a method to derive $g_{\text{out}}$ for logistic channels , but the actual formula for $g_{\text{out}}$ is not given in their paper. To make our paper self-contained, we outline the derivation of $g_{\text{out}}$ for logistic channels. In order to calculate $g_{out}(k,y,\Theta)$ , we need to find $f(w|k,y,\Theta)$  and calculate $\mathbb{E}[w|k,y,\Theta]$. For logistic channels , $$\label{eq:logit_cond} \begin{split} & f(w|k,y,\Theta)=\\ &\frac{1}{\widetilde{Z}}\times \l[\frac{\delta(y-1)}{1+\operatorname{e}^{-w}}+\delta(y)\frac{\operatorname{e}^{-w}}{1+\operatorname{e}^{-w}}\r]\frac{1}{\sqrt{2\pi\Theta}} \operatorname{e}^{-\frac{1}{2\Theta}(w-k)^2}, \end{split}$$ where $\widetilde{Z}$ is a normalization factor. Therefore, it is difficult to calculate $\mathbb{E}[w|k,y,\Theta]$. Instead of calculating $\mathbb{E}[w|k,y,\Theta]$ by brute force, Byrne and Schniter [@ByrneSchniter2015ArXiv] use a mixture of Guassian cumulative distribution functions (CDF’s) to approximate the sigmoid function $\frac{1}{1+\text{exp}(-aw)} \approx\sum_{u=1}^{u_{\text{max}}} \alpha_u \Phi(\frac{w}{\sigma_u/a})$ [@Stefanski1991], where $u_{\text{max}}$ is the maximum number of Gaussian CDF’s one wants to use, $\Phi(\frac{w}{\sigma_u/a})$ denotes the Gaussian CDF whose standard deviation is $\frac{\sigma_u}{a}$, and $\alpha_u$ is the weight. Following Byrne and Schniter [@ByrneSchniter2015ArXiv], we define the $i$-th moment $$\int w^i \mathcal{N}(w;k,\Theta) \Phi\l(\frac{w}{\sigma_u/a}\r)dz=T_i(u),$$ where $\mathcal{N}(w;k,\Theta)$ is the pdf of an RV $w$ with mean $k$ and variance $\Theta$, and $\Phi(\cdot)$ is the CDF of a standard Gaussian RV. Defining $\eta_u=\frac{k}{\sqrt{\l(\frac{\sigma_u}{a}\r)^2+\Theta}}$, we obtain $$\begin{aligned} T_0(u)&=&\Phi(\eta_u),\nonumber\\ T_1(u)&= &k\Phi(\eta_u)+\frac{\Theta\phi(\eta_u)}{\sqrt{\l(\frac{\sigma_u}{a}\r)^2+\Theta}},\nonumber\end{aligned}$$ $$\begin{aligned} T_2(u)=\frac{(T_1(u))^2}{\Phi(\eta_u)}+\Theta \Phi(\eta_u)-\frac{\Theta^2\phi(\eta_u)}{\l(\frac{\sigma_u}{a}\r)^2+\Theta}\l(\eta_u+\frac{\phi(\eta_u)}{\Phi(\eta_u)}\r),\nonumber\end{aligned}$$ where $\phi(\eta_u)$ is the pdf of a standard Gaussian RV at $\eta_u$. Hence, the normalization factor $\widetilde{Z}$ in  can be derived, $$\begin{split} \widetilde{Z}=& \int \l[\frac{\delta(y-1)}{1+\operatorname{e}^{-aw}}+\delta(y)\frac{\operatorname{e}^{-aw}}{1+\operatorname{e}^{-aw}}\r]\frac{\operatorname{e}^{-\frac{1}{2\Theta}(w-k)^2}}{\sqrt{2\pi\Theta}} dz\\ = &\left\{ \begin{array}{ll} \sum_{u=1}^{u_{\text{max}}} \alpha_u \Phi(\eta_u),\ &y=1,\\ 1-\sum_{u=1}^{u_{\text{max}}} \alpha_u \Phi(\eta_u),\ &y=0. \end{array} \right.\\ \end{split}$$ We can further obtain the expression for $\mathbb{E}[w|k,y,\Theta]$ in . Hence, we can calculate $g_{\text{out}}$ . Apart from $g_{\text{out}}$, we also need to find the partial derivative of $g_{\text{out}}$ , which according to Rangan [@RanganGAMP2011ISIT] satisfies $$\label{eq:deri_g_out} -\frac{\partial}{\partial k}g_{\text{out}}(k,y,\Theta)=\frac{1}{\Theta}\l(1-\frac{\text{var}(w|k,y,\Theta)}{\Theta}\r),$$ where $\text{var}(w|k,y,\Theta)=\mathbb{E}[w^2|k,y,\Theta]-[\mathbb{E}[w|k,y,\Theta]]^2$. Note that $\mathbb{E}[w^2|k,y,\Theta]$ can be derived in the same way as  and the result is given below, $$\mathbb{E}[w^2|k,y,\Theta]=\left\{ \begin{array}{ll} \displaystyle k^2+\Theta+\frac{ \displaystyle\sum_{u=1}^{u_{\text{max}}} \alpha_u \xi_u}{\displaystyle\sum_{u=1}^{u_{\text{max}}} \alpha_u \Phi(\eta_u)},\ &y=1,\\ \displaystyle k^2+\Theta-\frac{\displaystyle\sum_{u=1}^{u_{\text{max}}} \alpha_u \xi_u}{\displaystyle 1-\sum_{u=1}^{u_{\text{max}}} \alpha_u \Phi(\eta_u)},\ &y=0, \end{array} \right.\\$$ where $$\xi_u=\frac{2k\Theta\phi(\eta_u)}{\sqrt{\l(\frac{\sigma_u}{a}\r)^2+\Theta}}-\frac{\Theta^2 \eta_u\phi(\eta_u)}{\l(\frac{\sigma_u}{a}\r)^2+\Theta}.$$ Inverting the MMSE {#app:inverMMSE} ------------------ For the MMV problem with i.i.d. matrices and joint Bernoulli-Gaussian signals, Zhu et al. provide an information-theoretic characterization of the MMSE by using replica analysis [@ZhuBaronKrzakala2017IEEE]. Suppose that we have already obtained the MMSE for an MMV problem. This appendix briefly shows how to invert the MMSE expression in order to obtain the equivalent scalar channel noise variance $\Delta_v$. The optimal denoiser for the pseudo data is $\widetilde{\x}_n=\mathbb{E}[\x_n|\q_n]=f_{a_n}(\Delta_v,\q_n)$, where $f_{a_n}(\Delta_v,\q_n)$ is given in . We then express MMSE expression using $\mathbb{E}[\x_n|\q_n]$ as follows, $$\text{MMSE}=\mathbb{E}[(\widetilde{\x}_n-\x_n)^2]=J\rho-\mathbb{E}[(\mathbb{E}[\x_n|\q_n])^2].\label{eq:mmse1}$$ We calculate $\mathbb{E}[(\mathbb{E}[\x_n|\q_n])^2]$ in , where the $J$-dimensional integral can be simplified by a change of coordinates. Then, we plug  into , and express the MMSE as a function of $\Delta_v$. Finally, we numerically solve $\Delta_v$ for any given MMSE. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Jin Tan for providing valuable insights into achieving metric-optimal performance in signal estimation. 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[^2]: Junan Zhu is with Bloomberg L.P., New York, NY 10017, and Dror Baron is with the Department of Electrical and Computer Engineering, NC State University, Raleigh, NC 27695. E-mail: {jzhu9, barondror}@ncsu.edu. [^3]: In SMV, our algorithm closely resembles the one proposed by Tan and coauthors [@Tan2012SSP; @Tan2014], with the difference being Tan and coauthors rely on relaxed belief propagation (relaxed BP, an MP algorithm) [@Rangan2010CISS] and do not provide rigorous proofs for the optimality of their algorithm. [^4]: Our proof is based on the approximate message passing algorithm [@DMM2009], while Tan and coauthors [@Tan2012SSP; @Tan2014] rely on relaxed BP and do not provide rigorous proofs. [^5]: Byrne and Schniter [@ByrneSchniter2015ArXiv] describe without detail how to derive $g_{\text{out}}(\cdot)$  for i.i.d. parallel logistic channels ; we present a detailed derivation for completeness in Appendix \[app:logit\], and do not claim it as a contribution. [^6]: Free energy is a term brought from statistical physics and is used to describe the interaction between the signals and the measurement matrices in linear models [@ZhuBaronCISS2013; @ZhuBaronKrzakala2017IEEE; @Krzakala2012probabilistic]. When the free energy has two fixed points, (G)AMP is not optimal with i.i.d. Gaussian matrices, and neither is Algorithm \[algo:metric\_opt\_MMV\]. We refer interested readers to the literature for detailed discussions [@ZhuBaronCISS2013; @ZhuBaronKrzakala2017IEEE; @Krzakala2012probabilistic]. [^7]: Pseudo-Lipschitz is a concept discussed in Bayati and Montanari [@Bayati2011]: For $k\geq 1$, we say a function $\phi: \mathbb{R}^m \rightarrow \mathbb{R}$ is [*pseudo-Lipschitz*]{} of order $k$ if there exists a constant $L>0$ such that for all $\x,\y\in \mathbb{R}^m$: $|\phi(\x)-\phi(\y)| \leq L(1+\|\x\|^{k-1}+\|\y\|^{k-1})\|\x-\y\|$. [^8]: Our work with Krzakala [@ZhuBaronKrzakala2017IEEE] focuses on a diagonal covariance matrix for $\x_n$ in . A recent work by Hannak et al. [@Hannak2017] extends our work [@ZhuBaronKrzakala2017IEEE] to non-diagonal covariance matrices for $\x_n$. Following Hannak et al. [@Hannak2017], we can extend the performance limits analysis in this paper to non-diagonal covariance matrices for $\x_n$. [^9]: Algorithm \[algo:metric\_opt\_MMV\] also applies to problems with other types of matrices, as long as the entries in the measurement matrices scale with $\frac{1}{\sqrt{N}}$. However, when the matrices are not i.i.d., there is no easy way to [*theoretically*]{} characterize the MMSE, the equivalent scalar channel noise variance $\Delta_v$, and the metric-optimal error. Such a theoretic characterization is sometimes necessary, because the MMSE behaves differently under different noise variances $\Delta_z$  and measurement rates $R$ [@ZhuBaronCISS2013; @ZhuBaronKrzakala2017IEEE]. [^10]: We do not have a replica analysis for logistic channels. In order to compute the theoretic MMAE, we use the average $\Delta_v$ from all the 50 simulations for each setting and calculate the MMAE with . [^11]: Details can be found in Krzakala et al. [@Krzakala2012probabilistic] and Barbier and Krzakala [@BarbierKrzakala2017IT]. [^12]: Note that the entries of the signal estimated by OMP are not exactly 0’s and 1’s. Hence, in order to provide meaningful results, we threshold the OMP estimates before calculating the Hamming distance. [^13]: Recall that Algorithm \[algo:metric\_opt\_MMV\] runs Algorithm \[algo:AMP\_MMV\] as a subroutine.

--- abstract: 'We consider an optimization problem for spatial power distribution generated by an array of transmitting elements. Using ultrasound hyperthermia cancer treatment as a motivating example, the signal design problem consists of optimizing the power distribution across the tumor and healthy tissue regions, respectively. The models used in the optimization problem are, however, invariably subject to errors. deposition as well as inefficient treatment. To combat such unknown model errors, we formulate a robust signal design framework that can take the uncertainty into account using a worst-case approach. This leads to a semi-infinite programming (SIP) robust design problem which we reformulate as a tractable convex problem, potentially has a wider range of applications.' author: - 'Nafiseh Shariati, Dave Zachariah, Johan Karlsson, Mats Bengtsson' bibliography: - 'bibliokthNafis.bib' title: Robust Optimal Power Distribution for Hyperthermia Cancer Treatment --- Introduction {#sec:introduction} ============ Local hyperthermia is a noninvasive technique for cancer treatment in which targeted body tissue is exposed to high temperatures to damage cancer cells while leaving surrounding tissue unharmed. This technique is used both to kill off cancer cells in tumors and as a means to enhance other treatments such as radiotherapy and chemotherapy. Hyperthermia has the potential to treat many types of cancer, including sarcoma, melanoma, and cancers of the head and neck, brain, lung, esophagus, breast, bladder, rectum, liver, appendix, cervix, etc. . Hyperthermia treatment planning involves modeling patient-specific tissue, using medical imaging techniques such as microwave, ultrasound, magnetic resonance or computed tomography, and calculating the spatial distribution of power deposited in the tissue to heat it [@13:; @Paulides]. There exist two major techniques to concentrate the power in a well-defined tumor region: electromagnetic and ultrasound, each with its own limitations. The drawback of electromagnetic microwaves is poor penetration in biological tissue, while for ultrasound the short acoustic wavelength renders the focal spot very small. Using signal design methods, however, one can improve the spatial power deposition generated by an array of acoustic transducers. Specifically, standard phased array techniques do not make use of combining a diversity of signals transmitted at each transducer. When this diversity exploited it is possible to dramatically improve the power distribution in the tumor tissue, thus improving the effectiveness of the method and reducing treatment time . Given a set of spatial coordinates that describe the tumor region and the healthy tissue, respectively, the transmitted waveforms can be designed to optimize the spatial power distribution while subject to certain design constraints. One critical limitation, however, is the assumption of an ideal wave propagation model from the transducers to a given point in the tissue. Specifically, model mismatches may arise from hardware imperfections, tissue inhomogeneities, inaccurately specified propagation velocities, etc. Thus the actual power distribution may differ substantially from the ideal one designed by an assumed model. This results in suboptimal clinical outcome due to loss of power in the tumor region and safety issues due to the possible damage of healthy tissue. These considerations motivate developing robust design schemes that take such unknown errors into account. In this paper we derive a robust optimization method that only assumes the unknown model errors to be bounded. The power is then optimized with respect to ‘worst-case’ model errors. By using a worst-case model, we provide an optimal signal design scheme that takes into account all possible, bounded model errors. Such a conservative approach is warranted in signal design for medical applications due to safety and health considerations. Our method further generalizes the approach in [@08:Guo] by obviating the need to specify a fictitious tumor center point. The framework developed here has potential use in wider signal design applications where the resulting transmit power distributions are subject to model inaccuracies. More specifically, the design problem formulated in this paper and the proposed robust scheme can be exploited to robustify the spatial power distribution for applications that an array equipped with multiple elements is used to emit waveforms in order to deliver power to an area of interest in a controlled manner. The core of this study is built upon exploiting waveform diversity which has been introduced in multiple-input multiple-output (MIMO) radar literature [@Stoica2007a], and later has been applied for local hyperthermia cancer treatment improvement in [@08:Guo]. In the MIMO radar field, robustness studies have been carried out in different applications under varying design parameter uncertainties, cf., [@07:Yang; @11:Grossi]. Recently, in [@14:Shariati2014c], we have studied the robustification of the waveform diversity methodology for MIMO radar applications. It should be highlighted that in this paper a more generic problem formulation has been studied with respect to those of [@14:Shariati2014c], where a new application area is considered to illustrate the performance of our proposed robust design. In the array processing literature, beamforming under array model errors has also spawned extensive work, cf., [@04:Li; @08:Yan; @08:Kim; @12:Khabbazibasmenj]. For hyperthermia therapy, the need for robust solutions when optimizing for phase and amplitude of conventional phased array, has been investigated in [@deGreef2010a] considering perfusion uncertainties, and in [@deGreef2011a] considering dielectric uncertainties. The authors emphasize on the role of uncertainty in such designs (hyperthermia planning) since it influences the calculation of power distribution, and correspondingly temperature distribution. The paper is organized as follows: In Section \[sec:system model\], we describe the system model and the relevant variables. In Section \[sec:problem formulation\], the signal design problem is presented. First, we consider the state-of-the-art method based on ‘waveform diversity’ [@08:Guo; @Stoica2007a; @JLi2009a], then we generalize the design problem by introducing a deterministic and bounded set of possible model errors which results in an infinite number of constraints. Importantly we show that this seemingly intractable problem can be equivalently formulated as a tractable convex optimization problem. In Section \[sec:numerical\_results\], we evaluate the design scheme. We evaluate the performance of our proposed robust power distribution scheme specifically for local hyperthermia breast cancer treatment. This example application is motivated by the alarming statistics pointing to breast cancer as one of the leading causes of death among women worldwide [@CancerReport2014UK; @CancerReport2014US; @CancerReport2013French][^1]. The case of no model mismatch is investigated first, and then the robust design scheme is applied where its power distribution in the worst-case model is evaluated and compared to the nonrobust formulation. *Notation:* Boldface (lower case) is used for column vectors, $\mathbf{x}$, and (upper case) for matrices, $\mathbf{X}$. $ \| \a \|_{\mathbf{W}} \triangleq \sqrt{\a^H {\mathbf{W}} \a}$ where ${\mathbf{W}} \succ {\mathbf{0}}$. ${\mathbf{x}}^T$ and ${\mathbf{x}}^H$ denote transpose and Hermitian transpose. ${\mathbf{R}} \succeq {\mathbf{0}}$ signifies positive semidefinite matrix and ${\mathbf{R}}^{1/2}$ a matrix square-root, e.g., Hermitian. The set of complex numbers is denoted by $\mathcal{C}$. *Abbreviations:* Semi-infinite programming (SIP); multiple-input multiple-output (MIMO); Semidefinite program (SDP); linear matrix inequality (LMI). system model {#sec:system model} ============ We consider an array of $M$ acoustic transducers to heat target points. These transducers are located at known positions $\th_m$, for $m = 1,2,\dots,M$, around the tissue at risk, cf., [@08:Guo; @14:Shariati2014c]. We parameterize an arbitrary point in 3D space using Cartesian coordinates $\mathbf{r} = [x \: y \: z]^T$. Let $x_m(n)$ denote the baseband representation of narrowband discrete-time signal transmitted at the $m$th transducer, at sample $n = 1, \dots, N$. Then the baseband signal received at a generic location $\r$ equals the superposition of signals from all $M$ transducers, i.e., $$\label{eq:baseband_recieved} \begin{aligned} y(\r,n) &= \sum_{m=1}^M a_m(\r) x_m(n) , \hspace{.25cm} n = 1, \ldots, N& \\ &= \a^H(\r) \x(n) , \hspace{.25cm} n = 1, \ldots, N,& \end{aligned}$$ where the $m$th signal is attenuated by a factor $a_m({\mathbf{r}})$ which depends on the properties of the transducers, the carrier wave and the tissue. This factor is modeled as $$\label{eq:a_m(r)} a_m(\r) = \frac{e^{-j2\pi f_c \tau_m(\r)}}{\|\th_m - \r\|^{\frac{1}{2}}},$$ where $f_c$ is the carrier frequency, and $\tau_m(\r) = \frac{\|\th_m - \r\|}{c}$ is the required time for any signal to arrive at location $\r$ where $c$ is the sound speed inside the tissue. Note that the root-squared term in the denominator in represents the distance dependent propagation attenuation of the acoustic waveforms. In , the narrowband signals are represented in vector form $\mathbf{x}(n) = [x_1(n) \: \ldots \: x_m(n) \: \ldots \: x_M(n)]^T \in \mathcal{C}^{M \times 1}$ and $\a({\mathbf{r}}) \triangleq [a_1(\r) \: \ldots \: a_m(\r) \: \ldots a_M(\r)]^T \in \mathcal{C}^{M \times 1}$ is the array steering vector as a function of $\r$. At a generic location $\r$ in the tissue, the power of the transmitted signal, i.e., *the transmit beampattern*, is given by $$\label{eq:transmit_beampattren} p(\r) = \mathbb{E} \{ | y(\r,n) |^2 \} = \a^H(\r)\R \a(\r),$$ where $$\R \triangleq \mathbb{E} \{ \x(n) \x^H(n)\}$$ is the $M \times M$ covariance matrix of the signal $\x(n)$. As equation suggests, the transmit beampattern is dependent on the waveform covariance matrix $\R$ and the array steering vector $\a(\r)$. In the following we analyze how one can form and control the beampattern by optimizing the covariance matrix $\R$, so as to heat up the tumor region of the tissue while keeping the power deposition in the healthy tissue minimal. In this work, we consider schemes which allow for the lowest possible power leakage to the healthy area. Once an optimal covariance matrix $\R$ has been determined, the waveform signal $\x(n)$ can be synthesized accordingly. One simple approach is $\x(n) = \R^{1/2} \mathbf{w}(n)$, where $\mathbf{w}(n)$ is a sequence of independent random vectors with mean zero and covariance matrix $\mathbf{I}$. For detailed discussion see [@Stoica2007b; @08:Fuhrmann][@12:He ch. 14]. A significant challenge to this approach, however, is that the *true* steering vector $\a(\r)$ in does not exactly match the model in for a host of reasons: array calibration imperfections, variations in transducing elements, tissue inhomogeneities, inaccurately specified propagation velocity, etc. We will therefore consider the aforementioned design problem subject to model uncertainties in the array steering vector at any given point $\r$. We refer to this approach as robust waveform diversity. problem formulation {#sec:problem formulation} =================== The waveform-diversity-based technique [@04:Fuhrmann; @Stoica2007a; @Stoica2007b; @08:Guo; @08:Fuhrmann; @14:Shariati2014c] have been used for designing beampatterns subject to practical constraints. In general, we aim to control and shape the spatial power distribution at a set of target points while simultaneously minimizing power leakage in the remaining area. By exploiting a combination of different waveforms in , the degrees of freedom increase for optimizing the beampattern under constraints. After reviewing the standard waveform diversity approach, we focus on the practical scenario where the assumed array steering vector model is subject to perturbations. In the subsequent section, the proposed robust technique is evaluated by numerical simulations, comparing the performance with and without robustified solution under perturbed steering vectors. Waveform Diversity based Ultrasound System ------------------------------------------ In the MIMO radar literature, sidelobe minimization is a beampattern design problem that has been addressed by using the waveform diversity methodology, cf., [@04:Fuhrmann; @Stoica2007a; @Stoica2007b; @08:Fuhrmann]. This design problem can be thought of as an optimization problem where the probing waveforms covariance matrix $\R$ is the optimization variable to be chosen under positive semi-definiteness assumption and with a constraint on the total power. The waveform-diversity-based scheme for ultrasound system has been introduced and explained in detail in [@08:Guo] based on the transmit beampattern design technique for MIMO radar systems [@04:Fuhrmann; @Stoica2007a]. In the following we consider the practical power constraint where all array elements have the same power. Therefore, the covariance matrix $\R$ belongs to the following set $\mathcal{R}$: $$\mathcal{R} \triangleq \{\mathbf{R} \hspace{.1cm}|\hspace{.1cm} \mathbf{R}\succeq {\mathbf{0}} , R_{mm} = \frac{\gamma}{M}, m=1,2,...,M \},$$ where $\gamma$ is the total transmitted power and $R_{mm}$ is the $m$th diagonal element of $\R$ corresponding to the power emitted by $m$th transducer. The healthy tissue and the tumor regions are represented by two sets of discrete control points $\r$: $$\begin{split} \Omega_S &= \{ \r_1,\r_2,\ldots,\r_{N_S}\} \\ \Omega_T &= \{ \r_1,\r_2,\ldots,\r_{N_T} \}, \end{split}$$ where $N_S$ and $N_T$ denote the number of points in the healthy tissue region and the tumor regions, respectively. Without loss of generality, let $\r_0$ be a representative point which is taken to be the center of the tumor region $\Omega_T$. The objectives for this optimization problem can be summarized as follows: Design the waveform covariance matrix ${\mathbf{R}}$ so as to - maximize the gap between the power at the tumor center $\r_0$ and the power at the control points $\r$ in the healthy tissue region $\Omega_S$; - while guaranteeing a certain power level for control points $\r$ in the tumor region $\Omega_T$. Mathematically, this problem is formulated as (see [@08:Guo]) $$\label{eq:sidelobe min} \begin{aligned} \underset{\R,t}{\textrm{max}} \hspace{.5cm} &t& \\ \textrm{s.t.} \hspace{.5cm} &\a^H(\r_0) \R \a(\r_0) - \a^H(\r) \R \a(\r) \geq t, \forall \r \in \Omega_S &\\ & \a^H(\r) \R \a(\r) \geq (1 - \delta) \a^H(\r_0) \R \a(\r_0), \forall \r \in \Omega_T& \\ & \a^H(\r) \R \a(\r) \leq (1 + \delta) \a^H(\r_0) \R \a(\r_0), \forall \r \in \Omega_T&\\ &\R \in \mathcal{R}& \end{aligned}$$ where $t$ denotes the gap between the power at $\r_0$ and the power at the control points $\r$ in the healthy region $\Omega_S$. The parameter $\delta$ is introduced here to control the required certain power level at the control points in the tumor region. For instance, if we set $\delta = 0.1$, then we aim for having power at the tumor region $\Omega_T$ to be within 10% of $p(\r_0)$, i.e., the power at the tumor center. This is an SDP problem which can be solved efficiently in polynomial time using any SDP solver, e.g., `CVX` [@cvx2013; @gb08]. Robust Waveform Diversity based Ultrasound System ------------------------------------------------- The convex optimization problem and consequently its optimal solution, i.e., the optimal covariance matrix ${\mathbf{R}}$, are functions of the steering vectors $\a(\r)$. In practice, however, the assumed steering vector model used to optimize ${\mathbf{R}}$ is inaccurate. Hence using *nominal* steering vectors $\ahat(\r)$ based on an ideal model, in lieu of the unknown *true* steering vectors $\a(\r)$ in , may result in undesired beampatterns with low power at the tumor region and damaging power deposition in the healthy tissue region. Such health considerations in medical applications motivate an approach that is robust with respect to the worst-case model uncertainties. In order to formulate the robust design problem mathematically, we parameterize the steering vector uncertainties as follows. Let the true steering vector for the transducer array be $\mathbf{a(\r)} = \mathbf{\hat{a}(\r)} + \mathbf{\tilde{a}(\r)}$ where $\mathbf{\tilde{a}(\r)}$ is an unknown perturbation from the nominal steering vector. The deterministic perturbation at any generic point $\r$ belongs to uncertainty set $\mathcal{E}_{\r}$ that is bounded $$\begin{split} \mathcal{E}_{\r} \triangleq \{ \mathbf{\tilde{a}}(\r) \hspace{.2cm} | \hspace{.2cm} \| \mathbf{\tilde{a}}(\r) \|_{\mathbf{W}}^2 \leq \epsilon_{\r} \}, \end{split}$$ where $\mathbf{W}$ is an $M \times M$ diagonal weight matrix with positive elements. The weight matrix $\mathbf{W}$ can be derived based on the type of uncertainty. Using $\mathbf{W}$, the set $\mathcal{E}_{\r}$ indicates an ellipsoidal region. The bound $\epsilon_{\r}$ for the set can be a constant or a function of $\r$, i.e., $\epsilon_{\r} = f(\r)$. This set enables parameterization of element-wise uncertainties in the nominal steering vector $\mathbf{\hat{a}(\r)}$ at each $\r$. Besides this consideration, we generalize the problem formulation further by setting a uniform bound (power level) $P$ across the tumor region $\Omega_T$ as an optimization variable to which the power of all the control points in the healthy region $\Omega_S$ are compared. This is in contrast to and the robust formulation in [@14:Shariati2014c], where the power levels of all the healthy grid points $\Omega_S$ are compared with the power of only a single reference point at fictitious tumor center $\r_0$. There is no need to limit our problem to a single point as a reference power level. Rather, the desired tightness of the power level across $\Omega_T$ is specified by the parameter $0 \leq \delta < 1$. This generalization also improves the efficiency when it comes to solving the robust design problem. With these considerations, the robust beampattern design problem can be formulated as$$\label{eq:robust beampattern problem} \begin{aligned} \underset{\R,t,P}{\textrm{max}} \hspace{.3cm} &t \hspace{.3cm} \textrm{subject to}&\\ &\hspace{-1cm} P - \left(\ahat(\r) \!\!+\!\! \atil(\r)\right)^H \R \left(\ahat(\r) \!\!+\!\! \atil(\r)\right) \geq t, \forall \atil(\r) \in \mathcal{E}_{\r}, \r \in \Omega_S&\\ &\hspace{-1.3cm}\left(\ahat(\r) \!\!+\!\! \atil(\r)\right)^H \R \left(\ahat(\r) \!\!+\!\! \atil(\r)\right) \geq (1-\delta)P, \forall \atil(\r) \in \mathcal{E}_{\r}, \r \in \Omega_T&\\ &\hspace{-1.3cm}\left(\ahat(\r) \!\!+\!\! \atil(\r)\right)^H \R \left(\ahat(\r) \!\!+\!\! \atil(\r)\right) \leq (1+\delta)P, \forall \atil(\r) \in \mathcal{E}_{\r}, \r \in \Omega_T&\\ &\hspace{-1cm}\R \in \mathcal{R},& \end{aligned}$$ where $t$ is the gap between the desired power level set across $\Omega_T$ and power deposition in the healthy tissue $\Omega_S$, similar to . Note that we take into account every possible perturbation $\atil(\r) \in \mathcal{E}_{\r}$. In contrast to the optimization problem which is a tractable convex problem, the robust problem is an SIP problem. For a given $\R$ in , there are infinite number of constraints in terms of $\atil(\r)$ to satisfy which makes the problem non-trivial. However, in the following theorem, extending the approach in [@14:Shariati2014c], we reformulate the robust power deposition problem as a convex SDP problem whose solution is the optimally robust covariance matrix. \[theo:robustSDP\] The robust power deposition for an M-element transducer array with the probing signal covariance matrix $\R \in \mathcal{R}$ and the perturbation vector $\atil(\r) \in \mathcal{E}_{\r}$, i.e., the solution of , is given as a solution to the following SDP problem $$\label{eq:robustSDP} \begin{aligned} &\underset{\R,t,P,\beta_i,\beta_{j,1},\beta_{j,2}}{\textrm{max}} \hspace{.3cm} t \hspace{.5cm} \textrm{subject to}& \\ &\Omega_S\!\!:\!\!\left[ \begin{array}{cc} \beta_i \mathbf{W} - \R & -\R \ahat(\r_i) \\ -\ahat(\r_i)^H \R & P - t - \ahat(\r_i)^H \R \ahat(\r_i) - \beta_i \epsilon_{\r_i} \end{array} \right] \succeq {\mathbf{0}}, &\\ &\Omega_T\!\!:\!\!\left[ \begin{array}{cc} \beta_{j,1} \mathbf{W} + \R & \R \ahat(\r_j) \\ \ahat(\r_j)^H \R & \ahat(\r_j)^H \R \ahat(\r_j) \!\!-\!\! (1-\delta)P \!\!-\!\! \beta_{j,1} \epsilon_{\r_j} \end{array} \right] \succeq {\mathbf{0}}, &\\ &\Omega_T\!\!:\!\!\left[ \begin{array}{cc} \beta_{j,2} \mathbf{W} - \R & -\R \ahat(\r_j) \\ -\ahat(\r_j)^H \R & (1+\delta)P \!\!-\!\! \ahat(\r_j)^H \R \ahat(\r_j) \!\!-\!\! \beta_{j,2} \epsilon_{\r_j} \end{array} \right] \succeq {\mathbf{0}}, &\\ &\R \in \mathcal{R},\beta_i,\beta_{j,1},\beta_{j,2} \geq 0, i=1,\ldots,N_S, j=1,\ldots,N_T.& \end{aligned}$$ *Proof:* See Appendix \[app A\]. Observe that the notations $\Omega_S$ and $\Omega_T$ indicate that the corresponding linear matrix inequalities (LMIs) should be satisfied for the points $\r_i \in \Omega_S$ and $\r_j \in \Omega_T$, respectively. Note that the robust SDP problem in this paper, which is stated in Theorem \[theo:robustSDP\], can be solved more efficiently than the SDP problem in [@14:Shariati2014c] since the matrices $\R$ and $\mathbf{W}$ in the current formulation have half of the size of the matrices involved in the latter problem. This occurs due to the generalization of the robust problem by using the uniform power level as a benchmark. Note that other robust problems with similar objectives can also be addressed using the above approach which are outlined in the following subsection. Alternative robust formulations {#subsec:alternative} ------------------------------- Similar robust problems to that of can be formulated in many different ways. For example, by restricting the power level outside the tumor in a weighted fashion. $$\begin{aligned} \min_{t,R}&& t \hspace{.3cm} \mbox{subject to}\\ && \hspace{-1.3cm}(\ahat(\r)\!\!+\!\!\atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le t w(\r), \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm S}\\ &&\hspace{-1.3cm} (\ahat (\r)\!\!+\!\!\atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \ge (1-\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm T}\\ && \hspace{-1.3cm}(\ahat (\r)\!\!+\!\!\atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le (1+\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm T}\\ &&\hspace{-1.3cm}R\in \mathcal{R}\end{aligned}$$ where $P,\delta$ are fixed and $w(\r)$ is a weighting function constructed, e.g., so that the energy bound close to the tumor is less restrictive. One could also construct problems that minimize the sum of the energy in the non-tumor area where $t(\r)$ denotes the energy at $\r$: $$\begin{aligned} \min_{t(\r),R}&& \sum_{\r\in\Omega_{\rm S}}t(\r) \hspace{.3cm} \mbox{subject to} \\ && \hspace{-1.3cm} (\ahat(\r) \!\!+\!\! \atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le t(\r), \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm S}\\ && \hspace{-1.3cm} (\ahat(\r) \!\!+\!\! \atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \ge (1-\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm T}\\ && \hspace{-1.3cm} (\ahat(\r) \!\!+\!\! \atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le (1+\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm T}\\ &&\hspace{-1.3cm} R\in \mathcal{R}.\end{aligned}$$ Both of the alternative formulations described above can be addressed following the steps derived in Appendix \[app A\] by using $\mathcal{S}$-lemma since we are still dealing with quadratic constraints. In the next section, we illustrate the reference performance of a nominal scenario where the steering vectors are perfectly known. Then we observe how much power can leak to the healthy tissue and cause damages when subject to uncertain steering vectors. Finally, we evaluate the proposed robust scheme in terms of improving the power deposition along our stated design goals. Numerical Results {#sec:numerical_results} ================= To illustrate the performance of the proposed robust scheme, we consider a 2D model of the organ at risk. Here, similar to [@08:Guo], we focus on the ultrasonic hyperthermia treatment for breast cancer where a 10-cm-diameter semi-circle is assumed to model breast tissues with a 16-mm-diameter tumor embedded inside. The tumor center is located at $\r_0 = [0 \hspace{.1cm} 34]^T$ mm. Fig. \[fig:2D\_model\] shows this schematic model. We consider a curvilinear array with $M=51$ acoustic transducers and half wavelength element spacing. Acoustic waveforms used to excite the array have the carrier frequency of $500$ kHz. The acoustic wave speed for the breast tissue is considered $1500$ m/s. ![A schematic 2-D breast model with an $16$-mm embedded tumor at $(0,34)$ as a reference geometry. A curvilinear ultrasonic array with $51$ transducers is located near to the organ at risk. The ultrasonic array is used for hyperthermia treatment.[]{data-label="fig:2D_model"}](schematic_2D_model_col "fig:"){width="\columnwidth" height="5.5cm"}\ To characterize (discretize) the healthy tissue region $\Omega_S$ and the tumor region $\Omega_T$, two grid sets with the spacing $4$mm are considered. For optimization, a rectangular surface of the dimension $ 64 \times 42$ in mm is assumed symmetric around the tumor to model the healthy region $\Omega_S$, while the grid points belonging to the circular tumor region are excluded from this surface and they model $\Omega_T$. Overall, $174$ and $13$ number of control points are considered to characterize $\Omega_S$ and $\Omega_T$ in order to optimize the array beampattern. The total transmitted power is constrained to $\gamma = 1$. For simplicity, the uncertainty set $\mathcal{E}_{\r}$ is modeled with ${\mathbf{W}} = {\mathbf{I}}_M$ with $\epsilon_{\r} \equiv \epsilon$ for all $\r$ where $\epsilon = 0.25$. Furthermore, the tightness of the desired power level in the across tumor region, $\delta$, is set to $0.7$. Note that for the small values of $\delta$ and/or large values of $\epsilon$, the problem may turn infeasible. In general, the feasibility of the problem depends on the value of the tightness bound $\delta$ relative to the size of the existing uncertainty in the system, i.e., the volume of the uncertainty set $\epsilon$, and the number of grid points $N_S$ and $N_T$ used to control the beampattern at the area of interest. When $\delta$ is too small, the desired power level across $\Omega_T$ is close to uniform and there may not exist enough degrees of freedom for the design problem to have a solution. For reference, the optimal covariance matrix when no uncertainty is taken into account, $\mathbf{R}_{nr}$, is obtained by solving problem using only nominal steering vectors $\ahat(\r)$, i.e., $\atil(\r) \equiv {\mathbf{0}}$. The optimal robust covariance matrix, denoted $\R^\star$, is obtained by solving where $\atil(\r) \in \mathcal{E}_{\r}$. For performance evaluation, we consider the power deposition in the tissue under the worst-case perturbations of the steering vectors. This scenario provides a lower bound to the achievable performances of all steering vector perturbations $\atil(\r)$ which belong to the deterministic uncertainty set $\mathcal{E}_{\r}$. In other words, for the points $\r$ in the healthy region $\Omega_S$, the worst-case performance is rendered by the steering vectors which provide the highest power, whereas for the points $\r$ in the tumor region $\Omega_T$, those steering vectors which attain the lowest power are the ones which contribute in the worst-case performance. They are collectively referred to as the *worst steering vectors*. Therefore, for a given $\R$, either $\R_{nr}$ or $\R^\star$, the worst steering vectors for the control points $\r$ in $\Omega_S$ and $\Omega_T$, are obtained by maximizing and minimizing the transmit beampattern , respectively. Observe that finding the worst steering vectors for the points in the tumor region $\Omega_T$ equals solving the following convex minimization problem at each $\r \in \Omega_T$, i.e., $$\underset{\|\atil(\r)\|^2 \leq \epsilon}{\textrm{min}} \hspace{.2cm} (\ahat(\r) + \atil(\r))^H \R (\ahat(\r) + \atil(\r))$$ using `CVX` [@cvx2013; @gb08]. Whereas, for finding the worst steering vectors for the points in the healthy region $\Omega_S$, we obtain a local optimum for the following non-convex maximization problem at each $\r \in \Omega_S$, i.e., $$\underset{\|\atil(\r)\|^2 \leq \epsilon}{\textrm{max}} \hspace{.2cm} (\ahat(\r) + \atil(\r))^H \R (\ahat(\r) + \atil(\r)),$$ using semidefinite relaxation techniques from [@Beck06strongduality]. We evaluate the designed beampatterns plotting the spatial power distribution in decibel scale, i.e., $20 \log_{10} (p(\r))$. Two different scenarios are considered, namely, *nominal* and *perturbed*, to evaluate the proposed robust power distribution scheme for the ultrasonic array. In the first scenario, nominal, we assume that the array steering vectors are precisely modeled, i.e., $\atil(\r) = \mathbf{0}$. In Fig. \[fig:nominal\], the beampattern generated by the array is plotted for the nominal scenario. This figure represents how power is spatially distributed over the organ at risk in an idealistic situation. Here, the covariance matrix of the waveforms is optimized under the assumption that the steering vectors are accurately modeled by , and the performance is evaluated using exactly the same steering vectors without any perturbations. The power is noticeably concentrated in the tumor region and importantly the power in the healthy tissue is several decibels lower. ![Power distribution (transmit beampattern in dB) for the nominal scenario, i.e., using $\R_{nr}$ and $\atil(\r) \equiv 0$.[]{data-label="fig:nominal"}](nominal_05_07lines "fig:"){width="\columnwidth" height="5.5cm"}\ In the second scenario, perturbed, the idealistic assumptions are relaxed and model uncertainties and imperfections are taken into account. The second scenario represents the case where the true steering vectors are perturbed versions of the nominal steering vectors $\ahat(\r)$, i.e., the true steering vector equals $\ahat(\r) + \atil(\r)$ where $\atil(\r) \in \mathcal{E}_{\r}$. The perturbation vectors $\atil(\r)$ are unknown but deterministically bounded. In the following we illustrate the worst-case performance, i.e., using the worst steering vectors to calculate the power distribution at each point. We start by illustrating the beampattern for the non-robust covariance matrix $\R_{nr}$ under the worst steering vectors. Fig. \[fig:NonRobustWorst\] shows how steering vector errors can degrade the array performance. Notice that in the worst-case, there is a substantial power leakage that occurs in the healthy tissue surrounding the tumor compared to Fig. \[fig:nominal\]. While, in Fig. \[fig:RobustWorst\], the robust optimal covariance matrix $\R^\star$, i.e., the solution to , is used to calculate the power for the worst steering vectors. Comparing Fig. \[fig:NonRobustWorst\] and Fig. \[fig:RobustWorst\], we see that by taking model uncertainties into account it is possible to obtain a noticeable increase in power in the tumor region for the worst case, and importantly, dramatic reductions of power deposited in the healthy tissue. ![Power distribution (transmit beampattern in dB) for the perturbed scenario, i.e., using $\R_{nr}$ and $\atil(\r) \in \mathcal{E}_{\r}$.[]{data-label="fig:NonRobustWorst"}](worst_nr_05_07lines "fig:"){width="\columnwidth" height="5.5cm"}\ ![Power distribution (transmit beampattern in dB) for the perturbed scenario, i.e., using $\R^\star$ and $\atil(\r) \in \mathcal{E}_{\r}$.[]{data-label="fig:RobustWorst"}](worst_r_05_07lines "fig:"){width="\columnwidth" height="5.5cm"}\ To finalize the numerical analysis, we provide a quantitative description for the performance of our proposed scheme summarized in Table \[table:power\]. It shows the average power calculated in dB received at the tumor region $\Omega_T$ and at the healthy region $\Omega_S$. Scenarios $\Omega_T$ $\Omega_S$ ----------------------- ------------ ------------ Nominal, $\R_{0}$ $-16.54$ $-29.78$ Perturbed, $\R_{0}$ $-36.40$ $-11.69$ Perturbed, $\R^\star$ $-27.17$ $-17.43$ : Average power for different regions[]{data-label="table:power"} Conclusion ========== The robust transmit signal design for optimizing spatial power distribution of an multi-antenna array is investigated. A robustness analysis is carried out to combat against inevitable uncertainty in model parameters which results in performance degradation. Such degradation occurs in practice quite often due to relying on imperfect prior and designs based upon them. Particularly, in this paper, the transmit signal design is based on exploiting the waveform diversity property but where errors in the array steering vector are taken into account. These errors are modeled as belonging to a deterministic set defined by a weighted norm. Then, the resulting robust signal covariance optimization problem with infinite number of constraints is translated to a convex problem which can be solved efficiently, by using the $\mathcal{S}$-procedure. Designs that are robust with respect to the worst case are particularly vital in biomedical applications due to health risks and possible damage. Herein we have focused on local hyperthermia therapy as one of the cancer treatments to be used either individually or along with other treatments such as radio/chemotherapy. Specifically, we consider hyperthermia treatment of breast cancer motivated by the fact that breast cancer is a major global health concern. The proposed robust signal design scheme aims to reduce unwanted power leakage into the healthy tissue surrounding the tumor while guaranteeing certain power level in the tumor region itself. We should emphasize on the fact that the robust design problem formulation and the analysis carried out herein yielding to the robust waveforms are general enough to be exploited whenever spatial power distribution is a concern to be addressed in real world scenarios dealing with uncertainties, e.g., for radar applications. Numerical examples representing different scenarios are given to illustrate the performance of the proposed scheme for hyperthermia therapy. We have observed significant power leakage into the healthy tissue that can occur if the design is based on uncertain model parameters. Importantly, we have shown how such damaging power deposition can be avoided using the proposed robust design for optimal spatial power distribution. Acknowledgement =============== The authors would like to acknowledge Prof. Jian Li for providing an implementation of examples from [@08:Guo]. \[sec:appendix\] Proof of Theorem \[theo:robustSDP\] {#app A} ----------------------------------- We start the proof by first stating the $\mathcal{S}$-Procedure lemma which helps us to turn the optimization problem with infinitely many quadratic constraints into a convex problem with finite number of LMIs. \[lem 6\]($\mathcal{S}$-Procedure [@Beck2009a Lemma 4.1]): Let $f_k(\x): \mathbb{C}^n \rightarrow \mathbb{R}$, $k = 0,1$, be defined as $f_k(\x) = \x^H\mathbf{A}_k \x + 2 \textrm{Re} \{\mathbf{b}_k^H \x \} + c_k$, where $\mathbf{A}_k = \mathbf{A}_k^H \in \mathbb{C}^{n \times n}, \mathbf{b}_k \in \mathbb{C}^n$, and $c_k \in \mathbb{R}$. Then, the statement (implication) $f_0(\x) \geq 0$ for all $\x \in \mathbb{C}^n$ such that $f_1(\x) \geq 0$ holds if and only if there exists $\beta \geq 0$ such that[^2] $$\left[ \begin{array}{cc} \mathbf{A}_0 & \mathbf{b}_0 \\ \mathbf{b}_0^H & c_0 \end{array}\right] - \beta \left[ \begin{array}{cc} \mathbf{A}_1 & \mathbf{b}_1 \\ \mathbf{b}_1^H & c_1 \end{array}\right] \succeq 0,$$ if there exists a point $\mathbf{\hat{x}}$ with $f_1(\mathbf{\hat{x}}) > 0$. The constraints in the optimization problem can be rewritten as the following functions of $\atil(\r)$ for $\r \in \Omega_S$ and $\r \in \Omega_T$. For notation simplicity we only specify the set from which the control points are drawn, and we also drop $\r$. $$\Omega_S: \begin{cases} f_0 = -\atil^H \R \atil - 2 \textrm{Re}(\ahat^H \R \atil) - \ahat^H \R \ahat - t + P \geq 0 \\ f_1 = - \atil^H \mathbf{W} \atil + \epsilon_{\r} \geq 0 \end{cases}$$ $$\Omega_T: \begin{cases} f_0 = \atil^H \R \atil + 2 \textrm{Re}(\ahat^H \R \atil) + \ahat^H \R \ahat - (1-\delta)P \geq 0 \\ f_1 = - \atil^H \mathbf{W} \atil + \epsilon_{\r} \geq 0 \end{cases}$$ $$\Omega_T: \begin{cases} f_0 = -\atil^H \R \atil - 2 \textrm{Re}(\ahat^H \R \atil) - \ahat^H \R \ahat + (1+\delta)P \geq 0 \\ f_1 = - \atil^H \mathbf{W} \atil + \epsilon_{\r} \geq 0 \end{cases}$$ Now, according to the $\mathcal{S}$-Procedure lemma, each pair of the quadratic constraints above is replaced with an LMI for each grid points in the pre-defined sets. In other words, all these quadratic constraints are satisfied simultaneously if we find $\beta_i$ for $i=1,\ldots,N_S$, $\beta_{j,1}$ and $\beta_{j,2}$ for $j=1,\ldots,N_T$ for which the mentioned LMIs in Theorem \[theo:robustSDP\] holds. Thus, the problem boils down to the SDP problem with $2N_T + N_S$ LMIs of the size $(M+1) \times (M+1)$ as the constraints. $\Box$ [^1]: Breast cancer is the most common cancer in the UK [@CancerReport2014UK]. The risk of being diagnosed with breast cancer is $1$ in $8$ for women in the UK and US [@CancerReport2014UK; @CancerReport2014US]. Breast cancer is also stated to be a leading cause of cancer death in the less developed countries [@CancerReport2013French]. [^2]: Note that $\mathcal{S}$-Procedure is lossless in complex space for the case of at most two constraints [@01:Jonsson].

--- abstract: 'We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall-Kapovich-Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index subgroups that are ascending HNN extensions.' address: | Department of Mathematical Sciences\ University of Arkansas\ Fayetteville, AR *Web address: <https://mutanguha.com/>* author: - Jean Pierre Mutanguha bibliography: - 'refs.bib' title: '*Irreducibility of a free group endomorphism is a mapping torus invariant*' --- Introduction ============ Suppose S is a hyperbolic surface of finite type and $f:S \to S$ is a [*pseudo-Anosov*]{} homeomorphism, then the [*mapping torus*]{} $M_f$ is a complete finite-volume hyperbolic 3-manifold; this is Thurston’s hyperbolization theorem for 3-manifolds that fiber over a circle [@Thu82]. It is remarkable fact since the hypothesis is a dynamical statement about surface homeomorphisms but the conclusion is a geometric statement about 3-manifolds. In particular, since the converse holds as well, i.e., a hyperbolic 3-manifold that fibers over a circle will have a pseudo-Anosov [*monodromy*]{}, the property of a fibered manifold having a pseudo-Anosov monodromy is in fact a geometric invariant: if $f:S\to S$ and $f':S' \to S'$ are homeomorphisms whose mapping tori have [*quasi-isometric*]{} (q.i.) fundamental groups, then $f$ is pseudo-Anosov if and only if $f'$ is pseudo-Anosov. There are three types of invariants that we study in geometric group theory: group invariants, which contain virtual/commensurability invariants, which contain geometric/q.i.-invariants; the geometric invariants are the most important and difficult to prove. In this paper, we exhibit geometric and commensurability invariants for free-by-cyclic groups inspired by Thurston’s hyperbolization theorem and our arguments will be general enough to also apply to ascending HNN extensions of free groups. There is a rough correspondence between the study of the outer automorphism group of a free group $\operatorname{Out}(F)$ and the study of the mapping class group of a hyperbolic surface $\operatorname{MCG}(S)$. Under this correspondence, surface groups are paired with free groups, surfaces with graphs, and 3-manifolds that fiber over a circle with free-by-cyclic groups. However, this correspondence is not perfect; pseudo-Anosov mapping classes have three possible analogous properties for free group automorphisms: [*induced by a pseudo-Anosov (on a punctured surface)*]{}, [*atoroidal*]{}, and [*irreducible*]{} (Section \[defs\]). We originally set out to prove that irreducibility was a group invariant of the automorphism’s mapping torus and, along the way, we proved more general statements for the first property and the composite property of being both irreducible and atoroidal. Our first result is that the first property is a geometric invariant: [Theorem]{}[geomqi]{} Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that the mapping tori $F *_\phi$ and $F' *_\psi$ are quasi-isometric. Then $\phi$ is induced by a pseudo-Anosov if and only if $\psi$ is induced by a pseudo-Anosov. Thus starting with just a free group automorphism $\phi$ induced by a pseudo-Anosov and a quasi-isometry between $F \rtimes_\phi \mathbb Z$ and $F' *_\psi$, we find that $\psi$ is induced by a surface homeomorphism too. The proof is short but uses deep geometric theorems: Thurston’s hyperbolization [@Thu82] and Schwartz rigidity [@Sch95]. Since pseudo-Anosovs have dynamics that are very similar to those of irreducible and atoroidal automorphisms, it is likely that the latter property is a geometric invariant too. Suppose $\phi : F \to F$ and $\psi : F' \to F'$ are free group automorphisms such that $F \rtimes_\phi \mathbb Z$ and $F' \rtimes_\psi \mathbb Z$ are quasi-isometric. Then $\phi$ is irreducible and atoroidal if and only if $\psi$ is irreducible and atoroidal. Our main result is that being irreducible and atoroidal is a commensurability invariant, which lends credence to the conjecture; again, the argument works for endomorphisms. [Theorem]{}[nongeomcomm]{} Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi$ and $F' *_\psi$ are commensurable and neither one of the endomorphisms has an image contained in a proper free factor of their domain. Then $\phi$ is irreducible and atoroidal if and only if $\psi$ is irreducible and atoroidal. The hypothesis on the images is necessary: Let $\phi:F_2\to F_2$ be the endomorphism on a free group of rank $2$ given by $\phi(a) = ab$ and $\phi(b) = ba$. Then $\phi$ is irreducible and atoroidal [@JPM Example 1.2]. Now let $F_2$ be a proper free factor of the free group $F_3$ generated by $\{ a,b,c \}$. Extend $\phi$ to $\psi:F_3 \to F_3$ by setting $\psi(c) \in F_2$; then $F_3*_{\psi} \cong F_2*_\phi$, but $\psi$ is reducible. The proof of Theorem \[nongeomcomm\] follows immediately from an algebraic characterization of $F*_\phi$ that detects exactly when $\phi: F \to F$ is irreducible and atoroidal. [Theorem]{}[grpinv]{} Suppose $\phi: F \to F$ is a free group injective endomorphism whose image is not contained in a proper free factor of $F$ and let $G = F*_\phi$. Then $\phi$ is irreducible and atoroidal if and only if $G$ has no finitely generated noncyclic subgroups with infinite index and vanishing Euler characteristic. These results imply that irreducibility is a group invariant, our original motivation: [Corollary]{}[irredgrp]{} Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi \cong F' *_\psi$ and neither one of the endomorphisms has an image contained in a proper free factor. Then $\phi$ is irreducible if and only if $\psi$ is irreducible. That irreducibility is a group invariant was an open problem [@DKL17c Question 1.4]. In a series of papers [@DKL15; @DKL17b; @DKL17c], Dowdall-Kapovich-Leininger studied the dynamics of ([*word-hyperbolic*]{}) free-by-cyclic groups and the main result of the third paper answered this problem under an extra condition that we now discuss: Fix a free-by-cyclic group $G$. The [*BNS-invariant*]{} $\mathcal C(G)$ is an open cone (with convex components) in $H^1(G; \mathbb R) \cong \operatorname{Hom}(G, \mathbb R)$. By rational rays in $H^1(G; \mathbb R)$, we refer to projective classes of homomorphisms $G \to \mathbb R$ with discrete/cyclic image. Without defining the BNS-invariant, we shall state its most relevant property for our purposes: a rational ray in $H^1(G; \mathbb R)$ is [*symmetric*]{}, i.e., is in $- \mathcal C(G) \cap \mathcal C(G)$ if and only if the corresponding class of homomorphisms $[p] : G \to \mathbb R$ have finitely generated kernel $K$; in this case, $ K $ is free for cohomological reasons [@FH99; @Bie81; @St68], $G \cong K \rtimes_{\phi} \mathbb Z$ for some free group automorphism $\phi: K \to K$, and the natural projection $K \rtimes_{\phi} \mathbb Z \to \mathbb Z$ is in the projective class $[p]$. Fix a symmetric rational ray $r_0$ in $\mathcal C(G)$, and let $\phi_0: K_0 \to K_0$ be the corresponding free group automorphism. The presentation complex for $K_0 \rtimes_{\phi_0} \mathbb Z$ has a natural semi-flow with respect to the [*stable direction*]{} $\mathbb Z_+$. Dowdall-Kapovich-Leininger show in [@DKL17b] that getting from $r_0$ to any symmetric rational ray in the same component of $\mathcal C(G)$ amounts to reparametrizing this semi-flow (hence the convexity of the component) and with a careful analysis of this semi-flow, they are able to relate the monodromy stretch factors and rank of kernels for all symmetric rays in the same component. In the third paper [@DKL17c], they conclude this analysis by showing that being irreducible and atoroidal is an invariant of a component of the BNS-invariant; that is, if $\phi_1: K_1 \to K_1$ and $\phi_2: K_2 \to K_2$ are free group automorphisms corresponding to symmetric rays in the same component of $\mathcal C(G)$, then $\phi_1$ is irreducible and atoroidal if and only if $ \phi_2 $ is too. Since this result relied heavily on the analysis of the semi-flow for a component of the BNS-invariant, the technique cannot be extended to work for symmetric rational rays in separate components. Furthermore, their result does not apply to any asymmetric rational rays, i.e., it does not apply to nonsurjective injective endomorphisms. Theorem \[nongeomcomm\] addresses both of these concerns. Masai-Mineyama have also proven a different special case of Theorem \[nongeomcomm\] that they call [*fibered commensurability*]{} [@MM17]: suppose $\phi: F \to F$ and $\psi: F' \to F'$ are free group automorphisms and let $K \le F, K' \le F'$ be finite index subgroups with an isomorphism $\epsilon: K \to K'$ such that there are the outer classes of the isomorphisms $\epsilon\left.\phi^i\right|_{K}: K \to K'$ and $\left.\psi^j \epsilon\right|_{K}:K \to K'$ are the same for some $i, j \ge 1$; a priori, we restrict ourselves to $i, j$ such that $\phi^i(K) = K$ and $\psi^j(K') = K'$, i.e., the maps $\epsilon\left.\phi^i\right|_{K}$ and $\left.\psi^j \epsilon\right|_{K}$ make sense. Masai-Mineyama prove that in this case, $\phi$ is irreducible and atoroidal if and only if $\psi$ is too. In other words, being irreducible and atoroidal is a fibered commensurability invariant. However, compared to commensurability, this equivalence is very restrictive since a typical isomorphism of finite index subgroups of free-by-cylic groups will not preserve the fibers one starts with. Feighn-Handel’s theorem that mapping tori of free group endomorphisms are coherent is the main tool that allows us to avoid the obstacles in the two approaches discussed above. We shall explicitly use the [*preferred presentation*]{} that their algorithm produces for any finitely generated subgroup of a mapping torus (Theorem \[rewrite\]). We conclude the introduction by noting that very little is known about geometric invariants of free-by-cyclic groups in relation to their monodromies. Here are the geometric invariants we know of: Brinkmann [@Bri00; @Bri02] showed 1) a free-by-cyclic group is word-hyperbolic (a geometric invariant) if and only if the monodromy is atoroidal and 2) a word-hyperbolic free-by-cyclic group has a menger curve [*Gromov boundary*]{} if and only if the monodromy fixes no free splitting of the fiber; Macura [@Mac02] proved that two polynomially-growing free group automorphisms that have quasi-isometric mapping tori must have polynomial growth of the same degree and, conjecturally, these mapping tori cannot be quasi-isometric to a mapping torus of an exponentially growing free group automorphism. [**Outline.**]{} We give the standard definitions and statements of results that are most relevant to the rest of the article in Section \[defs\]. Section \[geom\] contains the proof of Theorem \[geomqi\] while Section \[nongeom\] contains that of Theorem \[nongeomcomm\]. Finally, we briefly discuss the q.i.-invariance conjecture in Section \[qi\]. In Appendix \[app\], we prove a folk theorem and its converse: a free group endomorphism is irreducible with a nontrivial periodic conjugacy class if and only if it is induced by a pseudo-Anosov on a punctured surface with one orbit of punctures. **Acknowledgments:** I want to thank Ilya Kapovich for patiently checking my initial proof (and its numerous iterations) and Chris Leininger for sharing with me his work-in-progress in collaboration with Spencer Dowdall and I. Kapovich as it relates to my result. The appendix was born out of an insightful discussion with Saul Schleimer on a bus ride and it would not have been written up without Mladen Bestvina’s encouragement. Last but not least, I thank my advisor Matt Clay for his constant support. Definitions and Preliminaries {#defs} ============================= In this paper, free groups $F$ are assumed to have finite rank at least $2$. We will study the ascending HNN extension of a free group and how its properties relate to those of the defining endomorphism. Let $A \le F$ be a subgroup of a free group and $\phi : A \to F$ be an injective homomorphism, then we define the [**HNN extension**]{} of $F$ over $A$ to be: $$F*_A = \left\langle\, F, t~|~t^{-1} a t = \phi(a), \forall a \in A \,\right\rangle$$ An HNN extension has a natural map $F*_A \to \mathbb Z$ that maps $F \mapsto 0$ and $ t \mapsto 1$; we shall refer to this map as the [**natural fibration**]{}. For the rest of this paper, we restrict ourselves to HNN extensions defined over free factors. When A = F , then we call $F*_F = F*_\phi$ an [**ascending HNN extension**]{} or a [**mapping torus**]{} of $\phi : F \to F$. The latter terminology stems from the fact that the injective endomorphism $\phi$ can be topologically represented by a graph map on the rose whose topological mapping torus has a fundamental group isomorphic to $F*_\phi$ . Following this analogy, we shall call $F$ the [**fiber**]{} and $\phi$ the [**monodromy**]{} of the mapping torus. Finally, when $\phi : F \to F$ is an automorphism, we call $F*_\phi = F \rtimes_\phi \mathbb Z$ a [**free-by-cyclic group**]{}. The following are the properties of monodromies that we will study. An endomorphism $\phi: F \to F$ is [**reducible**]{} if there is a free factorization $A_1 * \cdots * A_k * B $ of $F$ where $B$ may be trivial if $k \ge 2$, and a sequence of elements, $(x_i)_{i=1}^k$, in $F$ such that $\phi(A_i) \le x_i A_{i+1} x_i^{-1}$ where the indices are considered$\mod k$; the collection $\{A_1, \ldots, A_k \}$ is a [**$\boldsymbol{\phi}$-invariant proper free factor system**]{}. An endomorphism is [**irreducible**]{} if it is not reducible. It is [**atoroidal**]{} if there does not exist a nontrivial element $a \in F$, element $x \in F$, and integer $n \ge 1$ such that $\phi^n(a) = x a x^{-1}$, equivalently, $i_x \phi^n(a) = a$ for some inner automorphism $i_x$. Feighn-Handel used the following proposition to show the coherence of ascending HNN extensions of free groups; the proposition and the next lemma allow us to rewrite presentations of ascending HNN extension subgroups of ascending HNN extension groups so that fibers of the subgroup lie in fibers of the ambient group. \[rewrite\] Suppose $\Phi:\mathbb F \to \mathbb F$ is an injective endomorphism of a free group (of possibly infinite rank) and $G = \mathbb F*_\Phi = \left\langle\, \mathbb F, t~|~t^{-1} x t = \Phi(x), \forall x\in\mathbb F \,\right\rangle$. Let $p: G \to \mathbb Z$ be the natural fibration that maps $\mathbb F \mapsto 0$ and $t \mapsto 1$. If $H \le G$ is finitely generated and $\left.p\right|_H$ is not trivial and not injective, then there is an isomorphism $\iota: F*_A \to H$, natural fibration $\pi: F*_A \to \mathbb Z$, and injective map $n:\mathbb Z \to \mathbb Z$ such that $n(1) \ge 1$ and $\left.p\right|_H \iota = n \pi$, where $F*_A$ is an HNN extension of a finitely generated free group $F$ over a free factor $A \le F$. The next lemma will be used to show certain subgroups of a mapping torus are also mapping tori if they have vanishing Euler characteristic. \[hwlem\] Let $F$ be a free group of finite rank $m$ and $A \le F$ a free factor of rank $n$. Then $\chi(F*_{A}) = n - m$. Choose a basis for $A$ and extend it to a basis for $F$. Then the natural finite presentation complex $X$ is aspherical, i.e., $X$ is a $K(\pi,1)$ space for $F*_{A}$, and $$\chi(F*_{A}) = \chi(X) = 1 - (1+m) + n = n-m. \qedhere$$ Finally, we give a characterization of irreducible endomorphisms with nontrivial periodic conjugacy classes. As the techniques used in the proof of the following theorem are not related to the rest of the paper, we postpone the proof to the appendix (Proposition \[eg\] and Theorem \[thmBH2\]). \[thmBH\] Let $\phi:F \to F$ be an infinite-order endomorphism. Then $\phi$ is irreducible and has a periodic nontrivial conjugacy class if and only if it is induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^b$ that acts transitively on the boundary components. This theorem allows us to partition injective endomorphisms of free groups with interesting dynamics into two categories: - automorphisms induced by pseudo-Anosov maps. - irreducible and atoroidal endomorphisms. Pseudo-Anosov Monodromies {#geom} ========================= The first result in this section is a straightforward application of Stallings’ fibration theorem and Nielsen-Thurston classification; it is the first half of the proof that irreducibility of the monodromy is a group invariant of the mapping torus. The subsequent generalizations are given to motivate the q.i.-invariance conjecture. \[geomgrp\] Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi \cong F' *_\psi$. Then $\phi$ is induced by a pseudo-Anosov if and only if $\psi$ is induced by a pseudo-Anosov. Without loss of generality, suppose $\phi$ is induced by a pseudo-Anosov on a compact surface with boundary. Therefore, $G = F \rtimes_\phi \mathbb Z$ is the fundamental group of a compact $3$-manifold that fibers over a circle. Bieri-Neumann-Strebel [@BNS] showed that the BNS-invariant of such a compact $3$-manifold group $G$ is symmetric, which implies that $\psi$ is an automorphism and $G \cong F' \rtimes_\psi \mathbb Z$. By Stallings’ fibration theorem [@St61], $\psi$ is induced by a homeomorphism of a compact surface with boundary. Any invariant essential multicurve of the $\psi$-inducing homeomorphism would determine a non-peripheral $\mathbb Z^2$ subgroup of $G$. But since $\phi$ was induced by a pseudo-Anosov, the only $\mathbb Z^2$ subgroups of $G$ are the peripheral ones. Thus $\psi$ is induced by an infinite-order irreducible homeomorphism. By Nielsen-Thurston classification, $\psi$ is induced by a pseudo-Anosov. Since the number of orbits of boundary components for a pseudo-Anosov is the number of boundary components in the mapping torus, this proposition combines with Theorem \[thmBH\] to give: \[irredtorgrp\] Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi \cong F' *_\psi$. Then $\phi$ is irreducible and has a periodic nontrivial conjugacy class if and only if $\psi$ is irreducible and has a periodic nontrivial conjugacy class. Surprisingly, the analogous statement for commensurable groups is much harder to prove. The difficulty lies in showing that if the restriction of an iterate $\psi^n$ to a finite index subgroup is induced by a pseudo-Anosov, then so is $\psi$. We could use the theory of train tracks to adapt the argument in Appendix \[app\] and get a comparatively elementary proof; we opt to use Thurston’s hyperbolization [@Thu82] and Mostow’s rigidity [@Mar74] to keep the exposition short. \[geomcomm\] Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi$ and $F' *_\psi$ are commensurable. Then $\phi$ is induced by a pseudo-Anosov if and only if $\psi$ is induced by a pseudo-Anosov. Let $H \le F*_\phi$ and $H' \le F'*_\psi$ be isomorphic finite index subgroups. Without loss of generality, suppose $\phi$ is induced by a pseudo-Anosov homeomorphism on a punctured surface. Then $\phi$ is an automorphism and $F*_\phi = F \rtimes_\phi \mathbb Z$. By Thurston’s hyperbolization theorem, $F \rtimes_\phi \mathbb Z$, and hence $H' \cong H$, is a fundamental group of a complete finite-volume hyperbolic 3-manifold $M$. Assume $H' \trianglelefteq F' *_\psi$, then Mostow’s rigidity implies the finite group $Q = (F' *_\psi)/H'$ acts on $M$ by isometries. The action is free since $F' *_\psi$ is torsion-free. Thus $F' *_\psi$ is the fundamental group of the hyperbolic 3-manifold $M/Q$. By symmetry of the BNS-invariant and Stallings’ fibration theorem, $\psi$ is induced by a surface homeomorphism. As $\psi$ has a hyperbolic mapping torus, it is induced by a pseudo-Anosov. As mentioned before the proposition, the proof can be replaced by an argument using the theory of train tracks. However, the next theorem is a geometric statement and there is no apparent way around Thurston’s hyperbolization theorem. In fact, we will also need Schwartz’ rigidity [@Sch95] to reduce the theorem to the previous proposition. \[geomqi\]Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are injective endomorphisms such that $F *_\phi$ and $F' *_\psi$ are quasi-isometric. Then $\phi$ is induced by a pseudo-Anosov if and only if $\psi$ is induced by a pseudo-Anosov. Suppose $\phi$ is induced by a pseudo-Anosov. By Thurston’s hyperbolization theorem, $F*_\phi$ is the fundamental group of a complete finite-volume hyperbolic 3-manifold with cusps. In particular, Schwartz proved such groups are [*q.i.-rigid*]{}. As $F*_\phi$ and $F'*_\psi$ are quasi-isometric torsion-free groups, q.i.-rigidity of $F*_\phi$ implies they are commensurable. Thus $\psi$ is induced by a pseudo-Anosov by Proposition \[geomcomm\]. This proof underscores how difficult it is to prove the q.i.-invariance conjecture since there is no common model like $\mathbb H^3$ when studying irreducible and atoroidal endomorphisms. Irreducible and Atoroidal Monodromies {#nongeom} ===================================== The goal of this section is to prove that being irreducible and atoroidal is a commensurability invariant. We proved the following result in previous work [@JPM]; it essentially characterizes being an irreducible and atoroidal endomorphism. \[invSbgrp\] Suppose $\phi : F \to F$ is an irreducible and atoroidal endomorphism. If nontrivial $K \le F$ is finitely generated and $i_x\phi^n(K) \le K$ for some $n \ge 1$ and inner automorphism $i_x$, then $[F : (i_x \phi^n)^{-k}(K)] < \infty$ for some $k \ge 0$. The key idea in this section is to use this proposition to characterize in terms of the mapping torus exactly when a monodromy is irreducible and atoroidal. To this end, we need the following property to deal with nonsurjective monodromies. An injective endomorphism $\phi:F \to F$ is [**minimal**]{} if the image $\phi(F)$ is not contained in a proper free factor of $F$. Automorphisms and irreducible endomorphisms are clearly minimal. Minimality is preserved by taking powers and composing with (inner) automorphisms (See [@JPM Proposition 5.4]). We now have enough to state and prove the main result: \[grpinv\] Suppose $\phi: F \to F$ is a minimal injective endomorphism and let $G = F*_\phi$. Then $\phi$ is irreducible and atoroidal if and only if $G$ has no finitely generated noncyclic subgroups with infinite index and vanishing Euler characteristic. If $\phi$ is not atoroidal, then $G$ has a $\mathbb Z^2$ subgroup that necessarily has infinite index. If $\phi$ is not irreducible, then there exists a proper free factor $A \le F$, $x \in F$, and $n \ge 1$ such that $\phi^n(A) \le xAx^{-1}$. Then, using normal forms, $A *_{i_x \phi^n} \cong \langle A, t^n x \rangle \le G$. Suppose $[G: \langle A, t^n x \rangle] < \infty$, then $[F: F \cap \langle A, t^n x \rangle] < \infty$. Set $K = F \cap \langle A, t^n x \rangle = \cup_k(i_x \phi^n)^{-k}(A)$. As $[F: K] < \infty$, K is finitely generated and there exists a $k_0 \ge 1$ such that $K = (i_x \phi^n)^{-k_0}(A)$. The statements $[F:K] < \infty$, $K = (i_x \phi^n)^{-k_0}(A)$, and $A$ is a proper free factor of $F$ imply $F = (i_x \phi^n)^{-k_0}(A)$, which contradicts the minimal assumption on $\phi$. Therefore, $[G: \langle A, t^n x \rangle] = \infty$ as needed. This concludes the reverse direction. For the forward direction, suppose $\phi$ is irreducible and atoroidal and let $H \le G$ be a finitely generated noncyclic group with $\chi(H) = 0$. We need to show $[G: H] < \infty$. Let $p:G=F*_\phi \to \mathbb Z$ be the natural fibration. Note that $\ker p = \cup_i t^i F t^{-i}$. Then $\left.p\right|_H$ is not trivial since $\ker p$ is locally free yet $H$ is finitely generated but not free: it is not cyclic and $\chi(H) = 0$. Also $\left.\ker p\right|_H = H \cap \ker p$ is not trivial as $H \not \cong \mathbb Z$. As $H$ is finitely generated and $\left.p\right|_H$ is not trivial and not injective, by Proposition \[rewrite\], there is an isomorphism $\iota: F_m*_A \to H$, natural fibration $\pi: F_m*_A \to \mathbb Z$, and injective map $n:\mathbb Z \to \mathbb Z$ such that $n(1) \ge 1$ and $\left.p\right|_H \iota = n \pi$, where $F_m*_A$ is an HNN extension of a finitely generated free group $F_m$ over a free factor $A \le F_m$. As $\chi(H) = 0$, $A$ is not a proper free factor by Lemma \[hwlem\]. Therefore, $H \cong F_m*_{F_m}$. As $F_m$ is finitely generated, there is an $i \ge 0$ such that $K = \iota(F_m) \le t^i F t^{-i} \le \ker p$. Fix large enough $i$, then $K \le t^i F t^{-i}$ is a finitely generated nontrivial subgroup such that $i_x\bar\phi^n(K) \le K$, where $n = n(1)$, $\bar \phi$ is the natural extension of $\phi$ to $t^i F t^{-i}$, and $x \in t^i F t^{-i}$. As $\phi$ is irreducible and atoroidal, so is $\bar \phi$. By Proposition \[invSbgrp\], $(i_x \bar\phi^n)^{-k}(K)$ has finite index in $t^i F t^{-i}$ for some $k \ge 0$. Therefore, $H = \langle (t^n x)^k K (t^n x)^{-k}, t^n x \rangle$ has finite index in $G = \langle t^i F t^{-i}, t \rangle$. The following lemma shows that the property of having a finitely generated noncyclic subgroups with infinite index and vanishing Euler characteristic is a commensurability invariant. This implies the invariance of irreducibility for atoroidal endomorphisms. Let $G$ be a finitely generated torsion-free group and $G' \le G$ be a finite index subgroup. Then $G$ has an infinite index finitely generated noncyclic subgroup with vanishing Euler characteristic if and only if $G'$ has such a subgroup. The reverse direction is obvious. Suppose $H \le G$ is an infinite index finitely generated noncyclic subgroup with vanishing Euler characteristic. Then $H \cap G'$ has finite index in $H$. In particular, it is finitely generated with infinite index in $G$ and hence $G'$. Furthermore, $[ H: H \cap G'] < \infty$ implies $H \cap G'$ has vanishing Euler characteristic too and it is noncyclic since the only virtually cyclic torsion-free group is $\mathbb Z$, yet $H$ is not cyclic. \[nongeomcomm\]Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are minimal injective endomorphisms such that $F *_\phi$ and $F' *_\psi$ are commensurable. Then $\phi$ is irreducible and atoroidal if and only if $\psi$ is irreducible and atoroidal. Finally, we combine this theorem with Corollary \[irredtorgrp\] to get: \[irredgrp\]Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are minimal injective endomorphisms such that $F *_\phi \cong F' *_\psi$. Then $\phi$ is irreducible if and only if $\psi$ is irreducible. Q.I.-Invariance Conjecture {#qi} ========================== Very little is known about the geometry of mapping tori whose monodromies are irreducible and atoroidal. Brinkmann proved that atoroidal automorphisms of free groups have word-hyperbolic mapping tori [@Bri00] and gave an explicit description for when such mapping tori have non-trivial splittings over cyclic subgroups [@Bri02]: an atoroidal automorphism of $F$ has a mapping torus that splits over $\mathbb Z$ if and only if the (outer) automorphism fixes a [*free splitting*]{} of $F$; in particular, mapping tori of irreducible and atoroidal automorphisms never split over $\mathbb Z$. By work of Kapovich-Kleiner [@KK00], word-hyperbolic free-by-cyclic groups that do not split over $\mathbb Z$ have Gromov boundary homeomorphic to a menger curve. Unfortunately, the topology of the boundary is not sufficient to detect irreducible and atoroidal monodromies as there are reducible and atoroidal automorphisms that do not fix a free splitting. The Gromov boundary of a word-hyperbolic group can be given a [*visual metric*]{} that is unique up to [*quasi-symmetry*]{} and its quasi-symmetry class is a complete geometric invariant of the group, i.e., word-hyperbolic groups are quasi-isometric if and only their visual boundaries are quasi-symmetric [@Pau96]. Although a reducible and an irreducible automorphism can have word-hyperbolic mapping tori with homeomorphic boundaries, the conjecture asserts that the visual boundaries cannot be quasi-symmetric! {#app} The following is a folk theorem that was used to construct examples of [*fully irreducible*]{} automorphisms, i.e., [*irreducible with irreducible powers (iwip)*]{}, along with examples that are infinite-order irreducible but not fully irreducible [@BH92 Example 1.4]. At the end of the appendix, we will show that the latter examples are complete/exhaustive (Corollary \[irredIWIP\]). \[eg\] If $\phi:F \to F$ is an automorphism induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^{b \ge 1}$ that acts transitively on the boundary components, then $\phi$ is irreducible and it is fully irreducible if and only if $b = 1$. Let $f : \Sigma \to \Sigma$ be the inducing pseudo-Anosov homeomorphism. Suppose $\phi$ was reducible, i.e., there exists a $\phi$-invariant proper free factor system $A_1 * \cdots * A_k$ of $F$. Thus $i_j \phi^k(A_j) \le A_j$ for some inner automorphisms $i_j$. Let $\hat \Sigma\to \Sigma$ be the cover corresponding to $A_j \le F \cong \pi_1(\Sigma)$; the inclusion implies $f^k$ lifts to a map $\hat f: \hat \Sigma \to \hat \Sigma$. Furthermore, up to homotopy, $\hat f$ preserves the core of $\hat \Sigma$, a compact subsurface that supports $A_j$. Let $\gamma \subset \hat \Sigma$ be a peripheral simple closed curve of the core. After replacing $f$ with a power, we can assume $\gamma$ is an $f$-invariant simple closed curve. However, the projection of $\gamma$ to $\Sigma$ may not be a simple closed curve. Using the [*LERF*]{} property of $F \cong \pi_1(\Sigma)$, construct a finite cover $\bar \Sigma$ such that the projection of $\gamma$ to $\Sigma$ lifts to a simple closed curve $\bar \gamma$ in $\bar \Sigma$. Since $\phi$ is an automorphism, after passing to a power, we can assume $f$ lifts to a homeomorphism $\bar f: \bar \Sigma \to \bar \Sigma$. This map is pseudo-Anosov since the $f$-invariant measured foliations on $\Sigma$ lift to $\bar f$-invariant measured foliations on $\bar \Sigma$. After passing to power again, we can assume $\bar \gamma$ is $\bar f$-invariant. But the only invariant nontrivial simple closed curves of a pseudo-Anosov homemorphism are the peripheral curves. Thus, the projection of $\gamma$ to $\Sigma$ is a (power of a) peripheral curve. Since this holds for any boundary component $\gamma$ of the core of $\hat \Sigma$, it must be that $A_j$ is a proper free factor corresponding to a boundary component of $\Sigma$. But $f$ acts transitively on the boundary components, hence there is a one-to-one correpondence between the proper free factors $A_1, \ldots, A_k$ and the boundary components of $\Sigma$. This is a contradiction, as all boundary components of a surface can not be simultaneously realized as proper free factors of the surface group. Therefore, $\phi$ is irreducible. If $b=1$, then all powers of $f$ are pseudo-Anosovs that act transitively on the boundary component. Thus, all powers of $\phi$ are irreducible. If $b \ge 2$, then any boundary component of $\Sigma$ determines a periodic proper free factor. For the rest of the appendix, we will prove the converse. Bestvina-Handel use the following proposition to prove that fully irreducible automorphisms with periodic nontrivial conjugacy classes are induced by pseudo-Anosov homeomorphisms of surfaces with one boundary component [@BH92 Proposition 4.5]. Suppose $\phi:F \to F$ is fully irreducible and there exists $k \ge 1$ and nontrivial conjugacy class $[c]$ of $F$ such that $[\phi^k(c)] = [c]$. Then some iterate of $\phi$ has an irreducible train track representative with exactly one (unoriented) indivisible Nielsen loop which covers each edge of the graph twice. In particular, if $k$ is minimal, then $k \le 2$ with equality if and only if $[\phi(c)] = [c^{-1}]$. We start by extending this proposition to irreducible endomorphisms. For brevity, we assume familiarity with Bestvina-Handel’s argument [@BH92 Section 3]. The main change is we work with periodic indivisible Nielsen paths (of a fixed period) rather than indivisible Nielsen paths. The argument in the final paragraph is different too: unlike Bestvina-Handel’s argument where periodic proper free factors are enough to contradict fully irreducibility, we need to construct a fixed proper free factor system to contradict irreducibility. \[propStab\] Suppose endomorphism $\phi:F \to F$ is irreducible, it has infinite-order, and there exists $k \ge 1$ and nontrivial conjugacy class $[c]$ of $F$ such that $[\phi^k(c)] = [c]$. Then there exists an irreducible train track representative $f:\Gamma \to \Gamma$ with exactly one $f$-orbit of (unoriented) periodic indivisible Nielsen paths that make up an $f$-orbit of (unoriented) periodic Nielsen loops that collectively cover each edge of $\Gamma$ twice. Represent $\phi$ by an expanding irreducible train track map $f:\Gamma \to \Gamma$. Let $\sigma$ be a loop representing $[c]$. By hypothesis, $f^k(\sigma) \simeq \sigma$. Break $\sigma$ into maximal legal segments $\sigma_0, \ldots, \sigma_s$, and note that, since $f$ is a train track, tightening the loop $f^k(\sigma)$ produces a cyclic permutation of the maximal legal segments $\sigma_i, \ldots \sigma_{s+i}$ for some $i$. Then $f^{ks}(\sigma)$ will give the original maximal legal segments $\sigma_0, \ldots, \sigma_s$. In particular, as $f$ is expanding, each segment $\sigma_j$ has a $f^{ks}$-fixed point and $\sigma$ is an $f^{ks}$-Nielsen loop, or equivalently, $\sigma$ is an $f$-periodic Nielsen loop. The goal is to replace $f$ with another train track representative such that the $f$-orbit of $\sigma$ is a set of (unoriented) loops that collectively cover each edge of $\Gamma$ twice. For the rest of the proof, a periodic indivisible Nielsen path (piNp) will refer to an indivisible Nielsen path of the fixed $ks$-iterate. It follows from the bounded cancellation lemma that there are finitely many piNps. The $f$-orbit of $\sigma$ breaks into piNps; just like indivisible Nielsen paths, a piNp $\rho$ can be uniquely written as a concatenation of two legal paths $\rho = \alpha \beta$. By an $f$-orbit of unoriented piNps, we mean the minimal sequence $\{ \rho_0 = \rho, \rho_1 = f(\rho), \ldots, \rho_m = f^m(\rho) \}$ such that $f^{m+1}(\rho) = \rho \text{ or } \bar \rho$ for some piNp $\rho$. If we write each $\rho_i$ in the $f$-orbit of $\rho$ as $\rho_i = \alpha_i \beta_i$, then we get $f(\alpha_{i-1}) = \alpha_i \tau_i$ and $f(\beta_{i-1}) = \bar \tau_i \beta_i$, with the exception at the end where possibly (due to reversal of orientation) $f(\alpha_m) = \bar \beta_0 \tau_0$ and $f(\beta_m) = \bar \tau_0 \bar \alpha_0$. Since $f$ is expanding, at least one of the paths $\tau_i$ is nontrivial. We will now describe the process of folding an $f$-orbit of piNps. Given an $f$-orbit of piNps, $\{\alpha_0\beta_0, \ldots, \alpha_m\beta_m\}$, we know that at least one of the corresponding $\tau_i$ is nontrivial. Then we can fold at the turn between $\alpha_{i-1}$ and $\beta_{i-1}$. As more than one of the $\tau_i$ may be nontrivial, the full folds take precedence over the partial folds. We say $f$ is [**stable**]{} if the process of folding its $f$-orbits of piNps can always be done by full folds. If $f$ has no piNps, then it is vacuously stable. Since full folds do not increase the number of vertices in the graph, the process of folding a stable representative produces finitely many [*projective (equivalences) classes*]{} of graphs. $\phi$ has a stable representative. Suppose $f$ was not stable. Then after doing some preliminary full folds, $f$ has an $f$-orbit of piNps whose only possible folds are partial folds. Fold this orbit enough times so that every turn ${\bar \alpha_i, \beta_i}$ is at a trivalent vertex. Now apply a homotopy so that the segments $\{\alpha_0\beta_0, \ldots, \alpha_m\beta_m\}$ are isometrically and cyclically permuted. Since $\phi$ is irreducible, these segments form an invariant forest which we can collapse to create a representative with strictly fewer piNps. As there were finitely many piNps to begin with, this process will terminate with a stable representative. This ends the claim. A stable representative $f:\Gamma \to \Gamma$ has at most one $f$-orbit of piNps. Assume has at least one $f$-orbit of piNps $\{\rho_i\}_{i=0}^m$. Let $\Gamma$ have a [*Perron-Frobenius eigenmetric*]{} and denote with $\operatorname{vol}(\Gamma)$ the sum of all the edge lengths. Folding an $f$-orbit of piNps $\{\rho_i\}_{i=0}^m$ produces a graph $\Gamma'$ with $\operatorname{vol}(\Gamma') = \operatorname{vol}(\Gamma) - x$ and an $f'$-orbit of piNps $\{\rho_i'\}_{i=0}^m$ of $\Gamma'$ with $\operatorname{vol}(\{\rho_i'\}_{i=0}^m) = \operatorname{vol}(\{\rho_i\}_{i=0}^m) - 2x$ for some $x > 0$; the volume of a collection of paths is the sum of their lengths. Since there are finitely many projective classes of graphs, the graph $\Gamma$ and $f$-orbit $\{\rho_i\}_{i=0}^m$ must satisfy the [**critical equation**]{} $\operatorname{vol}\left(\{\rho_i\}_{i=0}^m\right) = 2 \operatorname{vol}(\Gamma)$. Note that this equation holds for any $f$-orbit of piNps. Fix one such $f$-orbit $\{\rho_i\}_{i=0}^m$ and suppose there were an $f$-orbit $\{r_i\}_{i=0}^n$. If $\{\rho_i\}_{i=0}^m$ and $\{r_i\}_{i=0}^n$ did not share all their illegal turns, then folding $\{\rho_i\}_{i=0}^m$ would eventually decrease $\operatorname{vol}(\Gamma)$ while leaving $\operatorname{vol}(\{r_i\}_{i=0}^n)$ the same. This would break the critical equation for $\{r_i\}_{i=0}^n$, contradicting stability. Therefore, all $f$-orbits of piNps have the same set of illegal turns. Since each fold in a stable representative is a full fold, there cannot be two distinct $f$-orbits of piNps that share the same illegal turns and maintain the critical equation throughout all the folds. This ends the second claim. Stable representative $f:\Gamma \to \Gamma$ has exactly one $f$-orbit of piNps that make up an $f$-orbit of periodic Nielsen loops that collectively cover each edge of $\Gamma$ twice. The process of folding an $f$-orbit of piNps preserves periodic Nielsen loops. Since we started with a representative in which $\sigma$ is a periodic Nielsen loop, we know $f$ has a periodic Nielsen loop in $\Gamma$. In particular, the loop splits into piNps and therefore $f$ has exactly one $f$-orbit of piNps by the previous claim. This orbit makes up an $f$-orbit of periodic Nielsen loops containing $\sigma$. Let $\{s_i\}_{i=0}^n$ be a minimal length $f$-orbit of periodic Nielsen loops formed by concatenating paths in the $f$-orbit of piNps. To maintain the critical equation, folding $\{s_i\}_{i=0}^n$ must eventually reduce the lengths of all edges; therefore, every edge of $\Gamma$ appears at least once in $\{s_i\}_{i=0}^n$. Suppose some edge of $\Gamma$ appeared exactly once in $\{s_i\}_{i=0}^n$, say in $s_n$. Then the loop $s_n$ determines a cyclic free factor $C_n$ of $F$ such that $F = B * C_n$ where all the other loops $s_i~(i \neq n)$ determine elements in $B$. As $\phi$ is injective, $s_1^{\pm1} = [f(s_n)]$ determines a cyclic free factor of $\phi(F) \cap B$; since $s_n$ and $s_1$ can be simultaneously realized as free factors of $\phi(F)$ and $\phi$ is injective, the loops $s_{n-1}$ and $s_n$ determine cyclic free factors $C_{n-1}, C_n$ of $F$ that can be simultaneously realized, i.e., $F = B' * C_{n-1} * C_n$. Note that we use preimages to get the free factors of $F$ since we did not assume $\phi$ was an automorphism. Iterate this process to show $\{s_i\}_{i=0}^n$ determines a $\phi$-fixed proper free factor system $\{C_0, \ldots C_n \}$. This contradicts the irreducibility assumption on $\phi$. Thus every edge of $\Gamma$ appears at least twice in $\{s_i\}_{i=0}^n$ and the critical equation implies every edge appears exactly twice. This concludes the third claim and proof of the proposition. The proof of the next theorem follows that of Bestvina-Handel [@BH92 Proposition 4.5]. We give an outline with a modification that extends the argument to injective endomorphisms. \[thmBH2\] Suppose an endomorphism $\phi:F \to F$ is irreducible, it has infinite-order, and there exists $k \ge 1$ and nontrivial conjugacy class $[c]$ of $F$ such that $[\phi^k(c)] = [c]$. Then $\phi$ is an automorphism induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^{b \ge 1}$ that acts transitively on the boundary components. Apply Proposition \[propStab\] to get an irreducible train track $f:\Gamma \to \Gamma$ with an $f$-orbit of (unoriented) periodic indivisible Nielsen loops that collectively cover each edge of $\Gamma$ twice. Let $b \ge 1$ be the number of these periodic Nielsen loops. For each periodic Nielsen loop, attach an annulus by gluing one end along the loop and call the resulting space $M$; $M$ contains $\Gamma$ as a deformation retract, so $\pi_1(M) \cong F$. Since $f$ transitively permutes the periodic Nielsen loops (possibly reversing orientation, up to homotopy), the map $f$ extends to a map $g:M \to M$ that transitively permutes the components of $\partial M$ such that $g_* = f_* = \phi$; so $g$ is $\pi_1$-injective. Since the loops collectively covered each edge twice, the space $M$ is a surface except at finitely many singularities. Use the blow-up trick and the irreducibility of $\phi$ to conclude $M$ is a surface. Thus $g$ is a $\pi_1$-injective map of a surface that transitively permutes the boundary components. Let $D(M)$ be the closed hyperbolic surface obtained by gluing two copies of $M$ along their boundary components. The map $g$ induces a $\pi_1$-injective map $g \cup_\partial g : D(M) \to D(M)$ such that $(g \cup_\partial g)_* = \phi *_\partial \phi$. But closed hyperbolic surfaces have coHopfian fundamental group (classification of surfaces), therefore $\phi *_\partial \phi$ is an automorphism. This implies $\phi$ is an automorphism and the map $g$, a homotopy equivalence, is homotopic to a homeomorphism. Assume $g$ is a homeomorphism. Any $g$-invariant collection of disjoint essential simple closed curves of $g$ determines a reduction of $\phi$, thus the irreducibility of $\phi$ implies $g$ is an infinite-order irreducible homeomorphism. By Nielsen-Thurston classification, the map $g$ is isotopic to a pseudo-Anosov homeomorphism of $M = \Sigma_g^b$ that acts transitively on the boundary components. \[irredIWIP\] If $\phi:F \to F$ is an infinite-order irreducible endomorphism that is not fully irreducible, then $\phi$ is induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^{b \ge 2}$ that acts transitively on the boundary components. It follows from Proposition \[invSbgrp\] that for atoroidal endomorphisms, irreducible implies fully irreducible (see also [@DKL15; @JPM]). So $\phi$ has a nontrivial periodic conjugacy class. By Theorem \[thmBH2\], $\phi$ is induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^{b \ge 1}$ that acts transitively on the boundary components. If $\phi$ is not fully irreducible, then $b \ge 2$ by Proposition \[eg\].

--- abstract: | Zero-Knowledge Proof-of-Identity from trusted public certificates (e.g., national identity cards and/or ePassports; eSIM) is introduced here to permissionless blockchains in order to remove the inefficiencies of Sybil-resistant mechanisms such as Proof-of-Work (i.e., high energy and environmental costs) and Proof-of-Stake (i.e., capital hoarding and lower transaction volume). The proposed solution effectively limits the number of mining nodes a single individual would be able to run while keeping membership open to everyone, circumventing the impossibility of full decentralization and the blockchain scalability trilemma when instantiated on a blockchain with a consensus protocol based on the cryptographic random selection of nodes. Resistance to collusion is also considered. Solving one of the most pressing problems in blockchains, a zk-PoI cryptocurrency is proved to have the following advantageous properties: - an incentive-compatible protocol for the issuing of cryptocurrency rewards based on a unique Nash equilibrium - strict domination of mining over all other PoW/PoS cryptocurrencies, thus the zk-PoI cryptocurrency becoming the preferred choice by miners is proved to be a Nash equilibrium and the Evolutionarily Stable Strategy - PoW/PoS cryptocurrencies are condemned to pay the Price of Crypto-Anarchy, redeemed by the optimal efficiency of zk-PoI as it implements the social optimum - the circulation of a zk-PoI cryptocurrency Pareto dominates other PoW/PoS cryptocurrencies - the network effects arising from the social networks inherent to national identity cards and ePassports dominate PoW/PoS cryptocurrencies - the lower costs of its infrastructure imply the existence of a unique equilibrium where it dominates other forms of payment **Keywords**: zero-knowledge, remote attestation, anonymous credentials, incentive compatibility, dominant strategy equilibria, Nash equilibria, Price of Crypto-Anarchy, Pareto dominance, blockchain, cryptocurrencies author: - | David Cerezo Sánchez^^\ [david@calctopia.com]{} bibliography: - '15\_home\_amnesia\_Persistent\_articleHaMafteach\_bib.bib' title: '**Zero-Knowledge Proof-of-Identity**: [Sybil-Resistant, Anonymous Authentication on Permissionless Blockchains and Incentive Compatible, Strictly Dominant Cryptocurrencies]{}' --- Introduction ============ Sybil-resistance for permissionless consensus comes at a big price since it needs to waste computation using Proof-of-Work (PoW), in addition to assuming that a majority of the participants must be honest. In contrast, permissioned consensus is able to overcome these issues assuming the existence of a Public-Key Infrastructure[@aaba; @cpps; @pbft] otherwise it would be vulnerable to Sybil attacks[@the-sybil-attack]: indeed, it has been recently proved[@cryptoeprint:2018:302] that consensus without authentication is impossible without using Proof-of-Work. Proof-of-Stake, the alternative to PoW, is economically inefficient because participants must keep capital at stake which incentivise coin hoarding and ultimately leads to lower transaction volume. Another major challenge in permissionless blockchains is scalability, both in number of participants and total transaction volume. Blockchains based on Proof-of-Work are impossible to scale because they impose a winner-take-all contest between rent-seeking miners who waste enormous amounts of resources, and their proposed replacements based on Proof-of-Stake don’t exhibit the high decentralization desired for permissionless blockchains. The solution proposed in this paper prevents Sybil attacks without resorting to Proof-of-Work and/or Proof-of-Stake on permissionless blockchains while additionally guaranteeing anonymous identity verification: towards this goal, zero-knowledge proofs of trusted PKI certificates (i.e., national identity cards and/or ePassports) are used to limit the number of mining nodes that a single individual could run; alternatively, a more efficient solution based on mutual attestation is proposed and demonstrated practical \[sub:Performance-Evaluation\]. Counterintuitively, the blockchain would still be permissionless even though using government IDs because the term “permissionless” literally means “without requiring permission” (i.e., to access, to join, ...) and governments would not be authorizing access to the blockchain; moreover, the goal is to be open to all countries of the world \[sub:Worldwide-Coverage-and\], thus its openness is indistinguishable from PoW/PoS blockchains (i.e., the union of all possible national blockchains equals a permissionless, open and global blockchain). Coincidently, the latest regulations [@chinaIdentityRegulation; @chinaRules] point to the obligation to verify and use real-world identities on blockchains, and the banning of contaminant cryptocurrency mining[@chinaBanMining; @NDRCbanlist]. Blockchain research has focused on better consensus algorithms obviating that incentives are a central aspect of permissionless blockchains and that better incentive mechanisms would improve the adoption of blockchains much more that scalability improvements. To bridge this gap, new proofs are introduced to demonstrate that mining a new cryptocurrency based on Zero-Knowledge Proof-of-Identity would strictly dominate previous PoW/PoS cryptocurrencies, thus replacing them is proved to be a Nash equilibrium; additionally, the circulation of the proposed cryptocurrency would Pareto dominate other cryptocurrencies. Furthermore, thanks to the network effects arising from the network of users of trusted public certificates, the proposed cryptocurrency could become dominant over previous cryptocurrencies and the lower costs of its infrastructure imply the existence of a unique equilibrium where it dominates other forms of payment. Contributions ------------- The main and novel contributions are: - The use of anonymous credentials in permissionless blockchains in order to prevent Sybil attacks \[sec:Authentication-Protocols\]: previous works[@DBLP:journals/corr/KulynychITD17; @blockchainCA] considered the use of PKI infrastructures in blockchains (i.e., permissioned ledgers) but without transforming them into anonymous credentials in order to obtain the equivalent of a permissionless blockchain. Other works have considered anonymous credentials on blockchains[@cryptoeprint:2013:622; @coconut; @DBLP:conf/ccs/CamenischDD17; @quisquis; @indyAnonCreds], but requiring the issuance of new credentials and not reusing previously existing ones: verifying real-world identities and issuing their corresponding digital certificates is the most expensive part of any real-world deployment. - The practical implementation and its perfomance evaluation \[sub:Performance-Evaluation\] for national identity cards and ePassports. - Circumventing the impossibility of full decentralization \[sub:Circumventing-the-Impossibility\] and the blockchain scalability trilemma. - A protocol for an incentive-compatible cryptocurrency \[alg:Incentive-Compatible-Protocol\]: previous blockchains mint cryptocurrencies tied to the process of reaching a consensus on the order of the transactions, but the game-theoretic properties of this mechanism is neither clear nor explicit. - A proof that mining the proposed cryptocurrency is a dominant strategy over other PoW/PoS blockchains and a Nash equilibrium over previous cryptocurrencies \[sub:Strictly-Dominant-Cryptocurrenci\], in addition to an Evolutionary Stable Strategy \[sub:Evolutionary-Stable-Strategies\]. - The insight that the optimal efficiency of zk-PoI resides in that it’s implementing the social optimum, unlike PoW/PoS cryptocurrencies that have to pay the Price of (Crypto-)Anarchy \[sub:Price-of-Crypto-Anarchy\]. - A proof that the circulation of the proposed zk-PoI cryptocurrency Pareto dominates other PoW/PoS cryptocurrencies \[sub:Pareto-Dominance-on\]. - A proof that the proposed cryptocurrency could become dominant over previous ones due to stronger network effects and the lack of acceptance of previous cryptocurrencies as a medium of payment \[sub:On-Network-Effects\]. - Finally, the lower costs of its infrastructure imply the existence of a unique equilibrium where it dominates other forms of payment \[sub:Dominance-over-Cash\]. Related Literature ================== This section discusses how the present paper is significantly better and more innovative than previous approaches in order to fulfill the objective of providing a Sybil-resistant and permissionless blockchain with anonymous transaction processing nodes (i.e., miners). Moreover, it’s considerably cheaper than other approaches[@coconut; @pop] that would require the re-identification and issuing of new identities to the global population because the current proposal relies on the previously issued credentials of electronic national identity cards (3.5 billion issued at the time of publication) and electronic passports (1 billion issued at the time of publication). Proof of Space[@cryptoeprint:2013:796; @cryptoeprint:2013:805] reduces the energy costs of Proof-of-Work but it’s not economically efficient. Proof of Authority[@poa](PoA) maintains a public list of previously authorised nodes: the identities are not anonymised and the blockchain is not open to everyone (i.e., the blockchain is permissioned). Proof of Personhood[@pop](PoP) can be understood as an improvement over Proof of Authority in that identities are anonymised, but the parties/gatherings used to anonymise and incorporate identities into the blockchain don’t scale to national/international populations and could compromise Sybil resistance because it’s trivial to get multiple identities by using different disguises on different parties/gatherings (i.e., they need to be validated simultaneously and without disguises): however, the present paper produces Sybil-resistant, anonymised identities on a global scale for a permissionless blockchain. Moreover, Proof of Personhood[@pop] is endogenizing all the costly process of credential verification and issuing: by contrast, Zero-Knowledge Proof of Identity is exogenizing/outsourcing this costly process to governments, thus making the entire blockchain system cheaper. More recently, Private Proof-of-Stake protocols[@cryptoeprint:2018:1105; @cryptoeprint:2018:1132](PPoS) achieve anonymity, but the economic inefficiencies of staking capital still remain and the identities have no relation to the real world. A conceptually close work (“*Decentralized Multi-authority Anonymous Authentication for Global Identities with Non-interactive Proofs*”, [@cryptoeprint:2019:701]), concurrently developed, doesn’t reuse real-world certificates and therefore it would require that governments re-issue the cryptographic credentials of their citizens: therefore, it doesn’t consider neither Sybil-resistance nor blockchain integration. Pseudo-anonymous signatures[@pseudoanonymousSignatures] for identity documents provide an interesting technical solution to the problem of anonymous authentication using identity documents. However, the proposed schemes present a number of shortcomings that discourage their use in the present setting: some schemes are closely tied to particular countries (i.e., the German Identity Card[@cryptoeprint:2012:558; @bsitr03110; @cryptoeprint:2018:1148]), thus non-general purpose enough to include any country in the world, or flexible to adapt to future changes; they require interaction with an issuer during card initialization; they feature protocols for deanonymisation and revocation, not desired in the setting considered in this paper; the initial German scheme[@bsitr03110] could easily be subverted[@10.1007/978-3-319-40367-0_31] because the formalization of pseudo-anonymous signatures is still incipient[@10.1007/978-3-319-49151-6_17], and improvements are being worked out[@cryptoeprint:2014:067; @cryptoeprint:2016:070; @cryptoeprint:2018:1148]. Anonymous credentials, first envisioned by David Chaum[@10.1007/978-1-4757-0602-4_18], and first fully realised by Camenisch and Lysyanskaya[@cryptoeprint:2001:019] with follow-up work improving its security/performance[@cryptoeprint:2009:107; @cryptoeprint:2005:060; @cryptoeprint:2010:496; @10.1007/978-3-540-28628-8_4; @cryptoeprint:2012:298], are a centrally important building block in e-cash. The use of anonymous credentials to protect against Sybil attacks[@the-sybil-attack] has already been proposed in previous works[@DBLP:conf/wistp/AnderssonKMP08; @cryptoeprint:2007:384] although with different cryptographic techniques and for different goals. The main problem with anonymous credentials is that they require a first identification step to an issuing party[@indyAnonCreds] and that would compromise anonymity. This problem is shared with other schemes for pseudoanymization: for example, Bitnym[@sybilPseudonym] requires that a Trusted Third Party must check the real identity of a user before allowing the creation of a bounded number of valid genesis pseudonyms. Decentralized Anonymous Credentials[@cryptoeprint:2013:622] was first to show how to decentralise the issuance of anonymous credentials and integrate them within a blockchain (i.e., Bitcoin), but they do not re-use previously existing credentials and they still rely on Proof-of-Work for Sybil-resistance. Decentralized Blacklistable Anonymous Credentials with Reputation[@cryptoeprint:2017:389] introduce blacklistable reputation on blockchains, but users must also publish their real-world identity (i.e., non-anonymous). QuisQuis[@quisquis] introduces the novel primitive of updatable public keys in order to provide anonymous transactions in cryptocurrencies, but it doesn’t consider their Sybil-resistance. DarkID[@darkID] is a practical implementation of an anonymous decentralised identification system, but requires non-anonymous pre-authentication and doesn’t consider Sybil-resistance. A previous work[@whoami] on secure identity registration on distributed ledgers achieved anonymity from a credential issuer, but the pre-authentication is non-anonymous, it doesn’t consider Sybil-resistance and it doesn’t re-use real-world cryptographic credentials. Recently, anonymous credentials on standard smart cards have been proved practical[@cryptoeprint:2019:460], but in a different setting where the credential issuer and the verifier are the same entity. Previous works have also considered anonymous PKIs: for example, generating pseudonyms[@anonymousPKI] using a Certificate Authority and a separate Private Certificate Authority; however, this architecture is not coherent for a permissionless blockchain because both certificate authorities would be open to everyone and that would allow the easy linking of anonymous identities. Another recent proposal for a decentralised PKI based on a blockchain[@cryptoeprint:2018:853] does not provide anonymity, although it improves the work on cryptographic accumulators on blockchains started by Certcoin[@certcoin; @cryptoeprint:2015:718]; another proposals introduce privacy-aware PKIs on blockchains[@pbpki; @cryptoeprint:2019:527], but they are not Sybil-resistant and do not re-use certificates from other CAs. Previously, BitNym[@sybilPseudonym] introduced Sybil-resistant pseudonyms to Bitcoin, but a Trusted-Third Party must check the real identities of users before allowing the creation of a bounded number of valid genesis pseudonyms. ChainAnchor[@chainAnchor] wasn’t Sybil-resistant and used Direct Anonymous Attestation just for anonymous authentication, but not for mutual authentication: it worked on the permissioned model, explicitly not permissionless, and the GroupOwner initially knew the true identity of members; moreover, the Permissions Issuer is supposed not to collude with the Verifier, although it has reading access to the identity database. ClaimChain[@DBLP:journals/corr/KulynychITD17] improves the decentralised distribution of public keys in a privacy-preserving way with non-equivocation properties, but it doesn’t consider their Sybil-resistance because it’s more focused on e-mail communications. Blind Certificate Authorities[@WAPRS18; @cryptoeprint:2016:925] can simultaneously validate an identity and prove a certificate binding a public key to it, without ever learning the identity, which sounds perfect for the required scenario except that it requires 3 parties and it’s impossible to achieve in the 2-party setting; moreover, it doesn’t consider Sybil-resistance. Other approaches to anonymous identity include: Lightweight Anonymous Subscription with Efficient Revocation[@cryptoeprint:2018:290], although it doesn’t consider the real-world identity of users because it’s focused on the host and its Trusted Platform Module; One Time Anonymous Certificates[@onetimeanonymouscertificate] extends the X.509 standard to support anonymity through group signatures and anonymous credentials, although it doesn’t consider Sybil-resistance and their group signatures require that users hold two group secret keys, a requisite that is not allowed in the current scenario because the user is not trusted to store them on the national identity card (for the very same reasons, Linkable Ring Signatures[@cryptoeprint:2004:027] and Linkable Message Tagging[@cryptoeprint:2014:014] are not allowed as cryptographic tools whilst group signatures and Deniable Anonymous Group Authentication[@daga] would require a non-allowed setup phase). Opaak[@opaak] provides anonymous identities with Sybil-resistance based on the scarcity of mobile phone numbers: however, users must register by receiving an SMS message (i.e., the Anonymous Identity Provider knows the real identity of participants). Oblivious PRFs[@cryptoeprint:2018:733] are not useful in the permissionless blockchains because the secret key of the OPRF would be known by everyone, and the forward secrecy of the scheme that would provide security even if the secret key is known would not be of any use because the object identifiers ObjID would be easily predictable (i.e., derived from national identifiers). SPARTA[@sparta] provides pseudonyms through a distributed third-party-based infrastructure; however, it requires non-anonymous pre-registration. UnlimitID[@unlimitid] provides anonymity to OAuth/OpenID protocols, although users must create keypairs and keep state between and within sessions, a requisite that is not allowed in the current scenario. Another proposal for anonymous pseudonyms with one Trusted-Third Party[@Yilek_traceableanonymous] requires a division of roles between the TTP and the server that is not coherent in a permissionless blockchain. With Self-Certified Sybil-free Pseudonyms[@MKAP08; @DBLP:conf/wistp/AnderssonKMP08], the user must keep state (i.e., dispenser D) generated by the issuer during enrollment and the Sybil-free identification is based on unique featurs of the devices, not on the user identity. Another anonymous authentication using smart cards[@smartcardAnonAuth] is only anonymous from an eavesdropping adversary, not from the authentication server itself. TATA provides a novel way to achieve Sybil-resistant anonymous authentication: members of an induction group must interact and keep a list of who has already been given a pseudonym; therefore, a list of participants could be collected, but they can’t be linked to their real-world identities; it’s not clear how to bootstrap the initial set of trusted users to get them to blindly sign each other’s certificates. Self-sovereign identity solutions usually rely on identities from social networks, but their Sybil-resistance is very questionable because almost half of their accounts could be fake[@facebookNYT]: in spite of this, SybilQuorum[@sybilQuorum; @sybilQuorumArxiv] proposes the use of social network analysis techniques to improve their Sybil-resistance; other research projects consider privacy-preserving cryptographic credentials from federated online identities[@2014arXiv1406.4053M]. Regarding the game-theoretic aspects, most papers focus on attacking only one cryptocurrency (e.g., selfish mining[@DBLP:journals/corr/EyalS13], miner’s dilemma[@DBLP:journals/corr/Eyal14], fork after withholding[@DBLP:journals/corr/abs-1708-09790]). For a recent survey of these topics, see[@sokGameTheoryCryptocurrencies]. Exceptionally, “*Game of Coins*”[@DBLP:journals/corr/abs-1805-08979] considers the competition between multiple cryptocurrencies: a manipulative miner alters coin rewards in order to move miners to other cryptocurrencies of his own interest (with a fixed cost and a finite number of steps). However, in this paper, it’s the cryptocurrency issuer who changes the rewards in order to attract miners from other cryptocurrencies by producing the most efficient cryptocurrency to mine. **** PoW PoSpace PoS PPoS PoA PoP **zk-PoI** ------------------------ ---------- ---------- ---------- ---------- ---------- ---------- ------------ (Pseudo)-Anonymity $\times$ $\times$ **** Energy-Efficient $\times$ **** Economically Efficient $\times$ $\times$ $\times$ $\times$ $\times$ **** Permissionless $\times$ ([\*]{}) **** Proof-of-Personhood Considered Harmful (and Illegal) ---------------------------------------------------- To be considered lawful in the real world, Proof-of-Personhood (PoP, [@pop]) requires the concurrence of multiple unrestricted freedoms: assembly, association, and wearing of masks. However, in most countries these freedoms are limited: - freedom of assembly[@freedomAssembly] and association[@freedomAssociation]: most countries usually require previous notification and permission from the governing authorities, that may reject for multiple grounds including but not limited a breach of public order. Thus, PoP cannot be considered permissionless in these countries. - it’s forbidden to wear a mask in most countries[@antiMaskLaw], as required for the anonimity of PoP (“All parties are recorded for transparency, but attendees are free to hide their identities by dressing as they wish, including hiding their faces for anonymity.”, [@pop]). Thus, PoP won’t be anonymous in countries that outlaw the covering of faces. - promoters and organizers of PoP parties may themselves be committing a crime, due to incitement, conspiracy and complicity. The solution proposed in this paper it’s the only possible lawful one according to current regulations that require the use of national IDs to register on blockchains (AMLD5[@AMLD5], FATF[@fatfGuidance], Cyberspace Administration of China[@chinaRules; @chinaIdentityRegulation]). Building Blocks =============== Consensus based on the Cryptographic Random Selection of Transaction Processing Nodes\[sub:Modern-Consensus-based-on\] ---------------------------------------------------------------------------------------------------------------------- The new family of consensus algorithms based on the cryptographically random selection of transaction processing nodes[@cryptoeprint:2016:918; @cryptoeprint:2016:919; @cryptoeprint:2017:406; @cryptoeprint:2017:454; @dfinityConsensus] is characterised by: [|c|&gt;p[5cm]{}|c|]{} Consensus algorithm & Random selection method & Sybil resistance[\ ]{} OmniLedger & PVSS + collective BLS/BDN signatures [@cryptoeprint:2016:1067; @cryptoeprint:2017:406; @cryptoeprint:2018:483; @cryptoeprint:2019:676] & PoW/PoS[\ ]{} RapidChain & Performance improvements over OmniLedger[@cryptoeprint:2018:460] & PoS[\ ]{} Algorand & Cryptographic sortition by a unique digital signature & PoS[\ ]{} Dfinity & BLS threshold signature scheme[@Boneh01shortsignatures] & PoS[\ ]{} Snow White & Extract public keys based on the amount of currency owned & PoS[\ ]{} - Transaction processing workers/nodes are randomly selected from a larger group: in the case of Dfinity[@dfinityConsensus], an unbiasable, unpredictable verifiable random function (VRF) based on the BLS threshold signature scheme[@Boneh01shortsignatures] with the properties of uniqueness and non-interactivity; in the case of OmniLedger[@cryptoeprint:2017:406], the original proposal used a collective Schnorr threshold signature scheme[@DBLP:journals/corr/SytaTVWF15; @cryptoeprint:2016:1067; @cryptoeprint:2017:406], although it has been updated to collective BLS/BDN signatures[@cryptoeprint:2018:483] and now it uses MOTOR[@cryptoeprint:2019:676] instead of ByzCoin[@DBLP:journals/corr/Kokoris-KogiasJ16] with improvements for open and public settings; in the case of Algorand, secure cryptographic sortition is generated using an elliptic curve-based verifiable random function (ECVRF-ED25519-SHA512-Elligator2[@algorandVRF]); in the case of Snow White, cryptographic committee reconfiguration is done by extracting public keys from the blockchain based on the amount of currency owned. For a detailed comparison of random beacon protocols, see [@cryptoeprint:2018:319]. - Regular time intervals (also named epochs or rounds) on which randomly selected workers/nodes process the transactions. - Faster transaction confirmation and finality. - High scalability. - Decoupling Sybil-resistance from the consensus mechanism (PoW/PoS is about membership, not consensus). - PoW/PoS to protect against Sybil attacks: however, the present paper proposes the use of Zero-Knowledge Proof-of-Identity (i.e., more economically[@stakedPoS] and environmentally efficient[@natureEnergyCarbonCosts; @bitcoinCarbonFootprint]). X.509 Public Key Infrastructure\[sub:X.509-Public-Key\] ------------------------------------------------------- X.509 is an ITU-T standard[@x509] defining the format of public key certificates, itself based on the ASN.1 standard[@asn1]: these certificates underpin most implementations of public key cryptography, including SSL/TLS and smartcards. An X.509v3 certificate has the following structure: Certificates are signed creating a certificate chain: the root certificate of an organization is a self-signed certificate that signs intermediate certificates that themselves are used to sign end-entities certificates. To obtain a signed certificate, the entity creates a key pair and signs a Certificate Signing Request (CSR) with the private key: the CSR contains the applicant’s public key that is used to verify the signature of the CSR and a unique Distinguished Name within the organization. Then, one of the intermediate certificate authorities issues a certificate binding a public key to the requested Distinguished Name and that also contains information identifying the certificate authority that vouches for this binding. The certificate validation chain algorithm checks the validity of an end-entity certificate following the next steps: 1. The certificates are correct regarding the ASN.1 grammar of X.509 certificates. 2. The certificates are within their validity periods (i.e., non-expired). 3. If access to a Certificate Revocation List is granted, the algorithm checks that none of the certificates is included (i.e., the certificate has not been revoked). 4. The certificate chain is traversed checking that: 1. The issuer matches the subject of the next certificate in the chain. 2. The signature is valid with the public key of the next certificate in the chain. 5. The last certificate is a valid self-signed certificated trusted by the end-entity checker. Additionally, the algorithm could also check complex application policies (i.e., the certificate can be used for web server authentication and/or web client authentication). Electronic Passports\[sub:Electronic-Passport\] ----------------------------------------------- **** **Data Group** **Data Elements** ---------------- ------------------------------------ Document Types Issuing State or Organizaton Name (of Holder) Document Number Check Digit - Doc Number Nationality Date of Birth Check Digit - DOB Sex Date of Expiry or Valid Until Date Check Digit DOE/VUD Optional Data Check Digit - Optional Data Field Composite Check Digit Data Group 11 Personal Number Data Group 15 User’s Public Key Modern electronic passports feature NFC chips[@icaoDoc9303part11] that contain all their printed information in digital form, using a proprietary format set by International Civil Aviation Organization[@icaoDoc9303part10] and not X.509 certificates \[sub:X.509-Public-Key\] like the ones used in national identity cards: the relevant fields are contained within its Data Group 1 \[tab:Data-Group-1\] (i.e., the same information available within the Machine Readable Zone), and the Document Security Object contains a hash of all the Data Groups signed by a Document Signing Certificate issued every three months (also stored on the passports), itself signed by a Country Signing Certificate Authority (all the certificates are available online[@icaoPKD]). Additionally, the data within the NFC chips are cryptographically protected and it’s necessary to derive the cryptographic keys by combining the passport number, date of birth and expiry date (i.e., BAC authentication). Finally, note that the electronic identity cards of some countries can also work as ePassports (e.g., Spanish Identity Card -Documento Nacional de Identidad-). Verifiable Computation\[sub:Verifiable-Computation\] ---------------------------------------------------- A public verifiable computation scheme allows a computationally limited client to outsource to a worker the evaluation of a function $F\left(u,w\right)$ on inputs $u$ and $w$: other alternative uses of these schemes allow a verifier $V$ to efficiently check computations performed by an untrusted prover $P$. More formally, the following three algorithms are needed: (Public Verifiable Computation). A public verifiable computation scheme $VC$ consists of three polynomial-time algorithms $\left(\mbox{Keygen},\mbox{ Compute, Verify}\right)$ defined as follows: - $\left(EK_{F},VK_{F}\right)\leftarrow\mbox{Keygen}\left(F,1^{\lambda}\right)$: the key generation algorithm takes the function $F$ to be computed and security parameter $\lambda$; it outputs a public evaluation key $EK_{F}$ and a public verification key $VK_{F}$. - $\left(y,\pi_{y}\right)\leftarrow\mbox{Compute}\left(EK_{F},u,w\right)$: the prover runs the deterministic worker algorithm taking the public evaluation key $EK_{F}$, an input $u$ supplied by the verifier and an input $w$ supplied by the prover. It outputs $y\leftarrow F\left(u,w\right)$ and a proof $\pi_{y}$ of $y$’s correctness (as well as of prover’s knowledge of $w$). - $\left\{ 0,1\right\} \leftarrow\mbox{Verify}\left(VK_{F},u,w,y,\pi_{y}\right)$: the deterministic verification algorithm outputs $1$ if $F\left(u,w\right)=y$, and $0$ otherwise. A public verification computation scheme $VC$ must comply with the following properties of correctness, security, and efficiency: - Correctness: for any function $F$ and any inputs to $F$, if we run $\left(EK_{F},VK_{F}\right)\leftarrow\mbox{Keygen}\left(F,1^{\lambda}\right)$ and $\left(y,\pi_{y}\right)\leftarrow\mbox{Compute}\left(EK_{F},u,w\right)$ then we always get $\mbox{Verify}\left(VK_{F},u,w,y,\pi_{y}\right)=1$. - Efficiency: $\mbox{Keygen}$ is a one-time setup operation amortised over many calculations and $Verify$ is computationally cheaper than evaluating $F$. - Security: for any function $F$ and any probabilistic polynomial-time adversary $A$, we require that $$\mbox{Pr}\left[\left(\hat{u},\hat{w},\hat{y},\hat{\pi_{y}}\right)\leftarrow A\left(EK_{F},VK_{F}\right):F\left(\hat{u},\hat{w}\right)\neq\hat{y}\right]\leq\mbox{negl}\left(\lambda\right)$$ and $$1=\mbox{Verify}\left(VK_{F},\hat{u},\hat{w},\hat{y},\hat{\pi_{y}}\right)\leq\mbox{negl}\left(\lambda\right)$$ where $\mbox{negl}\left(\lambda\right)$ denotes a negligible function of inputs $\lambda$. Additionally, we require the public verification computation scheme $VC$ to be succinct and zero-knowledge: - Succinctness: the generated proofs $\pi_{y}$ are of constant size, that is, irrespective of the size of the function $F$ and inputs $u$ and $w$. - Zero-knowledge: the verifier learns nothing about the prover’s input $w$ beyond the output of the computation. Practical implementations are Pinocchio[@cryptoeprint:2013:279] and Geppeto[@cryptoeprint:2014:976], or Buffet[@cryptoeprint:2014:674] and Pequin[@pequin](a simplified version of Pepper[@Setty12makingargument]). ### Verifiable Validation of X.509 Certificates as Anonymous Credentials The algorithm for certificate chain validation chain in section \[sub:X.509-Public-Key\] can be implemented with the public verifiable computation scheme of section \[sub:Verifiable-Computation\] using zk-SNARKS to obtain a verifiable computation protocol so that a certificate holder is able to prove that he holds a valid X.509 certificate chain with a unique Distinguished Name, without actually sending the public key to the verifier and selectively disclosing the contents of the certificate: in other words, we re-use existing certificate chains and PKI infrastructure without requiring any modifications, turning X.509 certificates into anonymous credentials. A previous work already demonstrated the technical and practical viability of this approach[@cinderella-turning-shabby-x-509-certificates-into-elegant-anonymous-credentials-with-the-magic-of-verifiable-computation]: the only handicap was that the proof generation could take a long time (e.g., more than 10 minutes) and large keys (e.g., 1 Gbyte).. Recent research advances have improved[@cryptoeprint:2017:602] the initial setup of the zk-SNARK protocol used to generate the Common-Reference String (CRS) with an MPC protocol, such that it’s secure even if all participants are malicious (except one). And faster proving times could be obtained by efficiently composing the non-interactive proving of algebraic and arithmetic statements[@cryptoeprint:2018:557] since QAP-based zk-SNARKs are only efficient for arithmetic representations and not algebraic statements, but at the cost of increasing the proof size. In this paper, a practical implementation has been completed to check a certificate chain with an additional validation policy and written as C code for Pequin[@pequin], then compiled into a public evaluation and verification keys: unfortunately, it isn’t scalable to millions of users and/or the large circuits/constraints required to cover all the typologies of national identity cards/ePassports, thus an implementation based on TEE and mutual attestation is the preferred implementation \[sub:Detailed-Authentication-Protocol-Remote-Attestation\]. The only zero-knowledge proof system that could be scalable enough[@cryptoeprint:2018:691] works on a computer cluster, thus it doesn’t fit the setting of a single user authenticating on his own device, and a libsnark backend can’t handle more than 4 million gates requiring more than an hour of computation. Therefore, an implementation only using software means is still Work-In-Progress. Cryptographic Accumulators -------------------------- Firstly devised by Benaloh and de Mare[@10.1007/3-540-48285-7_24], a cryptographic accumulator [@cryptoeprint:2015:087] is a compact binding set of elements supporting proofs of membership and more space-efficient than storing all of the elements of the set; given an accumulator, an element, and a membership witness, the element’s presence in the accumulated set can be verified. Generally speaking, an accumulator consists of four polynomial-time algorithms: - $Generate\left(1^{k}\right)$: given the security parameter $k$, it instantiates the initial value of the empty accumulator. - $Add\left(a,y\right)\rightarrow\left(a',w\right)$: adds the element $y$ to the current state of the accumulator $a$ producing the updated accumulator value $a'$ and the membership witness $w$ for $y$. - $WitnessAdd\left(w,y\right)\rightarrow w'$: on the basis of the current state of a witness $w$ and the newly added value $y$, it returns an updated witness $w'$. - $Verify\left(a,y,w\right)\rightarrow\left\{ true,false\right\} $: verifies the membership of $y$ using its witness $w$ on the current state of accumulator $a$. The following are interesting security properties of accumulators: - Dynamic accumulators[@10.1007/3-540-45708-9_5]: accumulators supporting the removal of elements from the accumulator by means of a deletion algorithm $Removal()$ and a witness update algorithm $WitnessRemoval\left(\right)$. - Universality[@10.1007/978-3-540-72738-5_17]: accumulators supporting non-membership proofs, $NonWitnessAdd\left(\right)$, $NonWitnessRemoval\left(\right)$ and $NonVerify\left(\right)$. - Strong accumulators[@Camacho2012]: deterministic and publicly executable, meaning that it does not rely on a trusted accumulator manager. - Public checkable accumulators, the correctness of every operation can be publicly verified. Recent constructions of cryptographic accumulators specifically tailored for blockchains are: a dynamic, universal, strong and publicly checkable accumulator [@certcoin]; an asynchronous accumulator[@cryptoeprint:2015:718] with low frequency update and old-accumulator compatibility (i.e., up-to-date witnesses can be verified even against an outdated accumulator); a constant-sized, fair, public-state, additive, universal accumulator[@cryptoeprint:2018:853], and an accumulator optimised for batch and aggregation operations[@cryptoeprint:2018:1188]. Remote Attestation ------------------ In the terminology of Intel SGX, remote attestation is used to prove that an enclave has been established without alterations of any kind: in other words, remote parties can verify that an application is running inside an SGX enclave. Concretely, remote attestation is used to verify three properties: the identity of the application, that it has not been tampered with, and that it is running securely within an SGX enclave. Remote attestation is carried out in several stages: requesting a remote attestation from the challenger; performing a local attestation of the enclave; converting said local attestation to a remote attestation; returning the remote attestation to the challenger, and the challenger verifying the remote attestation to the Intel Attestation Service. A detailed technical description is outside of the scope of this paper: detailed descriptions can be found in the standard technical documentation[@cryptoeprint:2016:086; @officialAttestation; @sampleAttestation]. Recent attacks[@vanbulck2018foreshadow] can be used to extract the secret attestation keys used to verify the identity of an SGX enclave, and microcode updates must be installed[@intelL1TF] to prevent their exploitation: that is, it’s essential to check that parties to a remote attestation are using a safe and updated version. However, our protocols are inherently resistant to deniability attacks[@cryptoeprint:2018:424] because they are based on mutual attestation. As it would be shown in the next section \[sub:Detailed-Authentication-Protocol-Remote-Attestation\], remote attestation can be used as a more efficient substitute of verifiable computation. Authentication Protocols\[sec:Authentication-Protocols\] ======================================================== In this section, we describe authentication protocols for Sybil-resistant, anonymous authentication using Zero-Knowledge protocols \[sub:Detailed-Authentication-Protocol\] and remote attestation \[sub:Detailed-Authentication-Protocol-Remote-Attestation\]. Authentication Protocols using Zero-Knowledge\[sub:Detailed-Authentication-Protocol\] ------------------------------------------------------------------------------------- ### Security Goals The following security goals must be met for the system to be considered secure: 1. The registered miner’s key to the blockchain *opens, but no one can shut; he can shut, but no one can open* (*Isaiah 22:22*, [@isaiah2222]). For the security of the system to be considered equivalent to the currently available permissionless blockchains, anyone holding a valid public certificate should be able to register a pseudo-anonymous identity on the blockchain but no one should be able to remove it (i.e., uncensorable free entry is guaranteed). 2. Protection against malicious issuers: some certification authorities may turn against some citizens and try to cancel access to the permissionless blockchain or stole their funds. 1. Mandatory passphrase. An issuer may counterfeit a certificate with the same unique identifiers, thus possessing a valid certificate isn’t secure enough and a passphrase is deemed mandatory. 2. Non-bruteforceable. Operations must be computationally costly on the client side to prevent brute-forcing. 3. No OCSP checking. Prevention against malicious blacklisting. 3. Privacy: miner’s real identity can’t be learned by anyone. 4. Unique pseudonyms: from each identity card/ePassport, only one unique identifier can be generated. 5. Publicly verifiable: anyone should be able to verify the validity of the miner’s public key and its pseudonym. ### Zero-Knowledge Protocols (X.509)\[sub:Zero-Knowledge-Protocols-(X.509)\] **Anonymous miner registration of a new public key on a permissionless blockchain.** This protocol generates a unique pseudonym for each miner, and attaches a verifiable proof that its new public key to be stored on-chain is signed with a valid public certificate included on a recognised certification authorities list, and that the new public key is linked to the blockchain-specific pseudonym that is in turn uniquely linked to the citizen’s public key certificate. Miners holding a public key certificate must execute the following steps: 1. Create a deterministic public/secret key pair based on a secret passphrase (no need for verifiable computation): $$pk,sk=\mbox{Det\_KeyPairGen}\left(KDF\left(passphrase,hash(publicCert)\right)\right)$$ The generation algorithm must be determistic because the smartcard may be unable to store them and/or the miner may loose them (i.e., as in deterministic wallets). KDF is a password-based key derivation function (e.g., PBKDF2). 2. Obtain a signature of the previously generated public key $pk$ with the miner’s public key certificate (no need for verifiable computation, this operation could be executed on a smartcard): $$sign_{PK}=\mbox{PKCS\_Sign}\left(secretKey_{publicCert},pk\right)$$ 3. Check the validity of the certificate chain of the miner’s public key certificate as extracted from the smartcard: 1. Load the public key of the root certificate. 2. Hash and verify all intermediates, based on their certificate templates, and the public key of their parent certificate starting from the root certificate and following with the verified public key from the previous intermediate certificate template. 3. Hash and verify the miner’s public key certificate using the last verified public key returned from the previous step. 4. Check the time validity of the miner’s public key certificate. 5. Check that the miner’s public key certificate is contained on a list of trusted certification authorities. 4. Obtain the unique identifier from the miner’s public key: $$uniqueID=getID(publicCert)$$ Note that the unique identifier is usually contained on Serial Number of the certificate, or the Subject Alternative Name extension under different OIDs, depending on the country. 5. Generate a deterministic pseudonym using the blockchain identifier: $$\begin{aligned} signatureSecret & = & \mbox{PKCS\_Sign}\left(secretKey_{publicCert},\right.\\ & & \left."\mbox{PREFIXED\_COMMON\_STRING}"\right)\end{aligned}$$ $$\begin{aligned} pseudonym & = & Hash\left(signatureSecret||BlockchainIdentifier||uniqueID\right)\\ & & ||"\mbox{REG}"\end{aligned}$$ PKCS\_Sign is the deterministic PKCS\#1.5 signing algorithm executed on a prefixed string to obtain a unique, non-predictable secret based on the certificate’s owner. The obtained signature is appended to the blockchain identifer and the unique identifier, and then hashed to derive a unique pseudonym. Finally, the string “REG” is appended to differentiate this pseudonym from the one generated during a remove protocol and prevent replay attacks for removal reusing the generated zero-knowledge proof. 6. Verify the signature $sign_{PK}$ on the miner’s public key certificate $pk$: $$\mbox{PKCS\_Verify}\left(publicCert,sign_{PK}\right)$$ 7. As the $signatureSecret$ is calculated offline by the smartcard, it’s also necessary to verify it using the miner’s public key certificate $publicCert$: $$\mbox{PKCS\_Verify}\left(publicCert,signatureSecret\right)$$ 8. Generate the zero-knowledge proof $\pi$ (e.g., zk-SNARK, zk-STARK or zk-SNARG) of the miner’s public key certificate $pk$, the generated pseudonym and, signature $sign_{PK}$ such that all the previous conditions 3-7 hold. 9. Anonymously contact the permissionless blockchain: 1. optionally, check the miner’s real identity on a cryptographic accumulator: 1. establish a shared secret running a Diffie-Hellman key exchange between the prospective miner and the permissionless blockchain 2. send attributes of the miner’s real identity encrypted with the shared secret 3. execute the non-membership proof $NonWitnessAdd\left(w,y\right)$ on the cryptographic accumulator 2. register the generated pseudonym, the new public key $pk$, the signature $sign_{PK}$ and $\pi$: note that they don’t reveal the miner’s real identity ($publicCert$, $uniqueID$ and $signatureSecret$ are all keep as a secret). The registering node of the permissionless blockchain verifies $\pi$ before adding the new public key, the associated pseudonym, the signature $sign_{PK}$ and the succinct proof $\pi$: note that the miner is unable to register multiple pseudonyms, and he can only use one running node that would be signing messages with the generated secret key $sk$. Other nodes would be able to efficiently verify $\pi$ to confirm that the public key $pk$ is a signed by someone from an allowed certificate authority, and that the pseudonym is the miner’s unique alias for the blockchain.\ **Taking offline registrations from a permissionless blockchain.** This protocol takes offline a pseudonym and its associated public key $pk$ and signature $sign_{PK}$ from a permissionless blockchain. Miners must execute the following steps to take offline an identity from a permissionless blockchain: 1. Generate a zero-knowledge proof $\pi$ (e.g., zk-SNARK, zk-STARK or zk-SNARG) of the steps 3-7 of the previous protocol to prove secret knowledge of $sk$ and that he’s able to re-generate the pseudonym, but this time appending the string “OFF” to the pseudonym. 2. Anonymously contact the permissionless blockchain to take offline the generated pseudonym and all its associated data (including the cryptographic accumulator), attaching $\pi$. The registering node of the permissionless blockchain verifies $\pi$ before taking offline the pseudonym without learning the real identity of the miner (publicCert, uniqueID and signatureSecret remain secret). ### Zero-Knowledge Protocols (ePassports) Analogous to the zero-knowledge protocols for X.509 \[sub:Zero-Knowledge-Protocols-(X.509)\], but now considering the specific details of ePassports \[sub:Electronic-Passport\], which usually contain a unique keypair with the public key on Data Group 15 and the private key hidden within the chip: the Active Authentication protocol can be used to sign random challenges that can be verified with the corresponding public key. Some ePassports don’t feature Active Authentication, nonetheless a modified version of the following protocols could still be executed (see subsection \[sub:zkAbsence-of-AA\]). **Anonymous miner registration of a new public key on a permissionless blockchain.** This protocol generates a unique pseudonym for each miner, and attaches a verifiable proof that its new public key to be stored on-chain is signed with a valid public certificate included on the list of Country Signing Certificate Authorities, and that the new public key is linked to the blockchain-specific pseudonym that is in turn uniquely linked to the public key certificate of the passport holder. Miners holding a public key certificate must execute the following steps: 1. Create a deterministic public/secret key pair based on a secret passphrase (no need for verifiable computation): $$pk,sk=\mbox{Det\_KeyPairGen}\left(KDF\left(passphrase,hash(publicCert)\right)\right)$$ The *publicCert* is taken from the Data Group 15. KDF is a password-based key derivation function (e.g., PBKDF2). 2. Obtain a signature of the previously generated public key $pk$ with the miner’s public key certificate (no need for verifiable computation, this operation is executed within the ePassport’s chip using the Active Authentication protocol): $$sign_{PK}=\mbox{Sign}\left(secretKey_{publicCert},pk\right)$$ 3. Check the validity of the Data Security Object of the miner’s ePassport: 1. Load the public key of the Country Signing Certificate from a trusted source [@icaoPKD] and the Document Signing Certificate from the ePassport. 2. Hash all the Data Groups and check their equivalence to the Data Security Object. 3. Verify the signature of the Data Security Object using the Document Signing Certificate. 4. Verify the signature of the Document Signing Certificate using the Country Signing Certificate. 5. Check the time validity of the certificates. 4. Obtain the unique identifier of the ePassport: $$uniqueID=getID(DataGroups)$$ Note that the unique identifier is usually contained on the Data Element “Document Number” of the Data Group 1: as it’s legally valid for the same person to own multiple passports with different Document Numbers, some countries include a unique “Personal Number” on the Data Group 11. 5. Generate a deterministic pseudonym using the blockchain identifier: $$\begin{aligned} signatureSecret & = & \mbox{Sign}\left(secretKey_{publicCert},\right.\\ & & \left."\mbox{PREFIXED\_COMMON\_STRING}"\right)\end{aligned}$$ $$\begin{aligned} pseudonym & = & Hash\left(signatureSecret||BlockchainIdentifier||uniqueID\right)\\ & & ||"\mbox{REG}"\end{aligned}$$ Sign is the Active Authentication protocol executed within the ePassport’s chip, a deterministic signing algorithm executed on a prefixed string to obtain a unique, non-predictable secret based on the certificate’s owner. The obtained signature is appended to the blockchain identifer and the unique identifier, and then hashed to derive a unique pseudonym. Finally, the string “REG” is appended to differentiate this pseudonym from the one generated during a remove protocol and prevent replay attacks for removal reusing the generated zero-knowledge proof (e.g., zk-SNARK, zk-STARK or zk-SNARG). 6. Verify the signature $sign_{PK}$ on the miner’s public key certificate $pk$: $$\mbox{PKCS\_Verify}\left(publicCert,sign_{PK}\right)$$ The *publicCert* is taken from the Data Group 15. 7. As the $signatureSecret$ is calculated offline by the ePassport’s chip, it’s also necessary to verify it using the miner’s public key certificate $publicCert$: $$\mbox{PKCS\_Verify}\left(publicCert,signatureSecret\right)$$ The *publicCert* is taken from the Data Group 15. 8. Generate the zero-knowledge proof $\pi$ (e.g., zk-SNARK, zk-STARK or zk-SNARG) of the miner’s public key certificate $pk$, the generated pseudonym, and signature $sign_{PK}$ such that all the previous conditions 3-7 hold. 9. Anonymously contact the permissionless blockchain 1. optionally, check the miner’s real identity on a cryptographic accumulator: 1. establish a shared secret running a Diffie-Hellman key exchange between the prospective miner and the permissionless blockchain 2. send attributes of the miner’s real identity encrypted with the shared secret 3. execute the non-membership proof $NonWitnessAdd\left(w,y\right)$ on the cryptographic accumulator 2. register the generated pseudonym, the new public key $pk$, the signature $sign_{PK}$ and $\pi$: note that they don’t reveal the miner’s real identity ($publicCert$, $uniqueID$ and $signatureSecret$ are all keep as a secret). The registering node of the permissionless blockchain verifies $\pi$ before adding the new public key, the associated pseudonym, the signature $sign_{PK}$ and the succinct proof $\pi$: note that the miner is unable to register multiple pseudonyms, and he can only use one running node that would be signing messages with the generated secret key $sk$. Other nodes would be able to efficiently verify $\pi$ to confirm that the public key $pk$ is a signed by someone from an allowed certificate authority and that the pseudonym is the miner’s unique alias for the blockchain.\ **Taking offline registrations from a permissionless blockchain.** This protocol takes offline a pseudonym and its associated public key $pk$ and signature $sign_{PK}$ from a permissionless blockchain. Miners must execute the following steps to take offline an identity from a permissionless blockchain: 1. Generate a zero-knowledge proof $\pi$ (e.g., zk-SNARK, zk-STARK or zk-SNARG) of the steps 3-7 of the previous protocol to prove secret knowledge of $sk$ and that he’s able to re-generate the pseudonym, but this time appending the string “OFF” to the pseudonym. 2. Anonymously contact the permissionless blockchain to take offline the generated pseudonym and all its associated data (including the cryptographic accumulator), attaching $\pi$. The registering node of the permissionless blockchain verifies $\pi$ before taking offline the pseudonym without learning the real identity of the miner (publicCert, uniqueID and signatureSecret remain secret). ### Mapping to goals The previous protocols achieve the security goals: 1. The registered miner’s key to the blockchain *opens, but no one can shut; he can shut, but no one can open*. Only someone in possession of a valid public certificate can create a unique miner identity on the open blockchain and destroy it. Please note that the signing and verification of steps 2, 5, 6 and 7 are only needed if it’s required to check that the miner is the real owner of the smartcard/ePassport. 2. Protection against malicious issuers: the passphrase is mandatory, there’s no OCSP checking and the protocol is non-bruteforceable because it requires the generation of a proof $\pi$ for every passphrase that is going to be tried (&gt;60 secs per $\pi$). 3. Privacy: miner’s real identity can’t be learned by anyone because publicCert and uniqueID are keep secret. 4. Unique pseudonyms: from each identity card/ePassport, only one unique identifier can be generated because there’s only one uniqueID per citizen. 5. Publicly verifiable: using the proof $\pi$, anyone is able to validate the miner’s public key and its pseudonym. Additionally, cryptographic accumulators could be added to the protocols in order to prevent multiple registrations whenever an expired certificate is renovated. ### Absence of Active Authentication\[sub:zkAbsence-of-AA\] Signing using the secret key of the Active Authentication protocol provides an extra layer of security: it guarantess that the remote party executing the protocol owns a physical copy of the ePassport (i.e., it hasn’t stolen a copy of the public certificates from others). However, some ePassports don’t feature Active Authentication, requiring a simplified version of the previous protocols: - Steps 2,6 and 7 are removed. - Step 5 doesn’t calculate the signature. - The zero-knowledge$\pi$ is extended to Step 1, with a password-based key derivation function using less steps. Detailed Authentication Protocols using Mutual Attestation\[sub:Detailed-Authentication-Protocol-Remote-Attestation\] --------------------------------------------------------------------------------------------------------------------- ![\[fig:Legend:-(1)-Attestation\]Simplified overview of mutual attestation.](mutualAttestation.png) ### Security Goals The following security goals must be met for the system to be considered secure: 1. The registered miner’s key to the blockchain *opens, but no one can shut; he can shut, but no one can open* (*Isaiah 22:22*, [@isaiah2222]). For the security of the system to be considered equivalent to the currently available permissionless blockchains, anyone holding a valid public certificate should be able to register a pseudo-anonymous identity on the blockchain but no one should be able to remove it (i.e., uncensorable free entry is guaranteed). 2. Protection against malicious issuers: some certification authorities may turn against some citizens and try to cancel access to the permissionless blockchain or stole their funds. 1. Mandatory passphrase. An issuer may counterfeit a certificate with the same unique identifiers, thus possessing a valid certificate isn’t secure enough and a passphrase is deemed mandatory. 2. Non-bruteforceable. Operations must be computationally costly on the client side to prevent brute-forcing. 3. No OCSP checking. Prevent against malicious blacklisting. 3. Privacy: miner’s real identity can’t be learned by anyone. 4. Unique pseudonyms: from each identity card/ePassport, only one unique identifier can be generated. ### Mutual Attestation for X.509 Certificates\[sub:Mutual-Attestation-X509\] **Anonymous miner registration of a new public key on a permissionless blockchain.** This protocol generates a unique pseudonym for each miner, with a new public key linked to the blockchain-specific pseudonym that is in turn uniquely linked to the citizen’s public key certificate: the mutual attestation between the parties guarantees the correctness of the execution of both parties. The following are the steps to the protocol: 1. The client locally generates a signature secret using its secret key: $$\begin{aligned} signatureSecret & = & \mbox{PKCS\_Sign}\left(secretKey_{publicCert},\right.\\ & & \left."\mbox{PREFIXED\_COMMON\_STRING}"\right)\end{aligned}$$ 2. Mutual attestation between the authenticating client and the blockchain: the attestation is anonymous thanks to the use of unlinkable signatures (Enhanced Privacy ID -EPID-), and both parties obtain a temporary secret key to encrypt their communications. 3. Client’s attested code checks the validity of the certificate chain of the miner’s public key certificate as extracted from the smartcard: 1. Load the public key of the root certificate. 2. Hash and verify all intermediates, based on their certificate templates, and the public key of their parent certificate starting from the root certificate and following with the verified public key from the previous intermediate certificate template. 3. Hash and verify the miner’s public key certificate using the last verified public key returned from the previous step. 4. Check the time validity of the miner’s public key certificate. 5. Check that the miner’s public key certificate is contained on a list of trusted certification authorities. 4. If the previous step concluded satisfactorily, then the client’s attested code verifies the $signatureSecret$ using the miner’s public key certificate $publicCert$ because the $signatureSecret$ is calculated offline by the smartcard: $$\mbox{PKCS\_Verify}\left(publicCert,signatureSecret\right)$$ 5. If the previous step concluded satisfactorily, then the client’s attested code creates a deterministic public/secret key pair based on a secret passphrase: $$pk,sk=\mbox{Det\_KeyPairGen}\left(KDF\left(passphrase,hash(publicCert)\right)\right)$$ The generation algorithm must be deterministic because the smartcard may be unable to store them and/or the miner may lose them (i.e., as in deterministic wallets). KDF is a password-based key derivation function (e.g., PBKDF2). 6. The client’s attested code generates a deterministic pseudonym using the blockchain identifier: $$\begin{aligned} pseudonym & = & Hash\left(signatureSecret||BlockchainIdentifier||uniqueID\right)\\ & & ||"\mbox{REG}"\end{aligned}$$ and it obtains the unique identifier from the miner’s public key: $$uniqueID=getID(publicCert)$$ Note that the unique identifier is usually contained on Serial Number of the certificate, or the Subject Alternative Name extension under different OIDs, depending on the country. 7. Anonymously contact the attested encrypted database of the permissionless blockchain to register the generated pseudonym and the new public key $pk$: the uniqueID is also included using the temporary encrypted key, but it won’t be revealed to the host computer of the blockchain node because it will only be decrypted within the attested enclave. 8. The blockchain’s attested code checks within its encrypted database that the uniqueID has never been included: then, it proceeds to store the encrypted uniqueID (i.e., this time with a database secret key that only resides within the enclaves), the generated pseudonym and the new public key $pk$. 9. Then, the encrypted database’s attested code contacts the permissionless blockchain to register the generated pseudonym and its new public key $pk$. **Taking offline registrations from a permissionless blockchain.** This protocol takes offline a pseudonym and its associated public key $pk$ from a permissionless blockchain. To take offline an identity from a permissionless blockchain, miners must re-run the previous protocol to prove that the client is able to re-generate the pseudonym with the same certificate, but this time appending the string “OFF” to the pseudonym. The registering encrypted database of the permissionless blockchain verifies that the encrypted uniqueID is included in the database before taking offline the pseudonym from the permissionless blockchain without it learning the real identity of the miner. ### Mutual Attestation for ePassports\[sub:Mutual-Attestation-ePassports\] Analogous to the zero-knowledge protocols for X.509 \[sub:Mutual-Attestation-X509\], but now considering the specific details of ePassports \[sub:Electronic-Passport\], which usually contain a unique keypair with the public key on Data Group 15 and the private key hidden within the chip: the Active Authentication protocol can be used to sign random challenges that can be verified with the corresponding public key. Some ePassports don’t feature Active Authentication, nonetheless a modified version of the following protocols could still be executed (see subsection \[sub:attAbsence-of-AA\]). **Anonymous miner registration of a new public key on a permissionless blockchain.** This protocol generates a unique pseudonym for each miner, with a new public key linked to the blockchain-specific pseudonym that is in turn uniquely linked to the citizen’s ePassport: the mutual attestation between the parties guarantees the correctness of the execution of both parties. The following are the steps to the protocol: 1. The client locally generates a signature secret using its secret key: $$\begin{aligned} signatureSecret & = & \mbox{Sign}\left(secretKey_{publicCert},\right.\\ & & \left."\mbox{PREFIXED\_COMMON\_STRING}"\right)\end{aligned}$$ 2. Mutual attestation between the authenticating client and the blockchain: the attestation is anonymous thanks to the use of unlinkable signatures (Enhanced Privacy ID -EPID-), and both parties obtain a temporary secret key to encrypt their communications. 3. Client’s attested code checks the validity of the Data Security Object of the miner’s ePassport: 1. Load the public key of the Country Signing Certificate from a trusted source[@icaoPKD] and the Document Signing Certificate from the ePassport. 2. Hash all the Data Groups and check their equivalence to the Data Security Object. 3. Verify the signature of the Data Security Object using the Document Signing Certificate. 4. Verify the signature of the Document Signing Certificate using the Country Signing Certificate. 5. Check the time validity of the certificates. 4. If the previous step concluded satisfactorily, then the client’s attested code verifies the $signatureSecret$ using the miner’s public key certificate $publicCert$ because the $signatureSecret$ is calculated offline by the ePassport’s chip: $$\mbox{PKCS\_Verify}\left(publicCert,signatureSecret\right)$$ The publicCert is taken from the Data Group 15. 5. If the previous step concluded satisfactorily, then the client’s attested code creates a deterministic public/secret key pair based on a secret passphrase: $$pk,sk=\mbox{Det\_KeyPairGen}\left(KDF\left(passphrase,hash(publicCert)\right)\right)$$ The generation algorithm must be deterministic because the ePassport is unable to store them and/or the miner may lose them (i.e., as in deterministic wallets). KDF is a password-based key derivation function (e.g., PBKDF2). 6. The client’s attested code generates a deterministic pseudonym using the blockchain identifier: $$\begin{aligned} pseudonym & = & Hash\left(signatureSecret||BlockchainIdentifier||uniqueID\right)\\ & & ||"\mbox{REG}"\end{aligned}$$ and it obtains the unique identifier from the ePassport: $$uniqueID=getID(DataGroups)$$ Note that the unique identifier is usually contained on the Data Element “Document Number” of the Data Group 1: as it’s legally valid for the same person to own multiple passports with different Document Numbers, some countries include a unique “Personal Number” on the Data Group 11. Sign is the Active Authentication protocol executed within the ePassport’s chip, a deterministic signing algorithm executed on a prefixed string to obtain a unique, non-predictable secret based on the certificate’s owner. 7. Anonymously contact the attested encrypted database of the permissionless blockchain to register the generated pseudonym and the new public key $pk$: the uniqueID is also included using the temporary encrypted key, but it won’t be revealed to the host computer of the blockchain node because it will only be decrypted within the attested enclave. 8. The blockchain’s attested code checks within its encrypted database that the uniqueID has never been included: then, it proceeds to store the encrypted uniqueID (i.e., this time with a database secret key that only resides within the enclaves), the generated pseudonym and the new public key $pk$. 9. Then, the encrypted database’s attested code contacts the permissionless blockchain to register the generated pseudonym and its new public key $pk$. **Taking offline registrations from a permissionless blockchain.** This protocol takes offline a pseudonym and its associated public key $pk$ from a permissionless blockchain. To take offline an identity from a permissionless blockchain, miners must re-run the previous protocol to prove that the client is able to re-generate the pseudonym with the same certificate, but this time appending the string “OFF” to the pseudonym. The registering encrypted database of the permissionless blockchain verifies that the encrypted uniqueID is included in the database before taking offline the pseudonym from the permissionless blockchain without it learning the real identity of the miner. ### Mapping to goals The previous protocols achieve the security goals: 1. The registered miner’s key to the blockchain *opens, but no one can shut; he can shut, but no one can open*[@isaiah2222]. Only someone in possession of a valid public certificate can create a unique miner identity on the open blockchain and destroy it. The signing and verification operations of steps 1 and 4 are only needed if it’s required to check that the miner is the real owner of the smartcard/ePassport. 2. Protection against malicious issuers: the passphrase is mandatory, there’s no OCSP checking and the protocol is non-bruteforceable because it can be rate-limited. 3. Privacy: miner’s real identity can’t be learned by anyone because publicCert and uniqueID are keep secret. 4. Unique pseudonyms: from each identity card/ePassport, only one unique identifier can be generated because there’s only one uniqueID per citizen. The proposed solution depends on the security of Intel SGX (enclave and remote attestation protocols): in order to limit the impact of side-channels attacks on Intel SGX, mining nodes featuring the role of the Attested Encrypted DB will be restricted to trustworthy nodes. ### Performance Evaluation\[sub:Performance-Evaluation\] \# VMs Mean Time/Req. \#Req./Sec Time/Connections Total time -------- ---------------- ------------ ------------------ ------------ 1 VM 416 ms 4.76 210 ms 21 secs 4 VM 112 ms 16.5 59 ms 5.9 secs A load testing scenario featuring an Intel Xeon E3-1240 3.5 GHz and running 1 or 4 virtual machines was performed (with 5 users executing 100 requests per user). Operations like reading and/or signing from the smartcard were not included in the performance evaluation. The implementation will be open-sourced. ### Absence of Active Authentication\[sub:attAbsence-of-AA\] Signing using the secret key of the Active Authentication protocol provides an extra layer of security: it guarantess that the remote party executing the protocol owns a physical copy of the ePassport (i.e., it hasn’t stolen a copy of the public certificates from others). However, some ePassports don’t feature Active Authentication, requiring a simplified version of the previous protocols by removing steps 1 and 4. ### Removing Single-Points of Failure One of the shortcomings of relying on Intel’s Attestation Service (IAS) is that it becomes a single-point of failure: in practice, Intel would learn who is performing the attestation. For a public, permissionless blockchain it would be preferable to remove this trusted third party: to solve this problem, OPERA[@operaSGX] provides the first open, privacy-preserving attestation service to substitute Intel’s Attestation Service. Worldwide Coverage and Distribution\[sub:Worldwide-Coverage-and\] ----------------------------------------------------------------- ![\[fig:Legend:-(1)-National\]Legend: (1) National identity card is a mandatory smartcard; (2) National identity card is a voluntary smartcard; (3) No national identity card, but cryptographic identification is possible using an ePassport, driving license and/or health card; (4) Non-digital identity card.](identityCards.png) Fortunately, there is a unique cryptographic identifier for most people in the world: figure \[fig:Legend:-(1)-National\] shows a worldmap of the distribution of national identity cards. For some countries, there is no national identity card -code 3-, but some other unique cryptographic identifier is available (e.g., ePassport[@cryptoeprint:2009:200] and/or biometric passports as in figure \[fig:Availability-of-biometric\], social security card, driving license and/or health card). Transforming these unique identifiers into anonymous credentials enables the unique identification of individuals in a permissionless blockchain without revealing their true identities, making them indistinguishable: that is, authentication is not only anonymous but permissionless since there is no need to be pre-invited. Please note the enormous cost savings resulting from this approach compared to other anonymous credential[@cryptoeprint:2013:622; @coconut; @DBLP:conf/ccs/CamenischDD17] proposals that would require re-issuing new credentials: for example, consider that the UK’s national identity scheme was estimated at £5.4bn[@homeOfficeCostReport]. In some cases, an individual could obtain multiple cryptographic identifiers (e.g., multiple nationalities), but their number would still be limited and certainly less than the number of mining nodes that could be spawned on PoW permissionless blockchains. Additionally, the true identities provided by national identity cards could be used for other purposes, such as non-anonymous accounts identified by their legal identities. ![\[fig:Availability-of-biometric\]Availability of biometric passports. Source (ICAO, 2019)](icaopkd.jpg)  ### eSIM’s Public Key Infrastructure Latest specifications of SIM cards determine that SIM’s identity and data can be downloaded and remotely provisioned to devices[@eSIMwhitepaper]: instead of the traditional SIM card, there is an embedded SIM (i.e., eUICC[@eUICCtechnical]) that can store multiple SIM profiles containing the operator and subscriber data that would be stored on a traditional SIM card (e.g., IMSI, ICCID, ...). A novel public key infrastructure has been created in order to protect the distribution of these new eSIM profiles[@gsmaPKI]: every certified eSIM is signed by its certified manufacturer, with a certificate that is itself signed by the GSMA root certificate issuer[@gsmaRootCertificateIssuer]. Network operators must also get certified and obtain certificates for their Subscription Management roles. The eSIM’s PKI provides an alternative identification system for users where national identity cards and/or ePassports are difficult to obtain, as they must be unique and non-anonymous (4.13 and 4.1.5[@gsmaPKI]), but only when the mobile operator’s KYC processes can be considered trustworthy. ### Combining with Non-Zero-Knowledge Authentication ![\[fig:Mobile Authentication\]Mobile App using Non-Zero-Knowledge Authentication](readPassport.png "fig:")![\[fig:Mobile Authentication\]Mobile App using Non-Zero-Knowledge Authentication](transfer.png "fig:") The use of zero-knowledge techniques to authenticate miners doesn’t preclude non-zero-knowledge authentication on the same blockchain: figure \[fig:Mobile Authentication\] shows a mobile app[@calctopiaApp] using BAC authentication to read a National Identity Card and/or ePassport (left), and then transferring to another ePassport (right). All national identifiers are publicly addressable. Circumventing the Impossibility of Full Decentralization\[sub:Circumventing-the-Impossibility\] ----------------------------------------------------------------------------------------------- Most blockchains using PKIs are consortium blockchains, thus it has become widespread that they always are permissioned and centralised. However, the term “permissionless” literally means “without requiring permission” (i.e., to access, to join, ...), thus a blockchain with a PKI could be permissionless if it accepts any self-signed certificate (i.e., a behaviour conceptually equivalent to Bitcoin), or any certificate from any government in the world as described in the previous subsection \[sub:Worldwide-Coverage-and\]. In the same way, a blockchain using PKIs doesn’t imply that its control has become centralised, it means that it accepts identities from said PKIs as described in this paper: actually, decentralization in the blockchain context strictly means that the network and the mining are distributed in a large number of nodes, thus unrelated to authentication. A recent publication[@impossibleDecentralization] proves that it’s impossible for blockchains to be fully decentralised without real identity management (e.g., PoW, PoS and DPoS) because they cannot have a positive Sybil cost, defined as the additional cost that a player should pay to run multiple time nodes compared to the total cost of when those nodes are run by different players. To reflect the level of decentralization, they introduce the following definition: ($\left(m,\epsilon,\delta\right)$-Decentralization)[@impossibleDecentralization]. For $1\leq m$, $0\leq\epsilon$ and $0\leq\delta\leq100$, a system is $\left(m,\epsilon,\delta\right)$-decentralized if it satisfies the following: 1. The size of the set of players running nodes in the consensus protocol, $P$, is not less than $m$ (i.e., $\left|P\right|\geq m$). 2. Define $EP_{p_{i}}$ as the effective power of player $p_{i}$ as $\sum_{n_{i}\in N_{p_{i}}}\alpha_{n_{i}}$ where $N$ is the set of all nodes in the consensus protocol and $\alpha_{p_{i}}$ is the resource power of player $p_{i}$. The ratio between the effective power of the richest player, $EP_{max}$, and the $\delta-th$ percentile player, $EP_{\delta}$, is less than or equal to $1+\epsilon$ (i.e., $\left(EP_{max}/EP_{delta}\right)\leq1+\epsilon$). Ideally, the number $m$ should be as high as possible (i.e., too many players do not combine into one node); and for the most resourceful and the $\delta$-th percentile player running nodes, the gap between their effective power is small. Therefore, full decentralization is represented by $\left(m,0,0\right)$ for sufficiently large $m$. (Sufficient Conditions for Fully Decentralized Systems)[@impossibleDecentralization]. The four following conditions are sufficient to reach $\left(m,\epsilon,\delta\right)$-decentralization with probability 1. 1. (Give Rewards (GR-$m$)). Nodes with any resource power earn rewards. 2. (Non-delegation (ND-$m$)). It is not more profitable for too many players to delegate their resource power to fewer participants than to directly run their own nodes. 3. (No Sybil nodes (NS-$\delta$)). It is not more profitable for a participant with above the $\delta$-th percentile effective power to run multiple nodes than to run one node. 4. (Even Distribution (ED-($\epsilon,\delta$)). The ratio between the resource power of the richest and the $\delta$-th percentile nodes converges in probability to a value less than $\text{1+\ensuremath{\epsilon}}$. For any initial state, a system satisfying GR-$m$, ND-$m$, NS-$\delta$, and ED-$\left(\epsilon,\delta\right)$ converges in probability to $\left(m,\epsilon,\delta\right)$-decentralization. [@impossibleDecentralization] As it should be obvious by now, a blockchain using zk-PoI with strong identities from trusted public certificates (e.g., national identity cards and/or ePassports \[sub:Worldwide-Coverage-and\]) as described in this paper is the perfect candidate to achieve a fully decentralized blockchain. A blockchain using zk-PoI with strong identities from trusted public certificates (e.g., national identity cards and/or ePassports \[sub:Worldwide-Coverage-and\]) reaches $\left(m,\epsilon,\delta\right)$-decentralization with probability 1. A blockchain using zk-PoI with strong identities from trusted public certificates effectively limits the number of mining nodes to one per individual (ND-$m$), independently of how resourceful they are (NS-$\delta$, ED-($\epsilon,\delta$)), while keeping membership open to everyone (i.e., achieves a large number of participants (GR-$m$)). The presence of strong identities allows positive Sybil costs, thus the fulfillment of the sufficient conditions for fully decentralized systems[@impossibleDecentralization]. Preventing delegation (ND-$m$) is the most difficult condition to meet: - market-enforced: richest participants could buy rights-of-use of others’ identities, but the market value of said identities (e.g., the Net Present Value of future profits obtained from the exploitation of said identities by their real owners) should wipe away almost all the profits from these exchanges. - strictly-enforced: miners’ software could frequently check for the presence of the physical trusted public certificate (e.g., national identity cards and/or ePassports) and/or require them when transferring funds out of their accounts. A posterior revision of the paper[@impossibleDecentralizationv2] introduces new definitions that try to emphasize that Trusted Third Parties (i.e., Certificate Authorities) shouldn’t be used in decentralized blockchains: as described in this paper\[sub:Worldwide-Coverage-and\], using 400 CAs of national identity cards and ePassports from over the world is still being decentralized according to the original definition of decentralization, and certainly much more decentralized than Bitcoin/Ethereum that concentrate &gt;50% of their hashrate in 4-3 entities[@decentralizationBitCoinEthereum]. It’s very important not to mix concepts and misattribute qualities to different concepts: - permissionless doesn’t imply without identification - a decentralized consensus protocol doesn’t imply that it can’t use identification from TTPs: recent consensus protocols decouple Sybil-resistance from the consensus mechanism\[sub:Modern-Consensus-based-on\] Thus, permissionless, decentralized and identification(-less) are different qualities that shouldn’t be intermixed. The results of this paper hold in the *trusted* decentralized setting, while the results of [@impossibleDecentralizationv2] hold in the *trustless* decentralised setting. Resistance against Dark DAOs ---------------------------- Dark DAOs[@darkDaos] appear as a consequence of permissionless blockchains where users can create their own multiple identities and there’s no attributability of the actions. 1. When using real-world identities, it’s possible to establish the identity of the parties running the Dark DAO that are committing frauds (attributability) or at least, their pseudonyms: therefore, it’s possible to punish them. 2. To prevent Dark DAOs buying real-world identities, a smart contract can be setup that pays a reward for denouncing the promoters of the fraud: the whistleblowers would be paid a multiple of what they would get from the defrauders, thus making denunciation the preferred option. Then defrauders would be banned as in step 1. Resistance against Collusion and other Attacks ---------------------------------------------- In this sub-section, we consider different avenues for attack and provide detailed defense mechanisms: 1. Corrupt root certificate authorities 2. Attacks against consensus protocols 3. Resistance against collusion 4. On achieving collusion-freeness ### Corrupt Root Certificate Authorities Corrupt countries may be tempted to create fake identities or frequently renovate existing ones: these countries can be easily banned out by removing them from the list of valid authorities (i.e., root X.509 certificate and/or Country Signing Certificate). Bounties in cryptocurrency could be offered for whistleblowing any corrupt attack against the long-term existence of the blockchain. ### Attacks against Consensus Protocols Modern consensus protocols based on the cryptographically secure random choice of the leader (e.g., [@dfinityConsensus; @cryptoeprint:2017:406]) detect cheating by monitoring changes to the chain quality. The following table gathers cheating events for different consensus algorithms that could be detected and punished: [|c|&gt;p[7.5cm]{}|]{} Protocol & Cheater detection[\ ]{} & Equivocation: multiple blocks for same round with same rank.[\ ]{} & Equivocation: multiple blocks with the highest priority.[\ ]{} & All the blocks must be timely published.[\ ]{} & All the notarizations must be timely published within one round.[\ ]{} & Core validators can detect rogue validators.[\ ]{} & Withholders can be detected after multiple consecutive rounds.[\ ]{} & 5&gt;= failed RandHound views from a rogue validator.[\ ]{} ### Resistance against Collusion Consensus protocols already provide collusion-tolerance by design: an adversary controlling a high number of nodes, or equivalently all said nodes colluding for the same attack, must confront the difficulties introduced by shard re-assignment at the beginning of every new epoch. For the case of OmniLedger[@cryptoeprint:2017:406], the security of the validator assignment mechanism can be modeled as a random sampling problem using the binomial distribution, $$P\left[X\leq\left\lfloor \frac{n}{3}\right\rfloor \right]=\sum_{k=0}^{n}\left(\begin{array}{c} n\\ k \end{array}\right)m^{k}\left(1-m\right)^{n-k}$$ assuming that each shard has less than $\left\lfloor \frac{n}{3}\right\rfloor $ malicious validators. Then, the failure rate of an epoch is the union bound of the failures rates of individual shards, each one calculated as the cumulative distribution over the shard size $n$, with $X$ being the random variable that represents the number of times we pick a malicious node. An upper bound of the epoch failure event, $X_{E}$, is calculated as: $$P\left[X_{E}\right]\leq\sum_{k=0}^{l}\frac{1}{4^{k}}\cdot n\cdot P\left[X_{S}\right]$$ where $l$ is the number of consecutive views the adversary controls, $n$ is the number of shards and $P\left[X_{S}\right]$ is the failure rate of one individual shard. Finally, for $l\rightarrow\infty$, we obtain $$P\left[X_{E}\right]\leq\frac{4}{3}\cdot n\cdot P\left[X_{S}\right]$$ ### On Achieving Collusion-Freeness Start noticing that collusion-freeness is not about preventing malicious behaviour, only preventing that malicious players act as independently of each other as possible. Following a previous seminal work[@collusionFreeProtocols], collusion-freeness can only be obtained under very stringent conditions: (a) the game must be finite; (b) the game must be publicly observable; and (c) the use of private channels at the beginning of the game is essential, but forbidden during the execution of the protocol. Although blockchains are publicly observable, they are also an infinite game where parties can freely communicate between them using private channels at any time: therefore, collusion-freeness is impossible in the sense of [@collusionFreeProtocols]. Fortunately, there is a way to get around this impossibility result: forbid malicious/Byzantine behaviours requiring the use of mutual attestation for all the nodes, thus precluding any deviation from the original protocol. If mutual attestation is required for all the nodes, any infinite, partial-information blockchain game with publicly observable actions has a collusion-free protocol. As mutual attestation is already required for zk-PoI \[sub:Detailed-Authentication-Protocol-Remote-Attestation\], we would only be extending its use for the rest of the blockchain protocol. Incentive Compatible and Strictly Dominant Cryptocurrencies =========================================================== The success of cryptocurrencies is better explained by their incentive mechanisms rather than their consensus algorithms: a cryptocurrency with poor incentives (e.g., a cryptocurrency not awarding coins to miners) will not achieve any success; conversely, better incentives and much more inefficient consensus algorithm could still find some success. Much research has been focused on conceiving better consensus algorithms for decentralised systems and cryptocurrencies[@cryptoeprint:2016:918; @cryptoeprint:2016:919; @cryptoeprint:2017:406; @cryptoeprint:2017:454; @DBLP:journals/corr/Kokoris-KogiasJ16; @dfinityConsensus]: unfortunately, obtaining consensus mechanisms with better incentives and economic properties is an area that is lacking much research, and the combination of all the game-theoretic results contained in this section fills this gap for the sake of achieving a *focal point* (i.e., Schelling point[@schellingConflict]) in the multi-equilibria market of cryptocurrencies. Thus, a selective advantage is introduced by design over all the other cryptocurrencies, in explicit violation of the neutral model of evolution[@2017arXiv170505334E] in order to obtain an incentive compatible and strictly dominant cryptocurrency. Incentive-Compatible Cryptocurrency\[sub:Incentive-Compatible-Cryptocurre\] --------------------------------------------------------------------------- Shard-based consensus protocols have been recently introduced in order to increase the scalability and transaction throughput of public permissionless blockchains: however, the study of the strategic behaviour of rational miners within shard-based blockchains is very recent. Unlike Bitcoin, that grants all rewards to the most recent miner, block rewards and transactions fees must be proportionally shared between all the members of the sharding committee[@DBLP:journals/corr/Kokoris-KogiasJ16], and this includes incentives to remain live during all the lifecycle of the consensus protocol. Even so, existing sharding proposals[@cryptoeprint:2017:406; @cryptoeprint:2018:460] remain silent on how miners will be rewarded to enforce their good behaviour: as it’s evident, if all miners are equally rewarded without detailed consideration of their efforts, rational players will *free-ride* on the efforts of others. One significant difference introduced in this paper with respect to other shard-based consensus protocols is the use of Zero-Knowledge Proof-of-Identity as the Sybil-resistance mechanism: as we will see in the following sections, it’s a significant novelty because solving Proof-of-Work puzzles is the most computationally expensive activity of consensus protocols, thus it’s no longer dominated by computational costs. This makes the necessity for an incentive-compatible protocol even more acute: the preferred rational miner’s strategy is to execute the Proof-of-Work of the initial phase of the protocol for each epoch and to refrain from the transaction verification and consensus of subsequent phases of the protocol, but still selfishly claim the rewards as if they had participated. The substitution of costly PoW for cheap Zero-Knowledge Proof-of-Identity only increases the attractiveness of this rational strategy, that can only be counteracted by using an incentive-compatible protocol. ### A Nash Equilibrium for a Cryptocurrency on a Shard-Based Blockchain This section is based on a stylised version of a recent game-theoretic model[@2018arXiv180907307M], taking into consideration that there is no cost associated with committee formation to enter each shard since we are using Zero-Knowledge Proof-of-Identity, and not costly Proof-of-Work: instead, a penalty $p$ is imposed to defective and/or cheating miners. The following is a list of symbols: Symbol Definition ------------------- ------------------------------------------------------------- $k$ Number of shards $N$ Number of miners $x_{i}^{j}$ Vector of received transactions by miner $i$ in shard $j$ $y^{j}$ Vector of transactions submitted by shard $j$ to blockchain $c$ Minimum number of miners in each committee $\tau$ Required number of miners in shard for consensus $r$ Benefit for each transaction $b_{i}$ Benefit of miner $i$ after adding the block $c_{i}^{t}$ Total cost of computation for miner $i$ $c^{o}$ Total optional costs in each epoch $c^{v}$ Cost of transaction verification $c^{f}$ Fixed costs in optional cost $p$ Penalty cost $BR$ Block Reward $l_{j}$ Number of cooperative miners in each shard $L$ Total number of cooperative miners in all shards $C_{j}^{l_{j}}$ Set of all cooperative miners in shard $j$ $D_{j}^{n-l_{j}}$ Set of all defective miners in shard $j$ $C^{L}$ Set of all cooperative miners $D^{N-L}$ Set of all defective miners $s^{r}$ Signed receipt of a transaction \ Let $\mathbb{G}$ denote the shard-based blockchain game, defined as a triplet $\left(P,S,U\right)$ where $P=\left\{ P_{i}\right\} _{i=1}^{N}$ is the set of players, $S=\left\{ C,D\right\} $ is the set of strategies (Cooperate $C$, or Defect $D$) and $U$ is the set of payoff values. Each miner receives a reward if and only it has already cooperated with other miners within the shard, the payoff of cooperative miners in set $C^{l_{j}}$ is $$u_{i}\left(C\right)=\frac{BR}{kl_{j}}+\frac{r\left|y^{j}\right|}{l_{j}}-\left(c^{f}+\left|x_{i}^{j}\right|c^{v}\right)\label{eq:5.1}$$ We assume that the block reward $BR$ is uniformly distributed among shards and each cooperative miner can receive a share of it. A miner might be cooperative but all other miners may agree on a vector of transactions $y^{j}$ that is different from his own vector of transactions $x_{i}^{j}$ (i.e., $\left|x_{i}^{j}\right|\neq\left|y^{j}\right|$): nonetheless, transaction rewards are uniformly distributed among all cooperative miners in each shard, proportional to all the transactions submitted to the blockchain by each shard. The defective miners’ payoff can be calculated as $$u_{i}^{D}=-p^{m}$$ because the defective miners will have to pay a penalty and they will not receive any benefit (and it doesn’t incur in any mandatory cost such as solving PoW puzzles because we use cheap Zero-Knowledge Proof-of-Identity). There exists a cooperative Nash equilibrium profile in game $\mathbb{G}$ under the following conditions: \[thm:UniqueNashEqulibriumCryptocurrency\]Let $C_{j}^{l_{j}}$ and $D_{j}^{m-l_{j}}$ denote the sets of $l_{j}$ cooperating miners and $n-l_{j}$ defecting miners inside each shard $j$ with $n$ miners, respectively. $\left(C^{L},D^{N-L}\right)$ represents a Nash equilibrium profile in each epoch of game $\mathbb{G}$, if the following conditions are satisfied: 1. In all shards $j$, $l_{j}\geq\tau$. 2. If for a given miner $P_{i}$ in shard $j$, with $x_{i}^{j}=y^{j}$, then the number of transactions $\left|x_{i}^{j}\right|$ must be greater than $$\theta_{c}^{1}=\frac{c^{f}-\frac{BR}{kl_{j}}+p}{\nicefrac{r}{l_{j}}-c^{v}}$$ 3. If for a given miner $P_{i}$ in shard $j$, with $x_{i}^{j}\neq y^{j}$, then the number of transactions $\left|x_{i}^{j}\right|$must be smaller than $$\theta_{c}^{2}=\frac{\frac{BR}{kl_{j}}+\frac{r\left|y^{j}\right|}{l_{j}}-c^{f}-p}{c^{v}}$$ The first condition $l_{j}\geq\tau$ (i.e., the number of cooperative miners must be greater than $\tau$) must hold so that cooperative miners will receive benefits for transactions and block rewards. Let $l_{j}^{*}$ be the largest set of cooperative miners in each shard, where no miner in $D_{j}^{n-l_{j}}$ can join $C_{j}^{l_{j}}$ to increase its payoff: if miner $P_{i}^{j}$ is among the set of cooperative miners where $x_{i}^{j}=y^{j}$, then it would not unilaterally deviate from cooperation if: $$\frac{BR}{kl_{j}}+\frac{r\left|x_{i}^{j}\right|}{l_{j}}-\left(c^{f}+\left|x_{i}^{j}\right|c^{v}\right)\geq-p$$ which shows that $x_{i}^{j}\geq\theta_{c}^{1}$, whereas in the second condition, $$\theta_{c}^{1}=\frac{c^{f}-\frac{BR}{kl_{j}}-p}{\nicefrac{r}{l_{j}}-c^{v}}$$ But if $P_{i}^{j}$ is among the cooperators whose vector of transactions is different from the output of the shard, $x_{i}^{j}\neq y^{j}$, then it would not deviate from cooperation if: $$\frac{BR}{kl_{j}}+\frac{r\left|y^{j}\right|}{l_{j}}-\left(c^{f}+\left|x_{i}^{j}\right|c^{v}\right)\geq-p$$ which shows that $x_{i}^{j}<\theta_{c}^{2}$, whereas in the third condition, $$\theta_{c}^{2}=\frac{\frac{BR}{kl_{j}}+\frac{r\left|y^{j}\right|}{l_{j}}-c^{f}-p}{c^{v}}$$ Then if $l_{j}^{\text{*}}$ represents the largest set of cooperative miners in each shard, then $\left(C^{L},D^{N-L}\right)$ would be the unique cooperative Nash equilibrium of the game $\mathbb{G}$. As can be understood from the proof, cooperative miners have less incentive to cooperate when: 1) the number of participants $N$ increases; 2) the optional costs of computation increase ($c^{f}$ is in the numerator or $c^{v}$ in denominator of $\theta_{C}$); 3) or in general, when the number of transactions is not large enough compared to a fixed threshold. ### Incentive-Compatible Cryptocurrency on a Shard-Based Coordinated Blockchain In order to increase the incentives to cooperate rather than defect, an incentive-compatible protocol enforcing cooperation based on the previously presented Nash equilibrium is introduced here \[alg:Incentive-Compatible-Protocol\]: all miners should disclose their list of transactions to a coordinator, who then announces to each miner whether their cooperation would be in their interests based on being within the maximum subset of miners with a similar list of transactions (i.e., $x_{i}^{j}=y^{j}$), and then enforces their cooperation by checking their compliance and rewarding them properly. The protocol proceeds as follows: for the first function (i.e., *ShardTransactionsAssignment*), each miner receives a list of transactions $x_{i}^{j}$ to verify based on the epochRandomness and his pseudonymous identity and public key obtained by the Zero-Knowledge Proof-of-Identity. For the second function (i.e., *NodeSelection*), all miners calculate a hash $H\left(x_{i}^{j}\right)$ over their transaction list and send it to the coordinator. The coordinator finds the subset with the maximum number of miners with a common transaction list, thus calculating $\theta_{c}^{1}$, $\theta_{c}^{2}$, $l_{j}$ and $C_{j}^{l_{j}}$: in each epoch, the coordinator publicly defines the list of cooperative miners $C_{j}^{l_{j}}$ and defective miners $D_{j}^{n-l_{j}}$ using on . For the third function (i.e., *ShardParticipation*), all the transactions of each miner are verified and a signed consensus is reached. For the fourth function (i.e., *VerificationAndRewards*), the rewards are shared between the cooperative miners and denied to those miners in $C_{j}^{l_{j}}$ that didn’t cooperate. ### Improved Incentive-Compatible Cryptocurrency on a Shard-Based Blockchain Although the role of a coordinator is essential to BFT protocols[@DBLP:journals/corr/Kokoris-KogiasJ16], its expanded functionality in the previous incentive-compatible protocol is problematic: it introduces latency and network costs due to the new obligations to report to the coordinator; moreover, it creates new opportunities for malicious miners which may report false $H\left(x_{i}^{j}\right)$ or not follow coordinator’s instruction to cooperate or defect. The next incentive-compatible protocol significantly improves over the state of the art: the role of the coordinator is minimised, strengthing the protocol by removing the previous vulnerabilities and making it resistant to malicious miners. Information propagation[@bitcoinInformationPropagation] is an essential part of any blockchain, and gossiping transactions to neighbouring miners is one of its key features. In the new incentive-protocol protocol, we require that any broadcasted/gossiped/propagated transaction gets acknowledged with a signed receipt to its sender: then, senders must attach these receipts to the consensus leaders and verification nodes in order to ease detection of defective and/or cheating miners. . In other words, the signed receipts serve as snitches that denounce non-cooperative miners thus preventing that any reward gets assigned to them: at the same time, all miners are incentivised to participate in the denunciation in order to gain the rewards of non-cooperative miners and other *free-riders*. In order to save bandwidth, note that it’s not obligatory to send the full list of all signed transaction receipts to consensus leaders and/or verification nodes: only a random subset per each miner should be enough to catch defective miners. Additionally, the absence of signed receipts could be used to detect the need of a change of a consensus leader (i.e., “view-change”) in BFT protocols[@DBLP:journals/corr/Kokoris-KogiasJ16; @cryptoeprint:2017:406]. On Strictly Dominant Cryptocurrencies ------------------------------------- A cryptocurrency using Zero-Knowledge Proof-of-Identity as the Sybil-resistance mechanism strictly dominates PoW/PoS cryptocurrencies: a miner having to choose between mining different cryptocurrencies, one with no costs associated with its Sybil-resistance mechanism and distributing equally the rewards, and the others using costly PoW/PoS and thus featuring mining concentration, will always choose the first one. That is, mining equally distributed cryptocurrencies using Zero-Knowledge Proof-of-Identity is a dominant strategy; in other words, the strategy of mining Bitcoin and other similar cryptocurrencies is strictly dominated by the hereby described cryptocurrency. *Ceteris paribus*, this cryptocurrency will have better network effects, thus better long-term valuation. ### Strictly Dominant Cryptocurrencies and a Nash Equilibrium\[sub:Strictly-Dominant-Cryptocurrenci\] The intuition behind the preference to mine fully decentralised cryptocurrencies with the lowest expenditure (i.e., lowest CAPEX/OPEX implies higher profitability), thus the search for better hash functions[@cryptoeprint:2017:1168; @cryptoeprint:2016:989; @cryptoeprint:2015:430; @cryptoeprint:2016:027], is formally proved here and then applied to the specific case of the proposed cryptocurrency. (Power-Law Fee-Concentrated (PLFC) cryptocurrency)\[def:(Power-law-fee-concentrated-cryp\]. A cryptocurrency whose distribution of mining and/or transaction fees follows a power-law (i.e., a few entities earn most of the fees/rewards), usually due to the high costs of its Sybil-resistance mechanism. Proof-of-Work cryptocurrencies are Power-Law Fee-Concentrated: 90% of the mining power is concentrated in 16 miners in Bitcoin and 11 in Ethereum[@decentralizationBitCoinEthereum]. Proof-of-Stake cryptocurrencies are Power-Law Fee-Concentrated: miners earn fees proportional to the amount of money at stake, and wealth is Pareto-concentrated[@Pareto2014-PARMOP-2]. (Uniformly-Distributed Capital-Efficient (UDCE) cryptocurrency)\[def:(Uniformly-distributed-fee-uncon\]. A cryptocurrency whose distribution of mining and/or transaction fees is uniformly distributed among all the transaction processing nodes, and doesn’t require significant investments from the participating miners. The proposed cryptocurrency using Zero-Knowledge Proof-of-Identity is a Uniformly-Distributed Capital-Efficient cryptocurrency. (Game of Rational Mining of Cryptocurrencies)\[def:(Game-of-Rational\]. A rational miner ranks the cryptocurrencies according to their expected mining difficulty, and chooses to mine those with lowest expected difficulty. Awesome Miner[@awesomeMiner], MinerGate[@minerGate], MultiMiner[@multiMiner], MultiPoolMiner[@multipoolMiner], Smart-Miner[@smartMiner; @smartMinerPaper] and NiceHash Miner[@niceHash] are practical implementations of the Game of Rational Mining of Cryptocurrencies (although also considering their prices in addition to their difficulties). Specific calculators for mining profitability[@whatToMine; @2CryptoCalc; @CoinWarz; @CryptoCompare; @CryptoZone; @Crypto-CoinZ] could also be used for the similar purposes. Additionally, other papers[@modelMinerHashRate] provide models regarding optimal hash rate allocation. Let $u_{i}$ be the payoff or utility function for each miner, expressing his payoff in terms of the decisions or strategies $s_{i}$ of all the miners, $$u_{i}\left(s_{1},s_{2},\ldots,s_{n}\right)=u_{i}\left(s_{i},s_{-i}\right)$$ whese $s_{-i}$ are set of the strategies of the rest of miners, $$s_{-i}=\left(s_{1},s_{2},\ldots,s_{i-1},s_{i+1},\ldots,s_{n}\right)$$ A strategy $s_{1}$ *strictly dominates a strategy $s_{2}$ for miner $i$* if and only if, for any $s_{-i}$ that miner $i$’s opponents might use, $$u_{i}\left(s_{1},s_{-i}\right)>u_{i}\left(s_{2},s_{-i}\right)$$ That is, no matter what the other miners do, playing $s_{1}$ is strictly better than playing $s_{2}$ for miner $i$. Conversely, we say that the strategy *$s_{2}$ is strictly dominated by $s_{1}$*: a rational miner $i$ would never play a strictly dominated strategy. A strategy $s_{i}^{*}$ is a *strictly dominant strategy for miner $i$* if and only if, for any profile of opponent strategies $s_{-i}$ and any other strategy $s_{i}^{'}$ that miner $i$ could choose, $$u_{i}\left(s_{i}^{*},s_{-i}\right)>u_{i}\left(s_{i}^{'},s_{-i}\right)$$ We now demonstrate that mining *UDCE crypto-cryptocurrencies* \[def:(Uniformly-distributed-fee-uncon\] is a strictly dominant strategy with regard to *PLFC cryptocurrencies \[def:(Power-law-fee-concentrated-cryp\]* in the *Game of Rational Mining of Cryptocurrencies* \[def:(Game-of-Rational\] by showing that every miner’s expected profitability is higher in UDCE cryptocurrencies. \[thm:Uniformly-distributed-crypto-cur\]UDCE cryptocurrencies yield a strictly higher miner’s expected profitability compared to PLFC cryptocurrencies in the Game of Rational Mining of Cryptocurrencies. Let $N$ be the number of miners and $R$ the average daily minted reward per day: UDCE cryptocurrencies award an average of $\nicefrac{N}{R}$ units of cryptocurrency to every participant miner. For every miner on the long tail of the power distribution, the amount earned with UDCE cryptocurrencies is obviously higher than with PLFC cryptocurrencies. For the few miners that dominate PLFC cryptocurrencies, their profitability is lower because they have to account for the energy[@natureEnergyCarbonCosts; @bitcoinCarbonFootprint] and equipment costs in the case of PoW cryptocurrencies or the opportunity cost of staking capital in volatile PoS cryptocurrencies[@stakedPoS], meanwhile in UDCE cryptocurrencies their cost of mining is so negligible compared to PLFC cryptocurrencies that the balance of profitability is always tipped in their favour. \[def:The-process-to\]The process to solve games called *Iterated Deletion of Strictly Dominated Strategies (IDSDS)* is defined by the next steps: 1. For each player, eliminate all strictly dominated strategies. 2. If any strategy was deleted during Step 1, repeat Step 1. Otherwise, stop. If the process eliminates all but one unique strategy profile $s^{*}$, we say it is the *outcome of iterated deletion of strictly dominated strategies* or a *dominant strategy equilibrium*. \[def:pure-nash\]A strategy profile $s^{*}=\left(s_{1}^{*},s_{2}^{*},\ldots,s_{n}^{*}\right)$ is a *Pure-Strategy Nash Equilibrium* (PSNE) if, for every player $i$ and any other strategy $s_{i}^{'}$ that player $i$ could choose, $$u_{i}\left(s_{i}^{*},s_{-i}^{*}\right)\geq u_{i}\left(s_{i}^{'},s_{-i}^{*}\right)$$   \[def:strict-nash\]A strategy profile $s^{*}=\left(s_{1}^{*},s_{2}^{*},\ldots,s_{n}^{*}\right)$ is a *Strict Nash Equilibrium* (SNE) if, for every player $i$ and any other strategy $s_{i}^{'}$ that player $i$ could choose, $$u_{i}\left(s_{i}^{*},s_{-i}^{*}\right)>u_{i}\left(s_{i}^{'},s_{-i}^{*}\right)$$ Additionally, if a game is solvable by Iterated Deletion of Strictly Dominated Strategies, the outcome is a Nash equilibrium. \[thm:UDCE-cryptocurrency-dominates\]A UDCE cryptocurrency dominating PLFC cryptocurrencies is a Nash equilibrium. Mining a UDCE cryptocurrency is a strictly dominant strategy with regard to other miners mining PLFC cryptocurrencies because PLFC cryptocurrencies are strictly dominated by UCDE cryptocurrencies by Theorem \[thm:Uniformly-distributed-crypto-cur\], thus a rational miner will always prefer to miner the latter. Thus, by the application of Iterated Deletion of Strictly Dominated Strategies (IDSDS) to the Game of Rational Mining of Cryptocurrencies \[def:(Game-of-Rational\], each miner will eliminate mining PLFC cryptocurrencies in favor of mining an UDCE cryptocurrency, leaving this as the unique outcome: therefore, mining a UDCE cryptocurrency is a *dominant strategy equilibrium* by Definition \[def:The-process-to\] and a *Nash equilibrium* by Definition \[def:pure-nash\] or by Definition \[def:strict-nash\]. \[claim:(Uniqueness-of-Technical-Solution)\](Uniqueness of Technical Solution). The proposed technical solution using Zero-Knowledge Proof of Identity from trusted public certificates (i.e., national identity cards and/or ePassports) is the only practical and unique solution for a UCDE cryptocurrency. As demonstrated in the paper describing “The Sybil Attack”[@the-sybil-attack], Sybil attacks are always possible unless a trusted identification agency certifies identities. As National Identity Cards and ePassports are the only globally available source of trusted cryptographic identity (3.5B for National Identity Cards and 1B for ePassports), the only way to bootstrap a UCDE cryptocurrency is by using the proposed Zero-Knowledge Proof-of-Identity from trusted public certificates (National Identity Cards and/or ePassports). ### Strictly Dominant Cryptocurrencies and Evolutionary Stable Strategies\[sub:Evolutionary-Stable-Strategies\] Another interesting viewpoint to consider in the analysis of the cryptocurrency market is the one offered by behavioural ecology and its Evolutionary Stable Strategies \[def:ESS\]: each cryptocurrency can be considered a unique individual in a population, genetically programmed to play a pre-defined strategy. New cryptocurrencies are introduced into the population as individuals with different mutations that define their technical features (e.g., forking the code of a cryptocurrency to change the hashing algorithm, or a zk-PoI cryptocurrency). An Evolutionary Stable Strategy \[def:ESS\] is a strategy that cannot be invaded by any alternative strategy, that is, it can resist to the invasion of a mutant and it’s impenetrable to them: once it’s introduced and becomes dominant in a population, natural selection is sufficient to prevent invasions from new mutant strategies. \[def:ESS\]The pure strategy $s^{*}$ is an Evolutionary Stable Strategy[@logicAnimalConflict] if there exists $\epsilon_{0}>0$ such that: $$\left(1-\epsilon\right)\left(u\left(s^{*},s^{*}\right)\right)+\epsilon\left(u\left(s^{*},s^{'}\right)\right)>\left(1-\epsilon\right)\left(u\left(s^{'},s^{*}\right)\right)+\epsilon\left(u\left(s^{'},s^{'}\right)\right)$$ for all possible deviations $s^{'}$and for all mutation sizes $\epsilon<\epsilon_{0}$. There are two conditions for a strategy $s^{*}$ to be an Evolutionary Stable Strategy: for all $s^{*}\neq s^{'}$ either 1. $u\left(s^{*},s^{*}\right)>u\left(s^{'},s^{*}\right)$, that is, it’s a Strict Nash Equilibrium \[def:strict-nash\], **or**, 2. if $u\left(s^{*},s^{*}\right)=u\left(s^{'},s^{*}\right)$ then $u\left(s^{*},s^{'}\right)>u\left(s^{'},s^{'}\right)$ Mining a UDCE cryptocurrency is an Evolutionary Stable Strategy. Since mining a UCDE cryptocurrency is a strictly-dominant strategy and a Strict Nash Equilibrium \[thm:UDCE-cryptocurrency-dominates\], then it is an Evolutionary Stable Strategy because it fulfills its first condition \[def:ESS\]. Additionally, mining a UCDE cryptocurrency based on the global network of National Identity Cards and ePassports is an Evolutionary Stable Strategy over national variants/mutants due to Claim \[claim:(Uniqueness-of-Technical-Solution)\]. Thus, the Game of Rational Mining of Cryptocurrencies \[def:(Game-of-Rational\] is a “survival of the fittest” ecology, where the cheapest cryptocurrency to mine offering the higher profits rises above the others. ### Obviating the Price of Crypto-Anarchy\[sub:Price-of-Crypto-Anarchy\] The most cost efficient Sybil-resistant mechanism is the one provided by a trusted PKI infrastructure[@the-sybil-attack] and a centralised social planner would prefer the use of National Identity Cards and/or ePassports in order to minimise costs: instead, permissionless blockchains are paying very high costs by using PoW/PoS as Sybil-resistant mechanisms. In this paper, Zero-Knowledge Proof-of-Identity is introduced as a compromise solution between both approaches, thus obtaining a very efficient Sybil-resistant mechanism with the best of both worlds. In order to measure how the efficiency of a Sybil-resistant mechanism degrades due to the selfish behaviour of its agents (i.e., a fixed amount of block reward to be distributed among a growing and unbounded number of miners paying high energy costs, as in Bitcoin), we compare the ratio between the worst Nash equilibrium and the optimal centralised solution, a concept known as Price of Anarchy in game theory because it bounds and quantifies the costs of the selfish behavior of the agents. *The Price of Anarchy*[@DBLP:conf/stacs/KoutsoupiasP99]. Consider a game $G=\left(N,S,u\right)$ defined as a set of players $N$, strategy sets $S_{i}$ for each player and utilities $u_{i}:S\rightarrow\mathbb{R}$ (where $S=S_{1}x\ldots xS_{n}$ are also called the set of outcomes). Define a measure of efficiency of each outcome that we want to minimise, $Cost:S\rightarrow\mathbb{R}$, and let $Equil\subseteq S$ be the set of strategies in Nash equilibria. The *Price of Anarchy* is given by the following ratio: $$\mbox{Price of Anarchy}=\frac{\max_{s\in Equil}Cost\left(s\right)}{\min_{s\in S}Cost\left(s\right)}$$ The competition game between several blockchains and their cryptocurrencies can be reformulated[@altman:hal-01906954] as a congestion game[@congestionGames; @potentialGames] (hereby included for completeness), more amenable to the formulations commonly used for analyzing the Price of Anarchy (the necessity for the following definitions is already intuited in the Discussion of [@sokGameTheoryCryptocurrencies]): as the number of miners increases, it also exponentially decreases the chance that a given miner wins the block reward by being the first to solve the mining puzzle (i.e., the system becomes increasingly congested); it has been proved that free entry is solely responsible for determining the resource usage[@bitcoinMarketStructure], and that the difficulty is not an instrument that can regulate it. #### Miners, mining servers and crypto-currencies Denote by $\mathcal{N}\coloneqq\left\{ 1,2,\ldots,N\right\} $ the finite set of miners that alter the utilities of other miners if any of them change strategies and let $\mathcal{K}\coloneqq\left\{ 1,2,\ldots,K\right\} $ be the set of cryptocurrencies, each associated to exactly one puzzle that miners are trying to solve. Let $\mathcal{M}\coloneqq\left\{ 1,2,\ldots,M\right\} $denote the set of Edge computing Service Providers (ESPs), or mining servers used to offload the costly computational processing. #### Strategies Let $\mathcal{S}_{i}\subset\mathcal{K}x\mathcal{M}$ denote the set of ordered pairs (cryptocurrency, ESP) corresponding to ESPs that miner $i$ can rely on to solve the puzzles of a given cryptocurrency. A strategy for miner $i$ is denoted by $s_{i}\in\mathcal{S}_{i}$ corresponding to the cryptocurrency (puzzle) which a miner intends to solve using a given infrastructure. A strategy vector $s\coloneqq\left(s_{i}\right)_{i\in\mathcal{N}}$ produces a load vector $l\coloneqq\left(l_{k,m}\right)_{k,m}$, where $l_{k,m}$ denotes the number of users mining blockchain $k$ at ESP $m$. #### Rewards, costs, and utilities Let $\eta_{k}$ be the load of miners across all ESPs towards cryptocurrency $k.$ Then, $$\eta_{k}\coloneqq\sum_{m^{'}\in\mathcal{M}}l_{k,m'}\mu_{k,m'}$$ For a given load vector $l$, the time to solve the puzzle of the $k^{th}$ cryptocurrency is exponentially distributed with expectation $\nicefrac{1}{\eta_{k}}$. Let $q_{k}$ be the probability that puzzle $k$ is solved by time $T$, $$q_{k}=1-\mbox{exp}\left(-T\eta_{k}\right)$$ The probability that a given miner using ESP $m$ is the first to solve puzzle $k$ is $$p_{k,m}=1_{l_{k,m>0}}q_{k}\mu_{k,m}/\eta_{k}$$ where $1_{c}$ equals 1 if condition $c$ holds and 0 otherwise. For simplification, subscript $m$ can be dropped and we consider a single ESP. Then, the probability that a miner is the first to solve the puzzle is $$p_{k}\left(l_{k}\right)=\left(1-\mbox{exp}\left(-T\mu_{k}l_{k}\right)\right)/l_{k}$$ Let $U_{k,m}\left(l\right)$ denote the utility to a miner who tries to find the solution of the current puzzle associated to cryptocurrency $k$ using ESP $m$ and $\gamma_{k,m}$ denote the cost of mining blockchain $k$ at ESP $m$: $$U_{k,m}\left(l\right)=\begin{cases} p_{k,m}-\gamma_{k,m} & \mbox{if }p_{k,m}>\gamma_{k,m},\\ 0 & \mbox{otherwise} \end{cases}$$ and the utility of a tagged miner to mine a cryptocurrency $k$ when there are $l_{k}$ miners associated with the same cryptocurrency is $$U_{k}\left(l_{k}\right)=p_{k}-\gamma_{k},\mbox{ if }p_{k}-\gamma_{k}\geq0$$ \[thm:altmanTheo\][@altman:hal-01906954]If for all $i$ and $j$, $S_{i}=S_{j}$ and $s_{i}$ does not depend on $i$, then the Nash equilibrium is given by the solution of the following optimization problem, $$\begin{aligned} \mbox{argmin}_{s}\Phi\left(s\right) & \coloneqq & \sum_{k\in\mathcal{K}}\sum_{l=1}^{l_{k}}p_{k}\left(l\right)-\gamma_{k}\\ \mbox{subject to:} & & \sum_{k\in\mathcal{K}}l_{k}\leq N,l_{k}\geq0\end{aligned}$$ Let $NashCongestedEquil\subseteq S$ be the set of strategies given as solution of the optimization problem of Theorem \[thm:altmanTheo\], then the *Price of Crypto-Anarchy* is given by the following ratio: $$\mbox{Price of Crypto-Anarchy=\ensuremath{\frac{\mbox{max}_{\mbox{\ensuremath{s}}\in\mbox{\ensuremath{NashCongestedEquil}}}\mbox{\ensuremath{Cost\left(s\right)}}}{\mbox{\ensuremath{Cost\left(\mbox{zk-PoI}\right)}}}}}$$ In practice, the real-world costs of the Zero-Knowledge Proof of Identity can be considered almost zero because it’s subsidised by governments and thus exogenous to any blockchain system. Quite the opposite, the energy costs of PoW cryptocurrencies are notoriously high[@natureEnergyCarbonCosts; @bitcoinCarbonFootprint]: it is estimated that mining Bitcoin, Ethereum, Litecoin and Monero consumed an average of 17, 7, 7 and 14 MJ to generate one US\$, respectively; and that Bitcoin causes around 22 megatons in CO2 emissions annually[@bitcoinCarbonFootprint]. The trivial extension to Proof-of-Stake is left as an exercise to the reader, although it’s not as affordable as it may be seen: as of March 2019, an average of 40% of the cryptocurrency supply is staked at a total of \$4Bn between all PoS blockchains[@stakedPoS]. Actually, Proof-of-Stake is not strictly better than Proof-of-Work as the distribution of the market shares between both technologies has been shown to be indistinguishable (Appendix 3, [@2017arXiv170505334E]). ### Pareto Dominance on Currency Circulation\[sub:Pareto-Dominance-on\] For completeness, a stylised version of a model of competing currencies[@RePEc:nbr:nberwo:22157] is introduced here to prove that UDCE cryptocurrencies also dominate PLFC cryptocurrencies on their circulation (i.e., trading, speculating) due to their stronger network effects, and not only mining as previously proved. The key observation here is that by definition \[def:(Power-law-fee-concentrated-cryp\], the returns of mining PLFC cryptocurrencies is concentrated on a very limited number of miners and the newly minted cryptocurrency could be held for long periods of times: otherwise, if they didn’t expect that the held cryptocurrencies would appreciate in time, they would be mining another set of cryptocurrencies with better expectations. In direct contrast, the distribution of mining and/or transaction fees of UDCE cryptocurrencies is uniformly distributed by definition \[def:(Uniformly-distributed-fee-uncon\]: therefore, the returns of the holding strategy after minting them would be lower and their subsequent circulation much less restricted. Suppose an economy divided into periods, each period divided into two subperiods: in the first subperiod, a perishable good demanded by everyone is produced and consumed in a Centralised Market; in the second subperiod, buyers who only consume are randomly matched with sellers who only produce with probability $\sigma\in\left(0,1\right)$ in a Decentralised Market. Let $\beta\in\left(0,1\right)$ denote the discount factor, $\phi_{t}^{i}\in\mathbb{R}_{+}$ denote the value of a unit of currency $i\in\left\{ 1,\ldots,N\right\} $ in terms of the CM food and $\phi_{t}=\left(\phi_{t}^{1},\ldots,\phi_{t}^{N}\right)\in\mathbb{R}_{+}^{N}$ denote the vector of real prices. (Buyers). In a $\left[0,1\right]$-continuum of buyers, $x_{t}^{b}\in\mathbb{R}$ denotes the buyer’s net consumption of the CM good and $q_{t}\in\mathbb{R}_{+}$denotes the consumption of the DM good. The utility function of the buyer’s preferences is given by $$U^{b}\left(x_{t}^{b},q_{t}\right)=x_{t}^{b}+u\left(q_{t}\right)$$ with $u:\mathbb{R}_{+}\rightarrow\mathbb{R}$ continuously differentiable, increasing and strictly concave with $u'\left(0\right)=\infty$ and $u\left(0\right)=0$. Let $W^{b}\left(M_{t-1}^{b},t\right)$ denote the value function for a buyer who starts period $t$ holding a portfolio $M_{t-1}^{b}\in\mathbb{R}_{+}^{N}$ of cryptocurrencies in the CM and let $V^{b}\left(M_{t}^{b},t\right)$ denote the value function in the DM: the dynamic programming equation is $$W^{b}\left(M_{t-1}^{b},t\right)=\underset{\left(x_{t}^{b},M_{t}^{b}\right)\in\mathbb{R}\times\mathbb{R}_{+}^{N}}{\mbox{max}}\left[x_{t}^{b}+V^{b}\left(M_{t}^{b},t\right)\right]$$ subject to the budget constraint $$\phi_{t}\cdot M_{t}^{b}+x_{t}^{b}=\phi_{t}\cdot M_{t-1}^{b}.$$ The value for a buyer holding a portfolio $M_{t}^{b}$ in the DM is $$\begin{aligned} V^{b}\left(M_{t}^{b},t\right) & = & \sigma\left[u\left(q\left(M_{t}^{b},t\right)\right)+\beta W^{b}\left(M_{t}^{b}-d\left(M_{t}^{b},t\right),t+1\right)\right]\\ & & +\left(1-\sigma\right)\beta W^{b}\left(M_{t}^{b},t+1\right)\end{aligned}$$ and let $q^{*}\in\mathbb{R}$ denote the quantity satisfying $u'\left(q^{*}\right)=w'\left(q^{*}\right)$ so that $q^{*}$ gives the surplus-maximizing quantity that determines the efficient level of production in the DM. The solution to the bargaining problem is given by $$q\left(M_{t}^{b},t\right)=\begin{cases} m^{-1}\left(\beta\times\phi_{t+1}\cdot M_{t}^{b}\right) & \mbox{if}\,\phi_{t+1}\cdot M_{t}^{b}<\beta^{-1}\left[\theta w\left(q^{*}\right)+\left(1-\theta\right)u\left(q^{*}\right)\right]\\ q^{*} & \mbox{if}\,\phi_{t+1}\cdot M_{t}^{b}\geq\beta^{-1}\left[\theta w\left(q^{*}\right)+\left(1-\theta\right)u\left(q^{*}\right)\right] \end{cases}$$ and $$\phi_{t+1}\cdot d\left(M_{t}^{b},t\right)=\begin{cases} \phi_{t+1}\cdot M_{t}^{b} & \mbox{if}\,\phi_{t+1}\cdot M_{t}^{b}<\beta^{-1}\left[\theta w\left(q^{*}\right)\right.\\ & \left.+\left(1-\theta\right)u\left(q^{*}\right)\right]\\ \beta^{-1}\left[\theta w\left(q^{*}\right)+\left(1-\theta\right)u\left(q^{*}\right)\right] & \mbox{if}\,\phi_{t+1}\cdot M_{t}^{b}\geq\beta^{-1}\left[\theta w\left(q^{*}\right)\right.\\ & \left.+\left(1-\theta\right)u\left(q^{*}\right)\right] \end{cases}$$ with the function $m:\mathbb{\mathbb{R}}_{+}\rightarrow\mathbb{R}$ defined as $$m\left(q\right)\equiv\frac{\left(1-\theta\right)u\left(q\right)w'\left(q\right)+\theta w\left(q\right)u'\left(q\right)}{\theta u'\left(q\right)+\left(1-\theta\right)w'\left(q\right)}.$$ The optimal portfolio problem can be defined as $$\underset{M_{t}^{b}\in\mathbb{R}_{+}^{N}}{\mbox{max}}\left\{ -\phi_{t}\cdot M_{t}^{b}+\sigma\left[u\left(q\left(M_{t}^{b},t\right)\right)-\beta\times\phi_{t+1}\cdot d\left(M_{t}^{b},t\right)\right]+\beta\times\phi_{t+1}\cdot M_{t}^{b}\right\}$$ thus the optimal choice satisfies $$\phi_{t}^{i}=\beta\phi_{t+1}^{i}L_{\theta}\left(\phi_{t+1}\cdot M_{t}^{b}\right)\label{eq:1}$$ for every type $i\in\left\{ 1,\ldots,N\right\} $ together with the transversality condition $$\underset{t\rightarrow\infty}{\mbox{lim}}\beta^{t}\times\phi_{t}\cdot M_{t}^{b}=0\label{eq:2}$$ where $L_{\theta}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is given by $$L_{\theta}\left(A\right)=\begin{cases} \sigma\frac{u'\left(m^{-1}\left(\beta A\right)\right)}{w'\left(m^{-1}\left(\beta A\right)\right)}+1-\sigma & \mbox{if}\, A<\beta^{-1}\left[\theta w\left(q^{*}\right)+\left(1-\theta\right)u\left(q^{*}\right)\right]\\ 1 & \mbox{if}\, A\geq\beta^{-1}\left[\theta w\left(q^{*}\right)+\left(1-\theta\right)u\left(q^{*}\right)\right] \end{cases}$$   (Sellers). In a $\left[0,1\right]$-continuum of sellers, $x_{t}^{s}\in\mathbb{R}$ denotes the seller’s net consumption of the CM good and $n_{t}\in\mathbb{R}_{+}$denotes the seller’s effort level to produce the DM good. The utility function of the seller’s preferences is given by $$U^{s}\left(x_{t}^{s},n_{t}\right)=x_{t}^{s}-w\left(n_{t}\right)$$ with $w:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ continuously differentiable, increasing and weakly convex with $w\left(0\right)=0$. Let $W^{s}\left(M_{t-1}^{s},t\right)$ denote the value function for a seller who enters period $t$ holding a portfolio $M_{t-1}^{s}\in\mathbb{R}_{+}^{N}$ of cryptocurrencies in the CM and let $V^{s}\left(M_{t}^{s},t\right)$ denote the value function in the DM: the dynamic programming equation is $$W^{s}\left(M_{t-1}^{s}\right)=\underset{\left(x_{t}^{s},M_{t}^{s}\right)\in\mathbb{R}\times\mathbb{R}_{+}^{N}}{\mbox{max}}\left[x_{t}^{s}+V^{s}\left(M_{t}^{s},t\right)\right]$$ subject to the budget constraint $$\phi_{t}\cdot M_{t}^{s}+x_{t}^{s}=\phi_{t}\cdot M_{t-1}^{s}.$$ The value for a seller holding a portfolio $M_{t}^{s}$ in the DM is $$\begin{aligned} V^{s}\left(M_{t}^{s},t\right) & = & \sigma\left[-w\left(q\left(M_{t}^{b},t\right)\right)+\beta W^{s}\left(M_{t}^{s}+d\left(M_{t}^{b},t\right),t+1\right)\right]\\ & & +\left(1-\sigma\right)\beta W^{s}\left(M_{t}^{s},t+1\right)\end{aligned}$$   (Miners). In a $\left[0,1\right]$-continuum of miners of each type-$i\in\left\{ 1,\ldots,N\right\} $ token, $x_{t}^{i}\in\mathbb{R}_{+}$ denotes the miner’s consumption of the CM good and $\triangle_{t}^{i}\in\mathbb{R}_{+}$denotes the production of the type-$i$ token. The utility function of the miner’s preferences is given by $$U^{e}\left(x_{t}^{i},\Delta_{t}\right)=x_{t}^{i}-c\left(\Delta_{t}^{i}\right)$$ with the cost function $c:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ continuously differentiable, strictly increasing and weakly convex with $c\left(0\right)=0$. Let $M_{t}^{i}\in\mathbb{R}_{+}$ denote the per-capita supply of cryptocurrency $i$ in period $t$ and $\Delta_{t}^{i}\in\mathbb{R}$ denote the miner $i$’s net circulation of newly minted tokens in period $t$. To describe the miner’s problem to determine the money supply in the economy, we start assuming that all miners solve the same decision problem, thus the law of motion of type-$i$ tokens at all date $t\geq0$ is given by $$M_{t}^{i}=\Delta_{t}^{i}+M_{t-1}^{i}\label{eq:4}$$ where $M_{-1}^{i}\in\mathbb{R}_{+}$ denotes the initial stock. The budget constraint is $$x_{j}^{i}=\phi_{t}^{i}\Delta_{t}^{i},$$ and given that the miner takes prices $\left\{ \phi_{t}\right\} _{t=0}^{\infty}$ as given, the profit maximization of the cryptocurrency emission problem is solved by $$\Delta_{t}^{*,i}\in\underset{\Delta\in\mathbb{R}_{+}}{\mbox{arg max}}\left[\phi_{t}^{i}\Delta-c\left(\Delta\right)\right]\label{eq:3}$$   (Equilibrium). A perfect-foresight monetary equilibrium is an array $\left\{ M_{t},M_{t}^{b},\Delta_{t}^{*},\phi_{t}\right\} _{t=0}^{\infty}$ satisfying \[eq:1\], \[eq:2\], \[eq:3\] and \[eq:4\] for each $i\in\left\{ 1,\ldots,N\right\} $ at all dates $t\geq0$ and satisfying the following market-clearing condition $$M_{t}=M_{t}^{b}+M_{t}^{s}$$ Suppose that each miner $j$ starts with $M^{i}>0$ units of currency $i\in\left\{ 1,\ldots,N\right\} $: let $\delta$ denote the fraction $1-\delta$ of randomly selected miners in each location $j$ at each date $t\geq0$ who doesn’t offer their tokens to sellers because they are holding them in expectation of their appreciation (i.e., PLFC cryptocurrencies), so these tokens don’t circulate to other $j$ positions whenever sellers are relocated. Conversely, an equilibrium with the property that miners don’t restrict the circulation of recently mined tokens (i.e., UDCE cryptocurrencies) is as follows: the optimal portfolio choice implies the first-order condition $$\frac{u'\left(q\left(M_{t},t\right)\right)}{w'\left(q\left(M_{t},t\right)\right)}=\frac{1}{\beta\gamma_{t+1}^{i}}$$ for each currency $i$, where $\gamma_{t+1}\in\mathbb{R}_{+}$ represents the common return across all valued currencies between dates $t$ and $t+1$. The demand for real balances in each location is given by $$z\left(\gamma_{t+1};1\right)\equiv\frac{1}{\gamma_{t+1}}L_{1}^{-1}\left(\frac{1}{\beta\gamma_{t+1}}\right)$$ because $$\beta\gamma_{t+1}\sum_{i=1}^{N}b_{t}^{i}<\theta w\left(q^{*}\right)+\left(1-\theta\right)u\left(q^{*}\right)$$ and with $$L_{\delta}\left(A\right)=\begin{cases} \delta\frac{u'\left(m^{-1}\left(\beta A\right)\right)}{w'\left(m^{-1}\left(\beta A\right)\right)}+1-\delta & \mbox{if}\, A<\beta^{-1}w\left(q^{*}\right)\\ 1 & \mbox{if}\, A\geq\beta^{-1}w\left(q^{*}\right) \end{cases}$$ Because the market-clearing condition implies $$\sum_{i=1}^{N}\phi_{t}^{i}M^{i}=z\left(\gamma_{t+1};1\right)$$ the equilibrium sequence $\left\{ \gamma_{t}\right\} _{t=0}^{\infty}$ satisfies the law of motion $$z\left(\gamma_{t+1};1\right)=\gamma_{t}z\left(\gamma_{t};1\right)$$ because $$M_{t}^{i}=M_{t-1}^{i}=\Delta^{i}$$ for each $i$ and provided that $\gamma_{t}\leq t$, and the boundary condition $$\beta\gamma_{t}z\left(\gamma_{t};1\right)\leq w\left(q^{*}\right)$$ Suppose $u\left(q\right)=\left(1-\eta\right)^{-1}q^{1-\eta}$, with $0<\eta<1$, and $w\left(q\right)=\left(1+\alpha\right)^{-1}q^{1+\alpha}$ with $\alpha\geq0$. The dynamic system describing the equilibrium evolution of $\gamma_{t}$ is $$\gamma_{t+1}^{\frac{1+\alpha}{\eta+\alpha}-1}=\gamma_{t}^{\frac{1+\alpha}{\eta+\alpha}}\label{eq:28}$$ The allocation associated with the circulation of UDCE cryptocurrencies Pareto dominates the allocation with the associated the circulation of PLFC cryptocurrencies, on a stationary equilibrium with the property that the quantity traded in the Decentralised Market is given by $\hat{q}\left(1\right)\in\left(\hat{q}\left(\delta\right),q^{*}\right)$ satisfying $$\frac{u'\left(\hat{q}\left(1\right)\right)}{w'\left(\hat{q}\left(1\right)\right)}=\beta^{-1}\label{eq:29}$$ The sequence $\gamma_{t}=1$ for all $t\geq0$ satifies \[eq:28\]. Then, the solution to the optimal portfolio problem implies that the DM output must satisfy \[eq:29\]. The quantity $\hat{q}$ satisfies $$\frac{u'\left(\hat{q}\left(1\right)\right)}{w'\left(\hat{q}\left(1\right)\right)}=\delta\frac{u'\left(\hat{q}\left(\delta\right)\right)}{w'\left(\hat{q}\left(\delta\right)\right)}+1-\delta$$ Because $\delta\in\left(0,1\right)$, we have $\hat{q}\left(1\right)>\hat{q}\left(\delta\right)$, that is, the allocation associated with the circulation of UCDE cryptocurrencies -$\hat{q}\left(1\right)$- Pareto dominates the allocation associated with the circulation of PLFC cryptocurrencies -$\hat{q}\left(\delta\right)$-. ### On Network Effects\[sub:On-Network-Effects\] At the time of the release of this paper, cryptocurrencies have failed to provide an alternative to traditional payment networks due to a combination of high transaction fees, high finalization time and high volatility. The failure to find the favor of merchants is also their biggest weakness: they aren’t part of two-sided networks, and thus easily replaceable by any newer cryptocurrency better able to create them. Actually, the first-mover advantage of the most valued cryptocurrencies is lower than expected if any competing cryptocurrency leverages network effects from other different sources (e.g., Zero-Knowledge Proof-of-Identity from trusted PKI certificates). A simple model is introduced here to analyze the evolution of competing payment networks: consider the two-sided and incompatible payment networks of two cryptocurrencies, BTC and zk-PoI, each with their corresponding groups of merchants $m$ and customers $c$; let $m_{BTC}^{t},m_{zkPOI}^{t}$ denote the number of merchants at time $t$ and $c_{BTC}^{t},c_{zkPOI}^{t}$ the number of customers. A user joins the payment networks at each time step $t$, with $\lambda$ being the probability of being a customer and $1-\lambda$ of being a merchant: each merchant prefers to join BTC or zk-PoI depending on the number of customers in the same network, thus the probabilities to join one of the networks are given by $$\frac{c_{BTC}^{\beta}}{c_{BTC}^{\beta}+c_{zkPOI}^{\beta}},\frac{c_{zkPOI}^{\beta}}{c_{BTC}^{\beta}+c_{zkPOI}^{\beta}}$$ and conversely, for customers the probabilities are given by $$\frac{m_{BTC}^{\alpha}}{m_{BTC}^{\alpha}+m_{zkPOI}^{\alpha}},\frac{m_{zkPOI}^{\alpha}}{m_{BTC}^{\alpha}+m_{zkPOI}^{\alpha}}.$$ Note that some categories of users would prefer to use the expected number of users and not their current tally: forward-looking merchants that need to invest on equipment to access the payment network are within this group, thus they would prefer to use expected numbers, $$\frac{E\left(c_{BTC}^{\beta}\right)}{E\left(c_{BTC}^{\beta}\right)+E\left(c_{zkPOI}^{\beta}\right)},\frac{E\left(c_{zkPOI}^{\beta}\right)}{E\left(c_{BTC}^{\beta}\right)+E\left(c_{zkPOI}^{\beta}\right)}.$$ Each user can only join one payment network, modelling the fact that single-homing is preferred to multi-homing in the real-world, and the particular network is determined by the distribution of users on the other side at each time $t$. The parameters $\alpha,\beta>0$ are elasticities of demand with regard to the numbers of users on the other side of the payment network, effectively acting as measures of indirect network effects: $\alpha$ can be empirically estimated by observing joining customers over a small period of time and then calculating $$\alpha=\frac{\mbox{ln }\left(\nicefrac{m_{BTC}^{\alpha}}{\left(m_{BTC}^{\alpha}+m_{zkPOI}^{\alpha}\right)}\right)-\mbox{ln }\left(1-\left(\nicefrac{m_{BTC}^{\alpha}}{\left(m_{BTC}^{\alpha}+m_{zkPOI}^{\alpha}\right)}\right)\right)}{\mbox{ln }m_{BTC}-\mbox{ln }m_{zkPOI}}$$ and conversely$\beta$ can be empirically estimated by observing joining merchants and then calculating $$\beta=\frac{\mbox{ln }\left(\nicefrac{c_{BTC}^{\beta}}{\left(c_{BTC}^{\beta}+c_{zkPOI}^{\beta}\right)}\right)-\mbox{ln }\left(1-\left(\nicefrac{c_{BTC}^{\beta}}{\left(c_{BTC}^{\beta}+c_{zkPOI}^{\beta}\right)}\right)\right)}{\mbox{ln }c_{BTC}-\mbox{ln }c_{zkPOI}}.$$ (Dominance of the Zero-Knowledge Proof-of-Identity cryptocurrency). A new cryptocurrency could achieve dominance over previous cryptocurrencies, overcoming first-mover advantages, if the expected number of accepting customers would be much higher and the number of merchants using the previous cryptocurrencies is low. Note that the number of steps needed for a new cryptocurrency, $m_{zkPOI}$, to overtake the previous one, $m_{BTC}$, on the number of merchants, $m_{zkPOI}>m_{BTC}$, is given by $$\left(m_{BTC}+1\right)\cdot\left(1-\lambda\right)^{-1}$$ It’s possible for a new cryptocurrency to overtake a previous one on the number of merchants whenever $$E\left(c_{zkPOI}\right)-E\left(c_{BTC}\right)>\left(m_{BTC}+1\right)\cdot\left(1-\lambda\right)^{-1}$$ and since $m_{BTC}$ is a low number and $E\left(c_{zkPOI}\right)\gg E\left(c_{BTC}\right)$, it’s conceivable that the previous condition could hold. Now let’s consider the results of strong network effects on the final market shares of both payment networks by examining the following differential equations, $$\frac{d\left(\nicefrac{m_{BTC}}{m_{zkPOI}}\right)}{dt}=\left(1-\lambda\right)\frac{\left(\nicefrac{c_{BTC}}{c_{zkPOI}}\right)^{\beta}-\left(\nicefrac{m_{BTC}}{m_{zkPOI}}\right)}{\left(1+\left(\nicefrac{c_{BTC}}{c_{zkPOI}}\right)^{\beta}\right)m_{zkPOI}}$$ and $$\frac{d\left(\nicefrac{c_{BTC}}{c_{zkPOI}}\right)}{dt}=\lambda\frac{\left(\nicefrac{m_{BTC}}{m_{zkPOI}}\right)^{\alpha}-\left(\nicefrac{c_{BTC}}{c_{zkPOI}}\right)}{\left(1+\left(\nicefrac{m_{BTC}}{m_{zkPOI}}\right)^{\alpha}\right)c_{zkPOI}}$$ According to the signs of the previous derivatives, when $\alpha\cdot\beta>1$ and $t\rightarrow\infty$, the payment network with even a slight advantage over the other will end acquiring all the merchants and customers, for example $$\underset{t\rightarrow\infty}{\lim}m_{zkPOI}=\infty,\underset{t\rightarrow\infty}{\lim}c_{zkPOI}=\infty$$ $$\underset{t\rightarrow\alpha}{\lim}m_{BTC}=0,\underset{t\rightarrow\alpha}{\lim}c_{BTC}=0$$ but with $\alpha\cdot\beta<1$, the number of merchants and customers will equalize $$m_{zkPOI}=m_{BTC},c_{zkPOI}=c_{BTC}$$ thus highlighting the importance of network effects. ### Dominance over Cash and other Cryptocurrencies\[sub:Dominance-over-Cash\] The dominance of subsection \[sub:Strictly-Dominant-Cryptocurrenci\] is based on mining and subsection \[sub:Pareto-Dominance-on\] extends said dominance to the circulation of currencies: in this subsection, the dominance will be based on the lower costs of a payment network of the cryptocurrency using Zero-Knowledge Proof-of-Identity; therefore, there exists a unique equilibrium in which this payment system dominates. A recent paper[@coordinationElectronicPaymentInstrument] offers a model based on a version of Lagos-Wright[@lagosWright] to explain the substitution of cash by debit cards or any other non-deferred electronic payment system incurring a fixed cost $\Omega\left(z\right)\tau$ per each period $\tau$, the cost $\Omega\left(z\right)$ being financed by imposing fee $\omega$ on each payment where $\omega$ should satisfy $$\Omega\left(z\right)=S\left[\theta\omega+\left(1-\theta\right)\omega\right]$$ and where $S$ denotes the instantaneous measure of electronic payment transactions, $z$ is the state of development of the economy, $\theta\in\left[0,1\right]$ is the share of cost allocated to a buyer and $\left(1-\theta\right)$ is the share of cost allocated to a seller. In this model, an electronic payment system can achieve dominance over cash using the solution concept of iterative elimination of conditionally dominated strategies whenever the state of development of the economy $z$ is sufficiently high, and there exists a unique equilibrium in the model such that agents choose electronic payment transactions when $z$ is strictly higher than the limiting cut-off function $Z_{\infty}$ of the sequence of boundaries $Z_{0,}Z_{1,}\ldots$ of regions where an agent chooses electronic payment transactions regardless of the choices of other agents. In other words, it’s strictly dominant to choose electronic payments in an economy having sufficiently advanced information technology so that $\Omega$ is negligible. Since the cost function $\Omega\left(z\right)$ of a UCDE cryptocurrency based on Zero-Knowledge Proof-of-Identity is much cheaper than of PoW/PoS cryptocurrencies and other forms of electronic payment because its cards are already distributed (i.e., de facto subsidised by governments), there exists a unique equilibrium in the model [@coordinationElectronicPaymentInstrument] such that the agents choose the UCDE cryptocurrency using zk-PoI and it dominates the other forms of payment. Conclusion ========== Although all permissionless blockchains critically depend on Proof-of-Work or Proof-of-Stake to prevent Sybil attacks, their high resource consumption corroborates their non-scalability and act as a limiting factor to the general diffusion of blockchains. This paper proposed an alternative approach that not only doesn’t waste resources, it could also help in the real-world identity challenges faced by permissionless blockchains: the derivation of anonymous credentials from widely trusted public PKI certificates. Additionally, we study the better incentives offered by the proposed cryptocurrency based on our anonymous authentication scheme: mining is proved to be incentive-compatible and a strictly dominant strategy over previous cryptocurrencies, thus a Nash equilibrium over previous cryptocurrencies and an Evolutionary Stable Strategy; furthermore, zk-PoI is proved to be optimal because it implements the social optimum, unlike PoW/PoS cryptocurrencies that are paying the Price of (Crypto-)Anarchy. The circulation of the proposed cryptocurrency is proved to Pareto dominate other cryptocurrencies based on its negligible mining costs and it could also become dominant thanks to stronger network effects; finally, the lower costs of its infrastructure imply the existence of a unique equilibrium where it dominates other forms of payment.

--- abstract: | The past few years have seen a surge of interest in the field of probabilistic logic learning and statistical relational learning. In this endeavor, many probabilistic logics have been developed. ProbLog is a recent probabilistic extension of Prolog motivated by the mining of large biological networks. In ProbLog, facts can be labeled with probabilities. These facts are treated as mutually independent random variables that indicate whether these facts belong to a randomly sampled program. Different kinds of queries can be posed to ProbLog programs. We introduce algorithms that allow the efficient execution of these queries, discuss their implementation on top of the YAP-Prolog system, and evaluate their performance in the context of large networks of biological entities.\ *To appear in Theory and Practice of Logic Programming (TPLP)* author: - | Angelika Kimmig, Bart Demoen and Luc De Raedt\ Departement Computerwetenschappen, K.U. Leuven\ Celestijnenlaan 200A - bus 2402, B-3001 Heverlee, Belgium\ \ - | Vítor Santos Costa and Ricardo Rocha\ CRACS $\&$ INESC-Porto LA, Faculty of Sciences, University of Porto\ R. do Campo Alegre 1021/1055, 4169-007 Porto, Portugal\ \ bibliography: - 'tplp.bib' title: On the Implementation of the Probabilistic Logic Programming Language ProbLog --- \#1 Introduction {#sec:intro} ============ In the past few years, a multitude of different formalisms combining probabilistic reasoning with logics, databases, or logic programming has been developed. Prominent examples include PHA and ICL [@Poole:93; @Poole00], PRISM [@SatoKameya:01], SLPs [@Muggleton96], ProbView [@Lakshmanan], CLP($\cal BN$) [@Costa03:uai], CP-logic [@Vennekens], Trio [@Trio], probabilistic Datalog (pD) [@Fuhr00], and probabilistic databases [@DalviS04]. Although these logics have been traditionally studied in the knowledge representation and database communities, the focus is now often on a machine learning perspective, which imposes new requirements. First, these logics must be simple enough to be learnable and at the same time sufficiently expressive to support interesting probabilistic inferences. Second, because learning is computationally expensive and requires answering long sequences of possibly complex queries, inference in such logics must be fast, although inference in even the simplest probabilistic logics is computationally hard. In this paper, we study these problems in the context of a simple probabilistic logic, ProbLog [@DeRaedt07], which has been used for learning in the context of large biological networks where edges are labeled with probabilities. Large and complex networks of biological concepts (genes, proteins, phenotypes, etc.) can be extracted from public databases, and probabilistic links between concepts can be obtained by various techniques [@Sevon06]. ProbLog is essentially an extension of Prolog where a program defines a distribution over all its possible non-probabilistic subprograms. Facts are labeled with probabilities and treated as mutually independent random variables indicating whether or not the corresponding fact belongs to a randomly sampled program. The success probability of a query is defined as the probability that it succeeds in such a random subprogram. The semantics of ProbLog is not new: it is an instance of the distribution semantics [@Sato:95]. This is a well-known semantics for probabilistic logics that has been (re)defined multiple times in the literature, often in a more limited database setting; cf. [@Dantsin; @Poole:93; @Fuhr00; @Poole00; @DalviS04]. Sato has, however, shown that the semantics is also well-defined in the case of a countably infinite set of random variables and formalized it in his well-known distribution semantics [@Sato:95]. However, even though relying on the same semantics, in order to allow efficient inference, systems such as PRISM [@SatoKameya:01] and PHA [@Poole:93] additionally require all proofs of a query to be mutually exclusive. Thus, they cannot easily represent the type of network analysis tasks that motivated ProbLog. ICL [@Poole00] extends PHA to the case where proofs need not be mutually exclusive. In contrast to the ProbLog implementation presented here, Poole’s AILog2, an implementation of ICL, uses a meta-interpreter and is not tightly integrated with Prolog. We contribute exact and approximate inference algorithms for ProbLog. We present algorithms for computing the success and explanation probabilities of a query, and show how they can be efficiently implemented combining Prolog inference with Binary Decision Diagrams (BDDs) [@Bryant86]. In addition to an iterative deepening algorithm that computes an approximation along the lines of [@Poole93:jrnl], we further adapt the Monte Carlo approach used by [@Sevon06] in the context of biological network inference. These two approximation algorithms compute an upper and a lower bound on the success probability. We also contribute an additional approximation algorithm that computes a lower bound using only the $k$ most likely proofs. The key contribution of this paper is the tight integration of these algorithms in the state-of-the-art YAP-Prolog system. This integration includes several improvements over the initial implementation used in [@DeRaedt07], which are needed to use ProbLog to effectively query Sevon’s Biomine network [@Sevon06] containing about 1,000,000 nodes and 6,000,000 edges, as will be shown in the experiments. This paper is organised as follows. After introducing ProbLog and its semantics in Section 2, we present several algorithms for exact and approximate inference in Section 3. Section 4 then discusses how these algorithms are implemented in YAP-Prolog, and Section 5 reports on experiments that validate the approach. Finally, Section 6 concludes and touches upon related work. ProbLog {#sec:problog} ======= A ProbLog program consists of a set of labeled facts $p_i::c_i$ together with a set of definite clauses. Each ground instance (that is, each instance not containing variables) of such a fact $c_i$ is true with probability $p_i$, that is, these facts correspond to random variables. We assume that these variables are mutually independent.[^1] The definite clauses allow one to add arbitrary *background knowledge* (BK). Figure \[fig:Ex\] shows a small probabilistic graph that we shall use as running example in the text. It can be encoded in ProbLog as follows: $$\begin{array}{lllll} 0\ldotp8 :: \mathtt{edge(a,c)\ldotp} & ~~~~ & 0\ldotp7 :: \mathtt{edge(a,b)\ldotp} & ~~~~ & 0\ldotp8 :: \mathtt{edge(c,e)\ldotp} \\ 0\ldotp6 :: \mathtt{edge(b,c)\ldotp} & ~~~~ & 0\ldotp9 :: \mathtt{edge(c,d)\ldotp} & ~~~~ & 0\ldotp5 :: \mathtt{edge(e,d)\ldotp} \end{array}$$ Such a probabilistic graph can be used to sample subgraphs by tossing a coin for each edge. Given a ProbLog program $T=\{p_1::c_1,\cdots,p_n::c_n\} \cup BK$ and a finite set of possible substitutions $\{\theta_{j1}, \ldots \theta_{ji_j}\}$ for each probabilistic fact $p_j::c_j$, let $L_T$ denote the maximal set of *logical* facts that can be added to $BK$, that is, $L_T=\{c_1\theta_{11}, \ldots , c_1\theta_{1i_1}, \cdots, c_n\theta_{n1}, \ldots , c_n\theta_{ni_n}\}$. As the random variables corresponding to facts in $L_T$ are mutually independent, the ProbLog program defines a probability distribution over ground logic programs $L \subseteq L_T$: $$P(L|T)=\prod\nolimits_{c_i\theta_j\in L}p_i\prod\nolimits_{c_i\theta_j\in L_T\backslash L}(1-p_i)\ldotp$$ Since the background knowledge $BK$ is fixed and there is a one-to-one mapping between ground definite clause programs and Herbrand interpretations, a ProbLog program thus also defines a distribution over its Herbrand interpretations. Sato has shown how this semantics can be generalized to the countably infinite case; we refer to [@Sato:95] for details. For ease of readability, in the remainder of this paper we will restrict ourselves to the finite case and assume all probabilistic facts in a ProbLog program to be ground. We extend our example with the following background knowledge: $$\begin{array}{lll} \mathtt{path(X,Y)} & \mathtt{:-} & \mathtt{edge(X,Y)\ldotp} \\ \mathtt{path(X,Y)} & \mathtt{:-} & \mathtt{edge(X,Z), path(Z,Y)\ldotp} \end{array}$$ We can then ask for the probability that there exists a path between two nodes, say *c* and *d*, in our probabilistic graph, that is, we query for the probability that a randomly sampled subgraph contains the edge from *c* to *d*, or the path from *c* to *d* via *e* (or both of these). ![ Example of a probabilistic graph: edge labels indicate the probability that the edge is part of the graph.[]{data-label="fig:Ex"}](graph) Formally, the *success probability* $P_s(q|T)$ of a query $q$ in a ProbLog program $T$ is the marginal of $P(L|T)$ with respect to $q$, i.e. $$P_s(q|T) = \sum\nolimits_{L\subseteq L_T}P(q|L)\cdot P(L|T)\;, \label{eq:p_suc}$$ where $P(q|L) = 1$ if there exists a $\theta$ such that $L\cup BK\models q\theta$, and $P(q|L)=0$ otherwise. In other words, the success probability of query $q$ is the probability that the query $q$ is *provable* in a randomly sampled logic program. In our example, $40$ of the $64$ possible subprograms allow one to prove *path$(c,d)$*, namely all those that contain at least the edge from *c* to *d* or both the edge from *c* to *e* and from *e* to *d*, so the success probability of that query is the sum of the probabilities of these programs: $P_s(path(c,d)|T)=P(\{ab,ac,bc,cd,ce,ed\}|T)+\ldots +P(\{cd\}|T)=0\ldotp94$, where $xy$ is used as a shortcut for *edge$(x,y)$* when listing elements of a subprogram. We will use this convention throughout the paper. Clearly, listing all subprograms is infeasible in practice; an alternative approach will be discussed in Section \[sec:exact\]. A ProbLog program also defines the probability of a *specific* proof $E$, also called *explanation*, of some query $q$, which is again a marginal of $P(L|T)$. Here, an *explanation* is a minimal subset of the probabilistic facts that together with the background knowledge entails $q\theta$ for some substitution $\theta$. Thus, the probability of such an explanation $E$ is that of sampling a logic program $L\cup E$ that contains at least all the probabilistic facts in $E$, that is, the marginal with respect to these facts: $$P(E|T) = \sum\nolimits_{ L\subseteq (L_T\backslash E)} P(L\cup E |T) = \prod\nolimits_{c_i \in E}p_i\label{eq:deriv_px}$$ The *explanation probability* $P_x(q|T)$ is then defined as the probability of the most likely explanation or proof of the query $q$ $$P_x(q|T) = \max\nolimits_{E\in E(q)}P(E|T) = \max\nolimits_{E\in E(q)} \prod_{c_i \in E}p_i,\label{eq:p_exp}$$ where $E(q)$ is the set of all explanations for query $q$, i.e., all minimal sets $E\subseteq L_T$ of probabilistic facts such that $E \cup BK \models q$ [@Kimmig07]. In our example, the set of all explanations for *path$(c,d)$* contains the edge from *c* to *d* (with probability 0.9) as well as the path consisting of the edges from *c* to *e* and from *e* to *d* (with probability $0\ldotp8\cdot 0\ldotp5=0\ldotp4$). Thus, $P_x(path(c,d)|T)=0\ldotp9$. The ProbLog semantics is essentially a distribution semantics [@Sato:95]. Sato has rigorously shown that this class of programs defines a joint probability distribution over the set of possible least Herbrand models of the program (allowing functors), that is, of the background knowledge $BK$ together with a subprogram $L \subseteq L_T$; for further details we refer to [@Sato:95]. The distribution semantics has been used widely in the literature, though often under other names or in a more restricted setting; see e.g. [@Dantsin; @Poole:93; @Fuhr00; @Poole00; @DalviS04]. Inference in ProbLog {#sec:inference} ==================== This section discusses algorithms for computing exactly or approximately the success and explanation probabilities of ProbLog queries. It additionally contributes a new algorithm for Monte Carlo approximation of success probabilities. Exact Inference {#sec:exact} --------------- Calculating the *success probability* of a query using Equation (\[eq:p\_suc\]) directly is infeasible for all but the tiniest programs, as the number of subprograms to be checked is exponential in the number of probabilistic facts. However, as we have seen in our example in Section \[sec:problog\], we can describe all subprograms allowing for a specific proof by means of the facts that such a program has to contain, i.e., all the ground probabilistic facts used in that proof. As probabilistic facts correspond to random variables indicating the presence of facts in a sampled program, we alternatively denote proofs by conjunctions of such random variables. In our example, query *path(c,d)* has two proofs in the full program: *{edge(c,d)}* and *{edge(c,e),edge(e,d)}*, or, using logical notation, $cd$ and $ce \wedge ed$. The set of all subprograms containing *some* proof thus can be described by a disjunction over all possible proofs, in our case, $cd \vee (ce \wedge ed)$. This idea forms the basis for the inference method presented in [@DeRaedt07], which uses two steps: 1. Compute the proofs of the query $q$ in the logical part of the theory $T$, that is, in $BK \cup L_T$. The result will be a DNF formula. 2. Compute the probability of this formula. Similar approaches are used for PRISM [@SatoKameya:01], ICL [@Poole00] and pD [@Fuhr00]. The probability of a single given proof, cf. Equation (\[eq:deriv\_px\]), is the marginal over all programs allowing for that proof, and thus equals the product of the probabilities of the facts used by that proof. However, we cannot directly sum the results for the different proofs to obtain the success probability, as a specific subprogram can allow several proofs and therefore contributes to the probability of each of these proofs. Indeed, in our example, all programs that are supersets of *{edge(c,e),edge(e,d),edge(c,d)}* contribute to the marginals of both proofs and would therefore be counted twice if summing the probabilities of the proofs. However, for mutually exclusive conjunctions, that is, conjunctions describing disjoint sets of subprograms, the probability is the sum of the individual probabilities. This situation can be achieved by adding *negated* random variables to a conjunction, thereby explicitly excluding subprograms covered by another part of the formula from the corresponding part of the sum. In the example, extending $ce \wedge ed$ to $ce \wedge ed \wedge \neg cd$ reduces the second part of the sum to those programs not covered by the first: $$P_s(path(c,d)|T)=P(cd \vee (ce\wedge ed)|T)$$$$= P(cd|T)+P(ce\wedge ed\wedge\neg cd|T)$$$$= 0\ldotp9 + 0\ldotp8\cdot0\ldotp5\cdot(1-0\ldotp9)=0\ldotp94$$ However, as the number of proofs grows, disjoining them gets more involved. Consider for example the query *path(a,d)* which has four different but highly interconnected proofs. In general, this problem is known as the *disjoint-sum-problem* or the two-terminal network reliability problem, which is \#P-complete [@Valiant1979]. Before returning to possible approaches to tackle the disjoint-sum-problem at the end of this section, we will now discuss the two steps of ProbLog’s exact inference in more detail. ![SLD-tree for query *path$(c,d)$.*[]{data-label="fig:SLD"}](sld) Following Prolog, the first step employs SLD-resolution to obtain all different proofs. As an example, the SLD-tree for the query *?- path$(c,d)$.* is depicted in Figure \[fig:SLD\]. Each successful proof in the SLD-tree uses a set of ground probabilistic facts $\{p_1::c_1, \cdots, p_k::c_k\} \subseteq T$. These facts are necessary for the proof, and the proof is *independent* of other probabilistic facts in $T$. Let us now introduce a Boolean random variable $b_i$ for each ground probabilistic fact $p_i::c_i \in T$, indicating whether $c_i$ is in a sampled logic program, that is, $b_i$ has probability $p_i$ of being true.[^2] A particular proof of query $q$ involving ground facts $\{p_1::c_1, \cdots, p_k::c_k\} \subseteq T$ is thus represented by the conjunctive formula $b_1 \wedge \cdots \wedge b_k$, which at the same time represents the set of all subprograms containing these facts. Furthermore, using $E(q)$ to denote the set of proofs or explanations of the goal $q$, the set of all subprograms containing *some* proof of $q$ can be denoted by $\bigvee_{e \in E(q) } \, \bigwedge_{c_i \in e} b_i $, as the following derivation shows: $$\begin{aligned} \bigvee_{e \in E(q) } \, \bigwedge_{c_i \in e} b_i & = & \bigvee_{e \in E(q) } \left( \bigwedge_{c_i \in e} b_i \wedge \bigwedge_{c_i \in L_T \backslash e} (b_i \vee \neg b_i)\right)\\ & = & \bigvee_{e \in E(q) } \bigvee_{L \subseteq L_T\backslash e } \left( \bigwedge_{c_i \in e} b_i \wedge \left(\bigwedge_{c_i \in L} b_i \wedge \bigwedge_{c_i \in L_T \backslash (L\union e)} \neg b_i\right)\right)\\ & = & \bigvee_{e \in E(q) , L \subseteq L_T\backslash e } \left( \bigwedge_{c_i \in L \union e} b_i \wedge \bigwedge_{c_i \in L_T \backslash (L\union e)} \neg b_i\right)\\ & = & \bigvee_{ L \subseteq L_T, \exists\theta L\union BK \models q\theta } \left( \bigwedge_{c_i \in L } b_i \wedge \bigwedge_{c_i \in L_T \backslash L} \neg b_i\right)\end{aligned}$$ We first add all possible ways of extending a proof $e$ to a full sampled program by considering each fact not in $e$ in turn. We then note that the disjunction of these fact-wise extensions can be written on the basis of sets. Finally, we rewrite the condition of the disjunction in the terms of Equation (\[eq:p\_suc\]). This is possible as each subprogram that is an extension of an explanation of $q$ entails some ground instance of $q$, and vice versa, each subprogram entailing $q$ is an extension of some explanation of $q$. As the DNF now contains conjunctions representing fully specified programs, its probability is a sum of products, which directly corresponds to Equation (\[eq:p\_suc\]): $$\begin{aligned} \lefteqn{ P(\bigvee_{ L \subseteq L_T, \exists\theta L\union BK \models q\theta } \left( \bigwedge_{c_i \in L } b_i \wedge \bigwedge_{c_i \in L_T \backslash L} \neg b_i\right)) }\\ &=& \sum_{ L \subseteq L_T, \exists\theta L\union BK \models q\theta } \left( \prod_{c_i \in L } p_i \cdot \prod_{c_i \in L_T \backslash L} (1- p_i)\right)\\ &=& \sum_{ L \subseteq L_T, \exists\theta L\union BK \models q\theta } P(L|T)\end{aligned}$$ We thus obtain the following alternative characterisation of the success probability: $$P_s(q|T) = P\left( \bigvee_{e \in E(q) } \, \bigwedge_{c_i \in e} b_i \right) \label{eq:dnf}$$ where $E(q)$ denotes the set of proofs or explanations of the goal $q$ and $b_i$ denotes the Boolean variable corresponding to ground probabilistic fact $p_i::c_i$. Thus, the problem of computing the success probability of a ProbLog query can be reduced to that of computing the probability of a DNF formula. However, as argued above, due to overlap between different conjunctions, the proof-based DNF of Equation (\[eq:dnf\]) cannot directly be transformed into a sum of products. Computing the probability of DNF formulae thus involves solving the disjoint-sum-problem, and therefore is itself a \#P-hard problem. Various algorithms have been developed to tackle this problem. The pD-engine HySpirit [@Fuhr00] uses the inclusion-exclusion principle, which is reported to scale to about ten proofs. For ICL, which extends PHA by allowing non-disjoint proofs, [@Poole00] proposes a symbolic disjoining algorithm, but does not report scalability results. Our implementation of ProbLog employs Binary Decision Diagrams (BDDs) [@Bryant86], an efficient graphical representation of a Boolean function over a set of variables, which scales to tens of thousands of proofs; see Section \[sec:BDD\] for more details. PRISM [@SatoKameya:01] and PHA [@Poole:93] differ from the systems mentioned above in that they avoid the disjoint-sum-problem by requiring the user to write programs such that proofs are guaranteed to be disjoint. On the other hand, as the *explanation probability* $P_x$ exclusively depends on the probabilistic facts used in one proof, it can be calculated using a simple branch-and-bound approach based on the SLD-tree, where partial proofs are discarded if their probability drops below that of the best proof found so far. Approximative Inference ----------------------- As the size of the DNF formula grows with the number of proofs, its evaluation can become quite expensive, and ultimately infeasible. For instance, when searching for paths in graphs or networks, even in small networks with a few dozen edges there are easily $O(10^6)$ possible paths between two nodes. ProbLog therefore includes several approximation methods. ### Bounded Approximation The first approximation algorithm, a slight variant of the one proposed in [@DeRaedt07], uses DNF formulae to obtain both an upper and a lower bound on the probability of a query. It is closely related to work by [@Poole93:jrnl] in the context of PHA, but adapted towards ProbLog. The method relies on two observations. First, we remark that the DNF formula describing sets of proofs is *monotone*, meaning that adding more proofs will never decrease the probability of the formula being true. Thus, formulae describing subsets of the full set of proofs of a query will always give a lower bound on the query’s success probability. In our example, the lower bound obtained from the shorter proof would be $P(cd|T) = 0\ldotp9$, while that from the longer one would be $P(ce\wedge ed|T) = 0\ldotp4$. Our second observation is that the probability of a proof $b_1 \wedge \ldots\wedge b_n$ will always be at most the probability of an arbitrary prefix $b_1 \wedge \ldots\wedge b_i, i\leq n$. In our example, the probability of the second proof will be at most the probability of its first edge from $c$ to $e$, i.e., $P(ce|T) = 0\ldotp8 \geq 0\ldotp4$. As disjoining sets of proofs, i.e., including information on facts that are *not* elements of the subprograms described by a certain proof, can only decrease the contribution of single proofs, this upper bound carries over to a set of proofs or partial proofs, as long as prefixes for all possible proofs are included. Such sets can be obtained from an incomplete SLD-tree, i.e., an SLD-tree where branches are only extended up to a certain point. This motivates ProbLog’s *bounded approximation algorithm*. The algorithm relies on a probability threshold $\gamma$ to stop growing the SLD-tree and thus obtain DNF formulae for the two bounds[^3]. The lower bound formula $d_1$ represents all proofs with a probability above the current threshold. The upper bound formula $d_2$ additionally includes all derivations that have been stopped due to reaching the threshold, as these still *may* succeed. Our goal is therefore to grow $d_1$ and $d_2$ in order to decrease $P(d_2|T)-P(d_1|T)$. Given an acceptance threshold $\delta_p$, an initial probability threshold $\gamma$, and a shrinking factor $\beta\in(0,1)$, the algorithm proceeds in an iterative-deepening manner as outlined in Algorithm \[alg:delta\]. Initially, both $d_1$ and $d_2$ are set to <span style="font-variant:small-caps;">False</span>, the neutral element with respect to disjunction, and the probability bounds are $0$ and $1$, as we have no full proofs yet, and the empty partial proof holds in any model. $d_1 = $ <span style="font-variant:small-caps;">False</span>; $d_2 = $ <span style="font-variant:small-caps;">False</span>; $P(d_1|T) =0$; $P(d_2|T) = 1$; $p = $<span style="font-variant:small-caps;">True</span>; Expand current proof $p$ set $d_2 = $ <span style="font-variant:small-caps;">True</span> Compute $P(d_1|T)$ and $P(d_2|T)$ $\gamma := \gamma\cdot\beta$ return $[P(d_1|T),P(d_2|T)]$ It should be clear that $P(d_1|T)$ monotonically increases, as the number of proofs never decreases. On the other hand, as explained above, if $d_2$ changes from one iteration to the next, this is always because a partial proof $p$ is either removed from $d_2$ and therefore no longer contributes to the probability, or it is replaced by proofs $p_1,\ldots , p_n$, such that $p_i = p \land s_i$, hence $P(p_1 \lor \ldots \lor p_n|T) = P(p \land s_1\lor\ldots\lor p\land s_n|T) = P(p \land ( s_1\lor\ldots\lor s_n)|T)$. As proofs are subsets of the probabilistic facts in the ProbLog program, each literal’s random variable appears at most once in the conjunction representing a proof, even if the corresponding subgoal is called multiple times when constructing the proof. We therefore know that the literals in the prefix $p$ cannot be in any suffix $s_i$, hence, given ProbLog’s independence assumption, $P(p \land ( s_1\lor\ldots\lor s_n)|T) = P(p|T)P(s_1\lor\ldots\lor s_n|T) \leq P(p|T)$. Therefore, $P(d_2)$ monotonically decreases. As an illustration, consider a probability threshold $\gamma =0\ldotp9$ for the SLD-tree in Figure \[fig:SLD\]. In this case, $d_1$ encodes the left success path while $d_2$ additionally encodes the path up to *path$(e,d)$*, i.e., $d_1 = cd$ and $d_2 = cd \vee ce$, whereas the formula for the full SLD-tree is $d = cd \vee (ce \wedge ed)$. The lower bound thus is $0\ldotp9$, the upper bound (obtained by disjoining $d_2$ to $cd \vee (ce\wedge\neg cd)$) is $0\ldotp98$, whereas the true probability is $0\ldotp94$. Notice that in order to implement this algorithm we need to compute the probability of a set of proofs. This task will be described in detail in Section \[sec:implementation\]. ### K-Best Using a fixed number of proofs to approximate the probability allows better control of the overall complexity, which is crucial if large numbers of queries have to be evaluated, e.g., in the context of parameter learning. [@Gutmann08] therefore introduces the $k$-probability $P_k(q|T)$, which approximates the success probability by using the $k$-best (that is, the $k$ most likely) explanations instead of all proofs when building the DNF formula used in Equation (\[eq:dnf\]): $$P_k(q|T) = P\left( \bigvee_{e \in E_k(q) } \, \bigwedge_{b_i \in var(e)} b_i \right)\label{eq:p_k}$$ where $E_k(q)=\{e \in E(q)|P_x(e)\geq P_x(e_k)\}$ with $e_k$ the $k$th element of $E(q)$ sorted by non-increasing probability. Setting $k=\infty$ leads to the success probability, whereas $k=1$ corresponds to the explanation probability provided that there is a single best proof. The branch-and-bound approach used to calculate the explanation probability can directly be generalized to finding the $k$-best proofs; cf. also [@Poole:93]. To illustrate $k$-probability, we consider again our example graph, but this time with query *path$(a,d)$*. This query has four proofs, represented by the conjunctions $ac\wedge cd$, $ab\wedge bc \wedge cd$, $ac\wedge ce \wedge ed$ and $ab\wedge bc \wedge ce \wedge ed$, with probabilities $0\ldotp72$, $0\ldotp378$, $0\ldotp32$ and $0\ldotp168$ respectively. As $P_1$ corresponds to the explanation probability $P_x$, we obtain $P_1(path(a,d))=0\ldotp72$. For $k=2$, the overlap between the best two proofs has to be taken into account: the second proof only adds information if the first one is absent. As they share edge $cd$, this means that edge $ac$ has to be missing, leading to $P_2(path(a,d))=P((ac\wedge cd) \vee (\neg ac \wedge ab\wedge bc \wedge cd))=0\ldotp72+(1-0\ldotp8)\cdot 0\ldotp378=0\ldotp7956$. Similarly, we obtain $P_3(path(a,d))=0\ldotp8276$ and $P_k(path(a,d))=0\ldotp83096$ for $k\geq 4$. ### Monte Carlo {#sec:mc_method} As an alternative approximation technique, we propose a Monte Carlo method, where we proceed as follows. Execute until convergence: 1. Sample a logic program from the ProbLog program 2. Check for the existence of some proof of the query of interest 3. Estimate the query probability $P$ as the fraction of samples where the query is provable We estimate convergence by computing the 95% confidence interval at each $m$ samples. Given a large number $N$ of samples, we can use the standard normal approximation interval to the binomial distribution: $$\delta \approx 2\times\sqrt{\frac{P.(P-1)}{N}}$$ Notice that confidence intervals do not directly correspond to the exact bounds used in our previous approximation algorithm. Still, we employ the same stopping criterion, that is, we run the Monte Carlo simulation until the width of the confidence interval is at most $\delta_p$. A similar algorithm (without the use of confidence intervals) was also used in the context of biological networks (not represented as Prolog programs) by [@Sevon06]. The use of a Monte Carlo method for probabilistic logic programs was suggested already by [@Dantsin], although he neither provides details nor reports on an implementation. Our approach differs from the MCMC method for Stochastic Logic Programs (SLPs) introduced by [@Cussens00] in that we do not use a Markov chain, but restart from scratch for each sample. Furthermore, SLPs are different in that they directly define a distribution over all proofs of a query. Investigating similar probabilistic backtracking approaches for ProbLog is a promising future research direction. Implementation {#sec:implementation} ============== This section discusses the main building blocks used to implement ProbLog on top of the YAP-Prolog system. An overview is shown in Figure \[fig:problog\_imp\], with a typical ProbLog program, including ProbLog facts and background knowledge (BK), at the top. ![ProbLog Implementation: A ProbLog program (top) requires the ProbLog library which in turn relies on functionality from the tries and array libraries. ProbLog queries (bottom-left) are sent to the YAP engine, and may require calling the BDD library CUDD via SimpleCUDD.[]{data-label="fig:problog_imp"}](implementation "fig:")\[fig:problog\] The implementation requires ProbLog programs to use the `problog` module. Each program consists of a set of labeled facts and of unlabeled *background knowledge*, a generic Prolog program. Labeled facts are preprocessed as described below. Notice that the implementation requires all queries to non-ground probabilistic facts to be ground on calling. In contrast to standard Prolog queries, where one is interested in answer substitutions, in ProbLog one is primarily interested in a probability. As discussed before, two common ProbLog queries ask for the most likely explanation and its probability, and the probability of whether a query would have an answer substitution. We have discussed two very different approaches to the problem: - In exact inference, $k$-best and bounded approximation, the engine explicitly reasons about probabilities of proofs. The challenge is how to compute the probability of each individual proof, store a large number of proofs, and compute the probability of sets of proofs. - In Monte Carlo, the probabilities of facts are used to sample from ProbLog programs. The challenge is how to compute a sample quickly, in a way that inference can be as efficient as possible. ProbLog programs execute from a top-level query and are driven through a ProbLog query. The inference algorithms discussed above can be abstracted as follows: - Initialise the inference algorithm; - While probabilistic inference did not converge: - initialise a new query; - execute the query, instrumenting every ProbLog call in the current proof. Instrumentation is required for recording the ProbLog facts required by a proof, but may also be used by the inference algorithm to stop proofs (e.g., if the current probability is lower than a bound); - process success or exit substitution; - Proceed to the next step of the algorithm: this may be trivial or may require calling an external solver, such as a BDD tool, to compute a probability. Notice that the current ProbLog implementation relies on the Prolog engine to efficiently execute goals. On the other hand, and in contrast to most other probabilistic language implementations, in ProbLog there is no clear separation between logical and probabilistic inference: in a fashion similar to constraint logic programming, probabilistic inference can drive logical inference. From a Prolog implementation perspective, ProbLog poses a number of interesting challenges. First, labeled facts have to be efficiently compiled to allow mutual calls between the Prolog program and the ProbLog engine. Second, for exact inference, $k$-best and bounded approximation, sets of proofs have to be manipulated and transformed into BDDs. Finally, Monte Carlo simulation requires representing and manipulating samples. We discuss these issues next. Source-to-source transformation ------------------------------- We use the `term_expansion` mechanism to allow Prolog calls to labeled facts, and for labeled facts to call the ProbLog engine. As an example, the program: $$\begin{array}{l} \mathtt{0\ldotp715::edge('PubMed\_2196878','MIM\_609065')\ldotp}\\ \mathtt{0\ldotp659::edge('PubMed\_8764571','HGNC\_5014')\ldotp}\\ \end{array}$$ would be compiled as: $$\begin{array}{lll} \mathtt{edge(A,B)} &\mathtt{:-} & \mathtt{problog\_edge(ID,A,B,LogProb),}\\ & & \mathtt{grounding\_id(edge(A,B),ID,GroundID),}\\ & & \mathtt{add\_to\_proof(GroundID,LogProb)\ldotp}\\ & & \\ \multicolumn{3}{l}{\mathtt{problog\_edge(0,'PubMed\_2196878','MIM\_609065',-0\ldotp3348)\ldotp}} \\ \multicolumn{3}{l}{\mathtt{problog\_edge(1,'PubMed\_8764571','HGNC\_5014',-0\ldotp4166)\ldotp}} \\ \end{array}$$ Thus, the internal representation of each fact contains an identifier, the original arguments, and the logarithm of the probability[^4]. The `grounding_id` procedure will create and store a grounding specific identifier for each new grounding of a non-ground probabilistic fact encountered during proving, and retrieve it on repeated use. For ground probabilistic facts, it simply returns the identifier itself. The `add_to_proof` procedure updates the data structure representing the current path through the search space, i.e., a queue of identifiers ordered by first use, together with its probability. Compared to the original meta-interpreter based implementation of [@DeRaedt07], the main benefit of source-to-source transformation is better scalability, namely by having a compact representation of the facts for the YAP engine [@DBLP:conf/padl/Costa07] and by allowing access to the YAP indexing mechanism [@jit-index]. Proof Manipulation ------------------ Manipulating proofs is critical in ProbLog. We represent each proof as a queue containing the identifier of each different ground probabilistic fact used in the proof, ordered by first use. The implementation requires calls to non-ground probabilistic facts to be ground, and during proving maintains a table of groundings used within the current query together with their identifiers. Grounding identifiers are based on the fact’s identifier extended with a grounding number, i.e. $5\_1$ and $5\_2$ would refer to different groundings of the non-ground fact with identifier $5$. In our implementation, the queue is stored in a backtrackable global variable, which is updated by calling `add_to_proof` with an identifier for the current ProbLog fact. We thus exploit Prolog’s backtracking mechanism to avoid recomputation of shared proof prefixes when exploring the space of proofs. Storing a proof is simply a question of adding the value of the variable to a store. As we have discussed above, the actual number of proofs can grow very quickly. ProbLog compactly represents a proof as a list of numbers. We would further like to have a scalable implementation of *sets* of proofs, such that we can compute the joint *probability* of large sets of proofs efficiently. Our representation for sets of proofs and our algorithm for computing the probability of such a set are discussed next. Sets of Proofs -------------- Storing and manipulating proofs is critical in ProbLog. When manipulating proofs, the key operation is often *insertion*: we would like to add a proof to an existing set of proofs. Some algorithms, such as exact inference or Monte Carlo, only manipulate complete proofs. Others, such as bounded approximation, require adding partial derivations too. The nature of the SLD-tree means that proofs tend to share both a prefix and a suffix. Partial proofs tend to share prefixes only. This suggests using *tries* to maintain the set of proofs. We use the YAP implementation of tries for this task, based itself on XSB Prolog’s work on tries of terms [@RamakrishnanIV-99], which we briefly summarize here. Tries [@Fredkin-62] were originally invented to index dictionaries, and have since been generalised to index recursive data structures such as terms. Please refer to [@Bachmair-93; @Graf-96; @RamakrishnanIV-99] for the use of tries in automated theorem proving, term rewriting and tabled logic programs. An essential property of the trie data structure is that common prefixes are stored only once. A trie is a tree structure where each different path through the trie data units, the *trie nodes*, corresponds to a term described by the tokens labelling the nodes traversed. For example, the tokenized form of the term $f(g(a),1)$ is the sequence of 4 tokens: $f/2$, $g/1$, $a$ and $1$. Two terms with common prefixes will branch off from each other at the first distinguishing token. Trie’s internal nodes are four field data structures, storing the node’s token, a pointer to the node’s first child, a pointer to the node’s parent and a pointer to the node’s next sibling, respectively. Each internal node’s outgoing transitions may be determined by following the child pointer to the first child node and, from there, continuing sequentially through the list of sibling pointers. When a list of sibling nodes becomes larger than a threshold value (8 in our implementation), we dynamically index the nodes through a hash table to provide direct node access and therefore optimise the search. Further hash collisions are reduced by dynamically expanding the hash tables. Inserting a term requires in the worst case allocating as many nodes as necessary to represent its complete path. On the other hand, inserting repeated terms requires traversing the trie structure until reaching the corresponding leaf node, without allocating any new node. In order to minimize the number of nodes when storing proofs in a trie, we use Prolog lists to represent proofs. For example, a ProbLog proof $[3, 5\_1, 7, 5\_2]$ uses ground fact 3, a first grounding of fact 5, ground fact 7 and another grounding of fact 5, that is, list elements in proofs are always either integers or two integers with an underscore in between. Figure \[fig:trie\_proofs\] presents an example of a trie storing three proofs. Initially, the trie contains the root node only. Next, we store the proof $[3, 5\_1, 7, 5\_2]$ and six nodes (corresponding to six tokens) are added to represent it (Figure \[fig:trie\_proofs\](a)). The proof $[3, 5\_1, 9, 7, 5\_2]$ is then stored which requires seven nodes. As it shares a common prefix with the previous proof, we save the three initial nodes common to both representations (Figure \[fig:trie\_proofs\](b)). The proof $[3, 4, 7]$ is stored next and we save again the two initial nodes common to all proofs (Figure \[fig:trie\_proofs\](c)). ![Using tries to store proofs. Initially, the trie contains the root node only. Next, we store the proofs: (a) $[3, 5\_1, 7, 5\_2]$; (b) $[3, 5\_1, 9, 7, 5\_2]$; and (c) $[3, 4, 7]$.[]{data-label="fig:trie_proofs"}](trie_proofs) Binary Decision Diagrams {#sec:BDD} ------------------------ ![Binary Decision Diagram encoding the DNF formula $cd \vee (ce \wedge ed)$, corresponding to the two proofs of query *path(c,d)* in the example graph. An internal node labeled $xy$ represents the Boolean variable for the edge between $x$ and $y$, solid/dashed edges correspond to values true/false and are labeled with the probability that the variable takes this value.[]{data-label="fig:BDD"}](bdd) To efficiently compute the probability of a DNF formula representing a set of proofs, our implementation represents this formula as a reduced ordered Binary Decision Diagram (BDD) [@Bryant86], which can be viewed as a compact encoding of a Boolean decision tree. Given a fixed variable ordering, a Boolean function $f$ can be represented as a full Boolean decision tree, where each node on the $i$th level is labeled with the $i$th variable and has two children called low and high. Leaves are labeled by the outcome of $f$ for the variable assignment corresponding to the path to the leaf, where in each node labeled $x$, the branch to the low (high) child is taken if variable $x$ is assigned 0 (1). Starting from such a tree, one obtains a BDD by merging isomorphic subgraphs and deleting redundant nodes until no further reduction is possible. A node is redundant if the subgraphs rooted at its children are isomorphic. Figure \[fig:BDD\] shows the BDD for the existence of a path between *c* and *d* in our earlier example. We use SimpleCUDD[^5] as a wrapper tool for the BDD package CUDD[^6] to construct and evaluate BDDs. More precisely, the trie representation of the DNF is translated to a BDD generation script, which is processed by SimpleCUDD to build the BDD using CUDD primitives. It is executed via Prolog’s shell utility, and results are reported via shared files. $i := 1$ $S_{\wedge} := \{(C,P)|C $ leaf in $T$ and single child of its parent $P \}$ write $n_i = P\wedge C$ $T := \textsc{Replace}(T,(C,P),n_i)$ $i := i + 1$ $S_{\vee} := \{[C_1,\ldots,C_n]|$ leaves $C_j $ are all the children of some parent $P$ in $T\}$ write $n_i = C_1 \vee \ldots \vee C_n$ $T := \textsc{Replace}(T,[C_1,\ldots,C_n],n_i)$ $i := i + 1$ write $top = n_{i-1}$ During the generation of the code, it is crucial to exploit the structure sharing (prefixes and suffixes) already in the trie representation of a DNF formula, otherwise CUDD computation time becomes extremely long or memory overflows quickly. Since CUDD builds BDDs by joining smaller BDDs using logical operations, the trie is traversed bottom-up to successively generate code for all its subtrees. Algorithm \[alg:trie2bdd\] gives the details of this procedure. Two types of operations are used to combine nodes. The first creates conjunctions of leaf nodes and their parent if the leaf is a single child, the second creates disjunctions of all child nodes of a node if these child nodes are all leaves. In both cases, a subtree that occurs multiple times in the trie is translated only once, and the resulting BDD is used for all occurrences of that subtree. Because of the optimizations in CUDD, the resulting BDD can have a very different structure than the trie. The translation for query *path(a,d)* in our example graph is illustrated in Figure \[fig:trie2bdd\], it results in the following script: $$\begin{aligned} n1 & = & ce \wedge ed\\ n2 & = & cd \vee n1\\ n3 & = & ac \wedge n2\\ n4 & = & bc \wedge n2\\ n5 & = & ab \wedge n4\\ n6 & = & n3 \vee n5\\ top & = & n6\end{aligned}$$ After CUDD has generated the BDD, the probability of a formula is calculated by traversing the BDD, in each node summing the probability of the high and low child, weighted by the probability of the node’s variable being assigned true and false respectively, cf. Algorithm \[alg:calcprob\]. Intermediate results are cached, and the algorithm has a time and space complexity linear in the size of the BDD. If $n$ is the 1-terminal then return 1 If $n$ is the 0-terminal then return 0 let $h$ and $l$ be the high and low children of $n$ $prob(h) :=$ call <span style="font-variant:small-caps;">Probability</span>($h$) $prob(l) :=$ call <span style="font-variant:small-caps;">Probability</span>($l$) return $p_n \cdot prob(h) + (1-p_n) \cdot prob(l)$ For illustration, consider again Figure \[fig:BDD\]. The algorithm starts by assigning probabilities $0$ and $1$ to the $0$- and $1$-leaf respectively. The node labeled $ed$ has probability $0\ldotp5\cdot1+0\ldotp5\cdot0=0\ldotp5$, node $ce$ has probability $0\ldotp8\cdot0\ldotp5+0\ldotp2\cdot0=0\ldotp4$; finally, node $cd$, and thus the entire formula, has probability $0\ldotp9\cdot1+0\ldotp1\cdot0\ldotp4=0\ldotp94$. Monte Carlo {#monte-carlo} ----------- The Monte Carlo implementation is shown in Algorithm \[alg:mc\]. It receives a query $q$, an acceptance threshold $\delta_p$ and a constant $m$ determining the number of samples generated per iteration. At the end of each iteration, it estimates the probability $p$ as the fraction of programs sampled over all previous iterations that entailed the query, and the confidence interval width to be used in the stopping criterion as explained in Section \[sec:mc\_method\]. $c = 0$; $i = 0$; $p = 0$; $\delta = 1$; Generate a sample $P'$; $c:=c+1;$ $i:=i+1$; $p := c/i$ $\delta := 2\times\sqrt{\frac{p\cdot(p-1)}{i}}$ return $p$ Monte Carlo execution is quite different from the approaches discussed before, as the two main steps are **(a)** generating a sample program and **(b)** performing standard refutation on the sample. Thus, instead of combining large numbers of proofs, we need to manipulate large numbers of different programs or samples. Our first approach was to generate a complete sample and to check for a proof. In order to accelerate the process, proofs were cached in a trie to skip inference on a new sample. If no proofs exist on a cache, we call the standard Prolog refutation procedure. Although this approach works rather well for small databases, it does not scale to larger databases where just generating a new sample requires walking through millions of facts. We observed that even in large programs proofs are often quite short, i.e., we only need to verify whether facts from a small fragment of the database are in the sample. This suggests that it may be a good idea to take advantage of the independence between facts and generate the sample *lazily*: we verify whether a fact is in the sample only when we need it for a proof. YAP represents samples compactly as a three-valued array with one field for each fact, where $0$ means the fact was not yet sampled, $1$ it was already sampled and belongs to the sample, $2$ it was already sampled and does not belong to the sample. In this implementation: 1. New samples are generated by resetting the sampling array. 2. At every call to `add_to_proof`, given the current ProbLog literal $f$: 1. if $s[f] == 0 $, $s[f] = sample(f)$; 2. if $s[f] == 1$, succeed; 3. if $s[f] == 2$, fail; Note that as fact identifiers are used to access the array, the approach cannot directly be used for non-ground facts. The current implementation of Monte Carlo therefore uses the internal database to store the result of sampling different groundings of such facts. Experiments {#sec:experiments} =========== We performed experiments with our implementation of ProbLog in the context of the biological network obtained from the Biomine project [@Sevon06]. We used two subgraphs extracted around three genes known to be connected to the Alzheimer disease (HGNC numbers 983, 620 and 582) as well as the full network. The smaller graphs were obtained querying Biomine for best paths of length 2 (resulting in graph <span style="font-variant:small-caps;">Small</span>) or all paths of length 3 (resulting in graph <span style="font-variant:small-caps;">Medium</span>) starting at one of the three genes. <span style="font-variant:small-caps;">Small</span> contains 79 nodes and 144 edges, <span style="font-variant:small-caps;">Medium</span> 5220 nodes and 11532 edges. We used <span style="font-variant:small-caps;">Small</span> for a first comparison of our algorithms on a small scale network where success probabilities can be calculated exactly. Scalability was evaluated using both <span style="font-variant:small-caps;">Medium</span> and the entire <span style="font-variant:small-caps;">Biomine</span> network with roughly 1,000,000 nodes and 6,000,000 edges. In all experiments, we queried for the probability that two of the gene nodes mentioned above are connected, that is, we used queries such as `path(’HGNC_983’,’HGNC_620’,Path)`. We used the following definition of an acyclic path in our background knowledge: $$\begin{array}{lll} \mathtt{path(X,Y,A)} & \mathtt{:-} & \mathtt{path(X,Y,[X],A)},\\ \mathtt{path(X,X,A,A)\ldotp} & &\\ \mathtt{path(X,Y,A,R)} & \mathtt{:-} & \mathtt{X~\backslash ==~Y}, \\ & & \mathtt{edge(X,Z),} \\ & & \mathtt{absent(Z,A),} \\ & & \mathtt{path(Z,Y,[Z|A],R)\ldotp}\\ \end{array}$$ As list operations to check for the absence of a node get expensive for long paths, we consider an alternative definition for use in Monte Carlo. It provides cheaper testing by using the internal database of YAP to store nodes on the current path under key `visited`: $$\begin{array}{lll} \mathtt{memopath(X,Y,A)} & \mathtt{:-} & \mathtt{eraseall(visited)}, \\ && \mathtt{memopath(X,Y,[X],A)\ldotp}\\ \mathtt{memopath(X,X,A,A)\ldotp} & &\\ \mathtt{memopath(X,Y,A,R)} & \mathtt{:-} & \mathtt{X~\backslash ==~Y}, \\ & & \mathtt{edge(X,Z),} \\ & & \mathtt{recordzifnot(visited,Z,\_),}\\ & & \mathtt{memopath(Z,Y,[Z|A],R)\ldotp}\\ \end{array}$$ Finally, to assess performance on the full network for queries with smaller probabilities, we use the following definition of paths with limited length: $$\begin{array}{lll} \mathtt{lenpath(N,X,Y,Path)} & \mathtt{ :-} & \mathtt{lenpath(N,X,Y,[X],Path)\ldotp}\\ \mathtt{lenpath(N,X,X,A,A) } & \mathtt{ :-} & \mathtt{ N >= 0\ldotp}\\ \mathtt{lenpath(N,X,Y,A,P) } & \mathtt{ :-} & \mathtt{ X \backslash == Y},\\ && \mathtt{ N > 0},\\ && \mathtt{ edge(X,Z)},\\ && \mathtt{ absent(Z,A)},\\ && \mathtt{ NN\ is\ N-1},\\ && \mathtt{ lenpath(NN,Z,Y,[Z|A],P)\ldotp} \end{array}$$ All experiments were performed on a Core 2 Duo 2.4 GHz 4 GB machine running Linux. All times reported are in `msec` and do not include the time to load the graph into Prolog. The latter takes 20, 200 and 78140 `msec` for <span style="font-variant:small-caps;">Small</span>, <span style="font-variant:small-caps;">Medium</span> and <span style="font-variant:small-caps;">Biomine</span> respectively. Furthermore, as YAP indexes the database at query time, we query for the explanation probability of `path(’HGNC_620’,’HGNC_582’,Path)` before starting runtime measurements. This takes 0, 50 and 25900 `msec` for <span style="font-variant:small-caps;">Small</span>, <span style="font-variant:small-caps;">Medium</span> and <span style="font-variant:small-caps;">Biomine</span> respectively. We report $T_P$, the time spent by ProbLog to search for proofs, as well as $T_B$, the time spent to execute BDD programs (whenever meaningful). We also report the estimated probability $P$. For approximate inference using bounds, we report exact intervals for $P$, and also include the number $n$ of BDDs constructed. We set both the initial threshold and the shrinking factor to $0\ldotp5$. We computed $k$-probability for $k=1,2,\ldots,1024$. In the bounding algorithms, the error interval ranged between 10% and 1%. Monte Carlo recalculates confidence intervals after $m=1000$ samples. We also report the number $S$ of samples used. #### Small Sized Sample ----------- ------ ----- ------ ------ ----- ------ ----- ----- ------ path [**k**]{} 1 0 13 0.07 0 7 0.05 0 26 0.66 2 0 12 0.08 0 6 0.05 0 6 0.66 4 0 12 0.10 10 6 0.06 0 6 0.86 8 10 12 0.11 0 6 0.06 0 6 0.92 16 0 12 0.11 10 6 0.06 0 6 0.92 32 20 34 0.11 10 17 0.07 0 7 0.96 64 20 74 0.11 10 46 0.09 10 38 0.99 128 50 121 0.11 40 161 0.10 20 257 1.00 256 140 104 0.11 80 215 0.10 90 246 1.00 512 450 118 0.11 370 455 0.11 230 345 1.00 1024 1310 537 0.11 950 494 0.11 920 237 1.00 **exact** 670 450 0.11 8060 659 0.11 630 721 1.00 ----------- ------ ----- ------ ------ ----- ------ ----- ----- ------ : $k$-probability on <span style="font-variant:small-caps;">Small</span>. []{data-label="tab:1"} We first compared our algorithms on <span style="font-variant:small-caps;">Small</span>. Table \[tab:1\] shows the results for $k$-probability and exact inference. Note that nodes 620 and 582 are close to each other, whereas node 983 is farther apart. Therefore, connections involving the latter are less likely. In this graph, we obtained good approximations using a small fraction of proofs (the queries have 13136, 155695 and 16048 proofs respectively). Our results also show a significant increase in running times as ProbLog explores more paths in the graph, both within the Prolog code and within the BDD code. The BDD running times can vary widely, we may actually have large running times for smaller BDDs, depending on BDD structure. However, using SimpleCUDD instead of the C++ interface used in [@Kimmig08] typically decreases BDD time by at least one or two orders of magnitude. Table \[tab:2\] gives corresponding results for bounded approximation. The algorithm converges quickly, as few proofs are needed and BDDs remain small. Note however that exact inference is competitive for this problem size. Moreover, we observe large speedups compared to the implementation with meta-interpreters used in [@DeRaedt07], where total runtimes to reach $\delta=0\ldotp01$ for these queries were 46234, 206400 and 307966 `msec` respectively. Table \[tab:3\] shows the performance of the Monte Carlo estimator. On <span style="font-variant:small-caps;">Small</span>, Monte Carlo is the fastest approach. Already within the first 1000 samples a good approximation is obtained. The experiments on <span style="font-variant:small-caps;">Small</span> thus confirm that the implementation on top of YAP-Prolog enables efficient probabilistic inference on small sized graphs. ---------- ---------- --------------- ---------------- --------------- ------------- --------------- path $\delta$ 0.10 0  48  4 \[0.07,0.12\] 10     74    6 \[0.06,0.11\] 0     25  2 \[0.91,1.00\] 0.05 0  71  6 \[0.07,0.11\] 0     75    6 \[0.06,0.11\] 0   486  4 \[0.98,1.00\] 0.01 0  83  7 \[0.11,0.11\] 140  3364  10 \[0.10,0.11\] 60  1886  6 \[1.00,1.00\] ---------- ---------- --------------- ---------------- --------------- ------------- --------------- : Inference using bounds on <span style="font-variant:small-caps;">Small</span>. []{data-label="tab:2"} ---------- ------- ----- ------ ------- ----- ------ ------ ---- ------ path $\delta$ 0.10 1000 10 0.11 1000 10 0.11 1000 30 1.00 0.05 1000 10 0.11 1000 10 0.10 1000 20 1.00 0.01 16000 130 0.11 16000 170 0.11 1000 30 1.00 ---------- ------- ----- ------ ------- ----- ------ ------ ---- ------ : Monte Carlo Inference on <span style="font-variant:small-caps;">Small</span>. []{data-label="tab:3"} #### Medium Sized Sample For graph <span style="font-variant:small-caps;">Medium</span> with around 11000 edges, exact inference is no longer feasible. Table \[tab:1a\] again shows results for the $k$-probability. Comparing these results with the corresponding values from Table \[tab:1\], we observe that the estimated probability is higher now: this is natural, as the graph has both more nodes and is more connected, therefore leading to many more possible explanations. This also explains the increase in running times. Approximate inference using bounds only reached loose bounds (with differences $>0\ldotp 2$) on queries involving node `’HGNC_983’`, as upper bound formulae with more than 10 million conjunctions were encountered, which could not be processed. The Monte Carlo estimator using the standard definition of `path/3` on <span style="font-variant:small-caps;">Medium</span> did not complete the first $1000$ samples within one hour. A detailed analysis shows that this is caused by some queries backtracking too heavily. Table \[tab:3a\] therefore reports results using the memorising version `memopath/3`. With this improved definition, Monte Carlo performs well: it obtains a good approximation in a few seconds. Requiring tighter bounds however can increase runtimes significantly. ----------- ------ ---- ------ ------ --- ------ ------ ------ ------ path [**k**]{} 1 180 6 0.33 1620 6 0.30 10 6 0.92 2 180 6 0.33 1620 6 0.30 20 6 0.92 4 180 6 0.33 1630 6 0.30 10 6 0.92 8 220 6 0.33 1630 6 0.30 20 6 0.92 16 260 6 0.33 1660 6 0.30 30 6 0.99 32 710 6 0.40 1710 7 0.30 110 6 1.00 64 1540 7 0.42 1910 6 0.30 200 6 1.00 128 1680 6 0.42 2230 6 0.30 240 9 1.00 256 2190 7 0.55 2720 6 0.49 290 196 1.00 512 2650 7 0.64 3730 7 0.53 1310 327 1.00 1024 8100 41 0.70 5080 8 0.56 3070 1357 1.00 ----------- ------ ---- ------ ------ --- ------ ------ ------ ------ : $k$-probability on <span style="font-variant:small-caps;">Medium</span>. []{data-label="tab:1a"} ---------- ------- ------- ------ ------- ------- ------ ------ ------ ------ memo $\delta$ 0.10 1000 1180 0.78 1000 2130 0.76 1000 1640 1.00 0.05 2000 2320 0.77 2000 4230 0.74 1000 1640 1.00 0.01 29000 33220 0.77 29000 61140 0.77 1000 1670 1.00 ---------- ------- ------- ------ ------- ------- ------ ------ ------ ------ : Monte Carlo Inference using `memopath/3` on <span style="font-variant:small-caps;">Medium</span>. []{data-label="tab:3a"} #### Biomine Database The Biomine Database covers hundreds of thousands of entities and millions of links. On <span style="font-variant:small-caps;">Biomine</span>, we therefore restricted our experiments to the approximations given by $k$-probability and Monte Carlo. Given the results on <span style="font-variant:small-caps;">Medium</span>, we directly used `memopath/3` for Monte Carlo. Tables \[tab:1c\] and \[tab:3b\] show the results on the large network. We observe that on this large graph, the number of possible paths is tremendous, which implies success probabilities practically equal to 1. Still, we observe that ProbLog’s branch-and-bound search to find the best solutions performs reasonably also on this size of network. However, runtimes for obtaining tight confidence intervals with Monte Carlo explode quickly even with the improved path definition. ----------- --------- -------- ------ ----------- --------- ------ --------- -------- ------ path [**k**]{} 1 5,760 49 0.16 8,910 48 0.11 10 48 0.59 2 5,800 48 0.16 10,340 48 0.17 180 48 0.63 4 6,200 48 0.16 13,640 48 0.28 360 48 0.65 8 7,480 48 0.16 15,550 49 0.38 500 48 0.66 16 11,470 49 0.50 58,050 49 0.53 630 48 0.92 32 15,100 49 0.57 106,300 49 0.56 2,220 167 0.95 64 53,760 84 0.80 146,380 101 0.65 3,690 167 0.95 128 71,560 126 0.88 230,290 354 0.76 7,360 369 0.98 256 138,300 277 0.95 336,410 520 0.85 13,520 1,106 1.00 512 242,210 730 0.98 501,870 2,744 0.88 23,910 3,444 1.00 1024 364,490 10,597 0.99 1,809,680 100,468 0.93 146,890 10,675 1.00 ----------- --------- -------- ------ ----------- --------- ------ --------- -------- ------ : $k$-probability on <span style="font-variant:small-caps;">Biomine</span>. []{data-label="tab:1c"} ---------- ------ --------- ------ ------ ----------- ------ ------ ----------- ------ memo $\delta$ 0.10 1000 100,700 1.00 1000 1,656,660 1.00 1000 1,696,420 1.00 0.05 1000 100,230 1.00 1000 1,671,880 1.00 1000 1,690,830 1.00 0.01 1000 93,120 1.00 1000 1,710,200 1.00 1000 1,637,320 1.00 ---------- ------ --------- ------ ------ ----------- ------ ------ ----------- ------ : Monte Carlo Inference using `memopath/3` on <span style="font-variant:small-caps;">Biomine</span>. []{data-label="tab:3b"} Given that sampling a program that does not entail the query is extremely unlikely for the setting considered so far, we performed an additional experiment on <span style="font-variant:small-caps;">Biomine</span>, where we restrict the number of edges on the path connecting two nodes to a maximum of 2 or 3. Results are reported in Table \[tab:shortpath\]. As none of the resulting queries have more than 50 proofs, exact inference is much faster than Monte Carlo, which needs a higher number of samples to reliably estimate probabilities that are not close to $1$. ---------- ------- ----------- ------ ------- ----------- ------ ------- --------- ------ len $\delta$ 0.10 1000 21,400 0.04 1000 18,720 0.11 1000 19,150 0.58 0.05 1000 19,770 0.05 1000 20,980 0.10 2000 35,100 0.55 0.01 6000 112,740 0.04 16000 307,520 0.11 40000 764,700 0.55 exact - 477 0.04 - 456 0.11 - 581 0.55 0.10 1000 106,730 0.14 1000 105,350 0.33 1000 45,400 0.96 0.05 1000 107,920 0.14 2000 198,930 0.34 1000 49,950 0.96 0.01 19000 2,065,030 0.14 37000 3,828,520 0.35 6000 282,400 0.96 exact - 9,413 0.14 - 9,485 0.35 - 15,806 0.96 ---------- ------- ----------- ------ ------- ----------- ------ ------- --------- ------ : Monte Carlo inference for different values of $\delta$ and exact inference using `lenpath/4` with length at most $2$ (top) or $3$ (bottom) on <span style="font-variant:small-caps;">Biomine</span>. For exact inference, runtimes include both Prolog and BDD time.[]{data-label="tab:shortpath"} Altogether, the experiments confirm that our implementation provides efficient inference algorithms for ProbLog that scale to large databases. Furthermore, compared to the original implementation of [@DeRaedt07], we obtain large speedups in both the Prolog and the BDD part, thereby opening new perspectives for applications of ProbLog. Conclusions {#sec:conclusion} =========== ProbLog is a simple but elegant probabilistic logic programming language that allows one to explicitly represent uncertainty by means of probabilistic facts denoting independent random variables. The language is a simple and natural extension of the logic programming language Prolog. We presented an efficient implementation of the ProbLog language on top of the YAP-Prolog system that is designed to scale to large sized problems. We showed that ProbLog can be used to obtain both explanation and (approximations of) success probabilities for queries on a large database. To the best of our knowledge, ProbLog is the first example of a probabilistic logic programming system that can execute queries on such large databases. Due to the use of BDDs for addressing the disjoint-sum-problem, the initial implementation of ProbLog used in [@DeRaedt07] already scaled up much better than alternative implementations such as Fuhr’s pD engine HySpirit [@Fuhr00]. The tight integration in YAP-Prolog presented here leads to further speedups in runtime of several orders of magnitude. Although we focused on connectivity queries and Biomine in this work, similar problems are found across many domains; we believe that the techniques presented apply to a wide variety of queries and databases because ProbLog provides a clean separation between background knowledge and what is specific to the engine. As shown for Monte Carlo inference, such an interface can be very useful to improve performance as it allows incremental refinement of background knowledge, e.g., graph procedures. Initial experiments with Dijkstra’s algorithm for finding the explanation probability are very promising. ProbLog is closely related to some alternative formalisms such as PHA and ICL [@Poole:93; @Poole00], pD [@Fuhr00] and PRISM [@SatoKameya:01] as their semantics are all based on Sato’s distribution semantics even though there exist also some subtle differences. However, ProbLog is – to the best of the authors’ knowledge – the first implementation that tightly integrates Sato’s original distribution semantics [@Sato:95] in a state-of-the-art Prolog system without making additional restrictions (such as the exclusive explanation assumption made in PHA and PRISM). As ProbLog, both PRISM and the ICL implementation AILog2 use a two-step approach to inference, where proofs are collected in the first phase, and probabilities are calculated once all proofs are known. AILog2 is a meta-interpreter implemented in SWI-Prolog for didactical purposes, where the disjoint-sum-problem is tackled using a symbolic disjoining technique [@Poole00]. PRISM, built on top of B-Prolog, requires programs to be written such that alternative explanations for queries are mutually exclusive. PRISM uses a meta-interpreter to collect proofs in a hierarchical datastructure called explanation graph. As proofs are mutually exclusive, the explanation graph directly mirrors the sum-of-products structure of probability calculation [@SatoKameya:01]. ProbLog is the first probabilistic logic programming system using BDDs as a basic datastructure for probability calculation, a principle that receives increased interest in the probabilistic logic learning community, cf. for instance [@Riguzzi; @sato:ilp08]. Furthermore, as compared to SLPs [@Muggleton96], CLP($\cal BN$) [@Costa03:uai], and BLPs [@Kersting08], ProbLog is a much simpler and in a sense more primitive probabilistic programming language. Therefore, the relationship between probabilistic logic programming and ProbLog is, in a sense, analogous to that between logic programming and Prolog. From this perspective, it is our hope and goal to further develop ProbLog so that it can be used as a general purpose programming language with an efficient implementation for use in statistical relational learning [@Getoor07] and probabilistic programming [@DeRaedt-PILPbook]. One important use of such a probabilistic programming language is as a target language in which other formalisms can be efficiently compiled. For instance, it has already been shown that CP-logic [@Vennekens], a recent elegant probabilistic knowledge representation language based on a probabilistic extension of clausal logic, can be compiled into ProbLog [@Riguzzi] and it is well-known that SLPs [@Muggleton96] can be compiled into Sato’s PRISM, which is closely related to ProbLog. Further evidence is provided in [@DeRaedt-NIPSWS08]. Another, related use of ProbLog is as a vehicle for developing learning and mining algorithms and tools [@Kimmig07; @DeRaedt08MLJ; @Gutmann08; @Kimmig09; @DeRaedt-IQTechReport]. In the context of probabilistic representations [@Getoor07; @DeRaedt-PILPbook], one typically distinguishes two types of learning: parameter estimation and structure learning. In parameter estimation in the context of ProbLog and PRISM, one starts from a set of queries and the logical part of the program and the problem is to find good estimates of the parameter values, that is, the probabilities of the probabilistic facts in the program. [@Gutmann08] introduces a gradient descent approach to parameter learning for ProbLog that extends the BDD-based methods discussed here. In structure learning, one also starts from queries but has to find the logical part of the program as well. Structure learning is therefore closely related to inductive logic programming. The limiting factor in statistical relational learning and probabilistic logic learning is often the efficiency of inference, as learning requires repeated computation of the probabilities of many queries. Therefore, improvements on inference in probabilistic programming implementations have an immediate effect on learning. The above compilation approach also raises the interesting and largely open question whether not only inference problems for alternative formalisms can be compiled into ProbLog but whether it is also possible to compile learning problems for these logics into learning problems for ProbLog. Finally, as ProbLog, unlike PRISM and PHA, deals with the disjoint-sum-problem, it is interesting to study how program transformation and analysis techniques could be used to optimize ProbLog programs, by detecting and taking into account situations where some conjunctions are disjoint. At the same time, we currently investigate how tabling, one of the keys to PRISM’s efficiency, can be incorporated in ProbLog [@Mantadelis09; @Kimmig-SRL09]. ### Acknowledgements {#acknowledgements .unnumbered} We would like to thank Hannu Toivonen, Bernd Gutmann and Kristian Kersting for their many contributions to ProbLog, the Biomine team for the application, and Theofrastos Mantadelis for the development of SimpleCUDD. This work is partially supported by the GOA project 2008/08 Probabilistic Logic Learning. Angelika Kimmig is supported by the Research Foundation-Flanders (FWO-Vlaanderen). Vítor Santos Costa and Ricardo Rocha are partially supported by the research projects STAMPA (PTDC/EIA/67738/2006) and JEDI (PTDC/ EIA/66924/2006) and by Fundação para a Ciência e Tecnologia. [^1]: If the program contains multiple instances of the same fact, they correspond to different random variables, i.e. $\{p::c\}$ and $\{p::c, p::c\}$ are different ProbLog programs. [^2]: For better readability, we do not write substitutions explicitly here. [^3]: Using a probability threshold instead of the depth bound of [@DeRaedt07] has been found to speed up convergence, as upper bounds have been found to be tighter on initial levels. [^4]: We use the logarithm to avoid numerical problems when calculating the probability of a derivation, which is used to drive inference. [^5]: <http://www.cs.kuleuven.be/~theo/tools/simplecudd.html> [^6]: <http://vlsi.colorado.edu/~fabio/CUDD>

--- abstract: 'In this paper we consider a model for the diffusion of a population in a strip-shaped field, where the growth of the species is governed by a Fisher-KPP equation and which is bounded on one side by a road where the species can have a different diffusion coefficient. Dirichlet homogeneous boundary conditions are imposed on the other side of the strip. We prove the existence of an asymptotic speed of propagation which is greater than the one of the case without road and study its behavior for small and large diffusions on the road. Finally we prove that, when the width of the strip goes to infinity, the asymptotic speed of propagation approaches the one of an half-plane bounded by a road, case that has been recently studied in [@BRR1; @BRR2].' author: - | Andrea Tellini\ \ \ \ `andrea.tellini@ehess.fr` title: '**Propagation speed in a strip bounded by a line with different diffusion**[^1]' --- *In memory of Giuliano Bardi (1948–2012),\ the one who taught me what a derivative is.* **Keywords:** KPP equations, reaction-diffusion systems, 1D-2D systems, asymptotic speed of propagation. **2010 MSC:** 35K57, 35B40, 35K40, 35B53. Introduction {#sec1} ============ Recently, in [@BRR1], the system $$\label{11} \left\{ \begin{array}{lll} \!\!\!u_t(x,t)\!-\!Du_{xx}(x,t)\!=\!\nu v(x,L,t)\!-\!\m u(x,t) & \!\!\text{for } x\in{{\mathbb{R}}}, & \!\! t>0 \\ \!\!\!v_t(x,y,t)-d\D v(x,y,t)=f(v) & \!\!\text{for } \!(x,y)\!\in\!{{\mathbb{R}}}\!\times\!(-\infty,L), & \!\! t>0 \\ \!\!\!dv_y(x,L,t)=\m u(x,t)-\nu v(x,L,t) & \!\!\text{for } x\in{{\mathbb{R}}}, & \!\! t>0. \end{array} \right.$$ was introduced to model the evolution of the species $v(x,y,t)$ in a *field* ${{\mathbb{R}}}\times(-\infty,L)$ which is bounded at the level $y=L$ by a *road* where part of the same species, $u(x,t)$, diffuses with coefficient $D>0$, which in principle may be different from the diffusion coefficient in the field $d>0$. A reaction of Fisher-KPP type takes place in the field, i.e. $f\in{\mathcal}{C}^1([0,+\infty))$ satisfies $$\label{12} f(0)=0=f(1), \qquad 0<f(s)<f'(0)s \;\; \text{ in } (0,1), \qquad f<0 \text{ in } (1,+\infty).$$ On the contrary, no reaction occurs on the road, where the density of the species varies only because a fraction $\mu>0$ of the population jumps from the road to the field while a fraction $\nu>0$ of the population jumps from the field to the road. This model was motivated by empirical observations of wolves moving along seismic lines in Canada (see [@McK]) or insects like the *Aedes albopictus* (tiger mosquito) spreading in the United States along highways (see [@MM]). Another example of this phenomenon is the diffusion of diseases along commercial and transport networks (see [@TRH] and the references therein). In [@BRR1], the authors established the existence of an asymptotic speed of propagation (see Definition \[deasp\]) of the solution of , starting from a continuous, nonnegative, compactly supported initial datum $(u_0, v_0)\neq(0, 0)$, towards the unique steady state of the problem, as well as some qualitative properties of it. Denoting such speed by $c^*_{\infty}$, they showed that, if $D\leq 2d$, then $c^*_{\infty}={c_{\operatorname{KPP}}}$, where $$\label{13} {c_{\operatorname{KPP}}}=2\sqrt{df'(0)}$$ is the asymptotic speed of propagation of the classical Fisher-KPP equation $$v_t(x,y,t)-d\D v(x,y,t)=f(v(x,y,t))$$ in the half-plane (see [@KPP; @AW]), while, if $D> 2d$, then $c^*_{\infty}>{c_{\operatorname{KPP}}}$. This means that a large diffusion on the road speeds up the propagation of the population in the field. Moreover the authors showed that the spreading velocity increases to infinity as the diffusivity on the line grows to infinity. In [@BRR2] they also studied the influence that a drift term and a Fisher-KPP reaction also on the road have on the asymptotic speed of propagation. In this work we investigate the effect of the road on the propagation in a field which is no longer a half-plane but a strip $\O={{\mathbb{R}}}\times(0,L)$. On the other part of the boundary of the field we impose homogeneous Dirichlet boundary conditions, modeling in this way an unfavorable region at level $y=0$. The system we consider is therefore $$\label{14} \left\{ \begin{array}{lll} u_t(x,t)-Du_{xx}(x,t)=\nu v(x,L,t)-\m u(x,t) & \text{for } x\in{{\mathbb{R}}}, & t>0 \\ v_t(x,y,t)-d\D v(x,y,t)=f(v) & \text{for } (x,y)\in\O, & t>0 \\ dv_y(x,L,t)=\m u(x,t)-\nu v(x,L,t) & \text{for } x\in{{\mathbb{R}}}, & t>0 \\ v(x,0,t)=0 & \text{for } x\in{{\mathbb{R}}}, & t>0, \end{array}\right.$$ where $D,d,\mu,\nu,L$ are positive constants. Ascertaining the long time behavior of the solutions of the Cauchy problem associated to will be the first step in the study of the problem. The result is the following \[thltb\] Let $(u,v)$ denote the solution of starting from a nonnegative, not equal to $(0,0)$, bounded and continuous initial datum $(u_0, v_0)$. If $$\label{15} \frac{f'(0)}{d}\leq\left(\frac{\pi}{2L}\right)^2,$$ then $$\label{16} \lim_{t\to+\infty}(u(x,t),v(x,y,t))=(0,0)$$ locally uniformly in $\overline{\O}$, while if $$\label{17} \frac{f'(0)}{d}>\left(\frac{\pi}{2L}\right)^2$$ and, moreover, $$\label{18} \frac{f(s)}{s} \text{ is nonincreasing},$$ then $$\label{19} \lim_{t\to+\infty}(u(x,t),v(x,y,t))=\left(\frac{\nu}{\mu}V(L),V(y)\right)$$ locally uniformly in $\overline{\O}$, where $V(y)$ is the unique solution of $$\label{110} \left\{ \begin{array}{l} -dV''(y)=f(V(y)) \quad \text{for } y\in(0,L)\\ V'(L)=0, \quad V(0)=0. \end{array}\right.$$ A remarkable difference with respect to is that here the width of the strip $L$ plays a role in the existence of positive steady states. Moreover, condition was not necessary to guarantee uniqueness in [@BRR1], while here it is (see Remark \[re33\] below). From a biological point of view, Theorem \[thltb\] says that, if the strip is not sufficiently large, the influence of the unfavorable region $y=0$ drives the species to extinction. On the contrary, the species will persist if the strip is sufficiently large. In this latter case, a natural question is to study more deeply how the convergence to the steady state occurs. To this end we consider the following concept \[deasp\] We say that $c^*\in{{\mathbb{R}}}_+$ is an *asymptotic speed of propagation* (in the $x-$direction) for if, denoting by $(u, v)$ the solution of with a continuous, nonnegative, compactly supported initial datum $(u_0, v_0)\neq(0, 0)$, we have - for all $c > c^*$, $$\label{111} \lim_{t\to +\infty} \sup_{\substack{|x|\geq ct \\ y\in[0,L]}} |(u(x, t), v(x, y, t))| = 0,$$ - for all $0<c < c^*$, $$\label{112} \lim_{t\to +\infty}\sup_{\substack{|x|\leq ct \\ y\in[0,L]}} \left|(u(x, t), v(x, y, t)) - \left(\frac{\nu}{\mu}V(L), V(y)\right)\right|=0,$$ where $V(y)$ is the unique solution of . In this sense, the main result of the paper is the following \[thmain\] Problem admits an asymptotic speed of propagation, denoted by $c^*=c^*_L(D,d,\mu,\nu)$, such that: - $D\mapsto c^*(D)$ is increasing, - the following limits exist and are positive real numbers $$\lim_{D\to 0}c^*(D)=\ell_0, \qquad \lim_{D\to+\infty}\frac{c^*(D)}{\sqrt{D}}=\ell_\infty,$$ - for fixed $D,d,\mu,\nu$ we have $$\lim_{L\to+\infty}c^*_{L}(D,d,\mu,\nu)=c^*_{\infty}(D,d,\mu,\nu),$$ where $c^*_{\infty}(D,d,\mu,\nu)$ is the asymptotic speed of propagation of Problem . The paper is organized like follows: in Section \[sec2\] we recall some tools from [@BRR1; @BRR2] which will be indispensable throughout the rest of the paper; Section \[sec3\] provides the proof of Theorem \[thltb\]; in Sections \[sec4\] and \[sec5\] we construct $c^*$ and derive some properties that will allow us, in Section \[sec6\], to show that it satisfies Definition \[deasp\] and relation $(i)$ of Theorem \[thmain\]. In Section \[sec7\] we prove relation $(ii)$ of Theorem \[thmain\] and finally in Section \[sec8\] we study the influence of the road on the asymptotic speed of propagation, comparing with the case in which no road is present, and give the proof of Theorem \[thmain\]$(iii)$. Preliminary results {#sec2} =================== In this section we present some fundamental results that are contained in or follow easily from [@BRR1; @BRR2]. The existence of a solution for the Cauchy problem associated to with a continuous initial datum $(u_0,v_0)$ follows from an easy modification of [@BRR1 Appendix A] and uniqueness follows from a comparison principle which will be diffusely used throughout this paper and whose proof can be easily adapted from [@BRR1 Proposition 3.2]. Before stating it we point out that, as usual, by a *supersolution* (resp. *subsolution*) of we mean a pair $(u,v)$ satisfying System with $\geq$ (resp. $\leq$) instead of $=$. \[prcp1\] Let $(\un{u}, \un{v})$ and $(\ov{u},\ov{v})$ be, respectively, a subsolution bounded from above and a supersolution bounded from below of satisfying $\un{u}\leq\ov{u}$ and $\un{v}\leq\ov{v}$ at $t = 0$. Then, either $\un{u}<\ov{u}$ and $\un{v}<\ov{v}$ for all $t$, or there exists $T > 0$ such that $(\un{u}, \un{v})=(\ov{u},\ov{v})$ for $t\leq T$. We also need the following comparison principle regarding an extended class of generalized subsolutions and which is a particular instance of [@BRR1 Proposition 3.3]. \[prcp2\] Let $E\subset {{\mathbb{R}}}\times{{\mathbb{R}}}_+$ and $F\subset \O\times{{\mathbb{R}}}_+$ be two open sets, let $(u_1, v_1)$ be a subsolution of bounded from above and satisfying $$u_1 = 0 \text{ on } (\p E) \cap ({{\mathbb{R}}}\times{{\mathbb{R}}}_+), \quad v_1= 0 \text{ on } (\p F) \cap (\O\times{{\mathbb{R}}}_+),$$ and consider $$\un{u} :=\begin{cases} \max\{u_1, 0\} & \text{in $\ov{E}$} \\ 0 & \text{otherwise,} \end{cases} \qquad \un{v} :=\begin{cases} \max\{v_1, 0\} & \text{in $\ov{F}$} \\ 0 & \text{otherwise.} \end{cases}$$ If they satisfy $$\label{21} \begin{split} \un{v}(x, L, t) &\geq v_1(x, L, t) \quad \text{ for all } (x,t) \text{ such that } \un{u}(x, t) > 0, \\ \un{u}(x, t) &\geq u_1(x, t) \qquad \text{ for all } (x,t) \text{ such that } \un{v}(x, L, t) > 0, \end{split}$$ then, for any supersolution $(\ov{u},\ov{v})$ of bounded from below and such that $\un{u}\leq\ov{u}$ and $\un{v}\leq\ov{v}$ at $t = 0$, we have $\un{u}\leq\ov{u}$ and $\un{v}\leq\ov{v}$ for all $t > 0$. \[recp2\] The same result of Proposition \[prcp2\] holds for problems like with an additional drift term in the differential operator. As a consequence of the previous analysis, we will consider continuous nonnegative initial data throughout the rest of this work, since we are interested in nonnegative solutions of . Liouville-type result and long time behavior {#sec3} ============================================ In order to determine the long time behavior of the solutions of we need to study the solutions of the elliptic system associated to it, precisely $$\label{31} \left\{ \begin{array}{ll} -D U_{xx}(x)=\nu V(x,L)-\m U(x) & \text{for } x\in{{\mathbb{R}}}, \\ -d\D V(x,y)=f(V) & \text{for } (x,y)\in\O, \\ d V_y(x,L)=\m U(x)-\nu V(x,L) & \text{for } x\in{{\mathbb{R}}}, \\ V(x,0)=0 & \text{for } x\in{{\mathbb{R}}}. \end{array}\right.$$ Actually, Propositions \[pr31\] and \[pr34\] below suggest that we have to focus on solutions of which are $x-$independent. They are of the form $(U,V(y))$, where $V$ satisfies and, thanks to the first equation of , $U=\frac{\nu}{\m} V(L)$. The first result regarding the long time behavior is the following \[pr31\] Let $(u,v)$ be the solution of starting with a nonnegative, bounded initial datum $(u_0,v_0)$. Then, there exists a nonnegative, bounded solution $V_1$ of such that $$\limsup_{t\to+\infty}u(x,t)\leq U_1, \qquad \limsup_{t\to+\infty}v(x,y,t)\leq V_1(y)$$ locally uniformly in $\overline{\O}$, where $$U_1=\frac{\nu}{\m} V_1(L).$$ Observe preliminarily that, if we define, for $(x,y)\in\overline{\O}$, $$\overline{v}(x,y)=\max\left\{1,\left\|v_0\right\|_{\infty},\frac{\mu}{\nu}\left\|u_0\right\|_{\infty}\right\}, \qquad \overline{u}(x)=\frac{\nu}{\mu}\overline{v},$$ then $(\overline{u},\overline{v})$ is a strict supersolution of which is larger than $(u_0,v_0)$. Therefore, by Proposition \[prcp1\], we have that the solution of with $(\overline{u},\overline{v})$ as initial datum is decreasing and, thanks to parabolic estimates, converges locally uniformly in $\overline{\O}$ to a nonnegative stationary solution $(U_1,V_1)$ of , i.e. a solution of . Proposition \[prcp1\] also gives $$\limsup_{t\to+\infty} u(x,t)\leq U_1(x), \qquad \limsup_{t\to+\infty} v(x,y,t)\leq V_1(x,y).$$ From the invariance of Problem in the $x$ direction and the uniqueness of the associated Cauchy problem, translations in $x$ of a solution of with a certain initial datum coincide with the solution of starting from the translated initial datum. Since $(\ov{u},\ov{v})$ is $x-$independent, the $x-$invariance of $(U_1,V_1)$ follows. Obviously, admits the trivial solution $V=0$. In the following proposition we will show that is a necessary and sufficient condition for to possess positive bounded solutions. \[pr32\] Problem admits positive bounded solutions if and only if holds. Moreover, if we assume and , then Problem admits a *unique* positive bounded solution. We begin with the necessity of . Suppose it does not hold and Problem admits a positive solution $v$. Then, multiplying the differential equation of by $\sin(\frac{\pi}{2L}y)$ and integrating by parts in $(0,L)$, we get $$\begin{gathered} \int_0^Lf(v(y))\sin\left(\frac{\pi}{2L}y\right)\,dy=d\left(\frac{\pi}{2L}\right)^2\int_0^Lv(y)\sin\left(\frac{\pi}{2L}y\right)\,dy\geq \\ \geq f'(0)\int_0^Lv(y)\sin\left(\frac{\pi}{2L}y\right)\,dy>\int_0^Lf(v(y))\sin\left(\frac{\pi}{2L}y\right)\,dy,\end{gathered}$$ where, for the last relation, we have used the second assumption in . We have reached a contradiction and therefore no positive solution can exist. Now we pass to the sufficiency. First of all we show that any positive solution of must satisfy $v(y)<1$ in $[0,L]$. Indeed if there was a point $y_0\in(0,L]$ where $v(y_0)=1$, either it would be a maximum of $v$ in a (relative to $(0,L]$) neighborhood of $y_0$, but in such a case $v\equiv 1$ by the uniqueness of the Cauchy problem $-v''=f(v)$ with conditions $v(y_0)=1, v'(y_0)=0$, or there would be $y_1\in(0,L]$ with $v(y_1)=\max v>1$. But in this case, $-dv''(y_1)=f(v(y_1))<0$, which is impossible for a maximum. As a consequence of this result and , we have that $v''<0$ in $(0,L)$ and therefore $v'$ is decreasing. This means that $v'$ is positive in $(0,L)$, since $v'(L)=0$, i.e. $v$ is increasing in $(0,L)$. By multiplying the differential equation of by $v'$ and integrating in $(y,L)$, with $0<y<L$, we get $$d\frac{v'(y)^2}{2}=\int_{v(y)}^{v(L)}f(s)\,ds$$ and, recalling that $v'>0$, we have that any solution of must satisfy $$L=\int_0^L\frac{v'(y)\,dy}{\sqrt{\frac{2}{d}\int_{v(y)}^{v(L)}f(s)\,ds}}=\int_0^1\frac{d\xi}{\sqrt{\frac{2}{d}\int_{\xi}^{1}\frac{f(v(L)\eta)}{v(L)\eta}\eta\,d\eta}}$$ with $0<v(L)<1$. Therefore, if we define the function $$\label{33} M(\r):=\int_0^1\frac{d\xi}{\sqrt{\frac{2}{d}\int_{\xi}^{1}\frac{f(\r\eta)}{\r\eta}\eta\,d\eta}},$$ which is continuous in $(0,1)$ and measures the length of the interval necessary for a solution of $$\left\{ \begin{array}{l} -dV''(y)=f(V(y)) \quad \text{for } y\in(0,y_0)\\ V(0)=0, \quad V'(y_0)=0 \end{array}\right.$$ to attain its maximum value $\r$ (at $y_0$), the uniqueness of the Cauchy problem associated to the ordinary differential equation of , we have that any solution of $M(\r)=L$ provides a solution of . The function $M$ satisfies $$\lim_{\r\ua 1}M(\r)=+\infty$$ since a maximum equal to $1$ cannot be attained in a finite interval, as seen before. Moreover, thanks to , $$\lim_{\r\da 0}M(\r)=\sqrt{\frac{d}{f'(0)}}\int_0^1\frac{d\xi}{\sqrt{1-\xi^2}}=\sqrt{\frac{d}{f'(0)}}\frac{\pi}{2}<L$$ and, therefore, there exists $\bar{\r}\in(0,1)$ such that $M(\bar{\r})=L$, which provides us with a solution of . As far as uniqueness, it is easily seen that, under hypothesis , the function $M$ is increasing and therefore there exists a unique value of $\rho$ for which $M(\rho)=L$. \[re33\] If condition does not hold, Problem may exhibit more than one solution. Consider indeed $f(s)=s(-6s^3+9s^2-4s+1)$, which satisfies but not . With this choice, the function $M$ defined in , which is ${\mathcal}{C}^1[0,1)$, satisfies $$M'(\r)=\sqrt{\frac{15\, d}{2}}\int_0^1h(\r,\xi)\,d\xi$$ where $$\label{34} h(\r,\xi):=\frac{216\r^2(1-\xi^5)-270\r(1-\xi^4)+80(1-\xi^3)}{\left(-72\r^3(1-\xi^5)+135\r^2(1-\xi^4)-80\r(1-\xi^3)+30(1-\xi^2)\right)^{3/2}}$$ and, as a consequence, $M'(0)>0$, since $h(0,\xi)>0$ for every $\xi\in(0,1)$. On the other hand, for $\r=1/2$ the numerator in reduces to $$h_1(\xi)=-54\xi^5+135\xi^4-80\xi^3-1,$$ which satisfies $h_1(0)=-1$, is decreasing for $\xi\in(0,2/3)$ and increasing for $\xi>2/3$. Since $h_1(1)=0$, this implies that $h(1/2,\xi)<0$ for all $\xi\in(0,1)$ and, therefore, $M'(1/2)<0$. Recalling that $M(\r)\to+\infty$ as $\r\ua 1$, the previous analysis entails that $L$ can be chosen in such a way that Problem possesses at least $3$ solutions. The last result we need in order to prove Theorem \[thltb\] is the following \[pr34\] Assume and let $(u,v)$ be the solution of starting with a nonnegative, not equal to $(0,0)$, bounded initial datum. Then, there exists a positive bounded solution $V_2$ of such that $$U_2\leq \liminf_{t\to+\infty}u(x,t), \qquad V_2\leq \liminf_{t\to+\infty}v(x,y,t)$$ locally uniformly in $(x,y)\in\overline{\O}$, where $$U_2=\frac{\nu}{\mu}V_2(L).$$ Consider $$\label{35} (\un{u},\un{v}):=\left\{\begin{array}{ll} \cos(\o x)\left(1,\frac{\mu\sin(\b y)}{d\b\cos(\b L)+\nu\sin(\b L)}\right) &\text{if $(x,y)\in\left(-\frac{\pi}{2\o},\frac{\pi}{2\o}\right)\times\left[0,L\right]$} \\ (0,0) &\text{otherwise}. \end{array}\right.$$ We now show that $\b$ and $\o$ can be chosen so that $(\un{u},\un{v})$ satisfy $$\label{36} \left\{ \begin{array}{ll} -D\un{u}_{xx}(x)\leq\nu \un{v}(x,L)-\m \un{u}(x) & \text{for } x\in \left(-\frac{\pi}{2\o},\frac{\pi}{2\o}\right) \\ -d\D \un{v}(x,y)\leq(f'(0)-\d)\un{v} & \text{for } (x,y)\in\left(-\frac{\pi}{2\o},\frac{\pi}{2\o}\right)\times\left(0,L\right) \\ d\un{v}_y(x,L)=\m \un{u}(x)-\nu \un{v}(x,L) & \text{for } x\in \left(-\frac{\pi}{2\o},\frac{\pi}{2\o}\right) \\ \un{v}(x,0)=0 & \text{for } x\in \left(-\frac{\pi}{2\o},\frac{\pi}{2\o}\right), \end{array}\right.$$ for $0<\d<f'(0)$ and therefore, by the second relation of , there exists $\e_0$ such that, for all $0<\e<\e_0$, $\e(\un{u},\un{v})$ is a strict generalized subsolution of to which Proposition \[prcp2\] can be applied. Observe that, with choice , the last two equations of are satisfied and the first two inequalities reduce to $$D\o^2\leq \frac{-\mu d\b\cos(\b L)}{d\b\cos(\b L)+\nu\sin(\b L)}, \qquad d\o^2+d\b^2\leq f'(0)-\d.$$ Now, thanks to , it is possible to fix $\d$ in such a way that $$d\left(\frac{\pi}{2L}\right)^2<f'(0)-\d$$ and take $\frac{\pi}{2L}<\b<\frac{\pi}{L}$ in a neighborhood of $\frac{\pi}{2L}$ (we denote $\b\sim\frac{\pi}{2L}$), so that $$m:=\min\left\{\frac{f'(0)-\d}{d}-\b^2,\frac{-\mu d\b\cos(\b L)}{D\left(d\b\cos(\b L)+\nu\sin(\b L)\right)}\right\}>0.$$ As a consequence, if $\o^2\leq m$, $(\un{u},\un{v})$ satisfies . Moreover, reducing $\e$ if necessary, we can assume that $\e(\un{u},\un{v})<(u(x,1),v(x,y,1))$, because, thanks to Proposition \[prcp1\] and the Hopf lemma, we have that $$u(x,1)>0, \quad v_y(x,0,1)>0 \quad \text{ and } \quad v(x,y,1)>0 \quad \text{ for all } (x,y)\in\O,$$ and, in addition, $v(x,L,1)>0$ for every $x\in{{\mathbb{R}}}$, since if there was $x_0$ such that $v(x_0,L,1)=0$, then from the third equation in we would have $$dv_y(x_0,L,1)=\mu u(x_0,1)>0,$$ which is impossible, since, again by Proposition \[prcp1\], $v(x,y,1)\geq 0$ in $\ov{\O}$. By Proposition \[prcp2\], the solution of with $\e(\un{u},\un{v})$ as initial datum, converges, increasingly, to a stationary solution $(U_2,V_2)$ of locally uniformly in $\overline{\O}$ and moreover $$U_2\leq\liminf_{t\to+\infty}u(x,t) \quad \text{ and } \quad V_2\leq\liminf_{t\to+\infty}v(x,y,t).$$ As before we have, for all $(x,y)\in\O$ $$U_2(x)>0, \quad V_{2,y}(x,0)>0, \quad V_2(x,y)>0 \quad \text{ and } \quad V_2(x,L)>0$$ and, since $\e(\un{u},\un{v})$ is continuous and compactly supported and, thanks to the above-mentioned monotonicity, it does not touch $(U_2,V_2)$, we have that there exists $k>0$, $k\sim 0$, such that $\e(\un{u}(x-h),\un{v}(x-h,y))$, which still is a subsolution of lies below $(U_2(x),V_2(x,y))$ for all $h\in(-k,k)$. Anyway, by the uniqueness of the Cauchy problem associated to , the solution of with the translated subsolution as initial datum converges to the corresponding translation of $(U_2,V_2)$ and, by comparison, we have that $(U_2,V_2)$ is smaller than small translations in the $x$ direction of itself, which entails that the partial derivatives of $(U_2,V_2)$ with respect to $x$ are $0$. We are now able to give the *Proof of Theorem \[thltb\].* If holds, we obtain from Propositions \[pr31\] and \[pr32\], since Proposition \[prcp1\] guarantees that $(u,v)\geq(0,0)$. On the other hand, if and hold, follows from Propositions \[pr31\], \[pr32\] and \[pr34\]. Since we are interested in the speed of propagation towards positive steady states of , we will assume and throughout the rest of the paper, for to hold. Supersolutions in the moving framework {#sec4} ====================================== In this section we construct positive supersolutions to moving at appropriate speeds in the $x-$direction. This will be the key to find an upper bound for the asymptotic speed of propagation of Theorem \[thmain\] (see Section \[sec6\]). Observe that solutions of the linearized problem $$\label{41} \left\{ \begin{array}{lll} u_t(x,t)-Du_{xx}(x,t)=\nu v(x,L,t)-\m u(x,t) & \text{for } x\in{{\mathbb{R}}}, &\!\! t>0 \\ v_t(x,y,t)-d\D v(x,y,t)=f'(0)v(x,y,t) & \text{for } (x,y)\in\O, &\!\! t>0 \\ dv_y(x,L,t)=\m u(x,t)-\nu v(x,L,t) & \text{for } x\in{{\mathbb{R}}}, &\!\! t>0 \\ v(x,0,t)=0 & \text{for } x\in{{\mathbb{R}}}, &\!\! t>0, \end{array}\right.$$ provide us with supersolutions to , thanks to the second condition of . We start looking for solutions of of the form $$\label{42} (u(x,t),v(x,y,t))=e^{\a(x+ct)}(1,\g \sin (\b y))$$ with positive $\a,\g, c$ and $\b\in(0,\frac{\pi}{L})$, in order for $u$ and $v$ to be positive in $\O$. Notice that is a solution of if and only if $$\label{43} \left\{ \begin{array}{l} -D\a^2+c\a=\n\g\sin(\b L)-\m \\ -d\a^2+d\b^2+c\a=f'(0) \\ \g=\frac{\m}{d\b \cos(\b L)+\n \sin (\b L)}. \end{array}\right.$$ In order for $\g$ to be positive, $\b$ must lie in $(0,\ov{\b})$, where $\ov{\b}\in(\frac{\pi}{2L},\frac{\pi}{L})$ is the first positive value of $\b$ for which $$\label{44} d\b\cos(\b L)+\n \sin(\b L)=0.$$ By substituting the expression of $\g$ given by the third equation of into the first one, we get $$-D\a^2+c\a+\frac{\m}{1+\frac{\n\tan(\b L)}{d\b}}=0,$$ whose solutions are $$\label{eqdef} \a_D^{\pm}(c,\b)=\frac{1}{2D}\left(c\pm \sqrt{c^2+\chi(\b)}\right),$$ where we have set $$\chi(\b)=\frac{4\m D}{1+\frac{\n\tan(\b L)}{d\b}}=\frac{4\m dD\b\cos(\b L)}{d\b\cos(\b L)+\n \sin(\b L)}.$$ It is easy to see that $\chi(\b)$ is continuous, even, decreases for $\b\in(0,\ov{\b})$ and satisfies $$\lim_{\b\da 0} \chi(\b)=\frac{4\m D}{1+\frac{\n L}{d}}, \qquad \chi\left(\frac{\pi}{2L}\right)=0, \qquad \lim_{\b\ua\ov{\b}} \chi(\b)=-\infty,$$ where $\ov{\b}$ is the one defined in . Hence, for every $c>0$, there exists a unique $\tilde{\b}(c)\in(\frac{\pi}{2L},\ov{\b})$, for which $$\label{45} c^2=-\chi(\tilde{\b}(c)).$$ Moreover it satisfies $$\lim_{c\da 0} \tilde{\b}(c)=\frac{\pi}{2L}, \qquad \lim_{c\ua \infty} \tilde{\b}(c)=\ov{\b}.$$ As a consequence of these properties, for fixed $c>0$, $\a_D^{+}(c,\b)$ is a regular even function, which decreases in $(0,\tilde{\b}(c))$ and satisfies $$\begin{gathered} \a_D^+(c,0)\!=\!\frac{1}{2D}\left(c\!+\!\sqrt{c^2+\frac{4\m D}{1+\frac{\n L}{d}}}\right), \quad \a_D^{+}\left(c,\frac{\pi}{2L}\right)\!=\!\frac{c}{D}, \quad \a_D^{+}(c,\tilde{\b}(c))\!=\!\frac{c}{2D}, \label{eqstar} \\ \p_\b\a_D^+(c,0)=0, \qquad \lim_{\b\ua\tilde{\b}(c)} \p_\b\a_D^+(c,\b)=-\infty. \notag\end{gathered}$$ In addition it is easy to verify that $$\label{46} \p_c\a_D^+(c,\b)>0.$$ Since $\a_D^-$ is symmetric to $\a_D^+$ with respect to the line $\a=\frac{c}{2D}$, analogous properties can be established for $\a_D^-$. In particular observe that $$\a_D^{-}\left(c,\frac{\pi}{2L}\right)=0$$ and that, for fixed $\b\in(\frac{\pi}{2L},\ov{\b})$, we have $$\begin{aligned} \lim_{c\ua\infty} \a_D^{-}(c,\b)=0,\end{aligned}$$ $\a_D^{-}$ decreasing monotonically in $c$. As a consequence, the bounded region of the first quadrant in the $(\b,\a)-$plane delimited by the curve $$\label{47} \S_D(c):=\left\{(\b,\a_D^-(\b)):\b\in\left[\frac{\pi}{2L},\tilde{\b}(c)\right]\right\}\cup\left\{(\b,\a_D^+(\b)):\b\in\left(0,\tilde{\b}(c)\right]\right\},$$ which has been represented in Figure \[fig21\], invades monotonically, as $c\ua\infty$, the half-strip $\{(\b,\a): \b\in(0,\ov{\b}), \a>0\}$. ![The curve $\S_D(c)$ defined in .[]{data-label="fig21"}](fig1.pdf) As far as the monotonicity of the curves with respect to $D$, the other parameters being fixed, it follows from the definition of $\a_D^{\pm}$ that $$\label{48} \begin{split} D&\mapsto \a_D^+(c,\b) \; \text{ is decreasing for every $\b\in[0,\tilde{\b}(c)]$} \\ D&\mapsto \a_D^-(c,\b) \; \text{ is increasing for every $\b\in\left(\frac{\pi}{2L},\tilde{\b}(c)\right]$.} \end{split}$$ On the other hand (see Figure \[fig22\]), the second equation of represents an hyperbola whose branches are given by $$\label{49} \a_d^{\pm}(c,\b)=\frac{1}{2d}\left(c\pm\sqrt{c^2-\eta(\b)}\right),$$ where we have set, $$\eta(\b):=4d(f'(0)-d\b^2)={c_{\operatorname{KPP}}}^2-4d^2\b^2,$$ $c_{\operatorname{KPP}}$ being the one defined in . The function $\eta$ is decreasing and satisfies $$\eta(0)={c_{\operatorname{KPP}}}^2, \qquad \eta\left(\sqrt{\frac{f'(0)}{d}}\right)=0, \qquad \lim_{\b\ua\infty}\eta(\b)=-\infty.$$ Therefore, the functions $\a_d^{\pm}(c,\b)$ are defined for every $\b\in(0,+\infty)$ if $c\geq {c_{\operatorname{KPP}}}$, while, if $c\in(0,{c_{\operatorname{KPP}}})$ there exists $\hat{\b}(c)>0$ such that their domain (within the positive part of the real line) is $[\hat{\b}(c),+\infty)$, where $\hat{\b}(c)$ satisfies $$\label{410} c^2=\eta(\hat{\b}(c)).$$ As a consequence, $$\lim_{c\da 0} \hat{\b}(c)=\sqrt{\frac{f'(0)}{d}}, \qquad \lim_{c\ua {c_{\operatorname{KPP}}}} \hat{\b}(c)=0.$$ It can be easily seen, as $\a_d^-$ is the reflection of $\a_d^+$ about the line $\a=\frac{c}{2d}$, that $$\a_d^-\left(c,\sqrt{\frac{f'(0)}{d}}\right)=0, \qquad \lim_{\b\ua\infty}\a_d^+(c,\b)=+\infty,$$ and, in the proper domain of definition, $$\label{411} \p_\b\a_d^+(c,\b)>0, \qquad \p_c\a_d^+(c,\b)>0, \qquad \p_c\a_d^-(c,\b)<0.$$ Moreover, if $0<c<{c_{\operatorname{KPP}}}$, we have that $$\a_d^+(c,\hat{\b}(c))=\frac{c}{2d} \qquad \lim_{\b\da\hat{\b}(c)}\p_\b\a_d^{+}(c,\b)=+\infty,$$ while, for $c={c_{\operatorname{KPP}}}$, the hyperbolas degenerate into the straight lines with equations $$\label{ipdeg} \a_d^{\pm}({c_{\operatorname{KPP}}},\b)=\pm\b+\frac{{c_{\operatorname{KPP}}}}{2d}.$$ Finally, for $c>{c_{\operatorname{KPP}}}$, we have that $$\a_d^{\pm}(c,0)=\frac{c\pm\sqrt{c^2-{c_{\operatorname{KPP}}}^2}}{2d}, \qquad \p_\b\a_d^{\pm}(c,0)=0$$ and, for fixed $\b$, $$\lim_{c\ua\infty} \a_d^{-}(c,\b)=0.$$ As a consequence of all the aforementioned properties, if we set $\hat{\b}(c)=0$ for $c\geq {c_{\operatorname{KPP}}}$, which is consistent with the previous notation, and define $$\S_d(c):=\left\{(\b,\a_d^-(\b)):\b\in\left[\hat{\b}(c),\sqrt{\frac{f'(0)}{d}}\right]\right\}\cup\left\{(\b,\a_d^+(\b)):\b\geq\hat{\b}(c)\right\},$$ we have that the region of the first quadrant in the $(\b,\a)-$plane bounded by $\S_d(c)$ and containing the point $\left(\sqrt{\frac{f'(0)}{d}},\frac{c}{2d}\right)$ invades monotonically, as $c\ua\infty$, the first quadrant in the $(\b,\a)-$plane. All these features have been represented in Figure \[fig22\]. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![The hyperbola defined by the second equation of , for $0<c<{c_{\operatorname{KPP}}}$ (A), $c={c_{\operatorname{KPP}}}$ (B) and $c>{c_{\operatorname{KPP}}}$ (C).[]{data-label="fig22"}](fig221bis.pdf "fig:") ![The hyperbola defined by the second equation of , for $0<c<{c_{\operatorname{KPP}}}$ (A), $c={c_{\operatorname{KPP}}}$ (B) and $c>{c_{\operatorname{KPP}}}$ (C).[]{data-label="fig22"}](fig222bis.pdf "fig:") ![The hyperbola defined by the second equation of , for $0<c<{c_{\operatorname{KPP}}}$ (A), $c={c_{\operatorname{KPP}}}$ (B) and $c>{c_{\operatorname{KPP}}}$ (C).[]{data-label="fig22"}](fig223bis.pdf "fig:") ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Another candidate to construct supersolutions to for $c>{c_{\operatorname{KPP}}}$ are the functions $$\label{412} (u(x,t),v(x,y,t))=e^{\a(x+ct)}(1,\phi(y))$$ with $$\phi(y):=\g(e^{\b y}-e^{-\b y}),$$ where $\a,\b,\g$ are positive constants. By plugging into we are driven to $$\label{413} \left\{ \begin{array}{l} -D\a^2+c\a=\n\phi(L)-\m \\ -d\a^2-d\b^2+c\a=f'(0) \\ \g=\frac{\m}{d\b(e^{\b L}+e^{-\b L})+\n (e^{\b L}-e^{-\b L})}>0. \end{array}\right.$$ Using the expression of $\phi(L)$ given from the third equation of , the first one reduces to $$-D\a^2+c\a+\frac{\m d\b(e^{\b L}+e^{-\b L})}{d\b(e^{\b L}+e^{-\b L})+\n (e^{\b L}-e^{-\b L})}=0,$$ whose solutions are $$\label{tipo2} \tilde{\a}_D^{\pm}(c,\b)=\frac{1}{2D}\left(c\pm \sqrt{c^2+\tilde{\chi}(\b)}\right),$$ where we have set $$\tilde{\chi}(\b)=\frac{4\m D}{1+\frac{\n\tanh(\b L)}{d\b}}.$$ This function is positive and therefore $\tilde{\a}_D^{+}(c,\b)$ is defined for every $\b\in{{\mathbb{R}}}$. It can be easily seen that it is even and satisfies the following monotonicity conditions $$\label{417} \p_{\b}\tilde{\a}_D^+(c,\b)>0, \quad \p_{c}\tilde{\a}_D^+(c,\b)>0, \quad \p_{D}\tilde{\a}_D^+(c,\b)<0$$ for every positive $c,\b$. Moreover we have $\tilde{\a}_D^+(c,0)=\a_D^+(c,0)$, for every $c>0$. Naturally, similar properties hold for $\tilde{\a}_D^-$, taking into account that it is symmetric to $\tilde{\a}_D^+$ with respect to $\a=\frac{c}{2D}$. Finally, observe that, for every $\b\geq 0$, $$\lim_{c\ua+\infty}\tilde{\a}_D^+(c,\b)=+\infty$$ and, therefore, the regions of the first quadrant in the $(\b,\a)-$plane delimited by $$\label{418} \tilde{\S}_D(c):=\left\{(\b,\tilde{a}_D^-(c,\b)):\b>0\right\}\cup\left\{(\b,\tilde{a}_D^+(c,\b)):\b>0\right\}.$$ and containing $\left(\frac{c}{2D},1\right)$ invade monotonically the first quadrant of the plane $(\b,\a)$. ![The curves $\tilde{\a}_D^+$ and $\tilde{\a}_d^{\pm}$.[]{data-label="fig24"}](fig24.pdf) On the other hand, the second equation of describes, for $c>{c_{\operatorname{KPP}}}$, as we are assuming, a circle in the $(\b,\a)-$plane, with center at $\left(0,\frac{c}{2d}\right)$ and radius $$r(c)=\frac{\sqrt{c^2-{c_{\operatorname{KPP}}}^2}}{2d}$$ (see Figure \[fig24\]). Precisely, the part of the graph of the circle which lies in the first quadrant is given by $$\tilde{\S}_d(c):=\left\{(\b,\tilde{\a}_d^{\pm}(c,\b)):\b\in[0,r(c)]\right\},$$ where $$\label{419} \tilde{\a}_d^{\pm}(c,\b)=\frac{1}{2d}\left(c\pm\sqrt{c^2-{c_{\operatorname{KPP}}}^2-4d^2\b^2}\right).$$ The function $\tilde{\a}_d^-$ satisfies $\p_{\b}\tilde{\a}_d^-(c,\b)>0$, $\p_{c}\tilde{\a}_d^-(c,\b)<0$, for $c>{c_{\operatorname{KPP}}}$ and $\b\in(0,r(c))$, while its value at $\b=0$ satisfies $\tilde{\a}_d^-(c,0)=\a_d^-(c,0)$. Moreover, it can be easily seen that $$\tilde{\S}_d({c_{\operatorname{KPP}}})=\left\{\left(0,\frac{{c_{\operatorname{KPP}}}}{2d}\right)\right\}$$ and $$\lim_{c\ua+\infty}\tilde{a}_d^-(c,\b)=0.$$ As a consequence of these properties, the half-disks delimited by $\tilde{\S}_d(c)$ and contained in the first quadrant of the $(\b,\a)-$plane invade it monotonically as $c$ increases. With these ingredients we are able to give the following result \[pr41\] There exists $c^*:=c^*_L(D,d,\mu,\nu)$ such that, for every $c>c^*$, Problem admits supersolutions either of the form or , with positive $\a,\b,\g$. Moreover the function $D\mapsto c^*(D)$ is increasing. By the previous discussion, we have that, in order to find a solution of and therefore a supersolution to , it is sufficient to find an intersection between either $\S_D(c)$ and $\S_d(c)$ or $\tilde{\S}_D(c)$ and $\tilde{\S}_d(c)$ lying in the interior of the first quadrant of the $(\b,\a)-$plane. Due to the monotonicity properties of these curves with respect to $c$ shown above, $c^*$ will be the smallest value of $c$ for which such an intersection exists for all $c>c^*$. Let us start by examining the case $D$ small (relatively to $d$), in which we consider the curves $\S_D(c)$ and $\S_d(c)$. Recalling that we are assuming , it is clear from the above discussion that, for $c\sim 0$, they are disjoint, since so are their domain of definition. On the contrary, for sufficiently large $c$ such an intersection exists, since $$\a_d^-(c,0)<\a_D^+(c,0), \quad \a_d^-\left(c,\sqrt{\frac{f'(0)}{d}}\right)=0, \quad \a_D^-\left(c,\frac{\pi}{2L}\right)=0.$$ ------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------- ![The tangency between $\S_D(c)$ and $\S_d(c)$, for $D\leq D_{\operatorname{KPP}}$.[]{data-label="fig23"}](fig231.pdf "fig:") ![The tangency between $\S_D(c)$ and $\S_d(c)$, for $D\leq D_{\operatorname{KPP}}$.[]{data-label="fig23"}](fig232.pdf "fig:") ![The tangency between $\S_D(c)$ and $\S_d(c)$, for $D\leq D_{\operatorname{KPP}}$.[]{data-label="fig23"}](fig233.pdf "fig:") ![The tangency between $\S_D(c)$ and $\S_d(c)$, for $D\leq D_{\operatorname{KPP}}$.[]{data-label="fig23"}](fig234.pdf "fig:") ------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------- Actually, the first value of $c$, denoted by $\un{c}=\un{c}(D)$, for which an intersection can exist is the one for which the domains of $\a_D^{\pm}$ and $\a_d^{\pm}$ intersect in one point, i.e. the one for which $$\left\{ \begin{array}{l} c^2=-\chi(\b) \\ c^2=\eta(\b) \end{array}\right.$$ admits a solution. Indeed, as shown in Figure \[fig23\](A), the curves $\S_D(\un{c})$ and $\S_d(\un{c})$ are tangent if $D=d$, their common tangent being vertical. Therefore in this case $c^*=\un{c}$. If $D<d$, there exists $c^*$ satisfying $\un{c}<c^*<{c_{\operatorname{KPP}}}$ and such that $\a_D^-$ and $\a_d^+$ are tangent, as described in Figure \[fig23\](B). If $D>d$ the situation is more complex, because we have to take into account the change in the nature of $\a_d^-(c,\b)$ as $c$ crosses ${c_{\operatorname{KPP}}}$. From , it follows that $$\lim_{D\ua+\infty}\tilde{\b}(c_{\operatorname{KPP}},D)=\frac{\pi}{2L}$$ and, together with the first and third relations of , this implies that there exists a value of $D$, denoted by $D_{\operatorname{KPP}}$, for which, for $c=c_{\operatorname{KPP}}$, $\a_{D_{\operatorname{KPP}}}^+$ and the straight line $\a_d^-$ are tangent (see Figure \[fig23\](C)). Observe that, for $D=2d$, we have $$\a_{D}^+(c_{\operatorname{KPP}},0)>\a_{D}^+\left(c_{\operatorname{KPP}},\frac{\pi}{2L}\right)=\frac{c_{\operatorname{KPP}}}{D}=\frac{c_{\operatorname{KPP}}}{2d}=\a_{d}^-(c_{\operatorname{KPP}},0),$$ which implies that $D_{\operatorname{KPP}}>2d$. Thanks to , for $d<D<D_{\operatorname{KPP}}$ and $c=c_{\operatorname{KPP}}$, $\a_D^+$ and $\a_d^-$ will be secant, therefore in this case the tangency will occur between $\a_D^+$ and $\a_d^-$ for $\un{c}<c^*<{c_{\operatorname{KPP}}}$, as represented in Figure \[fig23\](D). ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![Different configurations of the tangency between $\S_D(c)$ and $\S_d(c)$ (continuous lines) or $\tilde{\S}_D(c)$ and $\tilde{\S}_d(c)$ (dashed lines), for $c>c_{\operatorname{KPP}}$.[]{data-label="fig26"}](fig261.pdf "fig:") ![Different configurations of the tangency between $\S_D(c)$ and $\S_d(c)$ (continuous lines) or $\tilde{\S}_D(c)$ and $\tilde{\S}_d(c)$ (dashed lines), for $c>c_{\operatorname{KPP}}$.[]{data-label="fig26"}](fig262.pdf "fig:") ![Different configurations of the tangency between $\S_D(c)$ and $\S_d(c)$ (continuous lines) or $\tilde{\S}_D(c)$ and $\tilde{\S}_d(c)$ (dashed lines), for $c>c_{\operatorname{KPP}}$.[]{data-label="fig26"}](fig263.pdf "fig:") ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Finally, when $D>D_{\operatorname{KPP}}$ we consider at the same time the pairs of curves $\S_D(c)$, $\S_d(c)$ and $\tilde{\S}_D(c)$, $\tilde{\S}_d(c)$. We define $c^*$ to be the smallest value of $c$ for which either the former two curves or the latter two are tangent in a positive $\b$ (see Figure \[fig26\](A) and (B) respectively). The last case to be examined is the one in which all the four curves touch for the first time, being tangent, at $\b=0$ for $c={c_{\operatorname{int}}}$, where ${c_{\operatorname{int}}}$ is $$\label{420} {c_{\operatorname{int}}}^2={c_{\operatorname{int}}}^2(D)=\frac{\left((d+\nu L)D{c_{\operatorname{KPP}}}^2+ 4\mu d^3\right)^2}{4d(D-d)(d+\nu L)\left((d+\nu L){c_{\operatorname{KPP}}}^2 + 4\mu d^2\right)}$$ (see figure \[fig26\](C)). In this case, thanks to and , we have that $\S_D(c)$ and $\S_d(c)$ intersect for every $c>{c_{\operatorname{int}}}$ and a certain $\b>0$. Therefore we set $c^*={c_{\operatorname{int}}}$. The monotonicity of the function $D\mapsto c^*(D)$ follows from the monotonicity of the curves $\a_D^{\pm}$ and $\tilde{\a}_D^+$, given by and the last relation of respectively. \[remark42\] - The proof of Proposition \[pr41\] shows indeed that it is possible to construct a supersolution to of type or not only for every $c>c^*$, but also for $c=c^*$, except for the case represented in Figure \[fig26\](C). Actually it is possible to construct a supersolution also in this case, when $c^*={c_{\operatorname{int}}}$, by taking $$\label{421} (u(x,t),v(x,y,t))=e^{\a(x+ct)}\left(1,\frac{\mu y}{d+\nu L}\right).$$ This can be heuristically seen by taking the limit of or as $\b\da 0$ (notice that $\g=\g(\b)\to 0$), while a formal proof consists in plugging into and observing that the resulting algebraic system in $\a,c$ has a solution for $c={c_{\operatorname{int}}}$. - From the last part of proof of Proposition \[pr41\], it arises the natural question of characterizing which of the curves first touch, either $\S_D(c)$ and $\S_d(c)$ or $\tilde\S_D(c)$ and $\tilde\S_d(c)$. This analysis can be performed following the same ideas of the proof of Theorem \[thmain\](iii) (see Section \[sec8\]), i.e. by studying, for $c={c_{\operatorname{int}}}$, the second derivatives of the curves at $\b=0$, and adapting a fine result given in [@RTV Proposition 4.1], which is based on the comparison principles and characterizes the total number of possible intersections between the curves. Since such analysis is quite technical in general (indeed some of the arguments provided in Section \[sec8\] do not adapt to the general situation) and not strictly necessary for the results of this work, we send the reader to [@RTV] for the details. Generalized subsolutions with compact support {#sec5} ============================================= In this section we construct stationary compactly supported generalized (in the sense of Proposition \[prcp2\]) subsolutions in a framework moving in the $x-$direction at slightly smaller speeds than $c^*$, the one of Proposition \[pr41\]. Provided that Proposition \[prcp2\] can be applied, this will provide a lower bound for the asymptotic speed of propagation and will be the second and last ingredient for the proof of Theorem \[thmain\] (see Section \[sec6\]). The result is the following \[pr51\] Let $c^*$ be the one constructed in Proposition \[pr41\]. Then, for every $c<c^*$, $c\sim c^*$, and $\d>0$, $\d\sim 0$, there exists $\e_0>0$ such that, for every $0<\e<\e_0$, $\e(\un{U}(x),\un{V}(x,y))$ is a compactly supported generalized (in the sense of Proposition \[prcp2\]) subsolution of $$\label{51} \left\{ \begin{array}{lll} u_t-Du_{xx}+c u_x=\nu v(x,L,t)-\m u & \text{for } x\in{{\mathbb{R}}}, & t>0 \\ v_t-d\D v+c v_x=f(v) & \text{for } (x,y)\in\O, & t>0 \\ dv_y(x,L,t)=\m u-\nu v(x,L,t) & \text{for } x\in{{\mathbb{R}}}, & t>0 \\ v(x,0,t)=0 & \text{for } x\in{{\mathbb{R}}}, & t>0, \end{array}\right.$$ satisfying , where $(\un{U}(x),\un{V}(x,y))$ is a truncation of a solution of $$\label{52} \left\{ \begin{array}{ll} -D U_{xx}(x)+c U_x(x)=\nu V(x,L)-\m U(x) & \text{for } x\in{{\mathbb{R}}}, \\ -d\D V(x,y)+c V_x(x,y)=(f'(0)-\d)V(x,y) & \text{for } (x,y)\in\O, \\ d V_y(x,L)=\m U(x)-\nu V(x,L) & \text{for } x\in{{\mathbb{R}}}, \\ V(x,0)=0 & \text{for } x\in{{\mathbb{R}}}. \end{array}\right.$$ Once the solution $(\un{U},\un{V})$ of will be constructed, the existence of $\e_0$ such that $\e(\un{U}(x),\un{V}(x,y))$ is a subsolution to for every $0<\e<\e_0$ follows immediately from . From the construction of $c^*$ carried out in the proof of Proposition \[pr41\], we know that, for $c<c^*$, no real solution of of type or exists. However we will show that exhibits, for $c<c^*$, $c\sim c^*$, complex solutions, which will be the starting point for the construction of $(\un U,\un V)$. Actually we will consider the case $\d=0$ and the existence of $(\un{U},\un{V})$ for $\d\sim 0$ will follow by a perturbation argument which consists in repeating the same construction of the proof of Proposition \[pr41\] considering the curves with $f'(0)$ replaced by $f'(0)-\d$ (observe that the dependence of the curves with respect to $f'(0)$ is continuous). We will give the details only in the case in which $c^*$ was constructed in Proposition \[pr41\] as the one for which $\a_D^+(c^*,\b)$ and $\a_d^-(c^*,\b)$ were tangent in a point $\b=\b^*>0$. The other cases related to supersolution like are analogous; the case of supersolutions like was treated in [@BRR1] and the one related to supersolutions like will follow from the case that we are going to present here, by passing to the limit as $\b^*\to 0$, like in Remark \[remark42\](i). Let us consider, for $(c,\b)$ in a neighborhood of $(c^*,\b^*)$, the function $$\label{53} g(c,\b)=\a_d^-(c,\b)-\a_D^+(c,\b).$$ Our goal is to find, for $c<c^*$, $c\sim c^*$, a root $\b\in{{\mathbb{C}}}\setminus {{\mathbb{R}}}$ of . In this way we will obtain a solution $(\a,\b,\g)=(\a_1+i\a_2,\b_1+i\b_2,\g_1+i\g_2)\in({{\mathbb{C}}}\setminus {{\mathbb{R}}})^3$ of . It is easily seen that $(\ov{\a},\ov{\b},\ov{\g})$ also solves and, therefore, by taking the real part in , we can set $$\begin{gathered} \label{54} \un{U}(x)=\max\{e^{\a_1x}\cos(\a_2x), 0\}, \\ \begin{split} \label{55} \un{V}(x,y)=&\max\{e^{\a_1x}(\sin(\b_1 y)\cosh(\b_2 y)(\g_1\cos(\a_2x)-\g_2\sin(\a_2x))+ \\ &- \cos(\b_1 y)\sinh(\b_2 y)(\g_1\sin(\a_2x)+\g_2\cos(\a_2x))), 0\}, \end{split}\end{gathered}$$ where $$\label{56} \g_1=\frac{\mu\t_1}{\t_1^2+\t_2^2}, \qquad \g_2=\frac{\mu\t_2}{\t_1^2+\t_2^2},$$ being $$\label{57} \begin{split} \t_1& = ( d\b_1\cos( \b_1 L ) + \nu\sin( \b_1 L ) )\cosh( \b_2 L )+d\b_2\sin( \b_1 L )\sinh( \b_2 L ) \\ \t_2& = ( d\b_1\sin( \b_1 L ) - \nu\cos( \b_1 L ) )\sinh( \b_2 L ) - d\b_2\cos( \b_1 L )\cosh( \b_2 L ). \end{split}$$ After the change of variables $$\xi=\b-\b^*, \qquad \tau=c-c^*$$ the search for zeros of is equivalent to the search for zeros of the function $$\label{58} h(\xi,\tau):=g(c^*+\tau,\b^*+\xi)$$ with $(\xi,\tau)$ in a neighborhood of $(0,0)$. Since $(c^*,\b^*)$ is the first contact point between $\a_D^+$ and $\a_d^-$, we have that there exists $n\in{{\mathbb{N}}}\setminus\{0\}$ such that $$h(0,0)=\dots=\p^{2n-1}_\xi h(0,0)=0 \quad \text{ and } a_{2n}:=\frac{\p^{2n}_\xi h(0,0)}{(2n)!}>0,$$ while from and it follows that $$a_1:=\p_\tau h(0,0)<0.$$ By considering the Taylor series of in a neighborhood of $(0,0)$ we have that $h(\xi,\tau)=0$ is equivalent to $$\label{59} a_1\tau+a_{2n}\xi^{2n}=p(\xi,\tau)\tau+o(\xi^{2n+1})$$ where $p(\xi,\tau)$ is a polynomial which is either identically $0$ or of degree at least $1$. Thanks to the signs of the coefficients determined above, we know that the left hand side of $$h_1(z):=a_{2n}z^{2n}+a_1\tau$$ has, for $\tau<0$, $2n$ complex roots $$z_j:=z_j(\tau)=\left(\frac{a_1 \tau}{a_{2n}}\right)^{\frac{1}{2n}}e^{i\frac{(2j-1)\pi}{2n}} \quad j=1,\dots,2n.$$ Consider now the ball $$B:=B_r(z_1)\subset{{\mathbb{C}}}\qquad \text{ with } r=\s|\tau|^{\frac{1}{2n}}, \quad \s\sim 0$$ From geometrical considerations we have that, on $\p B$, $$|h_1(z)|=\prod_{j=1}^{2n}|z-z_j|\geq r\prod_{j=2}^{2n}C_j|\tau|^{\frac{1}{2n}}>C|\tau|$$ while, the right hand side $\varphi$ of , considered as a function of $\xi$, satisfies, on $\p B$, $$|\varphi(z)|\leq|p(z,\tau)||\tau|+o(|z|^{2n+1})<\tilde{C}|\tau|^{1+\frac{1}{2n}}.$$ Therefore, by choosing $\tau$ negative and sufficiently small, we can make $|\varphi|<|h_1|$ on $\p B$ and Rouché’s theorem can be applied, guaranteeing the existence of complex roots of and therefore of for $c<c^*$, $c\sim c^*$. This same analysis also shows that $\b=\b_1+i\b_2=\b_1(c)+i\b_2(c)$ satisfies $$\label{510} \b_1(c)\to \b^*, \quad \b_2(c)\to 0 \qquad \text{ as } c\ua c^*.$$ As a consequence, from and we have that $$\cosh(\b_2 L)\to 1, \quad \sinh(\b_2 L)\to 0, \quad \g_1\to\frac{\mu}{d\b^*\cos(\b^*L)+\nu\sin(\b^*L)}>0, \quad \g_2\to 0,$$ as $c\ua c^*$, since $0<\b^*<\ov{\b}<\pi/L$. Moreover, when $n>1$, the second equations in systems and ensure that $\a_2\neq 0$, since both $\b_1$ and $\b_2$ are positive for $c\sim c^*$. This follows from by continuity in the case $\b^*\neq 0$ and, when $\b^*=0$, it can be shown directly by analyzing the equation for $\a$ which is obtained by plugging into . On the other hand, when $n=1$, it can be proved as in Section \[sec8\] (see ) that $\b^*>0$ and then $\a_2\neq 0$ follows as in [@BRR1 Lemma 6.1]. In conclusion, it is apparent from and that, by taking $c$ sufficiently close to $c^*$, it is possible to take a component of the sets $\{\un{U}>0\}$ and $\{\un{V}>0\}$ in such a way is satisfied, obtaining compactly supported generalized subsolutions satisfying . Asymptotic speed of propagation {#sec6} =============================== We are now able to give the proof of the first part of Theorem \[thmain\]. *Proof of Theorem \[thmain\].* To prove the first condition of Definition \[deasp\] we will use the supersolutions of constructed in Proposition \[pr41\] and Remark \[remark42\](i). We recall that they solve the linear system and are of type $$(\bar{u}(x,t),\bar{v}(x,y,t))=e^{\a(x+c^*t)}(1,\g\phi(y))$$ with $\a,\g>0$ and $\phi(y)>0$ in $(0,L]$ satisfies $\phi'(0)>0$. As a consequence, since $(u_0,v_0)$ has compact support, there exists $k>0$ such that $$(u_0(x),v_0(x,y))<k(\bar{u}(x,0),\bar{v}(x,y,0))$$ for every $(x,y)\in{{\mathbb{R}}}\times(0,L]$. Moreover $(0,0)$ is a strict subsolution of and from Proposition \[prcp1\] we have that $$\label{61} (0,0)\leq(u(x,t),v(x,y,t))<k(\bar{u}(x,t),\bar{v}(x,y,t))$$ for every $t>0$. Observe that $$(\bar{\bar{u}}(x,t),\bar{\bar{v}}(x,y,t))=e^{\a(-x+c^*t)}(1,\g\phi(y))$$ is also a supersolution of satisfying $(u_0(x),v_0(x,y))<k(\bar{\bar{u}}(x,0),\bar{\bar{v}}(x,y,0))$ for large $k$. Fix now $c>c^*$, $t>0$, $|x|\geq ct$ and $y\in[0,L]$. We distinguish the cases $x\leq -ct<0$ and $-x\leq -ct<0$. In the first case, it follows that $$e^{\a(x+c^*t)}\leq e^{\a(c^*-c)t}$$ and this, together with , implies $$(0,0)\leq(u(x,t),v(x,y,t))<k e^{\a(c^*-c)t}\left(1,\g\phi(y)\right)$$ for every $(x,y)\in{{\mathbb{R}}}\times(0,L]$ and follows. The second case can be treated by comparing $(u,v)$ with $(\bar{\bar{u}},\bar{\bar{v}})$ in a similar fashion. By adapting the arguments of the proof of Proposition \[pr34\] to the case of Problem , using the subsolution constructed in Proposition \[pr51\], it can be shown that $$\label{62} \lim_{t\to+\infty}(u(x+ct,t),v(x+ct,y,t))=\left(\frac{\nu}{\mu}V(L),V(y)\right)$$ locally uniformly in $\overline{\O}$, where $V(y)$ is the unique solution of . Property now follows from and by using [@RTV Lemma 4.4]. Properties $(ii)$ and $(iii)$ of Theorem \[thmain\] will be proved in the next sections. Limits for small and large diffusion on the road {#sec7} ================================================ In this section we analyze the behavior of $c^*=c^*(D)$ as the diffusion on the road $D$ tends to $0$ and to $+\infty$, giving the proof of Theorem \[thmain\]$(ii)$. The result regarding the first case is the following \[pr71\] We have that $$\label{71} \lim_{D\da 0} c^*(D)=\ell_0>0,$$ where $\ell_0$ is the asymptotic speed of propagation of the problem $$\label{72} \left\{ \begin{array}{lll} u_t(x,t)=\nu v(x,L,t)-\m u(x,t) & \text{for } x\in{{\mathbb{R}}}, & t>0 \\ v_t(x,y,t)-d\D v(x,y,t)=f(v) & \text{for } (x,y)\in\O, & t>0 \\ d v_y(x,L,t)=\m u(x,t)-\nu v(x,L,t) & \text{for } x\in{{\mathbb{R}}}, & t>0 \\ v(x,0,t)=0 & \text{for } x\in{{\mathbb{R}}}, & t>0. \end{array}\right.$$ Observe first of all that limit exists thanks to Theorem \[thmain\]$(i)$. Fix now $D<d$. By the discussion of Sections \[sec4\] and \[sec5\], we have that $c^*(D)$ is the one for which $\a_{D}^-(c,\b)$ given by is tangent to $\a_{d}^+(c,\b)$ defined in . Moreover we have that necessarily the tangency occurs for $\b^*>\frac{\pi}{2L}$ (see Figure \[fig23\](B)). By passing to the limit for $D\da 0$ in , we get that $\ell_0$ is the unique value of $c$ for which $\a_{d}^+(c,\b)$ is tangent to $$\label{73} \a_0^-(c,\b)=\frac{-\mu d\b\cos(\b L)}{c\left(d\b\cos(\b L)+\nu\sin(\b L)\right)}, \quad \b\in\left[\frac{\pi}{2L},\ov{\b}\right),$$ where $\ov{\b}$ is the one defined in . The existence of such $c$ follows from the fact that $$\begin{gathered} \a_0^-\left(c,\frac{\pi}{2L}\right)=0, \qquad \a_0^-(c,\b)>0 \text{ for } \b\in\left(\frac{\pi}{2L},\ov{\b}\right), \qquad \lim_{\b\ua\ov{\b}}\a_0^-(c,\b)=+\infty, \\ \p_\b\a_0^-(c,\b)>0, \qquad \p_c\a_0^-(c,\b)<0, \\ \lim_{c\da 0}\a_0^-(c,\b)=+\infty, \qquad \lim_{c\ua +\infty}\a_0^-(c,\b)=0, \end{gathered}$$ together with the properties of $\a_d^+(c,\b)$ already described in Section \[sec4\]. To see that $\ell_0$ coincides with the asymptotic speed of propagation of , it is sufficient, as in Section \[sec4\], to construct supersolutions of of the form for $c>\ell_0$ by intersecting the curves $\a_{d}^+(c,\b)$ and and to proceed like in Section \[sec5\] to construct compactly supported subsolutions to for every $c<\ell_0$. Of course one has to prove the corresponding comparison principles for system , which couples a strongly parabolic equation with a degenerate one. They essentially hold because, if there was a first contact point at a positive time between a supersolution and a subsolution, either it would be for the $v$ component, which is impossible since a classical comparison principle holds, or for the $u$ component at $y=L$. In such case, the time derivative of the difference between the super- and the subsolution would be negative, while the right-hand side of the first equation of would be positive, obtaining again a contradiction (for a more detailed treatment of the comparison principles for such degenerate system in a similar context see [@RTV Proposition 2.5]). We now pass to the case $D\to+\infty$. \[pr72\] We have that $c^*(D)$ is unbounded as $D\to+\infty$ and $$\label{74} \lim_{D\ua \infty} \frac{c^*(D)}{\sqrt{D}}=\ell_\infty>0$$ where $\ell_\infty$ is the asymptotic speed of propagation of the problem $$\label{75} \left\{ \begin{array}{lll} u_t(x,t)-u_{xx}(x,t)=\nu v(x,L,t)-\m u(x,t) & \text{for } x\in{{\mathbb{R}}}, & t>0 \\ v_t(x,y,t)-d v_{yy}(x,y,t)=f(v) & \text{for } (x,y)\in\O, & t>0 \\ dv_y(x,L,t)=\m u(x,t)-\nu v(x,L,t) & \text{for } x\in{{\mathbb{R}}}, & t>0 \\ v(x,0,t)=0 & \text{for } x\in{{\mathbb{R}}}, & t>0. \end{array}\right.$$ Recall from the proof of Proposition \[pr41\] that, for large $D$, $$\label{76} c^*(D)=\min\{c^{*,1}(D),c^{*,2}(D)\},$$ where $c^{*,1}$ is the first value of $c$ for which $\a_D^+(c,\b)$ and $\a_d^-(c,\b)$ intersect, being tangent and the same for $c^{*,2}$, considering $\tilde{\a}_D^+(c,\b)$ and $\tilde{\a}_d^-(c,\b)$. We will prove that holds both for $c^{*,1}(D)$ and $c^{*,2}(D)$ and therefore will follow from . We start with the case of $c^{*,1}(D)$ (for convenience, we will denote it by $c^{*}(D)$ when there is no possibility of confusion), which is increasing thanks to Proposition \[pr41\] and admits a limit as $D\ua\infty$. It is obvious (see Figure \[fig26\](A)) that $$\label{77} \a_d^-(c^*(D),\tilde{\b}(c^*(D),D))<\a_D^+(c^*(D),0),$$ where $\tilde{\b}(c^*(D),D)\in(\frac{\pi}{2L},\ov{\b})$ is the one defined in (we have pointed out explicitly the dependence on $D$). Relation can be written as $$\label{78} \frac{1}{d}\left(\!\!1\!-\!\sqrt{1-\frac{{c_{\operatorname{KPP}}}^2-4d^2\tilde{\b}^2(c^*(D),D)}{{c^*}^2(D)}}\right)\!<\frac{1}{D}\left(1+\sqrt{1+\frac{4\mu d}{d+\nu L}\frac{D}{{c^*}^2(D)}}\right).$$ Assume by contradiction that $c^*(D)$ is bounded. Then, from , we have that $$\label{79} \frac{c^*(D)^2}{D}=\frac{-4\mu d\tilde{\b}(c^*(D),D)\cos(\tilde{\b}(c^*(D),D) L)}{d\tilde{\b}(c^*(D),D)\cos(\tilde{\b}(c^*(D),D) L)+\nu\sin(\tilde{\b}(c^*(D),D) L)},$$ from which we get $$\lim_{D\ua\infty} \tilde{\b}(c^*(D),D)=\frac{\pi}{2L}.$$ By passing now to the limit as $D\ua\infty$ in , we get a contradiction and, therefore, $$\label{710} \lim_{D\ua\infty}c^*(D)=+\infty.$$ As the curves are tangent for the first time, we also have $\a_D^+(c^*,0)<\a_d^-(c^*,0)$, which reads $$\frac{1}{D}\left(1+\sqrt{1+\frac{4\mu d}{d+\nu L}\frac{D}{{c^*}^2}}\right)<\frac{1}{d}\left(1-\sqrt{1-\frac{{c_{\operatorname{KPP}}}^2}{{c^*}^2}}\right).$$ Using , we derive from the previous relation $$\label{711} \frac{1}{D}\left(1+\sqrt{1+\frac{4\mu d}{d+\nu L}\frac{D}{{c^*}^2}}\right)<{c^*}^{-2}\left(\frac{{c_{\operatorname{KPP}}}^2}{2d}+o(1)\right),$$ where, as usual, $o(1)$ denotes a quantity that goes to $0$ as $D\ua\infty$. Solving now for ${c^*}^2/D$, we obtain $$\frac{{c^*}^2}{D}<\frac{(d+\nu L)f'(0)^2}{(d+\nu L)f'(0)+\mu d}+o(1),$$ from which we conclude $$\label{712} \limsup_{D\ua\infty}\frac{{c^*}^2}{D}\leq\frac{(d+\nu L)f'(0)^2}{(d+\nu L)f'(0)+\mu d}.$$ We now observe that $$\label{713} \tilde{\b}(c^*(D),D)<\sqrt{\frac{f'(0)}{d}}$$ for every $D$, because otherwise $$\a_D^+(c^*(D),\tilde{\b}(c^*(D),D))>0\geq\a_d^-(c^*(D),\tilde{\b}(c^*(D),D))$$ and there would be another intersection between $\S_D(c^*)$ and $\S_d(c^*)$ apart from $\b^*$, which contradicts the construction of $c^*$. As a consequence of we have that $$\label{714} \liminf_{D\ua\infty}\tilde{\b}(c^*(D),D)\leq \limsup_{D\ua\infty}\tilde{\b}(c^*(D),D)\leq\sqrt{\frac{f'(0)}{d}}.$$ We now distinguish two cases. If $$\label{715} \liminf_{D\ua\infty}\tilde{\b}(c^*(D),D)=\sqrt{\frac{f'(0)}{d}},$$ we have from that $$\lim_{D\ua\infty}\tilde{\b}(c^*(D),D)=\sqrt{\frac{f'(0)}{d}}>\frac{\pi}{2L}$$ and, therefore, using and taking the limsup for $D\ua\infty$ in , we get that $\lim_{D\ua\infty}c^*(D)^2/D$ exists and is positive. On the other hand, if $$\label{716} \liminf_{D\ua\infty}\tilde{\b}(c^*(D),D)<\sqrt{\frac{f'(0)}{d}},$$ we consider relation , which, by using and , becomes $$2{c^*}^{-2}\left(f'(0)-d\tilde{\b}^2(c^*(D),D)+o(1)\right)<\frac{1}{D}\left(1+\sqrt{1+\frac{4\mu d}{d+\nu L}\frac{D}{{c^*}^2}}\right)$$ for large $D$. Solving for ${c^*}^2/D$, we now get $$\frac{{c^*}^2}{D}>\!\min\!\left\{\frac{(d+\nu L)(f'(0)-d\tilde{\b}^2(c^*(D),D))^2}{(d+\nu L)(f'(0)-d\tilde{\b}^2(c^*(D),D))+\mu d},2\left(f'(0)\!-\!d\tilde{\b}^2(c^*(D),D)\right)\right\}+o(1)$$ and, thanks to , we have $$\label{717} \liminf_{D\to+\infty}\frac{{c^*}^2}{D}>0.$$ Summing up, holds both in case and . This, together with , implies that ${c^*}^2/D$ is bounded and bounded away from $0$. It is therefore natural to perform in the change of variables $$\label{718} \hat{\a}=\sqrt{D}\a, \qquad \hat{c}=\frac{c}{\sqrt{D}},$$ obtaining $$\label{719} \left\{\begin{array}{l} -\hat{\a}^2+\hat{c}\hat{\a}+\frac{\mu}{1+\frac{\nu \tan(\b L)}{d \b}}=0 \\ -\frac{d\hat{\a}^2}{D}+\hat{c}\hat{\a}=f'(0)-d\b^2, \end{array}\right.$$ where $\hat{c}$ is bounded and bounded away from $0$. The first equation describes, in the plane $(\b,\hat{\a})$, the curve $\S_1(\hat{c})$ defined in , therefore the function $\hat{\a}_1^+(\hat{c},\b)$, to which we will be interested in, is bounded and bounded away from $0$ for all $c>c_{\operatorname{KPP}}$ and $\b$ in the proper domain of definition. Therefore, by taking the limit for $D\ua\infty$ in , we get $$\label{720} \left\{\begin{array}{l} -\hat{\a}^2+\hat{c}\hat{\a}+\frac{\mu}{1+\frac{\nu \tan(\b L)}{d \b}}=0 \\ \hat{\a}=\frac{f'(0)-d\b^2}{\hat{c}}. \end{array}\right.$$ The second equation is a concave parabola, symmetric with respect to the $\hat{\a}-$axis, passes through $\left(\sqrt{f'(0)/d},0\right)$ and whose vertex is $\left(0,f'(0)/\hat{c}\right)$. Now we pass to the case of $c^{*,2}$, which, as above, will be simply denoted by $c^*$. In this case, as it is apparent from Figure \[fig26\](B), we have $$\tilde{\a}_d^-(c^*(D),0)<\lim_{\b\to+\infty}\tilde{\a}_D^+(c^*(D),\b),$$ which reads as $$\label{721} \frac{1}{d}\left(1-\sqrt{1-\frac{{c_{\operatorname{KPP}}}^2}{{c^*}^2}}\right)<\frac{1}{D}\left(1+\sqrt{1+4\mu \frac{D}{{c^*}^2}}\right)$$ and gives that $$\lim_{D\to+\infty}c^*(D)=+\infty,$$ because, if the limit was finite, say $\ell$, passing to the limit in would lead to $$0<\frac{1}{d}\left(1-\sqrt{1-\frac{{c_{\operatorname{KPP}}}^2}{\ell^2}}\right)\leq 0,$$ which is impossible. Therefore, gives $${c^*}^{-2}\left(\frac{{c_{\operatorname{KPP}}}^2}{2d}+o(1)\right)<\frac{1}{D}\left(1+\sqrt{1+4\mu \frac{D}{{c^*}^2}}\right)$$ and, solving for ${c^*}^2/D$ and taking the liminf as $D\ua\infty$ we get $$\liminf_{D\to+\infty}\frac{{c^*}^2}{D}\geq\min\left\{\frac{f'(0)^2}{f'(0)+\mu},2f'(0)\right\}.$$ On the other hand, in this situation we also have that $\tilde{\a}_D^+(c^*(D),0)<\tilde{\a}_d^-(c^*(D),0)$, which provides us with and, therefore, with the upper bound for ${c^*}^2/D$ given by . By performing the change of variables in and passing to the limit for $D\to+\infty$ we obtain $$\label{722} \left\{\begin{array}{l} -\hat{\a}^2+\hat{c}\hat{\a}+\frac{\mu}{1+\frac{\nu \tanh(\b L)}{d \b}}=0 \\ \hat{\a}=\frac{f'(0)+d\b^2}{\hat{c}}. \end{array}\right.$$ The first equation describes, in the plane $(\b,\hat{\a})$, the curve $\tilde{\S}_1(\hat{c})$ defined in , while the second one is a parabola, which is symmetric to the one of the second equation of with respect to the line $\hat{\a}=f'(0)/\hat{c}$. With a similar reasoning as that of Section \[sec4\], it is easily seen that there is a smallest value of $\hat{c}$ for which either the two curves of or of are tangent, which provides us with $\ell_\infty$. To see that this limit coincides with the asymptotic speed of propagation of Problem it suffices to repeat the construction of Sections \[sec4\]-\[sec5\] for this problem, starting from supersolutions of type and and using comparison principles analogous to the ones of Section \[sec2\], which can be proved for by applying the parabolic maximum principle in $y$ on every slice, with fixed $x$ (see [@RTV] for a detailed proof in a similar context and [@D], in the context of travelling waves, for comparison principles related to this degenerate system with Neumann boundary conditions at $y=0$). Influence of the road and limit for large field {#sec8} =============================================== To examine the influence of the road in Problem , it is appropriate to compare its asymptotic speed of propagation with the one of the following problem $$\label{81} \left\{ \begin{array}{lll} v_t(x,y,t)-d\D v(x,y,t)=f(v) & \text{for } (x,y)\in\O, & t>0 \\ dv_y(x,L,t)=-\nu v(x,L,t) & \text{for } x\in{{\mathbb{R}}}, & t>0 \\ v(x,0,t)=0 & \text{for } x\in{{\mathbb{R}}}, & t>0, \end{array}\right.$$ which models a classical Fisher-KPP diffusion in the strip $\O$ and part of the population $v$ just leaves the field at level $y=L$. By using the same techniques of Section \[sec3\] it is possible to show that Problem admits a unique positive steady state if and only if $$\label{82} \frac{f'(0)}{d}>\ov{\b}^2>\left(\frac{\pi}{2L}\right)^2,$$ where $\ov{\b}=\ov{\b}(d,\nu,L)$ is, as in Section \[sec4\], the first positive value for which vanishes. By comparing with , it is apparent that one effect of the road is to enhance the persistence of the species, since the condition for persistence is less restrictive in the presence of the road. On the other hand, when holds, by taking $$\ov{v}(x,y,t)=e^{\a(x+ct)}\sin(\ov{\b} y)$$ as supersolution and following the lines of Sections \[sec4\]–\[sec6\], it is possible to show that Problem admits an asymptotic speed of propagation $$c_{\operatorname{KPP}}^{\operatorname{DR}}=2\sqrt{d\left(f'(0)-d\ov{\b}^2\right)}<c_{\operatorname{KPP}}$$ (here $\operatorname{DR}$ stands for the Dirichlet-Robin boundary conditions associated to the Fisher-KPP equation in ). Recalling the monotonicity property of $c^*(D)$ given by Theorem \[thmain\]$(i)$, we have that $c^*(D)>\ell_0$, where $\ell_0$ is the one constructed in Proposition \[pr71\] as the smallest value of $c$ for which the curves and $\a_d^+(c,\b)$ intersect. Observe that the former is defined for $\b<\ov{\b}$, while, recalling , $\a_d^+(c_{\operatorname{KPP}}^{\operatorname{DR}},\b)$ is defined for $\b\geq\ov{\b}$. This means that, for every $D$, $$\label{83} c^*(D)\geq\lim_{D\da 0}c^*(D)>c_{\operatorname{KPP}}^{\operatorname{DR}}.$$ Therefore, a second effect of the road is speeding up the propagation in the field. Moreover, from the second relation of Theorem \[thmain\]$(ii)$, we have that this effect can be arbitrarily enhanced, provided that $D$ is sufficiently large. We conclude with the proof of Theorem \[thmain\], considering the limit for large field. We will emphasize the dependence of $c^*$ on the width of the strip $L$, by writing $c^*_L$. *Proof of Theorem \[thmain\](iii).* We recall that in [@BRR1] it was proved that $$c^*_{\infty}(D)\begin{cases} ={c_{\operatorname{KPP}}}&\text{if } D\leq 2d \\ >{c_{\operatorname{KPP}}}&\text{if } D> 2d. \end{cases}$$ We distinguish the same two cases, starting with $D\leq 2d$. From and the discussion of Section \[sec4\] we have that $$\label{84} c_{\operatorname{KPP}}^{\operatorname{DR}}(L)< c^*_L<{c_{\operatorname{KPP}}}.$$ Recalling from that $\ov{\b}(L)\in\left(\frac{\pi}{2L},\frac{\pi}{L}\right)$, we have that implies $$\lim_{L\to+\infty}c^*_L=c_{\operatorname{KPP}}.$$ We now assume $D>2d$ and recall (see [@BRR1]) that $c^*_{\infty}$ is the value of $c$ for which the curve $$\tilde{\a}_D^{\infty}(c,\b):=\frac{1}{2D}\left(c+\sqrt{c^2+\frac{4\mu D}{1+\frac{\nu}{d\b}}}\right),$$ and the curve $\tilde{\a}_d^-(c,\b)$ defined in are tangent. In our case, a crucial role will be played by the behavior of the curves $\S_D(c)$, $\S_d(c)$, $\tilde\S_D(c)$, $\tilde\S_d(c)$ for $c={c_{\operatorname{int}}}$, where ${c_{\operatorname{int}}}$ is the one of . It is possible to show with direct computations that $D\mapsto {c_{\operatorname{int}}}(D)$ is increasing for $D>2d$. As a consequence, there exists $\tilde D$ such that ${c_{\operatorname{int}}}(D)>{c_{\operatorname{KPP}}}$ for every $D>\tilde D$. Due to the monotonicity of ${c_{\operatorname{int}}}(D)$, we deduce that $\tilde D$ is the unique value of $D$ for which $\a_D^+({c_{\operatorname{KPP}}},0)=\a_d^\pm({c_{\operatorname{KPP}}},0)$, which, taking into account and , provides $$\tilde D=\tilde D(L)=2d+\frac{4\mu d^3}{(d+\nu L){c_{\operatorname{KPP}}}^2}.$$ In particular, since we are taking $D>2d$, we have that, for $L$ large enough, $D>\tilde D(L)$ and, therefore, ${c_{\operatorname{int}}}>{c_{\operatorname{KPP}}}$. Moreover, recalling and , $$\label{85} \p_{\b\b}\left(\tilde{\a}_{D}^+({c_{\operatorname{int}}},0)-\tilde{\a}_{d}^-({c_{\operatorname{int}}},0)\right)=2d\left(\frac{\psi}{3\zeta\sqrt{{c_{\operatorname{int}}}^2+\frac{4\mu d D}{d+\nu L}}}-\frac{1}{\sqrt{{c_{\operatorname{int}}}^2-{c_{\operatorname{KPP}}}^2}}\right),$$ where we have set $\psi:=L^3\mu\nu$ and $\zeta:=(d+\nu L)^2$. It is possible to check, by direct computations, that has the same sign as $$\begin{gathered} (\psi^2-9\zeta^2) \left((d+\nu L)D{c_{\operatorname{KPP}}}^2+ 4\mu d^3\right)^2+\\ -4d(D-d)\left((d+\nu L){c_{\operatorname{KPP}}}^2 + 4\mu d^2\right)\left(36\zeta^2\mu d D+\psi^2(d+\nu L){c_{\operatorname{KPP}}}^2\right),\end{gathered}$$ which is positive for large $L$, since it is a polynomial of degree $8$ in $L$ with leading coefficient equal to $\left(\mu\nu^2c_{\operatorname{KPP}}^2(D-2d)\right)^2>0$. This implies that $$\tilde{\a}_{D}^+({c_{\operatorname{int}}},\b)>\tilde{\a}_{d}^-({c_{\operatorname{int}}},\b) \qquad \text{ for } \b>0, \quad \b\sim0$$ and, since the curve $\tilde\S_d({c_{\operatorname{int}}})$ intersects the $\a-$axis in another point, which lies above the intersection with $\tilde\S_D({c_{\operatorname{int}}})$, we obtain that $\tilde{\S}_D({c_{\operatorname{int}}})$ and $\tilde{\S}_d({c_{\operatorname{int}}})$ intersect, apart from $\b=0$, in a positive value of $\b$ too. On the other hand, using and defining $$\mathfrak{c}^2:=\lim_{L\to+\infty}{c_{\operatorname{int}}}^2(L)=\frac{D^2c_{\operatorname{KPP}}^2}{4d(D-d)},$$ it is easy to see that the curve $\S_D(c^*_L)$ introduced in approaches in the $(\b,\a)-$plane, as $L$ goes to $+\infty$, the vertical segment $\{0\}\times [0,\mathfrak{c}/D)$. As a consequence of these considerations and the discussion of Section \[sec4\], we have that, for large $L$, $c^*_L$ is obtained as the value for which $\tilde{\a}_D^+(c,\b)$ and $\tilde{\a}_d^-(c,\b)$ are tangent. Observe that $\tilde{\a}_D^+$ is decreasing in $L$ and, therefore, $c^*_L$ is increasing. In addition $\tilde{\a}_D^+>\tilde{\a}_D^{\infty}$, which entails that $c^*_L<c^*_\infty$. Finally, $\tilde{\a}_D^+$ tends, as $L\to+\infty$, to $\tilde{\a}_D^{\infty}$, together with its derivatives, locally uniformly in ${{\mathbb{R}}}_+\times{{\mathbb{R}}}_+$. This implies that $$\lim_{L\to+\infty}c^*_L=c^*_\infty$$ also in this case and concludes the proof. At a first glance, one may think that the result provided in Theorem \[thmain\](iii) gives a contradiction for $D\in(2d,\limsup_{L\to+\infty}D_{\operatorname{KPP}})$, since holds for $D<D_{\operatorname{KPP}}$ and one could repeat the argument of the first part of the proof to show that $c^*_L$ converges to ${c_{\operatorname{KPP}}}$ as $L\to+\infty$. However, this is not the case, as the previous interval is empty, since $$\label{86} \lim_{L\to+\infty}D_{\operatorname{KPP}}(L)=2d.$$ This follows as a byproduct of the proof of Theorem \[thmain\](iii), but for clarity we provide a direct proof here. Thanks to the monotonicity in $L$ and $D$ of $\S_D({c_{\operatorname{KPP}}})$ and by definition of $D_{\operatorname{KPP}}$, it follows easily that $L\mapsto D_{\operatorname{KPP}}(L)$ is decreasing and therefore the limit in exists. Let us denote it by $\ov D$ and assume by contradiction that it satisfies $\ov D>2d$. As shown in the last part of the proof of Theorem \[thmain\](iii), the curve $\S_{D_{\operatorname{KPP}}(L)}({c_{\operatorname{KPP}}})$ converges, as $L\to+\infty$, to the segment $\{0\}\times [0,{c_{\operatorname{KPP}}}/\ov D)$, which, since we are assuming $\ov D>2d$, lies at a positive distance from $\S_d({c_{\operatorname{KPP}}})$. As a consequence, there would be no intersection between the latter curve and $\S_{D_{\operatorname{KPP}}(L)}({c_{\operatorname{KPP}}})$ for large $L$ either, contradicting the definition of $D_{\operatorname{KPP}}$. Acknowledgement {#acknowledgement .unnumbered} =============== The author is deeply grateful to Professors Henri Berestycki, Jean-Michel Roquejoffre and Luca Rossi for the broad discussions that helped him in the preparation of this work. Moreover he wishes to thank all the people at CAMS (*Centre d’analyse et de mathématique sociales*, Paris), where most of the work was developed, in 2013, for the very warm treatment. [5]{} D.G. Aronson, H.F. Weinberger, *Multidimensional nonlinear diffusion arising in population genetics*, Adv. in Math. **30** (1978), 33–76. H. Berestycki, J.-M. Roquejoffre, L. Rossi, *The influence of a line with fast diffusion in Fisher-KPP propagation*, J. Math. Biology **66** (2013), 743–766. H. Berestycki, J.-M. Roquejoffre, L. Rossi, *Fisher-KPP propagation in the presence of a line: further effects*, Nonlinearity **26** (2013), 2623–2640. L. Dietrich, *Velocity enhancement of reaction-diffusion fronts by a line of fast diffusion*, to appear in Trans. Amer. Mat. Soc. (2015), preprint arXiv:1410.4738 \[math.AP\]. A.N. Kolmogorov, I.G. Petrovskii, N.S. Piskunov, *Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique*, Bull. Univ. État. Moscou Sér. Intern. A **1** (1937), 1–26. H.W. McKenzie, E.H. Merrill, R.J. Spiteri, M.A. Lewis, *How linear features alter predator movement and the functional response*, Interface focus **2** (2012) 205–216. C.G. Moore, C.J. Mitchell, *Aedes albopictus in the United States: ten-year presence and public health implications*, Emerging Infectious Diseases **3** (1997), 6 pages. L. Rossi, A. Tellini, E. Valdinoci, *The effect on Fisher-KPP propagation in a cylinder with fast diffusion on the boundary*, arXiv:1504.04698 \[math.AP\]. A.J. Tatem, D.J. Rogers, S.I. Hay, *Global transport networks and infectious disease spread*. Adv Parasitol **62** (2006), 293–343. [^1]: This work was mainly supported by the Spanish Ministry of Economy and Competitiveness through grants BES-2010-039030 and EEBB-I-13-05962. Partial founding was also provided by European Union’s ERC Grant Agreement n. 321186 - ReaDi - Reaction-Diffusion Equations, Propagation and Modelling and Research Project Stabilità asintotica di fronti per equazioni parabolicheof the University of Padova (2011) coordinated by Luca Rossi.

--- abstract: 'Active regions are open wounds in the Sun’s surface. Seismic oscillations from the interior pass through them into the atmosphere, changing their nature in the process to fast and slow magneto-acoustic waves. The fast waves then partially reflect and partially mode convert to upgoing and downgoing Alfvén waves. The reflected fast and downgoing Alfvén waves then re-enter the interior through the active regions that spawned them, infecting the surface seismology with signatures of the atmosphere. Using numerical simulations of waves in uniform magnetic fields, we calculate the upward acoustic and Alfvénic losses in the atmosphere as functions of field inclination and wave orientation as well as the Time-Distance ‘travel time’ perturbations, and show that they are related. Travel time perturbations relative to quiet Sun can exceed 40 seconds in 1 kG magnetic field. It is concluded that active region seismology is indeed significantly infected by waves leaving and re-entering the interior through magnetic wounds, with differing travel times depending on the orientation of the wave vector relative to the magnetic field. [This presages a new directional-time-distance seismology.]{}' author: - | Paul S. Cally[^1] and Hamed Moradi[^2]\ Monash Centre for Astrophysics and School of Mathematical Sciences, Monash University, Victoria, Australia 3800 bibliography: - 'fred.bib' title: Seismology of the Wounded Sun --- Sun: helioseismology – Sun: oscillations – Sun: magnetic fields Introduction ============ Classical local helioseismology [@GizBir05aa] uses Doppler velocity or intensity data from near the solar surface to probe the Sun’s interior: thermal, magnetic, and flow properties may all be inferred with varying degrees of reliability. The overlying atmosphere, the chromosphere and corona, are generally assumed to not be important in this process, though a separate ‘coronal seismology’ [@NakVer05aa; @De-05aa; @SteZaiNak12aa] using observations from coronal heights and aimed at determining the physical characteristics of coronal structures has developed in recent years. It is well-appreciated that the Sun’s internal waves manifest in the atmosphere as well, particularly in active regions [@BogJud06aa; @JefMcIArm06aa; @KhoCal13aa], but any back-reaction on the interior seismology has heretofore been largely ignored [though see @KhoCol09aa for an explanation of acoustic halos that relies on reflected fast waves]. We argue that this neglect can lead to serious errors in interpretation of seismic data. ![Schematic diagram ($x$-$z$ plane) depicting fast/slow conversion/transmission at the equipartition level $a=c$, and fast-to-Alfvén conversion in a nebulous region near the fast wave reflection height. The magnetic field lines are shown as background. Conversion to upward or downward propagating Alfvén waves depends in the relative orientation of the magnetic field and the wave propagation direction. Both panels should be regarded as projections, with the fast wave raypath having a component in the $y$-direction; otherwise, there is no fast-Alfvén coupling. (Adapted from @KhoCal12aa) \[fig:Alfschem\]](\figdir/Alfschem){width="\hsize"} The most widely used local helioseismic technique is Time-Distance Helioseismology [TD; @DuvJefHar93aa; @GizBir05aa]. By correlating observations of perturbations at different times and positions on the solar surface, a causative link is inferred and a travel time between pairs of points determined. Comparing these travel times with those calculated for a standard model, one infers the presence of wave-speed anomalies beneath the surface that may be due to such features as temperature variations or flows. Utilizing the full gamut of wave paths sampled by one’s data set, inversions may be performed for subsurface structure. However, ‘travel time’ is not necessarily as straightforward a concept as we might hope. Proper group travel time is difficult to measure with any precision, and in practice phase variations are used as a travel time proxy. This approach seems well-founded in quiet Sun, where magnetic effects may be ignored, but is subject to several uncertainties and complications in and around active regions dominated by magnetism in the surface layers and above. To set the scene, the quiet Sun’s internal $p$-modes ($p$ for pressure) are essentially acoustic waves travelling in an unmagnetised stratified medium, with dispersion relation $\omega^2-\omega_c^2-c^2k_h^2=c^2k_z^2$, where $\omega$ is the frequency, $c$ is the sound speed (which increases with depth), $k_h$ and $k_z$ are the horizontal and vertical components of the wave vector, and $\omega_c$ is the acoustic cutoff frequency [commonly defined by $\omega_c^2=(c^2/4H^2)(1-2\, dH/dz)$ where $H$ is the density scale height, but see @SchFle98aa for a critique of such formulæ]. Vertical propagation is therefore limited to the cavity where the left hand side is positive. At depth, where $c^2k_h^2\gg\omega_c^2$ typically, the lower turning point is around the Lamb depth $\omega^2=c^2 k_h^2$, whilst at the upper turning point $\omega^2=\omega_c^2+c^2 k_h^2\geq\omega_c^2$. The introduction of strong magnetic field in surface layers has several effects to alter this picture: 1. Magnetic field alters the density and thermal structure of the plasma by supplying additional magnetic forces, requiring the plasma pressure to adjust accordingly (the ‘indirect’ effect). Changes in the sound speed naturally affect wave travel times. 2. Magnetohydrodynamic (MHD) wave modes become available: fast, slow, and Alfvén. At depth, where the Alfvén speed $a$ is much less than the sound speed $c$, the $p$-modes are effectively fast waves. But on passing through the equipartition layer $a=c$ they partially transmit as acoustic waves (now slow) and partially convert to magnetically dominated fast waves [@Cal06aa; @SchCal06aa; @Cal07aa], depending on the attack angle between the wave vector and the magnetic field. In sunspot umbrae, $a=c$ is typically situated several hundred kilometres below the surface, whilst in penumbrae it is around the surface [@MatSolLag04aa]. In the regions surrounding sunspots, $a=c$ may crudely be equated with the level of the magnetic canopy in the low atmosphere. 3. The transmitted slow waves above $a=c$ are essentially field-guided acoustic waves. Unlike their non-magnetic cousins though, they may still propagate vertically at frequencies below the acoustic cutoff (around 5 mHz), provided $\omega>\omega_c\cos\theta$, where $\theta$ is the field inclination from the vertical [@BelLer77aa]; the so-called ‘magneto-acoustic portals’ [@JefMcIArm06aa]. This ramp effect has recently been shown computationally to be far more important than radiation in modifying the acoustic cutoff [@HegHanDe-11aa]. 4. Whether or not the slow wave escapes into the solar atmosphere in strong field regions, the fast wave certainly does. It is immune to the acoustic cutoff effect, but nevertheless has its own upper turning point substantially higher in the atmosphere. For simplicity, assume that this is high enough that $a\gg c$ and therefore ignore sound speed altogether. The dispersion relation is then $\omega^2=a^2(k_h^2+k_z^2)$, indicating reflection where $\omega^2=a^2k_h^2$. This is an important point: the fast wave reflects where its horizontal phase speed $\omega/k_h$ coincides with the local Alfvén speed. After it reflects, the fast wave re-enters the solar interior wave field, and therefore is part of the seismology of the Sun. Its journey through the atmosphere must therefore have some effect on wave timings. However, that is not the full story. The fast wave itself may partially mode-convert to upward or downward propagating Alfvén waves, depending on magnetic field orientation relative to the wave vector [@CalGoo08aa; @CalHan11aa; @HanCal12aa; @KhoCal11aa; @KhoCal12aa; @Fel12aa], thereby removing energy from the seismic field and potentially altering its phase (see the schematic diagram Figure \[fig:Alfschem\]). This conversion typically occurs in a broad region near the upper turning point of the fast wave, but only if the wave is propagating across the vertical plane containing the magnetic field lines; we let $\phi$ denote the angle between the vertical magnetic flux and wave propagation planes. Such a phase change could be wrongly interpreted in TD as a travel time shift [@Cal09ab; @Cal09aa]. So, there are many processes going on here: fast/slow conversion/transmission of upward travelling waves at $a=c$; slow wave reflection or transmission, depending on the acoustic cutoff and the magnetic field inclination $\theta$; fast-to-Alfvén conversion near the fast wave turning point; partial reflection of the fast wave to re-enter the seismic field; further fast/slow conversion/transmission as the fast wave passes downward through $a=c$. In totality, this is too complicated to model analytically. However, knowing the physics of the constituent parts, a full numerical simulation can provide valuable insight. The novel aspect explored here is to simultaneously calculate the TD travel time shifts $\delta\tau$ and upward acoustic and Alfvénic losses in a simulation as functions of $\theta$ and $\phi$ (for selected frequencies and horizontal wavenumbers). The considerations set out above indicate that active regions are truly wounds in the surface of the Sun. They allow waves and wave energy to escape from the interior cavity wherein they are normally trapped, but also allow their depleted, phase-shifted, and mode-converted remnants to re-enter and infect the local seismology that we rely on for information about the sub-surface. The purpose of this paper is to look for correspondences between acoustic and magnetic wave losses on the one hand, and ‘travel time’ shifts on the other, and thereby to verify that the seismology of the wounded Sun depends on propagation, transmission, and conversion in overlying active region atmospheres. [We introduce a directionally filtered time-distance approach[^3] to simulation data that is directly extensible to real helioseismic data, and thereby envisage a Directional-Time-Distance (DTD) seismology sensitive to magnetic field orientation. In Section \[sec:BVP\], the DTD results are verified using the Boundary Value approach of @Cal09ab.]{} [ The purpose of this paper is to demonstrate that the above mode conversion scenarios actually operate in the solar context; to survey how they depend on field inclination $\theta$ and wave orientation $\phi$; and to assess the utility of DTD in discerning directional magnetic effects. To this end, we explore only vertical atmospheric stratification, characterized by density scale heights of order 0.15 Mm, and ignore magnetic field inhomogeneity, which in sunspots has scale length of order many Mm. A subsequent paper will address wave simulations in realistic sunspot models using DTD. This is not to say though that flux tubes are not important to wave propagation, though probably they are more relevant to the corona where the far greater gravitational scale height is no longer the primary feature, leaving complex loop structures to dominate [@VanBraVer08aa]. Chromospheric sunspot fields, dominated be a single monopolar spot, may be expected to be smoother. ]{} Computational Details ===================== We employ the Seismic Propagation through Active Regions and Convection (SPARC) $6^{\rm th}$ order linear MHD code of @Han07aa, with uniform $B_0=500$ G or 1 kG magnetic fields of inclination $\theta$ from the vertical threading the Convectively Stabilized solar Model (CSM\_B) atmosphere of @SchCamGiz11aa. The ‘observational height’ for vertical velocities used in seismic calculations is 0.3 Mm, above the $a=c$ equipartition layer for both 500 G and 1 kG (0.24 Mm and 0.08 Mm respectively). A stochastic driving plane is placed at depth $z=-0.156$ Mm. The random sources produce a solar wave power spectrum with maximum power in the range 2 – 5.5 mHz, peaking at about 3.2 mHz. Being a linear code, the overall velocity normalization is arbitrary. The standard ${\leavevmode\kern.1em\raise.5ex\hbox{\the\scriptfont0 2}\kern-.1em/\kern-.15em\lower.25ex\hbox{\the\scriptfont0 3}}$ dealiasing rule is applied to avoid spectral blocking, with $|k_x|$ and $|k_y|$ limited to about 1.9 $\rm Mm^{-1}$. A $140\,{\rm Mm}\times140\,{\rm Mm}\times26.53\,{\rm Mm}$ box is spanned by a $128\times128\times265$ grid of spacings $\Delta x=\Delta y=1.09$ Mm covering heights $-25\,{\rm Mm} \le z \le 1.53\,{\rm Mm}$ with nonuniform $\Delta z$. Periodic lateral boundaries and absorbing PML layers at top (starting at $z_t=1.26$ Mm) and bottom complete the definition of the computational region. Without loss of generality, the field lines are assumed to lie parallel to the $x$-$z$ plane. As with most such codes, SPARC normally employs an Alfvén speed ‘limiter’ or ‘controller’ (or similar) in the atmosphere to avoid infeasibly small time steps required to satisfy the CFL numerical stability condition [*e.g.*. @Han08aa; @RemSchKno09aa; @CamGizSch11aa]. These commonly cap the Alfvén speed at 20–60 $\rm km\,s^{-1}$. We do not impose such a limiter though, as it corrupts the very processes of fast wave reflection and conversion that we seek to explore [@MorCal13aa]. This is why we adopt lower box heights than usual, so as to keep the peak Alfvén speed manageable. Numerical experiment has allowed us to select box heights sufficient to encompass fast-to-Alfvén conversion without mandating impossibly small time steps. For the 500 G case we use $\Delta t=0.1$ s, and half that for 1 kG. Data cubes are produced from the simulations consisting of vertical velocity values at each grid point at observation height $z_{\rm obs}=0.3$ Mm over typically $8{\leavevmode\kern.1em\raise.5ex\hbox{\the\scriptfont0 1}\kern-.1em/\kern-.15em\lower.25ex\hbox{\the\scriptfont0 2}}$ hours. Filtering is applied in Fourier space, both frequency and wavevector. This is more efficient than repeating the simulation for a large number of monochromatic drivers. Wavevector filtering uses a circular Gaussian ball filter with standard variation $\sigma=0.1$ $\rm Mm^{-1}$ centred at a particular horizontal wavevector ${{\bmath{k}}}={{\bmath{k}}}_h$ oriented angle $\phi$ from the $x$-direction. Horizontal wavenumber $k_h=|{{\bmath{k}}}_h|=0.5$, 0.75, or 1.0 $\rm Mm^{-1}$ determines the skip distance and phase speed, and $\phi$ represents the wave propagation direction. Frequency filtering is centred on 3 mHz and 5 mHz with a 0.5 mHz standard deviation. $k_h$ ($\rm Mm^{-1}$) $v_{ph}$ ($\rm km\,s^{-1}$) Skip Dist (Mm) ----------------------- ----------------------------- ---------------- 1.0 21.8 14.7 0.75 29.3 19.5 0.5 43.8 38.8 : Approximate phase speeds and skip distances associated with each of the three selected horizontal wavenumbers $k_h$ in the quiet solar model. \[tab:skip\] The resultant filtered data cubes are analysed in two ways. First, the acoustic (slow) and magnetic (Alfvén) wave energy fluxes are calculated at $z_f=1.2$ Mm, just below the PML layer, and plotted as contoured functions of field inclination $\theta=0^\circ$, $10^\circ$, $20^\circ$, …, $90^\circ$ and wave orientation $\phi=0^\circ$, $5^\circ$, $10^\circ$, …, $180^\circ$ for each frequency (3 mHz and 5 mHz). The fast wave is evanescent, and so contributes no flux. Similarly, Time-Distance travel time perturbations relative to quiet Sun (same atmospheric model but no magnetic field) are also plotted against $\theta$ and $\phi$.[^4] These are displayed and analysed in Section \[results\]. Results ======= In this section, wave energy fluxes at the top of the computational box will be compared with ‘travel time’ perturbations relative to the quiet Sun for a range of magnetic field inclinations $\theta$ and orientations $\phi$, field strengths (500 G and 1 kG), horizontal wavenumbers ($k_h=1$, 0.75, and 0.5 $\rm Mm^{-1}$), and wave frequencies (3 mHz and 5 mHz). Wave Energy Fluxes {#fluxes} ------------------ {width=".497\hsize"} {width=".497\hsize"}\ {width=".497\hsize"} {width=".497\hsize"} With magnetic field strengths of 500 G and 1 kG, Figure \[fig:Fz\] depicts the top vertical acoustic and magnetic wave energy fluxes at 3 mHz and 5 mHz as functions of field inclination from the vertical $\theta$ and wave orientation as selected by the wavevector space ball filter $\phi$. The (vector) fluxes are calculated according to $${{{\bmath{F}}}}_{\rm ac}=\langle p\,{{\bmath{v}}}\rangle\,, \quad {{{\bmath{F}}}}_{\rm mag}=\langle {{{\bmath{b}}}}{{\bmath{\times}}}({{\bmath{v}}}{{\bmath{\times}}}{{\bmath{B}}}_0)\rangle/\mu_0\,, \label{flux}$$ where ${{\bmath{v}}}$, $p$, and ${{{\bmath{b}}}}$ are the Eulerian velocity, pressure, and magnetic field perturbations, $\mu_0$ is the magnetic permeability, and the angled brackets indicate averaging over time, or over $x$ and $y$. For the most part, only vertical components of these fluxes will be plotted. Three different horizontal wavenumbers are selected: $k_h=1.0$, 0.75, and 0.5 $\rm Mm^{-1}$, corresponding to phase speeds and skip distances set out in Table \[tab:skip\]. The overall flux normalization is arbitrary, but is consistent with the solar-like spectrum of the stochastic driving plane across the frequencies, *i.e.*, having maximum power in the range 2 – 5.5 mHz with peak at about 3.2 mHz. The features in Figure \[fig:Fz\] are very much in accord with the simplified analyses of @CalGoo08aa and @KhoCal11aa. Specifically, there is negligible acoustic power at low field inclination because of the acoustic cutoff of a little over 5 mHz in the atmosphere. However, once $\omega > \omega_c\cos\theta$, acoustic waves may propagate upward (the ramp effect), and substantial flux is recorded at the top of the box. At 3 mHz, acoustic power peaks at $\theta=50^\circ$– $60^\circ$, whilst the peak is at around $30^\circ$ at 5 mHz. These peak powers may be understood in terms of the ‘attack angle’ $\alpha$ between the 3D wavevector and the magnetic field at the equipartition level [@Cal06aa; @SchCal06aa; @HanCal09aa], with most efficient acoustic transmission occurring at small $\alpha$ in the generalized ray approximation. This also explains the drop-off of acoustic flux with increasing $\phi$. The slight shift in the maximum of acoustic flux away from $\phi=0$ is consistent with the 3 mHz case of @KhoCal11aa. The magnetic flux also accords with previous (more idealized) modelling [@CalGoo08aa], peaking at higher $\theta$ than the acoustic flux (especially at 5 mHz), and at wave orientations of $60^\circ$ or more. The bias toward $\phi<90^\circ$ is in accord with the schematic scenario of Figure \[fig:Alfschem\] and the detailed cold plasma survey of @CalHan11aa. The corresponding downgoing Alfvén waves at $\phi>90^\circ$ are not registered by these figures. Strictly, there should be no Alfvénic flux at $\phi=0^\circ$ and $180^\circ$ because in those 2D cases the fast and Alfvén waves decouple. However, the finite width of the ball filter in wavevector space, the incomplete separation of the fast and Alfvén waves at $z_f=1.2$ Mm, and other numerical inexactitudes result in weak magnetic fluxes at those orientations. Practical time step constraints preclude us from using a taller box and higher $z_f$, where fast-to-Alfvén conversion is more complete, or applying magnetic fields greater than 1 kG, without invoking some form of Alfvén limiter. Another feature evident in Figure \[fig:Fz\] is increasing acoustic and magnetic flux with increasing $k_h$ in each case. This is due to the velocity power distribution in the stochastic driver, which is uniform in ${{\bmath{k}}}_h$. That leads to RMS power in $p$ and ${{{\bmath{b}}}}$ that increases linearly with wavenumber, and hence to a similar trend in flux. ![Representative examples of vertical wave energy fluxes (full curve: acoustic; dashed curve: magnetic; dotted curve: total) as a function of height. The top frame pertains to $\theta=60^\circ$, $\phi=80^\circ$, $B_0=1$ kG, and $k_h=1$ $\rm Mm^{-1}$ at 5 mHz, for which significant fast-to-Alfvén conversion is expected (see Figure \[fig:Fz\]). The bottom frame is for the corresponding vertical field case $\theta=0^\circ$ (for which $\phi$ is irrelevant). The full vertical lines indicates the location of the $a=c$ equipartition height, and the vertical dashed line shows the position of the stochastic driving plane.[]{data-label="fig:FzVz"}](figures/fz_th60ph80_5mHz_1kG "fig:"){width=".75\hsize"}\ ![Representative examples of vertical wave energy fluxes (full curve: acoustic; dashed curve: magnetic; dotted curve: total) as a function of height. The top frame pertains to $\theta=60^\circ$, $\phi=80^\circ$, $B_0=1$ kG, and $k_h=1$ $\rm Mm^{-1}$ at 5 mHz, for which significant fast-to-Alfvén conversion is expected (see Figure \[fig:Fz\]). The bottom frame is for the corresponding vertical field case $\theta=0^\circ$ (for which $\phi$ is irrelevant). The full vertical lines indicates the location of the $a=c$ equipartition height, and the vertical dashed line shows the position of the stochastic driving plane.[]{data-label="fig:FzVz"}](figures/fz_th0ph80_5mHz_1kG "fig:"){width=".75\hsize"} Figure \[fig:FzVz\] shows how the acoustic, magnetic, and total fluxes vary with height in two representative cases, the first with highly inclined magnetic field and the second with vertical field. Most fluxes have settled quite well to their asymptotic states by about $z_f=1.2$ Mm, indicating that our computational box is (for the most part) tall enough. There is also clearly a transfer of energy flux from acoustic to magnetic in the inclined field case over several hundred kilometres above $a=c$, as expected [@CalHan11aa]. This is absent in the vertical field case, as there the fast and Alfvén waves are decoupled. Another reason for the (moderate) differences between the flux maps presented here and those of @CalGoo08aa is the different way that waves are injected from below. In @CalGoo08aa, care is taken to inject only a pure fast wave at $z=-4$ Mm, where it is overwhelmingly acoustic. Any magnetic wave at the top of the computational box could therefore have only originated from mode transmission/conversion. The 2.5D simulations of @KhoCal11aa [@KhoCal12aa] similarly impose a deep fast wave driver (at $-5$ Mm). In the current SPARC simulations though, a shallow more realistic solar driver is placed at $z=-0.156$ Mm, and this excites both acoustic and magnetic oscillations directly (see Figure \[fig:FzVz\]). Consequently, some portion of the magnetic wave energy fluxes at the top may have travelled directly along the fast and Alfvén dispersion relation loci rather than tunnelling from the fast branch. Nevertheless, the correspondence between the flux maps presented here and in @CalGoo08aa is striking. ![Pure field-aligned acoustic and Alfvén fluxes $F_{\parallel\rm ac}$ (left column) and $F_{\rm Alf}$ (right column) for the case $B_0=1$ kG, $\nu=5$ mHz, $k_h=1$ 0.75, and 0.5 $\rm Mm^{-1}$ (top to bottom).[]{data-label="fig:Fpar"}](\figdir/Fparalf_5mHz_1kG){width="\hsize"} {width=".4\hsize"} {width=".4\hsize"}\ {width=".4\hsize"} {width=".4\hsize"} As the magnetic field inclination $\theta$ increases toward $90^\circ$, the vertical acoustic and magnetic fluxes obviously diminish since asymptotically, as $z_f\to\infty$, both are identically field-aligned. They therefore give a biased view of energy loss rates; even for purely horizontal magnetic field, the parallel acoustic and Alfvén waves take energy away (horizontally) from the sites of mode conversion. Using vector identities, it is easily shown that ${{{\bmath{F}}}}_{\rm mag} = p_{\rm mag}{{\bmath{v}}}+{{{\bmath{b}}}}{{\bmath{\cdot}}}{{\bmath{v}}}\,{{\bmath{B}}}_0/\mu_0$, where $p_{\rm mag}={{\bmath{B}}}_0{{\bmath{\cdot}}}{{{\bmath{b}}}}/\mu_0$. This comprises the magnetic flux contributions to both the magnetoacoustic and Alfvén waves. To select the ‘pure’ Alfvén contribution, we project both ${{{\bmath{b}}}}$ and ${{\bmath{v}}}$ in the asymptotic Alfvén wave polarization direction $ \hat{{{\bmath{E}}}}_{\rm Alf}=(\cos^2\theta\sin\phi,\cos\phi,\sin\theta\cos\theta\sin\phi)/\sqrt{1-\sin^2\theta\sin^2\phi} $ (this is the ${{\hat{\bmath{e}}}}_{\rm perp}$ of @KhoCal11aa rotated into our present frame and normalized) to obtain the components $b_{\rm Alf}$ and $v_{\rm Alf}$, and identify $F_{\rm Alf}=-\langle b_{\rm Alf} v_{\rm Alf} B_0\rangle/\mu_0$. Similarly, $F_{\parallel\rm ac}=\langle p\,{{\bmath{v}}}{{\bmath{\cdot}}}\hat{{\bmath{B}}}_0\rangle$ is the pure field-aligned acoustic flux. Figure \[fig:Fpar\] plots these pure parallel acoustic and Alfvén fluxes against $\phi$ and $\theta$. For horizontal field $\theta=90^\circ$ there is skew-symmetry in Alfvén flux about $\phi=90^\circ$, as there must be, with the Alfvén flux propagating in the positive direction for $\phi<90^\circ$ and the negative direction for $\phi>90^\circ$. As $\theta$ decreases though, the positive flux on $\phi<90^\circ$ is favoured since it follows the field lines sloping upward from the interaction region. On $\phi>90^\circ$ the census height $z_f$ is above the bulk of this interaction region, and so $F_{\rm Alf}$ does not yet exhibit substantial negative values – it will do so as we move lower in the atmosphere. Indeed, if we could reliably measure $F_{\rm Alf}$ on $\phi>90^\circ$ at lower altitudes, these figures would display greater (anti-) symmetry [ (see Section \[sec:BVP\])]{}. The parallel acoustic flux is very similar to the vertical fluxes displayed in the top right panel of Figure \[fig:Fz\], with an additional $\cos\theta$ factor. Travel Time Perturbations {#TD} ------------------------- ‘Travel time’ perturbations $\delta\tau$ (relative to quiet Sun) are displayed in Figure \[fig:tau\], on a $10^\circ\times10^\circ$ resolution grid. They show clear manifestations of the acoustic cutoff at $\theta=30^\circ$– $40^\circ$ for 5 mHz, and $\theta=50^\circ$– $60^\circ$ at 3 mH where greater inclination is needed to overcome it. Below these inclinations, $\delta\tau$ is small, and typically negative, suggesting a weak speed-up due to the magnetic field. At higher inclinations, substantial travel time perturbations are seen at all orientations $\phi$, most cleanly at 5 mHz. With only about $3{\leavevmode\kern.1em\raise.5ex\hbox{\the\scriptfont0 1}\kern-.1em/\kern-.15em\lower.25ex\hbox{\the\scriptfont0 2}}$ skips fitting into the computational box for $k_h=0.5$ $\rm Mm^{-1}$ (see Table \[tab:skip\]), the corresponding $\delta\tau$ measurements become very noisy at 3 mHz (not shown). Even the $k_h=0.75$ results appear unreliable at this lower frequency. The large positive $\delta\tau$ at around $\theta=50^\circ$ in the $B_0=500$ G, $\nu=3$ mHz, $k_h=0.75$ $\rm Mm^{-1}$ case should therefore be regarded with some suspicion; it requires confirmation in wider boxes. There are also clear variations in $\delta\tau$ with $\phi$ at inclinations $\theta$ sufficient for the ramp effect to take hold. In each case, negative travel time perturbations are substantial at small $\sin\phi$ but much reduced at ‘intermediate’ $\phi$, typically around $90^\circ$. Comparison with the flux figures suggests a strong link between both acoustic and magnetic wave energy losses and travel time lags. For magnetic field inclinations $\theta$ sufficient that $\omega>\omega_c\cos\theta$, the atmosphere is opened up to wave penetration and therefore to both types of mode conversion. With $\phi=0^\circ$, the fast magnetically-dominated waves emerging from $a=c$ return to the surface after reflection near $\omega/k_h=a$ with a different (advanced) phase compared to that of the simply reflecting acoustic waves in quiet Sun. However, if $\phi\ne0^\circ$ (or $180^\circ$), the fast wave loses more energy near its apex to the Alfvén wave, and seeming suffers a further phase retardation that we might interpret as partially cancelling the underlying negative travel time perturbation. Using acoustic holography to probe sunspot penumbral oscillations, @SchBraCal05aa also detected directionally dependent phase perturbations, finding that phase shift varies with the angle between the line-of-sight and the magnetic field, with equivalent phase travel time variations with viewing angle of order 30 s. These results seem similar in type and magnitude to those found here, though it is difficult to compare precisely since line-of-sight selects fast and slow waves differently, depending on their velocity polarizations with respect to magnetic field. Their result that phase change is minimal along the line of sight is consistent with our understanding that the atmospheric slow (acoustic) wave that will be preferentially selected by that viewing angle has the same phase as the fast wave incident from below on $a=c$ [there is no phase jump in the transmitted wave; see @TraKauBri03aa Eqn. (22)]. Comparison with BVP Results {#sec:BVP} --------------------------- {width=".8\hsize"} For the case of a uniform magnetic field and a horizontally invariant atmosphere, the wave propagation equations may be reduced to a $4^\mathrm{th}$ (2D) or $6^\mathrm{th}$ (3D) order system of ordinary differential equations, assuming an $\exp[{\rmn{i}}(k_x x+k_y y-\omega t)]$ dependence on horizontal coordinates and time. Boundary conditions may be applied so as to allow the ingress of only a pure fast wave at the bottom of the computational region (a few Mm below the solar surface), with outgoing radiation or evanescent (as appropriate) conditions on all other waves at top and bottom. This defines an ordinary differential Boundary Value Problem (BVP). Details are set out in @CalGoo08aa, @Cal09ab, and @NewCal10aa. Figure \[fig:BVP\] displays the results of such a calculation for one of the cases considered above: $B_0=1$ kG, $\nu=5$ mHz, $k_h=1$ $\rm Mm^{-1}$. Travel time perturbations (top left) are calculated according to the prescription of @Cal09ab. These are computed based on the phase of the reflected fast wave when it exits the box at the bottom, compared to the same for quiet Sun. No full seismic skip is constructed, but this assessment of the phase change through the surface layers amounts to the phase change per skip. It is therefore directly comparable to the time-distance results calculated above based on standard TD methodology, and indeed, the correspondence between the results of the two approaches is striking. Experiment with higher magnetic field strength (not shown) indicates that peak travel time perturbations scale roughly linearly with $B_0$, as expected from our Fig. \[fig:tau\] and from Fig. 7 of @Cal09ab. This is sufficient by $B_0=2$ kG for the phase difference $\delta\varphi$ to fold over the $360^\circ$ ambiguity, so that one would not be sure whether it were advanced or retarded. Since $\delta\tau=-\omega^{-1}\delta\varphi$, travel time perturbations may therefore artificially appear positive. Figure \[fig:BVP\] also displays the acoustic and magnetic fluxes (bottom row) calculated at the top boundary ($z=2$ Mm), though now they are normalized by the injected fast wave flux at the bottom. These are very similar to those obtained above from SPARC simulation, though the peaks of power are shifted a little in $\theta$ and $\phi$, probably because the injected flux in the simulations is not purely fast wave, since the stochastic driver is both shallow and indiscriminate in the type of waves it produces. The greater computational box height for the BVP may also contribute to the difference. It is seen that $F_{\rm ac}$ reaches over 55% and $F_{\rm mag}$ more than 40% in this case, showing that indeed, losses to the upper atmosphere can be substantial. Total fractional loss $\mathcal{L}$ is also shown (top right). This includes both upgoing and downgoing losses from the fast wave, a quantity difficult to estimate in the SPARC simulations. In other words, the remnant reflected fast wave that ultimately rejoins the interior seismic wavefield has been reduced in power by fraction $\mathcal{L}$, which can be as high as 80%. It is notable that, as expected, $\mathcal{L}$ is more nearly symmetric about $\phi=90^\circ$ than are the upward fluxes alone. This accords with the near-symmetry of $\delta\tau$. The smaller total losses around $\phi=90^\circ$ for fixed $\theta\ga30^\circ$ are clearly reflected in $\delta\tau$. On the other hand, the small losses for $\phi\approx0^\circ$ at high $\theta$ run counter to this nexus, presumably because acoustic and magnetic losses affect phase in different ways. In passing, we mention that the BVP travel time perturbation map for the case $B_0=0.5$ kG, $\nu=3$ mHz, $k_h=0.75$ $\rm Mm^{-1}$ (not shown) does not display the strong positive feature seen in the bottom left panel of Figure \[fig:tau\] at mid-values of $\theta$ and $\phi$ (there is a positive peak, but it is less than 2 s). This indicates that we must be careful with very noisy TD phase fits. Longer time series or broader wave vector filters may be required. Discussion and Conclusions ========================== ‘Travel time’ perturbations, as measured by the standard techniques of Time-Distance helioseismology, actually determine phase shifts rather than true (group) travel times. Shifts in phase trivially come about through changes in path and in propagation speed. If Fermat’s principle is valid, then travel time is stationary with respect to variations of path, and so wave speed will be the main culprit. This could consist of perturbations in sound speed (temperature), magnetic field (fast wave speed), and flow (doppler shift). Flow is isolated by contrasting travel times between two points in opposite directions. There is no background flow in our simulations. However, phase shifts are also produced by wave reflection and mode conversion [@Cal09aa; @TraKauBri03aa]. It is a difficult (and inexact) task to theoretically determine shifts due to a single mode conversion process, but our modelled system consists of several occurring sequentially. It is further complicated by the shallow wave generation layer that excites both acoustic and magnetic waves directly, thereby differing from the model of @CalGoo08aa where care was taken to inject only acoustic waves from the bottom of the computational region [ (see Section \[sec:BVP\])]{}. The stochastic driving layer is however more solar-relevant, and so is of more practical relevance. To enumerate, the mode transmissions/conversions consist of: fast-to-slow (acoustic) transmission at $a=c$; the resulting slow wave may then either escape upward or be reflected by the acoustic cutoff, depending on frequency and magnetic field inclination; the converted fast wave continues upward to be reflected near where its horizontal phase speed matches the Alfvén speed; conversion to upgoing and downgoing Alfvén waves occurs near this reflection point, depending on field inclination (with or against the wave direction); the downgoing fast wave again passes through $a=c$ and is split into fast and slow; the downgoing Alfvén wave impacts the surface. Each of these in turn leaves a signature on the phase. Altogether, this is far too complex to treat analytically. We are therefore left to analyse simulations by comparing wave fluxes with ‘travel time’ perturbations. Unfortunately, the plotted fluxes do not reveal the full story. Downgoing acoustic and magnetic losses are not easily determined, and in any case are complicated by the excitation layer. Furthermore, phase shifts are not simple functions of conversion or transmission coefficients. Consequently, we should not expect a simple relationship between wave fluxes at the top of the computational domain and travel time discrepancies calculated near the photosphere. Nevertheless, there is clearly a profound link. Especially at 5 mHz, substantial $\delta\tau$ is associated with large fluxes, at field inclinations $\theta$ sufficient to allow acoustic propagation. The link is perhaps clearer in Figure \[fig:compare\], in which the $\theta=30^\circ$ and $\theta=60^\circ$ fluxes and travel time perturbations with $B_0=1$ kG, $\nu=5$ mHz, $k_h=1$ $\rm Mm^{-1}$ are compared. For $\theta=30^\circ$, acoustic losses dominate, and produce travel time shifts of similar structure in $\phi$. Similarly, at $\theta=60^\circ$, magnetic losses dominate,[^5] and now their structure is mirrored in the $\delta\tau$ plot. Similar behaviour is seen at 3 mHz in Figure \[fig:compare3mHz\]. [ Of course, the results of the Boundary Value Problem (BVP) calculations of Section \[sec:BVP\] are ‘cleaner’ than those from TD, because they are for a monochromatic wave rather than a stochastically excited spectrum of oscillations, and so do not require filtering, nor do they need to be averaged over a finite time series. The TD approach though does presage extension to more realistic atmospheres and field geometries, and importantly is directly applicable to real solar data and not just simulations. The close correspondence between the results of the two methods argues strongly for the viability of directional TD probing of real solar magnetic regions. ]{} [ Modelling of helioseismic waves is usually carried out in the linear regime, since internal oscillation velocities are invariably highly subsonic. However, our extension of the domain of helioseismology into the chromosphere poses the question of whether linearity is still a valid assumption for the extended waves addressed here. For the most part, the velocities of photospheric oscillations associated with individual p-modes are at most only a few tens of $\rm cm\,s^{-1}$ and do not of themselves grow large enough in the upper chromosphere to be nonlinear. However, it is undoubtedly the case that, regarded as a spectrum of modes, atmospheric acoustic (slow) waves driven by the p-modes in sunspots do steepen and shock with height [@BogCarHan03aa], though the relationship between the photospheric and chromospheric power spectra can be understood broadly in terms of linear theory [@BogJud06aa]. However, this is irrelevant to our considerations. Below the (ramp adjusted) acoustic cutoff frequency, slow waves reflect at too low an altitude for this to occur, and at higher frequencies the slow wave is lost to the seismic wavefield whether it propagates forever upward or is thermalized due to nonlinearities; in either case, the change in phase of the fast wave has already occurred at lower altitudes. By mid-chromospheric heights the plasma $\beta$ is already low and the fast wave is essentially the ‘compressional’ Alfvén wave, travelling at (near) the Alfvén speed, which is now very large. We therefore do not expect significant nonlinearities in the fast waves. The 2D simulations of @BogCarHan03aa identify nonlinear steepening in fast waves only in the neighbourhood of a much more powerful driving piston than is relevant to our p-mode scenario.]{} Overall, how are our results to be interpreted? At small magnetic field inclination, insufficient to provoke the ramp effect, both fluxes and travel time perturbations relative to quiet Sun are small, suggesting that seismic waves are largely reflected before reaching heights at which they would become involved in mode conversion. In this respect, they do not behave very differently to the quiet Sun case. However, once the ramp effect kicks in at larger $\theta$, wave paths extend into the atmosphere and both ‘travel times’ (wave phases) and energy losses become substantial. This is so even at $\phi=0^\circ$ and $180^\circ$, where Alfveńic losses (essentially) vanish. Further large variations in $\delta\tau$, now positive, are then correlated with the Alfvénic losses as $\sin\phi$ increases. In summary, the current simulations suggest that fast-to-slow conversion at $a=c$ yields large negative travel time shifts, and that subsequent fast-to-Alfvén conversions produce positive shifts superimposed on and therefore partially cancelling the negative shifts. Importantly, $\delta\tau$ displays a very clear directional dependence. A complication of the current model is that mode conversion is happening at both ends of the skip path, because the atmosphere and magnetic field is horizontally invariant. For sunspot seismology though, normally one end is in the spot and the other in a quiet Sun annular pupil where there is no conversion. This scenario will be explored in a subsequent work. For the moment though, we must realize that the travel time measurements are affected by conversions at both ends of the path, whereas the flux calculations sample only one end. This probably explains why the $\delta\tau$ contour graphs are more symmetric about $90^\circ$ in $\phi$ than are the flux plots, a feature we would not expect to persist in the sunspot/pupil model. Nevertheless, with that caveat in mind, it appears that very significant ‘travel time’ discrepancies of several tens of seconds (depending on field strength, frequency, and wavenumber) are related to phase changes resulting from mode conversion and not true travel time changes (which is not to say that actual travel times have not changed as well due to the routing of fast and slow waves through the atmosphere). This should give pause to helioseismologists attempting to invert TD sunspot data for subsurface structure. ![Comparison of the upward acoustic and magnetic fluxes (left panels) with the travel time perturbations (right panels) against $\phi$ at $\theta=30^\circ$ (top row) and $\theta=60^\circ$ (bottom row), for the case $B_0=1$ kG, $\nu=5$ mHz, $k_h=1$ $\rm Mm^{-1}$, corresponding to the top left panels in each of Figures \[fig:Fz\] and \[fig:tau\]. Acoustic flux: full curve; magnetic flux: dashed curve.[]{data-label="fig:compare"}](figures/compare){width="\hsize"} ![Comparison of the upward acoustic and magnetic fluxes (left panel) with the travel time perturbations (right panel) against $\phi$ at $\theta=70^\circ$, for the case $B_0=1$ kG, $\nu=3$ mHz, $k_h=1$ $\rm Mm^{-1}$. Acoustic flux: full curve; magnetic flux: dashed curve.[]{data-label="fig:compare3mHz"}](figures/compare3mHz){width="\hsize"} This work was supported by an award under the Merit Allocation Scheme on the NCI National Facility at the ANU, as well as by the Multi-modal Australian ScienceS Imaging and Visualisation Environment (MASSIVE). A portion of the computations was also performed on the gSTAR national facility at Swinburne University of Technology. gSTAR is funded by Swinburne and the Australian Government’s Education Investment Fund. [^1]: E-mail: paul.cally@monash.edu [^2]: E-mail: hamed.moradi@monash.edu [^3]: A wedge filter was suggested for use in TD by @Gil00aa to measure either rotation or meridional circulation; see his fig. 4.4. @ChoLiaYan09aa employed a similar idea in the construction of an acoustic power map. [^4]: Travel times are calculated in the standard TD manner [@CouBirKos06aa], after transforming the ball-filtered data back to physical space. [^5]: The small negative values of $F_{\rm mag}$ near $\phi=0^\circ$ and $180^\circ$ are an indication that the top of the computational box is not quite high enough for the fluxes to have attained their asymptotic values.

[ **Irreducibility of the Cayley-Menger determinant,**]{}\ \   \ [Mowaffaq Hajja$^{1}$, Mostafa Hayajneh$^2$, Bach Nguyen$^3$, and Shadi Shaqaqha$^4$]{}\  \ (1): P. O. Box 1, Philadelphia University, 19392, Amman, Jordan\ mowhajja@yahoo.com\   \ (2), (4): Yarmouk University, Irbid, Jordan\ hayaj86@yahoo.com, shadish2@yahoo.com\   \ (3): Louisiana State University, Baton Rouge, LA, USA\ bnguy38@lsu.edu > [**Abstract.**]{} [If $S$ is a given regular $n$-simplex, $n \ge 2$, of edge length $a$, then the distances $a_1$, $\cdots$, $a_{n+1}$ of an arbitrary point in its affine hull to its vertices are related by the fairly known elegant relation $\phi_{n+1} (a,a_1,\cdots,a_{n+1})=0$, where $$\phi = \phi_t (x, x_1,\cdots,x_{n+1}) = \left( x^2+x_1^2+\cdots+x_{n+1}^2\right)^2 > - t\left( x^4+x_1^4+\cdots+x_{n+1}^4\right).$$ The natural question whether this is essentially the only relation is answered positively by M. Hajja, M. Hayajneh, B. Nguyen, and Sh. Shaqaqha in a recently submitted paper entitled [*Distances from the vertices of a regular simplex*]{}. In that paper, the authors made use of the irreducibility of the polynomial $\phi $ in the case when $n \ge 2$, $t=n+1$, $x= a \ne 0$, and $k = {\mathbb{R}}$, but supplied no proof, promising to do so in another paper that is turning out to be this one. It is thus the main aim of this paper to establish that irreducibility. In fact, we treat the irreducibility of $\phi$ without restrictions on $t$, $x$, $a$, and $k$. As a by-product, we obtain new proofs of results pertaining to the irreducibility of the general Cayley-Menger determinant that are more general than those established by C. D’Andrea and M. Sombra in [*Sib. J. Math. [**46**]{}, 71–76.*]{} ]{} [**Keywords:**]{} Cayley-Menger determinant; circumscriptible simplex; discriminant; homogeneous polynomial; irreducible polynomial; isodynamic simplex; isogonic simplex; orthocentric simplex; Pompeiu’s theorem; pre-kites; quadratic polynomial; regular simplex; symmetric polynomial; tetra-isogonic simplex; volume of a simplex Introduction {#111} ============ Let $S=[A_1, \cdots, A_{n+1}]$, $n \ge 2$, be a regular $n$-simplex of edge length $a$, and let $B$ be an arbitrary point in its affine hull. It is fairly well known that the distances $a_1,\cdots,a_{n+1}$ from $B$ to the vertices of $S$ satisfy the elegant relation $$\begin{aligned} \label{f-a} \left(a^2+a_1^2+\cdots+a_{n+1}^2\right)^2 &=& (n+1)\left(a^4+a_1^4+\cdots+a_{n+1}^4\right);\end{aligned}$$ see [@Bentin-1]. The natural question whether this is (essentially) the only relation was posed and answered in the affirmative in [@Shadi]. However, the proof rests heavily on a result whose proof was not included therein, but was postponed for another paper, namely this one. This result states that the polynomial obtained from (\[f-a\]) by thinking of $a_1,\cdots,a_{n+1}$ as indeterminates is irreducible over ${\mathbb{R}}$ when $a \ne 0$ and $n \ge 2$. The main goal of this paper is to prove this, and to actually give a complete treatment of the irreducibility of the more general polynomial $$\begin{aligned} \label{g} g &=& \left(a^2+x_1^2+\cdots+x_{n+1}^2\right)^2 - t\left(a^4+x_1^4+\cdots+x_{n+1}^4\right) \in k[x_1,\cdots,x_{n+1}]\end{aligned}$$ for any field $k$, any $t \in k$, and any $a \in k$ (including the case $a=0$). It turns out that if ${\mbox{char~}}k \ne 2$, then $g$ is irreducible except in the two cases $(n,a,t)=(2,0,2)$ and $(n,a,t)=(2,0,3)$. When $(n,a,t)=(2,0,2)$, $g$ is the Heron polynomial with the well known factorization given in (\[uuu\]). When $(n,a,t)=(2,0,3)$, $g$ is reducible if and only if $k$ contains a primitive third root of unity, in which case $g$ factors as in (\[vvvv\]). If ${\mbox{char~}}k = 2$, then $g$ reduces to the polynomial $(1-t)(a+x_1+\cdots+x_{n+1})^4.$ The cases when $n=0$ and $n=1$ are ignored because they have no geometric significance. Also, the case $n=1$ turned out to be rather lengthy and complex, but it gives rise to interesting number theoretic aspects. We do not include these cases in this paper. The polynomial obtained from $g$ by replacing $n+1$ with $n$ and $t$ with $n$ is essentially the Cayley-Menger determinant of a special $n$-simplex, named a prekite, that was introduced and studied in [@prekites], and calculated in Theorem 4.1 there. If we put $a=0$ in $g$, then the instances $t=0$, $t=n$, $t=n-1$, and $t=n-2$ are very closely related to the Cayley-Menger determinants, with respect to certain parametrizations, of orthocentric, isodynamic, circumscriptible, and tetra-isogonic $n$-simplices; see Theorem 5.2 of [@impurity]. Thus the results of this paper would help give complete factorizations of the Cayley-Menger determinants of the aforementioned five special families of $n$-simplices. This raises the question whether the Cayley-Menger determinant $M$ of a general $n$-simplex is irreducible. It is proved in Theorem \[CM\] that the answer is affirmative except in the single case $n=2$, in which case $M$ is the Heron polynomial given in (\[uuu\]). This result has been obtained earlier in [@Siberian] when $k$ is ${\mathbb{R}}$ or ${\mathbb{C}}$. We should note, however, that the result in [@Siberian] does not imply the results in this paper, and does not give any information about the irreducibility of the Cayley-Menger determinant of any of the five special families mentioned earlier. This is because the Cayley-Menger determinants of these special families are obtained from the general Cayley-Menger determinant $M$ given in (\[CMCM\]) by replacing each $x_{ij}^2$, $1 \le i < j \le n+1$, by quantities like $x_i+x_j$, $x_i x_j$, $(x_i+x_j)^2$, $x_i^2+x_ix_j+x_j^2$; see [@impurity Theorem  5.1, p. 54]. Such substitutions obviously do not preserve irreducibility. Furthermore, our treatment is not confined to the cases when $k = {\mathbb{R}}$ or ${\mathbb{C}}$. The paper is organized as follows. Section \[222\] contains preliminary facts that we shall need, and freely use, in the sequel. These are quite few, and come from the elementary theories of symmetric and of homogeneous polynomials. Section \[333\] contains proofs of the main results, namely Theorems \[DDD\], \[CCDD\], and \[CM\]. These proofs make use of Lemmas \[AAA\], \[BBB\], and \[CCC\]. Preliminaries {#222} ============= In this section, we put together definitions and simple facts pertaining to symmetric and homogeneous polynomials that we shall freely use throughout the paper. For ease of reference, we also include a simple theorem about the factorization of quadratics. Let $A$ be any commutative ring with identity, and let $B=A[x_1,\cdots,x_n]$ be the polynomial ring in the $n$ indeterminates $x_1,\cdots,x_n$. Then the symmetric group ${\mathcal{S}}_n$ acts as permutations on the indices of $x_1,\cdots,x_n$, and hence as $A$-automorphisms on $B$. A polynomial in $B$ is called [*symmetric*]{} if it is invariant under the action of (every element of) ${\mathcal{S}}_n$. It is well known that the ring $T=A[x_1,\cdots,x_n]^{{\mathcal{S}}_n}$ of symmetric polynomials is given by $$\begin{aligned} \label{symsym} T=A[x_1,\cdots,x_n]^{{\mathcal{S}}_n} &=&A[e_1,\cdots,e_n],\end{aligned}$$ where $e_1, \cdots, e_n$ are the so-called [*elementary symmetric polynomials*]{} defined by $$e_j = \sum_{1 \le i_1 < i_2 < \cdots < i_j \le n} x_{i_1} x_{i_2} \cdots x_{i_j};$$ see [@Lang Theorem 6.1, p. 191]. A non-zero polynomial $\Phi$ in $B$ is said to be [*homogeneous*]{} if all of its terms are of the same (total) degree $d$. More precisely, if $$\Phi (\lambda x_1, \cdots, \lambda x_n) = \lambda^d \Phi (x_1, \cdots, x_n)$$ for any variable $\lambda$. It is clear that every non-zero polynomial $\Phi \in B$ of (total) degree $n \ge 0$ can be written uniquely in the form $$\Phi = h^{(0)} + h^{(1)} + \cdots + h^{(n)},$$ where $h^{(j)}$ is either 0 or homogeneous of degree $j$, and $h^{(n)}$ is not zero. The last term $h^{(n)}$ is called the [*leading homogeneous component*]{} of $h$, and will be denoted by $\Phi^*$. Notice that $$\begin{aligned} \deg \Phi^* = \deg \Phi. \label{degh}\end{aligned}$$ If $A$ is an integral domain, then it is easy to see that $$\begin{aligned} (\Phi \Psi)^* = \Phi^* \Psi^*. \label{fgh}\end{aligned}$$ Then it follows from (\[degh\]) and (\[fgh\]) that if $f$ is reducible, then $f^*$ is reducible. It also follows that $$\begin{aligned} \label{hhh} \mbox{every factor of a homogeneous polynomial is homogeneous};\end{aligned}$$ see [@Walker Theorem 10.5, p. 28]. To see that this is not true if $A$ was not an integral domain, consider the factorization $x^2 = (x-2)(x+2)$ in ${\mathbb{Z}}_4[x]$. We end this section by proving a simple property of quadratic polynomials that we shall use later. \[Q\] Let $A$ be an integral domain with ${\mbox{char~}}A \ne 2$, and let $A[x]$ be the polynomial ring over $A$ in the indeterminate $x$. Let $H =ax^2+bx+c \in A[x]$, with $a \ne 0$, and let $\Delta = b^2 - 4 ac$ be its discriminant. Then - If $H$ is reducible, then $\Delta$ is a square (in $A$). - If $\Delta$ is a square, and if $2a$ is a unit, then $H$ is reducible. - If $H$ is a square, then $\Delta = 0$. - If $\Delta = 0$, and $a$ is a square and $2a$ is a unit, then $H$ is a square. [*Proof.*]{} To prove (i), suppose that $H$ is reducible. Then $H=(U x + V)(ux+v)$ for some $U, V, u, v \in A$. Multiplying out and equating coefficients, we obtain $Uu=a$, $Uv+Vu=b$, and $Vv=c$. Therefore $\Delta = b^2 - 4ac = (Uv+Vu)^2 - 4 (Uu)(Vv) =(Uv-Vu)^2$, which is a square, as desired. To prove (ii), suppose that $\Delta = b^2 - 4ac$ is a square, say $\Delta = \delta^2$, and that $a$ is a unit. Letting $\alpha = \left(-b+\delta\right)\left(2a\right)^{-1}$ and $\beta = \left(-b-\delta\right)\left(2a\right)^{-1}$, we can easily see that $H=a(x-\alpha)(x-\beta),$ and hence is reducible, as desired. To prove (iii), suppose that $H$ is a square, say $H=(ux+v)^2$, where $u, v \in A$. Then $a = u^2$, $b=2uv$, and $c=v^2$. Thus $b^2-4ac=0$. To prove (iv), suppose that $2a$ is a unit and that $a$ is a square, say $a=u^2$, and $b^2-4ac=0$. Then it is easy to check that $H=\left(ux+ b (2u)^{-1} \right)^2,$ a square. $\Box$ [ *The assumption that $2a$ is a unit in (ii) and (iv) above cannot be replaced by the weaker assumption that $a$ is a unit. For (ii), consider the example $A = {\mathbb{Z}}[\sqrt{5}]$ and $H=x^2+\sqrt{5} x + 1$. Then $a=\Delta = 1$, but $H$ is irreducible. In fact, if $H=(x+s)(x+t)$, then $st=1$, $s+t=\sqrt{5}$, and $(s-t)^2 = (s+t)^2-4st=1,$ and hence $s-t = \pm 1$. In view of the fact that $s+t=\sqrt{5}$, we obtain $2s=\sqrt{5}\pm 1$, contradicting the fact that $s \in A$. For (iv), let $D = \{ f(x) = c_0+c_1 x+\cdots +c_n x^n\in {\mathbb{Z}}[x] : c_1 \mbox{~is even}\},$ and let $H = x^2 + 2x + 1$. Then $\Delta = 0$, and $H$ is not a square in $D$.*]{} The main results {#333} ================ In this section, we establish, in Theorems \[CCDD\], \[DDD\], and \[CM\], the main irreducibility theorems of this paper. Lemmas \[AAA\], \[BBB\], and \[CCC\] are needed in the proofs. Lemma \[CBA\] disposes of the case when $t=0$ in (\[g\]). \[CBA\] Let $k$ be a field, and let $g=c_0 + c_1x_1^2+\cdots + c_n x_n^2 \in k[x_1,\cdots, x_n]$, where $n \ge 1$ and $c_1,\cdots,c_n \in k \setminus \{0\}$. For $0 \le i \le n$, let $t_i = c_i/c_n$, and let $h=t_0 + t_1x_1^2+\cdots + t_n x_n^2.$ Then $h$ (and hence $g$) is reducible if and only if $-t_j$ is a square for all $j$, $0 \le j \le n-1$ and if - either ${\mbox{char~}}k =2$, in which case $$\begin{aligned} h&=&\left(\sqrt{t_0} + \sqrt{t_1} x_1+ \cdots + \sqrt{t_{n-1}} x_{n-1} + x_{n} \right)^2,\end{aligned}$$ - or ${\mbox{char~}}k \ne 2$ and $n=1$, in which case, $$\begin{aligned} h&=&\left(x_1 - \sqrt{-t_0}\right)\left(x_1 + \sqrt{-t_0}\right), \end{aligned}$$ - or ${\mbox{char~}}k \ne 2$, $n=2$, and $t_0 = 0$, in which case $$\begin{aligned} h&=&\left(x_2 - \sqrt{-t_1}x_1 \right)\left(x_2 + \sqrt{-t_1} x_1\right). \end{aligned}$$ [*Proof.*]{} Suppose that $h$ is reducible. Then any factorization of $h$ must be of the form $$\begin{aligned} \label{Must-1} h &=&(a_0 + a_1 x_1+ \cdots+a_nx_n)(b_0 + b_1 x_1+ \cdots+b_nx_n).\end{aligned}$$ We may also assume that $a_n=b_n=1,$ since $a_n b_n = 1$. Comparing coefficients of $x_nx_i$, $1 \le i \le n-1$, and coefficients of $x_n$, we see that $b_i=-a_i$ for $0 \le i \le n-1$. Therefore (\[Must-1\]) can be rewritten as $$\begin{aligned} \label{Must-2} h &=&(a_0 + a_1 x_1+ \cdots+a_{n-1} x_{n-1}+ x_n)(-a_0 -a_1 x_1- \cdots -a_{n-1} x_{n-1} + x_n).\end{aligned}$$ Comparing coefficients of $x_i^2$, $1 \le i \le n-1$, and the constant terms, we see that $-t_i = a_i^2$ for $0 \le i \le n-1$. Thus $- t_i$ is a square for all $i$, $0 \le i \le n-1.$ Of course, $t_n = 1$. If char ($k$) = 2, then $h$ factors into $$\begin{aligned} h &=&(a_0 + a_1 x_1+ \cdots+a_{n-1} x_{n-1}+ x_n)^2.\end{aligned}$$ If char ($k$) $\ne 2$, and $n > 2$, then comparing the coefficients of $x_1x_2$ (in (\[Must-2\])), we obtain $-2 a_1 a_2 = 0$, contradicting the assumptions that $a_1^2 = -t_1 \ne 0$, $a_2^2 = -t_2 \ne 0$, and $2 \ne 0$. If (char ($k$) $\ne 2$, and) $n = 1$, then $h$ factors into $h = (a_0 + x_1)(-a_0 + x_1)$. If char ($k$) $\ne 2$, and $n = 2$, then comparing the coefficients of $x_1$, we obtain $-2 a_0 a_1 = 0$, and hence $a_0 = 0$, since $a_1^2 = -t_1 \ne 0$ and $2 \ne 0$. Therefore $h$ factors into $h = (a_1 x_1 + x_2)(-a_1 x_1 + x_2).$ This completes the proof. $\Box$ It follows, for example, that the polynomial $x_1^2 + \cdots + x_n^2$ is reducible over the field ${\mathbb{C}}$ of complex numbers if and only if $n=1$ or $n=2$. In particular, $x^2+y^2+z^2$ is irreducible over ${\mathbb{C}}$, a fact that appears as a problem in [@AMATYC]. In view of the lemma above, we may exclude the case $t=0$ in (\[g\]). Also, it is obvious that if ${\mbox{char~}}k = 2$, then $g = (1-t) (a+x_1+\cdots+x_n)^4$. That is why the lemmas and theorems below assume that $t\ne 0$ and ${\mbox{char~}}k \ne 2$. \[AAA\] Let $k$ be a field with ${\mbox{char~}}k \ne 2$, and let $$\begin{aligned} f &=& (x_1^2+\cdots+x_n^2)^2-t(x_1^4+ \cdots +x_n^4) \in k[x_1,\cdots,x_n], \end{aligned}$$ where $t \in k$ and $t \ne 0$, and where $n \ge 3$. Then $f$ has a factor that is symmetric in two of the variables if and only if $n=3$ and $t=2$. In this case, $$\begin{aligned} f &=& (x_1^2+x_2^2+x_3^2)^2-2(x_1^4+ x_2^4+ x_3^4)\nonumber \\&=& (x_1+x_2+x_3)(-x_1+x_2+x_3)(x_1-x_2+x_3)(x_1+x_2-x_3). \label{uuu} \end{aligned}$$ [*Proof.*]{} We shall freely use the fact that factors of a homogeneous polynomial are homogeneous; see (\[hhh\]). Let $$\begin{aligned} &&y = x_{n-1}, ~ z = x_n,~ u = y+z,~v=yz, \label{yzuv}\\ &&S_2=x_1^2+\cdots+x_{n-2}^2,~S_4=x_1^4+\cdots+x_{n-2}^4,\label{S2S4}\\ &&R=k[x_1,\cdots,x_n] = k[x_1,\cdots,x_{n-2},y,z],~ R_0=k[x_1,\cdots,x_{n-2},u,v]. \label{RR}\end{aligned}$$ Clearly, $R$ and $R_0$ are polynomial rings over $k$ in the $n$ respective variables, and $R_0$ is the subring of $R$ consisting of the $(y,z)$-symmetric polynomials, i.e., polynomials that are symmetric in the variables $y$ and $z$; see (\[symsym\]). Using the identities $$\begin{aligned} y^2+z^2&=&u^2-2v,\\ y^4+z^4&=&(y^2+z^2)^2-2v^2~=~(u^2-2v)^2-2v^2, $$ and the definition $$f=(S_2+y^2+z^2)^2-t(S_4+y^4+z^4),$$ we obtain (after few lines of computations, or by Maple) $$\begin{aligned} \label{fSuv} f&=&(4-2t)v^2 -4 ((1-t)u^2+S_2)v +S_2^2 + (1-t) u^4 + 2S_2 u^2 -t S_4.\end{aligned}$$ Letting $F \in R_0$ denote the right hand side of (\[fSuv\]), i.e., $$\begin{aligned} F&=& (4-2t)v^2 -4 ((1-t)u^2+S_2)v +S_2^2 + (1-t) u^4 + 2S_2 u^2 -t S_4, \label{F}\end{aligned}$$ we see that $f$ has a $(y,z)$-symmetric factor if and only if $F$ is reducible in $R_0$. From now on, we will be working in $R_0$. Suppose that $F$ is reducible. If $t=2$, then $F$ simplifies into $$\begin{aligned} F_0&=& -4 (-u^2+S_2)v +S_2^2 - u^4 + 2S_2 u^2 -2 S_4 \\ &=& 4 (u^2-S_2)v - (u^2-S_2)^2 + 2S_2^2 -2 S_4.\end{aligned}$$ Since $F_0$ is linear in $v$, it follows that every factor of $F_0$ is a factor of $(u^2-S_2)$ and of $-(u^2-S_2)^2+2S_2^2 -2 S_4$. But any factor of $u^2-S_2$ must contain $u$, while $2S_2^2 -2 S_4$ does not. Therefore $F_0$ is reducible if and only if $2S_2^2 -2 S_4 = 0$, i.e., if and only if $n=3$, in which case $S_4=x_1^4=(x_1^2)^2=S_2^2$. In this case $F_0$ is as given in (\[uuu\]). If $t \ne 2$, then $F$ is a quadratic in $v$ with coefficients in $k[x_1,\cdots,x_{n-2},u]$. Also, its leading coefficient $(4-2t)$ is a unit. By (i) and (ii) of Lemma \[Q\], it is reducible if and only if its discriminant $\Delta$ in $v$ is a square in $k[x_1,\cdots,x_{n-2},u]$. Using Maple, we find that $$\begin{aligned} \label{D} \Delta &=& 8t((t-1)u^4-2S_2u^2+(2-t)S_4+S_2^2).\end{aligned}$$ If $t=1$, then $\Delta = 8(-2S_2 u^2 + S_4 + S_2^2)$. By (iii) of Lemma \[Q\], this cannot be a square, since its discriminant in $u$ is $8^3S_2(S_4 + S_2^2) \ne 0$. Thus suppose that $$t \ne 1.$$ Then for $\Delta$ to be a square, we must have $$\begin{aligned} \label{DD} \Delta &=& 8t((t-1)u^4-2S_2u^2+(2-t)S_4+S_2^2) ~=~ (\alpha u^2 + \beta u + \gamma)^2, \end{aligned}$$ where $\alpha$, $\beta$, and $\gamma$ belong to $k[x_1,\cdots,x_{n-2}]$. Equating the coefficients of $u^4$ and $u^3$ in (\[DD\]), we see that $\alpha^2 = 8t (t-1) \ne 0$ and $2 \alpha \beta = 0$. Therefore $\beta =0$, and we have $$\begin{aligned} \label{DDDD} \Delta &=& 8t((t-1)u^4-2S_2u^2+(2-t)S_4+S_2^2) ~=~ (\alpha u^2 + \gamma)^2. \end{aligned}$$ Equating coefficients in (\[DDDD\]), we find that $$\alpha^2 = 8t(t-1),~~\gamma^2 = 8t ((2-t)S_4+S_2^2),~~2 \alpha \gamma = 8t (-2S_2).$$ It follows that $(t-1) ((2-t)S_4+S_2^2) = (-S_2)^2.$ This simplifies into $(2-t)((t-1)S_4 - S_2^2) = 0.$ Since $t \ne 2$, it follows that $S_2^2$ and $S_4$ are linearly dependent over $k$. This happens if and only if $n=3$, in which case $S_4=x_1^4=(x_1^2)^2=S_2^2$, and $t-1=1$. This is the case treated earlier. Thus $\Delta$ is not a square, a contradiction. This completes the proof. $\Box$ \[BBB\] Let $k$ be a field with ${\mbox{char~}}k \ne 2$, and let $$\begin{aligned} f &=& (x_1^2+\cdots+x_n^2)^2-t(x_1^4+ \cdots +x_n^4) \in k[x_1,\cdots,x_n], \end{aligned}$$ where $t \in k$ and $t \ne 0$, and where $n \ge 3$. Then $f$ has a linear factor if and only if $n=3$ and $t=2$. In this case, $f$ factors as in (\[uuu\]). [*Proof.*]{} If $(n,t)=(3,2)$, then $f$ factors as in (\[uuu\]), and we are done. So we assume that $(n,t)\ne (3,2)$, that $f$ has a linear factor, say $g=a_1x_1+\cdots+a_nx_n$, and we seek a contradiction. Clearly we may assume that $a_1 \ne 0$. By Lemma \[AAA\], $g$ is not symmetric in the variables $x_2$ and $x_3$. Therefore $a_2 \ne a_3$. If $g_1$ is the polynomial obtained from $g$ by interchanging $x_2$ and $x_3$, then $g_1$ is not an associate of $g$, and it is also a factor of $f$. Therefore $gg_1$ is a $(x_2,x_3)$-symmetric factor of $f$ (of degree 2). This contradicts Lemma \[AAA\]. $\Box$ \[CCC\] Let $k$ be a field with ${\mbox{char~}}k \ne 2$, and let $$\begin{aligned} f &=& (x_1^2+x_2^2+x_3^2)^2-t(x_1^4+x_2^4+x_3^4) \in k[x_1,x_2,x_3], \end{aligned}$$ where $t \in k$ and $t \ne 0$. If $t=2$, then $f$ factors as in (\[uuu\]). If $t \ne 2$, then $f$ is reducible if and only if $t=3$ and $k$ contains a primitive third root $\omega$ of 1. In this case, $$\begin{aligned} f&=&(x_1^2+x_2^2+x_3^2)^2 - 3(x_1^4+x_2^4+x_3^4) \nonumber \\ &=& (-2)(x_1^2+\omega x_2^2 + \omega^2 x_3^2)(x_1^2+\omega^2 x_2^2 + \omega x_3^2). \label{vvvv}\end{aligned}$$ Also, the factors in the right hand sides of (\[vvvv\]) are irreducible. [*Proof.*]{} There is nothing to prove in the case when $t=2$, since this is covered in Lemma \[BBB\]. Thus we assume that $t \ne 2$. Let $x_1, x_2, x_3$ be renamed as $x, y, z$, and suppose that $f$ is reducible and that $g$ is an irreducible factor of $f$. Since $t \ne 2$, Lemma \[BBB\] implies that $g$ is not linear. Thus $g$ is an irreducible quadratic. Thus $$\begin{aligned} g&=&ax^2+by^2+cz^2+\alpha yz + \beta zx + \gamma xy,\end{aligned}$$ where $a, b, c, \alpha, \beta, \gamma$ are in $k$. Also, $$\begin{aligned} f&=& (x^2+y^2+z^2)^2 - t(x^4+y^4+z^4)\\ &=& (1-t)(x^4+y^4+z^4)+2(x^2y^2+y^2z^2+z^2x^2).\end{aligned}$$ Letting $s$ be the permutation $s = (x \mapsto y \mapsto z \mapsto x)$, we see that $f$ is divisible by $g$, $s(g)$, and $s^2 (g)$. Since $\deg (g s(g) s^2(g)) = 6 > \deg f$, and since $g, s(g), s^2(g)$ are irreducible, it follows that two (and hence all) of the polynomials $g$, $s(g)$, and $s^2 (g)$ are associates (i.e., constant multiples of each other). Thus $g = \lambda s(g)$ for some $\lambda \in k$. Since $$\begin{aligned} s(g)&=&ay^2+bz^2+cx^2+\alpha zx + \beta xy + \gamma yz,\end{aligned}$$ it follows that $$b = \lambda a,~ c = \lambda b,~a = \lambda c,~ \beta = \lambda \alpha,~\gamma = \lambda \beta,~ \alpha = \lambda \gamma.$$ Thus $$a = \lambda^3 a,~\alpha = \lambda^3 \alpha.$$ If both $a$ and $\alpha$ are zero, then $a=b=c=\alpha = \beta = \gamma=0$, and $g=0$, a contradiction. Thus either $a \ne 0$ or $\alpha \ne 0$. In both cases $ \lambda^3 = 1$. If $\lambda = 1$, then $a=b=c$ and $\alpha = \beta = \gamma$, and hence $g$ is symmetric, contradicting Lemma \[AAA\]. Thus $\lambda \ne 1$ and $\lambda^3 = 1$. Therefore $\lambda$ is a primitive third root of 1 (and ${\mbox{char~}}k$ cannot be 3). Thus $g$ is of the form $$g = a (x^2+\lambda y^2+ \lambda^2 z^2) + \alpha (yz+\lambda zx+ \lambda^2 xy),$$ where $\lambda$ is a primitive third root of 1. Applying the permutation $y \mapsto z \mapsto y$ to $g$, we obtain another factor $$h = a (x^2+\lambda^2 y^2+ \lambda z^2) + \alpha (yz+\lambda^2 zx+ \lambda xy),$$ of $f$ that is not associate of $g$. Therefore $gh$ divides $f$, and has the same degree as $f$. Therefore $f = c gh$, where $c \in k$. Equating the coefficients of $x^3z$ and $x^2yz$ in the identity $f=cgh$, we obtain $-a\alpha c = 0$ and $2a\alpha c - \alpha^2 c = 0$. Thus $\alpha =0$ and $$g = a (x^2+\lambda y^2+ \lambda^2 z^2),~~ h = a (x^2+\lambda^2 y^2+ \lambda z^2).$$ Therefore $$\begin{aligned} gh &=& a^2 [(x^4+y^4+z^4) - (x^2y^2+y^2z^2+z^2x^2)].\end{aligned}$$ But $$\begin{aligned} f &=& (1-t) (x^4+y^4+z^4) +2(x^2y^2+y^2z^2+z^2x^2).\end{aligned}$$ Therefore it follows from $f = c gh$ that $(1-t)= ca^2$ and $-a^2c = 2$. Hence $1-t= -2$ and $t=3$. Therefore $$\begin{aligned} f&=&(x^2+y^2+z^2)^2-3(x^4+y^4+z^4) \\ &=&(-2)(x^2+\lambda y^2 + \lambda^2 z^2)(x^2+\lambda^2 y^2 + \lambda z^2),\end{aligned}$$ as desired. The two factors in (\[vvvv\]) are irreducible because $f$ has no linear factors, by Lemma \[BBB\]. This completes the proof. $\Box$ \[CCDD\] Let $k$ be a field with ${\mbox{char~}}k \ne 2$, and let $a$ be a non-zero element of $k$. Let $$\begin{aligned} g &=& (a^2 + x_1^2+\cdots+x_n^2)^2-t(a^4 +x_1^4+ \cdots +x_n^4) \in k[x_1,\cdots,x_n], \end{aligned}$$ where $t \in k$ and $t \ne 0$, and where $n \ge 3$. Then $g$ is irreducible. [*Proof.*]{} We start with the case $n=3$, and we rename $x_1$, $x_2$, $x_3$ as $x$, $y$, $z$. Thus we are to prove that the polynomial $$g = (a^2 + x^2 + y^2 + z^2)^2 - t(a^4 + x^4 + y^4 + z^4),~~a\ne 0, ~t \ne 0, ~{\mbox{char~}}k \ne 2,$$ is irreducible. The general case will follow easily as shown later. Suppose that $g$ is reducible, and that $g = \alpha \beta$, where $\alpha$ and $\beta$ are non-constant polynomials in $k[x,y,z]$. Let $G$, $A$, and $B$ be the leading homogeneous components of $g$, $\alpha$, and $\beta$, respectively. Then $$G=(x^2+y^2+z^2)^2-t(x^4+y^4+z^4),$$ and $G = AB$, and $A$ and $B$ are non-constant. Therefore $G$ is reducible. By Lemma \[CCC\], we have the following two cases: $$\begin{aligned} G&=&(x^2+y^2+z^2)^2-2(x^4+y^4+z^4)\nonumber \\ &=&(x+y+z)(-x+y+z)(x-y+z)(x+y-z), \label{GUV}\end{aligned}$$ and hence $$\begin{aligned} g&=&(a^2+x^2+y^2+z^2)^2-2(a^4+x^4+y^4+z^4).\end{aligned}$$ Suppose that one of the factors $\alpha$ or $\beta$, say $\alpha$, is linear. Then we may assume that either $\alpha=x+y+z+c$ or $\alpha=x+y-z+c$, where $c \in k$. In the first case, we plug $(x,y,z)=(x,-x,-c)$ in $g = \alpha \beta$, and in the second case, we plug $(x,y,z)=(x,-x,c)$. In both cases, we obtain $$\begin{aligned} 0&=&(a^2+2x^2+c^2)^2-2(a^4+2x^4+c^4) \\&=& 4x^2(a^2+c^2) +(-a^4-c^4+2a^2c^2)\\ &=& 4x^2(a^2+c^2)-(a^2-c^2)^2.\end{aligned}$$ Therefore $a^2+c^2=a^2-c^2=0$, and hence $a=c=0$, contradicting the assumption that $a \ne 0$. If $\alpha$ and $\beta$ are quadratic irreducible, then $A$ and $B$ are quadratic. Also $G=AB$ by (\[fgh\]). By (\[GUV\]), we may assume that $A$ or $B$, say $A$, is $(x+y+z)(-x+y+z)$. Thus $$\begin{aligned} \alpha&=&(x+y+z)(-x+y+z)+L,\end{aligned}$$ where $L$ is a linear polynomial. By applying the permutation $\sigma : x \mapsto y \mapsto x$ and then $\tau : x \mapsto z \mapsto x$, we see that the two (irreducible) polynomials $$\begin{aligned} \alpha_1=(x+y+z)(x-y+z)+\sigma (L) \mbox{~~and~~} \alpha_2=(x+y+z)(x+y-z)+ \tau (L)\end{aligned}$$ are also factors of $g$. Since the coefficients of $x^2$ in the polynomials $\alpha_1$ and $\alpha_2$ are the same (and equal 1), and since they are not equal, it follows that $\alpha_1$ and $\alpha_2$ are not associates. By considering the coefficients of $y^2$ and $z^2$, we see that no two of the polynomials $\alpha$, $\alpha_1$ and $\alpha_2$ are associates. Therefore $\alpha \alpha_1 \alpha_2$ divides $g$, a contradiction since $\deg g = 4 < 6$. Therefore $g$ cannot be reducible. $$\begin{aligned} G&=&(x^2+y^2+z^2)^2-3(x^4+y^4+z^4)\\ &=&(-2)(x^2+\omega y^2+ \omega^2 z^2)(x^2+\omega^2 y^2+ \omega z^2),\end{aligned}$$ where $\omega \in k$ is a primitive third root of 1, and where the quadratics on the right hand side are irreducible. Therefore $A$ and $B$ are the polynomials $$u(x^2+\omega y^2+\omega^2 z^2) \mbox{~and~} v(x^2+\omega^2 y^2+\omega z^2),$$ where $$u, v \in k,~uv=-2.$$ Hence $\alpha$ and $\beta$ are the polynomials $$u(x^2+\omega y^2+\omega^2 z^2 +L) \mbox{~and~} v(x^2+\omega^2 y^2+\omega z^2 +K),$$ where $L$ and $K$ are linear polynomials. Also, $$\begin{aligned} g&=&(a^2+x^2+y^2+z^2)^2-3(a^4+x^4+y^4+z^4) \nonumber\\ &=&(-2)(x^2+\omega y^2+ \omega^2 z^2+L)(x^2+\omega^2 y^2+ \omega z^2+K). \label{Kh1}\end{aligned}$$ Let $L_0$ and $K_0$ be the constant terms of $L$ and $K$, respectively. Plugging $x=y=z=0$ in (\[Kh1\]), we obtain $-2 L_0 K_0 = -2 a^4$, and hence $$\begin{aligned} L_0 K_0 &\ne& 0. \label{Kh2}\end{aligned}$$ Applying the permutation $\sigma : x \mapsto y \mapsto x$, we obtain $$\begin{aligned} g&=&(a^2+x^2+y^2+z^2)^2-3(a^4+x^4+y^4+z^4)\\ &=&(-2)(y^2+\omega x^2+ \omega^2 z^2+ \sigma (L))(y^2+\omega^2 x^2+ \omega z^2+\sigma (K)),\\ &=&(-2)(x^2+\omega^2 y^2+ \omega z^2+ \omega^2 \sigma (L))(x^2+\omega y^2+ \omega^2 z^2+\omega \sigma (K)).\end{aligned}$$ Since $k[x,y,z]$ is a unique factorization domain, it follows that $$\begin{aligned} \sigma (L) = \omega K,~ \sigma (K) = \omega^2 L. \label{Kh3}\end{aligned}$$ Similarly, if $\tau$ is the transposition $x \mapsto z \mapsto x$, then $$\begin{aligned} \tau (L) = \omega^2 K,~ \tau (K) = \omega L. \label{Kh4}\end{aligned}$$ Observing that the constant terms of $L$ and $K$ are unchanged under permutations on $x, y, z$, and using (\[Kh3\]) and (\[Kh4\]), we obtain $$\begin{aligned} L_0 = \omega K_0,~L_0 = \omega^2 K_0.\end{aligned}$$ It follows that $L_0 = K_0 =0$, contradicting (\[Kh2\]). Thus we have proved the case $n=3$. If $n \ge 3$, we let $h$ be obtained from $g$ by putting $x_j = 0$ for all $j \ge 4$. Since $h$ is irreducible by the case $n=3$, it follows that $g$ is irreducible. In fact, if $g=\alpha \beta$, and if $A$ and $B$ are obtained from $\alpha$ and $\beta$ by plugging $x_j = 0$ for all $j \ge 4$, then either $A$ is zero or $A$ is homogeneous of the same degree as $\alpha$. Since $h=AB$, and $h \ne 0$, it follows that $A\ne 0$ and therefore $h$ is reducible, a contradiction. This completes the proof. $\Box$ \[DDD\] Let $k$ be a field with ${\mbox{char~}}k \ne 2$, and let $$\begin{aligned} f &=& (x_1^2+\cdots+x_n^2)^2-t(x_1^4+ \cdots +x_n^4) \in k[x_1,\cdots,x_n], \end{aligned}$$ where $t \in k$ and $t \ne 0$, and where $n \ge 3$. Then $f$ is reducible in and only in the following two cases: - $n=3$ and $t=2$, in which case $f$ is as given in (\[uuu\]). - $n=3$, $t=3$, and $k$ contains a primitive third root $\omega$ of unity, in which case $f$ is as given in (\[vvvv\]). [*Proof.*]{} The case when $n=3$ was completely treated in Lemmas \[BBB\] and \[CCC\]. So we assume that $n \ge 4$. For any polynomial $F \in k[x_1,\cdots,x_n]$, let $F^* \in k[x_1,x_2,x_3]$ be the polynomial obtained from $F$ by putting $x_4=1$ and $x_j = 0$ for all $j \ge 5$. Then $f^*$ is the case $n=3$ and $a=1$ of Theorem \[CCDD\], and is hence irreducible. To show that $f$ is irreducible, we suppose that $f = gh$, and we show that $g$ or $h$ is a constant. Since $f^*=g^*h^*$, and since $f^*$ is irreducible, it follows that $g^*$ or $h^*$, say $g^*$, is a constant. Since $f^* \ne 0$, it follows that $g^*$ is a non-zero constant. Therefore $$4 = \deg (f^*) = \deg (g^*) + \deg (h^*)= 0 + \deg (h^*).$$ Hence $\deg (h^*) = 4$. Since $\deg (h) \ge \deg (h^*)$, it follows that $\deg (h) = 4$. Therefore $4 = \deg (f) = \deg (g) + \deg (h) = \deg (g) + 4$, and hence $g$ is a constant. Thus $f$ is irreducible, as desired. $\Box$ \[CM\] Let $k$ be a field with ${\mbox{char~}}k \ne 2$, and let $R=k[x_{ij} : 1 \le i < j \le n+1]$, $n \ge 2$, be the polynomial ring over $k$ in the set $X=\{x_{ij} : 1 \le i < j \le n+1\}$ of $(n+1)n/2$ indeterminates. For $1 \le i \le j \le n+1$, let us make the convention that $x_{i,j} = x_{j,i}$ and $x_{j,j} = 0$. Let $M$ be the Cayley-Menger determinant in $X$, i.e., $M$ is the $(n+2)\times (n+2)$ determinant whose entries $c_{i,j}$, $0 \le i, j \le n+1$, are given by $$\begin{aligned} c_{i,j} &=& \left\{ \begin{array}{lcl} 0 &\mbox{if}& i=j,\\ 1 &\mbox{if}& i \ne j \mbox{~and~} ij=0,\\ x_{i,j}^2 & \mbox{otherwise.} \end{array}\right. \end{aligned}$$ Thus $$\begin{aligned} \label{CMCM} M&=& \left| \begin{array}{cccccc} 0&1&1&1& \cdots\cdots & 1\\ 1&0&x_{1,2}^2&x_{1,3}^2& \cdots\cdots&x_{1,n+1}^2\\ 1&x_{2,1}^2&0&x_{2,3}^2& \cdots\cdots&x_{2,n+1}^2\\ 1&x_{3,1}^2&x_{3,2}^2& 0& \cdots\cdots&x_{3,n+1}^2\\ \cdots \cdots & \cdots \cdots & \cdots \cdots & \cdots \cdots & \cdots \cdots & \cdots \cdots \\ \cdots \cdots & \cdots \cdots & \cdots \cdots & \cdots \cdots & \cdots \cdots & \cdots \cdots \\ \cdots \cdots & \cdots \cdots & \cdots \cdots & \cdots \cdots & \cdots \cdots & \cdots \cdots \\ 1&x_{n+1,1}^2&x_{n+1,2}^2& x_{n+1,3}^2& \cdots \cdots & 0 \end{array} \right|.\end{aligned}$$ Then 1. $M$ is homogeneous of homogeneity degree $2n$, 2. $M$ is reducible if and only if $n=2$. In this case, if we let $$x_{1,2} = z,~x_{2,3} = x,~x_{1,3}=y,$$ then $$\begin{aligned} M&=& \left| \begin{array}{cccccc} 0&1&1&1\\ 1&0&z^2&y^2\\ 1&z^2&0&x^2\\ 1&y^2&x^2& 0 \end{array} \right| \nonumber \\ &=& -(x + y + z) (-x+y+z) (x - y + z) (x+y-z). \label{Heron-M}\end{aligned}$$ [*Proof.*]{} (i) Let $M_1$ be the matrix obtained from $M$ by replacing each $x_{i,j}$ in $M$ by $\lambda x_{i,j}$, let $M_2$ be obtained from $M_1$ by pulling out $\lambda^2$ as a common factor in each row of $M_1$ except the upper most one, and let $M_3$ be obtained from $M_2$ by pulling out $1/\lambda^2$ as a common factor in the left most column. Then $M_3=M$, and therefore $$M_1 = \lambda^{2(n+1)} M_2 = \lambda^{2(n+1)-2} M_3 = \lambda^{2n} M.$$ Hence $M$ is either 0 or homogeneous of homogeneity degree $2n$. To see that $M \ne 0$, it is easy to see that, and it follows from Lemma 3.3 of [@impurity], the determinant obtained from $M$ by plugging $x_{i,j}=1$ for every $x_{i,j}$ is $(-1)^{n-1} (n+1) \ne 0$. This completes the proof of (i). \(ii) The case $n=2$, i.e., the identity (\[Heron-M\]), is easy to check. So we assume that $$n\ge 3.$$ Let $V_0$, $V_1$, and $V$ be the sets of indeterminates defined by $$\begin{aligned} V_1 = \{ x_{i,j} : 1 \le i < j \le n\},~V_0 = \{ x_{n+1,j} : 1 \le j \le n\},~ V = V_0 \cup V_1, \label{VV0V1}\end{aligned}$$ and let $R_0$ and $R$ be the polynomial rings $$\begin{aligned} R_0 = k[V_0],~R = k[V]. \label{R0R}\end{aligned}$$ For any non-zero $\phi \in R$, let the total degree of $\phi$ be denoted by $\deg (\phi)$, and let the degree of $\phi$ in the indeterminates $V_0$ be denoted by $\deg_0 (\phi)$. Since the indeterminates $V_0$ appear only in the last row and last column of $M$, it follows that $$\begin{aligned} \deg_0 (M) &\le& 4. \label{delta}\end{aligned}$$ Now suppose that $M$ is reducible, say $$\begin{aligned} M &=& fg, \label{Mfg}\end{aligned}$$ where $f$ and $g$ are homogeneous polynomials in $R$ of degrees $t, s \ge 1$. We are to arrive at a contradiction. For any $\phi \in R$, define $\phi^* \in R_0 [x]$ to be the polynomial obtained from $\phi$ by replacing every $x_{i,j}$ and $x_{j,i}$, $1 \le i < j \le n$, by $x$. Then $$\begin{aligned} M^* &=& f^*g^*. \label{Mfgstar}\end{aligned}$$ By [@prekites Theorem 4.1], $M^*$ is the Cayley-Menger determinant of the $n$-pre-kite $$PK[n;x;x_{n+1,1},\cdots,x_{n+1,n}]$$ and is given by $$\begin{aligned} \label{Mstar=H} M^*&=& (-x^2)^{n-2} H, \mbox{~where~} \nonumber\\ H &=& n(x^4+x_{n+1,1}^4+\cdots+x_{n+1,n}^4) -(x^2+x_{n+1,1}^2+\cdots+x_{n+1,n}^2)^2.\end{aligned}$$ By Theorem \[DDD\], $H$ is irreducible. Since $M^* = f^* g^*$, it follows that $H$ divides one of the polynomials $f^*$ and $g^*$, say $g^*$. Therefore $$\begin{aligned} \deg_0 (g) = \deg_0 (g^*) \ge \deg_0 (H) = 4,\end{aligned}$$ and hence $$\begin{aligned} 4 \ge \deg_0 (M) = \deg_0 (f) + \deg_0 (g) \ge \deg_0 (f) + 4.\end{aligned}$$ Therefore $\deg_0 (f) = 0$, and hence $f$ does not contain any of the variables $x_{i,n+1}$, $1 \le i \le n$. Thus $f$ must contain at least one of the other variables, say $x_{1,2}$. Let $F$ be obtained from $f$ by replacing $x_{1,2}$ by $x_{1,n+1}$. Since $M$ is symmetric, and since $f$ divides $M$, it follows that $F$ divides $M$, and therefore $M=FG$, for some $G \in R$. Thus $M^*=F^*G^*=(-x^2)^{n-2}H$. We now show that this is a contradiction. Since $H$ is irreducible, it follows that either $H$ divides $F^*$ or $G^*$. The first case is impossible since $F^*$ does not contain $x_{2,n+1}$. The second case implies that $F^*$ divides $(-x^2)^{n-2}$, which is again impossible because $F^*$ contains $x_{1,n+1}$. This completes the proof. $\Box$ Remarks {#555} ======= In the mathematical literature, there have been some polynomials which have attracted special attention, either for the elegance, or for their popping up in diverse, seemingly unrelated, contexts. The polynomial $x^3+y^3+z^3-3xyz$ is the favorite polynomial alluded to in the title of [@favorite], and it is the subject of Remark 4 (p. 193) of [@EM-ec]. Remark 5 (p. 194) of [@EM-ec] is devoted to the polynomial $x^3-(a^2-b^2-c^2)x+2abc$. The Newton polynomial $x^3-(a^2+b^2+c^2)x+2abc$, too, has been a source of fascination, and has appeared in [@OMT], and is the subject of [@NewtonDE]. The polynomials $f$ and $g$ defined in (\[f-a\]) and (\[g\]) above also have their shares. The special case $$\begin{aligned} h=(a^2+x^2+y^2+z^2)^2-3(a^4+x^4+y^4+z^4)\end{aligned}$$ which describes the relation among the distances of the vertices of an equilateral triangle of side length $a$ to an arbitrary point in its plane is the tool that solves a problem that has appeared very frequently. The problem challenges readers to find $a$ given $x, y, z$ (or to find $z$ given $a$, $x$, $y$). The three known numbers are usually given to be $3,4,5$, and the fourth unknown number is found to be some irrational number that is not very pleasant, namely $\sqrt{25+12\sqrt{3}}$; see [@500 Problem 327, p. 162)], [@Wagon Chapter 1, Problem 27, p. 8; solution, p. 83–84]. A similar problem is solved in [@Graham §55, p. 34], where the given numbers are $80$, $100$, and $150$, and where the answer is again not pleasant. However, the problem appears in [@CMJ], where the given numbers are 5, 7, and $8$, and where the fourth number is found to be 3. One wonders whether there are integral quadruplets other than $(3, 5, 7, 8)$. This naturally leads to the Diophantine equation $$\begin{aligned} \label{Sastry} (w^2+x^2+y^2+z^2)^2-3(w^4+x^4+y^4+z^4)&=&0,\end{aligned}$$ which is studied in [@Sastry], where many (but not all) of its integer solutions are found. It would be an interesting challenge to try to find [*all*]{} the integer solutions of (\[Sastry\]), and even of the more general (\[g\]). [99]{} S. Abu-Saymeh and M. Hajja, Equicevian points on the altitude of a triangle, [*Elem. Math.*]{} [**67**]{} (2012), 187–195. S. Abu-Saymeh, M. Hajja, and M. Hayajneh, The open mouth theorem, or the scissors lemma, for orthocentric tetrahedra, [*J. Geom.*]{} [**103**]{} (2012), 1–16. E. J. Barbeau, M. S. Klamkin, and W. O. J. Moser, [*Five Hundred Mathematical Challenges*]{}, Spectrum Series, MAA, Washington, D. C., 1995. J. Bentin, Regular simplicial distances, [*Math. Gaz.*]{} [**79**]{} (1995), 106. C. D’Andrea and M. Sombra, The Cayley-Menger determinant is irreducible for $n \ge 4$, Sib. J. Math. [**46**]{}, 71–76. L. A. Graham, [*Ingenious Mathematical Problems and Methods*]{}, Dover, N. Y. 1959. M. Hajja and M. Hayajneh, Impurity of the corner angles in certain special families of simplices, [*J. Geom.*]{} **105** (2016), 539–560. M. Hajja, M. Hayajneh, and I. Hammoudeh, Pre-kites: Simplices having a regular facet, preprint. M. Hajja, M. Hayajneh, B. Nguyen, and Sh. Shaqaqha, Distances from the vertices of a regular simplex, preprint. M. Hajja and J. Sondow, Newton’s quadrilateral and his equation $x^3-(a^2+b^2+c^2)x + 2abc$, preprint. J. D. E. Konhauser, D. Velleman, and S. Wagon, [*Which Way Did the Bicycle Go? ... and Other Intriguing Mathematical Mysteries*]{}, The Dolciani Math. Expositions, No. 18, MAA, Washington, D. C., 1996. S. Lang, [*Algebra*]{}, Third Edition, Addison-Wesley Publishing Company, Reading, Massachusetts, 1993. D. MacHale, My favourite polynomial, [*Math. Gaz.*]{}, [**75**]{} (1991), 157–165. S. Rabinowitz, Problem N-3, [*AMATYC Review*]{} [**9**]{} (Spring 1988), 71. K. R. S. Sastry, Natural number solutions to $3(p^4+q^4+r^4+s^4)=(p^2+q^2+r^2+s^2)^2$, [*Mathematics and Computer Education*]{}, Winter 2000; 34, 1. N. Schaumberger, Problem 187, [*College Math. J.*]{} [**12**]{} (1981), 155; solution by Howard Eves, ibid, [**13**]{} (1982), 276–283. R. J. Walker, [*Algebraic Curves*]{}, Springer-Verlag, New York, 1978.

--- abstract: 'We consider the hydrodynamic theory of an active fluid of self-propelled particles with nematic aligning interactions. This class of materials has polar symmetry at the microscopic level, but forms macrostates of nematic symmetry. We highlight three key features of the dynamics. First, as in polar active fluids, the control parameter for the order-disorder transition, namely the density, is dynamically convected by active currents, resulting in a generic, model independent dynamical self-regulation that destabilizes the uniform nematic state near the mean-field transition. Secondly, curvature driven currents render the system unstable deep in the nematic state, as found previously. Finally, and unique to self-propelled nematics, nematic order induces local polar order that in turn leads to the growth of density fluctuations. We propose this as a possible mechanism for the smectic order of polar clusters seen in numerical simulations.' author: - 'Aparna Baskaran$^{1}$ and M. Cristina Marchetti$^{2}$' bibliography: - 'Baskaran-MCM-EPJE-2012.bib' title: 'Self-regulation in Self-Propelled Nematic Fluids' --- Introduction {#sec:intro} ============ Active materials are soft materials driven out of equilibrium by energy input at the microscale. This liberates the fluctuations from the constraints of equilibrium such as fluctuation-dissipation relations and reciprocity. As a consequence, several exotic emergent behaviors result, such as long range order in 2D [@Toner1995; @Toner1998], anomalous fluctuations [@Ramaswamy2003; @Narayan2007], dynamical structures and patterns [@Koch1994; @Budrene1991; @Budrene1995; @Matsushita1997]. In addition to serving as prototypical systems to explore emergent dynamical behavior, active materials also form the physical scaffold of biological systems in that active matter, when coupled to regulatory signaling pathways, provides a model for a variety of living systems, such as bacterial biofilms or the cytoskeleton of a cell. Active particles are generally elongated and form orientationally ordered states [@Ramaswamy2010]. The nature of the ordered state depends on both the symmetry of the individual particles and the symmetry of the aligning interactions (see Fig. \[Fig1\]). Physical realizations of *polar* active particles (characterized by a head and a tail) include bacteria, asymmetric vibrated granular rods, and polarized migrating cells. Polar active entities are often modeled as self-propelled (SP) particles, where the activity is incorporated via a self-propulsion velocity of the individual entities. *Apolar* (head-tail symmetric) active particles, often referred to as “shakers”, have also been considered in the literature. Realizations are symmetric vibrated rods [@Narayan2007]. It has also been argued that melanocytes, the cell that distribute pigments in our skins, may effectively behave as “shakers” [@Gruler1999; @Kemkemer2000]. The nature of the interaction is of course crucial in controlling the properties of the ordered state. Apolar active particles generally experience apolar interactions and the resulting ordered state has the symmetry of equilibrium nematic liquid crystals. The broken orientational symmetry identifies a direction ${\bf \hat{n}}$, but the ordered state is invariant for ${\bf \hat{n}}\rightarrow -{\bf \hat{n}}$. The properties of these *active nematic* fluids have been studied by several authors [@Simha2002; @Ramaswamy2003]. SP and polar particles may experience either polar interactions, i.e., ones that tend to align particles head to head and tail to tail, or interactions that are apolar, i.e, align particles regardless of their polarity. A well studied example of polar particles with polar interactions is provided by Vicsek-type models [@Vicsek1995; @Gregoire2004]. This class of active systems can order in polar states, characterized by a nonzero vector order parameter and mean motion, and will be referred to as *active polar* fluids. ![ Top - active particles of various microscopic symmetry: (a) Polar active particles with head/tail asymmetry resulting in polar interactions, as studied in [@Vicsek1995; @Gregoire2004], (b) Apolar active particles, as studied in [@Simha2002; @Ramaswamy2003], (c) Sels-propelled particles, with physical head-tail symmetry, resulting in apolar interactions. Bottom - ordered macroscopic states of active particles: polar active fluid (left) formed by polar particles (a) with polar interactions; active nematic fluid (center) formed by apolar particles (b) with apolar interactions; self-propelled active nematic fluid (right) formed by self-propelled particles (c) with apolar interactions.[]{data-label="Fig1"}](Fig0-v2.pdf){width="8cm"} One can envisage a third class of active fluids that consists of SP particles whose emergent macrostates are nematic in symmetry. A realization of this is self-propelled particles with physical interactions, such as steric repulsion or hydrodynamic interactions among swimmers in a suspension. It has been shown that these interactions lead to nematic, rather than polar order [@Baskaran2008a; @Baskaran2009]. What is key is the fact that interactions such as steric collisions or hydrodynamic couplings individually conserve momentum and hence cannot lead to the development of a macroscopic momentum for the system. The ordered state of these systems has nematic symmetry, but as we will see below its properties are distinct from those of active nematics composed of apolar particles. Here we refer to this third class of active systems as *self-propelled nematic* fluids. Further, it has recently become apparent that models of polar particles with apolar interactions may be relevant to a number of physical systems, including gliding myxobacteria [@Peruani2012], suspensions of self-catalythic Janus colloids [@Palacci2010] and motile epithelial cell sheets such as those studied in wound healing assays [@Saez2007; @Petitjean2010]. Self-propelled nematics therefore represent an important new class of active systems of direct experimental relevance. A useful theoretical framework for describing the collective behavior of active systems is a continuum model that generalizes liquid crystal hydrodynamics to include new terms induced by activity  [@Simha2002; @Kruse2004; @Voituriez2005; @Voituriez2006; @Marenduzzo2007; @Giomi2008; @Cates2008; @Sokolov2009; @Giomi2010; @Saintillan2010]. In this paper, we consider a minimal phenomenological continuum model of self-propelled nematics based on the equations derived by us from a microscopic model of SP hard rods. We show that in these systems it is important to explicitly retain the dynamics of both the collective velocity or polarization field of the particles and of the nematic order parameter to unfold the mechanisms at play in the formation of emergent structures. Further, we show that *all* active nematics (both consisting of shakers and SP particles) exhibit the phenomenon of dynamical self regulation, due to the fact that the parameter controlling the order-disorder transition, namely the density $\rho $ of active particles, is not externally tuned, as in systems undergoing equilibrium phase transitions, but it is dynamically coupled to the order parameter. This coupling is analogue to the one present in polar fluids [@Toner1995; @Toner1998; @Toner2005] and is a generic mechanism for emergent structure in all active systems, as demonstrated in our recent work [@Gopinath2012]. The layout of the paper is as follows. First, we construct the hydrodynamic description of a self-propelled nematic using symmetry considerations highlighting the key features that distinguish the dynamics of the system from that of an active nematic. Then we examine the linear stability of the homogeneous nematic state. We show that there exists three dynamical mechanisms responsible for emergent structures in active fluids with nematic symmetry. The first is a model-independent instability that occurs in the vicinity of the mean field order disorder transition due to the coupling between order parameter and mass transport which renders the dynamics of the system self regulating. We argue that this instability is the basis for the emergence of bands and phase separation found ubiquitously active systems  [@Chate2006; @Yang2010; @Ginelli2010]. The second is the well known instability of director fluctuations that arises from nonequilibrium curvature-induced fluxes and is closely related to the giant number fluctuations observed in these systems [@Simha2002; @Ramaswamy2003]. These two instabilities are common to both active nematics and self-propelled nematics, i.e., occur regardless of the symmetry of the microdynamics. Finally, we show that there exists a third instability unique to self-propelled nematic fluids due to the fact that in these systems, large scale nematic order can induce local polar order, which in turn destabilizes the density. This mechanism may be responsible for the smectic order of polar clusters observed recently in simulations of SP rods [@Yang2010; @Sam2012]. We conclude with a brief discussion. The Macroscopic Theory {#sec:macro} ====================== The hydrodynamic equations of a self-propelled nematic have been derived from systematic coarse-graining of specific microscopic models [@Baskaran2008a; @Baskaran2009]. Here we introduce these equations phenomenologically, with the goal of examining the dynamics without the limitations imposed by the specific parameter values obtained from a microscopic model or resulting from the choice of the closure used in the kinetic equation. We limit ourselves to overdamped systems in two dimensions. The hydrodynamic equations are then written in terms of three continuum fields: the conserved number density $\rho \left( \mathbf{r},t\right) $ of active units, the polarization density $\bm\tau({\bf r},t)=\rho({\bf r},t) \mathbf{P}\left( \mathbf{r},t\right) $, with $\mathbf{P}\left( \mathbf{r},t\right)$ a polarization order parameter, and the nematic alignment density tensor $Q_{ij}({\bf r},t)=\rho({\bf r},t)S_{ij}\left( \mathbf{r},t\right) $. The polarization ${\bf P}$ is directly proportional to the collective velocity of the active particles, while $S_{ij}$ is the conventional nematic order parameter tensor familiar from liquid crystal physics. For a uniaxial system in two dimensions, $Q_{ij}$ is a symmetric traceless tensor with only two independent components and can be written in terms of a unit vector ${\bf \hat{n}}$ as $Q_{ij}=Q(\hat{n}_i\hat{n}_j-\frac12\delta_{ij})$, where $Q=\rho S$; $S$ is the magnitude of the order parameter and the director ${\bf \hat{n}}$ identifies the direction of spontaneously broken symmetry in the nematic state. For simplicity most of the discussion below refers to the case where the active particles are modeled as long thin rods with repulsive interactions. Active Nematic Hydrodynamics {#subsec:active_nematic} ---------------------------- We first construct the dynamical equations of the system including single-particle convection terms induced by self propulsion, but assuming that self-propulsion does not modify the interaction between two rods. The hydrodynamic equations then take the form [@Baskaran2008] $$\begin{gathered} \partial _{t}\rho +v_{0}\nabla \cdot \bm{\tau }=D\nabla ^{2}\rho \label{1.1}\\ \partial _{t}\bm{\tau }+D_r\bm{\tau }+v_{0}\nabla \cdot \mathbf{Q}+ \frac{v_{0}}{2}\nabla \rho =D_\tau\nabla ^{2}\bm{\tau } \label{1.2}\end{gathered}$$ $$\begin{aligned} \partial _{t}Q_{ij}-&D_r\left[ \alpha\left( \rho \right) -\beta {\bf Q}{\bf :}{\bf Q}\right] Q_{ij}+v_{0}F_{ij} = D_{b}\nabla ^{2}Q_{ij}\notag\\ &+D_{s}\partial _{k}\left( \partial _{i}Q_{kj}+\partial _{j}Q_{ik}-\delta _{ij}\partial _{l}Q_{kl}\right)\notag\\ &\label{1.3} \end{aligned}$$ where ${\bf Q}{\bf :}{\bf Q}=Q_{kl}Q_{kl}$, $D_r$ is the rotational diffusion rate, and $F_{ij}=\left( \partial _{i}\tau _{j}+\partial _{j}\tau _{i}-\delta _{ij}\nabla \cdot \tau \right)$. All terms proportional to $v_0$ arise from one-particle convection due to self-propulsion and are the only consequence of activity in this simple model. The repulsive interactions among the particles generate the cubic homogeneous term (with $\beta>0$) on the right hand side of Eq.  and a change in sign of $\alpha(\rho)\sim \rho-\rho_c$ at a critical density $\rho_c$, controlling the transition between the isotropic and the nematic states.  [^1] Interactions also give density-dependent corrections to the various diffusion coefficients for density ($D$), polarization ($D_\tau$), splay ($D_s$) and bend ($D_b$) deformations of the nematic alignment tensor. We will ignore all such corrections in the following [^2]. Due to the fact that the interactions are purely nematic, the polarization $\bm \tau $ decays on short time scales $\sim D_r^{-1}$ for all strengths of activity. At long time, a hydrodynamic description can then be obtained by neglecting $\partial_t\bm\tau$ in Eq. , and using Eq.  to eliminate $\bm\tau$ from the other equations, with the result (to leading order in gradients) $$\begin{gathered} \partial _{t}\rho =D\nabla ^{2}\rho+{\cal D}_Q\bm\nabla\bm\nabla{\bf :}{\bf Q} \label{2.1}\end{gathered}$$ $$\begin{aligned} \partial _{t}Q_{ij}-&D_r\left[ \alpha\left( \rho \right) -\beta {\bf Q}{\bf :}{\bf Q}\right] Q_{ij} = D_{b}\nabla ^{2}Q_{ij}\notag\\ &+D_{s}\partial _{k}\left( \partial _{i}Q_{kj}+\partial _{j}Q_{ik}-\delta _{ij}\partial _{l}Q_{kl}\right)\notag\\ &+{\cal D}_\rho( \partial _{i}\partial _{j}-\frac12\delta _{ij}\nabla ^{2}) \rho &\label{2.2} \end{aligned}$$ \[2\] This procedure yields qualitative new terms proportional to ${\cal D}_Q$ and ${\cal D}_\rho$ that vanish in equilibrium. The term proportional to ${\cal D}_Q$ is the curvature induced density flux that has been discussed extensively by Ramaswamy and collaborators [@Ramaswamy2003] and shown to be responsible for giant number fluctuations in the ordered state of active nematic. The diffusive coupling proportional to ${\cal D}_\rho$ describes similar physics but has not been considered in earlier description of active nematic fluids. In addition, activity also yields corrections to the various diffusion coefficients. We have, however, implicitly neglected those here by retaining the same notation for these quantities as in Eqs. - to highlight the difference between these corrections that do not change the dynamics qualitatively and the new terms proportional to ${\cal D}_Q$ and ${\cal D}_\rho$. Although obtained here by considering a system of self-propelled particles, Eqs.  and have the same structure as the hydrodynamic equations of an active nematic, consisting of a collection of *apolar* active particles (shakers) with apolar interactions. This is an important point as it stresses that the qualitative differences between active and self-propelled nematic that have been observed in simulations must arise entirely from the dependence of the interaction on self propulsion $v_0$. Momentum-conserving interaction of self-propelled nematogens {#subsec:interaction} ------------------------------------------------------------ As shown in Ref. [@Baskaran2008a] and supported by simulations of self-propelled hard rods [@Peruani2006; @Ginelli2010; @Yang2010], self-propulsion does modify the repulsive interaction in a qualitative way. This modification results in local build-up of polarization in the nematic state, making it necessary to retain the dynamics of polarization density in the continuum model. ![ Illustration of momentum conserving collisions among self propelled particles. It can readily be shown that two rods as shown in (a), coming in with only their self-replenishing velocities, will acquire opposite angular momenta $\bm\omega _{1}\sim {\bf \hat{z}} \ell v_{0}\left[{\bf \hat{z}}\cdot\left( \mathbf{\hat{u}}\right]% _{1}\times \mathbf{\hat{u}}_{2}\right) $ and $\bm\omega _{2}\sim-{\bf \hat{z}} \ell v_{0} \left( \mathbf{\hat{u}}_{1}\cdot \mathbf{\hat{u}}_{2}\right)\left[{\bf \hat{z}}\cdot\left( \mathbf{\hat{u}}_{1}\times \mathbf{\hat{u}}_{2}\right)\right] $, where the vectors are defined in the image and in  [@Baskaran2010]. The collision will therefore induce rotations as indicated, promoting alignment of the two rods. On the other hand, two nearly antialigned rods as in (b) acquire angular momenta of the same sign, inducing rotation of both rods in the same directions, and leaving their relative angle unchanged. []{data-label="Fig1.1"}](Fig1.pdf){width="49.00000%"} The modification of the Onsager excluded volume interactions among hard rods due to self propulsion is worked out in Ref. [@Baskaran2010]. Here we simply give a qualitative description of this effect and we refer the reader to that work for the technical details. First we note that the presence of a self propulsion speed along the long axis of the nematogen, results in a breaking of the nematic symmetry of the the microdynamics, as shown in Fig. \[Fig1\]. On the other hand, since the interactions conserve momentum, this cannot lead to a macroscopic breaking of polar symmetry as this would amount to the appearance of a spontaneous macroscopic momentum from a zero momentum state. Hence, only a homogeneous ordered nematic state can occur and the associated mean field transition will be the same as in the case of the active nematic considered above, albeit with coefficients $ \alpha$ and $\beta$ renormalized by self-propulsion [@Baskaran2008a]. Even though the polar symmetry cannot be broken macroscopically, momentum conservation allows the nematic ordering to induce local polar order in the system. To illustrate this, let us consider hard rods in two dimensions undergoing energy-momentum conserving interactions. As shown in Fig. (\[Fig1.1\]), the angular momentum transfer due to the linear momentum from self-propulsion for a collision between two rods scales as $\omega \sim \cos \left( \theta _{1}-\theta _{2}\right) \sin \left( \theta _{1}-\theta _{2}\right) $. If the rods are nearly aligned head to head (as in Fig \[Fig1.1\].a), the effect of this angular momentum is to turn the rods towards each other, while if they are nearly aligned head to tail as in Fig \[Fig1.1\].b, the collision turns both rods in the same direction, leaving their relative angle unchanged. This mechanism effectively promotes head-to-head alignment. Since collisions among such nearly aligned nematogens will dominate the dynamics in the nematic state, the nematic order effectively induces polar order. Self-propelled Nematic Hydrodynamics {#subsec:sp_nematic} ------------------------------------ The fact that interactions among self-propelled nematogens tend to induce polar order is reflected in the hydrodynamic description by a number of new nonlinear terms that couple $\bm\tau$ and ${\bf Q}$, with coefficients that vanish in the limit $v_0=0$. The continuum equations for a self-propelled nematic that incorporate the above physics are given by $$\begin{gathered} \label{3.1} \partial _{t}\rho +v_{0}\nabla \cdot \bm{\tau }=D \nabla ^{2}\rho +\mathcal{D}_{Q}\nabla \nabla :\mathbf{Q}\\ % \partial _{t}\bm{\tau }+D_r\bm{\tau }+\gamma_1{\bf Q}{\bf :}{\bf Q}\bm\tau-\gamma_2\bm{\tau \cdot Q} +\lambda_1 \bm{\tau }\cdot \nabla \bm{\tau } \mathbf{=}-v_{0}\bm\nabla \cdot \mathbf{Q}-\frac{v_{0}}{2}\nabla \rho +\lambda _{2}\bm{\tau }\nabla \cdot \bm{\tau }+\frac{\lambda _{3}}{2}\nabla \tau ^{2}+D_{\tau}\nabla ^{2}\bm{\tau } \label{3.2}\\ % \partial _{t}Q_{ij}-D_r(\alpha-\beta {\bf Q}{\bf :}{\bf Q}) Q_{ij}+v_0F_{ij}+\lambda _{4}G_{ij} = \mathcal{D}_\rho( \partial _{i}\partial _{j}-\frac{\delta _{ij}}{2}% \nabla ^{2}) \rho +D_{s}\partial _{k}\left( \partial _{i}Q_{kj}+\partial _{j}Q_{ik}-\delta _{ij}\partial _{l}Q_{kl}\right) +D_{b}\nabla ^{2}Q_{ij} \label{3.3}\end{gathered}$$ \[3\] where again we have implicitly neglected active corrections to $D$, $D_\tau$, $D_s$ and $D_b$ to highlight the new, purely active terms. Activity enters in Eqs.  through the convective terms proportional to $ v_{0}$, the new terms with coefficients $\gamma_i$ and to $\lambda_i $, which vanish in equilibrium, as well as the terms proportional to ${\cal D}_Q$ and ${\cal D}_\rho$ that arise here from active corrections to interactions. Finally, the parameters $\alpha$ and $\beta$ controlling the mean field isotropic-nematic transition are also renormalized by activity. The homogeneous nonlinearities proportional to $\gamma_i$ in the polarization equation encode the fact that nematic order induces polar order. The latter is, however, only local as the equations do not a admit a homogeneous solution with nonzero $\bm\tau$. Further, the active modification of the interactions, yield the convective nonlinearities $\sim{\cal O}(\bm{\tau }\nabla \bm{\tau })$ that play a central role in the emergent physics of active polar fluids. Since the goal of this presentation is to highlight the mechanisms responsible for emergent structures, we simplify the equations by setting all of the equilibrium-like diffusion coefficients to be equal, i.e., $D=D_\tau=D_b=D_{0}$, with the exception of the splay relaxation constants $D_{s}$. In addition, we assume $\lambda_i=\lambda $ for all $i$’s and $\gamma_1=\gamma_2=\gamma$. Finally, we measure time in units of $1/D_r$ and lengths in units of $ \sqrt{D_{0}/D_r}$. The hydrodynamic equations then simplify to (in nondimensional form) $$\begin{gathered} \partial _{t}\rho +\overline{v}\bm\nabla \cdot \bm\tau=\nabla ^{2}\rho + \overline{D}_Q\bm\nabla \bm\nabla :\mathbf{Q} \label{4.1}\\ % \partial _{t}\bm{\tau }+\left(1+\gamma{\bf Q}{\bf :}{\bf Q}\right)\bm\tau-\gamma\bm{\tau \cdot Q} +\lambda \bm{\tau }\cdot \nabla \bm{\tau } =-\overline{v} \bm\nabla \cdot \mathbf{Q}-\frac{\overline{v}}{2}\bm\nabla \rho +\lambda \left( \bm{\tau }\nabla \cdot \bm{\tau }+\frac12\nabla \tau ^{2}\right) +\nabla ^{2}\bm{\tau } \label{4.2} \\ % \partial _{t}Q_{ij}-\left( \alpha-\beta{\bf Q}{\bf :}{\bf Q} \right) Q_{ij}+\overline{v} F_{ij}+\lambda G_{ij} = \overline{D}_\rho( \partial _{i}\partial _{j}-\frac12 \delta _{ij}\nabla ^{2}) \rho +\overline{D}_{S}\partial _{k}\left( \partial _{i}Q_{kj}+\partial _{j}Q_{ik}-\delta _{ij}\partial _{l}Q_{kl}\right) +\nabla ^{2}Q_{ij} \label{4.3}\end{gathered}$$ \[4\] with $\overline{D}_Q={\cal D}_Q/D_0$, $\overline{D}_\rho={\cal D}_\rho/D_0$ and $\overline{D}_s=D_s/D_0$. Finally, we assume $\alpha=\frac{\rho }{\rho _{c}}-1 $ and $ \beta $ independent of $\rho$. The effect of activity is assumed to affect the mean field phase transition only through the dependence of the critical density $\rho _{c}$ on the magnitude of self-propulsion speed. In this simplified form, the dynamics of the system is characterized by two central parameters: the mean density $\rho _{0}$ of active nematogens and the self-propulsion velocity $\overline{ v}=v_0/\sqrt{D_rD_0}$, which is effectively the Peclet number for this flow. The other parameters $\gamma$, $\lambda$, $\overline{D}_Q$, $\overline{D}_\rho$ and $\overline{D}_s$ are in general functions of $\rho _{0}$ and $\overline{v}$, although we will treat them here as independent parameters and simply fix their values. Linear Dynamics and Emergent Structures {#sec:emergent} ======================================= The dynamics of self propelled rod-like particles with steric repulsion has been studied extensively by numerical simulation of microscopic models [@Mishra2006; @Chate2006; @Yang2010; @Sam2012]. This work has revealed a rich variety of emergent structures, including bands of high density regions where the particles are ordered along the direction of the band, lane formation, migrating defect structures and low Reynolds number turbulence. Here we examine the minimal continuum model of self-propelled nematic given by Eqs.  to identify the generic dynamical mechanisms responsible for the emergence of these structures. As mentioned above, there are three important mechanisms for dynamical instabilities and parttern formation in these systems. To unfold the role of each of these mechanisms in controlling the large-scale dynamics of the system, we analyze the linear stability of the ordered nematic state in various special cases that best highlight a particular mechanism. The ordered nematic state has constant density $\rho_0$, zero mean polarization density, $\bm\tau_0=0$, and a finite value for the nematic alignment tensor. Choosing a coordinate system with the $x$ axis pointing along the direction of broken nematic symmetry, the alignment tensor in the uniform nematic state has components $Q^0_{xx}=-Q^0_{yy}=Q_0/2$ and $Q^0_{xy}=Q^0_{yx}=0$, with $Q_0=\sqrt{\alpha_0/\beta}$ and $\alpha_0=\alpha(\rho_0)$. We now examine the linear stability of this state in various regions of parameters by considering the dynamics of small fluctuations, $\delta\rho({\bf r},t)=\rho({\bf r},t)-\rho_0$, $\delta\bm\tau({\bf r},t)=\bm\tau({\bf r},t)$ and $\delta Q_{ij}({\bf r},t)=Q_{ij}({\bf r},t)-Q_{ij}^0$. We will generally work in Fourier space by introducing Fourier transforms of the fluctuations as $\phi^\alpha_{\bf k}(t)=\int_{\bf r}e^{i{\bf k}\cdot{\bf r}}\delta\phi_\alpha({\bf r},t)$, where $\delta\phi_\alpha=\left(\delta\rho,\bm\tau,\delta Q_{ij}\right)$. Dynamical Self-Regulation and Banding Instability {#subsec:self-reg} ------------------------------------------------- We first consider the linear dynamics of the system in the region just above the mean-field transition at $\rho_c$. For simplicity we only discuss spatial variations normal to the direction of broken symmetry, as these correspond to the most unstable modes, i.e., let $\mathbf{k}=k\mathbf{\hat{y}}$. Fluctuations in $\tau _{x}$ and $\delta Q_{xy}$ then decouple and are always stable. The dynamics of fluctuations in $\delta \rho $, $\tau _{y}$ and $\delta Q_{yy}$ is governed by three coupled equations. Fluctuations in $\tau _{y}$ are always quickly damped near the mean-field transition, while the decay rate of $\delta Q_{yy}$, controlled to leading order by $\alpha _{0}$, vanishes as $\rho _{0}\rightarrow \rho _{c}^{+}$. We therefore neglect fluctuations in $\tau _{y}$ and simply examine the coupled dynamics of $\delta \rho $ and $\delta Q\equiv \delta Q_{yy}$, given by $$\begin{gathered} \partial _{t}\delta \rho _{\mathbf{k}}=-k^{2}\delta \rho _{\mathbf{k}}- \overline{D}_Qk^{2}\delta Q_{\mathbf{k}} \label{6.1} \\ \partial _{t}\delta Q_{\mathbf{k}}=-\left[\frac{ \alpha _{0}}{2}+(1+\overline{D} _{s})k^{2}\right] \delta Q_{\mathbf{k}}-\frac{1}{2}(\alpha ^{\prime }Q_{0}+\overline{D}_\rho k^{2})\delta \rho _{\mathbf{k}} \label{6.2}\end{gathered}$$ where $\alpha ^{\prime }=\left( \frac{\partial \alpha }{\partial \rho }% \right) _{\rho =\rho _{0}}$, or $\alpha ^{\prime }=1/\rho _{c}$ with the chosen parameters. The decay of density and ordered parameter fluctuations is then controlled by two coupled hydrodynamic modes. One of the modes has a finite decay rate (proportional to $\alpha _{0}$) in the limit $k\rightarrow 0$ and is always stable. At small wavevector, the dispersion relation of the other mode is given by $$s_{y}(k)=-s_{2}k^{2}-s_{4}k^{4}+\mathcal{O}(k^{6})\;. \label{7}$$ with $s_{2}=1-\frac{\overline{D}\alpha ^{\prime }}{\sqrt{\alpha _{0}\beta }}$ and $s_{4}>0$. Near the transition where $\alpha _{0}\rightarrow 0$, $s_{2}<0 $ and $s_{4}\simeq \frac{2\overline{D}^{2}\alpha ^{^{\prime }2}}{\alpha _{0}^{2}\beta }$. As a result, $s_{y}(k)>0$ for a range of wavevectors, resulting in the unstable growth of density and order parameter fluctuations illustrated in Fig. \[Fig3\]. The fastest growing mode has wave vector $k_{0}=\sqrt{ -s_{2}/2s_{4}}\sim (\rho _{0}-\rho _{c})^{3/2}$. Including the coupling to $\tau_y$ will yield finite Peclet number corrections to the instability. Note that this instability is strongest in the vicinity of the order-disorder transition and is a manifestation of the fact that the dynamics of the system is self regulating, i.e., the control parameter associated with the phase transition, namely the density is dynamically coupled to the emergent ordering that results from the transition through the curvature induced fluxes. This is the dynamics that leads the system to be intrinsically phase separated [@Ramaswamy2003]. ![ The banding instability that occurs due to the self-regulating nature of the flow. The striped region is the parameter space in which this instability occurs.[]{data-label="Fig3"}](Fig3.pdf){width="6cm"} We recall that polar active fluids exhibit a similar instability for wave vectors parallel to the direction of mean order. In that case the mode that goes unstable is a propagating mode and the instability signals the onset of solitary waves consisting of alternating ordered and disordered bands extending in the direction normal to that of mean order and traveling along the direction of broken symmetry. These bands have been observed in simulations of the Vicsek model [@Gregoire2004; @Chate2008], as well as in numerical solutions of the nonlinear hydrodynamic equations for polar fluids  [@Bertin2006; @Bertin2009; @Mishra2010]. We have shown here that active nematics exhibit a similar instability, controlled by the interplay of of curvature currents ($\overline{D}_Q$) and the self-regulation due to the density dependence of $\alpha $. The instability occurs even for $\overline{v}=0$, i.e., is present in both active and self-propelled nematics. It occurs for wavevectors perpendicular to the direction of broken nematic symmetry and the mode that goes unstable is a diffusive one. It is therefore tempting to associate it with the emergence of the stationary bands consisting of alternating ordered (nematic) and disorders regions that have been seems in simulations of active systems with apolar interactions  [@Chate2006] and physical excluded volume interactions [@Yang2010],[@Sam2012]. Finally, but most importantly, this instability mechanism is *generic*, in the sense that it does not depend on microscopic parameters, but only on the presence of a dynamical feedback between density and active currents. Curvature Induced Flux {#subsec:curvature} ---------------------- Next we consider the region of small $\overline{v}$, $\lambda $ and $\gamma $, deep in the nematic phase. In this case, the long-wavelength dynamics is controlled by hydrodynamic modes associated with fluctuations in the density and the director $\mathbf{\hat{n}}$. This case has been considered in the literature already and is summarized here for completeness  [@Simha2002a; @Ramaswamy2003; @Baskaran2008]. For our choice of coordinates to linear order we have $\delta Q_{{xx}}=-\delta Q_{yy}=0$ and $\delta Q_{xy}=\delta Q_{yx}=Q_{0}\delta \hat{n}\left( \mathbf{r},t\right) $. Neglecting polarization fluctuations that decay on microscopic time scales, the linearized equations are given by $$\begin{gathered} \partial _{t}\delta \rho _{\mathbf{k}}=-k^{2}\delta \rho _{\mathbf{k}}-Q_{0}% \overline{D}k^{2}\sin 2\theta \delta \hat{n}_{\mathbf{k}}\;, \label{5.1} \\ \partial _{t}\delta \hat{n}_{\mathbf{k}}=-\frac{\overline{D}_\rho k^2}{2Q_{0}}\sin 2\theta \delta \rho _{\mathbf{k}}+\left[ \overline{D}_{s}+\cos 2\theta \right] k^{2}\delta \hat{n}_{\mathbf{k}}\;, \label{5.2}\end{gathered}$$ where $\theta $ is the angle between $\mathbf{k}$ and the direction of broken symmetry ($x$). If $\theta=0,\pi$, the two equations are decoupled and the modes are diffusive and stable. For general $\theta$ one of the hydrodynamic modes becomes unstable for $\overline{D}\overline{D}_\rho\sin^{2} 2\theta >2(\overline{D}_{s}+\cos 2\theta)$. This can be satisfied provided $\overline{D}_Q\overline{D}_\rho>2D_{s}$, i.e., the curvature driven fluxes exceed the restoring effects of diffusion. This instability has been discussed in detail elsewhere [@Baskaran2008]. Induced Polar Order {#subsec:polar} -------------------- Finally, we examine the effect of fluctuations with spatial variations along the direction of broken symmetry, i.e., $\mathbf{k}=k\mathbf{\hat{x}}$. The relevant coupled fluctuations in this case are $\delta \rho $, $\tau _{x}$ and $ \delta Q_{xx}$. For simplicity, we consider the regime of large Peclet number $\overline{v}$, where the linear dynamics is controlled by Euler order terms and neglect terms quadratic in the gradients, with the result $$\begin{gathered} \partial _{t}\delta \rho _{\mathbf{k}}+ik\overline{v}\tau _{x,\mathbf{k}}=0 \label{7.1} \\ \partial _{t}\tau _{x,\mathbf{k}}+\gamma _{e}\tau _{x,\mathbf{k}}=-ik% \overline{v}\delta Q_{xx,\mathbf{k}}-ik\frac{\overline{v}}{2}\delta \rho _{% \mathbf{k}} \label{7.2} \\ \partial _{t}\delta Q_{xx,\mathbf{k}}-\frac{\alpha _{0}}{2}\delta Q_{xx,% \mathbf{k}}=ik\left( \overline{v}+\frac{\lambda }{2}Q_{0}\right) \tau _{x,% \mathbf{k}}+\frac{\alpha ^{\prime }}{2}Q_{0}\delta \rho _{\mathbf{k}} \label{7.3}\end{gathered}$$where $\gamma _{e}=1+\frac{\gamma }{2}Q_{0}^{2}-\frac{\gamma }{2}Q_{0}$. As discussed earlier and highlighted in Fig. \[Fig1.1\], the anisotropy of small angle collisions in the nematic state enhances polar order by suppressing the decay rate of $\tau _{x,\mathbf{k}}$ from its bare value of $ 1$ (in units of $D_{r}^{-1}$) to $\gamma _{e}$. The dispersion relations of the hydrodynamic modes associated with Eqs.  are easily calculated at small wavevectors. Clearly, if $\gamma _{e}\leq 0$, In addition, as a consequence of this built up of polar order, the diffusive mode associated with the density fluctuations (i.e., the only truly hydrodynamic mode in these considerations), given by $$s_{x}(k)=-k^{2}\frac{\overline{v}^{2}}{2\gamma _{e}}\left( 1+\frac{% 2\alpha ^{\prime }}{\sqrt{\alpha _{0}\beta }}\right) \label{8}$$ becomes unstable. In general $\gamma _{e}$ depends on microscopic details of the model, but there is no reason to exclude a priori that it could change sign and indeed does for the case of long thin hard rods with excluded volume interactions [@Baskaran2008]. The linear analysis here is of limited utility because of the existence of a homogeneous instability but is shown here to indicate that the build up of polarization due to the momentum conserving nature of the interactions has a dramatic consequence on the dynamics of the system. This may indeed be the mechanism responsible for the smectic order observed within a single polar cluster in simulations of self-propelled rods [@Yang2010; @Sam2012]. Finally, we stress that the nonlinear homogeneous terms proportional to $\gamma $ and responsible for the renormalization of $\gamma _{e}$ always vanish in an equilibrium state because the nematic symmetry of such a state by definition forbids a nonzero uniform value of the mean polarization. Discussion {#sec:discussion} ========== We have considered in this paper the hydrodynamics of active overdamped fluids that can order in nematic states. These are collections of active particles that interact via apolar (nematic) aligning interactions, such as steric repulsion or medium-mediated hydrodynamic couplings. One can identify two classes of such fluids, depending on the properties of the individual active units. Active nematics consist of shaker particles that are themselves apolar. Self-propelled nematics are collections of particles that are physically head-tail symmetric (such as SP rods), but where a microscopic dynamical polarity is induced by self-propulsion . Although both systems form ordered states of nematic symmetry, their dynamical behavior is qualitatively different, as seen in recent simulations [@Peruani2006; @Ginelli2010; @Yang2010]. The hydrodynamic equations of active nematics have the form given in Eqs. [2]{}. We have shown that the same equations are also obtained by considering SP particles and neglecting the effect of self-propulsion on the interaction between active units, suggesting that the active nematic may be considered the zero Peclet number $\overline{v}$ limit of self-propelled nematic. In this case the only active term is the curvature current proportional to ${\cal D}_Q$ in Eq. . This non equilibrium coupling of orientation and flow induces instabilities of the ordered state that have been studied before in the literature [@Simha2002; @Ramaswamy2003; @Baskaran2008] and are also summarized in section. \[subsec:curvature\]. The curvature current is also key in controlling the banding instability arising from dynamical self-regulation discussed in section \[subsec:self-reg\]. In fact this instability, although not discussed before in the literature for overdamped active nematic, occurs in all active fluids of nematic symmetry, both for shakers and self-propelled particles. It arises from the density dependence of the parameter $\alpha(r\rho)$ that controls the mean-field transition and the fact that in active systems $\rho$ is not tuned from the outside, as in equilibrium, but is itself a dynamical variable convected by the order parameter. The hydrodynamic equations of self-propelled nematics given in Eqs.  (or Eqs.  in the dimensionless form studied here) contain many new active terms that arise from modifications of the two-body interaction due to self propulsion. These equations have also been derived by us for a specific microscopic model of self-propelled hard rods [@Baskaran2008a; @Baskaran2010], although the low order closure of the kinetic theory used in that work only gives terms up to quadratic in the hydrodynamic fields. Self-propelled nematics also exhibit both the curvature induced instability discussed in section \[subsec:curvature\] and the banding instability discussed in section \[subsec:self-reg\]. Both are of course modified at finite Peclet number due to additional convective contributions to the underlying mechanisms the details of which will be discussed elsewhere. In addition, self-propulsion yields a novel instability due to the built-up of local polar order discussed in section \[subsec:polar\]. This arises because in the nematic state most binary collisions involve nematogens that are nearly aligned or anti aligned, as shown in Fig. \[Fig1.1\]. When the nematogens are self-propelled, collisions of nearly aligned and nearly anti aligned pairs are not identical. Nearly aligned pairs tend to further align upon collisions, while nearly anti-aligned pairs are turned away from each other. As a result, local polar order is enhanced and the nematic state becomes unstable as discussed in section \[subsec:polar\]. It is tempting to associate this instability with the onset of “polar clusters" that have been observed ubiquitously in simulations of self-propelled rods [@Peruani2006; @Ginelli2010; @Yang2010], as well as in experiments in gliding myxobacteria [@Peruani2012]. MCM was supported by the National Science Foundation through awards DMR-0806511 and DMR-1004789. AB was supported by the Brandeis-MRSEC through NSF DMR-0820492. [^1]: Note that the cubic term was not derived in Ref. [@Baskaran2008], but is easily obtained by a higher order closure of the moment expansion of the kinetic equation. [^2]: [ Retaining the density dependence of the diffusion coefficients results in interesting emergent structures as shown by [@Cates2010]. Since we seek to focus on fundamental features that do not depend on the detailed structure of the hydrodynamic coefficients, we ignore this physically important feature.]{}

--- abstract: 'Renormalization factors for three- and four-quark operators, which appear in the low energy effective Lagrangian of the proton decay and the weak interactions, are perturbatively calculated in domain-wall QCD. We find that the operators are multiplicatively renormalizable up to one-loop level without mixing with any other operators that have different chiral structures. As an application, we evaluate a renormalization factor for $B_K$ at the parameters where previous simulations have been performed, and find one-loop corrections to $B_K$ are 1-5% in these cases.' address: | $^1$Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan\ $^2$Department of Physics, Washington University, St. Louis, Missouri 63130, USA\ author: - '$^1$Sinya Aoki, $^1$Taku Izubuchi, $^{2}$Yoshinobu Kuramashi [^1] and $^1$Yusuke Taniguchi' title: | [hep-th/9902008]{}\ [UTHEP-398]{}\ Perturbative renormalization factors of three- and four-quark operators for domain-wall QCD --- \#1\#2 Introduction ============ Calculation of hadron matrix elements of phenomenological interest represents an inevitable application of lattice QCD. In the past decade much efforts have been devoted for the calculation of three- and four-quark hadron matrix elements relevant to the proton decay amplitude and the weak interaction ones using the Wilson and the Kogut-Susskind(KS) quark actions. However, the satisfactorily precise measurement of the matrix elements has not been achieved so far because of the inherent defects in these quark actions: the explicit chiral symmetry breaking in the Wilson quark action causes the non-trivial operator mixing between different chiralities and for the KS quark it is hard to treat the heavy-light cases due to the flavor symmetry breaking. The domain-wall quark formulation in lattice QCD, which is based on the introduction of many heavy regulator fields, was proposed by Shamir[@Shamir93; @Shamir95] anticipating superior features over other quark formulations: no need of the fine tuning to realize the chiral limit and no restriction for the number of flavors. Recent simulation results seem to support the former feature non-perturbatively[@Blum-Soni; @Wingate; @Blum]. It is also perturbatively shown that the massless mode at the tree level still remains stable against the quantum correction[@Aoki-Taniguchi]. These advantageous features fascinate us to the application of the domain-wall quark for calculation of the three- and four-quark hadron matrix elements. In order to convert the matrix elements obtained by lattice simulations to those defined in some continuum renormalization scheme(${\it e.g.}, {{\overline {\rm MS}}}$), we must know the renormalization factors connecting the lattice operators to the continuum counterparts defined in some renormalization scheme. In this article we make a perturbative calculation of the renormalization factors for the three- and four-quark operators consisting of physical quark fields in the domain-wall QCD(DWQCD). This work is an extension of the previous paper[@AIKT98], in which we developed a perturbative renormalization procedure for DWQCD demonstrating the calculation of the renormalization factors for quark wave function, mass and bilinear operators. We focus on whether or not the renormalization of the three- and four-quark operators in DWQCD is free from the notorious operator mixing problem. In the Wilson case it is well known that the mixing problem is not adequately manipulated by the perturbation theory, leading to an “incorrect” value for the $B_K$ matrix element. This paper is organized as follows. In Sec. \[sec:model\] we briefly introduce the DWQCD action and the Feynman rules relevant for the present calculation to make this paper self-contained. In Sec. \[sec:4fermi\] our calculational procedure of the renormalization factors for the four-quark operators is described in detail. We also evaluate the renormalization factors for three-quark operators in Sec. \[sec:3fermi\]. In Secs. \[sec:4fermi\] and \[sec:3fermi\] numerical results for one-loop coefficients of the renormalization factors are given with and without the mean field improvement. In Sec. \[sec:bk\], using our results, we analyze a renormalization factor for $B_K$. Our conclusions are summarized in Sec. \[sec:concl\]. The physical quantities are expressed in lattice units and the lattice spacing $a$ is suppressed unless necessary. We take SU($N$) gauge group with the gauge coupling $g$ and the second Casimir $C_F = \displaystyle \frac{N^2-1}{2N}$, while $N=3$ is specified in the numerical calculations. Action and Feynman rules {#sec:model} ======================== We take the Shamir’s domain-wall fermion action[@Shamir93], $$\begin{aligned} S_{\rm DW} &=& \sum_{n} \sum_{s=1}^{N_s} \Biggl[ \frac{1}{2} \sum_\mu \left( {\overline{\psi}}(n)_s (-r+\gamma_\mu) U_\mu(n) \psi(n+\mu)_s + {\overline{\psi}}(n)_s (-r-\gamma_\mu) U_\mu^\dagger(n-\mu) \psi(n-\mu)_s \right) {\nonumber}\\&& + \frac{1}{2} \left( {\overline{\psi}}(n)_s (1+\gamma_5) \psi(n)_{s+1} + {\overline{\psi}}(n)_s (1-\gamma_5) \psi(n)_{s-1} \right) + (M-1+4r) {\overline{\psi}}(n)_s \psi(n)_s \Biggr] {\nonumber}\\&+& m \sum_n \left( {\overline{\psi}}(n)_{N_s} P_{+} \psi(n)_{1} + {\overline{\psi}}(n)_{1} P_{-} \psi(n)_{N_s} \right), \label{eqn:action}\end{aligned}$$ where $n$ is a four dimensional space-time coordinate and $s$ is an extra fifth dimensional or “flavor” index, the Dirac “mass” $M$ is a parameter of the theory which we set $0 < M < 2$ to realize the massless fermion at tree level, $m$ is a physical quark mass, and the Wilson parameter is set to $r=-1$. It is important to notice that we have boundaries for the flavor space; $1 \le s \le N_s$. In our one-loop calculation we will take $N_s\to\infty$ limit to avoid complications arising from the finite $N_s$. $P_{R/L}$ is a projection matrix $P_{R/L}=(1\pm\gamma_5)/2$. For the gauge part we employ a standard four dimensional Wilson plaquette action and assume no gauge interaction along the fifth dimension. In the DWQCD the zero mode of domain-wall fermion is extracted by the “physical” quark field defined by the boundary fermions $$\begin{aligned} q(n) = P_R \psi(n)_1 + P_L \psi(n)_{N_s}, {\nonumber}\\ {\overline{q}}(n) = {\overline{\psi}}(n)_{N_s} P_R + {\overline{\psi}}(n)_1 P_L. \label{eq:quark}\end{aligned}$$ We will consider the QCD operators constructed from this quark fields, since this field has been actually used in the previous simulations. Moreover our renormalization procedure is based on the Green functions consisting of only the “physical” quark fields, in which we have found that the renormalization becomes simple[@AIKT98]. Weak coupling perturbation theory is developed by expanding the action in terms of gauge coupling. The gluon propagator can be written as $$\begin{aligned} G_{\mu\nu}^{AB}(k) =\delta_{\mu\nu}\delta_{AB} \frac{1}{4\sin^2(k/2)+\lambda^2}\end{aligned}$$ in the Feynman gauge with the infrared cut-off $\lambda^2$, where $\sin^2(k/2) =\sum_\mu \sin^2(k_\mu/2)$. Quark-gluon vertices are also identical to those in the $N_s$ flavor Wilson fermion. We need only one gluon vertex for our present calculation: $$\begin{aligned} V_{1\mu}^A (k,p)_{st} &=& -i g T^A \{ \gamma_\mu \cos(-k_\mu/2 + p_\mu/2) -i r \sin(-k_\mu/2 + p_\mu/2) \} \delta_{st},\end{aligned}$$ where $k$ and $p$ represent incoming momentum into the vertex (see Fig. 1 of Ref.[@AIKT98]). $T^A$ $(A=1,\dots,N^2-1)$ is a generator of color SU($N$). The fermion propagator originally takes $N_s\times N_s$ matrix form in $s$-flavor space. In the present one-loop calculation, however, we do not need the whole matrix elements because we consider Green functions consisting of the physical quark fields. The relevant fermion propagators are restricted to following three types: $$\begin{aligned} && {\langle q(-p) {\overline{q}}(p) \rangle} = \frac{-i\gamma_\mu \sin p_\mu + \left(1-W e^{-\alpha}\right) m} {-\left(1-e^{\alpha}W\right) + m^2 (1-W e^{-\alpha})} \equiv S_q(p), \label{eqn:phys-prop} \\&& {\langle q(-p) {\overline{\psi}}(p,s) \rangle} = \frac{1}{F} \left( i\gamma_\mu \sin p_\mu - m \left(1 -W e^{-\alpha} \right) \right) \left( e^{-\alpha (N_s-s)} P_R + e^{-\alpha (s-1)} P_L \right) {\nonumber}\\&&\qquad +\frac{1}{F} \Bigl[ m \left(i\gamma_\mu \sin p_\mu -m \left(1-W e^{-\alpha}\right)\right) - F \Bigr] e^{-\alpha} \left( e^{-\alpha (s-1)} P_R + e^{-\alpha (N_s-s)} P_L \right), \\&& {\langle \psi(-p,s) {\overline{q}}(p) \rangle} = \frac{1}{F} \left( e^{-\alpha (N_s-s)} P_L + e^{-\alpha (s-1)} P_R \right) \left( i\gamma_\mu \sin p_\mu - m \left(1 - W e^{-\alpha} \right) \right) {\nonumber}\\&&\qquad +\frac{1}{F} \left( e^{-\alpha (s-1)} P_L + e^{-\alpha (N_s-s)} P_R \right) e^{-\alpha} \Bigl[ m \left(i\gamma_\mu \sin p_\mu -m\left(1- We^{-\alpha}\right) \right) - F \Bigr]\end{aligned}$$ with $$\begin{aligned} W &=& 1-M -r \sum_\mu (1-\cos p_\mu), \\ \cosh (\alpha) &=& \frac{1+W^2+\sum_\mu \sin^2 p_\mu}{2|W|}, \label{eq:alpha} \\ F &=& 1-e^{\alpha} W-m^2 \left(1-W e^{-\alpha}\right), \label{eq:F}\end{aligned}$$ where the argument $p$ in the factors $\alpha$ and $W$ is suppressed. In the perturbative calculation of Green functions the external quark momenta and masses are assumed to be much smaller than the lattice cut-off, so that we can expand the external quark propagators in terms of them. We have the following expressions as leading term of the expansion: $$\begin{aligned} \langle q\bar q \rangle (p) & = & \frac{1-w_0^2}{i{p\kern-1ex /}+ (1-w_0^2)m}, \\ \langle q \bar \psi_s \rangle (p) &=& \langle q\bar q \rangle (p) \left( w_0^{s-1}P_L + w_0^{N_s-s} P_R\right), \label{eqn:qpsi} \\ \langle \psi_s \bar q\rangle (p) &=& \left( w_0^{s-1}P_R + w_0^{N_s-s} P_L\right) \langle q\bar q \rangle (p), \label{eqn:psiq}\end{aligned}$$ where $w_0 = 1-M$. Renormalization factors for four-quark operators {#sec:4fermi} ================================================ We consider the following four-quark operators: $$\begin{aligned} {\cal O}_\pm & = & \frac{1}{2} \left[ (\bar q_1 \gamma_\mu^L q_2)(\bar q_3 \gamma_\mu^L q_4) \pm (\bar q_1 \gamma_\mu^L q_4)(\bar q_3 \gamma_\mu^L q_2) \right], \\ {\cal O}_1 & = & -C_F (\bar q_1 \gamma_\mu^L q_2)(\bar q_3 \gamma_\mu^R q_4) + (\bar q_1 T^A \gamma_\mu^L q_2)(\bar q_3 T^A \gamma_\mu^R q_4), \\ {\cal O}_2 & = & \frac{1}{2N} (\bar q_1 \gamma_\mu^L q_2)(\bar q_3 \gamma_\mu^R q_4) + (\bar q_1 T^A \gamma_\mu^L q_2)(\bar q_3 T^A \gamma_\mu^R q_4) ,\end{aligned}$$ where $\gamma_\mu^{L,R} = \gamma_\mu P_{L,R}$. Summation over repeated indices such as $\mu$ and $A$ is assumed. We note that $q_i$ $(i=1,2,3,4)$ are boundary quark fields in DWQCD. For the convenience of calculation we rewrite the above operators as $$\begin{aligned} {\cal O}_\pm & = & \frac{1}{2} \left[1{\widetilde{\otimes}}1 \pm 1{\widetilde{\odot}}1\right]^{ab;cd} \left[(\bar q_1^a \gamma_\mu^L q_2^b) (\bar q_3^c \gamma_\mu^L q_4^d)\right], \\ {\cal O}_1 & = & \frac{1}{2} \left[-N 1{\widetilde{\otimes}}1 + 1{\widetilde{\odot}}1\right]^{ab;cd} \left[(\bar q_1^a \gamma_\mu^L q_2^b) (\bar q_3^c \gamma_\mu^R q_4^d)\right], \\ {\cal O}_2 & = & \frac{1}{2} \left[1{\widetilde{\odot}}1\right]^{ab;cd} \left[(\bar q_1^a \gamma_\mu^L q_2^b) (\bar q_3^c \gamma_\mu^R q_4^d)\right], \end{aligned}$$ where $a,b,c,d$ are color indices, and ${\widetilde{\otimes}}$, ${\widetilde{\odot}}$ represent the tensor structures in the color space: $$\begin{aligned} \left[1 {\widetilde{\otimes}} 1\right]^{ab;cd} & \equiv & \delta_{ab}\delta_{cd}, \\ \left[1 {\widetilde{\odot}} 1\right]^{ab;cd} & \equiv & \delta_{ad}\delta_{cb}. \end{aligned}$$ To derive these formula, we have used the Fierz transformation for ${\cal O}_\pm$ and the formula $$\sum_A T^A{\widetilde{\otimes}} T^A = \frac{1}{2}\left[ -\frac{1}{N} 1{\widetilde{\otimes}} 1+1{\widetilde{\odot}} 1\right]$$ for ${\cal O}_{1,2}$ . We calculate the following Green function: $$\langle {\cal O}_\Gamma \rangle_{\alpha\beta;\gamma\delta}^{ij;kl} \equiv {\langle {\cal O}_\Gamma (q_1)_{\alpha}^i (\bar q_2)_{\beta}^j (q_3)_{\gamma}^k (\bar q_4)_{\delta}^l \rangle},$$ where $\Gamma = \pm, 1,2$. Spinor indices are labeled by $\alpha,\beta,\gamma,\delta$ and color ones by $i,j,k,l$. Truncating the external quark propagators from $\langle {\cal O}_\Gamma \rangle$, where we multiply $\langle {\cal O}_\Gamma \rangle$ by ${i{p\kern-1ex /}_i + (1-w_0^2)m}$, we obtain the vertex functions, which is written in the following form up to the one-loop level $$(1-w_0^2)^4\left(\Lambda_\Gamma\right)_{\alpha\beta;\gamma\delta}^{ij;kl} =(1-w_0^2)^4\left(\Lambda_\Gamma^{(0)} +\Lambda_\Gamma^{(1)}\right)_{\alpha\beta;\gamma\delta}^{ij;kl},$$ where the superscript $(i)$ refers to the $i$-th loop level and the trivial factor $(1-w_0^2)^4$ is factored out for the convenience. We suppress the external momenta $p_i$ since the renormalization factor does not depend on them. The tree level vertex functions $\Lambda_\Gamma^{(0)}$ are given by $$\begin{aligned} &\Gamma = \pm, &\qquad \frac{1}{2}\left[ \gamma_\mu^L \otimes \gamma_\mu^L\right]_{\alpha\beta;\gamma\delta} \left[1{\widetilde{\otimes}} 1 \pm 1{\widetilde{\odot}} 1\right]^{ij;kl}, \label{eq:lambda_pm_0}\\ &\Gamma = 1, &\qquad \frac{1}{2}\left[ \gamma_\mu^L \otimes \gamma_\mu^R \right]_{\alpha\beta;\gamma\delta} \left[-N 1{\widetilde{\otimes}} 1+1{\widetilde{\odot}} 1\right]^{ij;kl}, \label{eq:lambda_1_0}\\ &\Gamma = 2, &\qquad \frac{1}{2}\left[ \gamma_\mu^L \otimes \gamma_\mu^R \right]_{\alpha\beta;\gamma\delta} \left[1{\widetilde{\odot}} 1 \right]^{ij;kl}, \label{eq:lambda_2_0}\end{aligned}$$ where $\otimes$ acts on the Dirac spinor space representing $[\gamma_X \otimes \gamma_Y ]_{\alpha\beta;\gamma\delta} \equiv (\gamma_X)_{\alpha\beta}(\gamma_Y)_{\gamma\delta}$. The one-loop vertex corrections are illustrated by six diagrams in Fig. 1, the sum of which yields the one-loop level vertex function $$\Lambda_\Gamma^{(1)}= \int_{-\pi}^{\pi}\frac{d^4 k}{(2\pi)^4} \left(I_\Gamma^a+,\dots,+I_\Gamma^{c^\prime}\right).$$ In order to obtain the expressions for the integrands $I_\Gamma^a,\dots,I_\Gamma^{c^\prime}$ we should note that the internal quark propagators appearing in the diagrams are multiplied by the damping factor which comes from eqs.[(\[eqn:qpsi\])]{} and [(\[eqn:psiq\])]{}. The following formula are useful. $$\begin{aligned} \langle q \bar \psi_s\rangle \left( w_0^{s-1}P_L + w_0^{N_s-s} P_R\right) =\left( w_0^{s-1}P_R + w_0^{N_s-s} P_L\right) \langle \psi_s\bar q \rangle &=& \frac{i\gamma_\mu \sin p_\mu}{{\widetilde{F}}\cdot {\widetilde{F}}_0 }\equiv {\overline{G}}, \label{eq:G_b} \\ \langle q \bar \psi_s\rangle \left( w_0^{s-1}P_R + w_0^{N_s-s} P_L\right) =\left( w_0^{s-1}P_L + w_0^{N_s-s} P_R\right) \langle \psi_s\bar q \rangle &=& -\frac{1}{{\widetilde{F}}_0 } \equiv {\widetilde{G}}, \label{eq:G_t}\end{aligned}$$ where ${\widetilde{F}}=e^{-\alpha}-W$ and ${\widetilde{F}}_0 = e^\alpha-w_0$. Here we set $m=p_i=0$ for the internal propagator. The contribution from Fig. 1a takes the form $$\begin{aligned} I_\Gamma^a &=& \frac{1}{2} J_a^{AB} \left\{{\overline{V}}_\mu(k) {\overline{G}}(k) + {\widetilde{V}}_\mu (k){\widetilde{G}}(k) \right\} \Gamma_X \left\{{\overline{G}}(k) {\overline{V}}_\nu(k) + {\widetilde{G}}(k) {\widetilde{V}}_\nu(k)\right\} \otimes \Gamma_Y \, G_{\mu\nu}^{AB}(k),\end{aligned}$$ where $\Gamma_X = \gamma_\mu^L$, $\Gamma_Y = \gamma_\mu^L$ or $\gamma_\mu^R$, and the interaction vertices are $${\overline{V}}_\mu = -i g \gamma_\mu \cos(k_\mu/2),\quad {\widetilde{V}}_\mu = -rg \sin(k_\mu/2). \label{eq:V_bt}$$ The color factors are represented by $J_a^{AB}$, which are listed in Table \[tab:color\]. In a similar way the contributions from Fig. 1b and Fig. 1c are given by $$\begin{aligned} I_\Gamma^b &=& \frac{1}{2} J_b^{AB} \left\{{\overline{V}}_\mu(k) {\overline{G}}(k) + {\widetilde{V}}_\mu(k) {\widetilde{G}}(k)\right\} \Gamma_X\otimes \left\{{\overline{V}}_\nu (-k) {\overline{G}}(-k) + {\widetilde{V}}_\nu (-k){\widetilde{G}}(-k)\right\} \Gamma_Y \, G_{\mu\nu}^{AB}(k), \\ I_\Gamma^c &=& \frac{1}{2} J_c^{AB} \left\{{\overline{V}}_\mu(k) {\overline{G}}(k) + {\widetilde{V}}_\mu (k) {\widetilde{G}}(k)\right\} \Gamma_X\otimes \Gamma_Y \left\{{\overline{G}}(k){\overline{V}}_\nu(k) + {\widetilde{G}}(k) {\widetilde{V}}_\nu(k) \right\} \, G_{\mu\nu}^{AB}(k).\end{aligned}$$ After a little algebra the expressions of $I_\Gamma^{a,b,c}$ are reduced to $$\begin{aligned} I_\Gamma^a &=& \frac{1}{2} g^2 J_a^{AA} K \left[ T + A_{VA}\right]\left[\Gamma_X\otimes\Gamma_Y\right], \label{eq:vtx_a}\\ I_\Gamma^b &=& - \frac{1}{2} g^2 J_b^{AA} K \left[T \Gamma_X\otimes \Gamma_Y + \cos^2(k_\mu/2) \sin^2 k_\alpha (\gamma_\mu\gamma_\alpha\Gamma_X) \otimes (\gamma_\mu\gamma_\alpha\Gamma_Y) \right], \label{eq:vtx_b}\\ I_\Gamma^c &=& \frac{1}{2} g^2 J_c^{AA} K \left[T \Gamma_X \otimes \Gamma_Y + \cos^2(k_\mu/2) \sin^2 k_\alpha (\gamma_\mu\gamma_\alpha\Gamma_X) \otimes (\Gamma_Y\gamma_\alpha\gamma_\mu) \right], \label{eq:vtx_c}\end{aligned}$$ where $$\begin{aligned} && K =\displaystyle \frac{1}{{\widetilde{F}}^2{\widetilde{F}}_0^2 (4\sin^2(k/2)+\lambda^2)}, \label{eq:K}\\&& T=r^2 \sin^2(k/2) {\widetilde{F}}^2+r \sin^2 k {\widetilde{F}}, \label{eq:T}\\&& A_{VA}= \sum_\mu \cos^2(k_\mu/2) \sin^2 k_\mu. \label{eq:A_VA}\end{aligned}$$ In order to rewrite the second term of $I_\Gamma^{b,c}$ we apply the Fierz transformation: $$\begin{aligned} (\gamma_\mu\gamma_\alpha\gamma_\nu^L)\otimes (\gamma_\mu\gamma_\alpha\gamma_\nu^L) &=& -\gamma_\nu^L\odot\gamma_\nu^L = \gamma_\nu^L\otimes\gamma_\nu^L, \\ (\gamma_\mu\gamma_\alpha\gamma_\nu^L)\otimes (\gamma_\nu^L\gamma_\alpha\gamma_\mu) &=& -\gamma_\nu^L\odot\gamma_\nu^L (1-2\delta_{\alpha\nu})(1-2\delta_{\mu\nu}) \\&=& -\gamma_\nu^L\otimes\gamma_\nu^L (1-2\delta_{\alpha\nu}) (1-2\delta_{\mu\nu}) +2\gamma_\nu^L\otimes\gamma_\nu^L \delta_{\alpha\mu}, \\ (\gamma_\mu\gamma_\alpha\gamma_\nu^L)\otimes (\gamma_\mu\gamma_\alpha\gamma_\nu^R) &=& 2P_R\odot P_L \delta_{\mu\alpha} = \gamma_\nu^L \otimes \gamma_\nu^R \delta_{\mu\alpha}, \label{eq:lr1_fierz}\\ (\gamma_\mu\gamma_\alpha\gamma_\nu^L)\otimes (\gamma_\nu^R\gamma_\alpha\gamma_\mu) &=& 2P_R\odot P_L = \gamma_\nu^L \otimes \gamma_\nu^R . \label{eq:lr2_fierz}\end{aligned}$$ where $[\gamma_X \odot \gamma_Y ]_{\alpha\beta;\gamma\delta} \equiv (\gamma_X)_{\alpha\delta}(\gamma_Y)_{\gamma\beta}$, and summation over $\nu$ is taken. The Fierz transformation is again used for the second equality. We omit the tensor term in eq.(\[eq:lr1\_fierz\]) since it vanishes in the integral. Choosing $\Gamma_{X,Y} = \gamma_\mu^L$ in eqs.(\[eq:vtx\_a\]), (\[eq:vtx\_b\]) and (\[eq:vtx\_c\]) we first consider the case of ${\cal O}_{\pm}$. After simplifying the expressions of the color factors we obtain $$\begin{aligned} I_\pm^a &=& \frac{1}{2} g^2 K (T+A_{VA})\left[\gamma_\nu^L\otimes\gamma_\nu^L\right] \left[ (C_F\pm \frac{1}{2}) 1{\widetilde{\otimes}}1 \mp \frac{1}{2N} 1{\widetilde{\odot}}1 \right], \\ I_\pm^b &=& - \frac{1}{2} g^2 K (T+A_{SP})\left[\gamma_\nu^L\otimes\gamma_\nu^L\right] \left[ (-\frac{1}{2N}\pm \frac{1}{2}) 1{\widetilde{\otimes}}1 +(\frac{1}{2}\mp \frac{1}{2N}) 1{\widetilde{\odot}}1 \right], \\ I_\pm^c &=& \frac{1}{2} g^2 K (T+A_{VA})\left[\gamma_\nu^L\otimes\gamma_\nu^L\right] \left[ -\frac{1}{2N} 1{\widetilde{\otimes}}1 +(\frac{1}{2}\pm C_F) 1{\widetilde{\odot}}1 \right]. \end{aligned}$$ The total contribution becomes $$\begin{aligned} I_+^a+I_+^b+I_+^c &=& \frac{1}{2} g^2 K \left[\gamma_\nu^L\otimes\gamma_\nu^L\right]_{\alpha\beta;\gamma\delta} \left[1{\widetilde{\otimes}}1+1{\widetilde{\odot}}1\right]^{ij;kl} \nonumber \\ &\times & \{ T C_F + A_{VA}(C_F +\frac{1}{2} - \frac{1}{2N}) -A_{SP}(\frac{1}{2} - \frac{1}{2N}) \}, \label{eq:sumP}\end{aligned}$$ and $$\begin{aligned} I_-^a+I_-^b+I_-^c &=& \frac{1}{2} g^2 K \left[\gamma_\nu^L\otimes\gamma_\nu^L\right]_{\alpha\beta;\gamma\delta} \left[1{\widetilde{\otimes}}1-1{\widetilde{\odot}}1\right]^{ij;kl} \nonumber \\ &\times & \{ T C_F + A_{VA}(C_F -\frac{1}{2} - \frac{1}{2N}) +A_{SP}(\frac{1}{2} + \frac{1}{2N}) \}, \label{eq:sumM}\end{aligned}$$ where $$A_{SP}=\cos^2(k/2)\cdot \sin^2 k. \label{eq:A_SP}$$ We should note that the other three contributions $I_\pm^{a^\prime,b^\prime,c^\prime}$ from Fig. 1$a^\prime$, 1$b^\prime$ and 1$c^\prime$ are equal to $I_\Gamma^{a,b,c}$ respectively, therefore the factors $1/2$ in eqs. (\[eq:sumP\]) and (\[eq:sumM\]) disappear in the total contributions of all. Comparing the one-loop results to the tree level ones we obtain $$\begin{aligned} \Lambda_+ &=& \left[ 1 + g^2 \frac{N-1}{N}\left\{ <T> (N+1)+<A_{VA}>(N+2)-<A_{SP}> \right\}\right] \Lambda_+^{(0)}, \label{eq:1loop_+}\\ \Lambda_- &=& \left[ 1 + g^2 \frac{N+1}{N}\left\{ <T> (N-1)+<A_{VA}>(N-2)+<A_{SP}>\right\}\right] \Lambda_-^{(0)} \label{eq:1loop_-} \end{aligned}$$ with $$< X > = \int_{-\pi}^{\pi}\frac{d^4 k}{(2\pi)^4} K(k) X(k) \label{eq:integ_X}$$ for $X = T, A_{VA}, A_{SP}$. We remark that $C_F< T+A_{VA} >$ and $C_F< T+A_{SP} >$ correspond to the one-loop vertex corrections to the (axial) vector current and the (pseudo) scalar density which are expressed as $(T_{VA}-1)$ and $(T_{SP}-1)$ in Ref. [@AIKT98]. The expressions of eqs.(\[eq:1loop\_+\]) and (\[eq:1loop\_-\]) show an important property of ${\cal O}_{\pm}$ in the DWQCD formalism: the one-loop vertex corrections are multiplicative. This is contrary to the Wilson case, in which the mixing operators with different chiralities appears at the one-loop level[@pt_w4]. We next turn to the case of ${\cal O}_{1,2}$. For ${\cal O}_1$ the vertex corrections of eqs.(\[eq:vtx\_a\]), (\[eq:vtx\_b\]) and (\[eq:vtx\_c\]) with $\Gamma_X = \gamma_\mu^L$ and $\Gamma_Y = \gamma_\mu^R$ are written as $$\begin{aligned} I_1^a &=& \frac{1}{2} g^2 K (T+A_{VA}) \left[\gamma_\nu^L\otimes\gamma_\nu^R\right] \left[ (-N C_F+ \frac{1}{2}) 1{\widetilde{\otimes}}1 - \frac{1}{2N} 1{\widetilde{\odot}}1 \right], \\ I_1^b &=& - \frac{1}{2} g^2 K (T+A_{VA}) \left[\gamma_\nu^L\otimes\gamma_\nu^R\right] \left[ (\frac{1}{2}+ \frac{1}{2}) 1{\widetilde{\otimes}}1 +(-\frac{N}{2}- \frac{1}{2N}) 1{\widetilde{\odot}}1 \right], \\ I_1^c &=& \frac{1}{2} g^2 K (T+A_{SP}) \left[\gamma_\nu^L\otimes\gamma_\nu^R\right] \left[ \frac{1}{2} 1{\widetilde{\otimes}}1 +(-\frac{N}{2}+ C_F) 1{\widetilde{\odot}}1 \right]. \end{aligned}$$ The total contribution including those from Fig. $1a^\prime$, $1b^\prime$ and $1c^\prime$ is given by $$\begin{aligned} 2(I_1^a+I_1^b+I_1^c) &=& 2\frac{1}{2} g^2 K \left[\gamma_\nu^L\otimes\gamma_\nu^R\right]_{\alpha\beta;\gamma\delta} \left[- N 1{\widetilde{\otimes}} 1 +1{\widetilde{\odot}} 1\right]^{ij;kl} \nonumber \\&\times & \left[ T C_F + A_{VA}\frac{N}{2}-A_{SP}\frac{1}{2N} \right].\end{aligned}$$ Using the tree level result in eq.(\[eq:lambda\_1\_0\]) the vertex function up to the one-loop level is expressed as $$\begin{aligned} \Lambda_1 &=& \left[ 1 + g^2 \frac{1}{N}\left\{ <T> (N^2-1)+<A_{VA}>N^2-<A_{SP}>\right\}\right] \Lambda_1^{(0)}.\end{aligned}$$ This result shows that the operator ${\cal O}_1$ is multiplicatively renormalizable in DWQCD, which is in contrast with the Wilson case[@pt_w4]. In a similar way we write the vertex corrections for ${\cal O}_2$. $$\begin{aligned} I_2^a &=& \frac{1}{2} g^2 K (T+A_{VA}) \left[\gamma_\nu^L\otimes\gamma_\nu^R\right] \left[ \frac{1}{2} 1{\widetilde{\otimes}}1 - \frac{1}{2N} 1{\widetilde{\odot}}1 \right], \\ I_2^b &=& - \frac{1}{2} g^2 K (T+A_{VA}) \left[\gamma_\nu^L\otimes\gamma_\nu^R\right] \left[ \frac{1}{2} 1{\widetilde{\otimes}}1 - \frac{1}{2N} 1{\widetilde{\odot}}1 \right], \\ I_2^c &=& \frac{1}{2} g^2 K (T+A_{SP}) \left[\gamma_\nu^L\otimes\gamma_\nu^R\right] \left[ C_F 1{\widetilde{\odot}}1 \right]. \end{aligned}$$ The total contribution including those from Fig. $1a^\prime$, $1b^\prime$ and $1c^\prime$ becomes $$\begin{aligned} 2(I_2^a+I_2^b+I_2^c) &=& 2\frac{1}{2}g^2 K \left[\gamma_\nu^L\otimes\gamma_\nu^R\right]_{\alpha\beta;\gamma\delta} \left[1{\widetilde{\odot}} 1 \right]^{ij;kl} \nonumber \\ &\times & C_F\left[ T + A_{SP}\right],\end{aligned}$$ which leads to $$\begin{aligned} \Lambda_2 &=& \left[ 1 + g^2 \frac{N^2-1}{N}\left\{ <T> +<A_{SP}>\right\}\right] \Lambda_2^{(0)}.\end{aligned}$$ We again find that the vertex correction is multiplicative up to the one-loop level as opposed to the Wilson case[@pt_w4]. The contribution from the fermion self-energy has already been evaluated [@Aoki-Taniguchi; @AIKT98] and the total lattice renormalization factor is now obtained: $$Z_\Gamma^{lat} = (1-w_0^2)^2 Z_w^2 Z_2^2 V_\Gamma,$$ where $$\begin{aligned} Z_2 &=& 1 + \frac{g^2}{16\pi^2} C_F \left[ \log (\lambda a)^2 + \Sigma_1\right], \\ V_\Gamma &=& 1 + \frac{g^2}{16\pi^2} \left[ -\delta_\Gamma \log (\lambda a)^2 + v_\Gamma \right],\end{aligned}$$ $$\begin{aligned} v_+ &=&\frac{16\pi^2(N-1)}{N}\left[ <T> (N+1)+<<A_{VA}>>(N+2)-<<A_{SP}>> \right] +\delta_+ \log \pi^2, \\ v_- &=&\frac{16\pi^2(N+1)}{N}\left[ <T> (N-1)+<<A_{VA}>>(N-2)+<<A_{SP}>> \right] +\delta_- \log \pi^2, \\ v_1 &=&\frac{16\pi^2}{N}\left[ <T> (N^2-1)+<<A_{VA}>>N^2-<<A_{SP}>> \right] +\delta_1 \log \pi^2, \\ v_2 &=&\frac{16\pi^2(N^2-1)}{N}\left[ <T> + <<A_{SP}>> \right] +\delta_2 \log \pi^2 \end{aligned}$$ with $$\delta_\Gamma = \left\{ \begin{array}{ll} \displaystyle \frac{(N-1)(N-2)}{N} & \qquad \Gamma = + \\ & \\ \displaystyle \frac{(N+1)(N+2)}{N} & \qquad \Gamma = - \\ & \\ \displaystyle \frac{(N+2)(N-2)}{N} & \qquad \Gamma = 1 \\ & \\ \displaystyle \frac{4(N+1)(N-1)}{N} & \qquad \Gamma = 2 \end{array} \right.$$ The infrared singularity of $< A_X>$ is subtracted as $$<< A_X >> = \int_{-\pi}^{\pi}\frac{d^4k}{(2\pi)^4} \left[K(k) A_X(k) - c_X \frac{1}{(k^2)^2}\theta(\pi^2-k^2)\right]$$ with $c_{SP}=4$ and $c_{VA}=1$. Numerical values of $v_\Gamma$ are evaluated by two independent methods. In one method the momentum integration is performed by a mode sum for a periodic box of a size $L^4$ after transforming the momentum variable through $k_\mu = q_\mu -\sin q_\mu$. We employ the size $L=64$ for integrals. In the other method the momentum integration is carried out by the the Monte Carlo integration routine VEGAS, using 20 samples of 1000000 points each. We find that both results agree very well. Numerical values of $v_\Gamma$ are presented in Table \[tab:4fermi\] as a function of $M$. We have to also calculate the corresponding continuum wave-function renormalization factor and vertex corrections in the ${{\overline {\rm MS}}}$ scheme employing the same gauge and the same infrared regulator as the lattice case. For the present calculation it seems preferable to choose Dimensional Reduction(DRED) as the ultraviolet regularization, in which the loop momenta of the Feynman integrals are defined in $D<4$ dimensions while keeping the Dirac matrices in four dimensions. In the DRED scheme we can use the same calculational techniques for the vertex corrections as the lattice case thanks to applicability of the Fierz transformation for the Dirac matrices. For the wave-function renormalization factor a simple calculation gives $$Z_2^{{{\overline {\rm MS}}}} = 1+ \frac{g^2}{16\pi^2}C_F \left[ \log (\lambda/\mu)^2 -1/2\right],$$ where $\mu$ is a renormalization scale. This result leads to $\Sigma_1^{{{\overline {\rm MS}}}}=-1/2$. For the vertex corrections we obtain $$V_\Gamma^{{{\overline {\rm MS}}}} = 1+\frac{g^2}{16\pi^2} \delta_\Gamma \left[ -\log (\lambda/\mu)^2 +1\right],$$ giving $v_\Gamma^{{{\overline {\rm MS}}}}=\delta_\Gamma$. Here we should remark that the one-loop vertex corrections yield the evanescent operators which vanish in $D=4$ for the DRED scheme[@BW]. It is meaningless to give results without mentioning the definition of evanescent operators, because the constant terms at the one-loop level depend on the definition of the evanescent operators. Our choice is as follows: $$\begin{aligned} E^{\rm DRED}_{\pm}=& {\bar \delta}_{\mu\nu}\gamma_\mu(1-\gamma_5) \otimes\gamma_\nu(1-\gamma_5) -\frac{D}{4}\gamma_\mu(1-\gamma_5)\otimes\gamma_\mu(1-\gamma_5), \\ E^{\rm DRED}_{1,2}=& {\bar \delta}_{\mu\nu}\gamma_\mu(1-\gamma_5) \otimes\gamma_\nu(1+\gamma_5) -\frac{D}{4}\gamma_\mu(1-\gamma_5)\otimes\gamma_\mu(1+\gamma_5),\end{aligned}$$ where ${\bar \delta}_{\mu\nu}$ is the $D$-dimensional metric tensor which emerges inevitably in the evaluation of the Feynman integrals. Combining these results with the previous lattice ones we obtain $$\begin{aligned} {\cal O}_\Gamma^{{{\overline {\rm MS}}}}(\mu) & = &\frac{1}{(1-w_0^2)^2 Z_w^2} Z_\Gamma (\mu a ) {\cal O}_\Gamma^{lat} (1/a),\end{aligned}$$ where $$\begin{aligned} Z_\Gamma (\mu a) &=& \frac{ (Z_2^{{{\overline {\rm MS}}}})^2 V_\Gamma^{{{\overline {\rm MS}}}}} {(Z_2)^2 V_\Gamma} \nonumber \\ &=& 1 + \frac{g^2}{16\pi^2}\left[ (\delta_\Gamma - 2 C_F)\log (\mu a)^2 + z_\Gamma \right], \label{eq:4fermi} \\ z_\Gamma &=& v_\Gamma^{{{\overline {\rm MS}}}} - v_\Gamma + 2 C_F\{\Sigma_1^{{{\overline {\rm MS}}}} -\Sigma_1\} .\end{aligned}$$ Numerical values of $z_\Gamma$ are given in Table \[tab:total\] and the results for the mean-field improved one, $z_\Gamma^{MF}$ , are also given in Table \[tab:totalMF\]. Although the results for the DRED scheme are presented here, it is an easy task to obtain those for the Naive Dimensional Regularization(NDR) scheme. In Appendix B we summarize the finite parts of the wave-function renormalization factor and vertex corrections in the NDR scheme. Renormalization factors for three-quark operators {#sec:3fermi} ================================================= The three-quark operators relevant to the proton decay amplitude are given by $$\left({\cal O}_{PD}\right)_\delta =\varepsilon^{abc} \left((\bar q^C_1)^a \Gamma_X (q_2)^b\right) (\Gamma_Y (q_3)^c)_\delta, \label{eq:O_PD}$$ where $\bar q^C = -q^T C^{-1}$ with $C=\gamma_0\gamma_2$ is a charge conjugated field of $q$ and $\Gamma_X\otimes\Gamma_Y = P_R\otimes P_R, P_R\otimes P_L, P_L\otimes P_R, P_L\otimes P_L$. The summation over repeated color indices $a,b,c$ is assumed. We should note that the domain-wall fermion action [(\[eqn:action\])]{} is transformed identically into that with the conjugated field by using transpose and matrix $C$. The resultant action and Feynman rules for the conjugated field is obtained by the replacement that $$\begin{aligned} ig T^A &\rightarrow & -ig (T^A)^T , $$ where the superscript $T$ means the transposed matrix. In order to evaluate the vertex corrections we consider the following Green function: $$\langle \left({\cal O}_{PD}\right)_\delta \rangle_{\alpha\beta\gamma}^{ijk} \equiv {\langle \left({\cal O}_{PD}\right)_\delta (q^C_1)_\alpha^i(\bar q_2)_\beta^j (\bar q_3)_\gamma^k \rangle},$$ where $\alpha,\beta,\gamma$ and $i,j,k$ are spinor and color indices respectively. Truncating the external quark propagators of $\langle {\cal O}_{PD}\rangle$ we obtain the vertex function $$(1-w_0^2)^3\left(\Lambda_{PD}\right)_{\alpha\beta;\delta\gamma}^{ijk} =(1-w_0^2)^3\left(\Lambda_\Gamma^{(0)} +\Lambda_\Gamma^{(1)}\right)_{\alpha\beta;\delta\gamma}^{ijk},$$ where the trivial factor $(1-w_0^2)^3$ is factored out for the convenience. We suppress the external momenta $p_i$ since the renormalization factor does not depend on them. At the tree level the vertex function takes the form $$\Lambda_\Gamma^{(0)}=\varepsilon^{ijk} \left[ \Gamma_X \otimes \Gamma_Y \right]_{\alpha\beta;\delta\gamma}, \label{eq:PD_tree}$$ where $[\Gamma_X \otimes \Gamma_Y]_{\alpha\beta;\delta\gamma} \equiv (\Gamma_X)_{\alpha\beta}(\Gamma_Y)_{\delta\gamma}$. The one-loop vertex corrections are shown in Figs. 2a, 2b and 2c, the sum of which gives the one-loop level vertex function $$\Lambda_{PD}^{(1)}= \int_{-\pi}^{\pi}\frac{d^4 k}{(2\pi)^4} \left(I_{PD}^a+I_{PD}^b+I_{PD}^{c}\right).$$ Using the notations in eqs.(\[eq:G\_b\]), (\[eq:G\_t\]) and (\[eq:V\_bt\]) the integrands $I_{PD}^{a,b,c}$ are written as follows: $$\begin{aligned} I_{PD}^a &=& \varepsilon^{abk}(-T^A)^T_{ia}T^B_{bj} {\nonumber}\\ &&\times \left\{{\overline{V}}_\mu(k) {\overline{G}}(k) + {\widetilde{V}}_\mu (k){\widetilde{G}}(k)\right\} \Gamma_X \left\{{\overline{G}}(k) {\overline{V}}_\nu(k) + {\widetilde{G}}(k) {\widetilde{V}}_\nu(k)\right\} \otimes \Gamma_Y G_{\mu\nu}^{AB}(k), \\ I_{PD}^b &=& \varepsilon^{ibc}T^A_{bj}T^B_{ck} {\nonumber}\\ &&\times \Gamma_X\left\{{\overline{G}}(k) {\overline{V}}_\nu(k) + {\widetilde{G}}(k) {\widetilde{V}}_\nu(k)\right\} \otimes\Gamma_Y\left\{{\overline{G}}(-k) {\overline{V}}_\nu(-k) + {\widetilde{G}}(-k) {\widetilde{V}}_\nu(-k)\right\} G_{\mu\nu}^{AB}(k), \\ I_{PD}^c &=& \varepsilon^{ajc}(-T^A)^T_{ia}T^B_{ck} {\nonumber}\\ &&\times \left\{{\overline{V}}_\mu(k) {\overline{G}}(k) + {\widetilde{V}}_\mu (k) {\widetilde{G}}(k)\right\} \Gamma_X\otimes \Gamma_Y \left\{{\overline{G}}(k){\overline{V}}_\nu(k) + {\widetilde{G}}(k){\widetilde{V}}_\nu(k) \right\} G_{\mu\nu}^{AB}(k).\end{aligned}$$ A little algebra yields $$\begin{aligned} I_{PD}^a &=& g^2\frac{N+1}{2N}\varepsilon^{ijk} K \left[ T + A_{SP}\right]\left[\Gamma_X\otimes\Gamma_Y\right] , \\ I_{PD}^b &=& g^2\frac{N+1}{2N}\varepsilon^{ijk} K \left[T (\Gamma_X\otimes\Gamma_Y) + \cos^2 (k_\mu/2) \sin^2 k_\alpha (\Gamma_X\gamma_\alpha\gamma_\mu) \otimes (\Gamma_Y\gamma_\alpha\gamma_\mu) \right], \\ I_{PD}^c &=& g^2\frac{N+1}{2N}\varepsilon^{ijk} K \left[T (\Gamma_X \otimes\Gamma_Y) + \cos^2 (k_\mu/2) \sin^2 k_\alpha (\gamma_\mu\gamma_\alpha\Gamma_X) \otimes (\Gamma_Y\gamma_\alpha\gamma_\mu) \right],\end{aligned}$$ where $K$, $T$ and $A_{SP}$ are given in eqs.(\[eq:K\]), (\[eq:T\]) and (\[eq:A\_SP\]). It is noted that a sum of $I_{PD}^b$ and $I_{PD}^c$ becomes $$\begin{aligned} & &g^2\frac{N+1}{2N}\varepsilon^{ijk} K \left[2T (\Gamma_X \otimes\Gamma_Y) + \cos^2 (k_\mu/2) \sin^2 k_\alpha (\gamma_\mu\gamma_\alpha\Gamma_X+ \Gamma_X\gamma_\alpha\gamma_\mu) \otimes (\Gamma_Y\gamma_\alpha\gamma_\mu) \right] \nonumber \\ & = & g^2\frac{N+1}{2N}\varepsilon^{ijk} K \left[2T (\Gamma_X \otimes\Gamma_Y) + \cos^2 (k_\mu/2) \sin^2 k_\alpha (\{\gamma_\mu\gamma_\alpha +\gamma_\alpha\gamma_\mu \}\Gamma_X) \otimes (\Gamma_Y\gamma_\alpha\gamma_\mu) \right ]\end{aligned}$$ for $\Gamma_X = P_R$ or $P_L$, therefore no Fierz transformation is necessary to simplify the spinor structure of the total contribution. Finally we obtain $$\begin{aligned} I_{PD}^a+I_{PD}^b+I_{PD}^c &=& g^2\frac{N+1}{2N}\varepsilon^{ijk} K \left[\Gamma_X\otimes\Gamma_Y\right] \left[ 3T + A_{SP} + 2A_{VA} \right].\end{aligned}$$ Compared with the tree level result of eq.(\[eq:PD\_tree\]) we find that the vertex correction is multiplicative up to the one-loop level: $$\Lambda_{PD} = \left[ 1 + g^2 \frac{N+1}{2N}\left\{ 3<T> + <A_{SP}> + 2<A_{VA}> \right\}\right] \Lambda_{PD}^{(0)},$$ where $<X>$ $(X = T, A_{VA}, A_{SP})$ are defined in eq.(\[eq:integ\_X\]). We remark that in the Wilson case ${\cal O}_{PD}$ mixes with other operators which have different chiral structures under renormalization[@pt_w3]. Taking account of the contribution of the wave function the lattice renormalization factor for ${\cal O}_{PD}$ is expressed as $$Z_{PD}^{lat} = (1-w_0^2)^{3/2}Z_w^{3/2} Z_2^{3/2}V_{PD},$$ where $$\begin{aligned} V_{PD} &=& 1+ \frac{g^2}{16\pi^2}\left[-\delta_{PD}\log (\lambda a)^2 + v_{PD}\right],\end{aligned}$$ $$\begin{aligned} v_{PD} &=& \frac{16\pi^2 (N+1)}{2N}\left[ 3 <T> + <<A_{SP}>> + 2 <<A_{VA}>> \right]+\delta_{PD}\log\pi^2\end{aligned}$$ with $\delta_{PD}=\displaystyle\frac{6(N+1)}{2N}$. Numerical values for $v_{PD}$, evaluated as before, are given in Table \[tab:4fermi\] as a function of $M$. The corresponding continuum renormalization factors in the ${{\overline {\rm MS}}}$ scheme are calculated employing the DRED scheme as the regularization in the Feynman gauge with the fictitious gluon mass $\lambda$. The vertex correction for ${\cal O}_{PD}$ is $$\begin{aligned} V_{PD}^{{{\overline {\rm MS}}}} &=& 1+ \frac{g^2}{16\pi^2}\delta_{PD} \left[-\log (\lambda/\mu)^2 +1 \right],\end{aligned}$$ giving $v_{PD}^{{{\overline {\rm MS}}}}=\delta_{PD}$. We remark that in this case the evanescent operator does not appear at the one-loop level. Combining this result with the previous lattice one we finally obtain the relation between the operators ${\cal O}_{PD}^{{{\overline {\rm MS}}}}$ and ${\cal O}_{PD}^{lat}$: $$\begin{aligned} {\cal O}_{PD}^{{{\overline {\rm MS}}}}(\mu) & = &\frac{1}{(1-w_0^2)^{3/2} Z_w^{3/2}} Z_{PD} (\mu a ) {\cal O}_{PD}^{lat} (1/a),\end{aligned}$$ with $$\begin{aligned} Z_{PD} (\mu a) &=& \frac{ (Z_2^{{{\overline {\rm MS}}}})^{3/2} V_{PD}^{{{\overline {\rm MS}}}}} {(Z_2)^{3/2} V_{PD}} \nonumber \\ &=& 1 + \frac{g^2}{16\pi^2}\left[ (\delta_{PD} - 3 C_F/2)\log (\mu a)^2 + z_{PD} \right], \\ z_{PD} &=& v_{PD}^{{{\overline {\rm MS}}}} - v_{PD} + \frac{3}{2} C_F\{\Sigma_1^{{{\overline {\rm MS}}}} -\Sigma_1\}.\end{aligned}$$ We present numerical values for $z_{PD}$ in Table \[tab:total\] and those for the mean-field improved one, $z_{PD}^{MF}$, in Table \[tab:totalMF\]. Renormalization factor for $B_K$ {#sec:bk} ================================ As an application of results in the previous sections, we estimate a renormalization factor for the kaon $B$ parameter $B_K$, defined by $$B_K =\frac{\langle \overline{K}^0 \vert {\cal O}_+ \vert K^0 \rangle} {\frac{8}{3} \langle \overline{K}^0 \vert A_4 \vert 0 \rangle \langle 0 \vert A_4 \vert K^0 \rangle}$$ with $q_1=q_3 = s$ and $q_2=q_4= d$ in ${\cal O}_+$. Denoting the renormalization factor between the continuum $B_K$ at scale $\mu$ and the lattice one at scale $1/a$ as $Z_{B_K}( \mu a )$, we obtain $$Z_{B_K}( \mu a ) =\frac{(1-w_0^2)^{-2} Z_w^{-2} Z_+ (\mu a )} {(1-w_0)^{-2} Z_w^{-2} Z_A(\mu a)^2} =\frac{Z_+ (\mu a )}{Z_A(\mu a)^2},$$ where $$Z_+(\mu a) = 1 + \frac{g^2}{16\pi^2}\left[ -4\log (\mu a) + z_+ \right]$$ from eq. (\[eq:4fermi\]) in this paper, and $$Z_A(\mu a) = 1 + \frac{g^2 C_F}{16\pi^2} z_A$$ from Ref. [@AIKT98], so that $$Z_{B_K}(\mu a) = 1 +\frac{g^2}{16\pi^2}\left[ -4\log (\mu a) + z_+ -2 C_F z_A \right] .$$ Note that $z_A$ in Ref. [@AIKT98] is evaluated in the NDR scheme while the DRED scheme is used for $z_+$ in this paper. From the result in Appendix B we have $$z_A({\rm DRED}) = z_A({\rm NDR}) +1/2, \qquad z_+({\rm DRED}) = z_+({\rm NDR}) +3 .$$ In Ref.[@Blum-Soni] $B_K$ has been evaluated at $\beta = 5.85$, 6.0 with $ M= 1.7$ and $\beta = 6.3$ with $M=1.5$, using domain-wall QCD with the quenched approximation. Here we explicitly calculate $Z_{B_K}( \mu a )$ for these parameters. From Table \[tab:total\] and the previous result[@AIKT98], $z_+ = -41.854 (-42.399)$, $z_A = -17.039 (-16.827)$ and $z_+ -2C_F z_A = 3.583 (2.473)$ for $M=1.7 (1.5)$ in the DRED scheme, and $z_+ = -44.854 (-45.399)$, $z_A = -17.539 (-17.327)$ and $z_+ -2C_F z_A = 1.917 (0.8063)$ for $M=1.7 (1.5)$ in the NDR scheme. Taking $\mu =1/a$ and $g^2 = g^2_{\overline{MS}}(1/a)$, estimated by the formula $$\frac{1}{g^2_{\overline{MS}}}(1/a) = P\frac{\beta}{6} -0.13486$$ for the quenched QCD with $P$ being the average value of the plaquette, we have $Z_{B_K} = $ 1.053 (1.029), 1.049 (1.026) and 1.030 (1.010) at $\beta =$ 5.85, 6.0 and 6.3, respectively, in the DRED (NDR) scheme. Sizes of one-loop corrections for $B_K$ are not so large, $1-5\%$, at these $\beta$ values even without mean-field improvement, since the large contribution, which comes from a $(1-w_0) Z_w$ factor, cancels out in the ratio of ${\cal O}_+$ and $A_4^2$. If we employ the mean-field improvement by replacing $M\rightarrow {\widetilde{M}}=M+4(u-1)$ with $u = P^{1/4}$, we obtain $Z_{B_K} = $ 1.018 (0.994), 1.017 (0.994) and 1.009 (0.988) at $\beta =$ 5.85, 6.0 and 6.3, respectively, in the DRED (NDR) scheme. See Appendix A for some remarks. Note that there is no mean-field improvement factor for $B_K$ in actual simulations since, as mentioned before, it is defined by the ratio. Therefore the difference between values of $Z_{B_K}$ with and without mean-field improvement comes from higher order ambiguity in perturbation theory. Necessary informations for the analysis in this section are given in Table \[tab:bk\], together with values of $Z_{B_K}$. Conclusion {#sec:concl} ========== In this paper we have calculated the one-loop contributions for the renormalization factors of the three- and four-quark operators in DWQCD. We have demonstrated that the three- and four-quark operators in DWQCD can be renormalized without any operator mixing between different chiralities as opposed to the Wilson case. This desirable property in DWQCD would practically surpass the cost of the introduction of an unphysical fifth dimension. The numerical values for the finite parts $z_X$ with $X=\pm,1,2,PD$ settle in reasonable magnitude with the mean-field improvement, while unimproved values are rather large in general. In this work we do not treat the operators which yield the so-called “penguin” diagram. It seems feasible to carry out the calculation of their renormalization factors, which we leave to future investigation. Acknowledgments {#acknowledgments .unnumbered} =============== This work is supported in part by the Grants-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (Nos. 2373, 2375). T. I., Y. K. and Y. T. are supported by Japan Society for Promotion of Science. Appendix A: Mean-field improvement {#appendix-a-mean-field-improvement .unnumbered} ================================== The mean-field improvement[@MF] in our paper uses $$u= 1 - \frac{g^2 C_F}{2}T$$ with $T= 0.15493$, which is the value for the link in Feynman gauge. It may be better to use $u$ from $K_c$ or plaquette in DWQCD. In that case $$u = 1 - \frac{g^2 C_F}{2}(T + \delta T),$$ where $\delta T =0.00793$ for $K_c$, or $\delta T = -0.02993$ for plaquette. Accordingly we have to modify renormalization factors as follows: $$\begin{aligned} z_w^{MF}(T+\delta T) &=& z_w^{MF}(T)+\frac{2w_0}{1-w_0^2}16\pi^2\times 2\delta T, \\ z_2^{MF}(T+\delta T) &=&z_2^{MF}(T)+16\pi^2\times \delta T /2, \\ z_{\Gamma,\rm bilinear}^{MF}(T+\delta T) &=& z_{\Gamma,\rm bilinear}^{MF}(T) +16\pi^2 \times \delta T/2, \\ z_{\Gamma,\rm 4-quark}^{MF}(T+\delta T) &=& z_{\Gamma,\rm 4-quark}^{MF}(T) +16\pi^2 \times\delta T/2 \times 2 C_F, \\ z_{\Gamma,\rm 3-quark}^{MF}(T+\delta T) &=& z_{\Gamma,\rm 3-quark}^{MF}(T) +16\pi^2 \times\delta T/2 \times \frac{3}{2} C_F .\end{aligned}$$ Appendix B: Naive Dimensional Regularization(NDR) {#appendix-b-naive-dimensional-regularizationndr .unnumbered} ================================================= In this appendix we compile the finite part of the renormalization constant in the ${{\overline {\rm MS}}}$ subtraction scheme with the Naive Dimensional Regularization: $$\begin{aligned} \Sigma_1^{{{\overline {\rm MS}}}} & =& 1/2, \\ v_+^{{{\overline {\rm MS}}}} &=& \delta_+\times\{3/2 - \frac{2N+3}{N-2}\}, \\ v_-^{{{\overline {\rm MS}}}} &=& \delta_-\times\{3/2 - \frac{2N-3}{N+2}\}, \\ v_{ij}^{{{\overline {\rm MS}}}} & = & \left( \begin{array}{cc} \delta_1 \times\left\{3/2-\displaystyle \frac{2(N^2-5)}{N^2-4}\right\}, & \delta_2 \times 3/4 \\ \displaystyle\frac{1}{N}\times 3, & \delta_2\times 1/2 \end{array} \right), \\ v_{PD}^{{{\overline {\rm MS}}}} & = & \delta_{PD}\times 2/3,\end{aligned}$$ where $v_{ij}$ with $i,j=1,2$ is a matrix, which represents the mixing of the finite part for ${\cal O}_{1,2}$. The one-loop vertex corrections for ${\cal O}_{\Gamma}$ $(\Gamma=\pm,1,2)$ require to specify their evanescent operators, which originates from the property that the Fierz transformation can not be defined in the NDR scheme. We employ $$\begin{aligned} E^{\rm NDR}_{\pm}&=& \gamma_\rho\gamma_\delta\gamma_\mu(1-\gamma_5)\otimes \gamma_\mu(1-\gamma_5)\gamma_\delta\gamma_\rho -{(2-D)^2}\gamma_\mu(1-\gamma_5)\otimes\gamma_\mu(1-\gamma_5),\\ E^{\rm NDR}_{1,2}&=& \gamma_\rho\gamma_\delta\gamma_\mu(1-\gamma_5)\otimes \gamma_\mu(1+\gamma_5)\gamma_\delta\gamma_\rho -{D^2}\gamma_\mu(1-\gamma_5)\otimes\gamma_\mu(1+\gamma_5),\end{aligned}$$ where $D$ is the reduced space-time dimension. On the other hand, the evanescent operator does not appear in the one-loop vertex correction of ${\cal O}_{PD}$. For later convenience, values of the finite part of quark bilinear operators are also given here. For NDR scheme $$z_{V,A}^{{{\overline {\rm MS}}}} = 0, \qquad z_{S,P}^{{{\overline {\rm MS}}}} = 5/2, \qquad z_T^{{{\overline {\rm MS}}}} = 1/2,$$ while for DRED scheme $$z_{V,A}^{{{\overline {\rm MS}}}} = 1/2, \qquad z_{S,P}^{{{\overline {\rm MS}}}} = 7/2, \qquad z_T^{{{\overline {\rm MS}}}} = -1/2,$$ where the evanescent operators are $$\begin{aligned} E^{\rm DRED}_{\gamma_\mu}&=& {\bar \delta}_{\mu\nu}\gamma_\nu-\frac{D}{4}\gamma_\mu, \\ E^{\rm DRED}_{\gamma_\mu\gamma_5}&=& {\bar \delta}_{\mu\nu}\gamma_\nu\gamma_5-\frac{D}{4}\gamma_\mu\gamma_5.\end{aligned}$$ Y. Shamir, [[*[Nucl. Phys.]{}*]{} [**B406**]{} (1993) 90]{}. V. Furman and Y. Shamir, [[*[Nucl. Phys.]{}*]{} [**B439**]{} (1995) 54]{}. T. Blum and A. Soni, [[*[Phys. Rev.]{}*]{} [**D56**]{} (1997) 174]{} ; hep-lat/9706023 ; hep-lat/9712004 . T. Blum, A. Soni and M. Wingate, hep-lat/9809065. T. Blum, hep-lat/9810017 and references therein. S. Aoki and Y. Taniguchi, hep-lat/9711004 (to appear in PRD). S. Aoki, T. Izubuchi, Y. Kuramashi and Y. Taniguchi, hep-lat/9810020 (to appear in PRD). G. Martinelli, [[*[Phys. Lett.]{}*]{} [**B141**]{} (1984) 395]{}; C. Bernard, T. Draper and A. Soni, [[*[Phys. Rev.]{}*]{} [**D36**]{} (1987) 3224]{}. A. J. Buras and P. H. Weisz, [[*[Nucl. Phys.]{}*]{} [**B333**]{} (1990) 66]{}; S. Herrlich and U. Nierste, [[*[Nucl. Phys.]{}*]{} [**B455**]{} (1995) 39]{}. D. G. Richards, C. T. Sachrajda and C. J. Scott, [[*[Nucl. Phys.]{}*]{} [**B286**]{} (1987) 683]{}. G. P. Lepage and P. Mackenzie, [[*[Phys. Rev.]{}*]{} [**D48**]{} (1993) 2250]{}. $\Gamma$ $J_a^{AB}$ $J_b^{AB}$ $J_c^{AB}$ ---------- -------------------------------------------------------------------- ------------------------------------------------------------------ -------------------------------------------------------------------- $\pm$ $T^A T^B {\widetilde{\otimes}} 1 \pm T^A {\widetilde{\odot}} T^B$ $T^A {\widetilde{\otimes}} T^B \pm T^A {\widetilde{\odot}} T^B$ $T^A {\widetilde{\otimes}} T^B \pm T^A T^B {\widetilde{\odot}} 1$ $1$ $-N T^A T^B {\widetilde{\otimes}} 1 + T^A {\widetilde{\odot}} T^B$ $-N T^A {\widetilde{\otimes}} T^B + T^A {\widetilde{\odot}} T^B$ $-N T^A {\widetilde{\otimes}} T^B + T^A T^B {\widetilde{\odot}} 1$ $2$ $T^A {\widetilde{\odot}} T^B$ $T^A {\widetilde{\odot}} T^B$ $T^A T^B {\widetilde{\odot}} 1$ : Color factors for $I_\Gamma^{a,b,c}$ ($\Gamma=\pm,1,2$).[]{data-label="tab:color"} $M$ $V_+$ $V_- $ $V_1$ $V_2$ $V_{PD}$ ------ ------------ ----------- ------------- ----------- ---------- 0.05 13.9096(8) 10.847(8) 13.3992(19) 8.805(12) 8.646(5) 0.10 13.5696 11.537 13.2309 10.182 8.992 0.15 13.2941 12.098 13.0948 11.301 9.273 0.20 13.0548 12.587 12.9768 12.275 9.518 0.25 12.8391 13.029 12.8708 13.155 9.740 0.30 12.6404 13.438 12.7734 13.970 9.946 0.35 12.4542 13.822 12.6822 14.734 10.139 0.40 12.2775 14.188 12.5960 15.462 10.323 0.45 12.1083 14.539 12.5135 16.160 10.499 0.50 11.9449 14.880 12.4341 16.837 10.671 0.55 11.7861 15.212 12.3571 17.496 10.838 0.60 11.6307 15.538 12.2819 18.143 11.002 0.65 11.4779 15.859 12.2081 18.780 11.164 0.70 11.3269 16.178 12.1354 19.412 11.325 0.75 11.1770 16.495 12.0634 20.041 11.485 0.80 11.0275 16.813 11.9917 20.670 11.645 0.85 10.8779 17.131 11.9201 21.300 11.806 0.90 10.7276 17.452 11.8484 21.935 11.968 0.95 10.5760 17.777 11.7762 22.578 12.133 1.00 10.4225 18.107 11.7033 23.230 12.300 1.05 10.2659 18.437 11.6278 23.885 12.466 1.10 10.1076 18.790 11.5547 24.579 12.646 1.15 9.9443 19.139 11.4768 25.269 12.822 1.20 9.7768 19.505 11.3981 25.990 13.007 1.25 9.6037 19.880 11.3165 26.732 13.198 1.30 9.4244 20.272 11.2323 27.504 13.396 1.35 9.2375 20.680 11.1446 28.309 13.603 1.40 9.0419 21.109 11.0530 29.153 13.820 1.45 8.8361 21.560 10.9567 30.042 14.049 1.50 8.6183 22.037 10.8547 30.983 14.291 1.55 8.3863 22.547 10.7464 31.987 14.550 1.60 8.1375 23.093 10.6301 33.063 14.827 1.65 7.8685 23.683 10.5043 34.227 15.127 1.70 7.5747 24.328 10.3668 35.496 15.454 1.75 7.2502 25.038 10.2149 36.897 15.814 1.80 6.8864 25.832 10.0440 38.463 16.216 1.85 6.4706 26.737 9.8482 40.247 16.675 1.90 5.9812 27.794 9.6168 42.337 17.210 1.95 5.3749 29.093 9.3281 44.907 17.867 : Numerical values for $V_\Gamma$ ($\Gamma=\pm,1,2,PD$) as a function of $M$.[]{data-label="tab:4fermi"} $M$ $z_+$ $z_-$ $z_1$ $z_2$ $z_{PD}$ ------ ------------- ------------- ------------ ------------- ------------ 0.05 -49.908(10) -40.845(11) -48.397(8) -34.803(15) -32.144(7) 0.10 -49.332 -41.300 -47.994 -35.945 -32.314 0.15 -48.847 -41.651 -47.647 -36.853 -32.437 0.20 -48.416 -41.949 -47.338 -37.637 -32.539 0.25 -48.025 -42.215 -47.057 -38.341 -32.629 0.30 -47.663 -42.461 -46.796 -38.993 -32.713 0.35 -47.325 -42.693 -46.553 -39.605 -32.792 0.40 -47.007 -42.918 -46.326 -40.191 -32.870 0.45 -46.706 -43.137 -46.111 -40.758 -32.948 0.50 -46.419 -43.354 -45.908 -41.311 -33.027 0.55 -46.145 -43.571 -45.716 -41.855 -33.108 0.60 -45.883 -43.790 -45.534 -42.395 -33.191 0.65 -45.631 -44.012 -45.361 -42.933 -33.279 0.70 -45.388 -44.239 -45.196 -43.473 -33.370 0.75 -45.153 -44.472 -45.040 -44.017 -33.467 0.80 -44.927 -44.712 -44.891 -44.569 -33.570 0.85 -44.708 -44.961 -44.750 -45.130 -33.679 0.90 -44.496 -45.220 -44.617 -45.704 -33.795 0.95 -44.290 -45.491 -44.490 -46.292 -33.918 1.00 -44.091 -45.776 -44.372 -46.899 -34.051 1.05 -43.898 -46.070 -44.260 -47.517 -34.190 1.10 -43.710 -46.392 -44.157 -48.181 -34.347 1.15 -43.528 -46.723 -44.061 -48.853 -34.510 1.20 -43.352 -47.079 -43.973 -49.565 -34.688 1.25 -43.180 -47.457 -43.893 -50.308 -34.880 1.30 -43.014 -47.862 -43.822 -51.094 -35.088 1.35 -42.853 -48.296 -43.760 -51.924 -35.315 1.40 -42.697 -48.763 -43.708 -52.808 -35.561 1.45 -42.545 -49.269 -43.666 -53.751 -35.831 1.50 -42.399 -49.818 -43.635 -54.763 -36.127 1.55 -42.257 -50.417 -43.617 -55.857 -36.453 1.60 -42.119 -51.074 -43.611 -57.044 -36.813 1.65 -41.985 -51.800 -43.621 -58.343 -37.214 1.70 -41.854 -52.607 -43.646 -59.776 -37.663 1.75 -41.725 -53.513 -43.690 -61.372 -38.170 1.80 -41.595 -54.541 -43.753 -63.172 -38.748 1.85 -41.460 -55.727 -43.838 -65.237 -39.417 1.90 -41.311 -57.124 -43.946 -67.666 -40.207 1.95 -41.121 -58.840 -44.074 -70.652 -41.177 : Numerical values for $z_\Gamma$ ($\Gamma=\pm,1,2,PD$) as a function of $M$.[]{data-label="tab:total"} $M$ $z_+^{MF}$ $z_-^{MF}$ $z_1^{MF}$ $z_2^{MF}$ $z_{PD}^{MF}$ ------ ------------- ------------ ------------ ------------ --------------- 0.05 -17.287(10) -8.224(11) -15.776(8) -2.182(15) -7.679(7) 0.10 -16.712 -8.679 -15.373 -3.324 -7.848 0.15 -16.226 -9.030 -15.027 -4.232 -7.972 0.20 -15.796 -9.328 -14.718 -5.016 -8.074 0.25 -15.404 -9.594 -14.436 -5.720 -8.164 0.30 -15.042 -9.840 -14.175 -6.372 -8.247 0.35 -14.705 -10.073 -13.933 -6.98 -8.326 0.40 -14.386 -10.297 -13.705 -7.57 -8.404 0.45 -14.085 -10.516 -13.490 -8.13 -8.482 0.50 -13.799 -10.734 -13.288 -8.69 -8.561 0.55 -13.525 -10.951 -13.096 -9.23 -8.642 0.60 -13.262 -11.169 -12.913 -9.77 -8.726 0.65 -13.010 -11.391 -12.740 -10.3 -8.813 0.70 -12.767 -11.618 -12.575 -10.8 -8.905 0.75 -12.532 -11.851 -12.419 -11.3 -9.002 0.80 -12.306 -12.091 -12.270 -11.9 -9.104 0.85 -12.087 -12.340 -12.129 -12.5 -9.213 0.90 -11.875 -12.600 -11.996 -13.0 -9.329 0.95 -11.670 -12.871 -11.870 -13.6 -9.453 1.00 -11.470 -13.155 -11.751 -14.2 -9.585 1.05 -11.278 -13.449 -11.639 -14.8 -9.725 1.10 -11.089 -13.772 -11.536 -15.5 -9.882 1.15 -10.908 -14.103 -11.440 -16.2 -10.044 1.20 -10.731 -14.459 -11.352 -16.9 -10.223 1.25 -10.559 -14.836 -11.272 -17.6 -10.414 1.30 -10.393 -15.241 -11.201 -18.4 -10.623 1.35 -10.232 -15.675 -11.139 -19.3 -10.849 1.40 -10.076 -16.143 -11.087 -20.1 -11.096 1.45 -9.925 -16.648 -11.045 -21.13 -11.365 1.50 -9.778 -17.197 -11.014 -22.14 -11.661 1.55 -9.636 -17.796 -10.996 -23.23 -11.987 1.60 -9.498 -18.453 -10.991 -24.42 -12.347 1.65 -9.364 -19.179 -11.000 -25.72 -12.749 1.70 -9.233 -19.986 -11.025 -27.15 -13.198 1.75 -9.104 -20.892 -11.069 -28.75 -13.705 1.80 -8.975 -21.920 -11.132 -30.55 -14.283 1.85 -8.840 -23.106 -11.217 -32.61 -14.952 1.90 -8.690 -24.503 -11.326 -35.04 -15.742 1.95 -8.500 -26.219 -11.453 -38.03 -16.711 : Numerical value for $z_\Gamma^{MF}$ ($\Gamma=\pm,1,2,PD$) as a function of $M$.[]{data-label="tab:totalMF"} ---------------------------- --------- --------- --------- --------- --------- --------- $\beta$ $M$ $P$ $g^2_{\overline{MS}}(1/a)$ $u$ ${\widetilde{M}}$ DRED NDR DRED NDR DRED NDR $z_+$ -41.854 -44.854 -41.854 -44.854 -42.399 -45.399 $z_A$ -17.039 -17.539 -17.039 -17.539 -16.827 -17.327 $z_+ - 2C_F z_A$ 3.583 1.917 3.583 1.917 2.473 0.806 $Z_{B_K}(\mu a=1)$ 1.053 1.029 1.049 1.026 1.030 1.010 $z_+^{MF}$ -17.033 -20.033 -17.033 -20.033 -17.580 -20.580 $z_A^{MF}$ -6.853 -7.353 -6.853 -7.353 -6.864 -7.364 $z_+^{MF} - 2C_F z_A^{MF}$ 1.242 -0.425 1.242 -0.425 0.724 -0.943 $Z_{B_K}^{MF}(\mu a=1)$ 1.018 0.994 1.017 0.994 1.009 0.988 ---------------------------- --------- --------- --------- --------- --------- --------- : Renormalization factor for $B_K$($1/a$) at some parameters.[]{data-label="tab:bk"} (420,450)(0,20) (60,60)\[l\][${\rm a}^\prime$]{} (70,166)[4.5]{} (70,154)[4.5]{} (68,166)(8,226) (132,226)(72,166) (68,154)(8,94) (132,94)(72,154) (20,105)(120,105)[5]{}[8]{} (60,260)\[l\][a]{} (0,440)\[l\][$\alpha,i$]{} (115,440)\[l\][$\beta,j$]{} (0,285)\[l\][$\gamma,k$]{} (115,285)\[l\][$\delta,l$]{} (70,366)[4.5]{} (70,354)[4.5]{} (68,366)(8,426) (132,426)(72,366) (68,354)(8,294) (132,294)(72,354) (20,415)(120,415)[-5]{}[8]{} (200,60)\[l\][${\rm b}^\prime$]{} (210,166)[4.5]{} (210,154)[4.5]{} (208,166)(148,226) (272,226)(212,166) (208,154)(148,94) (272,94)(212,154) (260,213)(260,107)[-5]{}[8]{} (200,260)\[l\][b]{} (210,366)[4.5]{} (210,354)[4.5]{} (208,366)(148,426) (272,426)(212,366) (208,354)(148,294) (272,294)(212,354) (160,413)(160,307)[5]{}[8]{} (340,60)\[l\][${\rm c}^\prime$]{} (350,166)[4.5]{} (350,154)[4.5]{} (348,166)(288,226) (412,226)(352,166) (348,154)(288,94) (412,94)(352,154) (350,160)(30,51,230)[-5]{}[7]{} (340,260)\[l\][c]{} (350,366)[4.5]{} (350,354)[4.5]{} (348,366)(288,426) (412,426)(352,366) (348,354)(288,294) (412,294)(352,354) (350,360)(30,-50,130)[-5]{}[7]{} (420,250)(0,20) (60,60)\[l\][a]{} (0,240)\[l\][$\alpha,i$]{} (115,240)\[l\][$\beta,j$]{} (115,85)\[l\][$\gamma,k$]{} (70,166)[4.5]{} (70,154)[4.5]{} (68,166)(8,226) (132,226)(72,166) (132,94)(72,154) (20,215)(120,215)[-5]{}[8]{} (200,60)\[l\][b]{} (210,166)[4.5]{} (210,154)[4.5]{} (208,166)(148,226) (272,226)(212,166) (272,94)(212,154) (260,213)(260,107)[-5]{}[8]{} (340,60)\[l\][c]{} (350,166)[4.5]{} (350,154)[4.5]{} (348,166)(288,226) (412,226)(352,166) (412,94)(352,154) (350,160)(30,-50,130)[-5]{}[7]{} [^1]: On leave from Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization(KEK), Tsukuba, Ibaraki 305-0801, Japan

--- abstract: | We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of $6/7$ of the statement of the Robertson–Seymour–Thomas intrinsic linking criterion. Let us call a cell complex [*dichotomial*]{} if to every nonempty cell there corresponds a unique nonempty cell with the complementary set of vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere; there exist precisely two $3$-dimensional dichotomial cell complexes, and their $1$-skeleta are $K_5$ and $K_{3,3}$; and precisely six $4$-dimensional ones, and their $1$-skeleta all but one graphs of the Petersen family. In higher dimensions $n\ge 3$, we observe that in order to characterize those compact $n$-polyhedra that embed in $\R^{2n}$ in terms of finitely many “prohibited minors”, it suffices to establish finiteness of the list of all $(n-1)$-connected $n$-dimensional finite cell complexes that do not embed in $\R^{2n}$ yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the $n$-skeleta of $(2n+1)$-dimensional dichotomial cell complexes. The $2$-skeleta of $5$-dimensional dichotomial cell complexes include (apart from the three joins of the $i$-skeleta of $(2i+2)$-simplices) at least ten non-simplicial complexes. address: 'Steklov Mathematical Institute of the Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia' author: - 'Sergey A. Melikhov' title: Combinatorics of embeddings --- [^1] Introduction {#intro} ============ This introduction attempts to motivate the notions we eventually arrive at. The fast reader may want to first look at the Main Theorem and its corollaries in §\[dichotomial spheres\]; these involve only the (very short) definitions of linkless and knotless embeddings (in §\[graphs\]), of a cell complex (in §\[notation\]) and of an $h$-minor (in §\[h-minors\]). However most of the examples and remarks in §\[section:main\] do depend on much of the preceding development. The constructively minded reader might be best guided by §\[algorithmic\], which explains how non-algorithmic topological notions such as PL spheres and contractible polyhedra can be eliminated from our results and conjectures. All posets, and in particular simplicial complexes, shall be implicitly assumed to be finite. By a [*graph*]{} we mean a $1$-dimensional simplicial complex (so no loops or multiple edges). By a [*circuit*]{} in a graph we mean a connected subgraph with all vertices of degree $2$. Following the terminology of PL topology (as opposed to that of convex geometry), by a [*polyhedron*]{} we mean a space triangulable by a simplicial complex, and moreover endowed with a PL structure, i.e. a family of compatible triangulations (see e.g. [@Hu]); by our convention above, all polyhedra are compact. The polyhedron triangulated by a simplicial complex $K$ is denoted $|K|$. All [*maps*]{} between polyhedra shall be assumed piecewise linear, unless stated otherwise. Two embeddings $f$, $g$ of a polyhedron in a sphere are considered [*equivalent*]{} if they are related by an isotopy $h_t$ of the sphere (i.e. $h_0=\operatorname{id}$ and $h_1f=g$). Graphs ------ The complete graph $K_5$ and the complete bipartite graph $K_{3,3}$ (also known as the utilities graph) are shown in Fig. 1. They can be viewed as the $1$-skeleton $(\Delta^4)^{(1)}$ of the $4$-simplex and the join $(\Delta^2)^{(0)}*(\Delta^2)^{(0)}$ of two copies of the three-point set. A graph $G$ contains no subgraph that is a subdivision of $K_5$ or $K_{3,3}$ iff $|G|$ embeds in $S^2$. Known proofs of the ‘only if’ part involve exhaustion of cases. A relatively short proof was given by Yu. Makarychev [@Mak] and further developed in [@Sk-A1]. An interesting configuration space approach was suggested by Sarkaria [@Sa3] (but beware of an error, pointed out in [@Sk-M]). Given the considerable difficulty of the result, it is astonishing that besides Kuratowski’s own proof [@Ku] (announced in 1929), there were independent contemporary proofs: by L. S. Pontryagin (unpublished[^2], but acknowledged in Kuratowski’s original paper [@Ku]), and by O. Frink and P. A. Smith, announced in 1930 [@FS], [@Wh]. A useful reformulation of Kuratowski’s theorem was suggested by K. Wagner [@Wa]. The following non-standard definition is equivalent to the standard one (see Proposition \[nevo\] below). We call a graph $H$ a [*minor*]{} of a graph $G$, if $H$ is obtained from a subgraph $F$ of $G$ by a sequence of edge contractions, where an edge contraction as a simplicial map $f$ that shrinks one edge $\{v_1,v_2\}$ onto a vertex, provided that $\operatorname{lk}\{v_1\}\cap\operatorname{lk}\{v_2\}=\emptyset$. The latter condition is equivalent to saying that all point-inverses of $|f|$ are points, except for one point inverse, which is an edge. It is easy to see that $S^2$ modulo an arc is homeomorphic to $S^2$ (cf. Lemma \[trivial\]); hence if $|G|$ embeds in $S^2$ and $H$ is a minor of $G$, then $|H|$ embeds in $S^2$. This observation along with Kuratowski’s theorem immediately imply A graph $G$ has no minor isomorphic to $K_5$ or $K_{3,3}$ iff $|G|$ embeds in $S^2$. We are now ready for a more substantial application of minors. We call an embedding $g$ of a $1$-polyhedron $\Gamma$ in $S^m$ [*knotless*]{} if for every circuit $C\subset\Gamma$, the restriction of $g$ to $C$ is an unknot. We call an embedding $g$ of a polyhedron $X$ into $S^m$ [*linkless*]{} if for every two disjoint closed subpolyhedra of $g(X)$, one is contained in an $m$-ball disjoint from the other one. An $n$-polyhedron admitting no linkless embedding in $S^{2n+1}$ is also known in the literature (at least for $n=1$) as an “intrinsically linked” polyhedron.  For $n>1$, the notion of a linkless embedding in $S^{2n+1}$ can be reformulated in terms of linking numbers of pairs of [*co*]{}cycles (see Lemma \[circuits\] and Lemma \[is-linkless\]). When $n=1$, variations of the notion of “linking” (such as existence of a nontrivial two-component sublink, or of a two-component sublink with nonzero linking number) lead to inequivalent versions of the notion of a linkless embedding in $\R^3$. It is amazing, however, that they all become equivalent upon adding a quantifier: \[RST0\] Let $G$ be a graph. If $|G|$ admits an embedding in $S^3$ that links every pair of disjoint circuits with an even linking number, then $|G|$ admits a linkless, knotless embedding in $S^3$. This is a very powerful result; it implies, [*inter alia*]{}, that the Whitney trick can be made to work in dimension four in a certain limited class of problems (Theorem \[Whitney\]). Theorem \[RST0\] above as well as the following Theorem \[RST\] are essentially due to Robertson, Seymour and Thomas [@RST], [@RST']; they had a different formulation but its equivalence with the present one is relatively easy [@M2] (see also Lemma \[panels\] and Remark \[erratum\] below). \[RST\] Let $G$ be a graph. \(a) Two linkless, knotless embeddings of $|G|$ in $S^3$ are inequivalent iff they differ already on $|H|$ for some subgraph $H$ of $G$, isomorphic to a subdivision of $K_5$ or $K_{3,3}$. \(b) $|G|$ linklessly embeds in $S^3$ iff $G$ has no minor in the Petersen family. The [*Petersen family*]{} of graphs is shown in Fig. 1 (disregard the colors for now) and includes the Petersen graph $P$, the complete graph $K_6$, the complete tripartite graph $K_{3,3,1}$, the graph $K_{4,4}\but$(edge), and three further graphs which we denote $\Gamma_7$, $\Gamma_8$ and $\Gamma_9$. These seven graphs can be defined as all the graphs obtainable from $K_6$ by a sequence of $\nabla{\mathrm Y}$- and ${\mathrm Y}\nabla$-[*exchanges*]{}, which as their name suggests interchange a copy of the $1$-skeleton of a $2$-simplex $\Delta^2\subset\Delta^3$ with a copy of the triod, identified with the remaining edges of the $1$-skeleton of $\Delta^3$. Of course, (a) can be reformulated in terms of minors, for given a minor $H$ of $G$ and a knotless and linkless embedding of $|G|$ in $S^3$, the homeomorphism $S^3/\text{tree}\cong S^3$ (see Lemma \[trivial\]) yields a knotless and linkless embedding of $|H|$ in $S^3$, unique up to equivalence (by an orientation-preserving homeomorphism of $S^3$). Robertson and Seymour have proved, in a series of twenty papers spanning over 500 pages, that every minor-closed family of graphs is characterized by a finite set of forbidden minors (see [@Di] for an outline). Let us abbreviate “a subdivision of a subgraph” to a [*$\tau$-subgraph*]{}. There exists a $\tau$-subgraph-closed family of trees that is not characterized by a finite set of forbidden $\tau$-subgraphs (cf. [@Di §12, Exercise 5]). However, it is well-known and easy to see that every minor-closed family of graphs that is characterized by a finite set $S$ of forbidden minors is also characterized by a possibly larger but still finite set $S^+$ of forbidden $\tau$-subgraphs. (For each $G\in S$ and each $v\in G$, replace the star $\operatorname{st}(v,G)$ by a tree with $\deg v$ leaves and no vertices of degree $2$. Since for every $v$ there are only finitely many such trees, we obtain a finite set of graphs containing $S^+$.) It is easy to see that $\{K_5,\,K_{3,3}\}^+=\{K_5,\,K_{3,3}\}$, but $\Pi^+\ne\Pi$ where $\Pi$ is the Petersen family. Y. Colin de Verdiere introduced a combinatorially defined invariant $\mu(G)$ of the graph $G$, for which it is known that $\mu(G)\le 3$ iff $|G|$ is planar, and $\mu(G)\le 4$ iff $|G|$ admits a linkless embedding in $S^3$ (see [@LS], [@Iz]). Van der Holst conjectured that $\mu(G)\le 5$ iff $|\hat G|$ has zero $\bmod 2$ van Kampen obstruction, where $\hat G$ is the cell complex obtained by glueing up all circuits in $G$ by $2$-cells [@Ho]. His supporting evidence for this conjecture is that if $G$ is any of the 78 graphs (called the [*Heawood family*]{} in [@Ho]), related by a sequence of $\nabla$Y- and Y$\nabla$-exchanges to $K_7$ or $K_{3,3,1,1}$, then $\mu(G)=6$, and $|\hat G|$ has nonzero $\bmod 2$ van Kampen obstruction (so in particular does not embed in $S^4$); but if $H$ is a proper minor of $G$, then $\mu(G)\le 5$ and $|\hat H|$ has zero $\bmod 2$ van Kampen obstruction [@Ho]. On the other hand, since $K_7$ contains circuits of length $\ge 4$, $\hat K_7$ itself contains a [*proper*]{} subcomplex isomorphic to the $2$-skeleton of the $6$-simplex, and it is well-known (see [@M2]) that $|(\Delta^6)^{(2)}|$ still has a nonzero $\bmod 2$ van Kampen obstruction. The similar proper subcomplex of $\hat K_{3,3,1,1}$ is discussed in §\[dichotomial spheres\] below. Let $E_n$ stand for “the problem of embeddability of a certain $n$-polyhedron in $S^{2n}$”, and let $L_n$ stand for “the problem of linkless embeddability of a certain $n$-polyhedron in $S^{2n+1}$”; it is understood that the polyhedron is specified, but nevertheless omitted in the notation. In §\[commensurability\] we describe a reduction (if and only if) of every $E_n$ to an $L_n$; and of every $L_n$ to an $E_{n+1}$ (Theorem \[commensuration\]). The construction is geometric, i.e. does not involve configuration spaces. The case $n=1$ was also done by van der Holst [@Ho]. Let us mention other known relations between embeddability and linkless embeddability. (i) M. Skopenkov has derived the non-embeddability of a certain $n$-polyhedron in $S^{2n}$ from non-existence of linkless embeddings of its links of vertices in $S^{2n-1}$ (thereby reducing a certain $E_n$ to a few $L_{n-1}$’s) by a geometric argument [@Sk-M2]. (ii) Conversely, the author derived the linkless non-embeddability of the $n$-skeleton of the $(2n+3)$-simplex in $S^{2n+1}$ from the fact that the $(n+1)$-skeleton of the $(2n+4)$-simplex has nonzero van Kampen obstruction even modulo $2$ (thereby reducing a certain $L_n$ to a certain strengthened $E_{n+1}$ by an algebraic argument with configuration spaces) [@M2 Example 4.7]. (iii) An odd-dimensional version of the van Kampen obstruction $\theta_{2n}$ to an $E_n$ can be identified as a complete obstruction $\theta_{2n+1}$ to an $L_n$ [@M2]. Arguments which might relate to a possible common higher dimensional generalization of the Kuratowski and Robertson–Seymour–Thomas theorems could include, apart from those in §\[commensurability\], those in [@Sa0], [@Sa1] (see also [@Um1]), [@Sk-A2]. Unfortunately, the present paper gives little clue to understanding [*proofs*]{} of the Kuratowski and Robertson–Seymour–Thomas theorems, but it does attempt to provide a better understanding of their [*statements*]{}. To this end let us first look at a more statistically representative selection of cases. Self-dual complexes ------------------- MacLane and Adkisson proved that two embeddings of a $1$-polyhedron $\Gamma$ in the plane are inequivalent iff they differ already on a copy of $S^1$ or on a copy of the triod contained in $\Gamma$ [@AdM]. We restate this as follows. If $K$ is a simplicial complex, then two embeddings of $|K|$ in $S^2$ are inequivalent iff they differ already on $|L|$ for some subcomplex $L$ of $K$, isomorphic to $\Delta^2$ or $(\Delta^2)^{(0)}*\Delta^0$ or a subdivision of $\Delta^0\sqcup\partial\Delta^2$. It is quite obvious also that $|K|$ embeds in $S^1$ iff $K$ contains no subcomplex isomorphic to $\Delta^2$ or $(\Delta^2)^{(0)}*\Delta^0$ or a subdivision of $\Delta^0\sqcup\partial\Delta^2$. Halin and Jung were able to go one dimension higher and gave a list of prohibited subcomplexes for the problem of embedding of a $2$-polyhedron in the plane [@HJ]. We again restate it as applied to the problem of embedding a polyhedron of an arbitrary dimension in $S^2$: If $K$ is a simplicial complex, $|K|$ embeds in $S^2$ iff it does not contain a subcomplex, isomorphic to a subdivision of $K_5$, $K_{3,3}$, or one of the following complexes: $$\begin{gathered} HJ_0\bydef\Delta^3,\\ HJ_1\bydef\Delta^0\sqcup\partial\Delta^3,\\ HJ_2\bydef\Delta^0*(\Delta^0\sqcup\partial\Delta^2),\\ HJ_3\bydef\Delta^1*\partial\Delta^1*\emptyset\cup \partial\Delta^1*\emptyset*\Delta^0\cup\emptyset*\Delta^1*\emptyset,\\ HJ_4\bydef(\Delta^2)^{(0)}*\Delta^1,\\ HJ_5\bydef\Delta^2*\emptyset\cup(\Delta^2)^{(0)}*\partial\Delta^1.\\\end{gathered}$$ See [@MTW Appendix A] for pictures of $HJ_1$ through $HJ_5$. We note that $|HJ_0|$, $|HJ_3|$, $|HJ_4|$ and $|HJ_5|$ each contain a subpolyhedron homeomorphic to one of $|K_5|$, $|K_{3,3}|$, $|HJ_1|$ and $|HJ_2|$. This leads to a shorter list of prohibited [*subpolyhedra*]{} for the problem of embedding a polyhedron in $S^2$. A proof of this weak version of the Halin–Jung theorem is rather easy modulo the Kuratowski theorem (see [@Sk-A1]). A statement like that of the preceding theorem might look bewildering, and just like with the previously mentioned results, its proof does not seem to explain what is special about these particular complexes. But the following definition, going back to Schild [@Sch], does it, by $7/8$: A subcomplex of $\partial\Delta^n$ is called [*self-dual*]{} in $\partial\Delta^n$ if it contains precisely one face out of each pair $\Delta^k$, $\Delta^{n-k-1}$ of complementary faces of $\Delta^n$. Now observe that $K_5$, as well as each $HJ_i$, is self-dual as a subcomplex of $\partial\Delta^4$. Indeed, this is obvious for $HJ_0$. But each $HJ_{i+1}$, can be obtained from $HJ_i$ by exchanging a pair of complementary faces of $\partial\Delta^4$, except that $HJ_5$ is obtained in this way from $HJ_3$ not $HJ_4$. Also, $K_5$ is obtained in this way from $HJ_5$. It is not hard to check that the seven complexes $K_5$ and the $HJ_i$, are in fact all the self-dual subcomplexes of $\partial\Delta^4$ (up to isomorphism). Note that $K_{3,3}$ is missing in this picture (but see Example \[Halin-Jung revisited\] below). Similarly one can check that the three complexes $\Delta^2$, $(\Delta^2)^{(0)}*\Delta^0$ and $\Delta^0\sqcup\partial\Delta^2$ in the MacLane–Adkisson theorem are precisely all the self-dual subcomplexes of $\partial\Delta^3$. This is no coincidence; in fact, the following general result already implies that the prohibited subcomplexes that occur in the Kuratowski, MacLane–Adkisson and Halin–Jung theorems as well as in part (a) of the Robertson–Seymour–Thomas theorem [*must*]{} occur there (possibly along with some additional ones). \[self-dual\] Let $K=K_1*\dots*K_r$, where each $K_i$ is a self-dual subcomplex of $\partial\Delta^{m_i}$, where $m_1+\dots+m_r=m$. Then \(a) [(Schild, 1993)]{} $|K|$ does not embed in $S^{m-2}$; but $|L|$ embeds in $S^{m-2}$ for every proper subcomplex $L$ of $K$; \(b) every embedding $g$ of $|K|$ in $S^{m-1}$ is inequivalent to $hg$, where $h$ is an orientation-reversing homeomorphism of $S^{m-1}$; but for every proper subcomplex $L$ of $K$, the restrictions of $j$ and $hj$ to $|L|$ are equivalent, where $j$ denotes the inclusion of $|K|$ in $|\partial\Delta^{m_1}*\dots*\partial\Delta^{m_r}|=S^{m-1}$. A simple proof of the assertions on proper subcomplexes in (a) and (b) is given below. A simple proof of the non-embeddability in (a) and the inequivalence in (b) is given in §\[embeddings\], where we also elaborate on historic/logical antecedents of Theorem \[self-dual\]. Let $\sigma$ be a maximal simplex of $K$ that is not in $L$. Then $\sigma=\sigma_1*\dots*\sigma_r$, where each $\sigma_i$ is a maximal (and in particular nonempty) simplex of $K_i$. Since $K_i$ is self-dual, the complementary simplex $\tau_i$ to $\sigma_i$ is not in $K_i$. It follows that $L\subset(\partial\tau_1*\dots*\partial\tau_r)*\partial\sigma$. The latter is a combinatorial sphere, which is of codimension $r+1$ in the $(m+r-1)$-simplex $(\tau_1*\dots*\tau_r)*\sigma=\Delta^{m_1}*\dots*\Delta^{m_r}$. The preceding construction exhibits, within the sphere $S^{m-1}$ that contains $|K|$, an embedded copy of $S^{m-2}$ that contains $|L|$. Let $r{\colon}S^{m-1}\to S^{m-1}$ be the reflection in $S^{m-2}$. Then $hr^{-1}=hr$ is orientation preserving, and so is isotopic to the identity by the Alexander trick. Hence $hj$ is equivalent to $rj=j$. \[van Kampen\] The above construction can be easily extended (cf. [@M2 Example 3.5]) to yield, for any maximal simplex $\sigma$ of $K$, a map $f_\sigma{\colon}|K|\to S^{m-1}$ with precisely one double point, one of whose two preimages lies in the interior of $|\sigma|$. This implies that every proper subpolyhedron $P$ of $|K|$ embeds in $S^{2m-2}$. Indeed, since $P$ is compact, it is disjoint from some point in the interior of $|\sigma|$ for some maximal face $\sigma$ of $K$, and therefore from a disk $D$ in the interior of $|\sigma|$. Then $P$ is embedded in $S^{m-1}$ by $f_\sigma$ precomposed with an appropriate self-homeomorphism of $|K|$, fixed outside $|\sigma|$. We shall next see that Theorem \[self-dual\] is not as exciting as it might appear to be. Collapsible and cell-like maps ------------------------------ In this subsection we assume familiarity with collapsing and regular neighborhoods (see e.g. [@Hu]). The following fact is well-known. \[trivial\] If $Q$ is a collapsible subpolyhedron of a manifold $M$, then the quotient $M/Q$ is homeomorphic to $M$. Let $N$ be a regular neighborhood of $Q$ in $M$. Since $Q$ is collapsible, $N$ is a ball. On the other hand, $N/Q$ is homeomorphic rel $\partial N$ to $pt*\partial N$, which is also a ball. This yields a homeomorphism $N\to pt*\partial N$ keeping $\partial N$ fixed, which extends by the identity to a homeomorphism $M\to M/Q$. Let us call a map between polyhedra [*finite-collapsible*]{} if it is the composition of a sequence of quotient maps, each shrinking a collapsible subpolyhedron to a point. \[finite-collapsible\] Let $f{\colon}P\to Q$ be a finite-collapsible map between polyhedra. If $P$ embeds in $S^m$, then so does $Q$. We use this nearly trivial observation to give a simple proof of a result by Zaks [@Za] (see also [@Um2], [@Sa1 3.7.1]); as a byproduct, we also get a slightly stronger statement: \[Zaks\] For each $n\ge 2$ there exists an infinite list of pairwise non-homeomorphic compact $(n-1)$-connected $n$-polyhedra $P_i$ such that each $P_i$ does not embed in $S^{2n}$, but every its proper subpolyhedron does. Zaks’ series of examples satisfied the conclusion of Theorem \[Zaks\] except for being $(n-1)$-connected. Let $K$ be the $n$-skeleton of the $(2n+2)$-simplex and let $\sigma$ be an $(n-1)$-simplex of $K$. By inspection, it is a face of at least three $n$-simplices; let $\tau$ be one of them. Let $b$ be the barycenter of $\sigma$, and let $B_i$ (resp. $D_i$) be the star of $b$ in the $i$th barycentric subdivision of $\tau$ (resp. of $\sigma$). Let $C_i$ be the closure of $|\partial B_i|\but |D_i|$; it is a codimension one ball properly embedded in $|\tau|$, with boundary sphere embedded in $|\sigma|$. Given a positive integer $r$, let $P_r$ be the polyhedron obtained from $P_0\bydef|K|$ by shrinking each $C_i$ to a point $p_i$ for $i=1,\dots,r$. Then the quotient map $f_r{\colon}P_0\to P_r$ is finite-collapsible, and sends $|B_1|$ onto a collapsible polyhedron $X_r$. The quotient $P_r/X_r$ is homeomorphic to $P_0/|B_1|$ and therefore to $P_0$. Thus we obtain a finite-collapsible map $g_r{\colon}P_r\to P_0$. The links of the points $p_i$ in $P_r$ are homeomorphic to each other and not homeomorphic to the link of any other point in $P_r$. Consequently, $P_0,P_1,P_2,\dots$ are pairwise non-homeomorphic. Since $P_0$ does not embed in $S^{2n}$ (see Theorem \[self-dual\]), and $g_r$ is finite-collapsible, $P_r$ does not embed in $S^{2n}$. If $Q$ is a proper subpolyhedron of $P_r$, then $R\bydef f_r^{-1}(Q)$ is a proper subpolyhedron of $P_0$, and $f_r|_R{\colon}R\to Q$ is clearly finite-collapsible. Since $R$ embeds in $S^{2n}$ (see Remark \[van Kampen\]), so does $Q$. A map between polyhedra is called [*collapsible*]{}, resp. [*cell-like*]{} if every its point-inverse is collapsible, resp. contractible (and so, in particular, nonempty). The following generalization of Corollary \[finite-collapsible\] is a relatively easy consequence of well-known classical results. \[minors embed\] (a) If $f{\colon}P\to Q$ is a collapsible map between polyhedra, and $P$ embeds in a manifold $M$, then $Q$ embeds in $M$. \(b) If $f{\colon}P\to Q$ is a cell-like map between $n$-polyhedra, and $P$ embeds in an $m$-manifold $M$, where $m\ge n+3$, then $Q$ embeds in $M$. The proof is not hard and quite instructive, but as this introduction is getting a bit too involved we defer it until §\[collapsing\]. There we also elaborate on the following \[quasi-embedding\] If $f{\colon}X\to Y$ is a map between $n$-polyhedra whose nondegenerate point-inverses lie in a subpolyhedron of dimension $\le m-n-2$, and $X$ embeds in $S^m$, then $Y$ embeds in $S^m$. The given embedding of $X$ extends by general position to an embedding of $X\cup_Z MC(f|_Z)$, where $Z$ is the given subpolyhedron and $MC$ denotes the mapping cylinder. On the other hand, the point-inverses of the projection $X\cup_Z MC(f|_Z)\to Y$ are cones, so Theorem \[minors embed\](a) yields an embedding of $Y$. Let us call a polyhedron $Z$ an [*$h$-minor*]{} of a polyhedron $X$, if there exists a subpolyhedron $Y$ of $X$ and a cell-like map $Y\to Z$. (We note that composition of cell-like maps is obviously cell-like, cf. [@Sm]; see also [@Hat comment to Corollary 2.3] for a combinatorial proof.) By Theorem \[minors embed\](a), all $h$-minors of an $n$-polyhedron embeddable in $S^m$, $m-n\ge 3$, also embed in $S^m$. One could hope that using $h$-minors instead of subpolyhedra enables one to prevent Theorem \[Zaks\] from “driving us from the paradise” which Theorem \[self-dual\] might seem to promise. This is not so: using Corollary \[quasi-embedding\], it is easy to construct an infinite list $P_0,P_1,\dots$ of pairwise non-homeomorphic $n$-polyhedra, $n\ge 3$, such that each $P_i$ does not embed in $S^{2n}$, but every its proper (in any reasonable sense) $h$-minor embeds in $S^{2n}$. In fact, the original examples of Zaks [@Za] work (compare [@Ne 6.5]). However, all such examples (Zaks’ examples and their modifications constructed using Corollary \[quasi-embedding\]) are not going to be $(n-1)$-connected. Relevance of $(n-1)$-connected $n$-polyhedra is ensured by the following observation. \[connected\] Let $P$ be an $n$-polyhedron, $n\ne 2$, that embeds in $S^{2n}$. Then $P$ embeds in an $(n-1)$-connected $n$-polyhedron $Q$ such that $Q$ embeds in $S^{2n}$. We shall show that more generally if $P$ is an $n$-polyhedron, $n\ge 2$, with vanishing van Kampen obstruction, then $P$ embeds in a polyhedron $Q$ with vanishing van Kampen obstruction. The van Kampen obstruction is well-known to be complete for $n\ne 2$ (see [@M2] and Remark \[erratum2\] below). Let $K$ be some triangulation of $P$. By general position $P\cup |CK^{(n-2)}|$ embeds in $S^{2n}$. So without loss of generality $P$ is $(n-2)$-connected. Then $P$ is homotopy equivalent to a wedge of $(n-1)$-spheres. If $n=2$, $\pi_1(P)$ is finitely generated (and even finitely presented). If $n\ge 3$, $P$ is simply-connected, hence $\pi_{n-1}(P)$ is finitely generated over $\Z$ (see [@tD 20.6.2(3)] for a new simple proof of this classical result). Let $f_i{\colon}S^{n-1}\to P$ represent the free homotopy classes of some finite basis of $\pi_{n-1}(P)$. We may amend the $f_i$ so that the image of each $f_i$ meets each $n$-simplex of $K$. Let $Q$ be obtained by adjoining $n$-disks to $P$ along the $f_i$. By construction, $Q$ is $(n-1)$-connected. Moreover, due to our choice of the $f_i$, the van Kampen obstruction of $Q$ is zero. \[dual-collapsible\] The proof of Theorem \[minors embed\](a) works to establish it for the more general class of dual-collapsible maps. A map is [*dual-collapsible*]{} if it can be triangulated by a simplicial map $f{\colon}K\to L$ such that for each simplex $\tau$ of $L$, its dual cone $\tau^*$ has collapsible preimage $|f^{-1}(\tau^*)|$. It can be shown that composition of dual-collapsible maps is dual-collapsible (which seems not to be the case for collapsible maps, even though it happens to be the case for collapsible retractions [@Co2 8.6]). Many assertions in the present paper involving cell-like maps (directly or through $h$-minors), including the statement of the main theorem, hold for dual-collapsible maps. Edge-minors ----------- An [*edge contraction*]{} is a simplicial map $f{\colon}K\to L$ that sends every edge onto an edge, apart from one edge $\{v_1,v_2\}$ which it shrinks onto a vertex. We call $f$ admissible if $\operatorname{lk}\{v_1\}\cap\operatorname{lk}\{v_2\}=\operatorname{lk}\{v_1,v_2\}$. An equivalent condition is that $\{v_1,v_2\}$ is not contained in any “missing face” of $K$, i.e. in an isomorphic copy of $\partial\Delta^n$ in $K$ that does not extend to an isomorphic copy of $\Delta^n$ in $K$. We define a simplicial complex $L$ to be an [*edge-minor*]{} of a simplicial complex $K$ if $L$ is obtained from a subcomplex of $K$ by a sequence of admissible edge contractions. Yet another equivalent formulation of the admissibility condition is that every point-inverse of $|f|$ is collapsible. This has the following consequences: 1. If $f{\colon}K\to L$ is an edge contraction, and $\Lambda$ is a subcomplex of $L$, then $f|_{f^{-1}(\Lambda)}$ is either an edge contraction or a homeomorphism. 2. If $L$ is an edge-minor of $K$ and $|K|$ embeds in $S^m$, then $|L|$ embeds in $S^m$. E. Nevo considered a slightly different definition of a “minor” which we shall term a [*Nevo minor*]{} [@Ne]. It is similar to that of edge-minor, with the following amendment. We call an edge contraction $f{\colon}K\to L$, where $\dim K=n$, Nevo-admissible, if the $(n-2)$-skeleton $(\operatorname{lk}\{v_1\}\cap\operatorname{lk}\{v_2\})^{(n-2)}=\operatorname{lk}\{v_1,v_2\}$; an equivalent condition is that $\{v_1,v_2\}$ is not contained in any missing face of $K$ of (missing) dimension $\le n$. Note that in the case of graphs this is a vacuous condition. \[nevo\] The notions of Nevo minor and edge-minor are equivalent. The author learned from E. Nevo that the published version of his paper in fact contains this remark, which was also pointed out by his referee. We note that Proposition \[nevo\] along with assertion (2) above yields a proof of Conjecture 1.3 in [@Ne]: if $L$ is a Nevo minor of $K$, and $|K|$ embeds in $S^m$, then $|L|$ embeds in $S^m$. By assertion (1) above, it suffices to show that if $f{\colon}K\to L$ is a Nevo-admissible edge contraction, then $L$ is a minor of $K$. Suppose that $f$ shrinks an edge $e=\{v_1,v_2\}$. If $f$ is not admissible, $\operatorname{lk}(v_1)\cap\operatorname{lk}(v_2)$ is the union of $\operatorname{lk}(e)$ with a nonempty collection of $(n-1)$-simplices $\sigma_1,\dots,\sigma_k$. Let $K^+$ be the subcomplex of $K$ obtained by removing the $n$-simplices $\{v_0\}*\sigma_i$ from $K$. Then $f|_{K^+}{\colon}K^+\to L$ is an admissible edge contraction. Hence $L$ is a minor of $K$. \[Steinitz\] A simplicial complex of dimension $\le 2$ is an edge-minor of every its subdivision. This is proved in §\[edge-minors2\] using innermost circle arguments. Since every triangulation $T$ of $S^2$ is easily seen to be a subdivision of $\partial\Delta^3$, Lemma \[Steinitz\] implies the result of Steinitz (1934) that $\partial\Delta^3$ is a minor of $T$; see [@Zo] and [@Su]. In contrast, there exists a subdivision $S$ of $\partial\Delta^4$ such that $\partial\Delta^4$ is not an edge-minor of $S$; see [@Ne Example 6.1]. The preceding results now imply the following version of the Halin–Jung theorem: A simplicial complex $K$ has no edge-minors among the seven self-dual subcomplexes of $\partial\Delta^4$ along with $K_{3,3}$ iff $|K|$ embeds in $S^2$. Since a given (finite) simplicial complex only has finitely many minors, this yields an algorithm deciding embeddability of $|K|$ in $S^2$. A presumably faster, but more elaborate algorithm is discussed in [@MTW]. \[self-dual-addendum\] Let $K=K_1*\dots*K_r$, where each $K_i$ is a self-dual subcomplex of $\partial\Delta^{m_i}$, where $m_1+\dots+m_r=m$, and let $L$ be a proper edge-minor of $K$. Then \(a) $|L|$ embeds in $S^{m-2}$; \(b) the embeddings of $|L|$ in $S^{m-1}$ induced by $j$ and $hj$ are equivalent, where $j$ is the inclusion of $|K|$ in $S^{m-1}\bydef|\partial\Delta^{m_1}*\dots*\partial\Delta^{m_r}|$, and $h$ is an orientation-reversing homeomorphism of $S^{m-1}$. The proof, given in §\[edge-minors2\], is reminiscent of the above argument towards Theorem \[self-dual\]. Hemi-icosahedron and hemi-dodecahedron {#semi} -------------------------------------- The central symmetry of $\R^3$ in the origin yields a simplicial free involution on the boundary of a regular icosahedron centered at the origin. Its quotient by this involution is a simplicial complex $\R P^2_\triangle$ with $6$ vertices, triangulating the real projective plane. It is easy to see that its $1$-skeleton is the complete graph $K_6$, and for each pair of disjoint circuits in $K_6$, precisely one bounds a $2$-simplex in $\R P^2_\triangle$. Hence $\R P^2_\triangle$ is self-dual as a subcomplex of $\partial\Delta^5$ (compare [@Mat 5.8.5]). By Theorem \[self-dual\] this implies that the projective plane $\R P^2=|\R P^2_\triangle|$ does not embed in $S^3$, embeds in $S^4$, and every embedding of $\R P^2$ in $S^4$ is inequivalent to its reflection. In fact, every embedding of $\R P^2$ in $S^4$ with fundamental group of the complement isomorphic to $\Z/2$ is known to be topologically equivalent to either the standard embedding or its reflection [@La3]. A well-known $9$-vertex triangulation of $\C P^2$ [@KB] (see [@MY], [@BD1], [@MS], [@BD3] for other constructions) is easily seen to be self-dual as a subcomplex of $\partial\Delta^8$. Thus Theorem \[self-dual\] applies again. In this connection we note that every embedding of $\C P^2$ in $S^7$ is known to be equivalent to either the standard embedding or its reflection [@Sk-A3]. In fact, it is known that a combinatorial $n$-manifold with $v$ vertices is self-dual as a subcomplex of $\partial\Delta^{v-1}$ if and only if $2v=3n+6$ [@ArM] (if), [@Da] (only if). Combinatorial $n$-manifolds with $\frac{3n}2+3$ vertices can only occur in dimensions $n=0,2,4,8,16$ and with a $\bmod 2$ cohomology ring isomorphic to that of the respective projective plane [@BK1] (see also [@La2]). In dimensions $2$ and $4$ these are unique (up to a relabelling of vertices); an algebraic topology proof can be found in [@ArM] and a combinatorial one in [@BD2]. In dimension $8$, there exist three self-dual subcomplexes of $\partial\Delta^{14}$, all triangulating a certain $8$-manifold $\Ham P^2_{(?)}$ [@BK2]. The existence in dimension $16$ seems to be still open. Theorem \[self-dual\] now implies \[1.10\] The join of $r$ copies of $\R P^2$, $c$ copies of $\C P^2$ and $h$ copies of $\Ham P^2_{(?)}$ embeds in $S^{5r+8c+14h-1}$ and does not embed in $S^{5r+8c+14h-2}$. Every embedding $g$ of this join in $S^{5r+8c+14h-1}$ is inequivalent to $hg$, where $h$ is an orientation reversing homeomorphism of the sphere. We note that $K_6$ admits an embedding in $S^3$ that links any given disjoint circuits $|C|$, $|C'|$ in $|K_6|$ with any given odd linking number and does not link any other disjoint pair of circuits. Similar arguments work to prove the same assertion for any other graph of the Petersen family (cf. Remark \[hemi-exchanges\]). Indeed, one of $C$, $C'$ bounds a simplex in the semi-icosahedron and the other one can be identified with $\R P^1$. Thus $K_6$ is the $1$-skeleton of a triangulation of the Möbius band $\mu$ where $\partial\mu$ and the middle curve of $\mu$ are triangulated by $C$ and $C'$. Using various embeddings of $\mu$ in $S^3$, e.g. those corresponding to all half-odd-integer framings of the trivial knot, we obtain embeddings of $K_6$ in $S^3$ with $\operatorname{lk}(|C|,|C'|)$ equal to any given odd number and all other disjoint pairs of circuits geometrically unlinked. Similarly to the hemi-icosahedron $\R P^2_\triangle$ we have the hemi-dodecahedron $\R P^2_\bigstar$ (compare [@McS §6C]), whose $1$-skeleton is the Petersen graph $P$, and for each pair of disjoint circuits in $P$, precisely one bounds a cell in $\R P^2_\bigstar$. This suggests that we should not be too fixed on simplicial complexes; this will be our next concern. Main results {#section:main} ============ By a cell complex we mean what can be described as a finite CW-complex where each attaching map is a homeomorphism of the sphere onto a subcomplex. (We recall that we assume all maps between polyhedra to be PL by default.) Note that the empty set is not a cell in our notation. Cell complexes, and in particular simplicial complexes, are uniquely determined by their face posets, so we may alternatively view them as posets of a special kind. This view is inherent in the following combinatorial notation [@M3]. Combinatorial notation {#notation} ---------------------- Given a poset $P$ and a $\sigma\in P$, the [*cone*]{} $\fll\sigma\flr$ (resp. the [*dual cone*]{} $\cel\sigma\cer$) is the subposet of all $\tau\in P$ such that $\tau\le\sigma$ (resp. $\tau\ge\sigma$). The [*prejoin*]{} $P+Q$ of posets $P=(\mathcal P,\le)$ and $Q=(\mathcal Q,\le)$ is the poset $(\mathcal P\sqcup\mathcal Q,\le)$, where $\mathcal P$ and $\mathcal Q$ retain their original orders, and every $p\in\mathcal P$ is set to be less than every $q\in\mathcal Q$. The [*cone*]{} $CP=P+\{\hat 1\}$ and the [*dual cone*]{} $C^*P=\{\hat 0\}+P$, where $\{\hat 1\}$ and $\{\hat 0\}$ are just fancy notations for the one-element poset. The [*boundary*]{} $\partial Q$ of a cone $Q=CP$ is the original poset $P$, and the [*coboundary*]{} $\partial^*Q$ of a dual cone $Q=C^*P$ is again $P$. Given a finite set $S$, we denote the poset of all subsets of $S$ by $2^S$; and the poset $\partial^*2^S$ of all nonempty subsets by $\Delta^S$. When $S$ has cardinality $n+1$ and the nature of its elements is irrelevant, the [*(combinatorial) $n$-simplex*]{} $\Delta^S$ is also denoted $\Delta^n$. We call a poset $P$ a [*simplicial complex*]{} if it is a complete quasi-lattice (i.e. every subset of $P$ that has an upper bound in $P$ has a least upper bound in $P$; or equivalently every subset of $P$ that has a lower bound in $P$ has a greatest lower bound in $P$), and every cone of $P$ is order-isomorphic to a simplex. For a poset $P$ we distinguish its [*barycentric subdivision*]{} $P^\flat$ that is the poset of all nonempty chains in $P$, ordered by inclusion, and the [*order complex*]{} $|P|$ that is the polyhedron triangulated by the simplicial complex $P^\flat$. It should be noted that many fundamental homeomorphisms in combinatorial topology can be promoted to combinatorial isomorphisms by upgrading from the barycentric subdivision to the [*canonical subdivision*]{} $P^\#$ that is the poset of all order intervals in $P$, ordered by inclusion [@BBC], [@M3]. We call a poset $P$ a [*cell complex*]{} if for every $p\in P$ the order complex $|\partial\fll p\flr|$ is homeomorphic to a sphere. It is not hard to see that the so defined cell/simplicial complexes are precisely the posets of nonempty faces of the customary cell/simplicial complexes [@M3] (the case of cell complexes is trivial and well-known [@Mc], [@Bj]). General posets may be thought of as “cone complexes” [@vK2], [@Mc], [@M3], and their order-preserving maps may be thought of as “conical” maps. From now on, we switch to the new formalism. A few more auxiliary definitions follow. The [*dual*]{} of a poset $P=(\mathcal P,\le)$ is the poset $P^*\bydef(\mathcal P,\ge)$. We note that $2^S$ is isomorphic to its own dual (by taking the complement); and therefore so is $\partial\Delta^S=\partial(\partial^*2^S)$. A poset $Q=(\mathcal Q,\preceq)$ is a [*subposet*]{} of $P$ if $\mathcal Q$ is a subset of $\mathcal P$, and $p\preceq q$ iff $p\le q$ for all $p,q\in\mathcal Q$. We will often identify a poset with its underlying set by an abuse of notation. A subposet $Q$ of $P$ is a [*subcomplex*]{} of $P$ if the cone (in $P$) of every element of $Q$ lies in $Q$. Let $P=(\mathcal P,\le)$ and $Q=(\mathcal Q,\le)$ be posets. The [*product*]{} $P\x Q$ is the poset $(\mathcal P\x\mathcal Q,\preceq)$, where $(p,q)\preceq (p',q')$ iff $p\le p'$ and $q\le q'$. It is easy to see that $2^S\x 2^T\simeq 2^{S\sqcup T}$ naturally in $S$ and $T$. The [*join*]{} $P*Q\bydef\partial^*(C^*P\x C^*Q)$ is obtained from $(C^*P)\x (C^*Q)$ by removing the bottom element $(\hat0,\hat0)$. Thus $C^*(P*Q)\simeq C^*P\x C^*Q$, whereas $P*Q$ itself is the union $C^*P\x Q\cup P\x C^*Q$ along their common part $P\x Q$. From the above, $\Delta^S*\Delta^T\simeq\Delta^{S\sqcup T}$ naturally in $S$ and $T$. It follows that the join of simplicial complexes $K\subset\Delta^S$ and $L\subset\Delta^T$ is isomorphic to the simplicial complex $\{\sigma\cup\tau\subset S\sqcup T\mid\sigma\in K\cup\{\emptyset\}, \tau\in L\cup\{\emptyset\},\,\sigma\cup\tau\ne\emptyset\}\subset\Delta^{S\sqcup T}$. The join and the prejoin are related via barycentric subdivision: $(P+Q)^\flat\simeq P^\flat*Q^\flat$. Indeed, a nonempty finite chain in $P+Q$ consists of a finite chain in $P$ and a finite chain in $Q$, at least one of which is nonempty. Note that in contrast to prejoin, join is commutative: $P*Q\simeq Q*P$. Prejoin is associative; in particular, $C(C^*P)\simeq C^*(CP)$. [*h*]{}-Minors of cell complexes {#h-minors} -------------------------------- We call a cell complex $L$ an [*$h$-minor*]{} of a cell complex $K$, if there exists a subcomplex $M$ of $K$ and an order preserving map $f{\colon}M\to K$ such that on the level of order complexes, $|f|{\colon}|M|\to|K|$ is a cell-like map. (Note that cell-like maps include dual-collapsible maps, see Remark \[dual-collapsible\].) \[subdivision\] If a simplicial complex $L$ is an edge-minor of a simplicial complex $K$, then of course $L$ is an $h$-minor of $K$; but not vice versa. Indeed, let $K$ be a simplicial complex and $K'$ its simplicial subdivision such that $K'$ is not an edge-minor of $K$ (see [@Ne Example 6.1]); we may fix an identification of their order complexes. Let $f{\colon}K'\to K$ send every $\sigma\in K'$ to the minimal $\tau\in K$ such that $|\sigma|\subset|\tau|$. Then $f$ is an order preserving map such that $|f^{-1}(\fll\tau\flr)|$ is an $n$-ball for each $n$-simplex $\tau\in K$. (This understanding of a subdivision also arises in the study of PL transversality [@M3] and in that of combinatorial grassmanians [@Mn].) It is easy to see that $|f^{-1}(\fll\tau\flr)|$ collapses onto $|f^{-1}(\tau)|$ for each $\tau\in K$, and it follows that $|f|$ is a cell-like map. More generally, we say that an order preserving map $f{\colon}P'\to P$ between posets is a [*subdivision*]{} if $|f^{-1}(\fll\tau\flr)|$ is homeomorphic to $|Cf^{-1}(\partial\fll\tau\flr)|$ by a homeomorphism fixed on $|f^{-1}(\partial\fll\tau\flr)|$. It is easy to see that $|f|$ is a cell-like map and $|P'|$ is homeomorphic to $|P|$ [@M3] (see also [@Ak], [@DM 1.4] for a special case). In particular, a cell complex is an $h$-minor of every its simplicial subdivision. Another special case of taking an $h$-minor is zipping. Given a poset $P$ and a $\sigma\in P$ such that $\fll\sigma\flr$ is isomorphic to $Q+\Delta^1$ for some $Q$, by an isomorphism $h$, we say that $P$ [*elementarily zips*]{} to $P/h^{-1}(\Delta^1)$ (the quotient in the concrete category of posets over the category of sets, cf. [@AHS]). A [*zipping*]{} is a sequence of elementary zippings. It is not hard to see that if $K$ edge-contracts to $L$, then $K$ zips to $L$ (for instance, it takes two elementary zippings to zip a $2$-simplex onto a $1$-simplex). The author learned from E. Nevo that he has independently observed this fact, and that a definition of zipping appears in Reading’s paper [@Re]. In fact, we borrow the term “zipping” from that paper. E. Nevo also observed that if $K$ elementarily zips to $L$, then the barycentric subdivision $K^\flat$ edge-contracts to $L^\flat$ in two steps. We shall encounter a modification of zipping with $\Delta^1$ replaced by $C((\Delta^2)^{(0)})$, as well as zipping itself, in the proof of Proposition \[homology mfld\]. \[conjecture:main\] For each $n$ there exist only finitely many $n$-dimensional cell complexes $K$ such that $|K|$ is $(n-1)$-connected and does not embed in $S^{2n}$, but $|L|$ embeds in $S^{2n}$ for each proper $h$-minor $L$ of $K$. The author is not absolutely committed to this particular formulation, but he strongly feels that at least some reasonable modification of this conjecture should hold. Some variations are discussed in §\[algorithmic\]. Dichotomial spheres {#dichotomial spheres} ------------------- Some members of the list in the preceding conjecture are provided by the following result. Let $B$ be an $m$-dimensional [dichotomial]{} cell complex, that is a cell complex that together with each cell $A$ contains a unique cell $\bar A$ whose vertices are precisely all the vertices of $B$ that are not in $A$. Let $K$ be the $n$-skeleton of $B$, where $m=2n+1$ or $2n+2$, and let $L$ be any proper $h$-minor of $K$. Then: \(i) $B$ is uniquely determined by $K$. \(ii) $|B|\cong S^m$. \(iii) If $m=2n+1$, $n\ne 2$, then $|K|$ does not embed in $S^{2n}$, but $|L|$ does. (iii$'$) If $m=2n+1$, $n=2$, then $|K|$ has non-zero van Kampen obstruction, even modulo $2$ (so in particular does not embed in $S^4$) but $|L|$ has zero van Kampen obstruction. \(iv) If $m=2n+2$, then $|K|$ does not linklessly embed in $S^{2n+1}$, but $|L|$ does. \(v) If $m=2n+1$, then every embedding $g$ of $|K|$ in $S^{2n+1}$ is inequivalent with $hg$, where $h$ is an orientation-reversing homeomorphism of $S^{2n+1}$; but every two embeddings of $|K|$ in $S^{2n+1}$ (knotless when $n=1$) have equivalent “restrictions” to $|L|$. \(vi) Moreover, if $m=2n+1$, then every embedding of $|K|$ in $S^{2n+1}$ is linkless; and if $m=2n+2$, then every embedding of $|K|$ in $S^{2n+1}$ contains a link of a pair of disjoint $n$-spheres with an odd linking number. \(vii) If $M$ is a [self-dual]{} subcomplex of $B$ (that is, $M$ contains precisely one cell out of every pair $A$, $\bar A$ of complementary cells), then $|M|$ does not embed in $S^m$, and every embedding $g$ of $|M|$ in $S^{m+1}$ is inequivalent with $hg$, where $h$ is an orientation-reversing homeomorphism of $S^{m+1}$. The simplest example of a dichotomial complex is the boundary of a simplex. In particular, for $n=1$, the graphs in (iii) and (iv) include respectively the complete graphs $K_5$ and $K_6$. It is easy to see that there are no other dichotomial simplicial complexes, apart from the boundary of a simplex. The term “dichotomial” was suggested to the author by E. V. Shchepin. It is easy to construct the dichotomial $3$-sphere whose $1$-skeleton is $K_{3,3}$. In accordance with (i), we may start with $K_{3,3}$ itself. For every edge $\tau$ of $K_{3,3}$, the four edges disjoint from $\tau$ form a circuit; we glue up this circuit by a quadrilateral $2$-cell. (Note that circuits of length $6$ are [*not*]{} glued up by hexagonal $2$-cells.) For every vertex $\sigma$ of $K_{3,3}$, the edges disjoint from $\sigma$ form a $K_{3,2}$, to which we have attached three $2$-cells. Their union is a $2$-sphere; we glue it up by a $3$-cell. Let us verify that the resulting dichotomial cell complex $B_{3,3}$ is indeed a $3$-sphere. We shall identify each element of $B_{3,3}$ with a subcomplex of the barycentric subdivision of $\partial\Delta^2*\partial\Delta^2$. The atoms of $B_{3,3}$ are identified with the vertices of $\partial\Delta^2*\partial\Delta^2$, and the maximal elements of edges of $B_{3,3}$ are identified with the edges of $(\Delta^2)^{(0)}*(\Delta^2)^{(0)}$. The maximal element of the $2$-cell of $B_{3,3}$ disjoint with an edge $\sigma_1*\sigma_2$ of $K_{3,3}$ is identified with the disk $D_{\sigma_1\sigma_2}\bydef c*\partial\tau_1*\partial\tau_2$, where $\tau_i$ is the $1$-simplex in $\Delta^2$ disjoint from the vertex $\sigma_i$, and $c$ is the barycenter of the simplex $\tau_1*\tau_2$. The maximal element of the $3$-cell of $B_{3,3}$ disjoint with a vertex $\sigma*\emptyset$ (resp. $\emptyset*\sigma$) of $K_{3,3}$ is identified with the ball $E_{\sigma\emptyset}\bydef c*(D_{\sigma\sigma_1}\cup D_{\sigma\sigma_2}\cup D_{\sigma\sigma_3})$ (resp. $E_{\emptyset\sigma}\bydef c*(D_{\sigma_1\sigma}\cup D_{\sigma_2\sigma}\cup D_{\sigma_3\sigma})$), where $c$ is the barycenter of the $1$-simplex $\tau$ of $\Delta^2$ disjoint form $\sigma$, and $\sigma_1,\sigma_2,\sigma_2$ are the three vertices of $\Delta^2$. It is easy to see that the six balls $E_{\sigma\emptyset}$, $E_{\emptyset\sigma}$ cover the entire $(\partial\Delta^2*\partial\Delta^2)^\flat$. Similarly, it is not hard to construct the dichotomial $4$-sphere whose $1$-skeleton is the Petersen graph $P$. Firstly we glue up every circuit consisting of $5$ edges by a pentagonal $2$-cell. (Note that circuits of length $6$ are [*not*]{} glued up by hexagonal $2$-cells.) Then each $2$-cell already has its opposite $2$-cell. For each edge $\tau$ of $P$, the edges disjoint from $\tau$ form a graph that is a subdivision of the $1$-skeleton of the tetrahedron, and the four $2$-cells disjoint from $\tau$ can be identified with the $2$-simplices of this tetrahedron, with boundaries subdivided into pentagons. Thus these edges and $2$-cells cellulate a $2$-sphere; we glue it up by a $3$-cell. For every vertex $\sigma$ of $P$, the edges and the $2$-cells disjoint from $\sigma$ form a cell complex that subdivides $(\partial\Delta^2)+(\Delta^2)^{(0)}$. Then the edges, the $2$-cells and the $3$-cells disjoint from $\sigma$ form a cell complex that subdivides the $3$-sphere $(\partial\Delta^2)+(\partial\Delta^2)$. We glue it up by a $4$-cell. According to part (ii) of the Main Theorem, the resulting dichotomial cell complex $B_P$ is a $4$-sphere. Note that the hemi-dodecahedron is isomorphic to its self-dual subcomplex. It can be similarly verified by hand that the graphs of the Petersen family except $K_{4,4}\but$(edge) are $1$-skeleta of $4$-dimensional dichotomial cell complexes; whereas $K_{4,4}\but$(edge) is not. We describe a more industrial way of seeing this in §\[transforms\], where we also observe that $K_{4,4}\but$(edge) is the $1$-skeleton of a $4$-dimensional dichotomial poset, whose order complex admits a collapsible map onto $S^4$ (so in particular is homotopy equivalent to $S^4$). It should be possible to include this graph into the general framework of the Main Theorem by extending it to cone complexes whose cones are singular cells, with not ‘too many’ cells being ‘too singular’. There can be other approaches (see Example \[4.9\]). We note that the assertion on minors in part (iii) of the Main Theorem does not extend to cover arbitrary self-dual subcomplexes in part (vii). Indeed, by zipping an edge of the hemi-icosahedron we obtain a non-simplicial proper minor of the hemi-icosahedron which still cellulates $\R P^2$. The author believes that it should be possible to overcome this trouble, at least in the metastable range, by considering only minors that belong to a restricted class of cell complexes, such as ones whose cells have collapsible (or empty) pairwise intersections. (Note that two cells do not always intersect along a cell, and three cells do not always have a collapsible or empty intersection in the dichotomial $4$-sphere whose $1$-skeleton is the Petersen graph.) Taking into account the Kuratowski–Wagner and Robertson–Seymour–Thomas theorems and the obvious fact that a $0$-polyhedron admits a linkless embedding in $S^1$ iff it contains less than $4$ points, we obtain \[main corollary\] The only dichotomial complex in dimension two is $\partial\Delta^3$; there exist precisely two in dimension $3$, with $1$-skeleta $K_5$ and $K_{3,3}$, and precisely six in dimension $4$, with $1$-skeleta all graphs of the Petersen family excluding $K_{4,4}\but e$. Let us now review three constructions of new examples of dichotomial spheres. \(i) An $n$-dimensional join of the $i$-skeleta of dichotomial $(2i+1)$-spheres is the $n$-skeleton of some dichotomial $(2n+1)$-sphere dimensional cell complex (see Lemma \[5.3\] and Theorem \[construction\]). For instance, this yields, in addition to $\partial\Delta^6$, two dichotomial $5$-complexes, with $1$-skeleta $K_{1,1,1,1,1,3}$ and $K_{3,3,3}$ and with simplicial $2$-skeleta. \(ii) If $B$ is a dichotomial cell complex, it is easy to see, using part (ii) of Main Theorem, that $CB\cup_B B*pt$ is again a dichotomial cell complex. Applied to $B_{3,3}$, this construction produces the dichotomial $4$-sphere $B_{3,3,1}$ with $1$-skeleton $K_{3,3,1}$ (which belongs to the Petersen family). Applying this construction to the six dichotomial $4$-spheres, we get, apart from $\partial\Delta^6$, five dichotomial $5$-complexes whose $1$-skeleta are obtained from the graphs $\Gamma_7$, $\Gamma_8$, $\Gamma_9$, $K_{3,3,1}$, $P$ by adjoining an additional vertex and connecting it by edges to all the existing vertices. For instance, in the case of $B_{3,3,1}$ this yields a dichotomial sphere $B_{3,3,1,1}$ with $1$-skeleton $K_{3,3,1,1}$. A more general construction is: given a dichotomial $m$-sphere $B$ and a dichotomial $n$-sphere $B'$, the boundary sphere $\partial(CB*CB')$ of the join of the cones is a dichotomial $(m+n+2)$-sphere. (The previous paragraph treated the case where $B'$ is the dichotomial $(-1)$-sphere $\emptyset$.) Taking $B$ to be the dichotomial $3$-sphere with $1$-skeleton $K_{3,3}$ and $B'$ to be the dichotomial $0$-sphere, we arrive again at $B_{3,3,1,1}$. We note that the $2$-skeleton of $B_{3,3,1,1}$ is the union of the $2$-skeleton of $B_{3,3}$ with the $2$-skeleton of the join $K_{3,3}*\Delta^1$. Since the circuits of length $6$ in $K_{3,3}$ are not glued up by $2$-cells in $B_{3,3}$, they are similarly not glued up by $2$-cells in $B_{3,3,1,1}$ (although they of course bound [*subdivided*]{} $2$-cells in the $2$-skeleton of $B_{3,3,1,1}$). Thus the $2$-skeleton of $B_{3,3,1,1}$ is a proper subcomplex of van der Holst’s $2$-complex $\hat K_{3,3,1,1}$ (see §\[graphs\]). \(iii) A more interesting method, called ($\nabla$,Y)-transform, is introduced in §\[transforms\]. It is natural to expect that an appropriate generalization of this transform (or perhaps even the transform itself) would suffice to relate any two dichotomial spheres of the same dimension (compare the proof of Steinitz’ theorem in [@Zi Chapter 4]). It does relate with each other the two dichotomial $3$-spheres and all six dichotomial $4$-spheres. When applied to $\partial\Delta^6$, it produces, inter alia, dichotomial $5$-complexes with $1$-skeleta obtained by adjoining an additional vertex to any of the graphs $\Gamma_7$, $\Gamma_8$, $\Gamma_9$, $K_{4,4}\but$(edge), $P$ and connecting it to all existing vertices that are not marked red in Fig. 1. (The details are similar to Examples \[4.8\], \[4.9\].) To summarize, we have just used the easy (existence) part of Corollary \[main corollary\] to show the following \[2-skeleta\] There exist at least $13$ dichotomial $5$-spheres, distinct already on the level of their $1$-skeleta, of which $10$ have non-simplicial $2$-skeleta. The three simplicial $2$-skeleta, $(\Delta^6)^{(2)}$, $K_5*(\Delta^2)^{(0)}$ and $K_{3,3}*(\Delta^2)^{(0)}$, are well-known [@Gr], [@Sa1]. The non-simplicial ones are probably new, although some (all?) of their $1$-skeleta are in the Heawood family (see §\[graphs\]; it is clear that there in fact must be many more dichotomial $5$-spheres with $1$-skeleta in the Heawood family). By part (iii$'$) of the Main Theorem, none of the $13$ cell complexes in Corollary \[2-skeleta\] is a minor of any other one. But beware that some of these $2$-complexes have underlying polyhedra that are $h$-minors of each other (namely, some can be obtained from the others by shrinking $2$-simplices to triods). \[Halin-Jung revisited\] The self-dual subcomplexes of the dichotomial $3$-sphere with $1$-skeleton $K_{3,3}$ are, apart from $K_{3,3}$ itself: $$\begin{gathered} HJ'_4=I\x I\cup \{(0,0),(1,1)\}*a\cup\{(0,1),(1,0)\}*b,\\ HJ'_3=I\x I\cup_{0\x I\cup I\x 0}I\x I\cup \{(0,0)\}*a,\\ HJ'_2=S^2_\square\sqcup a,\\ HJ'_1=CS^2_\square,\end{gathered}$$ where $S^2_\square$ is the cellulation of $S^2$ obtained by glueing up all three circuits in $K_{3,2}$ by $2$-cells. Each $|HJ'_i|$ is homeomorphic with some $|HJ_j|$, but none of $HJ_i$, $HJ'_j$, $K_5$, $K_{3,3}$ is a subdivision of any other one. \[problem:main\] Given an $n\ge 5$, are there only finitely many of dichotomial $n$-spheres? Algorithmic issues {#algorithmic} ------------------ Due to the algorithmic nature of the van Kampen obstruction, the problem of embeddability of a compact $n$-polyhedron (given by a specific triangulation) in $\R^{2n}$ is algorithmically decidable for $n\ge 3$ [@MTW]. This suggests seeking a higher-dimensional Kuratowski embeddability criterion that would also provide an algorithm deciding the embeddability of the polyhedron. Let us thus discuss amendments needed to fit Problem \[problem:main\], Conjecture \[conjecture:main\], and the Main Theorem in the algorithmic framework. We do not address issues of complexity of algorithms here. 1\. The definition of a cell complex involves PL homeomorphism with $|\partial\Delta^n|$ which is not a fully algorithmic notion by S. P. Novikov’s theorem (see [@CL]). A standard workaround is to consider only cell complexes whose cells are shellable (see [@Bj]). This might potentially exclude some interesting examples, but the Main Theorem is still valid and Conjecture \[conjecture:main\] and Problem \[problem:main\] are still sensible. 1$'$. An alternative possibility is to generalize cell complexes to [*circuit complexes*]{}, where the boundary of every cone is a circuit; we call an $n$-dimensional poset $M$ an [*$n$-circuit*]{}, if $H^n(|M\but\cel p\cer|)=0$ for every $p\in M$. Then part (ii) of the Main Theorem has to be amended; its proof can be reworked to show that every $m$-dimensional dichotomial circuit complex has the integral homology of $S^m$. (Whether it must still be PL homeomorphic to $S^m$ is unknown to the author.) Other parts of the Main Theorem hold without changes, and so do their proofs; Conjecture \[conjecture:main\] and Problem \[problem:main\] stand. 2\. The condition of being $(n-1)$-connected is not algorithmically decidable already for $n=2$ by Adian’s theorem (see [@CL]). However, the proof of Theorem \[connected\] produces, for each $n$-dimensional cell complex $K$ such that $|K|$ embeds in $\R^{2n}$, $n\ge 3$, a cell complex $K^+$ containing $K$, whose $1$-skeleton lies in a collapsible subcomplex of the $2$-skeleton. On the other hand, if a cell complex $K$ satisfies (i) along with $H_i(|K|)=0$ for $i\le n-1$, then $|K|$ is $(n-1)$-connected by the Hurewicz theorem. The modification of Conjecture \[conjecture:main\] with the hypothesis that $|K|$ is $(n-1)$-connected replaced by the algorithmically decidable conditions (i) and (ii) still makes sense. 3\. The definition of an $h$-minor involves cell-like maps, which in turn involve the non-algorithmic notion of contractibility. An [*ad hoc*]{} solution is the following condition on the order-preserving map $f$: instead of requiring the geometric realization $|f|$ to be cell-like, we require the barycentric subdivision $f^\flat$ to be the composition of a sequence of admissible edge contractions (see §\[edge-minors\]). Drawbacks of this condition have been discussed in Example \[subdivision\], but it is working so that the Main Theorem remains valid and Conjecture \[conjecture:main\] remains meaningful. 4\. The fragmentary character of this subsection suggests that combinatorial topology is badly missing a coherent algorithmic development of foundations, which would include, [*inter alia*]{}, mutually compatible notions of an algorithmic cell complex, of an algorithmic subdivision, and of an algorithmic cell-like map. The author is working on such a project whose success is not yet obvious. Embeddings ========== Flores–Bier construction (simplified and generalized) {#embeddings} ----------------------------------------------------- In this subsection we prove the van Kampen–Flores–Grünbaum–Schild non-embedding theorem (see Theorem \[self-dual\]) and its generalization to dichotomial posets. Let $R$ be a poset and $P$, $Q$ be embedded in $R$. The [*deleted product*]{} $P\otimes Q$ is the embedded poset in $P\x Q$, consisting of all $(p,q)$ such that $\fll p\flr\cap\fll q\flr=\emptyset$. The [*deleted join*]{} $P\circledast Q=C^*P\otimes C^*Q$ (where $(\hat 0,\hat 0)$ is not subtracted like in the definition of $P*Q$ since it is already missing here). The [*deleted prejoin*]{} is not $P\oplus Q$ as one might guess, but $P\oplus Q^*=(\mathcal P\sqcup\mathcal Q,\preceq)$, where $P=(\mathcal P,\le)$, $Q=(\mathcal Q,\le)$, and $p\preceq q$ iff either - $p,q\in P$ and $p\le q$, or - $p,q\in Q$ and $p\ge q$, or - $p\in P$, $q\in Q$ and $\fll p\flr\cap\fll q^*\flr=\emptyset$. Similarly to the non-deleted case, $(P\oplus Q^*)^\flat\simeq (P\circledast Q)^\flat_{P\circledast\emptyset \cup\emptyset\circledast Q}\subset P^\flat*Q^\flat$ (using that $(Q^*)^\flat\simeq Q^\flat$). In particular, $|P\oplus Q^*|\cong |P\circledast Q|$. More specifically, it is not hard to see that $(P\circledast Q)^\flat$ is a subdivision of $(P\circledast Q)^\flat_{P\circledast\emptyset\cup\emptyset\circledast Q}\simeq (P\oplus Q^*)^\flat$. An advantage of the deleted prejoin $P\oplus Q^*$ over the would-be $P\oplus Q$ is revealed already by the slightly more delicate isomorphism $(P\oplus Q^*)^\#\simeq (P\circledast Q)^\#_{P\circledast\emptyset \cup\emptyset\circledast Q}\subset P^\#*Q^\#$, where the star can no longer be dropped. Associativity of join implies $(K\circledast K)*(L\circledast L)\simeq (K*L)\circledast (K*L)$. Thinking of the $n$-simplex $\Delta^n$ as the join of $n+1$ copies of a point, we get that $\Delta\circledast\Delta$ is isomorphic to the join of $n+1$ copies of $pt\circledast pt=S^0$, which is the boundary of the $(n+1)$-dimensional cross-polytope. A poset $Q$ will be called an [*$m$-obstructor*]{} if $|Q\circledast Q|$ with the factor exchanging involution is $\Z/2$-homeomorphic to $S^{m+1}$ with the antipodal involution. Much of what follows can be done without assuming the homeomorphism to be $\Z/2$-equivariant or to be with the genuine sphere, because it follows from the Smith sequences that every free involution on a polyhedral ($\Z/2$-)homology $m$-sphere has cohomological ($\bmod 2$) sectional category equal to $m$ (see [@M2]). \[5.1\] Let $[3]=\{0,1,2\}$ denote the three-point set. It is easy to see that $[3]\oplus[3]\simeq\partial\Delta^2$, whence $|[3]\circledast[3]|\cong|[3]\oplus[3]|\cong S^1$. From an explicit form of this homeomorphism it is easily seen to be equivariant. Thus $[3]$ is a $0$-obstructor. (In fact, this is a special case of a general fact, which will be given a more conceptual explanation in Example \[5.5\] and Theorem \[5.6\].) \[5.2\] If $Q$ is an $m$-obstructor, then \(a) $|Q|$ does not embed in $S^m$; \(b) every embedding $g$ of $|Q|$ in $S^{m+1}$ is inequivalent with $hg$, where $h$ is an orientation-reversing self-homeomorphism of $S^{m+1}$. Part (a) is due essentially to Flores [@F1] (see also [@Ro]). The following proof of (a) occurs essentially in [@VS]. The method of (b) yields another proof of (a), see [@M2 Example 3.3]. If $|Q|$ embeds in $S^m$, then the cone $|C^*Q|$ over $|Q|$ embeds in $B^{m+1}$. Since the homeomorphism $S^{m+1}\to |Q\circledast Q|=|C^*Q\otimes C^*Q|$ is equivariant, its composition with the projection $|C^*Q\otimes C^*Q|\subset |C^*Q\x C^*Q|\to|C^*Q|$ does not identify any pair of antipodes in $S^{m+1}$. The embedding $|C^*Q|\emb B^{m+1}$ then yields a map $S^{m+1}\to B^{m+1}$ identifying no pair of antipodes, which contradicts the Borsuk–Ulam theorem. For the proof of (b) we need the following definition. Given a poset $K$ and an embedding $g{\colon}|K|\emb\R^m$, define a map $|K\x K|\but\Delta_{|K|}\to S^{m-1}$ by $(x,y)\mapsto \frac{G(x)-G(y)}{||G(x)-G(y)||}$. This map is equivariant with respect to the factor exchanging involution on $|K\x K|\but\Delta_{|K|}$ and the antipodal involution on $S^{m-1}$. In particular, we have a $\Z/2$-map $\tilde g{\colon}|K\otimes K|\subset |K\x K|\but\Delta_{|K|}\to S^{m-1}$. An embedding $g{\colon}|Q|\emb S^{m+1}$ extends to an embedding $G{\colon}|C^*Q|\emb B^{m+2}$. Since $C^*Q\x C^*Q\simeq Q\circledast Q$, this yields a cohomology class $\tilde G^*(\Xi)\in H^{m+1}(|Q\circledast Q|)$, where $\Xi\in H^{m+1}(S^{m+1})$ is a generator (cf. [@M2 §3, subsection “1-Paramater van Kampen obstruction”]). The $\bmod 2$ reduction of $\tilde G^*(\Xi)$ is nonzero, since the Yang index of the factor exchanging involution on $|Q\circledast Q|\cong S^{m+1}$ is $m+1$ (see [@M2 §3, subsection “Unoriented van Kampen obstruction”]). The mirror symmetry $r$ in the equator $S^m\subset S^{m+1}$ extends to the mirror symmetry $R$ in $B^{m+1}\subset B^{m+2}$, which in turn corresponds to $r$ (i.e. $\widetilde{RG}=r\tilde G$). Since $r^*(\Xi)=-\Xi$ and $\tilde G^*(\Xi)$ is a nonzero integer, $RG$ is inequivalent to $G$. Hence $rg$ is inequivalent to $g$. Lemma \[5.2\] and Example \[5.1\] imply that the three-point set $[3]$ does not embed in $S^0$ and knots in $S^1$. However, this is not the end of the story. \[5.3\] If $K$ is a $k$-obstructor and $L$ an $l$-obstructor, then $K*L$ is an $(k+l+2)$-obstructor. We are given $\Z/2$-homeomorphisms $S^{k+1}\to|K\circledast K|$ and $S^{l+1}\to |L\circledast L|$. Their join is a $\Z/2$-homeomorphism $S^{k+l+3}\to |(K\circledast K)*(L\circledast L)|$. From the associativity of join $(K\circledast K)*(L\circledast L)\simeq (K*L)\circledast (K*L)$, which implies the assertion. Now from Example \[5.1\] and Lemma \[5.3\], the join $[3]*[3]$ is a $2$-obstructor. Thus the graph $K_{3,3}\bydef|[3]*[3]|$ does not embed in $S^2$. Moreover (by the proof of Lemma \[5.2\]), the cone over $K_{3,3}$ does not embed in $B^3$. The same argument establishes \[5.4\] The join of $n+1$ copy of the three-point set $[3]$ does not embed in $S^{2n}$. The more precise assertion that the join of $n+1$ copy of $[3]$ is an $n$-obstructor is due to Flores [@F1] (see also [@Ro]). An element $\sigma$ of a poset $P$ is called an [*atom*]{} of $P$, if $\fll\sigma\flr=\{\sigma\}$. The set of all atoms of $P$ will be denoted $A(P)$. A poset $P$ is called [*atomistic*]{}, if every its element is the least upper bound of some set of atoms of $P$. It is easy to see that $A(\fll\sigma\flr)=A(P)\cap\fll\sigma\flr$. Hence every element $\sigma$ of an atomistic poset is the least upper bound of $A(\fll\sigma\flr)$. (Beware that “atomic” has a different meaning in the literature on posets.) Let us call a poset $B$ [*dichotomial*]{}, if it is atomistic, and for each $\sigma\in B$ the set of atoms $A(B)\but A(\fll\sigma\flr)$ has the least upper bound, denoted $h(\sigma)$, in $B$. In other words the latter condition says that there exists an $h(\sigma)\in B$ such that $\fll\sigma\flr\cap\fll h(\sigma)\flr=\emptyset$ and at the same time $\fll\sigma\flr\cup\fll h(\sigma)\flr$ contains all the atoms of $B$. Clearly, $h(h(\sigma))=\sigma$, so there is defined an involution $h{\colon}B\to B$. Clearly, the composition $H{\colon}B\xr{h}B\xr{\operatorname{id}}B^*$ is order-preserving. In particular, every dichotomial poset $B$ is isomorphic to its dual $B^*$. If $K$ is a subcomplex of a dichotomial poset $B$, then $B\but K$ is a dual subcomplex of $B$. Hence $H(B\but K)$ is a dual subcomplex of $B^*$. Then $D(K)\bydef H(B\but K)^*$ is subcomplex of $B$. In particular, $D(K)=K$ iff $K$ is a fundamental domain of the involution $h$; in this case we say that $K$ is [*self-dual*]{} as a subcomplex of $B$. \[5.5\] The boundary of every simplex $\partial\Delta^S$ is dichotomial: $h(\sigma)=S\but\sigma$. If $n<m$ and $K=(\Delta^m)^{(n)}$ is the $n$-skeleton of (the boundary of) the $m$-simplex, then it is easy to see that $D(K)=(\Delta^m)^{(m-n-2)}$. In particular, the $n$-skeleton of the $(2n+2)$-simplex is self-dual in $\partial\Delta^{2n+2}$. \[5.6\] If $K$ is a subcomplex of a dichotomial poset $B$, then the deleted prejoin $K\oplus D(K)^*$ is isomorphic to $B$. Moreover, if $K=D(K)$, then the isomorphism $f$ is anti-equivariant with resect to the anti-involution $H$ and the factor-exchanging anti-involution $t$, i.e. the following diagram commutes: $$\begin{CD} B@>f>>K\oplus K^*\\ @VHVV@VtVV\\ B^*@>f^*>>(K^*\oplus K)^*. \end{CD}$$ The first assertion of Theorem \[5.6\] implies the following “Semi-combinatorial Alexander duality” theorems: \(i) $(K\circledast D(K))^\flat$ is a subdivision of $B^\flat$; \(ii) $|K\circledast D(K)|$ is homeomorphic to $|B|$. In the case where $B$ is the boundary of a simplex (see Example \[5.5\]), (ii) was originally proved by T. Bier in 1991 (see [@Mat]) and reproved in a more direct way by de Longueville [@dL]. The assertion of (i) is a special case of a result of Björner–Paffenholz–Söstrand–Ziegler [@BPSZ] (see also [@CD]). Let us map $K\oplus D(K)^*$ to $B$ by sending the first factor via the inclusion $K\subset B$ and the second factor via $H^{-1}{\colon}H(B\but K)\to B\but K$. The resulting bijection $f{\colon}K\oplus D(K)^*\to B$ is clearly an order-preserving embedding separately on the first factor and on the second factor. If $p\in K$ and $q\in D(K)^*$, then $p\le q$ in $K\oplus D(K)^*$ iff $\fll p\flr\cap\fll q^*\flr=\emptyset$. The latter is equivalent to $A(\fll p\flr)\cap A(\fll q^*\flr)=\emptyset$ (since $B$ is atomistic). Now $f(p)=p$, and $A(\fll f(q)\flr)$ is the complement of $A(\fll q^*\flr)$ in $A(B)$. Hence $A(\fll p\flr)\cap A(\fll q^*\flr)=\emptyset$ is equivalent to $A(\fll f(p)\flr)\subset A(\fll f(q)\flr)$. The latter is in turn equivalent to $\fll f(p)\flr\subset\fll f(q)\flr$ (since $B$ is atomistic), which is the same as $f(p)\le f(q)$. Thus $f$ is an isomorphism. In the case $K=D(K)$, the composition $K\oplus K^*\xr{f}B\xr{H}B^*$ is the the identity on the second factor and is $H$ on the first factor. The same is true of the composition $K\oplus K^*\xr{t}(K^*\oplus K)^*\xr{f^*}B^*$. We say that a dichotomial poset $B$ with its $*$-involution $H{\colon}B\to B^*$ is a [*dichotomial $m$-sphere*]{} if $|B|$ is $\Z/2$-homeomorphic to $S^m$ with the antipodal involution. Since $|K\circledast K|\cong|K\oplus K^*|$ equivariantly, we obtain \[5.7\] Let $B$ be a dichotomial $(m+1)$-sphere. Then every self-dual subcomplex of $B$ is an $m$-obstructor. In particular, the self-dual subcomplex $(\Delta^4)^{(1)}$ of the dichotomial $3$-sphere $\partial\Delta^4$ is a $2$-obstructor. Thus the graph $K_5\bydef(\Delta^4)^{(1)}$ does not embed in the plane. The same argument establishes \[5.8\] The $n$-skeleton of the $(2n+2)$-simplex does not embed in $S^{2n}$. The more precise assertion that the $n$-skeleton of the $(2n+2)$-simplex is an $n$-obstructor is due to Flores [@F2]. Bringing in Lemma \[5.3\], we immediately obtain the following generalization of Theorem \[5.8\], which includes Theorem \[5.4\] as well: \[5.9\] Every $n$-dimensional join of skeleta $(\Delta^{2n_i+2})^{(n_i)}$ does not embed in $S^{2n}$. The more precise assertion that $n$-dimensional $2n$-obstructors include $n$-dimensional joins of the form $F_{i_1}*\dots*F_{i_r}$, where each $F_i$ is the $i$-skeleton of the $(2i+2)$-simplex has a converse. Using matroid theory, Sarkaria has shown that these are the only simplicial complexes among $n$-dimensional $2n$-obstructors [@Sa1]. It would be interesting (in the light of Problem \[problem:main\]) to extend his methods to cell complexes. Construction of dichotomial spheres ----------------------------------- Our only example so far of a dichotomial poset is the boundary of simplex. To get more examples, we can move in the opposite direction and utilize Lemma \[5.3\]. We say that an atomistic poset $K$ has [*atomistic category $\ge n$*]{} if the set of atoms $A(K)$ is not contained in a union of $n$ cones of $K$. \[5.10\] If $K$ is atomistic and has atomistic category $\ne 1$, then $K\oplus K^*$ is dichotomial. The proof shows that if $K$ is atomistic and has atomistic category $1$, then $K\oplus K^*$ is not atomistic. The hard part is to show that $K\oplus K^*$ is atomistic. Since $K$ is a subcomplex of $K\oplus K^*$, we have $A(K)\subset A(K\oplus K^*)$. If $\sigma\in K$, then $\sigma$ is the least upper bound in $K$ of some set $S$ of atoms of $K$. If some $\tau\in K^*$ is an upper bound of $S$, then $\fll\tau^*\flr\cap S=\emptyset$. Hence $\fll\tau^*\flr\cap\fll\sigma\flr=\emptyset$, and so $\tau>\sigma$. Thus $\sigma$ is the least upper bound of $S$ in $K\oplus K^*$. If $\sigma\in K^*$, then it is an upper bound of $S\bydef A(K)\but A(\fll\sigma^*\flr_K)$. If $S$ is nonempty and $K$ is not a cone, then by the hypothesis $S$ has no upper bound in $K$. If $\tau\in K^*$ is another upper bound of $S$, then $\fll\tau^*\flr\cap S=\emptyset$. Hence $A(\fll\tau^*\flr)\subset A(\fll\sigma^*\flr)$. Since $K$ is atomistic, $\fll\tau^*\flr\subset\fll\sigma^*\flr$. Then $\tau^*\le\sigma^*$, so $\tau\ge\sigma$. Thus $\sigma$ is the least upper bound of $S$ in $K\oplus K^*$. In the case $S=\emptyset$ we have that $\sigma^*$ is an upper bound of $A(K)$. Since $\sigma^*$ is also the least upper bound of some subset of $A(K)$, the least upper bound of $A(K)$ exists and equals $\sigma^*$. Since $K$ is atomistic, this implies as above that the least upper bound of $K$ exists and equals $\sigma^*$. Then $\sigma\le\tau$ for each $\tau\in K^*$ and also $\sigma$ is incomparable with any element of $K$. Thus $\sigma$ an atom of $K\oplus K^*$ and so the least upper bound of a set of atoms. If $S\ne\emptyset$ and $K$ is a cone, in symbols $K=\fll\hat 1\flr$, then similarly to the above, $\sigma$ is the least upper bound of $S\cup\{\hat 1^*\}$ in $K\oplus K^*$. Thus $K\oplus K^*$ is atomistic. Moreover, we have proved that $A(K\oplus K^*)=A(K)$ if $K$ is not a cone, and if $K$ is a cone, then $A(K\oplus K^*)=A(K)\cup\{\hat 1^*\}$. Given a $\sigma\in K$, let $h(\sigma)=\sigma^*\in K^*$. If $K$ is not a cone, then by the above $A(\fll\sigma^*\flr)=A(K)\but A(\fll\sigma\flr)$. If $K$ is a cone, then $A(\fll\sigma^*\flr)=(A(K)\but A(\fll\sigma\flr))\cup\{\hat 1^*\}$. In either case $A(\fll\sigma^*\flr)=A(K\oplus K^*)\but A(\fll\sigma\flr)$, so $K\oplus K^*$ is dichotomial. If $P$ and $Q$ are posets, the atomistic category of $P*Q$ is clearly the maximum of the atomistic categories of $P$ and $Q$. The atomistic category of the $n$-skeleton of the $(2n+2)$-simplex is two, since the $2n+3$ vertices of the simplex cannot be covered by two $n$-simplices but can be covered by three. Hence all the joins in Theorem \[5.9\] have atomistic category two as well. \[5.11\] If $K$ is a join of the $n_i$-skeleta of the $(2n_i+2)$-simplices, then $K\oplus K^*$ is a dichotomial sphere. More generally: \[construction\] Every atomistic $2n$-obstructor that is an $n$-dimensional cell complex is the $n$-skeleton of some dichotomial $(2n+1)$-sphere. Let $K$ be the cell complex in question. Suppose that $K$ is a union of two cells, $K=C\cup D$. If $C\cap D=\emptyset$, then $K$ embeds in the $2n$-sphere $\partial C*D\cup D*\partial C$, contradicting Lemma \[5.2\]. Else let $C'$ be the union of all cells of $C$ that are disjoint from $D$. Then $C'\subset\partial C$. On the other hand, since $K$ is atomistic, it embeds in $C'*D$. Hence $K$ embeds in the $2n$-ball $\partial C*D$, again contradicting Lemma \[5.2\]. Proof of Main Theorem (beginning) --------------------------------- \[4.3\] Let $B$ be a dichotomial poset. The following are equivalent: \(i) $|B|$ is homeomorphic to a sphere; \(ii) $B^\flat$ is a combinatorial manifold; \(iii) $B$ a cell complex. Let $h{\colon}B\to B^*$ be the order-preserving isomorphism in the definition of a dichotomial poset. We note that (i)(ii) is obvious. Let $\sigma\in B$. Then $h(\cel\sigma\cer)=\fll h(\sigma)\flr^*$, so $\partial^*\cel\sigma\cer\simeq(\partial\fll h(\sigma)\flr)^*$. Let $[\sigma]\in B^\flat$ be the chain consisting of $\sigma$ only. Then $\operatorname{lk}([\sigma],B^\flat)\simeq (\partial\fll\sigma\flr+\partial^*\cel\sigma\cer)^\flat$. This is in turn isomorphic to $(\partial\fll\sigma\flr)^\flat*(\partial^*\cel\sigma\cer)^\flat\simeq (\partial\fll\sigma\flr)^\flat*(\partial\fll h(\sigma)\flr)^\flat$, which is a combinatorial sphere. Thus $B^\flat$ is a combinatorial manifold. If $\sigma\in B$, and $\fll\tau\flr$ is a maximal simplex in $\fll h(\sigma)\flr^\flat$, then the combinatorial sphere $\operatorname{lk}(\tau,B^\flat)=(\partial\cel h(\sigma)\cer)^\flat$ is the barycentric subdivision of $(\partial\cel h(\sigma)\cer)^*=h(\partial\fll\sigma\flr)$. Hence $|\partial\fll\sigma\flr|$ is a sphere. Suppose that $B^\flat$ is a combinatorial $m$-manifold, and let $\fll\sigma\flr$ be an $m$-cell of $B$. Then $h(\fll\sigma\flr)=\cel h(\sigma)\cer^*$. Let $\tau$ be a $d$-cell of $B$ not contained in $\fll\sigma\flr$. Then $h(\sigma)\in\fll\tau\flr$. The intersection $\cel h(\sigma)\cer\cap\fll\tau\flr$ is the interval $[\sigma,\tau]=\{\sigma\}+P+\{\tau\}$, where $S=\operatorname{lk}(\sigma,\partial\fll\tau\flr)$. Hence $[\sigma,\tau]^\flat\simeq\{[\sigma]\}*S^\flat*\{[\tau]\}$, where $S^\flat\simeq\operatorname{lk}([\sigma],(\partial\fll\tau\flr)^\flat)$ is a combinatorial $(d-2)$-sphere since $(\partial\fll\tau\flr)^\flat$ is a combinatorial $(d-1)$-sphere and so in particular a combinatorial $(d-1)$-manifold. Thus $\cel h(\sigma)\cer^\flat\cap\fll\tau\flr^\flat$ is a combinatorial $d$-ball which meets $(\partial\fll\tau\flr)^\flat$ in a combinatorial $(d-1)$-ball. Let $S$ be the subposet of $B$ consisting of all $\rho\in B$ that lie neither in $\fll\sigma\flr$ nor in $\cel h(\sigma)\cer$. Note that $h(S)=S^*$. Since $B^\flat$ is a combinatorial $m$-manifold and $\fll\sigma\flr^\flat$ is a combinatorial $m$-ball, $\fll S\flr^\flat$ is a combinatorial $m$-manifold with two boundary components, $\fll S\flr^\flat\cap\fll\sigma\flr^\flat$ and $\fll S\flr^\flat\cap\cel h(\sigma)\cer^\flat$. By the above, $\fll S\flr^\flat$ meets $\fll\tau\flr^\flat$ in a combinatorial $d$-ball $B_\tau$ meeting $(\partial\fll\tau\flr)^\flat$ in a combinatorial $(d-1)$-ball $D_\tau$. (We are using the relative combinatorial annulus theorem, which follows from the uniqueness of regular neighborhoods, see e.g. [@RS]). Collapsing each $|B_\tau|$ onto $|D_\tau|$ in an order of decreasing dimension, we obtain a collapse of $|\fll S\flr|$ onto $|\fll S\flr\cap\fll\sigma\flr|$. Hence $C\bydef\fll S\flr^\flat\cup\fll\sigma\flr^\flat$ is a regular neighborhood of $\fll\sigma\flr^\flat$, and therefore $C$ is a combinatorial ball. Thus $B^\flat$ is the union of two combinatorial balls $\cel h(\sigma)\cer^\flat$ and $C$ along an isomorphism of their boundaries. By the Alexander trick, it must be a combinatorial sphere. \[3.3b\] If $K$ is a poset and $L$ is its $h$-minor, there exists a $\Z/2$-map $H{\colon}|L\circledast L|\to |K\circledast K|$. Moreover, if $K$ is atomistic and $L$ is its proper $h$-minor, then $H$ is non-surjective. It suffices to consider the case where $K$ maps onto $L$ by a non-injective order-preserving map $f$ such that $|f|$ is cell-like. Let us define a map $g{\colon}|L|\to|K|$ such that $g(|C|)\subset |f^{-1}(C)|$ for each cone $C$ of $L$. Assume that $g$ is defined on cones of dimension $<d$, and let $C$ be a $d$-dimensional cone of $L$. Since $f$ is cell-like, $|f^{-1}(C)|$ is contractible. We then define $g$ on $|C|$ by mapping it via some null-homotopy of $g(|\partial C|)$ in $|f^{-1}(C)|$. Since $f$ is order-preserving, $f^{-1}(C)$ is a subcomplex of $K$ for each cone $C$ of $L$. Since $g(|C|)\subset |f^{-1}(C)|$ for each cone $C$ of $L$, and $f$-preimages of disjoint cones are disjoint, $g$ sends every disjoint pair of cones to a pair of disjoint unions of cones. Hence $(g*g)(|L\circledast L|)\subset |K\circledast K|$. To prove the non-surjectivity, note that $g$ in fact sends every disjoint pair of cones to a pair of unions of cones whose $|f|$-images are disjoint. Since $f$ is order-preserving and non-injective, there exist distinct $\sigma,\tau\in K$ whose $f$-images are the same. Since $K$ is atomistic, $\sigma$ and $\tau$ may be assumed to be its atoms. Then $(g*g)(|L\circledast L|)$ does not contain $|\{\sigma\}\x\{\tau\}\cup\{\tau\}\x\{\sigma\}|$, which lies in $|K\circledast K|$. \[quotient\] Let $K$ be an $n$-dimensional poset. Then there exists a $\Z/2$-map $p{\colon}|K\circledast K|\to S^0*|K\otimes K|$ such that $p$ and $p/t$ induce isomorphisms on $i$-cohomology with arbitrary (possibly twisted) coefficients for $i\ge n+2$ and an epimorphism for $i=n+1$. The map $p$ is the quotient map shrinking $|K*\emptyset|$ and $|\emptyset*K|$ to points. The relative mapping cylinder of $p$ collapses onto the pair of $(n+1)$-polyhedra $(|CK\sqcup CK|,|K\sqcup K|)$, and the assertion follows. \[is-linkless\] Let $K$ be an $n$-dimensional cell complex, $n\ge 2$. An embedding $g{\colon}|K|\emb B^{2n+1}$ is linkless iff for every pair of disjoint subcomplexes $L$ and $M$ of $K$, the map $\tilde g|_{|L\x M|}{\colon}|L\x M|\to S^{2n}$ is null-homotopic. Let $P$ and $Q$ be disjoint subpolyhedra of $|K|$. Let $P'$ be obtained by puncturing $K$ in some interior point of each $n$-cell $C$ of $K$ such that $|C|\not\subset P$; define $Q'$ similarly. Then $P'$ deformation retracts onto $|L|\cup|K^{(n-1)}|$ and $Q'$ deformation retracts onto $|M|\cup |K^{(n-1)}|$, where $L$ and $M$ are the maximal subcomplexes of $K$ such that $|L|\subset P$ and $|M|\subset Q$. Now the hypothesis implies that $\tilde g|_{P'\x Q'}$ is null-homotopic, and therefore $\tilde g|_{P\x Q}$ is null-homotopic. Then by the Haefliger–Weber criterion [@We], $g|_{P\sqcup Q}$ is equivalent to the embedding $h{\colon}P\sqcup Q\emb B^{2n+1}$, obtained by combining $e_1g|_P$ and $e_2g|_Q$, where $e_1,e_2{\colon}B^{2n+1}\to B^{2n+1}$ are embeddings with disjoint images. \[all-linkless\] Let $K$ be an $n$-dimensional cell complex. If $H^{2n+1}(|K\circledast K|)$ is cyclic, then every embedding of $|K|$ in $S^{2n+1}$ is linkless. Let $L$ and $M$ be disjoint subcomplexes of $K$, and let $G=H^{2n}(|L\x M|)$. Since $L$ and $M$ are disjoint, $H^{2n}(|L\x M\cup M\x L|)$ is isomorphic to $G\oplus G$ and also is an epimorphic image of $H^{2n}(|K\otimes K|)$. The latter group is cyclic by Lemma \[quotient\] (as long as $n>0$), so $G$ must be zero. If $n\ge 2$, the assertion follows from Lemma \[is-linkless\] and the Hopf classification theorem. For $n=1$, the proof of Lemma \[is-linkless\] works to show that if $P$ and $Q$ are disjoint subpolyhedra of $|K|$, then $H^2(P\x Q)=0$. But then either $P$ or $Q$ must be a forest, and the assertion follows. \[deleted\] Let $L$ be an $n$-dimensional cell complex. Then $H^{2n}(\overline{|L|},|L\otimes L|/t)=0$, where $\overline{|L|}=(|L|\x |L|\but\Delta_{|L|})/t$. If $C$ is a maximal cell of $L\x L$ that is not in $L\otimes L$, then $|C|$ meets the diagonal $\Delta_{|L|}$. Let us call an $n$-polyhedron $M$ an [*$n$-circuit*]{}, if $H^n(M\but\{x\})=0$ for every $x\in M$. \[proper minor\] Let $K$ be an $n$-dimensional atomistic cell complex such that $|K\circledast K|/t$ is a $(2n+1)$-circuit, and let $L$ be a proper $h$-minor of $K$ that is a cell complex. Then \(a) $|L|$ embeds in $S^{2n}$ if $n\ne 2$; \(b) every two embeddings (knotless if $n=1$) of $|K|$ in $S^{2n+1}$ become equivalent when “restricted” to $|L|$. It is interesting to compare this with the well-known result that an $n$-polyhedron $P$ embeds in $S^{2n}$ if $P$ itself is an $n$-circuit [@Sa-1], [@M2 8.2]. By Lemma \[3.3b\] we have a non-surjective $\Z/2$-map $f{\colon}|L\circledast L|\to|K\circledast K|$. Let $t$ denote the factor exchanging involution, let $\phi{\colon}|L\circledast L|/t\to\R P^\infty$ be its classifying map, and let $\xi\in H^{2n+1}(\R P^\infty;\,\Z_T)\simeq\Z/2$ be the generator, where $\Z_T$ denotes the twisted integer coefficients. Since $\phi$ factors up to homotopy through the non-surjective map $f/t$ into the $(2n+1)$-circuit $|K\circledast K|/t$, we have $\phi^*(\xi)=0$. By Lemma \[quotient\], there exists a $\Z/2$-map $p{\colon}|L\circledast L|\to S^0*|L\otimes L|$ such that $p/t$ induces an isomorphism on $(2n+1)$-cohomology, as long as $n>0$. Since $\phi$ factors up to homotopy as $p$ followed by a classifying map $\psi{\colon}(S^0*|L\otimes L|)/t\to\R P^\infty$ of $t$, we have $\psi^*(\xi)=0$. It follows (see [@CF (5.1)]) that $\chi^*(\zeta)=0$, where $\chi{\colon}|L\otimes L|/t\to\R P^\infty$ is a classifying map of $t$ and $\zeta\in H^{2n}(\R P^\infty;\,\Z)\simeq\Z/2$ is the generator. By Lemma \[deleted\], $\chi_+^*(\zeta)=0$, where $\chi_+{\colon}\overline{|L|}\to\R P^\infty$ is a classifying map of $t$. Now $\chi_+^*(\zeta)$ is the van Kampen obstruction $\theta(|L|)$; so $|L|$ embeds in $S^{2n}$ as long as $n\ge 3$ (see [@M2]). Let $g,h{\colon}|K|\to\R^{2n+1}$ be the given embeddings and $g',h'{\colon}|L|\to\R^{2n+1}$ their “restrictions”. We have $d(\tilde g',\tilde h')=d(\tilde gF,\tilde hF)=F^*d(\tilde g,\tilde h)$, where the $\Z/2$-map $F{\colon}|L\otimes L|\to|K\otimes K|$ is defined similarly to $f$. Now $(F/t)^*{\colon}H^{2n}(|K\otimes K|/t;\,\Z_T)\to H^{2n}(|L\otimes L|/t;\,\Z_T)$ is the zero map, since it is equivalent to $(f/t)^*{\colon}H^{2n+1}(|K\circledast K|/t;\,\Z)\to H^{2n+1}(|L\circledast L|/t;\,\Z)$ via Thom-isomorphisms. So $\tilde g'$ and $\tilde h'$ are equivalent as long as $n\ge 2$. It remains to consider the case $n=1$. Since $\phi^(\xi)=0$, there exists a $\Z/2$-map $|L\circledast L|\to S^2$ (this is similar to [@M2 proof of 3.2]). Hence by the Borsuk–Ulam theorem there exists no $\Z/2$-map $S^3\to|L\circledast L|$. Then by Lemma \[3.3b\], $L$ has no $2$-obstructor as a minor. In particular, by the preceding observations, $L$ has no minor isomorphic to $K_5$ or $K_{3,3}$. Then by Wagner’s version of the Kuratowski theorem, $|L|$ embeds in $S^2$. Also, given a knotless embedding of $|K|$ in $S^3$, it is also linkless by Lemma \[all-linkless\]. Its “restriction” to $|L|$ is also linkless and knotless, using Lemma \[trivial\] and the definition of a knotless embedding as an embedding $g$ such that $g(|C|)$ bounds an embedded disk in $S^3$ for every circuit $C$ of the graph. Hence the Robertson–Seymour–Thomas theorem implies that every two such “restrictions” are equivalent. Linkless embeddings =================== Proof of Main Theorem (conclusion) ---------------------------------- Let $K=(\mathcal K,\le)$ be an $n$-dimensional poset, for instance, an $n$-dimensional simplicial or cell complex. If $S$ is a subcomplex of $K$, let $\bar S$ be the subcomplex of $K$ consisting of all cones of $K$ disjoint from $S$. Note that $\bar{\bar S}=S$. Consider the set $\lambda_K$ of all subcomplexes $S$ of $K$ such that $H^n(|S|)\otimes H^n(|\bar S|)$ is nonzero (or, equivalently, not all maps $|S\x\bar S|\to S^{2n}$ are null-homotopic). Then $t_K{\colon}S\mapsto\bar S$ is a free involution on $\lambda_K$. \[4.4’\] If $L$ is the $n$-skeleton of a $(2n+2)$-dimensional dichotomial cell complex $B$, then each $S\in\lambda_L$ is the boundary of some $(n+1)$-cell of $B$. Let $K$ be the union of $L$ and a half of the $(n+1)$-cells of $B$, with precisely one cell from each complementary pair. Then $K$ is self-dual in $B$, so by Theorem \[5.6\], $K\oplus K^*$ is isomorphic to $B$. Hence by Lemma \[4.3\], $|K\oplus K^*|$ is a sphere, and therefore so is $|K\circledast K|$. By Lemma \[quotient\], $H^{2n+1}(|S*\bar S|)\simeq H^n(|S|)\otimes H^n(|\bar S|)\ne 0$, so by the Alexander duality the complement to $|S*\bar S|$ in the sphere $|K\circledast K|$ contains at least two connected components. Since $S$ and $\bar S$ are disjoint, and $|\bar S*S|$ is connected (regardless of whether any of $|S|$ and $|\bar S|$ is connected!), $|\bar S*S|$ lies one of these open components. The closure of that component is cellulated by a subcomplex $W$ of $K\circledast K$, and the closure of the union of the remaining components by a subcomplex $V$. Since $|V|$ has nonempty interior, $V$ contains at least one $(2n+2)$-cell. It must be of the form $C*D$, where $C$ is an $(n+1)$-cell of $K*\emptyset$ and $D$ is an $n$-cell of $\emptyset*K$, or vice versa. Let us consider the first case. If $v$ is a vertex of $S$ not contained in $C$, the join $C*v$ lies in $K\circledast K$. Then it is a cell of $K\circledast K$, and therefore lies either in $V$ or in $W$. Since $|\emptyset*v|$ lies in $|\bar S*S|$ and so in the interior of $|W|$, we have $C*v\subset W$. On the other hand, $C*\emptyset$ lies in $C*D$ and so in $V$. Hence $C*\emptyset$ lies in $V\cap W$, which is a subcomplex of $S*\bar S$. Thus $C\subset S$. But this cannot be since $S$ is $n$-dimensional and $C$ is an $(n+1)$-cell. Thus $C$, and therefore also $\partial C$, contains all vertices of $S$. But $B$ is atomistic, so its subcomplex $K$ is atomistic, and we have $S\subset\partial C$. Since $H^n(|S|)\ne 0$ and $|\partial C|$ is an $n$-sphere, $\partial C=S$. In the case where $V$ contains $C*D$, where $C$ is an $n$-cell of $K*\emptyset$ and $D$ is an $(n+1)$-cell of $\emptyset*K$, we can similarly show that $\partial D=\bar S$. This proves that either $S$ or $\bar S$ bounds a cell in $K$ (not just in $B$). Suppose that $S$ does not bound a cell in $K$. Then by the above, $\bar S$ bounds a cell $D$ in $K$. Let us amend $K$ by exchanging $D$ with its complementary $(n+1)$-cell, and let $K'$ denote the resulting subcomplex of $B$. Then $\bar S$ does not bound a cell in $K'$. Hence by the above, $S$ bounds a cell in $K'$. Let $\hat K$ be the poset $(\mathcal K\cup\lambda_K,\preceq)$, where $p\preceq q$ iff either $p,q\in K$ and $p\le q$ or $p\in K$, $q\in S$ and $p\in q$. Given a section $\xi{\colon}\lambda_K/t_K\to\lambda_K$ of the double covering $\lambda_K\to\lambda_K/t_K$, we have the subcomplex $K_\xi$ of $\hat K$ obtained by adjoining to $K$ all elements of $\xi(\lambda_K/t_K)$. Note that under the hypothesis of Lemma \[4.4’\], $\hat K$ and $K_\xi$ are cell complexes. If $K=K_6$, then $K_\xi$ can be chosen to be the semi-icosahedron, and if $K$ is the Petersen graph, then $K_\xi$ can be chosen to be the semi-dodecahedron (see §\[semi\]). \[3.3a\] If $K$ is a poset and $L$ is its (proper) $h$-minor, then each $L_\zeta$ is a (proper) $h$-minor of some $K_\xi$. Suppose that $f{\colon}K\to L$ is an order-preserving surjection such that $|f|$ is cell-like. Given subcomplexes $M$, $N$ of $L$ such that $t_L(M)=N$ and $M,N\in\lambda_L$, we have that $M'\bydef f^{-1}(M)$ and $N'\bydef f^{-1}(N)$ belong to $\lambda_K$ since $|f|$ restricts to a homotopy equivalence $|f|^{-1}(P)\to P$ for every subpolyhedron $P$ of $|L|$. Up to relabelling, we may assume that $\zeta(\{M,N\})=M$. Then we set $\xi(\{M',t_K(M')\})=M'$ and $\xi(\{N',t_K(N')\})=t_K(N')$. If $t_K(N')\ne M'$, then we label $M'$ as a “primary” element of the image of $\xi$. This defines $\xi$ on a subset of $\lambda_K/t_K$. We extend it to the remaining elements arbitrarily, and do not introduce any new labels. The subcomplex $K_\xi'$ of $K_\xi$ obtained by adjoining to $K$ all primary elements of $\xi(\lambda_K/t_K)$ then admits an order-preserving surjection $g$ onto $L_\zeta$ such that $g$ is an extension of $f$, and $|g|$ is cell-like. The remaining case where $L$ is a subcomplex of $K$ is similar (and easier). Let $K\hat\oplus K^*=K_\xi\oplus (K_\xi)^*$. This poset does not depend on the choice of $\xi$, since it is isomorphic to $\hat K\oplus K^*$ (and also to $K\oplus (\hat K)^*$). Moreover, it is easy to see that the involution on $K\hat\oplus K^*$ also does not depend on the choice of $\xi$. Lemma \[4.4’\] has the following \[4.4\] If $L$ is the $n$-skeleton of a $(2n+2)$-dichotomial cell complex $B$, then $L\hat\oplus L^*$ is anti-equivariantly isomorphic to $B$. By Lemma \[4.4’\], $L_\xi$ is a cell complex for every $\xi$. Hence $L_\xi$ is isomorphic to a $K$ as in the proof of Lemma \[4.4’\], and therefore by Theorem \[5.6\], $L_\xi\oplus L_\xi^*$ is anti-equivariantly isomorphic to $B$. Let $K\hat\circledast K$ be the union of $K\circledast K$ and cones of the form $C(S*\bar S)$, where $S\in\lambda_K$. In more detail, $K\hat\circledast K=(P\cup\lambda_K,\preceq)$, where $(P,\le)$ is the deleted join $K\circledast K=C^*K\otimes C^*K\subset K*K$, and $p\preceq q$ iff $p,q\in P$ and $p\le q$ or $p=(\sigma,\tau)\in P$, $q\in\lambda_K$, and either - $\sigma,\tau\ne\hat 1$ and $\sigma\in q$ and $\tau\in t_K(q)$, or - $\sigma=\hat 1$ and $\tau\in t_K(q)$, or - $\tau=\hat 1$ and $\sigma\in q$. Since $C(S*\bar S)$ is subdivided by $(CS)*\bar S$ (and also by $S*(C\bar S)$), we obtain that $K_\xi\circledast K_\xi$ is a subdivision of $K\hat\circledast K$, for each $\xi$. Then from Lemmas \[3.3b\] and \[3.3a\] we get the following \[3.3\] If $L$ is an $h$-minor of a poset $K$, there exists a $\Z/2$-map $H{\colon}|L\hat\circledast L|\to |K\hat\circledast K|$. Moreover, if $K$ is atomistic and $L$ is its proper $h$-minor, then $H$ is non-surjective. Since $K_\xi\circledast K_\xi$ (which does depend on $\xi$) is a subdivision of $K\hat\circledast K$, we also obtain that $|K\hat\circledast K|\cong |K\hat\oplus K^*|$, equivariantly. It is easy to find such a $\Z/2$-homeomorphism that does not depend on the choice of $\xi$. \[3.0\] Let $K$ be an $n$-dimensional cell complex. $|K|$ is linklessly embeddable in $S^{2n+1}$ iff there exists a $\Z/2$-map $|K\hat\circledast K|\to S^{2n+1}$. This is similar to [@M2 Theorem 4.2] but we give a more detailed proof here. A part of the proof is also parallel to a part of the proof of Lemma \[proper minor\]. Given a linkless embedding $g{\colon}|K|\emb S^{2n+1}$, we may extend it to an embedding $G{\colon}|C^*K|\emb B^{2n+2}$ and pick null-homotopies $H_S{\colon}|C^*S\sqcup C^*\bar S|\to B^{2n+2}$ of the links $g(|S\cup\bar S|)$ such that $H_S(|C^*S|)\cap H_S(|C^*\bar S|)=\emptyset$ and $H_S^{-1}(S^{2n+1})=|S\sqcup\bar S|$, for each $S\in\lambda_K$. Since each $H_S$ is homotopic through maps $H_t:|C^*S\sqcup C*\bar S|\to B^{2n+2}$ satisfying $H_t^{-1}(S^{2n+1})=|S\sqcup\bar S|$ (but not necessarily $H_t(|C^*S|)\cap H_t(|C^*\bar S|)=\emptyset$) to the restriction of $G$, the deleted product maps $\tilde G{\colon}|C^*K\otimes C^*K|\to S^{2n+1}$ and $\tilde H_S|_{\dots}{\colon}|C^*S\x C^*\bar S\sqcup C^*\bar S\x C^*S|\to S^{2n+1}$ have equivariantly homotopic restrictions to $|S*\bar S\sqcup\bar S*S|$. Hence $G$ extends to an equivariant map $|K\hat\circledast K|\to S^{2n+1}$ (using the homeomorphism $|C^*(S*\bar S)|\cong |C(S*\bar S)|$). Conversely, let $K\hat\otimes K$ be the union of $K\otimes K$ and cones of the form $C(S\x\bar S)$, where $S\in\lambda_K$. Then the quotient of $|K\hat\circledast K|$ obtained by shrinking $|K*\emptyset|$ and $|\emptyset*K|$ to points is $\Z/2$-homeomorphic to the suspension $S^0*|K\hat\otimes K|$. Then similarly to Lemma \[quotient\] there exists a $\Z/2$-map $S^0*|K\hat\otimes K|\to S^{2n+1}$. By the equivariant Freudenthal suspension theorem, $|K\otimes K|$ admits a $\Z/2$-map $\phi$ to $S^{2n}$ whose restriction to $|S\x\bar S|$ is null-homotopic for each $S\in\lambda$. By Lemma \[deleted\], $\phi$ extends equivariantly over $|K\x K|\but\Delta_{|K|}$. Hence by the Haefliger–Weber criterion ([@We]; alternatively, see [@M2 3.1] and Remark \[erratum2\] below), $|K|$ admits an embedding $g$ into $B^{2n+1}$ such that $\tilde g{\colon}|K\otimes K|\to S^{2n}$ is $\Z/2$-homotopic to $\phi$. In particular, the restriction of $\tilde g$ to $S\x\bar S$ is null-homotopic for each $S\in\lambda_K$. When $n=1$, this implies that any two disjointly embedded circles in $g(|K|)$ have zero linking number. Then by Theorem \[RST\](b), $|K|$ admits a linkless embedding in $S^3$. When $n\ge 2$, the embedding $g$ itself is linkless by Lemma \[is-linkless\]. Let $K$ be the $n$-skeleton of a $(2n+2)$-dimensional dichotomial cell complex. Then every embedding of $|K|$ in $S^{2n+1}$ contains a link of two disjoint $n$-spheres with an odd linking number. By Lemma \[4.4’\], every $S\in\lambda_K$ cellulates an $n$-sphere. If $g{\colon}|K|\emb\R^{2n+1}$ is an embedding that for every $S\in\lambda_K$ links $|S|$ and $|\bar S|$ with an even linking number, then by proof of the “only if” part of Theorem \[3.0\], $|K\hat\circledast K|$ admits an $\Z/2$-map to $S^{2n+2}$ of even degree. However no even degree self-map of a sphere can be equivariant with respect to the antipodal involution. (For the resulting self-map of the projective space lifts to the double cover, so must send $w_1$ of the covering to itself. However the top power of $w_1$ goes to zero if the degree is even.) Let $K$ be an $n$-dimensional atomistic cell complex such that $|K\hat\circledast K|$ is a $(2n+2)$-circuit. Then $|K|$ admits a linkless embedding in $S^{2n+1}$. The proof is similar to a part of the proof of Lemma \[proper minor\]. By Corollary \[3.3\] we have a non-surjective $\Z/2$-map $f{\colon}|L\hat\circledast L|\to|K\hat\circledast K|$. Let $t$ denote the factor exchanging involution, let $\phi{\colon}|L\hat\circledast L|/t\to\R P^\infty$ be its classifying map, and let $\xi\in H^{2n+2}(\R P^\infty;\,\Z)\simeq\Z/2$ be the generator. Since $\phi$ factors up to homotopy through the non-surjective map $f/t$ into the $(2n+2)$-circuit $|K\hat\circledast K|/t$, we have $\phi^*(\xi)=0$. Then $|L\hat\circledast L|$ admits a $\Z/2$-map into $S^{2n+1}$ (see [@M2 proof of 3.2]), and the assertion follows from Thorem \[3.0\]. Equivariant homotopy of the Petersen family {#Petersen topology} ------------------------------------------- \[3.4\] Let $\Delta$ be the $(2n+3)$-simplex and $L$ its $n$-skeleton. For each pair of complementary $(n+1)$-simplices, choose one, and let $L_\xi$ be the union of $L$ with the chosen $(n+1)$-simplices. Then $L_\xi$ is self-dual, hence by Theorem \[5.6\], $|L_\xi\circledast L_\xi|$ is $\Z/2$-homeomorphic to the $(2n+2)$-sphere. On the other hand, $|L_\xi\circledast L_\xi|$ is also $\Z/2$-homeomorphic to $|L\hat\circledast L|$ by the above. Hence by the Borsuk–Ulam theorem the latter admits no $\Z/2$-map to $S^{2n+1}$. Thus by Theorem \[3.0\], $L$ is not linklessly embeddable in $S^{2n+1}$. In particular, $K_6$ is not linklessly embeddable in $S^3$. This was first proved by Conway and Gordon, see also [@Sac1], [@Sac2]; the $n$-dimensional generalization is proved in [@LS Corollary 1.1], [@Ta] and [@M2 Example 4.7] (in all cases, by methods different from the above). A graph $H$ is said to be obtained by a $\nabla$Y-[*exchange*]{} from a graph $G$, if $G$ contains a subgraph $\nabla$ isomorphic to $\partial\Delta^2$, and $H$ is obtained from $G$ by removing the three edges of $\nabla$ and instead adjoining the triod Y$\bydef\Delta^0*(\nabla)^0$. We shall call a $\nabla$Y-exchange [*dangerous*]{} if $G$ contains no circuit disjoint from $\nabla$; and [*allowable*]{} if either (i) it is dangerous, or (ii) it is non-dangerous and $G$ contains precisely one circuit $C$ disjoint from $\nabla$, and $H$ contains no circuit disjoint from $C$. Let us consider an allowable $\nabla$Y-exchange $G\rightsquigarrow H$. The obvious map $|\nabla|\to|$Y$|$, sending the barycenters of the edges to the cone point and keeping $|\nabla^{(0)}|$ fixed, yields a map $f{\colon}|G|\to |H|$, which sends every pair of disjoint cells of $G$ to a pair of disjoint subcomplexes of $H$. Given an $H_\zeta$, let us choose a $G_\xi$ as follows. In the dangerous case we have a canonical bijection between $\lambda_G$ and $\lambda_H$, and we let $\xi$ correspond to $\zeta$ under this bijection. In the non-dangerous case, $\lambda_G$ contains two additional circuits, $\nabla$ and $C$, and we set $\xi(\{\nabla,C\})=\nabla$. Conversely, every $\xi$ satisfying the latter property uniquely determines a $\zeta$ for a non-dangerous exchange, and every $\xi$ whatsoever uniquely determines a $\zeta$ for a dangerous exchange. With such $\zeta$ and $\xi$, our $f$ extends to an $f_\zeta{\colon}|G_\xi|\to |H_\zeta|$, which in the non-dangerous case is collapsible. This $f_\zeta$ still sends every pair of disjoint cells of $G_\xi$ to a pair of disjoint subcomplexes of $H_\zeta$. Hence $f_\zeta*f_\zeta$ restricts to a $\Z/2$-map $F_\zeta{\colon}|G_\xi\circledast G_\xi|\to |H_\zeta\circledast H_\zeta|$. In the non-dangerous case, $F_\zeta$ is collapsible, and so is an equivariant homotopy equivalence. \[3.5\] There is the following diagram of $\nabla$Y-exchanges, which uses the notation of Fig. 1: $$\begin{CD} @.\!\!\!\!K_{3,3,1}\!\!\!\!@.@.@.\\ @.\hskip25pt\searrow\hskip-25pt@.@.@.\\ K_6@>>>\Gamma_7@>>>\Gamma_8@>>>\Gamma_9@>>>P\\ @.\hskip15pt_{\mathbf !}\searrow\hskip-30pt@.@.@.\\ @.@.\!\!\!\!K_{4,4}\!\but\! e\!\!\!\!@.@. \end{CD}\tag{$*$}$$ It can be verified by inspection that all these $\nabla$Y-exchanges are allowable, and that no further $\nabla$Y- or Y$\nabla$-exchange (allowable or not) can be applied to any graph in this diagram; thus these seven graphs are the Petersen family graphs. The only dangerous $\nabla$Y-exchange is marked by an “$\mathbf !$”. Write $L=K_6$, and select $L_\xi$ so that it contains a quadruple of $2$-cells whose pairwise intersections contain no edges. For instance, the hemi-icosahedron (see §\[semi\]) will do. Then the horizontal sequence of $\nabla$Y-exchanges in ($*$) can be made along these four $2$-cells, which compatibly defines a $G_\xi$ for each $G$ in this sequence. These also uniquely determine compatible $G_\xi$ for $G=K_{3,3,1}$ and $K_{4,4}\but$(edge). Then by the above, for each $G$ in the Petersen family, we obtain a $\Z/2$-map $|L\hat\circledast L|\to |G\hat\circledast G|$ (using the invertibility of the $\Z/2$-homotopy equivalence in the case of $K_{3,3,1}$). On the other hand, by the preceding example $|L\hat\circledast L|$ is $\Z/2$-homeomorphic to $S^4$, hence by the Borsuk–Ulam Theorem it admits no $\Z/2$-map to $S^3$. Thus $|G\hat\circledast G|$, for each $G$ in the Petersen family, admits no $\Z/2$-map to $S^3$. The (easy) “only if” direction in Theorem \[3.0\] now implies that none of the graphs of the Petersen family is linklessly embeddable in $S^3$, a fact originally observed by Sachs [@Sac1], [@Sac2]. Corollary \[3.3\] then implies the easy direction in the Robertson–Seymor–Thomas Theorem \[RST\](b). \[hemi-exchanges\] If we do choose $L_\xi$ to be the hemi-icosahedron, then all the non-dangerous $\nabla$Y and Y$\nabla$-exchanges in ($*$) yield cellulations of $\R P^2$ by the complexes $G_\xi$, in particular, by the hemi-dodecahedron $P_\xi$. \[K44-nonembed\] We shall see in the next subsection that the $F_\xi$ corresponding to all the non-dangerous $\nabla$Y-exchanges in ($*$) can be approximated by equivariant homeomorphisms (for those in the horizontal line this also follows from the Cohen–Homma Theorem \[Cohen-Homma\]). This cannot be the case for the dangerous one. Indeed if $G=K_{4,4}\but$(edge), then $G\circledast G$ alone contains a join of two triods, which is non-embeddable in $S^4$ since the link of its central edge is $K_{3,3}$, which does not embed in $S^2$. Combinatorics of the Petersen family {#transforms} ------------------------------------ Following van Kampen [@vK2] and McCrory [@Mc], we define a [*combinatorial manifold*]{} as a cell complex $K$ such that $K^*$ is also a cell complex (cf. [@M3]). If additionally both $K$ and $K^*$ are atomistic, we say that $K$ is an [*$\alpha$-combinatorial*]{} manifold. Now an [*homology ($\alpha$-)combinatorial manifold*]{} as an (atomistic) homology cell complex $K$ such that $K^*$ is also an (atomistic) homology cell complex. Here a poset $K$ is called an [*homology cell complex*]{} if for each $\sigma\in K$ there exists an $m$ such that $H_i(|\partial\fll\sigma\flr|)\simeq H_i(S^m)$, with integer coefficients. Let $K$ be a poset and let $C\in K$. We say that $K$ is Y$\nabla$[*-transformable*]{} at $C$ if $C$ is covered by precisely three elements $D_1$, $D_2$, $D_3$ such that $\partial\fll D_i^*\flr \subset\partial\fll D_{i+1}^*\flr\cup\partial\fll D_{i-1}^*\flr$, and $\fll D_{i-1}\flr\cap\fll D_{i+1}\flr=\fll C\flr$ for each $i$ (addition $\bmod 3$). \[homcomb\] Suppose that $K$ is an homology combinatorial $\alpha$-manifold and $C\in K$ is covered by precisely three elements $D_1$, $D_2$, $D_3$. Then $K$ is Y$\nabla$-transformable at $C$. The proof of property (ii) uses the hypothesis that $K^*$ is atomistic. If $E>D_i$, pick an $E'<E$ such that $E'$ covers $D$. The homology combinatorial sphere $\partial E$ is a pseudo-manifold, so $\fll C\flr$ is contained in precisely two homology $(c+1)$-cells in $\partial E$. One of them is $\fll D_i\flr$, so the other can only be $\fll D_{i+1}\flr$ or $\fll D_{i-1}\flr$. [*(ii).*]{} Since the link $\partial\cel C\cer^L$ of $C$ in $L\bydef\fll D_1\flr\cup\fll D_2\flr\cup\fll D_3\flr$ is the $3$-point set, which is not a homology sphere, $K$ cannot be of dimension $c+1$, and therefore $D_i$ is not maximal. Hence $C^*$ and $D_i^*$ are not atoms of $\fll C^*\flr$ and so all the atoms of $\fll C^*\flr$ lie in $\fll C^*\flr\but\{C^*,D_i^*\}$. By (i) the latter equals $\fll D_{i-1}^*\flr\cup\fll D_{i+1}^*\flr$. Hence $D_{i-1}^*$ and $D_{i+1}^*$ do not simultaneously belong to any cell of $K^*$ other than $\fll C^*\flr$. Suppose that $K$ is a poset that is Y$\nabla$-transformable at some $C\in K$. Then we define the Y$\nabla$[*-transform*]{} of $K$ at $C$ to be the poset $K_C$ that has one element $\hat B$ for each $B\in K$ and satisfies the following for each $A,A'\notin\{C,D_1,D_2,D_3\}$: - $\hat C>\hat D_i$, $i=1,2,3$; - $\hat A<\hat D_i$ iff $A<D_{i+1}$ or $A<D_{i-1}$; - $\hat A<\hat C$ iff $A<D_1$ or $A<D_2$ or $A<D_3$; - $\hat A>\hat D_i$ iff $A>D_{i+1}$ and $A>D_{i-1}$; - $\hat A>\hat C$ iff $A>D_1$ and $A>D_2$ and $A>D_3$; - $\hat A>\hat A'$ iff $A>A'$. This definition immediately implies that $(K_C)^*$ is Y$\nabla$-transformable at $(\hat C)^*$. A straightforward Boolean logic using that $K$ is Y$\nabla$-transformable at $C$ shows further that the Y$\nabla$-transform of $(K_C)^*$ at $(\hat C)^*$ is isomorphic to $K^*$. We say that a poset $L$ is $\nabla$Y[*-transformable*]{} at $E\in L$ if $L^*$ is Y$\nabla$-transformable at $E^*$. In that case the $\nabla$Y[*-transform*]{} of $L$ at $E$ is defined to be the poset $L^E\bydef ((L^*)_{E^*})^*$. Let $M$ be a triangulation of the Mazur contractible $4$-manifold. Let $M_1\cup M_2\cup M_3$ be the union of three copies of $M$ identified along an isomorphism of their boundaries. Then each $M_i\cup M_{i+1}$ (addition $\bmod 3$) is the double of $M$ and hence is homeomorphic to $S^4$. We glue it up by a $5$-cell $C_{i-1}$. Then $C_1\cap C_2=M_3$, so $C_1\cup C_2$ is simply-connected and acyclic, hence contractible. Its boundary $M_1\cup M_2\cong S^4$, so $(C_1\cup C_2)\cup C_3$ is a homotopy sphere, hence the genuine $5$-sphere. We glue it up by a pair of $6$-cells $D_1$, $D_2$. The resulting cell complex $K$ is homeomorphic to $S^6$, and hence its dual is also a cell complex (see [@M3]). Now $D_1^*$ is covered by precisely three elements $C_1^*$, $C_2^*$, $C_3^*$ of $K^*$. Let $L=(K^*)_{D_1^*}$ and let $D=\widehat{D_1^*}$. Then $D^*$ is a cell of $L^*$ such that $|\partial\fll D^*\flr|\cong|\partial M|$, is a nontrivial homology $3$-sphere. Note that $D_1^*$ is of codimension $6$ in $K^*$. \[homology mfld\] Let $K$ be a homology $\alpha$-combinatorial manifold, and suppose that $C\in K$ is covered by precisely three elements of $K$. Then the [Y]{}$\nabla$-transform $K_C$ is a homology $\alpha$-combinatorial manifold. If additionally $K$ is an $\alpha$-combinatorial manifold, and $\fll C\flr$ is of codimension $\le 5$ in $K$, then $K_C$ is an $\alpha$-combinatorial manifold. [*Case 1.*]{} Let us first prove that $|R|$ and $|A_i|$ are homology spheres, where $R=\partial\fll(\hat C)^*\flr$ and $A_i=\partial\fll(\hat D_i)^*\flr$. Let $S=\partial\fll C^*\flr$ and let $B_i=\fll D_i^*\flr$. If $\fll C\flr$ has codimension $k$ in $K$, then the homology $(k-1)$-sphere $|S|$ is the union of three homology $(k-1)$-balls $|B_i|$. Since $|\partial B_1|$ is a homology $(k-2)$-sphere, by the Alexander duality the closures of its complementary domains, including $|B_2\cup B_3|$, are homology balls. Then the Mayer–Vietoris sequence implies that $|B_2\cap B_3|$ is a homology $(k-2)$-ball, and in particular $|\partial(B_2\cap B_3)|$ is a homology $(k-3)$-sphere. Now $\partial(B_2\cap B_3)=B_1\cap B_2\cap B_3\simeq R$, and $(B_2\cap B_3)\cup_R CR\simeq A_1$. If $k\le 5$, then the homology $(k-3)$-sphere $|R|$ must be a genuine sphere; and if additionally $K$ is an $\alpha$-combinatorial manifold, then $|\partial B_2|$ is a genuine $(k-2)$-sphere, so by the Schönflies and Alexander theorems, $|R|$ bounds in it a genuine $(k-2)$-ball $|B_2\cap B_3|$. [*Case 2.*]{} Next, each $\fll\hat D_i\flr$ is an (homology) cell since it is isomorphic to the union of the (homology) cells $\fll D_{i-1}\flr$ and $\fll D_{i+1}\flr$ whose intersection is the (homology) cell $\fll C\flr$. Also, $\fll\hat C\flr$ is an (homology) cell since it is isomorphic to the union of the (homology) cells $\fll\hat D_{i-1}\flr$ and $\fll\hat D_{i+1}\flr$ whose intersection is isomorphic to $(\partial\fll D_i\flr)\but\{C\}$. The latter cellulates an (homology) ball, which is the closure of the complement to the homology ball $|\fll C\flr|$ in the homology sphere $|\partial\fll D_i\flr|$. [*Case 3.*]{} Finally, let us consider a poset $K_C'$ that has one element $\check B$ for each $B\in K$ and additional elements $\check F_1$, $\check F_2$, $\check F_3$ and $\check E_{12}$, $\check E_{13}$, $\check E_{21}$, $\check E_{23}$, $\check E_{31}$, $\check E_{32}$, and satisfies the following for each $A,A'\notin\{C,D_1,D_2,D_3\}$ and all $i,j\in\{1,2,3\}$, $i\ne j$: - $\check D_i>\check C>\check F_j$ and $\check D_i>\check E_{ij}>\check F_j$; - $\check C$ is incomparable with $\check E_{ij}$; - $\check A<\check E_{ij}$ iff $\check A<\check D_i$ iff $A<D_i$; - $\check A<\check F_j$ iff $\check A<\check C$ iff $A<C$; - $\check A>\check E_{ij}$ iff $\check A>\check F_j$ iff $A>D_{j+1}$ and $A>D_{j-1}$; - $\check A>\check D_i$ iff $\check A>\check C$ iff $A>D_1$ and $A>D_2$ and $A>D_3$; - $\check A>\check A'$ iff $A>A'$. It is easy to see that there exist subdivisions $\phi{\colon}K_C'\to K_C$ and $\psi{\colon}(K_C')^*\to K^*$. We note that $K_C'$ and $(K_C')^*$ are non-atomistic posets whose cones are cells, except for one cone $\fll C\flr$ in $K_C'$ isomorphic to $(\partial\fll C\flr)+T$ and one cone $\fll E^*\flr$ in $(K_C')^*$ isomorphic to $(\partial\fll(\hat C)^*\flr)+T$, where $T$ denotes the cone over $(\Delta^2)^{(0)}$. Then each of $\phi^*{\colon}(K_C')^*\to (K_C)^*$ and $\psi^*{\colon}K_C'\to K$ is obtained by taking the quotient by the copy of $T$ followed by three elementary zippings. Either from this, or because $\phi$ and $\psi$ are subdivisions, $|\phi^*|=|\phi|$ and $|\psi^*|=|\psi|$ are collapsible. On the other hand, for each $A\notin\{C,D_1,D_2,D_3\}$ we have $(\psi^*)^{-1}(\fll A\flr)=\fll\check A\flr=\phi^{-1}(\fll\hat A\flr)$ and $\psi^{-1}(\fll A^*\flr)=\fll(\check A)^*\flr=(\phi^*)^{-1}(\fll(\hat A)^*\flr)$. Since a collapsible map is a homology equivalence, we obtain that each cone of $K_C$ other than $\fll\hat C\flr$ and $\fll\hat D_i\flr$, and each cone of $(K_C)^*$ other than $\fll(\hat C)^*\flr$ and $\fll(\hat D_i)^*\flr$ is a homology cell. Then by the above, $K_C$ and $K_C^*$ are homology cell complexes. It is clear that they are atomistic. Moreover, for each $A\in K$ other than $C$, $D_1$, $D_2$, $D_3$, if $|\partial\fll A^*\flr|$ (resp. $|\partial\cel A^*\cer|$) is a sphere, then so is the subdivided $|\partial\fll(\check A)^*\flr|$ (resp.$|\partial\cel(\check A)^*\cer|$), and hence by the Cohen–Homma Theorem \[Cohen-Homma\] so is $|\partial\fll(\hat A)^*\flr|$ (resp.$|\partial\cel(\hat A)^*\cer|$). Suppose that $K$ is a dichotomial homology cell complex and $(A,\bar A)$ is a pair of its complementary elements such that $A$ is covered by precisely three elements and $\bar A$ covers precisely three elements. Consider first the Y$\nabla$-transform $K_{\bar A}$. Then the $\nabla$Y-transform $(K_{\bar A})^{\hat A}$ is again a dichotomial homology cell complex. We shall call it the ($\nabla$,Y)[*-transform*]{} of $K$ along $(A,\bar A)$, and also the (Y,$\nabla$)[*-transform*]{} of $K$ along $(\bar A,A)$. \[4.5\] The ($\nabla$,Y)-transform of $\partial\Delta^3$ along $(\Delta^2,\Delta^0)$ is isomorphic to $\partial\Delta^3$. \[4.6\] The ($\nabla$,Y)-transform of $\partial\Delta^4$ along $(\Delta^2,\Delta^1)$ is the dichotomial complex with $1$-skeleton $K_{3,3}$. \[4.8\] It is easy to see that the horizontal sequence of $\nabla$Y-exchanges in ($*$) lifts to a sequence of ($\nabla$,Y)-transforms of dichotomial cellulations of $S^4$, along pairs of the types $(\Delta^2,D_i)$ with $i=3,4,5,6$, where $D_i$ is a $2$-cell with $i$ edges in the boundary. The inverse (Y,$\nabla$)-transforms are along pairs of the type (vertex, $4$-cell). The Y$\nabla$-exchange $\Gamma_8\rightsquigarrow K_{3,3,1}$ lifts to a similar (Y,$\nabla$)-transform, whose inverse ($\nabla$,Y)-transform is of the type $(\Delta^2,D_4)$. Thus all $\Gamma_i$’s, $P$, and $K_{3,3,1}$ are the $1$-skeleta of their respective dichotomial cellulations of $S^4$. \[4.9\] Let us think of $K_6$ as the $1$-skeleton of the boundary of a top cell of $\partial\Delta^6$. The sequence of two $\nabla$Y-exchanges leading to $K_{4,4}\but$(edge) lifts to a sequence of two ($\nabla$,Y)-transforms of dichotomial cellulations of $S^5$, along pairs of the types $(\Delta^2,\Delta^3)$ and $(\Delta^2,\Sigma^3)$, where $\Sigma^3$ is isomorphic to a top cell of the dichotomial $3$-complex with $1$-skeleton $K_{3,3}$. Thus $K_{4,4}\but$(edge) is the $1$-skeleton of the boundary of a top cell of a dichotomial $5$-sphere. Suppose that $K$ is a poset and $\fll D_1\flr$, $\fll D_2\flr$, $\fll D_3\flr$ are its $(c+1)$-dimensional cones such that no two of them have a common $c$-dimensional subcone. The [*pseudo-[Y]{}$\nabla$-transform*]{} of $K$ at $(D_1,D_2,D_3)$ is the poset $L$ that has one element $\hat B$ for each $B\in K$, and an additional element $\hat C$, and satisfies the following (same as above) for each $A,A'\notin\{D_1,D_2,D_3\}$: - $\hat C>\hat D_i$, $i=1,2,3$; - $\hat A<\hat D_i$ iff $A<D_{i+1}$ or $A<D_{i-1}$; - $\hat A<\hat C$ iff $A<D_1$ or $A<D_2$ or $A<D_3$; - $\hat A>\hat D_i$ iff $A>D_{i+1}$ and $A>D_{i-1}$; - $\hat A>\hat C$ iff $A>D_1$ and $A>D_2$ and $A>D_3$; - $\hat A>\hat A'$ iff $A>A'$. We also say that $L^*$ is obtained by the $\nabla$Y[*-transform*]{} of $K^*$ at $(D_1^*,D_2^*,D_3^*)$. A [*pseudo-[($\nabla$,Y)]{}-transform*]{} of a dichotomial poset $K$ along $(A_1,A_2,A_3;\,\bar A_1,\bar A_2,\bar A_3)$ is now defined similarly to a ($\nabla$,Y)-transform. \[4.10\] It is easy to see that the dangerous $\nabla$Y-exchange $\Gamma_7\rightsquigarrow K_{4,4}\but$(edge) lifts to the pseudo-($\nabla$,Y)-transform along $(A_1,A_2,A_3;\,\bar A_1,\bar A_2,\bar A_3)$, where $A_i$ are the edges of the $\nabla$ and so $\bar A_i$ are their complementary $3$-cells. The resulting dichotomial poset $Q$ has $K_{4,4}\but$(edge) as its $1$-skeleton, and $|Q|$ is homeomorphic to the double mapping cone of the map $|\nabla*\nabla|\to |$Y$*$Y$|$. In particular, shrinking the pair of joins of two triods to points, we obtain a collapsible $\Z/2$-map $|Q|\to S^4$. Alternatively, it suffices to shrink $|$Y$*$Y$|\cong I*|K_{3,3}|$ to $pt*|K_{3,3}|$. Proofs for §\[intro\] ===================== Collapsible and cell-like maps {#collapsing} ------------------------------ The following result is due to T. Homma (see [@Bry] for references and for corrections of Homma’s other proofs in there) and M. M. Cohen [@Co]. \[Cohen-Homma\] If $M$ is a closed manifold, $X$ is a polyhedron, and $f{\colon}M\to X$ is a collapsible map, then $X$ is homeomorphic to $M$. We include a proof modulo Cohen’s simplicial transversality lemma (see [@Co], [@M3]) Let us triangulate $f$ by a simplicial map $F{\colon}L\to K$. Write $n=\dim M$. Given a simplex $\sigma$ of $K$, let $\sigma^*$ denote its dual cone, which is the join of the barycenter $\hat\sigma$ with the derived link $\partial\sigma^*$ (see e.g. [@RS 2.27(6)]). By Cohen’s simplicial transversality lemma, if $\sigma$ is a simplex of $K$ of dimension $n-i$, then $\sigma^*_f\bydef f^{-1}(\sigma^*)$ is an $i$-manifold with boundary $\partial\sigma^*_f\bydef f^{-1}(\partial\sigma^*)$, and this manifold collapses onto the point-inverse $f^{-1}(\hat\sigma)$. The latter is collapsible by our hypothesis, so $\sigma^*_f$ is an $i$-ball. Let $\sigma$ be a maximal simplex of $K$. Then $\partial\sigma^*=\emptyset$ and so $\sigma^*_f$ is a closed manifold. Since it also must be a ball of some dimension, this dimension is zero. Thus all maximal simplices of $K$ have dimension $n$; and if $\sigma$ is such a simplex, then $f$ restricts to a homeomorphism $\sigma^*_f\to\sigma^*$ between a $0$-ball and a point. Let $M_k$, resp. $X_k$ be the union of the manifolds $\sigma^*_f$, resp. of the cones $\sigma^*$, for all simplices $\sigma$ of $K$ of dimension $\ge n-k$. Assume inductively that there is a homeomorphism $g{\colon}M_k\to X_k$ that sends each $\sigma^*_f$ into $\sigma^*$. Given a simplex $\tau^{n-k-1}$ of $K$, by the assumption $g$ restricts to a homeomorphism $h_\tau{\colon}\partial\tau^*_f\to\partial\tau^*$. Since $\tau^*_f$ is a ball, it is homeomorphic to the cone over $\partial\tau^*_f$, so we can extend $h_\tau$ to a homeomorphism $\tau^*_f\to\tau^*$. At the end of this inductive construction lies a homeomorphism $M=M_n\to X_n=X$. If $f{\colon}P\to Q$ is a collapsible map between polyhedra, and $P$ embeds in a manifold $M$, then $Q$ embeds in $M$. Let us identify $P$ with a subpolyhedron of $M$. Then the adjunction space $M\cup_{f}Q$ is a polyhedron, and the quotient map $F{\colon}M\to M\cup_{f}Q$ is collapsible. Its target contains $Q$ and is homeomorphic to $M$ by the preceding theorem. If $\phi{\colon}P\to Q$ is a cell-like map between $n$-polyhedra, and $P$ embeds in an $m$-manifold $M$, where $m\ge n+3$, then $Q$ embeds in $M$. This can be deduced from known results, albeit in an awkward way. Using the theory of decomposition spaces [@Dav 23.2, 5.2], the quotient map $M\to M\cup_\phi Q$ can be approximated by topological homeomorphisms, and therefore $Q$ [*topologically*]{} embeds in $M$. Once again using the codimension three hypothesis, we can approximate the topological embedding by a PL one (see [@DV 5.8.1]). Below we give a proof avoiding topological embeddings; but we shall also see that they arose not accidentally, for we barely avoid using the $4$-dimensional topological Poincaré conjecture (=Freedman’s theorem). We use the notation in the proof of the Cohen–Homma theorem. We may assume that $\phi$ is triangulated by a simplicial map $\Phi{\colon}B\to A$, where $A$ and $B$ are subcomplexes of $K$ and $L$ and $\Phi$ is the restriction of $F$. Then the $\sigma^*_f$ are no longer balls but contractible manifolds with spines of codimension $\ge 3$. Because of the latter, their boundaries are simply connected and hence homotopy spheres. By the Poincaré conjecture those of dimensions $\ge 5$ are genuine spheres and then what they bound are genuine balls of dimensions $\ge 6$. The $3$-dimensional and $4$-dimensional contractible manifolds are also genuine balls since their codimension $\ge 3$ spines must be collapsible. Of the $5$-dimensional contractible manifolds with $2$-dimensional spines we only note that they would be genuine balls if either the Andrews–Curtis conjecture or the $4$-dimensional PL Poincaré conjecture were known to hold. Write $Q_k=X_k\cap Q$, and for a simplex $\sigma$ in $A$, let $\sigma^*_A=\sigma^*\cap A$ be the dual cone of $\sigma$ in $A$, which is the join of the barycenter $\hat\sigma$ with the derived link $\partial\sigma^*_A$. We claim that for each $k$ there is an embedding $g{\colon}Q_k\to M_k$ that sends each $\sigma^*_A$ into $\sigma^*_f$. There is nothing to prove for $k\le 2$. For $k=3$, we may have an $(n-3)$-simplex $\sigma$ in $A$, and then we need to embed its barycenter $\sigma^*_A$ into the $3$-ball $\sigma^*_f$; this is not hard. For $k=4$ and an $(n-4)$-simplex $\sigma$ in $A$, we have the finite set $\partial\sigma^*_A$ embedded in the $3$-sphere $\partial\sigma^*_f$, and we need to extend this embedding to an embedding of the $1$-polyhedron $\sigma^*_A$ into the $4$-ball $\sigma^*_f$; this is also not hard. For $k=5$ and an $(n-5)$-simplex $\sigma$ in $A$, we have the $1$-polyhedron $\partial\sigma^*_A$ embedded in the homotopy $4$-sphere $\partial\sigma^*_f$, and we need to extend this embedding to an embedding of the $2$-polyhedron $\sigma^*_A$ into the homotopy $5$-ball $\sigma^*_f$. This can be done: the boundary embedding extends to a map $\sigma^*_A\to\sigma^*_f$ since $\sigma^*_f$ is contractible, and then this map of a $2$-polyhedron in a $5$-manifold can be approximated by an embedding by general position. For $k\ge 6$ we simply use conewise extension just like we did for $k=1$ and in the proof of the Cohen–Homma lemma. Eventually we obtain an embedding $Q=Q_n\emb M_n=M$. Let $f{\colon}X\to Y$ be a map between $n$-polyhedra, where $X$ embeds in an $m$-manifold $M$. Then $Y$ embeds in $M$ if either \(a) the point-inverse of the barycenter of every $k$-simplex is collapsible for $k\ge m-n-1$ and collapses onto an $(m-n-2-k)$-polyhedron for $k\le m-n-2$; or \(b) $m-n\ge 3$, and $f$ is fiberwise homotopy equivalent to a $g{\colon}Z\to Y$ whose nondegenerate point-inverses lie in a subpolyhedron of dimension $\le m-n-2$. Let us triangulate $\phi$ by a simplicial map $\phi{\colon}K\to L$. Since a collapse may be viewed as a map with collapsible point-inverses (see [@Co2 §8]), $f$ is the composition of a collapsible map $g{\colon}X\to Z$ and a map $h{\colon}Z\to Y$ whose non-degenerate point-inverses lie in the subpolyhedron $Q\bydef h^{-1}(|L^{(m-n-2)}|)$ of dimension $\le m-n-2$. By Theorem \[minors embed\](a), $Z$ embeds in $M$. Since the mapping cylinder $MC(h|_Q)$ is of dimension $\le m-n-1$, by general position, this embedding extends to an embedding of $Z\cup MC(h|_Q)$ in $M$. The point-inverses of the projection $Z\cup MC(h|_Q)\to Y$ are cones, so applying theorem \[minors embed\](a) once again, we obtain that $Y$ embeds in $M$. Let $\phi{\colon}Z\to X$ be the given fiberwise homotopy equivalence over $Y$ and let $Q$ be the given subpolyhedron. Since $MC(g|_Q)$ is of dimension $\le m-n-1$, the original embedding of $X$ in $M$ extends to an embedding of $X\cup_\phi MC(g|_Q)$ in $M$. On the other hand, $X\cup_\phi MC(g|_Q)$ is a fiberwise deformation retract of $X\cup_\phi MC(g)$, which in turn fiberwise deformation retracts onto $Y$. Hence every point-inverse of the projection $X\cup_\phi MC(g|_Q)\to Y$ is contractible. Thus $Y$ embeds in $M$ by Theorem \[minors embed\](b). Edge-minors {#edge-minors2} ----------- Let $K'$ be a subdivision of the given $2$-complex $K$. We call a simplex $\sigma$ of $K'$ “old” if $|\sigma|=|\tau|$ for some simplex $\tau$ of $K$, and “new” otherwise. A subcomplex of $K'$ is said to be “old” if it consists entirely of old simplices, and “new” otherwise. Let $\partial\Delta^3$ be a missing tetrahedron in $K'$. Then $|\partial\Delta^3|$ has to be triangulated by a subcomplex of $K$, and therefore $\partial\Delta^3$ is old. Let $\partial\Delta^2$ be a missing triangle in $K'$. If $|\partial\Delta^2|$ lies in $|K^{(1)}|$, then it has to be triangulated by a subcomplex of $K^{(1)}$, and therefore $\partial\Delta^2$ is old. Thus every new missing triangle $\partial\Delta^2$ in $K'$ has a vertex in the interior of the combinatorial ball $\sigma'$ for some $2$-simplex $\sigma$ of $K$, and therefore $\partial\Delta^2$ is itself contained in $\sigma'$. Given a $2$-simplex $\sigma$ of $K$, every missing triangle $\partial\Delta^2$ of $K'$ contained in $\sigma'$ bounds a combinatorial disk in $\sigma'$. If two such disks intersect in at least one $2$-simplex, then one is contained in the other. Let $D_0$ be an innermost such disk in $\sigma'$. Since $\partial D_0$ is a missing triangle in $K'$, $D_0$ is not a $2$-simplex, and so contains at least one edge $\tau$ in its interior. By the minimality of $D_0$, $\tau$ is not contained in any missing triangle in $K'$. Since $\partial D_0$ is a complete graph, at least one vertex of $\tau$ lies in the interior of $D_0$, and hence in the interior of $\sigma'$. Then we may contract $e$. It remains to consider the case where $K'$ contains no new missing triangle. If $\tau$ is a new edge with at least one vertex in the interior of $\sigma'$ for some $2$-simplex $\sigma$ of $K$, we may contract $\tau$. If $\tau$ is a new edge such that $|\tau|$ lies in $|K^{(1)}|$, we may contract $\tau$. If there are no new edges of these two types, $K'=K$. By Theorem \[self-dual\] and assertion (2) above we may assume that $L$ is obtained from $K$ by a single edge contraction. Let $\sigma=\rho_1*\rho_2$ be the contracted edge. Let us first consider the case where $r=1$. Then the opposite $(m-2)$-simplex $\tau$ to $\sigma$ in $\Delta^m$ is not contained in $K$; so $K$ lies in $\sigma*\partial\tau$. The simplicial map $K\to L$ extends to an edge contraction $f{\colon}\sigma*\tau\to\rho*\tau$ of the entire $\Delta^m$, where $\rho$ is the $0$-simplex. We have $L=f(K)\subset\rho*\partial\tau$. This is a combinatorial $(m-2)$-ball, and $|\rho*\partial\tau|$ lies in $S^{m-2}\bydef|\tau\cup\rho*\partial\tau|$, which proves the assertion of (a). Moreover, it is not hard to see that the composition $g$ of the inclusion $|L|\subset S^{m-2}$ and the embedding $S^{m-2}\emb S^{m-1}$, $\rho\mapsto\rho_1$, is equivalent to the embedding $|L|\emb S^{m-1}$ induced by $j$. Choosing $h$ to be the reflection $S^{m-1}\to S^{m-1}$ in $S^{m-2}$, we have $hg=g$, which proves the assertion of (b). The case where $\sigma$ lies in a single factor of the join $K_1*\dots*K_r$, say in $K_1$, reduces to the case just considered, by observing that $K_2*\dots*K_r$ lies in the combinatorial sphere $\partial\Delta^{m_2}*\dots*\partial\Delta^{m_r}$ of dimension $m-1-m_1$. The remaining cases similarly reduce to the case where $r=2$ and $\sigma=\sigma_1*\sigma_2$, where $\sigma_i\in K_i$. Let $\tau_i$ be the opposite $(m_i-1)$-simplex to $\sigma_i$ in $\Delta^{m_i}$; then $K_i$ lies in $\sigma_i*\partial\tau_i$. Similarly to the above, we have $L\subset\rho*\partial\tau_1*\partial\tau_2$. This is a combinatorial $(m-2)$-ball, which lies in the combinatorial $(m-2)$-sphere $\tau_1*\partial\tau_2\cup\rho*\partial\tau_1*\partial\tau_2$, etc. Embeddability is commensurable with linkless embeddability {#commensurability} ========================================================== This section is concerned with relations between embeddings and linkless embeddings. It somewhat diverges from the combinatorial spirit of this paper, and instead contributes to a central theme of [@M2]; thus it is best viewed as an addendum to [@M2]. This said, the main result of this section also gives a geometric view of some examples and constructions mentioned in the introduction, and might contribute to an initial groundwork for a proof of a higher-dimensional Kuratowski(–Wagner) theorem. An embedding $g$ of a graph $|G|$ in $S^3$ is called [*panelled*]{} if for every circuit $Z$ in the graph, $g$ extends to an embedding of $|G\cup CZ|$. \[panels\] (a) [(Robertson–Seymour–Thomas [@RST])]{} A graph $G$ admits a panelled embedding in $S^3$ iff $|G|$ admits an embedding $g$ in $S^3$ such that for every two disjoint circuits $C$, $C'$ in the graph, $g$ links $|C|$ and $|C'|$ with an even linking number. \(b) [@M2 Lemma 4.1] An embedding of a graph in $S^3$ is panelled iff it is linkless and knotless. \[erratum\] The proof of (b) in [@M2] contains a minor inaccuracy: in showing that the embedding is linkless it explicitly treats the splitting of two disjoint subgraphs only in the case where one of them is connected; however, the argument works in the general case without any modifications. We also need a higher-dimensional analogue of Lemma \[panels\]. To this end, we recall from §\[embeddings\] that we call an $n$-polyhedron $M$ an [*$n$-circuit*]{}, if $H^n(M\but\{x\})=0$ for every $x\in M$. This implies that $H^n(M)$ is cyclic (since it is an epimorphic image of $H^n(M,M\but\{x\})$, where $x$ lies in the interior of an $n$-simplex of some triangulation of $M$) and that $H^n(P)=0$ for every proper subpolyhedron $P$ of $M$. An [*oriented*]{} $n$-circuit $M$ is endowed with a generator $\xi_M$ of $H^n(M)$. \[cohomology\] Let $P$ be a polyhedron. For every nonzero class $x\in H^n(P)$ there exists a singular $n$-circuit $f{\colon}M\to P$ such that $f^*(x)\ne 0$. Moreover, $f^*(x)$ is of the same order as $x$. We note that a more convenient and far-reaching geometric view of cohomology exists (see [@M2 §2], [@Fe], [@BRS]), but that is not what we need here. By the universal coefficient formula, $x$ maps to some $h{\colon}H_n(P)\to\Z$. If $x$ is of infinite order, then $h$ is nontrivial. Let us pick a $y\in H_n(P)$ with $h(y)\ne 0$. Then $y$ is representable by a singular oriented $\Z$-pseudo-manifold $f{\colon}M\to P$ such that $y=f_*([M])$ (see [@Fe Theorem 1.3.7]). Then $hf_*([M])\ne 0$, so $f^*(h){\colon}H_n(M)\to\Z$ is nontrivial. Hence by the naturality in the UCF, $f^*(x)\ne 0$. If $x$ is of a finite order, $m$ say, then $h$ is trivial. Hence $x$ comes from some extension $\Z\mono G\epi H_{n-1}(P)$ of order $m$ in $\operatorname{Ext}(H_{n-1}(P),\Z)$. Then there exists a $z\in H_{n-1}(P)$ of order $m$ that is covered by an element $\hat z\in G$ of infinite order. In particular, $z$ is the Bockstein image of some $y\in H_n(P;\,\Z/m)$. This $y$ is representable by a singular oriented $\Z/m$-pseudo-manifold $f{\colon}M\to P$ such that $y=f_*([M])$ (see [@BRS Chapter III], [@Do]). Then by the naturality of the Bockstein homomorphism, $z=f_*(\beta[M])$. Hence the induced extension $\Z\mono f^*G\epi H_{n-1}(M)$ is such that $\beta[M]$ is covered by an element of $f^*G$ of infinite order, which maps onto $\hat x$. Since $\beta[M]$ is of order $m$, the induced extension is itself of order $m$ in $\operatorname{Ext}(H_{n-1}(M),\Z)$. Hence by the naturality in the UCF, $f^*(x)$ is of order $m$. The [*linking number*]{} of an oriented singular $m$-circuit $f{\colon}M\to\R^{m+n+1}$ and an oriented singular $n$-circuit $g{\colon}N\to\R^{m+n+1}$ with disjoint images is the degree of the composition $M\x N\xr{f\x g}\R^{m+n+1}\x\R^{m+n+1}\but\Delta_{\R^{m+n+1}}\simeq S^{m+n}$, which is given by the formula $(m,n)\mapsto\frac{m-n}{||m-n||}$. This degree $\operatorname{lk}(f,g)$ lives in the cyclic group $H^{m+n}(M\x N)\simeq H^m(M)\otimes H^n(N)$. The linking number of an unoriented $m$-circuit and an unoriented $n$-circuit in $S^{m+n+1}$ is well-defined up to a sign. By an [*$n$-circuit with boundary*]{} we mean any $n$-polyhedron $M$ along with an $(n-1)$-dimensional subpolyhedron $\partial M$ such that $M/\partial M$ is a genuine $n$-circuit. If additionally $\partial M$ is an $(n-1)$-circuit and the coboundary map $H^{n-1}(\partial M)\to H^n(M)$ is an isomorphism, then we say that $\partial M$ [*bounds*]{} $M$. \[circuits\] Let $P$ be an $n$-polyhedron, $n\ge 2$, and $g{\colon}P\emb S^{2n+1}$ an embedding. \(a) $g$ is linkless iff every pair of singular $n$-circuits in $P$ with disjoint images have zero linking number under $g$. \(b) $g$ is linkless iff for every $(n+1)$-circuit $Z$ with boundary, every $f{\colon}\partial Z\to P$ and every function $\phi{\colon}\partial Z\to I$, the embedding $g\x\operatorname{id}_I{\colon}P\x I\emb S^{2n+1}\x I$ extends to an embedding of $P\x I\cup_{f\x\phi}Z$ in $S^{2n+1}\x I$. Let $Q$ and $R$ be disjoint subpolyhedra of $g(P)$. Since $Q$ is of codimension $\ge 3$ in the sphere, $M\bydef S^{2n+1}\but Q$ is simply-connected. Hence by the Alexander duality and the Hurewicz theorem $M$ is $(n-1)$-connected, and $\pi_n(M)\simeq H_n(M)\simeq H^n(Q)$. Therefore $$H^n(R;\,\pi_n(M))\simeq H^n(R;\,H^n(Q))\simeq H^n(R)\otimes H^n(Q)\simeq H^n(R\x Q).$$ The first obstruction to null-homotopy of the inclusion $R\subset M$ can be identified under this string of isomorphisms with the image of the generator of $H^{2n}(S^{2n})$ in $H^{2n}(R\x Q)$. If this image is nonzero, it is of the form $\sum r_i\otimes q_i$, where each $r_i\in H^n(R)$ and each $q_i\in H^n(Q)$, and each $r_i\otimes q_i$ is nonzero. Then by Lemma \[cohomology\] there are singular $n$-circuits $f{\colon}V\to Q$ and $g{\colon}W\to R$ such that $f^*(r_1)$ has the same order as $r_1$, and $g^*(q_1)$ has the same order as $q_1$. Then $f^*(r_1)\otimes g^*(q_1)$ is nonzero, and also it equals $\operatorname{lk}(f,g)$, contradicting the hypothesis. Hence by obstruction theory $R$ is null-homotopic in $M$. Then by engulfing (see e.g. [@Ze1 Lemma 2], [@Hu Chapter VII]), $M$ contains a ball that contains $R$. The if direction follows from (a), since the projection of the image of $Z$ onto $S^{2n+1}$ is disjoint from $g(P)$, except at $gf(\partial Z)$. Conversely, let $N$ be the second derived neighborhood of $f(\partial Z)$ in some triangulation $K$ of $P$. Given a linkless embedding $P\subset S^{2n+1}$, let $B$ be a codimension zero ball containing $f(\partial Z)$ and such that $B\cap P$ is contained in $N$. [*Step 1.*]{} By a simple engulfing argument, we may assume that the intersection of each simplex $\sigma$ of $K$ with the interior of $B$ is connected. In more detail, if $C$ and $D$ are two components of this intersection, pick some $c\in C$ and $d\in D$. Since the intersection of $\sigma$ with the interior of $N_i$ is connected, we can join $c$ and $d$ by an arc $J$ in that intersection. We may assume that $J$ meets $\partial B$ in finitely many points; arguing by induction, we may further assume that there are just two of them, $c'$ and $d'$. Let $J'$ be the segment of $J$ spanned by these points. Let us also join $c'$ and $d'$ by an arc $J''$ in $\partial B$ that meets $P$ only in its endpoints. The $1$-sphere $J'\cup J''$ bounds a $2$-disk $D$ in the closure of the complement to $B$ in $S^{2n+1}$ that meets $B\cup P$ only in $\partial D$. Then a small regular neighborhood $\beta$ of $D$ in the closure of the complement of $B$ is such that the ball $B\cup\beta$ still meets $P$ along a subpolyhedron of $N$. Moreover, replacing $B$ with $B\cup\beta$ decreases the number of components in the intersection of a simplex $\tau$ of $K$ with the interior of $B\cup\beta$ when $\tau=\sigma$, and keeps it the same when $\tau\ne\sigma$. [*Step 2.*]{} Let us write $Z'=Z\cup_{f\x\phi}(f\x\phi)(\partial Z)$. Pick a generic map $F{\colon}Z'\to B\x I$ extending the embedding $g|_{\partial Z'}$. Since $F$ is a codimension three map, and $Z'$ is a circuit, by the Penrose–Whitehead–Zeeman trick it can be replaced by an embedding $G$ that agrees with $F$ on $\partial Z'$ (compare [@Sa-1] and [@M2 §8]). In more detail, given a self-intersection $F(p)=F(q)$ of $F$, since $Z$ is a circuit, $p$ and $q$ can be joined by an arc $J$ in $Z'$ that contains only generic points. So a small regular neighborhood $R$ of $J$ in $Z'$ is a $2$-disk. Now $F(J)$ bounds a $2$-disk $D$ in $S^{2n+1}\x I$ that meets $F(Z')$ only in $\partial D$ and is disjoint from $g(P\x I)$. A small regular neighborhood $S$ of $D$ in $S^{2n+1}\x I$ is a ball disjoint from $g(P\x I)$ and we may assume that $F^{-1}(S)=R$. Then we redefine $F$ on $S$ by the conewise extension $R\to S$ of the boundary restriction $F|_{\partial R}{\colon}\partial R\to\partial S$. [*Step 3.*]{} By a further application of the same trick, using that the intersection of each $\sigma$ with the interior of $B$ is connected, we may further amend $G$ so that the image of the resulting embedding $G'$ meets $g(P\x I)$ only in $g(\partial Z')$. Indeed, given an intersection $F(p)=g(q)$ between $F$ and $g$, we may join $q$ to some point $r\in\partial Z'$ by an arc $J$ in $P\x I$ going only through generic points of $P\x I$ (except for $r$ itself) and such that $g(J)$ lies in the interior of $B$. We may then join $p$ and $r$ by an arc $J'$ in $Z'$ going only through generic points of $Z'$ (except for $r$ itself). Then a regular neighborhood of $J\cup J'$ is a cone, and the preceding construction works. We write $CP$ to denote the cone $pt*P$ over the polyhedron $P$. \[commensuration\] Let $P$ be an $n$-polyhedron, and let $Q$ be an $(n-1)$-dimensional subpolyhedron of $P$ such that the closure of every component of $P\but Q$ is an $n$-circuit with boundary. In part (a), assume further that every pair of disjoint singular $(n-1)$-circuits in $Q$ bounds disjoint singular $n$-circuits in $P$. \(a) $Q$ linklessly embeds in $S^{2n-1}$ iff $P\cup CQ$ embeds in $S^{2n}$. \(b) $P$ embeds in $S^{2n}$ iff $P\cup CQ$ linklessly embeds in $S^{2n+1}$. The case $n=1$ of (b) and much of the case $n=2$ of (a) were proved by van der Holst [@Ho], which the author discovered after writing up the proof below. The case $n=1$ in the “only if” assertion in (b) was also proved earlier in [@RST']. Given an embedding $P\cup CQ\subset S^{2n}$, let $B$ be a regular neighborhood of $CQ$ relative to $P$. Since $CQ$ link-collapses onto $Q$, this $B$ is a manifold [@HZ] (cf.[@Hus]), and therefore a ball (see [@HZ] or [@Co2]). Then $Q\subset\partial B$ is an embedding that is linkless (and knotless, if $n=2$) by Lemma \[circuits\](a) and Lemma \[panels\](a). Suppose we are given a linkless embedding $Q\subset S^{2n-1}$; if $n=2$, we may further assume that it is panelled by Lemma \[panels\](a). Let us extend it to the conical embedding of $CQ$ in $B^{2n}$ and to the vertical embedding of $Q\x I$ into a collar $S^{2n-1}\x I$ of $B^{2n}$ in $S^{2n}$. Let $M_1,\dots,M_r$ be the closures of the components of $P\but Q$. If $n=2$, each $\partial M_i$ bounds an embedded disk $D$ in $S^3$ that meets $Q$ only in $\partial D$. Then $Q\x\{\frac ir\}$ can be easily approximated by an embedded copy of $M_i$ lying in $S^3\x [\frac{i-1}r,\frac ir]$ and meeting $Q\x I$ only in $\partial M_i$. If $n\ge 3$, then Lemma \[circuits\](b) yields an embedding $g_i$ of each $M_i$ in $S^{2n-1}\x [\frac{i-1}r,\frac ir]$ extending the inclusion of $\partial M_i$ onto $\partial M_i\x\{\frac ir\}$ and disjoint from $Q\x I$ elsewhere. In either case, different $M_i$’s will be disjoint because of their heights in $S^{2n+1}\x I$. Let $\hat P=CQ\cup Q\x I\cup M_1\cup\dots\cup M_r$. The projection $\pi{\colon}Q\x I\to Q$ yields a collapsible map $S^4\to S^4\cup_\pi Q$. By the Cohen–Homma Theorem \[Cohen-Homma\], $S^4\cup_\pi Q$ is homeomorphic to $S^4$, and therefore $P=\hat P\cup_\pi Q$ embeds in $S^4$. Let us write $S^{2n+1}=\{\nu,\sigma\}*S^{2n}$. Given an embedding $P\subset S^{2n}$, it extends to the conical embedding $P\cup\nu*Q\subset \nu*P\subset\nu*S^{2n}\subset S^{2n+1}$. If $n=1$, then the latter is panelled, since every circuit $Z$ of $P\cup\nu*Q$ either lies in $P$ (and so bounds the disk $\sigma*Z$) or is of the form $\nu*(\partial J)\cup J$, where $J$ is an arc in $P$ (and so bounds the disk $\nu*J$). Now suppose that $n\ge 2$ and let $N$ and $S$ be disjoint subpolyhedra of $P\cup\nu*Q$. Without loss of generality $\nu\in N$. Then $H^n(S,\,S\cap P)\simeq H^n(S\cup P,\,P)=0$ due to $H^n(P\cup\nu*Q\but\nu*\emptyset,\,P)=0$. Let $\Sigma=\sigma*(S\cap P)$; then $H^n(S\cup\Sigma)\simeq H^n(S\cup\Sigma,\Sigma)=0$. By the Alexander duality, the open manifold $M\bydef S^{2n+1}\but (S\cup\Sigma)$ is homologically $n$-connected. But also it is simply-connected as the complement to the codimension $\ge 3$ subset $S\but\Sigma$ in the Euclidean space $S^{2n+1}\but\Sigma$. Hence $M$ is $n$-connected, and so the $n$-polyhedron $N$, which has codimension $\ge 3$ in $M$, can be engulfed into a ball in $M$ (see e.g.[@Ze1] or [@Hu Chapter VII]). If $n=1$, then the proof of Lemma \[panels\](b) in [@M2] works to extend the given panelled embedding of $P\cup CQ$ to an embedding $CP\emb S^3$. By considering the link of the cone vertex, we obtain an embedding of $P$ in $S^2$. If $n\ge 2$, let $M_1,\dots,M_r$ be the closures of the components of $P\but Q$. Pick a map $f_i{\colon}M_i\to C(\partial M_i)$ that restricts to the identification of $\partial M_i$ with the base of the cone. The mapping cylinder $MC(f_i)$ contains $\mu_i\bydef M_i\cup MC(f_i|_{\partial M_i})\cup C(\partial M_i)$, which is a copy of $M_i\cup C(\partial M_i)$. By Lemma \[circuits\](b), the natural embedding of $\mu_i$ in $MC(\operatorname{id}_B)$, which is a copy of $B\x I$, extends to an embedding $G_i{\colon}MC(f_i)\emb B\x I$ whose image meets $(P\cup CQ)\x I$ only in $\mu_i$. Combining the embeddings $G_i$ together, we obtain a map $F$ of $MC(f)$ into $S^{2n+1}\x I$, where $f{\colon}P\cup CQ\to CQ$ is obtained by combining the $f_i$. By construction, $F$ restricts to the natural embedding of $\mu\bydef P\cup MC(f|_Q)\cup CQ$, which is a copy of $P\cup CQ$, in $MC(\operatorname{id}_{S^{2n+1}})$, which is a copy of $S^{2n+1}\x I$. The only double points of $F$ are isolated, and occur between $MC(f_i)$ and $MC(f_j)$ for $i\ne j$. Since they all share the cone vertex in $\mu$, yet another application of the Penrose–Whitehead–Zeeman trick (in addition to those in Lemma \[circuits\](b)) enables one to replace $F$ by an embedding $G$ that agrees with $F$ on $\mu$ and still meets $(P\cup CQ)\x I$ only in $\mu$. Viewed as an embedding of one mapping cylinder in another, $G$ may be assumed to be level-preserving near the target base — in other words, at some interval $[1-\eps,1]$ of the parameter values (cf. [@RS proof of 4.23]). Hence $G$ restricts to a concordance (with parameter values in $[0,1-\eps]$) keeping $Q$ fixed between the inclusion $P\subset S^{2n+1}$ and an embedding of $P$ in the boundary of the second derived neighborhood $B$ of $CQ$ modulo $Q$ in an appropriate triangulation of $S^{2n+1}$. Since $CQ$ link-collapses onto $Q$, this $B$ is a manifold [@HZ] (cf.[@Hus]), and therefore a ball (see [@HZ] or [@Co2]). Since the image of this concordance meets $(P\cup CQ)\x [0,1-\eps]$ only in $P\x\{0\}\cup CQ\x [0,1-\eps]$, it may be viewed as a concordance of the entire $P\cup CQ$ keeping $CQ$ fixed. Then by the Concordance Implies Isotopy theorem we get an isotopy of $S^{2n+1}$ keeping $CQ$ fixed and taking $P$ into the $2n$-sphere $\partial B$. Appendix: Embedding 2-polyhedra in 4-sphere {#appendix-embedding-2-polyhedra-in-4-sphere .unnumbered} ------------------------------------------- It is well-known that if $K$ is a $2$-dimensional cell complex such that $|K^{(1)}|$ embeds in $S^2$, then $|K|$ embeds in $S^4$ (cf. [@2DH p. 44]); see [@Cu Theorem 4] for a related result. The following is essentially proved in [@Ho] (see also [@Gi]). If $K$ is a $2$-dimensional cell complex such that $|K^{(1)}\but\cel v\cer|$ linklessly embeds in $S^3$ for some vertex $v$ of $K$, then $|K|$ embeds in $S^4$. Let $L=K\but\cel v\cer$. Then $|K|$ embeds in $|L\cup L^{(1)}*\{v\}|$. By the hypothesis $|L^{(1)}|$ linklessly embeds in $S^3$, hence by Theorem \[commensuration\](a), $|L\cup L^{(1)}*\{v\}|$ embeds in $S^4$. It is a well-known open problem whether every contractible $2$-polyhedron embeds (i.e., PL embeds) in $S^4$; an affirmative solution is well-known to be implied by the Andrews–Curtis conjecture, cf. Curtis [@Cu §2]. (Indeed, by general position every $2$-polyhedron $P$ immerses in $I^4$, and therefore embeds in a $4$-manifold $M$. Let $N$ be the regular neighborhood of $P$ in $M$. If $P$ $3$-deforms to a point, then the double of $N$ is the $4$-sphere, see [@2DH Assertion (59) in Ch. I].) The most nontrivial ingredient of the proof of Theorem \[commensuration\] is Lemma \[panels\](a) of Robertson–Seymour–Thomas. The dependence of part (b) on this lemma is only apparent: it can be eliminated altogether if we replace “linklessly” by “linklessly (and knotlessly, when $n=1$)”. Part (a) depends on the Robertson–Seymour–Thomas lemma in an essential way. However, it does not use full strength of the lemma. The remaining power of their lemma is captured by the following striking result, which can also be deduced from the results of [@Ho]. \[Whitney\] Let $P$ be a $2$-polyhedron and $Q$ a $1$-dimensional subpolyhedron of $P$ such that the closure of every component of $P\but Q$ is a disk, and every two disjoint circuits in $Q$ bound disjoint singular surfaces in $P$. Then $P\cup CQ$ embeds in $S^4$ iff the $\bmod 2$ van Kampen obstruction of $P\cup CQ$ vanishes. This is saying basically that in a certain situation, the Whitney trick works in dimension $4$ in the PL category. Let us pick an embedding of $Q$ in $S^3=\partial B^4$, extend it to the conical embedding $CQ\to B^4$ and also to a map of $P$ into the other hemisphere of $S^4$. This defines a map $f{\colon}P\cup CQ\to S^4$, and we may assume that it only has transverse double points. Let $\bar P$ be the quotient of $P\x P\but\Delta_P$ by the factor exchanging involution $T$. Let $G$ be a triangulation of $Q$, and let $K$ be the cell complex extending the triangulation $CG$ of $CQ$ by adding the closures of the components of $P\but Q$. Let $\bar K\subset\bar P$ be the quotient by $T$ of the union of all products $\sigma\x\tau$, where $\sigma$ and $\tau$ are disjoint cells of $K$. Note that $H^4(\bar P;\,\Z/2)\simeq H^4(\bar K;\,\Z/2)$. For disjoint cells $\sigma,\tau$ of $K$, let $\sigma\boxtimes\tau$ denote the characteristic chain of the cell $(\sigma\x\tau\cup\tau\x\sigma)/T$ of $\bar K$. The van Kampen obstruction $\theta(P)\in H^4(\bar K;\,\Z/2)$ is represented by a cellular cocycle $c$ such that $c(\sigma\boxtimes\tau)$ is the parity of the number of intersections between $f(\sigma)$ and $f(\tau)$ (see [@M2]). By the hypothesis, $c$ is the coboundary of a cellular $1$-chain $b$. For each edge $\sigma$ of $K$ and each $2$-cell $\tau$ in $P$ disjoint from $\sigma$ and such that $b(\sigma\boxtimes\tau)\ne 0$, let us pick a copy of $S^2$ in a small neighborhood of $f(\sigma)$ in $S^4$, winding around $f(\sigma)$ with an odd linking number, and connect this sphere to $f(\tau)$ by a thin tube disjoint from the image of $f$. Next, for each edge $\sigma$ of $G$ and each $2$-simplex $C\tau$ of $CG$ disjoint from $\sigma$ let us do an equivalent, but fancier procedure. Let us pick a copy of $S^1$ in a small neighborhood of $f(\sigma)$ in $S^3$, winding around $f(\sigma)$ with and odd linking number (it might be easier to imagine the following steps if the linking number is $\pm 1$) and connect this loop to $f(\tau)$ within $S^3$ by a thin tube $S^0\x I$ disjoint from the image of $f$. We extend this modification of $f(\tau)$ conewise to $f(C\tau)$ and by a generic homotopy to a small neighborhood of $\tau$ mod $\partial\tau$ in $P$. Note that $f(CQ)$ stays within $B^4$ and remains embedded. The amended map $f'$ has the property that for every pair of disjoint cells $\sigma$, $\tau$ of $K$, the intersection number between $f'(\sigma)$ and $f'(\tau)$ is even. In addition, $f'$ still embeds $CQ$ in $B^4$, conically. Then all intersections between a $2$-cell $\sigma$ in $P$ and a $2$-simplex $C\tau$ can be pushed through the base of the cone; since their $\bmod 2$ algebraic number is zero, this will not change the $\bmod 2$ algebraic intersection numbers of $\sigma$ with other $2$-cells in $P$. Thus all the intersections of the resulting map $f''$ are between $2$-cells of $P$, which all lie in the upper hemisphere of $S^4$, and the intersection number between any pair of disjoint $2$-cells is even. Since $Q$ is still in $S^3$, we obtain that every two disjoint circuits in $Q$ have even linking number under $f''$. Now the assertion follows from Lemma \[panels\](a) and Theorem \[commensuration\](a). \[erratum2\] The above argument is parallel to a proof of the completeness of the van Kampen obstruction in higher dimensions, see e.g. [@M2 proof of 3.1]. We note some confusing typos in the final paragraph of that proof in [@M2]: “$f(p)=f(q)$” should read “$g(p)=g(q)$”, and more importantly “Then a small regular neighborhood of $f(J)$ ...” should read “Since $n>2$, the embedded $1$-sphere $g(J)$ bounds an embedded $2$-disk $D$ that meets $g(Y\cup\sigma_i)$ only in its boundary. Then a small regular neighborhood of $D$ ...” Acknowledgements {#acknowledgements .unnumbered} ================ The author is grateful to A. N. Dranishnikov, I. Izmestiev, E. Nevo, E. V. Shchepin, A. Skopenkov, M. Skopenkov, M. Tancer, S. Tarasov and M. Vyalyj for useful discussions. [00]{} [^1]: Supported by Russian Foundation for Basic Research Grant No. 11-01-00822 [^2]: Pontryagin’s autobiography dates it to the 1926/27 academic year, and mentions that it corrected a previous result by Kuratowski. Did Pontryagin hesitate to publish because he wanted to understand where $K_5$ and $K_{3,3}$ come from?

--- abstract: 'The development of the radio remnant of SN 1987A has been followed using the Australia Telescope Compact Array since its first detection in 1990 August. The remnant has been observed at four frequencies, 1.4, 2.4, 4.8 and 8.6 GHz, at intervals of 4 – 6 weeks since the first detection. These data are combined with the 843 MHz data set of Ball et al. (2001) obtained at Molonglo Observatory to study the spectral and temporal variations of the emission. These observations show that the remnant continues to increase in brightness, with a larger rate of increase at recent times. They also show that the radio spectrum is becoming flatter, with the spectral index changing from $-0.97$ to $-0.88$ over the 11 years. In addition, at roughly yearly intervals since 1992, the remnant has been imaged at 9 GHz using super-resolution techniques to obtain an effective synthesised beamwidth of about 05. The imaging observations confirm the shell-morphology of the radio remnant and show that it continues to expand at $\sim3000$ . The bright regions of radio emission seen on the limb of the shell do not appear to be related to the optical hotspots which have subsequently appeared in surrounding circumstellar material.' author: - 'R. N. Manchester $^{1}$' - 'B. M. Gaensler $^{2}$[^1]' - 'V. C. Wheaton $^{1,3}$' - 'L. Staveley-Smith $^{1}$' - 'A. K. Tzioumis $^{1}$' - 'N. S. Bizunok $^{2,4}$' - 'M. J. Kesteven $^{1}$' - 'J. E. Reynolds $^{1}$' title: 'Evolution of the Radio Remnant of SN 1987A: 1990 – 2001' --- =2em =17.5cm =24.6 cm =-2.5cm =-1.0cm =-1.0cm [$^1$ Australia Telescope National Facility, CSIRO, PO Box 76, Epping NSW 1710\ Dick.Manchester@csiro.au\ $^2$ Center for Space Research, Massachusetts Institute of Technology, Cambridge MA 02139, USA\ $^3$ School of Physics, University of Sydney, NSW 2006\ $^4$ Boston University, Boston MA 02215, USA\ ]{} [**Keywords:**]{} circumstellar matter — radio continuum: ISM — supernovae: individual (SN 1987A) — supernova remnants Introduction ============ As the closest observed supernova in nearly 400 years, SN 1987A in the Large Magellanic Cloud offers unique opportunities for detailed study of the evolution of a supernova and the birth of a supernova remnant. In the 14 years since the explosion was detected, it has been extensively studied in all wavelength bands from radio to gamma-ray using both ground- and space-based observatories. These observations have revealed a complex evolution of both the supernova itself and the supernova remnant which is developing as the ejecta and their associated shocks interact with the circumstellar material. Perhaps the most dramatic of these observations are the optical observations which show the beautiful triple-ringed structure surrounding and illuminated by the supernova (Burrows et al. 1995). The inner ring is believed to represent an equatorial density enhancement in the circumstellar gas at the interface between a dense wind emitted from an earlier red-giant phase of the supernova progenitor star, Sk$-69^{\circ}$202, and a faster wind emitted by this star in more recent times (Crotts, Kunkel & Heathcote 1995; Plait et al. 1995). In the past few years, there has been increasing evidence for interaction of the expanding ejecta or the associated shocks with the equatorial ring, with small regions of enhanced H$\alpha$ emission, or ‘hotspots’ appearing just inside the ring. The first of these was detected by Pun et al. (1997), but studies of ground-based and [*Hubble Space Telescope*]{} ([*HST*]{}) data by Lawrence et al. (2000) show that this spot was detectable as far back as March 1995 (day 2933 since the supernova[^2]). Lawrence et al. also present evidence for up to eight additional regions of enhanced emission from about day 4300 (December 1998). The most prominent are the original spot at position angle $29^{\circ}$ and a group of spots just south of east between position angles of $90^{\circ}$ and $140^{\circ}$. At radio wavelengths, the initial outburst was very short-lived (Turtle et al. 1987) compared to other radio supernovae (e.g. Weiler et al. 1998). This prompt outburst is attributed to shock acceleration of synchrotron-emitting electrons in the stellar wind close to the star at the time of the explosion (Storey & Manchester 1987; Chevalier & Fransson 1987). After about three years, in mid-1990, radio emission was again detected from the supernova, with the Molonglo Observatory Synthesis Telescope (MOST) at 843 MHz (Ball et al. 1995) and with the Australia Telescope Compact Array (ATCA) at 1.4, 2.4, 4.8 and 8.6 GHz (Staveley-Smith et al. 1992; Gaensler et al. 1997). This emission has increased more-or-less monotonically since its first detection. The spectral index over the observed frequency range has remained close to $-0.9$, indicating optically thin synchrotron emission. X-ray emission was observed to turn on at about the same time as the second phase of radio emission and also has increased in intensity since then (Gorenstein, Hughes & Tucker 1994; Hasinger, Aschenbach & Trümper 1996). Recent observations with the [*Chandra X-ray Observatory*]{} have resolved the X-ray emission into an approximately circular shell (Burrows et al. 2000). This second phase of increasing emission is attributed to the interaction of shocks driven by the ejecta with circumstellar material and is distinct from the interaction with a radially decreasing stellar wind which characterises radio supernovae. It therefore signifies the birth of the [*remnant*]{} of SN 1987A – the first observation of the birth of a supernova remnant. We use the name SNR 1987A for the remnant. By late-1992, SNR 1987A was sufficiently strong to image at 9 GHz using the ATCA (Staveley-Smith et al. 1993a). This image, which exploited super-resolution to resolve the source, showed that the remnant was roughly circular with a diameter of about 08, fitting inside the optical equatorial ring (Reynolds et al. 1995), and with bright lobes to the east and west, suggesting an annular structure with an axis similar to that of the optical emission. The eastern lobe was about 20% brighter than the western lobe. To follow the evolution of this structure, the remnant has been imaged at roughly yearly intervals since 1992. Despite the increasing brightness, the size of SNR 1987A is increasing only slowly. Gaensler et al. (1997) fitted a model consisting of a thin spherical shell to the $uv$-plane data and showed that, between 1992 and 1995, the average expansion velocity of the remnant was only $2800 \pm 400$ km s$^{-1}$. In contrast, the average expansion speed between 1987 and 1991 was about 35,000 km s$^{-1}$ (Jauncey et al. 1988). Comparison of four images obtained between 1992 and 1995 showed that the brightness of the lobes increased relative to that of the spherical shell and that the asymmetry between the east and west lobes increased markedly over this period (Gaensler et al. 1997). By 1995, the peak brightness of the eastern lobe was 1.8 times that of the western lobe. The only other young supernova which has been imaged with high resolution at radio wavelengths is SN 1993J in M81. Frequent observations using Very Long Baseline Interferometry (VLBI) techniques (e.g. van Dyk et al. 1994, Marcaide et al. 1997, Bartel et al. 2000) have shown that the radio emission is in the form of an expanding shell, the outline of which is nearly circular, but rather lumpy. Unlike SNR 1987A, SN 1993J remains in the radio supernova stage with decreasing flux density, making further imaging difficult. Recent observations of SN 1980K (Montes et al. 1998) and SN 1979C (Montes et al. 2000) have shown variations in the rate of decline of flux density or, in the case of SN 1979C, possibly small increases, indicating variations in the density of the circumstellar medium, but, like SN 1993J, these objects are best considered to be still in their radio supernova stage. The slow expansion velocity of SNR 1987A suggests that the expanding shock and the leading ejecta have encountered a significant density enhancement, greatly reducing their velocity (Chevalier 1992; Duffy, Ball & Kirk 1995; Chevalier & Dwarkadas 1995). These models assume spherical symmetry and hence do not account for the increasing asymmetry of the source. Furthermore, they do not predict the continuing increase in brightness of the remnant as observed by Ball et al. (1995) and discussed further below. An alternative explanation for the slow apparent expansion velocity is that the shock excites slowly moving clumps of circumstellar material and then moves on. Ball & Kirk (1992) modelled the emission observed up to day 1800 by shock heating of two clumps and obtained a good fit to the data up to that time. As discussed by Gaensler et al. (1997), it is not easy to account for the observed asymmetry of SNR 1987A. Models involving the annular structure of the circumstellar material (e.g. Chevalier & Dwarkadas 1995) have difficulty accounting for the degree of enhancement in the lobes and the east-west asymmetry. Similar difficulties are encountered in trying to explain the optical hotspots. Lawrence et al. (2000) conclude that “fingers or jets” in the distribution of ejecta from the supernova is the most plausible explanation. In a recent publication, Ball et al. (2001) have analysed the 843 MHz flux densities observed with MOST to 2000 May (day 4820). They find a transition from a declining rate of increase observed from about mid-1991 (day $\sim 1600$) to early-1995 (day $\sim 2900$), to a larger and constant rate of increase, $62.7 \pm 0.5 \;\mu$Jy day$^{-1}$, since then. This change in slope occurred at about the same time as the appearance of the first optical hotspot (Lawrence et al. 2000), suggesting a possible connection. In this paper, we extend the ATCA observational data base from 1995 to February 2001 (day 5100) and discuss these results in conjunction with those of Ball et al. (2001). The evolution of the radio flux densities is discussed in Section 2. The sequence of images is extended to late-2000 and discussed in relation to recent optical and X-ray imaging in Section 3. Future prospects are canvassed in Section 4. Evolution of Radio Flux Densities and Spectral Index ==================================================== Flux density monitoring observations of SNR 1987A are made using the ATCA at 4 – 6 week intervals using one of the 6-km array configurations. Observations are made simultaneously at two frequencies, either 1380 and 2496 MHz (2368 MHz before mid-1997) or 4790 and 8640 MHz. A 128-MHz bandwidth is observed at all frequencies. The two frequency pairs are observed alternately, with 20 min on SNR 1987A and 3 min on phase calibrators before and after the SNR observation, with a total observation time typically of 12 hours. All observations are made with a J2000 pointing and phase centre of R.A. $05^{\rm h}\;35^{\rm m}\;27\fs90$, Dec. $-69^{\circ}\;16'\;21\farcs6$, approximately $10''$ south of the SN 1987A position (Reynolds et al. 1995). The phase calibrators are PKS B0530–727, PKS B0407–658 and (at 4790/8640 MHz) PKS B0454-810. Flux calibration is relative to PKS B1934-638, assumed to have flux densities of 14.95, 11.14, 5.83 and 2.84 Jy at the four frequencies, respectively. Data are first checked for obvious interference or telescope problems, flagged if necessary and then are processed using [miriad]{}[^3] scripts to ensure consistency. Visibility data are calibrated in both phase and amplitude and images formed using baselines longer than 3 k$\lambda$. These images are cleaned and sources above a threshold identified. A table of source positions and integrated and peak flux densities is output for each frequency. At least at the two higher frequencies, SNR 1987A is resolved and so integrated flux densities are quoted. An unresolved source, J0536-6919, is present on all images with an observed flux density ranging between 80 mJy at 1.4 GHz and 6 mJy at 8.6 GHz and serves as a check on the system calibration. Its J2000 position determined from the 4.8 GHz observations since day 4000 is R.A. $05^{\rm h}\;36^{\rm m}\;04\fs789 \pm 0\fs005$, Dec. $-69^{\circ}\;18'\;44\farcs81 \pm 0\farcs02$. At 8.6 GHz it lies at about 1.5 primary beam radii from the pointing centre, so it’s flux density measurements at this frequency are not very reliable. Measured flux densities for SNR 1987A are given in Appendix 1. Figure \[fg:fluxes\] shows the measured flux densities of SNR 1987A at the four ATCA frequencies. Estimated uncertainties represent a combination of random noise and scale errors resulting from errors in the calibration. Except at 8.6 GHz, the scale errors are estimated from the scatter in the measured flux densities of J0536-6918 since day 4000. At 8.6 GHz, J0536-6918 is outside the half-power radius of the primary beam, and scale errors are taken to be 1.25 times the 4790 MHz scale errors. At the lower frequencies and at later times, the errors in the flux density estimations are dominated by the scale errors. Since, except at 8.6 GHz, the flux density of J0536-6918 is comparable to that of SNR 1987A and since there is no evidence for systematic changes in the flux density of J0536-6918, these scale errors can be reduced by dividing the flux densities by the normalised flux density of J0536-6918 from the same observation. Flux densities from day 3000 scaled in this way for 1.4, 2.4 and 4.8 GHz are shown in Figure \[fg:scflux\]. These plots show that the continued increase in flux density observed at 843 MHz by Ball et al. (2001) is also observed at higher frequencies. The increase in slope observed at about day 2900 in the MOST data is also seen in the ATCA data (Fig. \[fg:fluxes\]). Ball et al. (2001) stated that the MOST data after day 3000 were well fitted by a linear trend. However, we believe that there is significant evidence for an long-term increase in slope after day 3000 in both the MOST data set and the ATCA data sets, i.e., the rate of increase in flux density is increasing. Evidence for this is shown in Fig. \[fg:fluxvar\] which shows residual flux densities after fitting a second-order polynomial to the MOST and scaled ATCA flux densities from day 3000 and subtracting the linear component. These plots all show a systematic trend with the residual flux densities being, on average, negative at central times and positive at both early and late times. Fig. \[fg:fluxvar\] also shows the second-order term of the fit which in all cases is positive and of about 3-$\sigma$ significance. These data sets are essentially independent so there can be little doubt about the significance of the effect. To further quantify this effect, straight lines were fitted to the ATCA data sets from day 3000, both for the unscaled data and for the data (except at 8.6 GHz) scaled by the flux density of J0536-6918. Results of this fitting are given in Table \[tb:grad\]. The second and third columns give the gradient and $x$-intercept of the lines fitted to all data between day 3000 and day 5100 (cf. Fig. \[fg:fluxvar\]). The point of interest here is that there is a systematic increase of the date of intercept with frequency. Especially for the higher frequencies, the intercept is well after the date of first detection of the SNR. This shows that the higher frequencies have a higher relative rate of flux density increase and that the present rate of increase is much higher than that at early times. Note that this effect is seen in both the scaled and unscaled data, so it is not an artifact of the scaling procedure. As a further check, the data sets were split into two halves, from day 3000 to day 4050 and from day 4050 to day 5100, and straight lines fitted to both halves. Results are tabulated in columns 4 – 7 of Table \[tb:grad\]. The values of reduced $\chi^2$ for the combined fit were computed by treating the two lines as a single model describing the complete data set from day 3000 onward. These show that, independently at all frequencies, the change in gradient is significant at about the 2-$\sigma$ level, and that this gradient increase is present in the MOST data and in both the scaled and the unscaled ATCA data. [ccccccc]{} Frequency & Gradient & Intercept & Grad. ($<4050$) & Grad. ($>4050$) & Grad. change& Red. $\chi^2$\ (GHz) & ($\mu$Jy/day) & (day) & ($\mu$Jy/day) & ($\mu$Jy/day) & ($\mu$Jy/day)&\ \ 0.843 & $63.1\pm 0.6$ & $1640\pm 20$ & $61.4\pm 1.0$ & $66.9\pm 2.1$ & $5.5\pm 2.3$ & 0.5\ 1.4 & $40.9\pm 0.8$ & $1750\pm 50$ & $37.7\pm 1.7$ & $46.6\pm 2.4$ & $8.8\pm 2.9$ & 0.7\ 2.4 & $25.2\pm 0.5$ & $1780\pm 50$ & $20.7\pm 1.2$ & $28.0\pm 1.6$ & $7.3\pm 1.9$ & 0.7\ 4.8 & $15.2\pm 0.4$ & $2010\pm 55$ & $14.2\pm 0.6$ & $17.5\pm 1.2$ & $3.2\pm 1.4$ & 1.3\ 8.6 & $10.6\pm 0.4$ & $2160\pm 65$ & $10.3\pm 0.8$ & $ 13.3\pm 1.7$ & $3.0\pm 1.9$ & 0.8\ \ 1.4 & $41.2\pm 0.7$ & $1770\pm 45$ & $39.8\pm 2.0$ & $46.3\pm 2.0$ & $6.4\pm 2.8$ & 1.3\ 2.4 & $26.0\pm 0.5$ & $1840\pm 45$ & $22.8\pm 1.5$ & $27.8\pm 1.4$ & $5.0\pm 2.0$ & 0.4\ 4.8 & $15.3\pm 0.3$ & $2020\pm 45$ & $13.9\pm 0.8$ & $15.5\pm 1.0$ & $2.5\pm 1.3$ & 1.0\ Apparently significant short-term variations are seen, especially at 4.8 GHz, with a timescales of order 100 days. The clearest example is near the end of the 4.8 GHz data set and is best seen in Figure \[fg:fluxvar\]. This fluctuation is not obvious at lower frequencies, but could be masked by the low signal-to-noise ratio. If real, these short-term fluctuations imply significant interactions on a scale of 0.03 pc (01) or less, and furthermore, many such interactions. Figure \[fg:spec\_ind\] shows MOST flux densities from Ball et al. (2001) and from the ATCA at 1.4, 2.4, 4.8 and 8.6 GHz at five epochs spread through the data set. Values of the spectral index $\alpha$, where $S=\nu^{\alpha}$ and $\nu$ is the frequency, found by linear regression, are given on each plot. Quoted errors are 1$\sigma$. To test for systematic curvature in the spectrum, the spectral indices were calculated using the three lowest frequencies, $\alpha_1$ and separately using the three highest frequencies, $\alpha_2$. If there is systematic curvature in the spectrum, then the difference $\alpha_1 - \alpha_2$ should be significant, and roughly constant. In fact, the difference is typically small and of either sign, showing that there is no systematic curvature. The mean difference $\alpha_1 - \alpha_2$ across the entire data set is $-0.035$ compared to the rms fluctuation of $0.187$. The observation that the relative flux density gradient is both greater and increasing more rapidly at the higher frequencies (Table \[tb:grad\]) implies a systematic change in the spectral index. Figure \[fg:si\_time\] shows the computed spectral indices from 843 MHz to 8.4 GHz as a function of time. Because of the uneven and non-simultaneous sampling at the different frequencies, spline curves were fitted to all data sets except that at 1.4 GHz. Spectral indices were then computed by interpolating values at 843 MHz, 2.4, 4.8 and 8.6 GHz to times when the 1.4 GHz flux density was measured, and fitting power-law spectra to flux densities. The interpolation was unreliable near the start of the data set for 2.4 and 8.6 GHz because of sparse data, so only three points were fitted there. Apart from a few high points between days 1200 and 1400 near the start of the data set (which have large error bars), there is a more-or-less steady increase in spectral index, corresponding to a flattening of the radio spectrum, throughout the whole data set. It is possible that the variation consists of step changes, with the most obvious step times being around day 3000 and day 4700. At early times, the mean spectral index was about $-0.97$, but in the last year (2000) it was $-0.88$. The radio spectra are remarkably close to power law and show no sign of either positive curvature, as would be expected if either free-free absorption or synchrotron self-absorption were important, or negative curvature as is predicted by diffusive shock acceleration theories (e.g. Reynolds & Ellison 1992). The radio spectral index of about $-0.9$ is considerably steeper than the canonical $-0.5$ expected for synchrotron emission from relativistic electrons accelerated in strong shocks, for which the spectral index $\alpha = (1-s)/2$ and the electron energy index $s = (r+2)/(r-2)$, where $r$ is the compression ratio in the shock (Jones & Ellison 1991). For high Mach-number shocks in a monatomic non-relativistic gas, $r=4.0$, giving $s=2.0$ and $\alpha = -0.5$. A steeper radio spectrum implies a steeper electron energy distribution and a smaller compression ratio in the shock; $s \sim 2.8$ and $r \sim 2.7$ for $\alpha = -0.9$. Monte-Carlo modelling of non-linear diffusive shock acceleration, including the dynamical effects of the accelerated particles (e.g. Baring et al. 1999), suggest that young and hence fast shocks have a [*larger*]{} compression ratio than older shocks, resulting in a flatter synchrotron spectrum. Duffy, Ball & Kirk (1995) included the effect of accelerated ions on the shock structure and were able to model the observed spectrum with the shock expanding into a terminated stellar wind structure. However, this model predicts a declining brightness and a steepening spectrum for the radio emission, both of which are contrary to observation. The observed spectral index is also steeper than those for typical supernova remnants; no SNR in the Green (2000) catalogue has a spectral index definitely steeper than $-0.8$ and the mean spectral index for shell-type remnants is $-0.51$. It is interesting to speculate that the current flattening of the spectrum represents an evolution toward the typical index of $-0.5$. At the current rate, it will only take about 50 years to reach $-0.5$. The clump interaction model of Ball & Kirk (1992) predicts a declining flux density at later times. However, this model was based on only two clumps. The model could be extended to invoke interaction with an increasing number of clumps, maybe tens or hundreds at the present time. A more accurate description would involve a statistical hierarchy of effective clump sizes. This could account for both the gradient increase and the short-term fluctuations in the observed flux density. Both radio and optical evidence (e.g. Spyromilio, Stathakis & Meurer 1993, van Dyk et al. 1994, Spyromilio 1994) point to a clumpy circumstellar medium around other supernovae. 9 GHz Imaging Observations of SNR 1987A ======================================= Resolved Radio Images of SNR 1987A ---------------------------------- In previous papers we have shown that the ATCA’s diffraction-limited resolution at 9 GHz of 09 is sufficient to resolve the radio emission from SNR 1987A (Staveley-Smith et al. 1993b) and that super-resolution techniques can be used to improve the resolution to $\sim0\farcs5$. At this resolution, the radio emission forms a limb-brightened shell, with brightness enhancements on the eastern and western sides (Staveley-Smith et al. 1993a; Briggs 1994; Gaensler et al. 1997). We have continued to make regular observations of SNR 1987A at 9 GHz; observing parameters for all imaging observations are given in Table \[tab\_9ghz\]. We have analysed these data in the same manner as described by Gaensler et al. (1997), forming an image at each epoch and deconvolving the resulting image using a maximum entropy algorithm. From 1996 onwards, the source has been of sufficient signal-to-noise that phase self-calibration can be successfully applied to the data, resulting in a significant improvement in the accuracy of the complex gains for each antenna. ------------ ------------- -------- ------- ------------- ------------- ---------- -- Mean Epoch Observing Day Array Frequencies Time on Radius Date Number (MHz) Source (hr) ($''$) 1992.9 1992 Oct 21 2068 6C 8640,8900 15 0.66(2) 1993 Jan 04 2142 6A 8640,8900 13 0.66(2) 1993 Jan 05 2143 6A 8640,8900 5 0.62(2) 1993.6 1993 Jun 24 2314 6C 8640,8900 9 0.62(1) 1993 Jul 01 2321 6C 8640,8900 10 0.68(1) 1993 Oct 15 2426 6A 8640,9024 18 0.69(1) 1994.4 1994 Feb 16 2550 6B 8640,9024 9 0.69(2) 1994 Jun 27 2683 6C 8640,9024 21 0.659(7) 1994 Jul 01 2687 6A 8640,9024 10 0.659(9) 1995.7 1995 Jul 24 3074 6C 8640,9024 7 0.687(8) 1995 Aug 29 3111 6D 8896,9152 7 0.69(2) 1995 Nov 06 3178 6A 8640,9024 9 0.685(6) 1996.7 1996 Jul 21 3437 6C 8640,9024 14 0.688(4) 1996 Sep 08 3486 6B 8640,9024 13 0.684(4) 1996 Oct 05 3512 6A 8896,9152 8 0.692(6) 1998.0 1997 Nov 11 3914 6C 8512,8896 19 0.715(5) 1998 Feb 18 4013 6A 8896,9152 15 0.733(5) 1998 Feb 21 4016 6B 9024,8512 7 0.735(4) 1998.9 1998 Sep 13 4220 6A 8896,9152 12 0.721(3) 1998 Oct 31 4268 6D 9024,8512 11 0.729(5) 1999 Feb 12 4372 6C 8512,8896 10 0.737(3) 1999.7 1999 Sep 05 4578 6D 9152,8768 11 0.754(4) 1999 Sep 12 4585 6A 8512,8896 14 0.736(4) 2000.8 2000 Sep 28 4966 6A 8512,8896 10 0.756(2) 2000 Nov 12 5011 6C 8512,8896 11 0.764(3) ------------ ------------- -------- ------- ------------- ------------- ---------- -- : 9 GHz ATCA observations of SNR 1987A used for imaging. The radius listed is that obtained by fitting a thin spherical shell to each $u-v$ data-set (see text).[]{data-label="tab_9ghz"} The resulting series of images are shown in Figures \[fig\_movie\_natural\] and \[fig\_movie\_super\] under the conditions of diffraction-limited resolution and super-resolution, respectively. In these figures and Figure \[fig\_models\], the R.A. and Dec. offsets are with respect to J2000 R.A. $05^{\rm h}\;35^{\rm m}\;28\fs00$, Dec. $-69^{\circ}\;16'\;11\farcs1$ The diffraction-limited images indicate that the source is clearly extended, primarily in the east-west direction, and continues to brighten. In the super-resolved sequence of images, it can be seen that the shell-like morphology reported by Staveley-Smith et al. (1993a) and by Gaensler et al. (1997) is maintained throughout, with two bright regions on the east and west sides of the rim. Model Fits to the Radio Morphology ---------------------------------- Gaensler et al. (1997) showed that the size of SNR 1987A could be quantified at each epoch by approximating the morphology of SNR 1987A by a thin spherical shell of arbitrary position, flux and radius. The best-fit parameters are found by computing the Fourier-transform of this shell, subtracting this transform from the $u-v$ data, and then adjusting the properties of the model until the corresponding $\chi^2$ parameter is minimised (see also Staveley-Smith et al. 1993b). Using such an approach, Gaensler et al. (1997) were able to show that the radius of the supernova remnant increased from 065 at epoch 1992.9 to 068 at epoch 1995.7, corresponding to a (surprisingly low) mean expansion velocity of $2800\pm400$  (assuming a distance to the supernova of 50 kpc). Here we extend and expand on these attempts to quantify the changes in the radio remnant. We first fit a spherical shell to all subsequent data-sets. The resulting radii are listed in Table \[tab\_9ghz\], where the uncertainty in the last quoted digit is given in parentheses. A fit to these radii gives a linear expansion rate of $3500\pm100$ . The mean radius at each observing epoch is listed in Table \[tab\_params\] and plotted in Figure \[fig\_expand\]. The signal-to-noise of the data presented by Gaensler et al. (1997) was too low to justify more complex fits to the data. However, we are now in a position to compare a thin spherical shell to other models. We first note that other simple spherically-symmetric models, such as face-on rings and gaussians, produce significant residuals, and are clearly inconsistent with the data. In Figure \[fig\_models\] we show three possible fits to the data from epoch 1998.0. In each case, we have generated an input model (shown in the first column). We have then simulated an observation of the sky-distribution corresponding to this model as follows: we Fourier-transform the model, then multiply this transform by the transfer function of the ATCA observations from this epoch to produce a set of $u-v$ tracks identical to those of the real data. We then image these visibilities in the same way as for the real data, and similarly deconvolve and super-resolve the image, to give the result shown in the second column. We also subtract the $u-v$ data corresponding to the model from the observed data, and then image the resulting visibilities to produce the residual image shown in the third column. In the first row of Figure \[fig\_models\], we show the results of fitting a thin spherical shell to the 1998.0 data (cf. Gaensler et al. 1997) — the corresponding best-fit flux and radius are 18.4 mJy and 073. The resulting image successfully approximates the limb-brightened morphology seen in Figure \[fig\_movie\_super\], but lacks the enhancements in brightness seen on either side of the shell. The residual image, while having small amplitudes in an absolute sense, shows clear and systematic differences between this simple model and the data (maximum fractional residual of $\sim$20%). Given that the main difference between a thin shell model and the data seems to be the presence of the bright lobes on either side, we next generate a model involving a thin shell (of arbitrary flux, position and radius), and two point-sources of emission (each of arbitrary flux and position). For the 1998.0 data set, the best-fit parameters for this model are a shell of flux 15.0 mJy and radius 079, and point-sources of fluxes 3.1 and 0.4 mJy, one projected near the edge of, but inside, the eastern half of the shell, and the other similarly positioned on the western half. Note that the radius of the shell in this fit is 8% larger than that obtained by fitting to a shell alone. The resulting model, image and residual are all shown in the second row of Figure \[fig\_models\]. The image corresponding to the model now reasonably approximates the morphology seen in the real data, and the residuals are greatly reduced (maximum fractional residual of $\sim$5%) when compared to the case of a shell alone. We can further improve the fit between the model and the data by increasing the number of point sources in the model. In the bottom row of Figure \[fig\_models\], we show a fit to the 1998.0 data involving a spherical shell plus seven point sources. The large number of free parameters (25 in this case) makes finding an absolute minimum in $\chi^2$ very difficult — thus the “best fit” we have shown was found by trial-and-error. The shell has flux density 11.4 mJy and radius 072 (2% smaller than in the case of a shell alone); the point sources range in flux density between 0.3 and 3.5 mJy, and are distributed around the perimeter of the shell. The model image and residual image both indicate a very good match between the model and the data (maximum fractional residual $\sim$2%). We emphasise, though, that we make no claims to uniqueness for this solution. -------- ------------ ----------------- ----------------- ----------------- ----------------- ------------- -------------------- --------------------- ------------- -------------------- --------------------- Mean Total Flux Epoch Density $S_{\rm shell}$ $r_{\rm shell}$ $S_{\rm shell}$ $r_{\rm shell}$ $S_{\rm 1}$ $\delta{\rm RA}_1$ $\delta{\rm Dec}_1$ $S_{\rm 2}$ $\delta{\rm RA}_2$ $\delta{\rm Dec}_2$ (mJy) (mJy) ($''$) (mJy) ($''$) (mJy) ($''$) ($''$) (mJy) ($''$) ($''$) 1992.9 5.6(2) 5.2 0.668(9) $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ 1993.6 6.9(2) 6.6 0.659(6) 5.1 0.73(2) 1.2 0.26(3) -0.05(2) 0.4 -0.73(9) 0.06(5) 1994.4 7.5(1) 7.8 0.671(4) 6.2 0.75(2) 1.3 0.27(2) -0.03(1) 0.4 -0.69(7) 0.06(4) 1995.7 11.8(1) 11.0 0.684(4) 8.6 0.71(2) 1.9 0.35(3) -0.03(1) 0.5 -0.76(8) 0.04(5) 1996.7 15.5(1) 15.1 0.688(2) 12.4 0.764(7) 2.5 0.32(1) 0.03(1) 0.4 -0.69(6) 0.15(3) 1998.0 18.3(1) 18.4 0.731(2) 15.0 0.786(8) 3.1 0.34(1) -0.02(1) 0.4 -0.75(7) -0.02(5) 1998.9 21.7(1) 21.5 0.730(2) 17.2 0.778(6) 3.4 0.40(1) -0.02(1) 1.0 -0.81(3) -0.04(2) 1999.7 24.7(1) 24.4 0.742(2) 19.7 0.796(6) 3.9 0.41(1) -0.02(1) 1.0 -0.81(3) -0.05(2) 2000.8 30.8(1) 30.8 0.761(2) 24.9 0.804(4) 4.8 0.46(1) -0.01(1) 1.3 -0.82(2) -0.03(2) -------- ------------ ----------------- ----------------- ----------------- ----------------- ------------- -------------------- --------------------- ------------- -------------------- --------------------- : Parameters from model fits to the 9 GHz images[]{data-label="tab_params"} Given the difficulty in finding a multiple-component fit to the data by trial-and-error, and the non-uniqueness of this solution, we do not here present multi-parameter fits to other epochs such as those shown in the third row of Figure \[fig\_models\]. However, it seems clear that a shell alone is no longer the best simple model of the data, and that a shell with two point-sources is a significant improvement. We therefore have fitted all epochs with these two models, with all parameters free to vary in each case. The results of these fits are listed in Table \[tab\_params\] and plotted in Figure \[fig\_expand\]. Offsets of the two point sources in Table \[tab\_params\] are with respect to the reference position used for Figures 6, 7 & 9. The data from epoch 1992.9 is of low signal-to-noise, and no good fit using this more complex model cound be found. It can be seen from Table \[tab\_params\] that the radius of the shell in models which include two point-sources is 5–10% larger than that obtained in the shell-only model used by Gaensler et al. (1997). It is also clear that the point sources are moving outward with the expansion of the shell, the eastern one apparently at a larger relative rate than the shell. As for the shell model, we can fit these larger radii by a linear increase, to obtain an expansion speed of $2300\pm300$ , about 35% slower than in the case for the shell alone. However, we note that if multi-parameter fits like those in the third row of Figure \[fig\_models\] are carried out for other epochs, the resulting radii are $\sim$5% [*smaller*]{} than for the fits to a shell alone, and the resulting inferred expansion speed is $\sim3700$ . While we have not carried out the corresponding analysis here, we note that Staveley-Smith et al. (1993a) fitted the radio emission from SNR 1987A with a thick spherical shell of outer-to-inner radius ratio 1.25. This results in a shell diameter about 10% greater than for the thin-shell fit, consistent with the range of possible radii considered here. We therefore conclude that the radius of the remnant as determined from fitting to the $u-v$ data is uncertain by $\sim$10%, and that the resulting expansion velocity is uncertain by $\sim$30%. The main conclusions from earlier results — that the radio emission is originating from a region within the equatorial ring, and that the material producing this emission was initially moving rapidly but now has a very low rate of expansion — are unchanged, even when the assumption of spherical symmetry is relaxed. Comparison With Other Wavelengths and Discussion ------------------------------------------------ In Figure \[fig\_overlay\] we compare our 9 GHz ATCA data to recent observations with [*HST*]{} and with [*Chandra*]{}. In the upper panel, the super-resolved ATCA image from epoch 2000.8 is compared to the difference-image of optical emission around the supernova produced by Lawrence et al. (2000). The HST image has been registered on the radio frame with an accuracy of better than $0\farcs1$ using the position of the central star from Reynolds et al. (1995). It shows both the fading equatorial ring, and and at least seven hotspots on this ring, all brightening as the supernova shock begins to interact with dense circumstellar material. It can be seen from this optical/radio comparison that the conclusions made by Gaensler et al. (1997) are maintained: the bright radio lobes align with the major axis of the optical ring, with the brighter eastern lobe clearly more distant from the central star than the fainter western lobe. Within the uncertainties, the eastern radio lobe is coincident with some of the optical hotspots (but not the brighest one), just inside the optical ring. In the west, the optical emission lies outside the radio lobe, close to the lowest radio contour in Figure \[fig\_overlay\]. In fact, there is a good correspondence of the radio emission with optical emission [*within*]{} the optical ring (not visible in Figure \[fig\_overlay\]) corresponding to the reverse-shock, as reported by Garnavich, Kirshner & Challis (1999). Just as in the radio data, two optical lobes are seen, one to the east and one to the west of the supernova site. Also similar to the case in the radio, the eastern optical lobe is brighter than and further from the supernova than the western lobe. The lower panel of Figure \[fig\_overlay\] compares radio emission with a super-resolved image obtained by [*Chandra*]{} at epoch 1999.8 (Burrows et al. 2000). The X-ray and radio emission from the supernova remnant both take the form of limb-brightened shells; the images were aligned by placing the estimated centre of the X-ray remnant on the Reynolds et al. (1995) position (D. Burrows, private communication) and hence is less accurate than the optical–radio alignment. While there is a good correspondence between the brightest regions in each waveband, the X-ray maxima appear to lie outside the radio maxima. This is perhaps surprising since theoretical models (e.g. Borkowski, Blondin & McCray 1997) suggest that the radio emission is generated just inside the outer shock whereas the X-ray emission is generated by a reverse shock compressing and heating the denser ejecta as it propagates inwards. It is worth noting, however, that X-ray emission lying outside the radio emission and interpreted as coming from the outer shock has been detected in the SNR 1E 0102.2$-$7219 by Gaetz et al. (2000). The two-lobed radio morphology seen for SNR 1987A was apparent at least as early as 1992 (Figure \[fig\_movie\_super\]; Staveley-Smith et al 1993a), while optical hotspots did not begin to appear on the eastern side of the optical ring until 1995 and on the western side until 1998 (Lawrence et al. 2000). Furthermore, the earliest-appearing and brightest optical hotspot does not coincide in position angle with either of the two main radio lobes, nor with the brightest X-ray emission seen by [*Chandra*]{}(Figure \[fig\_overlay\]; Burrows et al. 2000). We therefore argue that the more rapid rate in the increase of radio emission beginning around day 3000, as reported by Ball et al. (2001) and confirmed in Section 2 above, is not related to the appearance of optical hotspots seen at around the same time. While Ball et al. (2001) have pointed out that the various optical hotspots turned on at about the right time for them to have been produced by the arrival of the radio-producing shock(s) in these regions, it seems clear that this had little effect on the radio morphology of the remnant. The best explanation for deviations from spherical symmetry in the radio morphology of SNR 1987A still seems to be that they result simply from regions of enhanced emission in the equatorial plane of the progenitor system, where circumstellar gas is expected to be both densest and closest to the progenitor star (cf. Gaensler et al. 1997). The fact that at both radio and optical wavelengths the eastern lobe is both brighter and more distant from the explosion site than the western lobe may represent an asymmetry in the distribution of ejecta. It is notable that the emission underlying the radio hotspots is well described by a spherical shell and that, in all of the models, the flux density of the shell dominates the total flux density. This is surprising given the strong equatorial enhancement evident in the optical data and implied for the circumstellar gas. One might expect eventually to see a faster expansion of the radio remnant in the polar directions (Blondin, Lundqvist & Chevalier 1996), but there is currently no evidence for this. Part of the radio emission from the lobes may be attributed to an equatorial enhancement but, since such an enhancement would be symmetric, it cannot account for all of the emission seen from the brighter eastern lobe. It is worth noting that the shell of SN 1993J also appears to be quite spherical (Marcaide et al. 1997, Bartel et al. 2000). Future Prospects ================ Results at all wavelengths suggest that there is an increasing interaction between the expanding ejecta and the circumstellar material. Hotspots around or just inside the emission-line ring are certainly becoming more numerous and prominent in the optical band. Neither the radio imaging nor the X-ray imaging has sufficient resolution to separately identify hot spots. However, the modelling of the radio $u-v$ data and the possible short-term fluctuations in the rise of the high-frequency radio flux densities suggest that compact radio hotspots do exist. The overall morphology and the time-evolution of the radio emission suggest that there is no detailed correspondence of the radio hotspots with the hotspots seen in the optical data and, in fact, that the radio hotspots are much more numerous and widespread. We have shown that the rate of increase of the radio emission from SNR 1987A has slowly increased over the past few years. It is possible that the rate of increase of the radio (and other) emission will dramatically increase when significant amounts of ejecta begin to interact with the dense circumstellar gas of the inner ring. Extrapolation of the radii and expansion speeds resulting model fits to the radio data suggest that this will happen in $2004\pm2$. The ATCA is presently being upgraded for observations in the 12 mm and 3 mm bands, with an expected commissioning date of mid-2002. While the radio remnant is unlikely to be detectable at 3 mm, we expect 12 mm observations to provide increased resolution. Extrapolating the present flux density increase and the currently observed power-law spectrum, we expect a flux density at 20 GHz in mid-2002 of about 17 mJy. The diffraction limited half-power beamwidth at 20 GHz will be about 04, somewhat less than the present super-resolved beamwidth at 9 GHz. Even in 2002, it is likely that super-resolution will be able to be applied to the 20 GHz data, giving a resolution of about 02, and this will certainly be true at later epochs if the flux density continues to rise. We hope and expect that this will reveal further detail in the radio images, allowing interesting comparisons with images obtained at other wavelengths and giving further insight into the physics of the radio emission process. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Lewis Ball for making available a pre-publication file of the MOST flux densities for SNR 1987A and for comments on an earlier version of the paper, Dave Burrows for providing the [*Chandra*]{} data, and Ben Sugerman and Peter Garnavich for providing [*HST*]{} data. We also thank the referees for helpful suggestions. The Australia Telescope is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO. BMG acknowledges the support of NASA through Hubble Fellowship grant HF-01107.01-98A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA under contract NAS 5–26555. 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[et al.]{} 1987, Nature, 327, 38 van Dyk, S. D., Weiler, K. W., Sramek, R. A., Rupen, M. P., & Panagia, N. 1994, ApJ, 432, L115 , K. W., [van Dyk]{}, S. D., [Montes]{}, M. J., [Panagia]{}, N., & [Sramek]{}, R. A. 1998, ApJ, 500, 51 Appendix {#appendix .unnumbered} ======== Table \[tb:fluxes\] gives flux densities measured at the four ATCA frequencies over the day range 2000 - 5100 (MJD 48449 to 51949). These are calibrated relative to the primary flux calibrator, PKS B1934-638, using standard techniques. This table, also including flux density measurements for the nearby unresolved source J0536-6918 and more recent measurements, is available at http://www.atnf.csiro.au/research/pulsar/snr/sn1987a/. -------- ------------------ -------- ------------------ -------- ----------------- -------- ------------------ Day Flux Density Day Flux Density Day Flux Density Day Flux Density Number (mJy) Number (mJy) Number (mJy) Number (mJy) 1243.9 $ 0.00\pm 1.20$ 1198.9 $ 0.00\pm 1.00$ 1243.6 $ 0.00\pm 0.25$ 1388.1 $ 1.33\pm 0.26$ 1270.7 $ 2.92\pm 1.20$ 1314.5 $ 3.27\pm 1.00$ 1269.7 $ 0.66\pm 0.25$ 1432.2 $ 1.12\pm 0.26$ 1386.4 $ 4.87\pm 1.21$ 1386.0 $ 3.45\pm 1.01$ 1305.5 $ 1.15\pm 0.25$ 1500.5 $ 1.74\pm 0.26$ 1490.0 $ 7.61\pm 1.22$ 1517.9 $ 6.13\pm 1.02$ 1333.7 $ 1.49\pm 0.26$ 1515.0 $ 2.34\pm 0.28$ 1517.9 $ 8.29\pm 1.23$ 1517.7 $ 6.20\pm 1.02$ 1366.3 $ 1.53\pm 0.26$ 1583.8 $ 2.62\pm 0.28$ 1517.7 $ 9.55\pm 1.23$ 1595.8 $ 5.55\pm 1.01$ 1385.2 $ 1.96\pm 0.26$ 1593.8 $ 2.21\pm 0.27$ 1595.8 $ 10.15\pm 1.24$ 1636.7 $ 8.71\pm 1.03$ 1401.4 $ 2.17\pm 0.26$ 1662.5 $ 2.92\pm 0.29$ 1636.7 $ 15.52\pm 1.29$ 1660.5 $ 9.15\pm 1.04$ 1402.4 $ 2.15\pm 0.26$ 1661.5 $ 2.50\pm 0.28$ 1660.5 $ 15.52\pm 1.29$ 1788.8 $ 10.67\pm 1.05$ 1403.4 $ 2.25\pm 0.27$ 1786.6 $ 3.92\pm 0.32$ 1788.8 $ 20.59\pm 1.35$ 1850.1 $ 13.07\pm 1.07$ 1404.3 $ 2.12\pm 0.26$ 1852.1 $ 4.37\pm 0.33$ 1850.1 $ 24.80\pm 1.41$ 1879.2 $ 15.03\pm 1.10$ 1405.3 $ 2.13\pm 0.26$ 1878.2 $ 4.14\pm 0.32$ 1879.2 $ 24.04\pm 1.40$ 1969.7 $ 17.95\pm 1.14$ 1407.1 $ 2.06\pm 0.26$ 1947.8 $ 4.70\pm 0.34$ 1969.7 $ 27.51\pm 1.46$ 2067.7 $ 19.00\pm 1.15$ 1408.3 $ 2.57\pm 0.27$ 1969.5 $ 4.93\pm 0.35$ 2068.4 $ 33.15\pm 1.56$ 2313.7 $ 23.25\pm 1.22$ 1409.3 $ 2.53\pm 0.27$ 1986.4 $ 4.64\pm 0.34$ 2313.7 $ 38.87\pm 1.67$ 2427.3 $ 23.37\pm 1.22$ 1410.3 $ 2.66\pm 0.27$ 2003.4 $ 5.09\pm 0.36$ 2427.3 $ 39.96\pm 1.70$ 2505.2 $ 25.91\pm 1.27$ 1431.2 $ 2.29\pm 0.27$ 2067.5 $ 5.48\pm 0.37$ 2505.2 $ 43.23\pm 1.77$ 2549.2 $ 26.56\pm 1.28$ 1446.2 $ 2.63\pm 0.27$ 2092.4 $ 5.92\pm 0.39$ 2549.2 $ 43.83\pm 1.78$ 2680.7 $ 28.40\pm 1.31$ 1460.1 $ 3.03\pm 0.28$ 2142.2 $ 5.43\pm 0.37$ 2680.7 $ 46.20\pm 1.83$ 2774.5 $ 29.92\pm 1.34$ 1516.9 $ 2.83\pm 0.27$ 2195.7 $ 4.20\pm 0.33$ 2774.5 $ 49.26\pm 1.90$ 2773.7 $ 30.40\pm 1.35$ 1524.9 $ 2.74\pm 0.27$ 2261.8 $ 6.04\pm 0.39$ 2773.7 $ 48.60\pm 1.89$ 2826.7 $ 28.40\pm 1.31$ 1586.8 $ 2.97\pm 0.28$ 2299.9 $ 7.14\pm 0.44$ 2857.7 $ 50.90\pm 1.94$ 2857.7 $ 29.30\pm 1.33$ 1594.8 $ 4.50\pm 0.31$ 2299.7 $ 5.80\pm 0.38$ 2874.3 $ 48.40\pm 1.88$ 2874.3 $ 25.20\pm 1.25$ 1635.6 $ 5.31\pm 0.33$ 2313.7 $ 6.92\pm 0.43$ 2919.2 $ 53.77\pm 2.01$ 2919.2 $ 33.11\pm 1.41$ 1662.5 $ 5.02\pm 0.32$ 2320.7 $ 6.90\pm 0.43$ 2919.3 $ 51.38\pm 1.95$ 2919.3 $ 34.20\pm 1.43$ 1746.8 $ 6.04\pm 0.35$ 2374.7 $ 5.38\pm 0.37$ 2975.8 $ 54.97\pm 2.04$ 2975.8 $ 33.55\pm 1.42$ 1786.8 $ 7.44\pm 0.39$ 2403.5 $ 7.14\pm 0.44$ 2975.7 $ 48.40\pm 1.88$ 2975.7 $ 32.50\pm 1.40$ 1852.1 $ 8.16\pm 0.41$ 2426.4 $ 7.68\pm 0.46$ 3001.7 $ 48.80\pm 1.89$ 3001.7 $ 33.11\pm 1.41$ 1878.2 $ 7.06\pm 0.38$ 2462.7 $ 6.50\pm 0.41$ 3000.7 $ 56.20\pm 2.07$ 3072.5 $ 34.60\pm 1.44$ 1947.8 $ 8.27\pm 0.41$ 2550.2 $ 7.47\pm 0.45$ 3072.5 $ 61.70\pm 2.21$ 3139.7 $ 36.60\pm 1.49$ 1969.5 $ 9.11\pm 0.44$ 2572.0 $ 5.93\pm 0.39$ 3072.5 $ 57.20\pm 2.09$ 3176.7 $ 35.90\pm 1.47$ 1986.4 $ 8.44\pm 0.42$ 2579.8 $ 6.92\pm 0.43$ 3139.7 $ 61.00\pm 2.19$ 3202.7 $ 39.20\pm 1.54$ 2003.4 $ 8.72\pm 0.43$ 2579.7 $ 6.50\pm 0.41$ 3176.7 $ 59.10\pm 2.14$ 3277.7 $ 41.60\pm 1.60$ 2067.2 $ 9.42\pm 0.45$ 2628.0 $ 9.00\pm 0.51$ -------- ------------------ -------- ------------------ -------- ----------------- -------- ------------------ : ATCA flux density measurements for SNR 1987A[]{data-label="tb:fluxes"} -------- ------------------ -------- ------------------ -------- ----------------- -------- ------------------ Day Flux Density Day Flux Density Day Flux Density Day Flux Density Number (mJy) Number (mJy) Number (mJy) Number (mJy) 3202.7 $ 64.20\pm 2.27$ 3325.7 $ 41.40\pm 1.59$ 2092.4 $ 9.94\pm 0.47$ 2627.7 $ 7.80\pm 0.46$ 3277.7 $ 66.20\pm 2.32$ 3414.8 $ 44.10\pm 1.66$ 2142.5 $ 8.56\pm 0.42$ 2647.7 $ 6.10\pm 0.39$ 3325.7 $ 67.10\pm 2.34$ 3455.7 $ 44.80\pm 1.68$ 2195.7 $ 9.10\pm 0.44$ 2754.8 $ 6.26\pm 0.40$ 3414.8 $ 71.80\pm 2.47$ 3515.4 $ 45.80\pm 1.70$ 2261.8 $10.86\pm 0.50$ 2753.7 $ 5.40\pm 0.37$ 3455.7 $ 71.80\pm 2.47$ 3579.2 $ 48.50\pm 1.77$ 2299.9 $11.50\pm 0.52$ 2774.5 $ 9.22\pm 0.52$ 3515.4 $ 73.30\pm 2.51$ 3633.1 $ 48.00\pm 1.75$ 2299.7 $11.60\pm 0.53$ 2826.7 $ 9.80\pm 0.55$ 3579.2 $ 77.50\pm 2.62$ 3679.0 $ 49.40\pm 1.79$ 2374.7 $10.82\pm 0.50$ 2919.2 $ 9.44\pm 0.53$ 3633.1 $ 79.10\pm 2.66$ 3714.0 $ 50.10\pm 1.81$ 2403.5 $12.42\pm 0.56$ 2919.3 $ 11.30\pm 0.62$ 3679.0 $ 79.60\pm 2.67$ 3744.7 $ 48.50\pm 1.77$ 2462.7 $12.80\pm 0.57$ 2975.8 $ 9.11\pm 0.52$ 3714.0 $ 79.30\pm 2.66$ 3771.6 $ 49.90\pm 1.80$ 2511.0 $11.98\pm 0.54$ 2975.7 $ 8.30\pm 0.48$ 3744.7 $ 84.40\pm 2.80$ 3833.6 $ 50.10\pm 1.81$ 2572.0 $13.99\pm 0.61$ 3000.7 $ 6.60\pm 0.41$ 3771.6 $ 81.90\pm 2.73$ 3900.3 $ 52.60\pm 1.87$ 2579.8 $13.62\pm 0.60$ 3001.7 $ 9.23\pm 0.52$ 3833.6 $ 84.20\pm 2.80$ 3945.4 $ 52.20\pm 1.86$ 2579.7 $13.10\pm 0.58$ 3072.5 $ 10.40\pm 0.58$ 3900.3 $ 86.00\pm 2.85$ 3987.1 $ 52.40\pm 1.86$ 2628.0 $13.53\pm 0.60$ 3073.6 $ 11.40\pm 0.62$ 3945.4 $ 88.10\pm 2.90$ 4015.1 $ 55.30\pm 1.94$ 2627.7 $14.10\pm 0.62$ 3110.6 $ 10.20\pm 0.57$ 3987.1 $ 93.80\pm 3.06$ 4058.5 $ 58.30\pm 2.01$ 2647.7 $12.90\pm 0.57$ 3176.7 $ 9.80\pm 0.55$ 4015.1 $ 89.30\pm 2.94$ 4100.8 $ 58.30\pm 2.01$ 2754.8 $14.17\pm 0.62$ 3202.7 $ 12.60\pm 0.68$ 4058.5 $ 87.70\pm 2.89$ 4169.5 $ 61.40\pm 2.10$ 2753.7 $13.61\pm 0.60$ 3277.7 $ 12.10\pm 0.65$ 4100.8 $ 96.70\pm 3.14$ 4222.5 $ 59.20\pm 2.04$ 2774.5 $14.63\pm 0.64$ 3325.7 $ 11.70\pm 0.64$ 4169.5 $ 99.60\pm 3.22$ 4291.4 $ 64.70\pm 2.18$ 2773.7 $14.10\pm 0.62$ 3414.8 $ 11.50\pm 0.63$ 4222.5 $ 97.50\pm 3.16$ 4373.1 $ 62.30\pm 2.12$ 2826.7 $14.50\pm 0.63$ 3436.7 $ 15.70\pm 0.82$ 4291.4 $104.50\pm 3.36$ 4424.0 $ 67.10\pm 2.25$ 2919.2 $16.47\pm 0.70$ 3455.7 $ 14.20\pm 0.75$ 4373.1 $108.30\pm 3.46$ 4460.9 $ 68.90\pm 2.30$ 2919.3 $16.10\pm 0.69$ 3485.5 $ 16.10\pm 0.84$ 4424.0 $107.40\pm 3.44$ 4539.8 $ 75.50\pm 2.48$ 2975.8 $15.28\pm 0.66$ 3512.4 $ 16.50\pm 0.86$ 4460.9 $113.20\pm 3.60$ 4571.7 $ 69.50\pm 2.31$ 2975.7 $16.50\pm 0.71$ 3515.4 $ 14.60\pm 0.77$ 4539.8 $112.60\pm 3.58$ 4684.9 $ 75.20\pm 2.47$ 3000.7 $15.10\pm 0.65$ 3579.2 $ 15.50\pm 0.81$ 4571.7 $116.10\pm 3.68$ 4728.9 $ 74.20\pm 2.44$ 3001.7 $16.84\pm 0.72$ 3633.1 $ 14.90\pm 0.79$ 4684.9 $121.20\pm 3.83$ 4768.1 $ 77.10\pm 2.52$ 3072.5 $17.80\pm 0.75$ 3679.0 $ 16.30\pm 0.85$ 4728.9 $121.40\pm 3.83$ 4799.9 $ 77.40\pm 2.53$ 3176.7 $16.90\pm 0.72$ 3714.0 $ 15.90\pm 0.83$ 4768.1 $131.30\pm 4.12$ 4837.9 $ 83.80\pm 2.71$ 3202.7 $20.20\pm 0.85$ 3744.7 $ 16.30\pm 0.85$ 4799.9 $124.90\pm 3.93$ 4850.8 $ 77.80\pm 2.54$ 3277.7 $21.60\pm 0.90$ 3771.6 $ 17.90\pm 0.93$ 4837.9 $126.40\pm 3.98$ 4870.7 $ 77.90\pm 2.54$ 3325.7 $20.70\pm 0.86$ 3833.6 $ 17.70\pm 0.92$ 4850.8 $128.70\pm 4.04$ 4936.5 $ 81.00\pm 2.63$ 3414.8 $21.30\pm 0.89$ 3900.3 $ 17.10\pm 0.89$ 4870.7 $127.00\pm 3.99$ 4997.3 $ 83.60\pm 2.70$ 3455.7 $23.10\pm 0.96$ 3945.4 $ 17.90\pm 0.93$ 4936.5 $133.40\pm 4.18$ 5024.8 $ 83.00\pm 2.68$ 3515.4 $24.80\pm 1.02$ 3987.1 $ 18.90\pm 0.98$ 4997.3 $136.90\pm 4.28$ 5050.1 $ 86.20\pm 2.77$ 3579.2 $25.40\pm 1.05$ 4015.1 $ 19.40\pm 1.00$ 5024.8 $135.00\pm 4.22$ 5093.0 $ 86.60\pm 2.78$ 3633.1 $25.10\pm 1.03$ 4100.8 $ 19.60\pm 1.01$ 5050.1 $135.90\pm 4.25$ 3679.0 $24.50\pm 1.01$ 4222.5 $ 19.10\pm 0.99$ 5093.0 $143.40\pm 4.47$ 3714.0 $24.50\pm 1.01$ 4291.4 $ 21.00\pm 1.08$ 3744.7 $27.00\pm 1.11$ 4373.1 $ 21.20\pm 1.09$ 3771.6 $27.10\pm 1.11$ 4424.0 $ 24.90\pm 1.27$ 3833.6 $26.50\pm 1.09$ 4460.9 $ 25.50\pm 1.30$ -------- ------------------ -------- ------------------ -------- ----------------- -------- ------------------ : – [*continued*]{} -------- -------------- -------- -------------- -------- ----------------- -------- ------------------ Day Flux Density Day Flux Density Day Flux Density Day Flux Density Number (mJy) Number (mJy) Number (mJy) Number (mJy) 3900.3 $29.20\pm 1.19$ 4539.8 $ 25.00\pm 1.27$ 3945.4 $29.80\pm 1.22$ 4571.7 $ 24.40\pm 1.25$ 3987.1 $31.90\pm 1.30$ 4684.9 $ 26.70\pm 1.36$ 4015.1 $30.20\pm 1.23$ 4728.9 $ 28.10\pm 1.43$ 4100.8 $32.00\pm 1.30$ 4768.1 $ 25.70\pm 1.31$ 4222.5 $32.00\pm 1.30$ 4799.9 $ 24.80\pm 1.26$ 4291.4 $35.30\pm 1.43$ 4837.9 $ 31.50\pm 1.59$ 4424.0 $34.10\pm 1.39$ 4850.8 $ 27.90\pm 1.42$ 4460.9 $38.60\pm 1.56$ 4870.7 $ 29.90\pm 1.52$ 4539.8 $37.60\pm 1.52$ 4936.5 $ 30.20\pm 1.53$ 4571.7 $39.60\pm 1.60$ 4997.3 $ 27.50\pm 1.40$ 4684.9 $40.10\pm 1.62$ 5024.8 $ 31.90\pm 1.61$ 4728.9 $43.70\pm 1.77$ 5050.1 $ 31.90\pm 1.61$ 4768.1 $42.70\pm 1.73$ 5093.0 $ 36.00\pm 1.82$ 4799.9 $39.70\pm 1.61$ 4837.9 $43.60\pm 1.76$ 4850.8 $45.50\pm 1.84$ 4870.7 $47.20\pm 1.90$ 4936.5 $48.10\pm 1.94$ 4997.3 $43.50\pm 1.76$ 5024.8 $47.70\pm 1.92$ 5050.1 $46.40\pm 1.87$ 5093.0 $49.90\pm 2.01$ -------- -------------- -------- -------------- -------- ----------------- -------- ------------------ : – [*continued*]{} [^1]: Hubble Fellow. Present address: Harvard-Smithsonian Centre for Astrophysics, Cambridge MA 02138, USA [^2]: Day number = MJD $-46849.3$ [^3]: See http://www.atnf.csiro.au/computing/software/miriad/

--- abstract: 'When a single cell senses a chemical gradient and chemotaxes, stochastic receptor-ligand binding can be a fundamental limit to the cell’s accuracy. For clusters of cells responding to gradients, however, there is a critical difference: even genetically identical cells have differing responses to chemical signals. With theory and simulation, we show collective chemotaxis is limited by cell-to-cell variation in signaling. We find that when different cells cooperate the resulting bias can be much larger than the effects of ligand-receptor binding. Specifically, when a strongly-responding cell is at one end of a cell cluster, cluster motion is biased toward that cell. These errors are mitigated if clusters average measurements over times long enough for cells to rearrange. In consequence, fluid clusters are better able to sense gradients: we derive a link between cluster accuracy, cell-to-cell variation, and the cluster rheology. Because of this connection, increasing the noisiness of individual cell motion can actually [*increase*]{} the collective accuracy of a cluster by improving fluidity.' author: - 'Brian A. Camley' - 'Wouter-Jan Rappel' title: 'Cell-to-cell variation sets a tissue-rheology-dependent bound on collective gradient sensing' --- Many cells follow signal gradients to survive or perform their functions, including white blood cells finding a wound, cells crossing a developing embryo, and cancerous cells migrating from tumors. Chemotaxis, sensing and responding to chemical gradients, is crucial in all of these examples [@swaney2010eukaryotic; @levine2013physics]. Chemotaxis is traditionally studied by exposing single cells to gradients – but cells often travel in groups, not singly [@hakim2017collective; @camley2017physical]. Collective cell migration is essential to development and metastasis [@friedl2009collective], and can have remarkable effects on chemotaxis. Even when single cells cannot sense a gradient, a cluster of cells may cooperate to sense it. While collective chemotaxis is our primary focus, this “emergent” gradient sensing is found in response to many signals, including soluble chemical gradients (chemotaxis) [@theveneau2010collective; @malet2015collective; @ellison2016cell], conditioned substrates (haptotaxis) [@winklbauer1992cell], substrate stiffness gradients (durotaxis) [@sunyer2016collective] and electrical potential (galvanotaxis) [@li2012cadherin; @lalli2015collective]. Cells can cooperate to sense gradients – but the physical principles limiting a cluster’s sensing accuracy are not settled. For single cells, the fundamental bounds on sensing chemical concentrations and gradients are well-studied [@berg1977physics; @kaizu2014berg; @hu2010physical; @hu2011geometry; @endres2009maximum; @endres2008accuracy; @fuller2010external; @ueda2007stochastic; @andrews2007information; @bialek2005physical], showing unavoidable stochasticity in receptor-ligand binding limits chemotactic accuracy. Is this true for cell clusters? Is a cell cluster simply equivalent to a larger cell? No! There is an essential difference between many clustered cells and a single large cell: even clonal populations of cells can have highly variable responses to signals, due to many factors, including intrinsic variations in regulatory protein concentrations [@swain2002intrinsic; @niepel2009non; @sigal2006variability]. These cell-to-cell variations (CCV) can be persistent over timescales much larger than the typical motility timescale of the cell [@sigal2006variability]. CCV has not been addressed in models of collective chemotaxis and it is not clear whether collective gradient sensing is limited by CCV or by stochastic receptor-ligand binding [@malet2015collective; @camley2016emergent; @camley2016collective; @varennes2016collective; @ellison2016cell; @mugler2016limits; @cai2016modeling]. Using a combination of analytics and simulations, we show that unless CCV is tightly controlled, collective guidance of a cluster of cells is limited by these variations: gradient sensing is biased toward cells with intrinsically strong responses. This bias swamps the effects of stochastic ligand-receptor binding. Cell clusters may reduce this error by time-averaging their gradient measurements only if the cells rearrange their positions, creating an unavoidable link between the [*mechanics*]{} of the cell cluster and its [*gradient sensing ability*]{}. As a result, surprising new tradeoffs arise: clusters must balance using motility to follow a biased signal with using motility to reduce error, and compromise between reducing noise and increasing cluster fluidity. Gradient sensing error is dominated by cell-to-cell variation, not receptor noise ================================================================================= {width="180mm"} We study a two-dimensional model of gradient sensing with CCV and ligand-receptor dynamics where cells sense a chemoattractant with concentration gradient ${\ensuremath{{ {\mathbf g}}}\xspace}$. Each cell at position ${\ensuremath{{ {\mathbf r}}}\xspace}$ measures local concentration, $c({\ensuremath{{ {\mathbf r}}}\xspace}) = c_0 (1 + {\ensuremath{{ {\mathbf g}}}\xspace}\cdot {\ensuremath{{ {\mathbf r}}}\xspace})$, via ligand-receptor binding, which is stochastic. This noise leads to unavoidable errors in the cluster’s estimate of ${\ensuremath{{ {\mathbf g}}}\xspace}$. In addition, even if concentration is perfectly sensed, each cell responds differently to a fixed $c$, which models known CCV in signal response [@wang2012diverse; @samadani2006cellular]. As a result, when the cluster combines measurements from its cells, it may develop a drift in the direction of stronger-responding cells ([Fig. ]{}\[fig:ccv\_illus\]). To combine these effects, we specify the “measured” signal in cell $i$, $M_i$, which is what the cluster believes the chemoattractant signal in cell $i$ to be, including ligand-receptor binding and CCV: $$M^i = \left[c({\ensuremath{{ {\mathbf r}}}\xspace}^i) + \delta c^i \eta^i\right]/\bar{c} + \Delta^i \label{eq:M}$$ where $\bar{c}$ is the mean concentration over the cluster, $\bar{c} = N^{-1} \sum_i c({\ensuremath{{ {\mathbf r}}}\xspace}^i)$, $\eta^i$ are uncorrelated Gaussian noises with zero mean and unit variance and $\Delta^i$ are uncorrelated Gaussian noises with zero mean and variance ${\ensuremath{\sigma_{\Delta}}}^2$. Stochastic fluctuations in ligand-receptor binding are taken into account in the term $\delta c^i$, where $(\delta c^i / c({\ensuremath{{ {\mathbf r}}}\xspace}^i))^2 = \frac{1}{n_r} \frac{(c^i+K_D)^2}{c^i K_D}$. This is the error in concentration sensing from a single snapshot of $n_r$ receptors with simple ligand-receptor kinetics and dissociation constant $K_D$ ([@kaizu2014berg], Appendix \[app:conc\]). [Eq. ]{}\[eq:M\] assumes that cell-cell variance additively corrupts the measurement of the concentration $c({\ensuremath{{ {\mathbf r}}}\xspace})$ [*after*]{} an adaptation to the overall level of signal across the cluster $\bar{c}$. This is natural if the primary cell-to-cell variation is [*downstream*]{} of adaptation, as found to be a reasonable model in [@wang2012diverse]. To determine gradient sensing accuracy, we perform maximum likelihood estimation (MLE) of ${\ensuremath{{ {\mathbf g}}}\xspace}$ in [Eq. ]{}\[eq:M\], as in past approaches for single cell gradient sensing [@hu2011geometry]. We obtain the MLE estimator ${\ensuremath{\hat{{ {\mathbf g}}}}\xspace}$ numerically ([*Methods*]{}, Appendix \[app:mle\]), and thus the uncertainty ${\ensuremath{\sigma_{\mathbf{g}}}}^2 \equiv {\ensuremath{\left\langle |\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} - {\ensuremath{{ {\mathbf g}}}\xspace}|^2 \right\rangle}}$ ([Fig. ]{}\[fig:ccv\_illus\]b, symbols), where $\langle \cdots \rangle$ is an average over CCV and ligand-receptor binding. For fixed and roughly circular (isotropic) cluster geometry, if the concentration change across the cluster is small, $g R_\textrm{cluster} \ll 1$, ${\ensuremath{\sigma_{\mathbf{g}}}}^2$ can be approximated by assuming $\delta c^i$ is constant across the cluster, resulting in $${\ensuremath{\left\langle |\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} - {\ensuremath{{ {\mathbf g}}}\xspace}|^2 \right\rangle}} \approx \frac{2}{\chi} \left({\ensuremath{\sigma_{\Delta}}}^2 + \frac{1}{n_r} \frac{(\bar{c}+K_D)^2}{\bar{c} K_D} \right) \label{eq:combined_error}$$ Here, $\chi = \frac{1}{2}\sum_i |{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}|^2$ is a shape parameter, and ${\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}= { {\mathbf r}}^i - { {\mathbf r}}_{\textrm{cm}}$ is cell position relative to cluster center of mass. Evaluating this expression reveals that it is an excellent approximation to the numerically-obtained uncertainty (dashed lines, [Fig. ]{}\[fig:ccv\_illus\]b). The approximate expression for the uncertainty, Eq. \[eq:combined\_error\], allows us to quantify the relative contribution of receptor-ligand fluctuations and CCV to the gradient sensing error. For background concentrations $\bar{c}$ near the receptor-ligand equilibrium constant $K_D$ and for typical receptor numbers in eukaryotic cells ($n_r \sim 10^5$ [@hesselgesser1998identification; @macdonald2008heterogeneity]), $\delta c / \bar{c}$ can be smaller than 0.01. Protein concentrations, on the other hand, often vary between cells to 10%-60% of their mean [@niepel2009non] – hence we estimate ${\ensuremath{\sigma_{\Delta}}}\approx 0.1-0.6$. Thus, we expect CCV to dominate gradient sensing error and that the error from concentration sensing and receptor binding can be neglected completely if ${\ensuremath{\sigma_{\Delta}}}> 0.1$ ([Fig. ]{}\[fig:ccv\_illus\]b). [Eq. ]{}\[eq:combined\_error\] also reveals that CCV masks the impact of changing background concentration. When ${\ensuremath{\sigma_{\Delta}}}= 0$, gradient sensing is limited by ligand-receptor fluctuations, and increases as $\bar{c}$ moves away from $K_D$ ([Fig. ]{}\[fig:ccv\_illus\]d) – accuracy decreases if either few receptors are bound, or if receptors are saturated. As CCV increases, ${\ensuremath{\sigma_{\mathbf{g}}}}^2$ no longer depends strongly on $\bar{c}$ ([Fig. ]{}\[fig:ccv\_illus\]d). Finally, Eq. \[eq:combined\_error\] shows that gradient sensing accuracy depends on the shape parameter $\chi$ and, therefore, on cluster size. For hexagonally packed clusters of cells with unit spacing[^1], a cluster with $Q$ layers has $N = 1 + 3Q + 3Q^2$ cells and $\chi(Q) = (5/8) Q^4 + (5/4) Q^3 + (7/8) Q^2 + (1/4) Q$ (Appendix \[app:Q\]), i.e. $\chi(Q) \sim Q^4 \sim N^2$. Clusters of increasing size then have an error that decreases as $1/N^2$ ([Fig. ]{}\[fig:ccv\_illus\]c); [this scaling is similar to earlier results for single cells (Appendix \[app:Q\]).]{} Reducing estimation error by time-averaging {#sec:timeaverage} =========================================== If a cluster made $n$ independent measurements, it could reduce ${\ensuremath{\sigma_{\mathbf{g}}}}^2$ by a factor of $n$. In single-cell gradient sensing, independent measurements can be made by averaging over time – improving errors by a factor $\sim T/\tau_{\textrm{corr}}$, where $T$ is the averaging time, and $\tau_{\textrm{corr}}$ the measurement correlation time. At first glimpse, time averaging seems unlikely to help with CCV, when correlation times for protein levels can be longer than cell division times, reaching 48 hours in human cells [@sigal2006variability]. However, since gradient sensing bias from CCV depends on the locations of strong- and weak-signaling cells within the cluster, time averaging can be successful if it is over a time long enough for the cluster to re-arrange. This is true even if, as we initially assume, CCV biases $\Delta^i$ are time-independent. We expect gradient sensing error with time averaging, ${\ensuremath{\sigma_{\mathbf{g},T}}}^2$, will decrease by a factor of $T/\tau_r$ from ${\ensuremath{\sigma_{\mathbf{g},0}}}^2$, where $\tau_r$ is a correlation time related to cell positions ([Fig. ]{}\[fig:timeaverage\]). Is this true, and how should we define $\tau_r$? Our earlier results suggest that CCV dominates the gradient sensing error. Ligand-receptor noise will also be even less relevant in the presence of time averaging, as the receptor relaxation time (seconds to minutes [@wang2007quantifying]) is much faster than that for cluster re-arrangement (tens of minutes or longer). We therefore completely neglect ligand-receptor binding fluctuations, allowing an analytical solution for the MLE estimator ${\ensuremath{\hat{{ {\mathbf g}}}}\xspace}$ ([*Methods*]{}, Appendix \[app:mle\]). ![. a) Schematic drawing of how cell-cell rearrangement can change bias due to CCV. Shades of gray indicate measured signal $M$; a cell with strong response (marked with $X$) moves through the cluster, leading to biases in gradient estimate (blue arrow). The characteristic relaxation time for this bias is $\tau_r$ (see text). b) This leads to a link between the timescale $\tau_r$, which is a measure of the cluster’s rheology, and chemotactic accuracy ${\ensuremath{\sigma_{\mathbf{g},T}}}$ (box). c) Different re-arrangement mechanisms will depend on cluster size in different ways (see text and Appendix \[app:mechanisms\]). []{data-label="fig:timeaverage"}](rearrangement_variation_drawing){width="90mm"} How much does time-averaging reduce error? If we average the MLE estimate $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}}$ over a time $T$ by applying a kernel $K_T(t)$, i.e., we define ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}(t) \equiv \int_{-\infty}^{\infty} {\ensuremath{\hat{{ {\mathbf g}}}}\xspace}(t') K_T(t-t') dt'$ and ${\ensuremath{\sigma_{\mathbf{g},T}}}^2 \equiv {\ensuremath{\left\langle |{\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}-{ {\mathbf g}}|^2 \right\rangle}}$, we can derive (Appendix \[app:timeaverage\]) $${\ensuremath{\sigma_{\mathbf{g},T}}}^2 = {\ensuremath{\sigma_{\mathbf{g},0}}}^2 \times \int_{-\infty}^\infty \frac{d\omega}{2\pi} |K_T(\omega)|^2 C_{rr}(\omega) \label{eq:snrt}$$ where $C_{rr}(t'-t'') \equiv \langle \boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}(t')\cdot\boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}(t'') \rangle / \langle |\boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}|^2 \rangle$ is the normalized cell position-position correlation function, $C_{rr}(\omega)$ its Fourier transform, and ${\ensuremath{\sigma_{\mathbf{g},0}}}^2 = 2 {\ensuremath{\sigma_{\Delta}}}^2/\chi$ is the error in the absence of time-averaging. To derive [Eq. ]{}\[eq:snrt\], we make two approximations: 1) the cluster has a constant and isotropic shape, and 2) re-arrangement of cell positions relative to the center of mass is independent of the particular values of $\Delta$. The first approximation is not necessary, but is a useful simplification; a generalized result is given in Appendix \[app:timeaverage\]. [The second approximation assumes that averaging over CCV and averaging over cell positions are independent. This decoupling approximation is necessary to characterize cluster fluidity and mechanics separately from the details of signaling. It excludes, e.g. models where cells with larger-than-average $\Delta$ sort out from the cluster. We will discuss potential errors due to this approximation later in the paper.]{} For exponential position-position correlation functions and averaging, $C_{rr}(t) = \exp(-t/\tau_r)$ and $K_T(t) = \theta(t) \frac{1}{T} e^{-t/T}$, where $\theta(t)$ is the Heaviside step function, [Eq. ]{}\[eq:snrt\] is simple: $${\ensuremath{\sigma_{\mathbf{g},T}}}^2 = \frac{{\ensuremath{\sigma_{\mathbf{g},0}}}^2}{1+T/\tau_r} \label{eq:snrt_simp}$$ In other words, gradient sensing accuracy can be improved by taking $T/\tau_r$ independent measurements in a time $T$. Crucial in this reduction is the position-position correlation time $\tau_r$ which depends on the cluster rearrangement mechanism. Two natural mechanisms are persistent cluster rotation and neighbor re-arrangements within the cluster ([Fig. ]{}\[fig:timeaverage\]c). These mechanisms may coexist, as when cells slide past one another during cluster rotation [@rappel1999self]. $\tau_r$ can depend on cluster size; for diffusive rearrangements, we expect $\tau_r \sim R^2/D_{\textrm{eff}}$, and for persistently rotating clusters, $\tau_{\textrm{r}} \sim R/v_{\textrm{cell}}$ (Appendix \[app:mechanisms\]). We have assumed that the CCV is time-independent over our scale of interest – consistent with the long memory found in [@sigal2006variability]. If $\Delta$ changes faster than the cluster re-arranges, our results can be straightforwardly modified. Generalizations of [Eq. ]{}\[eq:snrt\] and [Eq. ]{}\[eq:snrt\_simp\] to this case are provided in Appendix \[app:timeaverage\]. {width="180mm"} Tradeoffs in collective accuracy and motility: cluster rotation {#sec:rotate} =============================================================== Our central result ([Eq. ]{}\[eq:snrt\]) shows that clusters can improve their chemotactic accuracy by changing cell positions. The simplest mechanism to do this is cluster rotation, which occurs in border cell clusters [@combedazou2016myosin] and transiently in leukocyte clusters [@malet2015collective]. When should a cluster actively rotate in order to increase its accuracy? Rotation creates an important tradeoff: more work must be put into rotating, and therefore less into crawling up the gradient [^2] For constant work of motility, the maximum speed of a cluster of radius $R$ that rotates with angular speed $\Omega$ is ([*Methods*]{}) $$v(\Omega) = v_\textrm{max}\sqrt{1 - \frac{1}{2}\Omega^2\tau_{\textrm{rot}}^2} \label{eq:rot_tradeoff_main}$$ where $\tau_\textrm{rot} = R/v_\textrm{max}$ and $v_\textrm{max}$ the maximum cluster speed absent rotation. The cluster cannot rotate faster than $\Omega_\textrm{max} = \sqrt{2}/\tau_\textrm{rot}$. If a cluster follows its best estimate ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$ with speed $v(\Omega)$ given by [Eq. ]{}\[eq:rot\_tradeoff\_main\], it can improve its velocity in the gradient direction by rotating when the averaging time $T$ is long compared with $\tau_\textrm{rot}$ ([Fig. ]{}\[fig:rot\]a) [^3] We find that the optimal rotation speed $\Omega$ that maximizes the upgradient speed depends only on the signal-to-noise ratio without rotation, ${\ensuremath{\textrm{SNR}}}_0 \equiv \frac{1}{2}g^2/\sigma_{{\ensuremath{{ {\mathbf g}}}\xspace},0}^2$ and $T/\tau_\textrm{rot}$ ([Fig. ]{}\[fig:rot\]b, [*Methods*]{}). Mammalian cells have speeds in the range of microns/minute and radii of tens of microns, so to benefit from averaging ($T \gtrsim\tau_\textrm{rot}$), $T$ must be longer than tens of minutes. The timescale $\tau_\textrm{rot}$ and the signal-to-noise-ratio ${\ensuremath{\textrm{SNR}}}_0$ both depend on cluster size – larger clusters with more cells are both better gradient sensors and more difficult to drive to large angular speeds. As a consequence, the optimal $\Omega$ is highly cluster-size-dependent: as cluster size decreases, there is a continuous transition to nonzero optimal $\Omega$ if $T$ is sufficiently long ([Fig. ]{}\[fig:rot\]c). {width="170mm"} Linking chemotactic accuracy and fluidity {#sec:fluidity} ========================================= Clusters with more cell rearrangement are more accurate by [Eq. ]{}\[eq:snrt\_simp\]. To further quantify the consequences of cell rearrrangements, we model a cluster of cells as self-propelled particles that follow the cluster estimate ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$ with a noise characterized by angular diffusion $D_\psi$ and with cell-cell connections modeled as springs of strength $\kappa$ between Delaunay neighbors ([*Methods*]{}). We emphasize that the angular diffusion parameterized by $D_\psi$ is an additional source of noise: as $D_\psi$ increases, cells are less accurate in following the cluster’s estimate of the gradient. These two parameters are systematically varied to study the effects of cluster fluidity on chemotactic accuracy. Cluster fluidity improves cluster chemotaxis -------------------------------------------- Increasing cell-cell adhesion $\kappa$ makes clusters more ordered, moving between fluidlike and crystalline states ([Fig. ]{}\[fig:kappa\]a). As a consequence, rearrangement slows significantly ([Fig. ]{}\[fig:kappa\]c) with $\tau_r \sim \exp(\kappa/2)$ ($C_{rr}(t)$ is single-exponential). Cluster structure and size change when clusters fluidize ([Fig. ]{}\[fig:kappa\]a), which may in principle affect the shape parameter $\chi$, which also strongly affects the chemotactic accuracy ([Eq. ]{}\[eq:combined\_error\]). However, in our simulations $\chi$ is not significantly dependent on $\kappa$, changing by under 10% ([Fig. ]{}\[fig:kappa\]d). Averaging time $T$ also has only a weak effect on cluster shape and dynamics – changes in $\tau_r$ and $\chi$ when the averaging time $T$ is increased by orders of magnitude are small ([Fig. ]{}\[fig:kappa\]). This is consistent with our assumption decoupling the gradient estimate and cell rearrangements, suggesting clusters should obey the bound [Eq. ]{}\[eq:snrt\]. We can, using the results of Section \[sec:timeaverage\], predict the cluster chemotactic index, $\textrm{CI} \equiv \langle V_x / |{ {\mathbf V}}| \rangle$, where ${ {\mathbf V}}$ is the cluster velocity. Assuming ${ {\mathbf V}} \sim {\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$, we can compute $\textrm{CI}$ from ${\ensuremath{\sigma_{\mathbf{g},T}}}$ given by [Eq. ]{}\[eq:snrt\_simp\] ([*Methods*]{}). This requires parameters $\tau_r$ and $\chi$ (measured from simulations), and ${ {\mathbf g}}$, $T$, and ${\ensuremath{\sigma_{\Delta}}}$ (known). [We note that our approach, which extracts $\tau_r$ and $\chi$ from cell trajectories, could also be applied to experimental data; in that case, ${ {\mathbf g}}$ would still be known, but the extent of time-averaging ($T$) and the error due to CCV (${\ensuremath{\sigma_{\Delta}}}$) would have to be determined by fitting to the data.]{} This prediction should be an upper bound to the measured $\textrm{CI}$, because our model includes additional noise beyond the assumptions of [Eq. ]{}\[eq:snrt\_simp\], via $D_\psi$. As expected, cluster CI decreases significantly as clusters solidify and the relaxation time $\tau_r$ increases. The simulation data qualitatively follows the predicted upper bound ([Fig. ]{}\[fig:kappa\]b). When the averaging time $T$ is reduced below typical relaxation times, the CI significantly decreases. In addition, for this short time-averaging, changing cluster stiffness no longer strongly affects CI. Increasing single-cell stochasticity can increase cluster accuracy ------------------------------------------------------------------ {width="180mm"} Any mechanism that fluidizes the cluster can decrease the correlation time $\tau_r$. Because of this, [*increasing*]{} noise can improve cluster chemotactic accuracy ([Fig. ]{}\[fig:Dpsi\]). We increase single-cell angular noise $D_\psi$, and see an initial sharp increase in cluster chemotactic index as $D_\psi > 0$ ([Fig. ]{}\[fig:Dpsi\]b). At larger values of $D_\psi$, cluster CI decreases below the bound set by [Eq. ]{}\[eq:snrt\_simp\], as the additional noise added degrades the gradient-following behavior. Without significant time-averaging ($T = 0.2$), additional noise primarily impedes chemotactic accuracy. Why can extra noise $D_\psi$ help sensing? For $D_\psi = 0$, all cells follow the best estimate ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$ precisely, leading to an ordered cluster ([Fig. ]{}\[fig:Dpsi\]a) with $\tau_r$ effectively infinite. As $D_\psi$ is increased, the cluster fluidizes, and the relaxation time decreases strongly ([Fig. ]{}\[fig:Dpsi\]c) resulting in more independent measurements. As in [Fig. ]{}\[fig:kappa\], this fluidization is only relevant if the averaging time $T$ exceeds the relaxation time, so when $T = 0.2$, the effect of increasing $D_\psi$ is solely detrimental to chemotaxis. [In deriving our bound, we made two key approximations: cluster isotropy and decoupling. These approximations are exact for the rigid cluster rotation in Sec. \[sec:rotate\], but only approximate for this self-propelled particle model. As a consequence, at small $D_\psi$, simulated clusters have chemotactic indices slightly exceeding our predictions ([Fig. ]{}\[fig:Dpsi\]b). This error likely arises from emergent couplings between cluster shape and $\Delta_i$ – clusters may spread perpendicular to ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$, weakening the decoupling approximation ([Fig. ]{}\[fig:Dpsi\]a). The approximation of cluster isotropy can be removed (Appendix \[app:timeaverage\]), and does not resolve the violation of the bound ([Fig. ]{}\[fig:Dpsi\]b). Despite this potential error source, the model captures CI variation over a broad range of parameters (Appendix \[app:params\]).]{} Discussion ========== Our study results in several predictions and suggestions for experiments that investigate collective chemotaxis. For example, we predict that, when CCV is large, gradient sensing error is insensitive to background concentration ([Fig. ]{}\[fig:ccv\_illus\]c). This is consistent with recent measurements on developing organoids that show that the up-gradient bias is not strongly dependent on mean concentration [@ellison2016cell], though in contrast with results on lymphocyte clusters [@malet2015collective]. Furthermore, if CCV limits collective chemotaxis, clusters gradient sensing [*in vivo*]{} should have tightly regulated expression of proteins relevant to the signal response. Interestingly, measurements of zebrafish posterior lateral line primordium [@venkiteswaran2013generation] show tightly-controlled Sdf1 signaling, as measured by Cxcr4b internalization, suggesting that CCV may be small enough to allow for accurate gradient sensing. We also show that there is a direct link between fluidity and chemotaxis as shown by [Eq. ]{}\[eq:snrt\]. Verifying this expression in experiments requires simultaneous measurement of several quantities, including cluster size, cluster re-arrangement, and signal gradient. Therefore, care has to be taken when modifying experimental conditions as these might change several of these quantities simultaneously. Altering adhesion, for example, changes both cluster fluidity and spreading as shown in a recent study using neural crest clusters [@kuriyama2014vivo], creating a confounding factor. Nevertheless, these types of experiments may be successful in setting bounds on possible time-averaging and the link between fluidity and chemotaxis. Our results suggest that many recent experiments may need reinterpretation. Measured chemotactic accuracies can depend on cluster size [@theveneau2010collective; @malet2015collective; @cai2016modeling]; these results have been modeled without time averaging or CCV [@camley2016emergent; @malet2015collective; @cai2016modeling; @camley2016collective]. Our results show that rearrangement times $\tau_r$ also influence chemotaxis – and that $\tau_r$ depends on cluster size. Cluster relaxation dynamics are therefore an unexplored potential issue for interpreting collective gradient sensing experiments. Essential in the reduction of gradient sensing errors due to CCV is the existence of a biochemical or mechanical memory that can perform a time average over tens of minutes. There are several possibilities. First, memory could be external to the cluster – e.g. stored in extracellular matrix structure, or a long-lived trail [@lim2015neutrophil]. Secondly, supracellular structures like actin cables influence cell protrusion and leader cell formation [@reffay2014interplay], suggesting that collective directional memory could be kept by regulating actin cable formation and maintenance. Third, memory may be kept at the individual cell level by cells attempting to estimate their own bias level $\Delta^i$ and compensating for it. This contrasts with our straightforward average of the collective estimate ${\ensuremath{\hat{{ {\mathbf g}}}}\xspace}$, but could be an important alternative mechanism. Our results are critical for understanding the ubiquitous phenomenon of collective gradient sensing. The importance of CCV provides a new design principle: CCV must either be tightly controlled or mitigated by time-averaging. We also established a surprising link between a central [*mechanical*]{} property of a cluster – its rheology – and its sensing ability. This connects mechanical transitions like unjamming [@bi2016motility] to sensing, opening up new areas of study. In addition, our results show cluster accuracy depends strongly on cluster rearrangement mechanism. Finally, our results show that noise in cell motility can be beneficial for collective sensing. Methods ======= Maximum likelihood estimation of gradient direction in the presence of cell-cell variation and ligand-receptor noise {#maximum-likelihood-estimation-of-gradient-direction-in-the-presence-of-cell-cell-variation-and-ligand-receptor-noise .unnumbered} -------------------------------------------------------------------------------------------------------------------- We compute the maximum likelihood estimate of gradient direction given the measured signal at cell $i$, $M^i$, given by [Eq. ]{}\[eq:M\]. If the cluster of cells is in a shallow linear gradient, with concentration $c_0$ at the cluster’s center of mass ${\ensuremath{{ {\mathbf r}}}\xspace}_{\textrm{cm}} = N^{-1} \sum_i {\ensuremath{{ {\mathbf r}}}\xspace}^i$, then $c({\ensuremath{{ {\mathbf r}}}\xspace}) = c_0 \left[1 + { {\mathbf g}} \cdot ({\ensuremath{{ {\mathbf r}}}\xspace}-{\ensuremath{{ {\mathbf r}}}\xspace}_{\textrm{cm}}) \right]$ and thus $\bar{c} = c_0$. $M_i$ is then $M^i = 1 + {\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}+ (\delta c^i/c_0) \eta^i + \Delta^i$ with ${\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}= {\ensuremath{{ {\mathbf r}}}\xspace}-{\ensuremath{{ {\mathbf r}}}\xspace}_{\textrm{cm}}$ and $(\delta c^i/c^i)^2 = \frac{1}{n_r} \frac{(c^i+K_D)^2}{c^i K_D}$, i.e. $(\delta c^i/c_0)^2 = \frac{1}{n_r} (1+{\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}+K_D/c_0)^2\frac{1+{\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}}{K_D/c_0}$. $\Delta^i$ are uncorrelated between cells, with a Gaussian distribution of zero mean and standard deviation [$\sigma_{\Delta}$]{}, i.e. ${\ensuremath{\left\langle \Delta^i\Delta^j \right\rangle}} = {\ensuremath{\sigma_{\Delta}}}^2 \delta^{ij}$ with $\delta^{ij}$ the Kronecker delta function. As $\eta^i$ and $\Delta^i$ are both Gaussian, the sum of these variables is also Gaussian, and the likelihood of observing a configuration of measured signals $\{M^i\}$ as ${\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}}) = P({\ensuremath{\{M^{i}\}\xspace}}| {\ensuremath{{ {\mathbf g}}}\xspace})$, where $P({\ensuremath{\{M^{i}\}\xspace}}| {\ensuremath{{ {\mathbf g}}}\xspace})$ is the probability density function of observing the configuration [$\{M^{i}\}\xspace$]{}given parameters ${\ensuremath{{ {\mathbf g}}}\xspace}$. $${\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}}) = \prod_{i} \frac{1}{\sqrt{2\pi h^i}} \exp\left[-\frac{(M^i - \mu^i)^2}{2 h^i} \right]$$ where $\mu^i = 1 + {\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}$ is the mean value of $M^i$ and $h^i = (\delta c^i / c_0)^2 + {\ensuremath{\sigma_{\Delta}}}^2$ its variance. We want to apply the method of maximum likelihood by finding the gradient parameters $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}}$ that maximize this likelihood, i.e. $$\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} = \textrm{arg} \, \max\limits_{{\ensuremath{{ {\mathbf g}}}\xspace}} {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}})$$ However, because of the complex dependence of $h^i$ on the gradient ${\ensuremath{{ {\mathbf g}}}\xspace}$, this is not possible analytically. We perform this optimization numerically using a Nelder-Mead method (Matlab’s fminsearch), with an initial guess set by the maximum for $n_r \to \infty$ (i.e. neglecting concentration sensing noise), which can be found exactly. For numerical convenience, we maximize the log-likelihood $\ln {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}})$, $\ln {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}}) = -\frac{1}{2}\sum_{i} \ln h^i - \sum_i \frac{(M^i - \mu^i)^2}{2 h^i}$ up to an additive constant. In the limit of $n_r \to \infty$ (neglecting concentration noise), our model becomes a simple linear regression, and the log likelihood can be maximized analytically by finding [$\hat{{ {\mathbf g}}}$]{}such that $\partial_{{\ensuremath{{ {\mathbf g}}}\xspace}} \ln {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace}; {\ensuremath{\{M^{i}\}\xspace}})|_{{\ensuremath{\hat{{ {\mathbf g}}}}\xspace}} = 0$ (Appendix \[app:mle\]). The result is $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} = \mathcal{A}^{-1}\cdot \sum_i M^i {\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}$, where $\mathcal{A}_{\alpha\beta} \equiv \sum_i\delta r^i_\alpha \delta r^i_\beta$. This estimator is simplest in the limit of roughly circular (isotropic) clusters, where $\sum_i (\delta x^i)^2 \approx \sum_i (\delta y^i)^2 \gg \sum_i \delta x^i \delta y^i$. In this case, $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} = (\chi^{-1}) \sum_i M^i{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}$ where $\chi = \frac{1}{2}\sum_i |{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}|^2$. Cluster rotation dynamics {#cluster-rotation-dynamics .unnumbered} ------------------------- How much speed does a cluster lose by rotating? One possibility is to assume the power expended in generating motility is constant. Consider a circular cluster propelling itself over a surface, with the cells having velocity ${ {\mathbf v}}({\ensuremath{{ {\mathbf r}}}\xspace})$; we expect that the frictional force per unit area between the cluster and substrate will be ${ {\mathbf f}}_{\textrm{drag}} = -\xi { {\mathbf v}}$, where $\xi$ is a friction coefficient with the substrate. If all of the power available for motility is going into driving the cluster over the substrate, then we can write: $P = -\int d^2 r \, { {\mathbf v}}\cdot{ {\mathbf f}}_\textrm{drag} = \xi \int d^2 r |{ {\mathbf v}}|^2$. If the cluster is traveling as a rigid, circular cluster with its maximum possible velocity, ${ {\mathbf v}} = v_{\textrm{max}} \hat{{ {\mathbf x}}}$ then $P = \xi \pi R^2 v_\textrm{max}^2 \equiv \gamma_t v_\textrm{max}^2$ where $\gamma_t = \xi \pi R^2$ is the translational drag coefficient of the cluster. If, instead, the cluster puts its entire power into rigid-body rotation with ${ {\mathbf v}}({\ensuremath{{ {\mathbf r}}}\xspace}) = \Omega_{\textrm{max}} r (-\sin \theta,\cos \theta)$ (in polar coordinates), then $P = \xi \Omega_\textrm{max}^2 \int d^2 r r^2 = \xi\frac{\pi}{2} R^4 \Omega_{\textrm{max}}^2 \equiv \gamma_r \Omega_\textrm{max}^2$ where $\gamma_r = \frac{\xi \pi}{2} R^4$ is the rotational drag coefficient of the cluster. In general, the power dissipated if the cluster is moving rigidly with velocity ${ {\mathbf v}}$ and angular speed $\Omega$ is $P = \gamma_t v^2 + \gamma_r \Omega^2$ and hence, we find that the speed $v(\Omega$) that a cluster rotating with angular velocity $\Omega$ is able to travel to obtain is $$v(\Omega) = \sqrt{v_\textrm{max}^2 - \frac{\gamma_r}{\gamma_t} \Omega^2} \label{eq:tradeoff}$$ This quantifies one reasonable tradeoff between speed and angular velocity for a cluster. If the power available for cell motility is a small amount of the cell’s energy budget [@purcell1977life; @flamholz2014quantified; @katsu2009substantial], other tradeoffs may be more important and additional modeling will be necessary. We consider a circular cluster traveling towards its best estimate of the gradient with speed $v(\Omega)$ given by [Eq. ]{}\[eq:tradeoff\], and traveling in the direction of the estimator ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$. We can then determine when the cluster maximizes its mean velocity in the direction of the increasing gradient, which we choose to be $x$, ${\ensuremath{\left\langle v_x \right\rangle}}$, as a function of $\Omega$. This average is $${\ensuremath{\left\langle v_x \right\rangle}} = \sqrt{v_\textrm{max}^2 - \frac{\gamma_r}{\gamma_t} \Omega^2} \times {\ensuremath{\left\langle \frac{{\ensuremath{\hat{{ {\mathbf g}}}}_{T,x}\xspace}}{|{\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}|} \right\rangle}}$$ We know from our results above and in Appendix \[app:timeaverage\] that, for a fixed configuration, ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$ has a Gaussian distribution with mean ${ {\mathbf g}} = g \hat{{ {\mathbf x}}}$ and variance given by [Eq. ]{}\[eq:snrt\]. The average of $\frac{{\ensuremath{\hat{{ {\mathbf g}}}}_{T,x}\xspace}}{|{\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}|}$ depends only on the signal-to-noise ratio ${\ensuremath{\textrm{SNR}}}_T \equiv \frac{1}{2}(g^2/\sigma_{{ {\mathbf g}},T}^2)$, with ${\ensuremath{\left\langle \frac{{\ensuremath{\hat{{ {\mathbf g}}}}_{T,x}\xspace}}{|{\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}|} \right\rangle}} = C({\ensuremath{\textrm{SNR}}}_T^{-1/2})$. Given the angular velocity $\Omega$, we can work out the distribution of ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$ by [Eq. ]{}\[eq:snrt\]. We know ${\ensuremath{{ {\mathbf r}}}\xspace}(t)\cdot{\ensuremath{{ {\mathbf r}}}\xspace}(0) = |{\ensuremath{{ {\mathbf r}}}\xspace}(0)|^2\cos \Omega t$, and hence $C_{rr}(t) = \cos \Omega t$ and its Fourier transform $C_{rr}(\omega) = \pi \left[\delta(\omega-\Omega) + \delta(\omega+\Omega)\right]$, and thus ${\ensuremath{\sigma_{\mathbf{g},T}}}^2 = {\ensuremath{\sigma_{\mathbf{g},0}}}^2 \times |K_T(\Omega)|^2 = {\ensuremath{\sigma_{\mathbf{g},0}}}^2/(1+\Omega^2T^2)$. By rescaling to unitless parameters, we then find that $${\ensuremath{\left\langle v_x \right\rangle}}/v_\textrm{max} = \sqrt{1 - \frac{1}{2} \omega^2} \times C(\left[{\ensuremath{\textrm{SNR}}}_0 (1+\omega^2\widetilde{T}^2)\right]^{-1/2})$$ where $\omega = \Omega R / v_{\textrm{max}}$ is the unitless angular velocity, ${\ensuremath{\textrm{SNR}}}_0 = {\ensuremath{\sigma_{\Delta}}}^{-2} g^2 \chi$ is the usual SNR with no averaging, and $\widetilde{T} = T v_{\textrm{max}}/R$ is the ratio of the averaging time to the characteristic rotational time $R/v_{\textrm{max}}$, and $C(\sigma)$ is the function given by [Eq. ]{}\[eq:csigma\]. When ${\ensuremath{\textrm{SNR}}}_0$ is sufficiently small, and $\widetilde{T}$ sufficiently large, ${\ensuremath{\left\langle v_x \right\rangle}}/v_\textrm{max}$ has a maximum at finite $\omega$ ([Fig. ]{}\[fig:rot\]). In the limit of low [$\textrm{SNR}$]{}, $C(\sigma) \approx \sqrt{\pi/8} \sigma^{-1}$, and we find ${\ensuremath{\left\langle v_x \right\rangle}}/v_\textrm{max}$ is maximized by $\omega = \pm\sqrt{1-\frac{1}{2}\widetilde{T}^{-2}}$ when $\widetilde{T} > 1/\sqrt{2}$ and $\omega = 0$ otherwise. For the large ${\ensuremath{\textrm{SNR}}}$ limit, $C(\sigma) \approx 1-\sigma^2/2$ and rotation will increase the mean velocity in the direction of the gradient when $\widetilde{T}^2 > {\ensuremath{\textrm{SNR}}}_0/2-1/4$. More generally, it is possible to find the value of $\omega$ that maximizes ${\ensuremath{\left\langle v_x \right\rangle}}$ numerically. We show the complete phase diagram in [Fig. ]{}\[fig:rot\]b. Particle-based model of collective cell migration {#particle-based-model-of-collective-cell-migration .unnumbered} ------------------------------------------------- We use a minimal model of collective cell migration, describing cells as self-propelled particles connected by springs: $$\begin{aligned} \label{eq:velocity} \frac{d}{dt}{ {\mathbf r}}^i &= { {\mathbf p}}^i + \sum_{j\sim i} { {\mathbf F}}^{ij} \\ { {\mathbf p}}^i &= (\cos\theta^i,\sin\theta^i) \\ \theta^i &= \arctan(\hat{g}_{T,y}/\hat{g}_{T,x}) +\psi^i \\ \frac{d}{dt} \psi^i &= -\tau^{-1} \sin \psi^i+ \sqrt{2 D_\psi} \xi^i(t) \label{eq:psi}\end{aligned}$$ where $\xi^i(t)$ is a Gaussian Langevin noise with zero mean and $\langle \xi^i(t)\xi^j(t') \rangle = \delta^{ij}\delta(t-t')$, with $\delta^{ij}$ the Kronecker delta. In this model, the orientation of an individual cell $\theta^i$ is the cluster’s best estimate of the gradient direction, $\arctan(\hat{g}_{T,y}/\hat{g}_{T,x})$, plus a noise $\psi^i$ which varies from cell to cell. $\tau$ here controls the persistence of this noise, and $D_\psi$ its amplitude; when $D_\psi$ is increased, each individual cell is worse at following the estimate ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$. The cell-cell forces are $${ {\mathbf F}}^{ij} = -\kappa (|{\ensuremath{d^{ij}}\xspace}-\ell|){\ensuremath{\hat{{ {\mathbf r}}}^{ij}}\xspace}$$ where ${\ensuremath{d^{ij}}\xspace}= |{ {\mathbf r}}^i-{ {\mathbf r}}^j|$ and ${\ensuremath{\hat{{ {\mathbf r}}}^{ij}}\xspace}= ({ {\mathbf r}}^i-{ {\mathbf r}}^j)/{\ensuremath{d^{ij}}\xspace}$. The forces are only between neighboring cells $j\sim i$, where we define neighboring cells as any cells connected by the Delaunay triangulation of the cell centers ([Fig. ]{}\[fig:kappa\]a); this approach resembles that of [@meineke2001cell]. We use the Euler-Maruyama method to integrate Eqns. \[eq:velocity\]-\[eq:psi\]. Simulation units {#simulation-units .unnumbered} ---------------- We have chosen our parameters in the simulation and throughout the paper to be measured in units where the equilibrium cell-cell separation $\ell = 1$ (i.e. the cell diameter is unity), and the velocity of a single cell in the absence of cell-cell forces ${ {\mathbf v}} = { {\mathbf p}} = (\cos \theta,\sin\theta)$ has unit magnitude. For, e.g. neural crest cells, the cell diameters are of order 20 microns, and the cell speeds on the order of microns/minute – so a unitless time of $T$ corresponds to roughly $\textrm{20 minutes} \times T$ in real time. However, cell size and speed varies strongly from cell type to cell type, so we prefer to present these results in their unitless form so that they can be more easily converted. Computing chemotactic indices {#computing-chemotactic-indices .unnumbered} ----------------------------- ![. $C(\sigma)$ plotted numerically from definition in [Eq. ]{}\[eq:csigma\][]{data-label="fig:csigma"}](cos_sigma){width="90mm"} If we use the maximum likelihood method to make an estimate for the direction in which the cell moves, how do we translate between the uncertainty ${\ensuremath{\sigma_{\mathbf{g}}}}^2$ and the distribution of velocities? We found that the MLE estimate for the gradient is ${\ensuremath{\hat{{ {\mathbf g}}}}\xspace}= { {\mathbf g}} + { {\mathbf \Lambda}}$, with ${ {\mathbf \Lambda}}$ a Gaussian random variable with zero mean and variance ${\ensuremath{\left\langle \Lambda_x^2 \right\rangle}} = {\ensuremath{\left\langle \Lambda_y^2 \right\rangle}} = {\ensuremath{\sigma_{\mathbf{g}}}}^2/2$ – and similar results for the time-average ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$. One measure of this estimate’s accuracy is the [*instantaneous chemotactic index*]{} - or the cosine of the angle between the estimate and the gradient direction. To compute this, if ${ {\mathbf g}} = g \hat{{ {\mathbf x}}}$ without loss of generality, we find $$\begin{aligned} {\ensuremath{\left\langle \frac{{\ensuremath{\hat{{ {\mathbf g}}}}\xspace}_x}{|{\ensuremath{\hat{{ {\mathbf g}}}}\xspace}|} \right\rangle}} &= {\ensuremath{\left\langle \frac{g + \Lambda_x}{(g+\Lambda_x)^2 + \Lambda_y^2} \right\rangle}} \\ &= \int \frac{dx dy}{2\pi\sigma} \frac{1 + x}{\left[(1+x)^2+y^2\right]^{1/2}} e^{-\frac{(x^2+y^2)}{2\sigma^2}} \\ &\equiv C(\sigma) \label{eq:csigma}\end{aligned}$$ where $\sigma = {\ensuremath{\textrm{SNR}}}^{-1/2}$, with ${\ensuremath{\textrm{SNR}}}= \frac{1}{2} (g^2/{\ensuremath{\sigma_{\mathbf{g}}}}^2)$. These results carry over naturally to the time-averaged case if ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$ remains Gaussian – we find ${\ensuremath{\left\langle \frac{{\ensuremath{\hat{{ {\mathbf g}}}}\xspace}_{T,x}}{|{\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}|} \right\rangle}} = C({\ensuremath{\textrm{SNR}}}_T^{-1/2})$. The integral for $C(\sigma)$ can’t be solved analytically, but we can find asymptotic forms for $C(\sigma)$ or evaluate it numerically. For $\sigma \gg 1$, we find $C(\sigma) \approx \sqrt{\pi/8} \sigma^{-1}$, and $C(\sigma) \approx 1 - \frac{1}{2} \sigma^2$ for $\sigma \ll 1$. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Albert Bae and Monica Skoge for useful discussions, and many scientists from the Gordon Research Conference on Directed Cell Motility for interesting questions and reference suggestions. BAC also thanks Kristen Flowers for several useful suggestions. This work was supported by NIH Grant No. P01 GM078586. [52]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****,  ()]{} @noop [**** ()]{} @noop [****,  ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****, ()]{} @noop [ ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [ ()]{} @noop [ ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [ ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [ ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [SI Appendix]{} Review of concentration sensing accuracy {#app:conc} ======================================== With simple ligand-receptor kinetics, i.e. an on-rate of $k_\textrm{on} = k_+ c$ and an off-rate of $k_\textrm{off} = k_i$, the mean probability that each receptor will be occupied is $P_\textrm{on} = k_\textrm{on}/(k_\textrm{on}+k_\textrm{off}) = c({\ensuremath{{ {\mathbf r}}}\xspace})/(c({\ensuremath{{ {\mathbf r}}}\xspace})+K_D)$, where $K_D = k_-/k_+$ is the dissociation constant. The variance in the occupation probability for an individual receptor is then $P_\textrm{on} - P_\textrm{on}^2 = c({\ensuremath{{ {\mathbf r}}}\xspace}) K_D / (c({\ensuremath{{ {\mathbf r}}}\xspace})+K_D)^2$. By the central limit theorem, as the number of receptors $n_r$ becomes large, the number of [*occupied*]{} receptors on a cell will be a Gaussian distribution with mean $\bar{n} = n_r c({\ensuremath{{ {\mathbf r}}}\xspace})/(c({\ensuremath{{ {\mathbf r}}}\xspace})+K_D)$ and variance $\delta n^2 = n_r c({\ensuremath{{ {\mathbf r}}}\xspace}) K_D / (c({\ensuremath{{ {\mathbf r}}}\xspace})+K_D)^2$. Translating this number of occupied receptors into an uncertainty in the local concentration via $\delta c = \frac{dc}{d\bar{n}} \delta n = \frac{(c+K_D)^2}{K_D n_r} \delta n$ [@kaizu2014berg], we find $(\delta c / c)^2 = \frac{1}{n_r} \frac{(c+K_D)^2}{c K_D}$. Maximum likelihood estimates of gradient direction via collective guidance in the presence of cell-cell variation and ligand-receptor noise {#app:mle} =========================================================================================================================================== We begin with our model for the measured signal at cell $i$, $M^i$: $$M^i = \left[c({\ensuremath{{ {\mathbf r}}}\xspace}^i) + \delta c^i \eta^i\right]/\bar{c} + \Delta^i$$ where $\bar{c}$ is the mean concentration over the cluster of cells, $\bar{c} = N^{-1} \sum_i c({\ensuremath{{ {\mathbf r}}}\xspace}^i)$. If we assume that the cluster of cells is in a shallow linear gradient, with the concentration measured at the cluster’s center of mass ${\ensuremath{{ {\mathbf r}}}\xspace}_{\textrm{cm}} = N^{-1} \sum_i {\ensuremath{{ {\mathbf r}}}\xspace}^i$ being $c_0$, we have $c({\ensuremath{{ {\mathbf r}}}\xspace}) = c_0 \left[1 + { {\mathbf g}} \cdot ({\ensuremath{{ {\mathbf r}}}\xspace}-{\ensuremath{{ {\mathbf r}}}\xspace}_{\textrm{cm}}) \right]$ and thus $\bar{c} = c_0$. We can then write the measured signal $M^i$ as $$M^i = 1 + {\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}+ (\delta c^i/c_0) \eta^i + \Delta^i$$ with ${\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}= {\ensuremath{{ {\mathbf r}}}\xspace}-{\ensuremath{{ {\mathbf r}}}\xspace}_{\textrm{cm}}$ and $(\delta c^i/c^i)^2 = \frac{1}{n_r} \frac{(c^i+K_D)^2}{c^i K_D}$, i.e. $(\delta c^i/c_0)^2 = \frac{1}{n_r} (1+{\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}+K_D/c_0)^2\frac{1+{\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}}{K_D/c_0}$. We have assumed that $\Delta^i$ are uncorrelated between cells, with a Gaussian distribution of zero mean and standard deviation [$\sigma_{\Delta}$]{}, i.e. ${\ensuremath{\left\langle \Delta^i\Delta^j \right\rangle}} = {\ensuremath{\sigma_{\Delta}}}^2 \delta^{ij}$ with $\delta^{ij}$ the Kronecker delta. $M^i$, as the sum of the Gaussian variables $\eta^i$ and $\Delta^i$, is also Gaussian, and we can then write the likelihood of observing a configuration of measured signals $\{M^i\}$ as ${\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}}) = P({\ensuremath{\{M^{i}\}\xspace}}| {\ensuremath{{ {\mathbf g}}}\xspace})$, where $P({\ensuremath{\{M^{i}\}\xspace}}| {\ensuremath{{ {\mathbf g}}}\xspace})$ is the probability density function of observing the configuration [$\{M^{i}\}\xspace$]{}given parameters ${\ensuremath{{ {\mathbf g}}}\xspace}$. $${\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}}) = \prod_{i} \frac{1}{\sqrt{2\pi h^i}} \exp\left[-\frac{(M^i - \mu^i)^2}{2 h^i} \right]$$ where $\mu^i = 1 + {\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}$ is the mean value of $M^i$ and $h^i = (\delta c^i / c_0)^2 + {\ensuremath{\sigma_{\Delta}}}^2$ its variance. We want to apply the method of maximum likelihood by finding the gradient parameters $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}}$ that maximize this likelihood, i.e. $$\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} = \textrm{arg} \, \max\limits_{{\ensuremath{{ {\mathbf g}}}\xspace}} {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}})$$ However, because of the complex dependence of $h^i$ on the gradient ${\ensuremath{{ {\mathbf g}}}\xspace}$, analytically maximizing the likelihood is intractable. We instead perform this optimization numerically using a Nelder-Mead method (Matlab’s fminsearch), with an initial guess set by the maximum for $n_r \to \infty$ (i.e. neglecting concentration sensing noise), given by [Eq. ]{}\[eq:ghat\_simple\]. For numerical stability and convenience, we will usually instead maximize the log-likelihood $\ln {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}})$, which is $$\ln {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}}) = -\frac{1}{2}\sum_{i} \ln h^i - \sum_i \frac{(M^i - \mu^i)^2}{2 h^i}$$ up to an additive constant. In the limit of $n_r \to \infty$ (neglecting concentration noise), our model becomes a simple linear regression, and the log likelihood can be maximized analytically by finding [$\hat{{ {\mathbf g}}}$]{}such that $\partial_{{\ensuremath{{ {\mathbf g}}}\xspace}} \ln {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace}; {\ensuremath{\{M^{i}\}\xspace}})|_{{\ensuremath{\hat{{ {\mathbf g}}}}\xspace}} = 0$. This straightforwardly yields $$\sum_i (M^i - 1 - {\ensuremath{\hat{{ {\mathbf g}}}}\xspace}\cdot {\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}){\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}= 0$$ or, if we write ${\ensuremath{\hat{{ {\mathbf g}}}}\xspace}= {\ensuremath{\hat{g}_x}\xspace}\hat{{ {\mathbf x}}} + {\ensuremath{\hat{g}_y}\xspace}\hat{{ {\mathbf y}}}$, $$\nonumber \begin{pmatrix} \sum_{i} (\delta x^i)^2 & \sum_i \delta x^i \delta y^i \\ \sum_i \delta x^i \delta y^i & \sum_i (\delta y^i)^2 \end{pmatrix} \begin{pmatrix} {\ensuremath{\hat{g}_x}\xspace}\\ {\ensuremath{\hat{g}_y}\xspace}\end{pmatrix} = \begin{pmatrix} \sum_i (M^i - 1) \delta x^i \\ \sum_i (M^i - 1) \delta y^i \end{pmatrix}$$ We define $\mathcal{A}$ to be a matrix with elements $\mathcal{A}_{\alpha\beta} = \sum_i \delta r_\alpha \delta r_\beta$, where $\alpha,\beta$ run over the Cartesian coordinates $x,y$, $$\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} = (\mathcal{A}^{-1})\cdot \sum_i (M^i - 1) {\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}\label{eq:ghat_simple}$$ where we note that, as $\sum_i {\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}= 0$, we can also simply use $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} = (\mathcal{A}^{-1})\cdot \sum_i M^i{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}$. These estimators are simpler for roughly symmetric clusters, where $\sum_i (\delta x^i)^2 \approx \sum_i (\delta y^i)^2 \gg \sum_i \delta x^i \delta y^i$. In this case, $\mathcal{A}_{\alpha\beta} \approx \chi \delta_{\alpha\beta}$, where $\chi = \frac{1}{2}\sum_i |{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}|^2$ and $\delta_{\alpha\beta}$ is the Kronecker delta, and $$\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} = (\chi^{-1}) \sum_i (M^i - 1) {\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}$$ We can also compute the asymptotic covariance of these estimators, which arises from the Fisher information matrix $$\begin{aligned} \mathcal{I}_{\alpha\beta} &= {\ensuremath{\left\langle \left(\frac{\partial \ln {\ensuremath{\mathcal{L}}\xspace}}{\partial g_{\alpha}}\right) \left(\frac{\partial \ln {\ensuremath{\mathcal{L}}\xspace}}{\partial g_{\beta}}\right) \right\rangle}}\\ &= -{\ensuremath{\left\langle \frac{\partial^2 \ln {\ensuremath{\mathcal{L}}\xspace}}{\partial g_{\alpha} \partial g_{\beta}} \right\rangle}} \\ &= {\ensuremath{\sigma_{\Delta}}}^{-2} \sum_i \delta r^i_\alpha \delta r^i_\beta = {\ensuremath{\sigma_{\Delta}}}^{-2} \mathcal{A}_{\alpha\beta}\end{aligned}$$ where we use ${\ensuremath{\left\langle \cdots \right\rangle}}$ to indicate the average over the cell-to-cell systematic errors $\Delta_i$. The Fisher information matrix controls the variance of our maximum-likelihood estimator [@kay1993fundamentals], which is given by $$\begin{aligned} {\ensuremath{\left\langle ({\ensuremath{\hat{{ {\mathbf g}}}}\xspace}- { {\mathbf g}})_\alpha ({\ensuremath{\hat{{ {\mathbf g}}}}\xspace}- { {\mathbf g}})_\beta \right\rangle}} &= \left(\mathcal{I}^{-1}\right)_{\alpha\beta} \\ &= {\ensuremath{\sigma_{\Delta}}}^2 (\mathcal{A}^{-1})_{\alpha\beta}\end{aligned}$$ In particular, for ${\ensuremath{\sigma_{\mathbf{g}}}}^2 \equiv {\ensuremath{\left\langle |{\ensuremath{\hat{{ {\mathbf g}}}}\xspace}-{ {\mathbf g}}|^2 \right\rangle}}$, $${\ensuremath{\sigma_{\mathbf{g}}}}^2 = {\ensuremath{\sigma_{\Delta}}}^2 \textrm{tr} A^{-1} \label{eq:traceerr}$$ In the symmetric cluster limit, the matrix is diagonal, and the result is simply $\sigma_{\alpha}^{2} = {\ensuremath{\sigma_{\Delta}}}^{2}/\chi$ where $\sigma_{\alpha}$ is the standard deviation of the estimator $\hat{g}_\alpha$, and hence ${\ensuremath{\sigma_{\mathbf{g}}}}^2 = \sigma_x^2 + \sigma_y^2$, $${\ensuremath{\sigma_{\mathbf{g}}}}^{2} = 2{\ensuremath{\sigma_{\Delta}}}^{2}/\chi \label{eq:sigx}$$ Computation of $\chi$ for cells in hexagonally-packed cluster {#app:Q} ============================================================= ![ Illustration of $Q$-layer hexagonally packed cell clusters and computation of $\chi(Q)$.[]{data-label="fig:Q"}](qmer_chi_calculation){width="90mm"} When we keep a fixed cluster geometry, we choose to work with hexagonally-packed cell clusters, following our earlier work [@camley2016emergent]. We illustrate clusters with $Q = 1, 2,$ and $3$ layers in [Fig. ]{}\[fig:Q\]. How can we calculate $\chi(Q)$? Because of the isotropy of the cluster, it is easiest to work with $\chi = \frac{1}{2} \sum |{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}|^2 = \sum (\delta x^i)^2 = \sum (x^i)^2$. We know that for a single cell ($Q = 0$), $\chi(0) = 0$. We can then determine $\chi(Q)$ in terms of $\chi(Q-1)$ by computing $x^2$ for each of the cells in the outer layer, which we call $G(Q) = \sum_{\textrm{i in outside layer}} (x^i)^2$. Then, $\chi(Q) = \chi(Q-1) + G(Q) = \sum_{q=1}^{Q} G(q)$. To compute $G(Q)$, there are three relevant portions of the cluster, as drawn in [Fig. ]{}\[fig:Q\]. These are the two left and rightmost extreme cells at $x = \pm Q^2$ (two blue dashed boxes), the sides (four red boxes, solid lines), and the top and bottom edges (black dashed boxes). We then find $$\begin{aligned} \nonumber G(Q) &= 2 \times Q^2 + 4 \times \left[\sum_{j = 1}^{Q-1} (Q-j/2)^2\right] + 2 \times \left[\sum_{k = 0}^Q (Q/2-k)^2 \right] \\ \nonumber &= 2 \times Q^2 + 4 \times \left[ \frac{14 Q^3 - 15 Q^2 + Q}{24} \right] + 2 \times \left[ \frac{Q^3 + 3 Q^2 + 2 Q}{12}\right] \\ \nonumber &= \frac{5}{2} Q^3 + \frac{1}{2} Q\end{aligned}$$ and hence $\chi(Q) = \sum_{q=1}^{Q} G(q) = (5/8) Q^4 + (5/4) Q^3 + (7/8) Q^2 + (1/4) Q$, as we state in the main paper. How does $\chi$ scale with the cluster size? A cluster with $Q$ layers has $N(Q) = 1 + 3 Q + 3 Q^2$ cells, so $\chi \sim Q^4 \sim N^2$. For a roughly circular cluster of radius $R$, we’d then expect that $\chi \sim R^4$. This will obviously depend on the precise details of the cluster shape and cell-cell spacing, but if we can approximate the sum in $\chi = \frac{1}{2}\sum_i |{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}|^2$ as an integral, we find $\chi \approx \frac{1}{2}\rho_c \int d^2 r r^2 = \frac{\pi}{4} \rho_c R_{\textrm{cluster}}^{4}$, where $\rho_c$ is the number of cells per unit area in the cluster. [How does this scaling compare with earlier results for single cells? Ref. [@hu2011geometry] uses a maximum likelihood method to determine the accuracy limit for a single cell with $n_r$ receptors spaced around its radius, finding $\sigma_p^2 \sim \gamma / n_r$, where $\gamma = \frac{(\bar{c}+K_D)^2}{\bar{c}K_D}$ and $p \sim g \times R_\textrm{cell}$, with $R_\textrm{cell}$ the cell radius. Changing variables to ${ {\mathbf g}}$, this is $\sigma_g^2 \sim \frac{\gamma}{n_r R_\textrm{cell}^2}$. Our results, ignoring cell-to-cell variation, are that ${\ensuremath{\sigma_{\mathbf{g}}}}^2 \sim \frac{\gamma}{n_r \chi}$, with $\chi \sim \rho_c R^4$ for a circular cluster of radius $R$, and $n_r$ the number of receptors per cell. Then, as there are $N \sim \rho_c R^2$ cells in the cluster, the total number of receptors within the cluster is $n_t \sim N\times n_r$. Our result for ${\ensuremath{\sigma_{\mathbf{g}}}}$ (in the absence of time averaging and CCV) is then ${\ensuremath{\sigma_{\mathbf{g}}}}^2 \sim \frac{\gamma}{n_t R^2}$. This shows that, up to geometric factors, a cluster’s sensing bound is the same as a giant cell with the same [*total*]{} number of receptors and radius.]{} Detailed derivation of time-averaged gradient sensing error {#app:timeaverage} =========================================================== We will, in this section, completely neglect the ligand-receptor fluctuations in gradient sensing, as appropriate for the physically likely case ${\ensuremath{\sigma_{\Delta}}}> 0.1$. We show in Appendix \[app:mle\] that the maximum likelihood estimator of the gradient is then $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} = (\mathcal{A}^{-1})\cdot \sum_i (M^i - 1) {\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}$, where $\mathcal{A}_{\alpha\beta} = \sum_i \delta r_\alpha \delta r_\beta$, with $\alpha,\beta$ the Cartesian coordinates $x,y$. We now evaluate $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}}$ with the signal $M^i = 1 + {\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}\cdot{\ensuremath{{ {\mathbf g}}}\xspace}+ \Delta^i$: $$\begin{aligned} \hat{{\ensuremath{{ {\mathbf g}}}\xspace}} &= {\ensuremath{{ {\mathbf g}}}\xspace}+ (\mathcal{A}^{-1})\cdot\sum_i \Delta^i {\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}\\ &\equiv {\ensuremath{{ {\mathbf g}}}\xspace}+ {\ensuremath{\boldsymbol\Lambda\xspace}}\end{aligned}$$ We will treat a more general case than in the main body, allowing $\Delta_i$ to vary in a time-dependent manner, ${\ensuremath{\left\langle \Delta^i(t) \Delta^j(0) \right\rangle}} = {\ensuremath{\sigma_{\Delta}}}^2 C_{\Delta\Delta}(t) \delta^{ij}$ where $\delta^{ij}$ is the Kronecker delta function and $C_{\Delta\Delta}(t)$ characterizes the correlation of the CCV; within the main paper, we take $C_{\Delta\Delta} \to 1$, assuming CCV is persistent over all relevant time scales of the motion. Suppose the cell time-averages its maximum likelihood estimator with a time window [$T$]{}: $$\begin{aligned} {\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}(t) &= \int {\ensuremath{\hat{{ {\mathbf g}}}}\xspace}(t') K_T(t-t') \\ &= {\ensuremath{{ {\mathbf g}}}\xspace}+ {\ensuremath{\boldsymbol\Lambda_T\xspace}}\end{aligned}$$ where $K_T(t)$ is an averaging function with $K(t<0) = 0$ and ${\ensuremath{\boldsymbol\Lambda_T\xspace}}\equiv \int_{-\infty}^{\infty} {\ensuremath{\boldsymbol\Lambda\xspace}}(t') K_T(t-t') dt'$. We will often use a simple exponential average with $K(t) = \theta(t) \frac{1}{T} e^{-t/T}$, where $\theta(t)$ is the Heaviside step function. Clearly, for a single configuration of cells, ${\ensuremath{\left\langle {\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}\right\rangle}} = {\ensuremath{{ {\mathbf g}}}\xspace}$. We then would like to compute how much the variations in ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$ are reduced by the time-averaging, i.e. we compute $$\begin{aligned} {\ensuremath{\left\langle |{\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}-{\ensuremath{{ {\mathbf g}}}\xspace}|^2 \right\rangle}} \equiv {\ensuremath{\sigma_{\mathbf{g},T}}}^2 &= {\ensuremath{\left\langle |{\ensuremath{\boldsymbol\Lambda_T\xspace}}|^2 \right\rangle}} \\ &= {\ensuremath{\left\langle \int_{-\infty}^{\infty}dt'\int_{-\infty}^{\infty}dt'' {\ensuremath{\boldsymbol\Lambda\xspace}}(t')\cdot{\ensuremath{\boldsymbol\Lambda\xspace}}(t'') K_T(t-t') K_T(t-t'') \right\rangle}} \\ &= \int_{-\infty}^{\infty}dt''\int_{-\infty}^{\infty}dt'' K_T(t-t') K_T(t-t'') {\ensuremath{\left\langle \sum_{i,j} {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\gamma}(t') \Delta^i(t') \delta r_\gamma^i(t') {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\beta}(t'') \Delta^j(t'') \delta r_\beta^j(t'') \right\rangle}} \end{aligned}$$ where we have used Einstein summation notation in the last equation. We also note that the average ${\ensuremath{\left\langle \cdots \right\rangle}}$ now includes an average over time – the cell configurations are changing. We now want to perform the average over the $\Delta^i$. This is possible if the re-arrangement of the cell positions are independent of the particular values of $\Delta$, i.e. ${\ensuremath{\left\langle {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\gamma}(t') \Delta^i(t') \delta r_\gamma^i(t') {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\beta}(t'') \Delta^j(t'') \delta r_\beta^j(t'') \right\rangle}} \approx {\ensuremath{\left\langle \Delta^i(t')\Delta^j(t'') \right\rangle}} \langle {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\gamma}(t') \delta r_\gamma^i(t') {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\beta}(t'') \delta r_\beta^j(t'') \rangle$. This would be natural if, e.g. the cluster collectively chooses an estimated direction, but the re-arrangements are only due to local fluctuations, independent of $\Delta$. However, if each cell has a motility related to $\Delta^i$, this assumption may not be accurate. This approximation is also slightly violated if the cell cluster takes on a different shape in response to its estimate of the gradient location. This is an important approximation, but one that we suspect is unavoidable to create a measure of the correlation ${\ensuremath{\left\langle {\ensuremath{\boldsymbol\Lambda\xspace}}(t) {\ensuremath{\boldsymbol\Lambda\xspace}}(t') \right\rangle}}$ that does not depend on $\Delta$. With this decoupling approximation, we find: $$\begin{aligned} {\ensuremath{\sigma_{\mathbf{g},T}}}^2 &\approx \int_{-\infty}^{\infty}dt'\int_{-\infty}^{\infty}dt'' K_T(t-t') K_T(t-t'') \sum_{i,j}{\ensuremath{\left\langle \Delta^i(t')\Delta^j(t'') \right\rangle}} \langle {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\gamma}(t') \delta r_\gamma^i(t') {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\beta}(t'') \delta r_\beta^j(t'') \rangle\\ &= {\ensuremath{\sigma_{\Delta}}}^2 \int_{-\infty}^{\infty}dt'\int_{-\infty}^{\infty}dt'' K_T(t-t') K_T(t-t'') C_{\Delta\Delta}(t'-t'') \sum_i \langle {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\gamma}(t') \delta r_\gamma^i(t') {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\beta}(t'') \delta r_\beta^i(t'') \rangle \label{eq:inprogress}\end{aligned}$$ We emphasize that in the absence of averaging, $K_T(t-t') \to \delta(t-t')$, our result agrees with the results of Appendix \[app:mle\]. In this case, the right-hand-side of [Eq. ]{}\[eq:inprogress\] becomes ${\ensuremath{\sigma_{\Delta}}}^2 C_{\Delta\Delta}(0) \sum_i \langle {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\gamma}(t) \delta r_\gamma^i(t) {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\beta}(t) \delta r_\beta^i(t) \rangle = C_{\Delta\Delta}(0) {\ensuremath{\sigma_{\Delta}}}^2 \langle {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\alpha} \rangle$, as $\sum_i \delta r_\gamma^i \delta r_\beta^i = \mathcal{A}_{\gamma\beta}$. $C_{\Delta\Delta}(0) = 1$ by definition. This suggests we write $$\begin{aligned} \nonumber {\ensuremath{\sigma_{\mathbf{g},T}}}^2 = {\ensuremath{\sigma_{\mathbf{g},0}}}^2 &\times \\ \int_{-\infty}^{\infty}dt''&\int_{-\infty}^{\infty}dt'' K_T(t-t') K_T(t-t'') C_{\Lambda\Lambda}(t'-t'') \label{eq:conv}\end{aligned}$$ where $C_{\Lambda\Lambda}(t)$ is the normalized correlation function $C_{\Lambda\Lambda}(t'-t'') = C_{\Delta\Delta}(t-t')\times \sum_i \langle {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\gamma}(t') \delta r_\gamma^i(t') {\ensuremath{\mathcal{A}^{-1}}}_{\alpha\beta}(t'') \delta r_\beta^i(t'') \rangle / \langle {\ensuremath{\mathcal{A}^{-1}}}_{\mu\mu} \rangle$. The correlation function $C_{\Lambda\Lambda}$ can be calculated readily from the cell trajectories relative to the cluster center of mass $\delta {\ensuremath{{ {\mathbf r}}}\xspace}^i(t)$. However, in the limit of isotropic clusters of roughly constant shape, we can significantly simplify this form. For isotropic clusters of constant shape, $\mathcal{A}_{\alpha\beta}(t) = \chi \delta_{\alpha\beta}$ independent of time. Given this assumption, $C_{\Lambda\Lambda}(t'-t'') / C_{\Delta\Delta}(t'-t'') = \chi^{-1} \sum_i \langle \delta r_\alpha^i(t') \delta r_\alpha^i(t'') \rangle$ – i.e. $C_{\Lambda\Lambda}(t'-t'') = C_{\Delta\Delta}(t'-t'')C_{rr}(t'-t'')$, where $C_{rr}(t'-t'') \equiv \langle \boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}(t')\cdot\boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}(t'') \rangle / \langle |\boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}|^2 \rangle$. The double convolution in [Eq. ]{}\[eq:conv\] is simpler in Fourier space, $${\ensuremath{\sigma_{\mathbf{g},T}}}^2 = {\ensuremath{\sigma_{\mathbf{g},0}}}^2 \times \int_{-\infty}^\infty \frac{d\omega}{2\pi} |K_T(\omega)|^2 C_{\Lambda\Lambda}(\omega) \label{eq:gtvar}$$ where $K_T(\omega)$ is the Fourier transform of $K_T(t)$, $K_T(t) = \int \frac{d\omega}{2\pi} e^{i\omega t} K_T(\omega)$ and $C_{\Lambda\Lambda}(\omega)$ the Fourier transform of $C_{\Lambda\Lambda}(t)$. For $K_T(t-t') = \theta(t-t') \frac{1}{T} e^{-(t-t')/T}$, $K_T(\omega) = \frac{1}{1+i\omega T}$. In the common case that $C_{\Lambda\Lambda}(t) = \exp(-t/\tau_\Lambda)$, where $\tau_\Lambda$ is a characteristic correlation time, and $K_T(t-t') = \theta(t-t') \frac{1}{T} e^{-(t-t')/T}$, this is even simpler: $${\ensuremath{\sigma_{\mathbf{g},T}}}^2 = \frac{{\ensuremath{\sigma_{\mathbf{g},0}}}^2}{1 + T/\tau_\Lambda}$$ When we can approximate $C_{\Lambda\Lambda} \approx C_{rr}$, this is consistent with our intuition: it takes roughly a time of $\tau_r$ for the cells to re-arrange, and so in a time of $T$, the cluster can make $T/\tau_r$ independent measurements in an averaging time, and so it can decrease the measurement error by $T/\tau_r$. If the amount of CCV varies over time, the characteristic timescale is then $\tau_c = \frac{\tau_r \tau_\Delta}{\tau_r + \tau_\Delta}$ – the relaxation timescale that is relevant is the faster of the two timescales. Within the main article, we have assumed that this is always cell position rearrangement. Because of the complexity of the different results and regimes in this section, we provide a summary in Table \[tab:summary\]. [**Formula for ${\ensuremath{\sigma_{\mathbf{g},T}}}^2$**]{} [**Assumptions made**]{} [**Associated quantities**]{} ---------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ${\ensuremath{\sigma_{\mathbf{g},0}}}^2 \times \int_{-\infty}^\infty \frac{d\omega}{2\pi} |K_T(\omega)|^2 C_{\Lambda\Lambda}(\omega)$ Decoupling of averages over CCV and configurations ${\ensuremath{\sigma_{\mathbf{g},0}}}^2 \times \int_{-\infty}^\infty \frac{d\omega}{2\pi} |K_T(\omega)|^2 \left\{C_{\Delta\Delta}(t) C_{rr}(t)\right\}_\omega$ Decoupling of averages over CCV and configurations; isotropic clusters of roughly constant shape $C_{rr}(t'-t'') \equiv \langle \boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}(t')\cdot\boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}(t'') \rangle / \langle |\boldsymbol\delta{\ensuremath{{ {\mathbf r}}}\xspace}|^2 \rangle$, $\{f(t)\}_\omega$ is the Fourier transform of $f(t)$ to $\omega$. ${\ensuremath{\sigma_{\mathbf{g},0}}}^2/\left[1+T/\tau_\Lambda\right]$ Decoupling of averages over CCV and configurations, exponential time averaging, $C_{\Lambda\Lambda} = e^{-t/\tau_\Lambda}$ ${\ensuremath{\sigma_{\mathbf{g},0}}}^2/\left[1+T/\tau_{c}\right]$ Decoupling of averages over CCV and configurations, exponential time averaging, isotropic clusters of roughly constant shape, $C_{\Delta\Delta} = e^{-t/\tau_\Delta}$, $C_{rr} = e^{-t/\tau_r}$ $\tau_c = \frac{\tau_r \tau_\Delta}{\tau_r + \tau_\Delta}$ Characteristic time scales of different re-arrangement mechanisms {#app:mechanisms} ================================================================= The characteristic time of the position-position correlation, $\tau_r$ is critical in calculating chemotactic accuracy via [Eq. ]{}\[eq:snrt\_simp\]. What is $\tau_r$, and what does it depend on? This depends on the mechanism of re-arrangement. [**Persistent cluster rotation.**]{} If the cluster rigidly rotates with angular velocity $\Omega$, we can see that ${\ensuremath{{ {\mathbf r}}}\xspace}(t)\cdot{\ensuremath{{ {\mathbf r}}}\xspace}(0) = |{\ensuremath{{ {\mathbf r}}}\xspace}(0)|^2 \cos \Omega t$, and time-averaging over a timescale $T > \Omega^{-1}$ can improve the signal-to-noise ratio. How does $\Omega$ depend on the cluster and cell properties? As, for a cluster to actively rotate with velocity $\Omega$, cells at the edge must be able to crawl with speed $R \Omega$, where $R$ is the cluster radius, we expect that $\Omega \sim v_{\textrm{max}}/R$, where $v_{\textrm{max}}$ is the maximum speed a cell can crawl. Hence, the characteristic time over which ${\ensuremath{{ {\mathbf r}}}\xspace}$ changes is $\tau_{\textrm{rot}} \sim R/v_{\textrm{max}}$, as studied explicitly for rotating clusters in the main paper. [**Rotational diffusion of the cluster.**]{} If the cluster diffuses rotationally as a rigid body with angular diffusion rate $D_r$, $\langle {\ensuremath{{ {\mathbf r}}}\xspace}(t)\cdot{\ensuremath{{ {\mathbf r}}}\xspace}(0)\rangle = \langle |{\ensuremath{{ {\mathbf r}}}\xspace}(0)|^2\rangle e^{-D_r t}$. However, the scaling of the rotational diffusion coefficient with cluster size is not obvious, and will depend on the model. Rotational diffusion could arise from clusters that undergo collectively-driven rotation, but occasionally switch between moving in different directions over a timescale $\tau_{\textrm{switch}}$ – this is observed in small clusters of cells on micropatterns [@segerer2015emergence]. If this is the case, the effective rotational diffusion coefficient is just $D_r \sim \Omega^2 \tau_{\textrm{switch}}$ – so $D_r \sim v_{\textrm{max}}^2 \tau_{\textrm{switch}}/R^2$. By contrast, we have seen that in the absence of a collective aligning effect, rotational diffusion can be small or absent, depending on certain details about the underlying cell motility model [@camley2016emergent]. [**Cell re-arrangements.**]{} Cells within a tissue can often be described as undergoing a persistent random walk: they maintain a direction over a timescale $\tau_{\textrm{persist}}$, but lose their orientation beyond this time, leading to an effectively diffusive motion with effective diffusion coefficient $D_{\textrm{eff}}$ at long time scales [@szabo2010collective]; these numbers can be on the order of 10 microns$^2$/min. For cells to move from one half of the cluster to another by diffusion will then take a timescale $\tau_\textrm{diff}$ where $D_{\textrm{eff}} \tau_{\textrm{diff}} \sim R^2$. However, if cells are persistent over the timescale required to cross the cluster, re-arrangements could be accelerated – in this case, $\tau_{\textrm{rearrange}} \sim R/v_\textrm{cell}$. Naturally, if cell motion becomes subdiffusive, this will also change the dynamics of the cluster re-arrangement. This would all be captured in [Eq. ]{}\[eq:snrt\] once the cell-cell correlation function is determined. Bounds capture variation of CI over large range of parameters {#app:params} ============================================================= {width="180mm"} {width="180mm"} In this section, we show a larger range of variations in parameters, showing that the predicted CI from the bound captures simulated CI in our models well. While our primary results on computing the upper bound in Section \[sec:fluidity\] were performed with [Eq. ]{}\[eq:snrt\_simp\], it is also possible to compute ${\ensuremath{\sigma_{\mathbf{g}}}}^2$ without any assumptions about cluster isotropy, using the correlation function $C_{\Lambda\Lambda}(t)$ as seen in Appendix \[app:timeaverage\], and the no-time-averaging result of $\sigma_{\mathbf{g},0}^2$ via [Eq. ]{}\[eq:traceerr\]. We find that typically, in our simple collective cell motility simulations, that $\tau_{\Lambda}$ and $\tau_r$ are very close, as are $\textrm{tr} \mathcal{A}^{-1}$ and $2/\chi$ – unless the cluster becomes elongated or otherwise anisotropic. However, we do note that $\textrm{tr} \mathcal{A}^{-1} \ge 2/\chi$ for any configuration of cells, so this implies a larger uncertainty than the isotropic approximation. We compare the simulated CI and our predictions from the isotropic and full theories in [Fig. ]{}\[fig:compareall\]. [^1]: We measure in units of the cell diameter; see [*Methods*]{} [^2]: We note that this is most relevant if motility is a large portion of the cluster’s energy budget, a complex, cell-type-dependent question [@purcell1977life; @flamholz2014quantified; @katsu2009substantial] [^3]: The cluster’s directionality is always improved by rotating, so there is no tradeoff unless speed of motion matters.

--- abstract: 'Estimation of a deterministic quantity observed in non-Gaussian additive noise is explored via order statistics approach. More specifically, we study the estimation problem when measurement noises either have positive supports or follow a mixture of normal and uniform distribution. This is a problem of great interest specially in cellular positioning systems where the wireless signal is prone to multiple sources of noises which generally have a positive support. Multiple noise distributions are investigated and, if possible, minimum variance unbiased (MVU) estimators are derived. In case of uniform, exponential and Rayleigh noise distributions, unbiased estimators without any knowledge of the hyper parameters of the noise distributions are also given. For each noise distribution, the proposed order statistic-based estimator’s performance, in terms of mean squared error, is compared to the best linear unbiased estimator (BLUE), as a function of sample size, in a simulation study.' author: - | Kamiar Radnosrati\ Department of Electrical Engineering\ Linköping University\ Linköping, Sweden\ `kamiar.radnosrati@liu.se` Gustaf Hendeby\ Department of Electrical Engineering\ Linköping University\ Linköping, Sweden\ `gustaf.hendeby@liu.se` Fredrik Gustafsson\ Department of Electrical Engineering\ Linköping University\ Linköping, Sweden\ `fredrik.gustafsson@liu.se` bibliography: - 'Exploring-positive-noise-in-estimation-theory.bib' title: Exploring positive noise in estimation theory --- Introduction {#sec:introduction} ============ We consider the problem of estimating the mean $x$ observed in noise as $y_k=x+e_k$, for $k=1,2,\dots, N$, also known as “estimation of location" [@article:PI_kassam_85], where the noise $e_k$ has positive support. We will refer to such distributions as [*positive noise*]{}. Examples of distributions we will study include uniform, exponential, Rayleigh, Pareto. A bias compensated linear estimator as the sample mean has a variance that decays as $1/N$, while it is well-known from the statistical literature, see for example [@book:ET_kay_93; @book:TPE_lehmann], that the minimum has a variance that decays as $1/N^2$. The minimum is the simplest example of order statistics. Certain care has to be taken for the cases where the parameters in the distributions are unknown, in which case bias compensation becomes tricky. This paper derives all combinations of known/unknown parameters for order statistics/BLUE (best linear unbiased estimator) for some selected and common distribution that allow for analytical solutions. Problems involving positive noise can be motivated from applications where the arrival times of radio or sound waves are used. Such waves travel with the speed of the medium, and non line of sight conditions give rise to delayed arrival times. Physics does simply not allow for negative noise, only positive one. This case occur in a variety of applications such as target tracking using radar or lidar, and localisation using radio waves such as is done in for instance global satellite navigation systems [@article:ITVT_kok_15; @article:ITVT_chen_09; @article:ISPM_gustafsson_05; @article:IME_eling_12]. For example, the error histograms of time-of-arrival measurements collected from three separate cellular antennas are given in Figure \[fig:kista\]. For detailed description of hardware and the measurement campaign see [@conf:PIMRC_medbo_09] To deal with the estimation’s performance degradation in non-Gaussian error conditions, conventional estimation techniques which are developed based on Gaussian assumptions need to be adjusted properly. As discussed in [@article:ITSP_yin_13], “identify and discard”, “mathematical programming”, and “robust estimation” are the three broad categories of estimation methods which are robust against non-Gaussian errors. Robustness of the estimator has been a concern for many years in both research [@article:JASA_stigler_73] and different engineering topics  [@article:PI_kassam_85; @book:SDNGN_kassam; @article:SIAM_stewart_99; @book:NSP_arce] for a long time now. A more recent survey on this topic containing more references can be found in [@article:ISPM_zoubir_12]. The maximum likelihood estimator (MLE), developed under Gaussian assumptions, can be modified to become robust in presence of non-Gaussian noises. The authors in [@conf:ICML_eskin_00] first detect and then reject the outliers by learning the probability density function (PDF) of the measurements and develop a mixture model of outliers and clean data. A similar idea to k-nearest-neighbor approach is used in [@article:DMKD_chawla_10] to classify outliers as the data points that are less similar to a number of neighboring measurements. Surveys of advances in clustering the data into outliers and clean data can be found in [@article:AIR_hodge_04; @article:ITSP_yin_13; @conf:ICASSP_fritsche_09]. While these approaches might result in high estimation accuracy, they typically require large datasets [@article:ISPM_zoubir_12]. M-estimators [@book:RS_huber], in the contrary to identification-based methods, do not require pre-processing and can be used in non-Gaussian noise conditions. In principle, M-estimators can be seen as generalization of MLE and rely on solving a minimization problem of some loss function. For a detailed discussion on different loss functions, see [@book:RS_huber]. Since minimization problems are typically solved numerically based on the derivative of the loss function [@book:RSTM_maronna], they might converge to local minima. In this work, we strive to find minimum variance unbiased (MVU) estimators for the location of estimation problems for non-Gaussian noise distributions where multiple distributions with positive support are considered. In case where MVU is not found, we introduce unbiased order-statistic-based estimators and compare their variances against the BLUE. The MVU estimators without any knowledge of the hyper parameters of the noise distributions are also derived, if possible. Finally, we derive an estimator for the case in which the noise follows a mixture of normal and uniform distribution. The rest of this paper is structured as follows. In Section \[sec:marginal\_distribution\_of\_order\_statistics\] the marginal distribution of order statistics is introduced. In Section \[sec:location\_estimation\_problem\] the location estimation problem is formulated. The problem is then investigated for different noise distributions and estimators for each distribution are derived in Sections \[sec:uniform\_distribution\]–\[sec:other\_distributions\]. The proposed estimators are evaluated in a simulation study in Section \[sec:performance\_evaluation\] followed by the concluding remarks given in Section \[sec:conclusions\]. Marginal Distribution of Order Statistics {#sec:marginal_distribution_of_order_statistics} ========================================= The marginal distribution of order statistics, in this work is computed by differentiating the corresponding cumulative distribution function (CDF). In this section, we first introduce the minimum, also know as first or extreme, order statistic and then give the generalization to any statistics of order $k$. Let $F$ denote the common CDF of $N$ independent and identically distributed sample of random variables $y_1,\ldots,y_N$. We let $y^{(k)}$ denote the $k$:th order statistic of the sample, defined as the $k$:th smallest value of the set $\{y_i\}_{i=1}^N$. We define $f_{(k,N)}(y)$ as the marginal PDF of the $k$:th order statistics corresponding to a sample of size $N$. The PDF $f_{(k,N)}(y)$ is then calculated by differentiating $F_{(k,N)}(y)$ with respect to $y$. Marginal distribution of minimum order statistic {#subsec:marginal_distribution} ------------------------------------------------ To further illustrate the problem, consider fist an example in which we have drawn $N=5$ independent random variables $\{y_i\}_{i=1}^5$ each from a common distribution with PDF $f(y)$. Assume that we are interested in the PDF of the first order statistic, $f_{(1,5)}(y)$. The CDF $F_{(1,5)}(y)$ is defined as $P(y^{(1)} < y)$. We note that the minimum order statistic $Y^{(1)}$ would be less than $y$ if at least $1$ of the random variables $y_1, y_2, y_3, y_4, y_5$ are less than $y$. In other words, we need to count the number of ways that can happen such that at least one random variable is less than $y$. This leads to a binomial probability calculation. The ’success’ is considered to be the event $\{y_i < y\}$, $i = 1$ and we let $\zeta$ denote the number of successes in five trials, then $$\begin{aligned} F_{(1,5)}(y) &= P(y_{(1)}<y) = P(\zeta=1)+\ldots + P(\zeta=5),\\ f_{(1,5)}(y) &= \frac{\,d}{\,dy}F_{(1,5)}(y).\end{aligned}$$ To generalize the example, let $y_{(1)} < y_{(2)} < \ldots < y_{(N)}$ be the order statistics of $N$ independent observations from a continuous distribution with cumulative distribution function $F(y)$ and probability density function $f(y)=F'(y)$. The marginal PDF $f_{(1,N)}(y)$ of the minimum order statistic can be obtained by considering the event $\{Y_i \leq y\}, i = 1$ as a “success,” and letting $\zeta$ = the number of such successes in $N$ mutually independent trials. $\zeta$ is a binomial random variable with $N$ trials and probability of success $P(y_i\leq y)$. Hence, the CDF of the minimum order statistic is given by, $$\begin{aligned} F_{(1,N)}(y)=\sum_{n=1}^{N}P(\zeta=n). \label{eq:cdf_order_minimum} \end{aligned}$$ Noting that the probability mass function of this binomial distribution is given by, $$\begin{aligned} P(\zeta=n) = \begin{pmatrix}N\\n\end{pmatrix}[F(y)]^n[1-F(y)]^{N-n}. \label{eq:pmf_order_minimum} \end{aligned}$$ Substituting  into  and taking the last term out of the sum, we get $$\begin{aligned} F_{(1,N)}(y)=\sum_{n=1}^{N-1}\begin{pmatrix}N\\n\end{pmatrix}[F(y)]^n[1-F(y)]^{N-n}+[F(y)]^N. \label{eq:cdf_order_minimum_2} \end{aligned}$$ Differentiating  with respect to $y$ gives a telescoping sum of the form, $$\begin{aligned} f_{(1,N)}(y)&=\sum_{n=1}^{N-1}\frac{N!}{(n-1)!(N-n)!}[F(y)]^{n-1}f(y)[1-F(y)]^{N-n}\nonumber\\ &+ \sum_{n=1}^{N-1}\frac{N!}{n!(N-n-1)!}[F(y)]^n[1-F(y)]^{N-n-1}(-f(y)) \nonumber\\&+ N[F(y)]^{N-1}f(y),\end{aligned}$$ in which, except the first term, all other terms cancel each other out. Hence, the marginal probability density function of the minimum order statistic of a set of $N$ independent and identically random variables with common CDF $F(y)$ and PDF $f(y)$ is given by, $$\begin{aligned} f_{(1,N)}(y) = Nf(y)\left(1-F(y)\right)^{N-1}. \label{eq:density_order_minimum}\end{aligned}$$ Marginal distribution of general order statistic {#subsec:marginal_distribution_of_general_order_statistic} ------------------------------------------------ The marginal PDF $f_{(k,N)}(y)$ of the general order statistic $k$ can be obtained by generalizing the results of the previous section, and considering the event $\{y_i \leq y\}, i = 1, 2, \ldots, k$ as a “success,” and letting $\zeta$ = the number of such successes in $N$ mutually independent trials, $$\begin{aligned} F_{(k,N)}(y)=\sum_{n=k}^{N-1}\begin{pmatrix}N\\n\end{pmatrix}[F(y)]^n[1-F(y)]^{N-n}+[F(y)]^N. \label{eq:cdf_generic2} \end{aligned}$$ Differentiating  with respect to $y$ gives a telescoping sum of the form, $$\begin{aligned} f_{(k,N)}(y)&=\sum_{n=k}^{N-1}\frac{N!}{(n-1)!(N-n)!}[F(y)]^{n-1}f(y)[1-F(y)]^{N-n}\nonumber\\ &+ \sum_{n=k}^{N-1}\frac{N!}{n!(N-n-1)!}[F(y)]^n[1-F(y)]^{N-n-1}(-f(y)) \nonumber\\&+ N[F(y)]^{N-1}f(y),\end{aligned}$$ in which, except the first term, all other terms cancel each other. Hence, the marginal probability density function of the $k$:th order statistic of a set of $N$ independent and identically random variables with common CDF $F(y)$ and PDF $f(y)$ is given by, $$\begin{aligned} f_{(k,N)}(y) = Nf(y)\begin{pmatrix}N-1\\k-1\end{pmatrix}F(y)^{k-1}\left(1-F(y)\right)^{N-k}. \label{eq:density_order}\end{aligned}$$ Location Estimation Problem {#sec:location_estimation_problem} =========================== Consider the location estimation problem in which we have measurements $y_k$, $k=1,\ldots,N$ of the unknown parameter $x$. Assuming that the measurements are corrupted with additive noise $e_k\sim p_e(\theta)$, where $\theta$ denotes the parameter(s) of the noise distribution, the measurement model is given by $$\begin{aligned} y_k=x+e_k, \quad k=1,\ldots,N. \label{eq:generic_model}\end{aligned}$$ The BLUE for the estimation problem  is given by $$\begin{aligned} \hat{x}_{\mathrm{BLUE}}^{p_e(\theta)}(y_{1:N},\theta) &= \frac{1}{N}\sum_{k=1}^N y_k- \delta(\theta), \label{eq:mean_estimator}\end{aligned}$$ where $y_{1:N} = \{y_k\}_{k=1}^{N}$ and $\delta(\theta)=\mathbb{E}(e_k)$ is the bias compensation term. In the following sections, closed-form expressions for the mean squared error (MSE) of the BLUE estimator for multiple noise distributions with positive support are provided. Given hyperparameter $\theta$, the MVU estimator for each noise distribution $p_e$ is denoted by $\hat{x}_{\mathrm{MVU}}^{p_e}(y_{1:N},\theta)$. MVU estimators with unknown hyperparameter are denoted by $\hat{x}_{\mathrm{MVU}}^{p_e}(y_{1:N})$. If the MVU cannot be found, an unbiased order-statistics-based estimator is derived and denoted by $\hat{x}^{p_e}(y_{1:N},\theta)$ and $\hat{x}^{p_e}(y_{1:N})$ for known and unknown hyperparameter cases, respectively. For example, $\hat{x}^{\mathcal{U}}_{\mathrm{MVU}}(y_{1:N},\beta)$ denotes the MVU estimator when $e_k\sim\mathcal{U}[0,\beta]$ and $\beta$ is known. $\hat{x}^{\mathcal{U}}_{\mathrm{MVU}}(y_{1:N})$, on the other hand, corresponds to the MVU estimator of uniform noise with unknown hyper parameters of the distribution. Table \[tbl:notation\] summarizes the notation used throughout this work. --------------------------------------------------------- ------------------------------------------------------------ $\{y_k\}_{k=1}^N$ noisy measurements of the unknown parameter $x$ \[2mm\] $\left(y^{(m)}\right)_{m=1}^N$ ordered measurement sequence \[2mm\] $\theta$ parameters of the noise distribution \[2mm\] $\delta(\theta)$ bias compensation term \[2mm\] $\hat{x}^{p_e}_{\mathrm{BLUE}}(y_{1:N},\theta)$ BLUE when $e_k\sim p_e$ for known $\theta$ \[2mm\] $\hat{x}^{p_e}_{\mathrm{MVU}}(y_{1:N},\theta)$ MVU estimator when $e_k\sim p_e$ for known $\theta$ \[2mm\] $\hat{x}^{p_e}_{\mathrm{MVU}}(y_{1:N})$ MVU estimator when $e_k\sim p_e$ for unknown $\theta$ \[2mm\] $\hat{x}^{p_e}(y_{1:N},\theta)$ unbiased estimator when $e_k\sim p_e$ for known $\theta$ \[2mm\] $\hat{x}^{p_e}(y_{1:N})$ unbiased estimator when $e_k\sim p_e$ for unknown $\theta$ --------------------------------------------------------- ------------------------------------------------------------ : Notation. \[tbl:notation\] For each noise distribution, we also consider the minimum order statistic estimator, denoted by $\hat{x}^{p_e}_{\mathrm{min}}(y_{1:N})$. Let $\left(y^{(m)}_{1:N}\right)_{m=1}^{N}$ denote the ordered sequence obtained from sorting $y_{1:N}$ in an ascending order, $\hat{x}^{p_e}_{\mathrm{min}}(y_{1:N})$ is defined as $$\begin{aligned} \hat{x}^{p_e}_{\mathrm{min}}(y_{1:N}) = y_{1:N}^{(1)} \triangleq \min_{k} y_k.\end{aligned}$$ Noting that for any generic estimator $\hat{x}$ the MSE is given by $$\begin{aligned} \label{eq:mse_generic} {\mathrm{MSE}}(\hat{x}) = {\mathrm{Var}}(\hat{x}) + b^2(\hat{x}).\end{aligned}$$ The MSE for BLUE and MVU or any other bias compensated estimator coincides with the estimator’s variance. In case of $\hat{x}^{p_e}_{\mathrm{min}}$, the existing bias enters the MSE. In order to find the MVU estimator, the first step is to find the PDF $f(y_{1:N};\theta)$ with $\theta$ denoting the parameters of the distribution. If the PDF satisfies regularity conditions, the CRLB can be determined. Any unbiased estimator that satisfies CRLB is thus the MVU estimator. However, the considered PDFs do not satisfy the regularity conditions, $$\begin{aligned} {\mathbb{E}}\left[\frac{\partial\ln f(y_k;\theta)}{\partial \theta}\right]\neq0.\end{aligned}$$ Hence, the CRLB approach is not applicable. Instead, we rely on the RBLS theorem [@article:IJS_lehmann_1; @article:IJS_lehmann_2; @book:ET_kay_93], to find the MVU estimator. The RBLS theorem [@article:IJS_lehmann_1] states that for any unbiased estimator $\tilde{x}$ and sufficient statistics $T(y_{1:N})$, $\hat{x}={\mathbb{E}}(\tilde{x}\mid T(y_{1:N}))$ is unbiased and ${\mathrm{Var}}(\hat{x})\leq{\mathrm{Var}}(\tilde{x})$. Additionally, if $T(y_{1:N})$ is complete, then $\hat{x}$ is MVU. As shown in [@book:ET_kay_93], if the dimension of the sufficient statistics is equal to the dimension of the parameter, then the MVU estimator is given by $\hat{x}=g(T(y_{1:N}))$ for any function $g(\cdot)$ that satisfies $$\begin{aligned} {\mathbb{E}}(g(T)) = x.\end{aligned}$$ Hence, the problem of MVU estimator turns into the problem of finding a complete sufficient statistic. The Neyman-Fisher theorem [@article:fisher_22; @article:AMS_halmos_49] gives the sufficient statistic $T(y_{1:N})$, if the PDF can be factorized as follows $$\begin{aligned} f(y_{1:N};\Psi) =g(T(y_{1:N}),\Psi)h(y_{1:N}),\end{aligned}$$ where $\psi$ is the union of the noise hyper parameter(s) $\theta$ and $x$. The estimators in this work are derived in the order statistics framework. Uniform Distribution {#sec:uniform_distribution} ==================== As the first scenario, consider the case in which the additive noise $e_k$ has a uniform distribution with a positive support, $p_e(\theta)=\mathcal{U}[0,\beta]$, $\beta>0$ and $\theta=\beta$. The BLUE is given by $$\begin{aligned} \hat{x}_{\mathrm{BLUE}}^{\mathcal{U}}(y_{1:N},\beta) &= \frac{1}{N}\sum_{k=1}^N y_k- \frac{\beta}{2}. \label{eq:sample_mean_example} \end{aligned}$$ The MSE of BLUE for this case is given by $$\begin{aligned} {\mathrm{MSE}}\left(\hat{x}_{\mathrm{BLUE}}^{\mathcal{U}}(y_{1:N},\beta)\right) &= \frac{1}{N^2}\sum_{k=1}^{N}{\mathrm{Var}}\left(y_k-\frac{\beta}{2}\right)= \frac{\beta^2}{12N}. \label{eq:mse_blue_uniform} \end{aligned}$$ In order to find the MSE of the minimum order statistics estimator, $\hat{x}_{\mathrm{min}}^{\mathcal{U}}(y_{1:N}) $, we need to find the first two moments of the estimator. Let $\tilde{y}_k= \frac{1}{\beta}y_k$. Since $y_k\sim\mathcal{U}[x,x+b]$, then for any constant $\beta>0$, $\tilde{y_k}\sim\mathcal{U}[\frac{x}{\beta},\frac{x}{\beta}+1]$. Hence, $f(\tilde{y}_k)=1$ and $F(\tilde{y}_k)=\frac{1}{\beta}(y_k-x)$. From  we get, $$\begin{aligned} f&_{(k,N)}^{\mathcal{U}[0,\beta]}(\tilde{y}) = N\begin{pmatrix}N-1\\k-1\end{pmatrix}\left(\frac{\tilde{y}-x}{\beta}\right)^{k-1}\left(\frac{\beta-(\tilde{y}-x)}{\beta}\right)^{N-k}\nonumber\\ &= \frac{(N)!}{(k-1)!(N-k)!}\left(\frac{\tilde{y}-x}{\beta}\right)^{k-1}\left(\frac{\beta-(\tilde{y}-x)}{\beta}\right)^{N-k}. \end{aligned}$$ since $N\in\mathbb{N}^+$, $k\in\mathbb{N}^+$, and $k\in[1,N]$ we can the change the factorials to gamma functions, $$\begin{aligned} f_{(k,N)}^{\mathcal{U}[0,\beta]}(\tilde{y}) = \frac{\Gamma(N+1)}{\Gamma(k)\Gamma(N-k+1)} \left(\frac{\tilde{y}-x}{\beta}\right)^{k-1}\left(\frac{\beta-(\tilde{y}-x)}{\beta}\right)^{N-k}. \label{eq:order_uniform_general} \end{aligned}$$ The marginal distribution  is a generalized beta distribution, also known as four parameters beta distribution [@article:Mcdonal_JE_95]. The support of this distribution is from $0$ to $\beta>0$ and $f_{(k,N)}^{\mathcal{U}[0,\beta]}(\cdot)=\frac{1}{\beta}f_{(k,N)}^{\mathcal{U}[0,1]}(\cdot)$. The bias and variance of the general $k$:th order statistic estimator $\hat{x}_{(k)}^{\mathcal{U}}(y_{1:N})$ in case of uniform noise with support on $[0,\beta]$ are given by \[eq:moments\_uniform\_order\] $$\begin{aligned} {\mathbb{E}}(\hat{x}_{(k)}^{\mathcal{U}}(y_{1:N})) &= \frac{\beta k}{N+1},\\ {\mathrm{Var}}(\hat{x}_{(k)}^{\mathcal{U}}(y_{1:N})) &= \frac{k(N-k+1)\beta^2}{(N+1)^2(N+2)}. \end{aligned}$$ The first two moments of the minimum order statistic estimator are obtained by letting $k=1$ in  \[eq:moments\_uniform\_min\] $$\begin{aligned} b\left(\hat{x}_{\mathrm{min}}^{\mathcal{U}}(y_{1:N})\right) &= \frac{\beta}{N+1}\\ {\mathrm{Var}}\left(\hat{x}_{\mathrm{min}}^{\mathcal{U}}(y_{1:N})\right) &= \frac{N\beta^2}{(N+1)^2(N+2)}. \end{aligned}$$ The MSE of $\hat{x}_{\mathrm{min}}^{\mathcal{U}}(y_{1:N})$ is then given by $$\begin{aligned} {\mathrm{MSE}}\left(\hat{x}_{\mathrm{min}}^{\mathcal{U}}(y_{1:N})\right) = \frac{2\beta^2}{(N+1)(N+2)}. \label{eq:mse_min_uniform}\end{aligned}$$ MVU estimator {#subsec:mvu_estimator} ------------- In order to find the MVU estimator, we note that the PDF can be written in a compact form using the step function $\sigma(\cdot)$ as $$\begin{aligned} f(y_k;x,\beta) = \frac{1}{\beta}\left[\sigma(y_k-x) - \sigma(y_k-x-\beta)\right]. \label{eq:uniform_pdf_single} \end{aligned}$$ which gives $$\begin{aligned} f(y_{1:N};&x,\beta) = \frac{1}{\beta^N}\prod_{k=1}^{N}\left[\sigma(y_k-x) - \sigma(y_k-x-\beta)\right]\nonumber\\ &=\frac{1}{\beta^N}\left[\sigma( y^{(1)}_{1:N} -x) - \sigma(y^{(N)}_{1:N}-x -\beta)\right], \label{eq:unknown} \end{aligned}$$ where $y^{(N)}_{1:N}\triangleq\max_k y_k,\quad k=1,\ldots,N$. The expressions for the MVU estimator is derived for two different scenarios. We first assume that the hyper parameter $\beta$ of the noise distribution is known and then further discuss the unknown hyper parameter case. In the general case, let $\Psi = [x,\beta]^\top$ denote the unknown parameter vector, the Neyman-Fisher factorization gives $h(y_{1:N})=1$ and $$\begin{aligned} T(y_{1:N}) = \begin{bmatrix} y^{(1)}_{1:N} \\ y^{(N)}_{1:N} \end{bmatrix} = \begin{bmatrix} T_1(y_{1:N}) \\ T_2(y_{1:N}) \end{bmatrix}. \label{eq:uniform_ss}\end{aligned}$$ ### Known hyper parameter $\beta$ {#subsubsec:known_hyper_parameter_uniform} When the maximum support of the uniform noise $\beta$ is known, the dimensionality of the sufficient statistic is larger than that of the unknown parameter $x$. As discussed in [@book:ET_kay_93], the RBLS theorem can be extended to address this case if the form of a function $g(T_1(y_{1:N}),T_2(y_{1:N}))$ can be found that combines $T_1$ and $T_2$ into a single unbiased estimator of $x$. Let $Z = T_1(y_{1:N})+T_2(y_{1:N})=u+v$. Since $T_1$ and $T_2$ are dependent, $$\begin{aligned} \label{eq:dist_sum_gen} f_Z(z) = \int_{-\infty}^{\infty} f_{y^{(1)},y^{(N)}}(u,z-u)\,d_u, \end{aligned}$$ where $f_{y^{(1)},y^{(N)}}(u,z-u)$ is the joint density of minimum and maximum order statistics. As shown in [@book:OS_david_04], for $-\infty<u<v<\infty$, the joint density of two order statistics $y^{(i)}$ and $y^{(j)}$ is given by $$\begin{aligned} f_{y^{(i)},y^{(j)}}(u,v) = &\frac{N!}{(i-1)!(j-1-i)!(N-j)!}\nonumber\\&\times f_Y(u)f_Y(v)\left[F_Y(u)\right]^{i-1}\nonumber\\ &\times\left[F_Y(v)-F_Y(u)\right]^{j-1-i}\left[1-F_Y(v)\right]^{N-j}, \end{aligned}$$ that for the extreme orders, $i=1$ and $j=N$ can be simplified such that for $u<v$ $$\begin{aligned} \label{eq:order_joint_gen} f_{y^{(1)},y^{(N)}}(u,v) = N(N-1)\left[F_Y(v)-F_Y(u)\right]^{N-2} f_Y(u)f_Y(v). \end{aligned}$$ and zero otherwise. Substituting  into , we get $$\begin{aligned} f_Z(z) = \frac{1}{2}N\beta^{-N}(2x+2\beta-z)^{N-1}, \end{aligned}$$ for $2x+\beta<z<2(x+\beta)$ and $$\begin{aligned} f_Z(z) = -\frac{1}{2}N\beta^{-N}\frac{(z-2x)^{N}}{2x -z}, \end{aligned}$$ for $2x<z\leq2x+\beta$ and zero otherwise. It can be shown that $$\begin{aligned} {\mathbb{E}}(f_Z(z)) = 2x + \beta. \end{aligned}$$ Hence, noting that $\beta$ is known, the function $g(T_1(y_{1:N}),T_2(y_{1:N}))$ that gives an unbiased estimator should be of the form of $$\begin{aligned} \hat{x}_{\mathrm{MVU}}^{\mathcal{U}}(y_{1:N},\beta) &= g(T_1(y_{1:N}),T_2(y_{1:N}))\nonumber\\ &= \frac{1}{2}(y^{(1)}_{1:N}+ y^{(N)}_{1:N}) - \frac{\beta}{2}. \end{aligned}$$ The MSE of the MVU estimator is given by $$\begin{aligned} {\mathrm{MSE}}\left(\hat{x}_{\mathrm{MVU}}^{\mathcal{U}}(y_{1:N},\beta) \right) = \frac{\beta^2}{2N(N+3)+4}.\label{eq:mse_mvu_unif_known} \end{aligned}$$ Comparing to , the order statistics based MVU estimator outperforms the BLUE one order of magnitude. ### Unknown hyper parameter $\beta$ {#subsec:unknown_hyper_parameter_uniform} In this case, the MVU estimators for the parameter vector $\Psi=[x,\beta]^\top$ can be derived from sufficient statistics , $$\begin{aligned} \hat{\Psi}=g(T(y_{1:N})),\quad \mathrm{s.t.}\quad {\mathbb{E}}\left(g\left(T(y_{1:N})\right)\right) = \Psi.\end{aligned}$$ In this case, we have $$\begin{aligned} {\mathbb{E}}(T(y_{1:N})) = \begin{bmatrix} x+\frac{\beta}{N+1}\\\\x+\frac{N\beta}{N+1} \end{bmatrix} \label{eq:uniform_unknown_E_gen}\end{aligned}$$ To find the transformation that makes  unbiased, we define $$\begin{aligned} g(T(y_{1:N}))=\begin{bmatrix}\frac{1}{N-1}\left(NT_1(y_{1:N})-T_2(y_{1:N})\right)\\\\ \frac{N+1}{N-1}\left(T_2(y_{1:N})-T_1(y_{1:N})\right)\end{bmatrix} \end{aligned}$$ that gives $$\begin{aligned} {\mathbb{E}}\left(g(T(y_{1:N}))\right) = \begin{bmatrix}x\\\beta\end{bmatrix}. \end{aligned}$$ Finally, the MVU estimator of $x$ when the hyper parameter $\beta$ is unknown is given by $$\begin{aligned} \hat{x}_{\mathrm{MVU}}^{\mathcal{U}}(y_{1:N}) = \frac{N}{N-1}y^{(1)}_{1:N} - \frac{1}{N-1} y^{(N)}_{1:N}. \end{aligned}$$ and its MSE is $$\begin{aligned} {\mathrm{MSE}}\left(\hat{x}_{\mathrm{MVU}}^{\mathcal{U}}(y_{1:N})\right) = \frac{N\beta^2}{(N+2)(N^2-1)}. \end{aligned}$$ This is naturally slightly larger than  for finite $N$. Distributions in the exponential family {#sec:distributioons_in_the_exponential_family} ======================================= The exponential family of probability distributions, in their most general form, is defined by $$\begin{aligned} f(y;\theta) = h(y)g(\theta)\exp\left\{A(\theta)\cdot T(y)\right\},\end{aligned}$$ where $\theta$ is the parameter of the distribution, and $h(y)$, $g(\theta)$, $A(\theta)$, and $T(y)$ are all known functions. In this section, we only consider some example distributions of this family and show that the minimum order statistic estimator gets the same form of distribution as the noise distribution but with modified parameters. For the selected distributions, if possible, MVU estimators for both cases of known and unknown hyperparameter are derived. Otherwise, unbiased estimators with less variance than BLUE are proposed. Exponential distribution {#subsec:exponential_distribution} ------------------------ Exponential distributions are members of the gamma family with shape parameter 1; strongly skewed with no left sided tail ($y_k\in[x,\infty]$). Let $\beta>0$ denote the scale parameter, the PDF of an exponential distribution is then given by \[eq:exponential\] $$\begin{aligned} f^{\mathrm{Exp}}(y_k;x,\beta)=\left\{\begin{matrix} \frac{1}{\beta}\exp(-\frac{(y_k-x)}{\beta})&y_k\geq x, \\ 0&y_k< x. \end{matrix}\right. \label{eq:exponential_pdf} \end{aligned}$$ and the CDF, for $y\geq x$, is given by $$\begin{aligned} F^{\mathrm{Exp}}(y_k;x,\beta)=1-\exp(-\frac{(y_k-x)}{\beta}). \label{eq:exponential_cdf} \end{aligned}$$ For the BLUE estimator , from the properties of exponential distribution, we have $$\begin{aligned} \hat{x}_{\mathrm{BLUE}}^{\mathrm{Exp}}(y_{1:N},\beta) &= \frac{1}{N}\sum_{k=1}^{N}y_k - \beta, &\quad {\mathrm{MSE}}(\hat{x}_{\mathrm{BLUE}}^{\mathrm{Exp}}) &= \frac{\beta^2}{N}.\end{aligned}$$ Substituting  into , the marginal density of the $k$:th order statistic is given by $$\begin{aligned} f^{\mathrm{Exp}}_{(k,N)}(y;x,\beta) = \frac{N}{\beta}\begin{pmatrix}N-1\\k-1\end{pmatrix}\left(1-\exp(-\frac{(y-x)}{\beta})\right)^{k-1}\exp\left(-\frac{(N-k+1)(y-x)}{\beta}\right). \label{eq:exponential_order}\end{aligned}$$ The first order statistic density is then given by letting $k=1$ in  that results in another exponential distribution, $$\begin{aligned} f^{\mathrm{Exp}}_{(1,N)}(y;x,\beta) = f^{\mathrm{Exp}}(y;x,\bar{\beta}),\end{aligned}$$ where $\bar{\beta}=\frac{\beta}{N}$. Hence, the MSE of the minimum order statistics estimator is given by $$\begin{aligned} {\mathrm{MSE}}\left(\hat{x}_{\mathrm{min}}^{\mathrm{Exp}}(y_{1:N})\right) = \frac{2\beta^2}{N^2}.\end{aligned}$$ In order to find the MVU estimator, we re-write the PDF as $$\begin{aligned} f(y_{1:N};x,\beta) &= \frac{1}{\beta^N}\exp\left[-\frac{1}{\beta}\sum_{k=1}^{N}y_k\right]\exp\left[-\frac{N}{\beta}x\right]\times\sigma(y^{(1)}_{1:N} - x). \label{eq:exponential_pdf_2}\end{aligned}$$ ### Known hyper parameter $\beta$ {#known-hyper-parameter-beta} In case of the known hyper parameter $\beta$, the Neyman-Fisher factorization of PDF  gives $$\begin{aligned} T(y_{1:N}) &= y^{(1)}_{1:N}\\ h(y_{1:N}) &= \frac{1}{\beta^N}\exp\left[-\frac{1}{\beta}\sum_{k=1}^{N}y_k\right]. \label{eq:exponential_known_ss} \end{aligned}$$ The MVU estimator can then be obtained from a transformation of the minimum order statistic that makes it an unbiased estimator. Finally, in case of exponential noise with known hyper parameter of the distribution, the MVU estimator and its MSE are given by $$\begin{aligned} \hat{x}_{MVU}^{\mathrm{Exp}}(y_{1:N},\beta) &= y^{(1)}_{1:N} - \frac{\beta}{N} \\ {\mathrm{MSE}}\left(\hat{x}_{MVU}^{\mathrm{Exp}}(y_{1:N},\beta)\right) &= \frac{\beta^2}{N^2}. \end{aligned}$$ ### Unknown hyper parameter $\beta$ {#unknown-hyper-parameter-beta} If the hyper parameter $\beta$ is unknown, the factorization gives $$\begin{aligned} T(y_{1:N}) = \begin{bmatrix} y^{(1)}_{1:N}\\ \sum_{k=1}^{N} y_k \end{bmatrix} = \begin{bmatrix} T_1(y_{1:N}) \\ T_2(y_{1:N}) \end{bmatrix}. \label{eq:uniform_unknown_ss} \end{aligned}$$ Noting that sum of exponential random variables results in a Gamma distribution, we have $T_2(y_{1:N})\sim\Gamma(N,\beta)$. Hence, $$\begin{aligned} {\mathbb{E}}(T(y_{1:N})) = \begin{bmatrix} x+\frac{\beta}{N}\\\\N(x+\beta) \end{bmatrix}. \label{eq:exponential_unknown_E_gen} \end{aligned}$$ Following the same line of reasoning as in Section \[subsec:unknown\_hyper\_parameter\_uniform\], the unbiased estimator is given by the transformation $$\begin{aligned} g(T(y_{1:N}))=\begin{bmatrix}\frac{1}{N-1}\left(NT_1(y_{1:N})-\frac{1}{N}T_2(y_{1:N})\right)\\\\ \frac{1}{N-1}\left(T_2(y_{1:N})-NT_1(y_{1:N})\right)\end{bmatrix}, \end{aligned}$$ that gives $$\begin{aligned} {\mathbb{E}}\left(g(T(y_{1:N}))\right) = \begin{bmatrix}x\\\beta \end{bmatrix}. \end{aligned}$$ Finally, the MVU estimator when the hyper parameter $\beta$ is unknown, is given by $$\begin{aligned} \hat{x}_{\mathrm{MVU}}^{\mathrm{Exp}}(y_{1:N}) &= \frac{N}{N-1}y^{(1)}_{1:N} - \frac{1}{N(N-1)} \sum_{k=1}^{N} y_k \nonumber\\&= \frac{N}{N-1}y^{(1)}_{1:N}- \frac{1}{N-1} \bar{y}, \end{aligned}$$ where $\bar{y}$ is the sample mean. Assuming that $N$ is large $\min_k y_k$ and $\bar{y}$ are independent and the MSE of the estimator, asymptotically, is given by $$\begin{aligned} {\mathrm{MSE}}\left(\hat{x}_{\mathrm{MVU}}^{\mathrm{Exp}}(y_{1:N})\right) = \frac{\beta^2(N+1)}{N(N-1)^2}. \end{aligned}$$ Rayleigh distribution {#subsec:rayleigh_distribution} --------------------- One generalization of the exponential distribution is obtained by parameterizing in terms of both a scale parameter $\beta$ and a shape parameter $\alpha$. Rayleigh distribution is a special case obtained by setting $\alpha=2$ \[eq:rayleigh\] $$\begin{aligned} f^{\mathrm{Rayleigh}}(y_k;x,\beta)=\left\{\begin{matrix} \frac{y_k-x}{\beta^2}\exp(-\frac{(y_k-x)^2}{2\beta^2})&y_k> x, \\ 0&y_k\leq x. \end{matrix}\right. \label{eq:rayleigh_pdf} \end{aligned}$$ and the CDF, for $y_k>x$ is given by $$\begin{aligned} F^{\mathrm{Rayleigh}}(y_k;x,\beta)=1-\exp(-\frac{(y_k-x)^2}{2\beta^2}). \label{eq:rayleigh_cdf} \end{aligned}$$ Hence, the BLUE estimator , becomes $$\begin{aligned} \hat{x}_{\mathrm{BLUE}}^{\mathrm{Rayleigh}}(y_{1:N},\beta) &= \frac{1}{N}\sum_{k=1}^{N}y_k - \sqrt{\frac{\pi}{2}}\beta, \\ {\mathrm{MSE}}\left(\hat{x}^{\mathrm{Rayleigh}}_{\mathrm{BLUE}}(y_{1:N},\beta)\right) &= \frac{(4-\pi)\beta^2}{2N}. \end{aligned}$$ The marginal density of the $k$:th order statistic is given by $$\begin{aligned} f^{\mathrm{Rayleigh}}_{(k,N)}(y;x,\beta) =\left\{\begin{matrix} \frac{Ny}{\beta^2}\begin{pmatrix}N-1\\k-1\end{pmatrix}\left(1-\exp(-\frac{(y-x)^2}{2\beta^2})\right)^{k-1}\exp\left(-\frac{(N-k+1)(y-x)^2}{2\beta^2}\right)&y> x, \\ 0&y\leq x. \end{matrix}\right. \label{eq:rayleigh_order}\end{aligned}$$ Hence, the minimum order statistics density also is Rayleigh distributed $$\begin{aligned} f^{\mathrm{Rayleigh}}_{(1,N)}(y;x,\beta) = f^{\mathrm{Rayleigh}}(y;x,\bar{\beta}),\end{aligned}$$ where $\bar{\beta}=\frac{\beta}{\sqrt{N}}$. The MSE of the minimum order statistics is given by $$\begin{aligned} {\mathrm{MSE}}\left(\hat{x}^{\mathrm{Rayleigh}}_{\mathrm{min}}(y_{1:N})\right) = \frac{2\beta^2}{N}.\end{aligned}$$ The joint PDF of $N$ independent observations $y_{1:N}$ is given by \[eq:rayleigh\_pdf\_2\] $$\begin{aligned} f(y_{1:N};x,\beta) &= \frac{\prod_{k=1}^{N}(y_k-x)}{\beta^{2N}}\exp\left[\sum_{k=1}^{N}-\frac{(y_k-x)^2}{2\beta^2}\right]\sigma(y^{(1)}_{1:N}- x). \label{eq:rayleigh_pdf_2_1} \end{aligned}$$ Noting that $$\begin{aligned} \sum_{k=1}^{N}(y_k-x)^2 = \sum_{k+1}^{N}(y_k)^2 - 2x\sum_{k=1}^{N}y_k+Nx^2, \label{eq:rayleigh_pdf_2_2} \end{aligned}$$ the PDF becomes $$\begin{aligned} f(y_{1:N};x,\beta) &= \beta^{-2N}\prod_{k=1}^{N}(y_k-x)\exp\left[\frac{-1}{2\beta^2}\sum_{k=1}^{N}y_k^2\right]\nonumber\\&\times\exp\left[-\frac{Nx^2}{2\beta^2}\right]\exp\left[\frac{x}{\beta^2}\sum_{k=1}^{N}y_k\right]\sigma(y^{(1)}_{1:N} - x). \label{eq:rayleigh_pdf_2_3} \end{aligned}$$ ### Known hyper parameter $\beta$ {#known-hyper-parameter-beta-1} Since  cannot be factorized in the form of $f(y_{1:N};x,\beta) = g(T(y_{1:N}),x)h(y_{1:N})$, the RBLS theorem cannot be used. Hence, even if an MVU estimator exists for this problem, we may not be able to find it. Thus, in case of Rayleigh-distributed measurement noise, we propose unbiased estimators based on order statistics. If the hyper parameter of the distribution is known, the unbiased order statistic based estimator $\hat{x}^{\mathrm{Rayleigh}}(y_{1:N},\beta)$ is then given by, $$\begin{aligned} \hat{x}^{\mathrm{Rayleigh}}(y_{1:N},\beta) &= y^{(1)}_{1:N} - \frac{\sqrt{\pi}\beta}{\sqrt{2N}}, \\ {\mathrm{MSE}}\left(\hat{x}^{\mathrm{Rayleigh}}(y_{1:N},\beta)\right) &=\frac{(4-\pi)\beta^2}{2N}. \end{aligned}$$ which has the same variance as the BLUE estimator. ### Unknown hyper parameter $\beta$ {#unknown-hyper-parameter-beta-1} In case of unknown hyper parameters, as for the known case, no factorization that enables us to use the RBLS theorem can be found. In this case, we propose the following unbiased estimator $$\begin{aligned} \hat{x}^{\mathrm{Rayleigh}}(y_{1:N}) &= \frac{\sqrt{N}}{\sqrt{N}-1}y^{(1)}_{1:N} - \frac{1}{N(\sqrt{N}-1)}\sum_{k=1}^{N}y_k\nonumber\\&= \frac{1}{\sqrt N-1}(\sqrt{N}y^{(1)}_{1:N} - \bar{y}).\end{aligned}$$ Asymptotically, for large $N$, the sample mean and minimum order statistic are independent and the estimator MSE is given by $$\begin{aligned} {\mathrm{MSE}}\left(\hat{x}^{\mathrm{Rayleigh}}(y_{1:N})\right)=\frac{(1+N)(4-\pi)\beta^2}{2N(\sqrt{N}-1)^2}. \label{eq:minimum_orde_unknown_rayleigh}\end{aligned}$$ Weibull distribution {#subsec:weibull_distribution} -------------------- Weibull distribution is a generalization of the Rayleigh, distribution that is parameterized by two parameters–scale parameter $\beta$ and shape parameter $\alpha>0$. In fact Weibull distribution is obtained by relaxing the assumption $\alpha=2$ in the Rayleigh distribution and its density function is given by \[eq:weibull\] $$\begin{aligned} f^{\mathrm{Weibull}}(y_k;x,\beta,\alpha)=\left\{\begin{matrix} \frac{\alpha}{\beta}\left(\frac{y_k-x}{\beta}\right)^{\alpha-1}\exp(-(\frac{y_k-x}{\beta})^\alpha)&y_k> x, \\ 0&y_k\leq x. \end{matrix}\right. \label{eq:weibull_pdf} \end{aligned}$$ and the CDF, for $y_k\geq x$ is given by $$\begin{aligned} F^{\mathrm{Weibull}}(y_k;x,\beta,\alpha)=1-\exp(-(\frac{y_k-x}{\beta})^\alpha). \label{eq:weibull_cdf} \end{aligned}$$ The BLUE estimator, in case of Weibull-distributed measurement noises is given by $$\begin{aligned} \hat{x}_{\mathrm{BLUE}}^{\mathrm{Weibull}}(y_{1:N},\beta,\alpha) &= \frac{1}{N}\sum_{k=1}^{N}y_k - \beta\Gamma(1+\frac{1}{\alpha}), &\quad {\mathrm{MSE}}(\hat{x}_{\mathrm{BLUE}}^{\mathrm{Weibull}}(y_{1:N},\beta,\alpha)) &= \frac{\beta^2}{N}\left[\Gamma(\frac{\alpha+2}{\alpha})-\left(\Gamma(\frac{\alpha+1}{\alpha})\right)^2\right].\end{aligned}$$ The marginal density of the $k$:th order statistic is given by $$\begin{aligned} f^{\mathrm{Weibull}}_{(k,N)}(y;x,\beta,\alpha) = \frac{N\alpha}{\beta}\begin{pmatrix}N-1\\k-1\end{pmatrix}(\frac{y-x}{\beta})^{\alpha-1}\left(1-\exp^{-(\frac{y-x}{\beta})^\alpha}\right)^{k-1}\exp\left(-(N-k+1)(\frac{y-x}{\beta})^\alpha\right). \label{eq:weibull_order}\end{aligned}$$ Hence, the first order statistic density in case of $e_k\sim\mathrm{Weibull}(\beta,\alpha)$, is another Weibull distribution, $$\begin{aligned} f^{\mathrm{Weibull}}_{(1,N)}(y;x,\beta,\alpha) = f^{\mathrm{Weibull}}(y;x,\bar{\beta},\alpha),\end{aligned}$$ where $\bar{\beta}=\sqrt[-\alpha]{N}\beta$. This gives the MSE of the minimum order statistic estimator as $$\begin{aligned} {\mathrm{MSE}}(\hat{x}_{\mathrm{min}}^{\mathrm{Weibull}}(y_{1:N})) = \beta^2N^{\frac{-2}{\alpha}}\Gamma(\frac{\alpha+2}{\alpha})\end{aligned}$$ Given $N$ independent observations, the joint density is given by $$\begin{aligned} f^{\mathrm{Weibull}}(y_{1:N};x,\beta,\alpha)=(\frac{\alpha}{\beta})^N\prod_{k=1}^{N}\left(\frac{y_k-x}{\beta}\right)^{\alpha-1}\exp(-\sum_{k=1}^{N}(\frac{y_k-x}{\beta})^\alpha)\sigma(\min y_k - x) \label{eq:weibull_pdf_joint}\end{aligned}$$ Since  cannot be factorized using Neyman-Fisher factorization, RBLS is not applicable. Additionally, in this case, it is not possible to find an unbiased estimator when the hyper parameters $\alpha$ and $\beta$ are unknown. In case of known hyper parameters, the unbiased minimum order statistic estimator, however, can be computed. The unbiased estimator based on minimum order statistic is given by, $$\begin{aligned} \hat{x}^{\mathrm{Weibull}}(y_{1:N},\beta,\alpha) &= y^{(1)}_{1:N}- \beta N^{-\frac{1}{\alpha}}\Gamma(1+\frac{1}{\alpha}), &\nonumber\\ {\mathrm{MSE}}(\hat{x}^{\mathrm{Weibull}}(y_{1:N},\beta,\alpha)) &=\beta^2N^{\frac{-2}{\alpha}}\left[\Gamma(\frac{\alpha+2}{\alpha})-\left(\Gamma(\frac{\alpha+1}{\alpha})\right)^2\right]. \end{aligned}$$ An order-statistics-based unbiased estimator with unknown hyper parameters of the distribution could not be obtained. Other Distributions {#sec:other_distributions} =================== In this section, we further study the location estimation problem for two other noise distributions. In the rest, the Pareto distribution with positive support is first studied followed by the mixture of uniform and normal distribution. Pareto distribution ------------------- Let the scale parameter $\beta$ (necessarily positive) denote the minimum possible value of $y_k$, and $\alpha>0$ denote the shape parameter. The Pareto Type I distribution is characterized by $\beta$ and $\alpha$ \[eq:pareto\] $$\begin{aligned} f^{\mathrm{Pareto}}(y_k;x,\beta,\alpha)=\left\{\begin{matrix} \alpha\beta^\alpha (y_k-x)^{-(\alpha+1)}&y_k\geq x+\beta, \\ 0&y_k< x+\beta. \end{matrix}\right. \label{eq:pareto_pdf} \end{aligned}$$ and the CDF is given by $$\begin{aligned} F^{\mathrm{Pareto}}(y_k;x,\beta,\alpha)=1-\left(\frac{\beta}{y-x}\right)^\alpha. \label{eq:pareto_cdf} \end{aligned}$$ For the BLUE we get, $$\begin{aligned} \hat{x}_{\mathrm{BLUE}}^{\mathrm{Pareto}}(y_{1:N},\beta,\alpha) &= \frac{1}{N}\sum_{k=1}^{N}y_k - \frac{\alpha\beta}{\alpha-1},&\quad \alpha>1 \\ {\mathrm{MSE}}(\hat{x}_{\mathrm{BLUE}}^{\mathrm{Pareto}}(y_{1:N},\beta,\alpha)) &=\frac{\alpha\beta^2}{N(\alpha-1)^2(\alpha-2)},&\quad \alpha>2 \end{aligned}$$ The RBLS theorem cannot be used in case Pareto-distributed noises. We provide an unbiased estimator using minimum order statistics and its variance. The marginal density of the $k^{\mathrm{th}}$ order statistic, for $y\geq x+\beta$ is given by $$\begin{aligned} f^{\mathrm{Pareto}}_{(k,N)}(y;x,\beta,\alpha) = N\alpha\beta^\alpha (y-x)^{-(\alpha+1)}\begin{pmatrix}N-1\\k-1\end{pmatrix}\left(1-(\frac{\beta}{y-x})^\alpha\right)^{k-1}\left(\frac{\beta}{y-x}\right){\alpha(N-k)}. \label{eq:pareto_order}\end{aligned}$$ The minimum order statistic has the same form of distribution $$\begin{aligned} f^{\mathrm{Pareto}}_{(1,N)}(y;\beta,\alpha) = f^{\mathrm{Pareto}}(y;\beta,\bar{\alpha}),\end{aligned}$$ where $\bar{\alpha}=N\alpha$. The MSE of the minimum order statistic estimator is $$\begin{aligned} {\mathrm{MSE}}(\hat{x}_{\mathrm{min}}^{\mathrm{Pareto}}(y_{1:N})) = \frac{N\alpha\beta^2}{N\alpha-2}\end{aligned}$$ The unbiased estimator is thus given by, $$\begin{aligned} \hat{x}^{\mathrm{Pareto}}(y_{1:N},\beta,\alpha) &= y^{(1)}_{1:N} - \frac{N\alpha\beta}{N\alpha-1},&\quad N\alpha>1 \\ {\mathrm{MSE}}(\hat{x}^{\mathrm{Pareto}}(y_{1:N},\beta,\alpha)) &=\frac{N\alpha\beta^2}{(N\alpha-1)^2(N\alpha-2)},&\quad N\alpha>2 \end{aligned}$$ No unbiased estimator for unknown hyper parameter case could be found for Pareto distribution. Mixture of Normal and Uniform Noise Distribution ================================================ Suppose the error is distributed as $$e_k\sim \alpha\mathcal{N}(0,\sigma^2) + (1-\alpha) \mathcal{U}[0,\beta],$$ where $\alpha$ is the mixing probability of the mixture distribution. Define $f^{\mathrm{\mathcal{U},\mathcal{N}}}(y_k)$ as the probability density function of the considered mixture distribution given by $$\begin{aligned} \label{eq:mixture_pdf} f&^{\mathrm{\mathcal{U},\mathcal{N}}}(y_k;x,\alpha,\sigma^2,\beta) =\nonumber\\ &\begin{cases} \begin{aligned} & \frac{a}{\sqrt{2\pi\sigma^2}}\exp\left[-\frac{(y_k-x)^2}{2\sigma^2}\right] +\frac{1-a}{\beta} \end{aligned} & \parbox[t]{4cm}{\raggedright $0\leq y_k-x\leq \beta$}\\ \frac{a}{\sqrt{2\pi\sigma^2}}\exp\left[-\frac{(y_k-x)^2}{2\sigma^2}\right] & \parbox[t]{4cm}{\raggedright Otherwise.}\\ \end{cases} \end{aligned}$$ The BLUE, in case of the mixture of normal and uniform measurement noises is given by $$\begin{aligned} \hat{x}_{\mathrm{BLUE}}^{\mathrm{\mathrm{\mathcal{U},\mathcal{N}}}}(y_{1:N},\alpha,\beta,\sigma^2) &= \frac{1}{N}\sum_{k=1}^{N}y_k - \frac{\beta(1-\alpha)}{2}, \\ {\mathrm{MSE}}\left(\hat{x}_{\mathrm{BLUE}}^{\mathrm{\mathrm{\mathcal{U},\mathcal{N}}}}(y_{1:N},\alpha,\beta,\sigma^2)\right) &=\nonumber\\ &\frac{\beta^2 \left(1+(2-3\alpha)\alpha\right)+12\alpha\sigma^2}{12N}. \end{aligned}$$ Noting that at $y_k-x=0$ contributions of the uniform distribution and the mean (mode) of the normal distribution are added together,  is maximized at this point. The order statistics PDF for $0\leq y-x\leq \beta$ is given by $$\begin{aligned} f_{(k,N)}^{\mathrm{\mathrm{\mathcal{U},\mathcal{N}}}}&(y;\alpha,\beta,\sigma^2,x,k) =\nonumber\\ &N\begin{pmatrix}N-1\\k-1\end{pmatrix}\left(\frac{\alpha\exp(-\frac{(y-x)^2}{2\sigma^2})}{\sqrt{2\pi\sigma^2}}+\frac{1-\alpha}{\beta}\right)\nonumber\\ &\times\left(\frac{(1-\alpha)(y-x)}{\beta}+\frac{\alpha}{2}(1+\mathrm{Erf}\left[\frac{y-x}{\sqrt{2\sigma^2}}\right])\right)^{k-1}\nonumber\\ &\times\left(1+\frac{(\alpha-1)(y-x)}{\beta}-\frac{\alpha}{2}(1+\mathrm{Erf}\left[\frac{y-x}{\sqrt{2}\sigma}\right])\right)^{N-k},\end{aligned}$$ where $\mathrm{Erf}(\cdot) = \frac{2}{\sqrt{\pi}}\int_{0}^{\cdot}e^{-t^2}dt$ is the error function. In order to find the best order statistic estimator, we maximize the likelihood function $\ell(k\mid y=x,a,\beta,\sigma^2)$ $$\begin{aligned} \ell(k\mid y=x,\alpha,\beta,\sigma^2) &= N\begin{pmatrix}N-1\\k-1\end{pmatrix}2(2-\alpha)^{-k}(1-\frac{\alpha}{2})^N\alpha^{k-1}\nonumber\\&\times\left(\frac{1-\alpha}{\beta}+\frac{\alpha}{\sqrt{2\pi}\sigma}\right). \label{eq:best_order_likelihood_general} \end{aligned}$$ Noting that $\left(\frac{1-\alpha}{\beta}+\frac{\alpha}{\sqrt{2\pi}\sigma}\right)$ is always positive and independent of $k$, we extract it from the likelihood function. Simplifying  by means of manipulating the terms, we get $$\begin{aligned} 2(2-\alpha)^{-k}&=2^{1-k}(1-\frac{\alpha}{2})^{-k},\\ \alpha^{k-1}&=(2\frac{\alpha}{2})^{k-1}=2^{k-1}(\frac{\alpha}{2})^{k-1}. \end{aligned}$$ the likelihood function to be maximized can be re-written as $$\begin{aligned} \ell(k\mid y=x,\alpha,,\sigma^2) \propto \begin{pmatrix}N-1\\k-1\end{pmatrix}(\frac{\alpha}{2})^{k-1}(1-\frac{\alpha}{2})^{N-k}. \label{eq:best_order_likelihood_modified} \end{aligned}$$ In order to find the maximum likelihood estimate $\hat{k} = \operatorname*{arg\,max}_k \ell(k\mid y-x=0)$, we note that  is a binomial distribution with probability of success $\frac{\alpha}{2}$. Hence, the maximum of the function is given at the mode of the distribution, $$\begin{aligned} \hat{k}=\lfloor \frac{N\alpha}{2}\rfloor+1\quad \mathrm{or}\quad \lceil\frac{N\alpha}{2}\rceil.\end{aligned}$$ This gives the best order statistic estimator for the case when noise is a mixture of normal and uniform distribution as $$\begin{aligned} \hat{x}^{\mathrm{\mathrm{\mathcal{U},\mathcal{N}}}}(y_{1:N},\alpha)= y^{(\hat{k})}_{1:N}.\end{aligned}$$ distribution bias MSE ------------------------------------------ --------------------------------------------------------- ---------------------------------------------------------- $\mathcal{U}[0,\beta]$ $\frac{\beta}{N+1}$ $\frac{2\beta^2}{(N+1)(N+2)}$ \[2mm\] $\mathrm{Exp}(\beta)$ $\frac{\beta}{N}$ $\frac{2\beta^2}{N^2}$ \[2mm\] $\mathrm{Rayleigh}(\beta)$ $\frac{\sqrt{\pi}\beta}{\sqrt{2N}}$ $\frac{2\beta^2}{N}$ \[2mm\] $\mathrm{Weibull}(\beta,\alpha)$ $\beta N^{-\frac{1}{\alpha}}\Gamma(1+\frac{1}{\alpha})$ $\beta^2N^{-\frac{2}{\alpha}}\Gamma(1+\frac{2}{\alpha})$ \[2mm\] $\mathrm{Pareto}(\beta,\alpha)$ $\frac{N\alpha\beta}{N\alpha -1}$ $\frac{N\alpha\beta^2}{N\alpha -2}$ : Bias and MSE of minimum order statistics estimators $\hat{x}_{\mathrm{min}}^{p_e}$. \[tbl:estimators\_biased\] \[tbl:estimators\] Performance Evaluation {#sec:performance_evaluation} ====================== The estimators (both unbiased and the ones without bias compensation) derived in sections \[sec:distributioons\_in\_the\_exponential\_family\]–\[sec:other\_distributions\] for different noise distributions together with their MSE are summarized in Tables \[tbl:estimators\] and \[tbl:estimators\_biased\]. The biased minimum order statistics based estimators and their MSE are also The estimators derived for each noise distribution are compared against each other as a function of the sample size $N\in[2,\ldots,2000]$. Additionally, in order to verify the analytical derivations of the estimator variances, they are compared against the numerical variances obtained from $M=5000$ Monte Carlo runs. Simulation Setup ---------------- For each sample size, $N$ noisy measurements of the unknown parameter $x$ are generated. The hyper parameters of the noise distributions are randomly selected in each repetition. In order to have a fair comparison, the hyper parameters are randomly drawn such that the error densities are mostly in the same range for all scenarios. The noise realizations are generated from the six considered distributions with the following hyper parameters - Uniform noise: $\beta\sim\mathcal{U}[6,50]$ - Exponential noise: $\beta\sim\mathcal{U}[5,14]$ - Rayleigh noise: $\beta\sim\mathcal{U}[5,12]$ - Weibull noise:$\beta=1$, $\alpha\sim\mathcal{U}[5,10]$ - Pareto noise: $\beta=6$, $\alpha\sim\mathcal{U}[2.1,2.5]$ - Mixture noise: $\sigma\sim\mathcal{U}[1,9]$, $\beta\sim\mathcal{U}[1,50]$ The empirical CDF of the error values used in the simulations are presented in Figure \[fig:cdf\_error\]. The support of the noise values, as can be read from the figure, is $\bm{e}_m\in[0,60]$ unit. Let $\hat{x}_N^{(m)}$ denote the estimated value of the unknown parameter $x$ in the $m$:th repetition obtained from a sample of size $N$. For each noise distribution, the estimators’ performances are evaluated in terms of the obtained MSEs. The theoretical MSE of each estimator, as defined in Table \[tbl:estimators\] and Table \[tbl:estimators\_biased\], is compared against the numerical MSE obtained in simulations. We let ${\mathbb{E}}[\hat{x}_N] = \frac{1}{M}\sum_{m=1}^{M}\hat{x}_N^{(m)}$ and define $$\begin{aligned} \hat{b}_N &= {\mathbb{E}}[\hat{x}_N] - x\\ \hat{\sigma}^2_N &= \frac{1}{M}\sum_{m=1}^{M}(\hat{x}_N^{(m)}-{\mathbb{E}}[\hat{x}_N])^2. \end{aligned}$$ The numerical MSE for each sample size $N$ is then computed by $$\begin{aligned} \hat{\mathrm{MSE}}(\hat{x}_N) = \hat{\sigma}^2_N + \hat{b}_N^2. \end{aligned}$$ ![Analytical (marked with solid lines) and numerical (marked with crosses) MSE for uniform noise distribution as a function of the sample size $N$.[]{data-label="fig:uniform_blue_mvu"}](cdf_error.pdf){width="1.14\linewidth"} ![Analytical (marked with solid lines) and numerical (marked with crosses) MSE for uniform noise distribution as a function of the sample size $N$.[]{data-label="fig:uniform_blue_mvu"}](estimationMse_unif.pdf){width="\linewidth"} Simulation Results ------------------ Figure \[fig:uniform\_blue\_mvu\] presents the performance of the four estimators when the noise is uniformly distributed. The solid lines correspond to the theoretical MSEs and the crosses are the numerical MSEs obtained from $M=5000$ repetitions. Both MVU estimators, with and without any knowledge of the hyper parameters of the underlying noise, result in noticeably less MSE compared to the BLUE estimator. The minimum order statistics estimator also outperforms BLUE when measurements are corrupted with additive, uniformly distributed, noise. It can be further observed that if the hyper parameter $\beta$ is unknown, the MSE of the proposed estimator is negligibly larger than the case with known $\beta$. \ For the exponential noise distribution, as shown in Figure \[fig:exponential\_blue\_mvu\_all\], there is still a non-negligible difference between BLUE and the other three estimators in terms of estimators’ MSE. However, the two MVU estimators, specially for large values of $N$, behave similarly. In order to verify their performance for smaller sample sizes, Figure \[fig:exponential\_blue\_mvu\_zoom\] illustrates the variances of all estimators for $N\leq 20$. At the beginning, $N\in[2,4]$ the estimator with unknown hyperparameter has the largest MSE. However, for larger sample sizes, the two MVU estimators are almost equal and both have less MSE than the BLUE estimator. As in case of uniformly distributed measurement noise, the minimum order statistics estimator outperforms BLUE specially for large sample sizes. In case of Rayleigh noise distribution, as given in Table \[tbl:estimators\], the minimum order statistics estimator has the largest MSE while the BLUE and the proposed unbiased estimator with known hyper parameter, result in similar estimation variance. This can be verified also in the simulation results presented in Figure \[fig:rayleigh\_blue\_order\_all\]. For large sample sizes, $N>20$, these two estimators and the proposed estimator with unknown hyperparameter have similar values. However, for the smaller sample sizes, as illustrated in Figure \[fig:rayleigh\_blue\_order\_zoom\], the BLUE (and order statistic with known hyper parameter) estimator has smaller variance compared to the case with unknown hyper parameter. The minimum order statistics estimator results in larger MSE compared to the other three estimators in case of Rayleigh noise distribution. As Table \[tbl:estimators\] suggests, for Pareto and Weibull noise distributions, we only derived BLUE and an unbiased order statistics based estimators when the two hyperparameters of the distributions are known. For both noise distributions, the MSE of the two unbiased estimators as well as the MSE of the minimum order statistics estimator are compared and the results are presented in Figure. \[fig:pareto\_weibull\_blue\_order\]. In both cases, the proposed estimators outperform the BLUE in terms of variance. The minimum order statistics estimator results in a lower MSE than the BLUE for Weibull noise distributions. However, in case of Pareto noise, the BLUE has a better performance compared to the minimum order statistics estimator. In case of mixture noise distribution, we consider three different scenarios based on the mixing probabilities; two extreme cases with dominant contribution from uniform noise, $\alpha=0.01$, and dominant contribution from normal noise, $\alpha=0.99$, and the case with $\alpha=0.5$. Fig. \[fig:mixture\_pdf\] illustrates the histogram of the noise realizations of the considered mixture noise distributions $e_k\sim \alpha\mathcal{N}(0,8^2) + (1-\alpha)\mathcal{U}(0,60)$ and the fitted densities. The empirical CDFs of the errors for the three cases are presented in Figure \[fig:mixture\_cdf\]. In order to estimate the unknown parameter $x$, in each Monte Carlo run, we sort the measurements and then find the $(\lfloor\frac{N\alpha}{2}\rfloor+1)$:th component. Figure \[fig:mixture\_order\] presents the estimation MSE for the three different scenarios with different mixing probabilities. As the results indicate, when the main contribution of the noise is from uniform distribution, $\alpha=0.01$, BLUE outperforms the proposed estimator. In this case, a periodic behavior for the MSE can be observed. The jumps in the MSEe occur exactly at pints where $\lfloor\frac{N\alpha}{2}\rfloor+1$ switches from the $k$:th measurement to the $k+1$:th measurement. For instance, for $N\in[1,199]$, $\lfloor\frac{N\alpha}{2}\rfloor=0$, hence $\hat{x}=y^{(1)}$. However, at $N=200$, $\lfloor\frac{N\alpha}{2}\rfloor=1$, resulting in $\hat{x}=y^{(2)}$. The proposed estimator and the BLUE result in similar estimation MSE for $\alpha=0.99$, as shown in Figure \[fig:estimationMse\_mixture99\], in which the normal component is the dominant source of error. However, the most interesting results are obtained when both distributions have equal contributions in the measurement noise, [*i.e*]{} $\alpha=0.5$. In this case, as Figure \[fig:estimationMse\_mixture5\] suggests, the proposed estimator outperforms the BLUE. Conclusions {#sec:conclusions} =========== In this work, the location estimation problem was studied in which an unknown parameter was estimated from observations under additive noise. Multiple noise distributions were considered and, in some cases, MVU estimators were proposed. In other cases an unbiased estimator based on minimum order statistic was derived. Furthermore, if applicable, MVU and minimum order statistic estimators without any knowledge of the hyper parameters of the underlying noise distributions were provided. The results of all the estimators were compared with BLUE in terms of variance for various measurement sample sizes. The results indicate better performance of the proposed estimators compared to BLUE. Additionally, the location estimation problem under mixture of normal and uniform noise distribution was studied and the numerical MSE of the proposed estimator were evaluated. The simulation results indicate that for the extreme cases where either of the two components, Gaussian or uniform, are dominant, the proposed estimator cannot beat the BLUE. However, when the mixing probability is not in the extreme region, [*e.g*]{} larger than $1$ percent, the proposed estimator has a noticeably less MSE compared to the BLUE.

--- abstract: 'In this work we study neighborhoods of curves in surfaces with positive self-intersection that can be embeeded as a germ of neighborhood of a curve on the projective plane.' author: - 'M. Falla Luza' - 'P. Sad' title: Positive Neighborhoods of Curves --- Introduction ============ We study in this paper neighborhoods of compact, smooth, holomorphic curves of complex surfaces which have positive self intersection number. Our main purpuse is to give a condition that guarantees the existence of an embedding of a neighborhood of the curve into the projective plane. The first example of a result on this problem comes from [@FA]; in that paper the authors showed that if the curve has genus 0 and self intersection number equal to 1 then the existence of three different fibrations over it implies that some neighborhood is diffeomorphic to a neighborhood of the line in the projective plane. In this paper we consider curves of self-intersection $d^2$ with $d \geq 2$. Since a fibration over a curve of genus 0 is defined by a local submersion over $\Pp^1$ (that is, defined in a neighborhood of the curve), we may wonder if in the case of higher genus the existence of a number of local submersions is enough to guarantee an embedding into the projective plane $\Pp^2$. It is in fact a necessary condition. In order to discuss this, let us suppose that a curve $C$ contained in some surface $S$ can be embedded in $\mathbb P^2$ as a curve $C_0$ of degree $d \geq 2$ (we have of course to start with $C\cdot C= d^2$ in $S$). It is easy to find infinitely many submersions in a neighborhood of $C_0$. For example, we take two curves $\{A=0\}$ and $\{B=0\}$ of the same degree $l\in {\mathbb N}$ which cross each other in $l^2$ distinct points not in $C_0$. It can be seen that the map $A/B$, which is well defined outside $\{A=0\}\cap \{B=0\}$, has no multiple fibers so that it has only a finite number of critical points; if $C_0$ avoids all these points then $A/B$ is a submersion in some neighborhood of $C_0$ and the restriction of $A/B$ to $C_0$ is a ramified map from $C_0$ to $\Pp^1$ of degree $l.d$. We will be particularly interested in the case $l=1$, that is, $A=0$ and $B=0$ are lines whose common point is not in $C_0$; the submersion $A/B$ will be called a [*pencil submersion*]{} and the restriction of $A/B$ to $C_0$ is a ramified map of degree $d$ (any local submersion that leaves such a trace in $C_0$ is in fact a pencil submersion). We see that to be equivalent to a neighborhood of $C_0$, a neighborhood of $C$ has to carry also many submersions to $\Pp^1$, the surprising feature in [@FA] is that only three submersions are needed. The converse is not true as we can see in the following example. \[fake-example\] Consider the rational curve in $\mathbb P^2$ defined in affine coordinates by the equation $y^2=x^2(x+1)$; it is a smooth rational curve except for the node at the point $(0,0)$. We blow up first at a point in the curve different from $(0,0)$, and then we blow up at $(0,0)$. The strict transform is a smooth rational curve $C$ of self intersection number equal to 4 with many local submersions (which come from submersions constructed in the plane as above), but its neighborhood can not be embedded in the plane: given a submersion constructed using $l=1$ as above (before blow up’s), we notice that it induces a ramified map from $C$ to $\Pp^1$ of degree 3; but for a conic $C_0$ in the plane (which has of course self intersection number equal to 4), the ramified map induced by any local submersion is of even degree. A more refined question would be: can we obtain an embedding once it is assumed the existence of three local submersions in a neighborhood of $C$ whose restrictions to $C$ are meromorphic maps of degree (a multiple of) $d$? We give a partial negative answer in Section \[sec-examples\]. We introduce then an extra condition (also a necessary one). A curve $C$ that has an embedding $\phi:C \rightarrow C_0\subset \mathbb P^2$ carries naturally a special set of meromorphic maps ${\bf G}_{\phi}$ =$\{G_{|_{C_0}}\circ {\phi},\,\, G \,\,\,pencil\, submersion\}$. A set $\{F_i\}$ of submersions defined in a neighborhood of $C$ whose restrictions to $C$ have no common critical points is [**projective at C**]{} if ${F_i}_{|_C}\in {\bf G}_{\phi}$. The submersions are called [**independet**]{} if the singularities of the correspondent pencils on $\Pp^2$ are not aligned. We may state then our main result: \[main-thm\] The existence of a projective triple of independent submersions at $C$ implies the existence of an embedding of a neighborhood of $C$ into the projective plane. The submersions in the statement of the Theorem are supposed to produce different fibrations; we remark that if $F$ is a submersion over $\Pp^1$ and $T$ is a Moebius transformation, then $F$ and $T\circ F$ induce the same fibration. The fibers of a submersion define a regular foliation in a neighborhood of $C$, which is generically transverse to $C$ with tangency points at the critical points of the restriction to the curve; the submersion is a meromorphic first integral for the foliation. The converse does not hold, that is, this type of foliation may not have a first integral, see [@MEZ]. We mention that the study of neighborhoods of curves has already been pursued when the self-intersection is not positive as we can see in [@GRA], [@SAV] and [@UE]. This paper is organized as follows: Section \[sec-examples\] presents some examples and it is followed by Section \[sec-const-mer-maps\] where we discuss how to built meromophic maps starting from two different pencil submersions. This allows (Section \[sec-new-foliation\]) to show the existence of foliations defined in a neighborhood of the curve which have this curve as an invariant set, and finally in Section \[sec-proof-thm\] we prove our theorem. Examples {#sec-examples} ======== This Section has two parts. In the first part we give examples of surfaces containing smooth curves of self-intersection number $d^2$ which are not embeddable in the plane, although they are fibered by submersions whose restrictions to the curves are meromorphic functions of a degree multiple of $d$. Once this is done, we give examples which satisfy the extra condition of our Theorem but have only one or two fibrations and do not embed them in the plane. Separating branches and examples with 3 fibrations -------------------------------------------------- We will use the following construction. Let us consider a curve H with an ordinary singularity $P$ with $m$ branches $L_1,\dots,L_m$. For each branch $L_j$ we take a neighborhood $V_j$ which is biholomorphic to a bidisc $D_j$ by means of a biholomorphism $\phi_j:D_j\rightarrow V_j$ ; we assume that $\delta L_j \cap V_i =\emptyset$ for all $i\ne j$. We fix a neighborhood $V$ of $H\setminus \cup_1^m L_j$. Finally we take the disjoint union of $V$ with all th $D_j$, and glue $D_j$ to $V$ using the restriction of the map $\phi_j$ to $\phi_j^{-1}(V\cap V_j)$. In this way the union of the sets $V_j$, which contains $P$, is replaced by $m$ copies of the bidisc, and there is a new curve $H^{\prime}$ replacing $H$ inside a new surface without the ordinary singularity. As for the self-intersection number $H^{\prime}\cdot H^{\prime}$, we have that $H^{\prime}\cdot H^{\prime}= H\cdot H -m(m-1)= (H\cdot H -m^2) +m$. Also any holomorphic foliation $\mathcal F$ defined in $V\cup_1^m V_j$ induces naturally a holomorphic foliation in the new surface which is $\mathcal F$ in $V$ and $\phi_j^*(\mathcal F)$ in each $D_j$. We refer to this construction as [*separating branches of $H$ at P*]{}. ![Separating branches[]{data-label="fig:viaduto"}](viaduto.png) Let us consider then a smooth plane curve $C^{\prime}$ of degree $d^{\prime}$ and genus $g(C^{\prime})= \dfrac{(d^{\prime}-1)(d^{\prime}-2)}{2}$; it can be also immersed in the plane as a curve $C$ of degree $d$ for any $d>2g(C^{\prime})$ with a number $s$ of nodal points such that $d^2-3d-2s={d^{\prime}}^2 -3d^{\prime}$. We choose $d-d^{\prime}=k^2$ for some $k\in \mathbb N$ such that (i) $d^{\prime}$ divides $k^3-k^2$, (ii) $d^{\prime}$ does not divide $2k^2$; we choose also three pencils $d\left(\dfrac {u_j}{v_j}\right)=0$ of curves of degree $k$ whose sets of $k^2$ base points lie in the regular part of $C$ and are two by two disjoint. After blowing up at these $3k^2$ points and separating branches at the nodal points of $C$ we get a curve $\tilde C$ containing in some surface with self-intersection number equal to $d^2-3k^2-2s = {d^{\prime}}^2$; the maps $\dfrac {u_j}{v_j}$ become submersions whose restrictions to $\tilde C$ are meromorphic maps of degree $k.d- k^2$, which is a multiple of $d^{\prime}$ because of (i). A neighborhood of $\tilde C$ is not equivalent to a neighborhood of $C^{\prime}$ in the plane. In fact, let us take a linear pencil $\mathcal L$ in the plane with base point outside $C$ and transverse to the branches at each nodal point (this is before blow-up‘s and separation of branches). We have $2d=Tang(\mathcal L,C)+ \chi(C)$; since $tang(\mathcal L,C,P)=2$ for each nodal point $P$, we get $2d = 2s+\chi(C)+ tang(\mathcal L,C)$, where the last term counts the tangencies with the regular part of $C$. These tangencies persist when we blow up and separate branches; therefore, if a neighborhood of $\tilde C$ is equivalent to a neighborhood of $C^{\prime}$, we get in this neighborhood a foliation $\mathcal L^{\prime}$ with $Tang(\mathcal L^{\prime}, C^{\prime})=tang(\mathcal L,C)=2d-2s-\chi(C)$. It follows that $(deg(\mathcal L^{\prime})+2)d^{\prime}=2d-2s-\chi(C)+\chi(C^{\prime})=2d-2s+2g(C)-2g(C^{\prime})$ and since $g(C)-s=g(C^{\prime})$, we conclude that $(deg(\mathcal L^{\prime})+2)d^{\prime}=2d= 2d^{\prime}+2k^2$, a contradiction because of (ii). We remark that when $d=4, d^{\prime}=3$ or $d=3,d^{\prime}=2$ the construction can be done with $k=1$ because $d>2g(C^{\prime})$ (and obviously (i) is satisfied in both cases). When $d=4, d^{\prime}=3$ we have also that (ii) holds true. In the general case $d^{\prime}>3$ we may choose $k=d^{\prime}+1$ for example in order to get both (i) and (ii) satisfied. The special case $d=3,d^{\prime}=2$ (and $k=1$) can be treated with a small difference in what concerns the proof that the neighborhood of $\tilde C$ is not equivalent to a neighborhood of $C$: we select $\mathcal L$ as the pencil whose base point is the node point $P$ of $C$. Since $Tang(\mathcal L,C)= 6=tang(\mathcal L,C,P)$, we see that there is no other point of tangency between $\mathcal L$ and $C$. We get then $(deg(\mathcal L^{\prime})+2).2= 4+2$ (after separating the branches at $P$ we obtain 2 radial singularities belonging to $\tilde C$ and a fortiori to $C^{\prime}$) and therefore $deg(\mathcal L^{\prime})=1$. But is impossible for a foliation of degree 1 in the plane to have 2 radial singularities. It would be nice to have examples where the degree induced by the submersions on the curve is exactly $d^{\prime}$. Special Examples ---------------- A construction already presented in [@SA] of a pair (curve, surface) with a submersion whose restriction to the curve is a ramification map given a priori (we will say that the submersion is a [**lifting**]{} of a ramification map). We start with a line bundle of Chern class $n\in \mathbb N$ over a curve $C$; the lines of the bundle define a foliation $\mathcal L$ in the total space of the bundle. Let $f: C \rightarrow \Pp^1$ be a ramified map with simple critical points and let $p$ be one of these points. There exists an involution $i$ defined in a neighborhood of $p$ in $C$ by $f(q)= f(i(q))$ for $q$ close to $p$. We fix a neighborhood $U$ of $p$ and a holomorphic diffeomorphism $H:U \rightarrow {\mathbb D}\times {\mathbb D}$ such that: 1) $H(p)= (0,0)$; 2) $H(C \cap U)=\{(z_1,0)\in {\mathbb D}\times {\mathbb D}\}$; 3)$H$ takes $\mathcal L$ to the foliation $dz_1=0$ and 4) $h:= H|_{C \cap U}$ conjugates $i$ to the involution $z \longmapsto -z$, that is: $h(i(q))=-h(q)$. We take also a biholomorphism $\psi$ from ${\mathbb D}\times {\mathbb D}$ to a neighborhood of $(0,0)$ with the properties: 1) $\psi (0,0)=(0,0)$; 2) $\psi(z_1,0)= (z_1,0)$; 3) $\psi(\{1/2<|z_1|<1\}\times {\mathbb D})$ is saturated by leaves of the foliation $dZ_2-Z_1dZ_1=0$ and 4) $\psi$ is a holomorphic diffeomorphism when restricted to $\{1/2 < |z_1| <1\}\times {\mathbb D}$ that sends the foliation $dz_1=0$ to the foliation $dZ_2-Z_1dZ_1=0$. Put $H_1= \psi \circ H$. We remove from the total space of the line bundle the fibers over the points of $h^{-1}(\{|z_1|\le 1/2\})$ and glue $\psi({\mathbb D}\times {\mathbb D})$ to the remaining set using $H_1$. In this way we get a new holomorphic surface which contains $C$ (the same curve we started with) and a holomorphic foliation transverse to $C$ except at $p$, where the tangency is simple. Furthermore, the “local holonomy” of the new foliation at $p$ is exactly $i$. We repeat the same procedure for all critical points of $f$. At the end, we have a holomorphic surface that contains $C$ and a holomorphic foliation which is transverse to $C$ except at the critical points of $f$; we may even assume that the self-intersection number of $C$ is $n\in \mathbb N$. The map $f$ can be extended along the leaves (because of its compatibility with the involutions involved), producing the desired lifting. A similar construction can be made if the critical points are not simple. Let us give two examples of pairs (curve, surface) which are not embeddable in the projective plane. We have already noticed that, in order to be embeddable in the projective plane, all the submersions defined in the neighborhood of the curve must have as restrictions maps whose degrees are multiple of $d$ (here $d^2$ is the self-intersection number of the curve in the surface). This does not happen in the example given in the Introdution. We give now another example of a different nature. Take $C\subset \mathbb P^2$. We start by claiming that there exists a ramification $f:C\rightarrow \Pp^1$ of degree $(d-1)d$ such that the set of poles is not contained in any curve of degree $d-1$. In order to see this, let us start with a ramification map $f_0:C\rightarrow \Pp^1$ defined as the restriction of $\dfrac {1}{Q_0}$ to $C$, where $Q_0$ is a polynomial of degree $d-1$ which intersects $C$ transversely at $l=(d-1)d$ different points $P_1,\dots,P_l$. Let us consider nearby points $P_1^{\prime},\dots,P_l^{\prime}$ and apply Riemann-Roch’s theorem to $D=P_1^{\prime} +\dots +P_l^{\prime}$: $l(D)\ge (d-1)d-g+1$; if we want to have $l(D)>1$, we ask for $(d-1)d-g+1>1$, or $(d-1)d >\dfrac{(d-1)(d-2)}{2}$, which is always true when $d>1$. In fact, from the proof of Riemann-Roch’s theorem , since $(P_1^{\prime},\dots,P_l^{\prime})$ is close to $(P_1,\dots,P_l)$, we may choose a meromorphic function close to $f_0$, so its polar divisor is $D$. On the other hand, the points $(P_1^{\prime},\dots,P_l^{\prime})$ which belong to a curve of degree $d-1$ are contained in a subvariety of dimension $\dfrac{d(d+1)}{2}-1$, and all we have to do is check if $(d-1)d > \dfrac{d(d+1)}{2}-1$, which is obvious if $d \ge 3$ (we remark that there are not two different curves of degree $d-1$ passing through the $(d-1)d$ points $P_1^{\prime},\dots,P_l^{\prime}$). We select then $P_1^{\prime},\dots,P_l^{\prime}$ outside this subvariety in order to get the ramification map $f$ and take a lifting $F$ defined in a surface $S$. We prove then the statement: there is no embedding $\Phi: S\rightarrow {\mathbb P^2}$. In fact, the submersion $F\circ{\Phi^{-1}}$ defined in a neighborhood of $C_0 \subset {\mathbb P^2}$ extends to $\mathbb P^2$ as a meromorphic function (holomorphic in a neighborhood of $C_0$). We observe that, for $d\ge 3$, given two embeddings $\phi_i:C\rightarrow {\mathbb P^2}$, i=1,2 there exists an automorphism $T\in Aut(\mathbb P^2)$ such that $T(\phi_1(C))= \phi_2(C)$, see Appendix. Then the map $\Phi|_{C}:C \rightarrow C_0$ comes from a linear map on $\Pp^2$ and poles of $f$ are the intersection of $C$ with a curve of degree $d-1$, which is impossible. We present now an example of a non embeddable pair (curve, surface) with a set of two fibrations which is projective at the curve. We start with a projective, smooth curve $C$ and select two pencil submersions; let $f_1$ and $f_2$ be the associated ramification maps of $C$. The tangencies between the pencils are obviously pieces of the common line. We will replace one of these pieces by a non-invariant curve of tangencies between two new foliations. The idea is the same used above to realize ramification maps; the homeomorphism $\psi$ is going to be changed. The point $p$ this time is a point of tangency, and the coordinate chart $H$ sends the foliations associated to the submersions to two foliations $(dz_1=0, \mathcal H)$. We consider in $\mathbb C^2$ a couple of foliations ($dZ_1=0, \mathcal H^{\prime}: d(Z_1-Z_2(Z_2-Z_1))=0)$, which have $Z_1=2Z_2$ as non-invariant line of tangencies. The homeomorphism $\psi$ is choosed in order to satisfy: 1) $\psi(0,0)=(0,0)$ and $\psi(z_1,0)=(z_1,0)$; 2) $\psi|_{\{1/2<|z_1|<1\}\times \mathbb D}$ is a holomorphic diffeomorphism over its image that sends $(dz_1=0,\mathcal H)$ to $(dZ_1=0, \mathcal H^{\prime})$. We put again $H_1=\psi \circ H$,which is the new glueing map. We can see that $C^2$ does not change and so the germ of surface is not isomorphic to $(C, \Pp^2)$. We could also use in the construction the pair of foliations ($dZ_1=0, \mathcal H^{\prime}: d(Z_1-Z_2(Z_2-Z_1^{k+1}))=0)$ for $k\in \mathbb N$, but the curve $C$ will have self-intersection number equal to $d^2-k$. Constructing meromorphic maps {#sec-const-mer-maps} ============================= Let us once more describe the setting we are going to analyse. We have a curve $C$ contained in some surface $S$ with $C\cdot C= d^2$ and $d\in \mathbb N$. There exist three submersions $F$, $G$ and $H$ defined in $S$ and taking values in $\Pp^1$ which define foliations $\mathcal F$, $\mathcal G$ and $\mathcal H$ generically transversal to $C$ whose leaves are the levels curves. In order to simplify the exposition, we assume that all tangencies with $C$ are simple and distinct (when we look to the tangencies for any pair of foliations). We denote $f=F|_{C}$, $g=G|_{C}$ and $h=H|_{C}$, all of them ramification maps from $C$ to $\Pp^1$ whose ramification points correspond to the tangency points of the foliations (because $F$, $G$ and $H$ are submersions). Furthermore, we assume that $C$ embedds into $\mathbb P^2$ by a map $\phi:C\rightarrow C_0$; $C_0$ is a smooth algebraic curve of degree $d$. In order to complete the picture, we select pencil submersions $F_0$, $G_0$ and $H_0$ (with associated foliations ${\mathcal F}_0$, ${\mathcal G}_0$ and ${\mathcal H}_0$), with singular points not aligned, which restric to $C_0$ as $d$ to $1$ maps $f_0$, $g_0$ and $h_0$ to $\Pp^1$ and ask $\{f,g,h\}$ to be conjugated by $\phi$ to $\{f_0,g_0,h_0\}$: $f_0 \circ \phi = f$, $g_0 \circ \phi = g$ and $ h_0 \circ \phi = h$. We remark that $\phi(tang(\mathcal F,C))= tang (\mathcal F_0,C_0)$ once more because these tangency points are exactly the ramification points of $f$ and $f_0$ (we have also that $\phi(tang(\mathcal G,C))= tang (\mathcal G_0,C_0)$ and $\phi(tang(\mathcal H,C))= tang (\mathcal H_0,C_0)$). For simplicity, we will assume that $F_{|_C}, G_{|_C}$ and $H_{|_C}$ have only simple critical points. Any pair of foliations defined by projective submersions at $C$ are generically transverse to each other along $C$. Let $\mathcal F$ and $\mathcal G$ be two projective submersions at $C$. From [@BRU] we have $$tang(\mathcal F,\mathcal G)\cdot C= N_{\mathcal F} \cdot C + N_{\mathcal G}\cdot C + K_S \cdot C$$ where $tang(\mathcal F,\mathcal G)$ is the curve of tangencies between the foliations, $N_{\mathcal F}$ (resp. $N_{\mathcal G}$) is the normal bundle associated to $\mathcal F$ (resp. $\mathcal G$) and $K_S$ is the canonical bundle of $S$. Since $$\begin{aligned} N_{\mathcal F}\cdot C &=& \chi(C)+ tang ({\mathcal F},C)= 3d-d^2 + d^2-d\\ N_{\mathcal G}\cdot C &=& \chi(C)+ tang ({\mathcal G},C) =3d-d^2 + d^2-d\\ -K_S \cdot C_0 &=& \chi(C) + C\cdot C = 3d-d^2 + d^2\\\end{aligned}$$ we conclude that $ tang(\mathcal F,\mathcal G)\cdot C = d $ so that $\mathcal F$ and $\mathcal G$ are not tangent to each other along $C$. We observe that the Lemma is not true for $d=1$ (see [@FA]). In this Section we will see how to associate to a pair of submersions, say $F, G$, a meromorphic map $\Phi_{F,G}$. It is defined initially as a biholormorphism from a neghborhood of the set $C\setminus (A \cup \phi^{-1}(A_0))$ to a neighborhood of $C_0\setminus (A_0 \cup \phi(A))$, where $A= tang(\mathcal F, C)\cup tang (\mathcal G,C) \cup (tang(\mathcal F, \mathcal G)\cap C)$ and $A_0= tang(\mathcal F_0, C_0)\cup tang (\mathcal G_0,C_0) \cup (tang(\mathcal F_0, \mathcal G_0)\cap C_0)$. Given a point $p\in C\setminus (A \cup {\phi}^{-1}(A_0))$, the foliations $\mathcal F$ and $\mathcal G$ are transverse to each other and to $C$ in a neighborhood of this point and the foliations $\mathcal F_0$ and $\mathcal G_0$ are transverse to each other and to $C_0$ in a neighborhood of $\phi(p)$; therefore, for $q\in S$ close to $p$ we may associate the points $q_{\mathcal F}$ and $q_{\mathcal G}$ where the leaves of $\mathcal F$ and $\mathcal G$ intersect $C$. The leaves of $\mathcal F_0$ and $\mathcal G_0$ through $\phi(q_{\mathcal F})$ and $\phi(q_{\mathcal G})$ will intersect (by definition) at the point $\Phi_{F,G}(q)$. It can be seen that this maps extends biholomorphically to the points of $tang(\mathcal F,C)$ and $tang(\mathcal G,C)$, essentially because the foliations $\mathcal F$ and $\mathcal G$ are transverse to each other at those points. From now on we change $A$ and $A_0$ to $A= tang(\mathcal F, \mathcal G)\cap C$ and $A_0=tang(\mathcal F_0, \mathcal G_0)\cap C_0$ and analyse the behavior of $\Phi_{F,G}$ at points of $A \cup {\phi^{-1}}(A_0)$. We distinguish two cases (A) $\phi(p)\in tang(\mathcal F_0,\mathcal G_0,C_0)$. (B) $\phi(p)\notin tang(\mathcal F_0,\mathcal G_0,C_0)$. \[extension\] $\Phi_{F,G}$ extends meromorphically to a neighborhood of $C$. [**Case A:**]{} We may assume, choosing conveniently the coordinates $(x,y)$ around $p$ and affine coordinates $(X,Y)$, that - $p=(0,0)$, $C$ is $y=0$ and $\mathcal F$ is defined by $dx=0$; - $\phi(p)=(0,0)$, $\mathcal F_0$ is defined by $dX=0$, $\mathcal G_0$ is the radial pencil with $(0,1)$ as base point ($X=0$ is a common fiber of $\mathcal F_0$ and $\mathcal G_0$); - $C_0$ is defined by $Y=h(X)$ with $h(0)=0$, $h^{\prime}(0)=0$ and $\phi(x)= (x,h(x))$. The leaf of $\mathcal F$ (respec. $\mathcal G$) through a point $(x,y)$ crosses the $x$-axis at $x$ (respectively $\xi(x,y)$ for a holomorphic function $\xi$ such that $\xi(x,0)=x$. It follows that $$\Phi_{F,G}(x,y)= \left(x,1-\dfrac{x(1-h(\xi(x,y))}{\xi (x,y))}\right)= \left(x,\dfrac{u(x,y)}{\xi(x,y)}\right)$$ The expression defines a meromorphic map in a neighborhood of $(0,0)$. There are two possible cases: - $\bf A_1$: the germs $x$ and $\xi$ are relatively prime; the line of poles of $\Phi_{F,G}(x,y)$ is $\xi (x,y)=0$ and has multiplicity 1. We write $\xi(x,y)-x=y\,A_1(x,y)$ for some holomorphic function $A_1(x,y)$; the $\mathcal G$-fiber may be transversal to the $\mathcal F$-fiber (when $A_1(0,0)\neq 0$) or tangent to it (in which case $A_1(0,0)\neq 0$). - $\bf A_2$: the germs $x$ and $\xi$ have a common factor; write $\xi(x,y)=x(1+y\,A_2(x,y))$, thus $\Phi_{F,G}(x,y)$ is a holomorphic map ($\mathcal F$ and $\mathcal G$ have $x=0$ as a common fiber), but it may be non-injective (unless $A_2(0,0)\neq 0$). [**Case B:**]{} We assume: - $p=(0,0)$, $C$ is $y=0$ and $\mathcal F$ is defined by $dx=0$; - $\phi(p)=(0,0)$, $\mathcal F_0$ is defined by $dX=0$ and $\mathcal G_0$ is defined by $dY-dX=0$ (in affine coordinates); - $C_0$ is defined by $Y=h(X)$ with $h(0)=0$, $h^{\prime}(0)=0$ and $\phi(x)= (x,h(x))$. We have then $$\Phi_{F,G}(x,y)= (x,\xi(x,y)-x +h(\xi(x,y))$$ It follows that $\Phi_{F,G}$ is a holomorphic map in a neighborhood of $p$; writing $\xi(x,y)-x=y\,B(x,y)$, we see that $\Phi_{F,G}$ is a local biholomorphism when $B(0,0)\neq 0$, that is, the fibers of $\mathcal F$ and $\mathcal G$ are transversal at $p$. An important consequence for us is that the pull-back by $\Phi_{F,G}$ of a holomorphic foliation $\mathcal L$ on $\Pp^2$ is also a holomorphic foliation in $S$. In the next Section we describe the singularities of ${\Phi^*_{F,G}}(\mathcal L)$. New foliations on S {#sec-new-foliation} =================== Let us take a foliation $\mathcal L$ on $\mathbb P^2$ defined by $\omega= LdP-d.PdL=0$, where $P(X,Y)=\sum_{i+j \leq d} a_{ij}X^iY^j$ is a polynomial of degree $d$ such that $C_0=\{P=0\}$ (we may assume $a_{0d}\neq 0$) and $L$ is a linear polynomial such that $L=0$ is transverse to $C_0$ . The singularities of $\mathcal L$ contained in $C_0$ are supposed to be disjoint of $A_0\cup \phi(A)$. We proceed to compute the multiplicity $Z(\mathcal L^{*},C,p)$ along $C$ of $p$ as a singularity of $\mathcal L^{*}=\Phi_{F,G}^{*}(\mathcal L)$ at the points where $\Phi_{F,G}$ maybe fails to be a biholomorphism. In order to make the computation easier, we take $L(X,Y)=X+b$. \[Indices-Z\] With notation of the proof of Proposition \[extension\], we have - Case A1: $Z(\mathcal L^{*},C,p)= d+mult_0(A_1(x,0))$. - Case A2: $Z(\mathcal L^{*},C,p)= mult_0(A_2(x,0))$. - Case B: $Z(\mathcal L^{*},C,p)=mult_0(B(x,0))$. [**Case A1**]{}: $x$ and $\xi$ are relatively prime. It follows that $$P(\Phi_{F,G}(x,y))=\dfrac{yv(x,y)}{\xi(x,y)^d}$$ In fact, $P(\Phi_{F,G}(x,0))=0$ and $ P(X,Y)=a_{0d}Y^d + \sum_{j\leq d-1}a_{ij}X^iY^j $ and therefore $$P(\Phi_{F,G}(x,y))= a_{0d}\dfrac{u^d}{\xi^d}+ \dfrac{\sum_{d-j\geq 1}a_{ij}x^iu^j{\xi}^{d-j}}{\xi^d}$$ In particular, $v(x,0)=x^{d-1}A_1(x,0)+\dots$. We have also $L(\Phi_{F,G}(x,y))=x+b$, $b\neq 0$, so that $$\Phi_{F,G}^{*}\,\omega = \dfrac{1}{\xi^{d+1}}[(x+b){\xi}(yd\,v + vd\,y) - d.yv((x+b)d\xi + {\xi}d\,x)]$$ Therefore $\mathcal L^{*}$ is defined by $ (x+b){\xi}(yd\,v + vd\,y) - d.yv((x+b)d\xi + {\xi}d\,x)=0$ near the point $p$ and $$Z(\mathcal L^{*},C,p)= 1+mult_0(v(x,0))= d+mult_0(A_1(x,0))$$ We observe that $Z(\mathcal L^{*},C,p)>0$ when the case ${\bf A1}$ is present. [**Case A2**]{}: $\xi$ divides $x$ ($\mathcal F$ and $\mathcal G$ share the leaf passing through $p$). Let us write as before $\xi(x,y)= x(1+yA_2(x,y))$; it follows that $$\Phi_{F,G}(x,y)=(x, \dfrac {yA_2(x,y) +h(\xi(x,y))}{1+yA_2(x,y)})$$ Writing $P(\Phi_{F,G}(x,y))= yv(x,y)$, we see that $v(x,0)=A_2(x,0)+\dots$ and $$\Phi_{F,G}^{*}\,\omega = (x+b)(vdy + ydv) - d.yvdx$$ We conclude that $$Z(\mathcal L^{*},C,p)=mult_0(v(x,0))=mult_0(A_2(x,0))$$ Let us notice that $Z(\mathcal L^{*},C,p)=0$ implies that $A_2(0,0)\neq 0$, that is, $\Phi_{F,G}(x,y)$ is a local biholomorphism at $p$. [**Case B**]{}: $\phi(p)\notin tang(\mathcal F_0,\mathcal G_0) \cap C_0$. We have $$\Phi_{F,G}(x,y)=(x,\xi-x+h(\xi(x,y))$$ Writing $P(\Phi_{F,G}(x,y))= yv(x,y)$, we see that $v(x,0)=B(x,0)+\dots$ and $$\Phi_{F,G}^{*}\,\omega = (x+b)(vdy + ydv) - d.yvdx$$ We conclude that $$Z(\mathcal L^{*},C,p)=mult_0(v(x,0))_0=mult_0(B(x,0))$$ Again, $Z(\mathcal L^{*},C,p)=0$ implies that $B(0,0)\neq0$, that is, $\Phi_{F,G}(x,y)$ is a local biholomorphism at the point p. We intend now to see the implications of having two maps $\Phi_{F,G}$ and $\Phi_{F,H}$ simultaneously; the fibrations $\mathcal F$, $\mathcal G$ and $\mathcal H$ are associated to pencil submersions $\mathcal F_0$, $\mathcal G_0$ and $\mathcal H_0$. Let us call $B= tang(\mathcal F,\mathcal H)\cap C$ and $B_0= tang(\mathcal F_0,\mathcal H_0)\cap C_0$. We consider two foliations $\mathcal I$ and $\mathcal L$ on $\mathbb P^2$ like before. Remark that $Z(\mathcal{I}, C_0)= Z(\mathcal{L}, C_0)=d$. We will assume: 1) all singularities of $\mathcal I$ and $\mathcal L$ lie outside the set $K= A_0 \cup \phi(A)\cup B_0 \cup \phi(B)$; 2) all curves of tangencies between $\mathcal I$ and $\mathcal L$ cross $C_0$ outside the set $K$. We denote $\mathcal I^*= \Phi_{F,G}^*(\mathcal I)$ and $\mathcal L^*= \Phi_{F,H}^*(\mathcal L)$. We will use again the formulae from [@BRU] to compute numerical invariants associated to tangent lines between two foliations. We have: $$tang(\mathcal I,\mathcal L)\cdot C_0= N_{\mathcal I} \cdot C_0 + N_{\mathcal L}\cdot C_0 + K_{\mathbb P^2}\cdot C_0=2d^2 -d,$$ since $\mathcal I$ and $\mathcal H$ have degree $d-1$ and $K_{\mathbb P^2}\cdot C_0 =-3d$. Let us call $\mathcal Z_1(\mathcal I^*,C)$ ($\mathcal Z_1(\mathcal L^*,C))$ the set of points where $\Phi_{F,G}$ is not a local biholomorphism (respectively $\Phi_{F,H}$ is not a local biholomorphism). We define $Z_1(\mathcal I^*,C)$ as the sum of all indexes $Z(\mathcal I^*,C,p)$ at points of $\mathcal Z_1(\mathcal I^*,C)$ (we put $Z_1(\mathcal L^*,C)$ for the correspondent sum at points of $\mathcal Z_1(\mathcal L^*,C)$). As for the foliations $\mathcal I^*$ and $\mathcal L^*$, we have that $$\begin{aligned} tang(\mathcal I^*,\mathcal L^*)\cdot C&=& N_{\mathcal I^*} \cdot C + N_{\mathcal L^*}\cdot C + K_S \cdot C \\ &=& Z(\mathcal I^*,C) + Z(\mathcal L^*,C) + 2d^2-3d\\ &=& Z_1(\mathcal I^*,C) + Z_1(\mathcal L^*,C) + 2d^2-d\end{aligned}$$ We conclude therefore that $$tang(\mathcal I^*,\mathcal L^*)\cdot C = Z_1(\mathcal I^*,C) + Z_1(\mathcal L^*,C)+ tang(\mathcal I,\mathcal L)\cdot C_0$$ Observe that curves $C$ and $C_0$ appear as components of the tangency locus in both sides of the last equation, thus we cancel $d$ from the equation and consider, from now, tangency loci besides $C$ and $C_0$. This formula suggests that $tang(\mathcal I^*,\mathcal L^*)\cap C$ may be also computed looking at the points of $\phi^{-1}(tang(\mathcal I, \mathcal L)\cap C) \cup \mathcal Z_1(\mathcal I^*,C) \cup \mathcal Z_1(\mathcal L^*,C)$. Our aim is to prove that $\Phi_{F,G}$ and $\Phi_{F,H}$ are everywhere local biholomorphisms. First of all we have to associate the tangencies between $\mathcal I$ and $\mathcal L$ to tangencies between $\mathcal I^*$ and $\mathcal L^*$. There is a little difficulty here because $\mathcal I^*$ and $\mathcal L^*$ are obtained from $\mathcal I$ and $\mathcal L$ using different pull-back’s; the pre-image by $\phi$ of a point of tangence between $\mathcal I$ and $\mathcal L$ might not be a point of tangency between $\mathcal I^*$ and $\mathcal L^*$. We take the foliations $\mathcal I$ and $\mathcal L$ defined by the equations $LdP-d.PdL=0$ and $(L+ a)dP + d.PdL=0$; their curve of tangencies is defined by $dL\wedge dP=0$ (besides the curve $C_0$). When intersecting with $C_0$, these are the points of tangency of $\mathcal F_0$ with $C_0$. \[intersection-at-tangencies\] Let $\phi(p)$ be a point of tangency between $\mathcal F_0$ and $C_0$. Then $(tang(\mathcal I^*,\mathcal L^*), C)_p=1$. We may take local coordinates $(x,y)$ around $p$ and $(r,s)$ around $\phi(p)$ such that - $C=\{y=0\}$ and $C_0= \{s=0\}$. - $\mathcal F$ and $\mathcal F_0$ are defined by $d(y-x^2)=0$ and $d(s-r^2)=0$ respectively. The foliations $\mathcal I$ and $\mathcal L$ are defined as $ su\,d(s-r^2)-(s-r^2 + \delta)\,d(su)=0$ and $su\,d(s-r^2)-(s-r^2 + a+\delta)\,d(su)=0$, where $u$ is a holomorphic function such that $u(0,0)\ne 0$ and $\delta \neq 0$. Let us write $\Phi_{F,G}(x,y)= (f(x,y),yA(x,y))$ and $\Phi_{F,H}(x,y)= (g(x,y),yB(x,y))$; we have $A(0,0)\neq 0$, $B(0,0)\ne 0$ and $(f(x,0),0)=(g(x,0),0)= \phi(x)$. The foliations $\mathcal I^*=\Phi_{F,G}^*{\mathcal I}$ and $\mathcal L^*=\Phi_{F,H}^*{\mathcal L}$ are defined as $$\begin{aligned} yAu\,d(yA-f^2) - (yA-f^2 +\delta)\,d(yAu)&=&0,\\ yBu\,d(yB-g^2) - (yB-g^2 + a+ \delta)\,d(yBu)&=&0\end{aligned}$$ We see easily that the curve of tangencies is given by $ABau^2{\phi}{\phi^{\prime}}y + y^2(...)=0$, so that the component different from $C=\{y=0\}$ crosses $C$ at $p$ transversaly. We proceed now to examine the points of tangency between $\mathcal I^*$ and $\mathcal L^*$ that possibly appear at ${\mathcal Z}_1(\mathcal I^*,C)\cup {\mathcal Z}_1(\mathcal L^*,C)$. If we denote their number as $tang_1 (\mathcal I^*,\mathcal L^*)$, we have seen that $$tang_1 (\mathcal I^*,\mathcal L^*) = Z_1(\mathcal I^*,C)+ Z_1(\mathcal L^*,C).$$ In fact, we have seen that out of ${\mathcal Z}_1(\mathcal I^*,C)\cup {\mathcal Z}_1(\mathcal L^*,C)$ tangency curves correspond to each other when restricted to $C$ and $C_0$. [**We claim that this equality holds at each point of**]{} ${\mathcal Z}_1(\mathcal I^*,C)\cup {\mathcal Z}_1(\mathcal L^*,C)$. Let us consider some point $p\in {\mathcal Z}_1(\mathcal I^*,C)$; since $\Phi_{F,G}$ is not a local biholomorphism, we have as explained before the possibilities $\bf A1$, $\bf A2$ and $\bf B$, the first two occuring when $\phi(p)\in tang({\mathcal F}_0,{\mathcal G}_0 \cap C_0$. If $p$ is $\bf A1$ or $\bf A2$ for $\Phi_{F,G}$, then $p$ is $\bf B$ for $\Phi_{F,H}$ (in the same way, when $q\in {\mathcal Z}_1(\mathcal L^*,C)$ is $\bf A1$ or $\bf A2$ for $\Phi_{F,H}$, then $q$ is $\bf B$ for $\Phi_{F,G}$. It may happen also that $p$ is $\bf B$ for $\Phi_{F,G}$ and $\Phi_{F,H}$. The reason is that we are supposing the submersions $F$, $G$ and $H$ to be independent so that we are in case $\bf A$ for maps $\Phi_{F,G}$ and $\Phi_{F,H}$ simultaneously. [**Case 1: $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is ${\bf A1}$ for $\Phi_{F,G}$ and $\bf B$ for $\Phi_{F,H}$.**]{} The local equations for $\mathcal I^*$ and $\mathcal L^*$ at $p$ are $$\begin{aligned} (x+b)\xi(ydv+vdy)-d.yv \{(x+b)d\xi+{\xi}dx\}&=&0\\ (x+b^{'})(v^{'}dy+ydv^{'})-d.yv^{'}dx&=&0.\end{aligned}$$ The line of tangencies has equation $$(b-b^{'}){\xi}vv^{'}-(x+b)(x+b^{'})[vv^{'}{\xi}_x+{\xi}(v^{'}v_x-vv^{'}_x)]=0$$ We observe that $mult_0({\xi}vv^{'})=mult_0(v)+1+mult_0(v^{'})=Z(\mathcal I^*,C,p)+Z(\mathcal L^*,C,p)$ (it may happen $Z(\mathcal L^*,C,p)=0$). [**Case 2: $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is $\bf A2$ for $\Phi_{F,G}$ and $\bf B$ for $\Phi_{F,H}$.**]{} The local equations are $$\begin{aligned} (x+b)(ydv+vdy)-d.yvdx&=&0\\ (x+b^{'})(v^{'}dy+ydv^{'})-d.yv^{'}dx&=&0\end{aligned}$$ The line of tangencies has equation $$(b-b^{'})vv^{'}-(x+b)(x+b^{'})[v^{'}v_x-vv^{'}_x]=0$$ We remark that $mult_0(vv^{'})=mult_0(v)+mult_0(v^{'})=Z(\mathcal I^*,C,p)+Z(\mathcal L^*,C,p)$ (it may happen that $Z(\mathcal L^*,C,p)=0$). [**Case 3: $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is $\bf B$ for $\Phi_{F,G}$ and $\Phi_{F,H}$.**]{} The conclusion is the same as above: $mult_0(vv^{'})=mult_0(v)+mult_0(v^{'})=Z(\mathcal I^*,C,p)+Z(\mathcal L^*,C,p)$. (it may happen $Z(\mathcal L^*,C,p)=0$). The remaining cases (when $p\in {\mathcal Z}_1(\mathcal L^*,C,p)$): $p$ is $\bf A1$ for $\Phi_{F,H}$ and $\bf B$ for $\Phi_{F,G}$; $p$ is $\bf A2$ for $\Phi_{F,H}$ and $\bf B$ for $\Phi_{F,G}$;  $p$ is $\bf B$ for both $\Phi_{F,H}$ and $\Phi_{F,G}$ are entirely similar. We conclude from $tang_1 (\mathcal I^*,\mathcal L^*) = Z_1(\mathcal I^*,C)+ Z_1(\mathcal L^*,C)$ (and the fact that $b$, $b'$ are generic) that the terms $[vv^{'}{\xi}_x+{\xi}(v^{'}v_x-vv^{'}_x)]$ (first case) and $[v^{'}v_x-vv^{'}_x]$ (second and third cases) have the same multiplicities at $0$ as ${\xi}vv^{'}$ and $vv^{'}$ respectively; [**the claim is proved**]{}. Let us make explicit the relations between the several multiplicities involved before. - Case 1: we write $v(x,0)=ax^l+\dots$ and $v^{'}(x,0)=cx^m+\dots$. It follows that $[vv^{'}{\xi}_x+{\xi}(v^{'}v_x-vv^{'}_x)]=ac(1-m+l)x^{m+l}+\dots$. Since $mult_0({\xi}vv^{'})=m+l+1$ necessarily $m=l+1$. Using $v(x,0)=x^{d-1}A_1(x,0)$ (and $v^{'}(x,0)=B^{'}(x,0)$) we get: $$mult_0(A_1(x,0))+d = mult_0(B^{'}(x,0))$$ - Case 2: we write again $v(x,0)=ax^l+\dots$ and $v^{'}(x,0)=cx^m+\dots$. Then $[v^{'}v_x-vv^{'}_x)]=ac(l-m)x^{m+l-1}+\dots $. Since $mult_0(vv^{'})=m+l$, we see that $l=m$. Using $v(x,0)=A_2(x,0)$ and $v^{'}(x,0)=B^{'}(x,0)$ $$mult_0(A_2(x,0))=mult_0(B^{'}(x,0))$$ - Case 3: it is analogous to Case 2 and we find $$mult_0(B(x,0))=mult_0(B^{'}(x,0))$$ There are correspondent equalities when $p$ is $\bf A_1$ for $\Phi_{F,H}$ and $\bf B$ for $\Phi_{F,G}$($mult_0(A^{'}_1(x,0))+d=mult_0(B(x,0))$) or $p$ is $\bf A_2$ for $\Phi_{F,H}$ and $\bf B$ for $\Phi_{F,G}$($mult_0(A^{'}_2(x,0))=mult_0(B(x,0)$). Proof of Theorem \[main-thm\] {#sec-proof-thm} ============================= Let us take some $C^{\infty}$ pertubation $\tilde C$ of $C$ and look to the curves $\Phi_{F,G}(\tilde C)$ and $\Phi_{F,H}(\tilde C)$, which are $C^{\infty}$ pertubations of $C_0$; we ask $\tilde C$ to be a holomorphic smooth curve with $({\tilde C}.{C})_p=1$ when passing through each $p\in {\mathcal Z_1}(\mathcal I^*,C) \cup {\mathcal Z_1}(\mathcal L^*,C)$ and ask also that $\Phi_{F,G}$ and $\Phi_{F,H}$ be holomorphic along these (local) holomorphic curves. Let us observe again that $\Phi_{F,G}$ is not a local biholomorphism at a point $p\in {\mathcal Z_1}(\mathcal I^*,C)$ (and $\Phi_{F,H}$ is not a local biholomorphism at a point $p\in {\mathcal Z_1}(\mathcal L^*,C)$ as well). We proceed now to prove that for any $p\in {\mathcal Z}_1(\mathcal I^*,C)\cup {\mathcal Z}_1(\mathcal L^*,C)$ one has $$\sum_q(\Phi_{F,G}({\tilde C}).C_0)_q + (\Phi_{F,H}({\tilde C}).C_0)_q \geq 2$$ for $q$ close to $\phi(p)$. Observe that in principle this number should be equal to $({\tilde C}.C)_{p}+ ({\tilde C}.C)_{P}=2$. Let us go back to the cases we discussed in the last Section. - $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is $\bf A1$ for $\Phi_{F,G}$ and $\bf B$ for $\Phi_{F,H}$. The pertubation $\tilde C$ near $p$ has to be contained in some small sector around $C$, where $\Phi_{F,G}$ is holomorphic. Since $$\Phi_{F,G}=\left(x,\dfrac{yA_1(x,y)+xh(\xi(x,y))}{x+yA_1(x,y)}\right)$$ when we put $y={\epsilon}x$ (for $\tilde C$) we see that $\sum_{q} (\Phi_{F,G}({\tilde C}).C_0)_q$ is the number of solutions (near $\phi(p)$) to the equation $$\dfrac{\epsilon\,x\,A_1(x,\epsilon\,x)+xh(\xi(x,\epsilon\,x))}{x+\epsilon\,x\,A_1(x,\epsilon\,x)}=h(x)$$ which is $= mult_0(A_1(x,0))$. In order to estimate $\sum_{q} (\Phi_{F,H}({\tilde C}).C_0)_q$ we use $$\Phi_{F,H}(x,y)=(x,yB^{'}(x,y)+h(\xi(x,y)))$$ and we have to find the number of solutions of $$\epsilon\,x\,B^{'}(x,\epsilon\,x)+h(\xi(x,\epsilon\,x))=h(x)$$ (remember that now $p$ is of $B$ type for $\Phi_{F,H}$), which is readily seen to be $1+mult_0(B^{'}(x,0))$. We have seen before that $mult_0(B^{'}(x,0))= mult_0(A_1(x,0))+d$, so for $q$ close to $\phi(p)$: $$\sum_q(\Phi_{F,G}({\tilde C}).C_0)_q + (\Phi_{F,H}({\tilde C}).C_0)_q= 2mult_0(A_1(x,0))+d+1$$ which is strictly bigger than 2 when $d>1$. - $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is $\bf A2$ for $\Phi_{F,G}$ and $\bf B$ for $\Phi_{F,H}$. We have $$\Phi_{F,G}(x,y)=\left(x, \dfrac {yA_2(x,y) +h(\xi(x,y))}{1+yA_2(x,y)}\right)$$ and $$\Phi_{F,H}(x,y)=(x,yB^{'}(x,y)+h(\xi(x,y)))$$ Using again $y=\epsilon x$, we get $\sum_q(\Phi_{F,G}({\tilde C}).C_0)_q=1+mult_0A_2(x,0)$ and $\sum_q(\Phi_{F,H}({\tilde C}).C_0)_{\phi(q)}=1+mult_0(B^{'}(x,0))$ Therefore for $q$ close to $\phi(p)$: $$\sum_q(\Phi_{F,G}({\tilde C}).C_0)_q + (\Phi_{F,H}({\tilde C}).C_0)_q= 2+2mult_0(A_2(x,0))$$ - $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is $\bf B$ for $\Phi_{F,G}$ and $\bf B$ for $\Phi_{F,H}$. Similarly we find for $q$ close to $\phi(p)$: $$\sum_q(\Phi_{F,G}({\tilde C}).C_0)_q + (\Phi_{F,H}({\tilde C}).C_0)_q= 2+2mult_0(B(x,0))$$ The remaining cases are analogous. We conclude that Case ${\bf A1}$ never appears and that Cases $\bf A2$ and $\bf B$ are present only at points $p$ where $\Phi_{F,G}$ is a local biholomorphisms. Appendix: Automorphisms of a plane curve {#appendix} ======================================== We present a proof of the following theorem given by Jose Felipe Voloch. \[Theorem1\] Let $C$ be a smooth plane curve of degree $d\geq 3$, then every automorphism of $C$ is linear, i.e. it comes from an element of $Aut(\mathbb{P}^2)$. The case $d=3$ is a consequence of Legendre’s normal form, see [@HOU], so we focus on the case $d \geq 4$. Before proving the theorem we give some useful remarks and lemmas based on exercises $17$ and $18$ of [@AR]. We say that the set $S=\{p_1, \ldots, p_k\} \subseteq \mathbb{P}^2$ of distinct points **impose independent conditions on curves of degree $n$** if $h^0(\mathbb{P}^2, \mathcal{I}_S (n)) = h^0(\mathbb{P}^2, \mathcal{O}(n)) - k$. Any set of $n+1$ points impose independent conditions on curves of degree $n$. On the other hand ${n+2}$ points impose independent conditions if and only if they are not aligned. Take first $S=\{p_1, \ldots, p_{n+1}\}$ and denote $S_k =\{p_1, \ldots, p_k\}$. Taking the product of $n$ lines through another point we see that $H^0(\mathbb{P}^2, \mathcal{I}_{S_{i+1}} (n))$ is strictly contained in $H^0(\mathbb{P}^2, \mathcal{I}_{S_{i}} (n))$, therefore $h^0(\mathbb{P}^2, \mathcal{I}_{S} (n))= h^0(\mathbb{P}^2, \mathcal{O}(n)) - (n+1)$. Consider now a set $S=\{p_1, \ldots, p_{n+1}, p_{n+2} \}$. If they are over a line $L$ and $E$ is a curve of degree $n$ passing through $n+1$ of them Bezout’s theorem implies $L \subseteq E$. This shows that $S$ fails to impose independent conditions on curves of degree $n$. Suppose now that every curve of degree $n$ passing by $n+1$ points contains also the other point of $S$. If they are not aligned, we can take for example the curve $E$ formed by lines joining $p_{n+1}$ with points $p_1, \ldots, p_{n}$, thus $p_{n+2}$ must be on this curve and we can assume that $p_n$, $p_{n+1}$ and $p_{n+2}$ are aligned. If some $p_j$, $j=1, \ldots, n-1$ is not on this line we consider $E'$ obtained from $E$ replacing $\overline{p_{n+1}, p_j}$ by a generic line passing by $p_{n+1}$, thus $E'$ contains $(n+1)$ points but not $S$, contradiction. Let $D$ be an effective divisor on $C$ of degree $m$. We use previous lemma in order to study meromorphic functions on $C$ having $D$ as polar divisor. Changing the fiber if necessary we will assume from now that $D$ has not multiple points. We recall that $l(D)$ is the dimension of the space of meromorphic functions f such that $(f) +D \geq 0$ and $i(D)$ is the dimension of the space of holomorphic forms $\omega$ such that $(\omega) \geq D$. If $m \leq d-2$ then $l(D)=1$. Recall (see [@R]) that holomorphic $1-$forms on $C=\{P=0\}$ are generated by elements $\frac{x^i y^j}{P_y}dx $ with $i+j \leq d - 3$. By the previous lemma the dimension of the space of polynomials vanishing at $D$ is $g(C) - m$, thus $i(D) = g(C) -m$ and Riemann-Roch gives $l(D) = 1$. If $m=d-1$ and $l(D) \geq2$ then $D= E- p$ where $p \in C$ and $E \in |\mathcal{O}_C(1)|$. Once again Riemann-Roch theorem gives $i(D) = g-d + l(D) \geq g-(d-1)+1$ then points of $D$ do not impose independent conditions and they must be aligned. We conclude by noting that intersection of a line with $C$ is a divisor of degree $d$. Finally we have $|\mathcal{O}_C(1)|$ is the only linear system of degree $d$ and dimension $3$. Let $D \in |\mathcal{O}_C(1)|$ be an aligned divisor of degree $d$ on $C$. Then points of $D$ fails to impose independent conditions on curves of degree $d-3$ and $i(D) = g(C) - (d-2)$ or equivalently $l(D) = 3$. If $A$ is another effective divisor of degree $d$ and $l(A)=3$ then any subset of $d-1$ points are aligned. We conclude that $A \in |\mathcal{O}_C(1)|$ and is linearly equivalent to $D$. **Proof of Theorem \[Theorem1\]:** Let $\phi : C \rightarrow C$ be an automorphism of $C$. Last proposition implies that for any line $L$ on $\mathbb{P}^2$, points of $\phi(L \cap C)$ determine a line $L' \subseteq \mathbb{P}^2$, thus $\phi$ comes from an automorphism of $\check{\mathbb{P}}^2$ which corresponds to an element of $Aut(\mathbb{P}^2)$. [99]{} M. Falla Luza and F. Loray. *On the number of fibrations transverse to a rational curve in complex surfaces*, Comptes Rendus Mathématique, v. 354, pg. 470-474, 2016. P. Sad. *Projective transverse structures for some foliations*, Trends in Mathematics (Singularities in Geometry, Topology, Foliations and Dynamics, pg. 197-206, 2017), Birkhauser. E. Arbarello, M. Cornalba, P. Griffiths, J. Harris. Geometry of Algebraic Curves, Volume 1, Springer-Verlag, 1984. M. Brunella. Birational Geometry of Foliations. Springer-Verlag, New York, 2015. D. Husemöller. Elliptic Curves, Springer Graduate Texts in Mathematics, 1987. H. Grauert. *Über Modifikationen and exzeptionelle analytische Mengen*, Math. Ann. 146, pg. 331-368, 1962. V.I. Savelev. *Zero-type imbedding of a sphere into complex surfaces*, Moscow Univ. Math. Bull. 37, no. 4, pg. 34-39, 1982. T.Ueda. *On the neighborhood of a compact complex curve with topologically trivial normal bundle*, J. Math. Kyoto Univ. 22, pg. 583-607, 1982. R. Meziani and P. Sad. *Singularités nilpotentes et intégrales premières*, Publ. Mat. 51, N 1, 143 -161,2007. , *Quelques Aspects des Surfaces De Riemann* Birkhauser Verlag AG.

--- abstract: | Let $X$ be a $n\times p$ matrix and $l_1$ the largest eigenvalue of the covariance matrix $X^{*}X$. The “null case" where $X_{i,j}\sim {\cal N}(0,1)$ is of particular interest for principal component analysis. For this model, when $n, p{\rightarrow}\infty$ and $n/p {\rightarrow}\gamma \in \mathbb{R}_+^*$, it was shown in @imj that $l_1$, properly centered and scaled, converges to the Tracy-Widom law. We show that with the same centering and scaling, the result is true even when $p/n$ or $n/p{\rightarrow}\infty$, therefore extending the previous result to $\gamma \in \overline{\mathbb{R}}_+$. The derivation uses ideas and techniques quite similar to the ones presented in @imj. Following @sosh, we also show that the same is true for the joint distribution of the $k$ largest eigenvalues, where $k$ is a fixed integer. Numerical experiments illustrate the fact that the Tracy-Widom approximation is reasonable even when one of the dimension is small. author: - | Noureddine El Karoui[^1]\ *Department of Statistics,*\ *Stanford University* bibliography: - 'research.bib' title: 'On the largest eigenvalue of Wishart matrices with identity covariance when $n$, $p$ and $p/n {\rightarrow}\infty$' --- Introduction ============ Large scale principal component analysis (PCA) - concerning an $n\times p$ matrix $X$ where $n$ and $p$ are both large - is nowadays a widely used tools in many fields, such as image analysis, signal processing, functional data analysis and quantitative finance. Several examples come to mind, including Eigenfaces, subspace filtering, or @lalouxetal where PCA (as well as some random matrix theory) is used to try to improve on the naive solution to Markovitz’s portfolio optimization problem. Important progress has been made recently in our understanding of the statistical properties of PCA in such settings. Emblematic of this is work of @imj, which explains the properties of the square of the largest singular value of a random matrix $X$ under the “null model" where its entries are iid ${\cal N}(0,1)$. Specifically, if we denote the sample eigenvalues of $X'X$ by $l_1\geq \ldots \geq l_p$, call $$\begin{aligned} n_1&=\max{(n,p)}-1\;, \; \; \; \; p_1=\min{(n,p)} \;,\\ \mu_{np}&=(\sqrt{n_1}+\sqrt{p_1})^2 \; ,\\ \sigma_{np}&=(\sqrt{n_1}+\sqrt{p_1})\left(\frac{1}{\sqrt{n_1}}+\frac{1}{\sqrt{p_1}}\right)^{1/3} \; ,\end{aligned}$$ and $W_1$ the Tracy-Widom distribution (see **A0**), it was shown in @imj that \[ThJ\] If $n,p {\rightarrow}\infty$ and $n/p{\rightarrow}\gamma \in (0,\infty)$, $$\frac{l_1-\mu_{np}}{\sigma_{np}} \overset{\cal L}{\rightarrow}W_1 \;.$$ Building on @imj and using properties of determinantal point processes, @sosh showed that the same result holds for the $k$ largest eigenvalues, where $k$ is a fixed integer: their joint distribution converges to their Tracy-Widom counterpart. This is a very interesting development because the classical theory (e.g @anderson) was developed under the assumption that $p$ was fixed and $n$ grew to $\infty$, whereas in modern day applications both $p$ and $n$ are large. However, Johnstone’s assumption $n/p{\rightarrow}\gamma$ imposes a limit on the validity of his result which one would like to remove. In an actual data analysis, with given $p$ and $n$, $n={\mathrm{o}}(p)$ and $n\asymp p$ could be equally plausible. Furthermore, a specific $X$ of size $n\times p$ could arise in many triangular arrays settings, where we have $X_j$ of size $n_j\times p_j$, and the limitation $n_j/p_j{\rightarrow}\gamma$ finite might only hold in some triangular situations and not in others. Accordingly in this paper we weaken the assumption that $n/p {\rightarrow}\gamma$ finite and show that \[ThJextended\] If $n,p {\rightarrow}\infty$ and $n/p{\rightarrow}\infty$, $$\frac{l_1-\mu_{np}}{\sigma_{np}} \overset{\cal L}{\rightarrow}W_1 \;.$$ Moreover, with the same centering and scaling, the joint distribution of the $k$ largest eigenvalues converges in law to its Tracy-Widom counterpart. Dually, the same result holds if $n/p{\rightarrow}0$. Let us note that the remark we made about centering and scaling sequences after Theorem \[ThJ\] is still valid in this context. There is clearly a mathematical motivation for dealing with this problem: the result completes the picture about the properties of $l_1$ with large $p$ and $n$ and, in a sense, closes Theorem \[ThJ\]. But is it interesting from a statistical standpoint? The situation $p\gg n$ is indeed a fairly common one in modern statistics. Microarray data are a prototypical example: currently they usually have $p$ of the order of a few thousands and $n$ of the order of a few tens. One encounters $p\gg n$ or $n\gg p$ in many other instances: data collection mechanisms are now effective enough so as to, for example, collect and retain thousands of piece of information for millions of customers (transactional data), or millions of pieces of information for thousands of stocks (tick-by-tick data in Finance). Analyzing these very high dimensional datasets raises new challenges and is at the center of recent statistical work, both applied and theoretical. Microarray analysis in particular is a very active field, and has contributed a flurry of activity in non classical situations (very high dimensional data), raising theoretical questions and sometimes revisiting classical techniques or results. As illustrated for instance in @wrr, PCA or PCA-related methods are used for various tasks in the microarray context, from traditional dimensionality reduction procedures to gene grouping. Having a good understanding of the behavior of the singular values of gaussian “white noise" matrices could provide valuable insights for these applications. Recent work of @bl03 about the properties of naive Bayes and Fisher’s linear discriminant function when $p\gg n$ illustrates the impetus these dimensionality assumptions are also gaining in theoretical studies. Our work is part of the larger effort to investigate the properties of high dimensional data structures. Here it is done in a simple, “null" situation. We now present a few numerical experiments we realized to assess how big (or small) $n$ or $p$ should be for Theorems \[ThJ\] and \[ThJextended\] to be practically useful. Numerical experiments --------------------- @imj showed empirically that in that situation the Tracy-Widom approximation was reasonably satisfying, even for small matrices. Similarly, to try to assess its accuracy in our setup, we ran the following experiments in `Matlab`: we picked $n$ and $p$ and generated $10,000$ $n\times p$ matrices $X$ with entries iid ${\cal N}(0,1)$. Then we used standard routines (`normest` in `Matlab`) to compute their spectral norms and squared them to obtain a dataset of $l_1$-s. Following [@imj2], we adjust centering and scaling to $$\begin{aligned} \tilde{\mu}_{np}&=\sqrt{n-1/2}+\sqrt{p-1/2}\; ,\\ \tilde{\sigma}_{np}&=(\sqrt{n-1/2}+\sqrt{p-1/2})\left(\frac{1}{\sqrt{n-1/2}}+\frac{1}{\sqrt{p-1/2}}\right)^{1/3} \;.\end{aligned}$$ This leads to a very significant improvement in the quality of the Tracy-Widom approximation for our simulations. Simple manipulations (explained in section 2.2) show that we have some freedom in choosing the centering and scaling: if we replace $n$ by $n+a$ and $p$ by $p+b$ (where $a$ and $b$ are fixed real numbers) in the definitions of $\mu_{np}$ and $\sigma_{np}$, Theorem \[ThJ\] and Theorem \[ThJextended\] still hold. The particular choice used here is motivated by a careful theoretical analysis of the entries of $K_N$ mentioned in section 2.2. Table 1 summarizes the “quantile" properties of the empirical distributions we obtained and compare them to the Tracy-Widom reference. We used the same reference points as @imj. TW Quantiles TW 10$\times$1000 10$\times$ 4000 10$\times$ 10000 100$\times$4000 30$\times$5000 -------------- ----- ---------------- ----------------- ------------------ ----------------- ---------------- -3.9 .01 0.009 0.010 0.015 0.012 0.013 -3.18 .05 0.047 0.050 0.060 0.053 0.055 -2.78 .10 0.102 0.107 0.112 0.103 0.105 -1.91 .30 0.303 0.308 0.316 0.304 0.303 -1.27 .50 0.506 0.506 0.522 0.508 0.503 -0.59 .70 0.705 0.704 0.723 0.706 0.702 0.45 0.9 0.904 0.904 0.913 0.901 0.904 0.98 .95 0.953 0.951 0.958 0.951 0.953 2.02 .99 0.992 0.990 0.992 0.991 0.991 : **Quality of the Tracy-Widom Approximation for some large matrices:** the leftmost columns displays certain quantiles of the Tracy-Widom distribution. The second column gives the corresponding value of its cdf. Other columns give the value of the empirical distribution functions obtained from simulations at these quantiles. $\tilde{\mu}_{np}$ and $\tilde{\sigma}_{np}$ are the centering and scaling sequences. \ TW Quantiles TW 50$\times$5000 50$\times$20000 50$\times$50000 5$\times$200 5$\times$2000 5$\times$20000 -------------- ----- ---------------- ----------------- ----------------- -------------- --------------- ---------------- -3.9 .01 0.010 0.017 0.021 0.008 0.014 0.018 -3.18 .05 0.053 0.067 0.079 0.047 0.057 0.069 -2.78 .10 0.104 0.125 0.139 0.094 0.110 0.120 -1.91 .30 0.309 0.331 0.345 0.293 0.314 0.320 -1.27 .50 0.502 0.522 0.538 0.500 0.506 0.519 -0.59 .70 0.705 0.718 0.727 0.714 0.712 0.710 0.45 .90 0.899 0.905 0.911 0.911 0.906 0.907 0.98 .95 0.949 0.955 0.957 0.959 0.951 0.954 2.02 .99 0.991 0.992 0.992 0.994 0.992 0.992 : **Quality of the Tracy-Widom Approximation for some large matrices:** the leftmost columns displays certain quantiles of the Tracy-Widom distribution. The second column gives the corresponding value of its cdf. Other columns give the value of the empirical distribution functions obtained from simulations at these quantiles. $\tilde{\mu}_{np}$ and $\tilde{\sigma}_{np}$ are the centering and scaling sequences. We picked the dimensions according to two criteria: $100\times4000$, $30\times5000$, and $50\times5000$ were chosen to investigate “representative" microarray situations. We chose the other to have a range of ratios and estimate how valuable the Tracy-Widom approximation would be in situations that could be considered classical, i.e one small dimension (less than 10) and one large (several hundreds to several thousands). For the sake of completeness, we redid the simulations presented in @imj and present in Table 2 the results obtained with $\tilde{\mu}_{np}$ and $\tilde{\sigma}_{np}$ as centering and scaling. TW Quantiles TW 5$\times$5 10$\times$ 10 100$\times$ 100 5$\times$20 10$\times$ 40 100$\times$400 -------------- ------ ------------ --------------- ----------------- ------------- --------------- ---------------- -3.9 .01 0 0.002 0.008 0.001 0.004 0.008 -3.18 .05 0.003 0.018 0.043 0.019 0.032 0.044 -2.78 0.10 0.022 0.054 0.090 0.056 0.077 0.095 -1.91 .30 0.217 0.257 0.295 0.262 0.279 0.294 -1.27 .50 0.464 0.486 0.497 0.490 0.494 0.489 -.59 .70 0.702 0.703 0.700 0.702 0.707 0.702 0.45 .90 0.903 0.903 0.901 0.905 0.906 0.899 0.98 .95 0.949 0.950 0.950 0.952 0.953 0.949 2.02 .99 0.988 0.990 0.991 0.989 0.990 0.990 : **Quality of the Tracy-Widom Approximation (Continued):** the columns have the same meaning as in Table 1. The ratio $p/n$ is smaller than in Table 1 and the matrices are not as big, but the Tracy-Widom approximation is already acceptable for the upper quantiles. $\tilde{\mu}_{np}$ and $\tilde{\sigma}_{np}$ are the centering and scaling sequences. We see that the fit is good to very good for the upper quantiles ($.9$ and beyond) across the range of dimensions we investigated. The practical interest of this remark is clear: these are the quantiles one would naturally use in a testing problem. We note that it appears empirically that the problem gets harder when the ratio $r$ of the larger dimension to the smaller one ($p_1$ in our notation) gets bigger: the larger $r$, the larger $p_1$ should be for the approximation to be acceptable. Conclusions and Organization ---------------------------- From a technical standpoint, the method developed in @imj proves to be versatile, and, at least conceptually, relatively easy to adapt to the case where $n/p{\rightarrow}\infty$. Nevertheless, substantial technical work is needed to obtain Theorem \[ThJextended\]. Using the elementary fact (see e.g theorem 7.3.7 in @hj) that the largest eigenvalue of $X^{*}X$ is the same as the largest eigenvalue of $XX^{*}$, it will be sufficient to give the proof in the case $n/p{\rightarrow}\infty$. From a practical point of view, we show that the Tracy-Widom limit law does not depend of how the sequence $(n,p)$ is embedded. As long as both dimensions go to infinity, the properly re-centered and re-scaled largest eigenvalue converges weakly to this law.\ We can compare this with the “classical" situation where $p$ is held fixed, in which case the limiting joint distribution is known, too (see e.g @anderson, corollary 13.3.2). In this case, the centering is done around $n$ and the scaling is $\sqrt{n}$; elementary computations show that $(l_1-\mu_{np})/\sigma_{np}$ also has a non-degenerate limiting distribution (possibly changing with each $p$). Nevertheless, even with the classical centering, it is hard to evaluate the marginals in this context and the results are therefore difficult to use in practice. Our simulation results show that the Tracy-Widom approximation is reasonably good (for the upper quantiles) even when $p$ or $n$ are small. As remarked by @imj, Proposition 1.2, this implies that when doing PCA, one could develop (conservative) tests based on the Tracy-Widom distribution that could serve as alternatives to the scree plot or the Wachter plot.\ The paper is organized as follows: after presenting (Section 2) the main elements of the proof of Theorem \[ThJ\], we describe (Section 3) the strategy that will lead to the proof of Theorem \[ThJextended\]. We prove the two crucial points needed in Section 4. To make the paper self-contained, we give some background information about different aspects of the problem in the appendices. Several technical issues are also treated there in order to avoid obscuring the proof of the main result. Outline of Johnstone’s proof ============================ Before describing the backbone of the proof presented in @imj, we need to introduce a few notational conventions. In what follows, we will use $N$ instead of $p$ to be consistent with the literature. We also denote by ${\mathbf{AB}}$ (for “asymptotic behavior") the situation where $n,N, \text{ and } n/N {\rightarrow}\infty$. We will frequently index functions that depend on both $N$ and $n$ with only $N$. The reason for this is that it will allow us to keep the notations relatively light, and that we think of $n$ as being a function of $N$. Notations like $\textbf{E}_N$ and $\textbf{P}_N$ will denote expectation and probability under the measure induced by the matrices (of size $n(N)\times N$) we are working with.\ Finally, it is technically simpler to work with a matrix $X$ whose entries are standard complex Gaussians (i.e the real and imaginary parts are independent, and they are both ${\cal N}(0,1/2)$), rather than with entries that are ${\cal N}(0,1)$. When we mention the complex case, we refer to this situation. We now give a quick overview of the important points around which the proof of Theorem \[ThJ\] was articulated. At the core of several random matrix theory results lie the fact that the joint distribution of the eigenvalues of the random matrices of interest is known and can be represented as the Fredholm determinant of a certain operator (or a totally explicit function of it). Building on this, if we introduce a number $b$ that is $1$ in the real case and $2$ in the complex one, it turns out that one has the representation formula $$\label{freddet} {\mathbf{E}_{N}\left(\prod_{i=1}^N (1+f(l_i))\right)}=\left[\det({\mathrm{Id}}+S_Nf)\right]^{b/2} \; ,$$ where $S_N$ is an explicit kernel, depending of course upon the kind of matrices in which one is interested. Here, $f$ treated as an operator means multiplication by this function. It is clear that if $\chi_t=-\mathbf{1}\{x:x\geq t\}$, we have $$\mathbf{P}_N(l_1\leq t)=\left[\det({\mathrm{Id}}+S_N\chi_t)\right]^{b/2}\;.$$ The interested reader can find background information on this in @mehta, chapters 5 and 6, @tw98 or @deift, chapter 5, which in turn (p.109) points to @simonreed, section 17, vol 4, for background on operator determinants. We stress the fact that all these formulas are finite dimensional. From the last display, the strategy to show convergence in law in either Theorem \[ThJ\] or \[ThJextended\] is clear: fix $s_0$, show that under the relevant assumptions, $\mathbf{P}(l_{1,N}\leq s_0){\rightarrow}W_1(s_0)$, and use the fact that $W_1$ is continuous to conclude. Complex case ------------ We just saw that to find the asymptotic behavior of $l_1$ is equivalent to showing the convergence of the determinant of a certain operator. This task can be reduced to showing convergence in trace class norm of this operator (see @simonreed for background on this, e.g, Lemma XIII.17.4 (p.323)). Through work from @widom99, @imj exhibits an integral representation formula for his operator, and the original problem is essentially transformed into showing that certain integrals have a predetermined limit. In somewhat more detail, if we call $\alpha=n-N$, and $L_k^{\alpha}$ the $k$-th Laguerre polynomial associated with $\alpha$ (as in @szego, p.100), let $$\phi_k(x)=\sqrt{\frac{k!}{(k+\alpha)!}}x^{\alpha/2}{\mathrm{e}}^{-x/2}L_k^{\alpha}(x) \; ,$$ $\xi_k(x)=\phi_k(x)/x$, $a_N=\sqrt{Nn}$, and finally $$\left\{ \begin{array}{cll} \phi(x)&=(-1)^N\sqrt{\frac{a_N}{2}} (\sqrt{n}\xi_N(x)-\sqrt{N}\xi_{N-1}(x)) \; ,\\ \psi(x)&=(-1)^N\sqrt{\frac{a_N}{2}}(\sqrt{N}\xi_N(x)-\sqrt{n}\xi_{N-1}(x)) \; . \end{array} \right.$$ We note two things: first, there is a slight abuse of notation since $\phi$ and $\psi$ obviously depend on $n$ and $N$, but as in @imj, we choose to not carry these indices in the interest of readability. Also, $\phi$ and $\psi$ admit more “compact" representations, in terms of a single Laguerre polynomial, with a modified $\alpha$, or another degree. These are easy to derive using @szego, p.102, for instance. Nevertheless we choose to work (except in **A7**) with the previous representations because of the symmetries they present. The kernel $S_N$ mentioned in (\[freddet\]) has the representation (@imj, equation (3.6)) $$S_N(x,y)=\int_{0}^{\infty} \phi(x+z)\psi(y+z)+ \psi(x+z)\phi(y+z) dz \; .$$ Now let $\bar{S}$ be the Airy operator. Its kernel is $$\bar{S}(x,y)=\frac{{\mathrm{Ai}}(x){\mathrm{Ai}}'(y)-{\mathrm{Ai}}(y){\mathrm{Ai}}'(x)}{x-y}=\int_0^{\infty}{\mathrm{Ai}}(x+u){\mathrm{Ai}}(y+u) du \; ,$$ where Ai denotes the Airy function. It was shown in @tw94 that, viewing $\bar{S}$ as an operator on $L^{2}[s,\infty)$, one had $$\det({\mathrm{Id}}-\bar{S})=W_2(s) \; ,$$ where $W_2$ is the Tracy-Widom law “emerging" in the complex case (see **A0**). So the complex analog of theorem \[ThJ\] follows from the fact that, after defining $S_{\tau}(x,y)={\sigma_N}S_N(\mu_N+{\sigma_N}x,\mu_N+{\sigma_N}y)$, Johnstone managed to show, for all $s$, that $$\det({\mathrm{Id}}-S_{\tau}) {\rightarrow}\det({\mathrm{Id}}-\bar{S}) \; .$$ To do this, he introduced $\phi_{\tau}(s)=\sigma_N \phi(\mu_N+s\sigma_N)$, and similarly $\psi_{\tau}$. Note that we have $$S_{\tau}(x,y)=\int_{0}^{\infty} \phi_{\tau}(x+z)\psi_{\tau}(y+z)+ \psi_{\tau}(x+z)\phi_{\tau}(y+z) dz \; .$$ Since what we are interested in is really $S_{\tau} \chi_s$, for some fixed $s$, we will view $S_{\tau}$ as an operator acting on $L^2[s,\infty)$ in what follows.\ So the problem becomes to show that, as $n,N {\rightarrow}\infty$ $$\label{whattoshow1} \phi_{\tau}(s),\psi_{\tau}(s){\rightarrow}\frac{1}{\sqrt{2}}\mathrm{Ai}(s) \; ,$$ and that $\forall s_0 \in \mathbb{R}$, there exists $N_0(s_0)$ such that if $N>N_0$, we have on $[s_0,\infty)$, $$\label{whattoshow2} \phi_{\tau}(s),\psi_{\tau}(s)=O({\mathrm{e}}^{-s/2}) \; .$$ Once this is shown (we give more details on this later), we can show that $S_{\tau} {\rightarrow}\bar{S}$ in the trace class norm of operators on $L^2[s,\infty)$. A classical way to do it is described in the remark at the end of section 3 of @imj, which bounds the trace class norm of the difference of $S_{\tau}-\bar{S}$ in terms of the Hilbert-Schmidt norm of operators whose kernels are related to $\phi_{\tau}, \psi_{\tau}$ and ${\mathrm{Ai}}$. This leads to the conclusion that $$\det({\mathrm{Id}}-S_{\tau}) {\rightarrow}\det({\mathrm{Id}}-\bar{S}) \;,$$ since $\det$ is continuous with respect to trace class norm. Therefore, the largest eigenvalue of $X^*X$ has the behavior it was claimed it has. Real Case --------- In the real case, using arguments from @tw96 and @widom99, @imj gets a representation similar to (\[freddet\]), this time involving an operator with kernel a $2\times 2$ matrix (instead of scalar in the complex case). He is then able to relate it to the complex case problem - the matrix operator determinant can be computed as the product of two scalar operator determinants - and shows that the “reduced" variable he works with ought to have the same limit as it had in the Gaussian Orthogonal Ensemble case, which was studied in depth by Tracy and Widom. For the sake of completeness, we recall that in this situation $\alpha=n-1-N$ and $$\mathbf{P}_N(l_1\leq t)=\sqrt{\det({\mathrm{Id}}+K_N\chi_t)} \;.$$ $K_N$ has the representation (in the $N$ even case) $$K_N= \begin{pmatrix} S_N+ \psi\otimes \epsilon \phi & S_N D-\psi\otimes\phi \\ {\epsilon}S_N -{\epsilon}+{\epsilon}\psi \otimes {\epsilon}\phi & S_N + {\epsilon}\phi \otimes \psi \end{pmatrix} \; ,$$ where $D$ is the differential operator, ${\epsilon}$ is convolution with the kernel ${\epsilon}(x-y)$, and ${\epsilon}(x)=\text{sgn}(x)/2$. We note the slight change in $\alpha$ and replace $n$ by $n-1$ when we need to use the results or formulas derived in the complex case (for instance, the $S_N$ we just mentioned is $S_{n-1,N}$, and not $S_{n,N}$). We refer the reader to @ggk for a complement of information on operator determinants and to the end of section VIII in @tw96 for details on the technical problems that $K_N$ poses. From a purely technical standpoint, one critical issue is to evaluate the large $n,N$ limit of $c_{\phi}=\int_{0}^{\infty}\phi(x)dx/2$. If one can show that it is $1/\sqrt{2}$ when $N{\rightarrow}\infty$ through even values, then Johnstone’s considerations hold true all the way and we have the same conclusion as in Theorem \[ThJ\].\ We note that using the interlacing properties of the singular values (as mentioned for instance in @sosh, Remark 5; see also @hj, theorem 7.3.9), as well as the estimates of the difference (resp. ratio) between two consecutive terms of the centering (resp. scaling) sequence, the $N$ odd case follows immediately from the $N$ even case. To be more precise, we use the fact that $$\frac{\mu_{n,N}-\mu_{n,N-1}}{{\sigma_N}}={\mathrm{O}}(N^{-1/3}) \; \; {\rightarrow}0 \text{ as } N {\rightarrow}\infty$$ to check that the $N$ even terms lower and upper bounding the $N$ odd probability have the same limit. Note that the same relationship holds for $\mu_{n+a,N+b}$ and $\mu_{n,N}$, if $a$ and $b$ are fixed real numbers. Therefore, after doing the proof with centering sequence $\mu_{n+3/2,N+1/2}$ (which is technically simpler), we will be able to conclude that the theorem holds true for $\mu_{n,N}$. Last, to be able to use @sosh, Lemma 2, which gives the result we wish for the joint distribution of the $k$-largest eigenvalues, we will need to verify that the entries of the $2\times 2$ operator converge pointwise, and are bounded above in an exponential way. This is what is done in the proof of Lemma 1 of @sosh, and we will show in **A8** that the arguments given there can be extended to handle our situation. Further Remarks and Agenda ========================== Most of the work in @imj is done in closed form, and in the finite dimensional case. That has two advantages from our standpoint: as the limiting behavior is only investigated in the last “step", most of the arguments given there carry through for our problem, and the method certainly does. Therefore, our contribution is mostly technical; it follows very closely the ideas of @imj, providing solutions to technical problems appearing in the case we consider. Only at a few points could we not use the approach developed in @imj. This led us to an analysis of the complex case that is slightly different from the original one, but the core reasons for which the result holds are the same. In what follows, we first focus on showing that (\[whattoshow1\]) and (\[whattoshow2\]) hold true when $n,N$ and their ratio tend to infinity. This takes care of the complex case. We then turn to the problem of the asymptotic behavior of $c_{\phi}$, and the technical points we have to verify for @sosh results to hold. The following remarks outline the differences between the analysis we present here and the one done in @imj. Remarks on adaptation of the original proof ------------------------------------------- ### Complex case {#errorcontrolproblem} To show that (\[whattoshow1\]) and (\[whattoshow2\]) held true, @imj essentially reduced his problem to studying the solution of a “perturbed" Airy equation and used tools from @olver to carefully study it. One point that was used repeatedly was that the turning points of the equation were bounded away from one another when $n,N$ were large. This is not true anymore in the case we consider, and we show how to get around this difficulty. So we do not work with a perturbed Airy equation anymore, but rather with Whittaker functions, which have a close relationship to Laguerre polynomials, and their expansion in terms of parabolic cylinder functions (see **A9** for some background information on special functions). In @olver80, the case we are interested in was studied in detail, giving us most of the tools we need to show (\[whattoshow2\]). Using @olver75, we reinterpret the [parabolic cylinder functions]{} results in terms of Airy functions and derive the elements we need to complete the proof of (\[whattoshow1\]) and (\[whattoshow2\]). The reason for which we could not exactly follow the “original" method is related to the error control function called ${\cal V}(\zeta)$ in @imj. This function depends upon the parameter $\omega=2\lambda/\kappa$, which in the case $n/N{\rightarrow}\gamma \in \mathbb{R}$ is bounded away from 2. This essentially allows a uniform control over ${\cal V}$, and it is possible to show that this error control function is bounded as a function of $N$. Since the control is actually something like $\exp(\lambda_0 {\cal V}/\kappa)-1$, it tends to zero as $N{\rightarrow}\infty$. This gave @imj a way to get part of (\[whattoshow2\]).\ In our case, it seems that ${\cal V}$ would tend to $\infty$, at a rate that is nevertheless ${\mathrm{o}}(\kappa)$. As it seems easier and more promising to use @olver80 than to derive the growth of ${\cal V}$, we choose this approach. Nevertheless, this is the only (but crucial) technicality (in the complex case) that did not carry through by the method described in @imj under ${\mathbf{AB}}$. ### Real Case For the $c_{\phi}$ problem, we provide a closed form expression at given $n,N$ and show that in the limit is the “right" one as long as $n$ and $N$ tend to $\infty$.. This does not use the saddlepoint method, but relies on the availability of a generating function formula for Laguerre polynomials. The proof is done in **A7**.\ A simple modification to @imj would give the same result: in the display preceding (6.13) there, we could write $$h(t)=\sum_{k=0}^{\infty}c_k t^k = 2^{\alpha/2}\Gamma(\alpha/2)(1+t)(1-t^2)^{-(\alpha/2+1)}$$ and expand $(1-t^2)^{-(\alpha/2+1)}$. Multiplying by $1+t$ has a very simple effect on the series, and so $c_k$ is known explicitly. In **A8**, we show how to check that the conditions required for Soshnikov’s results to hold are indeed met. They are straightforward consequences of the analysis we will carry below. Since the real case is derived from the complex one after analyzing a few technical points, we verify these in the appendices and present here the study of the complex case. We now turn to the main problem we solve in this note: showing (\[whattoshow1\]) and (\[whattoshow2\]) under our set of assumptions. Complex case: study of asymptotics ================================== In this section, we work on the problem of showing pointwise convergence and uniform boundedness, setting the problem in a way similar to section 5 of @imj. We recall his notations, slightly modified to avoid confusions: $N_+=N+1/2$, $n_+=n+1/2$, $z=\mu_N+\sigma_N s$, with $\mu_N=(\sqrt{(N+\alpha)_+}+\sqrt{N_+})^{1/2}$ and $\sigma_N=(\sqrt{(N+\alpha)_+}+\sqrt{N_+})(1/\sqrt{N_+}+1/\sqrt{(N+\alpha)_+})^{1/3}$. For reasons that will be transparent later on, our aim is to show that $$\label{pointwisecv} F_N(z)=(-1)^N \sigma_N^{-1/2}\sqrt{N!/n!}\,z^{(\alpha+1)/2}{\mathrm{e}}^{-z/2}L_N^{\alpha_N}(z) {\rightarrow}{\mathrm{Ai}}(s), \; \forall s \in \mathbb{R} \; ,$$ and $$\label{uniformcv} F_N(z)={\mathrm{O}}({\mathrm{e}}^{-s})\; \text{uniformly in } \; [s_0,\infty), s_0 \in \mathbb{R} \; .$$ The scaling is slightly different from the original proof: $N^{-1/6}$ has been replaced by ${\sigma_N}^{-1/2}$. As in @imj, we focus on $w_N(z)=z^{(\alpha+1)/2}{\mathrm{e}}^{-z/2}L_N^{\alpha}(z)$, which satisfies $$\frac{d^2w}{dz^2}=\left(\frac{1}{4}-\frac{\kappa}{z}+\frac{\lambda^2-1/4}{z^2}\right)w \; ,$$ where $\kappa=N+(\alpha+1)/2$ and $\lambda=\alpha/2$. Remark that under ${\mathbf{AB}}{\overset{\text{def}}{\Leftrightarrow}}n,N,n/N {\rightarrow}\infty$, $\kappa \sim \lambda$. Our strategy is to reformulate the problem in terms of so-called Whittaker functions, denoted $W_{k,m}$, and to use the extensive available studies of these functions to show (\[whattoshow1\]) and (\[whattoshow2\]). @temme, formula (3.1) p.117 shows that $$w_N(z)=\frac{(-1)^N}{N!}W_{\kappa,\lambda}(z) \; .$$ From now on, we will closely follow @olver80. Let us remark that $$F_N(z)={\sigma_N}^{-1/2}\frac{1}{\sqrt{n!N!}}{W_{\kappa,\lambda}}(z) \;.$$ We fix $s_0 \in \mathbb{R}$, and we work only with $z=\mu_N+{\sigma_N}s$, where $s\geq s_0$. #### Preliminaries Following @olver80, we introduce $l=\kappa/\lambda$, $\beta=\sqrt{2(l-1)}$, and the turning points $x_1=2l-2\sqrt{l^2-1}$, $x_2=2l+2\sqrt{l^2-1}$, after the rescaling $x=z/\lambda$. We remark that the two turning points coalesce at 2 under the hypothesis ${\mathbf{AB}}$. In the new variable $x$, we have $$\frac{d^2W}{dx^2}=\left(\lambda^2 g(x) - \frac{1}{4x^2}\right)W \;,$$ where $g(x)=\frac{(x-x_1)(x-x_2)}{4x^2}$. Using the ideas explained in @imj, we shall be - eventually - interested in the asymptotics for $z=\mu_N+\sigma_N s$, or $x=z/\lambda=x_2+\sigma_N s/\lambda$ of $F_N(z)$. Let us now define an auxiliary variable $\upsilon$ by $$\begin{aligned} \int_{\beta}^{\upsilon}(\tau^2-\beta^2)^{1/2}d\tau &= \int_{x_2}^x g^{1/2}(t)dt \hspace{1cm} \text{if } x_2 \leq x < \infty \;, \\ \int_{-\beta}^{\upsilon}(\beta^2-\tau^2)^{1/2}d\tau &= \int_{x_1}^x (-g)^{1/2}(t)dt \hspace{1cm} \text{if } x_1 \leq x \leq x_2 \;.\end{aligned}$$ We limit $x$ to this range because of the technically important following point: ${\sigma_N}/\lambda$ tends to zero faster than $x_2-x_1$ does, and so, when $s$ is bounded below, $x$ will stay in the range $(x_1,\infty)$ for all $N$ greater than a certain $N_0$. This is shown in **A2**, along with the closely related fact that we can focus on $\upsilon\geq 0$. Our analysis is based on section 3 of @olver80, where he builds on @olver75, in which he expands Whittaker functions in terms of parabolic cylinder functions. The condition ${\upsilon}\geq 0$ is critical, since Olver’s expansions depend on the sign of $\upsilon$. Therefore, **A2** entitles us to focus on only one specific form of these. From (3.10) p.219 in @olver80, one has $$\begin{aligned} {W_{\kappa,\lambda}}(\lambda x)= (2\lambda)^{1/4}&\{\lambda(2+\beta^2/2)/e\}^{\lambda(1+\beta^2/4)} &\times\left(\frac{{\upsilon}^2-\beta^2}{x^2-4lx+4}\right)^{1/4}x^{1/2} \{U(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})+{\epsilon}_1(\lambda^2,\beta^2,{\upsilon})\} \; ,\end{aligned}$$ where, if ${\mathbf{E}}$ and ${\mathbf{M}}$ are the weight and modulus functions associated with $U$ in @olver75 (p.156), we have, according to @olver80 (3.11) p.219, $$\label{error} {\epsilon}_1(\lambda^2,\beta^2,{\upsilon})={\mathbf{E}}^{-1}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda}) {\mathbf{M}}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})\;\mathrm{O}(\lambda^{-2/3})$$ **uniformly** with respect to $\beta\in [0,B]$ and ${\upsilon}\in [0,\infty)$, $B$ being an arbitrary positive constant. We recall that the main relationship between $U$, ${\mathbf{E}}$ and ${\mathbf{M}}$ : for $b \leq 0$ and $x\geq 0$, $|U(b,x)|\leq {\mathbf{E}}^{-1}(b,x){\mathbf{M}}(b,x)$. We now show that we have uniform boundedness on $[s_0,\infty)$. The pointwise convergence result will be a straightforward consequence of the arguments we need to develop to solve this first problem. Uniform Boundedness ------------------- Following up on the previous displays, if $n,N$ are large enough so that ${\upsilon}\geq 0$, we have $$\begin{gathered} \label{Wbound} \left|{W_{\kappa,\lambda}}(\lambda x)\right|\leq (2\lambda)^{1/4}\{\lambda(2+\beta^2/2)/e\}^{\lambda(1+\beta^2/4)} \times\left(\frac{{\upsilon}^2-\beta^2}{x^2-4lx+4}\right)^{1/4}x^{1/2}{\mathbf{M}}{\mathbf{E}^{-1}}(1+\mathrm{O}(\lambda^{-2/3})) \;,\end{gathered}$$ where we omitted the argument $(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})$ for readability purposes. Our plan is now to transform this upper bound into a somewhat similar one, involving the modulus and weight function associated with the Airy function, which have the advantage of having only one parameter and known asymptotics. To carry out this program, we need to split the investigation into two parts: first $s\geq 0$ or ${\upsilon}\geq \beta$. This will allow us to find an $s_1 \geq 0$ such that $F_N(z)={\mathrm{O}}({\mathrm{e}}^{-s})$ on $[2s_1,\infty)$. In the second part, we will just have to consider the case $s\in [s_0,2s_1]$, and show that $F_N$ is merely uniformly bounded on this interval. ### Case $\mathbf{ s\geq 0} $ In order to use the results linking parabolic cylinder functions and the Airy function (proved in @olver59 and cited in @olver75), let us define yet another auxiliary variable, $\eta$, by $$\frac{2}{3}\eta^{3/2}\beta^2=\int_{x_2}^xg^{1/2}(t)dt \;.$$ Then, if we call ${{\cal E}}$ and ${{\cal M}}$ the weight and modulus functions associated with the Airy function, we have, as shown in **A3**: $$\begin{aligned} {\mathbf{E}^{-1}}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda}) &\leq {{\cal E}^{-1}}(\lambda^{2/3}\beta^{4/3}\eta)(1+{\mathrm{O}}((\lambda\beta^2)^{-1})) \; ,\\ {\mathbf{M}}(-\frac{1}{2}\lambda\beta^2,{\upsilon}\sqrt{2\lambda})&\leq \frac{\sqrt{2}\pi^{1/4}\left(\Gamma((1+\lambda\beta^2)/2)\right)^{1/2} \beta^{1/2}}{(\lambda\beta^2)^{1/12}}\\ &\times\left(\frac{\eta}{{\upsilon}^2-\beta^2}\right)^{1/4}{{\cal M}}(\lambda^{2/3}\beta^{4/3}\eta) \left(1+{\mathrm{O}}((\lambda\beta^2)^{-1})\right) \; .\end{aligned}$$ Whence, if we call $\theta\triangleq \lambda^{2/3}\beta^{4/3}\eta$, $$|F_N(\lambda x)|\leq K_{n,N}x^{1/2}(\eta/(x^2-4lx+4))^{1/4}{{\cal E}^{-1}}(\theta){{\cal M}}(\theta) \left(1+{\mathrm{O}}((\lambda\beta^2)^{-1}\vee \lambda^{-2/3})\right) \; .$$ In **A4**, we show that $K_{n,N}\sim 2^{2/3}(N/n)^{1/4}$ under ${\mathbf{AB}}$. From now on, $\Delta$ will denote a generic constant; its value may change from display to display. As long as $x\geq x_2$, or $s\geq 0$, we have $$|F_N(\lambda x)|\leq \Delta (N/n)^{1/4} x^{1/2}(\eta/(x^2-4lx+4))^{1/4}{{\cal E}^{-1}}(\theta){{\cal M}}(\theta)) \; .$$ Now using the fact that (see @olver, chap. 11) $x^{1/4}{{\cal M}}(x)\leq \Delta$, ${{\cal E}^{-1}}(x)\leq \Delta \exp(-2x^{3/2}/3)$ for $x\geq 0$ and $\lambda\beta^2=2N+1$, we get the new inequality $$|F_N(\lambda x)|\leq \Delta \left(\frac{N}{n}\right)^{1/4}N^{-1/6}\left(\frac{x^2}{x^2-4lx+4}\right)^{1/4}\exp(-(2\theta^{3/2})/3) \; .$$ In **A5.1**, we show that there exists $s_1$ such that if $s\geq 2 s_1$, $(2\theta^{3/2})/3 \geq s$. Also, as shown in **A6.1**, if $s\geq 0$, $g$ is positive and increasing in $x$ (or, equivalently, in $s$). Since the rational function of $x$ appearing in the previous display is just $(4g(x))^{-1/4}$, we can bound it by its value at $x(2s_1)$ on $[2s_1,\infty)$. In **A6.2**, we show that, at $s$ fixed, under ${\mathbf{AB}}$, we have $4g(x)\sim \beta{\sigma_N}s/\lambda$, and using the equivalents mentioned in **A1**, we have ${\sigma_N}\beta/\lambda\sim 4 N^{1/3}/n$, from which we conclude that $$\left(\frac{N}{n}\right)^{1/4}N^{-1/6}(4g(2s_1))^{-1/4}\sim N^{1/12}n^{-1/4}(8s_1N^{1/3}/n)^{-1/4}\sim (8s_1)^{-1/4} \; .$$ Therefore, if $N$ is large enough, $$\forall s\in[2s_1,+\infty)\;\;|F_N(\lambda x)| \leq \Delta \exp(-s)$$ ### Case $s\in[s_0,2s_1]$ {#casesinterval} Our aim now is just to show that $F_N$ as a function of $s$ is bounded on this interval; from this we shall immediately have that $F_N={\mathrm{O}}(\exp(-s))$ on this interval, and we will have a proof of (\[uniformcv\]).\ This part is comparatively simpler: we use equation (\[Wbound\]), in which we have ${\mathbf{E}^{-1}}\leq 1$, by definition (@olver75, p.156, (5.22)). Now using the display between (6.12) and (6.13) p.159 of the same article, we have for ${\lambda\beta^2}\geq 1$ and ${\upsilon}\geq 0$, $$\frac{{\mathbf{M}}(-\lambda\beta^2/2,{\upsilon}\sqrt{2\lambda})}{\left(\Gamma((1+\lambda\beta^2)/2)\right)^{1/2}}\leq \frac{\Delta\beta^{1/2}}{(\lambda\beta^2)^{1/12}}\left(\frac{\eta}{{\upsilon}^2-\beta^2}\right)^{1/4} \; .$$ Hence, $$|F_N(\lambda x)| \leq K_{n,N}\Delta\left(\frac{\eta}{x^2-4lx+4}\right)^{1/4}x^{1/2} \; .$$ However on this interval, $x{\rightarrow}2$, by **A5.2** $\eta=(\lambda\beta^2)^{-2/3}s+{\mathrm{o}}((\lambda\beta^2)^{-2/3})$, and by **A6.2** $(x^2-4lx+4)=4s{\sigma_N}\beta(1+{\mathrm{o}}(1))/\lambda$. Therefore, $$\frac{\eta}{x^2-4lx+4}\sim 2^{-2/3}2^{-4} n/N$$ on the whole interval, and, because of the asymptotic estimate of $K_{n,N}$ given in **A4**, $F_N$ is bounded uniformly in $N$ on the interval $[s_0,2s_1]$. We can thus conclude that $$\forall s_0, \; \; \exists N_0(s_0) \; N>N_0(s_0) ,\; \; \; F_N(s)={\mathrm{O}}_{s_0}({\mathrm{e}}^{-s}) \; \; \text{ on } [s_0,\infty) \; . $$ Pointwise convergence --------------------- Having studied in detail the uniform boundedness of $F_N$ makes the pointwise convergence problem easier. First, since we bounded above $F_N$ in terms of ${\mathbf{M}}$ and ${\mathbf{E}^{-1}}$, equation (\[error\]) shows that ${\epsilon}_1={\mathrm{O}}(\lambda^{-2/3}{\mathrm{e}}^{-s})$ on $[s_0,\infty)$. So for fixed $s$, it tends to zero as $N$ gets large. The pointwise limit of $F_N$ will be the pointwise limit of the parabolic cylinder function part of the expansion. We call this part $\wp F_N$, for “principal part". Using the relationship between $U$ and ${\mathrm{Ai}}$ that we mention in **A3**, we have, with $\theta=(\lambda\beta^2)^{2/3}\eta$, $$\wp F_N(\lambda x)=K_{n,N}x^{1/2}\left(\frac{\eta}{x^2-4lx+4}\right)^{1/4}({\mathrm{Ai}}(\theta)+{{\cal E}^{-1}}(\theta){{\cal M}}(\theta) \, {\mathrm{O}}((\lambda\beta^2)^{-1})) \; .$$ Since $x{\rightarrow}2$, $K_{n,N}\sim 2^{2/3}(N/n)^{1/4}$ and given the estimate we just mentioned for the ratio $\eta/(x^2-4lx+4)$, we have $$K_{n,N}\, x^{1/2}\left(\frac{\eta}{x^2-4lx+4}\right)^{1/4} \sim 1 \; .$$ In other respects, we show in **A5.2** that $\theta{\rightarrow}s$ under ${\mathbf{AB}}$. Finally, ${{\cal E}^{-1}}$ and ${{\cal M}}$ are bounded on $\mathbb{R}$, as shown in 11.2 (pp.394-397) of @olver. Hence ${{\cal E}^{-1}}(\theta){{\cal M}}(\theta) \, (\lambda\beta^2)^{-1} {\rightarrow}0 $ under ${\mathbf{AB}}$, and we can conclude that $\wp F_N(\lambda x){\rightarrow}{\mathrm{Ai}}(s)$; combining all the elements gives $$\forall s \in \mathbb{R}, \; \; \; \;F_N(\lambda x){\rightarrow}{\mathrm{Ai}}(s) \; \text{ under }{\mathbf{AB}}\; .$$ Asymptotics for $\mathbf{\phi_{\tau}} \text{ and } \mathbf{\psi_{\tau}}$ ------------------------------------------------------------------------ So far we have shown that $F_N(z)=(-1)^N {\sigma_N}^{-1/2}\sqrt{z}\phi_N(z){\rightarrow}{\mathrm{Ai}}(s)$, and that ${\mathrm{e}}^s F_N$ was bounded when $N>N_0$ and $s \geq s_0$.\ Our aim is to show (\[whattoshow1\]) and (\[whattoshow2\]). Let us write, as in @imj, $$\phi_{\tau}=\phi_{I,N}+\phi_{I\!I,N} \; ,$$ where $$\phi_{I,N}(z)=(-1)^N{\sigma_N}\sqrt{a_N n}\phi_N(z)/(\sqrt{2} z)= F_N(z)d_N (z/\mu_N)^{-3/2}\;.$$ #### Study of $\phi_{I,N}$ In the previous display, we have $d_N=({\sigma_N}/\mu_N)^{3/2}\sqrt{a_N n/2}$. As ${\sigma_N}\sim n^{1/2}N^{-1/6}$ and $\mu_N\sim n$, $({\sigma_N}/\mu_N)^{3/2}\sim n^{-3/4}N^{-1/4}$. Since $a_N=\sqrt{Nn}$, $a_Nn=n^{3/2}N^{1/2}$, and therefore $d_N{\rightarrow}1/\sqrt{2}$. But when $s$ is fixed, $z/\mu_N {\rightarrow}1$, so it follows that $$\text{Under }{\mathbf{AB}},\hspace{.5cm} \phi_{I,N}(\mu_N+{\sigma_N}s){\rightarrow}\frac{{\mathrm{Ai}}(s)}{\sqrt{2}} \;.$$ To bound $\phi_{I,N}$ for $N>N_0$ and $s\geq s_0$, we use, as in @imj, the uniform bound for $F_N$ and $(z/\mu_N)^{-3/2}\leq \exp(-3{\sigma_N}s/(2\mu_N))$, if $s\geq 0$. If $s\leq 0$, we have $(z/\mu_N)^{-3/2}\leq (1+s_0{\sigma_N}/\mu_N)^{-3/2}$, and since this converges to 1 under ${\mathbf{AB}}$, it is bounded if $N$ is large enough. So we have shown that, $$\phi_{I,N}(\mu_N+s{\sigma_N}) \left\{ \begin{array}{clc} {\rightarrow}& 2^{-1/2}{\mathrm{Ai}}(s)\;, \; \; N{\rightarrow}\infty \;, \\ \leq & M {\mathrm{e}}^{-s} \; \; \text{ on } [s_0,\infty) \text{ if } N>N_0(s_0) \; . \end{array} \right.$$ #### Study of $\phi_{I\!I,N}$ We use once again the same approach as in @imj. We have $$\phi_{I\!I,N}=u_N v_{N-1} \phi_{I,N-1} \; ,$$ where $u_N=({\sigma_N}/\sigma_{N-1})\sqrt{a_N/a_{N-1}}$ and $v_N=(N/n)^{1/2}$, and $n_{N-1}$ appearing in $\sigma_{N-1}$ is $n_N-1$ (for $\phi_{N-1}$ is defined in terms of $L_{N-1}^{\alpha_N}$ and we should therefore have the same $\alpha=n-N=(n-1)-(N-1)$). Remark that under ${\mathbf{AB}}$, $v_N {\rightarrow}0 $ and $u_N {\rightarrow}1$.\ Define $s'$ by $\mu_N+{\sigma_N}s=\mu_{N-1}+\sigma_{N-1}s'$. From $$s'=\frac{\mu_N-\mu_{N-1}}{\sigma_{N-1}}+\frac{{\sigma_N}}{\sigma_{N-1}}s \; ,$$ we deduce that $s'\geq s/2 $ on $[0,\infty)$, if $N$ is large enough: as a matter of fact, under ${\mathbf{AB}}$, $\mu_N -\mu_{N-1 }={\mathrm{O}}(\sqrt{n/N})$, $\sigma_{N} \sim n^{1/2}N^{-1/6}$, and ${\sigma_N}/\sigma_{N-1}{\rightarrow}1$, so it is larger than $1/2$ when $N$ is large enough. To summarize, we just showed that $$\phi_{I\!I,N}(\mu_N+s{\sigma_N})\leq M v_N {\mathrm{e}}^{-s/2} \; \; \text{for } s\in [0,\infty) \; ,$$ by applying the bound we got for $\phi_{I,N}$ to $\phi_{I,N-1}$ and $s'$ as the dummy variable. Here, we are implicitly using the fact that since $n/N{\rightarrow}\infty$, $(n-1)/(N-1)$ does too, and we can apply all the results we derived before. On the other hand, when $s \in[s_0,0]$, we can use the fact that $(\mu_N-\mu_{N-1})\geq 0$ and ${\sigma_N}/\sigma_{N-1} \leq 2$ to show that $s'\geq 2 s $ and hence $$\phi_{I\!I,N}(\mu_N+s{\sigma_N})\leq M v_N {\mathrm{e}}^{-2s}\leq M' v_N {\mathrm{e}}^{-s/2} \; \; \text{for } s\in [s_0,0] \; .$$ The conclusion is therefore that $$\phi_{I\!I,N}(\mu_N+s{\sigma_N}) \left\{\begin{array}{clc} {\rightarrow}& 0 \;, \; \; N{\rightarrow}\infty ,\\ \leq & \Delta {\mathrm{e}}^{-s/2} \; \; \text{ on } [s_0,\infty)\;, \text{ if } N>N_0(s_0) \;. \end{array} \right.$$ Hence we have shown that (\[whattoshow1\]) and (\[whattoshow2\]) held for $\phi_{\tau}$. The analysis for $\psi_{\tau}$ is similar. Appendices ========== This section is devoted to giving background information needed to understand the problem and make the paper relatively self-contained. We also establish many of the properties needed in the course of the proofs of equations (\[whattoshow1\]) and (\[whattoshow2\]) here.\ Before we start, let us mention a notation issue: $\alpha$ changes value depending on whether we treat the complex case or the real one. For the complex case $\alpha+N=n$, whereas for the real one $\alpha+N=n-1$. We frequently replace $\alpha+N$ by $n$ in what follows; this is because the proof of equations (\[whattoshow1\]) and (\[whattoshow2\]) is done in the complex case and applies to the real one by just changing $n$ into $n-1$ everywhere. When dealing with problems which are real case specific, we keep the notation $N+\alpha$. The definition of ${\mu_N}$ and ${\sigma_N}$ are also given in terms of $N+\alpha$ to highlight the adjustments needed when dealing with the real or the complex case. A0: Tracy-Widom distributions {#a0-tracy-widom-distributions .unnumbered} ----------------------------- We recall here the definition of the Tracy-Widom distributions. We split the description according to whether the entries of the matrix we are considering are real or complex.\ We first need to introduce the function $q$, defined as $$\left\{ \begin{array}{l} q''(x)=xq(x)+2q^3(x) \;,\\ q(x)\sim {\mathrm{Ai}}(x) \; \; \; \text{ as } x{\rightarrow}\infty \;. \end{array} \right.$$ **$\bullet$ Complex Case** The Tracy-Widom distribution appearing in the complex case, $W_2$, has cumulative distribution function $F_2$ given by $$F_2(s)=\exp\left(-\int_s^{\infty}(x-s)q^2(x) dx\right)\;.$$ The joint distribution is slightly more involved to define. Following @sosh, we do it through its $k$-point correlation functions, using its determinantal point process character (see e.g @soshDet).\ Let us first call $\bar{S}$ be the Airy operator. Its kernel is $$\bar{S}(x,y)=\frac{{\mathrm{Ai}}(x){\mathrm{Ai}}'(y)-{\mathrm{Ai}}(y){\mathrm{Ai}}'(x)}{x-y}=\int_0^{\infty}{\mathrm{Ai}}(x+u){\mathrm{Ai}}(y+u) du \;.$$ In the complex case, the $k$-point correlation functions have the property that $$\rho_k(x_1,\ldots,x_k)=\det_{1\leq i,j\leq k} \bar{S}(x_i,x_j) \;.$$ **$\bullet$ Real Case** The real counterpart of $W_2$, which is called $W_1$, has cdf $F_1$ with $$F_1(s)=\exp\left(-\frac{1}{2}\int_s^{\infty}q(x)+(x-s)q^2(x) dx\right) \;.$$ The $k$-point correlation functions satisfy $$\rho_k(x_1,\ldots,x_k)=\left(\det_{1\leq i,j\leq k} K(x_i,x_j)\right)^{1/2} \;,$$ where the $2\times 2$ matrix kernel of $K$ has entries (see @sosh, eq (2.18) to (2.21)) $$\begin{aligned} K_{1,1}(x,y)&=\bar{S}(x,y)+\frac{1}{2}{\mathrm{Ai}}(x)\int_{-\infty}^y {\mathrm{Ai}}(u) du \; ,\\ K_{2,2}(x,y)&=K_{1,1}(y,x)\; ,\\ K_{1,2}(x,y)&=-\frac{1}{2}{\mathrm{Ai}}(x){\mathrm{Ai}}(y)-\frac{\partial}{\partial y}\bar{S}(x,y)\;,\\ K_{2,1}(x,y)&=-\int_0^{\infty}dt\left(\int_{x+t}^{\infty}{\mathrm{Ai}}(v) dv \right){\mathrm{Ai}}(y+t) -{\epsilon}(x-y)+\frac{1}{2}\int_y^x{\mathrm{Ai}}(u)du+\frac{1}{2}\int_x^{\infty}{\mathrm{Ai}}(u)du\int_{-\infty}^y{\mathrm{Ai}}(v)dv \; .\end{aligned}$$ A1: Asymptotic behavior of some simple functions {#a1-asymptotic-behavior-of-some-simple-functions .unnumbered} ------------------------------------------------ In this appendix, we present some basic facts and identities that we used throughout the proof.\ We will make repeated use of the following observations: since ${\sigma_N}=(\sqrt{(N+\alpha)_+}+\sqrt{N_+})(1/\sqrt{N_+}+1/\sqrt{(N+\alpha)_+})^{1/3} $ and $ \lambda =\alpha/2$, under **AB** we have $$\begin{aligned} {\sigma_N}&\sim n^{1/2}N^{-1/6} \;,\\ \lambda&\sim n/2 \; .\end{aligned}$$ We also use several times the following identities: \[factequal\] With $\lambda=\alpha/2$, $\kappa=N+(\alpha+1)/2$, and $l=\kappa/\lambda$, $\beta=\sqrt{2(l-1)}$, we have $$\begin{aligned} \lambda\beta^2&=(2N+1) \;,\\ \beta &\sim 2\sqrt{N/n} \;.\end{aligned}$$ The first remark is simple algebra, and the second one comes from $\beta^2=2(l-1)=2(2N+1)/\alpha\sim 4N/n$ under ${\mathbf{AB}}$. We have the estimates: \[factequiv\] $x_2-x_1 \sim 8 \sqrt{N/n}$ and $\sigma_N/\lambda \sim 2 n^{-1/2}N^{-1/6}$ . The second one is obvious; the first one comes from the fact that $x_2-x_1 = 2\sqrt{2}\beta (l+1)^{1/2}$ as $x_{2,1}=2l \pm 2\sqrt{l^2-1}$. Using Fact \[factequal\] immediately gives the claimed result. Finally, we have the following estimates \[factfunctions\] $\beta{\sigma_N}/\lambda \sim 4 N^{1/3}/n$ and ${\sigma_N}^3/(\lambda\beta^2)\sim(n/N)^{3/2}/2$ . The result directly follows from the aforementioned estimates. A2: Working with ${\upsilon}\geq 0$ {#a2-working-with-upsilongeq-0 .unnumbered} ----------------------------------- Here we assume that $s\in[s_0,\infty)$. We also assume that $s<0$, for otherwise we can work with ${\upsilon}\geq \beta>0$. From **A1**, we have $|x-x_2|=|s|{\sigma_N}/\lambda \leq |s_0|{\sigma_N}/\lambda \ll x_2-x_1 $ by Fact \[factequiv\]. Now ${\upsilon}=0$ corresponds to $x_0\leq \bar{x}=(x_1+x_2)/2$: as a matter of fact, since $(x_2-x)(x_1-x)$ is symmetric around $\bar{x}$ and $1/x$ is obviously larger on $[x_1,\bar{x}]$ than it is on $[\bar{x},x_2]$, we have $$\int_{-\beta}^{{\upsilon}_{\bar{x}}} (\beta^2 - \tau^2) d\tau = \int_{x_1}^{\bar{x}}(-g(t))^{1/2}dt \geq \int_{\bar{x}}^{x_2}(-g(t))^{1/2}dt \; .$$ By symmetry, we also get $$\begin{aligned} \int_{-\beta}^{0} (\beta^2 - \tau^2) d\tau = \int_{0}^{\beta} (\beta^2 - \tau^2) d\tau & = \frac{1}{2} \int_{x_1}^{x_2}(-g(t))^{1/2}dt \\ &\leq \int_{x_1}^{\bar{x}}(-g(t))^{1/2}dt = \int_{-\beta}^{{\upsilon}_{\bar{x}}} (\beta^2 - \tau^2) d\tau \; ,\end{aligned}$$ and therefore, ${\upsilon}_{\bar{x}}>0$.\ However $\bar{x}$ is always smaller than $x(s_0)$ if $N$ is large enough. So we can limit our investigations to the case ${\upsilon}\geq 0$. A3: Relationship between ${\mathbf{E}^{-1}}$, ${{\cal E}^{-1}}$, ${\mathbf{M}}$ and ${{\cal M}}$ {#a3-relationship-between-mathbfe-1-cal-e-1-mathbfm-and-cal-m .unnumbered} ------------------------------------------------------------------------------------------------ We claim that if $s\geq 0$, and we define $\theta=(\lambda\beta^2)^{2/3}\eta$, the following inequalities hold true: $$\begin{aligned} {\mathbf{E}^{-1}}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})&\leq {{\cal E}^{-1}}(\theta)(1+{\mathrm{O}}(({\lambda\beta^2})^{-1})) \; ,\\ {\mathbf{M}}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})&\leq \frac{(4\pi)^{1/4}}{({\lambda\beta^2})^{1/12}}[\Gamma((1+\lambda\beta^2)/2)]^{1/2}\beta^{1/2}\left(\frac{\eta}{{\upsilon}^2-\beta^2}\right)^{1/4}\\ & \times{{\cal M}}(\theta)(1+{\mathrm{O}}(({\lambda\beta^2})^{-1})) \;.\end{aligned}$$ For the sake of simplicity we call $\Xi$ the part that precedes the sign “$\times$" in the last inequality.\ According to @olver75, equations (5.12) and (5.13), we have $$\begin{aligned} U(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})&=\Xi\left\{{\mathrm{Ai}}(\theta)+{{\cal M}}(\theta){{\cal E}^{-1}}(\theta)\,{\mathrm{O}}(({\lambda\beta^2})^{-1})\right\} \; ,\\ \bar{U}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})&=\Xi\left\{{\mathrm{Bi}}(\theta)+{{\cal M}}(\theta){{\cal E}^{-1}}(\theta)\,{\mathrm{O}}(({\lambda\beta^2})^{-1})\right\} \;.\end{aligned}$$ We have, if $s\geq 0$, $x\geq x_2$, so $$\label{vardef} 2/3\beta^2\eta^{3/2}=\int_{x_2}^x g^{1/2}(t)dt=\int_{\beta}^{{\upsilon}}(\tau^2-\beta^2)^{1/2}d\tau \;.$$ For the Airy function, the weight and modulus functions had different definition depending on whether the argument was bigger than the largest root, $c$, of ${\mathrm{Ai}}(z)={\mathrm{Bi}}(z)$ or not. Likewise, the definition of ${\mathbf{E}^{-1}}$ and ${\mathbf{M}}$ depends on the position of the argument with respect to the largest root of the equation $\bar{U}(b,x)=U(b,x)$, which is called $\rho(b)$ in @olver75. #### Where do the auxiliary variables lie when $\mathbf{s\geq 0}$? We claim that the answer is that $\theta \geq 0 > c$, and ${\upsilon}\sqrt{2\lambda}\geq \rho(-{\lambda\beta^2}/2)$.\ The first part of equation (\[vardef\]) implies that $\eta\geq 0$, so $\theta \geq c$, as $c<0$. This means that we can use the definition ${{\cal M}}^2=2{\mathrm{Ai}}{\mathrm{Bi}}$ and ${{\cal E}^{-1}}{{\cal M}}=2^{1/2}{\mathrm{Ai}}$. The second part implies that ${\upsilon}\geq \beta$; therefore, $2\lambda{\upsilon}^2\geq 2{\lambda\beta^2}\geq \rho(-{\lambda\beta^2}/2)^2$, since by @olver75, equation (5.21), $\rho(b)\leq 2(-b)^{1/2}$ when $b{\rightarrow}-\infty$. This means that we have similar relationships between ${\mathbf{E}^{-1}}$, ${\mathbf{M}}$, $U$, and $\bar{U}$, to the one we had in the Airy case, $\bar{U}$ playing the role of ${\mathrm{Bi}}$, and $U$ playing the role of ${\mathrm{Ai}}$. #### Consequences of their positions The interesting consequence of the previous remarks is that we can write, if $N$ is large enough, for all $s\geq 0$ $$\mathbf{E}^{-2}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})=\frac{U}{\bar{U}}=\frac{{{\cal M}}(\theta){{\cal E}^{-1}}(\theta)}{{{\cal M}}(\theta){\cal E}(\theta)} \frac{2^{-1/2}+{\mathrm{O}}(({\lambda\beta^2})^{-1})}{2^{-1/2}+{\mathrm{O}}(({\lambda\beta^2})^{-1})} \;.$$ In other words, we just proved that $\exists N_0$ such that $N>N_0$ implies, $\forall \, s\geq 0$ $${\mathbf{E}^{-1}}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda}) \leq {{\cal E}^{-1}}(\theta)(1+{\mathrm{O}}(({\lambda\beta^2})^{-1}) \;.$$ By the same arguments, we derive that $${\mathbf{M}}(-{\lambda\beta^2}/2,{\upsilon}\sqrt{2\lambda})\leq \Xi {{\cal M}}(\theta)(1+{\mathrm{O}}(({\lambda\beta^2})^{-1})\;.$$ A4: Asymptotic behavior of $K_{n,N}$ {#a4-asymptotic-behavior-of-k_nn .unnumbered} ------------------------------------ The aim here is to show that $$K_{n,N}\sim 2^{2/3} (N/n)^{1/4} \;.$$ $K_{n,N}$ has the following expression: $$K_{n,N}=\frac{(2\lambda)^{1/4}\{\lambda(2+1/2\beta^2)/e\}^{\lambda(1+\beta^2/4)}\sqrt{2}\pi^{1/4}[\Gamma((1+\lambda\beta^2)/2)]^{1/2}\beta^{1/2}}{(\lambda\beta^2)^{1/12}\sqrt{n!N!{\sigma_N}}} \; .$$ Since $\lambda\beta^2=(2N+1)$, $\Gamma((1+\lambda\beta^2)/2)=\Gamma(N+1)=N!$ .\ In other respects, let $A_n=\{\lambda(2+1/2\beta^2)/e\}^{\lambda(1+\beta^2/4)}/\sqrt{n!}$ . Note that $2\lambda+\lambda\beta^2/2=n-N+(2N+1)/2=n+1/2=n_+$. So $A_n=(n_+/e)^{n_+/2}/\sqrt{n!}$. Using Stirling’s formula, we get that $A_n\sim (n_+/n)^{n/2}(n_+/n)^{1/4}(2\pi {\mathrm{e}})^{-1/4} \sim (2\pi)^{-1/4}$.\ Now rewriting $$K_{n,N}=\frac{A_n(\lambda\beta^2)^{1/4}(8\pi)^{1/4}}{(\lambda\beta^2)^{1/12}\sqrt{{\sigma_N}}} \;,$$ we get that $K_{n,N}\sim 2^{2/3} (N/n)^{1/4}$, from using $A_n(8\pi)^{1/4}\sim \sqrt{2}$ and the second estimates of Fact \[factfunctions\] in **A1**. A5: Asymptotic properties of $\mathbf{\eta}$ {#a5-asymptotic-properties-of-mathbfeta .unnumbered} -------------------------------------------- This appendix is divided into two parts. We first show that there exists $s_1$ such that, if $s\geq 2s_1$, $$\begin{gathered} \frac{2}{3}{\lambda\beta^2}\eta^{3/2} \geq s \;.\tag{P1}\end{gathered}$$ Then we shall show: $$\begin{gathered} \text{uniformly in }s \in [a,b], \; \; \;(2N+1)^{2/3}\eta=s+{\mathrm{o}}(1) \;.\tag{P2}\end{gathered}$$ ### A5.1: Proof of P1 {#a5.1-proof-of-p1 .unnumbered} This is the argument that was used in **A8** of @imj. We repeat it for the sake of completeness.\ Let us first suppose that $s$ is given. Since $g(x)=(x-x_1)(x-x_2)/(4x^2)$, we have $${\sigma_N}^2g(x) = s\frac{\sigma_N^3}{\lambda}\frac{(x_2-x_1)+s{\sigma_N}/\lambda}{4(x_2+s{\sigma_N}/\lambda)^2} \sim s\frac{{\sigma_N}^32\sqrt{2}\beta(l+1)^{1/2}}{16 \lambda}\sim s\frac{\beta{\sigma_N}^3}{4\lambda}\;,$$ the first equivalent coming from the fact that when $s$ is fixed, $x_2-x_1 \gg s{\sigma_N}/\lambda$, and $x_2{\rightarrow}2$. The second is just $l{\rightarrow}1$ under ${\mathbf{AB}}$. Now using the first point of Fact \[factfunctions\] in **A1**, together with ${\sigma_N}^2\sim n N^{-1/3}$, we get that $(\beta{\sigma_N}^3)/(4\lambda){\rightarrow}1$. So at $s$ fixed, $${\sigma_N}^2 g(x){\rightarrow}s \;.$$ Having this information let us now pick $s_1=8$. If $N$ is large enough, we have ${\sigma_N}^2 g(x(s_1))\geq s_1/2=4$. For all (fixed) $N$ $g$ is an increasing function of $s$. Therefore for the same $N$ we will have $$\forall \, s\geq s_1 \; {\sigma_N}^2 g(x) \geq {\sigma_N}^2 g(x(s_1))\geq s_1/2=4 \; ,$$ and hence, since $s\geq s_1\geq 0$, $g$ is positive and we have $g^{1/2}(s)\geq 2/{\sigma_N}$. Therefore, $$\frac{2}{3}{\lambda\beta^2}\eta^{2/3}=\lambda \int_{x_2}^xg^{1/2}(t)dt \geq \int_{x(s_1)}^xg^{1/2}(t)dt \geq \frac{2\lambda}{{\sigma_N}}\frac{{\sigma_N}}{\lambda}(s-s_1)=2(s-s_1)\;.$$ Consequently, if $s\geq 2s_1$, we have (P1). ### A5.2: Proof of P2 {#a5.2-proof-of-p2 .unnumbered} Without loss of generality, we can suppose that $a$ and $b$ have the same sign, and $a\geq 0$. (If it is not the case, we can split $[a,b]=[a,0]\bigcup[0,b]$, apply the reasoning on each of these, and get the claimed result for the original interval.)\ The idea is that on $[a,b]$, we have $$\frac{(x-x_2)(x_2+b{\sigma_N}/\lambda -x_1)}{4(x_2+a{\sigma_N}/\lambda)^2} \geq g(x)\geq \frac{(x-x_2)(x_2-x_1+a{\sigma_N}/\lambda)}{4(x_2+b{\sigma_N}/\lambda)^2} \; .$$ Now on both sides, the terms which are not $(x-x_2)$ are $(x_2-x_1)(1+{\mathrm{o}}(1))=4\beta(1+{\mathrm{o}}(1))$, again because ${\sigma_N}/\lambda \ll \beta$. So if we integrate the square root of the previous inequality between $x_2$ and $x(s)$, we get $$\begin{aligned} 2/3(s{\sigma_N}/\lambda)^{3/2}2\sqrt{\beta}(1+{\mathrm{o}}(1))/4 \geq 2/3 \eta^{3/2}\beta^2 \geq 2/3(s{\sigma_N}/\lambda)^{3/2}2\sqrt{\beta}(1+{\mathrm{o}}(1))/4 \;,\end{aligned}$$ or $$\frac{1}{2}s^{3/2}({\sigma_N}^3\beta/\lambda)^{1/2}(1+{\mathrm{o}}(1)) \geq \eta^{3/2}\lambda\beta^2 \geq \frac{1}{2}s^{3/2}({\sigma_N}^3\beta/\lambda)^{1/2}(1+{\mathrm{o}}(1)) \;.$$ The conclusion follows from **A1**, Fact \[factfunctions\], whose first point, along with the estimate of ${\sigma_N}$ mentioned there, shows that ${\sigma_N}^3\beta/\lambda\sim 4$. We note that (P2) also gives us pointwise convergence of $({\lambda\beta^2})^{2/3}\eta$ to $s$. A6: Properties of $\mathbf{g}$ {#a6-properties-of-mathbfg .unnumbered} ------------------------------ We first show that $g$ is increasing - at $N$ fixed - as a function of $s$, if $s\geq 0$. Then we give an estimate of $4x^2g(x)$ as $N{\rightarrow}\infty$ and $s\in[a,b]$. ### A6.1: $\mathbf{g}$ is increasing on $s\geq 0$ {#a6.1-mathbfg-is-increasing-on-sgeq-0 .unnumbered} Since $g(t)=(t-x_2)(t-x_1)/(4t^2)=(t^2-4lt+4)/(4t^2)$, we have $$g'(t)=\frac{l}{t^2}-\frac{2}{t^3}=\frac{lt-2}{t^3} \;.$$ Now $lx_2=2l^2+2l\sqrt{l^2-1}\geq 2$, since $l=1+(2N+1)/\alpha\geq 1$. But $lx\geq lx_2$ when $s\geq 0$, and the assertion is proved. ### A6.2: On the asymptotic behavior of $\mathbf{4x^2g(x)}$ for $\mathbf{s\in [a,b]}$ {#a6.2-on-the-asymptotic-behavior-of-mathbf4x2gx-for-mathbfsin-ab .unnumbered} This estimate is motivated by the fact that in the course of the proof of the main result, we have to deal with an expression of the form $$\frac{\eta}{x^2-4lx+4} \;.$$ We already studied in detail $\eta$ as a function of $s$ and $N$. We now focus on $x^2-4lx+4$. Recalling that $x^2-4lx+4=(x-x_2)(x-x_1)$ and $x=x_2+s{\sigma_N}/\lambda$, we have $$x^2-4lx+4=s\frac{{\sigma_N}}{\lambda}(x_2-x_1+s\frac{{\sigma_N}}{\lambda})=s\frac{{\sigma_N}}{\lambda}(x_2-x_1+{\mathrm{o}}(\beta)) \;,$$ because the first estimate in Fact \[factfunctions\] shows that ${\sigma_N}/\lambda = {\mathrm{o}}(\beta)$, and since $s \in [a,b]$, the previous statement holds true uniformly on this interval. Now $x_2-x_1\sim 4\beta$ under ${\mathbf{AB}}$, and therefore, uniformly on $[a,b]$, $$x^2-4lx+4=s\frac{{\sigma_N}}{\lambda}4\beta(1+{\mathrm{o}}(1)) \; ,$$ as was claimed in \[casesinterval\]. Also, since $x=x_2+s{\sigma_N}/\lambda$, and $x_2=2+(2l+2)^{1/2}\beta+\beta^2$, $$4g(x)=s\frac{{\sigma_N}}{\lambda}\beta(1+{\mathrm{o}}(1)) \;.$$ A7: Limit of $c_{\phi}$ {#a7-limit-of-c_phi .unnumbered} ----------------------- Recall that under the notation of @imj, $$\sqrt{2} c_{\phi}=\frac{1}{2}\sqrt{a_N}\left(\sqrt{N+\alpha}\int \xi_N - \sqrt{N} \int \xi_{N-1}\right)$$ where $\xi_k(x)=x^{\alpha/2-1}e^{(-x/2)}L_k^{(\alpha)}(x)\sqrt{\frac{k!}{(k+\alpha)!}}$. We are interested in\ $$\begin{aligned} v_{k,\alpha}&=\sqrt{k+\alpha}\int \xi_k - \sqrt{k} \int \xi_{k-1} \\ &=\sqrt{\frac{k!}{(k+\alpha-1)!}}\int_0^{\infty}x^{\alpha/2-1}e^{(-x/2)} \left(L^{\alpha}_k(x)-L^{\alpha}_{k-1}(x)\right) dx \\ \text{(by \citet{szego} 5.1.13 p.102)} &= \sqrt{\frac{k!}{(k+\alpha-1)!}}\int_0^{\infty}x^{\alpha/2-1}e^{(-x/2)} L^{\alpha-1}_k(x) dx \\ &=\sqrt{\frac{k!}{(k+\alpha-1)!}} I_{k,\alpha} \;.\end{aligned}$$ Now using @szego 5.1.9 p.101, $$\sum_{k=0}^{\infty}w^k L_k^{\alpha-1}(x)=(1-w)^{-\alpha} \exp\left(-\frac{xw}{1-w}\right) \;.$$ So if $F(\alpha)=\int_0^{\infty} \left(\sum_{k=0}^{\infty}w^k L_k^{\alpha-1}(x)\right)x^{\alpha/2-1}e^{(-x/2)} dx$, we have: $$\begin{aligned} F(\alpha)&=(1-w)^{-\alpha}\int_0^{\infty} x^{\alpha/2-1}e^{(-x/2)} {\mathrm{e}}^{-xw/(1-w)}dx\\ &=(1-w)^{-\alpha}\Gamma(\alpha/2)\left(\frac{2(1-w)}{1+w}\right)^{\alpha/2}\\ &=2^{\alpha/2}(1-w^2)^{-\alpha/2}\Gamma(\alpha/2) \;.\end{aligned}$$ Now if $x\geq 0,\hspace{.3cm} |L_n^{\alpha-1}(x)| \leq L_n^{\alpha-1}(-x)$, by 5.1.6 in @szego, and hence $$\left|\sum_{k=0}^{+\infty}w^k L_k^{\alpha-1}(x)\right|\leq \sum_{k=0}^{+\infty}|w|^k L_k^{\alpha-1}(-x)=(1-|w|)^{-\alpha}\exp\left(\frac{x|w|}{1-|w|}\right) \;.$$ Therefore, as long as $w \in(-1/3,1/3)$, we can switch orders of summation, and get $$\sum_{k=0}^{\infty}w^k I_{k,\alpha}=2^{\alpha/2}(1-w^2)^{-\alpha/2}\Gamma(\alpha/2) \;.$$ But $(1-w)^{-\alpha/2}\Gamma(\alpha/2)=\sum_{k=0}^{+\infty}\frac{\Gamma(\alpha/2+k)}{k!}w^k$, since the right-hand side converges without any difficulty on $(-1/3,1/3)$, and hence $$\sum_{k=0}^{\infty}w^kI_{k,\alpha}=\sum_{m=0}^{\infty}\frac{2^{\alpha/2}\Gamma(\alpha/2+m)}{m!}w^{2m}\;.$$ So we have $$\forall k\in 2\mathbb{N}, \hspace{.5cm} I_{k,\alpha}=\frac{2^{\alpha/2}\Gamma((\alpha+k)/2)}{(k/2)!} \;.$$ Now $v_{k,\alpha}=\sqrt{\frac{k!}{(k+\alpha-1)!}}I_{k,\alpha}=2^{\alpha/2}\frac{\Gamma((\alpha+k)/2)}{\sqrt{(k+\alpha-1)!}}\frac{\sqrt{k!}}{(k/2)!}$. Since $\Gamma(z)\sim (z/e)^z \sqrt{2\pi/z}$, we have $$\begin{aligned} \frac{\Gamma((\alpha+k)/2)}{\sqrt{\Gamma(k+\alpha)}}&\sim 2^{-(\alpha+k)/2}(\alpha+k)^{-1/4}(2\pi)^{1/4} \sqrt{2} \;, \\ \frac{\sqrt{k!}}{(k/2)!}&\sim 2^{k/2}(\pi k)^{-1/4}2^{1/4} \;,\end{aligned}$$ which in turn leads to $$\begin{aligned} v_{k,\alpha}&\sim 2^{\alpha/2}(k(\alpha+k))^{-1/4}2^{-(\alpha+k)/2}2^{k/2} \sqrt{2} \sqrt{2}\\ &\sim 2 (k(\alpha+k))^{-1/4} \\ &\sim 2/\sqrt{a_k} \;.\end{aligned}$$ Hence, as $N$ is even, $\sqrt{2}c_{\phi}=v_{N,\alpha} \sqrt{a_N}/2 {\rightarrow}1 $. A8: On @sosh Lemma 1 {#a8-on-lemma-1 .unnumbered} -------------------- For @sosh Lemma 1 to hold true in our case, we have to check two things. First that not only does ${\sigma_N}\phi({\mu_N}+{\sigma_N}s){\rightarrow}{\mathrm{Ai}}(s)/\sqrt{2}$, but also that this is true for the derivative: $$\label{eq:sosh1} {\sigma_N}^2 \phi'({\mu_N}+{\sigma_N}s) {\rightarrow}\frac{1}{\sqrt{2}} {\mathrm{Ai}}'(s) \; .\tag{S1}$$ We also have to verify that ${\sigma_N}^2 \phi'({\mu_N}+{\sigma_N}s)$ is bounded above by $\Delta(s_0) \exp(-\Delta s)$ on $[s_0,\infty)$, where $\Delta$ is a positive constant. We need to verify this for $\psi$ as well, but the techniques are similar, so we will verify it only for $\phi$. The second point that we need to check is that $$\label{eq:sosh2} \int_0^{\infty}\left(\int_0^{z}\phi(u) du \; \psi(y+z)\right) dz {\rightarrow}0 \; \text { as } N {\rightarrow}\infty \; .\tag{S2}$$ ### A8.1: Proof of (\[eq:sosh1\]) {#a8.1-proof-of-eqsosh1 .unnumbered} It is easy to see that all we need to work on are the properties of $g_N(s)=F_N({\mu_N}+{\sigma_N}s)$; if we can show that ${\sigma_N}F_N'({\mu_N}+{\sigma_N}s) {\rightarrow}{\mathrm{Ai}}'(s)$, and that it is bounded by $\Delta(s_0) {\mathrm{e}}^{-\Delta s}$ on $[s_0,\infty)$, we will be done.\ We have very easily that $$-{\sigma_N}F_N'({\mu_N}+{\sigma_N}s_1)=\int_{s_1}^{\infty} {\sigma_N}^2 \left. \frac{d^2 F_N}{ds^2} \right|_{{\mu_N}+{\sigma_N}u} du \; .$$ So the strategy is clear: we want to show that the integrand in the right-hand side is bounded by an integrable function and that it converges pointwise to ${\mathrm{Ai}}''(u)=u{\mathrm{Ai}}(u)$.\ However, $$\left. \frac{d^2 F_N(x)}{dx^2} \right|_{{\mu_N}+{\sigma_N}u} = \left[ \frac{1}{4}-\frac{{\kappa_N}}{{\mu_N}+{\sigma_N}u}+\frac{\lambda^2-1/4}{({\mu_N}+{\sigma_N}u)^2}\right] F_N({\mu_N}+{\sigma_N}u) \;,$$ and since we already know that $F_N({\mu_N}+{\sigma_N}s){\rightarrow}{\mathrm{Ai}}(s)$, we first need to check that, pointwise, $${\sigma_N}^2 \left[ \frac{1}{4}-\frac{{\kappa_N}}{{\mu_N}+{\sigma_N}s}+\frac{\lambda^2-1/4}{({\mu_N}+{\sigma_N}s)^2}\right] {\rightarrow}s \;.$$ In turn, this reduces to showing that $$\begin{gathered} {\sigma_N}^2 \left[ \frac{1}{4}-\frac{{\kappa_N}}{{\mu_N}}+\frac{\lambda^2-1/4}{{\mu_N}^2}\right] {\rightarrow}0 \;, \text{ and }\\ \frac{{\sigma_N}^3}{{\mu_N}}\left[\frac{{\kappa_N}}{{\mu_N}}-2 \frac{\lambda^2-1/4}{{\mu_N}^2}\right] {\rightarrow}1 \;.\end{gathered}$$ The first result comes from the remarkable equality ${\kappa_N}/{\mu_N}-\lambda^2/{\mu_N}^2=1/4$, which follows from the fact that if we call $x=\sqrt{N_+/(N+\alpha)_+}$, we have ${\kappa_N}/{\mu_N}=.5-x/(1+x)^2$ and $\lambda^2/{\mu_N}^2=.25-x/(1+x)^2$. Using these estimates, we see that ${\kappa_N}/{\mu_N}-2(\lambda/{\mu_N})^2 =x/(1+x)^2 \sim \sqrt{N_+/n_+}$, from which we conclude that the second result holds.\ Note that if we changed the centering and scaling (replacing $n$ by $\tilde{n}=n+\alpha$ and $N$ by $\tilde{N}=N+\beta$), by studying the first expression in this case as a “perturbation" of the study we just did, and using the fact that $\mu_{\tilde{N}}-{\mu_N}={\mathrm{O}}(\sqrt{n/N})$, one could show that the first expression is then ${\mathrm{O}}(N^{-1/3})$, and so the result would hold. We also have corresponding results for the second expression. This shows that we have some freedom in the centering and scaling we pick. It is also needed to show that $${\sigma_N}^2 \phi'({\mu_N}+{\sigma_N}s){\rightarrow}{\mathrm{Ai}}(s) \;,$$ since in our splitting of $\phi$, the second part $\phi_{II,N}$ corresponds to parameters $(n-1,N-1)$, but is centered and scaled using ${\mu_N}$ and ${\sigma_N}$, defined with $(n,N)$. To show that the sequence of functions we are interested in is bounded above by an integrable function, we split $[s_0,\infty)$ into $[s_0,\sqrt{n}]$ and $[\sqrt{n},\infty)$. On the first interval, we can apply the previous results since ${\sigma_N}s/{\mu_N}$ is small compared to 1. So in particular the whole integrand will be smaller that $\Delta(s_0) (1+|s|)^2 \exp(-s/2)$, after taking into account the properties of $F_N$. On the other hand, on $[\sqrt{n},\infty)$, ${\sigma_N}^2 \leq s^2$, and the denominators involving $s$ are bigger than ${\mu_N}$ and ${\mu_N}^2$ respectively, which gives immediately that the integrand is less than $\Delta(s_0) s^2 \exp(-s/2)$. From this we conclude that the integrand is less than $\Delta(s_0) \exp(-s/4)$, for instance, and that therefore the derivative we are interested in is too. It then follows easily that (\[eq:sosh1\]) is true, and we also showed that the left-hand side of (\[eq:sosh1\]) is dominated on $[s_0,\infty)$ and for $N>N_0(s_0)$ by $\Delta(s_0){\mathrm{e}}^{-s/4}$. ### A8.2: Proof of (\[eq:sosh2\]) {#a8.2-proof-of-eqsosh2 .unnumbered} The approach laid out in @sosh p.1044 works after some modifications. We first write $$\int_0^{\infty}\left(\int_0^{z}\phi(u) du \; \psi(y+z)\right) dz = \int_0^{n^{5/8}}\left(\int_0^{z}\phi(u) du \; \psi(y+z)\right) dz+ \int_{n^{5/8}}^{\infty}\left(\int_0^{z}\phi(u) du \; \psi(y+z)\right) dz \;.$$ Then we can check, via a third order asymptotic development in $x$ of the right-hand side of equation (2.10) in @olver80, that equation (2.18) therein is still true in our case, since, with his notations, $x_N\leq n^{-3/8}$. Therefore, the analysis carried out after equation (3.21) of the same reference applies, and after integration of the expansion following (3.22) adapted to our situation, we can show that $$\int_0^{n^{5/8}} \phi(u) du = {\mathrm{O}}(n^{-n/16})$$ With this estimate and this splitting of $[0,\infty)$, the rest of Soshnikov’s argument holds true and therefore (\[eq:sosh2\]) can be verified. A9: A quick look at special functions {#a9-a-quick-look-at-special-functions .unnumbered} ------------------------------------- In this note, we mentioned three types of special functions, Airy, Whittaker, and parabolic cylinder functions. We recall their definition in this appendix, as well as the main ideas behind some of the transformations Olver used. To justify their introduction, let us say that they play a special role because it is possible, in the setting we were in, to write the functions we studied as a perturbation of the differential equations these functions satisfy. ### A9.1: Airy function {#a9.1-airy-function .unnumbered} Let us consider the following second order differential equation: $$\frac{d^2w}{dx^2}=xw \;.$$ #### General remark: Recessive solutions Since these functions are used to get asymptotic expansions, it makes sense to define the independent solutions with respect to their behavior at $+\infty$. Usually, independent solutions $w_1$ and $w_2$ are sought, so that $w_2={\mathrm{o}}(w_1)$ at a particular point of the (extended) real line. In our cases, it will be $\infty$. $w_2$ is called the *recessive* solution. That leaves the problem underdetermined, but with this in mind, one can then give enough constraints so the problem is fully determined, and solve in terms of recessive and dominant solutions. For a more precise definition of recessivity, see @olver, p.155. In the case of the Airy function, we have for example: (from @olver, 11.1, p.392) $$\begin{aligned} {\mathrm{Ai}}(x)&= \frac{1}{\pi}\int_0^{\infty}\cos(t^3/3+xt)dt \\ {\mathrm{Bi}}(x)&= \frac{1}{\pi}\int_0^{\infty}\{\exp(-t^3/3+xt)+\sin(t^3/3+xt)\}dt\end{aligned}$$ ### A9.2: Whittaker functions {#a9.2-whittaker-functions .unnumbered} These are solution of the following differential equation $$\label{eq:whittaker} \frac{d^2 W}{dx^2}=\left(\frac{1}{4}-\frac{\kappa}{z}+\frac{\lambda^2 -1/4}{z^2}\right)W \;.$$ ${W_{\kappa,\lambda}}$, the recessive solution at $\infty$, is obtained by requiring $${W_{\kappa,\lambda}}(x) \sim e^{-x/2}x^{\kappa} \; \text{as } x{\rightarrow}\infty \;.$$ The other solution is $M_{\kappa,\lambda}$, which is required to satisfy $$M_{\kappa,\lambda}(x) \sim x^{\lambda+1/2} \; \text{as } x{\rightarrow}0^+ \;.$$ For more detail on these, see @olver, p.260, or @olver80. ### A9.3: Parabolic cylinder functions {#a9.3-parabolic-cylinder-functions .unnumbered} According to @olver59, equation (2.9) p.133, [parabolic cylinder functions]{} satisfy (in the case we are interested in) $$\frac{d^2 W}{dx^2}=\left(\frac{1}{4}x^2+a\right)W \;.$$ $U(a,x)$ is chosen to satisfy $$U(a,x) \sim x^{-a-1/2}{\mathrm{e}}^{-x^2/4} \; \text{as } x{\rightarrow}+\infty \;.$$ On the other hand, $\bar{U}$ satisfies $$\bar{U}(a,x) \sim (2/\pi)^{1/2} \Gamma(1/2-a)x^{a-1/2}{\mathrm{e}}^{x^2/4} \; \text{as } x{\rightarrow}+\infty\;.$$ $\bar{U}$’s definition is actually fairly complicated, and can be found in @olver59, equation (2.12) or in @olver75, section 5.1. ### A9.4: On the usage of these functions {#a9.4-on-the-usage-of-these-functions .unnumbered} As we mentioned earlier, these functions play a central role because it is relatively easy to transform the equations in which we are interested into one of the three mentioned above, or a perturbation of it. Then a range of techniques are available to study the effect of the perturbation, and one can sometimes, and obviously in the case we examine, get asymptotic expansions in terms of the “non-perturbed" solutions. Since these functions are quite well known, information can be gathered about the function of original interest this way.\ For example, in @imj, section 5, after the scaling $\xi=x/\kappa$, the Whittaker equation (\[eq:whittaker\]) becomes $$\frac{d^2W}{d\xi^2}=\left(\kappa^2\frac{(\xi-\xi_1)(\xi-\xi_2)}{4\xi^2}-\frac{1}{4\xi^2}\right)W \;.$$ Using the Liouville-Green transformation $\zeta(d\zeta/d\xi)^2=(\xi-\xi_1)(\xi-\xi_2)/(4\xi^2)$, with $w=(d\zeta/d\xi)^{-1/2} W$, one has $$\frac{d^2w}{d\zeta^2}=\{\kappa^2 \zeta + \psi(\zeta)\} w \;.$$ This is a perturbation of the (scaled) Airy equation, for ${\mathrm{Ai}}(\kappa^{2/3} \zeta)$ and ${\mathrm{Bi}}(\kappa^{2/3} \zeta)$ are solutions of $d^2w/d\zeta^2=\kappa^2 \zeta w$.\ $w$ is not $W$, but it can be related to it, and it is through this mean that Johnstone did his original analysis. As $\psi$ is a relatively involved function of $\xi$ and $\zeta$, we do not explicit it, but just mention that the understanding of $\psi$ is key to getting the uniform bound (\[whattoshow2\]). For more on this, see @imj or @olver, theorem 11.3.1 p.399. The problem we encountered (and mentioned in \[errorcontrolproblem\]) about the error control function is exactly here: we could not get enough information about the behavior of $\psi$ under ${\mathbf{AB}}$, so we slightly changed approach and turned to other studies. In @olver80, Olver starts with equation (\[eq:whittaker\]), where the dummy variable was $z$. Writing $x=z/\lambda$ and $l=\kappa/\lambda$, he gets $$\frac{d^2 W}{dx^2}=\left(\lambda^2g(x)-\frac{1}{4x^2}\right)W \;.$$ As he aims to expand the solution in terms of parabolic cylinder functions, he changes variables another time, by writing $$W=\left(\frac{dx}{d\zeta}\right)^{1/2} w \; \; \;, \; \left(\frac{d\zeta}{dx}\right)^2=\frac{x^2-4lx+4}{4x^2(\zeta^2-\beta^2)} \;,$$ with $\beta=\{2(l-1)\}^{1/2}$. Hence, he gets $$\frac{d^2w}{d\zeta^2}=\{\kappa^2 (\zeta^2-\beta^2) + \psi(\kappa,\beta,\zeta)\} w \;,$$ with $\psi(\kappa,\beta,\zeta)=-\dot{x}^2/(4x^2)+\dot{x}^{1/2}d^2(\dot{x}^{-1/2})/d\zeta^2$. His @olver75 is a study of this type of equations, and in particular of the control of the deviation of the solution of the previous equation to the corresponding parabolic cylinder function. In @olver80, he studies very explicitly the abstract estimate he gets in @olver75 in the case of Whittaker functions. We use this repeatedly in our study, as it is essential to get the crucial property (\[whattoshow2\]). [^1]: **Acknowledgements:** The author is grateful to Pr. Iain Johnstone for many discussions, key references, and guidance and to Pr. David Donoho for his helpful comments and support. Supported in part by NSF -0140698 and -008584 (ITR). **AMS 2000 SC:** Primary 62E20, Secondary 62H25. **Key words and Phrases :** Principal Component Analysis, largest singular value, Tracy-Widom distribution, Fredholm determinant, Random Matrix Theory, Wishart Matrices. **Contact :** `nkaroui@stanford.edu`

--- abstract: | In the days immediately following the contested June 2009 Presidential election, Iranians attempting to reach news content and social media platforms were subject to unprecedented levels of the degradation, blocking and jamming of communications channels. Rather than shut down networks, which would draw attention and controversy, the government was rumored to have slowed connection speeds to rates that would render the Internet nearly unusable, especially for the consumption and distribution of multimedia content. Since, political upheavals elsewhere have been associated with headlines such as “High usage slows down Internet in Bahrain” and “Syrian Internet slows during Friday protests once again,” with further rumors linking poor connectivity with political instability in Myanmar and Tibet. For governments threatened by public expression, the throttling of Internet connectivity appears to be an increasingly preferred and less detectable method of stifling the free flow of information. In order to assess this perceived trend and begin to create systems of accountability and transparency on such practices, we attempt to outline an initial strategy for utilizing a ubiquitious set of network measurements as a monitoring service, then apply such methodology to shed light on the recent history of censorship in Iran. , national Internet, Iran, throttling, M-Lab author: - 'Collin Anderson[^1]' bibliography: - 'paper.bib' subtitle: Detecting Throttling as a Mechanism of Censorship in Iran title: Dimming the Internet --- Introduction {#sec:introduction} ============ *“Prison is like, there’s no bandwidth.”*[ - Eric Schmidt]{}[^2] The primary purpose of this paper is to assess the validity of claims that the international connectivity of information networks used by the Iranian public has been subject to substantial throttling based on a historical and correlated set of open measurements of network performance. We attempt to determine whether this pattern is the result of administrative policy, as opposed to the service variations that naturally occur on a network, particularly one subject to the deleterious effects of trade restrictions, economic instability, sabotage and other outcomes of international politics. Overall, we outline our initial findings in order to provoke broader discussion on what we perceive to be the growing trend of network performance degradation as a means of stifling the free flow of information, and solicit feedback on our claims in order to create universally applicable structures of accountability. Furthermore, to the fullest extent possible, we focus our assessment on that which is quantitatively measurable, and limit attempts to augur the political or social aspects of the matters at hand. As in any other closed decision-making system, a wealth of rumors dominate the current perception of the actions taken by the government and telecommunications companies. Where these rumors are mentioned, they are discussed in order to test validity, and not cited as evidence. This paper is not intended to be comprehensive, and we err on the side of brevity where possible. Toward these ends, our contributions are threefold: 1. outline a methodology for the detection of the disruption of network performance and infer purposeful intent based on indicators, differentiated from normal network failures; 2. begin to identify potential periods of throttling, based on available historical data; 3. attempt to enumerate those institutions that are not subject to interference. The experiments described herein are motivated toward collecting initial, open-ended data on an opaque phenomenon; where possible our results and code are publicly available for outside investigation at: > <http://github.com/collina/Throttling> The remainder of this paper is structured as follows. In the following section, we identify the infrastructural properties of networks relevant to our line of inquiry that enable states and intermediaries to control access to content and service performance. We describe the dataset core to our investigation in Section \[sec:setup\] and Section \[sec:mathishard\] describes a mixed methods approach used to interpret measurements and extract broader information on the nature of the network. Finally, these techniques are applied in Section \[sec:findings\] toward identifing periods of significant interest in the connectivity of Iran-originating users. We conclude by enumerating the outstanding questions and future research directions. Domestic Network Structure Considerations {#sec:infrastructure_considerations} ========================================= After the declaration that the incumbent president, Mahmoud Ahmadinejad, had won a majority in the first round of voting, supporters of reformist candidates rallied against what was perceived to be election fixing in order to preserve the status quo of the power structure of the state. Already well-acquainted with bypassing Internet filtering using circumvention and privacy tools, such as VPNs and Tor, government blocks on YouTube and Facebook were minor impediments for activists to share videos and news in support of their cause. Unused to large-scale challenges against the legitimacy and integrity of the system, the government appears to have responded by ordering the shutdown of mobile phone services, increased filtering of social media sites and the disruption of Internet access [@bailey2011censorship]. Iran’s telecommunications infrastructure and service market differs substantially from the regulatory environment of broadcast television, wherein the state maintains an absolute monopoly on authorized transmissions [@imp:control]. The current on network ownership was initially shaped by the Supreme Council of the Cultural Revolution in 2001 under the directives ““Overall Policies on Computer-based Information-providing Networks” and “Regulations and Conditions Related to Computerized Information Networks” [@cso_trc; @ihrdc:ctrl]. Internet Service Providers (ISPs), which offer last-mile network connectivity, are privatized, but subject to strict licensing requirements and communications laws that hold companies liable for the activities of their customers.[^3] In addition to administrative requirements for filtering according to a nebulous and growing definition of subjects deemed ‘criminal,’ ISPs are forced to purchase their upstream connectivity from government-controlled international gateways, such as the state-owned telecommunications monopoly, the Telecommunication Company of Iran (TCI), which appear to implement an auxiliary, and often more sophisticated, censorship regime on traffic in transit across its network. Even amongst privately-owned networks, perspective entrepreneurs appear to be encouraged or coerced into ownership consortiums with the ever present set of Bonyads, charitable trusts often connected with the Iranian ideological establishment [@turkcell]. While a substantial amount has been written on controls imposed on content, the salient principles to our domain of research are the obligations of independent providers to the state’s administrative orders and the infrastructural centrality of two entities in Iran’s network, the TCI’s subsidiary, the Information Technology Company (ITC, AS12880), and the Research Center of Theoretical Physics & Mathematics (IPM, AS6736). As a result, all traffic to foreign-based hosts, and likely a majority of connections internally, route through entities with either direct or informal relationships to the government, as demonstrated in Figure \[fig:pathways\]. ![International Pathways to Iranian Hosts (Traceroute)[]{data-label="fig:pathways"}](resources/m_lab_ips_2013_02_annonated.png){width="\textwidth"} The centralization of international communications gateways and domestic peering (the linkages between networks) enables anticompetitive and potentially undemocratic practices that would be more administratively difficult and economically expensive in an open and multi-stakeholder telecommunications market. This design is not specific to Iran alone nor is it an indicator of a government’s desire to control citizen access. Centralization often resembles a commonplace model of public-sector infrastructure development or state revenue generation from telecommunications surcharges [@roberts2011mapping]. With the introduction of the Internet, regulators reflexively extended their mandate to include online communications, as the most common forms of physical connectivity often utilize telephony networks, as well as bringing competitive services such as voice-over-IP. When the network monitoring company Renesys addressed this topic in response to the Internet shutdowns of Egypt and Syria, they framed the dangers and fragility of the centralization of gateways as “the number of phone calls (or legal writs, or infrastructure attacks) that would have to be performed in order to decouple the domestic Internet from the global Internet,” naming 61 countries at ‘severe risk’ for disconnection [@renesys:couldit]. Although disconnection, failure and filtering are more perceivable forms of disruption, the same principles of risk and exposure apply to the degradation of connections. Throttling is not on its own a form of censorship or intent to stifle expression. In the context of Iran, a scarcity of available bandwidth and inadequate infrastructure has created a demonstrable need for limiting resource-intensive services and prioritizing real-time communications traffic, especially in rural markets [@hassani2010qos]. In other cases, often under the terms “quality of service” or “traffic shaping,” throttling is a means of providing higher performance to less bandwidth-intensive applications, through initially provisioning of faster speeds to a connection that is then slowed after a threshold is reached. These practices have spurred heated debate between civil society and telecommunications providers in the United States and Europe within the framework of “network neutrality,” pitting the core principle that communications on the Internet be treated equally against claims by companies that the bandwidth demands of online services exceeds current availability. However, the allegations of throttling we attempt to address differ substantially from these debates in scope and execution, and fit into a history of interference with the free flow of information, offline and online, and recurrent security intrusions on the end-to-end privacy of the communication of users, often originating from government-affiliated actors [@foxitdiginotar]. Setup {#sec:setup} ===== Degradations in network performance and content accessibility can be the product of a number of phenomenon and externalities, localized to one point or commonly experienced across a wider range of Internet users. In order to accurately and definitively measure broadly-targeted degradation, such as throttling, it is necessary to obtain data from a diversity of hosts, in terms of connection type, physical location, time of usage and nature of usage. A number of tools have been developed to actively probe qualities of infrastructure directly relevant to throttling and disruption [@dischinger2010glasnost; @filasto2012ooni; @kreibich2010netalyzr]. These techniques compare whether traffic flows of differing types sent at an identical rates are received differently, thereby comparing against an established baseline to give a clear indication of potential discrimination. Where such data has been core to network neutrality debates, for countries such as Iran, government opacity on broadband deployment means that domestic civil society and private parties have had few opportunities to embrace quantitative data to push for policy changes. Our research interest biases observations that are ubiquitous, recurrent and not necessitating the intervention of users, in order to plot historical trends and account for localized aberrations, even if at the cost of precision or confidence. As a result, to meet our operational needs, we resort to the use of measurements that, while not specifically designed to detect throttling, broadly assess relevant characteristics of the network in a manner that may indicate changes in the performance and nature of the host’s connectivity. In order to collect a statistically significant set of network performance datapoints, we utilize the data collected by client-initiated measurements of the Network Diagnostic Tool (NDT) hosted by Measurement Lab (M-Lab) [@dovrolis2010measurement], which contains both a client-to-server (C2S) and server-to-client (S2C) component. The throughput test consists of a simple ten second transfer of data sent as fast as possible through a newly opened connection from a M-Lab server to a NDT client. In addition to measuring the throughput rate, the NDT test also enables the collection of diagnostic data that can assess factors such as latency, packet loss, congestion, out-of-order delivery, network path and bottlenecks on the end-to-end connectivity between client to server. ![Unique Clients and Tests from Iran (Weekly), Jan 2010 - Jan 2013[]{data-label="fig:clientstests"}](resources/IR_Tests_2010-2013.png){width="\textwidth"} M-Lab’s added value is both methodological and institutional. Founded and administered by a consortium of non-governmental organizations, private companies and academic institutions, including New America Foundation’s Open Technology Institute, PlanetLab, and Google, M-Lab is an open data platform focused on the collection of network measurements related to real-world broadband connectivity, and is not in itself a human rights or political cause. M-Lab’s data is both non-partisan and widely-accepted, having been used by telecommunications regulators and development agencies in Austria, Cyprus, the European Commission, Greece and the United States. With the inclusion of NDT as a connectivity diagnostic test in version 2.0 of the Bittorrent file-sharing client $\mu$Torrent, M-Lab gained a significant increase in NDT measurements and a wider audience of users[@blog:utorrent]. Since 2009, M-Lab has collected over 725 Terabytes of data based on around 200,000 tests conducted per day. More relevant to our purposes, in the month of January 2013, M-Lab collected 2,925 tests from 2,158 clients from Iran that we consider valid under the definitions and parameters described in Section \[sec:mathishard\]. Figure \[fig:clientstests\] demonstrates the rapid growth of network measurements originating from Iran after the inclusion of NDT in $\mu$Torrent, with unique clients representing the number of IP addresses seen and tests describing the total number of tests. While M-Lab contains measurements from 2009, we consider data beginning in 2010 to take advantage of the critical mass of clients that resulted from the inclusion. During the process of preparing and running our experiments, we took special care to not violate any laws or, considering the diminishing opportunities for international collaboration, expose individuals within Iran to potential harm. All our experiments were in accordance to any applicable terms of service and within reasonable considerations of network usage, taking care to not engage in behavior that would be considered intrusive. The contribution of measurements of a network to NDT or M-Lab does not denote political activities on the part of the user, particularly as the use of Bittorrent file-sharing has broad appeal and apolitical implications. Moreover, use of Bittorrent or NDT do not appear to violate Iranian law, especially given the copyright and intellectual property framework of Iran. All data originating from M-Lab is openly available to the public, and collected in a manner that does not reveal potentially identifiable user information, outside of the client’s IP address. Parameters & Calculations {#sec:mathishard} ========================= NDT provides a multiplicity of metrics and diagnostic information that describe the test session. As outsiders attempting to assess the behavior and actions of network intermediaries, we are relegated to beginning from assumptions and hypotheses founded on general principles of networking and how others accomplish throttling. Additionally, we are limited in the lessons to be gained from the corpus of research on assessing violations of network neutrality, as we struggle to establish control measurements and perform the active probing necessary for proper comparative analysis. Most methods for measuring broad or application-specific throttling assume that connections are initially allocated a higher level of throughput, which is then throttled or terminated upon identification by a network intermediary or exceeding a bandwidth quota. Therefore, our primary form of detection of abnormal network conditions is comparative assessment of selected indicators based on historical trends and incongruities between subgroups of clients. In order to allow this comparative assessment through baselines that serve as controls, we define a set of measurements and aggregate clients in a consistent and non-abitrary manner. For the introductory purposes of this study, and based on cursory analysis of existing M-Lab data which was generated during suspected throttling events, we substantially rely on NDT’s measurements of round trip time, packet loss, throughput and network-limited time ratio as potential identicators of network disruption.[^4] Round Trip Time (RTT) : ($MinRTT, MaxRTT, \frac{SumRTT}{CountRTT}$)\ The time taken for the round trip of traffic between the server to client, computed as the difference between the time a packet is sent and the time an acknowledgement is received, also known as latency. There are a diversity of causes for latency, including ‘insertion latency’ (the speed of the network link), the physical distance of the path taken, ‘queue latency’ (time spent in the buffer of network routers), and ‘application latency.’ The last of these components, application latency, is accounted for within the NDT test under the metrics of *Receiver and Sender-limited time windows*. We are interested in latency that occurs due to network properties, primarily the queue, insertion and path latencies. Since the time taken should not change dramatically, fluctuations indicate a meaningful change in connectivity, such as a network outage that increases latency due to traffic to taking a longer route and telecommunications equipment having taken on additional load. NDT also provides different approaches to this measurement, which allow alternative perspectives on the network. The minimum round trip time record (MinRTT) mostly occurs before the network reached a point of congestion, and therefore is generally not thought to be indicative of real performance. Alternatively, the average of RTTs, through the division of the sum of all round trip times by the number of trips may more closely approximate latency, but is also vulnerable to outlier values. [@findingthelatency] Packet Loss : ($\frac{CongSignals}{SegsOut}, \frac{SegsRetrans}{DataSegsOut}$)\ The transmission of traffic across a route is not guaranteed to be reliable, and network systems are designed to cope with and avoid failure. These mechanisms include maintaining an internal timer that will give up on traffic the system has sent, alerts from the network that congestion is occurring, and notification from the other end that data is missing. Packet loss for our purposes is defined as the the number of transmission failures that occurred, due to all forms of congestion signals recorded under NDT’s test, including *fast retransmit*, *explicit congestion notifications* and *timeouts*. In order to address this relative to the amount of extensiveness of the test, packet loss is measured as a probability against the number of packets sent. Network-Limited Time Ratio : ($\frac{TimeCwnd}{TimeRwin + TimeCwnd + TimeSnd}$)\ The network stack of operating systems maintain internal windows of the amount of traffic that has been sent and not acknowledged by the other party. This enables the system to avoid over-saturating a network with traffic and to detect when a failure has occurred in communications. The NDT test attempts to send enough traffic to create congestion on the network, where the transmitting end of traffic exceeds this window of unacknowledged traffic and is waiting for clearance from the other side to continue sending. The Network-Limited Time Ratio is calculated as the percentage of the time of the test spent in a ‘Congestion Limited’ state, where sending of traffic by the client or server was limited due to the congestion window. Network Throughput : ($\frac{HCThruOctetsAcked*8}{TimeRwin + TimeCwnd + TimeSnd}$)\ The NDT test attempts to send as much data as quickly as possible between the client and an M-Lab server for a discrete amount of time in order to stress the capacity of the network link. For the upload performance, this is calculated based of the amount of data received from the client, and with the download being the number of sent packets that were acknowledged as received. The throughput rate is then calculated against the time that the test lasted. Measurements of connection properties allow for the indirect inference of broader network conditions, particularly when applied in a comparative fashion. For example, others have noted the varying degrees of correlation between round trip time and the total load presented on a network[@biaz2003round]. While these studies hold a higher correlation on a slow link than on a fast one, the former of which appears to more accurately describe the domestic connectivity of Iran. In this scenario, it may be possible to identify periods where an increase in load has created a bottleneck in the network or a build-up in the traffic queue on network devices. Therefore, we assume that the decrease of throughput, or increase of loss, will be associated with an increase in latency where traffic congestion is occurring. Additionally, while most inter-network routing protocols will attempt to route traffic over the best path to a destination, which can differ based on load balancing and service agreements, a change in round trip time may indicate a change in the path traversed by the data sent from the client to M-Lab. Due to the variability in network conditions that can affect latency, we use two measurements: the minimum RTT recorded in the session and the uniform average over the entire test. Aggregation and Comparative Methods ----------------------------------- In order to assess the general performance of the domestic network, client measurements are aggreggated across higher-level groupings and evaluated based on their median value. Where multiple tests are performed by a client during an evaluation period, the most performant measurement is used in order to mitigate potential bias in samples. Through giving preference to faster measurements, we intentionally bias our pool of data against our hypothesis and assume that less favorable numbers are aberrations. Furthermore, we prefer calculations that accomodate for outlier values, such as false positives that occur in geolocation services when foreign-hosted networks are registered to domestic entities. Finally, we assume the natural shifts in consumer behavior or infrastructure development that may affect measurements, such as the adoption of mobile broadband, are gradual and upward trends. In practice this assumption appears to not only hold, but we are led to question whether the availability of high-speed connectivity has declined, due to administrative limitations imposed on consumer providers and delays in the development of mobile data licensing. For our purposes, we discretely aggregate measurements across three dimensions related to the character of the tests or location of the client, National: : Measurements are grouped on a country level. Aggregation for large geographic areas or service providers may represent a diverse strata of connectivity types, such as ADSL, dialup, WiMAX and fibre. Internet Service Providers and Address Prefixes: : We use the Autonomous System Number (ASN) as a proxy for ISPs. Within a research methodology, aggregation based on the ASN provides a larger pool of clients at the cost of being less granular than address prefixes. Autonomous Systems are generally defined as a set of routers under a single technical administration, which keep an understanding of global network routes to direct traffic and announce their ownership of blocks of IP addresses (address prefixes). There is a limited pool of available numbers for the labeling of Autonomous Systems, and not every network has the need to advertises its own set of routing policies, particularly where directly connected networks have the same upstream connection. Therefore, a large ISP, such as Afranet, will generally maintain one or more Autonomous Systems, bearing the responsibility to maintain announcements and peering, for the connectivity leased to other smaller ISPs, government agencies, educational institutions or commercial organizations. Additionally, since traffic paths across networks are constructed using centrally allocated components of ASNs and address prefixes, both are registered and externally queriable. Control Groups: : We attempt to identify logical, coherent groups of networks and clients based on common characteristics, such as the nature of the end user or performance. Control group measurements differ from service providers because they are defined as narrowly as the data allows, rather than existing segmentations. Such groups, particularly when defined by network degradation, are possibly deterministic, however, these often produce surprising and mixed results, as described in Section \[sec:findings\]. Furthermore, the grouping of networks or entities, such as state agencies or educational institutions, during one incident is testable under other circumstances and may serve as a control for future monitoring. Finally, detection of significant events generally follows one of two themes. These mechanisms are designed to highlight precipitous changes of service quality as a warning system to flag potential events, however, they are not the holistic determinant of interest. Threshold: : A maximum and minimum threshold of reasonable values are established based on previous trends. Since our dataset is frequently limited to a small amount of clients that are subject to varying network conditions and unrelated externalities, producing wide fluctuations in measurements, a Poisson distribution is established based on a rolling average. Detection of an abnormal event occurs when the trend breaches these bounds [@danezis2011anomaly]. Variance: : Internet Service Providers offer a diversity of products with varying levels of performance across different markets, leading to variations in qualities, such as connection speed and reliability. This technological and commercial variation is exacerbated by informal differences, such the scrutiny placed on the documentation necessary to obtain faster broadband implementation of administrative orders (elaborated in Section \[sec:controlgroups\]) and ability to acquire network infrastructure, despite scarcity created by sanctions and exchange rates. While these differences may change across time, there should be a consistent trend of diversity within a free market. We evaluate the variation that occur in service quality amongst our subgroups such as ISPs or prefixes. Therefore we presume that when external limitations are not imposed, variation will be high, while the contrapositive holds that at a time of control, the variation will be low. We use the classic variance of the average of the squared deviations from the mean to accomodate proportional change. Analysis based on the variance of performance measurements day over day across short historical periods can be applied to a single network or client as between ISPs. We would anticipate that if an administrative ceiling were imposed on throughput speed, particularly at a limit below the potential capacity of the network, the variation would near zero, as the tests would cap out at the maximum available bandwidth. We would also expect that trend line of such an incident to follow a peak and valley model, where a sudden decrease or increase leads to a spike in variance when the limit is imposed or lifted, with low variance during the throttling. Figure \[fig:nov11:thuvar\], demonstrates that in practice this hypothesis holds mixed results. While peaks do occur, the variance, particularly the relative variance, remains high. As these mechanisms serve as a warning system to direct further investigation, rather than being a sole determinant of interest, we are less concerned about its robustness. Additionally, for the purposes of identifications and coding of events, we generally call attention to extremely abnormal values, relative to the normal trends. Once a significant change in service is identified, it is subject to correlation against other metrics. More constrained boundaries simply lead to more research cost and false positives. We assume throttling designed to stifle expression or access is not a subtle event. Limitations ----------- While the integration of NDT into $\mu$Torrent has enabled the massive proliferation of points of observation across a variety of geographic locations and network conditions, we remain relegated to user-contributed data collection based on a limited set of volunteers. Changes in connectivity or quality of service cannot directly be infered as an administratively-imposed censorship event. Additionally, the Network Diagnostic Test’s data collection does not occur in isolation of other externalities that are likely to affect performance. Cross traffic, local and upstream network activities from other applications or users, independent of the test can bias results through introducing additional latency or failures as the test mechanism competes for bandwidth and computing resources. As an end user is positioned at the border of the network, they cannot independently account for the conditions outside of their control that may impact the results of this test. Therefore, like others before, we attempt to mitigate and account for externalities based on interpretations derived from observable data and general networking principles [@salamatian2003cross; @weinsberg2011inferring]. ![Diurnal Patterns of Throughput Measurements, Iran, Jan - Mar 2013[]{data-label="figure:mathishard:diurnal"}](resources/IR-Throughput_Diurnal_2013.png){width="\textwidth"} The extent of externalities often vary based on diurnal patterns of use and in response to specific incidents. Using the performance metric of throughput, Figure \[figure:mathishard:diurnal\] demonstrates that observed network speed is higher in the early hours of the morning, Iran time (GMT +4:30), than during the day. These trends are mirrored in the performance across the week, and within other measurements, such as packet loss. It would appear that Iran’s network does not handle the additional load of office hours and evening use gracefully. M-Lab-based services attempt to perform testing under favorable conditions by selecting the measurement server by geographic proximity. These servers are physically located across the world, including in Australia, Austria, Czech Republic, France, Germany, Greece, Japan, Ireland, Italy, the Netherlands, New Zealand, Norway, Spain, Slovenia, Sweden, Taiwan, the United Kingdom and the United States. Although one of these M-Lab hosts countries directly peers with Iran,[^5] the selection mechanism predominantly directed NDT clients to Greece (of the 2921 tests in January 2013, 50.39%), United States (22.08%), United Kingdom (16.40%) and France (9.28%), with the remaining three countries constituting 1.85% of tests. It may be possible that if there were M-Lab servers in Turkey or Azerbajian, NDT tests would result in higher throughput measurements, however, we derive our conclusions from relative changes, rather than absolute numbers. Addditionally, users that are connected to anti-filtering tools during the test are likely to be recorded by M-Lab as originating from the country that the tool routes its traffic through, and therefore not included in our sample. We also constrain our expectations on the types of throttling or disruption that NDT will detect. As a diagnostic of direct connectivity between hosts, the test is based on traffic patterns that likely have yet to inspire scrutiny from intermediaries. Other protocols, such as those employed in VPN tunnels, HTTP proxies, Tor, voice-over-IP and streaming media, have at varying times been claimed to be subject to targeted interference, based on port or deep packet inspection. NDT’s measurement methodology is unlikely to detect more sophisticated discimination against specific forms or destination of traffic. Lastly, we assume that intermediaries have not sought to interfere or game the data collection of M-Lab through artifically biasing measurements from hosts. Findings {#sec:findings} ======== Using median country-level throughput, evaluated based on the most performant measurement per client per day, we find two significant and extended periods of potential throttling within our dataset, occuring *November 30 2011 - August 15 2012* (a 77% decrease in download throughput) and *October 4 - November 22 2012* (a 69% decrease). We identify an additional eight to nine short-term instances where the throughput or variance between providers underwent a precipituous change, triggering the attention of detection mechanisms. These events are correlated with a reduction of service quality across all networks, often more significantly impacting home consumers than commercial institutions. In most cases, these changes mirror more overt increases of interference of communications channnels. Lastly, within available indicators or traffic routes, we do not find evidence that these fluctuations are the result of externalities, such as changes to international connectivity or domestic network use. Periods of Significant Interest {#sec:throttling_events} ------------------------------- Figures \[findings:throughput\] and \[findings:variance\] outline the periods of time where fluctuations in values and variance of throughput exceeded thresholds, respectively. The performance measurements and indicators outlined in the prior section during these two extended periods of interest are documented in the graphs of Figures \[findings:throttling\_events:oct\_2012\] and \[findings:throttling\_events:nov\_2011\]. Although our methodology has not taken a deterministic view from prior awareness, these two major events mirror our prior understanding of periods of disruption. While we could not find instances where M-Lab or similar tests were used in the commission of news reports on Iran’s Internet, our results often mirror claims such as “The Internet In Iran Is Crawling, Conveniently, Right Before Planned Protests”[@tnw:crawling]. We also find potential events as detected by changes in performance surrounding holidays, notable protests events, international political upheaval and important anniversaries, such as Nowruz, the Arab Spring and 25 Bahman (Persian calendar date, early-to-middle February). These also often parallel more overt forms of disruption, such as the filtering of secure Google services (September 24 - October 1 2012) and significant jamming of international broadcasts (January 31 - February 7 2011, October 2012). Figure \[findings:tablesevents\] demonstrates the reverse methodology, correlating reported incidents of public protests or opposition rallies with NDT measurements.[^6] The dates identified focus on country-wide mobilizations, rather than localized events such as protests by Iran’s Ahvaz Arab minority population in Khuzestan during April 2011. While it is probable that localized throttling occurs, due to limitations in sampling, it may not be possible to detect such actions until more nodes of measurement are available. In addition to the throughput for the primary or initial day of the event, the table identifies the data trends of the week and the time period of the month before and after. This timeframe takes into consideration the measures applied by the state to stifle mobilizations for publicly-announced events and mitigate further political activities. These reactions are reflected both online and offline, during Winter 2011, in reaction to plans to protest on 25 Bahman, former Presidential candidates Mir-Hossein Mousavi and Mehdi Karroubi were detained and kept in house arrest by security forces for their role in the reformist politics. These mobilizations are compared against a two month window as a baseline of the general capacity of the network at that time. We find a direct correlation between the precipituous decline in connectivity for February 2012’s anniversary of the detention of Mousavi and Karroubi, as well as October 2012’s currency protests. During all events, variation between the average RTT of two month mean and those recorded during the event stayed within a 20% threshold. While we anticipated 10 February 2010 as a potential throttling event, we find that three days later, the date of our minimum measurement during the period, was associated with both street protests and the hacking of opposition news sites [@rsf:internetenemies]. Similarly, while our weekly minimum for the first anniversary of the 2009 Presidental election falls on the anticipated day, the decrease in performance does not exceed a reasonable threshold of variation. Finally, March 2010 stands out as a strong case of a negative correlation across the two-month context, however, it may be reasonsable to consider the patterns of network use established previously and this period’s proximity to the Nowruz holiday. Contrarily, if lower-end consumer users are subjected to more aggressive throttling, and as a result decide stay off the Internet until speeds improve, the median of the national throughput would increase. Against our two-month baseline, there was a 16% decrease in NDT clients on May 16 and a weekly minimum of 41% decrease. Thus, this undergirds the need to detect both precipitous decreases *and increases* in performance. [|L|c|L l l|c|c|c|]{} Event & Day Of & & Wk-Min & & Wk-Mean & 2-Month\ 2010-02-11 & 0.18 & 0.14 & 2010-02-14 & -34.3% & 0.20 & 0.19\ 2010-03-16 & 0.26 & 0.19 & 2010-03-13 & +7.3% & 0.22 & 0.17\ 2010-06-12 & 0.16 & 0.16 & 2010-06-12 & -21.6% & 0.19 & 0.20\ 2011-02-14 & 0.18 & 0.15 & 2011-02-17 & -18.9% & 0.18 & 0.18\ 2011-02-20 & 0.22 & 0.15 & 2011-02-17 & -21.1% & 0.20 & 0.18\ 2011-03-01 & 0.18 & 0.12 & 2011-03-04 & -52.3% & 0.17 & 0.19\ 2011-03-08 & 0.16 & 0.16 & 2011-03-08 & -14.0% & 0.18 & 0.19\ 2012-02-14 & 0.03 & 0.03 & 2012-02-14 & -102.9% & 0.07 & 0.06\ 2012-10-03 & 0.25 & 0.09 & 2012-10-04 & -86.2% & 0.20 & 0.16\ We anticipate based on the network principles and diurnal patterns previously established that network load will be directly correlated with higher roundtrip times. In Figures \[findings:throttling\_events:oct\_2012:rtt\] and \[findings:throttling\_events:nov\_2011:rtt\], there appears to be no such relationship between the round trip time of the clients’ traffic and measurements of service quality. It is less likely that these changes were the product of heavy use. Generally, sudden drops in service quality can be attributable to changes in domestic networks or the availability of upstream providers, due to a multitude of factors such as physical damage or electronic attacks. During the interval of throttling identified as the beginning of October 2012 and described in Figure \[findings:throttling\_events:oct\_2012\], the main international gateway provided by Information Technology Company (AS12880), experienced routing failures to networks connected through Telecom Italia Sparkle (AS6762)[@renesys:bulletin_oct_2012; @renesys:bulletin_oct_2012_2]. However, Iran’s international gateways are amongst the most unstable on the Internet, with frequent, short periods of routing failures even during normal operations [@nanog:bgp_update]. Therefore, it is important to differentiate relatively routine failures from protracted and wide-cutting outages. Additionally, as latency is a partial product of the psychical distance of a network path, changes in distances of traffic traversing paths to the global Internet should show as changes in latency. It remains unclear whether these reported disruptions were due to connectivity failures, or downtime due to maintance and application of changes to the network. In the case of the October currency crisis event, these reported failures were short-lived as normal service appears to have been restored within minutes, and little change in latency is measured. Despite the centralization of domestic peering through the key points of control, Iran has a diversity of physical pathways connecting the country to the global Internet, creating upstream redundancy. A clear demonstration of the effect, and minimal impact for our purposes, of infrastructural failure occurs within our October 2012 event, when an attack against a natural gas pipeline by the Kurdistan Workers Party caused damage to infrastructure providing connectivity through Turkcell Superonline [@renesys:blasts]. While these changes were detected in Figure \[findings:throttling\_events:oct\_2012:throughput\], they were within the tolerance levels established during the ongoing throttling event. This event also appears to be reflected in a marginal increase of the average and minimum round trip times of Figure \[findings:throttling\_events:oct\_2012:rtt\], as clients compete over diminished resources and traverse potentially longer routes to M-Lab servers. Thus far, we have primarily focused on two extended periods of time for analysis in order to explore the technical metrics outlined in Section \[sec:setup\] in an environment subject to false positives. In Figure \[findings:tablesvalues\], we enumerated those events that triggered our detection mechanism, including shorter term periods without vetting their veracity. As discussed throughout the paper, the deeper we narrow our evaluation to a network-level granularity, the more subject we are to the limitation of the manner in which our data was collected. The longer a detected abnormality lasts, the higher confidence we can assert our results are not aberrant testing, that independent mechanisms are causing peculiarities in network that should not otherwise occur. ![Throughput Variance on Daily Medians per ASN, November 2011 - January 2012[]{data-label="fig:nov11:thuvar"}](resources/IR-Throughput_Variance_2011-11-02-2012-01-31.png){width="\textwidth"} Applying these lessons, it would appear that a number of false positives, general the result of wide fluctuations in measures, trigger the detection mechanism of one metric but do not register on elsewhere. We remain especially interested in the reported incidents February 2010, March 2010, Feburary 2012, and January 2013. Additionally, we manually identify early April 2010 as an interesting period based on low variance between ISPs, the amount of time spent in a network-congested state and a rapid change in the daily variance of throughput measurements on the top five networks. Other remaining periods of interest consist of short timeframes that bear noteworthy links between metrics, but we cannot confidently assert are meaningful, including several periods in June 2010, late October 2010 and July - August 2011. ![Throughput Variance Amongst Iranian ASNs, February 2011 - May 2011[]{data-label="fig:nov11:var"}](resources/IR-Variance_2010-02-02-2010-05-31.png){width="\textwidth"} Control Groups {#sec:controlgroups} -------------- While throttling and disruption of international connectivity may be useful for intermediaries that intend to stifle the free flow information for the general public, Internet-based communications have grown to be a core component of state operations, diplomatic functionalities and international business transactions. The identification and segmentation of critical networks provides a means to mitigate the deleterious effects of communications loss. Traffic prioritization or exemptions to disruptions for white-listed users and protocols should be trivally easy in any modern network appliance. The utilization of such features for non-censoring purposes on Iranian networks is already well documented [@hassani2010qos]. Therefore, we expect that high value networks, such as government ministries and banks, would potentially be spared the majority of disruptions if possible. Conversely, from the perspective of inferring meaning from raw data, we can posit a diversity of circumstances for why a particular network or client would be less aversely affected by country-level disruptions. Since all connections appear to route through the same intermediaries, those that continue to have normal service likely have been purposefully excluded. However, it remains unclear as to whether throttling occurs at the international gateway or is left to be implemented by the service providers. It may hold that throttling is mandated administratively through the legal authority of the telecommunications regulators, but implemented technically by the service providers. Comments from former staff of Iranian ISPs have indicated that bandwidth restrictions have previously been enforced through an order delivered over the phone or by fax. These claims have gone on to allege that some ISPs, generally smaller and regional providers, delay or limit compliance as they attempt to balance the demands of the state with the possibility of losing customers over poor quality of service.[^7] If certain ISPs are more likely to delay implementation of throttling orders, this may add an indicator to our detection and establish performance disruptions based on intent. In such a scenario, after a mandate is distributed, we would anticipate seeing that larger ISPs enter into periods of throttling before smaller providers. Were the granularity of our dataset to allow for it, this question would potentially be answered by demonstrating that the majority of networks witnessed changes within a very close promixity of each other due to central coordination of implementation, as opposed to the delays and differing interpretations that may come with diffuse, independent implementation. This would likely require a data source that is statistically valid when reduced to an hourly basis, rather than our daily aggregation in Figure \[findings:throttling\_events:nov\_2011:asn\]. Additionally, this hypothesis requires prefix-level evaluation, and thus inevitably runs into the limitations of passive, crowdsourced datacollection, as smaller networks will have fewer users running NDT tests, thus rendering assessment less truthworthy or responsive. This claim also assumes that smaller ISPs are not subject to the throttling of upstream domestic peers. ![Throughput, Aggregated based on ASN, November 2011 - December 2011[]{data-label="findings:throttling_events:nov_2011:asn"}](resources/IR-ASN_Throughput_November_2011.png){width="\textwidth"} As M-Lab’s NDT data contains the IP address for clients, we are able to identify those networks within higher tiers of performance, which could then be used to create a control group or baseline in order to track service changes. Using the two significant throttling events described in Section \[sec:throttling\_events\], we then identify networks based on a threshold of 95^th^ percentile of throughput rates, and thus networks that may have priority during disruptions. In order to test this hypothesis, we compare the median throughput rates of our privileged subset against the national median. Since the sample of clients is based on a minority of domestic Internet users and more susceptable to fluctations or misattributions, we are more interested in the trends of these users and the identification of the types of networks they belong to. Figure \[fig:vip:comparative\] demonstrates the relationship between these higher tier services before and after a suspected throttling event. Based on our assumptions of the narrowness of exemption rules, these clients are aggregated within the most restrictive IP prefix available through Team Cymru’s IP to ASN Mapping service [@cymruip]. Appendix Figures \[appendix:vip:before\] and \[appendix:vip:during\] enumerate the number of clients that recorded measurements within the higher percentiles for the country, based on address prefix. As the number of addresses in a prefix varies according to how they were assigned originally, the immediate value of such data is limited. A proper evaluation of the trends of networks requires an understanding of scale and utilization of the network. For example, Parsonline’s *91.98.0.0/15* and *91.99.32.0/19* address prefixes are large consumer IP pools of over 135,000 addresses, across a range of connectivity methods and customer types. It would therefore be less interesting if these networks contained a substantial number of high percentile clients than if a smaller ISP with less customers or peculiar ownership to perform well. Also within this set is Mobin Net, the nationally licensed WiMAX data monopoly, which appears to provide service from 128kbps to 2Mbps packages, far beyond most ADSL offerings. ASN Owner $\Delta$ $\Delta$ (+2) $\Delta$ (+10) Oct 2012 --------- ------------------------------------------ ---------- --------------- ---------------- ---------- AS12660 Sharif University of Technology -74.64% -70.46% -2.43% -58.62% AS12880 Information Technology Company (ITC) -95.77% -93.26% -84.94% -91.57% AS16322 Parsonline -94.26% -91.83% -67.05% -86.46% AS25124 DATAK Internet Engineering -90.74% -93.42% -76.66% -87.23% AS25184 Afranet -87.73% -78.46% -32.25% -68.23% AS29068 University of Tehran Informatics Center -79.99% -90.31% -47.37% -69.43% AS31549 Aria Rasana Tadbir -94.46% -93.19% -82.86% -91.60% AS39074 Sepanta Communication Development -89.39% -90.92% -75.06% -91.60% AS39308 Andishe Sabz Khazar -90.34% -76.92% -82.14% -80.96% AS39501 Neda Gostar Saba Data Transfer Company -94.29% -89.80% -70.38% -86.13% AS41881 Fanava Group -79.30% -83.64% -83.98% -73.19% AS43754 AsiaTech Inc. -89.12% -89.49% -82.57% -86.36% AS44244 Irancell -87.68% -88.40% -69.52% -77.57% AS44285 Shahrad Net Company Ltd. -91.81% -85.17% -80.06% -62.23% AS48159 Telecommunication Infrastructure Company -94.72% -94.76% -89.54% -87.06% AS49103 Asre Enteghal Dadeha -95.51% -91.48% -71.45% -69.02% AS50810 Mobin Net Communication Company -95.50% -94.63% -80.26% -91.10% Thus, for our purposes in assessing the trends of networks, we rely on statistical measurements of relative changes. In the interest of stronger sampling, we rely on larger ASN aggregation and define a threshold for consideration to those that have performed measurements for at least half as many days as the time period. Figure \[finding:nov2011\_asn\_diff\] demonstrates the recovery of network throughput by comparing the mean values of the two month period preceding the November 2011 incident with, the two months immediately after, February to April 2012, August to October 2012, the comparative degradation of performance during the October 2012 event. Accordingly, between the two months preceding and the first two month following the November 2011 event, every network under consideration experienced more than a 74% drop in througput. Even within those networks (ASNs) that do not meet our qualifications, only one experienced an increase in throughput performance immediately after the November 30 2011 disruption, the prefix 80.191.96.0/19 run by the ITC, which according to reverse DNS records on the block appears to provide commercial hosting services and connectivity for academic institutions, such as Shiraz University. ![Throughput for Sharif University (AS12660), Oct 2011 - Jan 2013[]{data-label="fig:sharif_throttle"}](resources/Sharif-Variance_2011-09-05-2013-02-04){width="\textwidth"} While no major network appeared to have escaped the events of November 2011 or October 2012, clear trends exist in the level of disruption and the rate of return to normality. Academic institutions, as evinced in Figure \[appendix:vip:before\], have historically had access to faster connectivity prior to the November event [@upen:faa]. Figure \[fig:sharif\_throttle\], the trends for Sharif University of Technology, demonstrates that while the university has been significantly affected by network degradation, it recovered faster than other networks. The networks owned by the Fanava Group, University of Tehran Informatics Center, and Sharif University were the only three to experience less than a 80% decline in throughput. Eight months later, only Sharif, University of Tehran and Afranet had begun to return to their normal levels. This mirrors Figure \[fig:vip:comparative\], where the trend of the mean of the $95^{th}$ percentile of tests was impacted by the throttling at the initial event. However, this subset recovered to an approximation of prior values more quickly than others. Considering its large consumer dialup and ADSL Internet offerings, on first glance Afranet’s strong recovery is unexpected in comparison to other ISPs, such as Parsonline. While there are no upfront indicators of the type of connection used for a specific client, it is possible to infer the nature of the source or the network it is associated with, from indirect means. First, the registration and announcement of routing information may provide labeling of the use, such as “Shabake Almas Abi,” hinting that the client originated from a smaller ISP that Afranet is the upstream provider for, or “AFR@NET company, Tehran, Dialup pool,” which is most likely a consumer address pool. When we perform reverse DNS queries on the address and prefix, the answers may reveal the owner of the network through the domains pointing to the space. However, any prefix returning more than a marginal number of responses likely indicates that the network is used for commercial purposes, as consumers generally do not host sites on home connections, especially where addresses change dynamically. In our recover period of August to October 2012, the majority of clients performing NDT tests on Afranet originated from the blocks of 217.11.16.0/20 and 80.75.0.0/20, which appear to host the infrastructure or provide connectivity for prominent commercial entities, such as Iranohind and Saipa Automotive, or 79.175.144.0/20 and 31.47.32.0/20, which we suspect to be smaller ISPs or hosting providers. No clients originated from the home and dial up address pools that were identified at 78.109.192.0/20 and 79.175.176.0/24. Conclusions and Further Questions {#sec:conclusions} ================================= Absent independent, quantative evidence of claims, Iranian public officials have argued that, “in spite of negative ads and fallacies …recent numbers prove that Internet speed is very satisfactory in Iran,” defying the everyday experience of the public [@citlab2012]. In this paper, we sought to establish a historical, quantitative dataset used to describe a phenomenon that thus far has existed solely in the realm of rumors and anecdote. Immediately upon the most shallow evaluation of the trends, we find frequent and prolonged changes to the service quality of clients originating from Iran. We attempt to account for these changes based on more quotidian explanations of upstream connectivity degradation, domestic infrastructure failure, or increased network traffic. While we do find noteworthy incidences of publicly-reported network outages and diurnal patterns of service quality, these do not account for the length and timing of disruption, or the extent of impact. In order to test the hypothesis under consideration, we present a number of testable assumptions about how artificial throttling would manifest within our measurements, grounded in an understanding of the technological and administrative principles at work. While we quickly run into frustrations arising from the scope and breadth of our dataset, we are also able to derive an initial set of answers. When we are limited in the confidence of results due to the sample size, origin or consistency of information, then we can narrow our investigation based on the correlation of multiple analyses. Rather than detecting based on simple indicators of throughput or variance, we are required to look at a range of measurements. By its nature, throttling is opaque occurrence and technical measurements can rarely infer intent, however, the service disruptions documented herein cannot be accounted for within normal expectations of network operations. We establish that the periods of disruption identified are widely applied across all networks but vary in magnitude and recovery, lasting from only a few days to several months. These differences parallel the purpose of the networks, thus implying special consideration of the socioeconomic impact in application of the disruption. Finally, apart from citing specific historical and infrastructural circumstances, this paper attempts to describe the first steps of a broader framework to account for the stability and accessibility of international connectivity in states that impose limitations on free expression and access to information. As noted by others, Iran is within a large cohort of countries that have centralized international communications transit to a limited set of gateways. Remaining Questions ------------------- In order to continue the development of such a monitoring and accountability tool, we anticipate the integration of NDT tests with independent sources of complementary data and solicit input toward a number of outstanding questions: - **What remaining TCP/IP and NDT indicators apply to throttling?** Thus far, our indicators have relied largely on a few metrics available from NDT that we anticipated would be directly associated with disruption. This subset does not represent the full suite of measurements and properties available from M-Lab. Of particular interest remains the raw network stream captures retained from the individual NDT tests. We anticipate using the *time to live* (TTL) IP property, a counter that decrements for each router it traverses in order to detect routing loops, to detect changes in the network, as well as monitoring fields that may be manipulated by an intermediary attempting to prioritize traffic, such as the ToS field. - **What could we learn from vendor documentation?** The principles undergirding our hypothesis and analysis have largely been derived from the documentation available for the Linux *traffic control* subsystem, which provides the means for the Linux operation system to perform throttling and shaping on egress network traffic. This documentation is especially useful given the wealth of peer-reviewed publications on its implementation concepts, configuration and performance, as well as the proliferation of Linux-based devices at the core of modern public networks. Similar features exist for Cisco devices, under quality of service traffic classes such as *rate-limit* and *traffic-shape* [@ciscoqos], and for Huawei within the *traffic behavior* definitions [@huaweiqos]. Few telecommunications equipment vendors produce network stacks that are written from scratch or ignore the basic principles used for managing traffic flows, such as ‘token bucket’ mechanisms. Furthermore, open source reporting indicates that equipment from both of the prior mentioned manufacturers are core to Iran’s Internet [@wapo:intranet]. Therefore, performance testing, emulation of environments, identification of instruments of implementation and the development of more precise methodologies that are closer to real world conditions can potentially be accomplished using off-the-shelf equipment or documentation if the quality of data allows. - **What is the correlation between the disruption events and the measured increase of packet loss or latency?** We have previously asserted that the decrease in performance without a corresponding increase in loss or latency constitutes an abnormal network condition that may represent the imposition of rate limits. However, we do not hold directly that throttling cannot be associated or accomplished through these means. We hold both scenarios as loose relationships, and infact the literature on *traffic control* describes artifical loss as a means of throttling. Routers may police the rate of traffic flows by dropping packets once an assigned buffer is filled. Thus, the relationship between throttling and the factors of loss and latency is nebulous. In the case under consideration, the former is particularly pressing. While during three days in January and February 2010, aggregated measurements register a 100% increase in round trip time within twenty four hour periods, this instability is unmatched within our three year dataset and potentially dismissible due to the small sample size during that time. The rapid degradation of Iran’s connectivity in late November 2011 is associated with over 30% packet loss, a nearly 1000% increase over the preceeding days, only a few days within this period registered less that 10% loss. The October 2012 throttling event fits this theme with a consistent rate of about 10% loss. The only metric of latency that matches the throttling event is the pre-congestion round trip time, which registers a 49% increase, and maximum RTT, -10.5% decrease, in November 2011, but is not paralleled by the measurements of average and minimum time that we focus on. - **At which level of network infrastructure does throttling occur?** As we note throughout discussions on Iran’s infrastructure and our findings, rumors and references in public documents have pinpointed some level of throttling occuring on the part of the end-consumer Internet providers. This is to be expected considering that ISPs retain legal responsibilities to police criminal content, with the TCI serving an auxiliary role running its own filtering and deep packet inspection. Thus far, NDT has not clearly provided the granularity of data required to answer this question under the rubric outlined in Section 5.2. Although these records show variations in the extent of disruption, at any level in the path, an administrator would be able to differentiate the rules for handling traffic from different networks. These questions also reflect the frustrations in determining the most narrow application of exemptions to disrupt, as we search for IP prefixes, ASN and even cities that have been less aversely affected by disruption. - **Is the technical application of disruption rules consistent across domestic ISPs, instances of throttling, and countries?** Thus far we have avoid asserting a set of properties that we believe are direct and exclusive evidence of intermediary throttling. We have noted at length the difficulties of establishing such confidence within the nature of the test and the opacity of administration. However, we also anticipate variances in implementation. Between any service provider or, more expansively, countries, differences of technical capacity and infrastructure will lead to different opportunities or approaches. The example of Bittorrent throttling by American and European ISPs provides an independent and more thoroughly investigated illustration of the diversity of means available to manipulate the connections that pass through a network, and demonstrates the role of specific equipment in various strategies. Additionally, intent matters. If an intermediary is primarily concerned with disrupting streaming media, Internet telephony or anti-filtering connections, then dropping packets or sending connection resets may be a more efficient approached. Moreover, in such a case, ISPs may even offer their users an initial burst of fast speeds, that are then throttled down. Thus, a universal and explicit formula, as opposed to statistical inference and manual inspection, is unlikely to ever be possible. - **What is the most appropriate criteria to filter tests for consistency?** We impose conditions regarding the length and integrity of the records used in our assessment, in order to filter out misleading or error prone measurements. These limitations are: tests that lasted longer than 9 seconds and less than an hour, and exchanged at least 1 packet and less than 120,000 packets. Additionally, it may be useful to impose restrictions based on the M-Lab server, for purposes of consistency in routes and network conditions. However, such a decision would require tests to ensure this does not impede m-labs ability to accommodate changes in international routes. We do not consider upstream tests solely for the sake of brevity, although this direction may be equally important should the throttling of upload connect be a means of curtailing the outward flow of media. - **What is the future relevance of NDT in monitoring the next generation of throttling?** Iran’s strategies of censorship have followed a historical trend of increasing precision of disruption. Whereas the June 2009 elections corresponded with a multiple week outage of SMS services, by early Spring 2013 keyword filtering on political slogans or terms associated with controversial issues had become a normal occurrance. The blocking of SSL in February 2012 had shifted to the blocking of SSL to selection networks and the redirection of secure traffic through the interception of DNS requests \[IIIP 1\] . Similiarly, reports of throttling appear more specific, such as SSL or multimedia traffic in general, or SSL to services such as Google \[IIIP 3\]. The more that a censoring intermediary narrows its understanding and limiting of offensive traffic to a ‘black list,’ a strategy that has substantial political and economic value, the less that these disruptions will be reflected within the NDT test. However, a countervaling pressure is presented by the adoption of sophisticated strategies by anti-filtering tools to disguise or randomize their network traffic in a manner that makes deep packet inspection increasingly difficult and costly, such as the obfsproxy mechanism employed by Tor and Psiphon. Without the ability to confidently distinguish normal traffic from privacy-perserving connections that it cannot control, censors may be forced back into more to the broad throttling regime or shifting to a ‘white list’ strategy that NDT would detect. - **How do we best filter tests for consistency and accuracy?** We limit the records used in our assessment in order to filter out misleading or error prone measurements. Limit the measurements to those downstream tests that lasted longer than 9 seconds and less than an hour, and exchanged at least 1 packet and less than 120,000 packets. Additionally, it may be useful to further limit based on the m-lab server, for purposes of consistency in routes and network conditions. However, such a descision would require tests to ensure this does not impede m-labs ability to accomodate changes in international routes. We do not consider upstream tests solely for the sake of brevity. This direction may be equally important should the throttling of upload connect be a means of curtailing the outflow flow of media. Acknowledgements ================ This research would not have been possible if not for the substantial contributions of a number of individuals who I am privileged to even know and deeply regret not being able to acknowledge in name; the chilling effect of censorship and state intimidation is not limited to the borders of a country. Fortunately, it is possible to recognize Briar Smith, Meredith Whittaker and Philipp Winter for their technical, professional and moral support, making this matter of curiosity into something more professional, which I hope contributes to the ability of the public to protect such an important medium of expression. Appendix {#sec:appendix} ======== SHARIF-EDU-NET Sharif University of Technology, Tehran,Iran 213.233.160.0/19 165 ------------------------------------------------------------- ------------------ ----- NGSAS Neda Gostar Saba Data Transfer Company 188.158.0.0/16 126 UT-AS University of Tehran Informatics Center 80.66.176.0/20 112 SHARIF-EDU-NET Sharif University of Technology, Tehran,Iran 81.31.160.0/19 73 ASIATECH-AS AsiaTech Inc. 79.127.32.0/20 60 PARSONLINE PARSONLINE Autonomous System 91.98.0.0/15 53 IR-ASRETELECOM-AS Asre Enteghal Dadeha 188.34.0.0/17 49 NGSAS Neda Gostar Saba Data Transfer Company 188.159.0.0/16 48 PARSONLINE PARSONLINE Autonomous System 91.99.0.0/19 45 PARSONLINE PARSONLINE Autonomous System 91.99.0.0/16 42 DNET-AS Damoon Rayaneh Shomaj Company LLC 86.57.120.0/21 40 PARSONLINE PARSONLINE Autonomous System 91.98.160.0/19 38 DCI-AS Information Technology Company (ITC) 217.218.0.0/17 37 IR-ASRETELECOM-AS Asre Enteghal Dadeha 188.34.48.0/20 35 PARSONLINE PARSONLINE Autonomous System 91.99.32.0/19 34 IRANGATE Rasaneh Esfahan Net 212.50.244.0/22 33 DCI-AS Information Technology Company (ITC) 78.39.0.0/17 30 DCI-AS Information Technology Company (ITC) 78.39.128.0/17 30 DCI-AS Information Technology Company (ITC) 85.185.0.0/17 30 ASK-AS Andishe Sabz Khazar Autonomous System 95.82.40.0/21 27 DCI-AS Information Technology Company (ITC) 2.176.0.0/16 56 ------------------------------------------------------------- ------------------ ---- MOBINNET-AS Mobin Net Communication Company 178.131.0.0/16 40 PARSONLINE PARSONLINE Autonomous System 91.98.0.0/15 36 PARSONLINE PARSONLINE Autonomous System 91.99.32.0/19 33 NGSAS Neda Gostar Saba Data Transfer Company 188.158.0.0/16 25 PARSONLINE PARSONLINE Autonomous System 91.98.160.0/19 25 PARSONLINE PARSONLINE Autonomous System 91.99.0.0/19 25 TIC-AS Telecommunication Infrastructure Company 2.185.128.0/19 25 DCI-AS Information Technology Company (ITC) 2.181.0.0/16 24 IUSTCC-AS Iran University Of Science and Technology 194.225.232.0/21 24 TIC-AS Telecommunication Infrastructure Company 2.180.32.0/19 23 DCI-AS Information Technology Company (ITC) 46.100.128.0/17 22 PARSONLINE PARSONLINE Autonomous System 91.99.0.0/16 22 DCI-AS Information Technology Company (ITC) 2.185.0.0/16 21 NGSAS Neda Gostar Saba Data Transfer Company 188.158.96.0/21 21 SHARIF-EDU-NET Sharif University of Technology, Tehran,Iran 213.233.160.0/19 21 DCI-AS Information Technology Company (ITC) 217.218.0.0/17 20 DCI-AS Information Technology Company (ITC) 2.182.32.0/19 19 PARSONLINE PARSONLINE Autonomous System 91.98.208.0/20 19 UT-AS University of Tehran Informatics Center 80.66.176.0/20 19 FANAVA-AS Fanava Group 95.38.32.0/19 18 [^1]: This project received grant funding from the Center for Global Communication Studies at the University of Pennsylvania’s Annenberg School for Communcation and Google Research. [^2]: Remarks before Mobile World Congress. February 28, 2012 [^3]: While ISPs are the public face of Internet access to the Iranian public, these companies are only one component of a broader domestic telecommunications infrastructure responsible for delivering domestic and international traffic. ISPs interconnect with each other, known as *peering*, to provide accessibility to hosts within their network and share routes for traffic between others. Not all ISPs are consumer-facing, with some infrastructure companies acting as dedicated Internet exchange points (IXPs) between networks. [^4]: For the purpose of space we have abbrievated the NDT-recorded variables of SndLimTimeRwin, SndLimTimeCwnd, SndLimTimeSnd [^5]: Based on Hurricane Electric’s public data of the peering of AS12880, of these countries, Iran is only directly connected with Italy [^6]: We identify as incidents of public protests, the following dates: 2010-02-11, 2010-03-16, 2010-06-12, 2011-02-14, 2011-02-20, 2011-03-01, 2011-03-08, 2012-02-14, 2012-10-01 [^7]: We note these anecdotal stories as a theoretical condition capable of being tested, not as evidence or an asserted mechanism of implementation. Publicly-disclosable documentation of the ISP role in censorship from either tool-makers or former staff has been difficult to source due to the security issues.

--- abstract: | We consider packing LP’s with $m$ rows where all constraint coefficients are normalized to be in the unit interval. The $n$ columns arrive in random order and the goal is to set the corresponding decision variables irrevocably when they arrive so as to obtain a feasible solution maximizing the expected reward. Previous $(1 - \epsilon)$-competitive algorithms require the right-hand side of the LP to be $\Omega (\frac{m}{\epsilon^2} \log \frac{n}{\epsilon})$, a bound that worsens with the number of columns and rows. However, the dependence on the number of columns is not required in the single-row case and known lower bounds for the general case are also independent of $n$. Our goal is to understand whether the dependence on $n$ is required in the multi-row case, making it fundamentally harder than the single-row version. We refute this by exhibiting an algorithm which is $(1 - \epsilon)$-competitive as long as the right-hand sides are $\Omega (\frac{m^2}{\epsilon^2} \log \frac{m}{\epsilon})$. Our techniques refine previous PAC-learning based approaches which interpret the online decisions as linear classifications of the columns based on sampled dual prices. The key ingredient of our improvement comes from a non-standard covering argument together with the realization that only when the columns of the LP belong to few 1-d subspaces we can obtain small such covers; bounding the size of the cover constructed also relies on the geometry of linear classifiers. General packing LP’s are handled by perturbing the input columns, which can be seen as making the learning problem more robust. author: - | Marco Molinaro\ Carnegie Mellon - | R. Ravi\ Carnegie Mellon bibliography: - 'online-lp.bib' title: Geometry of Online Packing Linear Programs --- Introduction ============ Traditional optimization models usually assume that the input is known a priori. However, in most applications, the data is either revealed over time or only coarse information about the input is known, often modeled in terms of a probability distribution. Consequently, much effort has been directed towards understanding the quality of solutions that can be obtained without full knowledge of the input, which led to the development of online and stochastic optimization [@borodinbook; @stocprogbook]. Emerging problems such as allocating advertisement slots to advertisers and yield management in the internet are of inherent online nature and have further accelerated this development [@agrawal]. Linear programming is arguably the most important and thus well-studied optimization problem. Therefore, understanding the limitations of solving linear programs when complete data is not available is a fundamental theoretical problem with a slew of applications, including the ad allocation and yield management problems above. Indeed, a simple linear program with one uniform knapsack constraint, the Secretary Problem, was one of the first online problems to be considered and an optimal solution was already obtained by the early 60’s [@Dynkin; @GilbertMosteller]. Although the single knapsack case is currently well-understood under different models of how information is revealed [@BabaioffSurvey], much less is known about problems with multiple knapsacks and only recently algorithms with solution guarantees have been developed [@feldman; @agrawal; @Devanur11]. [*The Model.*]{} We study online packing LP’s in the *random permutation model*. Consider a fixed but unknown LP with $n$ columns $a^1, a^2, \ldots, a^n \in [0,1]^m$, whose associated variables are constrained to be in $[0,1]$, and $m$ packing constraints: $$\begin{aligned} {\ensuremath{\textrm{OPT}}}= \max \sum_{t = 1}^n \pi_t x_t \notag \\ \sum_{t = 1}^n a^t x_t \le B \label{eq:LP} \tag{LP}\\ x_t \in [0,1] \,. \notag \end{aligned}$$ Columns are presented in uniformly random order, and when a column is presented we are required to irrevocably choose the value of its corresponding variable. We assume that the number of columns $n$ is known.[^1] The goal is to obtain a feasible solution while maximizing its value. We use ${\ensuremath{\textrm{OPT}}}$ to denote the optimum value of the (offline) LP. By scaling down rows as necessary, we assume without loss of generality that all entries of $B$ are the same, which we also denote with some overload of notation by $B$. Due to the packing nature of the problem, we also assume without loss of generality that all the $\pi_t$’s are non-negative and all the $a^t$’s are non-zero: we can simply ignore columns which do not satisfy the first property and always set to 1 the variables associated to the remaining columns which do not satisfy the second property. Finally, we assume that the columns $a^t$’s are in *general position*: for all $p \in \mathbb{R}^m$, there are at most $m$ different $t \in [n]$ such that $\pi_t = p a^t$. Notice that perturbing the input randomly by a tiny amount achieves this property with probability one, while the effect of the perturbation is absorbed in our approximation guarantees [@DevanurHayes09; @agrawal]. Applications {#applications .unnumbered} ------------ Write about applications on online revenue management and resource allocation (like ads) to motivate the problem. [*Related work.*]{} The random permutation model has grown in popularity [@GoelMehta08; @DevanurHayes09; @BabaioffSurvey] since it avoids strong lower bounds of the pessimistic adversarial-order model [@BuchbinderMOR] while still capturing the lack of total information a priori. Different online problems have already been studied in this model, including bin-packing [@kenyon], matchings [@KVV; @GoelMehta08], the AdWords Problem [@DevanurHayes09] and different generalizations of the Secretary Problem [@BabaioffSurvey; @weightsSecretary; @submodularSecretary; @soto; @sungjin]. Closest to our work are packing problems with a single knapsack constraint. In [@kleinberg], Kleinberg considered the $B$-Choice Secretary Problem, where the goal is to select at most $B$ items coming online in random order to maximize profit. The author presented an algorithm with competitive ratio $1 - O(1/\sqrt{B})$ and showed that $1 - \Omega(1/\sqrt{B})$ is best possible. Generalizing the $B$-Choice Secretary Problem, Babaioff et al. [@babaioff] considered the online knapsack problem and presented a $(1/10e)$-competitive algorithm. Notice that in both cases the competitive ratio does not depend on $n$. Despite all these works, the first result for more general online packing LP’s here was only recently obtained by Feldman et al. [@feldman] and Agrawal et al. [@agrawal]. The first paper presents an algorithm that obtains with high probability a solution of value at least $(1 - \epsilon) {\ensuremath{\textrm{OPT}}}$ whenever $B \ge \Omega(\frac{m \log n}{\epsilon^3})$ and ${\ensuremath{\textrm{OPT}}}\ge \Omega(\frac{\pi_{\max} m \log n}{\epsilon})$, where $\pi_{\max}$ is the largest profit. In the second paper, the authors present an algorithm which obtains a solution of expected value at least $(1 - \epsilon) {\ensuremath{\textrm{OPT}}}$ under the weaker assumptions $B \ge \Omega\left(\frac{m}{\epsilon^2} \log \frac{n}{\epsilon}\right)$ or ${\ensuremath{\textrm{OPT}}}\ge \Omega\left(\frac{\pi_{\max} m^2}{\epsilon^2} \log \frac{n}{\epsilon}\right)$. One other way of stating this result is that the algorithm obtains a solution with competitive ratio $1 - O(\sqrt{\frac{m \log(n) \log B}{B}})$; notice that the guarantee degrades as $n$ increases. The current lower bound on $B$ to allow $(1 - \epsilon)$-competitive algorithms is $B \ge \frac{\log m}{\epsilon^2}$, also presented in [@agrawal]. We remark that these algorithms actually work for more general allocation problems, where a set of columns representing various options arrive at each step and the solution may choose at most one of the options. Both of the above algorithms use a connection between solving the online LP and PAC-learning [@cucker] a linear classification of its columns, which was initiated by Devanur and Hayes [@DevanurHayes09] in the context of the AdWords problem. Here we further explore this connection and our improved bounds can be seen as a consequence of making the learning algorithm more robust by suitably changing the input LP. Robustness is a topic well-studied in learning theory [@devroye; @partha], although existing results do not seem to apply directly to our problem. We remark that a component of robustness more closely related to the standard PAC-learning literature is used in [@DevanurHayes09]. In recent work, Devanur et al. [@Devanur11] consider the weaker *i.i.d. model* for the general allocation problem. While in the random permutation model one can think that columns are sampled without replacement, in the i.i.d. model they are sampled with replacement. Making use of the independence between samples, Devanur et al. substantially improve requirement on $B$ to $\Omega(\frac{\log (m/\epsilon)}{\epsilon^2})$ while showing that the lower bound $\Omega\left(\frac{\log m}{\epsilon^2}\right)$ still holds in this model. We remark, however, that these models can present very different behaviors: as a simple example, consider an LP with $n$ columns, $m = 1$ constraints and budget $B = 1$, where only one of the columns has $\pi_1 = a^1 = 1$ and all others have $\pi_i = a^i = 0$; in the random permutation model the expected value of the optimal solution is 1, while in the i.i.d. model this value is $1 - (1 - 1/n)^n \rightarrow 1 - 1/e$. The competitiveness of the algorithm of [@Devanur11] under the permutation model is still unknown and was left as an open problem by the authors. [*Our results.*]{} Our focus is to understand how large $B$ is required to be in order to allow $(1 - \epsilon)$-competitive algorithms. In particular, the requirements for $B$ in the above algorithms degrade as the number of columns in the LP increases, while the the lower bound does not. With the trend of handling LP’s with larger number of columns (e.g. columns correspond to the keywords in the ad allocation problem, which in turn correspond to visits of a search engine’s webpage), this gap is very unsatisfactory from a practical point of view. Furthermore, given that guarantees for the single knapsack case do not depend on the number of columns, it is important to understand if the multi-knapsack case is fundamentally more difficult. In this work, we give a precise indication of why the latter problem was resistant to arguments used in the single knapsack case, and overcome this difficulty to exhibit an algorithm with dimension-independent guarantee. We show that a modification of the [<span style="font-variant:small-caps;">DPA</span>]{}algorithm from [@agrawal] that we call *Robust [DPA]{}* obtains a $(1 -\epsilon)$-competitive solution for online packing LP’s with $m$ constraints in the random permutation model whenever $B \ge \Omega(\frac{m^2}{\epsilon^2} \log \frac{m}{\epsilon})$. Another way of stating this result is that the algorithm has competitive ratio $1 - O(m \sqrt{\log B}/\sqrt{B})$. Contrasting to previous results, our guarantee does not depend on $n$ and in the case $m = 1$ matches the bounds for the $B$-Choice Secretary Problem up to lower order terms. We finally remark that we can replace the requirement $B \ge \Omega(\frac{m^2}{\epsilon^2} \log \frac{m}{\epsilon})$ by ${\ensuremath{\textrm{OPT}}}\ge \Omega(\frac{\pi_{\max} m^3}{\epsilon^2} \log \frac{m}{\epsilon})$ exactly as done in Section 5.1 of [@agrawal]. [*High-level outline.*]{} As mentioned before, we use the connection between solving an online LP and PAC-learning a good linear classification of its columns; in order to obtain the improved guarantee, we focus on tightening the bounds for the generalization error of the learning problem. More precisely, solving the LP can be seen as classifying the columns into 0/1, which corresponds to setting their associated variable to 0/1. Consider a family $\mathcal{X} \subseteq \{0,1\}^n$ of linear classifications of the columns. Our algorithms sample a set $S$ of columns and learn a classification $x^S \in \mathcal{X}$ which is “good” for the columns $S$ (i.e., obtains large proportional revenue while not filling up the proportionally scaled budget too much). The goal is to upper bound the probability that $x^S$ is not good for the whole LP; this is typically done via a union bound over the classifications in $\mathcal{X}$ [@DevanurHayes09; @agrawal]. To obtain improved guarantees, we refine this bound using an argument akin to covering: we consider *witnesses* (Section \[sec:witness\]), which are representatives of groups of ‘similar’ bad classifications that can be used to bound the probability that *any* classification in the group is learned; for that we need to use a non-standard measure of similarity between classifications which is based on the budget of the LP. The problem is that, when the columns $(\pi_t, a^t)$’s do not lie in a two-dimensional subspace of $\mathbb{R}^{m}$, the set $\mathcal{X}$ may contain a large number of mutually dissimilar bad classifications; this is a roadblock for obtaining a small set of witnesses. In stark contrast, when these columns do lie in a two-dimensional subspace (e.g., $m = 1$), these classifications have a much nicer structure which indeed allows a small set of witnesses. This indicates that the latter learning problem is intrinsically more robust than the former, which seem to precisely capture the increased difficulty in obtained good bounds for the multi-row case. Motivated by this discussion we first consider LP’s whose columns $a^t$’s lie in *few* one-dimensional subspaces (Section \[sec:otp\]). For each of these subspaces, we are able to approximate the classifications induced in the columns lying in the subspace by considering a small subset of the induced classifications; patching together these partial classifications gives us a witness set for $\mathcal{X}$. However, this strategy as stated does not make use of the fact that the subspaces are embedded in an $m$-dimensional space, and hence leads to large witness sets. By establishing a connection between the “useful” patching possibilities with faces of a hyperplane arrangement in $\mathbb{R}^m$ (Lemma \[lemma:sizeP\]), we are able to make use of the dimension of the host space and exhibit witness sets of much smaller sizes, which leads to improved bounds. For a general packing LP, we perturb the columns $a^t$’s to make them lie in few one-dimensional subspaces that form an ‘$\epsilon$-net’ of the space, while not altering the feasibility and optimality of the LP by more than a $(1 \pm \epsilon)$ factor (Section \[sec:rotp\]). Finally, we tighten the bound by using the idea of periodically recomputing the classification, following [@agrawal] (Section \[sec:rdpa\]). OTP for almost 1-dim columns {#sec:otp} ============================ In this section we describe and analyze the algorithm [<span style="font-variant:small-caps;">OTP</span>]{}(One-Time Pricing) over LP’s whose columns are contained in few 1-dimensional subspaces of $\mathbb{R}^{m}$. The overall goal is to find an appropriate dual (perhaps infeasible) solution $p$ for and use it to classify the columns of the LP. More precisely, given $p \in \mathbb{R}^m$, we define $x(p)_t = 1$ if $\pi_t > p a^t$ and $x(p)_t = 0$ otherwise. Thus, $x(p)$ is the result of classifying the columns $(\pi_t, a^t)$’s with the homogeneous hyperplane in $\mathbb{R}^{m+1}$ with normal $(-1, p)$. The motivation behind this classification is that it selects the columns which have positive reduced cost with respect to the dual solution $p$, or alternatively, it solves to optimality the Lagrangian relaxation using $p$ as multipliers. [*Sampling LP’s.*]{} In order to obtain a good dual solution $p$ we use the (random) LP consisting on the first $s$ columns of with appropriately scaled right-hand side. $$\begin{aligned} \tag{$(s, \delta)$-LP} \label{eq:sdLP} \max & \sum_{t = 1}^{s} \pi_{\sigma(t)} x_{\sigma(t)} \\ &\sum_{t = 1}^{s} a^{\sigma(t)} x_{\sigma(t)} \le \frac{s}{n} \delta B \notag \\ &x_{\sigma(t)} \in [0,1] \ \ \ \ t = 1, \ldots, s \notag . \end{aligned}$$ $$\begin{aligned} \tag{$(s, \delta)$-Dual} \label{eq:sdDual} \min \ & \frac{s}{n} \delta B \sum_{i = 1}^{m} p_i + \sum_{t = 1}^s \alpha_{\sigma(t)} \\ & p a^{\sigma(t)} + \alpha_{\sigma(t)} \ge \pi_{\sigma(t)} \ \ \ \ t = 1, \ldots, s \notag \\ & p \ge 0 \notag \\ & \alpha \ge 0 \notag. \end{aligned}$$ Here $\sigma$ denotes the random permutation of the columns of the LP. We use ${\ensuremath{\textrm{OPT}}}(s,\delta)$ to denote the optimal value of $(s, \delta)$-LP and ${\ensuremath{\textrm{OPT}}}(s)$ to denote the optimal value of $(s, 1)$-LP. The static pricing algorithm [<span style="font-variant:small-caps;">OTP</span>]{}of [@agrawal] can then be described as follows.[^2] 1. Wait for the first $\epsilon n$ columns of the LP (indexed by $\sigma(1), \sigma(2), \ldots, \sigma(\epsilon n)$) and solve $(\epsilon n, 1 - \epsilon)$-Dual. Let $(p, \alpha)$ be the obtained dual optimal solution. 2. Use the classification given by $p$ as above by setting $x_{\sigma(t)} = x(p)_{\sigma(t)}$ for $t = \epsilon n + 1, \epsilon n + 2, \ldots$ for as long as the solution obtained remains valid. From this point on set all further variables to zero. Note that by definition this algorithm outputs a feasible solution with probability one. Our goal is then to analyze the quality of the solution produced, ultimately leading to the following theorem. \[thm:otp\] Fix $\epsilon \in (0,1]$. Suppose that there are $K \ge m$ 1-dim subspaces of $\mathbb{R}^m$ containing the columns $a^t$’s and that $B \ge \Omega\left(\frac{m}{\epsilon^3} \log \frac{K}{\epsilon}\right)$. Then algorithm [<span style="font-variant:small-caps;">OTP</span>]{}returns a feasible solution with expected value at least $(1 - 5\epsilon) {\ensuremath{\textrm{OPT}}}$. Let $S = \{\sigma(1), \ldots, \sigma(\epsilon n)\}$ be the (random) index set of the columns sampled by [<span style="font-variant:small-caps;">OTP</span>]{}. We use $p^S$ to denote the optimal dual solution obtained by [<span style="font-variant:small-caps;">OTP</span>]{}; notice that $p^S$ is completely determined by $S$. To simplify the notation, we also use $x^S$ to denote $x(p^S)$. Notice that, for all the scenarios where $x^S$ is feasible, the solution returned by [<span style="font-variant:small-caps;">OTP</span>]{}is identical to $x^S$ with its components $x^S_{\sigma(1)}, \ldots, x^S_{\sigma(\epsilon n)}$ set to zero. Given this observation and the fact that ${\mathbb{E}}[\sum_{t \leq \epsilon n} \pi_{\sigma(t)} x^S_{\sigma(t)}] \leq \epsilon {\ensuremath{\textrm{OPT}}}$, one can prove that the following lemma implies Theorem \[thm:otp\]. \[lemma:goodOtp\] Fix $\epsilon \in (0,1]$. Suppose that there are $K \ge m$ 1-dim subspaces of $\mathbb{R}^m$ containing the columns $a^t$’s and that $B \ge \Omega\left(\frac{m}{\epsilon^3} \log \frac{K}{\epsilon}\right)$. Then with probability at least $(1 - \epsilon)$, $x^S$ is a feasible solution for with value at least $(1 - 3\epsilon) {\ensuremath{\textrm{OPT}}}$. Connection to PAC learning -------------------------- We assume from now on that $B \ge \Omega(\frac{m}{\epsilon^3} \log \frac{K}{\epsilon})$. Let $\mathcal{X} = \{x(p) : p \in \mathbb{R}^m_+\} \subseteq \{0,1\}^n$ denote the set of all possible linear classifications of the LP columns which can be generated by [<span style="font-variant:small-caps;">OTP</span>]{}. With slight overload in the notation, we identify a vector $x \in \{0,1\}^n$ with the subset of $[n]$ corresponding to its support. Given a scenario, we say that $x^S$ is *bad* if it does not satisfy the properties of Lemma \[lemma:goodOtp\], namely $x^S$ is either infeasible or has value less than $(1 - 3\epsilon) {\ensuremath{\textrm{OPT}}}$. We say that $x^S$ is *good* otherwise. As noted in previous work, since our decisions are made based on reduced costs it suffices to analyze the *budget occupation* (or complementary slackness) of the solution in order to understand its *value*. To make this precise, given $x \in \{0,1\}^n$ let $a_i(x) = \sum_{t \in x} a_i^t$ be its occupation of the $i$th budget and let $a^S_i(x) = \frac{1}{\epsilon}\sum_{t \in x \cap S} a_i^t$ be its appropriately scaled occupation of $i$th budget in the sampled LP (recall $|S| = \epsilon n$). \[lemma:approximateCS\] Consider a scenario where $x^S$ satisfies: (i) for all $i \in [m]$, $a_i(x^S) \le B$ and (ii) for all $i \in [m]$ with $p^S_i > 0$, $a_i(x^S) \ge (1 - 3 \epsilon) B$. Then $x^S$ is good. Moreover, since we are making decisions based on the *optimal* reduced cost for the sampled LP, our solution satisfies the above properties for the sampled LP. \[lemma:sampleCS\] In every scenario, $x^S$ satisfies the following: (i) for all $i \in [m]$, $a_i^S(x^S) \le (1 - \epsilon)B$ and (ii) for every $i \in [m]$ with $p^S_i > 0$, $a_i^S(x^S) \ge (1 - 2\epsilon) B$. Given that $a_i(x) = {\mathbb{E}}[a^S_i(x)]$ for all $x$, the idea is to use concentration inequalities to argue that the conditions in Lemma \[lemma:approximateCS\] hold with good probability. Although concentration of $a^S_i(x)$ for *fixed* $x$ can be achieved via Chernoff-type bounds, the quantity $a^S_i(x^S)$ has undesired correlations; obtaining an effective bound is the main technical contribution of this paper. For a given scenario, we say that $x \in \mathcal{X}$ can be *badly learned for budget $i$* if either (i) $a_i^S(x) \le (1 - \epsilon) B$ and $a_i(x) > B$ or (ii) $a^S_i(x) \ge (1 - 2\epsilon) B$ and $a_i(x) < (1 - 3 \epsilon) B$. Essentially these are the classifications which look good for the sampled $(\epsilon n, 1- \epsilon)$-LP but are actually bad for . Putting Lemmas \[lemma:approximateCS\] and \[lemma:sampleCS\] together and unraveling the definitions gives that $$\Pr\left(x^S \textrm{ is bad}\right) \le \Pr\left(\bigvee_{i \in [m], x \in \mathcal{X}} x \textrm{ can be badly learned for budget } i\right). \label{eq:badlyLearned}$$ Notice that the right-hand side of this inequality does not depend on $x^S$, it is only a function of how skewed $a_i^S(x)$ is as compared to its expectation $a_i(x)$. Usually the right-hand side in the previous equation is upper bounded by taking a union bound over all its terms [@agrawal]. Unfortunately this is too wasteful: when $x$ and $x'$ are “similar” there is a large overlap between the scenarios where $a_i^S(x)$ is skewed and those where $a_i^S(x')$ is skewed. In order to obtain improved guarantees, we introduce in the next section a new way of bounding the right-hand side of the above expression. Similarity via witnesses {#sec:witness} ------------------------ First, we partition the classifications which can be badly learned for budget $i$ into two sets, depending on why they are bad: for $i \in [m]$, let $\mathcal{X}_i^+ = \{x \in \mathcal{X} : a_i(x) > B\}$ and $\mathcal{X}_i^- = \{x \in \mathcal{X} : a_i(x) < (1 - 3 \epsilon) B\}$. In order to simplify the notation, given a set $x$ we define $\operatorname{skewm}_i(\epsilon, x)$ to be the event that $a^S_i(x) \le (1 - \epsilon) B$ and $\operatorname{skewp}_i(\epsilon, x)$ to be the event that $a^S_i(x) \ge (1 - 2\epsilon) B$. Notice that if $x \in \mathcal{X}_i^+$, then $\operatorname{skewm}_i(\epsilon, x)$ is the event that $a_i^S(x)$ is significantly smaller than its expectation (skewed in the minus direction), while for $x \in \mathcal{X}_i^-$ $\operatorname{skewp}_i(\epsilon, x)$ is the event that $a_i^S(x)$ is significantly larger than its expectation (skewed in the plus direction). These definitions directly give the equivalence $$\label{eq:witness1} \Pr\left(\bigvee_{i,x \in \mathcal{X}} x \textrm{ can be badly learned for budget } i\right) = \Pr\left(\bigvee_{i, x \in \mathcal{X}_i^+} \operatorname{skewm}_i(\epsilon, x) \vee \bigvee_{i, x \in \mathcal{X}_i^-} \operatorname{skewp}_i(\epsilon, x)\right).$$ In order to introduce the concept of witnesses, consider two sets $x,x'$, say, in $\mathcal{X}_i^+$. Take a subset $w \subseteq x \cap x'$; the main observation is that, since $a^t \ge 0$ for all $t$, for all scenarios we have $a_i^S(w) \le a_i^S(x)$ and $a_i^S(w) \le a_i^S(x')$. In particular, the event $\operatorname{skewm}_i(\epsilon, x) \vee \operatorname{skewm}_i(\epsilon,x')$ is contained in $\operatorname{skewm}(\epsilon, w)$. The set $w$ serves as a witness for scenarios which are skewed for either $x$ or $x'$; if additionally $a_i(w)$ reasonably larger than $(1 - \epsilon) B$, we can then use concentration inequalities over $\operatorname{skewm}_i(\epsilon, w)$ in order to bound probability of $\operatorname{skewm}(\epsilon, x) \vee \operatorname{skewm}(\epsilon,x')$. This ability of bounding multiple terms of the right-hand side of simultaneously is what gives an improvement over the naive union bound. \[def:witness\] We say that $\mathcal{W}_i^+$ is a *witness set* for $\mathcal{X}_i^+$ if: (i) for all $w \in \mathcal{W}_i^+$, $a_i(w) \ge (1 - \epsilon/2) B$ and (ii) for all $x \in \mathcal{X}_i^+$ there is $w \in \mathcal{W}_i^+$ contained in $x$. Similarly, we say that $\mathcal{W}_i^-$ is a *witness set* for $\mathcal{X}_i^-$ if: (i) for all $w \in \mathcal{W}_i^-$, $a_i(w) \le (1 - 3 \epsilon/2) B$ and (ii) for all $x \in \mathcal{X}_i^-$ there is $w \in \mathcal{W}_i^-$ containing $x$. As indicated by the previous discussion, given witness sets $\mathcal{W}_i^+$ and $\mathcal{W}_i^-$ for $\mathcal{X}_i^+$ and $\mathcal{X}_i^-$, we directly get the bound $$\begin{aligned} \label{eq:witness2} \Pr\left(\bigvee_{i, x \in \mathcal{X}_i^+} \operatorname{skewm}(\epsilon, x) \vee \bigvee_{i, x \in \mathcal{X}_i^-} \operatorname{skewp}(\epsilon, x)\right) \le \Pr\left(\bigvee_{i, w \in \mathcal{W}_i^+} \operatorname{skewm}(\epsilon, w) \vee \bigvee_{i, w \in \mathcal{W}_i^-} \operatorname{skewp}(\epsilon, w)\right). \end{aligned}$$ Putting together the last three displayed equations and using Chernoff-type bounds, we can get an upper estimate on the probability that $x^S$ is bad in terms of the size of witnesses sets. \[lemma:badWitness\] Suppose that, for all $i \in [m]$, there are witness sets for $\mathcal{X}_i^+$ and $\mathcal{X}_i^-$ of size at most $M$. Then $\Pr(x^S \textrm{ is bad }) \le 8mM \exp\left(-\frac{\epsilon^3 B}{33}\right)$. One natural choice of a witness set for, say, $\mathcal{X}_i^+$ is the collection of all of its minimal sets; unfortunately this may not give a witness set of small enough size. But notice that a witness set need not be a subset of $\mathcal{X}_i^+$ (or even $\mathcal{X}$). Allowing elements outside $\mathcal{X}_i^+$ gives the flexibility of obtaining witnesses which are associated to multiple “similar” minimal elements of $\mathcal{X}_i^+$, which is effective in reducing the size of witness sets. Small witness sets for almost 1-dim columns {#sec:goodWitness} ------------------------------------------- Given the previous lemma, our task is to find small witness sets. Unfortunately, when the $(\pi_t, a^t)$’s lie in a space of dimension at least 3, $\mathcal{X}_i^+$ and $\mathcal{X}_i^-$ may contain many ($\Omega(n)$) disjoint sets (see Figure \[fig:disjoint3d\]), which shows that in general we cannot find small witness sets directly. This sharply contrasts with the case where the $(\pi_t, a^t)$’s lie in a 2-dimensional subspace of $\mathbb{R}^{m + 1}$, where one can show that $\mathcal{X}$ is a union of 2 chains with respect to inclusion. In the special case where the $a^t$’s lie in a 1-dimensional subspace of $\mathbb{R}^m$, we show that $\mathcal{X}$ is actually a single chain (Lemma \[lemma:chain\]) and therefore we can take $\mathcal{W}_i^+$ as *the* minimal set of $\mathcal{X}_i^+$ and $\mathcal{W}_i^-$ as *the* maximal set of $\mathcal{X}_i^-$. Due to the above observations, we focus on LP’s whose $a^t$’s lie in few 1-dimensional subspaces. In this case, $\mathcal{X}_i^+$ and $\mathcal{X}_i^-$ are sufficiently well-behaved so that we can find small (independent of $n$) witness sets. \[lemma:witness2Dim\] Suppose that there are $K \ge m$ 1-dimensional subspaces of $\mathbb{R}^m$ which contain the $a^t$’s. Then there are witness sets for $\mathcal{X}_i^+$ and $\mathcal{X}_i^-$ of size at most $(O(\frac{K}{\epsilon} \log \frac{K}{\epsilon}))^m$. Assuming the hypothesis of the lemma, partition the index set $[n]$ into $C_1, C_2, \ldots, C_K$ such that for all $j \in [K]$ the columns $\{a^t\}_{t \in C_j}$ belong to the same 1-dimensional subspace. Equivalently, for each $j \in [K]$ there is a vector $c^j$ of $\ell_\infty$-norm 1 such that for all $t \in C_j$ we have $a^t = \|a^t\|_{\infty} c^j$. An important observation is that now we can order the columns (locally) by the ratio of profit over budget occupation: without loss of generality assume that for all $j \in [K]$ and $t, t' \in C_j$ with $t < t'$, we have $\frac{\pi_t}{\|a^t\|_{\infty}} \ge \frac{\pi_{t'}}{\|a^{t'}\|}_{\infty}$.[^3] Given a classification $x$, we use $x|_{C_j}$ to denote its projection onto the coordinates in $C_j$; so $x|_{C_j}$ is the induced classification on columns with indices in $C_j$. Similarly, we define $\mathcal{X}|_{C_j} = \{x|_{C_j} : x \in \mathcal{X}\}$ as the set of all classifications induced in the columns in $C_j$. The most important structure that we get from working with 1-d subspaces, which is implied by the local order of the columns, is the following. \[lemma:chain\] For each $j \in [K]$, the sets in $\mathcal{X}|_{C_j}$ are prefixes of $C_j$. To simplify the notation fix $i \in [m]$ for the rest of this section, so we aim at providing witness sets for $\mathcal{X}_i^+$ and $\mathcal{X}_i^-$. The idea is to group the classifications according to their budget occupation caused by the different column classes $C_j$’s. To make this formal, start by covering the interval $[0, B + m]$ with intervals $\{I_\ell\}_{\ell \in L}$, where $I_0 = [0, \frac{\epsilon B}{4K})$ and $I_\ell = [\frac{\epsilon B}{4K} (1 + \frac{\epsilon}{4})^{\ell - 1}, \frac{\epsilon B}{4K} (1 + \frac{\epsilon}{4})^\ell)$ for $\ell > 0$ and $L = \{0, \ldots, \lceil \log_{1 + \epsilon/4} \frac{8K}{\epsilon} \rceil\}$ (note that since $B \geq m$, we have $B + m \leq 2B$). Define $\mathcal{B}_{i,j}^\ell$ as the set of partial classifications $y \in \mathcal{X}|_{C_j}$ whose budget occupation $a_i(y)$ lies in the interval $I_\ell$. For ${v}\in L^K$ define the family of classifications $\mathcal{B}_i^{{v}} = \{(y^1, y^2, \ldots, y^K) : y^j \in \mathcal{B}_{i,j}^{{v}_j}\}$. The $\mathcal{B}_i^{{v}}$’s then provide the desired grouping of the classifications. Note that the $\mathcal{B}_i^{{v}}$’s may include classifications not in $\mathcal{X}$ and may not include classifications in $\mathcal{X}$ which have occupation $a_i(.)$ greater than $B + m$. Now consider a non-empty $\mathcal{B}_i^{{v}}$. Let $\underline{w}_i^{{v}}$ be the inclusion-wise smallest element in $\mathcal{B}_i^{{v}}$. Notice that such unique smallest element exists: since $\mathcal{X}|_{C_j}$ is a chain, so is $\mathcal{B}_{i,j}^{{v}_j}$, and hence $\underline{w}_i^{{v}}$ is the product (over $j$) of the smallest elements in the sets $\{\mathcal{B}_{i,j}^{{v}_j}\}_j$. Similarly, let $\overline{w}_i^{{v}}$ denote the largest element in $\mathcal{B}_i^{{v}}$. Intuitively, $\underline{w}^{{v}}_i$ and $\overline{w}^{{v}}_i$ will serve as witnesses for all the sets in $\mathcal{B}_i^{{v}}$. Finally, define the witness sets by adding the $\underline{w}_i^{{v}}$ and $\overline{w}_i^{{v}}$’s of appropriate size corresponding to meaningful $\mathcal{B}_i^{{v}}$’s: set $\mathcal{W}_i^+ = \{\underline{w}_i^{{v}} : {v}\in L^K, \mathcal{B}_i^{{v}} \cap \mathcal{X} \neq \emptyset, a_i(\underline{w}_i^{{v}}) \ge (1 - \epsilon/2) B\}$ and $\mathcal{W}_i^- = \{\overline{w}_i^{{v}} : {v}\in L^K, \mathcal{B}_i^{{v}} \cap \mathcal{X} \neq \emptyset, a_i(\overline{w}_i^{{v}}) \le (1 - 3\epsilon/2) B\}$. It is not too difficult to see that, say, $\mathcal{W}_i^+$ is a witness set for $\mathcal{X}_i^+$: If $x \in \mathcal{X}_i^+$ belongs to some $\mathcal{B}_i^{{v}}$, then $\underline{w}_i^{{v}}$ belongs to $\mathcal{W}_i^+$ and is easily shown to be a witness for $x$. However, if $x$ does not belong to any $\mathcal{B}_i^{{v}}$, by having too large $a_i(x)$, the idea is to find $x' \subseteq x$ which belongs to some $\mathcal{B}_i^{{v}}$ *and* to $\mathcal{X}$, and then use $\underline{w}_i^{{v}}$ as a witness for $x$. We note that considering $B_i^{{v}}$’s for side lengths at most $B + m$ and only adding witnesses for $B_i^{{v}}$’s which intersect $\mathcal{X}$ are crucially used for bounding the size of $\mathcal{W}_i^+$ and $\mathcal{W}_i^-$. \[lemma:wWitness\] The sets $\mathcal{W}_i^+$ and $\mathcal{W}_i^-$ are witness sets for $\mathcal{X}_i^+$ and $\mathcal{X}_i^-$. [*Bounding the size of witness sets.*]{} Clearly the witness sets $\mathcal{W}_i^+$ and $\mathcal{W}_i^-$ have size at most $|L|^K$. Although this size is independent of $n$, it is still unnecessarily large since it only uses locally (for each $C_j$) the fact that $\mathcal{X}$ consists of linear classifications; in particular, it does not use the dimension of the ambient space $\mathbb{R}^m$. Now we sketch the argument for an improved bound, and details are provided in the appendix. First notice that the partial classification $x(p)|_{C_j}$ is completely defined by the value $pc^j$. Thus, if $J \subseteq [K]$ is such that the directions $\{c^j\}_{j \in J}$ form a basis of $\mathbb{R}^m$ then knowing $pc^j$ for all $j \in J$ completely determines the whole classification $x(p)$. Similarly, if we know that $x(p)|_{C_j} \in \mathcal{B}_i^{{v}_j}$ for all $j \in J$, then for each $j \notin J$ we should have fewer possible $\mathcal{B}_i^{{v}_j}$’s where the partial classification $x(p)|_{C_j}$ can belong to; this indicates that some of the sets $\{\mathcal{B}_i^{{v}}\}_{{v}\in L^K}$ do not contain any element from $\mathcal{X}$, which implies a reduced size for the witness sets. In order to capture this idea, we focus on the space of dual vectors $p$ and define the sets $P^\ell_j = \{p \in {{\ensuremath{\mathbb{R}}}}_+^m : x(p)|_{C_j} \in \mathcal{B}_{i,j}^{\ell}\}$ and $P^v = \{p \in {{\ensuremath{\mathbb{R}}}}_+^m : x(p) \in \mathcal{B}_i^v\}$. Notice that $P^v = \cap_j P^{v_j}_j$ and that $\mathcal{B}^v_i$ is empty iff $P^v$ is. The main step is to show that each $P^\ell_j$ is a polyhedron with “few” facets, which uses the definition of $x(p)$ and Lemma \[lemma:chain\]. We then consider the arrangement of the hyperplanes which are facet-defining for the $P_j^\ell$’s and conclude that the $P^v$’s are given by unions of the cells in this arrangement; classical bounds on the number of cells in a hyperplane arrangement in ${{\ensuremath{\mathbb{R}}}}^m$ then allow us to upper bound the number of nonempty $P^v$’s. This gives the following. \[lemma:sizeP\] At most $(O(\frac{K}{\epsilon} \log \frac{K}{\epsilon}))^m$ of the $\mathcal{B}_i^{{v}}$’s contain an element from $\mathcal{X}$. This lemma implies that $\mathcal{W}_i^+$ and $\mathcal{W}_i^-$ each has size at most $(O(\frac{K}{\epsilon} \log \frac{K}{\epsilon}))^m$, which then proves Lemma \[lemma:witness2Dim\]. Finally, applying Lemma \[lemma:badWitness\] we conclude the proof of Lemma \[lemma:goodOtp\]. Robust [<span style="font-variant:small-caps;">OTP</span>]{} {#sec:robustOTP} ============================================================ \[sec:rotp\] In this section we consider with columns that may not belong to few 1-dimensional subspaces. Given the results of the previous section we would like to perturb the columns of this LP so that it belongs to few 1-dim subspaces, and such that an approximate solution for this perturbed LP is also an approximate solution for the original one. More precisely, we obtain a set of vectors $Q \subseteq \mathbb{R}^m$ and transform each column $a^t$ into a column $\tilde{a}^t$ which is a scaling of a vector in $Q$, and we let the rewards $\pi_t$ remain unchanged. The crucial observation is that the solutions of an LP are robust to slight changes in the the constraint matrix. \[lemma:robust\] Consider real numbers $\pi_1, \ldots, \pi_n$ and vectors $a^1, \ldots, a^n$ and $\tilde{a}^1, \ldots, \tilde{a}^n$ in $\mathbb{R}^m_+$ such that $\|\tilde{a}^t - a^t\|_{\infty} \le \frac{\epsilon}{m + 1} \|a^t\|_{\infty}$. If $x$ is an $\epsilon$-approximate solution for with columns $(\pi_t, \tilde{a}^t)$ and right-hand side $(1 - \epsilon) B$, then $x$ is a $2\epsilon$-approximate solution for the LP . [*Perturbing the columns.*]{} To simplify the notation, set $\delta = \frac{\epsilon}{m+1}$; for simplicity of exposition we assume that $1/\delta$ is integral. When constructing $Q$ we want the rays spanned by the each of its vectors to be “uniform” over $\mathbb{R}^m_+$. Using $\ell_\infty$ as normalization, let $Q$ be a $\delta$-net of the unit $\ell_\infty$ sphere, namely let $Q$ be the vectors in $\{0, \delta, 2\delta, 3\delta, \ldots, 1\}^m$ which have $\ell_\infty$ norm 1. Note that $|Q| = (O(\frac{m}{\epsilon}))^m$. Given a vector $a^t \in {{\ensuremath{\mathbb{R}}}}^m$ we let $\tilde{a}^t = \|a^t\|_\infty q^t$, where $q^t$ is the vector in $Q$ closest (in $\ell_\infty$) to $\frac{a^t}{\|a^t\|_\infty}$. By definition of $Q$, for every vector $v \in \mathbb{R}^m$ with $\|v\|_\infty = 1$ there is a vector $q \in Q$ with $\|v - q\|_{\infty} \le \delta$. It then follows from positive homogeneity of norms that the $\tilde{a}^t$’s satisfy the property required in Lemma \[lemma:robust\]: $\|a^t - \tilde{a}^t\|_{\infty} \le \delta \|a^t\|_{\infty}$. [*Algorithm Robust [<span style="font-variant:small-caps;">OTP</span>]{}.*]{} One way to think of the algorithm Robust [<span style="font-variant:small-caps;">OTP</span>]{}is that it works in two phases. First, it transforms the vectors $a^t$ into $\tilde{a}^t$ as described above. Then it returns the solution obtained by running the algorithm [<span style="font-variant:small-caps;">OTP</span>]{}over the LP with columns $(\pi_t, \tilde{a}^t)$ and right-hand side $(1 - \epsilon)B$. Notice that this algorithm can indeed be implemented to run in an online fashion. Putting together the discussion in the previous paragraphs and the guarantee of [<span style="font-variant:small-caps;">OTP</span>]{}for almost 1-dim columns given by Theorem \[thm:otp\] with $K = |Q| = (O(\frac{m}{\epsilon}))^m$, we obtain the following theorem. \[thm:rotp\] Fix $\epsilon \in (0,1]$ and suppose $B \ge \Omega\left(\frac{m^2}{\epsilon^3} \log \frac{m}{\epsilon}\right)$. Then algorithm Robust [<span style="font-variant:small-caps;">OTP</span>]{}returns a solution to the online with expected value at least $(1 - 10\epsilon) {\ensuremath{\textrm{OPT}}}$. Robust [<span style="font-variant:small-caps;">DPA</span>]{} {#sec:rdpa} ============================================================ In this section we describe our final algorithm, which has an improved dependence on $1/\epsilon$. Following [@agrawal], the idea is to update the dual vector used in the classification as new columns arrive: we use the first $2^i \epsilon n$ columns to classify columns $2^i \epsilon n + 1, \ldots, 2^{i + 1} \epsilon n$. This leads to improved generalization bounds, which in turn give the reduced dependence on $1/\epsilon$. The algorithm Robust [<span style="font-variant:small-caps;">DPA</span>]{}(as the algorithm [<span style="font-variant:small-caps;">DPA</span>]{}) can be seen as a combination of solutions to multiple sampled LP’s, obtained via a modification of [<span style="font-variant:small-caps;">OTP</span>]{}denoted by $(s,\delta)$-[<span style="font-variant:small-caps;">OTP</span>]{}. [*Algorithm $(s,\delta)$-[<span style="font-variant:small-caps;">OTP</span>]{}.*]{} This algorithm aims at solving the program $(2s,1)$-LP and can be described as follows: it finds an optimal dual solution $(p, \alpha)$ for $(s, (1 - \delta))$-LP and sets $x_{\sigma(t)} = x(p)_{\sigma(t)}$ for $t = s + 1, s + 2, \ldots, t' \le 2s$ such that $t'$ is the maximum one guaranteeing $\sum_{t = s + 1}^{2s} a^{\sigma(t)} x_{\sigma(t)} \le \frac{s}{n} B$. The analysis of $(s,\delta)$-[<span style="font-variant:small-caps;">OTP</span>]{}is similar to the one employed for [<span style="font-variant:small-caps;">OTP</span>]{}. The main difference is that this algorithm tries to approximate the value of the *random* LP $(2s,1)$-LP. This requires a partition of the bad classifications which is more refined than simply splitting into $\mathcal{X}_i^+$ and $\mathcal{X}_i^-$, and witness sets need to be redefined appropriately. Nonetheless, using these ideas we can prove the following guarantee for $(s,\delta)$-[<span style="font-variant:small-caps;">OTP</span>]{}. Again let $S = \{\sigma(1), \sigma(2), \ldots, \sigma(s)\}$ be the random index set of the first $s$ columns of the LP, let $T = \{\sigma(s+1), \sigma(s+2), \ldots, \sigma(2s)\}$ and $U = S \cup T$. We use $\pi_U$ to denote the vector $(\pi_t)_{t \in U}$. \[lemma:sDeltaOtp\] Suppose that there are $K \ge m$ 1-dim subspaces of $\mathbb{R}^m$ containing the columns $a^t$’s. Fix an integer $s$ and a real number $\delta \in (0,1/10)$ such that $\frac{\delta^2 s B}{n} \ge \Omega(m \ln \frac{K}{\delta})$. Then algorithm $(s,\delta)$-[<span style="font-variant:small-caps;">OTP</span>]{}returns a solution $x$ satisfying $a_i^T(x) \le B$ for all $i \in [m]$ with probability 1 and with expected value ${\mathbb{E}}[\pi_U x] \ge (1 - 3 \delta) {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(2s)] - {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(s)] - \delta^2 {\ensuremath{\textrm{OPT}}}$. [*Algorithm Robust [<span style="font-variant:small-caps;">DPA</span>]{}.*]{} In order to simplify the description of the algorithm, we assume in this section that $\log (1/\epsilon)$ is an integer. Again the algorithm Robust [<span style="font-variant:small-caps;">DPA</span>]{}can be thought as acting in two phases. In the first phase it converts the vectors $a^t$ into $\tilde{a}^t$, just as in the first phase of Robust [<span style="font-variant:small-caps;">OTP</span>]{}. In the second phase, for $i = 0, \ldots, \log (1/\epsilon) - 1$, it runs $(\epsilon 2^i n, \sqrt{\epsilon/2^i})$-[<span style="font-variant:small-caps;">OTP</span>]{}over with columns $(\pi_t, \tilde{a}^t)$ and right-hand side $(1 - \epsilon) B$ to obtain the solution $x^i$. The algorithm finally returns the solution $x$ consisting of the “union” of $x^i$’s: $x = \sum_i x^i$. Note that the second phase corresponds exactly to using the first $\epsilon 2^i n$ columns to classify the columns $\epsilon 2^i n + 1, \ldots, \epsilon 2^{i+1} n$. This relative increase in the size of the training data for each learning problem allow us to reduce the dependence of $B$ on $\epsilon$ in each of the iterations, while the error from all the iterations telescope and are still bounded as before. Furthermore, notice that Robust [<span style="font-variant:small-caps;">DPA</span>]{}can be implemented to run online. The analysis of Robust [<span style="font-variant:small-caps;">DPA</span>]{}reduces to that of $(s,\delta)$-[<span style="font-variant:small-caps;">OTP</span>]{}. That is, using the definition of the parameters of $(s,\delta)$-[<span style="font-variant:small-caps;">OTP</span>]{}used in Robust [<span style="font-variant:small-caps;">DPA</span>]{}and Lemma \[lemma:sDeltaOtp\], it is routine to check that the algorithm produces a feasible solution which has expected value $(1 - \epsilon) {\ensuremath{\textrm{OPT}}}$. This is formally stated in the following theorem. \[thm:expValueDPA\] Fix $\epsilon \in (0,1/100)$ and suppose that $B \ge \Omega(\frac{m^2}{\epsilon^2} \ln \frac{m}{\epsilon})$. Then the algorithm Robust [<span style="font-variant:small-caps;">DPA</span>]{}returns a solution to the online LP with expected value at least $(1 - 50\epsilon) {\ensuremath{\textrm{OPT}}}$. Open problems ============= A very interesting open question is whether the techniques introduced in this work can be used to obtain improved algorithms for generalized allocation problems [@feldman]. The difficulty in this problem is that the classifications of the columns are not linear anymore; they essentially come from a conjunction of linear classifiers. Given this additional flexibility, having the columns in few 1-dimensional subspaces does not seem to impose strong enough properties in the classifications. It would be interesting to find the appropriate geometric structure of the columns in this case. Of course a direct open question is to improve the lower or upper bound on the dependence on the right-hand side $B$ to obtain $(1 - \epsilon)$-competitive algorithms. One possibility is to investigate how much the techniques presented here can be pushed and what are their limitations. Another possibility is to analyze the performance of the algorithm from [@Devanur11] under the random permutation model. ![Case $m = 2$, columns $(\pi_t, a^t)$ equal to $(1,\sin(\frac{\pi}{4} + \delta t), \cos(\frac{\pi}{4} + \delta t))$ for sufficiently small $\delta > 0$, represented by black dots. Each segment $\{t, t + 1, \ldots, t + j\}$ can be linearly classified and hence belongs to $\mathcal{X}$. Furthermore, all segments $\{j 2 B, \ldots, (j + 1) 2 B\}$ belong to $\mathcal{X}_i^+$, which then contains $\Omega(\frac{n}{B})$ disjoint sets. Similar analysis holds for $\mathcal{X}_i^-$.[]{data-label="fig:disjoint3d"}](disjoint3d.pdf) Bernstein inequality for sampling without replacement ===================================================== \[lemma:addChernoff\] Let $Y = \{Y_1, \ldots, Y_n\}$ be a set of real numbers in the interval $[0,1]$ and let $0 < \epsilon < 1$. Let $S$ be a random subset of $Y$ of size $s$ and let $Y_S = \sum_{i \in S} Y_i$. Setting $\mu = \frac{1}{n} \sum_i Y_i$ and $\sigma^2 = \frac{1}{n}\sum_i (Y_i - \mu)^2$, we have that for every $\tau > 0$ $$\begin{aligned} \Pr(|Y_S - s \mu| \ge \tau) \le 2 \exp \left( -\frac{\tau^2}{2 s \sigma^2 + \tau} \right) \end{aligned}$$ Notice that, since the $Y_i$’s belong to the interval $[0,1]$, we can upper bound the variance by the mean as follows: $$\sigma^2 \le \frac{1}{n} \sum_i |Y_i - \mu| \le \frac{1}{n} \left( \sum_i |Y_i| + \sum_i |\mu| \right) = 2 \mu.$$ This gives the following corollary. \[cor:multiChernoff\] Consider the conditions of the previous lemma. Then for all $\tau > 0$ $$\begin{aligned} \Pr(|Y_S - s \mu| \ge \tau) \le 2 \exp \left( -\frac{\tau^2}{4 s \mu + \tau} \right). \end{aligned}$$ Proof of Lemmas \[lemma:approximateCS\] and \[lemma:sampleCS\] ============================================================== [Lemma \[lemma:approximateCS\]]{} Fix a scenario $\sigma$ for the duration of the proof. By assumption $x^S$ is feasible for , so it suffices to show that it attains value at least $(1-3\epsilon) {\ensuremath{\textrm{OPT}}}$. For that, consider with a modified right-hand side: $$\begin{aligned} \max \sum_{t = 1}^n \pi_t x_t \notag\\ \sum_{t = 1}^n a^t_i x_t \le a_i(x^S) \ \ \ \ \forall i \in [m] \tag{modLP} \label{eq:modLP} \\ x \in [0,1]^n. \notag \end{aligned}$$ Consider the Lagrangian relaxation $L(p, x) = \sum_{t = 1}^{\epsilon n} \pi_t x_t - \sum_{i = 1}^m p_i (\sum_{t = 1}^{\epsilon n} a^t_i x_t - a_i(x^S))$. Notice that $x^S$ is an optimal solution for $\max_{x \in [0,1]^n} L(p^S, x)$, which is at least the [$\textrm{OPT}$]{}(\[eq:modLP\]), the optimum value of LP . Since $x^S$ is clearly feasible for , it follows that $x^S$ is an optimal solution for the latter. Now let $x^*$ be an optimal solution for . Since $a_i(x^S) \ge (1 - 3 \epsilon) B$ for all $i$, and since $a^t \ge 0$ for all $t$, it follows that $(1 - 3 \epsilon) x^*$ is feasible for . By linearity of the objective function we get that [$\textrm{OPT}$]{}(\[eq:modLP\]) $ \ge (1 - 3\epsilon) \sum_{t = 1}^n \pi_t x^*_t = (1 - 3 \epsilon) {\ensuremath{\textrm{OPT}}}$ and the result follows. [Lemma \[lemma:sampleCS\]]{} Fix a scenario $\sigma$ for the duration of the proof. Let $x^*$ be an optimal solution for $(\epsilon n, (1 - \epsilon))$-LP in complementary slackness with $p^S$. If $p^S a^t > \pi_t$, the corresponding constraint in the dual is loose and by complementary slackness we get $x^*_t = 0$. If $p^S a^t < \pi_t$, then for dual feasibility we have $\alpha^*_t > 0$ and by complementary slackness we have $x^*_t = 1$. From the definition of $x^S$ we get that $x^S \le x^*$ and, since the $a^t$’s are non-negative, the feasibility of $x^*$ implies that $a_i^S(x^S) \le (1-\epsilon)B$ for all $i \in [m]$. Moreover, from our assumption that the input is in general position we get that there are at most $m$ values of $t$ such that $p^S a^t = \pi_t$. Therefore, $x^S$ and $x^*$ differ in at most $m$ positions and from primal complementary slackness we get that whenever $p^S > 0$, $a_i^S(x^S) \ge a_i^S(x^*) - m = (1 - \epsilon) B - m \ge (1 - 2\epsilon) B$, where the last inequality follows from the fact that $B \ge \frac{1}{\epsilon}$. This concludes the proof of the lemma. Proof of Lemma \[lemma:badWitness\] =================================== The following simple inequalities will be helpful. \[obs:ratio\] For $\epsilon, \alpha, \beta \ge 0$, $\frac{1-\alpha \epsilon}{1 + \beta \epsilon} \ge 1 - (\alpha + \beta) \epsilon$ and $\frac{1-\alpha \epsilon}{1 - \beta \epsilon} \le 1 - (\alpha - \beta) \epsilon$. Combining equations , and and union bounding over all terms in the disjunction, we have that $$\begin{aligned} \Pr\left(x^S \textrm{ is bad}\right) \le \sum_{i, w \in \mathcal{W}_i^+} \Pr\left(\operatorname{skewm}(\epsilon,w)\right) + \sum_{i, w \in \mathcal{W}_i^-} \Pr\left(\operatorname{skewp}(\epsilon,w)\right). \end{aligned}$$ Thus, it suffices to show that for all $w \in \mathcal{W}_i^+$ (respectively $w \in \mathcal{W}_i^-$), the event $\operatorname{skewm}(\epsilon, w)$ (resp. $\operatorname{skewp}(\epsilon, w)$) occurs with probability at most $2\exp\left(-\frac{\epsilon^3 B}{33}\right)$. Take $w \in \mathcal{W}_i^+$. By definition of this set, $a_i(w) \ge (1 - \frac{\epsilon}{2})B$, so the event $\operatorname{skewm}(\epsilon,w)$ is contained in the event that $a_i^S(w) \le (1 - \epsilon) a_i(w)/(1 - \frac{\epsilon}{2})$, which is contained in the event $a_i^S(w) \le (1 - \frac{\epsilon}{2}) a_i(w)$. Using Corollary \[cor:multiChernoff\] with $\tau = \epsilon^2 a_i(w) /2$, we obtain that $\Pr(\operatorname{skewm}(\epsilon, w)) \le 2\exp\left(-\frac{\epsilon^3 B}{33}\right)$. Similarly, take $w \in \mathcal{W}_i^-$, such that $a_i(w) \le (1 - \frac{3\epsilon}{2})B$. It is easy to check that the event $\operatorname{skewp}(\epsilon,w)$ is contained in $a_i^S(w) \ge (1 + \frac{\epsilon}{2})a_i(w)$, so using Corollary \[cor:multiChernoff\] with $\tau = \epsilon^2 B/2$ we get that $\Pr(\operatorname{skewm}(\epsilon, w)) \le 2\exp\left(-\frac{\epsilon^3 B}{33}\right)$. This concludes the proof of the lemma. Proof of Lemma \[lemma:chain\] ============================== Fix $j \in [K]$. Consider a set $x \in \mathcal{X}$ and let $p$ be a dual vector such that $x(p) = x$. Let $t'$ be the last index of $C_j$ which belongs to $x|_{C_j}$; this implies that $\pi_{t'} > p a^{t'} = p c^j \|a^{t'}\|_{\infty}$, or alternatively $\frac{\pi_{t'}}{\|a^{t'}\|_{\infty}} > p c^j$. By the ordering of the columns, for all $t \in C_j$ smaller than $t'$ we have $\frac{\pi_t}{\|a^t\|_{\infty}} \ge \frac{\pi_{t'}}{\|a^{t'}\|_{\infty}} > p c^j$ and hence $t \in x|_{C_j}$. By definition of $t'$ it follows that $x|_{C_j} = \{t \in C_j : t \le t'\}$, a prefix of $C_j$; this concludes the proof. Proof of Lemma \[lemma:wWitness\] ================================= We prove that $\mathcal{W}_i^+$ is a witness set for $\mathcal{X}_i^+$; the proof that $\mathcal{W}_i^-$ is a witness set for $\mathcal{X}_i^-$ is analogous. First, we claim that for all $x \in \mathcal{X}_i^+$, there is $x' \in \mathcal{X}$ such that $x' \subseteq x$ and $a_i(x') \in [B, B + m]$. To see this, let $p$ be such that $x = x(p)$. For $\lambda \ge 0$, define $p^\lambda = p + \lambda e_i$, where $e_i$ denotes the $i$th canonical vector. We have that $a_i(x(p^0)) > B$ (since $x(p) \in \mathcal{X}_i^+$) and $a_i(x(p^\infty)) = 0$ (since columns with $a_i^t > 0$ will at have at some point $p^\lambda a^t \ge \pi_t$). Due to the assumption that the input is in general position, whenever $a_i(x(p^\lambda))$ is discontinuous (as a function of $\lambda \ge 0$) the right and the left limits differ by at most $m$. It then follows that there is $\lambda \ge 0$ such that $a_i(x(p^\lambda)) \in [B, B + m]$, and since $x(p^\lambda) \subseteq x$ for all $\lambda \ge 0$ the claim follows. So take a classification $x \in \mathcal{X}_i^+$ and let $x'$ be as above. The fact that $a_i(x') \le B + m$ and the non-negativity of the $a^t$’s imply that there is an $\ell \in L^K$ such that $x' \in \mathcal{B}_i^\ell$. Since $\underline{w}^\ell$ is the unique smallest set in $\mathcal{B}_i^\ell$, clearly $x' \subseteq \underline{w}^\ell$. To show that $\underline{w}^\ell \in \mathcal{W}_i^+$, it suffices to argue that $a_i(\underline{w}^\ell) \ge (1 - \epsilon/2) B$. Since $\underline{w}^\ell, x' \in \mathcal{B}_i^\ell$, for all $j$ such that $\ell_j > 0$ we have $a_i(\underline{w}^\ell|_{C_j}) \ge a_i(x'|_{C_j}) / (1 + \frac{\epsilon}{4})$. Moreover, for $j$ such that $\ell = 0$ we have $a_i(x(p)|_{C_j}) < \frac{\epsilon B}{4K}$. Adding over all $j \in [K]$ gives $$\begin{aligned} a_i(\underline{w}^\ell) \ge \left( \frac{1}{1 + \frac{\epsilon}{4}} \right) \left[ a_i(x(p)) - \sum_{j : \ell_j = 0} a_i(x(p)|_{C_j}) \right] \ge \frac{B}{1 + \frac{\epsilon}{4}} - \frac{\epsilon B}{4} \ge \left(1 - \frac{\epsilon}{2}\right) B, \end{aligned}$$ where the third inequality follows from Observation \[obs:ratio\]. Thus, $\underline{w}^\ell \in \mathcal{W}_i^+$. Since this property holds for all $x \in \mathcal{X}_i^+$, we conclude that $\mathcal{W}_i^+$ is a witness set for $\mathcal{X}_i^+$. Proof of Lemma \[lemma:sizeP\] ============================== Recall the definitions of $P^{{v}}$ (for ${v}\in L^K$) and $P_j^{\ell}$ (for $j \in [m]$, $\ell \in L$). It suffices to prove that at most $(O(\frac{K}{\epsilon} \log \frac{K}{\epsilon}))^m$ of the families $P^{{v}}$’s are non-empty. Since $x(p) \in \mathcal{B}_i^{{v}}$ if and only if for all $j \in [K]$ we have $x(p)|_{C_j} \in \mathcal{B}_{i,j}^{{v}_j}$, it follows that $P^{{v}} = \bigcap_j P_j^{{v}_j}$. Let $\tau^\ell_j$ denote the first index in $C_j$ such that the prefix $\{t \in C_j : t \le \tau^\ell_j\}$ occupies the budget $i$ to an extent in $I_\ell$. Using Lemma \[lemma:chain\] and the fact that the $a^t$’s are non-negative, we get that $\mathcal{B}_{i,j}^\ell$ is the set of all prefixes of $C_j$ which contain $\tau_j^\ell$ but do not contain $\tau_j^{\ell + 1}$. Moreover, notice that the set $x(p)|_{C_j}$ contains $\tau_j^\ell$ if and only if $\pi_{\tau_j^\ell} > p a^{\tau_j^\ell}$. It then follows from these observations we can express the set $P_j^\ell$ using linear inequalities: $P_j^\ell = \{p \in \mathbb{R}^m_+ : \pi_{\tau_j^\ell} > p a^{\tau_j^\ell}, \pi_{\tau_j^{\ell + 1}} \le p a^{\tau_j^{\ell + 1}}\}$. Since $P^{{v}} = \bigcap_j P_j^{{v}_j}$, we have that $P^{{v}}$ is given by the intersection of halfspaces defined by hyperplanes of the form $\pi_{\tau_j^\ell} = p a^{\tau_j^\ell}$ and $p_k = 0$ ($k \in [m]$). So consider the arrangement given by all hyperplanes $\{\pi_{\tau_j^\ell} = p a^{\tau_j^\ell}\}_{j \in [K], \ell \in L}$ and $\{p_i = 0\}_{i = 1}^m$. Given a face $F$ in this arrangement and a set $P^{{v}}$, either $F$ is contained in $P^{{v}}$ or these sets are disjoint. Since the faces of the arrangement cover $\mathbb{R}^m$, it follows that each non-empty $P^{{v}}$ contains at least one of these faces. Notice that the arrangement is defined by $K|L| + m \le O(\frac{Km}{\epsilon} \log \frac{K}{\epsilon})$ hyperplanes, where the last inequality uses the fact that $\log (1 + \frac{\epsilon}{4}) \ge \epsilon \log (1 + \frac{1}{4})$ holds (by concavity) for $\epsilon \in [0,1]$. It is known that an arrangement with $h \ge m$ hyperplanes in $\mathbb{R}^m$ has at most $\left(\frac{e h}{m}\right)^m$ faces (see Section 6.1 of [@matousek] and page 82 of [@matousekNesetril]). Using the conclusion of the previous paragraph, we get that there are at most $(O(\frac{K}{\epsilon} \log \frac{K}{\epsilon}))^m$ non-empty $P^{{v}}$’s and the result follows. Proof of Lemma \[lemma:robust\] =============================== Let LP1 denote the LP with columns $(\pi_t, \tilde{a}^t)$ and right-hand side $(1-\epsilon)B$ and LP2 denote the LP with columns $(\pi_t, a^t)$ and right-hand side $B$. Let $x$ be an $\epsilon$-approximate solution for LP1. Notice that we can upper bound $\|a^t - \tilde{a}^t\|_{\infty}$ as a function of $\|\tilde{a}^t\|_{\infty}$: $$\begin{aligned} \|\tilde{a}^t\|_{\infty} \ge \|a^t\|_{\infty} - \|a^t - \tilde{a}^t\|_{\infty} \ge \frac{m}{\epsilon} \|a^t - \tilde{a}^t\|_{\infty}, \end{aligned}$$ where the first inequality follows from triangle inequality. That is, we have $\|a^t - \tilde{a}^t\|_{\infty} \le \frac{\epsilon}{m} \|\tilde{a}^t\|_{\infty}$. Given this bound, it is easy to see that $x$ is feasible for LP2: $$\begin{aligned} \sum_t a_i^t x_t \le \sum_t (\tilde{a}_i^t + \|a_i^t - \tilde{a}_i^t\|) x_t \le (1 - \epsilon) B + \sum_t \|a^t - \tilde{a}^t\|_{\infty} x_t \le (1 - \epsilon) B + \frac{\epsilon}{m} \sum_t \|\tilde{a}^t\|_{\infty} x_t \le B, \end{aligned}$$ where the last inequality uses the fact that $\sum_t \|\tilde{a}^t\|_{\infty} x_t \le \|\tilde{a}^t\|_1 x_t \le mB$, since $x$ is a feasible solution and the $\tilde{a}^t$’s are non-negative. In order to show that $x$ is a $2\epsilon$-approximate solution for LP2, it suffices to show that the optimum of LP1 is at least $1/(1+\epsilon)$ times the optimum of the LP2, since then $x$ will be within a factor of $(1-\epsilon)/(1+\epsilon) \ge (1 - 2\epsilon)$ the optimum of LP2. So let $x^*$ be an optimal solution for LP2. Using the same argument as before, it is easy to see that $x^*/(1+\epsilon)$ is feasible for LP1; this concludes the proof of the lemma. Proof of Lemma \[lemma:sDeltaOtp\] ================================== The proof uses the same ideas used in the analysis of [<span style="font-variant:small-caps;">OTP</span>]{}, although some definitions need to be changed slightly. Recall that $S = \{\sigma(1), \sigma(2), \ldots, \sigma(s)\}$, $T = \{\sigma(s + 1), \sigma(s + 2), \ldots, \sigma(2s)\}$ and $U = S \cup T$. Again we use $p^S$ to denote the dual vector used by $(s, \delta)$-[<span style="font-variant:small-caps;">OTP</span>]{}for its classification, and set $x^S = x(p^S)$. With slight abuse in the notation, we often see $x^S$ as a (possibly infeasible) solution for $(2s,1)$-LP, which means that we truncate the vector $x^S$ to the first $2s$ coordinates $x^S_{\sigma(1)}, \ldots, x^S_{\sigma(2s)}$. As before, we focus on proving the following lemma; the proof that this lemma implies Lemma \[lemma:sDeltaOtp\] is presented at the end of this section. \[lemma:goodSDOTP\] Suppose that there are $K \ge m$ 1-dim subspaces of $\mathbb{R}^m$ containing the columns $a^t$’s. Fix an integer $s$ and a real number $\delta \in (0,1/10)$ such that $\frac{\delta^2 s B}{n} \ge \Omega(m \ln \frac{K}{\delta})$. Then with probability at least $(1 - \delta^2)$, $x^S$ satisfies $a_i^T(x^S) \le B$ for all $i \in [m]$ and has value $\pi_U x^S \ge (1 - 3\delta) {\ensuremath{\textrm{OPT}}}(2s)$. In a given scenario, we now say that $x^S$ is *bad* if $a_i^T(s^S) > B$ for some $i \in [m]$ or if $\pi_U x^S (1 - 3\delta) {\ensuremath{\textrm{OPT}}}(2s)$. In this scenario, now a classification $x \in \mathcal{X}$ can be *badly learned for budget $i$ due to infeasibility* if $a_i^S(x) \le (1 - \delta) B$ and $a_i^T(x) > B$; $x$ can be *badly learned for budget $i$ due to value* if $a_i^S(x) \ge (1 - 2\delta)B$ and $a_i^U(x) < (1 - 3\delta)B$. Then $x$ can be *badly learned for budget $i$* if it falls into any of the above cases. The following is the appropriate modification of Lemma \[lemma:approximateCS\] for our current setting, and can be proved exactly in the same way. \[lemma:approximateCS2\] Consider a scenario where $x^S$ satisfies the following: (i) for all $i \in [m]$, $a_i^T(x^S) \le B$ and (ii) for all $i \in [m]$ with $p^S_i > 0$, $a_i^U(x^S) \ge (1 - 3 \delta) B$. Then $x^S$ is good. Due to our definitions, this lemma implies that inequality still hold. #### Witness sets. In the analysis of [<span style="font-variant:small-caps;">OTP</span>]{}, each $x \in \mathcal{X}$ could be badly learned for budget $i$ due to either infeasibility or (exclusively) due to value, which motivated the definitions of $\mathcal{X}_i^+$ and $\mathcal{X}_i^-$. Now the same $x$ can be badly learned for budget $i$ due to both conditions. Therefore, we introduce two different partition of $\mathcal{X}$, which tells *why* a classification is unlikely to be badly learned due to the appropriate condition. That is, we define $\mathcal{X}_i^+ = \{x \in \mathcal{X} : a_i(x) > (1 - \delta) B + \frac{\delta B}{2}\}$ and $\mathcal{Y}_i^+ = \{x \in \mathcal{X} : a_i(x) \le (1 - \delta) B + \frac{\delta B}{2}\}$ as the partition associated to the infeasibility condition and $\mathcal{X}_i^- = \{x \in \mathcal{X} : a_i(x) < (1 - 2\delta) B - \frac{\delta B}{2}\}$ and $\mathcal{Y}_i^- = \{x \in \mathcal{X} : a_i(x) \ge (1 - 2\delta) B - \frac{\delta B}{2}\}$ as the partition associated to the value condition. For example, $\mathcal{X}_i^-$ is the set of classifications which are unlikely to be infeasible because of a small $a_i(.)$ value. Also, note that these classifications are all based on the total budget occupation rather than on the budget occupation in the first $2s$ columns only. Given this more refined tagging of elements in $\mathcal{X}$, we also need to redefine witness sets. We say that $(\mathcal{W}_i^+, \mathcal{W}_i^-, \mathcal{Z}_i^+, \mathcal{Z}_i^-)$ are *witness sets* for $(\mathcal{X}_i^+, \mathcal{X}_i^-, \mathcal{Y}_i^-, \mathcal{Y}_i^+)$ respectively if they satisfy the following: $$\begin{aligned} w \in \mathcal{W}_i^+ \Rightarrow a_i(w) \ge (1 - \delta) B + \frac{\delta B}{4}, x \in \mathcal{X}_i^+ \Rightarrow \exists w \in \mathcal{W}_i^+ : w \subseteq x \\ w \in \mathcal{Z}_i^+ \Rightarrow a_i(w) \ge (1 - 2\delta) B - \frac{3\delta B}{4}, x \in \mathcal{Y}_i^- \Rightarrow \exists w \in \mathcal{Z}_i^+ : w \subseteq x \\ w \in \mathcal{W}_i^- \Rightarrow a_i(w) \le (1 - 2\delta) B - \frac{\delta B}{4}, x \in \mathcal{X}_i^- \Rightarrow \exists w \in \mathcal{W}_i^+ : x \subseteq w \\ w \in \mathcal{Z}_i^- \Rightarrow a_i(w) \le (1 - \delta) B + \frac{3 \delta B}{4}, x \in \mathcal{Y}_i^+ \Rightarrow \exists w \in \mathcal{W}_i^+ : x \subseteq w \, . \end{aligned}$$ Again to simplify the notation, given a set $x$ we define $\operatorname{skewm}_i^S(\delta, x)$ to be the event that $a^S_i(x) \le (1 - \delta) B$, $\operatorname{skewp}^S_i(\delta, x)$ to be the event that $a^S_i(x) \ge (1 - \delta) B$ and similarly replacing the set $S$ by the sets $T$ and $U$. The following expression, which is the analogous to -, establishes the connection between the events where classifications can be badly learned and witness sets: $$\begin{aligned} \bigvee_{x \in \mathcal{X}} \{\textrm{$x$ can be badly learned for budget $i$}\} \subseteq & \left(\bigvee_{w \in \mathcal{W}_i^+} \operatorname{skewm}^S(\delta, w) \right) \vee \left( \bigvee_{w \in \mathcal{Z}_i^+} \operatorname{skewm}^{U}(3\delta, w) \right) \notag\\ &\vee \left( \bigvee_{w \in \mathcal{W}_i^-} \operatorname{skewp}^S(2\delta, w) \right) \vee \left( \bigvee_{w \in \mathcal{Z}_i^-} \operatorname{skewp}^{T}(0, w)\right). \label{eq:witnessSDOTP} \end{aligned}$$ To see that this expression holds, take $x \in \mathcal{X}$. Suppose that $x \in \mathcal{X}_i^+$ and let $w \in \mathcal{W}_i^+$ be contained in $x$. Then the event $\{\textrm{$x$ can be badly learned for budget $i$ due to infeasibility}\}$ is contained in $\operatorname{skewm}^S(\delta, w)$. Similarly, if $x \in \mathcal{Y}_i^+$ let $w \in \mathcal{Z}_i^-$ contain $x$; then the event $\{\textrm{$x$ can be badly learned for budget $i$ due to infeasibility}\}$ is contained in $\operatorname{skewm}^T(0, w)$. The reasoning for the event $\{\textrm{$x$ can be badly learned for budget $i$ due to value}\}$ is similar. The following is analogous to Lemma \[lemma:badWitness\]. \[lemma:badWitnessOTP\] Suppose that, for all $i \in [m]$, there are witness sets for $(\mathcal{X}_i^+, \mathcal{X}_i^-, \mathcal{Y}_i^+, \mathcal{Y}_i^-)$ of size at most $M$. Then $\Pr(x^S \textrm{ is bad }) \le 8mM \exp\left(-\frac{\delta^2 s B}{136 n}\right)$. #### Good witness sets. We now construct witness sets of size at most $(O(\frac{K}{\delta} \log \frac{K}{\delta}))^m$, so Lemma \[lemma:goodSDOTP\] will follow directly from Lemma \[lemma:badWitnessOTP\]. The development mirrors that of Section \[sec:goodWitness\]. Let $C_1, C_2, \ldots, C_K$ be a partition of the index set $[n]$ such that for all $j$, the columns $\{a^t\}_{t \in C_j}$ belong to the same 1-dimensional subspace. Cover the interval $[0, B+m]$ with intervals $\{I_\ell\}_{\ell \in L}$, where $I_0 = [0, \frac{\delta B}{8 K})$ and $I_\ell = [\frac{\delta B}{8K} (1 + \frac{\delta}{8})^{\ell - 1}, \frac{\delta B}{8K} (1 + \frac{\delta}{8})^\ell)$ for $\ell > 0$ and $L = \{0, \ldots, \lceil \log_{1 + \delta/8} \frac{16K}{\delta} \rceil + 1\}$. Define $\mathcal{B}_{i,j}^\ell$ as the set of classifications $x \in \mathcal{X}|_{C_j}$ whose occupation $a_i(x)$ lies in the interval $I_\ell$. Finally, for $\ell \in L^K$, define the family of boxes $\mathcal{B}_i^\ell = \prod_j \mathcal{B}_{i,j}^{\ell_j}$. Given $\ell \in L$, let $\underline{w}^\ell(j)$ be the smallest set in $\mathcal{X}|_{C_j}$ which has $a_i(\underline{w}^\ell(j)) \in I_\ell$ and for $\ell \in L^K$ define the set $\underline{w}^{\ell}$ as the union of the sets $\underline{w}^{\ell_j}(j)$’s (or equivalently, as the concatenation of the vectors $\underline{w}^{\ell_j}(j)$’s). Similarly, for $\ell \in L$ let $\overline{w}^\ell(j)$ be the largest set in $\mathcal{X}|_{C_j}$ which has $a_i(\overline{w}^\ell(j)) \in I_\ell$ and for $\ell \in L^K$ define the set $\overline{w}^\ell$ as the union of the sets $\overline{w}^{\ell_j}(j)$’s. Now we construct the witness sets as before. Set $\mathcal{W}_i^+ = \{\underline{w}^\ell : a_i(\underline{w}^\ell) \ge (1 - \delta)B + \frac{\delta B}{4}, \mathcal{B}_i^\ell \cap \mathcal{X} \neq \emptyset\}$, set $\mathcal{Z}_i^+ = \{\underline{w}^\ell : a_i(\underline{w}^\ell) \ge (1 - 2\delta)B - \frac{3\delta B}{4}, \mathcal{B}_i^\ell \cap \mathcal{X} \neq \emptyset\}$, set $\mathcal{W}_i^- = \{\overline{w}^\ell : a_i(\overline{w}^\ell) \le (1 - 2\delta)B - \frac{\delta B}{4}, \mathcal{B}_i^\ell \cap \mathcal{X} \neq \emptyset\}$ and finally set $\mathcal{Z}_i^- = \{\overline{w}^\ell : a_i(\overline{w}^\ell) \le (1 - \delta)B + \frac{3\delta B}{4}, \mathcal{B}_i^\ell \cap \mathcal{X} \neq \emptyset\}$. Following the same steps as in the proof of Lemma \[lemma:wWitness\], one can check that $(\mathcal{W}_i^+, \mathcal{W}_i^-, \mathcal{Z}_i^+, \mathcal{Z}_i^-)$ are *witness sets* for $(\mathcal{X}_i^+, \mathcal{X}_i^-, \mathcal{Y}_i^+, \mathcal{Y}_i^-)$. Moreover, the proof of Lemma \[lemma:sizeP\] can be used to show that, for a fixed $i \in [m]$, at most $(e \frac{K}{\delta} \log \frac{K}{\delta} )^m$ of the $\mathcal{B}_i^\ell$’s contain an element of $\mathcal{X}$, which then imposes the same upper bound on the size of the witness sets. This concludes the proof of Lemma \[lemma:goodSDOTP\]. [Lemma \[lemma:sDeltaOtp\]]{} Let $x$ be the solution returned by $(s, \delta)$-[<span style="font-variant:small-caps;">OTP</span>]{}and let $\mathcal{E}$ denote the event that $x^S$ is good. For any scenario in $\mathcal{E}$, we have $x_{\sigma(t)} = x^S_{\sigma(t)}$ for all $t = s+1, s+2, \ldots, 2s$. Therefore, we get that $$\begin{aligned} {\mathbb{E}}\left[\sum_{t = 1}^{2s} \pi_{\sigma(t)} x_{\sigma(t)}\right] &\ge {\mathbb{E}}\left[\sum_{t = 1}^{2s} \pi_{\sigma(t)} x_{\sigma(t)} \mid \mathcal{E} \right] \Pr(\mathcal{E}) \notag \\ & \ge {\mathbb{E}}\left[\sum_{t = 1}^{2s} \pi_{\sigma(t)} x_{\sigma(t)}^S \mid \mathcal{E}\right] \Pr(\mathcal{E}) - {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(s) \mid \mathcal{E}] \Pr(\mathcal{E}) \notag \\ &\ge {\mathbb{E}}\left[\sum_{t = 1}^{2s} \pi_{\sigma(t)} x_{\sigma(t)}^S \mid \mathcal{E}\right] \Pr(\mathcal{E}) - {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(s)]. \label{eq:sDeltaOpt1} \end{aligned}$$ To lower bound the first term in the right hand side we use again the definition of $\mathcal{E}$: $$\begin{aligned} {\mathbb{E}}\left[\sum_{t = 1}^{2s} \pi_{\sigma(t)} x_{\sigma(t)}^S \mid \mathcal{E} \right] \ge (1 - 3 \delta) {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(2s) \mid \mathcal{E}] \Pr(\mathcal{E}) \end{aligned}$$ and $$\begin{aligned} {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(2s)] = {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(2s) \mid \mathcal{E}] \Pr(\mathcal{E}) + {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(2s) \mid \overline{\mathcal{E}}] \Pr(\overline{\mathcal{E}}) \le {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(2s) \mid \mathcal{E}] \Pr(\mathcal{E}) + \delta^2 {\ensuremath{\textrm{OPT}}}, \end{aligned}$$ where the last inequality uses Lemma \[lemma:goodSDOTP\]. Combining the previous two inequalities give that ${\mathbb{E}}\left[\sum_{t = 1}^{2s} \pi_{\sigma(t)} x_{\sigma(t)}^S \mid \mathcal{E}\right] \ge (1 - 3 \delta) {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(2s)] - \delta^2 {\ensuremath{\textrm{OPT}}}$, and the result follows from equation . Proof of Theorem \[thm:expValueDPA\] ==================================== Let LP1 denote the LP with columns $(\pi_t, \tilde{a}^t)$ and right-hand side $\tilde{B} = (1-\epsilon)B$ and LP2 denote the LP with columns $(\pi_t, a^t)$ and right-hand side $B$. We show that Robust [<span style="font-variant:small-caps;">DPA</span>]{}returns a $(1 - 21.5\epsilon)$-approximation for LP1, and the theorem will follow from Lemma \[lemma:robust\]. First we show that the returned solution $x$ is feasible for LP1. By definition of the algorithm, $a_j(x^i) \le \epsilon 2^i \tilde{B}$ for all $i,j$. By linearity, $a_j(x) = \sum_i a_j(x^i) \le \epsilon \tilde{B} \sum_{i = 0}^{\log(1/\epsilon) - 1} 2^i \le \tilde{B}$. In order to verify the value of the returned solution, we first show that $\frac{\delta^2 s B}{n} \ge \Omega(m \ln \frac{K}{\delta})$ in every call to $(s,\delta)$-[<span style="font-variant:small-caps;">OTP</span>]{}made by Robust [<span style="font-variant:small-caps;">DPA</span>]{}. As in Section \[sec:robustOTP\], the columns $\tilde{a}^t$’s belong to at most $K = O(\frac{m}{\epsilon})^m$ 1-dim subspaces. Since $B \ge \Omega(\frac{m^2}{\epsilon^2} \ln \frac{m}{\epsilon})$, we have that for each $i = 0, \ldots, \log(1/\epsilon)-1$ setting $s = \epsilon 2^i n$ and $\delta = \sqrt{\epsilon/2^i}$ satisfies the expression $\frac{\delta^2 s B}{n} \ge \Omega(m \ln \frac{K}{\delta})$. Then applying Lemma \[lemma:sDeltaOtp\] we get that for all $i = 0, \ldots, \log(1/\epsilon)-1$, ${\mathbb{E}}[\pi x^i] \ge (1 - 3 \sqrt{\frac{\epsilon}{2^i}}) {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(\epsilon 2^{i+1} n)] - {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(\epsilon 2^i n)] - \frac{\epsilon {\ensuremath{\textrm{OPT}}}}{2^i}$. By linearity of the objective value and of expectations $$\begin{aligned} {\mathbb{E}}[\pi x] = \sum_i {\mathbb{E}}[\pi x^i] \ge - {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(\epsilon n)] - \sum_{i = 0}^{\log(1/\epsilon) - 2} \left(3 \sqrt{\frac{\epsilon}{2^i}}\right) {\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(\epsilon n 2^{i + 1})] + (1 - 3 \sqrt{2} \epsilon - \epsilon) {\ensuremath{\textrm{OPT}}}. \end{aligned}$$ Lemma 2.4 of [@agrawal] states that ${\mathbb{E}}[{\ensuremath{\textrm{OPT}}}(s)] \le \frac{s}{n} {\ensuremath{\textrm{OPT}}}$ for all $s \ge 0$. Employing this observation, we get $$\begin{aligned} {\mathbb{E}}[\pi x] \ge {\ensuremath{\textrm{OPT}}}- \epsilon {\ensuremath{\textrm{OPT}}}\left[3 \sqrt{2} + 2 + 3 \sqrt{\epsilon} \sum_{i = 0}^{\log(1/\epsilon) - 2} 2^{i/2 + 1} \right]. \end{aligned}$$ Since the summation in the expression can be upper bounded by $\frac{2 \sqrt{2}^{\log(1/\epsilon)}}{\sqrt{2} - 1} \le \frac{5}{\sqrt{\epsilon}}$, we get that ${\mathbb{E}}[\tilde{\pi} x] \ge (1 - 21.5 \epsilon) {\ensuremath{\textrm{OPT}}}$. This concludes the proof of the theorem. [^1]: Actually knowing $n$ up to $(1 \pm \epsilon)$ factor is enough. This assumption is required to allow algorithms with non-trivial competitive ratio [@DevanurHayes09]. [^2]: To simplify the exposition, we assume that $\epsilon n$ is an integer. [^3]: Notice that this ratio is well-defined since by assumption $a^t \neq 0$ for all $t \in [n]$.

--- abstract: | The purpose of this paper is to study the weak solutions of the fractional elliptic problem $$\label{000} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+\epsilon g(u)=k\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad &{\rm in}\quad\ \ \bar\Omega,\\[3mm] \phantom{(-\Delta)^\alpha +\epsilon g(u)} u=0\quad &{\rm in}\quad\ \ \bar\Omega^c, \end{array}$$ where $k>0$, $\epsilon=1$ or $-1$, $(-\Delta)^\alpha$ with $\alpha\in(0,1)$ is the fractional Laplacian defined in the principle value sense, $\Omega$ is a bounded $C^2$ open set in ${\mathbb{R}}^N$ with $N\ge 2$, $\nu$ is a bounded Radon measure supported in $\partial\Omega$ and $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ is defined in the distribution sense, i.e. $$\langle\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha},\zeta\rangle=\int_{\partial\Omega}\frac{\partial^\alpha\zeta(x)}{\partial \vec{n}_x^\alpha}d\nu(x), \qquad \forall\zeta\in C^\alpha({\mathbb{R}}^N),$$ here $\vec{n}_x$ denotes the unit inward normal vector at $x\in\partial\Omega$. In this paper, we prove that (\[000\]) with $\epsilon=1$ admits a unique weak solution when $g$ is a continuous nondecreasing function satisfying $$\int_1^\infty (g(s)-g(-s))s^{-1-\frac{N+\alpha}{N-\alpha}}ds<+\infty.$$ Our interest then is to analyse the properties of weak solution when $\nu=\delta_{x_0}$ with $x_0\in\partial\Omega$, including the asymptotic behavior near $x_0$ and the limit of weak solutions as $k\to+\infty$. Furthermore, we show the optimality of the critical value $\frac{N+\alpha}{N-\alpha}$ in a certain sense, by proving the non-existence of weak solutions when $g(s)=s^{\frac{N+\alpha}{N-\alpha}}$. The final part of this article is devoted to the study of existence for positive weak solutions to (\[000\]) when $\epsilon=-1$ and $\nu$ is a bounded nonnegative Radon measure supported in $\partial\Omega$. We employ the Schauder’s fixed point theorem to obtain positive solution under the hypothesis that $g$ is a continuous function satisfying $$\int_1^\infty g(s)s^{-1-\frac{N+\alpha}{N-\alpha}}ds<+\infty.$$ --- [**Existence, Non-existence, Uniqueness of solutions\ for semilinear elliptic equations involving\ measures concentrated on boundary**]{} Huyuan Chen[^1] Hichem Hajaiej[^2] [: Fractional Laplacian; Radon measure; Dirac mass; Green kernel; Schauder’s fixed point theorem.]{} [: 35R11, 35J61, 35R06. ]{} Introduction {#sec:intro} ============ Motivation ---------- In 1991, a fundamental contribution of semilinear elliptic equations involving measures as boundary data is due to Gmira and Véron in [@GV], which studied the weak solutions for $$\label{1.1.1} \arraycolsep=1pt \begin{array}{lll} -\Delta u+g(u)=0\quad &{\rm in}\quad \Omega, \\[2mm]\phantom{-----} u=\mu\quad&{\rm on}\quad \partial\Omega, \end{array}$$ where $\Omega$ is a bounded $C^2$ domain in ${\mathbb{R}}^N$ and $\mu$ is a bounded Radon measure defined in $\partial\Omega$. A function $u$ is said to be a weak solution of (\[1.1.1\]) [*if $u\in L^1(\Omega)$, $g(u)\in L^1(\Omega,\rho_{\partial\Omega} dx)$ and $$\label{1.1.1.0} \int_\Omega [u(-\Delta)\xi+ g(u)\xi]dx=\int_{\partial\Omega}\frac{\partial\xi(x)}{\partial\vec{n}_x}d\mu(x),\quad \forall\xi\in C^{1.1}_0(\Omega),$$ where $\rho_{\partial\Omega}(x)={\rm dist}(x,\partial\Omega)$ and $\vec{n}_x$ denotes the unit inward normal vector at point $x$.*]{} Gmira and V´eron proved that problem (\[1.1.1\]) admits a unique weak solution when $g$ is a continuous and nondecreasing function satisfying $$\label{14.04} \int_1^\infty [g(s)-g(-s)]s^{-1-\frac{N+1}{N-1}}ds<+\infty.$$ Furthermore, the weak solution of (\[1.1.1\]) is approached by the classical solutions of (\[1.1.1\]) replacing $\mu$ by a sequence of regular functions $\{\mu_n\}$, which converge to $\mu$ in the distribution sense. Then this subject has been vastly expanded in recent works, see the papers of Marcus and Véron [@MV1; @MV2; @MV3; @MV4], Bidaut-Véron and Vivier [@BV] and reference therein. A very challenging question consists in studying the analogue elliptic problem involving fractional Laplacian defined by $$(-\Delta)^\alpha u(x)=\lim_{\varepsilon\to0^+} (-\Delta)_\varepsilon^\alpha u(x),$$ where $$(-\Delta)_\varepsilon^\alpha u(x)=-\int_{{\mathbb{R}}^N\setminus B_\varepsilon(x)}\frac{ u(z)- u(x)}{|z-x|^{N+2\alpha}} dz$$ for $\varepsilon>0$. The main difficulty comes from how to define the boundary type data. Given a Radon measure $\mu$ defined in $\partial\Omega$, it is ill-posed that $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+g(u)=0\quad &{\rm in}\quad \Omega, \\[2mm]\phantom{------\ } u=\mu\quad&{\rm on}\quad \partial\Omega, \\[2mm]\phantom{------\ } u=0\quad&{\rm in}\quad \bar\Omega^c. \end{array}$$ Indeed, let $\{\mu_n\}$ be a sequence of regular functions defined in $\partial\Omega$ converging to the measure $\mu$ and a surprising result is that there is just zero solution for $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+g(u)=0\quad &{\rm in}\quad \Omega, \\[2mm]\phantom{------\ } u=\mu_n\quad&{\rm on}\quad \partial\Omega, \\[2mm]\phantom{------\ } u=0\quad&{\rm in}\quad \bar\Omega^c, \end{array}$$ which is in sharp contrast with Laplacian case, where (\[1.1.1\]) replacing $\mu$ by $\mu_n$ admits a unique nontrivial solution. On the other hand, it is also not proper to pose $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+g(u)=0\quad &{\rm in}\quad \Omega, \\[2mm]\phantom{------\ } u=\mu\quad&{\rm in}\quad \Omega^c \end{array}$$ with $\mu$ being a Radon measure in $\Omega^c$ concentrated on $\partial\Omega$. In fact, letting functions $\{\mu_n\}\subset C^1_0(\Omega^c)$ converging to $\mu$, the solution $u_n$ of $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+g(u)=0\quad &{\rm in}\quad \Omega, \\[2mm]\phantom{------\ } u=\mu_n\quad&{\rm in}\quad \Omega^c, \end{array}$$ is equivalent to the solution of $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+g(u)=G_{\mu_n}\quad &{\rm in}\quad \Omega, \\[2mm]\phantom{------\ } u=0\quad&{\rm in}\quad \Omega^c, \end{array}$$ where $$G_{\mu_n}(x)=\int_{\Omega^c}\frac{\mu_n(y) }{|x-y|^{N+2\alpha}}dy,\qquad x\in\Omega,$$ see [@CFQ]. It could be seen that $$\int_\Omega [u_n(-\Delta)^\alpha\xi+ g(u_n)\xi]dx=\int_{\Omega}G_{\mu_n}\xi dx,\quad \forall\xi\in C^2_0(\Omega),$$ Then the limit of $\{u_n\}$ as $n\to\infty$ wouldn’t be a weak solution as we desired, similar to (\[1.1.1.0\]). Therefore, a totally different point of view has to be found to propose the fractional elliptic problem involving measure concentrated on boundary. Our idea is inspired by the study of elliptic equations with fractional Laplacian and Radon measure inside of $\Omega$ in [@CV1], where the authors considered the equations $$\label{1.22} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+h(u)=\nu\quad & \rm{in}\quad\Omega,\\[2mm] \phantom{ (-\Delta)^\alpha +h(u)} u=0\quad & \rm{in}\quad \Omega^c \end{array}$$ for $\nu\in\mathfrak{M}(\Omega,\rho_{\partial\Omega}^\beta)$ with $ \beta\in[0,\alpha]$ the space of Radon measure $\nu$ in $\Omega$ satisfying $$\int_\Omega\rho_{\partial\Omega}^\beta(x) d|\nu(x)|<+\infty.$$ A function $u$ is said to be a weak solution of (\[1.22\]), if $u\in L^1(\Omega)$, $h(u)\in L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)$ and $$\int_\Omega [u(-\Delta)^\alpha\xi+ h(u)\xi]dx=\int_{\Omega}\xi(x)d\nu(x),\qquad \forall\xi\in \mathbb{X}_\alpha,$$ where $\mathbb{X}_{\alpha}\subset C({\mathbb{R}}^N)$ with $\alpha\in(0,1)$ denotes the space of functions $\xi$ satisfying: - 1. ${\rm supp}(\xi)\subset\bar\Omega$; <!-- --> 1. $(-\Delta)^\alpha\xi(x)$ exists for all $x\in \Omega$ and $|(-\Delta)^\alpha\xi(x)|\leq C$ for some $C>0$; <!-- --> 1. there exist $\varphi\in L^1(\Omega,\rho^\alpha_{\partial\Omega} dx)$ and $\varepsilon_0>0$ such that $|(-\Delta)_\varepsilon^\alpha\xi|\le \varphi$ a.e. in $\Omega$, for all $\varepsilon\in(0,\varepsilon_0]$. A unique weak solution of (\[1.22\]) is obtained when the function $h$ is continuous, nondecreasing and satisfies $$\int_1^{\infty}(h(s)-h(-s))s^{-1-k_{\alpha,\beta}}ds<+\infty,$$ where $$k_{\alpha,\beta}=\left\{ \arraycolsep=1pt \begin{array}{lll} \frac{N}{N-2\alpha},\quad &{\rm if}\quad \beta\in[0,\frac{N-2\alpha}N\alpha],\\[2mm] \frac{N+\alpha}{N-2\alpha+\beta},\qquad &{\rm if}\quad \beta\in(\frac{N-2\alpha}N\alpha,\alpha]. \end{array} \right.$$ Motivated by the above results, we may approximate $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ by a sequence measures defined in $\Omega$ and consider the limit of corresponding weak solutions. To this end, for a bounded Radon measure defined in $\bar\Omega$ with support in $\partial\Omega$, we observe that $$\langle\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha},\xi\rangle=\int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial \vec{n}_x^\alpha}d\nu(x), \quad \xi\in \mathbb{X}_\alpha,$$ and $$\frac{\partial^\alpha\xi(x)}{\partial \vec{n}_x^\alpha}=\lim_{s\to0^+}\frac{\xi(x+s\vec{n}_x)-\xi(x)}{s^\alpha} =\lim_{s\to0^+}\xi(x+s\vec{n}_x)s^{-\alpha},$$ so $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ could be approximated by measures $\{t^{-\alpha}\nu_t\}$ with support in $\{x\in\Omega: \rho_{\partial\Omega}(x)=t\}$ generated by $\nu$, see Section 2 for details. Then we consider the limit of weak solutions as $t\to0^+$ for the problem: $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+g(u)=t^{-\alpha}\nu_t\quad &{\rm in}\quad \Omega, \\[2mm]\phantom{------\ } u=0\quad&{\rm in}\quad \Omega^c. \end{array}$$ Here the limit of these weak solutions (if it exists) is called a weak solution of the following fractional elliptic problem with measure concentrated on boundary $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+g(u)=\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad &{\rm in}\quad\ \ \bar\Omega,\\[3mm] \phantom{------\ } u=0\quad &{\rm in}\quad\ \ \bar\Omega^c. \end{array}$$ This will be our main focus in this paper. Statement of our problem and main results ----------------------------------------- Let $\alpha\in(0,1)$, $g:{\mathbb{R}}\to{\mathbb{R}}$ be a continuous function, $\Omega$ be a bounded smooth domain in ${\mathbb{R}}^N$ with $N\ge 2$ and denote by $\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$ the bounded Radon measure in $\bar\Omega$ with the support in $\partial\Omega$. Our purpose in this article is to investigate the existence, non-existence and uniqueness of weak solutions to semilinear fractional elliptic problem $$\label{eq 1.1} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+\epsilon g(u)=k\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad &{\rm in}\quad\ \ \bar\Omega,\\[3mm] \phantom{------\ \ } u=0\quad &{\rm in}\quad\ \ \bar\Omega^c, \end{array}$$ where $\epsilon=1$ or $-1$, $k>0$, $(-\Delta)^\alpha$ is the fractional Laplacian and denote $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ with $\nu\in\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$ by $$\langle\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha},\xi\rangle=\int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial \vec{n}_x^\alpha}d\nu(x), \qquad \xi\in \mathbb{X}_\alpha,$$ with $\vec{n}_x$ being the unit inward normal vector at $x$. We call $g$ the absorption nonlinearity if $\epsilon=1$, otherwise it is called as source nonlinearity. Before starting our main theorems we make precise the notion of weak solution used in this article. \[weak solution GV\] We say that $u$ is a weak solution of (\[eq 1.1\]), if $u\in L^1(\Omega)$, $g(u)\in L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)$ and $$\int_\Omega [u(-\Delta)^\alpha\xi+\epsilon g(u)\xi]dx=k\int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial\vec{n}_x^\alpha}d\nu(x),\qquad \forall\xi\in \mathbb{X}_\alpha.$$ We notice that $\mathbb{X}_\alpha\supset C_0^2(\Omega)$ is the test functions space when we study semilinear fractional elliptic equations involving measures, which plays the same role as $C^{1,1}_0(\Omega)$ for dealing with second order elliptic equations with measures, see [@CV1; @CV2; @CV3; @CY]. Moreover, it follows from [@RS Proposition 1.1] that $\xi $ is $C^\alpha$ ($\alpha$-Hölder continuous) in ${\mathbb{R}}^N$ if $\xi\in \mathbb{X}_\alpha$. Denote by $G_\alpha$ the Green kernel of $(-\Delta)^\alpha$ in $\Omega\times\Omega$ and by $\mathbb{G}_\alpha[\cdot]$ the Green operator defined as $$\mathbb{G}_\alpha[\frac{\partial^\alpha\nu}{\partial\vec{n}^\alpha}](x)=\lim_{t\to0^+}\int_{\partial\Omega} G_\alpha(x,y+t\vec{n}_y)t^{-\alpha}d\nu(y).$$ Now we are ready to state our first result for problem (\[eq 1.1\]). \[teo 1\] Assume that $\epsilon=1$, $k>0$, $\nu\in\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$ and $g$ is a continuous nondecreasing function satisfying $g(0)\ge0$ and $$\label{g1} \int_1^\infty [g(s)-g(-s)]s^{-1-\frac{N+\alpha}{N-\alpha}}ds<+\infty.$$ Then $(i)$ problem (\[eq 1.1\]) admits a unique weak solution $u_\nu$; $(ii)$ the mapping $\nu\to u_\nu$ is increasing and $$\label{1.33} -k\mathbb{G}_\alpha[\frac{\partial^\alpha\nu_-}{\partial\vec{n}^\alpha}](x)\le u_\nu(x)\le k\mathbb{G}_\alpha[\frac{\partial^\alpha\nu_+}{\partial\vec{n}^\alpha}](x),\qquad x\in\Omega,$$ where $\nu_+,\nu_-$ are the positive and negative decomposition of $\nu$ such that $\nu=\nu_+-\nu_-$; $(iii)$ if we assume additionally that $g$ is $C^{\beta}$ locally in ${\mathbb{R}}$ with $\beta>0$, then $u_\nu$ is a classical solution of $$\label{1.1} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+g(u)=0\quad & {\rm in}\quad \Omega,\\[2mm] \phantom{------\ } u=0\quad & {\rm in}\quad \Omega^c\setminus {\rm supp}(\nu). \end{array}$$ We remark that\ $(i)$ the second equality in (\[1.1\]) is understood in the sense that $u=0$ in $\Omega^c\setminus {\rm supp}(\nu)$ and $u$ is continuous at every point in $\partial\Omega\setminus {\rm supp}(\nu)$;\ $(ii)$ the uniqueness requires the nondecreasing assumption on nonlinearity $g$, while the existence also holds without the nondecreasing assumption on $g$;\ $(iii)$ (\[g1\]) is called as integral subcritical condition with critical value $\frac{N+\alpha}{N-\alpha}$, similar integral subcritical conditions see the references [@BV; @CV1; @CV2; @V]. Applied Theorem \[teo 1\] when $\nu=\delta_{x_0}$ with $x_0\in\partial\Omega$, problem (\[eq 1.1\]) admits a unique nonnegative weak solution when $g$ satisfies the hypotheses in Theorem \[teo 1\]. Our second goal is to study the further properties of the weak solution. \[teo 2\] Assume that $\epsilon=1$, $k>0$, $\nu=\delta_{x_0}$ with $x_0\in\partial\Omega$, $g$ is a nondecreasing function in $C^\beta$ locally in ${\mathbb{R}}$ with $\beta>0$ satisfying $g(0)\ge0$ and (\[g1\]). Let $u_k$ be the weak solution of (\[eq 1.1\]), then $(i)$ $$\label{b k} \lim_{t\to0^+} \frac{u_k(x_0+t\vec{n}_{x_0})}{ \mathbb{G}_\alpha[\frac{\partial^\alpha\delta_{x_0}}{\partial\vec{n}^\alpha}](x_0+t\vec{n}_{x_0})}=k.$$ $(ii)$ if additionally $g(s)=s^p$ with $p\in(1+\frac{2\alpha}{N},\frac{N+\alpha}{N-\alpha})$, then the limit of $\{u_k\}$ as $k\to\infty$ exists in ${\mathbb{R}}^N\setminus\{x_0\}$, denoting $u_\infty$. Moreover, $u_\infty$ is a classical solution of $$\label{1.3} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+u^p=0\quad & {\rm in}\quad \Omega,\\[2mm] \phantom{(-\Delta)^\alpha +u^p} u=0\quad & {\rm in}\quad \Omega^c\setminus \{x_0\} \end{array}$$ and satisfies $$\label{b k 01} c_1 \le u_\infty(x_0+t\vec{n}_{x_0})t^{\frac{2\alpha}{p-1}}\le c_2,\qquad \forall t\in(0,\sigma_0),$$ where $c_2>c_1>0$ and $\sigma_0>0$ small enough. $(iii)$ if we assume more that $g(s)=s^p$ with $p\in(0,1+\frac{2\alpha}{N}]$, then $$\lim_{k\to\infty}u_k(x)=+\infty,\qquad \forall x\in \Omega.$$ We notice that the limit of $\{u_k\}$ as $k\to\infty$ blows up every where in $\Omega$ when $g(u)=u^p$ with $1<p\le1+ \frac{2\alpha}{N}$. This phenomena is different from the Laplacian case, which is caused by the nonlocal characteristic of the fractional Laplacian. Theorem \[teo 1\] and Theorem \[teo 2\] show the existence and properties of weak solutions to (\[eq 1.1\]) in the subcritical case. One natural question is what happens in the critical case, i.e., $g(s)=s^p$ with $p\ge \frac{N+\alpha}{N-\alpha}$. The results are given by: \[teo 4\] Assume that $\epsilon=1$, $k>0$, $\Omega=B_1(e_N)$ with $e_N=(0,\cdots, 0,1)$, $\nu=\delta_{0}$ and $g(s)=s^p$ with $p= \frac{N+\alpha}{N-\alpha}$. Then problem (\[eq 1.1\]) doesn’t admit any weak solution. In general, the nonexistence of weak solution is obtained by capacity analysis for second order differential elliptic equations involving measures, see [@V] and references therein. However, it is a very tough job to attain the nonexistence in the capacity framework by the nonlocal characteristic and the weak sense of $\frac{\partial^\alpha\delta_0}{\partial \vec{n}^\alpha}$, which is weaker than Radon measure. In the proof of Theorem \[teo 4\], we make use of the self-similar property in the half space. The last goal of this paper is to consider the fractional elliptic problem (\[eq 1.1\]) with source nonlinearity, that is, $\epsilon=-1$. In the last decades, semilinear elliptic problems with source nonlinearity and measure data $$\label{eq01} \arraycolsep=1pt \begin{array}{lll} -\Delta u=g(u)+k\nu\quad &{\rm in}\quad\ \ \Omega,\\[3mm] \phantom{-\Delta } u=\mu\quad &{\rm on}\quad\ \ \partial\Omega, \end{array}$$ have attracted numerous interests. There are three basic methods to obtain weak solutions. The first one is to iterate $$u_{n+1}=\mathbb{G}_1[g(u_n)]+k\mathbb{G}_1[\nu],\quad \forall n\in{\mathbb{N}}$$ and look for a function $v$ satisfying $$v\ge \mathbb{G}_1[g(v)]+k\mathbb{G}_1[\nu].$$ When $g$ is a pure power source, the existence results could be found in the references [@BC; @BV; @BY; @KV; @V]. The second method is to apply duality argument to derive weak solution when the mapping $r\mapsto g(r)$ is nondecreasing, convex and continuous, see Baras-Pierre [@BP]. These two methods are very difficult to deal with for a general source nonlinearity. Recently, Chen-Felmer-Véron in [@CFV] introduced a new method to solve problem (\[eq01\]) when $g$ is a general nonlinearity, where the authors employed Schauder’s fixed point theorem to obtain the uniform bound and then to approach the weak solution. Here we develop the latter method to attain weak solution of (\[eq 1.1\]) with $\epsilon=-1$ and the main results state as follows. \[teo 3\] Let $\epsilon=-1$, $k>0$ and $\nu\in\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$ nonnegative with ${\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}=1$. $(i)$ Suppose that $$\label{06-08-2} g(s)\le c_3s^{p_0}+\epsilon,\quad \forall s\ge0,$$ for some $p_0\in(0,1]$, $c_3>0$ and $\epsilon>0$. Assume more that $c_3$ is small enough when $p_0=1$. Then problem (\[eq 1.1\]) admits a nonnegative weak solution $u_\nu$ satisfying $$\label{1.5} u_\nu(x)\ge \mathbb{G}_\alpha[\frac{\partial^\alpha\nu}{\partial\vec{n}^\alpha}](x),\qquad \forall x\in\Omega.$$ $(ii)$ Suppose that $$\label{1.1+++} g(s)\le c_4s^{p_*}+\epsilon,\quad \forall s\in[0,1]$$ and $$\label{1.4} g_\infty:=\int_1^{\infty} g(s)s^{-1-\frac{N+\alpha}{N-\alpha}}ds<+\infty,$$ where $c_4,\epsilon>0$ and $p_*>1$. Then there exist $k_0,\epsilon_0>0$ depending on $c_4, p_* $ and $ g_\infty$ such that for $k\in[0,k_0)$ and $\epsilon\in(0,\epsilon_0)$, problem (\[eq 1.1\]) admits a nonnegative weak solution $u_\nu$ satisfying (\[1.5\]). We remark that $(i)$ it does not require any restrictions on parameters $c_3, \epsilon, k$ when $p_0\in(0,1)$ or on parameters $ \epsilon, \sigma$ when $p_0=1$; $(ii)$ the integral subcritical condition (\[1.4\]) has the same critical value with (\[g1\]). The rest of the paper is organized as follows. In Section 2 we study the properties of $\frac{\partial^\alpha \nu}{\partial\vec{n}^\alpha}$. Section 3 is devoted to prove Theorem \[teo 1\]. In Section 4 we analyse the properties of the weak solution for problem (\[eq 1.1\]) when $\nu$ is Dirac mass. The nonexistence of weak solution in the critical case is addressed in Section 5. Finally we give the proof of Theorem \[teo 3\] in Section 6. General measure concentrated on boundary ======================================== In this section, we first build the one-to-one connection between the Radon measure space $\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$ and the bounded Radon measure space $\mathfrak{M}^b(\partial\Omega)$. On the one hand, for any $\mu\in \mathfrak{M}^b(\partial\Omega)$, we denote by $\tilde \mu$ the measure generated by $\mu$ extending inside $\Omega$ by zero, that is, $$\tilde \mu(E):=\mu(E\cap\partial\Omega),\qquad \forall E\subset \bar\Omega\ {\rm Borel\ set},$$ then $\tilde\mu\in \mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$. On the other hand, let $\tilde \mu\in \mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$, we see that $$\tilde \mu(E)=\tilde \mu(E\cap\partial\Omega),\qquad \forall E\subset \bar\Omega\ {\rm Borel\ set}.$$ Denote by $\mu$ a Radon measure such that $\mu(F):=\tilde \mu(F),\ F\subset \partial\Omega\ {\rm Borel\ set}$. Then $\tilde \mu(E)=\mu(E\cap\partial\Omega)$ for any Borel set $E\subset\bar\Omega$ and $${\|\tilde \mu\|}_{\mathfrak{M}^b(\bar\Omega)}={\|\mu\|}_{\mathfrak{M}^b(\partial\Omega)}.$$ Now we make an approximation of $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ by a sequence Radon measure concentrated on one type of manifolds inside of $\Omega$. Indeed, we observe that there exists $\sigma_0>0$ small such that $$\Omega_t:=\{x\in \Omega,\ \rho_{\partial\Omega}(x)>t\}$$ is a $C^2$ domain in ${\mathbb{R}}^N$ for $t\in[0,\sigma_0]$ and for any $x\in\partial\Omega_t$, there exists a unique $x_\partial\in\partial\Omega$ such that $|x-x_\partial|=\rho_{\partial\Omega}(x)$. Conversely, for any $x\in\partial\Omega$, there exists a unique point $x_t\in\partial\Omega_t$ such that $|x-x_t|=\rho_{\partial\Omega_t}(x)$, where $t\in(0,\sigma_0)$ and $\rho_{\partial\Omega_t}(x)={\rm dist}(x,\partial\Omega_t)$. Then for any Borel set $E\subset \partial\Omega$, there exists unique $E_t\subset \partial\Omega_t$ such that $E_t=\{x_t: x\in E\}$. In what follows, we always assume that $t\in[0,\sigma_0]$. Denote by $\nu_t$ a Radon measure generated by $\nu$ as $$\nu_t(E_t)=\nu(E),$$ and then $\nu_t$ is a bounded Radon measure with support in $\partial\Omega_t$ and $$\nu_t(E)=\nu_t(E\cap\partial\Omega_t),\qquad \forall E\subset\bar\Omega\ {\rm Borel\ set}.$$ In the distribution sense, we have that $$\label{2.1.2} \langle\nu_t,f\rangle=\int_{\partial\Omega_t}f(x)d\nu_t(x)=\int_{\partial\Omega}f(x+t\vec{n}_x)d\nu(x),\qquad \forall f\in C_0(\Omega).$$ Then we observe that $$\label{2.1.0} \{x_t: x\in{\rm supp}(\nu)\}={\rm supp}(\nu_t) \quad{\rm and}\quad{\|\nu_t\|}_{\mathfrak{M}^b(\bar\Omega)}={\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}.$$ Now we are able to show an approximation of $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$. \[pr 2.1\] The sequence of Radon measures $\{t^{-\alpha}\nu_t\}_t$ converges to $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ as $t\to0^+$ in the following distribution sense: $$\lim_{t\to0^+}\int_{\partial\Omega_t}\xi(x)t^{-\alpha}d\nu_t(x)=\int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial \vec{n}^\alpha_x}d\nu(x),\qquad \forall\xi\in \mathbb{X}_\alpha.$$ [**Proof.**]{} It follows from [@RS Proposition 1.1], that $\xi\in C^\alpha({\mathbb{R}}^N)$ if $\xi\in \mathbb{X}_\alpha$. This together with the fact that supp$(\xi)\subset\bar\Omega$, $\frac{\partial^\alpha\xi(x)}{\partial \vec{n}_x^\alpha}$ is well-defined for any $x\in\partial\Omega$ and for $x_t\in\partial\Omega_t$, implies that there exists a unique $x\in\partial\Omega$ such that $$x_t=x+t\vec{n}_x\quad{\rm and}\quad |x-x_t|=\rho_{\partial\Omega}(x_t),$$ then $$\xi(x+t\vec{n}_x)t^{-\alpha}=\frac{\xi(x+t\vec{n}_x)-\xi(x)}{t^\alpha},$$ which implies that $$\xi(\cdot+t\vec{n})t^{-\alpha}\to \frac{\partial^\alpha\xi(\cdot)}{\partial \vec{n}^\alpha}\quad {\rm as}\quad t\to0^+\quad {\rm in}\quad C(\bar\Omega).$$ Along with (\[2.1.2\]), we have that $$\arraycolsep=1pt \begin{array}{lll} |\int_{\partial\Omega_t}\xi(x)t^{-\alpha}d\nu_t(x)-\int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial \vec{n}_x^\alpha}d\nu(x)| \\[3mm]\phantom{-----} =|\int_{\partial\Omega}\xi(x+t\vec{n}_x)t^{-\alpha}d\nu(x)-\int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial \vec{n}_x^\alpha}d\nu(x)| \\[3mm]\phantom{-----} \le\int_{\partial\Omega}|\xi(x+t\vec{n}_x)t^{-\alpha}-\frac{\partial^\alpha\xi(x)}{\partial \vec{n}_x^\alpha}|d|\nu(x)| \\[3mm]\phantom{-----}\to0\quad{\rm as}\ t\to0^+, \end{array}$$ which ends the proof. $\Box$ We note that Proposition \[pr 2.1\] shows that $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ is approximated by a sequence Radon measure with support in $\Omega$ in the distribution sense and this provides a new method to derive weak solution of (\[eq 1.1\]) by considering the limit of the weak solutions to $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+\epsilon g(u)=kt^{-\alpha}\nu_t\quad & {\rm in}\quad \Omega,\\[2mm] \phantom{(-\Delta)^\alpha +\epsilon g(u)} u=0\quad &{\rm in}\quad \Omega^c. \end{array}$$ To end this section, we give a upper bound for $\mathbb{G}_\alpha[\frac{\partial^\alpha|\nu|}{\partial \vec{n}^\alpha}]$ . \[lm 2.1\] Let $\nu\in\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$, then there exists $c_5>0$ such that $$\mathbb{G}_\alpha[\frac{\partial^\alpha|\nu|}{\partial \vec{n}^\alpha}](x)\le \int_{\partial\Omega}\frac{c_5}{|x-y|^{N-\alpha}}d|\nu|(y),\qquad x\in\Omega.$$ [**Proof.**]{} From [@BV Theorem 1.1], there exists $c_5>0$ independent of $t$ such that for any $(x,y)\in \Omega\times\partial\Omega_t$, $x\neq y$, $$\label{annex 01} G_\alpha(x,y)\le c_5\frac{\rho_{\partial\Omega}^\alpha(y)}{|x-y|^{N-\alpha}}=\frac{c_5 t^{\alpha}}{|x-y|^{N-\alpha}}.$$ Then for $x\in\Omega$, $$\begin{aligned} \mathbb{G}_\alpha[\frac{\partial^\alpha|\nu|}{\partial \vec{n}^\alpha}](x) &=& \lim_{t\to0^+}\int_{\partial\Omega_t} G_\alpha(x,y)t^{-\alpha}d|\nu_t|(y)\\ &\le & \lim_{t\to0^+}\int_{\partial\Omega_t} \frac{c_5}{|x-y|^{N-\alpha}}d|\nu_t|(y) \\&=&\int_{\partial\Omega} \frac{c_5}{|x-y|^{N-\alpha}}d|\nu|(y).\end{aligned}$$ We complete the proof. $\Box$ Absorption Nonlinearity {#sec:existence} ======================= In this section, our goal is to prove the existence and uniqueness of weak solution for fractional elliptic problem (\[eq 1.1\]) with $\epsilon=1$. To this end, we first consider the properties of weak solution of $$\label{2.0.6} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+g_n(u)=kt^{-\alpha}\nu_t\quad & {\rm in}\quad \Omega,\\[2mm] \phantom{(-\Delta)^\alpha +g_n(u)} u=0\quad &{\rm in}\quad \Omega^c, \end{array}$$ where $t\in(0,\sigma_0)$, $\nu_t$ is given in (\[2.1.2\]) and $\{g_n\}$ are a sequence of $C^1$ nondecreasing functions defined on ${\mathbb{R}}$ such that $g_n(0)=g(0)\ge 0$, $$\label{06-08-0} | g_n|\le g,\quad \sup_{s\in{\mathbb{R}}}|g_n(s)|=n\quad{\rm and}\quad \lim_{n\to\infty}{\|g_n-g\|}_{L^\infty_{loc}({\mathbb{R}})}=0.$$ The existence and uniqueness of weak solution to (\[2.0.6\]) is stated as follows. \[pr 1\] Assume that $k>0$, $\alpha\in(0,1)$, $g_n$ is a $C^1$ nondecreasing function satisfying $g_n(0)\ge 0$ and (\[06-08-0\]). Then for $t\in(0,\sigma_0)$, problem (\[2.0.6\]) admits a unique weak solution $u_{n,k\nu_t}$ such that $$|u_{n,k\nu_t}|\le k\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]\quad{\rm a.e.\ in}\quad \Omega$$ and $$\label{12-08-0} {\|g_n(u_{n,k\nu_t})\|}_{L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)}\le ck{\|\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]\|}_{L^1(\Omega)},$$ where $c>0$ independent of $t$, $k$ and $n$. Furthermore, for any fixed $n\in{\mathbb{N}}$, $t\in(0,\sigma_0)$ and $k>0$, the mapping $\nu\mapsto u_{n,k\nu_t}$ is increasing. [**Proof.**]{} For any $t>0$, we observe that $kt^{-\alpha}\nu_t$ is a bounded Radon measure in $\Omega$ and $g_n$ is bounded, then it follows from [@CV1 Theorem 1.1] that problem (\[2.0.6\]) admits a unique weak solution $u_{n,k\nu_t}$. Moreover, $kt^{-\alpha}\nu_t$ is increasing with respect to $\nu_t$ and $\nu_t$ is increasing with respect to $\nu$ by the definition of $\nu_t$, then applying [@CV1 Theorem 1.1], we have that for any fixed $t\in(0,\sigma_0)$ and $k>0$, the mapping $\nu\mapsto u_{n,k\nu_t}$ is increasing. $\Box$ To simplify the notation, we always write $u_{n,k\nu_t}$ by $u_{n,t}$ in this section. In order to consider the limit of $\{u_{n,t}\}$ as $t\to0^+$, we introduce some auxiliary lemmas which are the key steps to obtain $\{g_n(u_{n,t})\}$ uniformly integrable with respect to $t$. For $\lambda>0$, let us set $$\label{Slambda} S_\lambda=\{x\in \Omega:\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|](x)>\lambda\}\quad{\rm and}\quad m(\lambda)=\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx.$$ \[lm 0\] For $\nu\in\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$ and any $t\in(0, \sigma_0)$, there exists $c_6>0$ independent of $t$ such that $$\label{annex -0} m(\lambda)\le c_6\lambda^{-\frac{N}{N-\alpha}}.$$ [**Proof.**]{} For $\Lambda>0$ and $y\in\partial\Omega_t$ with $t\in(0, \sigma_0/2)$, we denote $$A_\Lambda(y)=\{x\in\Omega\setminus\{y\}: G_\alpha(x,y)>\Lambda\}\ \ {\rm {and}}\quad m_\Lambda(y)=\int_{A_\Lambda(y)}\rho_{\partial\Omega}^\alpha(x) dx.$$ For any $(x,y)\in\Omega\times\partial\Omega_t$, $x\neq y$, it infers by (\[annex 01\]) that $$\begin{aligned} A_\Lambda(y)\subset \left\{x\in\Omega\setminus\{y\}: \frac{c_5 t^\alpha}{|x-y|^{N-\alpha}}>\Lambda\right\}\subset B_r(y),\end{aligned}$$ where $r=(\frac{c_5t^\alpha}{\Lambda})^{\frac1{N-\alpha}}$. Thus, $\rho_{\partial\Omega}(x)\le R_0$ for some $R_0>0$ such that $\Omega\subset B_{R_0}(0)$ and $$\label{annex 1xhw} m_\Lambda(y)\le R_0^\alpha\int_{B_r(y)}dx\le c_7t^{\frac{N\alpha}{N-\alpha}}\Lambda^{-\frac{N}{N-\alpha}},$$ where $c_7>0$ independent of $t$. For $y\in\partial\Omega_t$, we have that $$\begin{aligned} \int_{S_\lambda} G_\alpha(x,y)\rho_{\partial\Omega}^\alpha(x)dx\le \int_{A_\Lambda(y)}G_\alpha(x,y)\rho_{\partial\Omega}^\alpha(x)dx+\Lambda\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx.\end{aligned}$$ By integration by parts, we obtain $$\begin{aligned} \int_{A_\Lambda(y)}G_\alpha(x,y)\rho_{\partial\Omega}^\alpha(x)dx=\Lambda m_\Lambda(y)+ \int_\Lambda^\infty m_s(y)ds\le c_8t^{\frac{N\alpha}{N-\alpha}}\Lambda^{1-\frac{N}{N-\alpha}},\end{aligned}$$ where $c_8>0$ independent of $t$. Thus, $$\begin{aligned} \int_{S_\lambda} G_\alpha(x,y)\rho_{\partial\Omega}^\alpha(x)dx\le c_8t^{\frac{N\alpha}{N-\alpha}}\Lambda^{1-\frac{N}{N-\alpha}}+\Lambda \int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx.\end{aligned}$$ Choose $\Lambda= t^\alpha(\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{-\frac{N-\alpha}{N}}$ and then $$\begin{aligned} \int_{S_\lambda} G_\alpha(x,y)\rho_{\partial\Omega}^\alpha(x)dx\le c_9t^\alpha (\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{\frac{\alpha}{N}},\end{aligned}$$ where $c_9=c_8+1$. Therefore, $$\begin{aligned} \int_{S_\lambda} \mathbb{G}_\alpha [t^{-\alpha}|\nu_t|](x)\rho_{\partial\Omega}^\alpha(x)dx&=&\int_\Omega\int_{S_\lambda} G_\alpha(x,y)\rho_{\partial\Omega}^\alpha(x)dxt^{-\alpha} d|\nu_t(y)| \\&\le &c_9\int_\Omega d|\nu_t(y)|(\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{\frac{\alpha}{N}} \\&\le& c_9\|\nu\|_{\mathfrak{M}^b(\bar\Omega)} (\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{\frac{\alpha}{N}}.\end{aligned}$$ As a consequence, $$\begin{aligned} \lambda m(\lambda)\le c_9\|\nu\|_{\mathfrak{M}^b(\bar\Omega)}m(\lambda)^{\frac{\alpha}{N}},\end{aligned}$$ which implies (\[annex -0\]). This ends the proof. $\Box$ From Lemma \[lm 0\], it implies that $$\label{24-08-0} {\|\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]\|}_{M^{\frac{N}{N-\alpha}}(\Omega,\rho_{\partial\Omega}^\alpha dx)}\le c_9\|\nu\|_{\mathfrak{M}^b(\bar\Omega)},$$ where $M^{\frac{N}{N-\alpha}}(\Omega,\rho_{\partial\Omega}^\alpha dx)$ is Marcinkiewicz space with exponent $\frac{N}{N-\alpha}$. The definition and properties of Marcinkiewicz space see the references [@BBC; @CC; @CV1; @V]. In next lemma, the uniformly regularity plays an important role in our approximation of weak solution. \[lm 1\] Assume that $u_{t}$ is a weak solution of (\[2.0.6\]) replacing $g_n$ by $g$, a continuous nondecreasing function satisfying $g(0)\ge 0$. Then for any compact subsets $\mathcal{K}\subset\Omega$, there exist $t_0>0$, $\beta>0$ small and $c_{10}>0$ independent of $t$ such that for $t\in(0,t_0]$, $$\label{2.0.8} {\|u_{t}\|}_{C^{\beta}(\mathcal{K})}\le c_{10} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}.$$ Moreover, if $g$ is $C^\beta$ locally in ${\mathbb{R}}$, then there exists $c_{11}>0$ independent of $t$ such that $$\label{2.0.9} {\|u_{t}\|}_{C^{2\alpha+\beta}(\mathcal{K})}\le c_{11} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}.$$ [**Proof.**]{} We observe from Proposition \[pr 1\] that $$\label{3--1} | u_t|\le \mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]\quad{\rm a.e.\ in}\quad \Omega.$$ For compact set $\mathcal{K}$ in $\Omega$, there exists $t_0>0$ such that $$\mathcal{K}_{5t_0}\subset \Omega,$$ where $\mathcal{K}_r:=\{x\in{\mathbb{R}}^N:{\rm dist}(x,\mathcal{K})<r\}$ with $r>0$. Then $\mathcal{K}_{4t_0}\cap \partial\Omega_t=\O$ for any $t\in(0,t_0]$ and $${\|g(u_t)\|}_{L^\infty(\mathcal{K}_{3t_0})}\le {\|g(\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|])\|}_{L^\infty(\mathcal{K}_{3t_0})}.$$ Since $t^{-\alpha}\nu_t$ is a bounded Radon measure in $\Omega$, there exists a sequence $\{f_n\}\subset C^2_0(\Omega)$ such that $f_n$ converges to $t^{-\alpha}\nu_t$ in the distribution sense and for some $N_{t_0}>0$ such that for $n\ge N_{t_0}$, supp$(f_n)\cap \mathcal{K}_{3t_0}=\O$. We may assume that $g$ is $C^\beta$ locally in ${\mathbb{R}}$. (In fact, we can choose a sequence of nondecreasing functions $\{g_n\}\subset C^\beta({\mathbb{R}})$ such that $g_n(0)\ge0$, $|g_n(s)|\le |g(s)|$ for $s\in{\mathbb{R}}$ and $g_n\to g$ locally in ${\mathbb{R}}$ as $n\to\infty$.) Let $w_n$ be the classical solution of $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+g_n(u)=f_n\quad & {\rm in}\quad\Omega,\\[2mm] \phantom{ (-\Delta)^\alpha + g_n(u)} u=0\quad & {\rm in}\quad \Omega^c. \end{array}$$ By the uniqueness of weak solution to (\[2.0.6\]), we obtain that, up to some subsequence, $$\label{12-08.2} u_t=\lim_{n\to\infty} w_n\quad {\rm a.e.\ in}\quad \Omega.$$ We observe that $0\le w_n= \mathbb{G}_\alpha[f_n]-\mathbb{G}_\alpha[g(w_n)]\le \mathbb{G}_\alpha[f_n]$ and $\mathbb{G}_\alpha[f_n]$ converges to $\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]$ uniformly in any compact set of $\Omega\setminus \partial\Omega_t$ and in $L^1(\Omega)$, then there exists $c_{11}>0$ independent of $n$ and $t$ such that $${\|w_n\|}_{L^\infty(\mathcal{K}_{3t_0})}\le {c_{11}}{\|\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]\|}_{L^\infty(\mathcal{K}_{3t_0})},\quad {\|w_n\|}_{L^1(\Omega)}\le {c_{11}}{\|\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]\|}_{L^1(\Omega)}.$$ By [@CV3 Lemma 3.1], for $\beta\in(0,2\alpha)$, there exists $c_{12}>0$ independent of $n$ and $t$, such that $$\arraycolsep=1pt \begin{array}{lll} {\|w_n\|}_{C^{\beta}(\mathcal{K}_{2t_0})} \le c_8[{\|w_n\|}_{L^1(\Omega)}+{\|g(w_n)\|}_{L^{\infty}(\mathcal{K}_{3t_0})}+{\|w_n\|}_{L^{\infty}(\mathcal{K}_{3t_0})}]\\[3mm] \phantom{} \le c_{12}[\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{L^1(\Omega)}+{\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|}_{L^\infty( \mathcal{K}_{3t_0})}+\|g(\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|])\|_{L^\infty( \mathcal{K}_{3t_0})}]. \end{array}$$ It follows by [@RS Corollary 2.4] that there exist $c_{13},c_{14}>0$ such that $$\label{2.0.10.0} \arraycolsep=1pt \begin{array}{lll} {\|w_n\|}_{C^{2\alpha+\beta}(\mathcal{K})} \le {c_{13}}[{\|w_n\|}_{L^1(\Omega)}+{\|g(w_n)\|}_{C^{\beta}(\mathcal{K}_{2t_0})} +{\|w_n\|}_{C^{\beta}(\mathcal{K}_{2t_0})}]\\[3mm] \phantom{------\ } \le c_{14}[\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{L^1(\Omega)}+{\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|}_{L^\infty( \mathcal{K}_{3t_0})}\\[3mm] \phantom{-------\ }+\|g\|_{C^\beta([0,\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{L^\infty( \mathcal{K}_{3t_0})}])} \|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{C^\beta( \mathcal{K}_{3t_0})}]. \end{array}$$ Therefore, together with (\[12-08.2\]) and the Arzela-Ascoli Theorem, it follows that $u_t\in C^{2\alpha+\epsilon}(\mathcal{K})$ for $\epsilon\in(0,\beta)$. Then $w_n\to u_t$ and $f_n\to 0$ uniformly in any compact subset of $\Omega\setminus\partial\Omega_t$ as $n\to\infty$. It infers by [@CV3 Lemma 3.1] that $$\arraycolsep=1pt \begin{array}{lll} {\|u_t\|}_{C^{\beta}(\mathcal{K})} \le c_{8}[{\|u_t\|}_{L^1(\Omega)}+{\|g(u_t)\|}_{L^{\infty}(\mathcal{K}_{3t_0})}+{\|u_k\|}_{L^{\infty}(\mathcal{K}_{3t_0})}] \\[2mm]\phantom{-}\le c_{12}[\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{L^1(\Omega)}+{\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|}_{L^\infty( \mathcal{K}_{3t_0})}+\|g(\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|)]\|_{L^\infty( \mathcal{K}_{3t_0})}]. \end{array}$$ We next claim that $\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|_{L^1(\Omega)}$, ${\|\mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|]\|}_{L^\infty( \mathcal{K}_{3t_0})}$ are uniformly bounded. In fact, for $x\in\mathcal{K}$ and $y\in\partial\Omega_t$ with $t\in(0,t_0)$, we have that $|x-y|\ge 3t_0$. By (\[annex 01\]), it implies that $$\label{2.1.3} \arraycolsep=1pt \begin{array}{lll} \mathbb{G}_{\alpha}[t^{-\alpha}|\nu_t|](x)\le \int_{\partial\Omega_t}\frac{c_5}{|x-y|^{N-\alpha}}d|\nu_t(y)| \\[3mm]\phantom{------} \le c_5t_0^{\alpha-N} {\|\nu_t\|}_{\mathfrak{M}^b(\bar\Omega)} = c_5t_0^{\alpha-N} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)} \end{array}$$ and $$\label{2.1.4} \arraycolsep=1pt \begin{array}{lll} {\|\mathbb{G}_{\alpha}[t^{-\alpha}\nu_t]\|}_{L^1(\Omega)}\le \int_{\Omega}\int_{\partial\Omega_t}\frac{c_5}{|x-y|^{N-\alpha}}d|\nu_t(y)|dx \\[3mm]\phantom{------}=\int_{\partial\Omega_t}\int_{\Omega}\frac{c_5}{|x-y|^{N-\alpha}}dxd|\nu_t(y)| \le c_{15} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}, \end{array}$$ which implies that $${\|u_t\|}_{C^{\beta}(\mathcal{K})}\le c_{15}{\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)},$$ where $c_{15}>0$ independent of $t$. Moreover, if $g$ is $C^\beta$ locally in ${\mathbb{R}}$, similar to (\[2.0.10.0\]) it implies by (\[2.1.3\]) and (\[2.1.4\]) that $${\|u_t\|}_{C^{2\alpha+\beta}(\mathcal{K})}\le c_{16}{\|\nu\|}_{\mathfrak{M}^b(\bar \Omega)},$$ where $c_{16}>0$ independent of $t$. We conclude by Theorem 2.2 in [@CFQ] that $u_t$ is a classical solution of $$\label{2.2.1} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+g(u)=0\quad & {\rm in} \quad\Omega\setminus\partial\Omega_t,\\[2mm] \phantom{ (-\Delta)^\alpha + g(u)} u=0\quad & {\rm in} \quad \Omega^c. \end{array}$$ This ends the proof.$\Box$ \[pr 2.01\] Assume that $k>0$ and $\{g_n\}$ are a sequence of $C^1$ nondecreasing functions defined on ${\mathbb{R}}$ such that $g_n(0)=g(0)$ and (\[06-08-0\]). Then problem $$\label{10-08-0} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+ g_n(u)=k\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}\quad &{\rm in}\quad\ \ \bar\Omega,\\[3mm] \phantom{(-\Delta)^\alpha + g_n(u)} u=0\quad &{\rm in}\quad\ \ \bar\Omega^c \end{array}$$ admits a unique weak solution $u_{n}$ satisfying $$\label{10-08-1} -k\mathbb{G}_\alpha[\frac{\partial^\alpha\nu_-}{\partial\vec{n}^\alpha}](x)\le u_n(x)\le k\mathbb{G}_\alpha[\frac{\partial^\alpha\nu_+}{\partial\vec{n}^\alpha}](x),\qquad x\in\Omega,$$ where $\nu_+,\nu_-$ are the positive and negative decomposition of $\nu$ such that $\nu=\nu_+-\nu_-$.\ Furthermore, $$\label{12-08-1} {\|g_n(u_{n})\|}_{L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)}\le k{\|\mathbb{G}_\alpha[\frac{\partial^\alpha|\nu|}{\partial\vec{n}^\alpha}]\|}_{L^1(\Omega)}$$ and $u_n$ is a classical solution of (\[1.1\]) replacing $g$ by $g_n$. [**Proof.**]{} [*To prove the existence of weak solution.*]{} Since $\nu_t$ is a bounded Radon measure with supp$(\nu_t)\subset\partial \Omega_t$ for $t\in(0,\sigma_0)$, then by Proposition \[pr 1\], we have that problem (\[2.0.6\]) admits a unique weak solution $u_{n,t}$ such that $$\label{24-08-1} |u_{n,t}|\le \mathbb{G}_\alpha [t^{-\alpha}|\nu_t|]\quad{\rm a.e.\ in}\quad \Omega,\qquad \int_{\Omega}|g_n(u_{n,t})| \rho_{\partial\Omega}^\alpha dx\le k{\|\mathbb{G}_\alpha [t^{-\alpha}|\nu_t|]\|}_{L^1(\Omega)}$$ and $$\label{2.1.1--} \int_\Omega [u_{n,t}(-\Delta)^\alpha\xi+g_n(u_{n,t})\xi]dx=\int_{\partial\Omega_t}t^{-\alpha}\xi(x)d\nu_t(x),\quad \forall\xi\in \mathbb{X}_\alpha.$$ For any compact set $\mathcal{K} \subset \Omega$, there exists $t_0\in(0,\sigma_0)$ such that $$\mathcal{K} \subset \Omega_t\quad {\rm and}\quad {\rm dist}(\mathcal{K},\partial\Omega_{t})\ge t_0,\quad \forall t\in(0,t_0].$$ By Lemma \[lm 1\], we observe that for some $\beta\in(0,\alpha)$ $${\|u_{n,t}\|}_{C^\beta(\mathcal{K})}\le c_5t_0^{-N+2\alpha} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}.$$ Therefore, up to some subsequence, there exists $u_n$ such that $$\lim_{t\to0^+}u_{n,t}=u_n\quad{\rm a.e.\ in}\quad \Omega.$$ Then $g_n(u_{n,t})$ converges to $g_n(u_n)$ almost every in $\Omega$ as $t\to0^+$. By (\[24-08-1\]) and (\[24-08-0\]), we have that $\{u_{n,t}\}_t$ is relatively compact in $L^1(\Omega)$, up to subsequence, $$u_{n,t}\to u_n\quad {\rm in}\ \ L^1(\Omega)\quad {\rm as}\ t\to0^+$$ and then $$g_n(u_{n,t})\to g_n(u_n)\quad {\rm in}\ \ L^1(\Omega,\rho^\alpha_{\partial\Omega}dx)\quad {\rm as}\ t\to0^+.$$ By Proposition \[pr 2.1\], $$\begin{aligned} \int_{\partial\Omega_t}t^{-\alpha}\xi(x)d\nu_t(x) \to \int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial\vec{n}_x^\alpha}d\nu(x)\quad {\rm as}\ t\to0^+,\end{aligned}$$ Passing to the limit as $t\to 0^+$ in the identity (\[2.1.1–\]), it implies that $$\int_\Omega [u_n(-\Delta)^\alpha\xi+g_n(u_n)\xi]dx=k\int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial\vec{n}_x^\alpha}d\nu(x),\quad \forall\xi\in\mathbb{ X}_\alpha.$$ This implies that $u_n$ is a weak solution of (\[10-08-0\]). We see that (\[12-08-1\]) follows by (\[12-08-0\]) and Lemma \[lm 0\]. Moreover, by the facts that $u_n=\lim_{t\to0^+} u_{n,t}$ and $$-k\mathbb{G}_\alpha[t^{-\alpha}\nu_-]\le u_{n,t}\le k\mathbb{G}_\alpha[t^{-\alpha}\nu_+]\quad {\rm in}\quad \Omega,$$ we have that (\[10-08-1\]) holds. [*To prove that $u_n=0$ in $\Omega^c\setminus{\rm supp}(\nu)$.* ]{} Let $x_0\in\partial\Omega\setminus{\rm supp}(\nu)$ and $x_s=x_0+s\vec{n}_{x_0}$ with $s\in(0,\sigma_0)$. We only have to prove that $\lim_{s\to0^+}u(x_s)=0$. From [@BV Theorem 1.1], for any $(x,y)\in \Omega\times \partial\Omega_t$, $x\neq y$, $$\label{annex 010} G_\alpha(x,y)\le c_5\frac{\rho^\alpha_{\partial\Omega}(y)\rho^\alpha_{\partial\Omega}(x)}{|x-y|^N }=c_5\frac{\rho^\alpha_{\partial\Omega}(x)t^{\alpha}}{|x-y|^N }.$$ For some $s_0>0$ and any $s\in(0,s_0)$, we observe that ${\rm dist}(x_s,{\rm supp}(\nu))\ge \frac{1}{2}{\rm dist}(x_0,{\rm supp}(\nu))$ and $$\begin{aligned} \mathbb{G}_\alpha[t^{-\alpha}|\nu_t|](x_s)&\le& c_5\int_{\partial\Omega}\frac{\rho^\alpha_{\partial\Omega}(x_s)}{|x_s-y|^N}d|\nu|(y) \\&=& c_5s^\alpha\int_{\partial\Omega\setminus{\rm supp}(\nu)}\frac{1}{|x_s-y|^N}d|\nu|(y) \\&\le& c_52^N s^\alpha{\rm dist}(x_0,{\rm supp}(\nu))^{-N}{\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)} \\&\to&0\quad {\rm as}\quad s\to0^+.\end{aligned}$$ Together with the facts that $$\label{facts} u_n=\lim_{t\to0^+}u_{n,t}\quad {\rm and}\quad |u_{n,t}|\le \mathbb{G}_\alpha[t^{-\alpha}|\nu_t|],$$ we derive that $u_n=0$ in $\Omega^c\setminus{\rm supp}(\nu)$. [*To prove the uniqueness of weak solution.*]{} Let $u_1,u_2$ be two weak solutions of (\[10-08-0\]) and $w=u_1-u_2$. Then $(-\Delta)^\alpha w=g_n(u_2)-g_n(u_1)$ and $g_n(u_2)-g_n(u_1)\in L^1(\Omega,\rho^\alpha_{\partial\Omega}dx)$. By Kato’s inequatlity, see Proposition 2.4 in [@CV1], for $\xi\in\mathbb{X}_\alpha$, $\xi\ge0$, we have that $$\begin{aligned} \int_\Omega |w|(-\Delta)^\alpha \xi dx+\int_\Omega[g_n(u_1)-g_n(u_2)]{\rm sign}(w)\xi dx\le0.\end{aligned}$$ Combining with $\int_\Omega[g_n(u_1)-g_n(u_2)]{\rm sign}(w)\xi dx\ge0$, then we have $$w=0\quad {\rm a.e.\ in}\ \ \Omega.$$ [*Regularity of $u_n$.*]{} Since $g_n$ is $C^1$ in ${\mathbb{R}}$, then by (\[2.0.9\]), we have $$\label{2.0.10} {\|u_n\|}_{C^{2\alpha+\beta}(\mathcal{K})}\le c_{17} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)},$$ for any compact set $\mathcal{K}$ and some $\beta\in(0,\alpha)$. Then $u_n$ is $C^{2\alpha+\beta}$ locally in $\Omega$. Together with the fact that $u_{n,t}$ is classical solution of (\[2.2.1\]), we derive by Theorem 2.2 in [@CFQ] that $u_n$ is a classical solution of (\[1.1\]). $\Box$ For $\lambda>0$, let us define $$\label{Slambda} \tilde S_\lambda=\{x\in \Omega:\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}](x)>\lambda\}\quad{\rm and}\quad \tilde m(\lambda)=\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx.$$ \[lm 00\] For $\nu\in\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$, then there exist $\lambda_0>1$ and $c_{18}>0$ such that for any $\lambda\ge \lambda_0$, $$\label{annex 0} \tilde m(\lambda)\le c_{18}\lambda^{-\frac{N+\alpha}{N-\alpha}}.$$ [**Proof.**]{} From Lemma \[lm 2.1\], we see that $$\mathbb{G}_\alpha[\frac{\partial^\alpha|\nu|}{\partial \vec{n}^\alpha}](x)\le \int_{\partial\Omega}\frac{c_5}{|x-y|^{N-\alpha}}d|\nu(y)|,\qquad x\in\Omega.$$ For $\Lambda>0$ and $y\in\partial\Omega$, we denote $$\tilde A_\Lambda(y)=\{x\in\Omega: \frac{c_5}{|x-y|^{N-\alpha}}>\Lambda\}\ \ {\rm {and}}\quad \tilde m_\Lambda(y)=\int_{\tilde A_\Lambda(y)}\rho_{\partial\Omega}^\alpha(x) dx.$$ For any $(x,y)\in\Omega\times\partial\Omega$, it infers by(\[annex 01\]) that $$\begin{aligned} \tilde A_\Lambda(y)\subset B_{r_0}(y),\end{aligned}$$ where $r_0=(\frac{c_5}{\Lambda})^{\frac1{N-\alpha}}$. Since $\Omega$ is $C^2$, there exists $\Lambda_0>1$ such that for $\Lambda>\Lambda_0$ such that $$\rho_{\partial\Omega}(x)\le |x-y|,\quad \forall x\in \tilde A_\Lambda(y)$$ and $$\label{annex 1xhw} \tilde m_\Lambda(y)\le \int_{ B_{r_0}(y)}|x-y|^\alpha dx\le c_{19}\Lambda^{-\frac{N+\alpha}{N-\alpha}}.$$ For $y\in\partial\Omega$, we have that $$\begin{aligned} \int_{\tilde S_\lambda} \frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx\le \int_{\tilde A_\Lambda(y)}\frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx+\Lambda\int_{\tilde S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx.\end{aligned}$$ By integration by parts, we obtain $$\begin{aligned} \int_{\tilde A_\Lambda(y)}\frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx&=&\Lambda \tilde m_\Lambda(y)+ \int_\Lambda^\infty\tilde m_s(y)ds \\&\le& c_{20} \Lambda^{1-\frac{N+\alpha}{N-\alpha}},\end{aligned}$$ where $c_{20}>0$. Thus, $$\begin{aligned} \int_{\tilde S_\lambda}\frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx\le c_{20}\Lambda^{1-\frac{N+\alpha}{N-\alpha}}+\Lambda \int_{\tilde S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx.\end{aligned}$$ Since $S_{\tilde \lambda_1}\subset S_{\tilde \lambda_2}$ if $\lambda_1\ge \lambda_2$ and $$\lim_{\lambda\to0^+}\int_{\tilde S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx=0,$$ then there exists $\lambda_0>0$ such that $$\left(\int_{\tilde S_{\lambda_0}} \rho_{\partial\Omega}^\alpha(x)dx\right)^{-\frac{N-\alpha}{N+\alpha}}\ge\Lambda_0$$ and for $\lambda\ge \lambda_0$, we may choose $\Lambda= (\int_{\tilde S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{-\frac{N-\alpha}{N+\alpha}}\ge \Lambda_0$ and then $$\begin{aligned} \int_{\tilde S_\lambda} \frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx\le c_{21}(\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{\frac{2\alpha}{N+\alpha}},\end{aligned}$$ where $c_{21}=c_{20}+1$. Therefore, $$\begin{aligned} \int_{\tilde S_\lambda}\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}](x)\rho_{\partial\Omega}^\alpha(x)dx&\le &\int_{\partial\Omega}\int_{\tilde S_\lambda} \frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx d|\nu(y)| \\&\le &c_{21}\int_{\partial\Omega} d|\nu(y)|(\int_{\tilde S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{\frac{2\alpha}{N+\alpha}} \\&\le& c_{21}\|\nu\|_{\mathfrak{M}^b(\bar\Omega)} (\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{\frac{2\alpha}{N+\alpha}}.\end{aligned}$$ As a consequence, $$\begin{aligned} \lambda \tilde m(\lambda)\le c_{21}\|\nu\|_{\mathfrak{M}^b(\bar\Omega)}\tilde m(\lambda)^{\frac{2\alpha}{N+\alpha}},\end{aligned}$$ which implies (\[annex 0\]). This ends the proof. $\Box$ To estimate the nonlinearity in $L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)$, we have to introduce an auxiliary lemma as follows. \[lm 08-09\] Assume that $g:{\mathbb{R}}_+\mapsto{\mathbb{R}}_+$ is a continuous function satisfying $$\label{p} \int_1^{\infty} g(s)s^{-1-p}ds<+\infty$$ for some $p>0$. Then there is a sequence real positive numbers $\{T_n\}$ such that $$\lim_{n\to\infty}T_n=\infty\quad{\rm and}\quad \lim_{n\to\infty}g(T_n)T_n^{-p}=0.$$ Assume additionally that $g$ is nondecreasing, then $$\lim_{T\to\infty} g(T)T^{-p}=0.$$ [**Proof.**]{} The first argument see [@CV2 Lemma 3.1] and second see [@CV1 Lemma 3.1].$\Box$ Now we are ready to prove Theorem \[teo 1\]. [**Proof of Theorem \[teo 1\].**]{} [*To prove the existence of weak solution.*]{} Take $\{g_n\}$ a sequence of $C^1$ nondecreasing functions defined on ${\mathbb{R}}$ satisfying $g_n(0)=g(0)$ and (\[06-08-0\]). By Proposition \[pr 2.01\], problem (\[10-08-0\]) admits a unique weak solution $u_{n}$ such that $$|u_{n}|\le \mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]\quad{\rm a.e.\ in}\quad \Omega$$ and $$\label{2.1.1000} \int_\Omega [u_n(-\Delta)^\alpha\xi+g_n(u_n)\xi]dx=k\int_{\partial\Omega}\frac{\partial^\alpha \xi(x)}{\partial \vec{n}_x^\alpha}d\nu(x),\quad \forall\xi\in \mathbb{X}_\alpha.$$ For any compact set $\mathcal{K} \subset \Omega$, we observe from Lemma \[lm 1\] that for some $\beta\in(0,\alpha)$, $${\|u_n\|}_{C^\beta(\mathcal{K})}\le c_{22} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}.$$ Therefore, up to some subsequence, there exists $u_\nu$ such that $$\lim_{n\to\infty}u_n=u_\nu\quad{\rm a.e.\ in}\ \Omega.$$ Then $ g_n(u_n)$ converge to $g(u_\nu)$ a.e. in $\Omega$ as $n\to\infty$. By Lemma \[lm 00\] and (\[12-08-1\]), we have that $$u_n\to u_\nu\ {\rm in}\ L^1(\Omega),\quad {\|g_n(u_n)\|}_{L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)}\le c_{23}{\|\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]\|}_{L^1(\Omega)}$$ and $$\tilde m(\lambda)\leq c_{18}\lambda^{-\frac{N+\alpha}{N-\alpha}} \ \quad {\rm for}\ \ \ \lambda>\lambda_0,$$ where $$\tilde m(\lambda)=\int_{\tilde S_\lambda}\rho_{\partial\Omega}^{\alpha}(x)dx \quad{\rm with}\quad \tilde S_\lambda=\{x\in\Omega: \mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]>\lambda\}.$$ For any Borel set $E\subset\Omega$, we have that $$\displaystyle\begin{array}{lll} \displaystyle\int_{E}|g_n(u_n)|\rho_{\partial\Omega}^\alpha(x) dx\le \int_{E\cap\tilde S^c_{\frac{\lambda}{k}}}g\left(k\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]\right)\rho_{\partial\Omega}^\alpha(x) dx+\int_{E\cap \tilde S_{\frac{\lambda}{k}}}g\left(k\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]\right)\rho_{\partial\Omega}^\alpha(x) dx \\[4mm]\phantom{\int_{E}|g(u_t)|\rho^{\alpha}_{\partial\Omega}(x)dx} \displaystyle\leq \tilde g\left(\frac{\lambda}{k}\right)\int_E\rho^{\alpha}_{\partial\Omega}(x)dx+\int_{\tilde S_{\frac{\lambda}{k}}}\tilde g\left(k\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]\right)\rho^{\alpha}_{\partial\Omega}(x)dx \\[4mm]\phantom{\int_{E}|g(u_t)|\rho^{\alpha}_{\partial\Omega}(x)dx} \displaystyle\leq \tilde g\left(\frac{\lambda}{k}\right)\int_E\rho^{\alpha}_{\partial\Omega}(x)dx+\tilde m\left(\frac{\lambda}{k}\right) \tilde g\left(\frac{\lambda}{k}\right)+\int_{\frac{\lambda}{k}}^\infty\tilde m(s)d\tilde g(s), \end{array}$$ where $\tilde g(r)=g(|r|)-g(-|r|)$. On the other hand, $$\int_{\frac{\lambda}{k}}^\infty \tilde g(s)d\tilde m(s)=\lim_{T\to\infty}\int_{\frac{\lambda}{k}}^T \tilde g(s)d \tilde m(s).$$ Thus, $$\displaystyle\begin{array}{lll} \displaystyle \tilde m\left(\frac{\lambda}{k}\right) \tilde g\left(\frac{\lambda}{k}\right)+ \int_{\frac{\lambda}{k}}^T \tilde m(s)d\tilde g(s) \le c_{24}\tilde g\left(\frac{\lambda}{k}\right)\left(\frac{\lambda}{k}\right)^{-\frac{N+\alpha}{N-\alpha}}+c_{24}\int_{\frac{\lambda}{k}}^T s^{-\frac{N+\alpha}{N-\alpha}}d\tilde g(s) \\[4mm]\phantom{-----\ \int_{\lambda}^T \tilde g(s)d\omega(s)}\displaystyle \leq c_{25}T^{-\frac{N+\alpha}{N-\alpha}}\tilde g(T)+\frac{c_{24}}{\frac{N+\alpha}{N-\alpha}+1}\int_{\frac{\lambda}{k}}^T s^{-1-\frac{N+\alpha}{N-\alpha}}\tilde g(s)ds. \end{array}$$ By assumption (\[g1\]) and Lemma \[lm 08-09\] with $p=\frac{N+\alpha}{N-\alpha}$, $T^{-\frac{N+\alpha}{N-\alpha}}\tilde g(T)\to 0$ when $T\to\infty$, therefore, $$\tilde m\left(\frac{\lambda}{k}\right) \tilde g\left(\frac{\lambda}{k}\right)+ \int_{\frac{\lambda}{k}}^\infty \tilde m(s)\ d\tilde g(s)\leq \frac{c_{24}}{\frac{N+\alpha}{N-\alpha}+1}\int_{\frac{\lambda}{k}}^\infty s^{-1-\frac{N+\alpha}{N-\alpha}}\tilde g(s)ds.$$ Notice that the above quantity on the right-hand side tends to $0$ when $\lambda\to\infty$. The conclusion follows: for any $\epsilon>0$ there exists $\lambda>0$ such that $$\frac{c_{24}}{\frac{N+\alpha}{N-\alpha}+1}\int_{\frac{\lambda}{k}}^\infty s^{-1-\frac{N+\alpha}{N-\alpha}}\tilde g(s)ds\leq \frac{\epsilon}{2}.$$ For $\lambda$ fixed, there exists $\delta>0$ such that $$\int_E\rho_{\partial\Omega}^\alpha(x) dx\leq \delta\Longrightarrow \tilde g\left(\frac{\lambda}{k}\right)\int_E\rho_{\partial\Omega}^\alpha(x) dx\leq\frac{\epsilon}{2},$$ which implies that $\{g_n\circ u_n\}$ is uniformly integrable in $L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)$. Then $g_n\circ u_n\to g\circ u_\nu$ in $L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)$ by Vitali convergence theorem. Passing to the limit as $n\to +\infty$ in the identity (\[2.1.1000\]), it implies that $$\int_\Omega [u_\nu(-\Delta)^\alpha\xi+g(u_\nu)\xi]dx=\int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial\vec{n}^\alpha_x}d\nu(x),\quad \forall\xi\in\mathbb{ X}_\alpha.$$ Then $u_\nu$ is a weak solution of (\[eq 1.1\]). Moreover, it follows by the fact $$-k\mathbb{G}_\alpha[\frac{\partial^\alpha \nu_-}{\partial \vec{n}^\alpha}]\le u_n\le k\mathbb{G}_\alpha[\frac{\partial^\alpha \nu_+}{\partial \vec{n}^\alpha}]\quad {\rm in}\ \Omega.$$ which, together with $u_\nu=\lim_{n\to+\infty} u_n$, implies (\[1.33\]). The arguments including $u_n=0$ in $\Omega^c\setminus{\rm supp}(\nu)$, uniqueness and regularity follow the proof of Proposition \[pr 2.01\]. $\Box$ The proof of the existence of weak solution is divided into two steps: the first step is to get weak solution $u_n$ to (\[eq 1.1\]) with truncated nonlinearity $g_n$ and then to prove the limit of $\{u_n\}$ as $n\to\infty$ is our desired weak solution. This is due to the estimate in Lemma \[lm 0\] where we only could get exponent $\frac{N}{N-\alpha}$ and in the second step, we make use of Lemma \[lm 00\], the critical exponent of the nonlinearity $g$ could be up to $\frac{N+\alpha}{N-\alpha}$. Isolated singularity on boundary {#sec:Dirac Mass} ================================ For simplicity, we assume that $x_0=0$ and $\vec{n}_0$ is the unit inward normal vector at the origin in what follows and $u_k$ is the weak solution of (\[eq 1.1\]). Weak singularity ---------------- In this subsection, we prove Theorem \[teo 2\] part $(i)$. The regularity refers to Theorem \[teo 1\] in the case that $\nu=\delta_{0}$ with $0\in\partial\Omega$ and our main work is to prove (\[b k\]). We start our analysis with an auxiliary lemma. \[lm 3.1\] Under the hypotheses of Theorem \[teo 2\] part $(i)$, we assume more that $x_s=s\vec{n}_{0}\in\Omega$ for $s>0$ small, then there exists $c_{26}>1$ such that $$\frac1{c_{26}}s^{-N+\alpha}\le \mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x_s)\le c_{26}s^{-N+\alpha}$$ and $$\lim_{s\to0^+}\mathbb{G}_\alpha[g(\mathbb{G}_\alpha[k\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x_s))]s^{N-\alpha}=0.$$ [**Proof.**]{} It follows by Lemma \[lm 2.1\] with $\nu=\delta_0$ that $$\label{4.3-1} \mathbb{G}_\alpha[\frac{\partial^\alpha \delta_0}{\partial \vec{n}^\alpha}](x)\le \frac{c_5}{|x|^{N-\alpha}}, \qquad \forall x\in\Omega,$$ in particular, $$\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_0}{\partial \vec{n}^\alpha}](x_s)\le \frac{c_5}{s^{N-\alpha}}.$$ Let $y_t=t\vec{n}_0$ with $t\in(0,s/2)$, then $$|y_t-x_s|=s-t> \frac s2=\frac12\max\{\rho_{\partial\Omega}(y_t),\rho_{\partial\Omega}(x_s)\}$$ and apply [@BV Theorem 1.2] to derive that there exists $c_{27}>0$ such that $$\label{11.04.2} G_\alpha(x_s,y_t) \ge c_{27}\frac{\rho^\alpha_{\partial\Omega}(y_t)\rho^\alpha_{\partial\Omega}(x_s)}{|x_s-y_t|^{N}}.$$ Thus, $$\mathbb{G}_\alpha[t^{-\alpha}\delta_{y_t}](x_s)\ge \frac{c_{27} s^{\alpha}}{|s-t|^{N}},$$ which implies that $$\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x_s)\ge \frac{c_{27}}{s^{N-\alpha}}.$$ $(ii)$ By (\[4.3-1\]) and monotonicity of $g$, we have that $$\begin{aligned} \mathbb{G}_\alpha[g(\mathbb{G}_\alpha[k\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}])](x_s)s^{N-\alpha} &\le& \int_\Omega G_\alpha(x_s,y)g\left(\frac{c_5k}{|y|^{N-\alpha}}\right)dy s^{N-\alpha} \\ &\le & \int_\Omega\frac{c_5}{|x_s-y|^{N-\alpha}} g\left(\frac{c_5k}{|y|^{N-\alpha}}\right)dy s^{N-\alpha} \\ &=& c_5 s^{N-\alpha}\left[\int_{ B_{\frac s2}(x_s)}\frac{|y|^\alpha}{|x_s-y|^{N-\alpha}} g\left(\frac{c_5k}{|y|^{N-\alpha}}\right)dy\right. \\&& \left.+ \int_{ \Omega\setminus B_{\frac s2}(x_s)}\frac{|y|^\alpha}{|x_s-y|^{N-\alpha}} g\left(\frac{c_5k}{|y|^{N-\alpha}}\right)dy\right] \\&:=&A_1(s)+A_2(s).\end{aligned}$$ For $y\in B_{\frac s2}(x_s)$, we have $\frac s2\le |y|\le \frac{3s}2$ and by applying Lemma \[lm 08-09\], we derive that $$\begin{aligned} A_1(s)&\le&c_5s^{N+\alpha}g\left(\frac{2^{N-\alpha}c_5k}{s^{N-\alpha}}\right)\int_{ B_{1/2}(\vec{n}_0)} \frac{|z|^\alpha}{|\vec{n}_0-z|^{N-\alpha}}dz \\&=&c_5r^{-\frac{N+\alpha}{N-\alpha}}g\left(2^{N-\alpha}c_5rk\right)\int_{ B_{1/2}(\vec{n}_0)} \frac{|z|^\alpha}{|\vec{n}_0-z|^{N-\alpha}}dz \\&\to&0\quad{{\rm as}}\ \ r\to+\infty,\end{aligned}$$ where $r=s^{\alpha-N}$. We next claim that $A_2(s)\to0$ as $s\to0^+$. In fact, for $y\in B_{\frac s2}(0)$, we see that $|x_s-y|> s/2$ and $$\begin{aligned} s^{N-\alpha} \int_{ B_{\frac s2}(0)}\frac{|y|^\alpha}{|x_s-y|^{N-\alpha}} g\left(\frac{c_5k}{|y|^{N-\alpha}}\right)dy&\le& 2^{N-\alpha}\int_{ B_{\frac s2}(0)} |y|^\alpha g\left(\frac{c_5k}{|y|^{N-\alpha}}\right) dy \\&=&c_{28}\int_0^{\frac s2} r^\alpha g\left(\frac{c_5k}{r^{N-\alpha}}\right)r^{N-1} dr \\&=&\frac{c_{28}}{N-\alpha}\int_{s^{-\frac1{N-\alpha}}}^\infty \tau^{-1-\frac{N+\alpha}{N-\alpha}} g\left( c_5k\tau\right) d\tau \\&\to&0\quad{{\rm as}}\ \ s\to0^+,\end{aligned}$$ where the converging used (\[g1\]). For $y\in \Omega\setminus \left(B_{\frac s2}(0)\cup B_{\frac s2}(x_s)\right)$, we have that $|y-x_s|>\frac14 |y|$ and $$\begin{aligned} &&s^{N-\alpha} \int_{\Omega\setminus \left(B_{\frac s2}(0)\cup B_{\frac s2}(x_s)\right)}\frac{|y|^\alpha}{|x_s-y|^{N-\alpha}} g\left(\frac{c_5k}{|y|^{N-\alpha}}\right)dy \\&&\qquad\le s^{N-\alpha}\int_{B_R(0)\setminus B_s(0)} |y|^{2\alpha-N} g\left(\frac{c_5k}{|y|^{N-\alpha}}\right) dy \\&&\qquad=c_{29}s^{N-\alpha}\int_s^R \tau^{2\alpha-1} g( c_5k\tau^{\alpha-N}) d\tau \\&&\qquad=c_{29}\frac{s^{2\alpha-1} g(c_5ks^{\alpha-N})}{(N-\alpha)s^{\alpha-N-1}}\qquad \quad{\rm (L'Hospital's\ Rule) } \\&&\qquad=\frac{c_{29}}{N-\alpha}s^{N+\alpha} g(c_5ks^{\alpha-N}) \\&&\qquad\to0\quad{{\rm as}}\ \ s\to0^+,\end{aligned}$$ for some $R>0$ such that $\Omega \subset B_R(0)$ and $c_{29}>0$. Then $$\begin{aligned} A_2(s)\to 0\quad{{\rm as}}\ \ s\to0^+.\end{aligned}$$ Therefore, $$\label{4.4} \lim_{s\to0^+}\mathbb{G}_\alpha[g(\mathbb{G}_\alpha[k\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}])](x_s)s^{N-\alpha}=0.$$ The proof ends. $\Box$ [**Proof of Theorem \[teo 2\] $(i)$.**]{} The existence, uniqueness and regularity follow by Theorem \[teo 1\]. We only need to prove (\[b k\]) to complete the proof. We observe that $$\begin{aligned} k\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x_s) \ge u_k(x_s) &\ge& k\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x_s)- \mathbb{G}_\alpha[g(u_k)](x_s) \\&\ge& k\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x_s)- \mathbb{G}_\alpha[g(k\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}])](x_s),\end{aligned}$$ where $s>0$ small. Together with Lemma \[lm 3.1\], (\[b k\]) holds.$\Box$ Strong singularity for $p\in(1+\frac{2\alpha}{N},\frac{N+\alpha}{N-\alpha})$ ---------------------------------------------------------------------------- In this subsection, we consider the limit of $\{u_k\}$ as $k\to\infty$, where $u_k$ is the weak solution of $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+u^p=k\frac{\partial^\alpha\delta_0}{\partial \vec{n}^\alpha}\quad &{\rm in}\quad\ \ \bar\Omega,\\[3mm] \phantom{-----\ } u=0\quad &{\rm in}\quad\ \ \bar\Omega^c, \end{array}$$ here $0\in\partial\Omega$ and $p\in(1+\frac{2\alpha}{N},\frac{N+\alpha}{N-\alpha})$. From Theorem \[teo 1\] $(iii)$, we know that $u_k$ is a classical solution of $$\label{4.101} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+u^p=0\quad &{\rm in}\quad\ \ \Omega,\\[3mm] \phantom{-----\ } u=0\quad &{\rm in}\quad\ \ \Omega^c\setminus\{0\}. \end{array}$$ In order to study the limit of $\{u_k\}$ as $k\to\infty$, we have to obtain a super solution of (\[4.101\]). To this end, we consider the function $$\label{4.2} w_p(x)=|x|^{-\frac{2\alpha}{p-1}},\qquad x\in{\mathbb{R}}^N\setminus\{0\}.$$ \[lm 4.1\] Assume that $p\in(1+\frac{2\alpha}{N},\frac{N+\alpha}{N-\alpha})$ and $w_p$ is defined in (\[4.2\]). Then there exists $\lambda_0>0$ such that $\lambda_0 w_p$ is a super solution of (\[4.101\]). [**Proof.**]{} For $p\in(1+\frac{2\alpha}{N},\frac{N+\alpha}{N-\alpha})$, we have that $-\frac{2\alpha}{p-1}\in (-N,-N+2\alpha)$ and from [@FQ], it shows that there exists $c(p)<0$ such that $$(-\Delta)^\alpha w_p(x)=c(p)|x|^{-\frac{2\alpha}{p-1}-2\alpha},\quad x\in{\mathbb{R}}^N\setminus\{0\},$$ thus, taking $\lambda_0=|c(p)|^{\frac{1}{p-1}}$, we derive that $$(-\Delta)^\alpha (\lambda_0 w_p)+ (\lambda_0w_p)^p=0\quad{\rm in}\quad {\mathbb{R}}^N\setminus\{0\}.$$ Together with $\lambda_0 w_p>0$ in $\Omega^c$, $\lambda_0 w_p$ is a super solution of (\[4.101\]). The proof ends. $\Box$ We observe that the super solution $\lambda_0w_p$ constructed in Lemma \[lm 4.1\] could control the asymptotic behavior of $u_\infty$ near the origin, but for $\partial\Omega\setminus\{0\}$, $\lambda_0w_p$ does not provide enough information for us. To control the behavior of $u_\infty$ on $\partial\Omega\setminus\{0\}$, we have to construct new super solutions. For any given $y_0\in\partial\Omega\setminus\{0\}$, we denote $\eta_0:{\mathbb{R}}^N\to[0,1]$ a $C^2$ functions such that $$\label{4.5} \eta_0(x)=\left\{\arraycolsep=1pt \begin{array}{lll} 0,\quad &x\in B_r(y_0),\\[2mm] 1,\quad &x\in {\mathbb{R}}^N\setminus B_{2r}(y_0), \end{array} \right.$$ where $r=\frac{|y_0|}{8}$. \[lm 4.2\] Assume that $p\in(1+\frac{2\alpha}{N},\frac{N+\alpha}{N-\alpha})$ and $w_{\lambda,j}=\lambda\tilde w_p+j \eta_1$, where $\lambda,j>0$, $\tilde w_p=w_p\eta_0$ in ${\mathbb{R}}^N$ and $\eta_1=\mathbb{G}_\alpha[1]$. Then there exist $\lambda_1>0$ and $j_1>0$ depending on $|y_0|$ such that $w_{\lambda_1,j_1}$ is a super solution of (\[4.101\]). [**Proof.**]{} For $x\in \Omega\setminus B_{4r}(y_0)$, we have that $\tilde w_p(x)=w_p(x)$ and $$\begin{aligned} (-\Delta)^\alpha \tilde w_p(x)&=& -\lim_{\epsilon\to0^+}\int_{{\mathbb{R}}^N\setminus B_\epsilon(x)}\frac{\tilde w_p(z)-w_p(x)}{|z-x|^{N+2\alpha}}dz \\&=& (-\Delta)^\alpha w_p(x)-\lim_{\epsilon\to0^+}\int_{{\mathbb{R}}^N\setminus B_\epsilon(x)}\frac{\tilde w_p(z)-w_p(z)}{|z-x|^{N+2\alpha}}dz \\&\ge& (-\Delta)^\alpha w_p(x)-\int_{B_{2r}(y_0)}\frac{w_p(z)}{|z-x|^{N+2\alpha}}dz \\&\ge& c(p)|x|^{-\frac{2\alpha}{p-1}-2\alpha} -c_{30}r^{-\frac{2\alpha}{p-1}-2\alpha},\end{aligned}$$ where $c_{30}>0$ and the last inequality used the facts $|z-x|\ge 2r$ and $w_p(z)\le r^{-\frac{2\alpha}{p-1}}$. For $x\in B_{2r}(0)\setminus\{0\}$, take $\lambda=\lambda_0$ from Lemma \[lm 4.1\] and $j\ge c_{30}\lambda_0r^{-\frac{2\alpha}{p-1}-2\alpha}$, then we have $$\begin{aligned} (-\Delta)^\alpha w_{\lambda,j}(x)+ w_{\lambda,j}^p(x) &\ge & -c(p)\lambda_0|x|^{-\frac{2\alpha}{p-1}-2\alpha} +w_p^p(x)\ge 0.\end{aligned}$$ We observe that there exists $c_{31}>0$ dependent of $r$ such that $$|(-\Delta)^\alpha \tilde w_p|\le c_{31}\quad{\rm in}\quad \Omega\setminus B_{2r}(0),$$ then take $j\ge c_{31}\lambda_0$, we have that $$(-\Delta)^\alpha w_{\lambda_0, j}\ge 0,\quad \forall x\in \Omega\setminus B_{2r}(0).$$ Therefore, letting $\lambda_1=\lambda_0$ and $j_1=\max\{c_{31}\lambda_0, c_{30}\lambda_0r^{-\frac{2\alpha}{p-1}-2\alpha}\}$, we have that $$(-\Delta)^\alpha w_{\lambda_1,j_1}+ w_{\lambda_1,j_1}^p\ge 0\quad{\rm in}\quad \Omega.$$ The proof ends. $\Box$ Let $x_s=s\vec{n}_0\in\Omega$ and a set $$A_r=\bigcup_{s\in (0,r)}B_{\frac s8}(x_s).$$ It is obvious that $A_r$ is a cone with the vertex at the origin. \[lm 4.3\] Assume that $p\in(0,\frac{N+\alpha}{N-\alpha})$, then there exists $c_{32}>0$ such that for any $x\in A_{r_0}$, $$\label{4.6} \mathbb{G}_\alpha[(\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}])^p](x) \le \left\{\arraycolsep=1pt \begin{array}{lll} c_{32} |x|^{-(N-\alpha)p+2\alpha}\quad &{\rm if}\quad p\in (\frac{2\alpha}{N-\alpha}, \frac{N+\alpha}{N-\alpha}),\\[1.5mm] -c_{32} \ln |x| \quad &{\rm if}\quad p= \frac{2\alpha}{N-\alpha},\\[1.5mm] c_{32}\quad &{\rm if}\quad p\in(0,\frac{2\alpha}{N-\alpha}). \end{array} \right.$$ [**Proof.**]{} Since $\partial\Omega$ is $C^2$, then for $r_0\in(0, 1/2)$ small enough, we observe that for any $x\in B_{\frac s8}(x_s)$ with $s\in(0,r_0)$, $$\frac {3s}{4}\le \rho_{\partial\Omega}(x)\le \frac {5s}{4}$$ and for any $t\in(0,\frac s8)$, $$|x-x_t|\ge \frac{5s}8 \ge \frac12\max\{\rho_{\partial\Omega}(x),\rho_{\partial\Omega}(x_t)\}.$$ Then it follows by [@BV Theorem 1.1, Theorem 1.2] that there exists $c_{33}>1$ such that $$\frac1{c_{33}} s^{\alpha-N} t^\alpha\le G_\alpha(x,x_t)\le c_{33}s^{\alpha-N} t^\alpha, \quad \forall x\in B_{\frac s8}(x_s).$$ Thus, there exists $c_{34}>0$ independent of $s,t$ such that $$\frac1{c_{34}}s^{-N+\alpha}\le \mathbb{G}_\alpha[t^{-\alpha}\delta_{x_t}](x)\le c_{34}s^{-N+\alpha}, \quad \forall x\in B_{\frac s8}(x_s),$$ which implies that $$\label{11.04.1} \frac1{c_{34}}s^{-N+\alpha}\le \mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x)\le c_{34}s^{-N+\alpha},\quad \forall x\in B_{\frac s8}(x_s).$$ From Lemma \[lm 2.1\], it shows that for any $x\in\Omega$, $$\label{11.04.3} \mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x)\le c_5 |x|^{-N+\alpha},\qquad \forall x\in \Omega.$$ It follows by (\[annex 01\]) and (\[11.04.3\]) that $$\arraycolsep=1pt \begin{array}{lll} \mathbb{G}_\alpha[(\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}])^p](x_s)\le c_5^p\int_\Omega G_\alpha(x_s,y)\frac{1}{|y|^{(N-\alpha)p}}dy \\[3mm]\phantom{--------\ } \le c_5^{p+1} \int_\Omega\frac{|y|^\alpha}{|x_s-y|^{N-\alpha}} \frac{1}{|y|^{(N-\alpha)p}}dy \\[3mm]\phantom{--------\ } = c_5^{p+1} s^{2\alpha-(N-\alpha)p}\int_{\tilde \Omega_s} \frac1{|\vec{n}_0-z|^{N-\alpha}} \frac{1}{|z|^{(N-\alpha)p-\alpha}}dz \\[3mm]\phantom{--------\ } = c_5^{p+1}s^{2\alpha-(N-\alpha)p}\left[\int_{\tilde \Omega_s\cap B_{1/2}(\vec{n}_0)} \frac1{|\vec{n}_0-z|^{N-\alpha}} \frac{1}{|z|^{(N-\alpha)p-\alpha}}dz\right. \\[3mm]\phantom{----------------} \left.+ \int_{\tilde\Omega_s\cap B_{\frac12}^c(\vec{n}_0)} \frac1{|\vec{n}_0-z|^{N-\alpha}} \frac{1}{ |z|^{(N-\alpha)p-\alpha}}dz\right] \\[3mm]\phantom{--------\ } :=c_5^{p+1}s^{2\alpha-(N-\alpha)p}[I_1(s)+I_2(s)], \end{array}$$ where $\Omega_s=\{sz:\ z\in\Omega\}$. We observe that $$I_1(s)\le c_{35}\int_{ B_{1/2}(\vec{n}_0)}\frac1{|\vec{n}_0-z|^{N-\alpha}} dz\le c_{36}$$ and since $(N-\alpha)p-\alpha<N$ by $p\in(0,\frac{N+\alpha}{N-\alpha})$, then $$\begin{aligned} I_2(s)&\le& c_{37}\int_{\tilde \Omega_s}\frac{1}{|z|^{(N-\alpha)p-\alpha}(1+|z|)^{N-\alpha}} dz \\&\le& c_{37}\int_{B_{\frac Rs}(0)\setminus B_{\frac12}(0)}\frac{1}{|z|^{(N-\alpha)p-2\alpha+N}} dz \\&\le& \left\{\arraycolsep=1pt \begin{array}{lll} c_{38} s^{(N-\alpha)p-2\alpha}\quad &{\rm if}\quad p\in (\frac{2\alpha}{N-\alpha}, \frac{N+\alpha}{N-\alpha}),\\[1.5mm] -c_{38} \ln s \quad &{\rm if}\quad p= \frac{2\alpha}{N-\alpha},\\[1.5mm] c_{38}\quad &{\rm if}\quad p\in(0,\frac{2\alpha}{N-\alpha}), \end{array} \right.\end{aligned}$$ where $c_{35},c_{36},c_{37},c_{38}>0$ and $R>0$ such that $\Omega \subset B_R(0)$. Then (\[4.6\]) holds. $\Box$ [**Proof of Theorem \[teo 2\] $(ii)$.**]{} For $p\in(1+\frac{2\alpha}{N},\frac{N+\alpha}{N-\alpha})$, we have that $$-\frac{2\alpha}{p-1}\in(-N,-N+\alpha)$$ and it follows by Lemma \[lm 2.1\] that $$u_k(x)\le k\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_0}{\partial \vec{n}^\alpha}](x)\le \frac{c_5k}{|x|^{N-\alpha}},\quad x\in\Omega.$$ Then $\lim_{x\in\Omega,|x|\to0}\frac{u_k(x)}{ w_p(x)}=0$ and we claim that $$u_k\le \lambda_0w_p\quad{\rm in}\quad\Omega.$$ In fact, if it fails, then there exists $z_0\in\Omega$ such that $$(u_k-\lambda_0w_p)(z_0)=\inf_{\Omega}(u_k-\lambda_0w_p)={\rm ess}\inf_{{\mathbb{R}}^N}(u_k-\lambda_0w_p)<0.$$ Then we have $(-\Delta)^\alpha(u_k-\lambda_0w_p)(z_0)<0$, which contradicts the fact that $$(-\Delta)^\alpha(u_k-\lambda_0w_p)(z_0)=\lambda_0w_p^p(z_0)-u_k^p(z_0)>0.$$ By monotonicity of the mapping $k\to u_k$, there holds $$u_\infty(x):=\lim_{k\to\infty} u_k(x),\quad x\in{\mathbb{R}}^N\setminus\{0\},$$ which is a classical solution of (\[4.2\]) and $$u_\infty(x)\le \lambda_0w_p(x)= \lambda_0 |x|^{-\frac{2\alpha}{p-1}},\quad \forall x\in\Omega.$$ By applying Lemma \[lm 4.2\], we obtain that $u_\infty$ is continuous up to the boundary except the origin. Finally, we claim that there exists $c_{39}>0$ and $t_0<\sigma_0$ such that $$\label{13-08-0} u_\infty(x_t)\ge c_{39}t^{-\frac{2\alpha}{p-1}},\quad \forall t\in(0,t_0),$$ where $x_t=t\vec{n}_0\in\Omega$. Indeed, let $r_k=(\sigma^{-1} k)^{\frac{p-1}{(N-\alpha)p-N-\alpha}}$, where $\sigma>0$ will be chosen later, then $k=\sigma r_k^{\frac{(N-\alpha)p-N-\alpha}{p-1}}$ and for $x\in A_{r_0}\cap \left[B_{r_k}(0)\setminus B_{\frac{r_k}{2}}(0)\right]$, we apply Lemma \[lm 4.3\] with $p\in(1+\frac{2\alpha}{N},\frac{N+\alpha}{N-\alpha})$ that $$\begin{aligned} u_k(x)&\ge & k\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x)-k^p\mathbb{G}_\alpha[(\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}])^p](x) \\ &\ge& c_5k|x|^{\alpha-N}[1-c_{40}k^{p-1}|x|^{(\alpha-N)p+\alpha+N}] \\&\ge& c_5\sigma r_k^{-\frac{2\alpha }{p-1}}[1-c_{40}\sigma^{p-1} r_k^{p-1}(r_k/2)^{(\alpha-N)p+\alpha+N}] \\&\ge& c_5\sigma r_k^{-\frac{2\alpha }{p-1}}[1-c_{40}\sigma^{p-1} 2^{(N-\alpha)p-\alpha-N}] \\&\ge& \frac{c_5\sigma}{2}|x|^{-\frac{2\alpha }{p-1}},\end{aligned}$$ where we choose $\sigma$ such that $c_{40}\sigma^{p-1} 2^{(N-\alpha)p-\alpha-N}=\frac12$. Then for any $x\in A_{r_0}\cap B_{r_k}^c(0)$, there exists $k>0$ such that $x\in A_{r_0}\cap [B_{r_k}(0)\setminus B_{\frac{r_k}2}(0)]$ and then $$u_\infty(x)\ge u_k(x)\ge \frac{c_5\sigma}{2}|x|^{-\frac{2\alpha }{p-1}},\qquad \forall x\in A_{r_0}\cap B_{r_k}^c(0).$$ This ends the proof. $\Box$ The limit of $\{u_k\}$ blows up when $p\in(0,1+\frac{2\alpha}{N}]$ ------------------------------------------------------------------ In this subsection, we derive the blow-up behavior of the limit of $\{u_k\}$ when $p\in(0,1+\frac{2\alpha}{N}]$. To this end, we first do precise estimate for $u_k$. \[lm 3.2\] Assume that $g(s)=s^p$ with $p\in(1,\frac{N}{N-\alpha}]$ and $u_k$ is the solution of (\[eq 1.1\]) obtained by Theorem \[teo 1\]. Then there exist $c_{41}>0$, $r_0\in(0,\frac14)$ and $\{r_k\}_k\subset(0,r_0)$ satisfying $r_k\to0$ as $k\to\infty$ such that $$\label{4.3.1} u_k(x)\ge\frac{c_{41}|x|^{-N}}{-\ln(|x|)},\qquad \forall x\in A_{r_0}\cap B_{r_k}^c(0).$$ [**Proof.**]{} [*To prove (\[4.3.1\]) in the case of $p\in (\frac{2\alpha}{N-\alpha},1+\frac{2\alpha}{N})$.*]{} Let $r_j=j^{-\frac1\alpha}$ with $j\in(k_0,k)$, then $j=r_j^{-\alpha}$. Applying Lemma \[lm 4.3\] with $p\in (\frac{2\alpha}{N-\alpha},1+\frac{2\alpha}{N})$ and (\[11.04.1\]), we have that for $x\in A_{r_0}\cap \left[B_{r_j}(0)\setminus B_{\frac{r_j}2}(0)\right]$, $$\begin{aligned} u_j(x)&\ge & j\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x)-j^p\mathbb{G}_\alpha[(\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}])^p](x) \\ &\ge& c_{34}^{-1}jr_j^{\alpha-N}-c_{32}j^p|x|^{(\alpha-N)p+2\alpha} \\&\ge& c_{34}^{-1}r_j^{-N}-c_{32}r_j^{-\alpha p-(N-\alpha)p+2\alpha} \\&\ge& \frac{1}{2c_{34}}|x|^{-N},\end{aligned}$$ where the last inequality holds since $-\alpha p-(N-\alpha)p+2\alpha>-N$ and $r_j\to0$ as $j\to\infty$. Then for any $x\in A_{r_0}\cap B_{r_k}^c(0)$, there exists $j\in (k_0,k)$ such that $x\in A_{r_0}\cap [B_{r_j}(0)\setminus B_{\frac{r_j}2}(0)]$ and then $$u_k(x)\ge u_j(x)\ge \frac{1}{2c_{34}}|x|^{-N},\qquad \forall x\in A_{r_0}\cap B_{r_k}^c(0).$$ [*To prove (\[4.3.1\]) in the case of $p\in(0,\frac{2\alpha}{N-\alpha}]$.*]{} Let $r_j=j^{-\frac1\alpha}$ with $j\in(k_0,k)$, then $j=r_j^{-\alpha}$ and for $x\in A_{r_0}\cap \left[B_{r_j}(0)\setminus B_{\frac{r_j}2}(0)\right]$, we have that $$\begin{aligned} u_j(x)&\ge & j\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x)-j^p\mathbb{G}_\alpha[(\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}])^p](x) \\ &\ge& c_{34}^{-1}j|x|^{\alpha-N}-c_{32}j^p \\&\ge& c_{34}^{-1}r_j^{-N}-c_{32}r_j^{-\alpha p} \\&\ge& \frac{1}{2c_{34}}|x|^{-N},\end{aligned}$$ where the last inequality holds since $-\alpha p>-N$ and $r_j\to0$ as $j\to\infty$. For any $x\in A_{r_0}\cap B_{r_k}^c(0)$, there exists $j\in (k_0,k)$ such that $x\in A_{r_0}\cap [B_{r_j}(0)\setminus B_{\frac{r_j}2}(0)]$ and then $$u_k(x)\ge u_j(x)\ge \frac{1}{2c_{34}}|x|^{-N},\qquad \forall x\in A_{r_0}\cap B_{r_k}^c(0).$$ [*To prove (\[4.3.1\]) in the case of $p=1+\frac{2\alpha}{N}$.*]{} Let $\rho_j=j^{-\frac1\alpha}$ and $r_j=\frac{\rho_j}{[-\log(\rho_j)]^{\frac1\alpha}}$, then $j=\rho_j^{-\alpha}$ and applied Lemma \[lm 4.3\] for $x\in A_{r_0}\cap \left[B_{r_j}(0)\setminus B_{\frac{r_j}{2}}(0)\right]$, $$\begin{aligned} u_j(x)&\ge & j\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}](x)-j^p\mathbb{G}_\alpha[(\mathbb{G}_\alpha[\frac{\partial^\alpha \delta_{0}}{\partial \vec{n}^\alpha}])^p](x) \\ &\ge& c_{34}^{-1}j|x|^{\alpha-N}-c_{32}j^p|x|^{(\alpha-N)p+2\alpha} \\&\ge& c_{34}^{-1}\rho_j^{-N} (-\log \rho_j)^{\frac{N-\alpha}{\alpha}}-c_{42} \rho_j^{-N} (-\log \rho_j)^{\frac{(N-\alpha)p-2\alpha}{\alpha}} \\&=& c_{34}^{-1}\rho_j^{-N} (-\log \rho_j)^{\frac{N-\alpha}{\alpha}}\left[1-c_{42}(-\log \rho_j)^{\frac{(N-\alpha)p-2\alpha}{\alpha}-\frac{N-\alpha}{\alpha}}\right] \\&\ge& c_{34}^{-1}\frac{r_j^{-N}}{-\log \rho_j}[1-c_{42}(-\log \rho_j)^{\frac{(N-\alpha)p-N-\alpha}{\alpha}}] \\&\ge& \frac{c_{34}|x|^{-N}}{-2\log |x|},\end{aligned}$$ where $c_{42}>0$ and we used the facts that $\log(\rho_j)\le c\log r_j\le c\log |x|$ and $\frac{(N-\alpha)p-N-\alpha}{\alpha}<0$. Then for any $x\in A_{r_0}\cap B_{r_k}^c(0)$, there exists $j\in (k_0,k)$ such that $x\in A_{r_0}\cap [B_{r_j}(0)\setminus B_{\frac{r_j}2}(0)]$ and then $$u_k(x)\ge \frac{c_{34}|x|^{-N}}{-2\log |x|},\qquad x\in A_{r_0}\cap B_{r_k}^c(0).$$ The proof ends. $\Box$ [**Proof of Theorem \[teo 2\] $(iii)$.**]{} It derives by Lemma \[lm 3.2\] that $$\label{3.2.3} \pi_k:=\int_{B_{r_0}(0)}u_k(x)\ge c_{41}\int_{A_{r_0}\cap B_{r_k}^c(0)}\frac{|x|^{-N}}{-\log|x|}dx\to\infty\quad {\rm as}\ k\to\infty.$$ Fix $y_0\in \Omega\setminus \bar B_{r_0}(0)$, it follows by Lemma 2.4 in [@CY] that problem $$\label{5.1} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+u^p=0 \quad & {\rm in}\quad B_{\varrho_0}(y_0),\\[2mm] \phantom{ (-\Delta)^\alpha +u^p} u=0 \quad & {\rm in}\quad {\mathbb{R}}^N \setminus (B_{\varrho_0}(y_0)\cup B_{r_0}(0)),\\[2mm] \phantom{ (-\Delta)^\alpha +u^p} u=u_k \quad & {\rm in}\quad B_{r_0}(0) \end{array}$$ admits a unique solution $w_k$, where $\varrho_0=\min\{\rho_{\partial\Omega}(y_0),|y_0|-r_0\}$. By Lemma 2.2 in [@CY], $$\label{4.1.3} u_{k}\ge w_k\quad {\rm in}\quad B_{\varrho_0}(y_0).$$ Let $\tilde w_k=w_k-u_k\chi_{B_{r_0}(0)},$ then $\tilde w_k=w_k$ in $B_{\varrho_0}(y_0)$ and for $x\in B_{\varrho_0}(y_0)$, $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha \tilde w_k(x) = -\lim_{\epsilon\to0^+}\int_{B_{\varrho_0}(y_0)\setminus B_\epsilon(x)}\frac{w_k(z)-w_k(x)}{|z-x|^{N+2\alpha}}dz \\[3mm]\phantom{--------}+\lim_{\epsilon\to0^+}\int_{B_{\varrho_0}^c(y_0)\setminus B_\epsilon(x)}\frac{w_k(x)}{|z-x|^{N+2\alpha}}dz \\[3mm]\phantom{------} =-\lim_{\epsilon\to0^+}\int_{{\mathbb{R}}^N\setminus B_\epsilon(x)}\frac{w_k(z)-w_k(x)}{|z-x|^{N+2\alpha}}dz +\int_{B_{r_0}(0)}\frac{u_k(z)}{|z-x|^{N+2\alpha}}dz \\[3mm]\phantom{------}\ge(-\Delta)^\alpha w_k(x)+c_{42}\pi_k, \end{array}$$ where $c_{42}=(|y_0|+r_0)^{-N-2\alpha}$ and the last inequality follows by the fact of $$|z-x|\le |x|+|z|\le |y_0|+r_0\quad {\rm for}\ z\in B_{\frac14}(0),\ x\in B_{\frac14}(y_0).$$ Therefore, $$\begin{aligned} (-\Delta)^\alpha \tilde w_k(x)+\tilde w_k^p(x) &\ge& (-\Delta)^\alpha w_k(x)+w_k^p(x)+ c_{42}\pi_k \\ &=&c_{42}\pi_k, \qquad x\in B_{\varrho_0}(y_0), \end{aligned}$$ that is, $\tilde w_k$ is a super solution of $$\label{4.1.2} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha u+u^p=c_{42}\pi_k \quad & {\rm in}\quad B_{\varrho_0}(y_0),\\[2mm] \phantom{ (-\Delta)^\alpha +u^{p,}} u=0 \quad & {\rm in}\quad B_{\varrho_0}^c(y_0). \end{array}$$ Let $\eta_1$ be the solution of $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u=1 \quad & {\rm in}\quad B_{\varrho_0}(y_0),\\[2mm] \phantom{ (-\Delta)^\alpha } u=0 \quad & {\rm in}\quad B^c_{\varrho_0}(y_0). \end{array}$$ Then $(c_{42}\pi_k)^{\frac1p} \frac{\eta_1}{2\max_{{\mathbb{R}}^N}\eta_1}$ is sub solution of (\[4.1.2\]) for $k $ large enough. By Lemma 2.2 in [@CY], we have that $$\tilde w_k(x)\ge (c_{42}\pi_k)^{\frac1p} \frac{\eta_1(x)}{2\max_{{\mathbb{R}}^N}\eta_1},\quad \forall x\in B_{\varrho_0}(y_0),$$ which implies that $$w_k(y)\ge c_{43} (c_{42}\pi_k)^{\frac1p},\qquad \forall y\in B_{\frac{\varrho_0}{2}}(y_0),$$ where $c_{43}=\min_{x\in B_{\varrho_0}(y_0)}\frac{\eta_1(x)}{2\max_{{\mathbb{R}}^N}\eta_1}$. Therefore, (\[4.1.3\]) and (\[3.2.3\]) imply that $$\lim_{k\to\infty}u_{k}(y)\ge \lim_{k\to\infty}w_k(y)=\infty,\qquad\forall y\in B_{\frac{\varrho_0}{2}}(y_0).$$ Similarly, we can prove $$\lim_{k\to\infty}u_{k}(y)\ge \lim_{k\to\infty}w_k(y)=\infty,\qquad\forall y\in \Omega.$$ The proof ends.$\Box$ Nonexistence in the critical case ================================= In this section, we prove the nonexistence in the critical case. To this end, we consider the weak solution to elliptic problem $$\label{14-08-0} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+u^{\frac{N+\alpha}{N-\alpha}}=k\frac{\partial^\alpha\delta_0}{\partial e_N^\alpha}\quad &{\rm in}\quad\ \ \overline{{\mathbb{R}}^N_+},\\[3mm] \phantom{(-\Delta)^\alpha +u^p} u=0\quad &{\rm in}\quad\ \ {\mathbb{R}}^N_-, \end{array}$$ where ${\mathbb{R}}^N_+={\mathbb{R}}^{N-1}\times{\mathbb{R}}_+$ and $e_N=(0,\cdots,0,1)$. \[weak definition\] A function $u\in L^1({\mathbb{R}}^N,\mu dx)$ is a weak solution of (\[14-08-0\]) if $u^p\in L^1({\mathbb{R}}^N,\rho^\alpha \mu dx)$ and $$\label{weak sense} \int_{{\mathbb{R}}^N_+} [u(-\Delta)^\alpha\xi+u^{\frac{N+\alpha}{N-\alpha}}\xi]dx=\frac{\partial^\alpha \xi(0)}{\partial e_N^\alpha},\quad \forall\xi\in \mathbb{X}_{\alpha,{\mathbb{R}}^N_+},$$ where $\mu(x)=\frac1{1+|x|^{N+2\alpha}}$, $\rho(x)=\min\{1,\rho_{\partial\Omega}(x)\}$ and $\mathbb{X}_{\alpha,{\mathbb{R}}^N_+}\subset C({\mathbb{R}}^N)$ is the space of functions $\xi$ satisfying: \(i) the support of $\xi$ is a compact set in $\bar{\mathbb{R}}^N_+$; \(ii) $(-\Delta)^\alpha\xi(x)$ exists for any $x\in{\mathbb{R}}^N_+$ and there exists $c>0$ such that $$|(-\Delta)^\alpha\xi(x)|\leq c\mu(x),\quad \forall x\in{\mathbb{R}}^N_+;$$ \(iii) there exist $\varphi\in L^1({\mathbb{R}}^N_+,\rho^\alpha dx)$ and $\varepsilon_0>0$ such that $|(-\Delta)_\varepsilon^\alpha\xi|\le \varphi$ a.e. in ${\mathbb{R}}^N_+$, for all $\varepsilon\in(0,\varepsilon_0]$. Let $\mathbb{G}_{\alpha,{\mathbb{R}}^N_+}$ the Green’s function on ${\mathbb{R}}^N_+\times{\mathbb{R}}^N_+$ and $$\label{5.2} \Gamma_\alpha(x)=\lim_{t\to0}t^{-\alpha}\mathbb{G}_{\alpha,{\mathbb{R}}^N_+}(x,te_N).$$ \[lm 5.1\] Let $\Gamma_\alpha$ defined in (\[5.2\]), then $$\label{5.1.1} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha \Gamma_\alpha=\frac{\partial^\alpha \delta_0}{\partial e_N^\alpha}\qquad &{\rm in}\quad\bar {\mathbb{R}}^N_+,\\[2mm] \phantom{--- } \Gamma_\alpha=0 \quad & {\rm in} \quad {\mathbb{R}}^N_-. \end{array}$$ Moreover, $$\label{5.3} \Gamma_\alpha(x)=|x|^{-N+\alpha}\Gamma_\alpha\left(\frac{x}{|x|}\right),\qquad x\in{\mathbb{R}}^N,$$ and $$\Gamma_\alpha\left(\frac{x}{|x|}\right)\ \left\{\arraycolsep=1pt \begin{array}{lll} >0 \quad & {\rm if}\quad x\in {\mathbb{R}}^N_+,\\[2mm] =0 \quad & {\rm if}\quad x\not\in {\mathbb{R}}^N_+. \end{array} \right.$$ [**Proof.**]{} We observe that $$(-\Delta)^\alpha_x t^{-\alpha}\mathbb{G}_{\alpha,{\mathbb{R}}^N_+}(x,te_N)=t^{-\alpha}\delta_{te_N}$$ and $$\lim_{t\to0^+}\langle t^{-\alpha}\delta_{te_N},\xi\rangle=\frac{\partial^\alpha \xi(0)}{\partial e_N^\alpha},\qquad \forall \xi\in\mathbb{X}_{\alpha,{\mathbb{R}}^N_+}.$$ Then (\[5.1.1\]) holds in the weak sense. By the regularity results, $\Gamma_\alpha$ is a solution of $$\label{5.1.2} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha \Gamma_\alpha=0\qquad &{\rm in}\quad{\mathbb{R}}^N_+,\\[2mm] \phantom{--- } \Gamma_\alpha=0 \quad & {\rm in} \quad\overline{{\mathbb{R}}^N_-}\setminus\{0\}. \end{array}$$ Let $\Gamma_{\alpha,\lambda}(x)=\lambda^{N-\alpha}\Gamma_{\alpha}(\lambda x)$ and $\xi_\lambda(x)=\xi(x/\lambda)$ for $\xi\in\mathbb{X}_{\alpha,{\mathbb{R}}^N_+}$, then we have that $$\begin{aligned} \int_{{\mathbb{R}}^N_+}\Gamma_{\alpha,\lambda}(-\Delta)^\alpha \xi dx&=&\lambda^\alpha\int_{{\mathbb{R}}^N_+}\Gamma_{\alpha}(z)(-\Delta)^\alpha \xi_\lambda(x)dx, \\ &=& \lambda^\alpha \frac{\partial^\alpha \xi_\lambda(0)}{\partial e_N^\alpha},\end{aligned}$$ which implies that $$\int_{{\mathbb{R}}^N_+}\Gamma_{\alpha,\lambda}(-\Delta)^\alpha \xi dx=\frac{\partial^\alpha \xi(0)}{\partial e_N^\alpha}.$$ By the uniqueness, we derive that $$\lambda^{N-\alpha}\Gamma_{\alpha}(\lambda x)=\Gamma_{\alpha}(x),$$ which, choosing $\lambda=\frac1{|x|}$, implies (\[5.3\]). The last argument is obvious. $\Box$ \[teo 4.1\] Let $k>0$, then problem (\[14-08-0\]) has no any weak solution. [**Proof.**]{} If there exists a weak solution $u_k$ to (\[14-08-0\]), then we observe that $$u_k>0\qquad {\rm in}\quad {\mathbb{R}}^N_+.$$ By Maximum Principle, we have that $$\label{5.5} u_k\le k\Gamma_\alpha\qquad {\rm in}\quad {\mathbb{R}}^N.$$ Denoting $$u_\infty=\lim_{k\to\infty}u_k\qquad {\rm in}\quad {\mathbb{R}}^N.$$ We claim that $$\label{3.2.1} u_\infty(x)=|x|^{\alpha-N}u_\infty(\frac{x}{|x|}),\quad \forall x\in{\mathbb{R}}^N\setminus\{0\}.$$ Indeed, let $$\tilde u_\lambda(x)=\lambda^{N-\alpha}u_k(\lambda x),\quad \forall x\in {\mathbb{R}}^N\setminus\{0\}.$$ By direct computation, we have that for $x\in{\mathbb{R}}^N_+$, $$\begin{aligned} (-\Delta)^\alpha \tilde u_\lambda(x) +\tilde u_\lambda^{\frac{N+\alpha}{N-\alpha}}(x) &=&\lambda^{N+\alpha}[(-\Delta)^\alpha u_k(\lambda x) + u_k^{\frac{N+\alpha}{N-\alpha}}(\lambda x)] \nonumber \\ &=&0.\label{13-09-5}\end{aligned}$$ Moreover, for $f\in C_0^1({\mathbb{R}}^N_+)$, $$\begin{aligned} \langle(-\Delta)^\alpha \tilde u_\lambda +\tilde u_\lambda^{\frac{N+\alpha}{N-\alpha}}, f\rangle&=&\lambda^{N+\alpha}\int_{{\mathbb{R}}^N} [(-\Delta)^\alpha u_k(\lambda x) + u_k^{\frac{N+\alpha}{N-\alpha}}(\lambda x)]f(x)dx\nonumber \\&=&\lambda^{ \alpha }\int_{{\mathbb{R}}^N} [(-\Delta)^\alpha u_k(z) + u_k^{\frac{N+\alpha}{N-\alpha}}(z)]f\left(\frac{z}{\lambda}\right)dz\nonumber \\ &=&\lambda^{\alpha}k\frac{\partial^\alpha f(0)}{\partial e_N^\alpha}.\end{aligned}$$ Thus, $$\label{3.2.2} (-\Delta)^\alpha \tilde u_\lambda +\tilde u_\lambda^{\frac{N+\alpha}{N-\alpha}}=\lambda^{\alpha}k\frac{\partial^\alpha \delta_0}{\partial e_N^\alpha}\quad {\rm in}\ \ {\mathbb{R}}^N_+.$$ We observe that $\lim_{|x|\to\infty}\tilde u_\lambda(x)=0$ and $u_{k\lambda^{\alpha }}$ is the unique weak solution of (\[14-08-0\]) with $k$ replaced by $\lambda^\alpha k$, then for $x\in {\mathbb{R}}^N\setminus\{0\}$, $$\label{21-10-1} u_{k\lambda^{\alpha}}(x)=\tilde u_\lambda (x)=\lambda^{N-\alpha}u_k(\lambda x)$$ and letting $k\to\infty$ we have that $$u_{\infty}(x)=\lambda^{N-\alpha}u_\infty(\lambda x),\qquad \forall x\in {\mathbb{R}}^N\setminus\{0\},$$ which implies (\[3.2.1\]) by taking $\lambda=|x|^{-1}$. Combine (\[5.3\]), (\[5.5\]) and (\[21-10-1\]), then we have that $$\begin{aligned} u_{k\lambda^{\alpha}}(x) \le \lambda^{N-\alpha}k\Gamma_\alpha(\lambda x) = k\Gamma_\alpha(x),\quad \forall x\in {\mathbb{R}}^N. \end{aligned}$$ Thus, $$u_\infty(x)\le k\Gamma_\alpha(x),\quad \quad \forall x\in {\mathbb{R}}^N.$$ By arbitrary of $k$, it implies that $$u_\infty\equiv0,$$ then $u_1\equiv 0$ in ${\mathbb{R}}^N$, which is impossible. $\Box$ [**Proof of Theorem \[teo 4\].**]{} Without loss generality, we let $k=1$, $0\in\partial\Omega$ and $e_N$ is the unit normal vector pointing inside of $\Omega$ at $0$. If $$\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+u^{\frac{N+\alpha}{N-\alpha}}=\frac{\partial^\alpha\delta_0}{\partial e_N^\alpha}\quad &{\rm in}\quad\ \ \bar\Omega,\\[3mm] \phantom{(-\Delta)^\alpha +u^{\frac{N+\alpha}{N-\alpha}}} u=0\quad &{\rm in}\quad\ \ \bar\Omega^c \end{array}$$ admits a solution weak $v_1$, we claim that there is a weak solution of (\[14-08-0\]), then the contradiction is obtained from Theorem \[teo 4.1\]. In fact, we may assume that $$\Omega=B_1(e_N)\quad{\rm and}\quad B_m=B_m(me_N).$$ Then $$\Omega\subset B_m\subset B_{m+1}\quad {\rm and}\quad \lim_{m\to\infty} B_m={\mathbb{R}}^N_+.$$ Let $$v_m(x)=m^{\alpha-N}v_1(\frac xm),\quad x\in{\mathbb{R}}^N.$$ By direct computation, $v_m$ is a weak solution of $$\label{15-08-0-0} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+u^{\frac{N+\alpha}{N-\alpha}}=\frac{\partial^\alpha\delta_0}{\partial e_N^\alpha}\quad &{\rm in}\quad\ \ \bar B_m,\\[3mm] \phantom{(-\Delta)^\alpha +u^{\frac{N+\alpha}{N-\alpha}}} u=0\quad &{\rm in}\quad\ \ \bar B_m^c, \end{array}$$ [*We next show that $v_m\le v_{m+1}$ in ${\mathbb{R}}^N$.* ]{} From Proposition \[pr 1\], $$\label{08-14-00} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+u^{\frac{N+\alpha}{N-\alpha}}=t^{-\alpha}\delta_{te_N}\quad &{\rm in}\quad\ \ B_m,\\[3mm] \phantom{(-\Delta)^\alpha +u^{\frac{N+\alpha}{N-\alpha}}} u=0\quad &{\rm in}\quad\ \ B_m^c \end{array}$$ admits a unique weak solution, denoting $v_{m,t}$. Choose a sequence nonnegative functions $\{f_{m,i}\}_{i\in{\mathbb{N}}}\subset C^1({\mathbb{R}}^N)$ with support $B_1(e_N)$ such that $f_{m,i}\rightharpoonup t^{-\alpha}\delta_{te_N}$ as $i\to\infty$ in the distribution sense. Let $v_{m,i,t}$ be the unique solution of $$\label{08-14-00-0} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+u^{\frac{N+\alpha}{N-\alpha}}=f_{m,i}\quad &{\rm in}\quad\ \ B_m,\\[3mm] \phantom{(-\Delta)^\alpha +u^{\frac{N+\alpha}{N-\alpha}}} u=0\quad &{\rm in}\quad\ \ B_m^c \end{array}$$ and by Maximum Principle, see [@CY Lemma 2.3], derive that $$v_{m,i,t}\le \tilde v_{m+1,i,t}\quad{\rm in}\quad {\mathbb{R}}^N.$$ Together with the facts that $v_{m,i,t}\to v_{m,t}$ a.e. in ${\mathbb{R}}^N$ and $v_{m+1,i,t}\to v_{m+1,t}$ a.e. in ${\mathbb{R}}^N$ as $i\to\infty$, we obtain that $$\label{3.2} v_{1,t}\le v_{m,t}\le v_{m+1,t}\quad{\rm a.e.\ in}\ \ {\mathbb{R}}^N$$ and $$\int_{B_m}v_{m,t}^{\frac{N+\alpha}{N-\alpha}}\rho^\alpha dx< {\|\mathbb{G}_{\alpha, B_m}[f_{m,i}]\|}_{L^1(\Omega,\ \rho^\alpha dx)},$$ which implies that $$\label{08-14-0-1} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+u^{\frac{N+\alpha}{N-\alpha}}=\frac{\partial^\alpha \delta_0}{\partial e_N^\alpha}\quad &{\rm in}\quad\ \ \bar B_m,\\[3mm] \phantom{(-\Delta)^\alpha +u^{\frac{N+\alpha}{N-\alpha}}} u=0\quad &{\rm in}\quad\ \ \bar B_m^c \end{array}$$ admits a solution $v_m$ for any $m\in{\mathbb{N}}$ and $$\label{3.3} v_{m}\le v_{m+1}\quad {\rm a.e.\ in}\ \ {\mathbb{R}}^N.$$ We observe that $$\label{3.4} 0\le v_{m}\le \mathbb{G}_{\alpha, B_{m}}[\frac{\partial^\alpha \delta_0}{\partial e_N^\alpha}]\le \frac{c_5}{|x|^{N-\alpha}} \quad {\rm a.e.\ in}\ \ {\mathbb{R}}^N$$ and $$\int_{B_m}v_{m}^{\frac{N+\alpha}{N-\alpha}}\rho^\alpha dx< {\|\mathbb{G}_{\alpha, B_m}[\frac{\partial^\alpha \delta_0}{\partial e_N^\alpha}]\|}_{L^1(B_m,\ \rho^\alpha dx)}.$$ By (\[3.3\]) and (\[3.4\]), we see that the limit of $\{v_m\}$ exists, denoted it by $w_1$. Hence, $$\label{2.1.1} 0\le w_1\le \mathbb{G}_{\alpha,{\mathbb{R}}^N_+}[\frac{\partial^\alpha \delta_0}{\partial e_N^\alpha}]\quad {\rm a.e.\ in}\ \ {\mathbb{R}}^N$$ and $$\int_{{\mathbb{R}}^N_+}w_1^{\frac{N+\alpha}{N-\alpha}}\rho^\alpha dx< {\|\mathbb{G}_{\alpha,{\mathbb{R}}^N_+}[\frac{\partial^\alpha \delta_0}{\partial e_N^\alpha}]\|}_{L^1({\mathbb{R}}^N_+,\rho^\alpha\mu dx)},$$ which implies that $w_1\in L^1({\mathbb{R}}^N,\ \mu dx)$. Thus, $v_m\to w_1$ in $L^1({\mathbb{R}}^N,\ \rho^\alpha \mu dx)$ as $m\to\infty$. For $\xi\in \mathbb{X}_{\alpha,{\mathbb{R}}^N_+}$, there exists $N_0>0$ such that for any $m\ge N_0$, $${\rm supp}(\xi)\subset \bar B_m,$$ which implies that $\xi\in \mathbb{X}_{\alpha,B_m}$ and then $$\label{3.6} \int_{{\mathbb{R}}^N_+} [v_m(-\Delta)^\alpha\xi+v_m^{\frac{N+\alpha}{N-\alpha}}\xi]dx= \frac{\partial^\alpha \xi(0)}{\partial e_N^\alpha}.$$ By [@CY Lemma 3.1 ], $$|(-\Delta)^\alpha\xi(x)|\le \frac{c_{9}{\|\xi\|}_{L^\infty(\Omega)}}{1+|x|^{N+2\alpha}},\quad \forall x\in {\mathbb{R}}^N_+.$$ Thus, $$\label{3.7} \lim_{m\to\infty}\int_{{\mathbb{R}}^N_+} v_m(x)(-\Delta)^\alpha\xi(x) dx=\int_{{\mathbb{R}}^N_+} w_1(x)(-\Delta)^\alpha\xi(x) dx.$$ By (\[2.1.1\]) and increasing monotonicity of $v_m$, for any $n\ge N_0$, $$\label{3.8} \lim_{m\to\infty}\int_{{\mathbb{R}}^N_+} v_m^{\frac{N+\alpha}{N-\alpha}}\xi(x) dx=\int_{{\mathbb{R}}^N_+} w_1^{\frac{N+\alpha}{N-\alpha}}\xi(x) dx.$$ Combining (\[3.7\]), (\[3.8\]) and taking $m\to\infty$ in (\[3.6\]), we obtain that $$\label{3.10} \int_{{\mathbb{R}}^N_+} \left[w_1(-\Delta)^\alpha\xi+w_1^{\frac{N+\alpha}{N-\alpha}}\xi\right]dx=\frac{\partial^\alpha \xi(0)}{\partial e_N^\alpha}.$$ Since $\xi\in \mathbb{X}_{\alpha,{\mathbb{R}}^N_+}$ is arbitrary, $w_1$ is a weak solution of (\[14-08-0\]).$\Box$ Forcing nonlinearity ==================== This section is devoted to consider problem (\[eq 1.1\]) when $\epsilon=-1$, we call it as forcing case. In order to derive the existence of weak solution to (\[eq 1.1\]) with forcing nonlinearity, we first introduce the following propositions. \[general\] [@CFV Proposition 2.2] Let $\alpha\in(0,1]$, $\beta\in[0,\alpha]$ and $\nu\in\mathfrak{M}(\Omega,\rho^\beta_{\partial\Omega})$, then there exists $c_{44}>0$ such that $$\label{annex 00} \|\mathbb{G}_\alpha[\nu]\|_{M^{p_\beta^*}(\Omega,\rho^\beta_{\partial\Omega} dx)}\le c_{44}\|\nu\|_{\mathfrak{M}(\Omega,\rho^\beta_{\partial\Omega})},$$ where $p_\beta^*=\frac{N+\beta}{N-2\alpha+\beta}$. \[pr5\] [@CFV Proposition 2.3] Let $\alpha\in(0,1]$ and $\beta\in [0, \alpha]$, then the mapping $f\mapsto \mathbb G_\alpha[f]$ is compact from $L^{1}(\Omega,\rho^\beta_{\partial\Omega} dx)$ into $L^{q}(\Omega)$ for any $q\in [1,\frac{N}{N+\beta-2\alpha})$. Moreover, for $q\in [1,\frac{N}{N+\beta-2\alpha})$, there exists $c_{45}>0$ such that for any $f\in L^{1}(\Omega,\rho^{\beta}_{\partial\Omega}dx)$ $$\label{power1} {\|\mathbb G_\alpha[f]\|}_{L^q(\Omega)}\leq c_{45}{\|f\|}_{L^{1}(\Omega,\rho^{\beta}_{\partial\Omega}dx)}.$$ For $\nu\in\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$, $\nu_t$ is given in section 2.2 for $t\in(0,\sigma_0)$. Let $t_j=\frac1j\in(0,\sigma_0/4)$ if $j\ge j_0$ for some $j_0>0$. Choose $\{\tilde\nu_n\}_n\subset C_0^1(\Omega)$ a sequence of nonnegative functions such that supp$(\tilde\nu_n)\subset\Omega_{t_{j_0}-2^{-n}}\setminus\Omega_{t_{j_0}+2^{-n}}$ and $\tilde\nu_{n}\to\nu_{t_{j_0}} $ in the duality sense with $C(\bar \Omega)$. Denote $$\nu_{n,j}(x)=\left\{ \arraycolsep=1pt \begin{array}{lll} \tilde \nu_{n}(x+t_j\vec{n_x}),\quad& {\rm if}\quad x\in \Omega_{t_{j_0}-2^{-n}}\setminus\Omega_{t_{j_0}+2^{-n}}, \\[2mm] 0,&{\rm if\ not.} \end{array} \right.$$ \[lm 6.1\] Up to subsequence, we have that $\nu_{n,j_n}\to\nu $ in the duality sense with $C(\bar \Omega)$, that is, $$\label{06-08} \lim_{n\to\infty}\int_{\bar \Omega}\zeta \nu_{n,j_n }dx=\int_{\bar \Omega}\zeta d\nu,\qquad\forall \zeta\in C(\bar \Omega).$$ Moreover, $${\rm supp}(\nu_n)\subset \Omega_{\frac{t_n}2}\setminus \Omega_{2t_n}.$$ [**Proof.**]{} For any fixed $j$ and $\zeta\in C(\bar\Omega)$, we observe that $$\begin{aligned} \lim_{n\to\infty}\int_{\bar \Omega}\zeta \nu_{n,j}dx = \int_{\Omega}\zeta d\nu_{t_j}\end{aligned}$$ and pass $j\to\infty$, we derive that $$\begin{aligned} \lim_{j\to\infty}\lim_{n\to\infty}\int_{\bar \Omega}\zeta \nu_{n,j}dx = \int_{\Omega}\zeta d\nu.\end{aligned}$$ The second argument is obvious by the definition of $\nu_{n,j}$. $\Box$ Sub-linear ------------ In this subsection, we are devoted to prove the existence of weak solution to (\[eq 1.1\]) when the source nonlinearity is sub-linear. [**Proof of Theorem \[teo 3\] $(i)$.**]{} Let $\{\nu_n\}$ be a sequence of nonnegative functions such that $\nu_{n}\to\nu $ in sense of duality with $C(\bar\Omega)$, see Lemma \[lm 6.1\]. By the Banach-Steinhaus Theorem, we may assume that ${\|\nu_n\|}_{L^1(\Omega)}\le {\|\nu\|}_{\mathfrak M^b (\Omega)}=1$ for all $n$. We consider a sequence $\{g_n\}$ of $C^1$ nonnegative functions defined on ${\mathbb{R}}_+$ such that $g_n(0)=g(0)$, $$\label{06-08-1} g_n\le g_{n+1}\le g,\quad \sup_{s\in{\mathbb{R}}_+}g_n(s)=n\quad{\rm and}\quad \lim_{n\to\infty}{\|g_n-g\|}_{L^\infty_{loc}({\mathbb{R}}_+)}=0.$$ We set $$M(v)={\|v\|}_{L^{1}(\Omega)}.$$ [*Step 1. To prove that for $n\geq 1$, $$\label{002.3} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u= g_{n}(u)+kt_n^{-\alpha}\nu_n\quad & {\rm in}\quad\Omega,\\[2mm] \phantom{ (-\Delta)^\alpha } u=0\quad & {\rm in}\quad \Omega^c \end{array}$$ admits a nonnegative solution $u_n$ such that $$M(u_n)\le \bar\lambda,$$ where $\bar\lambda>0$ independent of $n$.* ]{} To this end, we define the operators $\{\mathcal{T}_n\}$ by $$\mathcal{T}_nu=\mathbb{G}_\alpha\left[g_n(u)+k t_n^{-\alpha}\nu_n\right],\qquad \forall u\in L^1_+(\Omega),$$ where $L^1_+(\Omega)$ is the positive cone of $L^1(\Omega)$. By (\[power1\]) and (\[06-08-2\]), we have that $$\label{23-05-0} \arraycolsep=1pt \begin{array}{lll} M(\mathcal{T}_nu)\le c_{45}{\|g_n(u)+k t_n^{-\alpha}\nu_n\|}_{L^1 (\Omega,\rho_{\partial\Omega}^{\alpha}dx)} \\[2mm] \phantom{---- } \le c_3c_{45} \int_{\Omega}u^{p_0}\rho^\alpha(x)dx+c_{46}(k+\epsilon) \\[2mm] \phantom{---- } \le c_3c_{47}\int_{\Omega}u^{p_0} dx+c_{46}(k+\epsilon) \\[2mm] \phantom{---- } \le c_3 c_{48}(\int_{\Omega}u dx)^{p_0}+c_{46}(k+\epsilon) \\[2mm] \phantom{---- } =c_3c_{48}M(u)^{p_0}+c_{46}(k+\epsilon), \end{array}$$ where $c_{47},c_{48}>0$ independent of $n$. Therefore, we derive that $$M(\mathcal{T}_nu)\le c_3c_{48} M(u)^{p_0}+c_{45}(k+\epsilon).$$ If we assume that $M(u)\le \lambda$ for some $\lambda>0$, it implies $$M(\mathcal{T}_nu)\le c_3c_{48} \lambda^{p_0}+c_{45}(k+\epsilon).$$ In the case of $p_0<1$, the equation $$c_3c_{48}\lambda^{p_0 }+c_{45}(k+\epsilon)=\lambda$$ admits a unique positive root $\bar\lambda$. In the case of $p_0=1$, for $c_3>0$ satisfying $c_3c_{48}<1$, the equation $$c_3c_{48}\lambda+c_{45}(k+\epsilon)=\lambda$$ admits a unique positive root $\bar\lambda$. For $M(u)\le \bar\lambda$, we obtain that $$\label{07-05-5jingxuan} M(\mathcal{T}_nu)\le c_3c_{48}\bar\lambda^{p_0}+c_{45}(k+\epsilon)= \bar\lambda.$$ Thus, $\mathcal{T}_n$ maps $L^1(\Omega)$ into itself. Clearly, if $u_m\to u$ in $L^1(\Omega)$ as $m\to\infty$, then $g_n(u_m)\to g_n(u)$ in $L^1(\Omega)$ as $m\to\infty$, thus $\mathcal{T}_n$ is continuous. For any fixed $n\in{\mathbb{N}}$, $\mathcal{T}_nu_m=\mathbb{G}_\alpha\left[g_n(u_m)+k \nu_n\right]$ and $\{g_n(u_m)+k \nu_n\}_m$ is uniformly bounded in $L^1(\Omega,\rho_{\partial\Omega}^\beta dx)$, then it follows by Proposition \[pr5\] that $\{\mathbb{G}_\alpha\left[g_n(u_m)+k t_n^{-\alpha}\nu_n\right]\}_m$ is pre-compact in $L^1(\Omega)$, which implies that $\mathcal{T}_n$ is a compact operator. Let $$\displaystyle\begin{array}{lll}\displaystyle \mathcal{G}=\{u\in L^1_+(\Omega): \ M(u)\le \bar\lambda \}, \end{array}$$ which is a closed and convex set of $L^1(\Omega)$. It infers by (\[07-05-5jingxuan\]) that $$\mathcal{T}_n(\mathcal{G})\subset \mathcal{G}.$$ It follows by Schauder’s fixed point theorem that there exists some $u_n\in L^1_+(\Omega)$ such that $\mathcal{T}_nu_n=u_n$ and $M(u_n)\le \bar\lambda,$ where $\bar\lambda>0$ independent of $n$. We observe that $u_n$ is a classical solution of (\[002.3\]). Let open set $O$ satisfy $ O\subset \bar O\subset \Omega$. By [@RS Proposition 2.3], for $\theta\in(0,2\alpha)$, there exists $c_{49}>0$ such that $${\|u_n\|}_{C^{\theta}(O)}\le c_{49}\{{\|g(u_n)\|}_{L^\infty(\Omega)}+kt_n^{-\alpha}{\|\nu_n\|}_{L^{\infty}(\Omega)}\},$$ then applied [@RS Corollary 2.4], $u_n$ is $C^{2\alpha+\epsilon_0}$ locally in $\Omega$ for some $\epsilon_0>0$. Then $u_n$ is a classical solution of (\[002.3\]). Moreover, from [@CV2 Lemma 2.2], we derive that $$\label{5.60000} \int_\Omega u_n(-\Delta)^\alpha\xi dx=\int_\Omega g(u_n)\xi dx+k\int_\Omega\xi t_n^{-\alpha}\nu_ndx,\quad \forall\xi\in \mathbb{X}_{\alpha}.$$ [*Step 2. Convergence.* ]{} We observe that $\{g_n( u_n)\}$ is uniformly bounded in $L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)$, so is $\{\nu_n\}$. By Proposition \[pr5\], there exist a subsequence $\{u_{n_k}\}$ and $u$ such that $u_{n_k}\to u$ a.e. in $\Omega$ and in $L^1(\Omega)$, then by (\[06-08-2\]), we derive that $g_{n_k}(u_{n_k}) \to g( u)$ in $L^1(\Omega)$. Pass the limit of (\[5.60000\]) as $n_k\to \infty$ to derive that $$\int_\Omega u(-\Delta)^\alpha\xi=\int_\Omega g(u)\xi dx+k\int_\Omega\frac{\partial^\alpha\xi}{\partial \vec{n}^\alpha} d\nu,\quad \forall \xi\in\mathbb{X}_\alpha,$$ thus $u$ is a weak solution of (\[eq 1.1\]) and $u$ is nonnegative since $\{u_n\}$ are nonnegative. $\Box$ Integral subcritical --------------------- In this subsection, we prove the existence of weak solution to (\[eq 1.1\]) when the nonlinearity is integral subcritical. [**Proof of Theorem \[teo 3\] $(ii)$.**]{} Let $\{\nu_n\}\subset C^1(\bar \Omega)$ be a sequence of nonnegative functions given as the above and ${\|\nu_n\|}_{L^1(\Omega)}\le 2{\|\nu\|}_{\mathfrak M ^b(\bar\Omega)}=1$ for all $n$. We consider a sequence $\{g_n\}$ of $C^1$ nonnegative functions defined on ${\mathbb{R}}_+$ satisfying $g_n(0)=g(0)$ and (\[06-08-1\]). We set $$M_1(v)={\|v\|}_{M^{\frac{N+\alpha}{N-\alpha}}(\Omega,\rho^\alpha_{\partial\Omega} dx)}\quad{\rm and}\quad M_2(v)={\|v\|}_{L^{p_*}(\Omega)},$$ where $p_*$ is (\[1.4\]). We may assume that $p_*\in(1, \frac{N}{N-\alpha})$. In fact, if $p_*\ge \frac{N}{N-\alpha}$, then for any given $p\in(1, \frac{N}{N-\alpha})$, (\[1.4\]) implies that $$g(s)\le c_{4}s^p+\epsilon,\quad \forall s\in[0,1].$$ [*Step 1. To prove that for $n\geq 1$, $$\label{2.3} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u= g_{n}(u)+kt_n^{-\alpha}\nu_n\quad & {\rm in}\quad\Omega,\\[2mm] \phantom{ (-\Delta)^\alpha_n } u=0\quad & {\rm in}\quad \Omega^c \end{array}$$ admits a nonnegative solution $u_n$ such that $$M_1(u_n)+M_2(u_n)\le \bar\lambda,$$ where $\bar\lambda>0$ independent of $n$.* ]{} To this end, we define the operators $\{\mathcal{T}_n\}$ by $$\mathcal{T}_nu=\mathbb{G}_\alpha\left[g_n(u)+kt_n^{-\alpha} \nu_n\right],\qquad \forall u\in L^1_+(\Omega).$$ By Proposition \[general\], we have $$\begin{aligned} M_1(\mathcal{T}_nu) &\le& c_{44}{\|g_n(u)+kt_n^{-\alpha} \nu_n\|}_{L^1 (\Omega,\rho^{\alpha}_{\partial\Omega}dx)}\nonumber\\[2.5mm] &\le & c_{44} [{\|g_n(u)\|}_{L^1(\Omega,\rho^{\alpha}_{\partial\Omega}dx)}+k].\label{06-08-10}\end{aligned}$$ In order to deal with ${\|g_n(u)\|}_{L^1(\Omega,\rho^{\beta}_{\partial\Omega}dx)}$, for $\lambda >0$ we set $$S_\lambda=\{x\in\Omega:u(x)>\lambda\}\quad {\rm and}\quad \omega(\lambda)=\int_{S_\lambda}\rho^\alpha_{\partial\Omega} dx,$$ $$\label{chenyuhang1} \displaystyle\begin{array}{lll} \displaystyle{\|g_n(u)\|}_{L^1(\Omega,\rho^{\alpha}_{\partial\Omega}dx)}\le \int_{S^c_1}g(u)\rho^{\alpha}_{\partial\Omega}dx+\int_{S_1} g(u)\rho^{\alpha}_{\partial\Omega}dx. \end{array}$$ We first deal with $\int_{S_1} g(u)\rho^{\alpha}dx$. In fact, we observe that $$\int_{S_1} g(u)\rho^{\alpha}_{\partial\Omega}dx=\omega(1) g(1)+\int_1^\infty \omega(s)dg(s),$$ where $$\int_1^\infty g(s)d\omega(s)=\lim_{T\to\infty}\int_1^T g(s)d\omega(s).$$ It infers by Proposition \[pr 1\] and Proposition \[general\] that there exists $c_{50}>0$ such that $$\label{2.4} \omega(s)\leq c_{50}M_1(u)^{\frac{N+\alpha}{N-\alpha}}s^{-\frac{N+\alpha}{N-\alpha}}$$ and by (\[1.4\]) and Lemma \[lm 08-09\] with $p=\frac{N+\alpha}{N-\alpha}$, there exist a sequence of increasing numbers $\{T_j\}$ such that $T_1>1$ and $T_j^{-\frac{N+\alpha}{N-\alpha}} g(T_j)\to 0$ when $j\to\infty$, thus $$\displaystyle\begin{array}{lll} \displaystyle \omega(1) g(1)+ \int_1^{T_j} \omega(s)d g(s) \le c_{50}M_1(u)^{\frac{N+\alpha}{N-\alpha}} g(1)+c_{50}M(u)^{\frac{N+\alpha}{N-\alpha}}\int_1^{T_j} s^{-\frac{N+\alpha}{N-\alpha}}d g(s) \\[4mm]\phantom{-----}\displaystyle \leq c_{50}M_1(u)^{\frac{N+\alpha}{N-\alpha}}{T_j}^{-\frac{N+\alpha}{N-\alpha}} g(T_j)+\frac{c_{50}M_1(u)^{\frac{N+\alpha}{N-\alpha}}}{\frac{N+\alpha}{N-\alpha}+1}\int_1^{T_j} s^{-1-\frac{N+\alpha}{N-\alpha}} g(s)ds. \end{array}$$ Therefore, $$\label{06-08-11} \displaystyle\begin{array}{lll} \int_{S_1} g(u)\rho^{\alpha}dx=\omega(1)g(1)+ \int_1^\infty \omega(s)\ dg(s) \\[3mm]\phantom{------} \leq \frac{c_{50}M_1(u)^{\frac{N+\alpha}{N-\alpha}}}{\frac{N+\alpha}{N-\alpha}+1}\int_1^\infty s^{-1-\frac{N+\alpha}{N-\alpha}} g(s)ds \\[3mm]\phantom{------} \displaystyle \le c_{50}g_\infty M_1(u)^{\frac{N+\alpha}{N-\alpha}}, \end{array}$$ where $c_{50}>0$ independent of $n$. We next deal with $ \int_{S^c_1}g(u)\rho^{\alpha}_{\partial\Omega}dx$. For $p_*\in(1, \frac{N}{N-2\alpha+\beta})$, we have that $$\label{4.1} \displaystyle\begin{array}{lll} \int_{S^c_1}g(u)\rho^{\alpha}_{\partial\Omega}dx\le c_{4}\int_{S_1^c}u^{p_*}\rho^{\alpha}_{\partial\Omega}dx+\epsilon\int_{S_1^c}\rho^{\alpha}_{\partial\Omega}dx \\[3mm]\phantom{------} \le c_{4}c_{51}\int_{\Omega}u^{p_*}dx+c_{51}\epsilon \\[3mm]\phantom{------} \leq c_{4}c_{51}M_2(u)^{p_*} +c_{51}\epsilon, \end{array}$$ where $c_{51}>0$ independent of $n$. Along with (\[06-08-10\]), (\[chenyuhang1\]), (\[06-08-11\]) and (\[4.1\]), we derive $$\label{05-09-4} M_1(\mathcal{T}_nu)\le c_{44}c_{50}g_\infty M_1(u)^{\frac{N+\alpha}{N-\alpha}}+c_{44}c_{4}c_{51}M_2(u)^{p_*}+c_{44}c_{51}\epsilon+c_{44}k.$$ By [@NPV Theorem 6.5] and (\[power1\]), we derive that $$\displaystyle\begin{array}{lll} M_2(\mathcal{T}_nu)\le c_{45}{\|g_n(u)+k \nu_n\|}_{L^1 (\Omega,\rho_{\partial\Omega}^{\alpha}dx)}, \end{array}$$ which along with (\[chenyuhang1\]), (\[06-08-11\]) and (\[4.1\]), implies that $$\label{4.3} M_2(\mathcal{T}_nu)\le c_{45}c_{50}g_\infty M_1(u)^{\frac{N+\alpha}{N-\alpha}}+c_{45}c_{4}c_{51}M_2(u)^{p_*}+c_{45}c_{51}\epsilon+c_{45}k.$$ Therefore, inequality (\[05-09-4\]) and (\[4.3\]) imply that $$M_1(\mathcal{T}_nu)+M_2(\mathcal{T}_nu)\le c_{52}g_\infty M_1(u)^{\frac{N+\alpha}{N-\alpha}}+c_{53}c_4M_2(u)^{p_*}+c_{54}\epsilon+c_{54}k,$$ where $c_{52}=(c_{44}+c_{45})c_{50}$, $c_{21}=(c_{44}+c_{45})c_{51}$ and $c_{54}=c_{44}+c_{45}$. If we assume that $M_1(u)+M_2(u)\le \lambda$, implies $$M_1(\mathcal{T}_nu)+M_2(\mathcal{T}_nu)\le c_{52}g_\infty\lambda^{\frac{N+\alpha}{N-\alpha}}+c_{21}\lambda^{p_*}+c_{21}\epsilon+c_{54}k.$$ Since $\frac{N+\alpha}{N-\alpha},\ p_*>1$, then there exist $k_0>0$ and $\epsilon_0>0$ such that for any $k\in(0,k_0]$ and $\epsilon\in(0,\epsilon_0]$, the equation $$c_{52}g_\infty\lambda^{\frac{N+\alpha}{N-\alpha}}+c_{21}\lambda^{p_*}+c_{21}c_3\epsilon+c_{54}k=\lambda$$ admits the largest root $\bar\lambda>0$. We redefine $M(u)=M_1(u)+M_2(u)$, then for $M(u)\le \bar\lambda$, we obtain that $$\label{2.2} M(\mathcal{T}_nu)\le c_{52}g_\infty\bar\lambda^{\frac{N+\alpha}{N-\alpha}}+c_{21}\bar\lambda^{p_*}+c_{21}\epsilon+c_{54}k= \bar\lambda.$$ Especially, we have that $${\|\mathcal{T}_nu\|}_{L^1(\Omega)}\le c_8M_1(\mathcal{T}_nu)|\Omega|^{\frac{2\alpha}{N+\alpha}}\le c_{23} \bar\lambda\quad{\rm if}\quad M(u)\le \bar\lambda.$$ Thus, $\mathcal{T}_n$ maps $L^1(\Omega)$ into itself. Clearly, if $u_m\to u$ in $L^1(\Omega)$ as $m\to\infty$, then $g_n(u_m)\to g_n(u)$ in $L^1(\Omega)$ as $m\to\infty$, thus $\mathcal{T}_n$ is continuous. For any fixed $n\in{\mathbb{N}}$, $\mathcal{T}_nu_m=\mathbb{G}_\alpha\left[g_n(u_m)+k \nu_n\right]$ and $\{g_n(u_m)+k \nu_n\}_m$ is uniformly bounded in $L^1(\Omega,\rho^\alpha dx)$, then it follows by Proposition \[pr5\] that $\{\mathbb{G}_\alpha\left[g_n(u_m)+k \nu_n\right]\}_m$ is pre-compact in $L^1(\Omega)$, which implies that $\mathcal{T}_n$ is a compact operator. Let $$\displaystyle\begin{array}{lll}\displaystyle \mathcal{G}=\{u\in L^1_+(\Omega): \ M(u)\le \bar\lambda \} \end{array}$$ which is a closed and convex set of $L^1(\Omega)$. It infers by (\[2.2\]) that $$\mathcal{T}_n(\mathcal{G})\subset \mathcal{G}.$$ It follows by Schauder’s fixed point theorem that there exists some $u_n\in L^1_+(\Omega)$ such that $\mathcal{T}_nu_n=u_n$ and $M(u_n)\le \bar\lambda,$ where $\bar\lambda>0$ independent of $n$. In fact, $u_n$ is a classical solution of (\[2.3\]). Let $O$ an open set satisfying $ O\subset \bar O\subset \Omega$. By [@RS Proposition 2.3], for $\theta\in(0,2\alpha)$, there exists $c_{55}>0$ such that $${\|u_n\|}_{C^{\theta}(O)}\le c_{55}\{{\|g(u_n)\|}_{L^\infty(\Omega)}+kt_n^{-\alpha}{\|\nu_n\|}_{L^{\infty}(\Omega)}\},$$ then applied [@RS Corollary 2.4], $u_n$ is $C^{2\alpha+\epsilon_0}$ locally in $\Omega$ for some $\epsilon_0>0$. Then $u_n$ is a classical solution of (\[2.3\]). Moreover, $$\label{5.6} \int_\Omega u_n(-\Delta)^\alpha\xi dx=\int_\Omega g(u_n)\xi dx+k\int_\Omega\xi \nu_ndx,\quad \forall\xi\in \mathbb{X}_{\alpha}.$$ [*Step 2. Convergence.* ]{} Since $\{g_n( u_n)\}$ and $\{\nu_n\}$ are uniformly bounded in $L^1(\Omega,\rho^\beta_{\partial\Omega} dx)$, then by Propostion \[pr5\], there exist a subsequence $\{u_{n_k}\}$ and $u$ such that $u_{n_k}\to u$ a.e. in $\Omega$ and in $L^1(\Omega)$, and $g_{n_k}(u_{n_k}) \to g( u)$ a.e. in $\Omega$. Finally we prove that $g_{n_k}( u_{n_k})\to g( u)$ in $L^1(\Omega,\rho^\beta_{\partial\Omega} dx)$. For $\lambda >0$, we set $S_\lambda=\{x\in\Omega:|u_{n_k}(x)|>\lambda\}$ and $\omega(\lambda)=\int_{S_\lambda}\rho^{\alpha}_{\partial\Omega}dx$, then for any Borel set $E\subset\Omega$, we have that $$\label{chenyuhang1000} \displaystyle\begin{array}{lll} \displaystyle\int_{E}|g_{n_k}(u_{n_k})|\rho^\beta_{\partial\Omega} dx=\int_{E\cap S^c_{\lambda}}g(u_{n_k})\rho^\beta_{\partial\Omega} dx+\int_{E\cap S_{\lambda}} g(u_{n_k})\rho^\beta_{\partial\Omega} dx\\[4mm]\phantom{\int_{E}|g(u_{n_k})|\rho^{\alpha_{\partial\Omega}}dx} \displaystyle\leq \tilde g(\lambda)\int_E\rho^\beta_{\partial\Omega} dx+\int_{S_{\lambda}} g(u_{n_k})\rho^\beta_{\partial\Omega} dx\\[4mm]\phantom{\int_{E}g(u_{n_k})\rho^\beta_{\partial\Omega} dx} \displaystyle\leq \tilde g(\lambda)\int_E\rho^\beta_{\partial\Omega} dx+\omega(\lambda) g(\lambda)+\int_{\lambda}^\infty \omega(s)d g(s), \end{array}$$ where $\tilde g(\lambda)=\max_{s\in[0,\lambda]}g(s)$. On the other hand, $$\int_{\lambda}^\infty g(s)d\omega(s)=\lim_{T_m\to\infty}\int_{\lambda}^{T_m} g(s)d\omega(s).$$ where $\{T_m\}$ is a sequence increasing number such that $T_m^{-\frac{N+\alpha}{N-\alpha}} g(T_m)\to 0$ as $m\to\infty$, which could obtained by assumption (\[1.4\]) and Lemma \[lm 08-09\] with $p={\frac{N+\alpha}{N-\alpha}}$. It infers by (\[2.4\]) that $$\displaystyle\begin{array}{lll} \displaystyle \omega(\lambda) g(\lambda)+ \int_{\lambda}^{T_m} \omega(s)d g(s) \le c_{50} g(\lambda)\lambda^{-\frac{N+\alpha}{N-\alpha}}+c_{56}\int_{\lambda}^{T_m} s^{-\frac{N+\alpha}{N-\alpha}}d g(s) \\[4mm]\phantom{-----\ \int_{\lambda}^{T_m} g(s)d\omega(s)}\displaystyle \leq c_{56}T_m^{-\frac{N+\alpha}{N-\alpha}}g(T_m)+\frac{c_{56}}{\frac{N+\alpha}{N-\alpha}+1}\int_{\lambda}^{T_m} s^{-1-\frac{N+\alpha}{N-\alpha}} g(s)ds, \end{array}$$ where $c_{56}=c_{50}\frac{N+\alpha}{N-\alpha}$. Pass the limit of $m\to\infty$, we have that $$\omega(\lambda) g(\lambda)+ \int_{\lambda}^\infty \omega(s)\ d g(s)\leq \frac{c_{56}}{\frac{N+\alpha}{N-\alpha}+1}\int_{\lambda}^\infty s^{-1-\frac{N+\alpha}{N-\alpha}} g(s)ds.$$ Notice that the above quantity on the right-hand side tends to $0$ when $\lambda\to\infty$. The conclusion follows: for any $\epsilon>0$ there exists $\lambda>0$ such that $$\frac{c_{56}}{\frac{N+\alpha}{N-\alpha}+1}\int_{\lambda}^\infty s^{-1-\frac{N+\alpha}{N-\alpha}} g(s)ds\leq \frac{\epsilon}{2}.$$ Since $\lambda$ is fixed, together with (\[chenyuhang1\]), there exists $\delta>0$ such that $$\int_E\rho^{\alpha}_{\partial\Omega}dx\leq \delta\Longrightarrow g(\lambda)\int_E\rho^{\alpha}_{\partial\Omega}dx\leq\frac{\epsilon}{2}.$$ This proves that $\{g\circ u_{n_k}\}$ is uniformly integrable in $L^1(\Omega,\rho^\beta_{\partial\Omega} dx)$. Then $g\circ u_{n_k}\to g\circ u$ in $L^1(\Omega,\rho^\beta_{\partial\Omega} dx)$ by Vitali convergence theorem. 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Huyuan Chen Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China and Institute of Mathematical Sciences, New York University Shanghai, Shanghai 200120, PR China Hichem Hajaiej Institute of Mathematical Sciences, New York University Shanghai, Shanghai 200120, PR China [^1]: hc64@nyu.edu [^2]: hh62@nyu.edu